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https://arxiv.org/abs/1706.03298 | Polynomial Relations Between Matrices of Graphs | We derive a correspondence between the eigenvalues of the adjacency matrix $A$ and the signless Laplacian matrix $Q$ of a graph $G$ when $G$ is $(d_1,d_2)$-biregular by using the relation $A^2=(Q-d_1I)(Q-d_2I)$. This motivates asking when it is possible to have $X^r=f(Y)$ for $f$ a polynomial, $r>0$, and $X,\ Y$ matrices associated to a graph $G$. It turns out that, essentially, this can only happen if $G$ is either regular or biregular. | \section{Introduction}
For $G$ a simple graph, we let $A$ denote the adjacency matrix of $G$ and $D$ the diagonal matrix of vertex degrees of $G$. We define the signless Laplacian matrix $Q$ of $G$ by $Q=D+A$, the Laplacian matrix $L$ of $G$ by $L=D-A$, and when $G$ has no isolated vertices, we define the normalized Laplacian matrix $\NL$ of $G$ by $D^{-1/2}LD^{-1/2}$. For more detailed information about these matrices see, for example, \cite{chungButler}.
A graph $G$ is said to be $(d_1,d_2)$-biregular (sometimes called semiregular) if $V_1\cup V_2$ is a partition of the vertices of $G$ such that no two vertices of $V_i$ are adjacent to one another (so $G$ is bipartite), and for all $v\in V_i,\ d_v=d_i$. A graph is said to be biregular if it is $(d_1,d_2)$-biregular for some $d_1,d_2$.
When $G$ is biregular it is possible to directly relate the eigenvalues of $A$ and $Q$ through the following formula of \cite{signless}.
\begin{thm}\label{T-signlessPolynomial}
If $G$ is $(d_1,d_2)$-biregular with $|V_i|=n_i,\ n_1\ge n_2$, and $\lambda_1,\ldots,\lambda_{n_2}$ are the $n_2$ largest eigenvalues of $A$ in decreasing order, then
\[
Q_G(x)=x(x-d_1-d_2)(x-d_1)^{n_1-n_2}\prod_{i=2}^{n_2}((x-d_1)(x-d_2)-\lambda_i^2),
\]
where $Q_G(x)$ denotes the characteristic polynomial of $Q$.
\end{thm}
Theorem~\ref{T-signlessPolynomial} was proved in \cite{signless} by using arguments involving the line graph of $G$. In this paper we present an independent derivation of this theorem using the relation
\[
A^2=(Q-d_1I)(Q-d_2I).
\]
Given this derivation, it is natural to ask what other graphs $G$ satisfy $A^r=f(Q)$ for some polynomial $f$ and positive integer $r$. More generally, one can ask what $G$ satisfy $X^r=f(Y)$ when $X$ and $Y$ are matrices associated to the graph $G$. Of the cases we consider, the only graphs found to have this property are graphs that are either regular or biregular. We summarize our results in the following theorem, where we note that the first part of the theorem is clear from the definitions of $Q,\ L$ and $\NL$.
\begin{thm}
Let $G$ be a connected graph, $r$ a positive integer and $f$ a polynomial.
\begin{itemize}
\item If $G$ is regular, then $X^r=f(Y)$ can occur if $X,\ Y$ are any of $A,\ Q,\ L$ or $\NL$. Moreover, all of these matrices can be related to one another by a linear equation.
\item If $G$ is $(d_1,d_2)$-biregular, then $X^r=f(Y)$ can occur if $X=A$ and $Y=Q,\ L$, or $\NL$, or when $X=\NL$ and $Y=A$. Specifically, we have
\begin{align*}
(Q-d_1I)(Q-d_2I)&=(L-d_1I)(L-d_2I)=A^2\\
\mathcal{L}&=I-\frac{1}{\sqrt{d_1d_2}}A.
\end{align*}
Moreover, if $X,\ Y$ are any other pair from $A,\ Q,\ L, \NL$ then no such relation exists.
\item If $G$ is not regular or biregular, then $X^r=f(Y)$ can not hold if $X,\ Y$ are any distinct matrices of $A,\ Q,\ L,\ \NL$, except possibly for the case $A^r=f(\NL)$.
\end{itemize}
\end{thm}
We establish the following conventions. Whenever $X^r=f(Y)$ is written it is assumed that $f$ is a polynomial and $r$ is a positive integer. We assume throughout this paper that $G$ is a connected graph, though we emphasize this point in the statement of our theorems. $\mathbf{1}$ will denote the vector of all 1's. $m_i(u,v)$ will denote the number of walks of length $i$ between the vertices $u$ and $v$ in the graph $G$. We note that $(A^i)_{uv}=m_i(u,v)$ (see Theorem 1.1 of \cite{stanleyBook}, for example). For a matrix $M$ we let $\Eig{M}{\lambda}$ denote the eigenspace of $M$ with corresponding eigenvalue $\lambda$. $V(G)$ and $E(G)$ will denote the set of vertices and the set of edges of the graph $G$ respectively.
The structure of the paper is as follows. In Section~\ref{S-rel} we derive Theorem~\ref{T-signlessPolynomial} and in Section~\ref{S-tree} we apply this theorem to count the number of spanning trees of biregular graphs. In Sections \ref{S-Q} and \ref{S-NL} we establish necessary conditions for $G$ to satisfy $X^r=f(Y)$ with $X,\ Y$ equal to $A,\ Q,\ L$, and $\NL$. Lastly, in Section~\ref{S-gen} we briefly explore the more general question of establishing relations of the form $f(X)=g(Y)$ where $f$ and $g$ are both polynomials and $X$ and $Y$ are matrices associated to a graph $G$.
\section{Relating Eigenvalues of $A$ and $Q$}\label{S-rel}
\begin{prop}\label{P-keyrel}
If $G$ is a $(d_1,d_2)$-biregular graph, then
\[
A^2=(Q-d_1I)(Q-d_2I).
\]
\end{prop}
\begin{proof}
Let $Q'=(Q-d_1I)(Q-d_2I)$. By definition, $Q'_{uv}$ is equal to the dot product of the $u$th row of $Q-d_1 I$ with the $v$th column of $Q-d_2I$. From these definitions we have that \begin{align*}
Q'_{uv}=(d_u-d_1)m_1(u,v)+(d_v-d_2)m_1(v,u)+&\sum_{w\ne u,v}m_1(u,w)m_1(w,v)\\ =(d_u+d_v-d_1-d_2)m_1(u,v)+&\sum_{w\ne u,v}m_1(u,w)m_1(w,v).
\end{align*} If $u,v\in V_i,$ then $m_1(u,v)=0$ and we are left with $\sum_{w\ne u,v}m_1(u,w)m_1(w,v)=m_2(u,v)$. If, say, $u\in V_1,v\in V_2$, then $m_1(u,w)m_1(w,v)=0$ for all $w\ne u,v$, and further, $d_u+d_v-d_1-d_2=0$. Thus in this case $Q'_{uv}=0=m_2(u,v)$ (since the graph is bipartite and $u,v$ belong to different partition classes). We conclude that for all $u,v$ that $Q'_{uv}=m_2(u,v)=(A^2)_{uv}$, completing the proof.
\end{proof}
We note that through an analogous computation one can show that if $G$ is biregular, then \[A^2=(L-d_1I)(L-d_2I).\]
We also note that every result in this section remains valid, except for straightforward changes of some signs, if one replaces $Q$ with $L$.
\begin{cor}\label{C-QtoA}
Let $G$ be a $(d_1,d_2)$-biregular graph. If $q_1,\ldots,q_n$ are the eigenvalues of $Q$, then $(q_1-d_1)(q_1-d_2),\ldots,(q_n-d_1)(q_n-d_2)$ are the eigenvalues of $A^2$.
Moreover, if $\lambda$ is an eigenvalue of $A^2$ and $q_1,q_2$ are the solutions to the equation $\lambda=(x-d_1)(x-d_2)$, then $\Eig{Q}{q_1}\oplus\Eig{Q}{q_2}=\Eig{A^2}{\lambda}$.
\end{cor}
\begin{proof}
Let $V'=\{v_1,\ldots,v_n\}$ be a basis of eigenvectors of $Q$ with $Qv_i=q_i v_i$, which exists because $Q$ is symmetric and hence diagonalizable. Then \[A^2v_i=(Q-d_1I)(Q-d_2I)v_i=(q_i-d_1)(q_i-d_2)v_i,\] so $v_i$ will be an eigenvector of $A^2$ with the desired eigenvalue. From this it is also clear that a basis for $\Eig{Q}{q_1}\oplus \Eig{Q}{q_2}$ is also a basis for $\Eig{A}{\lambda}$ when $q_1,q_2$ are the solutions to $\lambda=(x-d_1)(x-d_2)$.
\end{proof}
From Corollary~\ref{C-QtoA} it is possible to translate from the eigenvalues of $Q$ to the eigenvalues of $A$ when $G$ is biregular. Namely, because $G$ is bipartite, $A$'s spectrum will be symmetric about 0 (see Proposition 3.4.1 of \cite{brouwer}), so knowing the eigenvalues (with multiplicity) of $A^2$ is equivalent to knowing the eigenvalues (with multiplicity) of $A$.
What is less obvious is that the converse of the above statement is true. That is, given the eigenvalues of $A$ when $G$ is biregular, one can compute the eigenvalues of $Q$. Certainly we know that if $\lambda^2$ is an eigenvalue of $A^2$ then $Q$ must have an eigenvalue $q$ satisfying $(q-d_1)(q-d_2)=\lambda^2$, but if $d_1\ne d_2$ then it is not clear which root of this equation correctly corresponds to the eigenvalue in $Q$ (if $d_1=d_2$ then $G$ is regular and there is only one root to choose). To figure out the multiplicities of the eigenvalues of $Q$ we will need the following lemma.
\begin{lem}\label{L-nonint}
Let $G$ be $(d_1,d_2)$-biregular with $d_1\ne d_2$ and let $v$ be an eigenvector of $A$ with eigenvalue $\lambda\ne 0$. Then $v$ is not an eigenvector of $Q$.
\end{lem}
\begin{proof}
Assume that $Qv=\mu v$. Then $Dv=(Q-A)v=(\mu-\lambda)v$, so $v$ is also an eigenvalue of $D$. This implies that $\mu-\lambda=d_i$ for $i=1$ or 2, and hence the set $V'=\{u:v_u\ne 0\}$ lies entirely in the corresponding $V_i$. But if $u\in V'$ then $(Av)_u=0$, as all of the neighbors of $u$ belong to the other partition class and hence are given 0 weight in $v$. Since $v_u\ne 0$ and $\lambda v_u=(Av)u=0$, we must have $\lambda=0$, contradicting the assumption that this is not the case.
\end{proof}
\begin{lem}\label{L-split}
For $G$ a $(d_1,d_2)$-biregular graph, let $\lambda^2\ne 0$ be an eigenvalue of $A^2$ with $\dim \Eig{A^2}{\lambda^2}=m$ and let $q_1,q_2$ be the two roots of the equation $\lambda^2=(x-d_1)(x-d_2)$. Then $\dim \Eig{Q}{q_1}=\dim \Eig{Q}{q_2}=m/2$.
\end{lem}
\begin{proof}
Note first that since $\lambda^2\ne 0$ and $G$ is bipartite, $m$ is even and $m/2$ is an integer. Moreover, $\dim \Eig{A}{\lambda}=\dim \Eig{A}{-\lambda}=m/2$ and $\Eig{A}{\lambda}\oplus\Eig{A}{-\lambda}= \Eig{A^2}{\lambda^2}$. By Corollary~\ref{C-QtoA} we have $\Eig{Q}{q_1}\oplus \Eig{Q}{q_2}= \Eig{A^2}{\lambda^2}$. If $\dim \Eig{Q}{q_i}>m/2$, then we must have $\Eig{Q}{q_i}\cap \Eig{A}{\lambda}\ne \{0\}$ by a dimensionality argument, but this can't happen by Lemma~\ref{L-nonint} and from the assumption that $\lambda\ne 0$. We conclude that $\dim \Eig{Q}{q_i}\le m/2$, and since $\dim \Eig{Q}{q_1}+\dim \Eig{Q}{q_2}=m$, we must have $\dim \Eig{Q}{q_1}=\dim \Eig{Q}{q_2}=m/2$.
\end{proof}
\begin{lem}\label{L-zeroes}
If $G$ is a biregular graph with $|V_1|=n_1,\ |V_2|=n_2,\ n_1\ge n_2$ and $\dim \Eig{A}{0}=m$, then $\dim \Eig{Q}{d_1}=n_1-n_2+k,\ \dim \Eig{Q}{d_2}=k$ where $k=\frac{m-(n_1-n_2)}{2}$.
\end{lem}
\begin{proof}
Let $Z_1$ denote the set of null-vectors of $A$ whose non-zero coordinates lie entirely in $V_1$, and similarly define $Z_2$. It is not difficult to see that $Z_1\oplus Z_2=\Eig{A}{0}$. Moreover, if $v\in Z_i$ then \[d_iv=Dv=Qv-Av=Qv,\] so $Z_i\sub \Eig{Q}{d_i}$. Since $d_1,\ d_2$ are the unique roots of $0=(x-d_1)(x-d_2)$, it follows from Corollary~\ref{C-QtoA} that \[\Eig{Q}{d_1}\oplus \Eig{Q}{d_2}=\Eig{A}{0}=Z_1\oplus Z_2,\] and this implies that $Z_i=\Eig{Q}{d_i}$. Thus it will be sufficient to prove that $\dim Z_1-\dim Z_2=n_1-n_2$.
Let $M$ be the $n_1\times n_2$ sub-matrix of $A$ whose rows are indexed by $V_1$ and whose columns are indexed by $V_2$. Let $r$ be the rank of this matrix. Then the null-space of $M$ has dimension $n_2-r$. Moreover if $v\in Z_2$, one can construct a vector $v'$ in the null space of $M$ by setting $v'_u=v_u$. It isn't difficult to see that the correspondence between $v$ and $v'$ is a bijection between vectors of $Z_2$ and null-vectors of $M$, and moreover this mapping implies that $\dim Z_2=n_2-r$. The same argument on $M^T$ shows that $\dim Z_1=n_1-r$, and hence that $\dim Z_1-\dim Z_2=n_1-n_2$, proving the statement.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{T-signlessPolynomial}]
The characteristic polynomial of $Q$ is the monic polynomial whose roots are the eigenvalues of $Q$ with corresponding multiplicity. For each positive eigenvalue $\lambda$ of $A$, the two roots of $(x-d_1)(x-d_2)-\lambda^2$ will be eigenvalues of $Q$ by Lemma~\ref{L-split}. Note that this will account for all of the eigenvalues of $Q$ except for the eigenvalues $d_1$ and $d_2$. Also note that all of the positive eigenvalues of $A$ are included in the $n_2$ largest eigenvalues of $A$ because $G$ is bipartite.
If $A$ has $n_2-k$ positive eigenvalues, then it must have $n_1-n_2+2k$ eigenvalues equal to 0, meaning $Q$ has $d_1$ as an eigenvalue with multiplicity $n_1-n_2+k$ and $d_2$ with multiplicity $k$ by Lemma~\ref{L-zeroes}. Thus if $\lambda_1,\ldots,\lambda_{n_2}$ are the $n_2$ largest eigenvalues of $A$, the eigenvalues of $Q$ agree with the roots of
\begin{align*}
(x-d_1)^{n_1-n_2}&\left(\prod_{i=1}^k (x-d_1)(x-d_2)\right)\left(\prod_{i=1}^{n_2-k}((x-d_1)(x-d_2)-\lambda_i^2)\right)\\
&=(x-d_1)^{n_1-n_2}\prod_{i=1}^{n_2}((x-d_1)(x-d_2)-\lambda_i^2),
\end{align*}
so this must equal $Q_G(x)$.
We lastly note the following fact stated in \cite{signless}: if $G$ is a connected $(d_1,d_2)$-biregular graph, then $\sqrt{d_1d_2}$ will be the largest eigenvalue of $A$, and the two roots of $d_1d_2=(x-d_1)(x-d_2)$ are $x=0$ and $x=d_1+d_2$. Thus to get the exact form as written in Theorem~\ref{T-signlessPolynomial} we simply pull out the factor $(x-d_1)(x-d_2)-\lambda_1^2=x(x-d_1-d_2)$ from the product.
\end{proof}
\section{Spanning Trees}\label{S-tree}
We provide an application of Theorem~\ref{T-signlessPolynomial}, namely that of counting spanning trees of biregular graphs. Our main tool will be the Matrix-Tree theorem, a proof of which can be found in \cite{stanleyBook}.
\begin{thm}[Matrix-Tree theorem]
Let $\mu_1=0,\mu_2,\ldots,\mu_n$ denote the eigenvalues of the Laplacian matrix $L$ of $G$. The number of spanning trees of $G$ is equal to \[\frac{\mu_2\cdot \mu_3\cdots \mu_n}{|V(G)|}.\]
\end{thm}
If $G$ is bipartite then $Q$ and $L$ will have the same spectrum (see Proposition~1.3.10 of \cite{brouwer}). Thus if we can compute the eigenvalues of $A$ when $G$ is biregular, we can use our previous results to obtain the eigenvalues of $Q$, and hence of $L$, in order to compute the number of spanning trees of $G$ by the Matrix-Tree theorem.
\begin{thm}\label{T-application}
If $G$ is a $(d_1,d_2)$-biregular graph with $|V_i|=n_i,\ n_1\ge n_2,$ and $\lambda_1,\ldots,\lambda_{n_2}$ are the largest eigenvalues of $A$, then the number of spanning trees of $G$ will be
\[
\frac{(d_1+d_2)d_1^{n_1-n_2}\prod_{i=2}^{n_2}(d_1d_2-\lambda_i^2)}{n_1+n_2}.
\]
\end{thm}
\begin{proof}
By the Matrix-Tree theorem, the number of spanning trees of $G$ will be equal to the product of the $n-1$ largest eigenvalues of $L$ divided by $n_1+n_2$. Since $G$ is bipartite, this is equivalent to taking the product of the eigenvalues of $Q$ after ignoring a 0 eigenvalue and dividing by $n_1+n_2$, and this will simply be $\frac{Q_G(x)}{(n_1+n_2)x}$ evaluated at $x=0$. By using this and Theorem~\ref{T-signlessPolynomial}, one arrives at the desired result.
\end{proof}
Let $C_{n}$ denote the $n$-cube, i.e. the graph whose vertices are $n$-length bit strings and two strings are adjacent if their hamming distance is 1. Define $C_{n,k}$ to be the subgraph of $C_n$ induced by all vertices of $C_n$ that have either $k-1$ or $k$ 1's.
\begin{thm}
The number of spanning trees of $C_{n,k}$ when $k\le n/2$ is
\[
\frac{(n+1)k^{{n\choose k}-{n\choose k-1}}\prod_{i=1}^{k-1}((k-i)(i+n-k+1))^{{n\choose k-i}-{n\choose k-i-1}}}{{n\choose k}+{n\choose k-1}}.
\]
\end{thm}
\begin{proof}
From the definition of $C_{n,k}$ it is clear that this graph is $(k,n-k+1)$-biregular with $|V_1|={n\choose k},\ |V_2|={n\choose k-1}$, and we have $|V_1|\ge |V_2|$ since $k\le n/2$. It was proven in Theorem~2.12 of \cite{stanleyPaper} that the squares of the $|V_2|$ largest eigenvalues of the adjacency matrix of $C_{n,k}$ are $i(n-2k+i+1)$ for $1\le i\le k$, each having multiplicity ${n\choose k-i}-{n\choose k-i-1}$. The result follows after applying Theorem~\ref{T-application} and observing that \[k(n-k+1)-i(n-2k+i+1)=(k-i)(i+n-k+1).\]
\end{proof}
More generally, let $C_n(q)$ be the lattice of subspaces of an $n$-dimensional vector space over the finite field $\F_q$. Let $C_{n,k}(q)$ denote the graph whose vertices are the elements of $C_n(q)$ of dimensions $k$ and $k-1$ with two vertices being adjacent if one is a subspace of the other (thus this is the Hasse graph of $C_n(q)$ induced by the elements of rank $k$ and $k-1$). Let
\begin{align*}
\q{n}&=1+q+\cdots+q^{n-1}=\frac{q^n-1}{q-1},\\ \qchoose{n}{k}&=\frac{(q^n-1)(q^{n-1}-1)\cdots(q^{n-k+1}-1)}{(q^k-1)(q^{k-1}-1)\cdots(q-1)}.
\end{align*}
\begin{thm}
The number of spanning trees of $C_{n,k}(q)$ when $k\le n/2$ is
\[
\frac{(\q{k}+\q{n-k+1})(\q{k})^{\qchoose{n}{k}-\qchoose{n}{k-1}}\prod_{i=1}^{k-1}(\q{k}\q{n-k+1}-\gamma_i)^{\qchoose{n}{k-i}-\qchoose{n}{k-i-1}}}{\qchoose{n}{k}+\qchoose{n}{k-1}},
\]
where $\gamma_i=\q{i}(q^{k-i}\q{n-2k}+q^{n-k-i}\q{i+1})$.
\end{thm}
\begin{proof}
$C_{n,k}(q)$ is $(\q{k},\q{n-k+1})$-biregular with $|V_1|=\qchoose{n}{k},\ |V_2|=\qchoose{n}{k-1}$, and $|V_1|\ge |V_2|$. It was proven in Theorem~2.12 of \cite{stanleyPaper} that the squares of the $|V_2|$ largest eigenvalues of the adjacency matrix of $C_{n,k}(q)$ are, for $1\le i\le k$, \begin{align*}r_{k-1}+r_{k-2}+\cdots+r_{k-i},\textrm{ with}\\ r_i=\q{n-i}-\q{i}=q^i\q{n-2i},\end{align*} each with multiplicity $\qchoose{n}{k-i}-\qchoose{n}{k-i-1}$. We wish to put these expressions into a closed form.
We have
\[\sum_{s=1}^i r_{k-s}=\sum_{s=1}^i q^{k-s}\q{n+2s-2k}=\sum_{s=1}^{i}q^{k-s}(\q{n-2k}+q^{n-2k}\q{2s}),\]
so it will be sufficient to find closed forms for the sums $\sum_{s=1}^{i}q^{k-s}\q{n-2k}$ and $q^{n-k}\sum_{s=1}^iq^{-s}\q{2s}$. The first sum can be written as \begin{align*}
\q{n-2k}\sum_{s=1}^iq^{k-s}&=\q{n-2k}\sum_{s=1}^i q^{k-i+s-1}=q^{k-i}\q{n-2k}\sum_{s=1}^iq^{s-1}\\ &=q^{k-i}\q{n-2k}\q{i}.
\end{align*}
For the second sum,
\begin{align*}
&q^{n-k}\sum_{s=1}^iq^{-s}\q{2s}=q^{n-k}\sum_{s=1}^i q^{-s}\frac{q^{2s}-1}{q-1}=\frac{q^{n-k}}{q-1}\sum_{s=1}^i q^s-q^{-s} \\&=\frac{q^{n-k}}{q-1}\left(\frac{q(q^i-1)}{q-1}-\frac{q^{-i}(q^i-1)}{q-1}\right) =\frac{q^{n-k-i}(q^{i+1}-1)(q^{i}-1)}{(q-1)^2}\\ &=q^{n-k-i}\q{i+1}\q{i}.
\end{align*}
Thus in total the squares of the eigenvalues are of the form \[q^{k-i}\q{n-2k}\q{i}+q^{n-k-i}\q{i+1}\q{i}=\q{i}(q^{k-i}\q{n-2k}+q^{n-k-i}\q{i+1})=\gamma_i,\] and plugging this into Theorem~\ref{T-application} gives the desired result.
\end{proof}
\section{Relations Involving $A,\ Q,$ and $L$}\label{S-Q}
When $G$ is biregular, we proved that there exists a relation of the form $A^r=f(Q)$ that allows us to translate between eigenvalues of $A$ and eigenvalues of $Q$, and if $G$ is $d$-regular, the relation $A=Q-dI$ gives an analogous result. One might hope that there exists some notion of ``tripartite'' graphs for which a similar result holds. However, it turns out that the only graphs that can satisfy $A^r=f(Q)$ are the regular and biregular graphs.
The general idea in proving that $X^r=f(Y)$ implies that the underlying graph $G$ has a certain property $P$ is as follows. We first show that if $X$ and $Y$ share a certain eigenvector $v$, then $G$ must have property $P$. We then use the following three lemmas to show that if $X^r=f(Y)$, then $X$ and $Y$ both have $v$ as an eigenvector.
We note that $Q,\ L,$ and $\NL$ have nonnegative spectrum (see \cite{chungButler}, for example), and that $A$'s spectrum is real, so $A^2$ has nonnegative spectrum.
\begin{lem}\label{L-same}
Let $X$ be a diagonalizable matrix with nonnegative spectrum (such as $A^2,\ L,\ Q,$ or $\NL$). Assume that $X^r=f(Y)$ for some matrix $Y$. If $v$ is an eigenvector of $Y$, then $v$ is an eigenvector of $X$.
\end{lem}
\begin{proof}
If $V'=\{v_1,\ldots,v_n\}$ is a basis of eigenvectors of $X$ with $Xv_i=\mu_iv_i$, then $X^r v_i=\mu_i^r v_i$ for all $i$, so $V'$ will also be a basis of eigenvectors of $X^r$. It follows that $\Eig{X^r}{\mu}=\bigoplus_{\mu_i^r=\mu} \Eig{X}{\mu_i}$ for all eigenvalues $\mu$ of $X^r$. As $\mu_i\ge 0$ for all $i$ by assumption, we must have $\Eig{X^r}{\mu}=\Eig{X}{\mu^{1/r}}$ for all eigenvalues of $X^r$. Thus any eigenvector of $X^r$ is also an eigenvector of $X$. But if $v$ is an eigenvector of $Y$ with eignevalue $\lambda$, then $X^rv=f(Y)v=f(\lambda)v$. Thus $v$ is an eigenvector of $X^r$, and hence of $X$.
\end{proof}
\begin{lem}\label{L-onedim}
Let $X,\ Y$ be diagonalizable matrices such that $X^r=f(Y)$, and assume that there exists a $\mu$ such that $\Eig{X}{\mu}=\Eig{X^r}{\mu^r}$ with $\dim \Eig{X}{\mu}=1$. If $v\in \Eig{X}{\mu}$, then $v$ is an eigenvector of $Y$.
\end{lem}
\begin{proof}
Let $V'=\{v_1,\ldots,v_n\}$ be a basis of eigenvectors of $Y$. This will also be a basis of eigenvectors of $X^r$, so there exists a vector $v_i\in V'$ such that $v_i\in \Eig{X^r}{\mu^r}= \Eig{X}{\mu}$. Since $\dim \Eig{X}{\mu}=1$, we conclude that $v_i$ is a scaler multiple of $v$, and hence $v$ is also an eigenvector of $Y$.
\end{proof}
One can strengthen the previous lemma if both matrices have nonnegative spectrum.
\begin{lem}\label{L-bothNonneg}
Let $X,\ Y$ be diagonalizable matrices with nonnegative spectrum and assume that there exists a $\mu$ such that $\dim \Eig{X}{\mu}=1$ with $v\in \Eig{X}{\mu}$. If either $X^r=f(Y)$ or $Y^r=f(X)$, then $v$ will be an eigenvector of $Y$.
\end{lem}
\begin{proof}
The case $X^r=f(Y)$ follows from Lemma~\ref{L-onedim} after one notes that $\Eig{X^r}{\mu^r}=\Eig{X}{\mu}$ because the spectrum of $X$ is nonnegative. The case $Y^r=f(X)$ follows from Lemma~\ref{L-same} because $Y$ has nonnegative spectrum.
\end{proof}
We recall the Perron-Frobenius theorem.
\begin{thm}[Perron-Frobenius]\label{T-pf}
Let $M$ be an irreducible matrix with nonnegative entries. If $\Lambda$ is the largest eigenvalue of $M$, then it has multiplicity one and there exists an eigenvector $\pf$ with $M\pf=\Lambda \pf$ such that every entry of $\pf$ is positive.
\end{thm}
If $G$ is connected (which we always assume to be the case), then Theorem~\ref{T-pf} applies to $A$ and $Q$.
It turns out that the key lemmas needed to prove necessary conditions for $A^r=f(Q)$ are the same lemmas needed to prove necessary conditions for $X^r=f(Y)$ when $X$ and $Y$ are \textit{any} two matrices of $A,\ Q,$ and $L$, so we shall generalize our notation to deal with all of these cases at the same time.
To this end, we will say that $(N,P)$ is a \textit{Laplacian pair} if $N$ is a nonnegative irreducible diagonalizable matrix, $P$ is a diagonalizable matrix with nonnegative spectrum, and $N+aP=bD$ for some $a,b\in \R\setminus\{0\}$. We note that $(A,Q),\ (A,L)$ and $(Q,L)$ are all Laplacian pairs, since we have $A-Q=-D,\ A+L=D,\ Q+L=2D$, and the other conditions are all clearly satisfied.
Given a Laplacian pair $(N,P)$, we will let $\Lambda$ refer to the largest eigenvalue of $N$ and $\pf$ will refer to its corresponding positive eigenvector as is guaranteed by the Perron-Frobenius theorem.
\begin{lem}\label{L-HW}
Let $(N,P)$ be a Laplacian pair. If $\pf$ is also an eigenvector of $P$, then $G$ is regular.
\end{lem}
\begin{proof}
Assume that $P\pf=\mu \pf $ for some $\mu$. Then $bD\pf=(aP+N)\pf=(a\mu+ \Lambda)\pf$, so $\pf$ is also an eigenvector of $D$. But the only way for $\pf$ to be an eigenvector of $D$ is if each of its non-zero coordinates have the same degree in $G$, and since every coordinate of $\pf$ is non-zero, this implies that $G$ is regular.
\end{proof}
\begin{thm}\label{T-pair}
If $(N,P)$ is a Laplacian pair and $P^r=f(N)$, then $G$ is regular.
\end{thm}
\begin{proof}
If $P^r=f(N)$, then $\pf$ will be an eigenvector of $P$ by Lemma~\ref{L-onedim} (as $\pf\in \Eig{N}{\Lambda},\ \dim\Eig{N}{\Lambda}=1$, and $P$ has nonnegative spectrum by definition of $(N,P)$ being a Laplacian pair). $G$ being regular then follows from Lemma~\ref{L-HW}.
\end{proof}
\begin{cor}
If $G$ is connected and $Q^r=f(A),\ L^r=f(A)$, or $L^r=f(Q)$, then $G$ is regular.
\end{cor}
\begin{proof}
$(A,Q),\ (A,L)$, and $(Q,L)$ are all Laplacian pairs, so this immediately follows from Theorem~\ref{T-pair}.
\end{proof}
\begin{thm}\label{T-LQ}
If $G$ is connected and $Q^r=f(L)$, then $G$ is regular.
\end{thm}
\begin{proof}
Let $\pf$ be the positive eigenvector of $Q$ guaranteed by the Perron-Frobenius theorem. If we have $Q^r=f(L)$, then we conclude that $\pf$ is an eigenvector of $L$ by Lemma~\ref{L-bothNonneg}. But $\pf$ being an eigenvector of both $L$ and $Q$ implies that $G$ is regular by Lemma~\ref{L-HW}.
\end{proof}
We now focus on Laplacian pairs with $N=A$.
\begin{lem}\label{L-odd bip}
If $(A,P)$ is a Laplacian pair and $A^r=f(P)$ with either $r$ odd or $G$ not bipartite, then $G$ is regular.
\end{lem}
\begin{proof}
Since $A$ has real spectrum it will always be the case that $\Eig{A^r}{\mu}=\Eig{A}{\mu^{1/r}}$ if $r$ is odd, and $\Eig{A^r}{\mu}=\Eig{A}{\mu^{1/r}}\oplus\Eig{A}{-\mu^{1/r}}$ if $r$ is even. If $r$ is odd, then in particular we have $\Eig{A}{\Lambda}=\Eig{A^r}{\Lambda^r}$. If $G$ is not bipartite then $-\Lambda$ is not an eigenvalue of $A$ (see Proposition~3.4.1 of \cite{brouwer}), and hence for all $r$ we have $\Eig{A}{\Lambda}=\Eig{A}{\Lambda}\oplus \Eig{A}{-\Lambda}=\Eig{A^r}{\Lambda^r}$. As $\dim \Eig{A}{\Lambda}=1$ with $\pf\in \Eig{A}{\Lambda}$, we conclude in either case that $\pf$ is an eigenvector of $P$ by Lemma~\ref{L-onedim}, so $G$ must be regular by Lemma~\ref{L-HW}.
\end{proof}
For a bipartite graph $G$ with vertex partition $V_1\cup V_2$, let $\mathbf{1}'$ be defined by $\mathbf{1}'_v=1$ if $v\in V_1$ and $\mathbf{1}'_v=-1$ if $v\in V_2$.
\begin{lem}\label{L-1}
If $G$ is a bipartite graph with vertex partition $V_1\cup V_2$ and either $\mathbf{1}'$ or $\mathbf{1}$ is an eigenvector of $A^2$, then $G$ is biregular.
\end{lem}
\begin{proof}
\[(A^2\mathbf{1}')_v=\sum_{u\in V(G)}(\mathbf{1}'_u) m_2(u,v)=(\mathbf{1}'_v)\sum_{u\in V(G)} m_2(u,v),\] as every vertex that $v$ can reach in two steps belongs to the same partition class as $v$. Thus $\mathbf{1}'$ will be an eigenvector of $A^2$ iff $\sum_{u\in V(G)} m_2(u,v)$ is equal to the same value for all $v$, and it is clear that this is also an equivalent condition for $\mathbf{1}$ being an eigenvector of $A^2$. We note that \[\sum_{u\in V(G)} m_2(u,v)=\sum_{uv\in E(G)}d_u,\] as every walk of length two starting from $v$ is characterized by walking along an edge to some $u$ and then taking one of the $d_u$ edges connected to $u$. Thus $\mathbf{1}'$ or $\mathbf{1}$ is an eigenvector of $A^2$ iff $\sum_{uv\in E(G)}d_u$ is the same value for all $v$.
Assume that there exists a $\lambda$ such that $\lambda= \sum_{uv\in E} d_u$ for all $v$. Let $v$ be a vertex with minimum degree $d$, and let $v'$ be a vertex with maximum degree $D$. Then
\[
\lambda=\sum_{uv\in E} d_u\le d\cdot D\le \sum_{uv'\in E}d_u=\lambda,
\]
where the first inequality follows from the fact that each of the $d$ terms in the sum can have value at most $D$, and the second from the fact that each of the $D$ terms in the sum have value at least $d$. Since both sides of the inequality are equal, both inequalities must in fact be equalities. We conclude that if a vertex in $G$ has degree $d$ then all of its neighbors have degree $D$, and conversely if a vertex in $G$ has degree $D$ then all of its neighbors will have degree $d$. Since $G$ is assumed to be connected, it follows that all vertices must have degree $d$ or $D$. Moreover, all the vertices of $V_1$ have the same degree, and similarly all the vertices of $V_2$ have the same degree. Thus $G$ is biregular.
\end{proof}
\begin{thm}\label{T-two}
If $G$ is connected and $A^r=f(Q)$ or $A^r=f(L)$, then $G$ is regular or biregular.
\end{thm}
\begin{proof}
Let $P$ stand for either $Q$ or $L$, and assume that $A^r=f(P)$. If $r$ is odd or $G$ is not bipartite, then $G$ must be regular by Lemma~\ref{L-odd bip}, so we will assume that $G$ is bipartite and $r=2k$ for some $k$. In this case we have $(A^2)^k=f(P)$, so by Lemma~\ref{L-same} any eigenvector of $P$ will also be an eigenvector of $A^2$. If $P=Q$ and if $G$ is bipartite, then it is easy to see that $\mathbf{1}'$ will be an eigenvector of $P$, and hence of $A^2$. If $P=L$, then $\mathbf{1}$ is an eigenvector of $P$ and hence of $A^2$. In either case we conclude that $G$ is biregular by Lemma~\ref{L-1}.
\end{proof}
\section{Relations Involving $\NL$}\label{S-NL}
From the definition of $\mathcal{L}$ it is immediate that if $G$ is $d$-regular or $(d_1,d_2)$-biregular then $\mathcal{L}=I-\frac{1}{d}A$ or $\mathcal{L}=I-\frac{1}{\sqrt{d_1d_2}}A$, so when $G$ is regular or biregular it is possible to have $A^r=f(\NL)$ and $\NL^r=f(A)$.
We note the following (see \cite{chungButler}). If $G$ is connected then $\Eig{\NL}{0}$ has dimension 1 and is spanned by $D^{1/2}\mathbf{1}$. If $G$ is connected then $\Eig{L}{0}$ has dimension 1 and is spanned by $\mathbf{1}$.
\begin{lem}\label{L-NL}
If $D^{1/2}\mathbf{1}$ is an eigenvector of $A$, then $G$ is regular or biregular.
\end{lem}
\begin{proof}
$D^{1/2}\mathbf{1}$ being an eigenvector of $A$ is equivalent to the statement that there exists a $\lambda$ such that $\sum_{uv\in E(G)}\sqrt{d_u}=\lambda \sqrt{d_v}$ for all $v$, or equivalently that for all $v$ $\frac{1}{\sqrt{d_v}}\sum_{uv\in E(G)}\sqrt{d_u}$ is the same value, $\lambda$. Assume that this condition holds and let $v$ be a vertex of minimal degree $d$ and $v'$ a vertex of maximum degree $D$. Then
\[
\lambda=\frac{1}{\sqrt{d}}\sum_{uv\in E}\sqrt{d_u}\le \sqrt{d D}\le \frac{1}{\sqrt{D}}\sum_{uv\in E}\sqrt{d_u}=\lambda,
\]
since the first sum has $d$ terms that are at most $\sqrt{D}$ and the second has $D$ terms that are at least $\sqrt{d}$. We conclude that the inequalities are equalities, and hence that all vertices have degree $d$ or $D$, and that every neighbor of a vertex with degree $d$ has degree $D$ and vice versa. If $d=D$ we conclude that $G$ is regular. If $d\ne D$ we can partition vertices into those with degree $d$ and those with degree $D$, and this shows that $G$ is bipartite and hence biregular.
\end{proof}
\begin{thm}\label{T-NLA}
If $G$ is connected and $\NL^r=f(A)$, then $G$ is regular or biregular.
\end{thm}
\begin{proof}
$\Eig{\NL}{0}=\Eig{\NL^r}{0},\ \dim \Eig{\NL}{0}=1$ and $D^{1/2}\mathbf{1}\in \Eig{\NL}{0}$. Thus if $\NL^r=f(A)$, then $D^{1/2}\mathbf{1}$ will be an eigenvector of $A$ by Lemma~\ref{L-onedim}, and this implies that $G$ is either regular or biregular by Lemma~\ref{L-NL}.
\end{proof}
\begin{lem}\label{L-NLQ}
If $D^{1/2}\mathbf{1}$ is an eigenvector of $Q$, then $G$ is regular.
\end{lem}
\begin{proof}
$D^{1/2}\mathbf{1}$ being an eigenvector of $A$ is equivalent to the statement that there exists a $\lambda$ such that $d_v\sqrt{d_v}+\sum_{uv\in E}\sqrt{d_u}=\lambda \sqrt{d_v}$ for all $v$, or equivalently that $d_v+\frac{1}{\sqrt{d_v}}\sum_{uv\in E}\sqrt{d_u}$ is the same for all $v$. Assume that this condition holds and let $v$ be a vertex of minimal degree $d$ and $v'$ a vertex of maximum degree $D$. Then
\[
\lambda=d+\frac{1}{\sqrt{d}}\sum_{uv\in E}\sqrt{d_u}\le d+ \sqrt{d D}\le D+\sqrt{d D}\le D+\frac{1}{\sqrt{D}}\sum_{uv\in E}\sqrt{d_u}=\lambda,
\]
since the first sum has $d$ terms that each have value at most $\sqrt{D}$ and the second has $D$ terms that each have value at least $\sqrt{d}$. Thus every inequality must be an equality, and in particular this implies that $d=D$, so $G$ is regular.
\end{proof}
\begin{lem}\label{L-NLL}
If $\mathbf{1}$ is an eigenvector of $\NL$, then $G$ is regular.
\end{lem}
\begin{proof}
We have that $(\NL\mathbf{1})_v=1-\frac{1}{\sqrt{d_v}}\sum_{uv\in E} \frac{1}{\sqrt{d_u}}$, and that $\mathbf{1}$ is an eigenvector of $\NL$ only if this value is equal to the same value $\lambda$ for all $v$. Assume this is true and let $v$ be a vertex of maximum degree $D$. We then have that \[\lambda=1-\frac{1}{\sqrt{D}}\sum_{uv\in E} \frac{1}{\sqrt{d_u}}\le 1-\frac{1}{\sqrt{D}}\frac{D}{\sqrt{D}}=0,\] since the sum is minimized when each of the terms is equal to $1/\sqrt{D}$. But $\lambda \ge 0$ (because the spectrum of $\NL$ is nonnegative), so this inequality must be an equality. This implies that every vertex of maximum degree is adjacent only to vertices of maximum degree, and since $G$ is connected, we conclude that $G$ is regular of degree $D$.
\end{proof}
\begin{thm}
If $G$ is connected and $\NL^r=f(Q)$ or $Q^r=f(\NL)$, then $G$ is regular.
\end{thm}
\begin{proof}
Either case implies that $D^{1/2}\mathbf{1}$ is an eigenvector of $Q$ by Lemma~\ref{L-bothNonneg}, and this implies that $G$ is regular by Lemma~\ref{L-NLQ}.
\end{proof}
\begin{thm}
If $G$ is connected and $\NL^r=f(L)$ or $L^r=f(\NL)$, then $G$ is regular.
\end{thm}
\begin{proof}
Either case implies that $\mathbf{1}$ is an eigenvector of $\NL$ by Lemma~\ref{L-bothNonneg}, and this implies that $G$ is regular by Lemma~\ref{L-NLL}.
\end{proof}
Of relations involving the four matrices $A,\ Q,\ L,$ and $\NL$, the only remaining case is $A^r=f(\NL)$. Unfortunately, we do not have a complete characterization for this case, though experimental data suggests the following conjecture.
\begin{conj}\label{con-full}
If $G$ is connected and $A^r=f(\NL)$ for some polynomial $f$ and $r>0$, then $G$ is regular or biregular.
\end{conj}
We present some partial results related to this conjecture.
\begin{prop}\label{P-odd}
If $A^r=f(\NL)$ with $r$ odd, then $G$ is regular or biregular.
\end{prop}
\begin{proof}
If $r$ is odd then $\Eig{A^r}{\mu^r}=\Eig{A}{\mu}$ for all $\mu$. If $A^r=f(\NL)$, then $D^{1/2}\mathbf{1}$ is an eigenvector of $f(\NL)$, and hence of $A^r$, and hence of $A$, implying that $G$ is regular or biregular by Lemma~\ref{L-NL}.
\end{proof}
We note the following conjecture, which again experimental data suggests is true.
\begin{conj}\label{con-square}
If $G$ is connected and $D^{1/2}\mathbf{1}$ is an eigenvector of $A^2$, then $G$ is regular or biregular.
\end{conj}
\begin{prop}
If Conjecture~\ref{con-square} is true, then Conjecture~\ref{con-full} is true.
\end{prop}
\begin{proof}
The case of Conjecture~\ref{con-full} when $r$ is odd is proved in Proposition~\ref{P-odd}. If $r$ is even then we have $(A^2)^k=f(\NL)$, so $D^{1/2}\mathbf{1}$ will be an eigenvector of $A^2$ by Lemma~\ref{L-same}. Conjecture~\ref{con-square} being true then implies that $G$ is regular or biregular as desired.
\end{proof}
\section{General Polynomial Relations}\label{S-gen}
A more general question one can ask is about the existence of nontrivial polynomials $f$ and $g$ such that $f(X)=g(Y)$ for $X,\ Y$ matrices of a graph $G$. By nontrivial we mean that $f$ and $g$ are not of the form $f=up+c,\ g=vq+c$ where $p,\ q$ are the minimal polynomials of $X$ and $Y$ respectively, $c$ is a constant, and $u,\ v$ are arbitrary polynomials. When this occurs we have the following correspondence between eigenvalues of $X$ and eigenvalues of $Y$.
\begin{prop}\label{P-general}
Let $X$ and $Y$ be diagonalizable matrices with $f(X)=g(Y)$ for polynomials $f$ and $g$. If $\lambda_1,\ldots,\lambda_n$ are the eigenvalues of $X$ and $\mu_1,\ldots,\mu_n$ are the eigenvalues of $Y$, then $\{f(\lambda_1),\ldots,f(\lambda_n)\}=\{g(\mu_1),\ldots,g(\mu_n)\}$.
\end{prop}
Note that this result holds even if $f$ and $g$ are trivial, but the conclusion isn't particularly interesting.
\begin{proof}
Let $Z=f(X)=g(Y)$ and let $V'=\{v_1,\ldots,v_n\}$ be a basis of eigenvectors of $X$ with $Xv_i=\lambda_iv_i$. Then $Zv_i=f(X)v_i=f(\lambda_i)v_i$, so $Z$ will have eigenvalues $\{f(\lambda_1),\ldots,f(\lambda_n)\}$. A symmetric argument shows that $Z$ will have eigenvalues $\{g(\mu_1),\ldots,g(\mu_n)\}$, so these sets must be equal.
\end{proof}
For example, if $P_4$ denotes the path on 4 vertices then one can compute that
\[
A(A^2-2I)=Q^3-5Q^2+6Q-I.
\]
One can also compute that the eigenvalues of $A$ are $\frac{1+\sqrt{5}}{2},\ \frac{1-\sqrt{5}}{2},\ \frac{-1+\sqrt{5}}{2},\ \frac{-1-\sqrt{5}}{2}$, and that the eigenvalues of $Q$ are $2+\sqrt{2},\ 2-\sqrt{2},\ 2,\ 0$. If $f(x)=x(x^2-1)$ and $g(x)=x^3-5x^2+6x-1$, then
\begin{align*}
f(\frac{1+\sqrt{5}}{2})=f(\frac{1-\sqrt{5}}{2})&=g(2+\sqrt{2})=g(2-\sqrt{2})\\
f(\frac{-1+\sqrt{5}}{2})=f(\frac{-1-\sqrt{5}}{2})&=g(2)=g(0),
\end{align*}
which agrees with Proposition~\ref{P-general}.
On the other hand, if $G$ denotes the graph which has the following adjacency matrix, then one can prove that there exists no nontrivial relation $f(A)=g(Q)$.
\[
A_{G}=\begin{bmatrix}
0 & 1 & 1 & 1 & 1\\
1 & 0 & 1 & 1 & 1\\
1 & 1 & 0 & 1 & 0\\
1 & 1 & 1 & 0 & 0\\
1 & 1 & 0 & 0 & 0
\end{bmatrix}.
\]
The idea of the proof is as follows. One observes that the minimal polynomial of $A$ has degree 4, which implies that every power of $A$ can be expressed as a polynomial of $A$ that has degree at most 3. Thus if a nontrivial polynomial $f$ exists such that $f(A)=g(Q)$, it can be chosen to be of degree 3 or smaller. The minimal polynomial of $Q$ is also of degree 4, so we again conclude that if $g$ exists it can be chosen to have degree at most 3. In total, if $f,g$ exist then one can express them as a linear combination of matrices from the set $\{I,A,A^2,A^3,Q,Q^2,Q^3\}$. However, one can verify that this collection of matrices (thought of as $5^2$-dimensional vectors) are linearly independent, so there exist no nontrivial polynomials such that $f(A)=g(Q)$.
There does not seem to be an obvious characterization of graphs that satisfy $f(X)=g(Y)$, nor does there seem to be a characterization of what these polynomials $f$ and $g$ look like when this occurs, but we have not investigated this question very thoroughly. It also does not appear that one can refine Proposition~\ref{P-general} in such a way that, given the eigenvalues of $X$ and the relation $f(X)=g(Y)$, one can compute the eigenvalues of $Y$ in general, but there may exist special classes of relationships like $X^r=f(Y)$ for which this refinement is possible.
One direction for future study would be to answer questions of the following type: let $P$ be a property that a graph can have (such as being $(d_1,d_2)$-biregular or being isomorphic to $P_n$) and two matrices of graphs $X$ and $Y$. Can one give an explicit (nontrivial) relation $f(X)=g(Y)$ for all graphs satisfying $P$? If so, can one use this explicit relation to directly relate the eigenvalues of $X$ and $Y$ for graphs satisfying $P$? For example, we have the theorem that if $G$ is $(d_1,d_2)$-biregular, then $A^2=(Q-d_1I)(Q-d_2I)$. An example of another problem of this type is as follows:
\begin{quest}
Are there (non-trivial) functions $f_n,\ g_n$ such that $f_n(A)=g_n(Q)$ when $G=P_n$ for all $n$? If so, can one give an explicit (nice) construction of such functions?
\end{quest}
Another direction to explore would be to generalize results like Theorem~\ref{T-two}, stating that the only graphs satisfying $A^r=f(Q)$ are those that are regular or biregular. One could instead ask the following question: given matrices of graphs $X$ and $Y$ and a family of ordered pairs of polynomials $\mathcal{F}=\{(f,g)\}$, does there exist a (nice) property $P$ such that the only graphs satisfying $f(X)=g(Y)$ for some $(f,g)\in \mathcal{F}$ are those satisfying $P$? For example, we have the following result.
\begin{prop}\label{P-deg2}
If $f(A)=g(L)$ where $f$ is a polynomial of degree at most 2 with nonnegative coefficients and $g$ is an arbitrary polynomial, then $G$ is regular or biregular. Moreover, if $f$ can't be chosen to be $f(x)=x^2$, then $G$ is regular.
\end{prop}
\begin{proof}
If these polynomials exist, choose them such that $f$ is monic and has no constant term. Let $c$ denote the constant term of $g$. If $f(x)=x$ then $A\mathbf{1}=g(L)\mathbf{1}=c\mathbf{1}$ (since $L\mathbf{1}=0$), and this implies that $G$ is $c$-regular. If $f(x)=x^2+ax$, then $A^2\mathbf{1}+aA\mathbf{1}=f(A)\mathbf{1}=g(L)\mathbf{1}=c\mathbf{1}$. We conclude that $a d_v+\sum_{uv\in E(G)}d_u=c$ for all $v$ by using the same logic as in Lemma~\ref{L-1}. If $v$ is a vertex of minimum degree $d$ and $v'$ is a vertex of maximum degree $D$ we have (noting that $a\ge 0$)
\[
c=ad+\sum_{uv\in E(G)}d_u\le ad+dD\le aD+dD\le aD+\sum_{uv'\in E(G)}d_u=c,
\]
so we conclude that all inequalities are equalities. If $a\ne 0$ this implies that $d=D$, making $G$ regular. If $a=0$ and $d\ne D$, then one can partition the vertices of $G$ into those with degree $d$ and those with degree $D$, making $G$ biregular.
\end{proof}
We note that the assumption that $f$ have nonnegative coefficients can not be relaxed. Indeed, let $G'$ be defined by the adjacency matrix
\begin{equation}\label{eq}
A_{G'}=\begin{bmatrix}
0 & 1 & 1 & 1 & 1\\
1 & 0 & 0 & 0 & 1\\
1 & 0 & 0 & 1 & 0\\
1 & 0 & 1 & 0 & 0\\
1 & 1 & 0 & 0 & 0
\end{bmatrix}.
\end{equation}
One can check that in this case $3A^2-3A=-L^3+9L^2-20L+12I$.
We have analogous results for the signless Laplacian.
\begin{prop}\label{P-Qdeg2}
If $f(Q)=g(L)$ where $f$ is a polynomial of degree at most 2 with nonnegative coefficients and $g$ is an arbitrary polynomial, then $G$ is regular.
\end{prop}
\begin{proof}
If these polynomials exist, choose them such that $f$ is monic and has no constant term and let $c$ denote the constant term of $g$. To proceed as in Proposition~\ref{P-deg2}, we will need to understand how $\mathbf{1}$ interacts with $Q$ and $Q^2$. It is clear that $(Q\mathbf{1})_v=2d_v$. For $Q^2$ we have
\[
Q^2=(A+D)^2=A^2+AD+DA+D^2
\]
We know that $(A^2\mathbf{1})_v=\sum_{uv\in E(G)}d_u$, and it isn't difficult to see that \[(AD\mathbf{1})_v=\sum_{uv\in E(G)}d_u,\ (DA\mathbf{1})_v=d_v^2,\ (D^2\mathbf{1})_v=d_v^2.\] Thus in total we have $(Q^2\mathbf{1})_v=2d_v^2+2\sum_{uv\in E(G)}d_u$.
If $f(x)=x$, then $Q\mathbf{1}=f(L)\mathbf{1}=c\mathbf{1}$, and this implies that $G$ is $c/2$-regular. If $f(x)=x^2+ax$ then we conclude that $Q^2\mathbf{1}+aQ\mathbf{1}=c\mathbf{1}$. By comparing the $v$th coordinates of both sides, we see that $d_v^2+ad_v+\sum_{uv\in E(G)}d_u=c/2$ and this holds for all $v$. If $v$ denotes a vertex with minimum degree $d$ and $v'$ a vertex with maximum degree $D$ then
\begin{align*}
c/2=d^2+ad+&\sum_{uv\in E(G)}d_u\le d^2+ad+dD \le\\ D^2+aD+dD&\le
D^2+aD+\sum_{uv'\in E(G)}d_u=c/2,
\end{align*}
so the inequalities must be equalities and we conclude that $d=D$, making $G$ regular.
\end{proof}
Again the condition that $f$ have nonnegative coefficients can not be weakened. Indeed, if $G$ is a $(d_1,d_2)$-biregular graph then $(Q-d_1I)(Q-d_2I)=(L-d_1I)(L-d_2I)$. While biregular graphs are the most obvious counterexample, they are not the only ones. For example, if we consider $G'$ as defined in \eqref{eq}, then one can show that $3Q^2-21Q=-2L^3+15L^2-25L-24I$.
One can also ask whether polynomials $f$ and $g$ exist such that $f(X)=g(Y)$ when $X$ is a matrix associated to a graph $G$ and $Y$ is not. One such example is $Y=J$, the $n\times n$ matrix whose entries are all 1. Note that $J$ has rank 1, so the only non-trivial polynomials of $J$ are of the form $cI+dJ$ with $d\ne 0$. Thus if $f$ and $g$ exist such that $f(X)=g(J)$, one can always choose $g(J)=J$.
\begin{lem}\label{L-Jreg}
If $f(A)=J$ for some polynomial $f(x)$, then $G$ is regular and connected.
\end{lem}
Note that all the matrices that we considered earlier had the property that $X_G=X_{G_1}\oplus X_{G_2}$ whenever $G$ was the disjoint union of the graphs $G_1$ and $G_2$. This meant that the relation $f(X_G)=g(Y_G)$ held iff $f(X_{G'})=g(Y_{G'})$ held for any connected component $G'$ of $G$. This is not the case when considering $J$, so we emphasize here the fact that $G$ must be connected.
\begin{proof}
Assume that such an $f$ exists. If vertex $i$ and vertex $j$ belong to different components of $G$, then for all $r$, $A^r_{ij}=0$, which implies that there exists no polynomial such that $f(A)=J$. It follows that $G$ must be connected.
If $\pf$ is the positive eigenvector of $A$ guaranteed by the Perron-Frobenius theorem, then
\[
J\pf=f(A)\pf=f(\Lambda)\pf,
\]
so $\pf$ is an eigenvector of $J$, but the only positive eigenvectors of $J$ are scaler multiples of $\mathbf{1}$, so $\pf=c\mathbf{1}$ for some $c$, which means $G$ must be regular.
\end{proof}
Let $m_A(x)$ denote the minimal polynomial of $A$. If $G$ is a $k$-regular graph, let $m'_A(x)=m_A(x)/(x-k)$. Note that $m'_A(x)=\prod_i(x-\lambda_i)$, where the $\lambda_i$ range over all distinct eigenvalues of $A$ that are not equal to $k$.
\begin{lem}\label{L-divide}
If $f(A)=J$, then $f(x)\mid m'_A(x)$.
\end{lem}
\begin{proof}
By Lemma~\ref{L-Jreg} we can assume that $G$ is $k$-regular for some $k$. Let $v$ be an eigenvector of $A$ with eigenvalue $\lambda\ne k$. We have
\[
Jv=f(A)v=f(\lambda)v,
\]
so $v$ is an eigenvector of $J$ with eigenvalue $f(\lambda)$. But $v\ne c\mathbf{1}$ by assumption of $\lambda\ne k$, so it must be that $v$ is a null-vector of $J$ and $f(\lambda)=0$. As every distinct eigenvalue of $A$ not equal to $k$ is a root of $f$, we conclude that $f(x)\mid m'_A(x)$.
\end{proof}
\begin{thm}
A polynomial $f(x)$ exists such that $f(A)=J$ iff $G$ is connected and regular. Moreover, if $f$ is chosen to have minimum degree, then $f(x)=cm'_A(x)$ for some $c\ne 0$.
\end{thm}
\begin{proof}
Lemma~\ref{L-Jreg} gives the forward direction, so assume $G$ is connected and $k$-regular. This implies that the null-space of $A-kI$ has dimension 1 and is spanned by $\mathbf{1}$. We also have \[(A-kI)m'_A(A)=m_A(A)=0.\] These two facts imply that every column of $m'_A(A)$ is a scaler multiple of $\mathbf{1}$. As $m'_A(A)$ is a symmetric matrix, we must have $m'_A(A)=c\mathbf{1}\mathbf{1}^T=cJ$ for some $c\ne 0$, so $f(x)=\frac{1}{c}m'_A(x)$ gives the desired polynomial.
Finally, assume $f(A)=J$ where $f(x)$ is chosen to have minimum degree. From the above proof we know that $\deg(f)$ can be at most $\deg(m'_A)$, but by Lemma~\ref{L-divide} we must have $\deg(f)\ge \deg(m'_A)$. We conclude that $\deg(f)=\deg(m'_A)$ and that $f(x)=cm'_A(x)$ for some $c\ne 0$.
\end{proof}
The above statement implies that if $G$ is a connected, $k$-regular graph such that $A$ has $r+1$ distinct eigenvalues, then there exists $c,d\ne 0$ such that $cm'_A(x)$ is a monic polynomial of degree $r$, and $cm'_A(A)=dJ$. The $r=2$ case corresponds to connected strongly regular graphs, which are usually defined combinatorially as is done so in \cite{AGT}, for example. Given any connected strongly regular graph, one can derive an equation of the form $cm'_A(A)=dJ$ by having the coefficients of the polynomial be defined in terms of combinatorial parameters of the graph. It would be interesting to know if this process could be reversed in general. That is, can one always interpret the coefficients of the equation $cm'_A(A)=dJ$ in terms of certain parameters of the underlying graph, and can these parameters be used to give a combinatorial description of connected regular graphs with precisely $r+1$ eigenvalues?
\section{Acknowledgments}
The author would like to thank Richard Stanley for suggesting this research topic, as well as his assistance with the general structure of the paper.
\bibliographystyle{plain}
| {
"timestamp": "2017-09-07T02:08:59",
"yymm": "1706",
"arxiv_id": "1706.03298",
"language": "en",
"url": "https://arxiv.org/abs/1706.03298",
"abstract": "We derive a correspondence between the eigenvalues of the adjacency matrix $A$ and the signless Laplacian matrix $Q$ of a graph $G$ when $G$ is $(d_1,d_2)$-biregular by using the relation $A^2=(Q-d_1I)(Q-d_2I)$. This motivates asking when it is possible to have $X^r=f(Y)$ for $f$ a polynomial, $r>0$, and $X,\\ Y$ matrices associated to a graph $G$. It turns out that, essentially, this can only happen if $G$ is either regular or biregular.",
"subjects": "Combinatorics (math.CO); Spectral Theory (math.SP)",
"title": "Polynomial Relations Between Matrices of Graphs",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9899864302631447,
"lm_q2_score": 0.7154240018510026,
"lm_q1q2_score": 0.7082600537170475
} |
https://arxiv.org/abs/1609.00840 | The Second Discriminant of a Univariate Polynomial | We define the second discriminant $D_2$ of a univariate polynomial $f$ of degree greater than $2$ as the product of the linear forms $2\,r_k-r_i-r_j$ for all triples of roots $r_i, r_k, r_j$ of $f$ with $i<j$ and $j\neq k, k\neq i$. $D_2$ vanishes if and only if $f$ has at least one root which is equal to the average of two other roots. We show that $D_2$ can be expressed as the resultant of $f$ and a determinant formed with the derivatives of $f$, establishing a new relation between the roots and the coefficients of $f$. We prove several notable properties and present an application of $D_2$. | \section{Introduction}
\label{sec:introduction}
The discriminant of a univariate polynomial $f=f(x)$ may be defined as a function of the coefficients of $f$ in $x$, whose vanishing is a necessary and sufficient condition for $f$ to have multiple roots for $x$. The term {\em discriminant} was used early by Sylvester in \cite{S1851O} and it will be referred to as the \emph{first discriminant} hereinafter. The first discriminant of $f$ contains information about the nature of the roots\footnote{For example, if the discriminant of a cubic polynomial $f$ with real coefficients is positive, then $f$ has no complex root \cite{G1990S}.} of $f$ and has played a fundamental role in the study of polynomial equations. It has many remarkable properties \cite{C1993R,GKZ1994D} and has been used in diverse areas ranging from algebraic geometry and Galois theory to bifurcation analysis and number theory.
To define the first discriminant $D_1$ of $f$, one considers the simple form $r_i-r_j$ for any pair of roots $r_i, r_j$ of $f$ with $i\neq j$ and takes the product of all such forms as $D_1$, which can be expressed as the resultant of $f$ and its derivative. In this paper, we define the {\em second discriminant} $D_2$ of $f$ (of degree greater than $2$) as the product of the linear forms
$2\,r_k-r_i-r_j$ for all triples of roots $r_i, r_k, r_j$ of $f$ with $i<j$ and $j\neq k$, $k\neq i$.
More concretely, let
\begin{equation}\label{eqF}
f=x^n+a_{n-1}x^{n-1}+\cdots +a_1x+a_0
\end{equation}
be any univariate polynomial
of degree $n\geq 3$ in $x$ with real or complex coefficients. Let $r_1,\ldots,r_n$ be the $n$ roots of $f$ for $x$ over ${\Bbb C}$, the field of complex numbers. By a \emph{symmetric triple} of roots, we mean a triple $(r_i,r_k,r_j)$ of roots of $f$ with $i<j$ and $j\neq k$, $k\neq i$ such that $r_k=(r_i+r_j)/2$.
Then, obviously, $D_2=0$ if and only if $f$ has a symmetric triple of roots.
We will show that $D_2$ can be expressed as the resultant of $f$ and a determinant formed with the derivatives of $f$, and thus as a polynomial in $a_0,\ldots,a_{n-1}$ with rational coefficients.
Several other properties of $D_2$ will also be proved, highlighting the geometric interest of the symmetric triples of roots.
The second discriminant $D_2$ complements the well-known first discriminant $D_1$ of $f$ in depicting the structural properties such as distribution, position, and configuration of the roots of $f$.
In the following section, the second discriminant $D_2$ for an arbitrary univariate polynomial $f$ of degree $n$ is defined formally in terms of the roots of $f$;
some simple properties of $D_2$ are then proved. In Sections \ref{sec:Construction} and \ref{sec:Properties}, we show that $D_2$ as a polynomial in the coefficients of $f$ is irreducible of total degree $3\,(n-1)(n-2)/2$. In Sections \ref{sec:resultant} and \ref{sec:ideals}, we elaborate $D_2$ with resultants and ideals from the perspective of modern algebra, which leads to different ways for the construction of $D_2$. In Section \ref{sec:determinant}, we provide exact formulas for the degrees of some determinant polynomials involved in the construction of $D_2$. Finally, an application of $D_2$ to the classification of root configurations is presented and the paper is concluded with some remarks in Section \ref{sec:ApplicationRemarks}.
\section{Symmetric Triples of Roots and the Second Discriminant}
\label{sec:Definition}
Let $f\in \mathbb{C}[x]$ be as in \eqref{eqF} with $\deg(f, x)=n\geq 3$ and $r_1, \ldots, r_n$ be the $n$ roots of $f$ over $\mathbb{C}$ as above. Consider any two roots $r_i$ and $r_j$. We call $(r_i+r_j)/2$ the \emph{average} of $r_i$ and $r_j$. For any triple $\bm{r}=(r_i, r_k, r_j)$, where
\[f(r_i)=f(r_j)=f(r_k)=0 \quad\mbox{and}\quad i< j, j\neq k, k\neq i,\]
if $r_k$ is the average of $r_i$ and $r_j$, i.e., $r_k=(r_i+r_j)/2$, then $\bm{r}$ is called a \emph{symmetric triple} of roots of $f$. We are interested in the condition under which $f$ has symmetric triples of roots.
Recall that the first discriminant of $f$ may be defined as
\[
D_1= \prod_{1\leq i<j\leq n}(r_i-r_j)^2=\pm\prod_{\scriptsize{\begin{array}{c}
1\leq i,j\leq n\\
i\neq j
\end{array}}}(r_i-r_j).
\]
$D_1=0$ if and only if $f$ has a multiple root.
To obtain the condition under which $f$ has a symmetric triple of roots, we define the second discriminant $D_2$ of $f$ as follows:
\begin{equation}\label{eq:D2}
D_2 = \prod_{
\scriptsize{\begin{array}{c}
1\leq i,j,k\leq n\\
i< j,j\neq k,k\neq i
\end{array}}
}(2\,r_k-r_i-r_j),
\end{equation}
a symmetric polynomial of total degree $n(n-1)(n-2)/2$ in $r_1,\ldots,r_n$. For the sake of simplicity, we shall write $i< j\neq k$ for the range of $i,j,k$ determined by $1< i,j,k\leq n$ and $i< j$, $j\neq k$, $k\neq i$.
\begin{remark}\em
$D_1=0$ does not imply $D_2=0$, and vice versa.
\end{remark}
\begin{proposition}
\begin{enumerate}[$($a$)$]
\item If $D_1=0$ and $D_2\neq 0$, then any root of $f$ has multiplicity not greater than $2$.
\item If $D_1\neq 0$ and $D_2=0$, then there exist pairwise distinct $r_i, r_j, r_k$ with $i<j\neq k$ such that $2\,r_k-r_i-r_j=0$.
\end{enumerate}
\end{proposition}
\begin{proof}
(a) Suppose that $f$ has a root with multiplicity $m>2$ under the condition $D_1=0$ and $D_2\neq 0$, e.g., $r_i=r_j=r_k~(i< j\neq k)$. This is then a special case of
\[r_k=(r_i+r_j)/2,\]
so $D_2=0$, which leads to contradiction.
(b) $D_1\neq 0$ implies that $r_i, r_j, r_k$ are pairwise distinct for any $i<j\neq k$ and $D_2=0$ implies the existence of $r_i, r_j, r_k$ with $i<j\neq k$ such that $2\,r_k-r_i-r_j=0$.
\end{proof}
\begin{theorem}
$D_2=0$ if and only if $f$ has a symmetric triple of roots.
\end{theorem}
\begin{proof}
($\Longrightarrow$) $D_2=0$ implies that there exist $r_i$, $r_j$, $r_k$ such that $r_k=(r_i+r_j)/2$. Thus $(r_i, r_k, r_j)$ is a symmetric triple of roots which we seek for.
($\Longleftarrow$) Suppose that $(r_i, r_k, r_j)$ is a symmetric triple of roots that $f$ has. Then $r_k=(r_i+r_j)/2$. It follows that
\[D_2=\prod_{i< j\neq k}(2\,r_k-r_i-r_j)=0.\]
\vspace{-0.5cm}
\end{proof}
The second discriminant $D_2$ defined above is a polynomial in the roots $r_1,\ldots,r_n$ of $f$. This polynomial is symmetric with respect to the roots, so $D_2$ can be expressed as another polynomial in the coefficients $a_0,\ldots,a_{n-1}$ of $f$. We will provide explicit formulas and simple algorithmic approaches for the construction of the polynomial in $a_i$, together with several properties about $D_2$.
\section{Expression of the Second Discriminant}
\label{sec:Construction}
In this section, we show that the second discriminant $D_2$ of $f$ can be expressed as a polynomial in $a_0,\ldots,a_{n-1}$, the coefficients of $f$. The expression of $D_2$ we have discovered as the resultant of $f$ and the determinant of a shifting matrix formed with the derivatives $f^{(1)}, \ldots, f^{(n)}$ of $f$, given in the following theorem, appears pretty amazing. It is puzzling how and why the derivatives of $f$ get occurred in $H$ so structurally. We will answer this question in Lemma~\ref{lem:resH}
by linking $H$ to the resultant of two other polynomials derived from $f$.
As usual, denote by $\det(M)$ the determinant of any square matrix $M$ and
by $\res(f, g, x)$ the Sylvester resultant of any two polynomials $f$ and $g$ with respect to $x$.
\begin{theorem}\label{thm:D2} The second discriminant $D_2$ of $f$ is equal to the
resultant of $f$ and a determinant $H$ formed with the derivatives of $f$ with respect to $x$. More precisely,
\[
D_2=\res(f, H, x),
\]
where $H$ is the $(n-2)$th leading principal minor of the following matrix
\begin{equation}\label{matM}
M=\left(
\begin{array}{cccccccc}
\frac{f^{(2)}}{2!} & \frac{f^{(4)}}{4!} & \frac{f^{(6)}}{6!} & \cdots \!\!\!&\!\!\! \cdots & \frac{f^{(2 l )}}{(2 l )!} & \cdots \!\!\!&\!\!\! \cdots\\\smallskip
\frac{f^{(1)}}{1!} & \frac{f^{(3)}}{3!} & \frac{f^{(5)}}{5!} & \cdots \!\!\!&\!\!\! \cdots & \frac{f^{(2 l -1)}}{(2 l -1)!} & \cdots \!\!\!&\!\!\! \cdots\\\smallskip
0 & \frac{f^{(2)}}{2!} & \frac{f^{(4)}}{4!} & \cdots \!\!\!&\!\!\! \cdots & \frac{f^{(2 l -2)}}{(2 l -2)!} & \cdots \!\!\!&\!\!\! \cdots\\\smallskip
0 & \frac{f^{(1)}}{1!} & \frac{f^{(3)}}{3!} & \cdots \!\!\!&\!\!\! \cdots & \frac{f^{(2 l -3)}}{(2 l -3)!} & \cdots \!\!\!&\!\!\! \cdots\\\smallskip
0 & 0 & \frac{f^{(2)}}{2!} &\cdots \!\!\!&\!\!\! \cdots &\frac{f^{(2 l -3)}}{(2 l -3)!} & \cdots \!\!\!&\!\!\! \cdots\\\smallskip
\vdots & \vdots & \vdots & \ddots \!\!\!&\!\!\! &\vdots & \ddots \!\!\!&\!\!\! \\[-12pt]
\vdots & \vdots & \vdots & \!\!\!&\!\!\!\hspace{-2pt}\raisebox{0.10cm}{\mbox{$\ddots$}} & \vdots & \!\!\!&\!\!\!\hspace{-2pt}\raisebox{0.10cm}{\mbox{$\ddots$}}
\end{array}
\right)
\end{equation}
and $f^{(\imath)}$ denotes the $\imath$th derivative of $f$.
\end{theorem}
Note that
\[\res(f,H,x)=\prod_{k=1}^nH(r_k),\]
where $r_1,\ldots,r_n$ are the $n$ roots of $f$ as before.
To prove Theorem~\ref{thm:D2}, we only need to show that for each $k$, $H(r_k)$ is the product of $2\,r_k-r_i-r_j$ for all $i,j$ with $i< j\neq k$. The proof will be divided into two parts. In the first part, it is shown that for any $i,j$ with $i< j\neq k$, $2\,r_k-r_i-r_j$ is a divisor of $H(r_k)$ (see Lemmas \ref{lem:rootH} and \ref{lem:division}). The second part is devoted to proving that the leading term of $H(r_k)$ with respect to $r_k$ is $(2\,r_k)^{\frac{(n-1)(n-2)}{2}}$ (see Lemma \ref{lem:initerm}).
\begin{lemma}\label{lem:rootH}
If $r_k=(r_i+r_j)/2$ for $i< j\neq k$, then $H(r_k)=0$.
\end{lemma}
\begin{proof}
It suffices to show that the lemma holds for $k=1$, $i=2$, and $j=3$.
Denote by $\Omega_{l}^{\gamma}$ the set of all $\gamma$-tuples obtained from $(l,\ldots,n)$ by deleting $n-\gamma$ components, where $l$ is a positive integer not greater than $n$.
Let $b_\imath=x-r_\imath$ for $\imath=1,\ldots,n$. By calculus, it is easy to verify that
\begin{equation}\label{eq:Fkr}
\frac{f^{(\jmath)}}{\jmath\,!}=\left\{
\begin{array}{cl}
\sum_{(\imath_1,\ldots,\imath_{n-\jmath})\in\Omega_1^{n-\jmath}}{b_{\imath_1}\cdots b_{\imath_{n-\jmath}}},&\jmath=0,\ldots, n-1;\\[8pt]
1,&\jmath=n.
\end{array}
\right.
\end{equation}
Let $c_\imath=r_1-r_\imath$ for $\imath=2,\ldots,n$ and suppose that $r_1=(r_2+r_3)/2$. Then $c_2+c_3=0$. Substituting $x=r_1$ into \eqref{eq:Fkr} and observing that any term ${b_{\imath_1}\cdots b_{\imath_{n-\jmath}}}$ involving $x-r_1$ vanishes at $x=r_1$, we have
\begin{align*}
\frac{f^{(\jmath)}}{\jmath\,!}\Big|_{x=r_1}&=\sum\limits_{(\imath_1,\ldots,\imath_{n-\jmath})\in\Omega_1^{n-\jmath}}{b_{\imath_1}\cdots b_{\imath_{n-\jmath}}}\Big|_{x=r_1}\\
&=t_{n-\jmath}+c_2c_3t_{n-\jmath-2}+c_2t_{n-\jmath-1}+c_3t_{n-\jmath-1}\\
&=c_2c_3t_{n-\jmath-2}+(c_2+c_3)t_{n-\jmath-1}+t_{n-\jmath}\\
&=-c_2^2t_{n-\jmath-2}+t_{n-\jmath},
\end{align*}
where
\[t_{n-\jmath}=\left\{
\begin{array}{cl}
\sum\limits_{(\imath_1,\ldots,\imath_{n-\jmath})\in\Omega_4^{n-\jmath}}{c_{\imath_1}\cdots c_{\imath_{n-\jmath}}}&\mbox{if}\,~3\leq \jmath\leq n-1;\\[8pt]
1&\mbox{if}\,~\jmath=n;\\[6pt]
0&\mbox{if}\,~\jmath\leq 2\mbox{~or~} \jmath\geq n+1.
\end{array}
\right.
\]
Substitution of $t_{n-\jmath}$ into $H(r_1)$ yields
\[\begin{array}{l}\medskip
H(r_1)=\left|
\begin{array}{cccccc}\smallskip
-c_2^2t_{n-4} & -c_2^2t_{n-6}+t_{n-4} & -c_2^2t_{n-8}+t_{n-6} & \cdots \!\!\!&\!\!\! \cdots & -c_2^2t_{n-2\jmath-2}+t_{n-2\jmath} \\ \smallskip
-c_2^2t_{n-3} & -c_2^2t_{n-5}+t_{n-3} & -c_2^2t_{n-7}+t_{n-5} & \cdots \!\!\!&\!\!\! \cdots & -c_2^2t_{n-2\jmath-1}+t_{n-2\jmath+1} \\ \smallskip
0 & -c_2^2t_{n-4} & -c_2^2t_{n-6}+t_{n-4} & \cdots \!\!\!&\!\!\! \cdots & -c_2^2t_{n-2\jmath}+t_{n-2\jmath+2} \\ \smallskip
0 & -c_2^2t_{n-3} & -c_2^2t_{n-5}+t_{n-3} & \cdots \!\!\!&\!\!\! \cdots & -c_2^2t_{n-2\jmath+1}+t_{n-2\jmath+3} \\ \smallskip
0 & 0 & -c_2^2t_{n-4}+t_{n-2} & \cdots \!\!\!&\!\!\! \cdots & -c_2^2t_{n-2\jmath+2}+t_{n-2\jmath+4} \\ \smallskip
\vdots &\vdots & \vdots & \ddots \!\!\!&\!\!\! &\vdots \\[-13pt]
\vdots &\vdots & \vdots & \!\!\!&\!\!\! \hspace{-4pt}\raisebox{0.12cm}{\mbox{$\ddots$}} &\vdots \\[6pt]\smallskip
0 & 0 & 0 & \cdots \!\!\!&\!\!\! \cdots & 0 \\ \smallskip
0 & 0 & 0 & \cdots \!\!\!&\!\!\! \cdots & 0 \\ \smallskip
0 & 0 & 0 & \cdots \!\!\!&\!\!\! \cdots & 0 \\ \smallskip
0 & 0 & 0 & \cdots \!\!\!&\!\!\! \cdots & 0
\end{array}
\right.\\
\qquad\qquad\qquad\quad\qquad\qquad\left.
\begin{array}{cccccccc}
\smallskip
\cdots \!\!\!&\!\!\! \cdots & 0 & \cdots \!\!\!&\!\!\! \cdots & 0 & 0 & 0\\
\smallskip
\cdots \!\!\!&\!\!\! \cdots & 0 & \cdots \!\!\!&\!\!\! \cdots & 0 & 0 & 0\\
\cdots \!\!\!&\!\!\! \cdots & 0 & \cdots \!\!\!&\!\!\! \cdots & 0 & 0 & 0\\
\smallskip
\cdots \!\!\!&\!\!\! \cdots & 0 & \cdots \!\!\!&\!\!\! \cdots & 0 & 0 & 0 \\
\smallskip
\cdots \!\!\!&\!\!\! \cdots & 0 & \cdots \!\!\!&\!\!\! \cdots & 0 & 0 & 0 \\
\smallskip
\ddots \!\!\!&\!\!\! & \vdots &\ddots \!\!\!&\!\!\! & \vdots & \vdots & \vdots \\[-12pt]
\smallskip
\!\!\!&\!\!\! \hspace{-4pt}\raisebox{0.1cm}{\mbox{$\ddots$}} & \vdots & \!\!\!&\!\!\! \hspace{-4pt}\raisebox{0.1cm}{\mbox{$\ddots$}} & \vdots & \vdots & \vdots \\[6pt]
\smallskip
\cdots \!\!\!&\!\!\! \cdots & -c_2^2t_{2\jmath-6}+t_{2\jmath-4} &\cdots \!\!\!&\!\!\! \cdots & -c_2^2+t_2 & 1 & 0\\
\smallskip
\cdots \!\!\!&\!\!\! \cdots & -c_2^2t_{2\jmath-5}+t_{2\jmath-3} & \cdots \!\!\!&\!\!\!\cdots & -c_2^2t_1+t_3 & t_1 & 0\\
\smallskip
\cdots \!\!\!&\!\!\! \cdots & -c_2^2t_{2\jmath-4}+t_{2\jmath-2} & \cdots \!\!\!&\!\!\!\cdots & -c_2^2t_2+t_4 & -c_2^2+t_2 & 1\\
\smallskip
\cdots \!\!\!&\!\!\! \cdots & -c_2^2t_{2\jmath-3}+t_{2\jmath-1} & \cdots \!\!\!&\!\!\!\cdots & -c_2^2t_3+t_5 & -c_2^2t_1+t_3 & t_1
\end{array}
\right|.
\end{array}
\]
\vskip 16cm
For each $\imath=n-2,\ldots,2$, add the $\imath$th column multiplied by $c_2^2$ to the $(\imath-1)$th column of $H(r_1)$ iteratively. It follows that
\[
H(r_1)=\left|
\begin{array}{ccccccccccc}
0 & t_{n-4} & t_{n-6} & \cdots \!\!\!&\!\!\! \cdots & t_{n-2\jmath} & \cdots \!\!\!&\!\!\! \cdots & 0 & 0\\
0 & t_{n-3} & t_{n-5} & \cdots \!\!\!&\!\!\! \cdots & t_{n-2\jmath+1} & \cdots \!\!\!&\!\!\! \cdots& 0 & 0\\
0 & 0 & t_{n-4} & \cdots \!\!\!&\!\!\! \cdots & t_{n-2\jmath+2} & \cdots \!\!\!&\!\!\! \cdots& 0 & 0\\
0 & 0 & t_{n-3} & \cdots \!\!\!&\!\!\! \cdots & t_{n-2\jmath+3} & \cdots \!\!\!&\!\!\! \cdots& 0 & 0\\
\vdots &\vdots & \vdots & \ddots \!\!\!&\!\!\! & \vdots &\ddots \!\!\!&\!\!\!& \vdots & \vdots\\[-10pt]
\vdots &\vdots & \vdots & \!\!\!&\!\!\!\hspace{-4pt}\raisebox{0.1cm}{\mbox{$\ddots$}} & \vdots &\!\!\!&\!\!\!\hspace{-4pt}\raisebox{0.1cm}{\mbox{$\ddots$}} & \vdots & \vdots\\
0 & 0 & 0 & \cdots \!\!\!&\!\!\! \cdots & \vdots & \cdots \!\!\!&\!\!\!\cdots & 1 & 0\\[-6pt]
0 & 0 & 0 & \cdots \!\!\!&\!\!\! \cdots & \vdots & \cdots \!\!\!&\!\!\!\cdots & t_1 & 0\\[-6pt]
0 & 0 & 0 & \cdots \!\!\!&\!\!\! \cdots & \vdots & \cdots \!\!\!&\!\!\!\cdots & t_2 & 1\\[-6pt]
0 & 0 & 0 & \cdots \!\!\!&\!\!\! \cdots & \vdots & \cdots \!\!\!&\!\!\!\cdots & t_3 & t_1
\end{array}
\right|=0.
\]
\vspace{-0.5cm}
\end{proof}
\begin{lemma}\label{lem:division}
For any $i< j\neq k$, the linear form $2\,r_k-r_i-r_j$ divides $H(r_k)$.
\end{lemma}
\begin{proof}
Let $r_k=(r_i+r_j)/2$, where $i$ and $j$ are arbitrary but fixed. By Lemma~\ref{lem:rootH}, $H(r_k)=0$. It follows that
\[[r_k-(r_i+r_j)/2]\mid H(r_k), \mbox{~~~or~~~} (2\,r_k-r_i-r_j)\mid H(r_k)\]
over ${\Bbb Q}$ (the field of rational numbers).
\end{proof}
Note that $u\mid v$ stands for ``$u$ divides $v$'' as usual.
Let $c_\imath=r_1-r_\imath$ for $\imath=2,\ldots,n$. It is easy to verify that
\begin{equation}
\frac{f^{(\jmath)}}{\jmath\,!}\Bigg|_{x=r_1}=t^*_{n-\jmath},
\end{equation}
where
\[
t^*_{n-\jmath}=\left\{
\begin{array}{cl}
\sum\limits_{(\imath_1,\ldots,\imath_{n-\jmath})\in\Omega_2^{n-\jmath}}{c_{\imath_1}\cdots c_{\imath_{n-\jmath}}}&\mbox{if}\,~1\leq \jmath\leq n-1;\\[10pt]
1&\mbox{if}\,~\jmath=n.
\end{array}
\right.
\]
Let $M(r_1)$ be the matrix obtained from $M$ in \eqref{eq:D2} by replacing $x$ with $r_1$. Then
\begin{equation*}
M(r_1)=\left(
\begin{array}{cccccccc}
t^*_{n-2} & t^*_{n-4} & t^*_{n-6} & \cdots \!\!\!&\!\!\! \cdots & t^*_{n-2\jmath} & \cdots \!\!\!&\!\!\! \cdots\\
t^*_{n-1} & t^*_{n-3} & t^*_{n-5} & \cdots \!\!\!&\!\!\! \cdots & t^*_{n-2\jmath+1} & \cdots \!\!\!&\!\!\! \cdots\\
0 & t^*_{n-2} & t^*_{n-4} & \cdots \!\!\!&\!\!\! \cdots & t^*_{n-2\jmath+2} & \cdots \!\!\!&\!\!\! \cdots \\
0 & t^*_{n-1} & t^*_{n-3} & \cdots \!\!\!&\!\!\! \cdots & t^*_{n-2\jmath+3} & \cdots \!\!\!&\!\!\! \cdots\\
\vdots & \vdots & \vdots &\ddots \!\!\!&\!\!\! &\vdots & \ddots \!\!\!&\!\!\!\\[-10pt]
\vdots & \vdots & \vdots & \!\!\!&\!\!\! \hspace{-4pt}\raisebox{0.1cm}{\mbox{$\ddots$}} &\vdots &\!\!\!&\!\!\!\hspace{-4pt}\raisebox{0.1cm}{\mbox{$\ddots$}}
\end{array}
\right).
\end{equation*}
Since $C_{n-1}^{n-\jmath}r_1^{n-\jmath}$ is the leading coefficient of $t^*_{n-\jmath}$ with respect to $r_1$,
$t^*_{n-\jmath}$ can be written as
\[
t^*_{n-\jmath}=C_{n-1}^{n-\jmath}r_1^{n-\jmath}+\mathcal{O}(r_1^{n-\jmath}),\]
where $\mathcal{O}(r_1^{n-\jmath})$ denotes terms of degree less than $n-\jmath$ in $r_1$.
Now let $M_n(r_1)$ be the $(n-2)$th leading principal minor of the matrix obtained from $M(r_1)$ by replacing each entry $t^*_{n-\jmath}$ with $C_{n-1}^{n-\jmath}r_1^{n-\jmath}$. Then
\[
M_n(r_1)=\Big|
\pt{m}_1(r_1),\ldots,\pt{m}_\imath(r_1),\ldots,\pt{m}_{n-2}(r_1)
\Big|,
\]
where
\[
\pt{m}_\imath(r_1)=\left\{
\begin{array}{l}
\Big(C_{n-1}^{n-2\imath}r^{n-2\imath}_1,\ldots,C_{n-1}^{n-2\imath+\jmath}r^{n-2\imath+\jmath}_1,\ldots, \underbrace{0,\ldots\ldots\ldots\ldots,0}_{\max(n-2\imath-2,0)\mbox{~terms}}\Big)^T\\[25pt]
\qquad\qquad\qquad\qquad\mbox{if}\,~\imath\leq \left\lceil \dfrac{n-2}{2}\right\rceil\mbox{~and~}0\leq \jmath\leq \min(n-3,2\,\imath-1);\\[25pt]
\Big(\underbrace{0,\ldots\ldots\ldots,0}_{2\imath-n\mbox{~terms}},C_{n-1}^{0}r_1^0,\ldots,C_{n-1}^{\jmath}r_1^\jmath,\ldots\Big)^T\qquad\\[25pt]
\qquad\qquad\qquad\qquad\mbox{if}\,~\imath> \left\lceil \dfrac{n-2}{2}\right\rceil\mbox{~and~}0\leq \jmath\leq 2\,n-2\,\imath-3.
\end{array}
\right.
\]
Therefore, $M_n(r_1)$ has the following form:
\[\begin{array}{l}\medskip
M_n(r_1)=\left|
\begin{array}{cccccccc}\smallskip
C_{n-1}^{n-2}r_1^{n-2} & C_{n-1}^{n-4}r_1^{n-4} & C_{n-1}^{n-6}r_1^{n-6} & \cdots \!\!\!&\!\!\! \cdots & C_{n-1}^{n-2\jmath}r_1^{n-2\jmath} & \cdots \!\!\!&\!\!\! \cdots \\ \smallskip
C_{n-1}^{n-1}r_1^{n-1} & C_{n-1}^{n-3}r_1^{n-3} & C_{n-1}^{n-5}r_1^{n-5} & \cdots \!\!\!&\!\!\! \cdots & C_{n-1}^{n-2\jmath+1}r_1^{n-2\jmath+1} & \cdots \!\!\!&\!\!\! \cdots \\ \smallskip
0 & C_{n-1}^{n-2}r_1^{n-2} & C_{n-1}^{n-4}r_1^{n-4} & \cdots \!\!\!&\!\!\! \cdots & C_{n-1}^{n-2\jmath+2}r_1^{n-2\jmath+2} & \cdots \!\!\!&\!\!\! \cdots \\ \smallskip
0 & C_{n-1}^{n-1}r_1^{n-1} & C_{n-1}^{n-3}r_1^{n-3} & \cdots \!\!\!&\!\!\! \cdots & C_{n-1}^{n-2\jmath+3}r_1^{n-2\jmath+3} & \cdots \!\!\!&\!\!\! \cdots \\ \smallskip
0 & 0 & C_{n-1}^{n-2}r_1^{n-2} &\cdots \!\!\!&\!\!\! \cdots & C_{n-1}^{n-2\jmath+4}r_1^{n-2\jmath+4} \\
\vdots & \vdots & \vdots &\ddots \!\!\!&\!\!\! & \vdots &\ddots \!\!\!&\!\!\! \\ [-9pt]
\vdots & \vdots & \vdots & \!\!\!&\!\!\! \hspace{-4pt}\raisebox{0.1cm}{\mbox{$\ddots$}} & \vdots & \!\!\!&\!\!\! \hspace{-4pt}\raisebox{0.1cm}{\mbox{$\ddots$}}
\end{array}
\right|.
\end{array}
\]
Apparently, the above expression for $M_n(r_1)$ remains valid when $r_1$ is substituted by $r_k$ for any $k>1$.
\begin{lemma}\label{lem:initerm}
$M_n(r_k)=(2\,r_k)^{\frac{(n-1)(n-2)}{2}}$ for $k=1,\ldots,n$.
\end{lemma}
\begin{proof}
We prove the lemma for $k=1$. The proof applies for any $k\neq 1$. Substitution of $C_n^\imath=C_{n-1}^{\imath}+C_{n-1}^{\imath-1}$ into $\pt{m}_\imath(r_k)$ yields
\[
\pt{m}_\imath(r_k)=\left\{
\begin{array}{l}
\Big((C_{n-2}^{n-2\imath}+C_{n-2}^{n-2\imath-1})r_k^{n-2\imath},\ldots,(C_{n-2}^{n-2\imath+\jmath}+C_{n-2}^{n-2\imath+\jmath-1})r_k^{n-2\imath+\jmath},\\
\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad \ldots, \underbrace{0,\ldots\ldots\ldots\ldots,0}_{\max(n-2\imath-2,0)\mbox{~terms}}\Big)^T\\[25pt]
\qquad\qquad\qquad\mbox{if}\,~\imath\leq \left\lceil \dfrac{n-2}{2}\right\rceil\mbox{~and~}0\leq \jmath\leq \min(n-3,2\,\imath-1);\\[10pt]
\Big(\underbrace{0,\ldots\ldots\ldots,0}_{2\imath-n\mbox{~terms}},C_{n-2}^{0}r_k^0,\ldots,(C_{n-2}^{\jmath}+C_{n-2}^{\jmath-1})r_k^\jmath,\ldots\Big)^T\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad\\[25pt]
\qquad\qquad\qquad\qquad\mbox{if}\,~\imath> \left\lceil \dfrac{n-2}{2}\right\rceil\mbox{~and~}0\leq \jmath\leq 2\,n-2\,\imath-3,
\end{array}
\right.
\]
where $C_{n-2}^\jmath=0$ for $\jmath<0$ and $\jmath>n-2$, and $\lceil \gamma \rceil$ denotes the smallest integer that is not less than the rational number $\gamma$. For any positive integer $l$, denote by ${\rm co}_l$ the $l$th column and by ${\rm ro}_l$ the $l$th row of this matrix. Then
\begin{align*}
M_n(r_k) & \xlongequal[\imath=n-3,\ldots,1]{{\rm co}_\imath+{\rm co}_{\imath+1}\cdot r_k^2}\\ &\left|
\begin{array}{cccccccc}
\sum\limits_{\imath=0}^{n-2}C_{n-2}^{\imath}r_k^{n-2} & \sum\limits_{\imath=0}^{n-4}C_{n-2}^{\imath}r_k^{n-4} & \cdots \!\!\!&\!\!\! \cdots & \sum\limits_{\imath=1}^{n-2\jmath}C_{n-2}^{\imath}r_k^{n-2\jmath} & \cdots \!\!\!&\!\!\! \cdots\\[8pt]
\sum\limits_{\imath=0}^{n-2}C_{n-2}^{\imath}r_k^{n-1} & \sum\limits_{\imath=0}^{n-3}C_{n-2}^{\imath}r_k^{n-3} & \cdots \!\!\!&\!\!\! \cdots & \sum\limits_{\imath=1}^{n-2\jmath+1}C_{n-2}^{\imath}r_k^{n-2\jmath+1} & \cdots \!\!\!&\!\!\! \cdots\\[8pt]
\sum\limits_{\imath=0}^{n-2}C_{n-2}^{\imath}r_k^{n} &\sum\limits_{\imath=0}^{n-2}C_{n-2}^{\imath}r_k^{n-2} & \cdots \!\!\!&\!\!\! \cdots &\sum\limits_{\imath=1}^{n-2\jmath+2}C_{n-2}^{\imath}r_k^{n-2\jmath+2} & \cdots \!\!\!&\!\!\! \cdots\\[8pt]
\sum\limits_{\imath=0}^{n-2}C_{n-2}^{\imath}r_k^{n+1} & \sum\limits_{\imath=0}^{n-2}C_{n-2}^{\imath}r_k^{n-1} & \cdots \!\!\!&\!\!\! \cdots
&\sum\limits_{\imath=1}^{n-2\jmath+3}C_{n-2}^{\imath}r_k^{n-2\jmath+3} &\cdots \!\!\!&\!\!\! \cdots\\
\vdots & \vdots & \ddots \!\!\!&\!\!\! & \vdots &\ddots \!\!\!&\!\!\!\\[-10pt]
\vdots & \vdots & \!\!\!&\!\!\!\hspace{-4pt}\raisebox{0.1cm}{\mbox{$\ddots$}} & \vdots &\!\!\!&\!\!\!\hspace{-4pt}\raisebox{0.1cm}{\mbox{$\ddots$}}
\end{array}
\right|
\xlongequal[\imath=2,\ldots,n-2]{{\rm ro}_{\imath}-{\rm ro}_{\imath-1}\cdot r_k}\\
&\left|
\begin{array}{cccccccc}
\sum\limits_{\imath=0}^{n-2}C_{n-2}^{\imath}r_k^{n-2} & \sum\limits_{\imath=0}^{n-4}C_{n-2}^{\imath}r_k^{n-4} & \sum\limits_{\imath=0}^{n-6}C_{n-2}^{n-6}r_k^{n-6} & \cdots \!\!\!&\!\!\! \cdots & \sum\limits_{\imath=0}^{n-2\jmath}C_{n-2}^{n-2\jmath}r_k^{n-2\jmath} & \cdots \!\!\!&\!\!\! \cdots\\[8pt]
0 & C_{n-2}^{n-3}r_k^{n-3} & C_{n-2}^{n-5}r_k^{n-5} & \cdots \!\!\!&\!\!\! \cdots & C_{n-2}^{n-2\jmath+1}r_k^{n-2\jmath+1} & \cdots \!\!\!&\!\!\! \cdots\\[8pt]
0 &C_{n-2}^{n-2}r_k^{n-2} & C_{n-2}^{n-4}r_k^{n-4} & \cdots \!\!\!&\!\!\! \cdots & C_{n-2}^{n-2\jmath+2}r_k^{n-2\jmath+2} & \cdots \!\!\!&\!\!\! \cdots\\[8pt]
0 &0 & C_{n-2}^{n-3}r_k^{n-3} & \cdots \!\!\!&\!\!\! \cdots & C_{n-2}^{n-2\jmath+3}r_k^{n-2\jmath+3} & \cdots \!\!\!&\!\!\! \cdots\\[8pt]
0 & 0 & C_{n-2}^{n-2}r_k^{n-2} & \cdots \!\!\!&\!\!\! \cdots & C_{n-2}^{n-2\jmath+4}r_k^{n-2\jmath+4} & \cdots \!\!\!&\!\!\! \cdots\\
\vdots & \vdots & \vdots & \ddots \!\!\!&\!\!\! & \vdots& \ddots \!\!\!&\!\!\! \\[-10pt]
\vdots & \vdots & \vdots & \!\!\!&\!\!\! \hspace{-4pt}\raisebox{0.1cm}{\mbox{$\ddots$}} &\vdots & \!\!\!&\!\!\!\hspace{-4pt}\raisebox{0.1cm}{\mbox{$\ddots$}}
\end{array}
\right|\\
=&\sum_{\imath=0}^{n-2}C_{n-2}^{\imath}r_k^{n-2}\cdot M_{n-1}(r_k)=(2\,r_k)^{n-2}\cdot M_{n-1}(r_k).
\end{align*}
Since $M_3(r_k)=2\,r_k$, it is easy to verify that
\[M_n(r_k)=(2\,r_k)^{\frac{(n-1)(n-2)}{2}}\]
by induction.
\end{proof}
\begin{proof}[Proof of Theorem 2]
By Lemma \ref{lem:division}, \[\left[\prod\nolimits_{i< j\neq k}^n{(2\,r_k-r_i-r_j)}\right]~\bigg|~
H(r_k).\]
Hence there exists a polynomial $P=P(r_1,\ldots,r_n)$ such that
\[
H(r_k)=P\,\prod_{i< j\neq k}^n{(2\,r_k-r_i-r_j)}.
\]
Observe that both of the leading terms of $\prod_{i< j\neq k}^n{(2\,r_k-r_i-r_j)}$ and $H$ with respect to $r_k$ is equal to $(2\,r_k)^{\frac{(n-1)(n-2)}{2}}$. This implies that $P=1$, so that
\[H(r_k)=\prod_{i< j\neq k}^n{(2\,r_k-r_i-r_j)}.\]
Therefore,
\[\res(f, H, x)=\prod_{k=1}^nH(r_k)=\prod_{i< j\neq k}(2\,r_k-r_i-r_j)=D_2.\]
\vspace{-0.5cm}
\end{proof}
\section{Irreducibility and Degree of the Second Discriminant}
\label{sec:Properties}
Using Theorem \ref{thm:D2}, one can easily verify that
\begin{enumerate}[(1)]
\item for $n=3$, $D_2=-2\,a_2^3+9\,a_1a_2-27\,a_0$;
\item for $n=4$, $D_2$ is an irreducible polynomial of total degree $9$, and more explicitly:
\begin{align*}
\!\!\!\!\!\!\!\!D_2\,=\,&216\,a_0a_3^8-72\,a_1a_2a_3^7+16\,a_2^3a_3^6-2304\,a_0a_2a_3^6+72\,a_1^2a_3^6+672\,a_1a_2^2a_3^5-144\,a_2^4
a_3^4\\
&+5310\,a_0a_1a_3^5+7446\,a_0a_2
^2a_3^4-2346\,a_1^2a_2a_3^4-1278\,a_1
a_2^3a_3^3+324\,a_2^5a_3^2\\
&-9675\,a_
0^2a_3^4\!-\!28950\,a_0a_1a_2a_3^3-6804
\,a_0a_2^3a_3^2+1658\,a_1^3a_3^3+
9423\,a_1^2a_2^2a_3^2\!-\!1296\,a_1a_2^
4a_3\\
&+\!51600\,a_0^2a_2a_3^2\!+\!31890\,a_0a_
1^2a_3^2\!+\!19440\,a_0a_1a_2^2a_3\!+\!1296\,
a_0a_2^4\!-\!17262\,a_1^3a_2a_3\!+\!972\,a_1
^2a_2^3\\
&-120000\,a_0^2a_1a_3-28800\,a_0
^2a_2^2-5040\,a_0a_1^2a_2+9261\,a_1^4
+160000\,a_0^3;
\end{align*}
\item for $n=5$, $D_2$ is an irreducible polynomial of total degree $18$, consisting of 521 terms, in $a_0,\ldots,a_4$.
\end{enumerate}
In what follows, we prove that $D_2$ is an irreducible polynomial of total degree $3\,(n-1)(n-2)/2$ in $a_0,\ldots,a_{n-1}$ for any $n\geq3$. For this purpose, let $s_i=\sum r_{k_1}\cdots r_{k_i}$ be the sum of all the possible, distinct products of $i$ elements taken from $r_1,\ldots, r_n$ for $i=1,\ldots,n$. The sum $s_i$ of products is called the {\em elementary symmetric polynomial} of degree $i$ in $r_1,\ldots, r_n$. It is easy to show that the Vieta formula $a_{n-i}=(-1)^is_i$ holds for $i=1,\ldots,n$.
\begin{proposition}\label{prop:irreducibility}
$D_2(a_0,\ldots,a_{n-1})\in\mathbb{Q}[a_0,\ldots,a_{n-1}]$ is irreducible over $\mathbb{Q}$.
\end{proposition}
\begin{proof}
Let $P\in \mathbb{Q}[a_0,\ldots,a_{n-1}]$ be a nonconstant irreducible polynomial and suppose that $P\mid D_2$. We show that $D_2\mid P$.
Substituting Vieta's formula $a_{n-i}=(-1)^i\sum r_{k_1}\cdots
r_{k_i}$ into $P$ and $D_2$, we obtain two symmetric polynomials $\bar{P}$ and $\bar{D}_2$ in $\mathbb{Q}[r_1,\ldots,r_n]$, respectively. Then $\bar{P}=0$ is equivalent to $P=0$, and so is $\bar{D}_2$ to $D_2$. Since $P$ is nonconstant, so is $\bar{P}$. As $P\mid D_2$, $\bar{P}\mid \bar{D}_2$; so $\bar{P}$ contains at least one irreducible factor of $\bar{D}_2$, say $2\,r_1-r_2-r_3$. Therefore, every $2\,r_k-r_i-r_j$ is a factor of $\bar{P}$ because $\bar{P}$ is symmetric with respect to $r_1,\ldots,r_n$. It follows that $\bar{D}_2\mid \bar{P}$. Hence $\bar{D}_2$ and $\bar{P}$ differ only by a nonzero constant factor, and so do $D_2$ and $P$. Therefore, $D_2\mid P$ and thus $D_2$ is irreducible over $\mathbb{Q}$.
\end{proof}
For simplicity, we write $\bm{a}$ for $(a_0, a_1,\ldots,a_{n-1})$ and $\deg(F,\bm{a})$ for the total degree of $F$ in $\bm{a}$ from now on.
\begin{proposition}\label{prop:degD2}
$\deg(D_2,\bm{a})=3\,(n-1)(n-2)/2$.
\end{proposition}
\begin{proof}
Set $B_0=D_2$. For $i=1,\ldots, n$, let $C_i$ be the homogeneous part of $B_{i-1}$ of the highest total degree in $\bm{a}^+=(a_0,\ldots,a_{n-1},r_1)$ and let $B_i$ be obtained from $C_i$ by substituting Vieta's formula $a_{n-i}=(-1)^i\sum r_{k_1}\cdots r_{k_i}=U_{n-i}r_1+V_{n-i}$, where $U_{n-i}\neq0$ and $\deg(U_{n-i},r_1)=\deg(V_{n-i},r_1)=0$. Then
\begin{align*}
C_i=S_{n-i}a_{n-i}^{N_i}+T_{n-i}&=S_{n-i}(U_{n-i}r_1+V_{n-i})^{N_i}+T_{n-i}=B_i,\\
S_{n-i}U_{n-i}^{N_{i}}r_1^{N_{i}}&=C_{i+1},
\end{align*}
where $N_i=\deg(C_i,a_{n-i})$, $S_{n-i}$ is the leading coefficient of $C_i$ with respect to $a_{n-i}$, and $S_{n-i}U_{n-i}\neq0$.
Therefore, the total degrees of $C_i$, $B_i$, $C_{i+1}$ in $\bm{a}^+$ remain the same for $i=1,\ldots,n$, so $\deg(C_1,\bm{a}^+)=\deg(C_n,\bm{a}^+)$. Note that $C_n$ is the leading term of $D_2$, expressed in terms of the roots $r_1,\ldots,r_n$ as in \eqref{eq:D2}, with respect to $r_1$ and $\deg(C_n, \bm{a}^+)=\deg(C_n, r_1)=3\,(n-1)(n-2)/2$. Thus $\deg(D_2,\bm{a})=\deg(C_1,\bm{a}^+)=3\,(n-1)(n-2)/2$ and the proposition is proved.
\end{proof}
\section{The Second Discriminant with Resultants}\label{sec:resultant}
The following three polynomials will play a significant role in this and later sections:
\begin{align*}
& f_1(x,y)=\dfrac{f(y)-f(x)}{y-x},\quad f_2(x,y)=\dfrac{f\left(\dfrac{x+y}{2}\right)-f(x)}{\dfrac{y-x}{2}},\\[-2pt] &f_3(x,y)=\dfrac{f(y)-2\,f\left(\dfrac{x+y}{2}\right)+f(x)}{\dfrac{(y-x)^2}{2}}.
\end{align*}
The rational functions on the right-hand side of the above equalities can all be simplified to polynomials in $x$ and $y$.
\begin{proposition}\label{prop:resD1D2}
$\res(f, \res(f_1, f_2, y), x)=0$ if and only if $D_1D_2=0$.
\end{proposition}
\begin{proof}
($\Longleftarrow$) Let
\[R_1(x)=\res(f_1, f_2, y), \quad R_2=\res(f, R_1, x).\]
We want to show that, if $D_1D_2=0$, then there exist $r_i$ and $r_j$ such that
\begin{equation}\label{eq:F12}
f(r_i)=f(r_j)=0,\quad f_1(r_i,r_j)=0,\quad f_2(r_i, r_j)=0.
\end{equation}
For this purpose, first suppose that $D_1=0$. Then there exist $r_i=r_j, i\neq j$, such that $f(r_i)=f(r_j)=0$ and $f'(r_i)=f'(r_j)=0$ (where $'$ is the derivation operator).
Note that
\[f_1(x,y)=\sum_{k=0}^{n-1}\frac{f^{(k+1)}(x)}{(k+1)!}(y-x)^k,\quad f_2(x,y)=\sum_{k=0}^{n-1}\frac{f^{(k+1)}(x)}{(k+1)!}\left(\frac{y-x}{2}\right)^k.\]
Substitution of $x=r_i$ and $y=r_j$ into the above expressions shows that \eqref{eq:F12} holds in this case.
Now suppose that $D_2=0$ and $D_1\neq0$. Then there exist $r_i\neq r_j$ such that $f(r_i)=f(r_j)=0$ and $f\left(\frac{r_i+r_j}{2}\right)=0$. It follows that
\[f_1(r_i,r_j)=\dfrac{f(r_j)-f(r_i)}{r_j-r_i}=0,\quad f_2(r_i,r_j)=\dfrac{f\left(\dfrac{r_i+r_j}{2}\right)-f(r_i)}{\dfrac{r_j-r_i}{2}}=0.\]
Thus \eqref{eq:F12} holds as well.
In any case, $f_1(r_i,y)$ and $f_2(r_i,y)$ have a common zero $r_j$ for $y$. Therefore,
\[R_1(r_i)=\res(f_1(r_i, y), f_2(r_i,y),y)=0.\]
Hence $f(x)$ and $R_1(x)$ have a common root $r_i$ for $x$. This implies that $R_2=0$.
($\Longrightarrow$) $\res(f, \res(f_1, f_2, y), x)=0$ implies that there exist $r_i$ and $r_j$, $i<j$, such that
\[f(r_i)=0,\quad f_1(r_i, r_j)=f_2(r_i, r_j)=0.\]
Thus $f(r_j)=f_1(r_i,r_j)(r_j-r_i)+f(r_i)=0$.
If $r_j=r_i$, then $f(x)$ has a multiple root and thus $D_1=0$.
Otherwise,
\[f_2(r_i,r_j)=2\left[f\left(\frac{r_i+r_j}{2}\right)-f(r_i)\right]\Big/(r_j-r_i)=0\]
implies that $f\left(\frac{r_i+r_j}{2}\right)=0$, so $f$ has three roots, which form a symmetric triple. Therefore $D_2=0$.
\end{proof}
Using similar ideas, we can prove the following proposition, which shows
how to construct $D_2$ via resultant computation twice.
\begin{proposition}\label{prop:resD2}
$\res(f, \res(f_1, f_3, y), x)=0$ if and only if $D_2=0$.
\end{proposition}
\begin{proof}
($\Longleftarrow$) Let
\[F(x)=\res(f_1, f_3, y),\quad E=\res(f, F, x).\]
We show that, if $D_2=0$, then there exist $r_i$ and $r_j$ such that
\begin{equation}\label{eq:F13}
f(r_i)=f(r_j)=0,\quad f_1(r_i,r_j)=0,\quad f_3(r_i, r_j)=0.
\end{equation}
First suppose that there exist $r_i=r_j=r_k$, $i< j\neq k$, such that $f(r_i)=f(r_j)=f(r_k)=0$. Then $f'(r_i)=f''(r_i)=0$.
Note that
\begin{align*}
f_1(x,y)&=\sum_{k=0}^{n-1}\frac{f^{(k+1)}(x)}{(k+1)!}(y-x)^k,\\
f_3(x,y)&=\dfrac{f_1-f_2}{\dfrac{y-x}{2}}=\dfrac{\sum\limits_{k=0}^{n-1}\dfrac{f^{(k+1)}(x)}{(k+1)!}(y-x)^k-\sum\limits_{k=0}^{n-1}\dfrac{f^{(k+1)}(x)}{(k+1)!}\left(\dfrac{y-x}{2}\right)^k}{\dfrac{y-x}{2}}\\[3pt]
&=\sum_{k=0}^{n-2}\left(2-\dfrac{1}{2^{k}}\right)\dfrac{f^{(k+2)}(x)}{(k+2)!}\left(y-x\right)^k.
\end{align*}
Substitution of $x=r_i$ and $y=r_j$ into the above expressions shows that \eqref{eq:F13} holds in this case.
Suppose otherwise that there exist $r_i\neq r_j$ such that $f(r_i)=f(r_j)=0$ and $f\left({(r_i+r_j)}/{2}\right)=0$. Then it follows from $D_2=0$ that
\[f_1(r_i,r_j)=\dfrac{f(r_j)-f(r_i)}{r_j-r_i}=0,\quad f_3(r_i,r_j)=\dfrac{f(r_j)-2\,f\left(\dfrac{r_i+r_j}{2}\right)+f(r_i)}{\dfrac{(r_j-r_i)^2}{2}}=0,\]
so \eqref{eq:F13} holds as well.
In any case, $f_1(r_i,y)$ and $f_3(r_i,y)$ have a common zero $r_j$ for $y$. Therefore,
\[F(r_i)=\res(f_1(r_i, y), f_3(r_i,y),y)=0.\]
Hence $f(x)$ and $F(x)$ have a common root $r_i$ for $x$. This implies that $E=0$.
($\Longrightarrow$) $\res(f, \res(f_1, f_3, y), x)=0$ implies that there exist $r_i$ and $r_j$, $i< j$, such that
\[f(r_i)=0,\quad f_1(r_i, r_j)=f_3(r_i, r_j)=0.\]
Moreover, $f(r_i)=0$ and $f_1(r_i,r_j)=0$ imply that $f(r_j)=0$.
Consider first the case when $r_i=r_j$. In this case, $D_1=0$ and thus $f'(r_i)=0$. The following calculation shows that $f''(r_i)=0$:
\begin{align*}
f''(r_i)&=~\lim_{x\rightarrow r_i}\dfrac{f'\left(\dfrac{x+r_i}{2}\right)-f'(r_i)}{\dfrac{x-r_i}{2}}\\
&=~\lim_{x\rightarrow r_i}\dfrac{\dfrac{f(x)-f\left(\dfrac{x+r_i}{2}\right)}{\dfrac{x-r_i}{2}}-\dfrac{f\left(\dfrac{x+r_i}{2}\right)-f(r_i)}{\dfrac{x-r_i}{2}}}{\dfrac{x-r_i}{2}}\\
&=~2\,\lim_{x\rightarrow r_i}\dfrac{f(x)+f(r_i)-2\,f\left(\dfrac{x+r_i}{2}\right)}{\dfrac{(x-r_i)^2}{2}}\\[2pt]
&=~2\lim_{x\rightarrow r_i}f_3(x,r_i)=~2\,f_3(r_i,r_j)=0.
\end{align*}
Therefore, there exists an $r_k$ such that $k\neq i$, $k\neq j$ and $r_k=r_i=r_j$, which implies that $2\,r_k-r_i-r_j=0$. Hence $D_2=0$.
Now consider the case when $r_j\neq r_i$. In this case,
\[f_3(r_i,r_j)=\left[f(r_i)+f(r_j)-2\,f\left(\dfrac{r_i+r_j}{2}\right)\right]\bigg/\dfrac{(r_j-r_i)^2}{2}=0\]
implies that $f\left({(r_i+r_j)}/{2}\right)=0$, so $x={(r_i+r_j)}/{2}$ is a root of $f$. Therefore $D_2=0$.
\end{proof}
Since $D_2$ is irreducible over ${\Bbb Q}$, there exist a positive integer $q$ and a nonzero constant $c\in{\Bbb Q}$ such that
\[D_2^q=c\cdot\res(f, \res(f_1, f_3, y), x).\]
In what follows, we prove that $q=2$. For simplicity, we write $F$ for $\res(f_1, f_3, y)$ and $E$ for $\res(f, \res(f_1, f_3, y), x)$.
\begin{theorem}\label{thm:qIs2}
$E=c\, D_2^2$, where $c$ is a nonzero rational number.
\end{theorem}
The proof of this theorem requires Lemmas \ref{lem:Fr1} and \ref{lem:deg_res_ff1f3}, of which the latter shows that $\deg(E, \bm{a})\leq 3\,(n-1)(n-2)+2\,(n-2)$.
\begin{lemma}\label{lem:Fr1}
For any $k,j$ with $1<k\neq j$, $r_1-2\,r_k+r_j$ divides $F(r_1)$.
\end{lemma}
\begin{proof}
It suffices to show that $F(r_1)=0$ when $r_1=2\,r_k-r_j$ for any fixed $k,j$ satisfying $1<k\neq j$.
According to the theory of resultants \cite[pp. 228f]{M1993A}, there exist polynomials $A_1(x,y)$ and $A_3(x,y)$ such that
\[F(x)=A_1(x,y)f_1(x,y)+A_3(x,y)f_3(x,y).\]
Suppose that $r_j\neq r_1$. Since $f(r_1)=f(r_k)=f(r_j)=0$, substitution of $x=r_1$ and $y=r_j$ into $f_1$ and $f_3$ yields
\begin{align*}
f_1(r_1,r_j)&=\dfrac{f(r_j)-f(r_1)}{r_j-r_1}=0,\\
f_3(r_1,r_j)&=\dfrac{f(r_j)-2\,f\left(\dfrac{r_1+r_j}{2}\right)+f(r_1)}
{\dfrac{(r_j-r_1)^2}{2}}=\dfrac{f(r_j)-2\,f(r_k)+f(r_1)}
{\dfrac{(r_j-r_1)^2}{2}}=0.
\end{align*}
Suppose otherwise that $r_j=r_1$. Then $r_k=(r_1+r_j)/2=r_1$, which implies that $x=r_1$ is a root of $f$ with multiplicity greater than $2$. Thus $f(r_1)=f'(r_1)=f''(r_1)=0$. It follows that
\begin{align*}
f_1(r_1,r_j)&=\sum_{k=0}^{n-1}\frac{f^{(k+1)}(r_1)}{(k+1)!}(r_j-r_1)^k=f'(r_1)=0,\\
f_3(r_1,r_j)&=\sum_{k=0}^{n-2}\left(2-\dfrac{1}{2^{k}}\right)\dfrac{f^{(k+2)}(r_1)}{(k+2)!}\left(r_j-r_1\right)^k=\dfrac{f''(r_1)}{2!}=0.
\end{align*}
Hence, in both cases we have $f_1(r_1,r_j)=f_3(r_1,r_j)=0$. Therefore
\[F(r_1)=A_1(r_1,r_j)f_1(r_1,r_j)+A_3(r_1,r_j)f_3(r_1,r_j)=0,\]
so $r_1-2\,r_k+r_j$ divides $F(r_1)$.
\end{proof}
\begin{proof}[Proof of Theorem \ref{thm:qIs2}]
By Lemma \ref{lem:Fr1}, $(r_1-2\,r_k+r_j)\mid F(r_1)$ for arbitrarily chosen $k,j$ with $1<k\neq j$. Hence
\[\prod_{1<k\neq j}{(r_1-2\,r_k+r_j)}\mid F(r_1).\]
It follows from the theory of resultants \cite[p.\,398]{GKZ1994D} that
\[E=\prod_{i=1}^n F(r_i)=\prod_{i< j\neq k}{(r_i-2\,r_k+r_j)^2}\cdot K=D_2^2\cdot K\]
for some polynomial $K$ in $r_1,\ldots,r_n$. By Proposition~\ref{prop:resD2}, there exist a nonzero constant $c$ and an integer $q\geq2$ such that $E=c\, D_2^q$.
On the other hand, by Lemma \ref{lem:deg_res_ff1f3}, $\deg(E,\bm{a})\leq 3\,(n-1)(n-2)+2\,(n-2)$; by Proposition~\ref{prop:degD2}, $\deg(D_2,\bm{a})= {3\,(n-1)(n-2)}/{2}$. Under these constraints, the only possibility for $E=c\, D_2^q$ to hold is that $K$ is a constant and $q=2$.
\end{proof}
\section{The Second Discriminant with Ideals}\label{sec:ideals}
In searching for explicit representations of $D_2$ in terms of the coefficients of $f$, we have discovered the amazingly structured matrix $M$ formed with the derivatives of $f$ shown in \eqref{matM}. In what follows, we establish an inherent connection between the $(n-2)$th leading principal minor $H$ of $M$ and $\res(f_1, f_3, y)$, which reveals the hidden mystery for the structure of $M$.
Let $\langle f_1,\ldots,f_m\rangle$ denote the ideal generated by $f_1,\ldots,f_m$ in a ring of polynomials. The polynomials $f_1,\ldots,f_m$ are called the generators of the ideal.
\begin{lemma}\label{lem:equiv_ideal}
\begin{align*}
~\qquad\qquad&\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\left\langle f(x),f(y), f\left(\frac{x+y}{2}\right),w(x-y)-1\right\rangle\\
&=\left\langle f(x),f(y)-f(x), f\left(\frac{x+y}{2}\right)-f(x),w(x-y)-1\right\rangle\\[1pt]
&=\left\langle f(x),f_1(x,y),f_2(x,y),w(x-y)-1\right\rangle\\[4pt]
&=\left\langle f(x),f_1(x,y),f_3(x, y),w(x-y)-1\right\rangle.
\end{align*}
\end{lemma}
\begin{proof}
Let the four ideals in the above identity be denoted successively by $\ideal{I}_1, \ldots, \ideal{I}_4$. It is obvious that $\ideal{I}_1=\ideal{I}_2$. We only need to show that $\ideal{I}_2=\ideal{I}_3$ and $\ideal{I}_3=\ideal{I}_4$.
\begin{enumerate}[(1)]
\item Since $f(y)-f(x)=f_1\cdot(y-x)$ and $f\left(\dfrac{x+y}{2}\right)-f(x)=f_2\cdot\dfrac{y-x}{2}$, we have $\ideal{I}_2\subset \ideal{I}_3$. On the other hand,
\begin{align*}
f_1&=-w\left[f(y)-f(x)\right]-[w(x-y)-1]\frac{f(y)-f(x)}{y-x},\\
f_2&=-2\,w\left[f\left(\dfrac{x+y}{2}\right)-f(x)\right]-2\,[w(x-y)-1]\frac{f\left(\dfrac{x+y}{2}\right)-f(x)}{y-x},
\end{align*}
so $\ideal{I}_3\subset \ideal{I}_2$.
\item $\ideal{I}_3\subset \ideal{I}_4$ follows from $f_2=f_1+\dfrac{x-y}{2}\cdot f_3$. $\ideal{I}_4\subset \ideal{I}_3$ can be easily deduced from $f_3=-2\,wf_1+2\,wf_2-f_3[w(x-y)-1]$.
\end{enumerate}
Therefore $\ideal{I}_1=\ideal{I}_2=\ideal{I}_3=\ideal{I}_4$.
\end{proof}
\begin{lemma}\label{lem:resH}
Let
\[g_1(x,y)=f_1(x-y,x+y), \quad g_3(x,y)=f_3(x-y,x+y),\quad G=\res(g_1,g_3,y).\] Then
$G=H^2$, where $H$ is as in Theorem~\ref{thm:D2}.
\end{lemma}
\begin{proof}
For any rational number $\gamma$, denote by $\lfloor \gamma \rfloor$ the biggest integer that is not greater than $\gamma$. Taking Taylor expansion for $g_1$ and $g_3$ at $x$, we have
\begin{align*}
g_1(x,y)&=\dfrac{f(x+y)-f(x-y)}{2\,y}\\
&=\dfrac{f(x+y)-f(x)}{2\,y}-\dfrac{f(x-y)-f(x)}{2\,y}\\
&=\dfrac{1}{2}\sum_{k=1}^n\left[1-(-1)^k\right]\dfrac{f^{(k)}(x)}{k!}y^{k-1}\\
&=\sum_{k=0}^{\lfloor\frac{n-1}{2}\rfloor}\dfrac{f^{(2\,k+1)}(x)}{(2\,k+1)!}y^{2\,k}
\end{align*}
and
\begin{align*}
g_3(x,y)&=\dfrac{f(x+y)+f(x-y)-2\,f(x)}{2\,y^2}\\
&=\dfrac{1}{2\,y}\cdot\left[\dfrac{f(x+y)-f(x)}{y}+\dfrac{f(x-y)-f(x)}{y}\right]\\
&=\dfrac{1}{2}\sum_{k=2}^n\left[1+(-1)^k\right]\dfrac{f^{(k)}(x)}{k!}y^{k-2}\\
&=\sum_{k=0}^{\lfloor\frac{n}{2}\rfloor-1}\dfrac{f^{(2\,k+2)}(x)}{(2\,k+2)!}y^{2\,k}.
\end{align*}
Let $g^*_1$ and $g^*_3$ be obtained from $g_1$ and $g_3$ by replacing $y^2$ with $z$.
Then
\[\res(g_1^*,g_3^*,z)=
\begin{array}{c@{\hspace{-5pt}}l}
\left|\begin{array}{ccccc}
\frac{f^{(n-1)}}{(n-1)!}&\frac{f^{(n-3)}}{(n-3)!}&\frac{f^{(n-5)}}{(n-5)!}&\cdots\!\!\!&\!\!\!\cdots\\
0&\frac{f^{(n-1)}}{(n-1)!}&\frac{f^{(n-3)}}{(n-3)!}&\cdots\!\!\!&\!\!\!\cdots\\
\vdots&\vdots&\vdots&\ddots\!\!\!&\!\!\!\\[-10pt]
\vdots&\vdots&\vdots&\!\!\!&\!\!\!\hspace{-4pt}\raisebox{0.1cm}{\mbox{$\ddots$}}\\
\frac{f^{(n)}}{n!}&\frac{f^{(n-2)}}{(n-2)!}&\frac{f^{(n-4)}}{(n-4)!}&\cdots\!\!\!&\!\!\!\cdots\\
0&\frac{f^{(n)}}{n!}&\frac{f^{(n-2)}}{(n-2)!}&\cdots\!\!\!&\!\!\!\cdots\\
\vdots&\vdots&\vdots&\ddots\!\!\!&\!\!\!\\[-10pt]
\vdots&\vdots&\vdots&\!\!\!&\!\!\!\hspace{-4pt}\raisebox{0.1cm}{\mbox{$\ddots$}}
\end{array}\right|
&
\begin{array}{l}\left.\rule{0mm}{10mm}\right\}\left\lfloor{\dfrac{n}{2}}\right\rfloor-1\\
\\\left.\rule{0mm}{10mm}\right\}\left\lfloor{\dfrac{n-1}{2}}\right\rfloor
\end{array}
\end{array}
=\pm\, H.\]
Therefore,
\[G=\res(g_1,g_3,y)=[\res(g_1^*(x,z),g_3^*(x,z),z)]^2=H^2.\]
\vspace{-0.5cm}
\end{proof}
\begin{corollary}
$D_2\in\left\langle f(x), g_1(x,y),g_3(x,y)\right\rangle\cap\mathbb{Q}[a_0,\ldots,a_{n-1}]$.
\end{corollary}
\begin{proof}
Let ${\cal K}=\mathbb{Q}[a_0,\ldots,a_{n-1}]$.
By Lemma \ref{lem:resH},
\[\res(f,G,x)=\res(f, H^2,x)=D_2^2.\]
According to the theory of resultants, there exist $A_1(x,z),A_2(x,z)\in{\cal K}[x,z]$ such that
\[H=A_1(x,z)g_1^*(x,z)+A_2(x,z)g_3^*(x,z).\]
Similarly, there exist $B_1(x),B_2(x)\in{\cal K}[x]$ such that
\begin{align*}
D_2&=\res(f,H,x)=B_1(x)f(x)+B_2(x)H\\
&=B_1(x)f(x)+B_2(x)[A_1(x,z)g_1^*(x,z)+A_2(x,t)g_3^*(x,z)].
\end{align*}
Substituting $z=y^2$, one gets
\begin{align*}
D_2&=B_1(x)f(x)+A_1(x,y^2)B_2(x)g_1(x,y)+A_2(x,y^2)B_2(x)g_3(x,y)\\
&\in\langle f, g_1,g_3\rangle\cap{\cal K}.
\end{align*}
The corollary is proved.
\end{proof}
\begin{theorem}\label{thm:D2in}
$D_2\in\left\langle f(x), f_1(x,y),f_3(x,y)\right\rangle\cap\mathbb{Q}[a_0,\ldots,a_{n-1}]$.
\end{theorem}
\begin{proof}
Replace $x$ and $y$ in $f(x)$, $f_1(x-y,x+y)$, $f_3(x-y,x+y)$ by ${(Y+X)}/{2}$ and ${(Y-X)}/{2}$, respectively. Since $D_2\in \langle f(x), f_1(x-y,x+y),f_3(x-y,x+y)\rangle$ and $D_2$ does not involve $x$ and $y$,
\[D_2\in \left\langle f\left(\dfrac{X+Y}{2}\right),f_1(X,Y), f_3(X,Y)\right\rangle.\]
Furthermore, from \[f\left(\dfrac{X+Y}{2}\right)=\dfrac{1}{2}(Y-X)^2f_3(X,Y)+\dfrac{1}{2}(Y-X)f_1(X,Y)+f(X,Y),\] one can deduce
\[\langle f(X),f_1(X,Y), f_3(X,Y)\rangle=\left\langle f\left(\dfrac{X+Y}{2}\right),f_1(X,Y), f_3(X,Y)\right\rangle.\]
Therefore,
\[D_2\in\langle f(X),f_1(X,Y), f_3(X,Y)\rangle.\]
Substitution of $X=x$ and $Y=y$ back to the above expression, we have
\[D_2\in\langle f(x),f_1(x,y), f_3(x,y)\rangle.\]
The proof is complete.
\end{proof}
\begin{corollary}\label{cor:D2in}
\[D_2\in\left\langle f(x),f(y),f\left(\frac{x+y}{2}\right),w(x-y)-1\right\rangle\cap\mathbb{Q}[a_0,\ldots,a_{n-1}].\]
\end{corollary}
\begin{proof}
It follows from Lemma \ref{lem:equiv_ideal} and Theorem \ref{thm:D2in}.
\end{proof}
\begin{proposition}\label{propD2}
\[\langle D_2\rangle=\left\langle f(x),f(y), f\left(\frac{x+y}{2}\right),w(x-y)-1\right\rangle\cap \mathbb{Q}[a_0,\ldots,a_{n-1}].\]
\end{proposition}
\begin{proof}
Let $\ideal{I}_1$ and $\ideal{I}_4$ be as in the proof of Lemma \ref{lem:equiv_ideal}, which implies that
\[\ideal{I}_1\cap {\cal K}=\ideal{I}_4\cap {\cal K},\]
where ${\cal K}=\mathbb{Q}[a_0,\ldots,a_{n-1}]$. We proceed to show that
$\langle D_2\rangle=\ideal{I}_4\cap {\cal K}$.
Since $E=\res(f, \res(f_1, f_3, y), x)$, $E\in\ideal{I}_4\cap {\cal K}$. Let $(\bar{a}_0,\ldots,\bar{a}_{n-1},\bar{x},\bar{y},\bar{w})$ be any zero of
$\ideal{I}_4$ and $h$ be any polynomial in $\ideal{I}_4\cap {\cal K}$. Then $E(\bar{a}_0,\ldots,\bar{a}_{n-1})=h(\bar{a}_0,\ldots,\bar{a}_{n-1})=0$. By Theorem~\ref{thm:D2in}, $D_2(\bar{a}_0,\ldots,\bar{a}_{n-1})=0$,
so $D_2$ and $h$ have a nonconstant common divisor. As $D_2$ is irreducible over $\mathbb{Q}$, $D_2\mid h$.
On the other hand, by Corollary \ref{cor:D2in}
\[D_2\in \left\langle f(x), f(y), f\left(\dfrac{x+y}{2}\right),w(x-y)-1\right\rangle = \ideal{I}_1=\ideal{I}_4.\]
Since $D_2\mid h$ for any $h\in\ideal{I}_4\cap{\cal K}$, the intersection $\ideal{I}_4\cap{\cal K}$ is a principal ideal generated by $D_2$. Therefore,
\[\langle D_2\rangle=\ideal{I}_4\cap {\cal K}=\ideal{I}_1\cap {\cal K}.\]
\vspace{-0.5cm}
\end{proof}
\begin{proposition}\label{propD2a}
Let $s_i$ be the elementary symmetric polynomial of degree $i$ in $r_1,\ldots,r_n$ and $v_i=a_{n-i}-(-1)^{i}s_i$ for $i=1,\ldots,n$. Then
\begin{align*}
\Big\langle\prod_{
\scriptsize{i< j\neq k}}&(2\,r_k-r_i-r_j),~
v_1,\ldots,v_{n}\Big\rangle\cap \mathbb{Q}[a_0,\ldots,a_{n-1}]\\
=&\,\left\langle 2\,r_1-r_2-r_3,~
v_1,\ldots,v_{n}\right\rangle\cap \mathbb{Q}[a_0,\ldots,a_{n-1}]
=\left\langle D_2\right\rangle.
\end{align*}
\end{proposition}
\begin{proof}
Let ${\cal K}=\mathbb{Q}[a_0,\ldots,a_{n-1}]$ as before and
\begin{align*}
\ideal{J}_1=&\left\langle\prod_{
\scriptsize{i< j\neq k}}(2\,r_k-r_i-r_j),~
v_1,\ldots,v_{n}\right\rangle\cap {\cal K},\\
\ideal{J}_2=&\,\left\langle 2\,r_1-r_2-r_3,~
v_1,\ldots,v_{n}\right\rangle\cap {\cal K}.
\end{align*}
Proof of $\ideal{J}_1=\left\langle D_2\right\rangle$.
Note first that each $v_i~(1\leq i\leq n)$ is a polynomial monic and linear in $a_{n-i}$. Dividing $D_2$ by $v_n,\ldots,v_1$ with respect to $a_0,\ldots,a_{n-1}$ respectively, one can obtain a remainder $R$ in $r_1,\ldots, r_n$. Then there exist polynomials $A_1,\ldots,A_n\in\mathbb{Q}[a_0,\ldots,a_{n-1},r_1,\ldots,r_n]$ such that
\[D_2=A_1v_1+\cdots+A_nv_n+R.\]
Substituting $a_{n-i}=(-1)^is_i$ into the above formula and by Theorem \ref{thm:D2}, we have
\[R=\prod_{\scriptsize{i< j\neq k}}(2\,r_k-r_i-r_j).\]
Therefore, $D_2$ can be written as a linear combination of polynomials in $\ideal{J}_1$. This implies that
$D_2\in \ideal{J}_1$
and thus $\left\langle D_2\right\rangle\subset\ideal{J}_1$.
To show that $\ideal{J}_1\subset\left\langle D_2\right\rangle$, let $h$ be any polynomial in $\ideal{J}_1$.
Then the greatest common divisor $\gcd(h,D_2)$ of $h$ and $D_2$ is contained in the ideal $\ideal{J}_1$. As $D_2$ is irreducible over ${\Bbb Q}$, $\gcd(h,D_2)$ is either a nonzero constant, or equal to $D_2$. If $\gcd(h,D_2)$ is a nonzero constant, then $\ideal{J}_1$ is equal to the unit ideal, which is not possible because for any $r_1,\ldots,r_n$ satisfying
$\prod_{
\scriptsize{i< j\neq k}}(2\,r_k-r_i-r_j)=0$, there always exist
$a_0,\ldots,a_{n-1}$ such that $v_1=\cdots=v_{n}=0$, i.e., $\ideal{J}_1$ always has zeros. Therefore, $\gcd(h,D_2)=D_2$ and $D_2\mid h$. It follows that $h\in\left\langle D_2\right\rangle$.
Proof of $\ideal{J}_1=\ideal{J}_2$.
Since $\ideal{J}_1\subset\ideal{J}_2$ holds obviously, we only need to show that $\ideal{J}_2\subset\ideal{J}_1$.
Observe that
\[
\ideal{J}_1\supset\bigcap_{i< j\neq k}\langle 2\,r_k-r_i-r_j,\,
v_1,\ldots,v_{n}\rangle\cap {\cal K}.
\]
Since $\ideal{J}_1=\langle D_2\rangle$ is a prime ideal, there exist $i< j\neq k$ such that
\[\ideal{J}_1\supset\ideal{J}_{ijk}=\langle 2\,r_k-r_i-r_j,~
v_1,\ldots,v_{n}\rangle\cap {\cal K}\]
and $\ideal{J}_{ijk}$ is prime.
Note that $v_1,\ldots,v_n$ are symmetric in $r_1,\ldots,r_n$. Hence the primality of $\ideal{J}_{ijk}$ implies the primality of $\ideal{J}_{\imath\jmath\kappa}$ for all $\imath<\jmath\neq\kappa$. Therefore, all the $\ideal{J}_{\imath\jmath\kappa}$ are identical. Hence
\[\ideal{J}_1\supset\ideal{J}_2=\langle 2\,r_1-r_2-r_3,~
v_1,\ldots,v_{n}\rangle\cap {\cal K}.\]
\vspace{-0.5cm}
\end{proof}
As shown by Propositions~\ref{propD2} and \ref{propD2a}, there are several
ideals with different generators whose intersections with ${\cal K}$ are equal to
$\langle D_2\rangle$. The generator $D_2$ of the principal ideal $\langle D_2\rangle$, which is an \emph{elimination ideal} of ${\cal I}_1=\cdots={\cal I}_4$ or ${\cal J}_1={\cal J}_2$, can be obtained by computing the reduced lexicographical Gr\"obner basis of any of the ideals ${\cal I}_{\imath}$ and ${\cal J}_{\jmath}$ (see \cite[Lemma 6.8]{B1985G}).
\section{Degrees of Some Determinant Polynomials}\label{sec:determinant}
The two determinant polynomials $H$ and $F=\res(f_1, f_3, y)$, defined in Theorem \ref{thm:D2} and Proposition \ref{prop:resD2} respectively, can be used for the construction of the second discriminant $D_2$. In what follows, we provide some simple formulas for the exact degrees of $H$ and $F$ in $x$, which may be used for complexity analysis of $D_2$.
\begin{lemma}\label{lem:degree_bound_H}
$\deg(H,x)\leq (n-1)(n-2)/2$.
\end{lemma}
\begin{proof}
Let $g_1$, $g_3$ and $g_1^*$, $g_3^*$ be as Lemma \ref{lem:resH} and its proof. Then \[G=\res(g_1,g_3,y)=[\res(g_1^*(x,z),g_3^*(x,z),z)]^2=H^2,\] where $H$ is as in Theorem~\ref{thm:D2}.
Now consider
\[\Delta (\bm{a},x,y,\alpha)=\left|
\begin{array}{cc}\smallskip
g_1(x,y)&g_3(x,y)\\
g_1(x,\alpha)&g_3(x,\alpha)
\end{array}
\right|\bigg/(y-\alpha)\]
and let $\bm{\nu}=(x, y,\alpha)$ and $\tilde{n}=2\,\lfloor\frac{n-1}{2}\rfloor$.
It is easy to see that $\Delta$ is of degree
$2\,n-4$ in $\bm{\nu}$ and
$\deg(\Delta,\alpha)=\deg(\Delta,y)= \tilde{n}-1$, $\deg(\Delta,\bm{a})\leq 2$.
Let $\Delta$ be written as
\[\Delta=\sum_{\scriptsize\begin{array}{c}\scriptsize 0 \leq i\leq 2\,n-4\\ \scriptsize 0 \leq j,k\leq \tilde{n}-1
\end{array}}\delta_{ijk}x^iy^j\alpha^k.\]
Then for every term $x^iy^j\alpha^k$ occurring in $\Delta$, $i+j+k\leq \deg(\Delta, \bm{\nu})=2\,n-4$, so $i\leq 2\,n-4-j-k$.
Denote by \[B=(b_{j+1,k+1})=\left(\sum_{i=0}^{2\,n-4}\delta_{ijk}x^i\right)\] the $\tilde{n}\times \tilde{n}$ B\'ezout matrix of $g_1$ and $g_3$ with respect to $y$. It follows that $\deg(b_{jk},x)\leq 2\,n-2-j-k$. Let $(k_1,\ldots,k_{\tilde{n}})$ denote an arbitrary permutation of $(1,\ldots, \tilde{n})$ and $\bar{G}=\det(B)$.
Then
\[\begin{array}{rl}\smallskip
2\,\deg(H,x)\!\!\!&\displaystyle=\deg(H^2,x)=\deg(G,x)=\deg(\bar{G},x)\\
&\displaystyle\leq\max_{(k_1,\ldots,k_{\tilde{n}})}\deg(b_{1k_1}\cdots b_{\tilde{n}~k_{\tilde{n}-1}},x) =\max_{(k_1,\ldots,k_{\tilde{n}})}\sum_{j=1}^{\tilde{n}}\deg(b_{jk_j},x) \\ \medskip
&\displaystyle \leq \max_{(k_1,\ldots,k_{\tilde{n}})}\sum_{j=1}^{\tilde{n}}[2\,n-2-j-k_j] =(2\,n-2)\cdot\tilde{n}-\sum_{j=1}^{\tilde{n}}j-\min_{(k_1,\ldots,k_{\tilde{n}})}
\sum_{j=1}^{\tilde{n}}k_j \\ \medskip
&\displaystyle =(2\,n-2)\cdot\tilde{n}-(\tilde{n}+1)\cdot\tilde{n}=(n-1)(n-2).
\end{array}
\]
Therefore, $\deg(H,x)\leq(n-1)(n-2)/2$.
\end{proof}
Lemma \ref{lem:degree_bound_H} provides an upper bound for $\deg(H, x)$. In what follows, we show that the bound can be achieved for a particular polynomial. Thus the degree of $H$ constructed from the generic form of $f$ is equal to the bound.
\begin{lemma}\label{lem:degree_specialH}
For $a_0=\cdots=a_{n-1}=0$, $\deg(H, x)={(n-1)(n-2)}/{2}$.
\end{lemma}
\begin{proof}
When $a_0=\cdots=a_{n-1}=0$, ${H}$ becomes the $(n-2)$th leading principal minor of the following matrix
\[\left(
\begin{array}{ccccc}\smallskip
C_n^2x^{n-2}&C_n^4x^{n-4}&C_n^6x^{n-6}&\cdots\!\!\!&\!\!\!\cdots\\ \smallskip
C_n^1x^{n-1}&C_n^3x^{n-3}&C_n^5x^{n-5}&\cdots\!\!\!&\!\!\!\cdots\\ \smallskip
0&C_n^2x^{n-2}&C_n^4x^{n-4}&\cdots\!\!\!&\!\!\!\cdots\\ \smallskip
0&C_n^1x^{n-1}&C_n^3x^{n-3}&\cdots\!\!\!&\!\!\!\cdots\\
\vdots&\vdots&\vdots&\ddots\!\!\!&\!\!\!\\[-10pt]
\vdots&\vdots&\vdots&\!\!\!&\!\!\!\hspace{-5pt}\raisebox{0.1cm}{\mbox{$\ddots$}}
\end{array}
\right).\]
Simple calculation shows that
\begin{align*}
&\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\left|
\begin{array}{ccccc}\smallskip
C_n^2x^{n-2}&C_n^4x^{n-4}&C_n^6x^{n-6}&\cdots\!\!\!&\!\!\!\cdots\\ \smallskip
C_n^1x^{n-1}&C_n^3x^{n-3}&C_n^5x^{n-5}&\cdots\!\!\!&\!\!\!\cdots\\ \smallskip
0&C_n^2x^{n-2}&C_n^4x^{n-4}&\cdots\!\!\!&\!\!\!\cdots\\ \smallskip
0&C_n^1x^{n-1}&C_n^3x^{n-3}&\cdots\!\!\!&\!\!\!\cdots\\
\vdots&\vdots&\vdots&\ddots\!\!\!&\!\!\!\\[-10pt]
\vdots&\vdots&\vdots&\!\!\!&\!\!\!\hspace{-4pt}\raisebox{0.1cm}{\mbox{$\ddots$}}
\end{array}
\right|\\
\xlongequal[\substack{{\rm ro}_2\div x \\[2pt] \ldots \\[2pt] {\rm ro}_{n-2}\div x^{n-3}}]{\substack{{\rm co}_1\div x^{n-2}\\[2pt] {\rm co}_2\div x^{n-4}, {\rm co}_{n-2}\times x^{n-4}\\[2pt] {\rm co}_3\div x^{n-3}, {\rm co}_{n-3}\times x^{n-3}\\[2pt] \cdots}}
&\left|
\begin{array}{ccccc}\smallskip
C_n^2&C_n^4&C_n^6&\cdots\!\!\!&\!\!\!\cdots\\ \smallskip
C_n^1&C_n^3&C_n^5&\cdots\!\!\!&\!\!\!\cdots\\ \smallskip
0&C_n^2&C_n^4&\cdots\!\!\!&\!\!\!\cdots\\ \smallskip
0&C_n^1&C_n^3&\cdots\!\!\!&\!\!\!\cdots\\
\vdots&\vdots&\vdots&\ddots\!\!\!&\!\!\!\\[-10pt]
\vdots&\vdots&\vdots&\!\!\!&\!\!\!\hspace{-4pt}\raisebox{0.1cm}{\mbox{$\ddots$}}
\end{array}
\right|x^{(n-2)+1+2+\cdots+(n-3)}\\ \smallskip
=&\left|
\begin{array}{ccccc}\smallskip
C_n^2&C_n^4&C_n^6&\cdots\!\!\!&\!\!\!\cdots\\ \smallskip
C_n^1&C_n^3&C_n^5&\cdots\!\!\!&\!\!\!\cdots\\ \smallskip
0&C_n^2&C_n^4&\cdots\!\!\!&\!\!\!\cdots\\ [1pt]
0&C_n^1&C_n^3&\cdots\!\!\!&\!\!\!\cdots\\
\vdots&\vdots&\vdots&\ddots\!\!\!&\!\!\!\\[-10pt]
\vdots&\vdots&\vdots&\!\!\!&\!\!\!\hspace{-4pt}\raisebox{0.1cm}{\mbox{$\ddots$}}
\end{array}
\right|x^{\frac{(n-1)(n-2)}{2}}\doteq c_{n}x^{\frac{(n-1)(n-2)}{2}}.
\end{align*}
In what follows, we prove that $c_n\neq 0$.
Let
\begin{align*}
U&=C_n^2z^2+C_n^4z^4+\cdots+C_n^{2\left\lfloor\frac{n}{2}\right\rfloor}z^{2\left\lfloor\frac{n}{2}\right\rfloor},\\
V&=C_n^1z+C_n^3z^3+\cdots+C_n^{2\left\lfloor\frac{n+1}{2}\right\rfloor -1}z^{2\left\lfloor\frac{n+1}{2}\right\rfloor-1},
\end{align*}
and $\bar{U}$ and $\bar{V}$ be obtained from $U/z^2$ and $V/z$, respectively, by replacing $z^2$ with $t$. Then $c_n=\pm\,\res(\bar{U}, \bar{V}, t)$.
If $c_n=0$, then $U/z^2$ and $V/z$ have at least one common zero, say $\bar{z}$, where $\bar{z}\neq0$. Note that
\[U+V=(z+1)^n-C_n^0.\]
Substituting $z=\bar{z}$ into the above equation, we have
$(\bar{z}+1)^n-1=0$.
Similarly, $(U-V)|_{z=\bar{z}}=(\bar{z}-1)^n-1=0$.
Therefore, there exist two unit roots $u_1, u_2$ such that $\bar{z}+1=u_1$ and $\bar{z}-1=u_2$, which leads to $u_1-u_2=2$. In other words, $u_1$ and $u_2$ have the same imaginary part and the difference of their real parts is $2$. This can happen only when $u_1=1$ and $u_2=-1$. Therefore, $\bar{z}=0$, which leads to contradiction since $\bar{z}$ is nonzero. Hence the conclusion holds.
\end{proof}
The following theorem follows from Lemmas \ref{lem:degree_bound_H} and \ref{lem:degree_specialH}.
\begin{theorem}
$\deg(H, x)= (n-1)(n-2)/2$.
\end{theorem}
Similarly, we have the following theorem.
\begin{theorem}\label{thm:deg_res_f1f3}
$\deg(F, x)= (n-1)(n-2)$.
\end{theorem}
This theorem is established by proving the following two lemmas.
\begin{lemma}\label{lem:deg_res_f1f3}
$\deg(F,x)\leq (n-1)(n-2)$.
\end{lemma}
\begin{proof}
Let
\[\Delta (\bm{a},x,y,\alpha)=\left|
\begin{array}{cc}\smallskip
f_1(x,y)&f_3(x,y)\\
f_1(x,\alpha)&f_3(x,\alpha)
\end{array}
\right|\bigg/(y-\alpha)\]
and $\bm{\nu}=(x, y,\alpha)$.
It is easy to see that $\Delta$ is of degree $2\,n-4$ in $\bm{\nu}$ and
$\deg(\Delta,\alpha)=\deg(\Delta,y)= n-2$, $\deg(\Delta,\bm{a})\leq 2$.
Let $\Delta$ be written as
\[\Delta=\sum_{\scriptsize\begin{array}{c}\scriptsize 0 \leq i\leq 2n-4\\ \scriptsize 0 \leq j,k\leq n-2
\end{array}}\delta_{ijk}x^iy^j\alpha^k.\]
Then for every term $x^iy^j\alpha^k$ occurring in $\Delta$, $i+j+k\leq \deg(\Delta, \bm{\nu})=2\,n-4$, so $i\leq 2\,n-4-j-k$.
Denote by \[B=(b_{j+1,k+1})=\left(\sum_{i=0}^{2\,n-4}\delta_{ijk}x^i\right)\] the $(n-1)\times (n-1)$ B\'ezout matrix of $f_1$ and $f_3$ with respect to $y$. It follows that $\deg(b_{jk},x)\leq 2\,n-2-j-k$. Let $(k_1,\ldots,k_{n-1})$ denote an arbitrary permutation of $(1,\ldots, n-1)$ and $\bar{F}=\det(B)$. According to the theory of resultants \cite{B1779T}, $F=\res(f_1,f_3,y)=\pm\, \bar{F}$. Therefore,
\[\begin{array}{rl}\medskip
\deg(F,x)\!\!\!&\displaystyle=\deg(\bar{F},x)\leq\max_{(k_1,\ldots,k_{n-1})}\deg(b_{1k_1}\cdots b_{n-1,k_{n-1}},x) =\max_{(k_1,\ldots,k_{n-1})}\sum_{j=1}^{n-1}\deg(b_{jk_j},x) \\ \medskip
&\displaystyle \leq \max_{(k_1,\ldots,k_{n-1})}\sum_{j=1}^{n-1}(2\,n-2-j-k_j)
=(2\,n-2)(n-1)-\sum_{j=1}^{n-1}j-\min_{(k_1,\ldots,k_{n-1})}
\sum_{j=1}^{n-1}k_j \\
&\displaystyle =2\,(n-1)^2-(n-1)n=(n-1)(n-2).
\end{array}
\]
\vspace{-0.5cm}
\end{proof}
\begin{lemma}
For $a_0=\cdots=a_{n-1}=0$, $\deg(F,x)=(n-1)(n-2)$.
\end{lemma}
\begin{proof}
When $a_0=\cdots=a_{n-1}=0$, $f=x^n$.
We first prove that $x=0$ is equivalent to $F=0$.
($\Longrightarrow$) If $x=0$, then $f_1=y^{n-1}$ and $f_3=\left(2-{1}/{2^{n-2}}\right)y^{n-2}$. In this case, $f_1$ and $f_3$ have a common zero and thus $F=0$.
($\Longleftarrow$) Let $F=0$; then $f_1$ and $f_3$ have at least one common zero for $y$, say $\bar{y}$. Then
\[
f_1(x,\bar{y})=\dfrac{\bar{y}^n-x^n}{\bar{y}-x}=0, \quad f_3(x,\bar{y})=\dfrac{\bar{y}^n-2\,\left(\dfrac{\bar{y}+x}{2}\right)^n+x^n}{\dfrac{(\bar{y}-x)^2}{2}}=0.
\]
Suppose that $x\neq 0$ and let $\bar{t}=\bar{y}/x$. Then the above equalities imply that
\[\bar{t}^n=1,\quad\left(\frac{1}{2}+\frac{1}{2}\bar{t}\right)^n=1.\]
Therefore, there exist two unit roots $u_1$ and $u_2$ such that $\bar{t}=u_1$ and $(1+\bar{t})/2=u_2$, which implies that $u_2=(1+u_1)/2$. This can happen only when $u_1=u_2=1$; so $\bar{y}=x$. Thus
\[f_1(x,\bar{y})=\dfrac{y^n-x^n}{y-x}=x^{n-1}+x^{n-2}\bar{y}+\cdots+x\bar{y}^{n-2}+x\bar{y}^{n-1}=nx^{n-1}=0,\]
which implies that $x=0$. This contradicts the assumption that $x\neq0$. Therefore, $x=0$.
Since $x=0$ and $F=0$ are equivalent, there exist a nonzero constant $c$ and an integer $N\geq1$ such that $F=c\,x^N$. It remains to show that $N=(n-1)(n-2)$.
Let
\[\Delta (x,y,\alpha)=\left|
\begin{array}{cc}\smallskip
f_1(x,y)&f_3(x,y)\\
f_1(x,\alpha)&f_3(x,\alpha)
\end{array}
\right|\bigg/(y-\alpha)\]
and $\bm{\nu}=(x, y,\alpha)$.
It is easy to see that $\Delta$ is homogeneous of degree $2\,n-4$ in $\bm{\nu}$ and
$\deg(\Delta,\alpha)=\deg(\Delta,y)= n-2$.
Let $\Delta$ be written as
\[\Delta=\sum_{\scriptsize\begin{array}{c}\scriptsize 0 \leq i\leq 2n-4\\ \scriptsize 0 \leq j,k\leq n-2
\end{array}}\delta_{ijk}x^iy^j\alpha^k.\]
Then for every term $x^iy^j\alpha^k$ occurring in $\Delta$, $i+j+k= \deg(\Delta, \bm{\nu})=2\,n-4$, so $i=2\,n-4-j-k$.
Denote by \[B=(b_{j+1,k+1})=\left(\sum_{i=0}^{2\,n-4}\delta_{ijk}x^i\right)\] the $(n-1)\times (n-1)$ B\'ezout matrix of $f_1$ and $f_3$ with respect to $y$ and let $\bar{F}=\det(B)$. According to the theory of resultants \cite{B1779T}, $F=\res(f_1,f_3,y)=\pm\, \bar{F}$, so $\deg(\bar{F},x)\geq1$.
Note that for any entry $b_{jk}$ in $B$, either $b_{jk}=0$ or $\deg(b_{jk},x)= 2\,n-2-j-k$. Let $(k_1,\ldots,k_{n-1})$ denote an arbitrary permutation of $(1,\ldots, n-1)$.
Then either $b_{1k_1}\cdots b_{n-1,k_{n-1}}=0$, or
\begin{align*}
\deg(b_{1k_1}\cdots b_{n-1,k_{n-1}},x)&=\sum_{j=1}^{n-1}\deg(b_{jk_j},x)=\sum_{j=1}^{n-1}(2\,n-2-j-k_j)\\
&=(2\,n-2)(n-1)-\sum_{j=1}^{n-1}j-\sum_{j=1}^{n-1}k_j \\
&=2\,(n-1)^2-(n-1)n=(n-1)(n-2).
\end{align*}
Note that $\bar{F}\neq 0$, so $\deg(\bar{F},x)=(n-1)(n-2)$. It follows that $\deg(F,x)=(n-1)(n-2)$.
\end{proof}
The following lemma has been used for the proof of Theorem \ref{thm:qIs2}.
\begin{lemma}\label{lem:deg_res_ff1f3}
$\deg(E,\bm{a})\leq 3\,(n-1)(n-2)+2\,(n-2)$.
\end{lemma}
\begin{proof}
Let $N=\deg(F,x)$; then $N\leq(n-1)(n-2)$ according to Lemma \ref{lem:deg_res_f1f3}. Moreover, from the proof of Lemma \ref{lem:deg_res_f1f3} we know that $\deg(F,\bm{a})\leq 2\,(n-2)$. Since $E$ is a determinant formed with $n$ rows of $f$-coefficients and $N$ rows of $F$-coefficients, the degree of each $f$-coefficient is at most $1$, and the degree of each $F$-coefficient is at most $2\,(n-2)$, the degree of $E$ is at most $N\cdot 1+n\cdot 2\,(n-2)\leq 3\,(n-1)(n-2)+2\,(n-2)$. The proof is complete.
\end{proof}
From Proposition \ref{prop:degD2} and Theorem \ref{thm:qIs2} the following corollary follows.
\begin{corollary}
$\deg(E,\bm{a})= 3\,(n-1)(n-2)$.
\end{corollary}
The result of this corollary allows us to reduce the upper bound $3\,(n-1)(n-2)+2\,(n-2)$ of $\deg(E,\bm{a})$ to $3\,(n-1)(n-2)$, the exact degree of $E$ in $\bm{a}$, which is also the degree of $D_2^2$ in $\bm{a}$.
\begin{remark}
{\em The determinant polynomials $F$ and $H$ are both irreducible over $\mathbb{Q}[\bm{a}]$. The irreducibility of $H$ is obvious because $D_2=\res(f, H,x)$ is irreducible and that of $F$ can be proved by using the symmetry of $F(r_1)$ with respect to $r_2,\ldots,r_n$.\footnote{Let $F(\bm{a},x)=F_1(\bm{a},x)F_2(\bm{a},x)$ with $\deg(F_1,x)\neq 0$. In this equality, substitution of $x$ by $r_1$ and elimination of each $a_i$ by using Vieta's formula yield $\bar{F}(r_1,\ldots,r_n)=\bar{F}_1(r_1,\ldots,r_n)\bar{F}_2(r_1,\ldots,r_n)$, where $\bar{F}$, $\bar{F}_1$, and $\bar{F}_2$ are all symmetric with respect to $r_2,\ldots,r_n$.
From the proof of Theorem \ref{thm:qIs2}, one sees that $\bar{F}=c\prod_{k\neq j}(r_1-2\,r_k+r_j)$ for some constant $c$. Thus $\bar{F}_1$ has at least one divisor $r_1-2\,r_k+r_j$ for some $j\neq k$. The symmetry of $\bar{F}_1$ with respect to $r_2,\ldots,r_n$ implies that $\prod_{\scriptsize{1<k\neq j}}(r_1-2\,r_k+r_j)$ is also a divisor of $\bar{F}_1$. Therefore, $\bar{F}_1$ differs from $\bar{F}$ only by a nonzero constant, and so does $F_1$ from $F$. It follows that $F_2$ is a constant. This proves the irreducibility of $F$.}
Hence $F$ and $H$ do not have any common divisor. On the other hand, $G$ is obtained from $f_1$ and $f_3$ via linear transformation and resultant computation and $F$ is connected to $H$ via $G$ by the relations
\begin{align*}
\res(f,F,x)&=\res(f,G,x)=[\res(f, H, x)]^2,\\
\deg(F, x)&=\deg(G,x)=2\,\deg(H,x),
\end{align*}
and $G=H^2$. However, it is unclear whether there is any direct connection between $F$ and $G$. Note that $F$ and thus $D_2$ are constructed from $f$, $f_1$, and $f_3$ naturally; yet the occurrence of the sequences of odd derivatives and even derivatives of $f$ with respect to $x$ in the determinant expressions of $H$ and $G$ remains uninterpretable. Meaningful interpretations of the occurrence might be figured out by exploring direct connections between $F$ and $G$.}
\end{remark}
\section{Application and Remarks}
\label{sec:ApplicationRemarks}
In this section, we illustrate the usefulness of the second discriminant by an application (to the classification of root configurations for the cubic polynomial) and discuss the possibility of introducing discriminants of higher order.
The form $r_i-r_j$ in $D_1$ can be viewed as the vector from $r_j$ to $r_i$, considered as two points in the complex plane. Similarly, the form $2\,r_k-r_i-r_j$ in $D_2$ can be viewed as twice the vector from the middle point of $r_i$ and $r_j$ to $r_k$. The signs of $D_1$ and $D_2$ carry information about the distribution, position, and relative configuration of the roots $r_1,\ldots,r_n$ of $f$. Therefore, $D_1$ and $D_2$ can be used to explore such structural properties of the roots of $f$ without exactly computing them out.
For the cubic polynomial $f=x^3+a_2x^2+a_1x+a_0$, we have the following Lagrange formula with radicals for its three roots:
\[
r_1\,=\dfrac{-a_2+\omega^1c_1+\omega^2c_2}{3},\quad
r_2\,=\dfrac{-a_2+\omega^0c_1+\omega^2c_2}{3},\quad
r_3\,=\dfrac{-a_2+\omega^2c_1+\omega^1c_2}{3},\quad
\]
where $\omega=e^{\frac{2\pi}{3}{\rm i}}=-\frac{1}{2}+\frac{\sqrt{3}}{2}{\rm i}$ and
\[c_1=\sqrt[3]{(D_2+2\,\sqrt{-3\,D_1})/2},\quad c_2=\sqrt[3]{(D_2-2\,\sqrt{-3\,D_1})/2}.\]
Using the above formula, one can classify the roots of $f$ into 9 types of configurations according to the signs of $D_1$ and $D_2$ as shown in Table~1 (cf.\ \cite{ZWH2011S}).
\noindent\!\!\!\!\!\includegraphics[width=1.08\textwidth]{D2figure.eps}
The second discriminant can also be used in the root formula with
radicals and to classify the types of configurations of the four
roots for the general quartic polynomial. The classification in this case
is somewhat involved and will be presented in a forthcoming paper \cite{HWYZ2016S}.
The second discriminant of a univariate polynomial $f$, a concept we have introduced, is defined as the product of all possible linear forms $2\,r_k-r_i-r_j$ in the roots $r_i,r_k,r_j$ of $f$ with $i<j\neq k$, so its vanishing is a necessary and sufficient condition for $f$ to have a symmetric triple of roots, i.e., a triple $(r_i, r_k, r_j)$ of roots of $f$ such that $r_k=(r_i+r_j)/2$. We have shown that the second discriminant of $f$ can be expressed as the resultant of $f$ and a determinant formed with the derivatives of $f$ and it possesses several notable properties\footnote{Our experiments also show that, when $a_0,\ldots,a_{n-1}$ take integer values, $D_2\not\equiv 2 \mod 4$ for $n>3$.} and can be used to analyze the structure of the roots of $f$.
We may naturally consider the product of linear forms in $d$ roots of $f$ for any $n\geq d\geq 4$. The product should be symmetric with respect to the $n$ roots of $f$ and the linear form should be chosen such that its vanishing constrains the $d$ general roots of $f$
to form a degenerate configuration which is geometrically interesting.
Then one can try to establish conditions for $f$ to have $d$ roots forming the degenerate configuration.
For $n\geq d=4$, linear forms of interest in four roots $r_i, r_j, r_k, r_l$ of $f$ could be taken of the following type
\begin{equation}\label{d4av}
r_i+r_j-r_k-r_l,\quad \mbox{or} \quad 3\,r_l-r_i-r_j-r_k.\end{equation}
The former is twice the difference between the average of the two roots $r_i$ and $r_j$ and that of the two roots $r_k$ and $r_l$, while the latter
is three times the difference from the root $r_l$ to the average of the three
roots $r_i, r_j, r_k$. When the roots are considered as points in the complex
plane, the average of two or three roots may be interpreted as the middle point or the centroid of the two or three points, respectively.
Using the first linear form in \eqref{d4av}, one may define
\[D_3=\prod_{\scriptsize{\begin{array}{c}
{i\neq j\neq k\neq l}\\
i<j, k<l, i<k
\end{array}}}{(r_i+r_j-r_k-r_l)}.\]
For $n=4$, $D_3$ can be expressed as a polynomial in the coefficients of $f$ and this polynomial has been used in the root formula of $f$ with radicals. How to express $D_3$ as a polynomial in the coefficients of $f$ for arbitrary $n> 4$ and what properties $D_3$ may have are questions that remain for further investigation. Similar questions may be asked for $D_3$ defined by using the other linear form, and for $D_4$, $D_5$, \ldots, when they are properly defined.
It should be pointed out that the ideas and methodologies used in the study of $D_2$ provide a new approach to explore the properties of $D_1$. It may be generalized to investigate $D_3, D_4, \ldots$ and to discover other mysteries about the roots of $f$.
| {
"timestamp": "2016-09-06T02:02:31",
"yymm": "1609",
"arxiv_id": "1609.00840",
"language": "en",
"url": "https://arxiv.org/abs/1609.00840",
"abstract": "We define the second discriminant $D_2$ of a univariate polynomial $f$ of degree greater than $2$ as the product of the linear forms $2\\,r_k-r_i-r_j$ for all triples of roots $r_i, r_k, r_j$ of $f$ with $i<j$ and $j\\neq k, k\\neq i$. $D_2$ vanishes if and only if $f$ has at least one root which is equal to the average of two other roots. We show that $D_2$ can be expressed as the resultant of $f$ and a determinant formed with the derivatives of $f$, establishing a new relation between the roots and the coefficients of $f$. We prove several notable properties and present an application of $D_2$.",
"subjects": "Commutative Algebra (math.AC); Rings and Algebras (math.RA)",
"title": "The Second Discriminant of a Univariate Polynomial",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9899864296722662,
"lm_q2_score": 0.7154240018510026,
"lm_q1q2_score": 0.7082600532943188
} |
https://arxiv.org/abs/1412.3089 | On Schemmel Nontotient Numbers | For each positive integer $r$, let $S_r$ denote the $r^{th}$ Schemmel totient function, a multiplicative arithmetic function defined by \[S_r(p^{\alpha})=\begin{cases} 0, & \mbox{if } p\leq r; \\ p^{\alpha-1}(p-r), & \mbox{if } p>r \end{cases}\] for all primes $p$ and positive integers $\alpha$. The function $S_1$ is simply Euler's totient function $\phi$. We define a Schemmel nontotient number of order $r$ to be a positive integer that is not in the range of the function $S_r$. In this paper, we modify several proofs due to Zhang in order to illustrate how many of the results currently known about nontotient numbers generalize to results concerning Schemmel nontotient numbers. We also invoke Zsigmondy's Theorem in order to generalize a result due to Mendelsohn. | \section{Introduction}
Integers in the range of Euler's totient function $\phi$ are known as totient numbers, and positive integers that are not totient numbers are known as nontotient numbers. The study of nontotient numbers has burgeoned in the past sixty years due to contributors such as Schinzel, Ore, Selfridge, Mendelsohn, and Zhang.
\par
In 1869, V. Schemmel introduced a class of functions $S_r$, now known as Schemmel totient functions, that generalize Euler's totient function \cite{schemmel69}. For each positive integer $r$, $S_r$ is a multiplicative function that satisfies
\[S_r(p^{\alpha})=\begin{cases} 0, & \mbox{if } p\leq r \\ p^{\alpha-1}(p-r), & \mbox{if } p>r \end{cases}\]
for all primes $p$ and positive integers $\alpha$. We will make use of the fact that $S_r(x)\vert S_r(y)$ whenever $x$ and $y$ are positive integers such that $x\vert y$.
\par
For a positive integer $r$, we define a Schemmel totient number of order $r$ to be an integer in the range of the function $S_r$. Any positive integer that is not a Schemmel totient number of order $r$ is said to be a Schemmel nontotient number of order $r$. For convenience, we will let $G_r$ denote the set of Schemmel nontotient numbers of order $r$. Our goal is to generalize some of the results currently known about nontotient numbers to results concerning Schemmel nontotient numbers and to encourage further investigation of Schemmel nontotient numbers. In fairness to Ming Zhi Zhang, we note that many of the proofs presented here are merely adaptations of proofs given in \cite{Zhang93}.
\par
Many theorems deal with which nontotient numbers are divisible by certain powers of $2$, so we will explore two ways of generalizing such theorems. If $r$ is odd, then it is easy to see that all odd integers greater than $1$ are Schemmel nontotient numbers, Thus, when $r$ is odd, we will continue to pay attention to which Schemmel nontotient numbers are divisible by certain powers of $2$. On the other hand, if $r$ is even, then every even positive integer is a Schemmel nontotient number of order $r$. This follows from the fact that if $r>1$ and $S_r(n)>0$ for some positive integer $n$, then $n$ must be odd. Furthermore, it is easy to see that $S_r(n)$ is odd whenever $S_r(n)$ is positive, $n$ is odd, and $r$ is even. Thus, it is uninteresting to look at powers of $2$ dividing Schemmel nontotient numbers of order $r$ for even values of $r$. Instead, we will concentrate on values of $r$ for which $r+1$ is prime, and we will focus on the Schemmel nontotient numbers of order $r$ that are divisible by certain powers of $r+1$.
\par
For now, we prove one result for which the parity of $r$ is irrelevant.
\begin{theorem} \label{Thm1.1}
If $r$ and $m$ are positive integers, then there exist infinitely many primes $p$ such that $pm$ is a Schemmel nontotient number of order $r$.
\end{theorem}
\begin{proof}
Fix positive integers $r$ and $m$, and let the positive divisors of $m$ be $d_1,d_2,\ldots,d_s$. Let $q_1,q_2,\ldots,q_s$ be primes satisfying $\max(m,r)<q_1<q_2<\cdots<q_s$. By the Chinese Remainder Theorem and Dirichlet's theorem concerning the infinitude of primes in arithmetic progressions, there are infinitely many primes $p>\max(q_s,m+r)$ that satisfy $d_ip\equiv -r\imod{q_i}$ for all $i\in\{1,2,\ldots,s\}$. Fix one such prime $p$, and suppose, for the sake of finding a contradiction, that $S_r(x)=pm$ for some positive integer $x$. If $p^2\vert x$, then $p(p-r)\vert S_r(x)=pm$, which contradicts the fact that $p>m+r$. If $p^2\nmid x$, then there must exist some prime $q$ such that $p\vert q-r$ and $q\vert x$. Then there exists some integer $d$ such that $pd=q-r\vert S_r(x)=pm$. This implies that $d=d_i$ for some $i\in\{1,2,\ldots,s\}$. Then, because $p$ satisfies the congruence $d_ip\equiv -r\imod{q_i}$, we see that $q_i\vert pd+r=q$, which implies that $q_i=q=pd+r$. However, this implies that $q_i>p$, which contradicts the fact that $p>q_s\geq q_i$.
\end{proof}
\par
Throughout the remainder of this paper, we will let $\mathbb{N}$, $\mathbb{N}_0$, and $\mathbb{P}$ be the sets of positive integers, nonnegative integers, and prime numbers, respectively.
\section{Schemmel Nontotient Numbers of Order One Less than a Prime}
When $r+1$ is prime, it is particularly interesting to consider positive integers $k$ such that $(r+1)^\alpha k\in G_r$ for all nonnegative integers $\alpha$. To do so, we first establish the following two lemmata, the first of which generalizes a theorem due to Mendelsohn \cite{Mendelsohn76}.
\begin{lemma} \label{Lem2.1}
Let $m$ be a positive integer such that $m+1$ is not a power of $2$. If there exist positive integers $N,p_1,p_2$ such that $p_1$ and $p_2$ are distinct primes and $\ord_{p_1}(m)=\ord_{p_2}(m)=2^N$, then there exists an arithmetic progression $A$ with the following three properties:
\begin{enumerate}[(a)]
\item $A$ contains infinitely many prime terms.
\item The common difference of $A$ is a product of $N+1$ distinct primes.
\item If $x$ is a term of $A$ and $t$ is a nonnegative integer, then $m^tx+m-1$ is divisible by exactly one of the $N+1$ prime divisors of the common difference of $A$.
\end{enumerate}
\end{lemma}
\begin{proof}
Zsigmondy's Theorem tells us that, for each positive integer $n$, there exists some prime that divides $m^{2^n}-1$ and does not divide $m^k-1$ for all positive integers $k<2^n$. In other words, for each positive integer $n$, we may find a prime $q_n$ such that $\ord_{q_n}(m)=2^n$. Suppose there exist positive integers $N,p_1,p_2$ such that $p_1$ and $p_2$ are distinct primes and $\ord_{p_1}(m)=\ord_{p_2}(m)=2^N$. Without loss of generality, we may let $q_N=p_1$ and write $q_0=p_2$. Let $\displaystyle{M=\prod_{i=0}^N}q_i$, and consider the system of congruences
\begin{equation} \label{Eq2.1}
\begin{cases} x+m\equiv 1\imod{q_n}, & \mbox{if } n=0 \\ m^{2^{n-1}}x+m\equiv 1\imod{q_n}, & \mbox{if } n\in\{1,2,\ldots,N\}. \end{cases}
\end{equation}
The Chinese Remainder Theorem tells us that the positive solutions to \eqref{Eq2.1} are precisely the terms of an arithmetic progression $A=a,a+M,a+2M,\ldots$ for some positive integer $a<M$. In addition, any solution to \eqref{Eq2.1} is relatively prime to $M$ because $q_n\nmid m-1$ for all $n\in\{0,1,\ldots,N\}$. Therefore, Dirichlet's theorem concerning the infinitude of primes in arithmetic progressions guarantees that $A$ has infinitely many prime terms.
\par
Now, choose some term $x$ of $A$, and let $t$ be a nonnegative integer. We will show that $q_n\vert m^tx+m-1$ for precisely one $n\in\{0,1,\ldots,N\}$. First, let $n\in\{1,2,\ldots,N\}$. We may use the fact that $m^{2^{n-1}}x+m\equiv 1\imod{q_n}$ to conclude that $q_n\vert m^tx+m-1$ if and only if $m^t\equiv m^{2^{n-1}}\imod{q_n}$. Furthermore, because $\ord_{q_n}(m)=2^n$, we see that $m^t\equiv m^{2^{n-1}}\imod{q_n}$ if and only if $t\equiv 2^{n-1}\imod{2^n}$. Similarly, because $x+m\equiv 1\imod{q_0}$, we see that $q_0\vert m^tx+m-1$ if and only if $m^t\equiv 1\imod{q_0}$. Because $\ord_{q_0}(m)=2^N$, we find that $m^t\equiv 1\imod{q_0}$ if and only if $t\equiv 0\imod{2^N}$. If $t=0$, it is clear that $q_0$ is the only element of the set $Q=\{q_0,q_1,\ldots,q_N\}$ that divides $m^tx+m-1$, so we may assume $t>0$. If we write $t=2^{\beta}\mu$, where $\beta,\mu\in\mathbb{N}_0$ and $2\nmid\mu$, then $\beta$ completely determines which primes in $Q$ divide $m^tx+m-1$. If $\beta\geq N$, then $q_0$ is the only element of $Q$ that divides $m^tx+m-1$. If $\beta<N$, then $q_{\beta+1}$ is the only element of $Q$ that divides $m^tx+m-1$. This completes the proof.
\end{proof}
\begin{lemma} \label{Lem2.2}
If $r$, $\alpha$, and $p$ are nonnegative integers with $r+1,p\in\mathbb{P}$, then $(r+1)^{\alpha}p\in G_r$ if and only if $p\neq (r+1)^t+r$ and $(r+1)^tp+r\not\in\mathbb{P}$ for all nonnegative integers $t\leq\alpha$.
\end{lemma}
\begin{proof}
First, suppose $S_r(x)=(r+1)^{\alpha}p$ for some positive integer $x$. If $p^2\vert x$, then $p(p-r)\vert S_r(x)=(r+1)^{\alpha}p$, so $p-r=(r+1)^t$ for some nonnegative integer $t\leq\alpha$. On the other hand, if $p^2\nmid x$, then there exists some prime $q$ such that $p\vert q-r$ and $q\vert x$. This implies that $q-r\vert S_r(x)=(r+1)^{\alpha}p$, which means that $q-r=(r+1)^tp$ for some nonnegative integer $t\leq\alpha$. Thus, if $p\neq (r+1)^t+r$ and $(r+1)^tp+r\not\in\mathbb{P}$ for all nonnegative integers $t\leq\alpha$, then $(r+1)^{\alpha}p\in G_r$.
\par
To prove the converse, suppose $p=(r+1)^t+r$ or $(r+1)^tp+r\in\mathbb{P}$ for some nonnegative integer $t\leq\alpha$. If $p=(r+1)^t+r$ and $t>0$, then $S_r((r+1)^{\alpha-t+1}p^2)=S_r((r+1)^{\alpha-t+1})S_r(p^2)=(r+1)^{\alpha-t}p(p-r)=(r+1)^{\alpha}p$. If $p=(r+1)^t+r$ and $t=0$, then $S_r((r+1)^{\alpha+2})=(r+1)^{\alpha+1}=(r+1)^{\alpha}p$. If $(r+1)^tp+r\in\mathbb{P}$, then $S_r((r+1)^{\alpha-t+1}((r+1)^tp+r))=S_r((r+1)^{\alpha-t+1})S_r((r+1)^tp+r)=(r+1)^{\alpha}p$. Thus, if $(r+1)^{\alpha}p\in G_r$, then we must have $p\neq (r+1)^t+r$ and $(r+1)^tp+r\not\in\mathbb{P}$ for all nonnegative integers $t\leq\alpha$.
\end{proof}
\begin{theorem} \label{Thm2.1}
Suppose $r+1$ is a prime that is not a Mersenne prime. If there exist integers $N,p_1,p_2$ such that $p_1$ and $p_2$ are distinct primes and $\ord_{p_1}(r+1)=\ord_{p_2}(r+1)=2^N$, then there are infinitely many primes $p$ such that $(r+1)^{\alpha}p\in G_r$ for all nonnegative integers $\alpha$.
\end{theorem}
\begin{proof}
Suppose that there exist integers $N,p_1,p_2$ such that $p_1$ and $p_2$ are distinct primes and $\ord_{p_1}(r+1)=\ord_{p_2}(r+1)=2^N$. We will show that there are infinitely many primes $p$ such that $p\neq (r+1)^t+r$ and $(r+1)^tp+r\not\in\mathbb{P}$ for all nonnegative integers $t$, from which Lemma \ref{Lem2.2} will yield the desired result. We may use Lemma \ref{Lem2.1} to conclude that there exists an arithmetic progression $A$ that has infinitely many prime terms and has common difference $\displaystyle{M=\prod_{i=0}^N}q_i$, where $q_0,q_1,\ldots,q_N$ are distinct primes. Furthermore, Lemma \ref{Lem2.1} tells us that if $p>M$ is a prime term of $A$ and $t$ is any nonnegative integer, then $(r+1)^tp+r$ is composite because it is divisible by one of the $N+1$ prime divisors of $M$. Hence, it suffices to show that there are infinitely many prime terms $p$ of $A$ that are not of the form $(r+1)^t+r$.
\par
If we let $\pi(x;M,a)$ denote the number of prime terms of $A$ that are less than or equal to $x$, then the Prime Number Theorem extended to arithmetic progressions tells us that $\displaystyle{\pi(x;M,a)\sim\frac{1}{\phi(M)}\frac{x}{\log x}}$ as $x\rightarrow\infty$. Because the number of primes less than or equal to $x$ of the form $(r+1)^t+r$ is clearly of order $\displaystyle{o\left(\frac{x}{\log x}\right)}$, the proof is complete.
\end{proof}
\begin{theorem} \label{Thm2.2}
Suppose that $r+1$ is a prime that is not a Mersenne prime and that there exist integers $N,p_1,p_2$ such that $p_1$ and $p_2$ are distinct primes and $\ord_{p_1}(r+1)=\ord_{p_2}(r+1)=2^N$. Let $M$ be as in the proof of Theorem \ref{Thm2.1}. Suppose $B=p_1^{\alpha_1}p_2^{\alpha_2}\cdots p_s^{\alpha_s}$, where $p_1,p_2,\ldots,p_s$ are distinct primes that are each greater than $r$ and congruent to $1$ modulo $M$ and $\alpha_1,\alpha_2,\ldots,\alpha_s$ are positive integers. If $p>M$ is one of the infinitely many primes that satisfies $(r+1)^{\alpha}p\in G_r$ for all $\alpha\in\mathbb{N}_0$ and $p-r$ has a prime divisor $P$ that does not divide $(r+1)B$, then $(r+1)^{\alpha}Bp\in G_r$ for all nonegative integers $\alpha$.
\end{theorem}
\begin{proof}
Suppose $S_r(x)=(r+1)^{\alpha}Bp$ for some nonnegative integers $\alpha$ and $x$. The existence of $P$ guarantees that $p^2\nmid x$, so there is some prime $q$ such that $p\vert q-r$ and $q\vert x$. Then $q=pd+r$ for some positive integer $d$, so $pd=q-r\vert (r+1)^{\alpha}Bp$. This implies that there exist nonnegative integers $t,\gamma_1,\gamma_2,\ldots,\gamma_s$ such that $pd+r=(r+1)^tpp_1^{\gamma_1}p_2^{\gamma_2}\cdots p_s^{\gamma_s}+r\equiv(r+1)^tp+r\imod{M}$. By Lemma \ref{Lem2.1}, there exists a unique prime divisor $q_i$ of $M$ that divides $(r+1)^tp+r$, so $q_i\vert pd+r=q$. This implies that $q=q_i$, which contradicts the fact that $q=pd+r>Md+r>q_i$.
\end{proof}
\begin{theorem} \label{Thm2.3}
Suppose $r+1$ is a prime and $k$ is a positive integer such that $r+1\nmid k$ and $(r+1)^{\alpha}k\in G_r$ for all nonnegative integers $\alpha$. If $k_1$ and $k_2$ are relatively prime positive integers such that $k_1k_2=k$, then either $(r+1)^{\alpha}k_1\in G_r$ for all nonnegative integers $\alpha$ or $(r+1)^{\alpha}k_2\in G_r$ for all nonnegative integers $\alpha$.
\end{theorem}
\begin{proof}
Suppose, for the sake of finding a contradiction, that there exist nonnegative integers $k_1,k_2,\alpha_1,\alpha_2,x_1,x_2$ such that $\gcd(k_1,k_2)=1$, $k_1k_2=k$, $S_r(x_1)=(r+1)^{\alpha_1}k_1$, and $S_r(x_2)=(r+1)^{\alpha_2}k_2$. We may assume that $\alpha_1$ and $\alpha_2$ are minimal with respect to these properties. Suppose $p=(r+1)^t+r$ is prime for some positive integer $t$. If $p^1\parallel x_1$, then we may write $x_1=p\mu$, where $\mu\in\mathbb{N}$ and $p\nmid\mu$. We then have $S_r(x_1)=(p-r)S_r(\mu)=(r+1)^tS_r(\mu)=(r+1)^{\alpha_1}k_1$, so $t\leq\alpha_1$ and $S_r(\mu)=(r+1)^{\alpha_1-t}k_1$, which contradicts the minimality of $\alpha_1$. Thus, if $p\vert x_1$, then $p^2\vert x_1$. By the same token, if $p\vert x_2$, then $p^2\vert x_2$. Let us write $d=\gcd(x_1,x_2)$ so that $S_r(d)\vert\gcd(S_r(x_1),S_r(x_2))=(r+1)^{\min(\alpha_1,\alpha_2)}$. Then $S_r(d)=(r+1)^{\beta}$ for some nonnegative integer $\beta$, so we may write $d=(r+1)^{\gamma}\lambda$, where $\gamma\in\mathbb{N}_0$ and $\lambda$ is a (possibly empty) product of distinct primes of the form $(r+1)^t+r$ ($t\in\mathbb{N}$). If $p=(r+1)^t+r$ is prime for some $t\in\mathbb{N}$ and $p\vert\lambda$, then $p\vert x_1,x_2$. This implies that $p^2\vert x_1,x_2$, so $p^2\vert d$. However, this contradicts the fact that $\lambda$ is a product of distinct primes, so we conclude that $\lambda=1$. Either $(r+1)^{\gamma+1}\nmid x_1$ or $(r+1)^{\gamma+1}\nmid x_2$, so we may assume, without loss of generality, that $(r+1)^{\gamma+1}\nmid x_1$. Then $x_1=(r+1)^{\gamma}y$, where $y\in\mathbb{N}$ and $r+1\nmid y$. Because $(r+1)^{\alpha_1}k_1=S_r(x_1)=S_r((r+1)^{\gamma})S_r(y)=(r+1)^{\gamma-1}S_r(y)$, we find that $S_r(y)=(r+1)^{\alpha_1-\gamma+1}k_1$. However, $x_2$ and $y$ are relatively prime, so $S_r(x_2y)=S_r(x_2)S_r(y)=(r+1)^{\alpha_1+\alpha_2-\gamma+1}k_1k_2=(r+1)^{\alpha_1+\alpha_2-\gamma+1}k$. This contradicts the hypothesis that $(r+1)^{\alpha}k\in G_r$ for all nonnegative integers $\alpha$, so the proof is complete.
\end{proof}
\section{Schemmel Nontotient Numbers \\ of Odd Order}
\begin{theorem} \label{Thm3.1}
Let $r$ be an odd positive integer, and write $n=2p_1^{\alpha_1}p_2^{\alpha_2}\cdots p_s^{\alpha_s}$, where, for all $i,j\in\{1,2,\ldots,s\}$ with $i<j$, $p_i$ and $p_j$ are odd primes, $\alpha_i$ and $\alpha_j$ are positive integers, and $p_i<p_j$. Then $n$ is a Schemmel nontotient number of order $r$ if and only if $n+r$ is composite and $p_s-r\neq 2p_1^{\alpha_1}p_2^{\alpha_2}\cdots p_{s-1}^{\alpha_{s-1}}$.
\end{theorem}
\begin{proof}
Zhang has proven the case $r=1$ \cite{Zhang93}, so we may assume that $r\geq 3$. First, suppose $S_r(x)=n$ for some positive integer $x$. Note that $S_r(p^{\alpha})$ is even for any prime $p$ and positive integer $\alpha$. Therefore, we know that $\omega(x)=1$, so we may write $x=p^{\alpha}$ for some prime $p$ and positive integer $\alpha$. If $\alpha=1$, then $n+r=p$. If $\alpha>1$, then we must have $p=p_s$ and $\alpha-1=\alpha_s$ because $n=p^{\alpha-1}(p-r)$. This implies that we must have $p_s-r=p-r=2p_1^{\alpha_1}p_2^{\alpha_2}\cdots p_{s-1}^{\alpha_{s-1}}$. Hence, if $n+r$ is composite and $p_s-r \neq 2p_1^{\alpha_1}p_2^{\alpha_2}\cdots p_{s-1}^{\alpha_{s-1}}$, then $n\in G_r$.
\par
Conversely, suppose that $n+r$ is prime or $p_s-r=2p_1^{\alpha_1}p_2^{\alpha_2}\cdots p_{s-1}^{\alpha_{s-1}}$. If $n+r$ is prime, then $S_r(n+r)=n$, so $n\not\in G_r$. If $p_s-r=2p_1^{\alpha_1}p_2^{\alpha_2}\cdots p_{s-1}^{\alpha_{s-1}}$, then $S_r(p_s^{\alpha_s+1})=n$, so $n\not\in G_r$.
\end{proof}
\begin{theorem} \label{Thm3.2}
Suppose $r$ is an odd positive integer and $n=2^{\alpha}p_1^{\alpha_1}p_2^{\alpha_2}\cdots p_s^{\alpha_s}$, where $p_1,p_2,\ldots,p_s$ are distinct odd primes that are each greater than $r$, $\alpha,\alpha_1,\alpha_2,\ldots,\alpha_s$ are positive integers, and $2^tp_1^{\gamma}+r$ is composite for all $t\in\{1,2,\ldots,\alpha\}$ and $\gamma\in\{1,2,\ldots,\alpha_1\}$. For each $t\in\{1,2,\ldots,\alpha\}$ and $\gamma\in\{1,2,\ldots,\alpha_1\}$, let $q_{t,\gamma}$ be a prime divisor of $2^tp^{\gamma}+r$, and let $M$ be the least common multiple of all such $q_{t,\gamma}$. If $p_i\equiv 1\imod{M}$ for all $i\in\{2,3,\ldots,s\}$ and $p_1-r\nmid n$, then $n$ is a Schemmel nontotient number of order $r$.
\end{theorem}
\begin{proof}
Suppose $S_r(x)=n$ for some positive integer $x$. Because $p_1-r\nmid n$, we see that $p_1^2\nmid x$. Thus, there exists a prime $q$ such that $p_1\vert q-r$ and $q\vert x$. Then $q=p_1d+r$ for some positive integer $d$, and we have $p_1d\vert 2^{\alpha}p_1^{\alpha_1}p_2^{\alpha_2}\cdots p_s^{\alpha_s}$. We may write $p_1d+r=2^tp_1^{\gamma}p_2^{\gamma_2}\cdots p_s^{\gamma_s}+r\equiv 2^tp_1^{\gamma}+r\imod{M}$, so $q=p_1d+r\equiv 2^tp_1^{\gamma}+r\equiv 0\imod{q_{t,\gamma}}$. This implies that $q=q_{t,\gamma}$, which contradicts the fact that $q=p_1d+r\geq2^tp_1^{\gamma}+r>q_{t,\gamma}$.
\end{proof}
\begin{remark} \label{Rem3.1}
Even within the case $r=1$, Theorem \ref{Thm3.2} provides a slight generalization of Theorem $3$ in \cite{Zhang93}.
\end{remark}
The proof of the following theorem utilizes Mendelsohn's original observations concerning the orders of the number $2$ modulo certain primes \cite{Mendelsohn76}.
\begin{theorem} \label{Thm3.3}
Suppose $r$ is an odd positive integer not divisible by $3$, $5$, $17$, $257$, $641$, $65537$, or $6700417$. There exist infinitely many primes $p$ such that $2^{\alpha}p$ is a Schemmel nontotient number of order $r$ for all nonnegative integers $\alpha$.
\end{theorem}
\begin{proof}
Let use write $q_0=6700417$, $q_1=3$, $q_2=5$, $q_3=17$, $q_4=257$, $q_5=65537$, and $q_6=641$. Observe that $\ord_{q_n}(2)=2^n$ for each positive integer $n\leq 6$ and $\ord_{q_0}(2)=2^6$. Now consider the system of congruences
\begin{equation} \label{Eq3.1}
\begin{cases} x+r\equiv 0\imod{q_n}, & \mbox{if } n=0 \\ 2^{2^{n-1}}x+r\equiv 0\imod{q_n}, & \mbox{if } n\in\{1,2,\ldots,6\}. \end{cases}
\end{equation}
By the Chinese Remainder Theorem, the positive solutions to \eqref{Eq3.1} are precisely the terms of an arithmetic progression with common difference \\
$\displaystyle{M=\prod_{i=0}^6q_i}$. Furthermore, any solution to \eqref{Eq3.1} is relatively prime to $M$ because $q_n\nmid r$ for all $n\in\{0,1,\ldots,6\}$. Following an argument virtually identical to that used in the proof of Lemma \ref{Lem2.1}, we see that if $x$ is any solution to the system of congruences \eqref{Eq3.1}, then, for any nonnegative integer $t$, $2^tx+r$ is divisible by precisely one element of the set $\{q_0,q_1,\ldots,q_6\}$. Furthermore, there are infinitely many prime solutions to \eqref{Eq3.1} that are not of the form $2^t+r$ ($t\in\mathbb{N}_0$). Therefore, there are infinitely many primes $p$ such that $p\neq 2^t+r$ and $2^tp+r\not\in\mathbb{P}$ for all nonnegative integers $t$. Fix one such prime $p$ and suppose, for the sake of finding a contradiction, that $S_r(x)=2^{\alpha}p$ for some nonnegative integers $x$ and $\alpha$. If $p^2\vert x$, then $p-r\vert2^{\alpha}$, which is a contradiction. If $p^2\nmid x$, then there exists some prime $q$ such that $q\vert x$ and $q-r=2^tp$ for some $t\in\mathbb{N}_0$, which is also a contradiction.
\end{proof}
\begin{theorem} \label{Thm3.4}
Let $r$ and $k$ be odd positive integers such that $2^{\alpha}k\in G_r$ for all nonnegative integers $\alpha$. Let $k_1$ and $k_2$ be relatively prime positive integers such that $k_1k_2=k$. Either $2^{\alpha}k_1\in G_r$ for all nonnegative integers $\alpha$ or $2^{\alpha}k_2\in G_r$ for all nonnegative integers $\alpha$
\end{theorem}
\begin{proof}
Zhang proves the case $r=1$ as Theorem 4 in \cite{Zhang93}, so we will assume $r>1$. Suppose that there exist nonnegative integers $\alpha_1,\alpha_2,x_1,x_2$ such that $S_r(x_1)=2^{\alpha_1}k_1$ and $S_r(x_2)=2^{\alpha_2}k_2$, and assume that $\alpha_1$ and $\alpha_2$ are minimal with respect to these properties. Note that $x_1$ and $x_2$ must be odd because $r>1$. Write $d=\gcd(x_1,x_2)$. Then $S_r(d)\vert\gcd(S_r(x_1),S_r(x_2))=2^{\min(\alpha_1,\alpha_2)}$, which means that $d$ is a (possibly empty) product of distinct primes that are each $r$ more than a power of $2$. Suppose $t$ is a nonnegative integer such that $p=2^t+r$ is prime. If $p\vert x_1$, then we may write $x_1=p\mu$, where $\mu\in\mathbb{N}$ and $p\nmid\mu$. This implies that $2^{\alpha_1}k_1=S_r(x_1)=S_r(p)S_r(\mu)=2^tS_r(\mu)$, so $S_r(\mu)=2^{\alpha_1-t}k_1$, which contradicts the minimality of $\alpha_1$. We reach a similar contradiction if we assume $p\vert x_2$. Thus, $d=1$, so $S_r(x_1x_2)=S_r(x_1)S_r(x_2)=2^{\alpha_1+\alpha_2}k_1k_2=2^{\alpha_1+\alpha_2}k$. This contradicts the fact that $2^{\alpha}k\in G_r$ for all nonnegative integers $\alpha$.
\end{proof}
\section{Concluding Remarks}
Clearly, we have only scratched the surface of the topic of Schemmel nontotient numbers, so we encourage the reader to continue the exploration. Indeed, except for Theorem \ref{Thm1.1}, we have not even considered Schemmel nontotient numbers of order $r$ when $r+1$ is odd and composite. Furthermore, after acknowledging the restrictions listed in the hypothesis of Theorem \ref{Thm2.1}, we make the following conjecture.
\begin{conjecture} \label{Conj4.1}
If $r+1$ is prime, then there are infinitely many primes $p$ such that $(r+1)^{\alpha}p\in G_r$ for all nonnegative integers $\alpha$.
\end{conjecture}
| {
"timestamp": "2014-12-10T02:18:04",
"yymm": "1412",
"arxiv_id": "1412.3089",
"language": "en",
"url": "https://arxiv.org/abs/1412.3089",
"abstract": "For each positive integer $r$, let $S_r$ denote the $r^{th}$ Schemmel totient function, a multiplicative arithmetic function defined by \\[S_r(p^{\\alpha})=\\begin{cases} 0, & \\mbox{if } p\\leq r; \\\\ p^{\\alpha-1}(p-r), & \\mbox{if } p>r \\end{cases}\\] for all primes $p$ and positive integers $\\alpha$. The function $S_1$ is simply Euler's totient function $\\phi$. We define a Schemmel nontotient number of order $r$ to be a positive integer that is not in the range of the function $S_r$. In this paper, we modify several proofs due to Zhang in order to illustrate how many of the results currently known about nontotient numbers generalize to results concerning Schemmel nontotient numbers. We also invoke Zsigmondy's Theorem in order to generalize a result due to Mendelsohn.",
"subjects": "Number Theory (math.NT)",
"title": "On Schemmel Nontotient Numbers",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9899864296722662,
"lm_q2_score": 0.7154240018510026,
"lm_q1q2_score": 0.7082600532943188
} |
https://arxiv.org/abs/1402.5913 | The majority game with an arbitrary majority | The $k$-majority game is played with $n$ numbered balls, each coloured with one of two colours. It is given that there are at least $k$ balls of the majority colour, where $k$ is a fixed integer greater than $n/2$. On each turn the player selects two balls to compare, and it is revealed whether they are of the same colour; the player's aim is to determine a ball of the majority colour. It has been correctly stated by Aigner that the minimum number of comparisons necessary to guarantee success is $2(n-k) - B(n-k)$, where $B(m)$ is the weight of the binary expansion of $m$. However his proof contains an error. We give an alternative proof of this result, which generalizes an argument of Saks and Werman. | \section{Introduction}
Fix $n$ and $k \in \mathbf{N}$ with $k > n/2$. The $k$-majority game is played with~$n$
numbered balls
which are each coloured with one of two colours. It is given
that there are at least~$k$ balls of the majority colour.
On each turn the player selects two balls to compare,
and it is revealed whether they are of the same colour, or of
different colours. The player's objective is to determine a ball of the majority colour.
We write $K(n,k)$ for the minimum number of comparisons that will guarantee success.
We write $B(m)$ for the number of digits $1$ in the binary representation of $m \in \mathbf{N}_0$.
The object of this paper is to prove the following theorem.
\begin{theorem}\label{thm:main}
If $n$, $k \in \mathbf{N}$ and $k \le n$ then $K(n,k) = 2(n-k) - B(n-k)$.
\end{theorem}
This theorem has been stated previously, as Theorem 3 of \cite[page 14]{Aigner}. We believe, however, that there is a flaw in the proof offered there
of the lower bound for $K(n,k)$, i.e.~the fact that $2(n-k) - B(n-k)$ comparisons are necessary.
The error arises in Case (ii) of the proof of Lemma 1, in which
it is implicitly assumed that if it is optimal at some point
for the player to compare balls~$i$ and~$t$, then there exist two balls~$j$ and~$\ell$ which it is optimal
to compare on the next turn, irrespective
of the answer received when balls $i$ and~$t$ are compared. The proof of Theorem~3 requires an analogue of
Lemma~1, stated as Lemma~3, which inherits the same error. The argument for Lemma~1 of \cite{Aigner}
is based on Lemma~5.1 in~\cite{Wiener}, which contains the same flaw; the authors are grateful to Prof.~Aigner and Prof.~Wiener for confirming these errors.\footnote{Personal communications.}
In \cite{SaksWerman}, Saks and Werman
have shown that
$K(2m+1,m+1) = 2m - B(m)$.
(An independent proof, using an elegant argument on the game tree,
was later given by Alonso, Reingold and Schott \cite{AlonsoEtAl}.)
The original contribution of this paper is
to supply a correct proof that $2(n-k) - B(n-k)$ questions are necessary in the general case, by generalizing the argument of Saks and Werman~\cite{SaksWerman}.
We remark that an alternative setting for the majority problem replaces the $n$ balls
with a room of $n$ people. Each person
is either a knight, who always tells the truth, or a knave, who always lies.
The question `Person $i$, is person $j$ a knight?'
corresponds to a comparison between balls $i$ and $j$. (The asymmetry in the
form of the questions is therefore illusory.)
In \cite[Theorem 6]{Aigner}, Aigner gives a clever questioning
strategy which demonstrates that $2(n-k) - B(n-k)$ questions
suffice, even when knaves are replaced by spies (Aigner's unreliable people), who may answer as they see fit.
He subsequently uses his Lemma 3 to show that $2(n-k) - B(n-k)$ questions are also necessary;
our Theorem~\ref{thm:main} can be used to replace this lemma, and so repair
the gap in the proof of Theorem 6 of \cite{Aigner}.
We refer the reader to \cite{Aigner} and the recent preprint \cite{ChengEtAl}
for a number of results on further questions that arise in this setting.
\section{Preliminary reformulation}
We begin with a standard reformulation of the problem
that follows \hbox{\cite[\S 2]{Aigner}}. In the special case $n=2m+1$ and $k=m+1$, it
may also be found in \cite[\S 4]{SaksWerman} and \cite[\S 2, \S3]{Wiener}.
A position in a $k$-majority game corresponds to a graph on $n$ vertices, in
which there is an edge, labelled either `same' or `different', between vertices $i$ and $j$ if balls $i$ and $j$
have been compared. Each connected component of this graph
admits a unique bipartition into parts corresponding to balls
of the same colour.
If $C$ is a component with bipartition $\{X, Y\}$
where $|X| \ge |Y|$ then we define the \emph{weight} of $C$ to be $|X| - |Y|$.
Suppose that the graph has distinct components $C$ and $C'$ of weights $w$ and $w'$ respectively,
where $w \ge w'$,
and that balls in $C$ and $C'$ are compared.
In the new question graph,
$C$ and~$C'$ are united in a single component; it is easily checked that
this component has weight either $w+w'$ or $w-w'$, depending
on which parts of $C$ and~$C'$ the balls lie in and which answer is given.
Moreover, if a game position has components
of weights $w_1, \ldots, w_c$ where $w_1 \ge w_2 \ge \ldots \ge w_c$
then exactly $n-c$ comparisons have been made.
Finally, if $e = k-(n-k)$ is the minimum
excess of the majority colour over the minority colour,
then $w_1 + \cdots + w_c = 2s + e$ for some $s \in \mathbf{N}_0$ and
the balls in the larger part of the component of weight $w_i$
can consistently be of the minority colour if and only if
$w_1 \le s$. (This is an equivalent condition to equation (14) in \cite{Aigner}.)
These remarks show that the $k$-majority game can be reformulated
as a two player adversarial game
played on multisets of non-negative
integers, which we shall call \emph{positions}. As above, let $e = k - (n-k)$.
The starting
position is the multiset $\{1,\ldots, 1\}$ containing $n$ elements. In
each turn, two distinct multiset elements $w$ and $w'$, with $w \ge w'$,
are chosen by the \emph{Selector}, and the \emph{Assigner}
chooses to replace them with either $w+w'$ or $w-w'$.
The game ends as soon as a position $\{w_1,\ldots, w_c\}$ is reached such
that $w_1 \ge \ldots \ge w_c$ and
\[ w_1 \ge s+1, \]
where $s$ is determined by $2s+e = w_1+\cdots + w_c$.
Following \cite{SaksWerman}, we call
such positions \emph{final}. We
define the \emph{value} of a general position $M$ to be the
number of elements in a final position reached from $M$, assuming,
as ever, optimal play by both sides. We denote the value of $M$ by $V(M)$.
The result we require, that $2(n-k) - B(n-k)$ questions are
necessary to identify a ball of the majority colour
in the $k$-majority game, is equivalent
to the following proposition.
\begin{proposition}\label{prop:bound}
Let $n \in \mathbf{N}$ and let $k > n/2$. The value of the
starting position in the $k$-majority game
is at most $B(n-k) + k - (n-k)$.
\end{proposition}
\section{Generalized Saks--Werman statistics}
If $M$ is a position in a majority game and $N$ is a submultiset
of $M$ then we shall say that $N$ is a \emph{subposition} of $M$.
Let $\bar{N}$ denote the complement of $N$ in $M$ and let
$\wt{M}$ denote the sum of all the elements of $N$.
Let $\epsilon_M(N) = \wt{\bar{N}} - \wt{N}$.
For $e \in \mathbf{N}$ and a position $M$ such that $\wt{M}$ and
$e$ have the same parity, we define
\[ \delta_e(M) = \sum_N (-1)^{\wt{N}} \]
where the sum is over all subpositions $N$ of $M$ such that
$\epsilon_M(N) \ge e$.
Thus a subposition of $M$ contributes to $\delta_e(M)$ if and only
if it corresponds to a colouring of the balls
in which the
excess of the majority colour over the minority colour is at least $e$.
We note that when $e=1$
we have
$\delta_1(M) = -f_M(-1)$ where $f_M$
is the polynomial defined in \cite[page 386]{SaksWerman}.
(The reason
for working with
minority subpositions rather than majority subpositions, as in \cite{SaksWerman},
will be seen in the proof of Lemma~\ref{lemma:alt}.)
The following lemma is a generalization of \cite[Lemma 4.2]{SaksWerman}.
\begin{lemma}\label{lemma:conserved}
Let $M$ be a position and let $e$ have the same parity as $\wt{M}$.
Let $w, w' \in M$ be two elements of $M$ with $w \ge w'$.
Let $M^+$ and $M^-$ be the positions obtained from $M$ if $w$ and $w'$
are replaced with $w+w'$ and $w-w'$, respectively. Then
\[ \delta_e(M) = \delta_e(M^+) + (-1)^{w'} \delta_e(M^-).\]
\end{lemma}
\begin{proof}
Let $N$ be a subposition
of $M$ such that $\epsilon_M(N) \ge e$
and let $N^\star = N \backslash \{ w, w' \}$.
We consider four possible cases for $N$.
\begin{itemize}
\item[(a)] If $w \in N$ and $w' \in N$ then $\wt{N} = \wt{N^\star \cup \{w, w'\}}= \wt{N^\star \cup \{w + w'\}}$ and
$\epsilon_M(N) = \epsilon_{M^+}(N^\star \cup \{w + w'\})$.
\item[(b)] If $w \not\in N$ and $w' \not\in N$ then $\wt{N} = \wt{N^\star}$ and
$\epsilon_M(N) = \epsilon_{M^+}(N^\star)$.
\item[(c)] If $w \in N$ and $w' \not\in N$ then $\wt{N} = \wt{N^\star \cup \{w\}} = \wt{N^\star \cup \{w - w'\}} + w'$
and $\epsilon_M(N) = \epsilon_{M^-}(N^\star \cup \{w-w'\})$.
\item[(d)] If $w \not\in N$ and $w' \in N$ then $\wt{N} = \wt{N^\star \cup \{w'\}} = \wt{N^\star} + w'$ and
$\epsilon_M(N) = \epsilon_{M^-}(N^\star)$.
\end{itemize}
Thus $\delta_e(M^+) = \sum_{N} (-1)^{\wt{N}}$ where the sum is over all subpositions
$N$ of $M$ such that $\epsilon_M(N) \ge e$ and either (a) or (b) holds, and
$(-1)^{w'} \delta_e(M^-) = \sum_{N} (-1)^{\wt{N}}$ where the sum is over all subpositions
$N$ of $M$ such that $\epsilon_M(N) \ge e$ and either (c) or (d) holds.
\end{proof}
We now define a family of further statistics.
For $e\in \mathbf{N}$ and
a position $M$ such that $\wt{M}$ and $e$ have the same parity, define
$\delta_e^{(1)}(M) = \delta_e(M)$, and for $b \in \mathbf{N}$ such that $b \ge 2$ define recursively
\[ \delta^{(b)}_e(M) = \sum_{t \in \mathbf{N}_0} \delta^{(b-1)}_{e+2t}(M). \]
For an alternative expression for $\delta^{(b)}_e(M)$ see Lemma~\ref{lemma:alt}.
We note that if $e + 2t > \wt{M}$
then $\delta^{(b-1)}_{e+2t}(M) = 0$,
and so the sum defining $\delta^{(b-1)}_e(M)$ is finite. Since we only need positions whose sum of elements
is at most $n$, it follows by induction on $b$ that
$\delta^{(b)}_e$ is a linear combination of the statistics $\delta_d$
for $d \ge e$. Hence, if $M$, $M^+$, $M^-$ and $w$, $w'$
are as in Lemma~\ref{lemma:conserved}, we have
\[ \delta^{(b)}_e(M) = \delta^{(b)}_e(M^+) + (-1)^{w'} \delta^{(b)}_e(M^-)
\tag{$\star$} . \]
For $r \in \mathbf{N}$ let $P(r)$ denote the highest power of $2$ dividing $r$
and let \hbox{$P(0) = \infty$}.
Let
\[ SW^{(b)}_e(M) = e + P\bigl( \delta^{(b)}_e(M)\bigr) \]
where, as expected, we set $e + \infty = \infty$. Our key statistic is now
\[ SW_e(M) = SW^{(e)}_e(M). \]
We remark that if $\wt{M}$ is odd then $SW_1(M) = \Phi(M)$ where $\Phi(M)$
is as defined in \cite[page 386]{SaksWerman}.
In \S 5 below we prove Proposition~\ref{prop:bound} by
using $SW_e(M)$ to prove an upper bound on the value $V(M)$ of a position $M$.
The key properties of $SW_e(M)$ we require are
$(\star)$ and the values of $SW_e(M)$ at starting and final positions. Starting
positions are dealt with in the following lemma, whose proof uses the basic identity
$\sum_{r=0}^s (-1)^r \binom{n}{r} = (-1)^s \binom{n-1}{s}$; see \cite[Equation 5.16]{CMath}.
\begin{lemma}\label{lemma:start}
Let $M_{\mathrm{start}}$ be the multiset containing $1$ with multiplicity $n$.
Suppose that $n = 2s+e$ where $s$, $e\in \mathbf{N}$. Then for any $b \in \mathbf{N}$ such that $b \le n$ we have
\[ \delta_e^{(b)}(M_\mathrm{start}) = (-1)^s \binom{n-b}{s}.\]
\end{lemma}
\begin{proof}
When $b=1$ we have $\delta_e^{(1)} = \delta_e$.
A subposition $N$ of~$M_\mathrm{start}$
contributes to the sum defining $\delta_e(M)$ if and only if $\wt{N} \le s$.
Therefore
\[ \delta_e^{(1)}(M) = \sum_{r=0}^s (-1)^r \binom{n}{r} = (-1)^s \binom{n-1}{s}.\]
If $b \ge 2$ then, by induction, we have
\[ \delta_e^{(b)}(M) = \sum_{t \in \mathbf{N}_0} \delta_{e+2t}^{(b-1)}(M) =
\sum_{t=0}^s (-1)^{s-t} \binom{n-(b-1)}{s-t} = (-1)^s \binom{n-b}{s}
\]
again as required.
\end{proof}
It follows that if $M_\mathrm{start}$, $s$ and $e$ are as in Lemma~\ref{lemma:start}, then
$SW_e(M_\mathrm{start}) = e + P \bigl( \binom{2s}{s} \bigr)$.
It is well known that
$P\bigl( \binom{2t}{t} \bigr) = B(t)$ for any $t \in \mathbf{N}$. (Two different
proofs are given in [2] and [4].) Hence
\[ \tag{$\dagger$} SW_e(M_\mathrm{start}) = e + B(s). \]
\section{Final positions}
Let $e = k - (n-k)$.
In this section we show that if $M$ is a final position containing exactly $c$ elements then
$SW_e(M) \ge c$. The proof uses the \emph{hyperderivative} on the ring $\mathbf{Z}[x,x^{-1}]$ of integral
Laurent polynomials, defined on the monomial basis for $\mathbf{Z}[x,x^{-1}]$ by
\[ D^{(r)} x^p = \binom{p}{r} x^{p-r} \]
for $p \in \mathbf{Z}$ and $r \in \mathbf{N}_0$. (This extends the usual definition of the hyperderivative
for polynomial rings, given in \cite[page 303]{LidlNiederreiter}.) The key property we require is
the following small generalization of \cite[Lemma~6.47]{LidlNiederreiter}.
\begin{lemma}\label{lemma:Leibnizrule}
Let $f,g \in \mathbf{Z}[x,x^{-1}]$ be Laurent polynomials. Let $r \in \mathbf{N}$. Then
\[ D^{(r)}(fg) = \sum_{t=0}^r D^{(t)}(f) D^{(r-t)}(g). \]
\end{lemma}
\begin{proof}
By bilinearity it is sufficient to prove the lemma when $f= x^p$ and $g= x^{q}$ where $p$, $q \in \mathbf{Z}$. In this case the lemma follows from
\[ \binom{p+q}{r} = \sum_{t=0}^r \binom{p}{t} \binom{q}{r-t} \]
which is the Chu--Vandermonde identity; see \cite[Equation 5.22]{CMath}.
\end{proof}
We also need an alternative expression for $\delta^{(b)}_e(M)$.
Let $\alpha_r(M)$ be the number of subpositions $N$ of a position $M$ such that $\wt{N} = r$.
The proof of the next lemma uses the basic identity
$\sum_{d=r}^n \binom{d}{r} = \binom{n+1}{r+1}$; see \cite[Table~174]{CMath}.
\begin{lemma}\label{lemma:alt}
Let $M$ be a position such that $\wt{M} = 2s + e$. Then
\[ \delta^{(b)}_e(M) = \sum_{r=0}^{s} \binom{s+b-1-r}{b-1} (-1)^{r} \alpha_r(M). \]
\end{lemma}
\begin{proof}
When $b=1$, we have $\delta^{(1)}_e(M) = \delta_e(M) = \sum_{r=0}^s (-1)^r \alpha_r(M)$.
If $b \ge 2$, then by induction, we have
\begin{align*}
\delta_e^{(b)}(M) &= \sum_{t \in \mathbf{N}_0} \delta_{e+2t}^{(b-1)}(M) \\
&= \sum_{t=0}^s \sum_{r=0}^{s-t} \binom{s-t+b-2-r}{b-2} (-1)^r \alpha_r(M) \\
&= \sum_{r=0}^s
\sum_{t =0}^{s-r} \binom{s-t+b-2-r}{b-2}
(-1)^r \alpha_r(M) \\
&= \sum_{r=0}^s \binom{s+b-1-r}{b-1} (-1)^r \alpha_r(M)
\end{align*}
as claimed.
\end{proof}
Now let $M = \{w_1, \ldots, w_c\}$ be a final position in the $k$-majority game
where $w_1 \ge \ldots \ge w_c$. Let $\wt{M} = 2s+e$.
As remarked in \S 2, we have
\[ w_1 \ge s+1. \
Let
\[ g = x^{s+e-1} (1+x^{-w_2}) \ldots (1+x^{-w_c}). \]
and note that, since $w_2 + \cdots + w_c \le s+e-1$, $g$ is a polynomial. Let
\[ g = \sum_{r=0}^{s+e-1} \alpha'_r(M) x^{s+e-1-r}. \]
If $r \le s$ then a subposition $N$ of $M$ such that $\wt{N} = r$ cannot contain $w_1$. Therefore
$\alpha'_r(M) = \alpha_r(M)$ whenever $r \le s$.
It follows that
\[ (D^{(e-1)} g) (-1) = \sum_{r=0}^{s} \binom{s-r+e-1}{e-1} (-1)^{s-r} \alpha_r(M). \]
Hence, by Lemma~\ref{lemma:alt}, we have
\[ (D^{(e-1)} g) (-1) = \delta^{(e)}_e(M). \]
(The normal derivative would introduce an unwanted $(e-1)!$ at the point.)
It follows from Lemma~\ref{lemma:Leibnizrule}, applied to the original
definition of $g$, that $D^{(e-1)}(g)$ is
a linear combination, with coefficients in $\mathbf{Z}$, of polynomials of the form
\[ h = x^{s+e-1-a_1}
\prod_{i \in A} x^{-w_i-a_i}
\prod_{j \in B} (1+x^{-w_j})
\]
where $A$ is a subset of $\{2,\ldots, n\}$, $B = \{2,\ldots, n\}\backslash A$,
and $a_1 + \sum_{i \in A} a_i = e-1$ where each summand is non-negative.
It is clear that $h(-1) = 0$ unless $w_j$ is even for all $j \in B$, in which
case $h(-1) = \pm 2^{|B|}$. Since $|B| \ge (c-1) - (e-1) = c-e$, it follows
that $P\bigl( h(-1) \bigr) \ge c-e$ for all such polynomials $h$.
Hence
\[ \tag{$\ddagger$}
SW_e(M) = e + P\bigl( \delta^{(e)}_e(M)\bigr) = e + P\bigl( (D^{(e-1)} g) (-1) \bigr)
\ge c \]
as claimed at the start of this section.
\section{Proof of Proposition~\ref{prop:bound}}
We are now ready to prove Proposition~\ref{prop:bound}.
Let $e = k - (n-k)$ be the minimum excess of the majority colour over the minority colour
in the $k$-majority game with $n$ balls.
Let $M$ be a position. Suppose that an optimal
play for the Selector is to choose $w$ and $w' \in M$.
Since the Assigner wishes to minimize the number of elements in the final position, we have
\[ V(M) = \min \bigl( V(M^+), V(M^-) \bigr) \]
where $M^+$ and $M^-$ are as defined in Lemma~\ref{lemma:conserved}.
If $x, y \in \mathbf{Z}$ then $P(x+y) \ge \min \bigl( P(x), P(y) \bigr)$.
Hence ($\star$) in \S 3 implies that
\[ SW_e(M) \ge \min\bigl( SW_e(M^+), SW_e(M^-) \bigr). \]
By ($\ddagger$) at the end of \S 4, if $M$
is a final position containing $c$ elements then $SW_e(M) \ge c$.
In this case $V(M) = c$, so we have $SW_e(M) \ge V(M)$.
It therefore follows by induction that
\[ SW_e(M) \ge V(M) \]
for all positions $M$. It was seen in $(\dagger)$ at the end of \S 3 that
if $M_\mathrm{start}$ is the starting position then
$SW_e(M_{\mathrm{start}}) = B(n-k) + e$ and so
\[ B(n-k) + e \ge V(M_{\mathrm{start}}) \]
as required.
\section{Final remark}
We end by showing that the statistics $SW_e(M)$ do not
predict all optimal moves for the Assigner.
We need the following lemma in the case when $m$ is odd; it can
be proved
in a similar way to Lemma~\ref{lemma:start}.
\begin{lemma}\label{lemma:last}
For any $m \in \mathbf{N}$ we have
\[ SW_1\bigl( \{2,1^{2m-1}\} \bigr) = \begin{cases} 2 + B(m-1) + P(m-1) & \text{if $m$ is odd} \\
2 + B(m-1) - P(m) & \text{if $m$ is even.}
\end{cases} \]
\end{lemma}
Let $m \equiv 3$ mod $4$ and let $M = \{1^{2m+1}\}$ be the
starting position in the majority game with $n=2m+1$ and $k=m+1$.
The positions the Assigner can choose between on the first move in the game
are
$M^+ = \{2,1^{2m-1} \}$ and $M^- = \{1^{2m-1},0\}$.
By Lemma~\ref{lemma:last}, we have
\[ SW_1(M^+) = 2 + B(m-1) + P(m-1) = 3 + B(m-1) = 2 + B(m).\]
It is clear that removing a zero element from a position decreases
its $SW_1$ statistic by~$1$, so by ($\dagger$) at the end of \S 3 we have
\[ SW_1(M^-) = 1 + SW_1( \{1^{2m-1} \}) = 1 + 1 + B(m-1) = 1 + B(m).\]
Using the $SW_1$ statistic, the Assigner will therefore choose
$M^-$ on the first move.
However Lemma~\ref{lemma:last} implies that
$SW_1(\{2,1^{2m-3},0\}) = 1 + B(m)$, and we have already seen that
$SW_1\bigl( \{1^{2m-1}\}\bigr) = B(m)$.
It follows that playing to $M^+$ is also an optimal
move for the Assigner.
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| {
"timestamp": "2014-02-25T02:13:37",
"yymm": "1402",
"arxiv_id": "1402.5913",
"language": "en",
"url": "https://arxiv.org/abs/1402.5913",
"abstract": "The $k$-majority game is played with $n$ numbered balls, each coloured with one of two colours. It is given that there are at least $k$ balls of the majority colour, where $k$ is a fixed integer greater than $n/2$. On each turn the player selects two balls to compare, and it is revealed whether they are of the same colour; the player's aim is to determine a ball of the majority colour. It has been correctly stated by Aigner that the minimum number of comparisons necessary to guarantee success is $2(n-k) - B(n-k)$, where $B(m)$ is the weight of the binary expansion of $m$. However his proof contains an error. We give an alternative proof of this result, which generalizes an argument of Saks and Werman.",
"subjects": "Combinatorics (math.CO)",
"title": "The majority game with an arbitrary majority",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9899864292291072,
"lm_q2_score": 0.7154240018510026,
"lm_q1q2_score": 0.7082600529772722
} |
https://arxiv.org/abs/2202.06810 | Structured Codes of Graphs | We investigate the maximum size of graph families on a common vertex set of cardinality $n$ such that the symmetric difference of the edge sets of any two members of the family satisfies some prescribed condition. We solve the problem completely for infinitely many values of $n$ when the prescribed condition is connectivity or $2$-connectivity, Hamiltonicity or the containment of a spanning star. We also investigate local conditions that can be certified by looking at only a subset of the vertex set. In these cases a capacity-type asymptotic invariant is defined and when the condition is to contain a certain subgraph this invariant is shown to be a simple function of the chromatic number of this required subgraph. This is proven using classical results from extremal graph theory. Several variants are considered and the paper ends with a collection of open problems. | \section{Introduction}
Celebrated problems of extremal combinatorics may get an exciting new flavour when the presence of some special structure is imposed in the condition. A prominent example is the famous Simonovits-S\'os conjecture \cite{SimSos} proven by Ellis,
Filmus and Friedgut \cite{EFF},
which determines the maximum possible cardinality of a family
of graphs on $n$ labeled vertices in which
the intersection of any two members contains a triangle. (The result of \cite{EFF} shows, along with several far reaching generalizations, that the best is to take all graphs containing a given triangle, just as it was conjectured in \cite{SimSos}. This is clearly reminiscent of the Erd\H{o}s-Ko-Rado theorem \cite{EKR}.) As another example we can also mention the Ramsey type problem investigated in \cite{trif} that was also initiated by a question of
S\'os and can be considered as a graph version of the first unsolved case of the so-called perfect hashing problem. (For details we refer to \cite{trif}).
In this paper we study several problems we arrive to if the basic code distance problem (how many binary sequences of a given length can be given at most if any two differ in at least a given number of coordinates) is
modified so that we do not prescribe the minimum distance of any two
codewords but require that they differ in some specific structure. In particular, just as in the Simonovits-S\'os problem we seek the largest family of (not necessarily induced) subgraphs of a complete graph such that the symmetric difference of the edge sets of any two graphs in the family has some required property. We will consider properties like connectedness, Hamiltonicity, containment of a triangle and some more. Formally all these can be described by saying that the graph defined by the symmetric difference of the edge sets of any two of our graphs belongs to a prescribed family of graphs (namely those that are connected, contain a Hamiltonian cycle, or contain a triangle, etc.)
Let ${\cal F}$ be a fixed class of graphs. A graph family ${\cal G}$ on $n$ labeled vertices is called ${\cal F}$-good if for any pair $G,G'\in {\cal G}$ the graph $G\oplus G'$ defined by $$V(G\oplus G')=V(G)=V(G')=[n],$$ where $[n]=\{1,\dots,n\}$ and $$E(G\oplus G')=\{e: e\in (E(G)\setminus E(G'))\cup (E(G')\setminus E(G))\}$$ belongs to ${\cal F}$.
Let $M_{\cal F}(n)$ denote the maximum possible size of an ${\cal F}$-good family on $n$ vertices. We are interested in the value of $M_{\cal F}(n)$ for various classes ${\cal F}$. We will give exact answers or both lower and upper bounds in several cases.
We mention that codes where the codewords are described by graphs already appear in the literature. In \cite{Tonchev}, for example, Tonchev looked at the usual code distance problem restricted to codes whose codewords are characteristic vectors of edge sets of graphs. Gray codes on graphs are also considered, see \cite{Mutze}, where the graphs representing the codewords should have some similarity properties if they are consecutive in a certain listing. Problems analogous to
the present ones though restricted to special graph classes were also considered in \cite{KMS} and \cite{CFK}. A very interesting result along these lines is the one in \cite{Kose}.
\medskip
\par\noindent
The paper is organised as follows. In Section~\ref{gbound} we give a general upper bound that will turn out to be sharp in several of the cases we consider.
In Section~\ref{global} we consider classes ${\cal F}$ defined by some global criterion as connectivity or $2$-connectivity, Hamiltonicity or containing a full star, that is, a vertex of degree $n-1$. We determine $M_{\cal F}(n)$ for infinitely many values of $n$ and for all $n$ in the first and the last case. In most of the cases when we give sharp bounds it is via also solving the problem we call dual: we give the largest possible size of a graph family for which the symmetric difference of no two of its members satisfies the original requirement. The case of the full star is an exception in this sense, nevertheless we also solve the dual problem in that case for all even $n$ by using a celebrated lemma of Shearer.
In Section~\ref{local} we consider classes ${\cal F}$ defined by local conditions. This means that for certifying the condition it is enough to see just a special part of the graph pair at hand. A capacity-type asymptotic invariant is natural to define in these cases. It turns out that when the requirement is that the pairwise symmetric differences contain a certain subgraph then this asymptotic invariant depends only on the chromatic number of the graph to be contained. We also discuss the case when the above containment is required in an induced manner, and obtain similar results in this case. The final section contains a collection of open problems.
\section{A general upper bound}\label{gbound}
To bound $M_{\cal F}(n)$ for various graph classes ${\cal F}$ it will often be useful to also consider the related problem of constructing large graph families in which no pair satisfies the condition prescribed by ${\cal F}$.
\begin{defi}\label{dual}
For a class of graphs ${\cal F}$ let $D_{{\cal F}}(n)$ denote the maximum possible size of a graph family on $n$ labeled vertices (that is, each member of the family has $[n]=\{1,\dots,n\}$ as vertex set), the symmetric difference of no two members of which belongs to ${\cal F}$. Determining $D_{{{\cal F}}}(n)$ will be referred to as the dual problem of determining $M_{{\cal F}}(n)$.
\end{defi}
\medskip
\par\noindent
Note that denoting by $\overline{{\cal F}}$ the class containing exactly those graphs that do not belong to ${\cal F}$
we actually have $$D_{{\cal F}}(n)=M_{\overline{{\cal F}}}(n),$$ that is the requirement of having no symmetric difference in ${\cal F}$ is clearly the same as saying that all symmetric differences belong to the complementary family $\overline{{\cal F}}$. Nevertheless, we will use the $D_{{\cal F}}(n)$ notation to emphasize the dual nature of the problem in those cases.
\begin{lemma}\label{lem:ub}
For any graph class ${\cal F}$ we have $$M_{{\cal F}}(n)\cdot D_{{\cal F}}(n)\le 2^{n\choose 2}.$$
\end{lemma}
{\noindent \it Proof. }
Let us define a graph $H_{{\cal F}}$ whose vertices are all the possible (simple) graphs on the vertex set $[n]$. Connect two such vertices if and only if the corresponding pair of graphs have their symmetric difference belonging to ${\cal F}$. Then by definition we have $$M_{{\cal F}}(n)=\omega(H_{{\cal F}}) \ \ {\rm and} \ \ D_{{{\cal F}}}(n)=\alpha(H_{{\cal F}}),$$
where $\omega(H)$ and $\alpha(H)$ denote the clique number and the independence number of the graph $H$, respectively.
Observe that $H_{{\cal F}}$ is vertex-transitive, (in fact it is a Cayley graph
of the group $Z_2^{{n \choose 2}}$). Indeed, if $G_1$ and $G_2$ are two graphs forming vertices of $H_{{\cal F}}$ then taking the symmetric difference of all $n$-vertex graphs forming vertices of $H_{{\cal F}}$ with the graph $G_1\oplus G_2$ is an automorphism of $H_{{\cal F}}$ that maps $G_1$ to $G_2$.
Since a vertex-transitive graph $H$ always satisfies $\alpha(H)\omega(H)\le |V(H)|$ (this can be seen by using that the fractional chromatic number $\chi_f(H)$ always satisfies $\omega(H)\le\chi_f(H)$, while if $H$ is a vertex-transitive graph we also have $\chi_f(H)=\frac{|V(H)|}{\alpha(H)}$, cf. \cite{SchU}), the statement follows.
\ifhmode\unskip\nobreak\fi\quad\ifmmode\Box\else$\Box$\fi
\medskip
\par\noindent
The above lemma makes it possible to bound $M_{{\cal F}}(n)$ from above by bounding $D_{{{\cal F}}}(n)$ from below. In particular, whenever we construct two families of graphs ${\cal A}$ and ${\cal B}$ on $[n]$ such that $A,A'\in {\cal A}$ implies $A\oplus A'\in {\cal F}$ and $B,B'\in{\cal B}$ implies $B\oplus B'\notin{\cal F}$, while $|{\cal A}||{\cal B}|=2^{n\choose 2}$, then we know that $|{\cal A}|$ and $|{\cal B}|$ realize the optimal values $M_{{\cal F}}(n)$ and $D_{{{\cal F}}}(n)$ for such families. Below we will see several cases when this simple technique can indeed be used to obtain these optimal values. An exception to this phenomenon is also presented by Theorems~\ref{thm:fullstar} and \ref{thm:nostar}.
\medskip
\par\noindent
\begin{remark}\label{rem:altproof} {\rm It is worth noting that Lemma~\ref{lem:ub} can be proven in a different way, with no reference to the fractional chromatic number. Indeed, if $G_1,\dots,G_k$ is an ${\cal F}$-good family, while $T_1,\dots,T_m$ is a family satisfying the conditions of the dual problem, then all the symmetric differences of the form $G_i\oplus T_j$ are different, implying $km\le 2^{n\choose 2}$. This is true because if $G_i\oplus T_j$ and $G_r\oplus T_s$ would be the same for some $\{i,j\}\neq\{r,s\}$, then $(G_i\oplus T_j)\oplus(G_r\oplus T_s)$ would be the empty graph that could also be written (by commutativity and associativity of the symmetric difference) as $(G_i\oplus G_r)\oplus(T_j\oplus T_s)$. This would mean that $G_i\oplus G_r$ and $T_j\oplus T_s$ are two identical graphs. But if one of them is the empty graph, then the other cannot be empty and if both are nonempty, then one of them belongs to ${\cal F}$ while the other one does not, so this is impossible. $\Diamond$}
\end{remark}
\section{Global conditions}\label{global}
\subsection{Connectivity}
When we speak about the class of connected graphs in the following theorem, we mean
graphs with a single connected component, and hence without isolated vertices.
\begin{thm}~\label{thm:conn}
Let ${\cal F}_c$ denote the class of connected graphs. Then $$M_{{\cal F}_c}(n)=2^{n-1}.$$
\end{thm}
{\noindent \it Proof. }
First we give a very simple dual family ${\cal B}_c$. Let it consist of all graphs on $[n]$ in which the vertex labeled $n$ is isolated. Clearly $|{\cal B}_c|=2^{{n-1}\choose 2}$ (that is the number of all graphs on $[n-1]$) and $n$ is also isolated in the symmetric difference of any two of them, so no such symmetric difference is connected, This gives $D_{{{\cal F}_c}}(n)\ge 2^{{n-1}\choose 2}$ and thus by Lemma~\ref{lem:ub} we have
$$M_{{\cal F}_c}(n)\le 2^{{n\choose 2}-{n-1\choose 2}}=2^{n-1}.$$
Now we show that this upper bound can be attained.
Let the family ${\cal A}_c$ consist of all those graphs on $[n]$ that are the vertex-disjoint union of two complete graphs
(where each vertex belongs to one of them)
including the case when one of the two is on the empty set. Clearly, the number of these graphs is just half the number of subsets of $[n]$, that is exactly $2^{n-1}$. All we have to show is that the symmetric difference of any two of these graphs is connected. Choose two arbitrary graphs $G$ and $G'$ from our family. Let $G$ be the union of complete graphs on the complementary vertex sets $K$ and $L$, while $G'$ be the same on $K'$ and $L'$. Let $A=K\cap L', B=L'\cap L, C=L\cap K'$ and $D=K'\cap K$. It is possible that one, but only one of $A,B,C,D$ is empty. The edges of $G\oplus G'$ are all the edges of the complete bipartite graph with partite classes $A\cup C$ and $B\cup D$, so it must be connected.
\ifhmode\unskip\nobreak\fi\quad\ifmmode\Box\else$\Box$\fi
With just a little more consideration one can also treat the case of $2$-connectedness at least for even $n$.
\begin{thm}~\label{thm:2conn}
Let ${\cal F}_{2c}$ denote the class of $2$-connected graphs. Then if $n$ is even, we have $$M_{{\cal F}_{2c}}(n)=2^{n-2}.$$
\end{thm}
{\noindent \it Proof. }
The proof is a modification of the previous one, therefore we use the notation introduced there. The construction given there may result in symmetric differences that are not $2$-connected only if $A\cup C$ or $B\cup D$ contains only one element. For even $n$ this can be avoided if we consider only such graphs in our construction where the bipartition of $[n]$ defining the individual graphs has an even number of elements in both partite classes $K$ and $L$. This proves the lower bound.
For the upper bound we consider all graphs in which the vertex $n$ is either isolated or it has one fixed neighbor, say $n-1$. The symmetric difference of any two such graphs is not $2$-connected, since $n$ has at most one neighbor in it. The number of such graphs
is just twice the number of graphs in which $n$ is an isolated point,
that is, $2^{{{n-1}\choose 2}+1}$ proving the matching upper bound by Lemma~\ref{lem:ub}.
\ifhmode\unskip\nobreak\fi\quad\ifmmode\Box\else$\Box$\fi
\medskip
\par\noindent
\begin{remark}\label{rem:evenodd} {\rm The upper bound proven in Theorem~\ref{thm:2conn} clearly holds also for odd $n$ but we have not found a matching construction in general. For $n=3$ a triangle and the empty graph would do, still achieving the upper bound. But for larger odd $n$ the best we could do is to take only those graphs from our construction for which in the corresponding bipartition the smaller partition class has an odd number of elements if $n\equiv 1\ ({\rm mod}\ 4)$ and it has an even number of elements if $n\equiv 3\ ({\rm mod}\ 4)$. The number of graphs obtained this way is $2^{n-2}-{(n-2)\choose {(n-3)/2}}$. $\Diamond$}
\end{remark}
\medskip
\par\noindent
\begin{remark}\label{rem:linear} {\rm Changing the graphs to their complements in the proofs of Theorems~\ref{thm:conn} and \ref{thm:2conn} makes these graph families vector spaces over the $2$-element field, while they still satisfy the conditions as the symmetric differences do not change by complementation (or by taking the symmetric difference of all elements with any fixed graph which is the complete graph in case of complementation). $\Diamond$}
\end{remark}
\medskip
\par\noindent
It does not sound surprising that if we step further on to $k$-connectedness for $k>2$ then the problem becomes rather more complicated. Nevertheless, if we insist on linear codes, that is graph families closed under the symmetric difference operation then for $k=3$ we can still determine the largest possible cardinality for infinitely many values on $n$ using Hamming codes.
\begin{thm} \label{thm:3connlin}
Let ${\cal F}_{3c}$ be the class of $3$-connected graphs and let $M^{(\ell)}_{{\cal F}_{3c}}(n)$ denote the size of a largest possible linear graph family on vertex set $[n]$ any two members of which have a $3$-connected symmetric difference. If $n=2^k-1$ for some integer $k\ge 2$, then
$$M^{(\ell)}_{{\cal F}_{3c}}(n)=2^{n-k-1}.$$
\end{thm}
{\noindent \it Proof. }
First we prove that $D_{{{\cal F}}_{3c}}(n)\ge n2^{{n-1\choose 2}}$ holds in general. Consider the family of all graphs on vertex set $[n]$ in which the degree of vertex $n$ is at most $1$. There are exactly $n2^{{n-1\choose 2}}$ such graphs. The symmetric difference of any two of these graphs is at most $2$-connected, since the vertex $n$ has degree at most $2$ in all these symmetric differences. This proves the claimed inequality and by Lemma~\ref{lem:ub} this implies
$M_{{\cal F}_{3c}}(n)\le 2^{n-1}/n$.
\smallskip
\par\noindent
It is well-known that if a family of subsets of a finite set contains the empty set and is closed under the symmetric difference operation then the cardinality of this set must be a power of $2$. This follows immediately from linear algebra and the fact that such a family forms a vector space over $GF(2)$, cf. e.g. Lemma 3.1 in Kozlov's book \cite{Kozlov} where a simple combinatorial proof of this fact is also presented. Since a linear graph family code on $[n]$ can be viewed as a collection of subsets of $E(K_n)$, this implies that $M^{(\ell)}_{{\cal F}_{3c}}(n)$ is a power of $2$. Since we obviously have $M^{(\ell)}_{{\cal F}_{3c}}(n)\le M_{{\cal F}_{3c}}(n)$, the upper bound proved above implies $M^{(\ell)}_{{\cal F}_{3c}}(n)\le 2^d$ with $d=\lfloor\log_2 \frac{1}{n}2^{n-1}\rfloor$ giving $$M^{(\ell)}_{{\cal F}_{3c}}(n)\le 2^{n-k-1}$$ for $n=2^k-1, k\ge 2$, which proves the required upper bound.
\smallskip
\par\noindent
For the lower bound consider the Hamming code ${\cal C}_H(n)$ with length $n=2^k-1$ that exists for every $k\ge 2$. (For a nice quick account on Hamming codes see e.g. \cite{Berlekamp}.) It is a linear code with minimum distance $3$ that consists of $2^{n-k}$ binary codewords having the property that if ${\mbf c}=(c_1,\dots,c_n)$ belongs to the code then so does also $\bar{\bf c}=(\bar c_1,\dots,\bar c_n)$ where $\bar c_i=1-c_i$. For each codeword ${\mbf c}\in {\cal C}_H(n)$ consider the bipartition of $[n]$ into the subsets $K_{\msbf c},L_{\msbf c}$, where $K_{\msbf c}=\{i: c_i=0\}, L_{\msbf c}=\{i:c_i=1\}$ and the complete bipartite graph $G_{K_{\msbf c},L_{\msbf c}}$ with partite classes $K_{\msbf c},L_{\msbf c}$. Note that by the above mentioned property of Hamming codes we have ${\mbf c}\in {\cal C}_H(n)$ if and only if $\bar{{\mbf c}}\in {\cal C}_H(n)$ and thus since
$G_{K_{{\msbf c}},L_{{\msbf c}}}=G_{K_{\bar{\msbf c}},L_{{\bar{\msbf c}}}}$, we get $\frac{1}{2}|{\cal C}_H(n)|=2^{n-k-1}$ different complete bipartite graphs this way. All we have to prove is that the symmetric difference of any two of our graphs is $3$-connected. This is equivalent to show that if ${\bf c}'\neq {\bf c},\bar{\bf c}$, then the cardinality of both partite classes of $G_{K_{\msbf c},L_{\msbf c}}\oplus G_{K_{{\msbf c}'},L_{{\msbf c}'}}$, that is of $(K_{\msbf c}\cap K_{{\msbf c}'})\cup (L_{\msbf c}\cap L_{{\msbf c}'})$ and $(K_{\msbf c}\cap L_{{\msbf c}'})\cup (K_{{\msbf c}'}\cap L_{\msbf c})$ is at least $3$. However, this immediately follows from the fact that the codeword ${\bf c}'$ must differ from both ${\bf c}$ and $\bar{{\bf c}}$ in at least $3$ coordinates. This completes the proof. \ifhmode\unskip\nobreak\fi\quad\ifmmode\Box\else$\Box$\fi
\subsection{Hamiltonicity}
A graph is connected if and only if it contains a spanning tree. Next we consider what happens if we require the containment of specific spanning trees: a path in this subsection and a star in the next one.
\medskip
\par\noindent
\begin{thm}\label{thm:Hp}
Let ${\cal F}_{H{\rm p}}$ denote the class of graphs containing a Hamiltonian path. Then for infinitely many values of $n$ we have $$M_{{\cal F}_{H{\rm p}}}(n)=2^{n-1}.$$
In particular, this holds whenever $n=p$ or $n=2p-1$ for some odd prime $p$.
\end{thm}
\medskip
\par\noindent
To prove the above theorem we will refer to the following old conjecture that is known to be true in several special cases. To state it we need the notion of perfect $1$-factorization. It means the partition of the edge set of a graph into perfect matchings such that the union of any two of them is a Hamiltonian cycle.
\medskip
\par\noindent
{\bf Perfect $1$-factorization conjecture (P1FC)} (Kotzig~\cite{Kotzig}). {\em The complete graph $K_n$ has a perfect $1$-factorization for all even $n>2$.}
\medskip
\par\noindent
This conjecture is still open in general, however it is known to hold in several special cases, for example, whenever $n=p+1$ (Kotzig~\cite{Kotzig}) or $n=2p$ for some odd prime $p$ (Anderson~\cite{Anderson} and Nakamura~\cite{Nakamura}, cf. also Kobayashi~\cite{Kobayashi}). For a recent survey, see Rosa~\cite{Rosa}, according to which the smallest open case of the conjecture is $n=64$.
\medskip
\par\noindent
{\it Proof of Theorem~\ref{thm:Hp}.}
Since Hamiltonian paths are connected, it follows from the proof of Theorem~\ref{thm:conn} that $2^{n-1}$ is again an upper bound. Now we show that it is also a lower bound whenever the Perfect $1$-factorization conjecture holds for $n+1$. (Note that if the conjecture is true, then this means all odd numbers at least $3$, while for $1$ our statement is void.)
\medskip
\par\noindent
Let $n$ be an odd number for which $K_{n+1}$ has a perfect $1$-factorization ${\cal M}$ and $v$ a fixed vertex of $K_{n+1}$. Note that deleting the edge incident to $v$ from all matchings belonging to ${\cal M}$ we obtain $n$ matchings of $K_n$ such that the union of any two of them is a Hamiltonian path in $K_n:=K_{n+1}\setminus \{v\}$. Now consider all those subgraphs of $K_n$ that can be obtained as the union of an even number of these $n$ matchings. Clearly, the symmetric difference of any two of them is also the union of at least two of these matchings and thus contains a Hamiltonian path. The number of graphs obtained this way is $2^{n-1}$, matching the upper bound.
\ifhmode\unskip\nobreak\fi\quad\ifmmode\Box\else$\Box$\fi
\medskip
\par\noindent
The case of Hamiltonian cycles can be treated essentially the same way.
\medskip
\par\noindent
\begin{thm}\label{thm:Hc}
Let ${\cal F}_{H{\rm c}}$ denote the class of graphs containing a Hamiltonian cycle. For all even values of $n$ for which the P1FC holds, we have $$M_{{\cal F}_{H{\rm c}}}(n)=2^{n-2}.$$ In particular, this is the case if $n=p+1$ or $n=2p$ for some odd prime $p$.
\end{thm}
\medskip
\par\noindent
{\noindent \it Proof. }
Since Hamiltonian cycles are $2$-connected, it follows from the proof of Theorem~\ref{thm:2conn} that $2^{n-2}$ is again an upper bound.
\medskip
\par\noindent
Let $n$ be an even number for which the P1FC holds and let ${\cal M}$ be a perfect $1$-factorization of $K_n$. Note that ${\cal M}$ contains $n-1$ matchings
(indeed the edge-chromatic number of $K_n$ for even $n$ is $n-1$). Now consider the $2^{n-2}$ graphs we can obtain as the union of an even number of matchings from ${\cal M}$. Clearly, the symmetric difference of any two of them contains a Hamiltonian cycle.
\ifhmode\unskip\nobreak\fi\quad\ifmmode\Box\else$\Box$\fi
\medskip
\par\noindent
\begin{remark}\label{rem:borsuk} {\rm Since Hamiltonian cycles are $2$-connected graphs the proof of Theorem~\ref{thm:Hc} obviously gives an alternative proof of Theorem~\ref{thm:2conn} for those values of $n$ for which the Perfect $1$-factorization conjecture is known to hold. (The situation is similar for Theorems~\ref{thm:Hp} versus \ref{thm:conn}.) On the other hand, the construction in the proof of Theorem~\ref{thm:2conn} utterly fails to give a good lower bound for the value of $M_{{\cal F}_{Hc}}(n)$ investigated in Theorem~\ref{thm:Hc}. Indeed, the symmetric difference of two graphs in the construction given in the proof of Theorem~\ref{thm:2conn} contains a Hamiltonian cycle if and only if the sets denoted by $A\cup C$ and $B\cup D$ in that proof both have cardinality $\frac{n}{2}$ and this happens exactly when the partition classes of the partitions $(K,L)$ and $(K',L')$ are orthogonal in the sense
that representing these bipartitions by characteristic vectors consisting of $+1$ and $-1$ coordinates in the obvious way, we get a collection of vectors that are pairwise orthogonal. So their number cannot be more than just $n$ and we can give exactly $n$ such vectors if and only if an $n\times n$ Hadamard matrix exists. $\Diamond$}
\end{remark}
\subsection{Containing a spanning star}
We have seen in the previous subsection that if we want every symmetric difference to contain a spanning tree which is a path, then for infinitely many values of $n$ our family can be just as large as if we did not want more than just the connectedness of these symmetric differences. In this subsection we show that if the required spanning tree is a star, then the largest possible family is drastically smaller.
\begin{thm}\label{thm:fullstar}
Let ${\cal F}_{S}$ denote the class of graphs containing a spanning star, that is a vertex connected to all other vertices in the graph. Then we have
$$M_{{\cal F}_S}(n)
=\left\{\begin{array}{lll}n+1&&\hbox{if }
n\ {\rm is\ odd}\ \\n&&\hbox{if }n\ {\rm is\ even}.\end{array}\right.$$
\end{thm}
{\noindent \it Proof. }
First we prove the upper bound. Let $G_1,\dots,G_m$ be an ${\cal F}_S$-good family on the vertex set $[n]$.
Consider the complete graph $K_m$ whose vertices are labeled with the graphs $G_1,\dots,G_m$.
For each edge $\{G_i,G_j\}$ of this graph assign an element $h\in [n]$ for which $h$ is adjacent to all other elements of $[n]$ in the graph $G_i\oplus G_j$. By the definition of ${\cal F}_S$-goodness such a $h$ exists for every pair of our graphs. Now observe that if an element $a\in [n]$ is assigned to two distinct edges $e$ and $f$ of our graph $K_m$, then $e$ and $f$ must be independent edges. Indeed, if that was not the case then we would have $e=\{G_i,G_j\}, f=\{G_i,G_k\}$ for some $i,j,k\in [n]$ and $a$ would be a full-degree vertex (that is one, connected to all other vertices) in both of the graphs $G_i\oplus G_j$ and $G_i\oplus G_k$. But since $G_j\oplus G_k=(G_i\oplus G_j)\oplus (G_i\oplus G_k)$, that would mean that $a$ is an isolated vertex in $G_j\oplus G_k$, so no vertex of this latter graph can have full degree contradicting the ${\cal F}_S$-goodness of our family. Thus our assignment of vertices from $[n]$ to the edges of our $K_m$ partitions the edge set of $K_m$ into sets of independent edges (every partition class consisting of the edges with the same assigned label), in other words, it defines a proper edge-coloring of $K_m$. This means that the number of possible labels, which is $n$, should be at least as large as the edge-chromatic number $\chi_e(K_m)$ of $K_m$. Since the latter is $m-1$ for even $m$ and $m$ for odd $m$, turning it around we obtain that for odd $n$ we must have $m\le n+1$ and for even $n$ we must have $m\le n$.
\medskip
\par\noindent
Now we show that the upper bound we proved is sharp. First assume that $n$ is odd and consider a complete graph $K_{n+1}$ on the vertices $v_1,\dots,v_{n+1}$ along with an optimal edge-coloring $c: E(K_{n+1})\to [n]$ of this graph. This edge-coloring partitions $E(K_{n+1})$ into $n$ disjoint matchings $M_1,\dots,M_n$, where $M_j$ consists of the edges colored $j$ for every $j\in [n]$. Now we construct the graphs $G_1,\dots,G_{n+1}$ by telling for each potential edge $ij$ of the complete graph on $[n]$ which $G_k$'s will contain it and which ones will not. Consider the edge $ij$ and the union of the matchings $M_i$ and $M_j$ (note that these matchings are in the ''other'' complete graph on $n+1$ vertices). This union is a bipartite graph on the vertex set $\{v_1,\dots,v_{n+1}\}$ with two equal size partite classes $A$ and $B$. Let $ij$ be an edge of the graph $G_k$ if and only if $v_k\in A$. (So $ij$ will be an edge of exactly half of our graphs $G_1,\dots,G_{n+1}$.) Do this similarly for all edges of $K_n$, the complete graph on vertex set $[n]$.
This way we defined our $n+1$ graphs. We have to show that they form an ${\cal F}_S$-good family.
To this end consider two of our graphs, say $G_h$ and $G_k$. The edge $\{v_h,v_k\}$ has got some color in our coloring $c$, call this color $j$. This means that $\{v_h,v_k\}$ belongs to the matching $M_j$. We claim this means that $j\in [n]$ is a full-degree vertex of $G_h\oplus G_k$. The latter is equivalent to the statement that every edge $ji$ incident to the point $j$ appears in exactly one of the graphs $G_h$ and $G_k$. But this follows from the way we constructed our graphs: when we decided about the edge $ji$ we considered the matchings $M_i$ and $M_j$ and the bipartite graph their union defines. Since $\{v_h,v_k\}\in M_j$, the points $v_h$ and $v_k$ are always in different partite classes of this bipartite graph, so whichever was called $A$, exactly one of $v_h$ and $v_k$ belonged to it. Thus the edge $ij$ was declared to be an edge of exactly one of $G_h$ and $G_k$. Since this is so for every $i\neq j$, $j$ is indeed a full-degree vertex in $G_h\oplus G_k$.
\medskip
\par\noindent
Assume now that $n$ is even. Then $n-1$ is odd and we can construct graphs $G_1,\dots,G_n$ on vertex set $[n-1]=\{1,\dots,n-1\}$ as given in the previous paragraph. These are not yet good, however, since we have an $n$th vertex that does not appear yet in any of the graphs. Note that we have $n-1$ matchings $M_1,\dots,M_{n-1}$ involved in the construction so far whose indices are just the first $n-1$ vertices of our graphs. Think about the additional vertex $n$ as the index of an additional ``matching'' $M_n$ that has no edges at all. We decide about the involvement of the edges $ni$ ($i<n$) in our graphs analogously as we did for the earlier edges: Consider the bipartite graph $M_i\cup M_n$, that consists of just the edges of $M_i$, so it is a perfect matching on the vertex set $\{v_1,\dots,v_n\}$. Let the two partite classes defined by this perfect matching be $A$ and $B$ and add the edge $ni$ to the graph $G_h$ if and only if $v_h$ belongs to $A$. Now we can prove analogously to the odd case that the symmetric difference of any two of our graphs contains a vertex of degree $n-1$. Consider $G_h$ and $G_k$. The edge between $v_h$ and $v_k$ in the auxiliary complete graph belongs to exactly one of the matchings $M_j$ and every edge $ij$ is in exactly one of the graphs $G_h$ and $G_k$ if $i\in\{1,\dots,j-1,j+1,\dots,n\}$. This completes the proof.
\ifhmode\unskip\nobreak\fi\quad\ifmmode\Box\else$\Box$\fi
\medskip
\par\noindent
The only graph family code proven to be optimal and nonlinear (or not the coset of a linear code) in this paper is
the one appearing in the above Theorem~\ref{thm:fullstar}.
This is also the first case so far when the upper bound is proven without the use of Lemma~\ref{lem:ub}.
This suggests the question of what could be said about the dual problem in this case. The next theorem solves this dual problem for even values of $n$ also showing that Lemma~\ref{lem:ub} would not give a sharp upper bound for $M_{{\cal F}_S}(n)$.
\medskip
\par\noindent
\begin{thm} \label{thm:nostar}
If $n$ is even, then
$$D_{{{\cal F}}_S}(n)= 2^{{n\choose 2}-\frac{n}{2}}.$$
When $n$ is odd, then we have
$$2^{{n\choose 2}-\frac{n+1}{2}}\le D_{{{\cal F}}_S}(n)\le 2^{{n\choose 2}-\frac{n}{2}}.$$
\end{thm}
\medskip
\par\noindent
For the proof we will need the following celebrated
result from \cite{CFGS} (see also Corollary~15.7.7 in \cite{AS}).
\medskip
\par\noindent
{\bf Shearer's Lemma.} (\cite{CFGS})
{\it Let $S$ be a finite set and $A_1,\dots, A_m$ be subsets of $S$ such that every element of $S$ is contained in at least $k$ of the sets $A_1,\dots,A_m.$ Let ${\cal M}$ be a collection of subsets of $S$ and let ${\cal M}_i=\{T\cap A_i: T\in {\cal M}\}$ for $1\le i\le m$. Then
$$|{\cal M}|^k\le \prod_{i=1}^m |{\cal M}_i|.$$}
\medskip
\par\noindent
{\em Proof of Theorem~\ref{thm:nostar}.}
We will prove $$2^{{n\choose 2}-\lceil\frac{n}{2}\rceil}\le D_{{{\cal F}}_S}(n)\le 2^{{n\choose 2}-\frac{n}{2}}$$ that implies both the even and the odd case.
For the lower bound fix a subgraph $T$ of $K_n$ with the minimum number $\lceil\frac{n}{2}\rceil$ of edges such that no vertex is isolated and take all possible subgraphs of $K_n$ that contain all edges of $T$. The number of such subgraphs is $2^{{n\choose 2}-\lceil\frac{n}{2}\rceil}$ and no two of them has a symmetric difference that contains all edges incident to any fixed vertex. This proves the lower bound.
\smallskip
\par\noindent
For the upper bound consider a graph family ${\cal M}$ that satisfies the condition that no two of its elements have a symmetric difference with a vertex of degree $n-1$. For $i=1,\dots,n$ let $S_i$ be the set of $n-1$ edges (of $K_n$) incident to vertex $i$. Then for any $T, T'\in {\cal M}$ we cannot have $E(T')\cap S_i=S_i\setminus (E(T)\cap S_i),$
that is, $E(T)$ and $E(T')$ cannot be complementary on any $S_i$. So if ${\cal M}_i$ denotes the family of graphs obtained by taking the projection of all graphs from ${\cal M}$ to the edge set $S_i$, then $|{\cal M}_i|\le 2^{n-2}.$ Since each edge of $K_n$ appears in exactly two of the sets $S_i$, we can apply Shearer's Lemma to these sets with $k=2$. This gives $$|{\cal M}|^2\le \prod_{i=1}^n |{\cal M}_i|\le 2^{n(n-2)}.$$ Taking square roots we get the upper bound.
\ifhmode\unskip\nobreak\fi\quad\ifmmode\Box\else$\Box$\fi
\medskip
\par\noindent
Note that if we restrict attention to linear graph families for the dual problem treated in Theorem~\ref{thm:nostar}, then using again that the cardinality of such a family should be a power of $2$ (cf. the similar argument in the proof of Theorem~\ref{thm:3connlin}) we get that our lower bound is also sharp for odd values of $n$.
\section{Local conditions}\label{local}
In the previous section we investigated $M_{{\cal F}}(n)$ in cases when the required symmetric differences contain specific spanning subgraphs, therefore to check whether these conditions are satisfied we have to consider our graphs on the whole vertex set. Now we turn to families ${\cal F}$
defined by containing some fixed small finite graphs, so the
nature of these conditions will be local.
\subsection{General local conditions}
\begin{defi}\label{defi:local}
A graph class ${\cal L}$ defines a {\em local condition} if it has the property that whenever $H_1$ is an induced subgraph of $H_2$ and $H_1$ belongs to ${\cal L}$ then so does also $H_2$. In short, we will refer to such an ${\cal L}$ as a {\em local graph class}.
\end{defi}
\medskip
\par\noindent
Note that the above definition implies that whenever two graphs $F$ and $G$ are in the ${\cal L}$-good relation (that is, $F\oplus G\in{\cal L}$) then any $F'$ with $F'[U]\cong F$ and $G'$ with $G'[U]\cong G$ for some $U\subseteq V(F')=V(G')$ (that is, $F'$ and $G'$ induce subgraphs isomorphic to $F$ and $G$, respectively, on the same subset $U$ of their vertex set) are also in the ${\cal L}$-good relation. This means that if two graphs are in this relation then there is always some local certificate for this.
\medskip
\par\noindent
Here are some examples of local graph classes.
\smallskip
\par\noindent
1. ${\cal L}=\{H: L\subseteq H\}$ for some fixed finite simple graph $L$. That is ${\cal L}$ contains all graphs that contain a (not necessarily induced) subgraph isomorphic to $L$. When ${\cal L}$ is such a family we will use the simplified notation $M_L(n)$ for $M_{{\cal L}}(n)$.
\smallskip
\par\noindent
2. ${\cal L}=\{H: L\subseteq_{ind} H\}$ for some fixed finite simple graph $L$. That is ${\cal L}$ contains all graphs that
have
an induced subgraph isomorphic to $L$. When ${\cal L}$ is such a family we will use the simplified notation $M_{L,{\rm ind}}(n)$ for $M_{{\cal L}}(n)$.
\smallskip
\par\noindent
(Note that although the above two examples give different notions, the word ``induced'' is indeed needed in Definition~\ref{defi:local}.)
\smallskip
\par\noindent
3. ${\cal L}={\cal C}_{\rm odd}:=\{H: C_{2k+1}\subseteq H\ {\rm for\ some\ integer}\ 1\le k\}$, that is, ${\cal C}_{\rm odd}$ contains all graphs that contain
an odd cycle.
\smallskip
\par\noindent
4. For some fixed integers $h$ and $\ell$ we can define ${\cal L}_{h,\ell}=\{H: \exists U\subseteq V(H), |U|=h, |E(H[U])|=\ell\}$,
that is, ${\cal L}_{h,\ell}$ is the class of all graphs that have an induced subgraph on $h$ vertices with exactly $\ell$ edges.
\medskip
\par\noindent
In the following we prove some general results related to $M_{{\cal L}}(n)$ for local graph classes ${\cal L}$ and will further investigate the special
case belonging to our first example above in the next subsection.
In Subsection~\ref{subsect:triodd} we will focus on $M_{K_3}(n)$
and $M_{{\cal C}_{\rm odd}}(n)$. In the final subsection we discuss the
behaviour of the functions
$M_{L,{\rm ind}}(n)$ mentioned in the second example above.
\medskip
\par\noindent
The next proposition gives a straightforward upper bound on the value of $M_{{\cal L}}(n)$. It is in terms of $ex(n,{\cal L})$ that, as usually in extremal graph theory, denotes the maximum number of edges a graph on $n$ vertices can have without containing any $L\in {\cal L}$ as a subgraph.
\medskip
\par\noindent
\begin{prop}\label{thm:Wilex}
For any local graph class ${\cal L}$
$$M_{{\cal L}}(n)\le 2^{{n\choose 2}-ex(n,{\cal L})}.$$
\end{prop}
{\noindent \it Proof. }
Consider an $n$-vertex graph $H$ satisfying $|E(H)|=ex(n,{\cal L})$ and containing no subgraph isomorphic to any $L\in{\cal L}$. The family of all subgraphs of $H$ clearly satisfies the requirements of the dual problem of $M_{{\cal L}}(n)$ (no two graphs in that family can have a symmetric difference containing some $L\in {\cal L}$) and the family has size $2^{ex(n,{\cal L})}$. Thus the claimed upper bound follows from Lemma~\ref{lem:ub}. \ifhmode\unskip\nobreak\fi\quad\ifmmode\Box\else$\Box$\fi
\medskip
\par\noindent
Proposition~\ref{thm:Wilex} and our following results will justify the relevance of the following notion in our current setting.
\begin{defi}~\label{defi:rate}
The rate $R_{{\cal L}}(n)$ of an optimal graph family code on $n$ vertices satisfying the requirement prescribed by the local graph class ${\cal L}$ is defined as
$$R_{{\cal L}}(n):=\frac{2}{n(n-1)}\log_2M_{{\cal L}}(n).$$
\end{defi}
\medskip
\par\noindent
We will soon see that the value $\limsup_{n\to\infty}R_{{\cal L}}(n)$ is strictly positive for any ${\cal L}$ belonging to this section.
We will use the following theorem due to Wilson to show that the limit actually exists for all local graph classes.
\medskip
\par\noindent
{\bf Wilson's theorem.} (\cite{Wilson})
{\it For every finite simple graph $T$ there exists a threshold $n_0(T)$ such that if $n>n_0(T)$ and the following two conditions hold then the edge set of the complete graph $K_n$ can be partitioned into subgraphs each of which is isomorphic to $T$. The two conditions are:
\par\noindent
1. $n\choose 2$ is divisible by $|E(T)|$;
\par\noindent
2. $n-1$ is divisible by the greatest common divisor of the degrees of vertices in $T$.}
\medskip
\par\noindent
Note that the two conditions in the above theorem are obviously necessary. The decomposition of $K_n$ in the conclusion of the theorem is called a $T$-design when it exists, cf. \cite{ABB}.
\begin{thm}\label{thm:limes}
Let ${\cal L}$ be an arbitrary fixed local graph class. Then the value
$\lim_{n\to\infty}R_{{\cal L}}(n)$ exists and is bounded from below by $R_{{\cal L}}(n)$ for every $n$.
\end{thm}
{\noindent \it Proof. }
Let $n$ be an arbitrary natural number and let
${\cal G}=\{G_1,\dots,G_m\}$ be an optimal graph family code for ${\cal L}$ with $V(G_i)=[n], i\in\{1,\dots,m\}$, that is one with $m=M_{{\cal L}}(n)$. By Wilson's theorem a $K_n$-design exists for $K_N$, whenever $N$ is large enough and both $n-1$ divides $N-1$ and $n\choose 2$ divides $N\choose 2$. Take such an $N$ and consider the $K_n$-design on $K_N$ consisting of the subgraphs $K^{(1)},\dots,K^{(r)}$, where $r=\frac{N(N-1)}{n(n-1)}$ and each $K^{(i)}$ is isomorphic to $K_n$. Now let ${\cal G}_j:=\{G_1^{(j)},\dots,G_m^{(j)}\}$ be an optimal graph family code for ${\cal L}$ on $V(K^{(j)})$ for every $j\in\{1,\dots,r\}$. (Obviously, we can choose each ${\cal G}_j$ to be isomorphic to ${\cal G}$.) Now define a graph family code on $K_N$ for ${\cal L}$ as the collection of graphs that can be written in the form of $G_{\msbf a}:=\cup_{j=1}^r G_{a_j}^{(j)}$ where ${\mbf a}=(a_1,\dots,a_r)$ runs through all possible sequences satisfying $a_i\in\{1,\dots,m\}$ for every $i$. Since there are $m^r$ such sequences ${\mbf a}$, this way we have $m^r$ different graphs in our family.
They form indeed a graph family code for ${\cal L}$ since for any two of them,
$G_{\msbf a}$ and $G_{\msbf b}$ there is some $j$ for which
$a_j\neq b_j$ and thus $G_{\msbf a}\oplus G_{\msbf b}
\supseteq_{ind} G_{a_j}\oplus G_{b_j}\supseteq_{ind} L$ for some $L\in{\cal L}$.
This implies $M_{{\cal L}}(N)\ge m^r$ and thus $$R_{{\cal L}}(N)\ge \frac{2}{N(N-1)}\log_2m^r=\frac{2}{n(n-1)}\log_2 M_{{\cal L}}(n)=R_{{\cal L}}(n).$$
\smallskip
\par\noindent
The requirements for $N$ are satisfied if $N=kn(n-1)+1$ and $k$ is large enough. (Also for $N=kn(n-1)+n$ and large enough $k$ but considering the former is enough for our argument.) Since $M_{{\cal L}}(n)$ is clearly monotone nondecreasing in $n$ (as we can always ignore some vertices and consider a graph family code only on the rest), we can write that for any $kn(n-1)+1\le i\le (k+1)n(n-1)$ we have $M_{{\cal L}}(i)\ge m^r$ for $r=\frac{{{kn(n-1)+1}\choose 2}}{{n\choose 2}}$. Introducing the sequence $b_i:= m^r$ for $r=\frac{{{kn(n-1)+1}\choose 2}}{{n\choose 2}}$ whenever $kn(n-1)+1\le i\le (k+1)n(n-1)$ we can write $$\liminf_{i\to\infty}\frac{2}{i(i-1)}\log_2 M_{{\cal L}}(i)\ge \liminf_{i\to\infty}\frac{2}{i(i-1)}\log_2 b_i\ge$$ $$\liminf_{k\to\infty}\frac{1}{{{{(k+1)n(n-1)}}\choose 2}}\log_2 m^{\frac{{{kn(n-1)+1}\choose 2}}{{n\choose 2}}}=\liminf_{k\to\infty}\frac{{{kn(n-1)+1}\choose 2}}{{{{(k+1)n(n-1)}}\choose 2}}\frac{2}{n(n-1)}\log_2m=R_{{\cal L}}(n).$$
This proves that $\lim_{n\to\infty} R_{{\cal L}}(n)$ exists and is equal to $\sup_n R_{{\cal L}}(n)$.
\ifhmode\unskip\nobreak\fi\quad\ifmmode\Box\else$\Box$\fi
\medskip
\par\noindent
\begin{remark}\label{rem:fekete}
{\rm The above proof is similar to proving that the limit defining the Shannon capacity of graphs exists which is usually done using Fekete's Lemma. Here, however, there are some technical subtleties (because of the divisibility requirements for $N$) that made it simpler to present a full proof than to refer simply to Fekete's Lemma. $\Diamond$}
\end{remark}
\medskip
\par\noindent
In view of Theorem~\ref{thm:limes} the following definition is meaningful.
\smallskip
\par\noindent
\begin{defi}~\label{defi:dc}
The distance capacity (or {\em distancity} for short) of a local graph class ${\cal L}$ is defined as $$DC({\cal L}):=\lim_{n\to\infty} R_{{\cal L}}(n).$$
\end{defi}
\medskip
\par\noindent
Based on Tur\'an's celebrated theorem \cite{Turan} (cf. also e.g. in \cite{Diestel}) and the famous theorem of Erd\H{o}s and Stone~\cite{ErdStone}, Erd\H{o}s and Simonovits \cite{ErdSim} proved that if ${\cal L}$ is an arbitrary family of graphs, then
\begin{equation}\label{eq:EStS}
\lim_{n\to\infty}\frac{ex(n,{\cal L})}{{n\choose 2}}=1-\frac{1}{\chi_{\rm min}({\cal L})-1},
\end{equation}
where $\chi_{\rm min}({\cal L})=\min_{L\in{\cal L}}\chi(L)$ and $\chi(G)$ denotes the chromatic number of graph $G$.
(We assume that $\chi_{\rm min}({\cal L})\ge 2$. For the case when ${\cal L}$ contains some edgeless graph see Remark~\ref{rem:empty}.)
\medskip
\par\noindent
Note that Proposition~\ref{thm:Wilex} and the above result determining the order of magnitude of $ex(n,{\cal L})$ has the following immediate consequence for the distancity.
\medskip
\par\noindent
\begin{cor}\label{cor:DCub}
For any local graph class ${\cal L}$ with $\chi_{\rm min}({\cal L})\ge 2$ we have
$$DC({\cal L})\le \frac{1}{\chi_{\rm min}({\cal L})-1}.$$
\ifhmode\unskip\nobreak\fi\quad\ifmmode\Box\else$\Box$\fi
\end{cor}
\subsection{Containing a prescribed subgraph}\label{subsect:noninduced}
\medskip
\par\noindent
Now we focus on local graph classes mentioned in our first example after Definition~\ref{defi:local}: we have some fixed finite simple graph $L$ and
consider ${\cal L}=\{H: L\subseteq H\}$. As said above in this case we will use the notation $M_L(n)$ for $M_{{\cal L}}(n)$ and similarly, we will also denote $R_{{\cal L}}(n)$ and $DC({\cal L})$ by $R_L(n)$ and $DC(L)$, respectively. We prove that in this case the upper bound of Corollary~\ref{cor:DCub} is always sharp.
\medskip
\par\noindent
\begin{thm}\label{thm:DCL}
For any fixed graph $L$ we have $$DC(L)=\frac{1}{\chi(L)-1}.$$
\end{thm}
\medskip
\par\noindent
For the proof we will use a result by Erd\H{o}s, Frankl and R\"odl \cite{EFR} about the number $F_n(L)$ of graphs on $n$ labeled vertices containing no subgraph isomorphic to $L$.
\medskip
\par\noindent
{\bf Erd\H{o}s-Frankl-R\"odl theorem.} (\cite{EFR})
{\it Suppose $\chi(L)=r\ge 3$. Then
$$F_n(L)=2^{ex(n,K_r)(1+o(1))}.$$}
\par\noindent
Note that this gives
$$F_n(L)=2^{{n\choose 2}\left(1-\frac{1}{\chi(L)-1}+o(1)\right)}$$
by (\ref{eq:EStS}) (in fact, already directly by Tur\'an's theorem).
\medskip
\par\noindent
While the proof of the Erd\H{o}s-Frankl-R\"odl theorem is based on Szemer\'edi's Regularity Lemma, a similar result for bipartite $L$
easily follows from (\ref{eq:EStS}) (or from the K\H{o}v\'ari-S\'os-Tur\'an
Theorem \cite{KST}).
Indeed, it implies that if $L$ is bipartite then $F_n(L)<{{n\choose 2}\choose {\varepsilon {n\choose 2}}}$ for any $\varepsilon>0$
provided $n>n_0(\varepsilon}\def\ffi{\varphi)$,
and that implies the claimed statement. (To see the latter one can use the well-known fact, cf. e.g. Lemma 2.3 in \cite{CsK}, that $${t\choose {\alpha t}}=2^{t(h(\alpha)+o(1))},$$ where $h(x)=-x\log_2x-(1-x)\log_2(1-x)$ is the binary entropy function and $0\le \alpha\le 1$ is meant to be such that $\alpha t$ is an integer. Applying this for $t:={n\choose 2}$ and $\alpha=\varepsilon$ we obtain that for any $0<\varepsilon<1$ the number ${{n\choose 2}\choose {\varepsilon {n\choose 2}}}$ is more than $2^{\delta{n\choose 2}}$ for some positive $\delta$.)
\medskip
\par\noindent
{\it Proof of Theorem~\ref{thm:DCL}.}
It follows immediately from Corollary~\ref{cor:DCub} that the right hand side is an upper bound on the left hand side so we only have to prove the reverse inequality.
\smallskip
\par\noindent
To this end let $G_L$ denote the graph whose vertices are all possible graphs on $n$ labeled vertices and two are connected if and only if their symmetric difference does not contain $L$ as a subgraph. (Note that this is just the complementary graph of $H_{{\cal F}}$ used in the proof of Lemma~\ref{lem:ub} when ${\cal F}$ is set to be the local graph class ${\cal L}$ belonging to our problem.) Then $M_L(n)$ is equal to the independence number $\alpha(G_L)$ of $G_L$.
Clearly, $G_L$ is vertex-transitive (cf. the argument in the proof of Lemma~\ref{lem:ub} for $H_{{\cal F}}$), in particular, it is regular. Since the degree of its vertex representing the edgeless graph is just $F_n(L)$, we get (denoting the maximum degree of a graph $G$ by $\Delta(G)$) that
$$M_L(n)=\alpha(G_L)\ge\frac{|V(G_L)|}{\Delta(G_L)+1}=\frac{|V(G_L)|}{F_n(L)+1}=\frac{2^{{n\choose 2}}}{2^{{n\choose 2}\left(1-\frac{1}{\chi(L)-1}+o(1)\right)}}=2^{{n\choose 2}\left(\frac{1}{\chi(L)-1}+o(1)\right)}$$
by the Erd\H{o}s-Frankl-R\"odl theorem (and by the above discussion also for bipartite graphs). Putting this inequality into the definition of $DC(L)$ the required result follows.
\ifhmode\unskip\nobreak\fi\quad\ifmmode\Box\else$\Box$\fi
\medskip
\par\noindent
\begin{cor}\label{DCgen}
Let ${\cal G}$ be a set of graphs, each containing at least one edge,
and let ${\cal L}_{{\cal G}}$ be the local graph class containing all graphs that contain at least one $G\in{\cal G}$ as a subgraph. Then $$DC({\cal L}_{{\cal G}})=\frac{1}{\chi_{\rm min}({\cal L}_{{\cal G}})-1}=\frac{1}{\chi_{\rm min}({\cal G})-1}.$$ In particular, $$DC({\cal C}_{\rm odd})=DC(K_3)=\frac{1}{2}.$$
\end{cor}
{\noindent \it Proof. }
The second statement is clearly a special case of the first one, so it is enough to prove the latter. It is a straightforward consequence of Corollary~\ref{cor:DCub} that the left hand side is bounded from above by the right hand side. For the reverse inequality note the trivial fact that $DC({\cal L}_{{\cal G}})\ge DC(G)$ for any $G\in{\cal G}$. Applying this for some $G\in{\cal G}$ that
satisfies $\chi(G)=\min_{G\in{\cal G}}\chi(G)=\chi_{\rm min}({\cal L}_{{\cal G}})$ the statement follows from Theorem~\ref{thm:DCL}.
\ifhmode\unskip\nobreak\fi\quad\ifmmode\Box\else$\Box$\fi
\medskip
\par\noindent
\begin{remark}\label{rem:asydual}
{\rm It is straightforward from the foregoing that the above results also determine for any graph family ${\cal G}$ the asymptotic behaviour of the value $D_{{\cal L}_{{\cal G}}}(n)$ belonging to the dual problem. Indeed, by Lemma~\ref{lem:ub} and Corollary~\ref{DCgen} we have that
$\lim_{n\to\infty}\frac{1}{{n\choose 2}}\log D_{{\cal L}_{{\cal G}}}(n)\le 1-DC({\cal L}_{{\cal G}})=1-\frac{1}{\chi_{\min}({\cal G})-1}$ while a matching lower bound follows from the argument in the proof of Proposition~\ref{thm:Wilex}. Thus we have
$$\lim_{n\to\infty}\frac{2}{n(n-1)}\log D_{{\cal L}_{{\cal G}}}(n)=1-\frac{1}{\chi_{\min}({\cal G})-1}$$
for any graph family ${\cal G}$. This means that by taking all subgraphs of a graph with the largest possible number of edges without containing a subgraph from ${\cal G}$ we obtain asymptotically a largest family of graphs no two of which have any $G\in{\cal G}$ in their symmetric difference.
$\Diamond$}
\end{remark}
\subsection{Containing a triangle or an odd cycle} \label{subsect:triodd}
In this subsection we are investigating $M_L(n)$ for small values of $n$ and the simplest $3$-chromatic graph, which is the triangle $K_3$. We will also look at the analogous problem when $K_3$, the cycle of length 3 is replaced by the family of all odd cycles.
\medskip
\par\noindent
For $L=K_3$ the bound of Proposition~\ref{thm:Wilex} gives us $M_{K_3}(n)\le 2^{{{n\choose 2}-\lceil\frac{n}{2}\rceil\lfloor\frac{n}{2}\rfloor}}$. Below we show that this upper bound is tight whenever $n$ is at most $6$.
\medskip
\par\noindent
The first part of the following Proposition is very simple and we present it only for the sake of completeness.
\medskip
\par\noindent
\begin{prop}~\label{prop:triv34}
We have $M_{K_3}(3)=2$ and $M_{K_3}(4)=4$.
\end{prop}
{\noindent \it Proof. }
For $n=3$ the statement is trivial: take the empty graph and a triangle on three vertices, this $2$-element family already achieves the value of the upper bound which is $2$ for $n=3$.
\medskip
\par\noindent
For $n=4$ we give the following four graphs on the vertex set $\{1,2,3,4\}$ by their edge sets.
Let $$E(G_0)=\emptyset, E(G_1)=\{12,23,13,34\}, E(G_2)=\{23,34,24,14\}, E(G_3)=\{12,13,24,14\}.$$
It takes an easy checking that the symmetric difference of any two of these graphs contains a triangle. Since the upper bound in Proposition~\ref{thm:Wilex} is also $4$ in this case, this proves that $M_{K_3}(4)=4$.
\ifhmode\unskip\nobreak\fi\quad\ifmmode\Box\else$\Box$\fi
\medskip
\par\noindent
\begin{remark}\label{rem:lspace} {\rm Note that both of the above simple constructions are closed under the symmetric difference operation, that is they form a linear space over $GF(2)$ when the graphs are represented by the characteristic vectors of their edge sets. In fact, the second construction could also be presented as the vector space generated in this sense by any two of the graphs $G_1, G_2, G_3$. $\Diamond$}
\end{remark}
\medskip
\par\noindent
\begin{prop}\label{lem:MK3_5}
$$M_{K_3}(5)=16.$$
\end{prop}
{\noindent \it Proof. }
The value of the upper bound in Proposition~\ref{thm:Wilex}
gives $16$ for $n=5$, so we only have to prove that $16$ is also a lower bound.
To this end we will give a set of graphs forming a vector space in the sense of Remark~\ref{rem:lspace}. We will give this vector space by a set of generators, although in a somewhat redundant way. (Our reason to keep this redundancy is that the construction has more symmetry this way.)
\smallskip
\par\noindent
Think about the vertices $\{1,2,3,4,5\}$ as if they were given on a circle at the vertices of a regular pentagon in their natural order. Consider the graph with edge set $$E(G_1):=\{12,23,13,35\}.$$ Let $G_2, G_3, G_4, G_5$ be the four graphs we obtain from $G_1$ by rotating it along the circle containing the vertices so that vertex $1$ moves to $2$, $2$ to $3$, etc. Thus we have $$E(G_2)=\{23,34,24,41\}, E(G_3)=\{34,45,35,52),$$ $$E(G_4)=\{45,51,41,13\}, E(G_5)=\{51,12,52,24\}.$$
Now we consider the linear space the characteristic vectors of the edge sets of these five graphs $G_i, i\in\{1,2,3,4,5\}$ generate. These graphs can be defined as the elements of the family ${\cal G}=\{G_I: I\subseteq [5]\},$ where $$G_I=\oplus_{i\in I} G_i,$$ meaning that $V(G_I)=[5]$ and $E(G_I)$ contains exactly those edges that appear in an odd number of the graphs $G_i$ with $i\in I$.
\medskip
\par\noindent
Note that every edge of the underlying $K_5$ on $[5]$ appears in exactly two of the graphs $G_1,\dots,G_5$, therefore for $I=[5]$ we have that $G_I$ is the empty graph just as $G_{\emptyset}$ is. This implies that for every $I\subseteq [5]$ and $\overline{I}:=[5]\setminus I$ we have $G_I=G_{\overline{I}}$, thus every graph in our graph family has exactly two representations as $G_I$ for some $I\subseteq [5]$. (The two representations are given by $I$ and $\overline{I}$ as we have seen. It also follows that if $J\neq I,\overline{I}$ then $G_J\neq G_I$,
otherwise we would have $G_{J\oplus I}$ be the empty graph for
$J\oplus I\notin\{\emptyset, [5]\}$ contradicting that every edge appears exactly twice in the sets $E(G_i),\ i=1,\dots,5$.) Thus we have indeed $\frac{1}{2}2^5=16$ graphs in our family matching our upper bound for $n=4$.
\medskip
\par\noindent
We have to show that the symmetric difference of any two of our graphs contains a triangle. Since our construction is closed for the symmetric difference operation this is equivalent to say that all graphs in our family except the empty graph contains a triangle. Since $G_I=G_{\overline{I}}$ it is enough to prove that $G_I$ contains a triangle for all $1\le |I|\le 2, I\subseteq [5]$. This is easy to see when $|I|=1$. For subsets with $|I|=2$ it is enough to check this for $I=\{1,2\}$ and $I=\{1,3\}$ by the rotational symmetry of our construction. But these two cases are easy to check: $G_{\{1,2\}}$ contains the triangles on the triples of vertices $1,2,4$ and $1,3,4$, while $G_{\{1,3\}}$ contains the triangle on vertices $1,2,3$.
\ifhmode\unskip\nobreak\fi\quad\ifmmode\Box\else$\Box$\fi
\medskip
\par\noindent
\begin{prop}\label{lem:MK3_6}
$$M_{K_3}(6)=64.$$
\end{prop}
{\noindent \it Proof. }
The value of the upper bound given by Proposition~\ref{thm:Wilex} is $2^6$ for $L=K_3$ and $n=6$, so we need to prove only the lower bound.
\smallskip
\par\noindent
To this end we give a construction of $64$ graphs forming a graph family code on $[6]$ for $K_3$. The construction will have several similarities to that in Proposition~\ref{lem:MK3_5} though with somewhat less symmetry. But again our graphs will form a vector space in the sense of Remark~6 to be specified through a set of seven generators that altogether cover each one of the edges of the underlying $K_6$ exactly twice, so every member of our graph family will have exactly two representations by the generators just as in the proof of Proposition~\ref{lem:MK3_5}. Here are the details.
\smallskip
\par\noindent
Think about the $6$ vertices $1,\dots,6$ as being on a circle in the vertices of a regular hexagon in their natural order as we go around the circle. Our first four generator graphs are the following four edge-disjoint triangles (plus three isolated points) given by their edge sets as follows.
$$E(G_1)=\{12,23,13\}, E(G_2)=\{34,45,35\}, E(G_3)=\{56,16,15\}, E(G_4)=\{24,46,26\}.$$
The other three graphs are three $K_4$'s (plus two isolated vertices) that are rotations of each other, in particular,
$$E(G_5)=\{12,24,45,15,14,25\}, E(G_6)=\{23,35,56,26,25,36\},$$ $$E(G_7)=\{34,46,16,13,36,14\}.$$
It is easy to check that the above seven graphs cover each edge of the underlying $K_6$ exactly twice. Just as in the proof of Proposition~\ref{lem:MK3_5} this implies that the generated family of graphs of the form $$G_I=\oplus_{i\in I}G_i$$ where $I$ runs through all subsets of $[7]$ contains exactly two representations of this form for each of its members, namely $$G_I=G_J\ {\rm if\ and\ only\ if}\ J=[7]\setminus I.$$
Thus our family has $2^6=64$ members that matches our upper bound. Now we have to show that the symmetric difference of every pair of our graphs contains a triangle. Since the family is closed under symmetric difference this is equivalent to every $G_I$ except $G_{\emptyset}=G_{[7]}$ containing a triangle. To show this we consider the representation of each of our graphs as $G_I$ where $I$ contains at most one of the three $K_4$ generators, that is $|I\cap \{5,6,7\}|\le 1$. When $I\cap \{5,6,7\}=\emptyset$ but $I$ itself is nonempty then this is trivial as in such a case $G_I$ is the union of some of the edge-disjoint graphs $G_1,\dots,G_4$ each of which is a triangle itself. In case $|I\cap \{5,6,7\}|=1$, then by symmetry we may assume w.l.o.g. that $I\cap \{5,6,7\}=\{5\}$. Then if we also have $\{1,2\}\subseteq I$ then the triangles on vertices $1,3,4$ and $2,3,5$ (and two more) will be contained in $G_I$. So we may assume that at least one of $G_1$ and $G_2$ is not part of our representation of $G_I$ and by symmetry, we may assume $2\notin I$. But then to avoid the triangles on vertices $1,4,5$ and $2,4,5$ being in $G_I$ we need both $3\in I$ and $4\in I$. In this case, however, we will have the triangle on vertices $4,5,6$ present in $G_I$. This completes the proof. \ifhmode\unskip\nobreak\fi\quad\ifmmode\Box\else$\Box$\fi
\medskip
\par\noindent
Recall ${\cal C}_{\rm odd}$ be the class of all graphs containing an odd cycle. Since $ex(n,{\cal C}_{\rm odd})=ex(n,K_3)$ the upper bound of Proposition~\ref{thm:Wilex} is also $2^{{n\choose 2}-\lceil\frac{n}{2}\rceil\lfloor\frac{n}{2}\rfloor}$ for $M_{{\cal C}_{\rm odd}}(n)$. Since $K_3\cong C_3$ is an odd cycle, we obviously have $M_{K_3}(n)\le M_{{\cal C}_{\rm odd}}(n)$ and so by Propositions~\ref{prop:triv34}, \ref{lem:MK3_5} and \ref{lem:MK3_6} the previous upper bound is also sharp for $M_{{\cal C}_{\rm odd}}(n)$ when $n\in\{3,4,5,6\}$.
Although we could not prove that $M_{K_3}(7)$ is also equal to this upper bound, we can show this at least for $M_{{\cal C}_{\rm odd}}(7)$.
\medskip
\par\noindent
\begin{prop}\label{lem:MC7}
$$M_{{\cal C}_{\rm odd}}(7)=2^9.$$
\end{prop}
{\noindent \it Proof. }
The upper bound $2^{{n\choose 2}-\lceil\frac{n}{2}\rceil\lfloor\frac{n}{2}\rfloor}$ is equal to $2^9$, so it is enough to prove that this is also a lower bound. This we do similarly as in the proofs of Propositions~\ref{lem:MK3_5} and \ref{lem:MK3_6}.
\smallskip
\par\noindent
Again, we think about the seven vertices forming the set $[7]$ as the vertices of a regular $7$-gon around a cycle in their natural order. We define $7+3=10$ simple graphs $G_1,\dots,G_7$ and $G_8,\dots,G_{10}$ that will generate our family.
Let $G_1$ be the triangle with edges $12,24,14$ and $G_2,\dots,G_7$ be its six possible rotated versions, that is the triangles with edge sets $\{23,35,25\}, \{34,46,36\},\dots,\{17,13,37\}$, respectively. Note that these seven triangles cover all pairs of vertices exactly once, that is, they form a Steiner triple system. The three other graphs $G_8,G_9,G_{10}$ are three edge-disjoint seven-cycles, namely those with edge sets $$\{12,23,34,45,56,67,17\}, \{13,35,57,27,24,46,16\}, \{14,47,37,36,26,25,15\},$$ respectively. Note that these three graphs also cover all pairs of vertices exactly once and that the edge sets of a $G_i$ for $i\in [7]$ and $G_j$ with $j\in \{8,9,10\}$ intersect in exactly one element.
Since our ten graphs cover the edges of the underlying $K_7$ exactly twice, just as in the proofs of Propositions~\ref{lem:MK3_5} and \ref{lem:MK3_6} the generated family $$\oplus_{i\in I}G_i$$ as $I$ runs over all subsets of $\{1,\dots,10\}$ will have exactly $2^9$ distinct members each of which is represented by two subsets of $\{1,\dots,10\}$, some $I$ and its complement. All we are left to show for proving $M_{{\cal C}_{odd}}(7)\ge 2^9$
is that each such $G_I$ except $G_{\emptyset}=G_{[10]}$ contains an odd cycle. If $I\subseteq [7]$, this is obvious and so is also if $I\subseteq\{8,9,10\}$. When both $I\cap [7]$ and $I\cap\{8,9,10\}$ are nonempty, then we consider that representation $G_I$ which has $|I\cap [7]|\le 3$. If we have $|I\cap\{8,9,10\}|=1$ then whichever $7$-cycle we have (that is, whichever of $G_8,G_9,G_{10}$) it will have two consecutive edges that do not appear in either of the at most three triangles. If we take the first pair of such edges (as we go along our $7$-cycle in an appropriate direction) for which the previous one is an edge of one of our triangles (since we take at least one triangle and each triangle intersects each $7$-cycle, such an edge must exist), then the construction ensures that these two consecutive edges close up to a $K_3$ in our $G_I$. In case we have two $7$-cycles in our $G_I$ representation, then those create $7$ distinct $K_3$'s in their union. Each of our triangles
intersects exactly three of those seven $K_3$'s created, so if we have $|I\cap [7]|\le 2$ then at least one of these seven $K_3$'s remain untouched. Thus we are left with the case of two $7$-cycles and exactly three triangles. For this case let us switch to the complementary representation with four triangles and one $7$-cycle. By symmetry, we may assume that our $7$-cycle is $G_8$. If the four triangles are such that two consecutive edges of $G_8$ do not appear in any of them then we can finish the argument as before. If this is not the case, then the four triangles must leave three such edges of $G_8$ uncovered which form a matching. Because of symmetry we may assume that these are the edges $12, 34, 56$. This also tells us exactly which are the four triangles we have in the representation of $G_I$, namely those that contain the remaining four edges, that is, $G_2, G_4, G_6$ and $G_7$. In this case $G_I$ contains the $K_3$, for example, on the vertices $2,5,6$. Finally, if we have all the three $7$-cycles in our representation then the complementary representation has no $7$-cycle at all and this case we have already covered. This completes the proof. \ifhmode\unskip\nobreak\fi\quad\ifmmode\Box\else$\Box$\fi
\subsection{Containing a prescribed induced subgraph}\label{subsect:induced}
\medskip
\par\noindent
In this subsection we discuss local graph classes mentioned in the second example after Definition~\ref{defi:local}. Here we have a fixed finite simple graph $L$ and consider the family ${\cal L}$ of all graphs containing $L$ as an induced subgraph. Recall that in this case we let $M_{L,{\rm ind}}(n)$ denote $M_{{\cal L}}(n)$ and similarly, we denote $R_{{\cal L}}(n)$ and $DC({\cal L})$ by $R_{L,{\rm ind}}(n)$ and $DC(L,{\rm ind})$, respectively. In this section we
prove that requiring the subgraphs to be induced does not change
the answer from that of subsection ~\ref{subsect:noninduced}. The upper bound, of course, trivially carries over from the non-induced case, while the lower bound strengthens the one in Theorem \ref{thm:DCL}.
\medskip
\par\noindent
\begin{thm}
\label{thm:DCLindb}
For any fixed graph $L$ we have $$DC(L,{\rm ind})=\frac{1}{\chi(L)-1}.$$
\end{thm}
\begin{remark}\label{partition_number}
{\rm Note that despite the apparent similarity,
the proof of Theorem \ref{thm:DCL} does not carry over to show
Theorem \ref{thm:DCLindb}. Still, it is possible to
describe the asymptotic number of induced $L$-free graphs for a fixed graph $L$, by introducing the partition number $r(L)$. Define $r(L)$ as the largest integer $r$ so that there is some integer $s$, $0 \leq s \leq r$ such that the vertices of $L$ cannot be covered by $s$ cliques and $r-s$ independent sets. Results obtained independently by Alekseev \cite{Ale} and by Bollob\'as and Thomason \cite{BT1,BT2} imply that the
number of induced-$L$-free graphs on $n$ vertices
is $2^{(1-1/r(L))n^2/2 +o(n^2)}$. Note that $r(L) \geq \chi(L)-1$, as the vertices of $L$ cannot be covered by $\chi(L)-1$ independent sets. There are cases when equality holds and thus the required result follows (an example for that is $L=C_5$, the $5$-length cycle), but in general, $r(L)$ can be much larger than $\chi(L)-1$, as shown, for example, by any long (even or odd) cycle. Hence the proof of Theorem \ref{thm:DCL} does not give the required bound
for an arbitrary $L$.}
\end{remark}
The main tool in the proof of Theorem \ref{thm:DCLindb} is
Lemma \ref{lemma:kpartiteub}, which bounds the number of balanced $k$-partite induced-$L$-free graphs.
\begin{lemma}\label{lemma:kpartiteub}
For a fixed positive integer $k$ and a fixed graph $L$, consider the set of balanced $k$-partite graphs $G$ on $k$-classes $A_1, \dots, A_k$, each having size $n/k$. Among those graphs,
at most $2^{\big(1-\frac{1}{\chi(L)-1}\big)n^2/2+o(n^2)}$ do not
contain an induced subgraph isomorphic to $L$.
\end{lemma}
In what follows we will assume some familiarity with Szemer\'edi's regularity lemma and the terminology related to it. A good introduction to these notions (with a full proof of the lemma itself) can be found, for example, in Section 7.4 of Diestel's book \cite{Diestel}.
Before presenting the proof of Lemma \ref{lemma:kpartiteub}, whose
proof relies on the regularity lemma, we need to establish the following
auxiliary claim about finding induced subgraphs using regularity. Versions
of this result have been used before, for completeness we
include a simple proof.
\begin{lemma}\label{lemma:regularity}
For every $0<\delta<\frac{1}{2}$ and a graph $L$, there exist positive constants $\varepsilon}\def\ffi{\varphi=\varepsilon}\def\ffi{\varphi(\delta, L)$ and $n_0=n_0(\delta, L)$ with the following property.
Suppose $G$ is a graph whose vertices are partitioned into $\chi(L)$
independent sets, $V_1, V_2, \dots, V_{\chi(L)}$ of equal size which is
at least $n_0$. If the pairs $(V_i, V_j)$ are $\varepsilon}\def\ffi{\varphi$-regular and if their density satisfies $d(V_i, V_j)\in (\delta, 1-\delta)$ for all $i, j$, then $G$ contains an induced copy of $L$.
\end{lemma}
\begin{proof}
Let $l$ be the number of vertices of $L$, and let
$V(L)=U_1\cup \dots \cup U_{\chi(L)}$ be a partition of $L$ into $\chi(L)$ independent sets. The idea is to find a copy of $L$ in $G$ such that the vertices of $U_i$ come from $V_i$, for all $i$.
Having this goal in mind, refine the partition $V_1, \dots, V_{\chi(L)}$
by partitioning
every $V_i$ into $|U_i|$ smaller sets, $V_{i, 1}, \dots, V_{i, |U_i|}$,
of nearly equal sizes. Subdividing $V_i$ does not affect regularity of the new pairs significantly, and so the pairs $(V_{i, j}, V_{k, m})$ are still $l\varepsilon}\def\ffi{\varphi$-regular.
Similarly, the density of the pairs $(V_{i, j}, V_{k, m})$ is in the interval $(\delta-\varepsilon}\def\ffi{\varphi, 1-\delta+\varepsilon}\def\ffi{\varphi)$ if $i\neq k$, and zero otherwise.
Now, we use the following standard lemma (see e.g. Lemma 3.2 in \cite{AFKS}).
\medskip
\par\noindent
{\bf Lemma on regularity and induced subgraphs.} (\cite{AFKS})
{\it For every $0<\delta_0<1$ and $l\in \mathbb{Z}_{>0}$, there
exist positive constants $\varepsilon}\def\ffi{\varphi_0=\varepsilon}\def\ffi{\varphi_0(\delta_0, l)$ and $\mu_0=\mu_0(\delta_0, l)$ with the following property.
Suppose $L$ is a graph on $l$ vertices, $v_1, \dots, v_l$ and
$S_1, \dots, S_l$ is an $l$-tuple of disjoint vertex sets of a large graph $G$ such that every pair $S_iS_j$ is $\varepsilon}\def\ffi{\varphi_0$-regular
with density at least $\delta_0$ if $v_iv_j$ is an edge of $L$ and at most $1-\delta_0$ if $v_iv_j$ is not an edge of $L$. Then, $G$ contains at least $\mu_0 \prod_{i=1}^l |S_i|$ tuples $(w_1, \dots, w_l)\in S_1\times\dots \times S_l$ spanning an induced copy of $L$, where each $w_i$ corresponds to $v_i$.}
\medskip
\par\noindent
Note that identifying the sets $V_{i, j}$ with $S_i$ from this lemma satisfies the conditions, as the pairs $(V_{i, j}, V_{i, k})$ of density $0$ occur only when the corresponding vertices of $L$ belong to the same independent set $U_i$. Therefore, if we set $\delta_0=\delta-\varepsilon}\def\ffi{\varphi, \varepsilon}\def\ffi{\varphi_0=\varepsilon}\def\ffi{\varphi l$ and choose $\varepsilon}\def\ffi{\varphi$ small enough, we conclude that for large
enough $n_0$, $G$ contains at least one induced copy of $L$, as needed.
\end{proof}
\begin{proof} [Proof of Lemma \ref{lemma:kpartiteub}.]
The proof uses Szemer\'edi's regularity lemma to partition an arbitrary induced-$L$-free graph,
after which a standard "cleaning" argument shows that the
main contribution to the total number of such graphs comes from the number of possible bipartite graphs induced by regular pairs of density bounded away from $0$ and $1$. Then, Tur\'an's theorem provides an upper bound for the number of such pairs, completing the proof. The details follow.
Begin by fixing a small $\delta>0$ and the corresponding $\varepsilon}\def\ffi{\varphi=\varepsilon}\def\ffi{\varphi(\delta, L)>0$ from Lemma \ref{lemma:regularity}.
What we have in mind is a not yet fixed induced-$L$-free graph $G$ whose vertex set will be partitioned and we will calculate the number of ways we can connect pairs of vertices so that our graph becomes indeed induced-$L$-free.
By Szemer\'edi's regularity lemma, there is an $\varepsilon}\def\ffi{\varphi$-regular partition $V(G)=V_0\cup V_1\cup \dots\cup V_T$,
where $T$ is a constant, that is, it can be bounded from above
by a number $T_0(\varepsilon}\def\ffi{\varphi,k)$
which does not depend on $n$. We may also assume that this partition refines the original partition $V(G)=A_1\cup\dots\cup A_k$,
i.e., that every $V_i$ for $i\geq 1$ is a subset of some $A_j$.
(This follows from the standard proof of the regularity lemma, cf.
e.g. \cite{Diestel}, that works by iterating the refinement of some
original partition which we can choose to be the one given by
the partite classes $A_i$.)
We split the pairs $(V_i, V_j)$ into three classes - the irregular pairs, the $\varepsilon}\def\ffi{\varphi$-regular pairs with $d(V_i, V_j)\in [0, \delta]\cup [1-\delta, 1]$, and the $\varepsilon}\def\ffi{\varphi$-regular pairs with $d(V_i, V_j)\in (\delta, 1-\delta)$. We first show that the contribution of all pairs except those in the third class to the total number of graphs is negligible, and then we discuss the number of pairs in the third class.
There are at most $(T+1)^n=2^{o(n^2)}$ ways to distribute the vertices
into parts $V_0, \dots, V_{T}$, and there are
at most $2^{\varepsilon}\def\ffi{\varphi n^2}$ ways to choose the edges incident to $V_0$, as $|V_0|\leq \varepsilon}\def\ffi{\varphi n$. Note that there are no edges within the other parts $V_i$, so we have no additional choice here.
The number of ways to associate pairs to classes is at most
$3^{\binom{T}{2}}$, which is a constant (depending on $\varepsilon}\def\ffi{\varphi$).
Further, there are at most $2^{\big(\frac{n}{T}\big)^2}$ ways to choose the edges between each of the irregular pairs, and there are at
most $\varepsilon}\def\ffi{\varphi T^2$ irregular pairs. Hence, the number of choices for the edges between all irregular pairs is at most $2^{\varepsilon}\def\ffi{\varphi n^2}$.
Similarly, for the parts of density close to 0 or 1, we have at
most $2\sum_{i=0}^{\delta (n/T)^2}\binom{(n/T)^2}{i}\leq \frac{\delta n^2}{T^2}\binom{(n/T)^2}{\delta (n/T)^2}\leq \frac{\delta n^2}{T^2}(e\delta^{-1})^{\delta (n/T)^2}$, and there are at most $\binom{T}{2}$ such pairs.
Hence, the total number of choices for the edges in these pairs is
at most $e^{\delta (1-\log \delta) \binom{n}{2}+o(n^2)}$.
Finally, we need to bound the contribution from the pairs
$(V_i, V_j)$ with $d(V_i, V_j)\in (\delta, 1-\delta)$.
Define an auxiliary graph $G'$ on the vertex set $\{V_1, \dots, V_T\}$, in which $V_iV_j$ is an edge if and only if $(V_i, V_j)$
forms an $\varepsilon}\def\ffi{\varphi$-regular pair of density in $(\delta, 1-\delta)$. Lemma \ref{lemma:regularity} shows that
if $G$ contains no induced copy of $L$ then $G'$ cannot contain a subgraph isomorphic to $K_{\chi(L)}$.
Applying Tur\'an's theorem, we conclude that at most
$(1-\frac{1}{\chi(L)-1})T^2/2$ pairs among $V_1, \dots, V_T$
can be $\varepsilon}\def\ffi{\varphi$-regular and have density bounded away from
$0$ and $1$.
Hence, there are at most
$2^{(1-\frac{1}{\chi(L)-1})(T^2/2)\frac{n^2}{T^2}}
\leq 2^{(1-\frac{1}{\chi(L)-1})n^2/2+o(n^2)}$ choices for the edges in this case.
To complete the argument, we let $\delta, \varepsilon}\def\ffi{\varphi\to 0$ and note that the contribution of all pairs except those in the third class is negligible.
We conclude that the number of $k$-partite balanced induced-$L$-free graphs on $n$ vertices is at most
$2^{(1-\frac{1}{\chi(L)-1})n^2/2+o(n^2)}$, as stated. \end{proof}
Using Lemma \ref{lemma:kpartiteub}, the proof of Theorem \ref{thm:DCLindb} follows almost immediately.
\begin{proof}[Proof of Theorem \ref{thm:DCLindb}]
Analogously to what was done in the proof of Theorem \ref{thm:DCL}, now consider the graph
$G_{L,k}$ whose vertices are all balanced $k$-partite graphs
on $n$ vertices, in which two vertices are adjacent if and only if their symmetric difference contains no induced copy of $L$. As shown by Lemma \ref{lemma:kpartiteub}, the maximum degree in this graph is at most
$\Delta(G_{L,k})\leq 2^{\big(1-\frac{1}{\chi(L)-1}\big)n^2/2+o(n^2)}$. On the other hand, $G_{L,k}$ has
$2^{\binom{k}{2}\big(\frac{n}{k}\big)^2}
=2^{(1-\frac{1}{k}) n^2/2}
$ vertices. Hence, the size of its maximum independent set is at least
$$\alpha(G_{L,k})\geq\frac{|V(G_{L,k})|}{\Delta(G_{L,k})+1}\geq
2^{\big(\frac{1}{\chi(L)-1}-\frac{1}{k}\big)n^2/2+o(n^2)}.$$
As $k$ can be arbitrarily large, we conclude that
$DC(L,$ ind$)=\frac{1}{\chi(L)-1}$, completing the proof.
\end{proof}
In the case of bipartite graphs, one can prove a result analogous to Lemma \ref{lemma:kpartiteub} with much better bounds, as shown in \cite{ABBM}, and \cite{AMY} with an improved error term.
\begin{lemma}
\label{l91}
For every fixed bipartite graph $L$ there is some
$\varepsilon}\def\ffi{\varphi=\varepsilon}\def\ffi{\varphi(L)>0$ so that the number of bipartite graphs on two
classes of vertices $A$ and $B$, both of size $m$, that do not
contain an induced copy of $L$ is smaller than
$2^{m^{2-\varepsilon}\def\ffi{\varphi}}$.
\end{lemma}
\medskip
\noindent
The statement of this lemma, with a non-optimal value of $\varepsilon}\def\ffi{\varphi$,
follows from the results in the above mentioned papers that estimate the
number of bipartite graphs that do not contain an induced copy
of the universal bipartite graph $U(k)$ with $k$ vertices in one
vertex class
and $2^k$ in the other, connected in all possible ways to the vertices of
the first class. Every bipartite graph $L$ is an induced copy
of $U(k)$ for all sufficiently large $k$.
Although this is good enough for our purpose here,
we present next a shorter
new proof of the lemma, which gives a tight estimate up to a logarithmic
factor in the exponent for many graphs $L$.
\begin{lemma}
\label{l92}
Let $L$ be a bipartite graph $L$ with color classes of sizes
$s$ and $t \geq s$. Then the
number of bipartite graphs on two
classes of vertices $A$ and $B$, each of size $m$, that do not
contain an induced copy of $L$, is at most
$2^{c(s,t) m^{2-1/s} \log m }$.
\end{lemma}
The above is tight, up to the logarithmic term in the exponent,
for every pair $t \geq s$ where $t$ is sufficiently large as a function
of $s$. Indeed, as shown by the (projective) norm-graphs of
\cite{KRS}, \cite{ARS}, there is a bipartite graph with classes of
vertices of size $m$ each and with $\Omega(m^{2-1/s})$ edges that
contains no copy of the complete bipartite graph $K=K_{s,t}$, provided
$t>(s-1)!$ Every member of the
collection of all subgraphs of this graph contains
no (induced or non-induced) copy of $K$.
\begin{proof}
We apply the Sauer-Perles-Shelah Lemma (\cite{Sa}, \cite{Sh}) which states
that for any collection ${\cal C}$ of more than $\sum_{i=0}^d {q \choose i}$
functions from a set $Q$ of size $q$ to $\{0,1\}$ there is a subset
$D \subset Q$ of cardinality $d+1$ {\em shattered} by ${\cal C}$. That is,
for any function
$g:D \mapsto \{0,1\}$ there is an $f \in {\cal C}$ such that
$g(x)=f(x)$ for all $x \in D$. Let $L$ be a bipartite graph with classes
of vertices of sizes $t \geq s$ and let ${\cal G}$ be a collection of
graphs on the two vertex classes $A$ and $B$, where $|A|=|B|=m$.
Let $d=ex(2m,K_{s,t})$ be the maximum possible
number of edges in a
graph on $2m$ vertices that
contains no copy of the complete bipartite graph $K_{s,t}$.
By the K\H{o}v\'ari-S\'os-Tur\'an Theorem \cite{KST},
$d \leq b(s,t)m^{2-1/s}$.
By the Sauer-Perles-Shelah Lemma, if
$$
|{\cal G}| \geq 1+\sum_{i=0}^d {{m^2} \choose i}
$$
then the collection of graphs ${\cal G}$, viewed as a collection of
functions from the set of all edges of the complete
bipartite graph on the classes of vertices $A,B$
to $\{0,1\}$, shatters a set of $d+1$ edges. By the definition
of $d$ this set of edges
contains a complete bipartite graph with vertex classes of sizes
$s$ and $t$. Any subgraph of this graph (and in particular $L$)
is an induced subgraph of some member of ${\cal G}$. Since the right-hand-side
of the last inequality is smaller than
$$m^{2d} \leq m^{2b(s,t) m^{2-1/s}}=2^{c(s,t)m^{2-1/s} \log m},
$$
this completes the proof.
\end{proof}
\medskip
\par\noindent
The following is a counterpart of Corollary~\ref{DCgen} for the induced case.
\begin{cor}\label{DCindgen}
Let ${\cal G}$ be a set of graphs, each containing at least one edge,
and let ${\cal L}_{({\cal G},{\rm ind})}$ be the local graph class containing all graphs that contain at least one $G\in{\cal G}$ as an induced subgraph.
Then
$$DC({\cal L}_{({\cal G},{\rm ind})})= \frac{1}{\chi_{\rm min}({\cal G})-1}.$$
\end{cor}
{\noindent \it Proof. }
The upper bound is immediate from $DC({\cal L}_{({{\cal G}},{\rm ind})})\le DC({\cal L}_{{\cal G}})$ and Corollary~\ref{DCgen}. The lower bound follows, as in Corollary~\ref{DCgen}, from the fact that $DC({\cal L}_{({{\cal G}},{\rm ind})})
\geq DC(G, {\rm ind})$ for any $G\in {\cal G}$.
In particular, picking a graph $G$ with $\chi(G)=\chi_{\rm min}({\cal G})$ suffices.
\ifhmode\unskip\nobreak\fi\quad\ifmmode\Box\else$\Box$\fi
\begin{remark}\label{rem:indual}
{\rm It is straightforward that the above results imply a strengthening (as far as the upper bound is concerned) of the statement in Remark~\ref{rem:asydual}, namely that we have for the dual problem also in the induced case
$$\lim_{n\to\infty}\frac{2}{n(n-1)}\log D_{{\cal L}_{({{\cal G}},{\rm ind})}}(n)=1-\frac{1}{\chi_{\min}({\cal G})-1}.$$
This follows from Lemma~\ref{lem:ub} and Corollary~\ref{DCindgen}
just as the statement in Remark~\ref{rem:asydual} followed from Lemma~\ref{lem:ub} and Corollary~\ref{DCgen}.
$\Diamond$}
\end{remark}
\begin{remark}\label{rem:empty}
{\rm If $L$ is the
edgeless graph on $r \geq 2$ vertices then it is clear that
for every $n \geq r$ there is a family of
$2^{{n \choose 2}-{r \choose 2}}$ graphs on $[n]$ so that the symmetric
difference between any two contains an induced copy of $L$. Indeed
we simply take all graphs which agree on the ${r \choose 2}$ edges
of a fixed $r$-clique. Note that for every fixed $r$ this is a constant
fraction of all graphs on $n$ vertices, much larger than
$M_L(n)$ or $M_{L,{\rm ind}}(n)$ for any graph $L$ with at least
$2$ edges.
This (including the constant) is clearly tight for $r=2$ (the family
cannot contain a graph and its complement), and by the main result of
\cite{EFF} it is tight also for $r=3$ (the result in \cite{EFF} holds
for any family in which any two members agree on a triangle, not only
if any two intersect in a common triangle-the
equivalence of these two statements
is proved already in \cite{CFGS}). We do not know if this is tight
for larger values of $r$. Note that an equivalent formulation of the
question here is the determination of $M_{{\cal F}}(n)$ for ${\cal F}$ which
is the family of all graphs with independence number at least $r$.
$\Diamond$}
\end{remark}
\section{Open problems}
In this final section we collect some
related problems left open.
\begin{problem}
For what graph families ${\cal F}$ is it true that $M_{{\cal F}}(n)$ is achieved
by a linear graph family code, that is one that is closed under
the symmetric difference operation?
\end{problem}
Our results here include examples where this is the case as well as
ones in which it is not. Indeed in Theorem \ref{thm:fullstar} the
precise answer is $n$ or $n+1$, and if this is not a power of $2$
there is no optimal linear solution. Another family of examples in which
the optimal family cannot be achieved by a linear example is that in which
the family ${\cal F}$ is the family of all graphs with at most $2r$ edges,
where $r$ is chosen so that the sum
$$
\sum_{i=0}^r {{n \choose 2} \choose i}
$$ is not a power of $2$. Indeed, by a theorem of Kleitman
\cite{Kl} (for usual codes)
the size of the optimum family here is the size of the family
of all graphs with at most $r$ edges.
\medskip
The construction in the proof of Theorem~\ref{thm:conn} has the property that for any two of its graphs $G$ and $G'$ with an equal number of edges (that is trivially necessary for satisfying the condition in the next problem) their two {\em asymmetric} differences $$G\setminus G'=([n],E(G)\setminus E(G'))\ {\rm and}\ G'\setminus G=([n],E(G')\setminus E(G))$$ are isomorphic. This suggests the following question.
\begin{problem}
What is the maximum possible size of a graph family ${\cal A}$ of graphs on $n$ vertices satisfying that if $A,A'\in {\cal A}$ then $A\setminus A'$ and $A'\setminus A$ are isomorphic?
\end{problem}
\medskip
\par\noindent
A larger family
than the one we can obtain from the construction in the proof of Theorem~\ref{thm:conn} can be given by taking $\lfloor n/k\rfloor$ vertex-disjoint stars, each on $k$ vertices. This gives a lower bound that is superexponential in $n$ but we do not have any nontrivial upper bound.
\medskip
Theorems~\ref{thm:Hp} and \ref{thm:fullstar} show a huge difference between requiring a spanning path or a spanning star in the symmetric differences. One may wonder what happens ``in between''. Note that if we formulate this ``in betweenness'' so that we want to have a spanning tree with diameter at most $k$, then while with $k=2$ we are at Theorem~\ref{thm:fullstar} and with $k=n-1$ at Theorem~\ref{thm:Hp}, already for $k=3$ we get the same result as for $k=n-1$ by the construction in the proof of Theorem~\ref{thm:conn}. (This is simply because complete bipartite graphs contain spanning trees of diameter at most $3$.) So it seems plausible to formulate questions in terms of more specific ``natural'' sequences of spanning trees $T_1, T_2, \dots$. (In the problem below the notation $M_{T_n}(n)$ is meant to denote the largest possible cardinality of a family of graphs on vertex set $[n]$ such that the symmetric difference of any two of them contains $T_n$ as a subgraph.)
\medskip
\par\noindent
\begin{problem}
For what ``natural'' sequences $T_1, T_2,\dots, T_i,\dots$ of trees (with $T_i$ having exactly $i$ vertices for every $i$) will the value of $M_{T_n}(n)$ grow only linearly in $n$? A similar question is valid if $T_i$ is
replaced by
${\cal T}_i$, some ``natural'' family of $i$-vertex trees.
\end{problem}
\medskip
\par\noindent
Propositions~\ref{prop:triv34}, \ref{lem:MK3_5}, \ref{lem:MK3_6}, \ref{lem:MC7}
showed that the upper bound of Proposition~\ref{thm:Wilex} can be sharp for small values of $n$ for the requirement that a triangle or at least an odd cycle is contained in the symmetric differences. It would be interesting to know whether this can also happen for large values of $n$.
\medskip
\par\noindent
\begin{problem}
Is $$M_{K_3}(n)= 2^{{n\choose 2}-\lceil\frac{n}{2}\rceil\lfloor\frac{n}{2}\rfloor}$$ true always or at least for infinitely many values of $n$? Even if this is not so, does the analogous equality hold for $M_{{\cal C}_{\rm odd}}(n)$?
\end{problem}
Note that there are much better known estimates for the number of triangle-free
graphs on $n$ labeled vertices than the one we have used here, in fact, it
is known that almost all of these graphs are bipartite \cite{EKRo}. While this
improves the gap between the upper and lower bounds that follow from
our proofs for
$M_{K_3}(n)$, it is still far from determining its precise value.
\medskip
\par\noindent
The final problem we mention is related to the remark
in the end of the last subsection.
\medskip
\par\noindent
\begin{problem}
Is it true that for any fixed $r>3$ the maximum possible
cardinality of a family of graphs on $n$ labeled vertices in which
the symmetric difference between any two members has independence
number at least $r$,
is exactly a $1/{r \choose 2}$ fraction of
the number of all graphs on these vertices ?
\end{problem}
| {
"timestamp": "2022-04-05T02:03:14",
"yymm": "2202",
"arxiv_id": "2202.06810",
"language": "en",
"url": "https://arxiv.org/abs/2202.06810",
"abstract": "We investigate the maximum size of graph families on a common vertex set of cardinality $n$ such that the symmetric difference of the edge sets of any two members of the family satisfies some prescribed condition. We solve the problem completely for infinitely many values of $n$ when the prescribed condition is connectivity or $2$-connectivity, Hamiltonicity or the containment of a spanning star. We also investigate local conditions that can be certified by looking at only a subset of the vertex set. In these cases a capacity-type asymptotic invariant is defined and when the condition is to contain a certain subgraph this invariant is shown to be a simple function of the chromatic number of this required subgraph. This is proven using classical results from extremal graph theory. Several variants are considered and the paper ends with a collection of open problems.",
"subjects": "Combinatorics (math.CO); Information Theory (cs.IT)",
"title": "Structured Codes of Graphs",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9899864287859482,
"lm_q2_score": 0.7154240018510026,
"lm_q1q2_score": 0.7082600526602257
} |
https://arxiv.org/abs/2210.09954 | Decomposition and conformal mapping techniques for the quadrature of nearly singular integrals | Gauss-Legendre quadrature and the trapezoidal rule are powerful tools for numerical integration of analytic functions. For nearly singular problems, however, these standard methods become unacceptably slow. We discuss and generalize some existing methods for improving on these schemes when the location of the nearby singularity is known. We conclude with an application to some nearly singular surface integrals of viscous flow. | \section{Introduction}
\label{sec:intro}
It is usually easy to compute definite integrals when the integrand is analytic. Gauss-Legendre and Clenshaw-Curtis quadratures generally converge rapidly for aperiodic problems, while the humble trapezoid rule is equally powerful for measuring the area under one period of a periodic integrand.
Unhappily, this is not the case for the class of problems known as \emph{nearly singular integrals,} wherein the integrand fails to be analytic somewhere near but not on the interval of integration in the complex plane. The trapezoidal and Gauss-Legendre rules still provide geometric convergence -- that is, the logarithm of the error decreases linearly with the number of quadrature nodes -- but with an unacceptably small slope. Standard theorems relate this slope to the domain of analyticity of the integrand. In this paper we survey the existing methods of accelerating the convergence of these standard methods and we introduce several new ones, assuming that the location of the nearby singularity is known, but without using any additional information on the nature of the singularity.
While our motivation comes from the quadrature problem in the context of solving integral equations, some very similar issues also arise in the context of interpolation from samples of a nearly singular function. This is an important component of spectral methods for differential equations when the desired solution has abrupt fronts or peaks. In fact, some of the same acceleration techniques have been independently discovered by researchers in the two communities. For example, Johnston and Elliot's hyperbolic sine transformation \cite{johnston2005sinh} was also discovered by Tee and Trefethen \cite{tee2006rational}, while Jafari's transformation \cite{jafari2015new} is very similar to the repeated sinh method of \cite{elliott2008iterated}. One of our aims is to assemble the results from these two communities in one place.
We focus on two families of acceleration strategies. The first group of methods split the domain into carefully chosen subintervals and then solve subproblems on each of them. The other strategy that we consider is to change variables with a complex-analytic transformation so that the singularity lies farther away.
An early example of a splitting method for aperiodic problems was given by Ma and Kamiya \cite{ma2002distance}, who subdivide at the real part of the singularity, assuming that this lies within the integration interval and the problem is aperiodic (in fact they subsequently use an exponential change of variables for each of the new subproblems, thereby combining both of the principal strategies considered here).
Subsequently, Driscoll and Weideman \cite{driscoll2014optimal} gave a formula for the optimal splitting location in terms of the location of the singularity, again for the aperiodic case. Their formula is especially useful in cases where the singularity is near the endpoint of the interval in the complex plane or on the real line outside the interval, where simply using the real part of the singularity is ineffective or impossible. We develop an analogous method for periodic problems, replacing the trapezoid rule for a full period with individual Gauss-Legendre integrations on two subintervals of different sizes.
Splitting methods are extremely simple and, as we demonstrate, can yield dramatic improvements in the convergence rate. However, we can find even better convergence rates using methods that avoid dividing the available quadrature nodes among two or more subproblems. There are a vast array of possibilities for conformal mappings that distort the neighborhood of a real interval while fixing the endpoints and remaining real-valued and monotone within the interval. The Jacobi elliptic functions are a powerful tool for designing transformations that use all of the analyticity we have assumed; we list existing methods for periodic and aperiodic problems and give a new version for the aperiodic case when the singularity lies on the real line outside the integration interval. This is a small extension of results by Trefethen, Tee and Hale \cite{tee2006adaptive,tee2006rational,hale2008new,hale2009conformal}. However, the methods employing elliptic functions are less robust than some elementary alternatives when the integrand has other challenging features like additional distant singularities or rapid growth away from the real line. We therefore list or introduce some good elementary alternatives such as the sinh transformation for aperiodic problems \cite{johnston2005sinh,tee2006rational}, our iterated sine map for periodic problems, and our quadratic transformation for aperiodic problems with real singularity.
The organization of the paper is as follows. We state and comment on the standard theorems describing the convergence of the trapezoidal and Gauss-Legendre rules in
Sec. \ref{sec:thms}. We consider periodic problems in Sec. \ref{sec:periodic}. For aperiodic problems we first consider singularities off the real line in Sec. \ref{sec:apc} and then singularities that occur on the real line (but outside the interval of integration) in Sec. \ref{sec:apr}. We test all of the methods on a suite of integrands with various properties in Sec. \ref{sec:examples}. As an application, we then compute some nearly singular surface integrals arising in Stokes flow in Sec. \ref{sec:SLP}, followed by concluding remarks.
\section{Discussion of the standard theorems}
\label{sec:thms}
For a periodic integrand, the convergence rate of the trapezoid rule depends on the distance from the singularity to the real line.
\begin{theorem}[Geometric convergence of trapezoid rule \cite{trefethen2014exponentially}]
\label{thm:trap}
Suppose that a $2\pi$-periodic
function $f$
is analytic
and
satisfies $\lvert f(z)\rvert<M$
within the
strip $\lvert\Im(z)\rvert <\lambda$.
Then the difference between the integral $I = \int_{0}^{2\pi} f(x)\,dx $ and its $n$-point trapezoidal rule approximation $I_n =(2\pi/n)\sum_{j=1}^{n} f\left(2\pi j / n\right) $ satisfies
\begin{equation}
\left\lvert I-I_n \right\rvert
\le
\frac{4\pi M}{\exp(\lambda n)-1}.
\end{equation}
\end{theorem}
A similar theorem holds for Gauss-Legendre quadrature, with an ellipse replacing the infinite strip.
\begin{theorem}[Geometric convergence of Gauss-Legendre quadrature \cite{trefethen2019approximation}] \label{thm:gl}
Suppose that $f$ is analytic with $\lvert f(z)\rvert < M$ on the interior of the Bernstein ellipse $E_\rho$, whose foci are $\pm 1$ and whose semimajor and semiminor axis lengths sum to $\rho>1$. Let $C$ be the constant $C=64\rho^2/({15(1-\rho^{-2})})$, let $I = \int_{-1}^1 f(x)\,dx$ be the exact value of the integral, and let $I_n$ be its $n$-point Gauss-Legendre rule approximation. Then
\begin{equation}
\lvert I-I_n \rvert \le \frac{CM}{\rho^{2n}}.
\end{equation}
\end{theorem}
To employ this result in practice, we will often need to find the value of the ellipse parameter $\rho$ so that the ellipse passes through a given point $z\in\mathbb{C} \setminus [-1,1]$.
We quote af Klinteberg and Barnett's useful formula \cite{af2021accurate},
\begin{equation}
\rho = \rho(z) = \lvert z \pm \sqrt{z^2-1}\rvert ,\quad \textnormal{with sign chosen so that }\rho>1.
\label{eq:rhofromz}
\end{equation}
When we apply the preceding theorems, the twin goals of maximizing $\lambda$ or $\rho$ while minimizing $M$ are in tension. If $f$ is entire or has only distant singularities and $n$ is fixed, this leads to an optimization problem for the value of $\lambda$ or $\rho$ that will produce the strongest statement from the theorem. Of course, the relative importance of $M$ decreases as $n$ grows.
In the nearly singular case, where $\lambda\approx0$ or $\rho\approx1$, our priority is to improve the convergence rate even if this results in a large increase in $M$.
In this paper we are assuming knowledge about the locations of the singularities of the integrand, so $\lambda$ and $\rho$ are known. However, we make no assumptions about the nature of those singularities or about the growth of $\lvert f(z)\rvert$ away from the integration interval, so $M$ is unavailable.
Therefore we present a range of options instead of searching for an optimal strategy.
The relatively cautious methods we consider do not make use of all of the analyticity we have assumed for $f(z)$, and are therefore less vulnerable to the danger of fast growth in $\lvert f(z) \rvert$ away from the integration interval. In contrast, the more aggressive methods make use of all of the analyticity we have assumed (these generally involve the Jacobi elliptic functions, as we will see below). The aggressive methods will boast better theoretical convergence rates, although the errors may reach machine precision before they actually decrease at the advertised rate. In contrast, the more cautious methods have (slightly) smaller convergence rates but are more likely to actually achieve these rates for challenging integrands.
\section{Methods for periodic problems}
\label{sec:periodic}
We begin with the case where the integrand is $2\pi$-periodic. We suppose that $f$ is analytic except at the points $x=2\pi k \pm Bi$ for $k\in\mathbb{Z}$, or along branch cuts extending vertically from these points to infinity. The ordinary trapezoidal rule will be an effective choice if $B$ is large, since Theorem 1 predicts errors of size $e^{-Bn}$ with $n$ function evaluations. We therefore assume that $B$ is small but nonzero, so the integral is nearly singular, and we consider several methods to use our knowledge of the domain of analyticity of $f$ in order to improve on the performance of the trapezoid rule.
We begin with a decomposition method and then continue with several conformal mappings. Numerical examples demonstrating these techniques appear in Sec. \ref{sec:examples}.
\subsection{Subdivision}
\label{sec:psub}
One way to integrate over a full period of $f$ is to choose an appropriately small $\delta>0$ and then split the integral into subproblems on $[-\delta, \delta]$ and $[\delta, 2\pi-\delta]$. The subproblems are not periodic, so we apply Gauss-Legendre integration for each of them.
Rescaling the subintegrals linearly to $[-1,1]$, we have
\begin{align}
\int_{-\pi}^\pi f(x)\,dx &=
\delta \int_{-1}^1 f(\delta t)\,dt + (\pi - \delta) \int_{-1}^1 f\left( \pi + (\pi-\delta)w\right)\,dw.
\label{eq:per2glsub}
\end{align}
The two subintegrals are singular at $t=Bi/\delta $ and at $w=\frac{B i \pm \pi }{\pi-\delta}$, respectively. If we want the overall integration procedure to converge as quickly as possible and we assume $n$ is large, we should choose $\delta$ so that both subproblems have the same asymptotic convergence rate. Equivalently, both $Bi/\delta$ and $\frac{B i \pm \pi }{\pi-\delta}$ should lie on the same ellipse with foci $\pm1$ in the complex plane. Letting $m$ denote the semiminor axis length, we have the equations
\[\frac{0^2}{1+m^2} + \frac{(B/\delta)^2}{m^2} = 1,\qquad \frac{(\pi/(\pi-\delta))^2}{1+m^2} + \frac{(B/(\pi-\delta))^2}{m^2} = 1.
\]
After some simplification we arrive at the cubic equation
\(
0 = 2\delta^3 + 2B^2 \delta - B^2 \pi.
\)
This has a unique real solution which we write in terms of hyperbolic sines as follows:
\begin{equation}
\delta = \frac{2B}{\sqrt{3}}\sinh\left(\frac13 \arcsinh\left(\frac{3\pi\sqrt{3}}{4B}\right)\right).
\label{eq:perdelt}
\end{equation}
The error of the trapezoid rule on the original integral decays like $e^{-Bn}$ for large $n$, where $n$ is the number of quadrature nodes.
With subdivision as in \eqref{eq:per2glsub} and Gauss-Legendre quadrature with $n/2$ nodes on each subinterval,\footnote{We experimented with unequal distributions of nodes between the two subintervals but did not see more than modest improvements. For odd $n$, one should assign the extra node to the larger subinterval since errors there are multiplied by $(\pi-\delta)$ instead of $\delta$ in \eqref{eq:per2glsub}. } the total error will decay like $\rho^{-2(n/2)} = \rho^{-n}$, where $\rho = (B + \sqrt{B^2 + \delta^2}) / \delta$. Therefore, the splitting procedure is advantageous when $\log\rho > B$. We found numerically that this holds for $B<0.95$, although in practice, for finite values of $n$, it may be advisable to use the trapezoid rule for slightly smaller values of $B$.
\subsection{Conformal maps for periodic problems}
\label{sec:permaps}
A more powerful method for periodic problems is to compute the integral using the $n$-point trapezoidal rule following a transformation $x(t)$ that preserves the interval $[-\pi,\pi]$:
\begin{equation}
\int_{-\pi}^\pi f(x)\,dx = \int_{-\pi}^\pi f(x(t))x'(t)\,dt \approx \frac{2\pi}{n}\sum_{j=1}^n f\left(x\left(t_j\right)\right)x'\left(t_j\right)
\end{equation}
where $t_j=-\pi + {2\pi j}/{n}$.
In view of Theorem 1, the error will decay like $e^{-\lambda n}$ as long as the new integrand $f(x(t))x'(t)$ is periodic and analytic on the strip $S_\lambda = \{t\in\mathbb{C}:\lvert \Im (t) \rvert<\lambda\}$. We therefore seek a transformation $x(t)$ which carries a wide strip surrounding the real line in the complex $t$-plane into the domain of analyticity of $f$ in the complex $x$-plane, avoiding the branch cuts; another consideration is that the derivative $x'(t)$ should not have any singularities for $t\in S_\lambda$.
Two recent works have presented transformations with the general properties we seek: Tee constructed one using the Jacobi elliptic functions \cite{tee2006adaptive}, while Berrut and Elefante gave an elementary formula based on a M\"obius transformation \cite{berrut2020periodic}. After discussing these two methods, we propose a third, which we call the iterated sine map. We illustrate the conformal maps in Figure \ref{fig:cmp} with $B=0.3$ along with the resulting predicted convergence rates. We then give numerical examples comparing these three mappings, as well as the decomposition method from Section \ref{sec:psub} and the ordinary trapezoid rule, in Fig. \ref{fig:many_periodic_examples}.
\subsubsection{The Jacobi amplitude map}
Tee and Trefethen designed a conformal map that carries a strip onto the doubly slit region of analyticity of $f$ \cite{tee2006adaptive}. Here we present a new derivation of the same formula using differential equations rather than complex analysis. Our approach is based on the intuition that the factor $x'(t)$ should be small when $x$ is close to the singularity. To respect periodicity, we wrap the real line onto the unit circle and imagine the singularity as a nearby point $(1,0,B)$. We then assume that $x'(t)$ is proportional to the distance from $(\cos(x),\sin(x),0)$ to the singularity, leading to the boundary value problem
\begin{equation}
\frac{dx}{dt} = k \sqrt{(1-\cos( x))^2 + \sin^2( x) + B^2},\quad x(-\pi)=-\pi,\;\; x(\pi)=\pi
\label{eq:bvp2}
\end{equation}
where the value of $k$ is determined as part of the solution. The exact solution can be obtained from the Jacobi amplitude function $\am(t,m)$, using the relation
\begin{equation}
\frac{d}{dt}\am(t,m) = \sqrt{1 - m \sin^2\big(\am(t,m)\big)} = \dn(t,m).
\end{equation}
The solution of the BVP \eqref{eq:bvp2} is explicitly
\begin{equation}
x(t) = -\pi + 2 \am\left(\frac{\pi + t}{\pi}K\left(\frac{4}{4+B^2}\right),\frac{4}{4+B^2}\right)
\label{eq:JAM}
\end{equation}
where the real quarter-period $K(m)$ is defined for $m<1$ by
\begin{equation}
K(m) = \int_0^{\pi/2} \frac{d\theta}{\sqrt{1-m\sin^2\theta}}.
\end{equation}
In practice, we suggest computing the derivative $dx/dt$ using the Jacobi elliptic function $\dn(t,m)$ rather than the differential equation \eqref{eq:bvp2}:
\begin{equation}
x'(t) = \frac{2}{\pi}K\left(\frac{4}{4+p_y^2}\right) \dn\left(K\left(\frac{4}{4+p_y^2}\right)(t-1),\frac{4}{4+p_y^2}\right).
\end{equation}
This leads to an improved convergence rate of
\begin{equation}
\lambda = \pi\frac{K\left(B^2/(4+B^2)\right)}{K\left(4/(4+B^2)\right)}.
\end{equation}
The transformation \eqref{eq:JAM} is the sum of the identity map and a $2\pi$-periodic function, and it carries the rectangle $\{x+iy: \lvert x\rvert \le\pi, \lvert y \rvert < \lambda\}$ onto the doubly slit region $\{x+iy:\lvert x \rvert\le\pi\text{ and }\lvert y \rvert<p_y \textnormal{ if }x=0\}$, thereby using all of the analyticity that we have assumed for $f$. This is the most aggressive option for the periodic problem.
\subsubsection{The boundary correspondence map}
Berrut and Elefante recently suggested an alternative mapping that avoids the use of elliptic functions \cite{berrut2020periodic}. Adapting their method slightly, we get the formulas
\begin{align}
a &= \exp(B) - \sqrt{-1+\exp(2B)}\\
x(t) &= -i \log\left(\frac{\exp(it)+a}{1+a\exp(it)}\right)\\
x'(t)&=
\frac{1-a^2}{a^2+2 a \cos (t)+1}\textnormal{ for } t\in\mathbb{R}\\
\lambda_{\max} &= -\log(a).
\end{align}
This method is much better than the ordinary trapezoid rule, but it does not perform nearly as well as the Jacobi amplitude map, as one can see in Fig. \ref{fig:many_periodic_examples}. The conformal image of the strip (top right in Fig. \ref{fig:cmp}) is unbounded and does not closely approach the sides of the vertical branch cuts.
\subsubsection{The iterated sine map}
We searched for another conformal mapping with the goal of reproducing the good convergence rate of the Jacobi amplitude map with a simpler transformation.
An early candidate was the mapping $\phi(t) = t - a \sin(t)$ for $a\in[0,1)$.
This function increases monotonically, it carries $[-\pi,\pi]$ to itself, and it has a small derivative when $0=t=x(t)$ if $a\approx 1$.
These properties also hold for the compositions $\phi\circ \phi$ and $\phi\circ\phi\circ\phi$.
The mapping $x(t) = \phi(t)$ is not competitive with the Jacobi amplitude transformation, but with a suitable choice of parameter $a$, the mapping $x(t)=\phi(\phi(t))$ is. We call this the \emph{iterated sine map}:
\begin{align}
x(t) &= t - a\sin(t) - a\sin(t-a\sin(t))\\
x'(t) &= (1-a\cos t)\left(1-a\cos(t-a\sin t)\right).
\end{align}
The optimal value of the parameter $a$ depends on the location of the singularity $0\pm Bi$.
To find it, we study the imaginary part of $x(0+i\lambda)$ as a function of $\lambda$. This increases from $0$ when $\lambda =0$ to a local maximum when $\lambda=\arccosh(1/a)$. Setting the value of this local maximum equal to $B$, we have the equation
\begin{equation}
-\sqrt{1-a^2}+a \sinh \left(\sqrt{1-a^2}-\arcsech(a)\right)+\arcsech(a)=B.
\end{equation}
This is difficult or impossible to solve analytically for $a$. In searching for a good initial guess for Newton's iteration, we found that the optimal value of $a$ was very close to $1 + B/5 - B^{2/5}$, especially for $B \approx 0$. The approximation is good enough that we forego Newton iteration entirely and use $a = 1+B/5-B^{2/5}$, with the caveat that for $B>1.5$, one should abandon the iterated sine map and use the ordinary trapezoid rule without any change of variable. This mapping is illustrated at bottom left in Fig. \ref{fig:cmp}; it is a `cautious' method because it carries the strip into a relatively small bounded region, but its convergence rate is nearly as good as the Jacobi amplitude map. For $B>0.03$ the image of the strip overlaps the branch cut slightly, possibly impairing the convergence rate, but this problem disappears for $B<0.03$.
\begin{figure}
\centering
\includegraphics[width=\linewidth]{PeriodicConformalMaps.pdf}
\caption{Comparison of three conformal mappings for periodic problems when the integrand has a branch cut extending vertically from $0+0.3i$ (red lines). The three grids are the conformal images of three rectangles $\{x+iy: \lvert x \rvert\le\pi, 0\le y\le \lambda\}$. The Jacobi amplitude mapping carries a wide strip $(\lambda = 1.5)$ onto the full, unbounded domain of analyticity we have assumed for the integrand. The boundary correspondance map carries a thinner strip ($\lambda = 0.81$) into a subset of the region of analyticity, avoiding the sides of the branch cut but extending to infinity for $\lvert x \rvert>\pi/2$. The iterated sine map carries a wide strip $(\lambda = 1.46$) onto a bounded region which slightly overlaps the branch cut, an issue that disappears when the height $B$ of the singularity is less than $0.03$. At bottom right, we compare the improved convergence rates for these conformal maps as well as the splitting method and the ordinary trapezoid rule for $B\in[10^{-4},1]$.
}
\label{fig:cmp}
\end{figure}
\section{Methods for aperiodic problems with singularity off the real line}
\label{sec:apc}
We now consider the problem of integrating an aperiodic function $f(x)$ on $[-1,1]$, assuming that $f$ is analytic except for singularities at $A\pm Bi$ with $B>0$, or along branch cuts extending vertically from these points to infinity. Our goal is to use this information about the integrand to improve on standard Gauss-Legendre quadrature. As before, we describe a decomposition method and also several methods based on conformal maps. Here we do not propose any novel mappings, but we give some new results on the convergence properties of those previously described \cite{tee2006rational,johnston2005sinh,elliott2008iterated,jafari2015new}.
\subsection{Decomposition}
A simple strategy for integration on $[-1,1]$ is to carry out separate Gauss-Legendre integrations on $[-1,\delta]$ and $[\delta,1]$, where $\delta$ depends on $A\pm Bi$. Rescaling both subproblems back to $[-1,1]$, we obtain
\begin{equation}
\footnotesize{ \int_{-1}^1 f(x)\,dx
= \frac{\delta+1}{2}\int_{-1}^1 f\left(\frac{\delta-1}{2} + \frac{\delta+1}{2}t \right)\,dt + \frac{1-\delta}{2}\int_{-1}^1 f\left(\frac{1+\delta}{2} + \frac{1-\delta}{2}w \right)\,dw.}
\label{eq:glchop}
\end{equation}
Now the integrands on the right side of \eqref{eq:glchop} have singularities at the points $t^* = (2A-\delta+1+ 2Bi)/(1+\delta)$ and $w^* = ({2A-\delta-1+2Bi})/({1-\delta})$, respectively. To optimize the convergence rate, we must choose $\delta$ so that the Bernstein ellipses passing through $t^*$ and $w^*$ coincide. In particular, their semimajor axis lengths must be equal. Letting $a$ denote this common length, we have the equations
\begin{equation}
\frac{(2A-\delta+1)^2}{a^2(\delta+1)^2} + \frac{(2B)^2}{(a^2-1)(\delta + 1)^2} = 1 =
\frac{(2A-\delta-1)^2}{a^2(\delta+1)^2} + \frac{(2B)^2}{(a^2-1)(1-\delta )^2} .
\end{equation}
By eliminating $a$ we arrive at the quartic polynomial equation
\begin{equation}
2\delta (2A - \delta+1)^2 (A-\delta) + 4\delta B^2 (2A-\delta) = 2(\delta+1)^2(2A-\delta)(A-\delta).
\end{equation}
The real solution with $\lvert \delta \rvert <1$ is given by
\begin{equation}
\label{eq:apdelta}
\delta=\sgn(A)\left(
\lvert A \rvert - \frac{\sqrt{2}}{2}\sqrt{A^2-1 - B^2 + \sqrt{-4A^2 + \left(1+A^2+B^2\right)^2}}\right).
\end{equation}
This formula agrees with Equation 2.7 in \cite{driscoll2014optimal}, and gives a simple method for evaluating $\int_{-1}^1 f(x)\,dx$, given the location of the nearest singularity $x^* = A+Bi$: define $\delta$ using \eqref{eq:apdelta}, then evaluate the right-hand side of \eqref{eq:glchop} using $n/2$-point Gauss-Legendre quadrature on each subintegral.
Letting $\rho_{x^*}$ and $\rho_{t^*}=\rho_{w^*}$ be the Bernstein ellipse parameters for the original and decomposed problems, we see that the error decay rates of Theorem \ref{thm:gl} are $\rho_{x^*}^{-2n}$ and $\rho_{t^*}^n$, respectively. Therefore the splitting procedure will be worthwhile if $\sqrt{\rho_{t^*}} > \rho_{x^*}$.
\subsection{Conformal maps}
We now turn to strategies based on conformal mapping. Writing $t_j$ and $w_j$ for the nodes and weights of Gauss-Legendre quadrature on $[-1,1]$, we have
\begin{equation}
\int_{-1}^1 f(x)\,dx = \int_{-1}^1 f(x(t))x'(t)\,dt \approx \sum_{j=1}^n f(x(t_j))x'(t_j)w_j
\label{eq:GLtransform11}
\end{equation}
The new integrand $f(x(t))x'(t)$ should be analytic on a large Bernstein ellipse in the complex $t$-plane. In particular, the image of the ellipse should lie within the domain of analyticity of $f$ in the complex $x$-plane, and the derivative $x'(t)$ should not itself be singular, at least for $t$ within the Bernstein ellipse. We also require that $x(t)\in[-1,1]$ for $t\in[-1,1]$, and $x(\pm1)=\pm 1$.
See Fig. \ref{fig:conformalC} for illustrations of the three transformations when $A\pm Bi = 2/3 \pm i/3$ as well as a plot of the convergence rates for other values of $B$, and Fig. \ref{fig:many_np_c_examples} for numerical results.
\subsubsection{The hyperbolic sine map}
We give a new derivation of the sinh transformation \cite{johnston2005sinh,tee2006adaptive}, using a differential equation. Motivated by the desire to keep the derivative $x'(t)$ small when $x$ is near the singularity $A+Bi$, we consider the BVP
\begin{equation}
x'(t) = k\sqrt{(x-A)^2 + B^2},\quad x(-1)=-1,\;\;x(1)=1
\end{equation}
where the proportionality constant $k$ is to be found as part of the solution.
This can be solved by hand, yielding the transformation
\begin{align}
x(t) &= A + B \sinh\left( \frac{1-t}{2}\arcsinh\left(\frac{-1-A}{B}\right) + \frac{1+t}{2}\arcsinh\left(\frac{1-A}{B}\right) \right).
\end{align}
The derivative $x'(t)$ is an entire function, so the convergence rate will be limited by the analyticity of $f(x(t))$.
The value of $t$ for which $x(t)=A+Bi$ is
\begin{equation}
t^* = 1 + \frac{i \pi - 2 \arcsinh((1-A)/B)}{\arcsinh((1-A)/B + \arcsinh((1+A)/B}
\end{equation}
and we can use this in \eqref{eq:rhofromz} to obtain the improved value of the ellipse parameter $\rho$.
\subsubsection{The elliptic sine map}
By conformally mapping a Bernstein ellipse onto the doubly slit plane, Tee and Hale \cite{hale2009conformal} obtained the transformation $x(t)$ given by the equations
\begin{align}
c &= \frac{\sgn(A)}{\sqrt{2}} \sqrt{A^2+B^2+1 - \sqrt{(A^2+B^2+1)^2-4A^2}}\\
m &= \frac{\left(-B + \sqrt{B^2 + 1-c^2}\right)^4}{(1-c^2)^2}\\
h(t) &= m^{1/4} \sn\left(\frac{2K(m)}{\pi}\arcsin(t), m\right)\\
x(t)&= \frac{c}{m^{1/4}} - \frac{1-m^{1/2}}{2m^{1/4}} \left(\frac{1-c}{ h(t) - 1} + \frac{1+c}{h(t)+1}\right).
\end{align}
Note that we have condensed their notation. The intermediate conformal map $h(t)$, carrying an ellipse to the unit circle, is due to Schwarz \cite{szego1950conformal,schwarz1869}.
The enlarged Bernstein ellipse has parameter given by \cite{hale2009conformal}
\begin{equation}
\rho = \exp\left(\frac{\pi K(1-m)}{4 K(m)}\right).
\end{equation}
For convenience in computing the derivative $x'(t)$, we note that
\begin{equation}
h'(t) = \frac{2m^{1/4}K(m)}{\pi\sqrt{1-t^2}} \cn\left(\frac{2K(m)}{\pi}\arcsin(t), m\right)\dn\left(\frac{2K(m)}{\pi}\arcsin(t), m\right).
\end{equation}
This aggressive strategy takes full advantage of the assumed analyticity of the integrand and has a large convergence rate.
\subsubsection{Jafari-Varzaneh's mapping}
Jafari-Varzaneh and Hosseini used the composition of the hyperbolic sine mapping and another similar function to obtain a new mapping \cite{jafari2015new}. With \(\alpha = \frac12 \arcsinh\left(\frac{1-A}{B}\right) + \frac12 \arcsinh\left(\frac{1+A}{B}\right)\), their formulas can be condensed to
\begin{align}
x(t) &= A + B \sinh\left(\arcsinh\left(\frac{1-A}{B}\right)-\alpha
+ \frac{2\alpha L + \pi}{2} \tan\left(t\, \arctan\left(\frac{2\alpha}{2\alpha L + \pi}\right)\right)\right)
\end{align}
where $L$ is a tunable parameter between $0.2$ and $0.9$. In light of their statement that ``experiments show that different values of this parameter approximately give the same results,'' we choose to take $L=0.5$ in all cases.
We solved for the value of $t^*$ satisfying $x(t^*) = A+Bi$ and obtained
\begin{equation}
t^* = \arctan\left(\frac{2\alpha - 2\arcsinh\left(\frac{1-A}{B}\right)+i\pi)}{2\alpha L+\pi}\right)/\arctan\left(\frac{2\alpha}{2\alpha L+\pi}\right),
\label{eq:sinhmapsing}
\end{equation}
which can be used in \eqref{eq:rhofromz} to find a prediction of the new convergence rate. This strategy is more cautious than Tee's elliptic sine mapping but more aggressive than the hyperbolic sine tranformation.
\subsubsection{The iterated sinh map}
For some problems, Elliot and Johnston suggested applying the sinh mapping twice in succession \cite{elliott2008iterated}. To construct the mapping, we can use \eqref{eq:sinhmapsing} to obtain the singularity location after one sinh transformation and then apply another sinh transformation accordingly. For convenience, we simplify to obtain the formula
\(x(t) = A+B \sinh \left(\frac{\pi}{2} \sinh \left(\ell(t)\right)\right)\), where $\ell(t)$ is the linear function
\begin{equation}\ell(t) = \frac{t+1}{2} \arcsinh\left(\frac{2}{\pi} \arcsinh\left(\frac{1-A}{B}\right)\right)+\frac{t-1}{2} \arcsinh\left(\frac{2}{\pi} \arcsinh \left(\frac{A+1}{B}\right)\right).
\end{equation}
The new singularity lies at
\begin{equation}
t^* = 1+\frac{i \pi-2 \arcsinh\left(\frac{2}{\pi} \arcsinh((1-A)/B)\right)}{\arcsinh\left(\frac{2}{\pi} \arcsinh((1-A)/B)\right)
+\arcsinh\left(\frac{2}{\pi} \arcsinh((1+A)/B)\right)}
\label{eq:sinhsinhtstar}
\end{equation}
instead of $x^*=A\pm Bi$.
This map tends to wrap the original Bernstein ellipse onto a relatively large region surrounding the singularity (overlapping the branch cut if there is one). Elliot and Johnston observed that the iteration gives better results for what they term \emph{nearly strongly singular} problems, meaning integrals that become divergent when the singularity reaches the real line. In contrast, they present experiments showing that a single sinh transformation is better for \emph{nearly weakly singular} integrals (which become convergent improper integrals when the singularity reaches the integration interval). This distinction is outside our scope because it relies on information about the nature of the singularity in addition to its location.
\begin{figure}
\centering
\includegraphics[width=\linewidth]{ComplexTransformsAPeriodicC.pdf}
\caption{Comparison of four conformal mappings designed to avoid a singularity at $2/3 + i/3$. We plot the image of four Bernstein ellipses with different values of $\rho$. The mapping introduced by Tee, in terms of the Jacobi elliptic sine, permits a large value of $\rho$ and uses all of the assumed analyticity. The $\rho$-value for the map of Jafari-Varzaneh and Hosseini is nearly as large, and the image is bounded. The hyperbolic sine mapping has the smallest $\rho$-value of the four, and it uses far less of the assumed domain of analyticity.
The iterated or double sinh transformation achieves a large value of $\rho$, but the image of the ellipse extends far from the original interval and, while avoiding the singularity itself, wraps around it and significantly overlaps any branch cut extending from the singularity.
}
\label{fig:conformalC}
\end{figure}
\section{Methods for aperiodic problems with singularity on the real line}
\label{sec:apr}
We now consider the case where the integrand $f(x)$ is analytic except for an isolated singularity at some real $A$ with $\lvert A \rvert >1$, or possibly with a horizontal branch cut from $A$ to infinity. This situation is less common than the fully complex case, both for quadrature problems and for rational barycentric interpolation and Chebyshev spectral methods, and accordingly has received less attention. We describe several strategies for using the analyticity of $f$ in order to improve on Gauss-Legendre integration with $n$ nodes; to our knowledge these are all new.
\subsection{Subdivision}
\label{sec:arsub}
One simple way to take advantage of information on the location of the singularity is to carry out separate Gauss-Legendre integrations, each using $n/2$ nodes, on $[-1,\delta]$ and $[\delta,1]$, where $\delta$ is given by
\eqref{eq:apdelta}. With $B=0$, that formula simplifies to
\begin{equation}
\delta = \sgn(A)
\left(
\lvert A \rvert - \sqrt{A^2-1}
\right).
\end{equation}
As above, the improved convergence rate comes from putting the rescaled singularity $(2A-\delta+1)/(1+\delta)$ into \eqref{eq:rhofromz}.
\subsection{Conformal maps}
We again follow \eqref{eq:GLtransform11}, applying Gauss-Legendre integration following a change of variable. The mapping $x(t)$ must preserve the interval $[-1,1]$ and fix its endpoints, and ideally should carry a large Bernstein ellipse in the $t$-plane into the domain of analyticity of $f$. We introduce three possibilities and illustrate them in Fig. \ref{fig:complextransformsReal}; numerical examples appear in Fig. \ref{fig:many_np_r_examples}.
\subsubsection{Quadratic transformation}
There is a unique second-degree polynomial $x(t)$ which satisfies the boundary conditions $x(-1)=-1$ and $x(1)=1$, satisfies $x'(t)=0$ when $x=A$, and is strictly increasing for $-1\le t\le1$.
It is given by
\begin{equation}
x(t) = \frac{-1}{2} \sgn(A) \left(\lvert A \rvert -\sqrt{A^2-1}\right)(t^2-1) + t.
\end{equation}
The unique value of $t$ satisfying $x(t)=A$ is the semimajor axis length of the enlarged ellipse; this leads to the improved ellipse parameter
\begin{equation}
\rho = \lvert A \rvert + \sqrt{A^2-1} + \sqrt{(\lvert A \rvert + \sqrt{A^2-1})^2-1}.
\end{equation}
This relatively cautious strategy gave the best results for most of our test problems.
\subsubsection{Exponential transformation}
By solving the BVP
\begin{equation}
\frac{dx}{dt} = k\lvert x-A\rvert ,\quad x(-1)=-1,\;x(1)=1,
\end{equation}
where the value of $k$ is determined as part of the problem, we obtain an exponential change of variable:
\begin{equation}
x(t) = A + (1-A)\exp\left(\frac{1-t}{2} \log\left(\frac{A+1}{A-1}\right)\right)
\end{equation}
There is no value of $t$ satisfying $x(t)=A$, so the composition $f(x(t))$ is entire if $f$ has an isolated singularity. On the other hand, if $f$ has a branch cut from $A$ to infinity, then the composition is analytic only on the Bernstein ellipse with parameter $\rho = s + \sqrt{1+s^2}$ with
$s = 2\pi/\lvert\log( (A+1)/(A-1))\rvert.$
\subsubsection{Elliptic sine map}
We have already seen the Schwarz mapping carrying the interior of an ellipse onto the unit disk. We can compose this map with the transformation
$z\mapsto - (1-z)^2/(1+z)^2$ to obtain a mapping from the ellipse to the slit plane $\mathbb{C}\backslash[0,\infty)$.
Then, by scaling and translation, we can arrange for $x(\pm 1) = \pm 1$. The transformation and its derivative are given by
\begin{align}
c_1 &= 17-80A^2+64A^4\\
c_2 &= (4A^2-3)\lvert A \rvert \sqrt{A^2-1}\\
m &= c_1+16c_2 - 4 \sqrt{2}\sqrt{(A^2-1)(8+c_1(4A^2-1)) + c_1c_2}\\
h(t) &= m^{1/4} \sn\left(\frac{2K(m)}{\pi}\arcsin(t), m\right)\\
x(t)&= \frac{h(t)+2\sqrt{m}\left(1+h(t)+h(t)^2\right) + m\, h(t)}{(1+h(t))^2 \left(1+\sqrt{m}\right)m^{1/4}}\\
x'(t) &= -h'(t) \frac{(h(t)-1)\left(1-\sqrt{m}\right)^2}{(1+h(t))^3\left(1+\sqrt{m}\right)m^{1/4}}
\end{align}
while the new convergence rate is $\rho = \exp(0.25\pi K(1-m)/K(m))$. This aggressive strategy was more effective than the splitting method but less effective than the other conformal maps in our tests.
\begin{figure}
\centering
\includegraphics[width=\linewidth]{ComplexTransformsAPeriodicR.pdf}
\caption{Three conformal mappings that avoid a singularity on the real line, at $4/3.$ The Jacobi elliptic sine mapping (top left) carries a large ellipse ($\rho = 8.38$) onto $\mathbb{C}\backslash[4/3,\infty)$. The exponential transformation (top right) carries a smaller ellipse ($\rho = 6.61)$ onto a large but bounded region. The quadratic transformation (bottom left) carries a smaller ellipse ($\rho=4.19$) into a much smaller bounded region. All three methods improve substantially against ordinary Gauss-Legendre integration, which has $\rho = 2.21$ for $A=4/3$. At lower right we plot the relationship between $\rho-1$ and $A-1$ for these methods as well as the decomposition method of section \ref{sec:arsub}. }
\label{fig:complextransformsReal}
\end{figure}
\section{Numerical examples}
\label{sec:examples}
We now assess the performance of all of the strategies described above on a collection of nearly singular integrals of varying difficulty. We begin with a careful discussion of the results for periodic problems and then treat the aperiodic problems more briefly, since the themes are similar.
\subsection{Periodic examples} For periodic problems, we consider integrals of the form $\int_{-\pi}^\pi f_i(x;\epsilon)\,dx$ for $\epsilon\in\{10^{-1}$, $10^{-2}$, $10^{-3}\}$ where $f_i$ is one of the following:
\begin{align}
f_1(x;\epsilon) &= \log(\cosh(\epsilon)-\cos(x)) + (\cosh(\epsilon)-\cos(x))^{3/10}\\
f_2(x;\epsilon) &= \frac{1}{\sqrt{\cosh(\epsilon)-\cos(x)}}\\
f_3(x;\epsilon) &= \frac{\cos^2(6x)}{\sqrt{\cosh(\epsilon)-\cos(x)}}\\
f_4(x;\epsilon) &= \frac{\sqrt{\cosh(1)+\cos(x)}}{\sqrt{\cosh(\epsilon)-\cos(x)}}.
\end{align}
Because $\cosh(\epsilon) = \cos(\epsilon i)$, all of these integrands have branch cuts extending vertically to infinity from $x=2\pi k \pm \epsilon i$, for $k\in\mathbb{Z}$. The first three integrands have no singularities other than these branch cuts, while $f_4$ has additional branch cuts extending to infinity from $2\pi k + \pi\pm 1i$. The results, displayed in Fig. \ref{fig:many_periodic_examples}, suggest that the iterated sine mapping method usually reaches machine precision with a similar or smaller number of quadrature nodes $n$ compared to the Jacobi amplitude mapping, and that these two methods give much better results than the other three methods.
We now make more detailed comments about each row of the figure.
\begin{figure}
\centering
\includegraphics[width=\linewidth]{fig_many_periodic_examples.pdf}
\caption{We test five strategies for integration of periodic, nearly singular integrals on twelve problems of varying difficulty. The relative numerical errors (glyphs) generally decay with the predicted slopes (lines), with exceptions in the last two rows due to particular features of the integrands as discussed in the text. The predicted slopes depend only on the locations of the singularities, $\pm \epsilon i$, and are identical within columns of the figure. The iterated sine mapping (stars) gives the best results, except in the second row where the Jacobi amplitude transformation (pluses) is better. }
\label{fig:many_periodic_examples}
\end{figure}
For the first integrand $f_1$, a sum of logarithmic and fractional-power singularities, all five methods converge with the predicted slopes (top row of Fig. \ref{fig:many_periodic_examples}). The Jacobi amplitude mapping has the best slope, but the iterated sine mapping reaches machine precision at the same time or slightly earlier, with about $n=50$ quadrature nodes.
For the second integrand $f_2$, which has an inverse root singularity, the Jacobi amplitude mapping gives remarkably good results. This is something of a lucky accident particular to this integrand (if one multiplies $f_2$ by $\cos(x)$, the result, not pictured, is similar to the top row of Fig. \ref{fig:many_periodic_examples}). We also see that the boundary correspondence mapping converges twice as quickly as predicted when $n$ is odd, but converges at the expected rate for even $n$ (the precise $n$ sampled in Fig. \ref{fig:many_periodic_examples} are the multiples of 7 up to 147). For the smaller values of $B$ (middle and right column of Fig. \ref{fig:many_periodic_examples}), this is still much slower than the ISM or JAM results.
The third integrand $f_3$ is the product of $f_2$ and the entire function $\cos^2(6x)$, which grows rapidly away from the real line. Therefore we expect the constant $M$ in Theorem \ref{thm:trap} to play a more influential role in the third row of Fig. \ref{fig:many_periodic_examples} than in the second; in particular, a larger $n$ will be required before the errors decrease at the predicted rate. When $\epsilon=1/1000$, both the iterated sine mapping and the Jacobi amplitude mapping converge more slowly than predicted, and the iterated sine mapping significantly outperforms the Jacobi amplitude mapping. We also remark that the splitting method, while not converging as quickly as the best methods, does converge at the predicted rate, which is unsurprising given that it uses analyticity only in two thin ellipses which do not extend far from the real line.
Finally we turn to the fourth integrand, which has a different domain of analyticity. While both the JAM and BCM methods have extensions to the the case of multiple singularities \cite{tee2006adaptive,berrut2020periodic}, we choose to construct the mappings with reference only to the singularities at $0\pm \epsilon i$, ignoring the more distant ones at $\pi \pm i$.
This means that the results based on conformal mapping should converge somewhat more slowly than expected, a prediction confirmed by the last row of Fig. \ref{fig:many_periodic_examples}. However, for small $B$ the iterated sine mapping is much less impaired by this failure of analyticity than the Jacobi amplitude and boundary correspondence mappings. We can explain this difference by examining the illustrations of the three mappings in Fig. \ref{fig:cmp}. Indeed, on the lines $x=\pm\pi+\xi i$, the iterated sine mapping assumes analyticity only for a bounded range of $\xi$, while the other mappings assume analyticity for all $\xi$. This makes the singularity at $\pm \pi \pm1i$ more hazardous for the BCM and JAM methods, which rely more completely on the analyticity we have assumed.
\subsection{Aperiodic problems with nonreal singularity}
For this setting we modify the test problems slightly so that the integration interval is $[-1,1]$ and the singularity lies at $2/3 \pm \epsilon i$. Specifically, we use
\begin{align}
\label{eq:firstg}
g_1(x;\epsilon) &= -\log(\cosh(x-2/3)-\cos(\epsilon)) + (\cosh(x-2/3)-\cos(\epsilon))^{0.3}\\
g_2(x;\epsilon) &= \frac{1}{\sqrt{\cosh(x-2/3)-\cos(\epsilon)}}\\
g_3(x;\epsilon) &= \frac{\cos^2(6\pi x)}{\sqrt{\cosh(x-2/3)-\cos(\epsilon)}}\\
g_4(x;\epsilon) &= \frac{\sqrt{\cosh(x+2/3)-\cos(1)}}{\sqrt{\cosh(x-2/3)-\cos(\epsilon)}}.
\label{eq:lastg}
\end{align}
Because the integration interval is smaller by a factor of $\pi$, we take $\epsilon\in \{1/30$, $ 1/300$, $1/3000\}$ to obtain problems of comparable difficulty to the previous subsection.
As above, the third integrand is the product of the second integrand and an entire function, while the fourth integrand has an additional pair of singularities at $x=-2/3\pm i$. The results, given in Fig. \ref{fig:many_np_c_examples}, are very similar to the periodic examples. In particular, the mapping that uses the Jacobi elliptic functions to carry a Bernstein ellipse onto the full domain of analyticity of the integrand does not give the best results, even though its predicted convergence rate is the best; instead, the more cautious hyperbolic sine mapping appears to be the best general choice. For these problems, the iterated sinh map does not converge at the large rate suggested by \eqref{eq:sinhsinhtstar}; its performance is similar to Tee's elliptic mapping.
All of these examples have $A=2/3$. In other tests, not plotted, we found similar results for integrands with singularities at $1+\epsilon + \epsilon i$ for $\epsilon = 1/30, 1/300, 1/3000$. In particular, the sinh map again gave the best results overall.
\begin{figure}
\centering
\includegraphics[width=\linewidth]{fig_many_NONperiodic_C_examples_wss.pdf}
\caption{Integration of the functions \eqref{eq:firstg} - \eqref{eq:lastg} by the methods of Section \ref{sec:apc}. Here ``Tee'' refers to the Jacobi elliptic mapping, ``JV-H'' is the composite mapping introduced by Jafari-Varzaneh and Hosseini, ``SinhSinh'' and ``Sinh'' are the iterated and ordinary hyperbolic sine mappings, ``Split'' is the decomposition strategy, and ``G-L'' is standard Gauss-Legendre quadrature. Among these, the ordinary sinh method generally gives the best results. }
\label{fig:many_np_c_examples}
\end{figure}
\subsection{Aperiodic problems with real singularity}
We now integrate on $[-1,1]$ with singularity at $1 + \epsilon \in\mathbb{R}$. Specifically, we use
\begin{align}
\label{eq:firsth}
h_1(x;\epsilon) &= -\log(1+\epsilon-x) + (1+\epsilon-x)^{0.3}\\
h_2(x;\epsilon) &= \frac{1}{\sqrt{1+\epsilon-x}}\\
h_3(x;\epsilon) &= \frac{\cos^2(6\pi x)}{\sqrt{1+\epsilon-x}}\\
h_4(x;\epsilon) &= \frac{\sqrt{\cosh(x+2/3)-\cos(1)}}{\sqrt{1+\epsilon-x}}.
\label{eq:lasth}
\end{align}
We again take $\epsilon\in \{1/30, 1/300, 1/3000\}$.
The results, in Fig. \ref{fig:many_np_r_examples}, indicate that the quadratic transformation gives the best results for $h_2$, $h_3$, and $h_4$, while the exponential transformation is better for $h_1$.
\begin{figure}
\centering
\includegraphics[width=\linewidth]{fig_many_NONperiodic_R_examples.pdf}
\caption{Integration of the functions \eqref{eq:firsth} - \eqref{eq:lasth} the exponential, quadratic and Jacobi elliptic sine maps along with the decomposition method and ordinary Gauss-Legendre quadrature as discussed in Sec. \ref{sec:apr}. The quadratic transformation is best in the last three rows, while the exponential map is best in the first row. }
\label{fig:many_np_r_examples}
\end{figure}
\section{Application: evaluation of single-layer potentials in Stokes flow} \label{sec:SLP}
As an application of the preceding methods, we will evaluate some nearly singular surface integrals that arise in the study of viscous fluid flow. Specifically, we will consider the Stokes single-layer potential defined by
\begin{equation}
\mathcal{S}(\bm x) = \int_D \left(\frac{\bm f(\bm y)}{\lvert \bm x-\bm y \rvert } + \frac{\bm x - \bm y}{\lvert \bm x-\bm y \rvert ^3}((\bm x - \bm y)\cdot \bm f(\bm y))\right) \, dS_{\bm y}
\label{eq:slp}
\end{equation}
where $D$ is the surface of the slender fiber depicted in the right panel of Fig. \ref{fig:fiber}. The integral has a physical meaning: it is the velocity field that results when point forces of strength $\bm f$ are distributed over the surface $D$.
The integral is challenging when the \emph{observation point} $\bm x$ is near but not on the surface $D$. Although the surface has an irregular shape, the integral has a known exact solution when the density $\bm f$ is equal to the outward pointing surface normal: in this case the resulting velocity is zero everywhere, and the surface traction $\bm f$ is due solely to hydrostatic pressure. This arrangement allows us to test various quadrature strategies in a setting where the geometry is nontrivial but an exact solution (zero) is known. Note that we have omitted the usual factor of $1/(8\pi)$ in \eqref{eq:slp}.
The problem of computing nearly singular surface integrals arises in many situations when a linear PDE is solved via integral equations. For the Stokes PDE, a number of methods have been developed that take into account the particular form of the Stokes fundamental solutions. Prominent recent examples include singularity subtraction and local expansion \cite{tlupova2013nearly,tlupova2019regularized}, quadrature by expansion \cite{af2016fast}, and singularity swapping \cite{af2021accurate}. Here we attempt to address this challenge without using special knowledge about the particular form of the integrands.
\begin{figure}
\centering
\includegraphics[width=\linewidth]{fig_bagel.pdf}
\caption{To demonstrate our integration strategies we consider the problem of evaluating a Stokes single-layer potential where the target point is near but not on the surface of integration. The surface is a fiber whose centerline follows a closed path on the surface of a torus (left). We then let the target point range over a square domain that is punctured in three places by the fiber (right). The integrals become numerically challenging when the target point approaches the surface. To organize the integration we subdivide the fiber surface into sixteen panels, as depicted in green and orange at right. }
\label{fig:fiber}
\end{figure}
To describe the fiber surface, we begin by parameterizing the surface of a torus whose centerline has unit radius and whose circular cross sections have radius $0.4$:
\begin{equation}
\bm v(\theta,\phi) = (1+0.4\cos(\phi))\begin{pmatrix}
\cos\theta\\\sin\theta\\0\end{pmatrix}+0.4\sin\phi\begin{pmatrix}
0\\0\\1\end{pmatrix}.
\end{equation}
Then we construct a closed curve on this surface by letting $\theta$ and $\phi$ be periodic functions of a (non-arclength) parameter $s$:
\begin{equation}
\bm w(s) = \bm v\Big(s,2 \exp(\cos(s + 1)) \cos(2 s) + 2 s\Big).
\end{equation}
The path $\bm w$ appears on the surface of the torus in the left panel of Fig. \ref{fig:fiber}.
Let $\bm T(s)$ denote the unit tangent vector for the curve, $\bm T(s) = \bm w'(s) / \lvert\bm w'(s)\rvert$. To define the fiber surface we need vectors $\bm N(s)$ and $\bm B(s)$ so that $\{\bm T, \bm N,\bm B\}$ is an orthonormal frame. To avoid the derivatives involved in the standard Frenet definition, we plotted $\bm T(s)$ as a path on the unit sphere and noticed that it always remains far from the line through $\pm\bm p$, where $\bm p = \langle 10,3,6\rangle$. Thus we can complete the frame by putting
\begin{equation}
\bm N(s) = \frac{\bm T(s) \times \bm p}{ \lvert\bm T(s) \times \bm p\rvert},\qquad
\bm B(s) = \bm T(s) \times \bm N(s).
\end{equation}
Finally we define the fiber surface $D$ via
\begin{equation}
\bm y(s,t) = \bm w(s) + \epsilon\cos(t)\bm N(s) + \epsilon\sin(t)\bm B(s)
\end{equation}
where $s$ and $t$ both range from $0$ to $2\pi$, and the radius $\epsilon$ is a constant (we take $\epsilon = 0.05$).
The surface normal vector for the fiber is
\begin{equation}
\bm f(s,t) = \bm \nu(s,t) = \cos(t)\bm N(s) + \sin(t)\bm B(s)
\end{equation}
while the Jacobian or surface integration weight is
\begin{equation}
J(s,t) = \left\lvert\frac{\partial \bm y}{\partial s} \times \frac{\partial \bm y}{\partial t}\right\rvert = 0.1 (\lvert\bm w'(s)\rvert-0.1 \kappa_1 \cos(t) -0.1\kappa_2\sin(t))
\end{equation}
where $\kappa_{1,2}$ are defined by $\kappa_1 = \bm w'(s)\cdot \bm N(s)$ and $\kappa_2 = \bm w'(s)\cdot \bm B(s)$.
\subsection{Reference solution}
We first evaluate the integral using ordinary Gauss-Legendre and trapezoidal quadrature.
We subdivide the fiber surface into 16 panels using a heuristic that considers the panel lengths and their maximum curvatures; the panel endpoints are at $s\in\{ 0$, $0.58$, $ 1.21$, $ 1.83$, $2.35$, $2.76$, $3.19$, $3.86$, $4.26$, $4.65$, $5.04$, $5.24$, $5.41$, $5.57$, $5.75$, $ 6.05$, $2\pi\}$.
On each panel, we use the outer product of a Gauss-Legendre grid in $s$ and an equally spaced grid in $t$, using the same quadrature rule for every target point. The target points range over a square domain $\{(x,y,0):0<x<1, -5/8<y<3/8\}$ that is punctured in three locations by the fiber surface. The exact velocity is zero, and we depict the norm of the computed velocity at each target point as a contour plot in the first column of Fig. \ref{fig:slpcontours}. Predictably, we see that this combination of Gauss-Legendre and trapezoidal rules is effective only when the target is far from the fiber surface.
\subsection{Finding the singularities}
In order to improve on the reference solution using the methods described in this paper, we need to find the singularities of the inner (periodic) integrand in the complex $t$-plane for fixed $s$, as well as the singularities of the outer integrand in the complex $s$-plane. Although the outer integral in $s$ is also $2\pi$-periodic, we chose to integrate separately on each of the panels, leading to sixteen aperiodic subproblems.
We can find the singularities for the periodic, inner problem on paper as follows.
We begin by writing \eqref{eq:slp} as a double integral. With $\bm r(s,t) = \bm x - \bm y(s,t)$ we have
\begin{equation}
\mathcal{S}(\bm x) = \int_0^{2\pi}\int_0^{2\pi} \left(\frac{\bm f(\bm y(s,t))}{\lvert\bm r(s,t)\rvert} + \frac{\bm r(s,t)}{\lvert\bm r(s,t)\rvert^3}(\bm r(s,t)\cdot \bm f(\bm y(s,t)))\right) J(s,t)\, dt\,ds.
\label{eq:slpit}
\end{equation}
For fixed $s$, the vector $\bm r(s,t)$ traces out a circle at constant speed as $t$ varies. The denominators can therefore be written as trigonometric functions of $t$ and we can solve analytically for the complex value of $t$ where they vanish.
To do this we write
\begin{align*}
\lvert\bm r(s,t)\rvert^2 &= \big\lvert\bm x - \bm w(s) - \epsilon\cos(t)\bm N(s) - \epsilon \sin(t)\bm B(s)\big\rvert^2\\
&=\lvert\bm x - \bm w(s)\rvert^2 + \epsilon^2 -2\epsilon\Big(\cos (t) (\bm x-\bm w(s))\cdot\bm N(s) + \sin(t)(\bm x-\bm w(s))\cdot\bm B(s)\Big)\\
&= \lvert\bm x - \bm w(s)\rvert^2 + \epsilon^2 -{2\epsilon}{\sqrt{((\bm x-\bm w(s))\cdot\bm N(s))^2+((\bm x-\bm w(s))\cdot\bm B(s))^2}}\cos(t-\xi)
\end{align*}
where the angle $\xi$ is defined by
\begin{align*}
\cos(\xi) &= \frac{(\bm x-\bm w(s))\cdot\bm N(s)}{\sqrt{((\bm x-\bm w(s))\cdot\bm N(s))^2+((\bm x-\bm w(s))\cdot\bm B(s))^2}},\;\\
\sin(\xi) &= \frac{(\bm x-\bm w(s))\cdot\bm B(s)}{\sqrt{((\bm x-\bm w(s))\cdot\bm N(s))^2+((\bm x-\bm w(s))\cdot\bm B(s))^2}}.
\end{align*}
Therefore, $\bm r(s,t)$ vanishes when
\[
\cos(t-\xi) = \frac{\lvert\bm x - \bm w(s)\rvert^2 + \epsilon^2}{2\epsilon\sqrt{((\bm x-\bm w(s))\cdot\bm N(s))^2+((\bm x-\bm w(s))\cdot\bm B(s))^2}}
\]
and we find that the required value of $t$ is
\begin{equation}
t^* = \xi + \sqrt{-1}\arccosh\left(\frac{\lvert\bm x - \bm w(s)\rvert^2 + \epsilon^2}{2\epsilon\sqrt{((\bm x-\bm w(s))\cdot\bm N(s))^2+((\bm x-\bm w(s))\cdot\bm B(s))^2}}\right).
\label{eq:innertstar}
\end{equation}
This allows us to choose a quadrature for the inner integral using knowledge of the integrand's complex singularity, as in Section 2.
We will need some numerical rootfinding to locate the singularities of the outer integral. We note that the inner integral will diverge if the imaginary part of $t^*$ in \eqref{eq:innertstar} vanishes. Therefore, the complex singularities of the outer integrand are the solutions of the equation
\begin{align}
\Big(\lvert\bm x - \bm w(s)\rvert^2 + 0.1^2\Big)^2 = 0.04\Big(((\bm x-\bm w(s))\cdot\bm N(s))^2 +((\bm x-\bm w(s))\cdot\bm B(s))^2\Big). \label{eq:outersing}
\end{align}
To obtain these roots we find the eigenvalues of a $51\times 51$ Chebyshev colleague matrix. For each computed root we then find the corresponding Bernstein ellipse parameter $\rho$. If all computed $\rho$-values are greater than 2, we revert to ordinary Gauss-Legendre quadrature; otherwise we use the root with the smallest $\rho$-value to accelerate the quadrature in the outer integral.
\subsection{Improvement via decomposition and conformal mapping}
Overall, the surface quadrature procedure that we have outlined is laborious: for each target point and within each panel, we find a customized 1D quadrature rule for the aperiodic outer integral. Then, for each of the resulting outer quadrature nodes, we find a customized 1D quadrature rule for the periodic inner integral. An example of the resulting surface quadrature rule appears in Fig. \ref{fig:dotsontube}. Although our goal is to demonstrate the accuracy of the underlying quadratures rather than to address the fast generation of rules for many target points, we make one adjustment for the sake of efficiency: for each panel, we revert to the reference Gauss-Legendre / trapezoid scheme for all targets whose distance to the panel surface is greater than $7\epsilon$. This allows us to use the same quadrature rule simultaneously for many distant targets.
\begin{figure}
\centering
\includegraphics[width=\linewidth]{junefig_qrsurf.jpg}
\caption{A surface quadrature rule with $24\cdot24$ nodes (blue dots) on one panel, customized for a target point (pink dot) that is near but not on the surface. We generated this rule with the hyperbolic sine transformation in the centerline direction, $s$, and the iterated sine map in the circumferential direction, $t$ (calculated separately for each value of $s$). }
\label{fig:dotsontube}
\end{figure}
The second column of Fig. \ref{fig:slpcontours} shows the result of combining the periodic decomposition method for the inner integral with the aperiodic decomposition method in the outer integral. This leads to more correct digits than the reference solution when the target point is near the surface.
However, the third column, the result of using the iterated sine map for the inner integral and the hyperbolic sine map for the outer integral, is dramatically better. We chose the sinh and ISM methods because of their simplicity and superior performance on the tests of section \ref{sec:examples}. However, other combinations of the conformal mapping strategies, including the Jafari-Varzaneh map and the various options based on Jacobi elliptic functions, yield similar (but not better) results for this application. Note that we did not use the results of \ref{sec:apr} because the numerical rootfinding procedure always gave a nonzero imaginary part.
Our Fig. \ref{fig:slpcontours} should be compared with Figures 2, 3, and 8 of \cite{af2021accurate}, a related work where the authors make use of additional information about the nature of the singularities, not merely their location, and accordingly develop a more powerful but less general method for accelerating the quadrature of the nearly singular integrals. We note that they developed methods for finding and using multiple root pairs, but found in practice that this did not give better results than the methods based on a single root pair.
\begin{figure}
\centering
\includegraphics[width=\linewidth]{fig_june_key.pdf}
\caption{Contour plot of errors in the single-layer problem as the target point ranges over a square domain, punctured in three places by the slender fiber. The error bar reports $\log_{10}(\lvert\bm u(\bm x)\rvert)$ where $\bm u(\bm x)$ is the single-layer velocity induced at the target point $\bm x$ by a surface traction which equals the unit vector ($\bm u=\bm0$ if the integration is done correctly). The columns of the figure show different quadrature strategies, while the rows of the table show differing densities of quadrature nodes; for example, $n=32$ means that we use $32\cdot32$ nodes on each panel (there are always 16 panels).
}
\label{fig:slpcontours}
\end{figure}
\section{Conclusion}
\label{sec:conclusion}
We surveyed a number of possible strategies for accelerating the quadrature of nearly single integrals given knowledge of the location of the nearby singularity. We found that the splitting methods are less effective than the conformal maps. Among the many possible conformal maps, we can make suggestions for general use: we recommend the iterated sine map for periodic problems, the sinh map for aperiodic problems with complex singularity, and the quadratic map for aperiodic problems with real singularity.
A natural priority for future work is to ask what additional improvements are possible when we have information about the nature of the nearest singularity in addition to its location. The integrand $f(x) = ((x-0.3)^2+\epsilon^2)^p$ has the same domain of analyticity for $p=2.5$ and $p=-2.5$, and accordingly the Gauss-Legendre quadrature errors will \emph{eventually} decay at the same rate in either case. However, it is also true that Gauss-Legendre integration reaches machine precision much more quickly for the version with $p>0$. This statement that can be made more precise with the aid of the theorems of Sec. \ref{sec:thms}, but it remains to make use of this information to optimize the choice of conformal map.
A second issue arises when many integrals over the same surface, but with varying singularities, need to be computed. The strategy described here would use customized quadratures for each of the singularities, resulting in a possibly expensive interpolation operation. It would be valuable to partition the complex plane into regions such that any singularity within a region can be integrated to prescribed accuracy with a precomputed reference set of quadrature nodes and weights.
\subsection{Acknowledgements}
We thank Nick Trefethen for helpful feedback on a draft. We acknowledge support from the National Science Foundation (DMS-1907796) and from Macalester College.
\bibliographystyle{spmpsci}
| {
"timestamp": "2022-10-19T02:18:30",
"yymm": "2210",
"arxiv_id": "2210.09954",
"language": "en",
"url": "https://arxiv.org/abs/2210.09954",
"abstract": "Gauss-Legendre quadrature and the trapezoidal rule are powerful tools for numerical integration of analytic functions. For nearly singular problems, however, these standard methods become unacceptably slow. We discuss and generalize some existing methods for improving on these schemes when the location of the nearby singularity is known. We conclude with an application to some nearly singular surface integrals of viscous flow.",
"subjects": "Numerical Analysis (math.NA)",
"title": "Decomposition and conformal mapping techniques for the quadrature of nearly singular integrals",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9899864293768269,
"lm_q2_score": 0.7154239957834733,
"lm_q1q2_score": 0.7082600470761827
} |
https://arxiv.org/abs/1611.04708 | Combinatorial Identities for Generalized Stirling Numbers Expanding $f$-Factorial Functions and the $f$-Harmonic Numbers | We introduce a class of $f(t)$-factorials, or $f(t)$-Pochhammer symbols, that includes many, if not most, well-known factorial and multiple factorial function variants as special cases. We consider the combinatorial properties of the corresponding generalized classes of Stirling numbers of the first kind which arise as the coefficients of the symbolic polynomial expansions of these $f$-factorial functions. The combinatorial properties of these more general parameterized Stirling number triangles we prove within the article include analogs to known expansions of the ordinary Stirling numbers by $p$-order harmonic number sequences through the definition of a corresponding class of $p$-order $f$-harmonic numbers. | \section{Introduction}
\subsection{Generalized $f$-factorial functions}
\subsubsection*{Definitions}
For any function, $f: \mathbb{N} \rightarrow \mathbb{C}$, and fixed
non-zero indeterminates $x, t \in \mathbb{C}$, we introduce and define the
\emph{generalized $f(t)$-factorial function}, or alternately the
\emph{$f(t)$-Pochhammer symbol}, denoted by $(x)_{f(t),n}$,
as the following products:
\begin{align}
\label{eqn_xft_gen_PHSymbol_def}
(x)_{f(t),n} & =
\prod_{k=1}^{n-1} \left(x + \frac{f(k)}{t^k}\right).
\end{align}
Within this article,
we are interested in the combinatorial properties of the coefficients of the
powers of $x$ in the last product expansions which we consider to be
generalized forms of the \emph{Stirling numbers of the first kind} in
this setting.
Section \ref{subSection_Intro_GenSNumsDefs} defines generalized
Stirling numbers of both the first and second kinds and motivates the
definitions of auxiliary triangles by special classes of formal power series
generating function transformations and their corresponding
negative-order variants considered in the references
\citep{GFTRANS2016,GFTRANSHZETA2016}.
\subsubsection*{Special cases}
Key to the formulation of applications and interpreting the generalized
results in this article is the observation that the
definition of \eqref{eqn_xft_gen_PHSymbol_def} provides an effective
generalization of almost all other related factorial function variants
considered in the references when $t \equiv 1$.
The special cases of $f(n) := \alpha n+\beta$ for some
integer-valued $\alpha \geq 1$ and $0 \leq \beta < \alpha$ lead to the
motivations for studying these more general factorial functions in
\citep{GFTRANSHZETA2016}, and form the expansions of multiple
$\alpha$-factorial functions, $n!_{(\alpha)}$, studied in the
triangular coefficient expansions defined by
\citep{MULTIFACT-CFRACS,MULTIFACTJIS}.
The \emph{factorial powers}, or
\emph{generalized factorials of $t$ of order $n$ and increment $h$},
denoted by $t^{(n, h)}$, or the \emph{Pochhammer k-symbol} denoted by
$(x)_{n,h} \equiv p_n(h, t) = t(t+h)(t+2h)\cdots(t+(n-1)h)$, studied in
\citep{q-DIFFSq-FACTS,MULTIFACT-CFRACS,CK} form particular special cases,
as do the the forms of the generalized \emph{Roman factorials} and
\emph{Knuth factorials} for $n \geq 1$ defined in \citep{LOEBBINOM}, and the
\emph{$q$-shifted factorial functions} considered in
\citep{q-SHIFTEDFACTS,q-DIFFSq-FACTS}.
When $(f(n), t) \equiv (q^{n+1}, 1)$ these products are related to the
expansions of the finite cases of the \emph{$q$-Pochhammer symbol} products,
$(a; q)_n = (1-a)(1-aq)\cdots(1-aq^{n-1})$, and the corresponding
definitions of the generalized Stirling number triangles defined in
\eqref{eqn_genS1ft_rec_def} of the next subsection are precisely the
\emph{Gaussian polynomials}, or \emph{$q$-binomial coefficients},
studied in relation to the $q$-series expansions and
$q$-hypergeometric functions in \citep[\S 17]{NISTHB}.
\subsubsection*{New results proved in the article}
The results proved within this article, for example, provide new
expansions of these special factorial functions in terms of their
corresponding \emph{$p$-order $f$-harmonic number sequences},
\[
F_n^{(p)}(t) := \sum_{k \leq n} \frac{t^k}{f(k)^p},
\]
which generalize known expansions of Stirling numbers by the ordinary
\emph{$p$-order harmonic numbers},
$H_n^{(p)} \equiv \sum_{1 \leq k \leq n} k^{-r}$, in
\citep{STIRESUMS,MULTIFACTJIS,GFTRANS2016,GFTRANSHZETA2016}.
Still other combinatorial sums and properties satisfied by the
symbolic polynomial expansions of these special case factorial functions
follow as corollaries of the new results we prove in the next sections.
The next subsection precisely expands the generalized factorial expansions
of \eqref{eqn_xft_gen_PHSymbol_def} through the generalized class of
Stirling numbers of the first kind defined recursively by
\eqref{eqn_genS1ft_rec_def} below.
\subsection{Definitions of generalized $f$-factorial Stirling numbers}
\label{subSection_Intro_GenSNumsDefs}
We first employ the next recurrence relation to define the
generalized triangle of Stirling numbers of the first kind,
which we denote by $\gkpSI{n}{k}_{f(t)} := [x^{k-1}] (x)_{f(t),n}$,
or just by $\gkpSI{n}{k}_f$ when the
context is clear, for natural numbers $n, k \geq 0$
\citep[\cf \S 3.1]{MULTIFACTJIS} \footnote{
The bracket symbol $\Iverson{\mathtt{cond}}$ denotes
\emph{Iverson's convention} which evaluates to exactly one of the
values in $\{0, 1\}$ and where $\Iverson{\mathtt{cond}} = 1$
if and only if the condition $\mathtt{cond}$ is true.
}.
\begin{align}
\label{eqn_genS1ft_rec_def}
\gkpSI{n}{k}_{f(t)} & = f(n-1) \cdot t^{1-n} \gkpSI{n-1}{k}_{f(t)} +
\gkpSI{n-1}{k-1}_{f(t)} + \Iverson{n = k = 0}
\end{align}
We also define the corresponding generalized forms of the
\emph{Stirling numbers of the second kind}, denoted by
$\gkpSII{n}{k}_{f(t)}$, so that we can consider inversion relations and
combinatorial analogs to known identities for the ordinary triangles by the sum
\begin{align*}
\gkpSII{n}{k}_{f(t)} & =
\sum_{j=0}^{k} \binom{k}{j} \frac{(-1)^{k-j} f(j)^n}{t^{jn} \cdot j!},
\end{align*}
from which we can prove the following form of a particularly useful
generating function transformation motivated in the references
when $f(n)$ has a Taylor series expansion in integral powers of $n$ about zero
\citep[\cf \S 3.3]{MULTIFACTJIS} \citep[\cf \S 7.4]{GKP}
\citep{SQSERIESMDS,GFTRANSHZETA2016}:
\begin{align}
\label{eqn_S2ft_GFTrans_geom_series_exp}
\sum_{0 \leq j \leq n} \frac{f(j)^k}{t^{jk}} z^j & =
\sum_{0 \leq j \leq k} \gkpSII{k}{j}_{f(t)} z^j \times
D_z^{(j)}\left[\frac{1-z^{n+1}}{1-z}\right].
\end{align}
The negative-order cases of the infinite series transformation in
\eqref{eqn_S2ft_GFTrans_geom_series_exp} are motivated in
\citep{GFTRANSHZETA2016} where we define modified forms of the
Stirling numbers of the second kind by
\begin{align*}
\gkpSII{k}{j}_{f^{\ast}} & =
\sum_{1 \leq m \leq j}
\binom{j}{m} \frac{(-1)^{j-m}}{j! \cdot f(m)^k},
\end{align*}
which then implies that the transformed ordinary and exponential
zeta-like power series enumerating
generalized polylogarithm functions and the
$f$-harmonic numbers, $F_n^{(p)}(t)$,
are expanded by the following two series variants \citep{GFTRANSHZETA2016}:
\begin{align*}
\sum_{n \geq 1} \frac{z^n}{f(n)^k} & = \sum_{j \geq 0}
\gkpSII{k}{j}_{f^{\ast}} \frac{z^j \cdot j!}{(1-z)^{j+1}} \\
\sum_{n \geq 1} \frac{F_n^{(r)}(1) z^n}{n!} & =
\sum_{j \geq 0}
\gkpSII{k}{j}_{f^{\ast}} \frac{z^j \cdot e^{z} (j+1+z)}{(j+1)}.
\end{align*}
We focus on the combinatorial relations and sums involving the
generalized positive-order Stirling numbers in the next few sections.
\section{Generating functions and expansions by $f$-harmonic numbers}
\subsection{Motivation from a technique of Euler}
We are motivated by Euler's original technique for solving the
\emph{Basel problem} of summing the series,
$\zeta(2) = \sum_n n^{-2}$, and later more generally all
even-indexed integer zeta constants, $\zeta(2k)$, in closed-form by
considering partial products of the sine function \citep[pp. 38-42]{GAMMA}.
In particular, we observe that we have both an infinite product and a
corresponding Taylor series expansion in $z$ for $\sin(z)$ given by
\begin{align*}
\sin(z) & = \sum_{n \geq 0} \frac{(-1)^n z^{2n+1}}{(2n+1)!} =
z \prod_{j \geq 1} \left(1 - \frac{z^2}{j^2 \pi^2}\right).
\end{align*}
Then if we combine the form of the coefficients of $z^3$ in the
partial product expansions at each finite $n \in \mathbb{Z}^{+}$ with the
known trigonometric series terms defined such that
$[z^3] \sin(z) = -\frac{1}{3!}$ given on each respective side of the last
equation, we see inductively that
\begin{align*}
H_n^{(2)} = -\pi^2 \cdot [z^2] \prod_{1 \leq j \leq n}
\left(1 - \frac{z^2}{j^2 \pi^2}\right)
\qquad\longrightarrow\qquad
\zeta(2) = \frac{\pi^2}{6}.
\end{align*}
In our case, we wish to similarly enumerate the $p$-order $f$-harmonic numbers,
$F_n^{(p)}(t)$, through the generalized product expansions defined in
\eqref{eqn_xft_gen_PHSymbol_def}.
\subsection{Generating the integer order $f$-harmonic numbers}
We first define a shorthand notation for another form of
generalized ``\emph{$f$-factorials}'' that we will need in
expanding the next products as follows:
\begin{equation*}
n!_f := \prod_{j=1}^n f(j) \qquad \text{ and } \qquad
n!_{f(t)} := \prod_{j=1}^{n} \frac{f(j)}{t^j} =
\frac{n!_f}{t^{n(n+1)/2}}.
\end{equation*}
If we let $\zeta_p \equiv \exp(2\pi\imath / p)$ denote the
\emph{primitive $p^{th}$ root of unity} for integers $p \geq 1$, and define the
coefficient generating function,
$\widetilde{f}_n(w) \equiv \widetilde{f}_n(t; w)$, by
\begin{align*}
\widetilde{f}_n(w) & := \sum_{k \geq 2} \gkpSI{n+1}{k}_{f(t)} w^k =
\left(\prod_{j=1}^{n} \left(w+f(j) t^{-j}\right) -
\gkpSI{n+1}{1}_{f(t)}\right) w,
\end{align*}
we can factor the partial products in \eqref{eqn_xft_gen_PHSymbol_def} to
generate the $p$-order $f$-harmonic numbers in the following forms:
\begin{align}
\label{eqn_fkp_partialsum_fCf2_exp_forms}
\sum_{k=1}^{n} \frac{t^{kp}}{f(k)^p} & =
\frac{t^{pn(n+1) / 2}}{\left(n!_{f}\right)^p}
[w^{2p}]\left((-1)^{p+1} \prod_{m=0}^{p-1} \sum_{k=0}^{n+1}
\FcfII{f(t)}{n+1}{k} \zeta_p^{m(k-1)} w^k\right) \\
\notag
& =
\frac{t^{pn(n+1) / 2}}{\left(n!_{f}\right)^p}
[w^{2p}]\left(\sum_{j=0}^{p-1}
\frac{(-1)^{j} w^{j}\ p}{p-j} \FcfII{f(t)}{n+1}{1}^j
\widetilde{f}_n(w)^{p-j}\right) \\
\label{eqn_fkp_partialsum_fCf2_exp_forms_v2}
\sum_{k=1}^{n} \frac{t^{k}}{f(k)^p} & =
\frac{t^{n(n+1) / 2}}{\left(n!_{f}\right)^p}
[w^{2p}]\left((-1)^{p+1} \prod_{m=0}^{p-1} \sum_{k=0}^{n+1}
\FcfII{f\left(t^{1 / p}\right)}{n+1}{k} \zeta_p^{m(k-1)} w^k\right).
\end{align}
\begin{example}[Special Cases]
For a fixed $f$ and any indeterminate $t \neq 0$,
let the shorthand notation $\bar{F}_n(k) := \FcfII{f(t)}{n+1}{k}$. Then the
following expansions illustrate several characteristic forms of these
prescribed partial sums for the first several
special cases of \eqref{eqn_fkp_partialsum_fCf2_exp_forms}
when $2 \leq p \leq 5$:
\begin{align}
\label{eqn_pth_partial_coeff_sums_p234}
\sum_{k=1}^{n} \frac{t^{2k}}{f(k)^2} & =
\frac{t^{n(n+1)}}{(n!_{f})^2}\left(\bar{F}_n(2)^2 - 2 \bar{F}_n(1) \bar{F}_n(3)
\right) \\
\notag
\sum_{k=1}^{n} \frac{t^{3k}}{f(k)^3} & =
\frac{t^{3n(n+1) / 2}}{(n!_{f})^3}\left(\bar{F}_n(2)^3 -
3 \bar{F}_n(1) \bar{F}_n(2) \bar{F}_n(3) + 3 \bar{F}_n(1)^2 \bar{F}_n(4)\right) \\
\notag
\sum_{k=1}^{n} \frac{t^{4k}}{f(k)^4} & =
\frac{t^{4n(n+1)}}{(n!_{f})^4}\bigl(\bar{F}_n(2)^4 -
4 \bar{F}_n(1) \bar{F}_n(2)^2 \bar{F}_n(3) + 2 \bar{F}_n(1)^2 \bar{F}_n(3)^2 +
4 \bar{F}_n(1)^2 \bar{F}_n(2) \bar{F}_n(4) \\
\notag
& \phantom{= \frac{t^{4n(n+1)}}{(n!_{f})^4}\bigl( \quad \ }
- 4 \bar{F}_n(1)^3 \bar{F}_n(5)\bigr) \\
\notag
\sum_{k=1}^{n} \frac{t^{5k}}{f(k)^5} & =
\frac{t^{5n(n+1) / 2}}{(n!_{f})^5}\bigl(\bar{F}_n(2)^5 -
5 \bar{F}_n(1) \bar{F}_n(2)^3 \bar{F}_n(3) + 5 \bar{F}_n(1)^2 \bar{F}_n(2) \bar{F}_n(3)^2 +
5 \bar{F}_n(1)^2 \bar{F}_n(2)^2 \bar{F}_n(4) \\
\notag
& \phantom{\frac{t^{5n(n+1) / 2}}{(n!_{f})^5}\bigl( \ \quad} -
5 \bar{F}_n(1)^3 \bar{F}_n(3) \bar{F}_n(4) -
5 \bar{F}_n(1)^3 \bar{F}_n(2) \bar{F}_n(5) + 5 \bar{F}_n(1)^4 \bar{F}_n(6)\bigr).
\end{align}
For each fixed integer $p > 1$, the particular partial sums defined by the
ordinary generating function, $\widetilde{f}_n(w)$, correspond to a function in
$n$ that is fixed with respect to the lower indices for the triangular
coefficients defined by \eqref{eqn_genS1ft_rec_def}.
Moreover, the resulting coefficient expansions enumerating the
$f$-harmonic numbers at each $p \geq 2$ are isobaric in the sense that the
sum of the indices over the lower index $k$ is $2p$ in each individual
term in these finite sums.
\end{example}
\subsection{Expansions of the generalized coefficients by
$f$-harmonic numbers}
The \emph{elementary symmetric polynomials} depending on the
function $f$ implicit to the product-based definitions of the generalized
Stirling numbers of the first kind expanded
through \eqref{eqn_xft_gen_PHSymbol_def}
provide new forms of the known $p$-order harmonic number, or
\emph{exponential Bell polynomial}, expansions of the ordinary
Stirling numbers of the first kind enumerated in the references
\citep{STIRESUMS,COMBIDENTS,ADVCOMB,UC}.
Thus, if we first define the weighted sums of the $f$-harmonic numbers,
denoted $w_f(n, m)$, recursively according to an identity for the
Bell polynomials, $\ell \cdot Y_{n,\ell}(x_1, x_2, \ldots)$,
for $x_k \equiv (-1)^k F_n^{(k)}(t^k) (k-1)!$ as \citep[\S 4.1.8]{UC}
\begin{align*}
w_f(n+1, m) & := \sum_{0 \leq k < m} (-1)^{k} F_n^{(k+1)}(t^{k+1})
(1-m)_k w_f(n+1,m-1-k) + \Iverson{m = 1},
\end{align*}
we can expand the generalized coefficient triangles through these
weighted sums as
\begin{align}
\label{eqn_FcfII_wfnm_genHNum_weighted_sum_exps_rdef}
\FcfII{f(t)}{n+1}{k} & = \frac{n!_{f}}{(k-1)!}\
w_f(n+1, k) \\
\notag
& =
\sum_{j=0}^{k-2} \FcfII{f(t)}{n+1}{k-1-j}
\frac{(-1)^{j} F_n^{(j+1)}(t^{j+1})}{(k-1)} +
n!_{f(t)} \cdot \Iverson{k = 1}.
\end{align}
This definition of the weighted $f$-harmonic sums for the generalized
triangles in \eqref{eqn_genS1ft_rec_def} implies the special case expansions
given in the next corollary.
\begin{cor}[Weighted $f$-Harmonic Sums for the Generalized Stirling Numbers]
The first few special case expansions of the coefficient identities in
\eqref{eqn_FcfII_wfnm_genHNum_weighted_sum_exps_rdef} are stated for
fixed $f$, $t \neq 0$, and integers $n \geq 0$ in the following forms:
\begin{align}
\label{eqn_fCf_GenFHHarmonic_exps}
\FcfII{f(t)}{n+1}{2} & = \frac{n!_{f}}{t^{n(n+1) / 2}}\ F_n^{(1)}(t) \\
\notag
\FcfII{f(t)}{n+1}{3} & = \frac{n!_{f}}{2\ t^{n(n+1) / 2}}\left(
F_n^{(1)}(t)^2 - F_n^{(2)}(t^2)\right) \\
\notag
\FcfII{f(t)}{n+1}{4} & = \frac{n!_{f}}{6\ t^{n(n+1) / 2}}\left(
F_n^{(1)}(t)^3 - 3 F_n^{(1)}(t) F_n^{(2)}(t^2) + 2 F_n^{(3)}(t^3)
\right) \\
\notag
\FcfII{f(t)}{n+1}{5} & = \frac{n!_{f}}{24\ t^{n(n+1) / 2}}\left(
F_n^{(1)}(t)^4 - 6 F_n^{(1)}(t)^2 F_n^{(2)}(t^2) + 3 F_n^{(2)}(t^2)^2 +
8 F_n^{(1)}(t) F_n^{(3)}(t^3) - 6 F_n^{(4)}(t^4)\right).
\end{align}
\end{cor}
\begin{proof}
These expansions are computed explicitly using the recursive formula in
\eqref{eqn_FcfII_wfnm_genHNum_weighted_sum_exps_rdef}
for the first few cases of the lower triangle index $2 \leq k \leq 5$.
\end{proof}
We will return to the expansions of these coefficients
in \eqref{eqn_FcfII_wfnm_genHNum_weighted_sum_exps_rdef} to formulate new
finite sum identities providing functional relations between the
$p$-order $f$-harmonic number sequences in the next section.
\subsection{Combinatorial sums and functional equations for the
$f$-harmonic numbers}
The next several properties give interesting expansions of
the $p$-order $f$-harmonic numbers recursively over the parameter $p$
that can then be employed
to remove, or at least significantly obfuscate, the
current direct cancellation problem with these forms phrased by the
examples in \eqref{eqn_pth_partial_coeff_sums_p234} and
in \eqref{eqn_fCf_GenFHHarmonic_exps}.
\begin{prop}
For any fixed $p \geq 1$ and $n \geq 0$, we have the following
coefficient product identities generating the
$p$-order $f$-harmonic numbers, $F_n^{(p)}(t)$:
\begin{align}
\label{eqn_Fnpt_pvar_rform_exps}
F_n^{(p+1)}(t) & = F_n^{(p)}(t) +
\frac{(-1)^{p} t^{n(n+1) / 2}}{t^{\frac{pn(n+1)}{2(p+1)}} n!_{f}}
\FcfII{f(t^{1 / (p+1)})}{n+1}{p+2} \\
\notag
& \phantom{= F_n^{(p)}(t)\ } +
\sum_{j=0}^{p-1}
\frac{p\ (-1)^{j+1} t^{n(n+1) / 2}}{t^{\frac{jn(n+1)}{2p}}
(n!_{f})^{p-j} (p-j)} \left(
\sum_{\substack{0 \leq i_1, \ldots, i_{p-j} \leq j \\
i_1 + \cdots + i_{p-j} = j}}
\FcfII{f(t^{1 / p})}{n+1}{i_1 + 2} \cdots
\FcfII{f(t^{1 / p})}{n+1}{i_{p-j} + 2}\right) \\
\notag
& \phantom{= F_n^{(p)}(t)\ } +
\sum_{j=0}^{p-1} \sum_{i=0}^{j}
\frac{(p+1) t^{n(n+1) / 2} (-1)^{j}}{
t^{\frac{jn(n+1)}{2(p+1)}} (n!_{f})^{p+1-j} (p+1-j)}
\FcfII{f(t^{1 / (p+1)})}{n+1}{i+2} \times \\
\notag
& \phantom{= F_n^{(p)}(t) + \sum\sum\ } \times
\left(
\sum_{\substack{0 \leq i_1, \ldots, i_{p-j} \leq j-i \\
i_1 + \cdots + i_{p-j} = j-i}}
\prod_{m=1}^{p-j} \FcfII{f(t^{1 / (p+1)})}{n+1}{i_m + 2}\right).
\end{align}
\end{prop}
\begin{proof}
To begin with, observe the following rephrasing of the partial sums
expansions from equations \eqref{eqn_fkp_partialsum_fCf2_exp_forms} and
\eqref{eqn_fkp_partialsum_fCf2_exp_forms_v2} as
\begin{align*}
F_n^{(p+1)}(t) & = \frac{t^{n(n+1) / 2}}{(n!_{f})^{p+1}} \sum_{j=0}^{p}
\frac{(p+1)\ (-1)^j}{(p+1-j)}
\FcfII{f\left(t^{1 / (p+1)}\right)}{n+1}{1}^j
[w^{2p+2-j}] \widetilde{f}_n(w)^{p+1-j} \\
& =
\frac{(p+1) (-1)^{p} t^{n(n+1) / 2}}{t^{\frac{pn(n+1)}{2(p+1)}} n!_{f}}
\FcfII{f(t^{1 / (p+1)})}{n+1}{p+2} \\
& \phantom{= \quad \ } +
\sum_{j=0}^{p-1}
\frac{(p+1) (-1)^{j} t^{n(n+1) / 2}}{t^{\frac{jn(n+1)}{2(p+1)}}
(n!_{f})^{p+1-j} (p+1-j)} [w^{j}] \left(
\frac{\widetilde{f}_n(w)}{w^2}\right)^{p+1-j}.
\end{align*}
The coefficients involved in the partial sum forms for each sequence of
$F_n^{(p)}(t)$ are implicitly tied to the form of
$t \mapsto t^{1 / p}$ in the triangle definition of \eqref{eqn_genS1ft_rec_def}.
Given this distinction, let the generating function $\widetilde{f}$ be
defined equivalently in the more careful definition as
$\widetilde{f}_n(w) :\equiv \widetilde{f}_n(t;\ w)$.
The powers of the generating function $\widetilde{f}_n(w)$ from the
previous equations satisfy the coefficient term expansions
according to the next equation \citep[\cf \Section 7.5]{GKP}.
\begin{align*}
[w^{2p-j}] \widetilde{f}_n(w)^{p-j} & :=
[w^{2p-j}] \widetilde{f}_n(t;\ w)^{p-j} =
[w^{j}] \left(\frac{\widetilde{f}_n(t;\ w)}{w^2}\right)^{p-j} \\
& \phantom{:} =
\sum_{\substack{0 \leq i_1, \ldots, i_{p-j} \leq j \\
i_1 + \cdots i_{p-j} = j}}
\FcfII{f(t)}{n+1}{i_1 + 2} \cdots \FcfII{f(t)}{n+1}{i_{p-j} + 2}
\end{align*}
Then by taking the
difference of the harmonic sequence terms over successive indices
$p \geq 1$ and at a fixed index of $n \geq 1$, the stated recurrences for
these $p$-order sequences result.
\end{proof}
The generating function series over $n$ in the next proposition
is related to the forms of the \emph{Euler sums}
considered in \citep{STIRESUMS} and to the
context of the generalized zeta function transformations considered in
\citep{GFTRANSHZETA2016} briefly noted in the introduction.
We suggest the infinite sums over these generalized
identities for $n \geq 1$ as a
topic for future research exploration in the concluding remarks of
Section \ref{Section_Concl}.
\begin{prop}[Functional Equations for the $f$-Harmonic Numbers]
\label{prop_FHNum_fnal_eqn_and_coeff_exps}
For any integers $n \geq 0$ and $p \geq 2$, we have the following
functional relations between the $p$-order and $(p-1)$-order
$f$-harmonic numbers over $n$ and $p$:
\begin{align*}
F_{n+1}^{(p)}(t^p) & = F_n^{(p)}(t^p) +
\sum_{1 \leq j < p} \gkpSI{n+2}{p+1-j}_{f(t)}
\frac{(-1)^{p+1-j} t^{j(n+1)}}{f(n+1)^{j} (n+1)!_{f(t)}} +
\gkpSI{n+1}{p}_{f(t)} \frac{(-1)^{p+1}}{(n+1)!_{f(t)}} \\
& = F_n^{(p)}(t^p) + \frac{t^{(p-1)(n+1)}}{f(n+1)^{p-1}} +
\frac{(-1)^{p-1}}{(n+1)!_{f(t)}} \left(
\gkpSI{n+1}{p}_{f(t)} + \gkpSI{n+1}{p-1}_{f(t)} \right) \\
& \phantom{= F_n^{(p)}(t^p) \ } +
\gkpSI{n+2}{p}_{f(t)} \frac{(-1)^{p} t^{n+1}}{f(n+1) (n+1)!_{f(t)}} \\
& \phantom{= F_n^{(p)}(t^p) \ } +
\sum_{j=0}^{p-3} \gkpSI{n+2}{j+2}_{f(t)}
\frac{(-1)^{j+1} \left(f(n+1)t^{-(n+1)} - 1\right) t^{(p-1-j)(n+1)}}{
f(n+1)^{p-1-j} (n+1)!_{f(t)}}.
\end{align*}
\end{prop}
\begin{proof}
First, notice that \eqref{eqn_FcfII_wfnm_genHNum_weighted_sum_exps_rdef}
implies that we have the following weighted harmonic number sums for the
$p$-order $f$-harmonic numbers:
\begin{align*}
F_n^{(p)}(t^p) & = \sum_{1 \leq j < p} \gkpSI{n+1}{p+1-j}_{f(t)}
\frac{(-1)^{p+1-j} F_n^{(j)}(t^j)}{n!_{f(t)}} +
\gkpSI{n+1}{p+1}_{f(t)} \frac{p (-1)^{p+1}}{n!_{f(t)}}.
\end{align*}
Next, we use \eqref{eqn_genS1ft_rec_def} twice to expand the
differences of the left-hand-side of the previous equation as
\begin{align*}
\frac{t^{p(n+1)}}{f(n+1)^p} & = F_{n+1}^{(p)}(t^p) - F_n^{(p)}(t^p) \\
& =
\sum_{1 \leq j < p} \gkpSI{n+2}{p+1-j}_{f(t)}
\frac{(-1)^{p+1-j} F_{n+1}^{(j)}(t^j)}{(n+1)!_{f(t)}} -
\sum_{1 \leq j < p} \gkpSI{n+1}{p+1-j}_{f(t)}
\frac{(-1)^{p+1-j} F_{n}^{(j)}(t^j)}{n!_{f(t)}} \\
& \phantom{= \sum \quad \ } +
\gkpSI{n+2}{p+1}_{f(t)} \frac{p (-1)^{p+1}}{(n+1)!_{f(t)}} -
\frac{f(n+1)}{t^{n+1}} \gkpSI{n+1}{p+1}_{f(t)}
\frac{p (-1)^{p+1}}{(n+1)!_{f(t)}} \\
& =
\sum_{1 \leq j < p} \gkpSI{n+2}{p+1-j}_{f(t)}
\frac{(-1)^{p+1-j} t^{j(n+1)}}{f(n+1)^{j} (n+1)!_{f(t)}} -
\sum_{1 \leq j < p} \gkpSI{n+1}{p-j}_{f(t)}
\frac{(-1)^{p-j} F_n^{(j)}(t^j)}{(n+1)!_{f(t)}} \\
& \phantom{= \sum \quad \ } +
\gkpSI{n+1}{p}_{f(t)} \frac{p (-1)^{p+1}}{(n+1)!_{f(t)}} \\
& =
\sum_{1 \leq j < p} \gkpSI{n+2}{p+1-j}_{f(t)}
\frac{(-1)^{p+1-j} t^{j(n+1)}}{f(n+1)^{j} (n+1)!_{f(t)}} -
\gkpSI{n+1}{p}_{f(t)} \frac{(p-1) (-1)^{p+1}}{(n+1)!_{f(t)}} \\
& \phantom{= \sum \quad \ } +
\gkpSI{n+1}{p}_{f(t)} \frac{p (-1)^{p+1}}{(n+1)!_{f(t)}}.
\end{align*}
The second identity is verified similarly by combining the coefficient
terms as in the last equations and adding the right-hand-side
differences of the $(p-1)$-order $f$-harmonic numbers to the first identity.
\end{proof}
One immediate corollary that must by its importance be expanded in turn
explicitly in the next example
provides new expansions of the $p$-order harmonic numbers in terms of the
ordinary triangle of Stirling numbers of the first kind corresponding to the
case where $(f(n), t) \equiv (n, 1)$ in the previous proposition.
Similar expansions of identities related to the generalized
generating function transformations in \citep{GFTRANSHZETA2016} result for the
special cases of the proposition where
$(f(n), t) \equiv (\alpha n+\beta, t)$ for some application-dependent
prescribed $\alpha, \beta \in \mathbb{C}$
defined such that $-\frac{\beta}{\alpha} \notin \mathbb{Z}$.
Another special case worth noting and independently expanding
provides analogous relations between the
$q$-binomial coefficients implicit to the forms of the
\emph{$q$-binomial theorem} expanding the $q$-Pochhammer symbols,
$(a; q)_n$, for each $n \geq 0$ \citep[\cf \S 17.2]{NISTHB}.
\begin{example}[Stirling Numbers and Euler Sums]
\label{example_SpCase_S1HNum_FnalEqn_Ident}
For all integers $p \geq 3$ and fixed $n \in \mathbb{Z}^{+}$,
we have the following identity relating the successive differences of the
$p$-order harmonic numbers and the Stirling numbers of the first kind:
\begin{align}
\label{eqn_S1HNum_fnaleqn_exp_v1}
\frac{1}{n^p} & = \frac{1}{n^{p-1}} + \frac{(-1)^{p-1}}{n!}\left(
\gkpSI{n}{p} + \gkpSI{n}{p-1}\right) +
\gkpSI{n+1}{p} \frac{(-1)^p}{n \cdot n!} \\
\notag
& \phantom{=\frac{1}{n^{p-1}}\ } +
\sum_{j=0}^{p-3} \gkpSI{n+1}{j+2} \frac{(-1)^{j+1} (n-1)}{
n^{p-1-j} \cdot n!}.
\end{align}
The relation in \eqref{eqn_S1HNum_fnaleqn_exp_v1} certainly implies new
finite sum identities between the $p$-order harmonic numbers and the
Stirling numbers of the first kind, though the generating functions and
limiting cases of these sums provide more information on infinite sums
considered in several of the references.
With this in mind, we define the \emph{Nielsen generalized polylogarithm},
$S_{t,k}(z)$, by the infinite generating series over the $t$-power-scaled
Stirling numbers as \citep[\cf \S 5]{STIRESUMS}
\begin{align*}
S_{t,k}(z) & := \sum_{n \geq 1} \gkpSI{n}{k} \frac{z^n}{n^t \cdot n!}.
\end{align*}
We see immediately that \eqref{eqn_S1HNum_fnaleqn_exp_v1} provides strictly
enumerative relations between the polylogarithm function
generating functions, $\operatorname{Li}_p(z) / (1-z)$, for the
$p$-order harmonic numbers and the Nielsen polylogarithms.
Perhaps more interestingly, we also find new identities between the
Riemann zeta functions,
$\zeta(p)$ and $\zeta(p-1)$, and the special classes of \emph{Euler sums}
given by $S_{t,k}(1)$ for $t \in [2, p-1]$ and $k \in [2, p]$
defined as in the reference \citep[\S 5]{STIRESUMS}.
\end{example}
\section{Coefficient identities and
generalized forms of the Stirling convolution polynomials}
\subsection{Generalized Coefficient Identities and Relations}
There are several immediate for small-indexed columns of the triangle
defined by \eqref{eqn_genS1ft_rec_def} and
that can both be given immediately and that follow from an
inductive argument. The
next identities in \eqref{eqn_gen_FcfII_ftnk_gen_k_idents} are given
for general lower column index $k \geq 1$ by
\begin{align}
\label{eqn_gen_FcfII_ftnk_gen_k_idents}
\FcfII{f(t)}{n}{k} & =
[w^{k-1}]\left(\prod_{j=1}^{n-1} (w + f(j)\ t^{-j})\right)
\Iverson{n \geq 1} + \Iverson{n = k = 0} \\
\notag
& =
\sum_{\substack{
0 < i_1 < \cdots < i_{n-k} < n}}
f(i_1) \cdots f(i_{n-k}) \cdot t^{-(i_1 + \cdots + i_{n-k})},
\end{align}
which follows immediately by considering the first products of the form
$\prod_i (z + x_i)$ in the context of elementary symmetric polynomials for
these specific $x_i$.
\begin{prop}[Horizontal and Vertical Column Recurrences]
The generalized Stirling numbers of the first kind over the
first several special case columns for the
shifted upper index of $n+1$ in the expansions of
\eqref{eqn_genS1ft_rec_def} are given by the
next recurrence relations for all $n \geq 0$ and any $k \geq 2$.
\begin{align}
\label{eqn_FcfII_ftnk_spcase_cols_and_rdefs}
\FcfII{f(t)}{n+1}{1} & = \frac{n!_{f}}{t^{n(n+1) / 2}} \\
\notag
\FcfII{f(t)}{n+1}{k} & = \frac{n!_{f}}{t^{n(n+1) / 2}} \sum_{j=1}^{n}
\FcfII{f(t)}{j}{k-1} \frac{t^{j(j+1) / 2}}{j!_{f}},\
\text{ if $k \geq 2$}
\end{align}
\end{prop}
\begin{proof}
We begin by observing that by \eqref{eqn_genS1ft_rec_def}
when $k \equiv 1$, we have that
\begin{align*}
\gkpSI{n+1}{1}_{f(t)} & = \frac{f(n)}{t^n} \gkpSI{n}{1}_{f(t)} +
\gkpSI{n}{0}_{f(t)} \\
& =
\frac{f(n)}{t^n} \gkpSI{n}{1}_{f(t)} + \Iverson{n = 0},
\end{align*}
which implies the first claim by induction since
$\gkpSI{1}{1}_{f(t)} = 1$ and $\gkpSI{0}{1}_{f(t)} = 1$.
To prove the column-wise recurrence relation given in
\eqref{eqn_FcfII_ftnk_spcase_cols_and_rdefs},
we notice again by induction that for any functions
$g(n)$ and $b(n) \neq 0$, the sequence, $f_k(n)$, defined recursively by
\begin{align*}
f_k(n) & =
\begin{cases}
b(n) \cdot f_k(n-1) + g(n-1) & \text{ if $n \geq 1$ } \\
1 & \text{ if $n = 0$, }
\end{cases}
\end{align*}
has a closed-form solution given by
\begin{align*}
f_k(n) & = \left(\prod_{j=1}^{n-1} b(j)\right) \times \sum_{0 \leq j < n}
\frac{g(j)}{\prod_{i=1}^{j} b(j)}.
\end{align*}
Thus by \eqref{eqn_genS1ft_rec_def} the second claim is true.
\end{proof}
\subsection{Generalized forms of the Stirling convolution polynomials}
\begin{definition}[Stirling Polynomial Analogs]
\label{def_CvlPolyAnalogs}
For $x,n,x-n \geq 1$, we suggest the next two variants of the
generalized \emph{Stirling convolution polynomials}, denoted by
$\sigma_{f(t),n}(x)$ and $\widetilde{\sigma}_{f(t),n}(x)$, respectively,
as the right-hand-side coefficient definitions in the following equations:
\begin{align}
\label{eqn_fnx_poly_coeff_def}
\sigma_{f(t),n}(x) := \FcfII{f(t)}{x}{x-n} \frac{(x-n-1)!}{x!_{f}}
& \quad \iff \quad
\FcfII{f(t)}{n+1}{k} = \frac{(n+1)!_{f}}{(k-1)!}\ \sigma_{f(t),n+1-k}(n+1) \\
\notag
\widetilde{\sigma}_{f(t),n}(x) := \FcfII{f(t)}{x}{x-n} \frac{(x-n-1)!}{x!}
& \quad \iff \quad
\FcfII{f(t)}{n+1}{k} = \frac{(n+1)!}{(k-1)!}\ \widetilde{\sigma}_{f(t),n+1-k}(n+1).
\end{align}
\end{definition}
\begin{prop}[Recurrence Relations]
For integers $x,n,x-n \geq 1$, the analogs to the Stirling convolution
polynomial sequences defined by \eqref{eqn_fnx_poly_coeff_def} each
satisfy a respective recurrence relation stated in the next equations.
\begin{align}
\notag
f(x+1) \sigma_{f(t),n}(x+1) & = (x-n) \sigma_{f(t),n}(x) +
f(x)\ t^{-x} \cdot \sigma_{f(t),n-1}(x) + \Iverson{n = 0} \\
\label{eqn_fnx_snx_genCvlPolySeqs_recs}
(x+1) \widetilde{\sigma}_{f(t),n}(x+1) & = (x-n) \widetilde{\sigma}_{f(t),n}(x) +
f(x)\ t^{-x} \cdot \widetilde{\sigma}_{f(t),n-1}(x) + \Iverson{n = 0}
\end{align}
\end{prop}
\begin{proof}
We give a proof of the second identity since the first recurrence follows
almost immediately from this result.
Let $x,n,x-n \geq 1$ and consider the expansion of the left-hand-side of
\eqref{eqn_fnx_snx_genCvlPolySeqs_recs} according to
Definition \ref{def_CvlPolyAnalogs} as follows:
\begin{align*}
(x + 1) \widetilde{\sigma}_{f(t),n}(x + 1) & =
\gkpSI{x+1}{x+1-n}_{f(t)} \frac{(x-n)!}{x!} \\
& =
\left(f(x) t^{-x} \gkpSI{x}{x+1-n}_{f(t)} + \gkpSI{x}{x-n}_{f(t)}
\right) (x-n) \cdot \frac{(x-n-1)!}{x!} \\
& =
(x-n) \widetilde{\sigma}_{f(t),n}(x) +
f(x) t^{-x} \cdot \widetilde{\sigma}_{f(t),n-1}(x).
\end{align*}
For any non-negative integer $x$, when $n = 0$, we see that
$\gkpSI{x+1}{x+1}_{f(t)} \equiv 1$, which implies the result.
\end{proof}
\begin{remark}[A Comparison of Polynomial Generating Functions]
The generating functions for the Stirling convolution polynomials,
$\sigma_n(x)$, and the $\alpha$-factorial polynomials, $\sigma_n^{(\alpha)}(x)$,
from \citep{MULTIFACTJIS} each have the comparatively simple
special case closed-form generating functions given by
\begin{align}
\label{eqn_SPoly_def_and_GF}
x \sigma_n(x) & = \gkpSI{x}{x-n} \frac{(x-n-1)!}{(x-1)!} =
[z^n] \left(\frac{z e^{z}}{e^{z}-1}\right)^{x} &&
\text{ for } (f(n), t) \equiv (n, 1) \\
\notag
x \sigma_n^{(\alpha)}(x) & = \gkpSI{x}{x-n}_{\alpha} \frac{(x-n-1)!}{(x-1)!} =
[z^n] e^{(1-\alpha)z}
\left(\frac{\alpha z e^{\alpha z}}{e^{\alpha z}-1}\right)^{x} &&
\text{ for } (f(n), t) \equiv (\alpha n + 1 - \alpha, 1) \\
\notag
x \sigma_n^{(\alpha; \beta)}(x) & = \gkpSI{x}{x-n}_{(\alpha; \beta)}
\frac{(x-n-1)!}{(x-1)!} =
[z^n] e^{\beta z}
\left(\frac{\alpha z e^{\alpha z}}{e^{\alpha z}-1}\right)^{x} &&
\text{ for } (f(n), t) \equiv (\alpha n + \beta, 1).
\end{align}
The Stirling polynomial sequence in \eqref{eqn_SPoly_def_and_GF} is a
special case of a more general class of \emph{convolution polynomial}
sequences defined by Knuth in his article \citep{CVLPOLYS}.
These polynomial sequences are defined by a general sequence of
coefficients, $s_n^{\ast}$ with $s_0^{\ast} = 1$,
such that the corresponding polynomials, $s_n(x)$, are
enumerated by the power series over the original sequence as
\begin{equation*}
\sum_{n=0}^{\infty} s_n(x) z^n := S(z)^{x} \equiv
\left(1 + \sum_{n=1}^{\infty} s_n^{\ast} z^n\right)^{x}.
\end{equation*}
Polynomial sequences of this form satisfy a number of interesting
properties, and in particular, the next identity provides a
generating function for a variant of the
original convolution polynomial sequence over $n$ when
$t \in \mathbb{C}$ is fixed.
\begin{equation}
\label{eqn_CvlPoly_Stz_GF_rdef}
\mathcal{S}_t(z) := S\left(z \mathcal{S}_t(z)^t\right)
\quad \implies \quad
\frac{x s_n(x+tn)}{(x+tn)} = [z^n] \mathcal{S}_t(z)^{x}
\end{equation}
This result is also useful in expanding many identities
for the $t := 1$ case as given for the Stirling polynomial case in
\citep[\Section 6.2]{GKP} \citep{CVLPOLYS}.
A related generalized class of polynomial sequences is considered in
Roman's book defining the form of
\emph{Sheffer polynomial} sequences. \nocite{UC}
The polynomial sequences of this particular type, say with sequence
terms given by $s_n(x)$, satisfy the form in the following
generating function identity where $A(z)$ and $B(z)$ are
prescribed power series satisfying the initial conditions from the
reference \citep[\cf \Section 2.3]{UC}:
\begin{equation*}
\sum_{n=0}^{\infty} s_n(x) \frac{z^n}{n!} := A(z) e^{x B(z)}.
\end{equation*}
For example, the form of the generalized, or higher-order Bernoulli
polynomials (numbers) is a parameterized sequence whose generating
function yields the form of many other special case sequences,
including the Stirling polynomial case defined in equation
\eqref{eqn_SPoly_def_and_GF} \citep[\cf \Section 4.2.2]{UC}
\citep[\cf \Section 5]{MULTIFACTJIS}.
\end{remark}
\subsubsection*{An experimental procedure towards evaluating the
generalized polynomials}
We expect that the generalized convolution polynomial analogs defined in
\eqref{eqn_fnx_poly_coeff_def} above form a
sequence of finite-degree polynomials in $x$, for example, as in the
Stirling polynomial case when we have that
\begin{align*}
\gkpSI{x}{x-n} & = \sum_{k \geq 0} \gkpEII{n}{k} \binom{x+k}{2n},
\end{align*}
where $\gkpEII{n}{k}$ denotes the special triangle of
\emph{second-order Eulerian numbers} for $n, k \geq 0$ and where the
binomial coefficient terms in the previous equations each have a finite-degree
polynomial expansion in $x$ \citep[\S 6.2]{GKP}.
The previous identity also allows us to extend the
Stirling numbers of the first kind to \emph{arbitrary} real,
or complex-valued inputs.
Given the
relatively simple and elegant forms of the generating functions that
enumerate the polynomial sequences of the special case forms in
\eqref{eqn_SPoly_def_and_GF}, it seems natural to
attempt to extend these relations to the generalized polynomial
sequence forms defined by \eqref{eqn_fnx_poly_coeff_def}.
However, in this more general context we appear to have a
stronger dependence of the form and ordinary generating functions of these
polynomial sequences on the underlying function $f$.
Specifically, for the form of the first sequence in
\eqref{eqn_fnx_poly_coeff_def},
we suppose that the function $f(n)$ is arbitrary.
Based on the first several cases of these polynomials,
it appears that the generating function for the sequence can be
expanded as
\begin{align}
\label{eqn_fnx_poly_GF_ident_v1}
& f_n(x) := [z^n] F(z)^{x} \quad \text{ where } \quad
F(z) := \sum_{n=0}^{\infty} g_n(x) z^n \\
\notag
& \phantom{f_n(x) :} \implies
g_n(x) = \frac{\sum_{j=0}^{n-1} f(x)^n \numpoly_n(j;\ x) x^{n-1-j}
(1+x)^j f(x+1)^j}{n!\ t^{nx}\ \sum_{j=0}^{2n-1}
\denompoly_n(j;\ x) x^{2n-1-j}
(1+x)^j f(x+1)^j}\ \Iverson{n \geq 1} + \Iverson{n = 0}
\end{align}
where the forms $\numpoly_n(j;\ x)$ and $\denompoly_n(j;\ x)$ denote
polynomial sequences of finite non--negative integral degree indexed over the
natural numbers $n, j \geq 0$.
Similarly it has been verified for the first $16$ of each $n$ and $k$
that the following equation holds where the
terms $g_n(x)$ involved in the series for $F(z)$ are defined through the
form of the last equation.
\begin{equation*}
s_n(k) := f_{n-k}(n) \implies s_n(k) = [z^n] z^k F(z)^n =
\sum_{j=1}^{n-k} \binom{n}{j} [z^{n-k}] (F(z) - 1)^j + \Iverson{n = k}
\end{equation*}
Note that the coefficients defined through these implicit power series forms
must also satisfy an implicit relation to the particular values of the
polynomial parameter $x$ as formed through the last equations, which is
much different in construction than in the cases of the
special polynomial sequence generating functions remarked on above.
Other different expansions may result for special cases of the
function $f(n)$ and explicit values of the parameter $t$.
\section{Conclusions and future research}
\label{Section_Concl}
\subsection{Summary}
We have defined a generalized class of factorial product functions,
$(x)_{f(t),n}$, that generalizes the forms of many special and symbolic
factorial functions considered in the references.
The coefficient-wise symbolic polynomial expansions of these
$f$-factorial function variants define generalized triangles of
Stirling numbers of the first kind which share many analogs to the
combinatorial properties satisfied by the ordinary combinatorial
triangle cases.
Surprisingly, many inversion relations and other finite sum properties
relating the ordinary Stirling number triangles are not apparent by
inspection of these corresponding sums in the most general cases.
A study of ordinary Stirling-number-like sums, inversion relations, and
generating function transformations is not contained in the article.
We pose formulating these analogs in the most general coefficient cases as a
topic for future combinatorial work with the generalized Stirling number
triangles defined in
Section \ref{subSection_Intro_GenSNumsDefs}.
\subsection{Topics suggested for future research}
Another new avenue to explore with these sums and the generalized $f$-zeta
series transformations motivated in \citep{GFTRANS2016,GFTRANSHZETA2016} is
to consider finding new identities and expressions for the
Euler-like sums suggested by the generalized identity in
Proposition \ref{prop_FHNum_fnal_eqn_and_coeff_exps} and by the
special case expansions for the Stirling numbers of the first kind given in
Example \ref{example_SpCase_S1HNum_FnalEqn_Ident}.
In particular,
if we define a class of so-termed ``\emph{$f$-zeta}'' functions,
$\zeta_f(s) := \sum_{n \geq 1} f(n)^{-s}$, we seek analogs to these infinite
Euler sum variants expanded through $\zeta_f(s)$ just as the Euler sums are
expressed through sums and products of the \emph{Riemann zeta function},
$\zeta(s)$, in the ordinary cases from \citep{STIRESUMS}.
For example, it is well known that for real-valued $r > 1$
\begin{align*}
\sum_{n \geq 1} \frac{H_n^{(r)}}{n^r} & =
\frac{1}{2}\left(\zeta(r)^2 + \zeta(2r)\right),
\end{align*}
and moreover, summation by parts shows us that for any real
$r > 1$ and any $t \in \mathbb{C}^{\ast}$ such that we have a
convergent limiting zeta function series we have that
\begin{align*}
\sum_{n \geq 1} \frac{F_n^{(r)}(t^r) t^{rn}}{f(n)^{r}} & =
\lim_{n\longrightarrow\infty}\ \left\{
\left(F_n^{(r)}(t^r)\right)^2 - \sum_{0 \leq j < n}
\frac{F_j^{(r)}(t^r) t^{r(j+1)}}{f(j+1)^r}
\right\} \\
& = \lim_{n\longrightarrow\infty}\ \left\{
\left(F_n^{(r)}(t^r)\right)^2 - \sum_{0 \leq j < n}
\frac{F_{j+1}^{(r)}(t^r) t^{r(j+1)}}{f(j+1)^r} +
\sum_{0 \leq j < n} \frac{t^{2r(j+1)}}{f(j+1)^{2r}}
\right\},
\end{align*}
which similarly implies that
\begin{align*}
\sum_{n \geq 1} \frac{F_n^{(r)}(1)}{f(n)^r} &
\quad \overset{: \rightsquigarrow}{\longrightarrow} \quad
\frac{1}{2}\left(\zeta_f(r)^2 + \zeta_f(2r)\right).
\end{align*}
Additionally, we seek other analogs to known identities for the infinite
Euler-like-sum variants over the weighted $f$-harmonic number sums of the form
\begin{align*}
H_f\left(\varpi_1, \ldots, \varpi_k; s, t, z\right) & := \sum_{n \geq 1}
\frac{F_n^{\left(\varpi_1\right)}\left(t^{\varpi_1}\right) \cdots
F_n^{\left(\varpi_k\right)}\left(t^{\varpi_k}\right) z^{sn}}{f(n)^{s}},
\end{align*}
when $t = \pm 1$, or more generally for any fixed $t \in \mathbb{C}^{\ast}$,
and where the right-hand-side series in the previous equation
converges, say for $|z| \leq 1$.
\renewcommand{\refname}{References}
| {
"timestamp": "2017-03-31T02:01:39",
"yymm": "1611",
"arxiv_id": "1611.04708",
"language": "en",
"url": "https://arxiv.org/abs/1611.04708",
"abstract": "We introduce a class of $f(t)$-factorials, or $f(t)$-Pochhammer symbols, that includes many, if not most, well-known factorial and multiple factorial function variants as special cases. We consider the combinatorial properties of the corresponding generalized classes of Stirling numbers of the first kind which arise as the coefficients of the symbolic polynomial expansions of these $f$-factorial functions. The combinatorial properties of these more general parameterized Stirling number triangles we prove within the article include analogs to known expansions of the ordinary Stirling numbers by $p$-order harmonic number sequences through the definition of a corresponding class of $p$-order $f$-harmonic numbers.",
"subjects": "Combinatorics (math.CO)",
"title": "Combinatorial Identities for Generalized Stirling Numbers Expanding $f$-Factorial Functions and the $f$-Harmonic Numbers",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9899864273087514,
"lm_q2_score": 0.7154239957834733,
"lm_q1q2_score": 0.7082600455966319
} |
https://arxiv.org/abs/1101.1251 | Pseudo-Taylor expansions and the Carathéodory-Fejér problem | We give a new solvability criterion for the boundary Carathéodory-Fejér problem: given a point $x \in \mathbb{R}$ and, a finite set of target values $a^0,a^1,...,a^n \in \mathbb{R}$, to construct a function $f$ in the Pick class such that the limit of $f^{(k)}(z)/k!$ as $z \to x$ nontangentially in the upper half plane is $a^k$ for $k= 0,1,...,n$. The criterion is in terms of positivity of an associated Hankel matrix. The proof is based on a reduction method due to Julia and Nevanlinna. | \section{Introduction} \label {intro}
A theme of classical analysis is to ascertain whether a given finite sequence of complex numbers comprises the initial Taylor coefficients of an analytic function of a specified class on a domain $U$ about some point $x$ of $U$. In the case that $U$ is the upper halfplane
\[
\Pi \df \{ z\in\C: \im z > 0 \}
\]
and the specified class is the Pick class $\Pick$ we obtain the much-studied {\em Carath\'{e}odory-Fej\'{e}r problem} \cite{CF, bgr}. Here
$\Pick$ is defined to be the set of analytic functions $f$ on $\Pi$ such that $\im f \geq 0$ on $\Pi$. We can equally well ask the same question for a point $x \in \partial U$, the boundary of $U$, but then, since an analytic function on $U$ will not
in general have a Taylor expansion about every point in $\partial U$, there is a question as to how we should interpret ``Taylor coefficients". The simplest answer is just to restrict attention to functions which are analytic at the interpolation point $x$. We then arrive at the {\em boundary Carath\'{e}odory-Fej\'{e}r problem}:\\
\noindent {\bf Problem $\partial CF\Pick(\R)$} \quad {\em
Given a point $x\in \R$ and $a^0,a^1,\dots,a^n \in \R$, find a function $f$ in the Pick class such that $f$ is analytic at $x$ and
\beq \label{interpcond_anal}
\frac{f^{(k)}(x)}{k!} = a^k, \qquad k=0,1,\dots,n.
\eeq
}
In \cite{ALY10} we gave a new criterion for Problem $\partial CF\Pick(\R)$ to have a solution $f$. Roughly speaking, such an $f$ exists if and only if a certain Hankel matrix constructed from the $a^k$ is either positive definite or {\em southeast-minimally positive} (to be defined in Section \ref{weak} below). However, the requirement that the solution $f$ be analytic at $x$ is unnecessarily strong. There is a natural weaker notion of solution,
in which the role of Taylor coefficients is played by a generalization appropriate to points on the boundary of $\Pi$. We call these {\em pseudo-Taylor coefficients}\footnote{Although this notion has been in use for 90 years or more, we cannot find an agreed name for it, and so are herewith introducing one.}.
In this paper we show that the same existence criterion applies to solvability in this weaker sense. It follows that the problem has a solution in the original (analytic at $x$) sense if and only if it has a solution in the weaker sense.
As in \cite{ALY10}, our main tool is a technique of reduction of functions in the Pick class that is originally due to G. Julia \cite{Ju20} and was greatly strengthened by Nevanlinna \cite{Nev1922}. One virtue of this approach is that it is elementary; it does not depend on the theories of operators or Hilbert function spaces. The proof is by induction on $n$ in combination with Julia reduction and an identity for Hankel matrices. A point of the paper is that the methods of \cite{ALY10} remain valid for the more delicate problems associated with weak solutions. We shall use some results from that paper, and for convenience we often refer to \cite{ALY10} for proofs even of statements that are well established.
The present paper could be regarded as a correction of Nevanlinna's own treatment: we rectify an oversight that led him to an incorrect statement about solvability. A discussion of Nevanlinna's assertion and a counterexample are given in \cite[Section 10]{ALY10}.
The problem we study has an extensive history, which we discuss in \cite[Sections 1 and 10]{ALY10}. We mention in particular a very recent paper \cite{Bol10}.
The paper is organised as follows. In Section \ref{taylor}
we define weak solutions and the notion of a pseudo-Taylor expansion of $f \in \Pick$ about $x \in \R$.
In Section \ref{reduction} we describe Julia's reduction procedure and its inverse, and give important properties of these procedures.
In Section \ref{relax} we show that positivity of a Hankel matrix is necessary and sufficient for weak solvability of a relaxation of Problem $\partial CF\Pick(\R)$, in which the last of the interpolation conditions (\ref{interpcond_anal}) is relaxed (equality is replaced by an inequality).
In Section \ref{weak} we prove our main theorem: Problem $\partial CF\Pick$ has a solution in the weak sense if and only if its associated Hankel matrix is either positive definite or southeast-minimally positive.
We shall write the imaginary unit as $\ii$, in Roman font, to have $i$ available for use as an index. We denote the open unit disc by $\D$.
\section{Pseudo-Taylor expansions} \label{taylor}
Recall that for any domain $U$ and any $x\in\partial U$, a subset $S$ of $U$ {\em approaches $x$ nontangentially} if $x$ is in the closure of $S$ and the quotient $|z-x|/\dist(z, \partial U)$ is bounded for $z\in S$, and that $z\to x$ {\em nontangentially in $U$} if $z\to x$ and $z$ lies in some set $S$ that approaches $x$ nontangentially.
We shall use the notation $z\nt x$ to mean that $z \to x$ nontangentially in a given domain $U$.
For a function
$h$ analytic on $U$ and a positive integer $n$, we write
\[
h(z) = \ont ((z-x)^{n})
\]
to mean that $\frac{h(z)}{(z-x)^{n}} \to 0$ as $z \nt x$. A function $f$ analytic on $U$ will be said to have a {\em pseudo-Taylor expansion of order $n$ about $x\in\partial U$} if there exist $c^0, c^1, \dots,c^n \in \C$ such that
\beq \label{psn}
f(z) = c^0 + c^1 (z-x) + \dots + c^n (z-x)^{n} + \ont ((z-x)^{n}).
\eeq
We call $c^j$ the {\em $j$th pseudo-Taylor coefficient} of $f$ at $x$. In fact pseudo-Taylor coefficients are not well defined in complete generality (for example, if $U$ has a cusp at $x$), but it is easy to see that if there is a line segment in $U$ that approaches $x$ nontangentially then $c^j$ is uniquely determined. In particular, pseudo-Taylor expansions of a function in $\Pick$ about a point $x\in\R$ are unique when they exist.
Pseudo-Taylor expansions are of course a form of asymptotic expansion, but are sufficiently special to deserve a separate name.
Note the special case $n=1$: $f$ has a pseudo-Taylor expansion of order $1$ if and only if $f$ has a nontangential limit $a^0$ and an angular derivative $a^1$ at $x\in\partial U$, and then the expansion of $f$ is $a^0 + a^1(z-x) + \ont(z-x)$. See \cite{car54, S2} for the notion of angular derivative. Pseudo-Taylor coefficients can thus be regarded as generalizations of angular derivatives.
We define a function $f \in \Pick$ to be a {\em weak solution} of Problem $\partial CF\Pick(\R)$ if $f$ has a pseudo-Taylor expansion of order $n$ at $x$ and the $j$th pseudo-Taylor coefficient of $f$ at $x$ is $a^j$ for $j=0,1,\dots,n$. Thus $f$ is a weak solution if and only if
\beq \label{psTay_int}
f(z) = a^0 + a^1 (z-x) + \dots + a^n (z-x)^{n} + R_n(z)
\eeq
where
\beq \label{psTay2_int}
\frac{R_n(z)}{(z-x)^{n}} \to 0
\eeq
as $z \to x$ nontangentially in $\Pi$.
This is essentially the notion of solution used by R. Nevanlinna \cite{Nev1922}; we believe he was the first mathematician to study such problems\footnote{Actually Nevanlinna took the interpolation node $x$ to be $\infty$.}, and subsequent authors (e.g. \cite{BD,BolKh08,Geo05}) have used equivalent notions.
Pseudo-Taylor expansions behave very differently from Taylor expansions, as the following examples show.
\begin{example}\label{ex1}
\rm The function $f(z)= z/(1-z\log z)$ is in $\Pick$ and has the pseudo-Taylor expansion
\[
f(z) = z + \ont(z)
\]
but no pseudo-Taylor expansion of order $2$ about $0$.
\end{example}
\begin{example} \label{ex2} \rm
Let $\nu$ be a positive integer, $\nu \ge 4$, and let
$$
f_{\nu}(z)= - \sum_{k=1}^{\infty} \frac{1}{k^{\nu} z + k^{\nu-1}}, \;\; z \in \Pi.
$$
Then $f_{\nu}\in\Pick$ has the pseudo-Taylor expansion
\beq\label{expandfnu}
f_{\nu}(z)= -\zeta(\nu-1) +\zeta(\nu -2)z - \dots + (-1)^{\nu} \zeta(2)z^{\nu -3} + \ont(z^{\nu-3})
\eeq
of order $\nu-3$ about $0$, but has no pseudo-Taylor expansion of order $\nu-2$.
We justify this assertion in the Appendix.
\end{example}
\begin{example}\label{ex3} \rm
The function
\[
f(z) = -\frac{1}{\mathrm e}\sum_{k=1}^\infty \frac{1}{k!(z+\frac{1}{k})}
\]
is in $\Pick$ and has a pseudo-Taylor expansion of infinite order about $0$, to wit
\[
f(z) = -1+ 2z -5z^2 + 15z^3-52z^4+ \dots =\sum_{n=0}^\infty (-1)^{n+1} A_n z^n
\]
where $A_0=1$ and, for $n\geq 0$,
\[
A_{n+1} = 2A_n + \sum_{r=1}^n {n \choose r} A_{n-r}.
\]
That is, $f$ has
pseudo-Taylor expansions of all orders about $0$, but $f$ is clearly not analytic at $0$.
\end{example}
There are two other natural ways that a function $f$ could be regarded as a solution of a boundary interpolation problem without necessarily being analytic at the interpolation node. Let us say that $f\in\Pick$ is a {\em nontangential solution} of Problem $\partial CF\Pick$
if
\[
\lim_{z\nt x} \frac{f^{(k)}(z)}{k!} = a^k \qquad \mbox { for } k=0,1,\dots,n,
\]
and is a {\em radial solution} of Problem $\partial CF\Pick$ if
\[
\lim_{y\to 0+} \frac{f^{(k)}(x+\ii y)}{k!} = a^k \qquad \mbox { for } k=0,1,\dots,n.
\]
Fortunately it transpires that the notions of weak, nontangential and radial solution all coincide.
The following theorem is widely known; see for example \cite[VI-1]{S2} and \cite[Corollary 7.9]{BD} for the corresponding statement for functions analytic on the open disc $\D$.
\begin{theorem}\label{Taylor} Let $f$ be a function
analytic on $\Pi$, let $x \in \R$, let $n$ be a non-negative integer and let $a^j \in \C$ for $j=1,2, \dots, n$. Then the following statements are equivalent:
\begin{enumerate}
\item[\rm (i)] $f$ has the pseudo-Taylor expansion
\beq \label{pseudo-Taylor}
f(z) = a^0 + a^1 (z-x) + \dots + a^n (z-x)^{n} + \ont ((z-x)^n),
\eeq
of order $n$ about $x$;
\item[\rm (ii)] the derivatives $f^{(k)}$, $k=0,1,\dots,n$, have nontangential limits at $x$ and
\beq \label{ang_derv}
\lim_{z \nt x}\frac{f^{(k)}(z)}{k!} = a^k, \qquad k=0,1,\dots,n; \nn
\eeq
\item[\rm (iii)] the derivatives $f^{(k)}$, $k=0,1,\dots,n$, have radial limits at $x$ and
\beq \label{rad_derv}
\lim_{y\to 0+}\frac{f^{(k)}(x+\ii y)}{k!} = a^k, \qquad k=0,1,\dots,n. \nn
\eeq
\end{enumerate}
\end{theorem}
\begin{proof}
(ii) $\Rightarrow $(iii) is trivial. We prove
(iii) $\Rightarrow $(i).
The statement is true when $n=0$: this is precisely Lindel\"of's Principle \cite[Theorem 8.7.1]{Kr}. Let us assume it is true for $n-1$, where $n \ge 1$, and deduce that it holds for $n$.
Suppose that (iii) holds, and let $g = f'$. Then
\[
\lim_{y\to 0+}\frac{g^{(k)}(x+\ii y)}{k!}=\lim_{y\to 0+}\frac{f^{(k+1)}(x+\ii y)}{k!} = (k+1) a^{k+1}
\]
for $k=0,1,\dots,n-1$. By the inductive hypothesis applied to $g$,
for $z \in \Pi$,
\beq \label{n-1-pseudo-Taylor}
f'(z) =g(z)=
a^1 + 2 a^{2} (z-x) + \dots + n a^n (z-x)^{n-1} + \ont ((z-x)^{n-1}) \nn
\eeq
and hence
\beq \label{n-pseudo-Taylor} \nn
f'(z) - \left(
a^1 + 2 a^{2} (z-x) + \dots + n a^n (z-x)^{n-1} \right) = \beta(z)(z-x)^{n-1},
\eeq
for some function $\beta$ such that $\beta(z) \to 0$ as $z \nt x$.
We denote by $[x,z]$ the straight line segment joining $x$ and $z$ in $\Pi$.
Now
\begin{eqnarray}\label{0_derv}
&~&\frac{f(z) - a^0 - a^1 (z-x) - \dots - a^n (z-x)^{n} }{(z-x)^{n}} \nn \\
&=& \frac{1}{(z-x)^n} \; \int_{[x,z]} f'(\zeta) - a^1 - 2 a^{2} (\zeta-x)- \dots - a^n n (\zeta-x)^{n-1} ~d \zeta \nn \\
&=& \frac{1}{(z-x)} \; \int_{[x,z]} \beta(\zeta) \left(\frac{\zeta-x}{z-x}\right)^{n-1}~d \zeta.
\end{eqnarray}
The right hand side of (\ref{0_derv}) tends to $0$ as $z$ tends nontangentially to $x$.
Hence
\[
\lim_{z \nt x} \frac{f(z) -( a^0 + a^1 (z-x) + \dots + a^n (z-x)^{n} )}{(z-x)^{n}}=0.
\]
The statement follows by induction.
(i) $\Rightarrow$ (ii)~ Let $K >0$ and consider the nontangential approach region at $x$
\[
S_K = \{ z \in \Pi: |z -x| \le K \;\dist (z, \R) \}.
\]
For $z$ in $S_K$ let $\gamma_z$ denote the circle with center $z$ and radius $\tfrac{1}{2}\dist (z, \R)$. It is clear that
$\gamma_z$ lies in $\Pi$. Note that, for each $\zeta \in \gamma_z$,
we have $|\zeta -z|=\tfrac{1}{2} \dist (z, \R)$, and so
$$ \tfrac{1}{2}\;\dist (z, \R) \le \dist (\zeta, \R) \le \tfrac{3}{2}\;\dist (z, \R).$$
Thus, for each $\zeta \in \gamma_z$,
\begin{eqnarray} \label{gamma_z_to _x} \nn
|\zeta -x| & \le & |\zeta -z|+|z -x| = \tfrac{1}{2} \;\dist (z, \R) + |z -x| \le \tfrac{3}{2}|z -x|
\end{eqnarray}
and so
\begin{eqnarray} \label{on_gamma_z}
|\zeta -x| & \le & |\zeta -z|+|z -x| \le \tfrac{1}{2} \;\dist (z, \R) + K \;\dist (z, \R) \\
& \le &(2K +1) \; \dist (\zeta, \R) \nn.
\end{eqnarray}
Thus $\gamma_z$ lies in $S_{2K+1}$.
By equation (\ref{pseudo-Taylor}), for any $\zeta \in U$,
\beq \label{pseudo-Taylor-gamma} \nn
f(\zeta) = a^0 + a^1 (\zeta-x) + \dots + a^{n-1} (\zeta-x)^{n-1} + a^n (\zeta-x)^{n} + \beta(\zeta)(\zeta-x)^{n},
\eeq
where $\beta(\zeta) \to 0$ as $\zeta \nt x$.
For $0 < \varepsilon <1$, define the number $\alpha (\varepsilon)$ by
$$
\alpha (\varepsilon) = \sup \left\{ \left| \beta(\zeta)
\right|: \zeta \in S_{2K+1}, |\zeta -x| < \varepsilon \right\}. $$
For each $\zeta \in \gamma_z$ we have the estimate
\beq \label{beta}
|\beta(\zeta)| \le \alpha( |\zeta -x| ) \le \alpha( \tfrac{3}{2}|z -x| ).
\eeq
By Cauchy's formula for derivatives,
\begin{eqnarray*}
f^{(n)}(z) &=&\frac{n!}{2 \pi \ii} \int_{\gamma_z} ~\frac{f(\zeta)}{(\zeta -z)^{n+1}} ~d \zeta\\
&=&\frac{n!}{2 \pi \ii} \int_{\gamma_z} ~\frac{
a^0 + a^1 (\zeta -x) + \dots + a^n (\zeta -x)^{n} + \beta(\zeta)(z-x)^{n}}{(\zeta -z)^{n+1}} ~d \zeta\\
&=&\frac{n!}{2 \pi \ii} \int_{\gamma_z} ~\frac{
a^0}{(\zeta -z)^{n+1}} ~d \zeta + \frac{n!}{2 \pi \ii} \int_{\gamma_z}\frac{a^1 (\zeta -x)}{(\zeta -z)^{n+1}} ~d \zeta + \dots \\
& ~ & ~~~ + \frac{n!}{2 \pi \ii} \int_{\gamma_z}\frac{a^n (\zeta -x)^{n}}{(\zeta -z)^{n+1}} ~d \zeta + \frac{n!}{2 \pi \ii} \int_{\gamma_z}\frac{\beta(\zeta)(\zeta-x)^{n}}{(\zeta -z)^{n+1}} ~d \zeta \\
&=& n! a^n + \frac{n!}{2 \pi \ii} \int_{\gamma_z}\frac{\beta(\zeta)(\zeta-x)^{n}}{(\zeta -z)^{n+1}} ~d \zeta.\\
\end{eqnarray*}
By (\ref{on_gamma_z}) and (\ref{beta}),
the integrand in the preceding integral is bounded in modulus by
\begin{eqnarray*}
\left| \frac{\beta(\zeta)(\zeta-x)^{n}}{(\zeta -z)^{n+1}} \right|
& \le & \alpha( \tfrac{3}{2}|z -x|)\frac{((K +1) \;\dist (z, \R))^n}{(\tfrac{1}{2}\; \dist (z, \R))^{n+1}}\\
&= &\alpha( \tfrac{3}{2}|z -x|) \;2^{n+1} (K+1)^n \;(\dist (z, \R))^{-1}.
\end{eqnarray*}
for all $\zeta \in \gamma_z$. The length of $\gamma_z$ is equal to $\pi \dist (z, \R)$, and so the integral itself is bounded in modulus by
$\alpha( \tfrac{3}{2}|z -x|)\; 2^{n+1} \pi (K+1)^n $
which tends to $0$ as $z \to x$ in the region $S_K$. This proves that
\[
\lim_{z \nt x}\frac{f^{(n)}(z)}{n!} = a^n.
\]
\end{proof}
\section{Julia reduction and augmentation in the Pick class} \label{reduction}
In 1920 G. Julia \cite{Ju20}, in the course of proving the well known ``Julia's Lemma" for bounded analytic functions on the disc, introduced a technique for passing from a function in the Pick class to a simpler one and back again. He showed that if $f\in\Pick$ is analytic at $x$ then the reduction of $f$ at $x$ also belongs to
$\Pick$. Subsequently Nevanlinna \cite{Nev1} proved that
the conclusion remains under a weaker hypothesis than analyticity at $x$.
Whereas our earlier paper \cite{ALY10} needed only Julia's result, the present one depends crucially on Nevanlinna's (considerably more subtle) refinement.
We shall say that $ x \in \R$ is a {\em $B$-point for} $f \in \Pick$ if the Carath\'eodory condition
\beq \label{cc}
\liminf_{z\to x} \frac{\im f(z)}{\im z} < \infty
\eeq
holds (\cite{AMcCY10}). A part of the Carath\'eodory-Julia Theorem \cite{car54, S2} asserts that if $x\in\R$ is a $B$-point for $f\in\Pick$ then $f$ has a nontangential limit and an angular derivative at $x$. We shall denote these quantities by $f(x), \ f'(x)$ respectively. The theorem also tells us that $f'(x)>0$ if $f$ is not a constant function.
\begin{definition} \label{defreduce} \rm
(1) For any non-constant function $f\in\Pick$ and any $x\in\R$ such that $x$ is a $B$-point for $f$ we define the {\em reduction of $f$ at $x$} to be the function $g$ on $\Pi$ given by the equation
\beq \label{reducef}
g(z) = -\frac{1}{f(z)-f(x)} + \frac{1}{f'(x)(z-x)}.
\eeq
(2) For any $g\in\Pick$, any $x\in\R$ and any $a_0\in\R, a_1 > 0$, we define the {\em augmentation of $g$ at $x$ by $a_0, a_1$} to be the function $f$ on $\Pi$ given by
\beq \label{augmentg}
\frac{1}{f(z)-a_0} = \frac{1}{a_1(z-x)} -g(z).
\eeq
\end{definition}
Note that in (1), since $f(x)$ is real and $f$ is non-constant, the denominator $f(z)-f(x)$ is non-zero, by the maximum principle.
Furthermore $f$ defined by equation (\ref{augmentg}) is necessarily non-constant, for otherwise
\[
\im g(z) = \mathrm{const} + \frac{1}{a_1}\im \frac{1}{z-x},
\]
and the last term can be an arbitrarily large negative number for $z\in\Pi$, contrary to the choice of $g\in\Pick$.
Here are the crucial invariance properties of reduction and its inverse.
\begin{theorem} \label{propfg}
Let $x \in\R$.
\begin{enumerate}
\item[\rm(1)] If $x$ is a $B$-point for a non-constant function $f\in\Pick$ then the reduction $g$ of $f$ at $x$ also belongs to $\Pick$.
\item[\rm(2)] If $g\in\Pick$ and $a_0\in\R,\, a_1>0$ then the augmentation $f$ of $g$ at $x$ by $a_0,\, a_1$ belongs to $\Pick$, has a $B$-point at $x$ and satisfies $f(x)=a_0, \ f'(x) \leq a_1$.
Moreover
\beq\label{nopole}
f'(x) = a_1 \quad\mbox{ if and only if }\quad \lim_{y\to 0+} yg(x+\ii y) =0.
\eeq
\end{enumerate}
\end{theorem}
\begin{proof}
Nevanlinna proved the analogue of (1) for the case that $x = \infty$, but his proof is easily modified for finite $x$; details are in
\cite[Theorem 5.4]{AMcCY10}. Here is a bare outline. Let $a^1=f'(x)>0$. One shows that, for any $\varepsilon > 0$,
\beq\label{expgro}
-\im g(z) \leq \frac{\varepsilon}{a_1} |w|
\eeq
for all $w \in \Pi$ of sufficiently large modulus, where
$w = -1/(z-x)$.
Introduce the analytic function $F$ on $\Pi$ by
\[
F(w) = \e^{\ii g(z)} = \e^{\ii g(x-1/w)}.
\]
We have, for any $w \in \Pi$,
\[
|F(w)| = \e^{\re \ii g(z)} = \e^{-\im g(z)}.
\]
By inequality \ref{expgro}, $F$ has only exponential growth on $\Pi$. Apply the Phragm\'en-Lindel\"of Theorem (e.g. \cite[p. 218]{BakNew}) to show that $|F| \leq \e^{\de/a_1}$ on $\Pi+\ii\de$, for any $\delta > 0$. On letting $\de$ tend to zero we deduce that $|F| \leq 1$ on $\Pi$, and hence that $\im g \geq 0$ on $\Pi$. Thus $g\in\Pick$.
The proof of (2) is an exercise in the mapping properties of linear fractional transformations of the complex plane. Again, details are in \cite{AMcCY10}.
\end{proof}
\begin{remark} {\rm Let $g$ be a real rational function of degree $m$ and let $f$ be
the augmentation of $g$ at $x$ by $a^0, a^1>0$. Then $f$ is a real rational function of degree $m+1$.}
\end{remark}
Here, as usual, the degree of a rational function $f = \frac{p}{q}$ is defined to be the maximum of the degrees of $p$ and $q$, where $p$, $q$ are polynomials in their lowest terms.
Pseudo-Taylor expansions behave well with respect to reduction and augmentation, as we now show.
\begin{proposition}\label{2augment_weak} {\rm (1)} Let $g,G \in \Pick$ and $x\in\R$. Suppose that $G$ is analytic at $x$ and that, for some non-negative integer $N$,
\[
g(z) - G(z) = \ont((z-x)^N) \qquad \mbox{ as } z \to x.
\]
Then the augmentations $f, F$ of $g$, $G$ respectively at $x$ by $a^0 \in \R$ and $a^1 >0$ satisfy
\beq\label{fF}
f(z) -F(z) = (g(z) - G(z))(F(z)-a^0)(f(z)-a^0)
\eeq
and $f(z) -F(z) = \ont((z-x)^{N+2})$ as $z \to x$.
{\rm (2)} Let $f\in\Pick$ be a non-constant function, let $x\in\R$ be a $B$-point for $f$ and let $F$ be a polynomial such that
\beq\label{fFoN}
f(z) -F(z) = \ont((z-x)^{N})
\eeq
for some $N \ge 2$. Let $g$, $G$ be the reductions of $f, F$ respectively at $x$. Then $F'(x) >0$ and
\beq\label{simpleid}
f(z) -F(z) = (g(z) - G(z))(F(z)-F(x))(f(z)-F(x))
\eeq
for all $z \in \Pi$. Moreover, $g$ has a pseudo-Taylor expansion at $x$ of order $N-2$, and
\beq\label{gGo(N-2)}
g(z) -G(z) = \ont((z-x)^{N-2}).
\eeq
\end{proposition}
\begin{proof} (1) Note that
\begin{eqnarray*}
g(z) - G(z) &=& -\frac{1}{f(z)-a^0} + \frac{1}{a^1 (z-x)} - \left(-\frac{1}{F(z)-a^0} + \frac{1}{a^1 (z-x)} \right)\\
&=& \frac{f(z) -F(z)}{(F(z)-a^0)(f(z)-a^0)}.
\end{eqnarray*}
Therefore
\begin{eqnarray*}
f(z) -F(z) &=& (g(z) - G(z))(F(z)-a^0)(f(z)-a^0)\\
&=& \ont((z-x)^N)\ont((z-x))\ont((z-x))= \ont((z-x)^{N+2})
\end{eqnarray*}
as $z \to x$.\\
(2) By the Carath\'eodory-Julia Theorem, the nontangential limit and angular derivative
$ f(x)\in\R$ and $f'(x)>0$ exist. By equation (\ref{fFoN}), $F(x)=f(x)$ and $F'(x) = f'(x).$ Hence
\[
F(z)= f(x) + f'(x)(z-x) +o(z-x).
\]
The identity (\ref{simpleid}) is immediate as in Part (1), and we have
\begin{eqnarray*}
g(z) - G(z) &=& \frac{f(z) -F(z)}{(F(z)-f(x))(f(z)-f(x))}\\
&=& \frac{\ont((z-x)^{N})}{[f'(x) (z-x) + \ont(z-x)]^2}\\
&=& \ont((z-x)^{N-2}).
\end{eqnarray*}
Since $G$ is the reduction of a polynomial $F$, it is rational and is analytic at $x$. Thus it has an infinite Taylor expansion about $x$, and so $g$ has a pseudo-Taylor expansion of order $N-2$ about $x$.
\end{proof}
\begin{corollary}\label{fg_C-point_N} Let $x \in \R$, let $f\in\Pick$ be non-constant function and let $g$ be the reduction of $f$ at $x$. Let $N > 2$. Then $f$ has a pseudo-Taylor expansion of order $N$ about $x$ if and only if $g$ has a pseudo-Taylor expansion of order $N-2$ about $x$.
\end{corollary}
\begin{proof}
It follows from Proposition \ref{2augment_weak}.
\end{proof}
\begin{lemma}\label{2augment_relax_weak} Let $x\in \R$ and
let $f\in\Pick$ be a non-constant function. Suppose $f$ has a pseudo-Taylor expansion of order $N \ge 2$ at $x$
and let $F$ be a polynomial such that
\beq\label{fFoN_relax}
f(z) -F(z) = A (z-x)^N + \ont((z-x)^{N})
\eeq
for some $A \in \C$. Let $g$, $G$ be the reductions of $f, F$ respectively at $x$. Then $F'(x) >0$ and
\beq\label{gG_relax}
g(z) -G(z) = \frac{A}{F'(x)^2} (z-x)^{N-2} + \ont((z-x)^{N-2}).
\eeq
\end{lemma}
\begin{proof} As in Proposition \ref{2augment_weak}, $F(x)=f(x)$ and $F'(x)=f'(x) >0$. The
equation (\ref{simpleid}) implies
\begin{eqnarray*}
g(z) - G(z) &=& \frac{f(z) -F(z)}{(F(z)-F(x))(f(z)-f(x))}\\
&=& \frac{A (z-x)^N + \ont((z-x)^{N})}{[F'(x) (z-x) + \ont((z-x))]^2}\\
&=& \frac{A}{F'(x)^2} (z-x)^{N-2} +\ont((z-x)^{N-2}).
\end{eqnarray*}
The relation (\ref{gG_relax}) follows.
\end{proof}
Another ingredient of the proof of our main result is an identity for Hankel matrices, which shows that the reduction of power series corresponds to Schur complementation of Hankel matrices.
\begin{theorem} \label{congruent}
Let
\[
f=\sum_{j=0}^\infty f_j z^j, \qquad g=\sum_{j=0}^\infty g_j z^j
\]
be formal power series over $\C$ with $f_1\neq 0$ and $f_0,f_1,\dots,f_n \in\R$, and let $g$ be the reduction of $f$ at $0$. Then the $n\times n$ Hankel matrix
$$
H_n(g)=[g_{i+j-1}]_{i,j=1}^n
$$
is congruent to the Schur complement of the $(1,1)$ entry in the $(n+1)\times(n+1)$ Hankel matrix
$$
H_{n+1}(f)=[f_{i+j-1}]_{i,j=1}^{n+1}.
$$
Consequently $H_{n+1}(f)> 0$ if and only if $f_1>0$ and $H_n(g)>0$.
\end{theorem}
This is Corollary 3.4 of \cite{ALY10}. It is convenient to introduce some notation for the relationship described in the theorem. For any $n \times n$ matrix $A = [a_{ij}]$ with $a_{11} \neq 0$ we define
$\schur A$ to be the Schur complement of $[a_{11}]$ in $A$. Thus, for $f$ and $g$ as in Theorem \ref{congruent}, $\schur H_{n+1}(f)$ is congruent to $H_n(g)$.
\section{A relaxation of the boundary Carath\'{e}odory-Fej\'{e}r problem} \label{relax}
Solvability of Problem $\partial CF\Pick(\R)$ is best approached through a slight relaxation of the problem, in which the final interpolation condition ($f^{(n)}(x)/n!=a^n$) is replaced by an inequality (see for example \cite{Geo98,BD,ALY10}). The reason is that solvability of the relaxed problem, not the original one, corresponds to positivity of a Hankel matrix. We therefore consider:
\noindent {\bf Problem $\partial CF\Pick'(\R)$} \quad {\em
Given a point $x\in \R$ and $a^0,a^1,\dots,a^n \in \R$, find a function $f$ in the Pick class such that $f$ is analytic at $x$,
\beq \label{interpcondRel}
\frac{f^{(k)}(x)}{k!} = a^k, \qquad k=0,1,\dots,n-1, \;\;{\text and } \;\; \frac{f^{(n)}(x)}{n!} \le a^n.\\
\eeq
}
The terminology for the problem was introduced in \cite{ALY10}, but in this paper we are interested in functions $f$ that satisfy the interpolation conditions in a weak sense.
We define a function $f \in \Pick$ to be a {\em weak solution} of Problem $\partial CF\Pick'(\R)$ if $f$ has a pseudo-Taylor expansion of order $n$ at $x$ and the $j$th pseudo-Taylor coefficient of $f$ at $x$ is $a^j$ for $j=0,1,\dots,n-1$ and is no greater than $a^n$ for $j=n$.
Here is an alternative description of weak solutions. Suppose that $F$ is analytic at $x$ and
\[
F(z) = a^0 + a^1 (z-x) + \dots + a^{n} (z-x)^{n} + O((z-x)^{n+1}).
\]
Then a function $f \in \Pick$ is a weak solution of
Problem $\partial CF\Pick'(\R)$ if and only if, for some $A \le 0$,
\beq \label{fFAo}
f(z) -F(z) = A(z-x)^n + \ont ((z-x)^{n}).
\eeq
We say a function $f \in \Pick$ is a {\em nontangential solution} of Problem $\partial CF\Pick'(\R)$ if
\beq \label{quasi-solution}
\lim_{z \nt x}\frac{f^{(k)}(z)}{k!} = a^k, \qquad k=0,1,\dots,n-1, \;\;\text{ and } \;\; \lim_{z \nt x} \frac{f^{(n)}(z)}{n!} \le a^n,
\eeq
and we define a {\em radial solution} of Problem $\partial CF\Pick'(\R)$ in the obvious way.
We might expect that more problems $\partial CF\Pick'$ would admit weak solutions than true solutions. In fact, though, the crux of the problem is the analytic case. This assertion is justified by the following result.
Corresponding to the sequence $a=(a^0, a^1, \dots, a^n)$ and any positive integer $m$ such that $2m-1 \leq n$ we define the Hankel matrix $H_m(a)$ to be the $m\times m$ matrix
$[a_{i+j-1}]_{i,j=1}^m$. If $F$ is a function analytic at the interpolation node $x$, we shall write $H_m(F)$ to mean $H_m(f_0, \dots, f_{2m-1})$, where $f_j$ is the $j$th Taylor coefficient of $F$ at $x$.
\begin{theorem} \label{weakequiv} Let $n$ be an odd positive integer.
Then the following statements are equivalent:
\begin{enumerate}
\item[\rm(1)] Problem $\partial CF\Pick'(\R)$ has a weak solution;
\item[\rm(2)] Problem $\partial CF\Pick'(\R)$ has a solution which is analytic at $x$;
\item[\rm(3)] Problem $\partial CF\Pick'(\R)$ has a rational solution;
\item[\rm(4)] Problem $\partial CF\Pick'(\R)$ has a real rational solution;
\item[\rm(5)] Problem $\partial CF\Pick'(\R)$ has a nontangential solution;
\item[\rm(6)] Problem $\partial CF\Pick'(\R)$ has a radial solution;
\item[\rm(7)] The Hankel matrix $H_m(a)$ is positive, where $m = \tfrac 12 (n+1)$.
\end{enumerate}
\end{theorem}
\begin{proof}
By \cite[Theorem 6.1]{ALY10},
(2) $\Leftrightarrow$ (3) $\Leftrightarrow$ (4) $\Leftrightarrow$ (7), and obviously (4)$\Rightarrow$(1) and (4)$\Rightarrow$(5). By Theorem \ref{Taylor}, (5)$\Leftrightarrow$(6)$\Leftrightarrow$(1).
We must show that (1)$\Rightarrow$(7). Suppose that $f$ is a weak solution of Problem $\partial CF\Pick'(\R)$, with pseudo-Taylor expansion
\[
f(z)=\sum_{j=0}^n f_j(z-x)^j +\ont((z-x)^n).
\]
Thus $f_j=a^j$ for $j\leq n-1$ and $f_n \leq a_n$.
We can assume that $f$ is nonconstant.
Consider the case that $m=1=n$.
We have $\im f(x+\ii y) = a^1y + o(y)$, and hence
\[
\lim_{y \to 0+} \frac{\im f(x+iy)}{y} = a^1 < \infty.
\]
It follows from the Carath\'{e}odory-Julia theorem \cite{car54} that $a^1> 0$, which is to say that $H_1(a) > 0$. Thus
(1)$\Rightarrow$(7) when $m=1$.
Now consider $m \ge 2$ and suppose the implication (1)$\Rightarrow$(7)
valid for $m-1$.
Let
\[
F(z)= \sum_{j=0}^n f_j(z-x)^j,
\]
so that $f(z)-F(z)=\ont((z-x)^n)$.
Let $g, G$ be the reductions of $f, F$ respectively at $x$; then $g\in\Pick$. By Proposition \ref{2augment_weak}, $g$ has a pseudo-Taylor expansion of order $n-2=2m-3$ about $x$ and
\beq\label{g-G}
g(z)-G(z) = \ont((z-x)^{2m-3}).
\eeq
That is to say, $g$ is a weak solution of Problem $\partial CF\Pick'(\R)$ with data $G$ and with new ``$n$" equal to $n-2=2m-3$. Accordingly this last problem has a weak solution, and we may invoke the inductive hypothesis to assert that $H_{m-1}(G) \geq 0$.
By the Hankel identity, Theorem \ref{congruent},
$ H_{m-1}(G)$ is congruent to $\schur H_{m}(F)$. Since (again by the Carath\'{e}odory-Julia theorem) $f_1= a^1 >0$, it follows that $ H_{m}(F) \ge 0$.
Now $H_m(a)$ and $H_m(F)$ differ only in their southeast corner entries -- in fact
\[
H_m(a) = H_m(F) + \diag \{0,0,\dots,a^n -f_n\} \geq H_m(F) \ge 0.
\]
Thus (1)$\Rightarrow$(7), and the theorem follows by induction.
\end{proof}
We now consider the question of determinacy for Problem $\partial CF\Pick'(\R)$. In the analytic case, the problem is determinate if and only if the associated Hankel matrix is positive and singular \cite[Theorem 5.1]{ALY10}. In principle, there might be one analytic solution and many weak solutions of a problem, but in fact this does not happen.
\begin{theorem} \label{probprimeHD_weak}
Let $ x\in\R$, $a=(a^0,\dots,a^{2m-1})\in \R^{2m}$ for some $m \ge 1$. Problem $\partial CF\Pick'(\R)$ has a unique weak solution if and only if the associated Hankel matrix $H_m(a)$ is positive and singular.
\end{theorem}
\begin{proof} By \cite[Theorem 5.1]{ALY10}, if $H_m(a) > 0$ the Problem $\partial CF\Pick(\R)$ is indeterminate. Thus, by Theorem \ref{weakequiv}, necessity holds.
Suppose that $H_m(a)$ is positive and singular. We show that
Problem $\partial CF\Pick'(\R)$ has a unique weak solution.
Consider the case $m=1$. Here $a^1=0$, and the constant function equal to $a^0$ is a solution of Problem $\partial CF\Pick'(\R)$.
Let $f$ be any weak solution, so that $f \in \Pick$ and
\beq\label{m=1}
f(z) = a^0 + \ont ((z-x)).
\eeq
We have
$ \im f(x+\ii y)/y \to 0$ as $y \to 0+$. Hence
\beq \label{B_cond}
\alpha \stackrel{\rm def}{=} \liminf_{z\to x, z \in \Pi} \frac{\im f(z)}{\im z}\leq \lim_{y\to 0+} \frac{\im f(x+\ii y)}{y} =0.
\eeq
By the Carath\'{e}odory-Julia theorem \cite{car54}, if $f$ is nonconstant then $\alpha > 0$. Thus the only weak solution is the constant $a^0$. The assertion of the theorem is therefore true when $m=1$.
Suppose the assertion holds for some $m \ge 1$; we prove it holds for $m+1$.
Let $H_{m+1}(a)$ be positive and singular for some $a=(a^0,\dots,a^{2m+1})$. Let $F(z)=\sum_0^{2m+1} a^j (z-x)^j$.
Assume that functions
$f_1$ and $f_2$ in $\Pick$ are solutions of the problem $\partial CF\Pick'(\R)$ with data $x$ and $a$. Then, for some $A_1, A_2 \le 0$,
\beq\label{f1f2FoN_relax}
f_i(z) -F(z) = A_i(z-x)^{2m+1} + \ont((z-x)^{2m+1}),\;\; i=1,2.
\eeq
Let $g_1, g_2$, $G$ be the reductions of $f_1, f_2$, $F$ respectively at $x$. Then $g_1, g_2 \in \Pick$ and $G$ is a rational function that is analytic at $x$. By Lemma \ref{2augment_relax_weak},
\beq\label{g1g2G_relax}
g_i(z) -G(z) = \frac{A_i}{(a^1)^2} (z-x)^{2m-1} + \ont((z-x)^{2m-1}), \;\; i=1,2.
\eeq
Since $\frac{A_i}{(a^1)^2} \le 0$, it follows that $g_1$ and $g_2$ are weak solutions of Problem $\partial CF\Pick'(\R)$ with data $x$, $b$ where $b = (b^0,\dots,b^{2m-1})$ comprises the first $2m$ Taylor coefficients of $G$ about $x$.
By Theorem \ref{congruent}, the associated Hankel matrix of this problem, $H_m(b)$ is congruent to $\schur H_{m+1}(a)$. Since $H_{m+1}(a)$ is positive and singular, so is $H_m(b)$. By the inductive hypothesis, the problem has a unique solution, and so $g_1=g_2$. Since $f_1, \ f_2$ are both equal to the augmentation of this function $g_1=g_2$ at $x$ by $a^0$, $a^1$ we have $f_1=f_2$. Thus, by induction, the statement of Theorem \ref{probprimeHD_weak} holds for all $m\geq 1$.
\end{proof}
\section{Weak solutions of Problem $\partial CF\Pick$} \label{weak}
In this section we prove the main result of the paper, a criterion for the existence of a weak solution of $\partial CF\Pick(\R)$. As in \cite{ALY10} we deduce the result from the corresponding criterion for Problem $\partial CF\Pick'(\R)$, but there is a subtlety: the deduction depends on the condition for the uniqueness of solutions of Problem $\partial CF\Pick'(\R)$, and now this must be understood in the sense of uniqueness in the class of weak solutions. We shall therefore need to use Theorem \ref{probprimeHD_weak} above.
We shall say that the Hankel matrix $H_{m}(a)$
is {\em southeast-minimally positive} if $H_{m}(a) \ge 0$ and, for every $\varepsilon >0$, $H_{m}(a) - {\rm diag}\{0,0,\dots, \varepsilon\}$ is not positive.
We shall abbreviate ``southeast-minimally" to ``SE-minimally".
\begin{theorem} \label{main_theorem}
Let $n$ be an odd positive integer and let $a=(a^0,\dots,a^n)\in \R^{n+1}$.
The following statements are equivalent:
\begin{enumerate}
\item[\rm(1)] Problem $\partial CF\Pick(\R)$ has a weak solution;
\item[\rm(2)] Problem $\partial CF\Pick(\R)$ has a nontangential solution;
\item[\rm(3)] Problem $\partial CF\Pick(\R)$ has a radial solution;
\item[\rm(4)] Problem $\partial CF\Pick(\R)$ has a solution which is analytic at $x$;
\item[\rm(5)] the associated Hankel matrix $H_{m}(a)$, $n=2m -1$, is either positive definite or SE-minimally positive.
\end{enumerate}
Moreover, the problem has a {\em unique} weak solution if and only if $H_m(a)$ is SE-minimally positive, and in this case the solution is rational of degree equal to $\rank H_m(a)$.
\end{theorem}
\begin{proof}
By \cite[Theorem 7.1]{ALY10},
(4) $\Leftrightarrow$ (5). By Theorem \ref{Taylor}, (1) $\Leftrightarrow$ (2) $\Leftrightarrow$ (3). It is clear that (4) $\Rightarrow$ (1). We will show that (1) $\Rightarrow$ (5).
(1) $\Rightarrow$ (5). Suppose that Problem $\partial CF\Pick(\R)$ has a weak solution $f\in\Pick$ but that its Hankel matrix $H_m(a)$ is neither positive definite nor SE-minimally positive. {\em A fortiori} $f$ is a weak solution of Problem $\partial CF\Pick'(\R)$, and so, by Theorem \ref{weakequiv}, $H_m(a)\geq 0$. Since $H_m(a)$ is not positive definite, $H_m(a)$ is singular, and so, by Theorem \ref{probprimeHD_weak}, Problem $\partial CF\Pick'(\R)$ has the {\em unique} weak solution $f$. Since $H_m(a)$ is not SE-minimally positive there is some positive $a^{n}{'} < a^{n}$ such that $H_m(f)\geq 0$, where $H_m(f)$ is the matrix obtained when the $(m,m)$ entry $a^{n}$, $n=2m-1$, of $H_m(a)$ is replaced by $a^{n}{'}$. Again by Theorem \ref{weakequiv}, there exists $h\in\Pick$ such that
$$
\lim_{z \nt x}\frac{h^{(k)}(z)}{k!} = a^k, \qquad k=0,1,\dots,n-1, \;\;\text{ and } \;\; \lim_{z \nt x} \frac{h^{(n)}(z)}{n!} \le a^{n}{'}< a^{n}.
$$
In view of the last relation we have $h\neq f$, while clearly $h$ is a weak solution of Problem $\partial CF\Pick'(\R)$, as is $f$. This contradicts the uniqueness of the weak solution $f$. Hence if the problem is solvable then either $H_m(a)>0$ or $H_m(a)$ is SE-minimally positive.
\end{proof}
We note that a different solvability criterion is given by D. Georgijevi\'c in \cite{Geo98}: Problem $\partial CF\Pick(\R)$ is solvable if and only if $H_m(a) \geq 0$ and its rank is equal to the rank of each of its singular submatrices. His methods are quite different from ours.
There is also a version of Theorem \ref{main_theorem} for even $n$.
\begin{theorem}\label{main_theorem_even}
Let $n$ be an even positive integer.
the following statements are equivalent:
\begin{enumerate}
\item[\rm(1)] Problem $\partial CF\Pick(\R)$ has a weak solution;
\item[\rm(2)] Problem $\partial CF\Pick(\R)$ has a nontangential solution;
\item[\rm(3)] Problem $\partial CF\Pick(\R)$ has a radial solution;
\item[\rm(4)] Problem $\partial CF\Pick(\R)$ has a solution which is analytic at $x$;
\item[\rm(5)] either the associated Hankel matrix $H_{m}(a)$, $n=2m$, is positive definite or both $H_{m}(a)$ is SE-minimally positive and $a^{n}$ satisfies
\beq \label{form_a_2m_Main}
a^{n} = \left[ \begin{array}{cccc}
a^m & a^{m+1} & \dots &a^{m+r-1}\end{array}\right] H_r(a)^{-1}
\left[\begin{array}{c} a^{m+1} \\ a^{m+2}\\ \cdot \\ a^{m+r} \end{array}\right]
\eeq
where $r=\rank H_m(a)$.
\end{enumerate}
Moreover, the problem has a {\em unique} solution if and only if $H_m(a)$ is SE-minimally positive and $a^{n}$ satisfies equation {\rm (\ref{form_a_2m_Main})}.
\end{theorem}
\begin{proof}
By \cite[Theorem 7.1]{ALY10},
(4) $\Leftrightarrow$ (5). By Theorem \ref{Taylor}, (1) $\Leftrightarrow$ (2) $\Leftrightarrow$ (3). It is clear that (4) $\Rightarrow$ (2). We will show that (2) $\Rightarrow$ (5).
(2) $\Rightarrow$ (5). Suppose that Problem $\partial CF\Pick(\R)$ has a weak solution $f\in\Pick$ such that $\lim_{z \nt x} f^{(k)}(z)/k! = a^k$ for $k=0,1,\dots,2m$. This
$f\in\Pick$ is also a weak solution of Problem $\partial CF\Pick(\R)$ for $n= 2m-1$.
The Hankel matrix $H_m(a)$ for Problem $\partial CF\Pick(\R)$ with $n=2m$ and with $n= 2m-1$ is the same.
By Theorem \ref{main_theorem}, $H_m(a)$ is positive definite or SE-minimally positive.
In the case that $a^1=0$, the constant function $f(z)=a^0$ is the solution of $\partial CF\Pick(\R)$. Therefore, $a^2 =a^3= \dots = a^{2m}=0$. Thus $H_m(a)$ is SE-minimally positive and $a^{n}$ satisfies (\ref{form_a_2m_Main}).
If $H_m(a)$ is SE-minimally positive and $a^1>0$ then by \cite [Proposition 7.4]{ALY10}, $a^{n}$ satisfies (\ref{form_a_2m_Main}).
\end{proof}
\begin{remark} \rm
In \cite[Theorem 8.3]{ALY10} we gave a parametrization of all solutions of Problem $\partial CF\Pick(\R)$ in the indeterminate case. The parametrization expresses the general solution $f$ as a continued fraction, containing as parameter a free function $f_{m+1} \in\Pick$ that is analytic at $x$ (when $n=2m-1$). It is simple to modify this parametrization to describe all {\em weak} solutions of Problem $\partial CF\Pick(\R)$: one simply takes the parameter set to be the set of all $f_{m+1} \in\Pick$ such that
$\lim_{y\to 0+} yf_{m+1}(x+\ii y) =0$, with no requirement of analyticity at $x$. This is essentially Nevanlinna's parametrization \cite[Satz I, p. 11]{Nev1922}. There is a similar parametrization in the case of even $n$.
\end{remark}
We conclude with an observation about a natural generalization of our main theorem.
Since functions in $\Pick$ can have simple poles with negative residue at points of the real axis, it is natural to study a slightly more general problem than $\partial CF\Pick(\R)$, in which the $(-1)$th Laurent coefficient is also prescribed \cite{Geo98, ALY10}: \\
{\em Given $x\in\R$ and $a^{-1}, a^0, \dots, a^n \in \R$ with $a^1>0$, determine whether there is a function $f\in\Pick$ such that}
\beq\label{genweak}
f(z) = \frac{a^{-1}}{z-x} + a^0+a^1 (z-x)+ \dots+a^n (z-x)^n + \ont((z-x)^n).
\eeq
In fact this interpolation problem is equivalent to the problem $\partial CF\Pick(\R)$ obtained by simply suppressing the condition on the $(-1)$th Laurent coefficient.
\begin{proposition}
There exists $f\in\Pick$ such that equation {\rm (\ref{genweak})} holds if and only if $a^{-1} \leq 0$ and there exists an $F\in\Pick$ such that
\beq\label{Fsolves}
F(z) = a^0+a^1 (z-x)+ \dots+a^n (z-x)^n + \ont((z-x)^n).
\eeq
\end{proposition}
\begin{proof}
Sufficiency is easy: if $a^{-1} \leq 0$ and $F\in\Pick$ satisfies (\ref{Fsolves}) then the function
\[
f(z) = F(z) + a^{-1}/(z-x)
\]
belongs to $\Pick$ and satisfies (\ref{genweak}).
Conversely, suppose $f\in\Pick$ satisfies (\ref{genweak}), and let $F(z) = f(z)-a^{-1}/(z-x)$. Certainly $F$ satisfies (\ref{Fsolves}); our task is to show that $a^{-1} \leq 0$ and $F\in\Pick$. Since $f\in\Pick$, we have for any $y> 0$,
\[
0\leq \im f(x+\ii y) = \im \frac{a^{-1}}{\ii y} +o(1)= -\frac{a^{-1}}{y} + o(1),
\]
and hence $a^{-1} \leq 0$.
Observe that $0$ is a $B$-point for the function $-1/f$ (which lies in $\Pick$) if and only if
\beq\label{li}
\liminf_{z\to x} \frac{\im f(z)}{|f(z)|^2 \im z} < \infty,
\eeq
a relation which does hold, in view of the fact that $f(z)= a^{-1}/(z-x) +\ont(1)$, and we find that the $\liminf$ (\ref{li}) is $-1/a^{-1}$. Thus $-1/f$ has nontangential limit $0$ and angular derivative $-1/a^{-1}$ at $x$. Let $G$ be the reduction of $-1/f$ at $0$. By Theorem 3.2(1), Nevanlinna's refinement of Julia's lemma, $G\in\Pick$. But
\[
G(z)= -\frac{1}{-1/f(z)} +\frac{1}{(-1/a^{-1})(z-x)} = f(z) - \frac{a^{-1}}{z-x} = F(z).
\]
Thus $F\in\Pick$.
\end{proof}
\begin{corollary}
Let $n$ be an odd positive integer, $n=2m-1$, and let $a^{-1}, a^0, \dots, a^n \in\R$ with $a^1>0$. There exists a function $f$ in $\Pick$ such that equation {\rm (\ref{genweak})} holds if and only if $a^{-1} \leq 0$ and the Hankel matrix $H_m(a)$ is either positive definite or SE-minimally positive.
\end{corollary}
\section{Appendix} \label{appendix}
Here we justify the assertions made concerning Example \ref{ex2}. We show that, for any integer $\nu \geq 4$, the function
\beq \label{deffnu}
f_\nu(z) = -\sum_{k=1}^\infty \frac{1}{k^\nu z + k^{\nu-1}}
\eeq
belongs to the Pick class, has a pseudo-Taylor expansion of order $\nu-3$ given by equation (\ref{expandfnu}) and has no expansion of order $\nu-2$.
The series (\ref{deffnu}) converges locally uniformly in $\Pi$, and so $f$ is analytic in $\Pi$. Since each summand belongs to $\Pick$, so does $f_{\nu}$.
For $j = 0,1, \dots, \nu -3$, we obtain, with the aid of the Dominated Convergence Theorem,
\begin{eqnarray*}
\lim_{y \to 0+} \frac{f_{\nu}^{(j)}(\ii y)}{j!}& =&
\lim_{y \to 0+} \sum_{k=1}^{\infty}
\frac{(-1)^{j+1}}{k^{\nu} \left(\ii y + \tfrac{1}{k} \right)^{j+1}}\\
& =&(-1)^{j+1} \zeta(\nu - j -1).
\end{eqnarray*}
and so, by Theorem \ref{Taylor}, the expansion (\ref{expandfnu}) is indeed a pseudo-Taylor expansion of $f_\nu$ of order $\nu-3$.
However $f_{\nu}$ does not have a pseudo-Taylor expansion
of order $\nu-2$.
In view of Theorem \ref{Taylor}, this claim will follow if we can show that $f_{\nu}^{(\nu-2)}(\ii y)$ does not have a finite limit as $y \to 0+$. We have
\[
\frac{f_{\nu}^{(\nu -2)}(\ii y)}{(\nu -2)!} = (-1)^{\nu -1} \sum_{k=1}^{\infty} h_{\nu}(k, y)
\]
where, for $t \ge 1$ and $y > 0$,
\[
h_{\nu}(t, y)= \frac{1}{t^{\nu}\left(\ii y + \tfrac{1}{t} \right)^{\nu -1}} = \frac{1}{t \left(1 + \ii t y \right)^{\nu -1}}.
\]
Now
\begin{eqnarray*}
\int_{1}^{\infty} h_{\nu}(t, y) dt &=& \int_{1}^{\infty}\frac{dt}{t \left(1 + \ii t y \right)^{\nu -1}} = \int_{y}^{\infty}\frac{du}{u \left(1 + \ii u \right)^{\nu -1}}\\
&=& - \sum_{j=1}^{\nu -2} \frac{1}{j \left(1 + \ii y \right)^{j}}
- \log \frac{\ii y}{1 + \ii y} \to \infty
\end{eqnarray*}
as $y \to 0+$, while
\[
\left|\frac{\partial h_{\nu}}{\partial t} (t,y) \right| =
\frac{1}{t^2} \left|\frac{1+ \nu \ii t y}{ \left(1 + \ii t y \right)^{\nu}} \right|.
\]
Hence, for $t \in [k, k+1]$ and $y > 0$,
\[
\left|\frac{\partial h_{\nu}}{\partial t} (t,y) \right| \le \frac{C_{\nu}}{k^2},
\]
where
\[
C_{\nu} = \sup_{y >0, t \ge 1} \left|\frac{1+ \nu \ii t y}{ \left(1 + \ii t y \right)^{\nu}} \right| =
\sup_{\tau >0} \left|\frac{1+ \nu^2 \tau}{ \left(1 + \tau \right)^{\nu}} \right|^{1/2} < \infty
\]
(in fact $C_{\nu}^2 = {\nu}/{ \left(1 + \frac{1}{\nu} \right)^{\nu-1}}$). By the Mean Value Theorem, for $t \in [k, k+1]$,
\[
\left|h_{\nu}(t, y)- h_{\nu}(k, y)\right| \le \frac{C_{\nu}}{k^2},
\]
and so, for all $y > 0$,
\[
\left|\int_{1}^{\infty} h_{\nu}(t, y) dt - \sum_{k=1}^{\infty} h_{\nu}(k, y) \right| \le \sum_{k=1}^{\infty} \frac{C_{\nu}}{k^2}
=\frac{C_{\nu} \pi^2}{6}.
\]
Hence
\[
\lim_{y\to 0+} f_\nu^{(\nu-2)} (\ii y) = (-1)^{\nu-1}(\nu-2)! \lim_{y \to 0+} \sum_{k=1}^{\infty} h_{\nu}(k, y) = \infty.
\]
Thus, as claimed, $f_{\nu}$ does not have a pseudo-Taylor expansion
of order $\nu-2$ at $0$.
| {
"timestamp": "2011-01-07T02:01:56",
"yymm": "1101",
"arxiv_id": "1101.1251",
"language": "en",
"url": "https://arxiv.org/abs/1101.1251",
"abstract": "We give a new solvability criterion for the boundary Carathéodory-Fejér problem: given a point $x \\in \\mathbb{R}$ and, a finite set of target values $a^0,a^1,...,a^n \\in \\mathbb{R}$, to construct a function $f$ in the Pick class such that the limit of $f^{(k)}(z)/k!$ as $z \\to x$ nontangentially in the upper half plane is $a^k$ for $k= 0,1,...,n$. The criterion is in terms of positivity of an associated Hankel matrix. The proof is based on a reduction method due to Julia and Nevanlinna.",
"subjects": "Complex Variables (math.CV)",
"title": "Pseudo-Taylor expansions and the Carathéodory-Fejér problem",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9899864284905089,
"lm_q2_score": 0.7154239897159439,
"lm_q1q2_score": 0.7082600404353179
} |
https://arxiv.org/abs/1304.5287 | A variant of Hörmander's $L^2$ existence theorem for Dirac operator in Clifford analysis | In this paper, we give the Hörmander's $L^2$ theorem for Dirac operator over an open subset $\Omega\in\R^{n+1}$ with Clifford algebra. Some sufficient condition on the existence of the weak solutions for Dirac operator has been found in the sense of Clifford analysis. In particular, if $\Omega$ is bounded, then we prove that for any $f$ in $L^2$ space with value in Clifford algebra, there exists a weak solution of Dirac operator such that $$\bar{D}u=f$$ with $u$ in the $L^2$ space as well. The method is based on Hörmander's $L^2$ existence theorem in complex analysis and the $L^2$ weighted space is utilised. | \section{Introduction}
The development of function theories on Clifford algebras has proved a
useful setting for generalizing many
aspects of one variable complex function theory to higher dimensions.
The study of these function theories is referred to as Clifford analysis \cite{c,c2,qt,J5}, which is closely related to a number of studies made in
mathematical physics, and many applications in this area
have been found in recent years. In \cite{J4}, Ryan considered solutions of the polynomial Dirac operator, which afforded an integral representation. Furthermore, the author gave a Pompeiu representation for $C^1$-functions in a Lipschitz bounded domain. In \cite{J2}, the author presented a classification of linear, conformally invariant, Clifford-algebra-valued differential operators over $\mathbb{C}^n$, which comprised the Dirac operator and its iterates. In \cite{J6}, Qian and Ryan used Vahlen matrices to study the conformal covariance of various types of Hardy spaces over hypersurfaces in $\mathbb{R}^n$. In \cite{mz}, the discrete Fueter polynomials was introduced, which formed a basis of the space of discrete spherical monogenics. Moreover, the explicit construction for this discrete Fueter basis, in arbitrary dimension $m$ and for arbitrary homogeneity degree $k$ was presented as well.
In \cite{H}, the famous H\"ormander's $L^2$ existence and approximation
theorems was given for the $\bar{\partial}$ operator in pseudo-convex domains in $\mathbb{C}^n$. When $n=1$, the existence theorem of complex variable can be deduced. The aim of this paper is to establish a H\"ormander's $L^2$ theorem in $\R^{n+1}$ with Clifford analysis, and present sufficient condition on the existence of the weak solutions for Dirac operator in the sense of Clifford algebra.
Let $\A$ be a real Clifford algebra over an (n+1)-dimensional real vector space $\R^{n+1}$ and the corresponding norm on $\A$ is given by $|\lambda|_0=\sqrt{(\lambda,\lambda)_0}$ (see subsection \ref{sub1}). Let $\Omega$ be an open subset of $\R^{n+1}$, $L^2(\Omega,\A,\varphi)$ be a right Hilbert $\A$-module for a given function $\varphi\in C^2(\Omega, \R)$ with the norm given by Definition \ref{defn10}. (see subsection \ref{sub3}). $\overline{D}$ denotes the Dirac differential operator and the dual operator $\overline{D}^*_\varphi $ of $\overline{D}$ is given by (\ref{7}). For $x=(x_0,x_1,...,x_n)\in \R^{n+1}$, $\Delta=\sum_{i=0}^{n}\frac{\partial^2}{\partial x_i^2}$. Then we can obtain our main results as follows.
\begin{thm}\label{thm1}
Given $f\in L^2(\Omega,\A,\varphi)$, there exists $u\in L^2(\Omega,\A,\varphi)$ such that
\begin{equation}\label{eq:5.1}
\begin{split}\overline{D}u=f\end{split}
\end{equation} with
\begin{equation}\label{eq:5.2}
\begin{split}\|u\|^2=\int_\Omega|u|^2_0e^{-\varphi}dx\leq 2^{2n}c \end{split}
\end{equation} if
\begin{equation}\label{eq:5}
\begin{split}|(f,\alpha)_\varphi|^2_0\leq c\|\overline{D}^*_\varphi\alpha\|^2=c\int_\Omega|\overline{D}^*_\varphi\alpha|^2_0e^{-\varphi}dx,~\forall \alpha\in C^\infty_0(\Omega,\A).\end{split}
\end{equation}
Conversely, if there exists $u\in L^2(\Omega,\A,\varphi)$ such that (\ref{eq:5.1}) is satisfied with
\begin{equation}
\begin{split}\|u\|^2 \leq c \nonumber\end{split}
\end{equation}
Then we can get the inequality (\ref{eq:5}) for norm estimation.
\end{thm}
The factor $2^{2n}$ in (\ref{eq:5.2}) comes from the definition of the norm in Clifford analysis. If $n=1$, then the factor would disappear which gives a necessary and sufficient condition in the theorem.
From the above theorem, we give the following sufficient condition on the existence of weak solutions for Dirac operator.
\begin{thm}\label{thm2}
Given $\varphi\in C^2(\Omega,\mathbb{R})$ and $n> 1$; $\Delta\varphi\geq0$,~and~$\frac{\partial^2 \varphi}{\partial x_j\partial x_i}=0,~i\neq j,~1\leq i,j\leq n$ and $\frac{\partial^2 \varphi}{\partial x^2_i}\leq 0,~1\leq i\leq n$. Then for all $ f\in L^2(\Omega,\A,\varphi)$ with $\int_\Omega\frac{|f|^2_0}{\Delta\varphi}e^{-\varphi}dx=c<\infty$, there exists a $u\in L^2(\Omega,\A,\varphi)$ such that $$\overline{D}u=f$$ with
$$\|u\|^2=\int_\Omega|u|^2_0e^{-\varphi}dx\leq2^{2n}\int_\Omega\frac{|f|^2_0}{\Delta\varphi}e^{-\varphi}dx.$$
\end{thm}
\begin{rem}
Assuming $x=(x_0,x_1,...,x_n)\in \R^{n+1}$, it is easy to see that $\varphi(x)=x_0^2$ satisfies the conditions in Theorem \ref{thm2}. Another simple example would be
$$\varphi(x)=(n+1)x_0^2-\sum_{i=1}^{n}x_i^2.$$
It is obvious that $\Delta\varphi(x)=2$, $\frac{\partial^2 \varphi}{\partial x^2_i}=-2$, and $\frac{\partial^2 \varphi}{\partial x_j\partial x_i}=0,~i\neq j,~1\leq i,j\leq n$.
\end{rem}
\begin{cor}\label{cor1}
Given $\varphi\in C^2(\Omega,\mathbb{R}),$ and $\varphi(x)=\varphi(x_0)$ with $\varphi''(x_0)\geq0$. Then for all $ f\in L^2(\Omega,\A,\varphi)$ with $\int_\Omega\frac{|f|^2_0}{\varphi''}e^{-\varphi}dx=c<\infty$, there exists a $u\in L^2(\Omega,\A,\varphi)$ such that $$\overline{D}u=f$$ with
$$\|u\|^2=\int_\Omega|u|^2_0e^{-\varphi}dx\leq2^{2n}\int_\Omega\frac{|f|^2_0}{\varphi''}e^{-\varphi}dx.$$
\end{cor}
It is noticed that there is nothing to do with the boundary conditions of $\Omega$ in the above results. This phenomenon is totally different with the famous H\"ormander's $L^2$ existence theorems of several complex variables in \cite{H}. Then we can also have the following theorem on global solutions.
\begin{thm}\label{cor3}
Given $\varphi\in C^2(\R^{n+1},\mathbb{R})$ with all derivative conditions in Theorem \ref{thm1} satisfied. Then for all $ f\in L^2(\R^{n+1},\A,\varphi)$ with $\int_{\R^{n+1}}\frac{|f|^2_0}{\Delta\varphi}e^{-\varphi}dx=c<\infty$, there exists a $u\in L^2(\R^{n+1},\A,\varphi)$ satisfying $$\overline{D}u=f$$ with
$$\|u\|^2=\int_{\R^{n+1}}|u|^2_0e^{-\varphi}dx\leq2^{2n} \int_{\R^{n+1}}\frac{|f|^2_0}{\Delta\varphi}e^{-\varphi}dx.$$
\end{thm}
On the other hand, if the boundary of $\Omega$ is concerned, we consider a special kind of domain ${\Omega}_0=\{x\in\R^{n+1}:a\leq x_0\leq b\}$ for any $a,~b\in \R$ with $a<b$, then we can get the following theorem within $L^2$ space instead of $L^2$ weighted space.
\begin{thm}\label{thm5}
Let $ f\in L^2({\Omega_0},\A)$. Then there exists a $u\in L^2({\Omega_0},\A)$ such that $$\overline{D}u=f$$ with
$$\int_{\Omega_0}|u|^2_0dx\leq2^{2n}c(a,b)\int_{\Omega_0}{|f|^2_0}dx$$ and $c(a,b)$ is a factor depending on $a,~b$.
\end{thm}
\begin{proof}
Let $\varphi(x)=x_0^2$. It can be obtained that $L^2({\Omega_0},\A)=L^2({\Omega_0},\A,\varphi)$ for the boundary of $x_0$.
Then the theorem is proved with Theorem \ref{thm2}.
\end{proof}
\begin{rem}
In particular, any bounded domain $\Omega$ in $\R^{n+1}$ can be regarded as one type of $\Omega_0$. Therefore, it comes from Theorem \ref{thm5} that for any $f\in L^2(\Omega,\A)$, we can find a weak solution of Dirac operator $\overline{D}u=f$ with $u\in L^2(\Omega,\A)$.
\end{rem}
\section{Preliminaries}
To make the paper self-contained, some basic notations and results used in this paper are included.
\subsection{The Clifford algebra $\A$ }\label{sub1}
Let $\A$ be a real Clifford algebra over an (n+1)-dimensional real vector space $\R^{n+1}$ with orthogonal basis $e:=\{e_0,e_1,...,e_n\}$, where $e_0=1$ is a unit element in $\R^{n+1}$. Furthermore,
\begin{equation} \label{eq:1}
\left\{ \begin{aligned}
e_ie_j+e_je_i&= 0,~i\neq j \\
e_i^2&= -1,~i=1,...,n.
\end{aligned} \right.\nonumber
\end{equation}
Then $\A$ has its basis $$\{e_A=e_{h_1\cdots h_r}=e_{h_1}\cdots e_{h_r}: 1\leq h_1<...<h_r\leq n, 1\leq r\leq n\}.$$
If $i\in \{h_1,...,h_r\}$, we denote $i\in A$ and if $i\not\in \{h_1,...,h_r\}$, we denote $i\not\in A$. $A-{i}$ means $\{h_1,...,h_r\}\setminus\{i\}$ and $A+{i}$ means $\{h_1,...,h_r\}\cup\{i\}$.
So the real Clifford algebra is composed of elements having the type $a=\sum\limits_{A}x_Ae_A$, in which $x_A\in \R$ are real numbers.
For $a\in \A$, we give the inversion in the Clifford algebra as follows:
$a^*=\sum\limits_{A}x_Ae_A^*$
where $e_A^*=(-1)^{|A|}e_A$ and $|A|=n(A)$ is the $r\in\mathbb{Z}^+$ as $e_A=e_{h_1\cdots h_r}$. When $A=\emptyset$, $e_A=e_0$, $|A|=0$. Next, we define the reversion in the Clifford algebra, which is given by
$a^\dag=\sum\limits_{A}x_Ae_A^\dag$
where $e_A^\dag=(-1)^{(|A|-1)|A|/2}e_A.$ Now we present the involution which is a combination of the inversion and the reversion introduced above.
$$\bar{a}=\sum\limits_{A}x_A\bar{e}_A$$
where $\bar{e}_A=e_A^{*\dag}=(-1)^{(|A|+1)|A|/2}e_A.$ From the definition, one can easily deduce that $e_A\bar{e}_A=\bar{e}_Ae_A=1.$ Furthermore, we have $$\overline{\lambda\mu}=\bar{\mu}\bar{\lambda},~~\forall \lambda, \mu\in \A.$$
Let $a=\sum\limits_{A}x_Ae_A$ be a Clifford number. The coefficient $x_A$ of the $e_A$-component will also be denoted by $[a]_A$. In particular the coefficient $x_0$ of the $e_0$-component will be denoted by $[a]_0$, which is called the scalar part of the Clifford number $a$. An inner product on $\A$ is defined by putting for any $\lambda,\mu\in \A$, $(\lambda,\mu)_0:=2^n[\lambda\bar{\mu}]_0=2^n\sum\limits_{A}\lambda_A\mu_A$. The corresponding norm on $\A$ reads $|\lambda|_0=\sqrt{(\lambda,\lambda)_0}$.
We define a real functional on $\A$ that $\tau_{e_A}:\A \rightarrow \R$
$$\langle \tau_{e_A},\mu\rangle=2^n(-1)^{(|A|+1)|A|/2}\mu_A.$$ In the special case where $A=\emptyset$ we have
$$\langle \tau_{e_0},\mu\rangle=2^n\mu_0.$$
Let $\Omega$ be an open subset of $\R^{n+1}$. Then functions $f$ defined in $\Omega$ and with values in $\A$ are considered. They are of the form
$$f(x)=\sum_{A}f_A(x)e_A$$ where $f_A(x)$ are functions with real value. Let $\overline{D}$ denotes the Dirac differential operator
$$\overline{D}=\sum_{i=0}^{n}e_i\partial_{x_i},$$
its action on functions from the left and from the right being governed by the rules
$$\overline{D}f=\sum_{i,A}e_ie_A\partial_{x_i}f_A~\mbox{and}~f\overline{D}=\sum_{i,A}e_Ae_i\partial_{x_i}f_A.$$
$f$ is called left-monogenic if $\overline{D}f=0$ and it is called right-monogenic if $f\overline{D}=0$. The conjugate operator is given by
$$D=\sum_{i=0}^{n}\bar{e}_i\partial_{x_i}.$$ It can be found that $$\overline{D}D=D\overline{D}=\Delta$$ where $\Delta$ denotes the classical Laplacian in $\R^{n+1}$.
When $n=1$, one can think of $x_0$ as the real part and of $x_1$ as the imaginary part of the variable and to identify $e_1$ with $i$. the operator $\overline{D}$ then take the form $\overline{D}=\partial_{x_0}+i\partial_{x_1}$, which is similar with the operator $\bar{\partial}$ in complex analysis.
\subsection{Modules over Clifford algebras} This subsection is to give some general information concerning a class of topological modules over Clifford algebras. In the sequel definitions and properties will be stated for left $\A$-module and their duals, the passage to the case of right $\A$-module being straight-forward.
\begin{defn}{\bf (unitary left $\A$-module)}
Let $X$ be a unitary left $\A$-module, i.e. $X$ is abelian group and a law $(\lambda,f)\rightarrow\lambda f:\A\times X\rightarrow X$ is defined such that $\forall\lambda,\mu\in \A$, and $f,~g\in X$
\begin{enumerate}
\item [(1)] $(\lambda+\mu)f=\lambda f+\mu f$,
\item [(2)] $\lambda\mu f=\lambda(\mu f)$,
\item [(3)] $\lambda(f+g)=\lambda f+\lambda g$,
\item [(4)] $e_0 f=f$.
\end{enumerate}
Moreover, when speaking of a submodule $E$ of the unitary left $\A$-module $X$, we mean that $E$ is a non empty subset of $X$ which becomes a unitary left $\A$-module too when restricting the module operations of $X$ to $E$.
\end{defn}
\begin{defn}{\bf (left $\A$-linear operator)}
If $X,Y$ are unitary left $\A$-modules, then $T:X\rightarrow Y$ is said to be a left $\A$-linear operator,
if $\forall~f,~g\in X$ and $\lambda\in \A$
$$T(\lambda f+g)=\lambda T(f)+T(g).$$
The set of all $``T"$ is denoted by $L(X,Y)$. If $Y=\A,~L(X,\A)$ is called the algebraic dual of $X$ and denoted by $X^{*alg}$. Its elements are called left $\A$-linear functionals on $X$ and for any $T\in X^{*alg}$ and $f\in X$, we denote by $\langle T,f \rangle$ the value of $T$ at $f$.
\end{defn}
\begin{defn}\label{bounded}{\bf (bounded functional)}
An element $T\in X^{*alg}$ is called bounded, if there exist a semi-norm $p$ on $X$ and $c>0$ such that for all $f\in X$
$$|\langle T,f\rangle|_0\leq c\cdot p(f).$$
\end{defn}
\begin{thm}\label{Hahn}{\bf (Hahn-Banach type theorem)}\cite{c}
Let $X$ be a unitary left $\A$-module with semi-norm $p$, $Y$ be a submodule of $X$, and $T$ be a left $\A$-linear
functional on $Y$ such that for some $c>0,$ $$|\langle T,g\rangle|_0\leq c\cdot p(g),~~ \forall g\in Y$$
Then there exists a left $\A$-linear functional $\widetilde{T}$ on $X$ such that
\begin{enumerate}
\item [(1)] $\widetilde{T}\mid_Y=T$,
\item [(2)] for some $c^*>0$,~$|\langle \widetilde{T},f\rangle|_0\leq c^*\cdot p(f)$,~~$\forall f\in X$.
\end{enumerate}
\end{thm}
\begin{defn}{\bf (inner product on a unitary right $\A$-module)}
Let $H$ be a unitary right $\A$-module, then a function $(~,~):~H\times H\rightarrow \A$ is said to be a inner product on $H$ if for all $ f,g,h\in H$ and $\lambda\in \A$,
\begin{enumerate}
\item [(1)] $(f,g+h)=(f,g)+(f,h)$,
\item [(2)] $(f,g\lambda)=(f,g)\lambda$,
\item [(3)] $(f,g)=\overline{(g,f)}$,
\item [(4)] $\langle\tau_{e_0},(f,f)\rangle\geq0$ and $\langle\tau_{e_0},(f,f)\rangle=0~ \mbox{if and only if} ~f=0$,
\item [(5)] $\langle\tau_{e_0},(f\lambda,f\lambda)\rangle\leq|\lambda|^2_0\langle\tau_{e_0},(f,f)\rangle$.
\end{enumerate}\end{defn}
From the definition on inner product, putting for each $f\in H$
$$\|f\|^2=\langle\tau_{e_0},(f,f)\rangle,$$
then it can be obtained that for any $f,g\in H,$
\begin{equation} \label{eq:2}
\begin{split}
|\langle\tau_{e_0},~(f,g)\rangle|\leq\|f\|\|g\|,\|f+g\|\leq\|f\|+\|g\|.
\end{split}\nonumber
\end{equation}
Hence, $\|\cdot\|$ is a proper norm on $H$ turning it into a normed right $A$-module.
Moreover, we have the following Cauchy-Schwarz inequality.
\begin{prop}\cite{c}\label{prop1}
For all $ f,g\in H,$ $|(f,g)|_0\leq\|f\|\|g\|.$
\end{prop}
\begin{defn}{\bf (right Hilbert $\A$-module)}
Let $H$ be a unitary right $\A$-module provided with an inner product $(~,~)$. Then is it called a right Hilbert $\A$-module if it is complete for the norm topology derived from the inner product.
\end{defn}
\begin{thm}\label{Riesz}{\bf (Riesz representation theorem)}\cite{c}
Let $H$ be a right Hilbert $\A$-modules and $T\in H^{*alg}$. Then $T$ is bounded if and only if there exists a (unique) element $g\in H$ such that for all $f\in H$,
$$T(f):=\langle T,f\rangle=(g,f).$$
\end{thm}
\subsection{Hilbert space of square integrable functions}\label{sub3}
Now we extend the standard Hilbert space of square integrable functions to Clifford algebra. First, we denote $L^1(\Omega,\mu)$ and $L^2(\Omega,\mu)$ be the sets of all integrable or square integrable functions defined on the domain $\Omega\in \R^{n+1}$ with respect to the measure $\mu$. Then $L^1(\Omega,\A,\mu)$ and $L^2(\Omega,\A,\mu)$ are defined as the sets of functions $f:\Omega\rightarrow \A$ which are integrable or square integrable with respect to $\mu$, i.e., if $f=\sum\limits_Af_Ae_A$, then for each $A$, $f_A\in L^1(\Omega,\mu)$ and $f^2_A\in L^1(\Omega,\mu)$, respectively. Then {\bf one may easily check that $L^1(\Omega,\A,\mu)$ and $L^2(\Omega,\A,\mu)$ are unitary bi-$\A$-module, i.e., unitary left-$\A$-module and unitary right-$\A$-module}. Furthermore, for any $f,g\in L^2(\Omega,\A,\mu)$, $\bar{f}\in L^2(\Omega,\A,\mu)$ while $\bar{f}g\in L^1(\Omega,\A,\mu)$, where $\bar{f}(x)=\overline{f(x)}$ and $(\bar{f}g)(x)=\bar{f}(x)g(x),~x\in \Omega$. Consider as a right $\A$-module, define for $f,g\in L^2(\Omega,\A,\mu)$ that
$$(f,g)=\int_{\Omega}\bar{f}(x)g(x)d\mu.$$
Furthermore for any real linear functional $T$ on $\A$
$$\langle T,(f,g)\rangle=\langle T,\int_{\Omega}\bar{f}(x)g(x)d\mu\rangle=\int_{\Omega}\langle T,\bar{f}(x)g(x)\rangle d\mu.$$
Consequently, taking $T=\tau_{e_0}$ we find that
\begin{equation} \label{eq:3}
\begin{split}
\langle \tau_{e_0},(f,f)\rangle&=\langle \tau_{e_0},\int_{\Omega}\bar{f}(x)f(x)d\mu\rangle=\int_{\Omega}\langle \tau_{e_0},\bar{f}(x)f(x)\rangle d\mu\\&=\int_{\Omega}|f(x)|^2_0d\mu.
\end{split}\nonumber
\end{equation}
Hence, for all $f\in L^2(\Omega,\A,\mu)$, $\langle \tau_{e_0},(f,f)\rangle\geq 0$ and $\langle \tau_{e_0},(f,f)\rangle=0$ if and only if $f=0$ a.e. in $\Omega$. Then it is easy to see that under the inner product defined, all conditions for $L^2(\Omega,\A,\mu)$ to be a unitary right inner product $\A$-module are satisfied. Since $L^2(\Omega,\A,\mu)=\prod_{A}L^2(\Omega,\mu)$, we have that $L^2(\Omega,\A,\mu)$ is complete; in other words $L^2(\Omega,\A,\mu)$ is a right Hilbert $\A$-module, with the norm $$\|f\|^2=\langle \tau_{e_0},(f,f)\rangle=\int_{\Omega}|f(x)|^2_0d\mu$$ for $f\in L^2(\Omega,\A,\mu)$.
\begin{defn}\label{defn10}{\bf (weighted $L^2$ space)}
Similar with $L^2(\Omega,\A,\mu)$, we can define the weighted $L^2(H,\A,\varphi)$ for a given function $\varphi\in C^2(\Omega, \R)$. First, let
$$L^2(\Omega,\varphi)=\big\{f|f:\Omega\rightarrow \R,~\int_\Omega|f(x)|^2e^{-\varphi}~dx<+\infty\big\}.$$Then we denote
$$L^2(H,\A,\varphi)=\{f|f:\Omega\rightarrow \A,~f=\sum\limits_Af_Ae_A,~f_A\in L^2(\Omega,\varphi)\}.$$
Moreover, for all $f,g\in L^2(H,\A,\varphi)$, we define
$$(f,g)_\varphi=\int_\Omega \bar{f}(x)g(x)e^{-\varphi}dx.$$
Then it is also easy to see $L^2(\Omega,\A,\varphi)$ is a right Hilbert $\A$-module, with the norm
\begin{equation} \label{norm}
\begin{split}
\|f\|^2=\langle \tau_{e_0},(f,f)_\varphi\rangle=\int_{\Omega}|f(x)|^2_0e^{-\varphi}dx
\end{split}\nonumber
\end{equation}
for $f\in L^2(\Omega,\A,\varphi)$.
\end{defn}
\subsection{Cauchy's integral formula}
Let $M$ be an (n+1)-dimensional differentiable and oriented manifold contained in some open subset $\Sigma$ of $\R^{n+1}$. By means of the n-forms
$$d\hat{x}_i=dx_0\wedge\cdots\wedge dx_{i-1}\wedge dx_{x_{i+1}}\wedge \cdots \wedge dx_n,~i=0,1,...,n,$$
an $\A$-valued n-form is introduced by putting
$$d\sigma=\sum_{i=0}^{n}(-1)^ie_id\hat{x}_i,$$ similarly, denote
$$d\bar{\sigma}=\sum_{i=0}^{n}(-1)^i\bar{e}_id\hat{x}_i.$$
Furthermore the volume-element $$dx=dx_0\wedge\cdots\wedge dx_n$$ is used.
\begin{prop}{\bf (Stokes-Green Theorem)}\cite{c} If $f,g\in C^1(\Sigma,\A)$ then for any (n+1)-chain $\Omega$ on $M\subset \Sigma$,
$$\int_{\partial\Omega}fd\sigma g=\int_\Omega(f\overline{D})gdx+\int_\Omega f(\overline{D}g)dx,$$
$$\int_{\partial\Omega}fd\bar{\sigma} g=\int_\Omega(fD)gdx+\int_\Omega f(Dg)dx.$$
\end{prop}
\begin{rem}
Denote $C^\infty_0(\Omega,\R)$ as the set of all smooth real-valued functions with compact support in $\Omega$ and $C^\infty_0(\Omega,\A):=\{f|f:\Omega\rightarrow \A,~f=\sum\limits_Af_Ae_A,~f_A\in C^\infty_0(\Omega,\R)\}.$ If $f$ or $g\in C^\infty_0(\Omega,\A)$, then we have from the Stokes-Green theorem that
$$\int_\Omega(f\overline{D})gdx=-\int_\Omega f(\overline{D}g)dx,$$
$$\int_\Omega(fD)gdx=-\int_\Omega f(Dg)dx.$$
\end{rem}
\begin{lem}
If $u(x)\in C^1(\Omega,\A)$, then $\overline{\overline{D}u}=\bar{u}D$.
\end{lem}
\begin{proof}
Let $u(x)=\sum_{A}e_Au_A$. Then
\begin{equation} \label{eq:4}
\begin{split}
\overline{\overline{D}u}=\sum_{i,A}\overline{e_ie_A}\partial_{x_i}u_A
=\sum_{i,A}\bar{e}_A\bar{e}_i\partial_{x_i}u_A
=\bar{u}D.
\end{split}\nonumber
\end{equation}
\end{proof}
\begin{lem}\cite{c2}
If $u(x)=\sum_{A}e_Au_A$, $v(x)=\sum_{i=0}^{n}e_iv_i$, then
$$\overline{D}(uv)=(\overline{D}u)v+u(\overline{D}v)+\sum\limits^n_{j=1}(e_ju-ue_j)\partial_{x_j} v.$$
\end{lem}
\subsection{Weak solutions}
Let $L_{loc}^1(\Omega,\A):=\{f|f:\Omega\rightarrow \A, ~f=\sum\limits_Af_Ae_A,~f_A\in L_{loc}^1(\Omega,\R)\}$. Then we define the weak solution in the sense of Clifford algebra as follows.
\begin{defn}\label{defn1}{\bf ($\overline{D}$ solution in weak sense)}
If $f\in L_{loc}^1(\Omega,\A)$, $u:\Omega \rightarrow \A$ is a weak solution of
$$\overline{D}u=f ~(\mbox{or}~{D}u=f)$$
if for any $\alpha\in C^\infty_0(\Omega,\A)$,
$$\int_{\Omega}\alpha f dx=-\int_{\Omega}(\alpha\overline{D})udx~(\mbox{or}~\int_{\Omega}\alpha f dx=-\int_{\Omega}(\alpha {D})udx).$$
\end{defn}
It should be noticed that if $u$ is a weak solution of Dirac equation $\overline{D}u=0$, in addition, if $u$ is smooth in $\Omega$, then it is left-monogenic.
Now it is natural to give the definition of $\Delta$ solution in the weak sense.
\begin{defn}\label{defn1.1}{\bf ($\Delta$ solution in weak sense)}
If $f\in L_{loc}^1(\Omega,\A)$, $u:\Omega \rightarrow \A$ is a weak solution of
$$\Delta u=f$$
if for any $\alpha\in C^\infty_0(\Omega,\A)$,
$$\int_{\Omega}\alpha f dx=\int_{\Omega}({\Delta}\alpha)udx.$$
\end{defn}
\begin{thm}\label{thm4}
If $f\in L_{loc}^1(\Omega,\A)$, and $\overline{D}f=0$ in weak sense, then $f$ is left-monogenic at any point of $\Omega$.
\end{thm}
\begin{proof}: Since $\overline{D}f=0$ in weak sense, then $\Delta f=0$ in weak sense. By Weyl's lemma, $f$ is smooth in $\Omega$ and has
$\Delta f=0$ in classical sense, then of course $f$ is left-monogenic at any point of $\Omega$.
\end{proof}
\begin{rem}\label{rem1}
This is useful to deal with uniqueness of weak solutions.
for example, if $ u,~ v\in L_{loc}^1(\Omega,\A)$ are two weak solutions of $\overline{D }u=f$, then $ u=v+w$ with any $w$ left-monogenic.
\end{rem}
\begin{rem}
An important example of a left monogenic function is the generalized Cauchy kernel $$G(x)=\frac{1}{\omega_{n+1}}\frac{\overline{x}}{|x|^{n+1}},$$
where $\omega_{n+1}$ denotes the surface area of the unit ball in $\R^{n+1}$. This function obviously belongs to $L_{loc}^1(\Omega,\A)$ and is a fundamental solution of the Dirac equation in the classical sense at any point of $\R^{n+1}$ except 0. However, it is not a weak solution of the Dirac operator. In fact, if it satisfies $\overline{D}f=0$ in the weak sense, then from Theorem \ref{thm4}, it must
be left-monogenic in the any point of $\Omega$ which could include $0$. Therefore, we get a contradiction.
\end{rem}
For $f\in L^2(\Omega,\A,\varphi)$, $u:\Omega \rightarrow \A$. If $\overline{D}u=f$, based on the Stokes-Green theorem, we can define the dual operator $\overline{D}^*_\varphi$ of $\overline{D}$ under the inner product of $L^2(\Omega,\A,\varphi)$. For any $\alpha\in C^\infty_0(\Omega,\A)$,
\begin{equation}\label{7}
\begin{split}
(\alpha,f)_\varphi=&~\int_\Omega\bar{\alpha}fe^{-\varphi}dx=\int_\Omega\bar{\alpha}e^{-\varphi}fdx\\
=&~\int_\Omega(\bar{\alpha}e^{-\varphi})(\overline{D}u)dx\\
=&~-\int_\Omega\big((\bar{\alpha}e^{-\varphi})\overline{D}\big)udx\\
=&~-\int_\Omega\big((\bar{\alpha}e^{-\varphi})\overline{D}\big)e^\varphi ue^{-\varphi}dx\\
=&~\int_\Omega\overline{-e^{\varphi}D(\alpha e^{-\varphi})}ue^{-\varphi}dx\\
=&~(-e^{-\varphi}D(\alpha e^{-\varphi}),u)_\varphi\triangleq(\overline{D}^*_\varphi\alpha,u)_\varphi,
\end{split}
\end{equation}
where $\overline{D}^*_\varphi\alpha=-e^\varphi D(\alpha e^{-\varphi})=\alpha (D\varphi)-D\alpha$,
i.e. $$(\alpha,\overline{D}u)_\varphi=(\overline{D}^*_\varphi\alpha,u)_\varphi.$$ In the same way, we also have
$$(\overline{D}u,\alpha)_\varphi=(u,\overline{D}^*_\varphi\alpha)_\varphi.$$
\section{The proof of Theorem \ref{thm1}}
Now we are in the position of proving Theorem \ref{thm1}.
\begin{proof}($Sufficiency$) From the definition of dual operator and Cauchy-Schwarz inequality in Proposition \ref{prop1}, we have
\begin{equation}
\begin{split}
|(f,\alpha)_\varphi|^2_0
=&|(\overline{D}u,\alpha)_\varphi|^2_0
=|(u,\overline{D}^*_\varphi\alpha)_\varphi|^2_0\\
\leq&~\|u\|^2\cdot\|\overline{D}^*_\varphi\alpha\|^2\\
\leq&~c\cdot\|\overline{D}^*_\varphi\alpha\|^2.\nonumber
\end{split}
\end{equation}
~\\
($necessity$) We aim to prove the necessity with Riesz representation theorem. First, we denote the submodule
$$E=\{\overline{D}^*_\varphi\alpha,~\alpha\in C^\infty_0(\Omega,\A),~\varphi\in C^2(\Omega,\R)\}\subset L^2(\Omega,\A,\varphi).$$
Then we define a linear functional $L_f$ on $E$, i.e., $L_f\in E^{*alg}$ for a fixed $f\in L^2(\Omega,\A,\varphi)$ as follows,
$$\langle L_f,\overline{D}^*_\varphi\alpha\rangle=(f,\alpha)_\varphi=\int_\Omega\bar{f}\cdot\alpha\cdot e^{-\varphi}dx\in \A.$$
From (\ref{eq:5}), we have
$$|\langle L_f,\overline{D}^*_\varphi\alpha\rangle|_0=|(f,\alpha)_\varphi|_0\leq\sqrt{c}\cdot\|\overline{D}^*_\varphi\alpha\|,$$
which meas that $L_f$ is a bounded functional from Definition \ref{bounded}. By the Hahn-Banach type theorem in Theorem \ref{Hahn}, $L_f$ can be extended to a linear functional $\widetilde{L}_f$ on $L^2(\Omega,\A,\varphi)$, and with
\begin{equation}\label{eq:6}
\begin{split}|\langle \widetilde{L}_f,g\rangle|_0\leq\sqrt{c^*}\|g\|,~\forall g\in L^2(\Omega,\A,\varphi),\end{split}
\end{equation}
where $\sqrt{c^*}=\sqrt{c}\cdot|e_0|_0$, {since} $|e_A|_0=2^{n/2}$, then $c^*=2^{n}c$ from \cite{c}. Now we are in the position to use the Riesz representation theorem for the operator $\widetilde{L}_f$. From Theorem \ref{Riesz}, there exists a $u\in L^2(\Omega,\A,\varphi)$ such that
\begin{equation}\label{eq:7}
\begin{split}\langle \widetilde{L}_f,g\rangle=(u,g)_\varphi,~\forall g\in L^2(\Omega,\A,\varphi).\end{split}
\end{equation}
For $\forall \alpha\in C^\infty_0(\Omega,\A)$, let $g=\overline{D}^*_\varphi\alpha$. Then
\begin{equation}\label{}
\begin{split}
(f,\alpha)_\varphi=&\langle \widetilde{L}_f,\overline{D}^*_\varphi\alpha\rangle =(u,\overline{D}^*_\varphi\alpha)_\varphi=(\overline{D}u,\alpha)_\varphi,\nonumber
\end{split}
\end{equation}
which deduces that
$$\int_\Omega\bar{f}\alpha e^{-\varphi}dx=\int_\Omega\overline{(\overline{D}u)}{\alpha} e^{-\varphi}dx.$$ Conjugating both sides of above equation leads to
$$\int_\Omega\bar{\alpha}f \cdot e^{-\varphi}dx=\int_\Omega\bar{\alpha} (\overline{D})u e^{-\varphi}dx.$$
Let $\alpha=\bar{\alpha}e^{\varphi}$, it can be obtained that
$$\int_\Omega\alpha fdx=\int_\Omega\alpha (\overline{D}u)dx,~\forall \alpha\in C^\infty_0(\Omega,\A).$$
Therefore, $$\overline{D}u=f$$ is proved from the definition of weak solutions.
Next, we give the bound for the norm of $u$. Let $g=u=\sum_{A}e_Au_A\in L^2(\Omega,\A,\varphi)$, from (\ref{eq:6}) and (\ref{eq:7}), we get that
\begin{equation}\label{eq:8}
\begin{split}|(u,u)_\varphi|_0\leq\sqrt{c^*}\|u\|.\end{split}
\end{equation}
On the other hand,
\begin{equation}
\begin{split}
|(u,u)_\varphi|_0^2=&\big|\int_\Omega\bar{u}ue^{-\varphi}dx\big|^2_0\\
=&~2^n\cdot\big[\int_\Omega\bar{u}ue^{-\varphi}dx\cdot\overline{\int_\Omega\bar{u}ue^{-\varphi}dx}\big]_0\\
=&~2^n\big[\int_\Omega(\sum\limits_Au^2_A+\sum\limits_{A\neq B}\bar{e}_Ae_Bu_Au_B)e^{-\varphi}dx\cdot\overline{\int_\Omega(\sum\limits_Au^2_A+\sum\limits_{A\neq B}\bar{e}_Ae_Bu_Au_B)e^{-\varphi}dx}\big]_0\\
=&~2^n\big[(\int_\Omega\sum\limits_Au^2_Ae^{-\varphi}dx)^2+(\int_\Omega\sum\limits_{A\neq B}u_Au_Be^{-\varphi}dx)^2\big],
\end{split}\nonumber
\end{equation}
and
\begin{equation}
\begin{split}
\|u\|^2=&~\int_\Omega|u|^2_0e^{-\varphi}dx
=2^n\int_\Omega[\bar{u}u]_0e^{-\varphi}dx
=2^n\int_\Omega\sum\limits_Au^2_A\cdot e^{-\varphi}dx
\end{split}\nonumber
\end{equation}
So we have $\|u\|^4=2^{2n}\cdot(\int_\Omega\sum\limits_Au^2_A\cdot e^{-\varphi}dx)^2$. Hence,
$$|(u,u)_\varphi|_0^2=2^n[(\int_\Omega\sum\limits_Au^2_A\cdot e^{-\varphi}dx)^2+(\int_\Omega\sum\limits_{A\neq B}u_Au_Be^{-\varphi}dx)^2]\geq 2^{-n}\|u\|^4.$$
Combining with (\ref{eq:8}), it is obtained that
$$\|u\|^2\leq 2^{n/2}|(u,u)_\varphi|_0\leq2^{n/2}\sqrt{c^*}\|u\|,$$ and
$$\|u\|^2\leq 2^{2n} {c}.$$
The proof is completed.
\end{proof}
\section{The proof of Theorem \ref{thm2}}
It should be noticed that inequality (\ref{eq:5}) in Theorem \ref{thm1} is related with $\alpha\in C^\infty_0(\Omega,\A)$. In the following, we will give another sufficient condition that has nothing to do with the space $C^\infty_0(\Omega,\A)$. First, we need to compute the norm of $\|\overline{D}^*_\varphi\alpha\|$ for any $\alpha\in C^\infty_0(\Omega,\A).$
\begin{equation}
\begin{split}
\|\overline{D}^*_\varphi\alpha\|^2=&\int_\Omega|\overline{D}^*_\varphi\alpha|^2_0e^{-\varphi}dx\\
=&\int_\Omega\langle\tau_{e_0},\overline{\overline{D}^*_\varphi\alpha}\cdot\overline{D}^*_\varphi\alpha\rangle e^{-\varphi}dx\\
=&\langle\tau_{e_0},\int_\Omega\overline{\overline{D}^*_\varphi\alpha}\cdot\overline{D}^*_\varphi\alpha e^{-\varphi}dx\rangle\\
=&\langle\tau_{e_0},(\overline{D}^*_\varphi\alpha,\overline{D}^*_\varphi\alpha)_\varphi\rangle\\
=&\langle\tau_{e_0},(\alpha,\overline{D}\overline{D}^*_\varphi\alpha)_\varphi\rangle\\
=&\langle\tau_{e_0},(\alpha,\overline{D}(\alpha (D\varphi)-D\alpha))_\varphi\rangle\\
=&\langle\tau_{e_0},(\alpha,\overline{D}\alpha (D\varphi)+\alpha\Delta\varphi-\Delta\alpha+\sum\limits^n_{j=1}(e_j\alpha-\alpha e_j)\frac{\partial}{\partial x_j}(D\varphi))_\varphi\rangle\\
=&\langle\tau_{e_0},(\alpha,\overline{D}^*_\varphi(\overline{D}\alpha)+\alpha\Delta\varphi+\sum\limits^n_{j=1}(e_j\alpha-\alpha e_j)\frac{\partial}{\partial x_j}(D\varphi))_\varphi\rangle\\
=&\langle\tau_{e_0},(\alpha,\overline{D}^*_\varphi(\overline{D}\alpha))_\varphi+(\alpha,\alpha\Delta\varphi)_\varphi+(\alpha,\sum\limits^n_{j=1}(e_j\alpha-\alpha e_j)\frac{\partial}{\partial x_j}(D\varphi))_\varphi\rangle\\
=&\langle\tau_{e_0},(\alpha,\overline{D}^*_\varphi(\overline{D}\alpha))_\varphi\rangle+\langle\tau_{e_0},(\alpha,\alpha\Delta\varphi)_\varphi\rangle+\langle\tau_{e_0},(\alpha,\sum\limits^n_{j=1}(e_j\alpha-\alpha e_j)\frac{\partial}{\partial x_j}(D\varphi))_\varphi\rangle\\
=&I_1+I_2+I_3,\nonumber
\end{split}
\end{equation}
where
\begin{equation}
\begin{split}
I_1=&\langle\tau_{e_0},(\alpha,\overline{D}^*_\varphi(\overline{D}\alpha))_\varphi\rangle=\langle\tau_{e_0},(\overline{D}\alpha,\overline{D}\alpha)_\varphi\rangle=\|\overline{D}\alpha\|^2,\\
I_2=&\langle\tau_{e_0},(\alpha,\alpha\Delta\varphi)_\varphi\rangle=\int_{\Omega}|\alpha|^2_0\Delta\varphi e^{-\varphi}dx, \nonumber
\end{split}
\end{equation}
and
\begin{equation}
\begin{split}
I_3=&\langle\tau_{e_0},(\alpha,\sum\limits^n_{j=1}(e_j\alpha-\alpha e_j)\frac{\partial}{\partial x_j}(D\varphi))_\varphi\rangle\\
=&\langle\tau_{e_0},(\alpha,\sum\limits^n_{j=1}(e_j\alpha-\alpha e_j)\frac{\partial}{\partial x_j}(\sum_{i=0}^{n}\bar{e}_i\frac{\partial \varphi}{\partial x_i}))_\varphi\rangle\\
=&\langle\tau_{e_0},(\alpha,\sum\limits^n_{j=1}\sum_{i=0}^{n}(e_j\alpha \bar{e}_i-\alpha e_j \bar{e}_i)\frac{\partial^2 \varphi}{\partial x_j\partial x_i})_\varphi\rangle\\
=&\langle\tau_{e_0},\int_\Omega \bar{\alpha}\sum\limits^n_{j=1}\sum_{i=0}^{n}(e_j\alpha \bar{e}_i-\alpha e_j \bar{e}_i)\frac{\partial^2 \varphi}{\partial x_j\partial x_i} e^{-\varphi}dx\rangle\\
=&\int_\Omega\langle\tau_{e_0}, \bar{\alpha}\sum\limits^n_{j=1}\sum_{i=0}^{n}(e_j\alpha \bar{e}_i-\alpha e_j \bar{e}_i)\frac{\partial^2 \varphi}{\partial x_j\partial x_i}\rangle e^{-\varphi}dx.
\end{split}\nonumber
\end{equation}
{\bf It should be noticed that if $n=1$, i.e., the space $\R^2$ is considered, then $I_3=0.$}
Since for $1\leq i,j\leq n$ and $i\neq j$, $e_j \bar{e}_i=-e_j {e}_i=e_i{e}_j=- {e}_i\bar{e}_j$. For simplicity, let
\begin{equation}
\begin{split}
I_4=&\langle\tau_{e_0},\bar{\alpha}\sum\limits^n_{j=1}\sum\limits_{i=0}^{n}(e_j\alpha \bar{e}_i-\alpha e_j \bar{e}_i)\frac{\partial^2 \varphi}{\partial x_j\partial x_i}\rangle\\
=&\langle\tau_{e_0},\sum\limits^n_{j=1}\sum\limits_{i=1}^{n}(\bar{\alpha}e_j\alpha \bar{e}_i-\bar{\alpha}\alpha e_j \bar{e}_i)\frac{\partial^2 \varphi}{\partial x_j\partial x_i}\rangle+\langle\tau_{e_0},\sum\limits^n_{j=1}(\bar{\alpha}e_j\alpha \bar{e}_0-\bar{\alpha}\alpha e_j \bar{e}_0)\frac{\partial^2 \varphi}{\partial x_j\partial x_0}\rangle\\
=&\langle\tau_{e_0},\sum\limits_{i=1}^{n}(\bar{\alpha}e_i\alpha \bar{e}_i-\bar{\alpha}\alpha e_i \bar{e}_i)\frac{\partial^2 \varphi}{\partial x^2_i}\rangle+\langle\tau_{e_0},\sum\limits^n_{j\neq i}(\bar{\alpha}e_j\alpha \bar{e}_i)\frac{\partial^2 \varphi}{\partial x_j\partial x_i}\rangle\\
&+\langle\tau_{e_0},\sum\limits^n_{j=1}(\bar{\alpha}e_j\alpha \bar{e}_0-\bar{\alpha}\alpha e_j \bar{e}_0)\frac{\partial^2 \varphi}{\partial x_j\partial x_0}\rangle\\
=&\langle\tau_{e_0},\sum\limits_{i=1}^{n}(\bar{\alpha}e_i\alpha \bar{e}_i-\bar{\alpha}\alpha )\frac{\partial^2 \varphi}{\partial x^2_i}\rangle+\langle\tau_{e_0},\sum\limits^n_{j\neq i}(\bar{\alpha}e_j\alpha \bar{e}_i)\frac{\partial^2 \varphi}{\partial x_j\partial x_i}\rangle\\
&+\langle\tau_{e_0},\sum\limits^n_{j=1}(\bar{\alpha}e_j\alpha \bar{e}_0-\bar{\alpha}\alpha e_j \bar{e}_0)\frac{\partial^2 \varphi}{\partial x_j\partial x_0}\rangle\\
=&I_5+I_6+I_7.
\end{split}\nonumber
\end{equation}
Assume $\alpha=\sum\limits_A\alpha_Ae_A\in \A,~\bar{\alpha}=\sum\limits_A\alpha_A\bar{e}_A$, then for any $1\leq i\leq n,$
\begin{equation}
\begin{split}
\bar{\alpha}e_i\alpha\bar{e}_i=&~\sum\limits_A\alpha_A\bar{e}_Ae_i\cdot\sum\limits_A\alpha_Ae_A\bar{e}_i\\
=&~\sum\limits_A(-1)^{\frac{|A|(|A|+1)}{2}}\alpha_Ae_Ae_i\cdot\sum\limits_A(-1)\alpha_Ae_Ae_i
\end{split}\nonumber
\end{equation}
Therefore
\begin{equation}
\begin{split}
I_5=&\langle\tau_{e_0},\sum\limits_{i=1}^{n}(\bar{\alpha}e_i\alpha \bar{e}_i-\bar{\alpha}\alpha )\frac{\partial^2 \varphi}{\partial x^2_i}\rangle\\
=&\langle\tau_{e_0},\sum\limits_{i=1}^{n}(\bar{\alpha}e_i\alpha \bar{e}_i)\frac{\partial^2 \varphi}{\partial x^2_i}\rangle-\langle\tau_{e_0},\sum\limits_{i=1}^{n}(\bar{\alpha}\alpha )\frac{\partial^2 \varphi}{\partial x^2_i}\rangle\\
=&\langle\tau_{e_0},\sum\limits_{i=1}^{n}(\sum\limits_A(-1)^{\frac{|A|(|A|+1)}{2}}\alpha_Ae_Ae_i\cdot\sum\limits_A(-1)\alpha_Ae_Ae_i)\frac{\partial^2 \varphi}{\partial x^2_i}\rangle-\langle\tau_{e_0},\sum\limits_{i=1}^{n}(\bar{\alpha}\alpha )\frac{\partial^2 \varphi}{\partial x^2_i}\rangle\\
=&2^n\sum\limits_{i=1}^{n}(\sum\limits_A(-1)^{\frac{|A|(|A|+1)}{2}+1}\alpha_A^2 e_Ae_ie_Ae_i)\frac{\partial^2 \varphi}{\partial x^2_i}-\sum\limits_{i=1}^{n}|\alpha|^2_0\frac{\partial^2 \varphi}{\partial x^2_i}\\
=&2^n\sum\limits_{i=1}^{n}(\sum\limits_{i\not\in A}(-1)^{\frac{|A|(|A|+1)}{2}+1}\alpha^2_A\cdot\overline{e_Ae_i}\cdot e_Ae_i\cdot
(-1)^{\frac{(|A|+1)(|A|+2)}{2}}\\
&+\sum\limits_{i\in A}(-1)^{\frac{|A|(|A|+1)}{2}+1}\cdot\alpha^2_A\cdot\overline{e_{A-{i}}}\cdot
e_{A-{i}}\cdot(-1)^{\frac{(|A|-1)(|A|)}{2}})\frac{\partial^2 \varphi}{\partial x^2_i}-\sum\limits_{i=1}^{n}|\alpha|^2_0\frac{\partial^2 \varphi}{\partial x^2_i}\\
=&2^n\sum\limits_{i=1}^{n}(\sum\limits_{i\not\in A}(-1)^{\frac{|A|(|A|+1)}{2}+1+\frac{(|A|+1)(|A|+2)}{2}}\cdot\alpha^2_A\\
&+\sum\limits_{i\in A}(-1)^{\frac{|A|(|A|+1)}{2}+1+\frac{(|A|-1)(|A|)}{2}}\cdot\alpha^2_A)\frac{\partial^2 \varphi}{\partial x^2_i}-\sum\limits_{i=1}^{n}|\alpha|^2_0\frac{\partial^2 \varphi}{\partial x^2_i}\\
=&2^n\sum\limits_{i=1}^{n}(\sum\limits_{i\not\in A}(-1)^{|A|^2}\cdot\alpha^2_A+\sum\limits_{i\in A}(-1)^{|A|^2+1}\cdot\alpha^2_A)\frac{\partial^2 \varphi}{\partial x^2_i}-\sum\limits_{i=1}^{n}|\alpha|^2_0\frac{\partial^2 \varphi}{\partial x^2_i}\\
=&2^n\sum\limits_{i=1}^{n}(\sum\limits_{i\not\in A,|A|^2 ~\mbox{is odd}}(-2)\alpha^2_A+\sum\limits_{i\in A,|A|^2 ~\mbox{is even}}(-2)\alpha^2_A)\frac{\partial^2 \varphi}{\partial x^2_i}\\
=&-2^{n+1}\sum\limits_{i=1}^{n}(\sum\limits_{i\not\in A,|A|^2 ~\mbox{is odd}}\alpha^2_A+\sum\limits_{i\in A,|A|^2 ~\mbox{is even}}\alpha^2_A)\frac{\partial^2 \varphi}{\partial x^2_i}.
\end{split}
\end{equation}
To consider $I_7$, we first study $\bar{\alpha}e_j\alpha$ for any $1\leq j\leq n$. Without loss of generality, let $e_j=e_1,~\bar{\alpha}=\sum\limits_A\alpha_A\bar{e}_A,~
\alpha=\sum\limits_A\alpha_Ae_A$. Then $\bar{\alpha}e_1\alpha=(\sum\limits_A\alpha_A\bar{e}_A)e_1(\sum\limits_A\alpha_Ae_A)$.
When $e_A=e_1e_{h_2}e_{h_3}\cdots e_{h_r}$,~where $1<h_2<h_3<\cdots<h_r$ and $1<r\leq n.$
\begin{equation}\label{16}
\begin{split}
\alpha_A\bar{e}_Ae_1=&\alpha_{1h_2\cdots h_r}(-1)^{\frac{r(r+1)}{2}}\cdot e_1e_{h_2}e_{h_3}\cdots e_{h_r}\cdot e_1
\\=&\alpha_{1h_2\cdots h_r}(-1)^{\frac{r(r+1)}{2}+r}e_{h_2}e_{h_3}\cdots e_{h_r}\\
\alpha_Ae_Ae_1=&\alpha_{1h_2\cdots h_r}e_1e_{h_2}\cdots e_{h_r}\cdot e_1=\alpha_{1h_2\cdots h_r}(-1)^re_{h_2}\cdots e_{h_r}.
\end{split}
\end{equation}
When $e_A=e_1$,
\begin{equation}\label{165}
\begin{split}
\alpha_A\bar{e}_Ae_1=&\alpha_{1}\\
\alpha_Ae_Ae_1=&-\alpha_{1}.
\end{split}
\end{equation}
When $e_A=e_{h_2}e_{h_3}\cdots e_{h_r}$,~where $1<h_2<h_3<\cdots<h_r$ and $1<r\leq n.$
\begin{equation}\label{17}
\begin{split}
\alpha_A\bar{e}_Ae_1=&\alpha_{h_2\cdots h_r}(-1)^{\frac{(r-1)(r)}{2}}\cdot e_{h_2}e_{h_3}\cdots e_{h_r}\cdot e_1
\\=&\alpha_{h_2\cdots h_r}(-1)^{\frac{(r-1)(r)}{2}+r-1}e_1e_{h_2}\cdots e_{h_r}\\
\alpha_Ae_Ae_1=&\alpha_{h_2\cdots h_r}e_{h_2}\cdots e_{h_r}\cdot e_1=\alpha_{h_2\cdots h_r}(-1)^{r-1}e_1e_{h_2}\cdots e_{h_r}.
\end{split}
\end{equation}
When $e_A=e_0$,
\begin{equation}\label{175}
\begin{split}
\alpha_A\bar{e}_Ae_1=&\alpha_{0}e_1\\
\alpha_Ae_Ae_1=&\alpha_{0}e_1.
\end{split}
\end{equation}
To compute $I_7$, one needs to know the coefficient for $e_0$ of $\bar{\alpha}e_1\alpha-\bar{\alpha}\alpha e_1$. It means that we should find out the corresponding terms of $e_1e_{h_2}e_{h_3}\cdots e_{h_r}$ and $e_{h_2}\cdots e_{h_r}$ in $\bar{\alpha}e_1$ and $\alpha$, in $\bar{\alpha}$ and $\alpha e_1$.
{\bf Case a1.} For $\bar{\alpha}e_1\alpha$, from (\ref{17}), the corresponding terms of $e_1e_{h_2}e_{h_3}\cdots e_{h_r}$ with $1<h_2<h_3<\cdots<h_r$ and $1<r\leq n$ in $\bar{\alpha}e_1=(\sum\limits_A\alpha_A\bar{e}_A)e_1$ and $\alpha=\sum\limits_A\alpha_Ae_A$ are $\alpha_{h_2\cdots h_r}(-1)^{\frac{(r-1)(r)}{2}+r-1}e_1e_{h_2}\cdots e_{h_r}$ and $\alpha_{1h_2\cdots h_r}e_1e_{h_2}\cdots e_{h_r}$, respectively. Multiplying these terms leads to
\begin{equation}\label{18}
\begin{split}
(-1)&^{\frac{(r-1)(r)}{2}+r-1}e_1e_{h_2}\cdots e_{h_r}\cdot e_1e_{h_2}\cdots e_{h_r}\cdot\alpha_{1h_2\cdots h_r}\cdot\alpha_{h_2\cdots h_r}\\
=&~(-1)^{\frac{(r-1)(r)}{2}+r-1}(-1)^{\frac{(r)(r+1)}{2}}\cdot\overline{e_1\cdots e_{h_r}}\cdot e_1e_{h_2}\cdots e_{h_r} \cdot\alpha_{1h_2\cdots h_r}\alpha_{h_2\cdots h_r}\\
=&~(-1)^{\frac{(r)(r+1)}{2}+r-1+\frac{(r-1)(r)}{2}}\cdot\alpha_{1h_2\cdots h_r}\alpha_{h_2\cdots h_r}.
\end{split}
\end{equation}
On the other hand, for $\bar{\alpha}e_1\alpha$, from (\ref{16}), the corresponding terms of $e_{h_2}e_{h_3}\cdots e_{h_r}$ with $1<h_2<h_3<\cdots<h_r$ and $1<r\leq n$ in $\bar{\alpha}e_1$ and $\alpha$ are $\alpha_{1h_2\cdots h_r}(-1)^{\frac{r(r+1)}{2}+r}e_{h_2}e_{h_3}\cdots e_{h_r}$ and $\alpha_{h_2\cdots h_r}e_{h_2}\cdots e_{h_r}$, respectively. Multiplying these terms leads to
\begin{equation}\label{19}
\begin{split}
(-1)&^{\frac{(r)(r+1)}{2}+r}e_{h_2\cdots h_r}\cdot e_{h_2\cdots h_r}\cdot\alpha_{1h_2\cdots h_r}
\cdot\alpha_{h_2\cdots h_r}\\
=&~(-1)^{\frac{(r)(r+1)}{2}+r}(-1)^{\frac{(r-1)(r)}{2}}\cdot\overline{e_{h_2\cdots h_r}}\cdot e_{h_2\cdots h_r} \cdot\alpha_{1h_2\cdots h_r}\alpha_{h_2\cdots h_r}\\
=&~(-1)^{\frac{(r)(r+1)}{2}+r+\frac{(r-1)(r)}{2}}\cdot\alpha_{1h_2\cdots h_r}\alpha_{h_2\cdots h_r}.
\end{split}
\end{equation}
From (\ref{18}) and (\ref{19}), these two terms vanish.
{\bf Case a2.} For $\bar{\alpha}e_1\alpha$, from (\ref{175}), the corresponding terms of $e_1$ in $\bar{\alpha}e_1$ and $\alpha$ are $\alpha_{0}e_1$ and $\alpha_{1}e_1$, respectively. Multiplying these terms leads to
\begin{equation}\label{185}
\begin{split}
\alpha_{0}e_1\alpha_{1}e_1=-\alpha_{0}\alpha_{1}.
\end{split}
\end{equation}
On the other hand, for $\bar{\alpha}e_1\alpha$, from (\ref{165}), the corresponding terms of $e_{0}$ in $\bar{\alpha}e_1$ and $\alpha$ are $\alpha_{1}$ and $\alpha_{0}$, respectively. Multiplying these terms leads to $\alpha_{0}\alpha_{1}$. Combining with (\ref{185}), these two terms also vanish.
From Cases a1 and a2, one can obtain that the coefficient for $e_0$ of $\bar{\alpha}e_1\alpha$ equals zero, i.e.,
\begin{equation}\label{23}
\begin{split}\langle\tau_{e_0},\sum\limits^n_{j=1}(\bar{\alpha}e_j\alpha \bar{e}_0)\frac{\partial^2 \varphi}{\partial x_j\partial x_0}\rangle=0.\end{split}
\end{equation}
{\bf Case b1.} For $\bar{\alpha}\alpha e_1$, from (\ref{17}), the corresponding terms of $e_1e_{h_2}e_{h_3}\cdots e_{h_r}$ with $1<h_2<h_3<\cdots<h_r$ and $1<r\leq n$ in ${\alpha}e_1=(\sum\limits_A\alpha_A{e}_A)e_1$ and $\bar{\alpha}=\sum\limits_A\alpha_A\bar{e}_A$ are $\alpha_{h_2\cdots h_r}(-1)^{r-1}e_1e_{h_2}\cdots e_{h_r}$ and $\alpha_{1h_2\cdots h_r}\overline{e_1e_{h_2}\cdots e_{h_r}}$, respectively. Multiplying these terms leads to
\begin{equation}\label{25}
\begin{split}
(\alpha_{1h_2\cdots h_r}&\overline{e_1e_{h_2}\cdots e_{h_r}})\cdot(\alpha_{h_2\cdots h_r}e_{h_2}\cdots e_{h_r}\cdot e_1)\\
=&~(\alpha_{1h_2\cdots h_r}\overline{e_1e_{h_2}\cdots e_{h_r}})\cdot((-1)^{r-1}e_1e_{h_2}\cdots e_{h_r}\cdot \alpha_{h_2\cdots h_r})\\
=&~(-1)^{r-1}\alpha_{1h_2\cdots h_r}\cdot\alpha_{h_2\cdots h_r}.
\end{split}
\end{equation}
On the other hand, for $\bar{\alpha}\alpha e_1$, from (\ref{16}), the corresponding terms of $e_{h_2}e_{h_3}\cdots e_{h_r}$ with $1<h_2<h_3<\cdots<h_r$ and $1<r\leq n$ in ${\alpha}e_1$ and $\bar{\alpha}$ are $\alpha_{1h_2\cdots h_r}(-1)^re_{h_2}\cdots e_{h_r}$ and $\alpha_{h_2\cdots h_r}\overline{e_{h_2}\cdots e_{h_r}}$, respectively. Multiplying these terms leads to
\begin{equation}\label{26}
\begin{split}
(\alpha_{h_2\cdots h_r}&\overline{e_{h_2}\cdots e_{h_r}})\cdot(\alpha_{1h_2\cdots h_r}e_1\cdots e_{h_r}\cdot e_1)\\
=&~(\alpha_{h_2\cdots h_r}\overline{e_{h_2}\cdots e_{h_r}})\cdot((-1)^re_{h_2}\cdots e_{h_r}\cdot \alpha_{1h_2\cdots h_r})\\
=&~(-1)^r\alpha_{h_2\cdots h_r}\cdot\alpha_{1h_2\cdots h_r}.
\end{split}
\end{equation}
From (\ref{25}) and (\ref{26}), these two terms vanish.
{\bf Case b2.} For $\bar{\alpha}\alpha e_1$, from (\ref{175}), the corresponding terms of $e_1$ in ${\alpha}e_1$ and $\bar{\alpha}$ are $\alpha_{0}e_1$ and $\alpha_{1}\bar{e}_1$, respectively. Multiplying these terms leads to
\begin{equation}\label{1855}
\begin{split}
\alpha_{0}e_1\alpha_{1}\bar{e}_1=\alpha_{0}\alpha_{1}.
\end{split}
\end{equation}
On the other hand, for $\bar{\alpha}\alpha e_1$, from (\ref{165}), the corresponding terms of $e_{0}$ in ${\alpha}e_1$ and $\bar{\alpha}$ are $-\alpha_{1}$ and $\alpha_{0}$, respectively. Multiplying these terms leads to $-\alpha_{0}\alpha_{1}$. Combining with (\ref{1855}), these two terms also cancel.
From Cases b1 and b2, one can obtain that the coefficient for $e_0$ of $\bar{\alpha}e_1\alpha$ equals zero, i.e.,
\begin{equation}\label{24}
\begin{split}\langle\tau_{e_0},\sum\limits^n_{j=1}(\bar{\alpha}\alpha e_j\bar{e}_0)\frac{\partial^2 \varphi}{\partial x_j\partial x_0}\rangle=0.\end{split}
\end{equation}
{\bf Thus, $I_7=0$ from (\ref{23}) and (\ref{24}).}
To compute $I_6$, i.e., to get $[\bar{\alpha}e_i\alpha \bar{e}_j]_0$ for $i\neq j$, similar with the analysis of $I_7$, we should divide the vectors in $\bar{\alpha}e_i$ and $\alpha \bar{e}_j$ into four
cases.
{\bf Case c1.} $i\in A,~j\not\in A$ for $e_A$ in $\bar{\alpha}$ and $i\not\in B,~j\in B$ for $e_B$ in ${\alpha}$ with $A-{i}=B-{j}$.
For this case, firstly, we assume $e_A=e_{h_1\cdots h_{p(i)}\cdots h_r}$ and $h_{p(i)}=i$, $e_B=e_{h_1\cdots h_{p(j)}\cdots h_r}$ and $h_{p(j)}=j$.
We have
\begin{equation}\label{}
\begin{split}
\alpha_A\bar{e}_Ae_i=&\alpha_{A}(-1)^{\frac{r(r+1)}{2}}\cdot e_{h_1}\cdots e_i\cdots e_{h_r}\cdot e_i\\
=&\alpha_{A}(-1)^{\frac{r(r+1)}{2}+r-p(i)} e_{h_1}\cdots e_i^2\cdots e_{h_r},\\
=&\alpha_{A}(-1)^{\frac{r(r+1)}{2}+r-p(i)+1} e_{A-{i}},\\
\alpha_B{e}_B\bar{e}_j=&\alpha_{B} e_{h_1}\cdots e_j\cdots e_{h_r}\cdot \bar{e}_j\\
=&\alpha_{B}(-1)^{r-p(j)} e_{h_1}\cdots e_j\bar{e}_j\cdots e_{h_r},\\
=&\alpha_{B}(-1)^{r-p(j)} e_{B-{j}}.
\end{split}\nonumber
\end{equation}
Then
\begin{equation}\label{}
\begin{split}
\alpha_A\bar{e}_Ae_i\alpha_B{e}_B\bar{e}_j=&\alpha_{A}(-1)^{\frac{r(r+1)}{2}+r-p(i)+1} e_{A-{i}}\alpha_{B}(-1)^{r-p(j)} e_{B-{j}}\\
=&\alpha_{A}\alpha_{B}(-1)^{\frac{r(r+1)}{2}+r-p(i)+1+r-p(j)+\frac{r(r-1)}{2}} \overline{e_{A-{i}}} e_{B-{j}}\\
=&\alpha_{A}\alpha_{B}(-1)^{r^2+1-p(i)-p(j)}.
\end{split}
\end{equation}
{\bf Case c2.} $i\not\in A,~j\in A$ for $e_A$ in $\bar{\alpha}$ and $i\in B,~j\not\in B$ for $e_B$ in ${\alpha}$ with $A+{i}=B+{j}$.
We assume $e_A=e_{h_1\cdots h_{p(j)}\cdots h_r}$ and $h_{p(j)}=j$, $e_B=e_{h_1\cdots h_{p(i)}\cdots h_r}$ and $h_{p(i)}=i$.
We have
\begin{equation}\label{}
\begin{split}
\alpha_A\bar{e}_Ae_i=&\alpha_{A}(-1)^{\frac{r(r+1)}{2}}\cdot e_{h_1}\cdots e_j\cdots e_{h_r}\cdot e_i,\\
\alpha_B{e}_B\bar{e}_j=&\alpha_{B} e_{h_1}\cdots e_i\cdots e_{h_r}\cdot \bar{e}_j\\
=&-\alpha_{B} e_{h_1}\cdots e_i\cdots e_{h_r}\cdot{e}_j\\
=&\alpha_{B} e_{h_1}\cdots e_j\cdots e_{h_r}\cdot e_i.
\end{split}\nonumber
\end{equation}
Then
\begin{equation}\label{}
\begin{split}
\alpha_A\bar{e}_Ae_i\alpha_B{e}_B\bar{e}_j=&\alpha_{A}(-1)^{\frac{r(r+1)}{2}}\cdot e_{h_1}\cdots e_j\cdots e_{h_r}\cdot e_i\alpha_{B} e_{h_1}\cdots e_j\cdots e_{h_r}\cdot e_i\\
=&\alpha_{A}\alpha_{B}(-1)^{\frac{r(r+1)}{2}+\frac{(r+1)(r+2)}{2}} \overline{e_{h_1}\cdots e_j\cdots e_{h_r}\cdot e_i} e_{h_1}\cdots e_j\cdots e_{h_r}\cdot e_i\\
=&\alpha_{A}\alpha_{B}(-1)^{r^2+1}.
\end{split}\nonumber
\end{equation}
{\bf Case c3.} $i\in A,~j\in A$ for $e_A$ in $\bar{\alpha}$ and $i\not\in B,~j\not\in B$ for $e_B$ in ${\alpha}$ with $A-{i}=B+{j}$.
For this case, we assume $e_A=e_{h_1\cdots h_{p(i)}\cdots h_{p(j)}\cdots h_{r+2}}$ with $h_{p(i)}=i,~ h_{p(j)}=j$. Without loss of generality, we assume $i<j$. Furthermore, let $e_B=e_{h_1\cdots h_r}$.
We have
\begin{equation}\label{}
\begin{split}
\alpha_A\bar{e}_Ae_i=&\alpha_{A}(-1)^{\frac{(r+2)(r+3)}{2}}\cdot e_{h_1}\cdots e_i\cdots e_j\cdots e_{h_{r+2}}\cdot e_i\\
=&\alpha_{A}(-1)^{\frac{(r+2)(r+3)}{2}+r+2-h(i)}\cdot e_{h_1}\cdots e_j\cdots e_{h_{r+2}}\cdot e^2_i\\
=&\alpha_{A}(-1)^{\frac{(r+2)(r+3)}{2}+r+1-h(i)}\cdot e_{h_1}\cdots e_j\cdots e_{h_{r+2}}\\
=&\alpha_{A}(-1)^{\frac{(r+2)(r+3)}{2}+r+1-h(i)+r+2-h(j)}\cdot e_{h_1}\cdots e_{h_{r+2}}\cdot e_j,\\
\alpha_B{e}_B\bar{e}_j=&\alpha_{B} e_{h_1}\cdots e_{h_{r}}\cdot \bar{e}_j\\
=&-\alpha_{B} e_{h_1}\cdots e_{h_{r}}\cdot{e}_j.
\end{split}\nonumber
\end{equation}
Then
\begin{equation}\label{}
\begin{split}
\alpha_A\bar{e}_Ae_i\alpha_B{e}_B\bar{e}_j=&\alpha_{A}(-1)^{\frac{(r+2)(r+3)}{2}+r+1-h(i)+r+2-h(j)}\cdot e_{h_1}\cdots e_{h_{r+2}}\cdot e_j (-1)\alpha_{B} e_{h_1}\cdots e_{h_{r}}\cdot{e}_j \\
=&\alpha_{A}\alpha_{B} (-1)^{\frac{(r+2)(r+3)}{2}-h(i)-h(j)}\cdot e_{h_1}\cdots e_{h_{r+2}} \cdot e_j e_{h_1}\cdots e_{h_r}\cdot e_j\\
=&\alpha_{A}\alpha_{B} (-1)^{\frac{(r+2)(r+3)}{2}-h(i)-h(j)+\frac{(r+1)(r+2)}{2}}\cdot \overline{e_{h_1}\cdots e_{h_{r+2}} \cdot e_j} e_{h_1}\cdots e_{h_r}\cdot e_j\\
=&\alpha_{A}\alpha_{B} (-1)^{r^2-h(j)-h(i)}.
\end{split}\nonumber
\end{equation}
{\bf Case c4.} $i\not\in A,~j\not\in A$ for $e_A$ in $\bar{\alpha}$ and $i\in B,~j\in B$ for $e_B$ in ${\alpha}$ with $A+{i}=B-{j}$.
For this case, we assume $e_A=e_{h_1\cdots h_r}$, $e_B=e_{h_1\cdots h_{p(i)}\cdots h_{p(j)}\cdots h_{r+2}}$ with $h_{p(i)}=i,~ h_{p(j)}=j$ and $i<j$.
We have
\begin{equation}\label{}
\begin{split}
\alpha_A\bar{e}_Ae_i=&\alpha_{A}(-1)^{\frac{r(r+1)}{2}}\cdot e_{h_1}\cdots e_{h_r}\cdot e_i,\\
\alpha_B{e}_B\bar{e}_j=&\alpha_{B} e_{h_1}\cdots e_i\cdots e_j\cdots e_{h_{r+2}}\cdot \bar{e}_j\\
=&\alpha_{B} (-1)^{r+2-h(j)}\cdot e_{h_1}\cdots e_i\cdots e_{h_{r+2}}\cdot e_j \bar{e}_j\\
=&\alpha_{B} (-1)^{r+2-h(j)+r+2-h(i)-1}\cdot e_{h_1}\cdots e_{h_{r+2}}\cdot e_i \\
=&\alpha_{B} (-1)^{1-h(j)-h(i)}\cdot e_{h_1}\cdots e_{h_{r+2}}\cdot e_i \\
\end{split}\nonumber
\end{equation}
Then
\begin{equation}\label{}
\begin{split}
\alpha_A\bar{e}_Ae_i\alpha_B{e}_B\bar{e}_j=&\alpha_{A}(-1)^{\frac{r(r+1)}{2}}\cdot e_{h_1}\cdots e_{h_r}\cdot e_i\alpha_{B} (-1)^{1-h(j)-h(i)}\cdot e_{h_1}\cdots e_{h_{r+2}}\cdot e_i \\
=&\alpha_{A}\alpha_{B} (-1)^{\frac{r(r+1)}{2}+1-h(j)-h(i)}\cdot e_{h_1}\cdots e_{h_r}\cdot e_i\cdot e_{h_1}\cdots e_{h_{r+2}}\cdot e_i \\
=&\alpha_{A}\alpha_{B} (-1)^{\frac{r(r+1)}{2}+1-h(j)-h(i)+\frac{(r+1)(r+2)}{2}}\cdot \overline{e_{h_1}\cdots e_{h_r}\cdot e_i}\cdot e_{h_1}\cdots e_{h_{r+2}}\cdot e_i \\
=&\alpha_{A}\alpha_{B} (-1)^{r^2-h(j)-h(i)}.
\end{split}\nonumber
\end{equation}
Combining cases c1-c4, we have
\begin{equation}\label{}
\begin{split}
I_6=&\langle\tau_{e_0},\sum\limits^n_{j\neq i}(\bar{\alpha}e_j\alpha \bar{e}_i)\frac{\partial^2 \varphi}{\partial x_j\partial x_i}\rangle\\
=&\langle\tau_{e_0},\sum\limits^n_{j\neq i}\big((\sum_{A}\bar{e_A}\alpha_A)e_j(\sum_{B}{e_B}\alpha_B) \bar{e}_i\big)\frac{\partial^2 \varphi}{\partial x_j\partial x_i}\rangle\\
=&\langle\tau_{e_0},\sum\limits^n_{j\neq i}\big((\sum_{A}\bar{e_A}\alpha_A)e_i(\sum_{B}{e_B}\alpha_B) \bar{e}_j\big)\frac{\partial^2 \varphi}{\partial x_i\partial x_j}\rangle\\
=&\sum\limits^n_{j\neq i}\langle\tau_{e_0},(\sum_{A}\bar{e_A}\alpha_A)e_i(\sum_{B}{e_B}\alpha_B) \bar{e}_j\rangle\frac{\partial^2 \varphi}{\partial x_i\partial x_j}\\
=&\sum\limits^n_{j\neq i}\langle\tau_{e_0},(\sum_{A}\bar{e_A}\alpha_A)e_i(\sum_{B}{e_B}\alpha_B) \bar{e}_j\rangle\frac{\partial^2 \varphi}{\partial x_i\partial x_j}\\
=&2^n\sum\limits^n_{j\neq i}\Big(\sum_{i\in A,~j\not\in A;A-{i}=B-{j}}\alpha_{A}\alpha_{B}(-1)^{r^2+1-p(i)-p(j)}\\
&+\sum_{i\not\in A,~j\in A;A+{i}=B+{j}}\alpha_{A}\alpha_{B}(-1)^{r^2+1}\\
&+\sum_{i\in A,~j\in A;A-{i}=B+{j}}\alpha_{A}\alpha_{B} (-1)^{r^2-h(j)-h(i)}\\
&+\sum_{i\not\in A,~j\not\in A;A+{i}=B-{j}}\alpha_{A}\alpha_{B} (-1)^{r^2-h(j)-h(i)}\Big)\frac{\partial^2 \varphi}{\partial x_i\partial x_j}.
\end{split}\nonumber
\end{equation}
In all,
\begin{equation}
\begin{split}
I_3=&\int_\Omega I_4e^{-\varphi}dx\\
=&\int_\Omega (I_5+I_6+I_7)e^{-\varphi}dx\\
=&-2^{n+1}\int_\Omega \sum\limits_{i=1}^{n}(\sum\limits_{i\not\in A,|A|^2 ~\mbox{is odd}}\alpha^2_A+\sum\limits_{i\in A,|A|^2 ~\mbox{is even}}\alpha^2_A)\frac{\partial^2 \varphi}{\partial x^2_i}e^{-\varphi}dx\\
&+2^n\int_\Omega \sum\limits^n_{j\neq i}\Big(\sum_{i\in A,~j\not\in A;A-{i}=B-{j}}\alpha_{A}\alpha_{B}(-1)^{r^2+1-p(i)-p(j)}\\
&+\sum_{i\not\in A,~j\in A;A+{i}=B+{j}}\alpha_{A}\alpha_{B}(-1)^{r^2+1}\\
&+\sum_{i\in A,~j\in A;A-{i}=B+{j}}\alpha_{A}\alpha_{B} (-1)^{r^2-h(j)-h(i)}\\
&+\sum_{i\not\in A,~j\not\in A;A+{i}=B-{j}}\alpha_{A}\alpha_{B} (-1)^{r^2-h(j)-h(i)}\Big)\frac{\partial^2 \varphi}{\partial x_i\partial x_j} e^{-\varphi}dx.
\end{split}\nonumber
\end{equation}
Then
\begin{equation}\label{38}
\begin{split}
\|\overline{D}^*_\varphi\alpha\|^2=\|\overline{D}\alpha\|^2+\int_\Omega|\alpha|^2_0\Delta\varphi e^{-\varphi}dx
+I_3.
\end{split}
\end{equation}
If $\frac{\partial^2 \varphi}{\partial x_j\partial x_i}=0,~i\neq j,~1\leq i,j\leq n$ and $\frac{\partial^2 \varphi}{\partial x^2_i}\leq 0,~1\leq i\leq n$, we have $I_3\geq 0$, and
$$\|\overline{D}^*_\varphi\alpha\|^2\geq \int_\Omega|\alpha|^2_0\Delta\varphi e^{-\varphi}dx.$$
With the above analysis, we can prove Theorem \ref{thm2} easily.
\begin{proof}
It is sufficient to prove the theorem if condition (\ref{eq:5}) in Theorem \ref{thm1} is presented. By Cauchy-Schwarz inequality in Proposition \ref{prop1}, we have for any $\alpha\in C^\infty_0(\Omega,\A)$ that
\begin{equation}
\begin{split}
|({f},\alpha)_\varphi|^2_0=&\big|\int_\Omega\bar{f}\cdot\alpha e^{-\varphi}dx\big|^2_0\\
=&~\big|\int_\Omega\bar{f}\cdot
\frac{1}{\sqrt{\Delta\varphi}}\cdot\alpha\cdot\sqrt{\Delta\varphi}\cdot e^{-\varphi}dx\big|^2_0\\
\leq&~\big\|\bar{f}\frac{1}{\sqrt{\Delta\varphi}}\big\|^2\cdot\big\|\alpha\cdot\sqrt{\Delta\varphi}\big\|^2\\
=&~\int_\Omega\big|\frac{\bar{f}}{\sqrt{\Delta\varphi}}\big|^2_0e^{-\varphi}dx\cdot
\int_\Omega\big|\alpha\cdot\sqrt{\Delta\varphi}\big|^2_0e^{-\varphi}dx\\
\leq & c\|\overline{D}^*_\varphi\alpha\|^2.
\end{split}\nonumber
\end{equation}
The proof is completed with Theorem \ref{thm1}.
\end{proof}
It should be noticed that when $n=1$, $I_3=0$. Then it comes from equation (\ref{38}) that the H\"ormander's $L^2$ theorem in $\R^{2}$ could be described which equals the classical H\"ormander's $L^2$ theorem in $\mathbb{C}$.
\begin{cor}\label{thm3}
Given $\varphi\in C^2(\Omega,\mathbb{R})$ with $\Omega$ being an open subset of $\R^{2}$; $\Delta\varphi\geq0$. Then for all $ f\in L^2(\Omega,\A,\varphi)$ with $\int_\Omega\frac{|f|^2_0}{\Delta\varphi}e^{-\varphi}dx=c<\infty$, there exists a $u\in L^2(\Omega,\A,\varphi)$ such that $$\overline{D}u=f$$ with
$$\|u\|^2\leq\int_\Omega\frac{|f|^2_0}{\Delta\varphi}e^{-\varphi}dx.$$
\end{cor}
\section{Conclusion}
In this paper, based on the H\"ormander's $L^2$ theorem in complex analysis, the H\"ormander's $L^2$ theorem for Dirac operator in $\R^{n+1}$ has been obtained by Clifford algebra. When $n=1$, the result is equivalent to the classical H\"ormander's $L^2$ theorem in complex variable. Moreover, for any $f$ in $L^2$ space over a bounded domain with value in Clifford algebra, there is a weak solution of Dirac operator with the solution in the $L^2$ space as well. The potential applications of the results will be studied in our future work.
\begin{acknowledgements}
This work was supported by the National Natural Science Foundations of China (No. 11171255, 11101373) and Doctoral Program Foundation of the Ministry of Education of China (No. 20090072110053).
\end{acknowledgements}
| {
"timestamp": "2013-04-22T02:00:34",
"yymm": "1304",
"arxiv_id": "1304.5287",
"language": "en",
"url": "https://arxiv.org/abs/1304.5287",
"abstract": "In this paper, we give the Hörmander's $L^2$ theorem for Dirac operator over an open subset $\\Omega\\in\\R^{n+1}$ with Clifford algebra. Some sufficient condition on the existence of the weak solutions for Dirac operator has been found in the sense of Clifford analysis. In particular, if $\\Omega$ is bounded, then we prove that for any $f$ in $L^2$ space with value in Clifford algebra, there exists a weak solution of Dirac operator such that $$\\bar{D}u=f$$ with $u$ in the $L^2$ space as well. The method is based on Hörmander's $L^2$ existence theorem in complex analysis and the $L^2$ weighted space is utilised.",
"subjects": "Complex Variables (math.CV); Analysis of PDEs (math.AP)",
"title": "A variant of Hörmander's $L^2$ existence theorem for Dirac operator in Clifford analysis",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9899864276041906,
"lm_q2_score": 0.7154239836484143,
"lm_q1q2_score": 0.7082600337944526
} |
https://arxiv.org/abs/1701.01961 | Learning from MOM's principles: Le Cam's approach | We obtain estimation error rates for estimators obtained by aggregation of regularized median-of-means tests, following a construction of Le Cam. The results hold with exponentially large probability -- as in the gaussian framework with independent noise- under only weak moments assumptions on data and without assuming independence between noise and design. Any norm may be used for regularization. When it has some sparsity inducing power we recover sparse rates of convergence.The procedure is robust since a large part of data may be corrupted, these outliers have nothing to do with the oracle we want to reconstruct. Our general risk bound is of order \begin{equation*} \max\left(\mbox{minimax rate in the i.i.d. setup}, \frac{\text{number of outliers}}{\text{number of observations}}\right) \enspace. \end{equation*}In particular, the number of outliers may be as large as (number of data) $\times$(minimax rate) without affecting this rate. The other data do not have to be identically distributed but should only have equivalent $L^1$ and $L^2$ moments.For example, the minimax rate $s \log(ed/s)/N$ of recovery of a $s$-sparse vector in $\mathbb{R}^d$ is achieved with exponentially large probability by a median-of-means version of the LASSO when the noise has $q_0$ moments for some $q_0>2$, the entries of the design matrix should have $C_0\log(ed)$ moments and the dataset can be corrupted up to $C_1 s \log(ed/s)$ outliers. |
\section{Introduction}
Consider the problem of estimating minimizers of the integrated square-loss over a convex class of functions : $f^*\in\argmin_{f\in F}P(Y-f(X))^2$ based on a data set $(X_i,Y_i)_{i=1,\ldots, N}$. The labels $Y$ and $Y_i$'s are real-valued while the inputs $X$ and $X_i$'s take values in an abstract measurable space $\cX$.
Empirical Risk Minimizers (ERM) of \cite{MR1641250, MR0474638} and later on, their regularized versions replace the unknown distribution $P$ in the definition of $f^*$ by the empirical distribution $P_N$ based on the sample $(X_i,Y_i)_{i=1,\ldots, N}$. Given a function ${\rm reg}: F\to \R_+$, this produces regularized ERM defined by
\[
\hat f^{\text{RERM}}_N\in\argmin_{f\in F}\{P_N(Y-f(X))^2+{\rm reg}(f)\}\enspace.
\]
These estimators are optimal in i.i.d. subgaussian setups but suffer several drawbacks when data are heavy-tailed or corrupted by ``outliers", see \cite{MR3052407,HubRonch2009}. These issues are critical in many modern applications such as high-frequency trading, where heavy-tailed data are quite common or in various areas of biology such as micro-array analysis or neuroscience where data are sometimes still nasty after being preprocessed. To overcome the problem, various methods have been proposed. The most common strategy is to replace the square-loss function to make it less sensitive to outliers. For example, \cite{MR0161415} proposed a loss that
interpolates between square and absolute loss to produce an estimator between the unbiased (but non robust) empirical mean and the (more robust but biased) empirical median. Huber's estimators have been intensively studied asymptotically by \cite{MR0161415,HubRonch2009}, non-asymptotic results have also been obtained more recently by \cite{MR3217454,shahar_general_loss, FanLiWang2016} for example. An alternative approach has been proposed by \cite{MR3052407} and used in learning frameworks such as least-squares regression by \cite{MR2906886} and for more general loss functions by \cite{MR3405602}.
Another line of research to build robust estimators and robust selection procedures was initiated by \cite{MR0334381, MR856411} and further developed by \cite{MR2219712}, \cite{MR2834722} and \cite{BaraudBirgeSart}. It is based on \emph{comparisons} or \emph{tests} between elements of $F$. More precisely, the approach builds on tests statistics $T_N(g,f)$ comparing $f$ and $g$. These tests define the sets $\cB_{T_N}(f)$ of all $g$'s that have been preferred to $f$ and the final estimator $\hat f$ is a minimizer of the diameter of $\cB_{T_N}(f)$. The measure of diameter is directly related to statistical performances one seeks for the estimator. These methods mostly focus on Hellinger loss and are generally considered difficult to compute, see however \cite{MR3224300, MR3224298}.
In a related but different approach, \cite{LugosiMendelson2016} have recently introduced ``median-of-means tournaments". Median-of-means estimators of \cite{MR1688610, MR855970, MR702836} compare elements of $F$. A ``champion" is an element $\hat f$ such that $\cB_{T_N}(\hat f)$ is smaller than a computable upper bound on the radius of $\cB_{T_N}(f^*)$. They prove that the risk of any champion is controlled by this upper bound. An important message of this paper is that Le Cam's estimators are quite common in statistics, in particular in robust statistics. For example, Section~\ref{Sec:LFT} shows that any penalized empirical loss function can be obtained by Le Cam's approach and that Le Cam's estimators based on median-of-means tests are champions of median-of-means tournaments.
This paper studies estimators derived from Le Cam's procedure based on regularized median-of-means (MOM) tests (see Section~\ref{Sec:QOMProcesses}). Our estimators are therefore particular instances of champions of MOM's tournaments and another motivation is to push further the analysis of this particular champion. The main advantage of MOM's tests over Le Cam's original ones is that they allow for more classical loss functions than Hellinger loss. This idea is illustrated on the square-loss. Compared to Huber or Catoni's losses, this approach allows to control easily the risk of our estimators by using classical tools from empirical process theory, it also allows to tackle the problem of ``aggressive" outliers.
The closest work is certainly that of \cite{LugosiMendelson2016}, but we believe that our paper contains substantial improvements. We stress the intimate relationship between their estimator and Le Cam general construction and use this parallel to propose a much simpler estimator. Our risk bounds are always better and we extend their results to possibly corrupted data-sets.
To investigate robustness properties of median-of-means estimators, we partition the dataset into two parts. One is made of outliers data. They are indexed by $\cO\subset[N]$ of cardinality $|\cO|=K_o$. \textbf{On those data, absolutely nothing is assumed }: they may not be independent, have distributions $P_i$ totally different from $P$, with no moment at all, etc.. These are typically data polluting datasets like in the case of declarative data on internet or when something went wrong during the storage, compression or transfer which resulted in complete non sense data. They may also be observations met in biology as in the classical \textit{eQTL (Expression Quantitative Trait Loci and The Phenogen Database)} from \cite{eQTL}. Many other examples of datasets containing outliers could be provided, this includes frauds detection and terrorist activity as examples. Of course, outliers are not flagged in advance and the statistician is given no a priori information on which data is an outlier or not.
The other part of the dataset is made of data on which the MOM estimator rely on to estimate the oracle $f^*$. There should be enough information in those data so that the estimation of $f^*$ is possible, even in the presence of outliers provided they remain in a ``decent proportion''. We therefore call the non-outliers, the \textit{informative data}, those that bring information on $f^*$. We denote by $\cI\subset[N]$ the set indexing these data. We therefore end up with a partition of $[N]$ as $[N] = \cI \cup \cO$ which, again, is not known from the statistician.
The radii of the sets $\cB_{T_N}(f)$ are computed for regularization and $L^2_P$ norms. The regularization norm is chosen in advance by the statistician to promote sparsity or smoothness. It can be used freely in our procedure, but it doesn't ensure a small $L^2_P$ risk for the estimator. The $L^2_P$-norm is unknown in general since it depends on the distribution of $X$. Furthermore, the classical $L^2_{P_N}$-empirical metric fails to estimate the $L^2_P$ metric without subgaussian properties of the design vector $X$. Fortunately, it can be replaced by a median-of-means metric. To handle simultaneously both regularization and $L^2_P$ norms, we will also slightly extend Le Cam's principle.
Our first important result shows that the resulting estimator is well localized w.r.t. both regularization and $L^2_P$ norms.
Median-of-means estimators rely on a data splitting into $K$ blocks and this parameter drives the resulting statistical performances (cf. \cite{MR3576558}). To achieve optimal rates, $K$ should be ultimately chosen using parameters that depend on the oracle $f^*$ like its sparsity which is not in general available to the statistician. To bypass this problem, the strategy of \cite{MR1147167} is used as in \cite{MR3576558} to select $K$ adaptively and get a fully data-driven procedure.
There are four important features in our approach. First, all results are proved under weak $L^{2+\epsilon}$ moment assumptions on the noise. This is an almost minimal condition for the problem to make sense. The class $F$ is only assumed to satisfy a weak ``$L_2/L_1$'' comparison. Second, performances of the estimators are not affected by the presence of complete outliers, as long as their number remains comparable to \textit{(number of observations)$\times$(rates of convergence)}.
Third, all results are non-asymptotic and the regression function $x\mapsto\E[Y|X=x]$ is never assumed to belong to the class $F$. In particular, the noise $Y-f^*(X)$ can be correlated with $X$. Finally, even ``informative data", those that are not ``outliers", are not requested to be i.i.d. $\sim P$, but only to have close first and second moments for all $f\in F-\{f^*\}$. Nevertheless, the estimators are shown to behave as well as the ERM when the data are i.i.d. $\sim P$, $\E[Y|X=\cdot]\in F$, the noise $\zeta=Y-f^*(X)$ and the class $F$ are Gaussian and the noise is independent from the design.
\textbf{Example: sparse-recovery via MOM LASSO.}
As a proof of concept, theoretical properties are illustrated in the classical example of sparse-recovery in high-dimensional spaces using the $\ell_1$-regularization.
This example illustrates typical results that follow from our analysis in one of the most classical problem of high dimensional statistics (cf. \cite{MR2807761,MR3307991}).
The interested reader can check that it also applies to other procedures like Slope (cf. \cite{slope1,slope2}) and trace-norm regularization as well as kernel methods, for instance, by using the results in \cite{LM_reg_comp,LM_reg_comp_2}.
Recall this classical setup.
Let $X$ denote a random vector in $\R^d$ such that $\E\inr{X, t}^2=\norm{t}_2^2$ for all $t\in\R^d$ ($X$ is isotropic) and let $Y$ be a real-valued random vector. Let $t^*\in\argmin_{t\in\R^d}\E(Y-\inr{X, t})^2$.
Let $(X_i,Y_i)_{i\in[N]}$ denote independent data corrupted by outliers : no assumption is made on a subset $(X_i,Y_i)_{i\in \cO}$ of the dataset. Let $\cI=[N]\setminus \cO$ denote the indices of \textit{informative data} $(X_i,Y_i)_{i\in \cI}$: for all $i\in \cI$, $(X_i, Y_i)$ are independent with the same distribution $(X, Y)$.
For the sake of simplicity, we only consider the case of i.i.d. informative data in this example.
In high-dimensional statistics, $N\leq d$ but $t^*$ has only $s$ ($s < N$) non-zero coordinates. To estimate $t^*$, the $\ell_1$-norm $\norm{\cdot}_1$ is used for penalization to promote zero coordinates. The following result holds.
\begin{Theorem}\label{theo:lasso_classi}[Theorem~1.4 in \cite{LM_reg_comp}] Assume $t^*$ is $s$-sparse, $N\geq c_0 s \log(ed/s)$, $X$ is isotropic and
\begin{enumerate}
\item[i)] $|\cI|=N$ and $|\cO|=0$ (no outliers in the dataset),
\item[ii)] $ \zeta=Y-\inr{X, t^*} \in L_{q_0}$ for some $q_0>2$
\item[iii)] there exists $L>0$ such that for all $t\in\R^d$ and all $p\ge 2$, $\norm{\inr{X,t}}_{L_p}\leq L \sqrt{p}\norm{\inr{X, t}}_{L_2}$
\item[iv)] there exist $u_0>0$ and $\beta_0>0$ such that for all $t\in\R^d$,
\[\bP\left[|\inr{X, t}|\geq u_0\norm{\inr{X, t}}_{L_2}\right]\geq \beta_0\enspace.\]
\end{enumerate}
The LASSO estimator, defined by
\begin{equation*}
\tilde t \in\argmin_{t\in\R^d}\left(\frac{1}{N}\sum_{i=1}^N \left(Y_i-\inr{X_i, t}\right)^2 +c_1\norm{\zeta}_{L_{q_0}}\sqrt{\frac{\log(ed)}{N}}\norm{t}_1\right)
\end{equation*}satisfies for every $1\leq p\leq 2$,
\begin{equation*}
\norm{\tilde t - t^*}_p\leq c_4(L, u_0, \kappa_0)\norm{\zeta}_{L_{q_0}}s^{1/p}\sqrt{\frac{\log(ed)}{N}}\enspace,
\end{equation*}
with probability at least
\begin{equation}\label{eq:proba_lasso_classic}
1-\frac{c_2 \log^{q_0}N}{N^{q_0/2-1}} - 2 \exp\left(-c_3s \log(ed/s)\right)\enspace.
\end{equation}\end{Theorem}
This paper shows that Theorem~\ref{theo:lasso_classi} holds for a MOM version of the LASSO estimator under much weaker assumptions, with a better probability estimate than \eqref{eq:proba_lasso_classic}. More precisely, the following theorem is proved.
\begin{Theorem}\label{theo:mom_lasso}
Assume that $t^*$ is $s$-sparse, $N\geq c_0 s \log(ed/s)$, $X$ is isotropic and
\begin{enumerate}
\item[i')] $|\cI|\geq N/2$ and $|\cO|\leq c_1 s \log(ed/s)$ (the number of outliers may be proportional to the sparsity times $\log(ed/s)$),
\item[ii)] $ \zeta=Y-\inr{X, t^*} \in L_{q_0}$ for some $q_0>2$
\item[iii')] for every $1\leq p\leq C_0 \log(ed)$, $\norm{\inr{X,e_j}}_{L_p}\leq L \sqrt{p}\norm{\inr{X,e_j}}_{L_2}$ where $(e_j)_{j\in[d]}$ is the canonical basis of $\R^d$ and $C_0$ is some absolute constant,
\item [iv')] there exists $\theta_0$ such that $\norm{\inr{X, t}}_{L^1}\leq \theta_0\norm{\inr{X, t}}_{L^2}$, for all $t\in \R^d$,
\item [v)] there exists $\theta_m$ such that ${\rm var}(\zeta\inr{X, t})\leq \param{m}^2 \norm{t}_2^2$, for all $t\in\R^d$.
\end{enumerate}
There exists an estimator $\hat t$, called MOM-LASSO, satisfying for every $1\leq p\leq 2$,
\begin{equation*}
\norm{\hat t - t^*}_p\leq c_4(L, \theta_m)\norm{\zeta}_{L_{q_0}}s^{1/p}\sqrt{\frac{1}{N}\log\left(\frac{ed}{s}\right)}\enspace,
\end{equation*}
with probability at least
\begin{equation}\label{eq:proba_mom_lasso}
1-c_2 \exp(-c_3 s \log(ed/s))\enspace.
\end{equation}
\end{Theorem}
Theoretical properties of MOM LASSO outperform those of LASSO in several ways.
\begin{itemize}
\item Estimation rates achieved by MOM-LASSO are the actual minimax rates $s \log(ed/s)/N$, see \cite{BLT16}, while classical LASSO estimators achieve the rate $s \log(ed)/N$. This improvement is possible thanks to the adaptation step in MOM-LASSO.
\item the probability deviation in \eqref{eq:proba_lasso_classic} is polynomial -- $1/N^{(q_0/2-1)}$ in \eqref{eq:proba_lasso_classic} -- it is exponentially small for MOM LASSO. Exponential rates for LASSO hold only if $\zeta$ is subgaussian ($\norm{\zeta}_{L_p}\leq C \sqrt{p}\norm{\zeta}_{L_2}$ for all $p\geq2$).
\item MOM LASSO is insensitive to data corruption by up to $s$ times $\log(ed/s)$ outliers while only one outlier can be responsible of a dramatic breakdown of the performances of LASSO.
\item All assumptions on $X$ are weaker for MOM LASSO than for LASSO. In particular, condition \textit{v)} holds with $\param{m} = \norm{\zeta}_{L_4}$ if for all $t\in\R^d$, $\norm{\inr{X, t}}_{L_4}\leq \param{0} \norm{\inr{X, t}}_{L_2}$ -- which is a much weaker requirement than condition iii) for LASSO.
\end{itemize}
From a mathematical point of view, our results are based on a slight extension of the Small Ball Method (SBM) of \cite{MR3431642,Shahar-COLT} to handle non-i.d. data. SBM is also extended to bound both quadratic and multiplier parts of the quadratic loss. Otherwise, all arguments are standard, which makes the approach very attractive and easily reproducible in other frameworks of statistical learning.
The paper is organized as follows. Section~\ref{sec:setting} briefly presents the general setting and our main illustrative example. Section~\ref{Sec:LFT} presents Le Cam's construction of estimators based on tests. We also show why many learning procedures may be obtained by this approach. The construction of estimators and the main assumptions are gathered in Section~\ref{sec:EstimatorsAssumptions}. Our main theorems are stated in Section~\ref{sub:main_result} and proved in Section~\ref{sec:Proofs}.
\paragraph{Notation} For any real number $x$, let $\lfloor x\rfloor$ denote the largest integer smaller than $x$ and let $[x]=\{1,\ldots,\lfloor x\rfloor\}$ if $x\ge 1$. For any finite set $A$, let $|A|$ denote its cardinality. All along the paper, $(\cabs{i})_{i\in \N}$ denote absolute constants which may vary from line to line and $\param{\cdot}$, with various subscripts, denote real valued parameters introduced in the assumptions. Finally, for any set $\mathcal{G}$ for which it makes sense, for any $g\in\mathcal{G}$, $c\ge 0$ and $\mathcal{C}\subset\mathcal{G}$,
\[
g+c\mathcal{C}=c\cC+g=\{h : \exists g'\in \mathcal{C} \text{ such that }h=g+cg'\}\enspace.
\]
Let also $g+\cG=g+1\cG$. We also denote by $I(g\in\cC)$ the indicator function of the set $\cC$ which equals to $1$ when $g\in\cC$ and $0$ otherwise.
\section{Setting}\label{sec:setting}
Let $\mathcal X$ denote a measurable space and let $(X,Y),(X_i,Y_i)_{i\in[N]}$ denote random variables taking values in $\cX\times \R$, with respective distributions $P, (P_i)_{i\in[N]}$. Given a probability distribution $Q$, let $L^2_Q$ denote the space of all functions $f$ from $\cX$ to $\R$ such that $\norm{f}_{L^2_Q}<\infty$ where $\norm{f}_{L^2_Q} = \big(Q f^2\big)^{1/2}$. Let $F\subset L^2_{P}$ denote a convex class of functions $f:\cX\to\R$. Assume that $PY^2<\infty$ and let, for all $f\in F$,
\[
R(f)=P\big[(Y-f(X))^2\big],\quad f^*\in\argmin_{f\in F}R(f) \mbox{ and } \zeta=Y-f^*(X)\enspace.
\]
Let $\norm{\cdot}$ denote a norm defined onto a linear subspace $E$ of $L^2_P$ containing $F$.
\paragraph*{Example : $\ell_1$-regularization of linear functionals}
For every $t=(t_j)_1^d\in\R^d$ and $1\leq p\leq +\infty$, let
\begin{equation*}
F = \{\inr{\cdot, t}: t\in\R^d\}\; {\mbox{ and }} \; \norm{\inr{\cdot, t}} = \norm{t}_1,\; \text{where}\; \norm{t}_p=\left(\sum_{j=1}^d |t_j|^p\right)^{1/p}\enspace.
\end{equation*}
Let $f^*=\inr{\cdot, t^*}\in F$, where
\begin{equation*}
t^*\in\argmin_{t\in\R^d}\left\{P\big(Y-\inr{X, t}\big)^2\right\}\enspace.
\end{equation*}
Whenever it's necessary, $(e_1, \ldots, e_d)$ will denote the canonical basis of $\R^d$ and $B_p^d$ (resp. $S_p^{d-1}$) will denote the unit ball (resp. sphere) associated to $\norm{\cdot}_p$.
To ease readability in this example, we focus on rates of convergence, we do not consider the ``full'' non-i.i.d. setup and assume that $P=P_i$ for all $i\in\cI$. We write $L^q$ for $L^q_P$ to shorten notations.
\section{Learning from tests}\label{Sec:LFT}
\subsection{General Principle}
This section details the ideas underlying the construction of a MOM estimator using an extension of Le Cam's approach.
\paragraph{Basic idea}
By definition of the oracle $f^*$, one has
\[
f^*=\argmin_{f\in F}R(f)=\argmin_{f\in F}\sup_{g\in F}\{R(f)-R(g)\},\; \text{where} \; R(f)=P[(Y-f(X)^2]\enspace.
\]
As $T_{\text{id}}(g,f)=R(f)-R(g)$ depends on $P$, we estimate it by test statistics $T(g,f,(X_i,Y_i)_{i\in[N]})\equiv T_N(g,f)$ that is, real random variables such that
\begin{equation}\label{eq:DefTest}
T_N(f,g)+T_N(g,f)=0\enspace.
\end{equation}
These statistics are used to \emph{compare} $f$ to $g$, simply by saying that $g$ $T_N$-\emph{beats} $f$ iff $T_N(g,f)\ge 0$. In this paper, the statistics $T_N(g,f)$ are median-of-means estimators of $R(f)-R(g)$ (cf. \eqref{eq:beat_on_Bk} in Section~\ref{Sec:QOMProcesses}).
\paragraph{Le Cam's construction}
Let $(T_N(g,f))_{f,g\in F}$ denote a collection of test statistics and let $d(\cdot, \cdot)$ denote a pseudo-distance on $F$ measuring (or related to) the risk we want to control. Let for all $f\in F$,
\[
\mathcal{B}_{T_N}(f)=\{g\in F : T_N(g,f)\ge 0\}
\]be the set of all functions $g\in F$ that beat $f$. If $f$ is far from $f^*$, then $\mathcal{B}_{T_N}(f)$ is expected to have a large radius w.r.t. $d(\cdot, \cdot)$. We therefore introduce this radius as a criteria to minimize : for all $f\in F$, let $C_{T_N}(f) = \sup_{g\in \cB_{T_N}(f)}d(f,g)$.
By \eqref{eq:DefTest}, $f\in \mathcal{B}_{T_N}(g)$ or $g\in \mathcal{B}_{T_N}(f)$ (both happen if $T_N(f,g)=0$), hence $d(f,g)\le C_{T_N}(f)\vee C_{T_N}(g)$. In particular, for all $f\in F$,
\begin{equation}\label{GenRiskBound2}
d(f,f^*)\le C_{T_N}(f)\vee C_{T_N}(f^*)\enspace.
\end{equation}
Eq~\eqref{GenRiskBound2} suggests to define the estimator
\begin{equation}\label{Def:Testimators}
\hat f_{T_N}\in \argmin_{f\in F}C_{T_N}(f)=\argmin_{f\in F}\sup_{g\in \cB_{T_N}(f)}d(f,g)\enspace.
\end{equation}
This estimator satisfies, from Eq~\eqref{GenRiskBound2},
\begin{equation}\label{GenRiskBound}
d(\hat f_{T_N},f^*)\le C_{T_N}(f^*)\enspace.
\end{equation}
Risk bounds for $\hat f_{T_N}$ follow from \eqref{GenRiskBound} and upper bounds on the radii of $\mathcal{B}_{T_N}(f^*)$.
\begin{Remark}
More generally, one can compare only the elements of a subset $\mathcal F\subset F$, typically a maximal $\epsilon$-net by introducing for all $f\in \mathcal F $, the set
\begin{equation}\label{def:BGen}
\mathcal{B}_{T_N}(f,\mathcal F)=\{g\in \mathcal F : T_N(g,f)\ge 0\}
\end{equation} and then by minimizing the diameter of $\mathcal{B}_{T_N}(f,\mathcal F)$ over $\cF$. This usually improves the rates of convergence for constant deviation results when there is a gap in Sudakov's inequality of the localized sets of $F$ (cf. Section~5 in \cite{LM13} for more details). These results are not presented because we are interested in exponentially large deviation results for which our results are optimal.
\end{Remark}
\paragraph{Dealing with regularization : the link function}
Statistical performances of estimators and the radius of $\mathcal{B}_{T_N}(f^*)$ can be measured by two norms: the regularization norm $\|\cdot\|$ and $\|.\|_{L^2_P}$. As \eqref{Def:Testimators} allows only for one distance $d$, we propose the following extension of Le Cam approach to handle two metrics.
To introduce this extension, assume first that $d(f,g)=\|f-g\|_{L^2_{P}}$ can be computed for all $f, g\in F$ (this is the case if the distribution of the design is known). The next paragraph explains how to deal with the more common framework where this distance is unknown. Remark that
\[
C_{T_N}(f)=\sup_{g\in\mathcal B_{T_N}(f)}\|f-g\|=\min\left\{\rho\ge 0 : \sup_{g\in \mathcal B_{T_N}(f)}\|g-f\|\le \rho\right\}\enspace.
\]
The main point to extend Le Cam's approach to simultaneously control two norms is to design a link function $r(\cdot)$. In a nutshell, the values $r(\rho)$ is the $L^2_P$-minimax rate of convergence in a ball of radius $\rho$ for the regularization norm
(cf. \eqref{eq:defr} in Section~\ref{sec:Complexity} for a formal definition). Then one can define
\begin{equation*}
C_{T_N}^{(2)}(f)=\min\left\{\rho\ge 0 : \sup_{g\in \mathcal B_{T_N}(f)}\|g-f\|\le \rho\text{ and } \sup_{g\in \mathcal B_{T_N}(f)}d(f,g)\le r(\rho)\right\}\enspace.
\end{equation*}
Theorem \ref{theo:basic-combining-loss-and-reg} shows that while a minimizer $\hat f^{(1)}$ of $C_{T_N}$ has only a nice risk for $\norm{\cdot}$, a minimizer $\hat f^{(2)}$ of $C_{T_N}^{(2)}$ has both $\norm{\hat f^{(2)}-f^*}$ and $d(\hat f^{(2)},f^*)$ properly controlled.
\paragraph{Dealing with unknown norms : the isometry property}In general, $L_P^2$-distances cannot be directly computed and have to be estimated. To deal with this issue, one considers usually the empirical $L^2_{P_N}$ distance and prove that empirical and actual distances are equivalent outside a $L^2_P$-ball centered in $f^*$ (cf. for instance, remark after Lemma~2.6 in \cite{LM13}). Unfortunately this approach only works under strong concentration property that we want to relax in this paper.
The unknown $L_P^2$-metric is instead estimated by a median-of-means approach, that is, we use MOM estimators $d_N(f,g)$ of all $d(f,g)$ (cf. Section~\ref{sec:estimators}). The final estimator is therefore defined as a minimizer of
\begin{equation*}
C_{T_N}^{\prime\prime}(f)=\min\left\{\rho\ge 0 : \sup_{g\in \mathcal B_{T_N}(f)}\|g-f\|\le \rho\text{ and } \sup_{g\in \mathcal B_{T_N}(f)}d_N(f,g)\le r(\rho)\right\}\enspace.
\end{equation*}
\subsection{Examples}\label{sec:Example}
Le Cam's approach has been used by Birgé to define $T$-estimators (cf. \cite{MR2449129,MR2219712,MR3186748}) and by Baraud, Birgé and Sart to define $\rho$-estimators (cf. \cite{MR3565484,BaraudBirgeSart}). \cite{MR2834722,MR3224300} also built efficient estimator selection procedures with this approach. It also extends many common procedures in statistical learning theory, as shown by the following examples.
\paragraph{Example 1 : Empirical minimizers}
Assume $T_N(g,f)=\ell_N(f)-\ell_N(g)$ for some random function $\ell_N:F\to\R$ and denote by $\hat f=\arg\min_{f\in F}\ell_N(f)$ a minimizer of the corresponding criterion (provided that it exists and is unique).
Then it is easy to check that $\mathcal B_{T_N}(\hat f)=\{\hat f\}$, so its radius is null, while the radius of any other point $f$ is larger than $d(f,\hat f)>0$ (whatever the non-degenerate notion of pseudo-distance used for $d$).
It follows that $\hat f$ is the estimator \eqref{Def:Testimators}.
In particular, any possibly penalized empirical risk minimizer
\[
\hat f=\arg\min_{f\in F}\{P_N\ell_f+{\rm reg}(f)\}
\]
is obtained by Le Cam's construction with the tests
\[T_N(g,f)=P_N(\ell_f-\ell_g)+{\rm reg}(f)-{\rm reg}(g)\enspace.\]
These examples encompass classical empirical risk minimizers of \cite{MR1641250} but also their robust versions from \cite{MR0161415,MR2906886}.
\paragraph{Example 2 : median-of-means estimators}
Another, perhaps less obvious example is the median-of-means estimator \cite{MR1688610, MR855970, MR702836} of the expectation $PZ$ of a real valued random variable $Z$. Let $Z_1,\ldots,Z_N$ denote a sample and let $B_1,\ldots,B_K$ denote a partition of $[N]$ into bins of equal size $N/K$. The estimator $\text{MOM}_K(Z)$ is the (empirical) median of the vector of empirical means $\left( P_{B_k}Z=|B_k|^{-1}\sum_{i\in B_k}Z_i\right)_{k\in[K]}$. Recall that
\[
PZ=\argmin_{m\in\R}P(Z-m)^2=\argmin_{m\in\R}\max_{m'\in\R}P[(Z-m)^2-(Z-m')^2]\enspace.
\]
Define the MOM test statistic to compare any $m,m'\in \R$ by
\begin{align*}
T_N(m,m')&=\text{MOM}_K[(Z-m')^2-(Z-m)^2]\enspace.
\end{align*}
Basic properties of the median (recalled in Eq~\eqref{prop:Cone} and \eqref{prop:Opposes} of Section~\ref{Sec:QOMProcesses}) yield
\begin{align*}
T_N(m,m')&=(m')^2-m^2+\text{MOM}_K[-2Z(m'-m)]\\
&=(m')^2-2m'\text{MOM}_K(Z)-[m^2-2m\text{MOM}_K(Z)]\\
&=(m'-\text{MOM}_K(Z))^2-(m-\text{MOM}_K(Z))^2\enspace.
\end{align*}
Defining $\ell_N(m)=(m-\text{MOM}_K(Z))^2$, one has
\[
T_N(m,m')=\ell_N(m')-\ell_N(m)\enspace.
\]
As in the previous example, Le Cam's estimator based on $T_N$ is therefore the unique minimizer of $\ell_N$, that is $\text{MOM}_K(Z)$.
\paragraph{Example 3 : ``Champions" of a Tournament}\label{Ex:LugMen} \cite{LugosiMendelson2016} introduced median-of-means tournaments. More precisely, they used median-of-means tests to compare elements in $F$. These tests cannot be separated $T_N(f,g)\neq\ell_N(g)-\ell_N(f)$ in general. \cite{LugosiMendelson2016} assume that an upper bound $r^*$ on the radius $C_{T_N}(f^*)$ of $\mathcal B_{T_N}(f^*)$ (that holds with exponentially large probability) is known from the statistician and call ``champion" any element $\hat f$ of $F$ such that $C_{T_N}(\hat f)\le r^*$.
It is clear that, by definition the radius $C_{T_N}(\hat f_{T_N})$ of $\hat f_{T_N}$ is smaller than $C_{T_N}(f^*)$ and therefore smaller than $r^*$. This means that $\hat f_{T_N}$ is a ``champion" for this terminology. The main advantage of Le Cam's approach is that $r^*$ (which usually depends on some attribute of the oracle like the sparsity) is not required to \emph{build} the estimator $\hat f_{T_N}$.
\section{Construction of the regularized MOM estimators}\label{sec:EstimatorsAssumptions}
\subsection{Quantile of means processes and median-of-means tests}\label{Sec:QOMProcesses}
This section presents median-of-means (MOM) tests used in this work. Designing a family of tests $(T_N(g, f): f, g\in F)$ is one of the most important building blocks in Le Cam's approach together with the right choice of the metric measuring the diameters $\cB_{T_N}(f)$ for $f\in F$.
Start with a few notations. For all $\alpha\in[0,1]$, $\ell\ge 1$ and $z\in\R^\ell$, the set of $\alpha$-quantiles of $z$ is denoted by
\[
\mathcal Q_{\alpha}(z)=\left\{x\in \R : \frac1\ell\sum_{k=1}^\ell I(z_i\le x)\ge \alpha\quad\text{and}\quad\frac1\ell\sum_{k=1}^\ell I(z_i\ge x)\ge 1-\alpha\right\}\enspace.
\]
For a non-empty subset $B\subset [N]$ and a function $f:\cX\times\R\to\R$, let
\begin{equation*}
P_Bf=\frac{1}{|B|}\sum_{i\in B}f(X_i,Y_i) \mbox{ and } \overline{P}_Bf=\frac{1}{|B|}\sum_{i\in B} P_i f\enspace.
\end{equation*} Let $K\in[N]$ and let $(B_1,\ldots,B_K)$ denote an equipartition of $[N]$ into bins of size $|B_k|=N/K$. When $K$ does not divide $N$, at most $K-1$ data can be removed from the dataset. For any real number $\alpha\in [0,1]$ and any function $f:\cX\times\R\to\R$, the set of $\alpha$-quantiles of empirical means is denoted by
\[
\mathcal Q_{\alpha,K}(f)=\mathcal Q_{\alpha}\left((P_{B_k}f)_{k\in[K]}\right)\enspace.
\]
With a slight abuse of notations, we shall repeatedly denote by $Q_{\alpha,K}(f)$ any element in $\mathcal Q_{\alpha,K}(f)$ and write $Q_{\alpha,K}(f)=u$ if $u\in \mathcal Q_{\alpha,K}(f)$, $Q_{\alpha,K}(f)\ge u$ if $\sup\mathcal Q_{\alpha,K}(f)\ge u$, $Q_{\alpha,K}(f)\le u$ if $\inf\mathcal Q_{\alpha,K}(f)\le u$, and $Q_{\alpha,K}(f)+Q_{\alpha',K}(f')$ any element in the Minkowski sum $\mathcal Q_{\alpha,K}(f)+\mathcal Q_{\alpha',K}(f')$. Let also $\text{MOM}_K(f)=Q_{1/2,K}(f)$ denote an empirical median of the empirical means on the blocks $B_k$. Empirical quantiles satisfy for any $ c\ge 0$, $f,f':\cX\times\R\to\R$ and $\alpha\in[0,1]$,
\begin{gather}
\mathcal Q_{\alpha,K}(c f)=c \mathcal Q_{\alpha,K}(f)\enspace,\label{prop:Cone}\\
\mathcal Q_{\alpha,K}(-f)=-\mathcal Q_{1-\alpha,K}(f)\enspace,\label{prop:Opposes}\\
\sup \cro{{\mathcal Q}_{1/4,K}(f)+{\mathcal Q}_{1/4,K}(f')}\le \inf \mathcal Q_{1/2,K}(f+f')\enspace,\\
\sup \mathcal Q_{1/2,K}(f+f')\le \inf \cro{{\mathcal Q}_{3/4,K}(f)+{\mathcal Q}_{3/4,K}(f')}\enspace.
\label{prop:Sum}
\end{gather}
With some abuse of notations, we shall write these properties respectively
\begin{align*}
& Q_{\alpha,K}(c f)=c Q_{\alpha,K}(f),\qquad Q_{\alpha,K}(-f)=-Q_{1-\alpha,K}(f)\enspace,\\
Q_{1/4,K}(f)&+Q_{1/4,K}(f')\le \MOM{K}{f+f'}\le Q_{3/4,K}(f)+Q_{3/4,K}(f')\enspace.
\end{align*}
A \emph{regularization} parameter $\lambda>0$ is introduced to balance between data adequacy and regularization.
The (quadratic) loss and regularized (quadratic) loss are respectively defined on $F\times\cX\times\R$ as the real valued functions such that
\begin{equation*}
\ell_f(x,y) = (y-f(x))^2, \quad \ell^\lambda_f = \ell_f+\lambda \norm{f} ,\qquad \forall (f,x,y)\in F\times\cX\times\R\enspace.
\end{equation*}
To compare/test functions $f$ and $g$ in $F$, median-of-means tests between $f$ and $g$ are now defined by
\begin{equation}\label{eq:beat_on_Bk}
T_{K,\lambda}(g,f)=\MOM{K}{\ell^\lambda_f - \ell^\lambda_g}=\MOM{K}{\ell_f-\ell_g}+\lambda(\|f\|-\|g\|)\enspace.
\end{equation}
From \eqref{prop:Opposes}, $T_{K,\lambda}$ satisfies \eqref{eq:DefTest} and is a tests statistic in the sense of Section \ref{Sec:LFT}.
\subsection{Main assumptions}
Recall that $[N]=\cO\cup\cI$ and that $(X_i,Y_i)_{i\in \cO}$ is a set of outliers on which we make no assumption so these may be aggressive in any sense one can imagine. The remaining informative data $(X_i,Y_i)_{i\in \cI}$ need to bring enough information onto $f^*$. We therefore need some assumption on the sub-dataset $(X_i,Y_i)_{i\in \cI}$ and, in particular, some connexion between the distributions $P_i$ for $i\in\cI$ and $P$. These assumptions are pretty weaksince we only assume essentially that the $L^2_P, L^2_{P_i}$ and $L^1_{P_i}$ geometries are comparable in the following sense.
\begin{Assumption}\label{ass:Mom2F}
There exists $\param{r}\ge 1$ such that, for all $i\in\cI$ and $f\in F$,
\[
\norm{f-f^*}_{L^2_{P_i}}\le \param{r}\norm{f-f^*}_{L^2_P}\enspace.
\]
\end{Assumption}
Of course, Assumption~\ref{ass:Mom2F} holds in the i.i.d. framework, with $\param{r}=1$ and $\cI=[N]$.
The second assumption bounds the correlation between the noise function $\zeta:(y, x)\in\R\times \cX\to y-f^*(x)$ and the design on the shifted class $F-f^*$ in $L^2_Q$ for all $Q\in \{P,(P_i)_{i\in \cI}\}$.
\begin{Assumption}
\label{ass:margin}
There exists $\param{m}>0$ such that, for all $Q\in \{P,(P_i)_{i\in \cI}\}$ and $f\in F$,
\[
\text{var}_{Q}(\zeta(f-f^*))=Q\left[\zeta^2(f-f^*)^2-[Q(\zeta(f-f^*))]^2\right]\le \param{m}^2\norm{f-f^*}^2_{L^2_P}\enspace.
\]
\end{Assumption}
Let us give some examples where Assumption~\ref{ass:margin} holds. If the noise random variable $\zeta(Y, X)$ (resp. $\zeta(Y_i, X_i)$ for $i\in\cI$) has a variance conditionally to $X$ (resp. $X_i$ for $i\in\cI$) that is uniformly bounded then Assumption~\ref{ass:margin} holds. This is, for example, the case, when $\zeta(Y,X)$ (resp. $\zeta(Y_i, X_i)$ for $i\in\cI$) is independent of $X$ (resp. $X_i$ for $i\in\cI$) and has finite $L^2$-moment with $\param{m}=\max_{Q\in P,\{P_i\}_{i\in \cI}}\norm{\zeta}_{L^2_Q}$. It also holds without independence under higher moment conditions.
For example, assume $\sigma=\max_{Q\in P,\{P_i\}_{i\in \cI}}\norm{\zeta}_{L^4_Q}<\infty$ and, for every $f\in F$, $\norm{f-f^*}_{L^4_Q}\leq \theta_1 \norm{f-f^*}_{L^2_P}$ then by Cauchy-Schwarz inequality,
$\sqrt{{\rm var}_Q(\zeta(f-f^*))} \leq \norm{\zeta(f-f^*)}_{L^2_Q}\leq \norm{\zeta}_{L^4_Q}\norm{f-f^*}_{L^4_Q}\leq \theta_1 \sigma\norm{f-f^*}_{L^2_P}$ and so Assumption~\ref{ass:margin} holds for $\param{m} = \theta_1 \sigma$.
\begin{Assumption}\label{ass:small-ball}
There exists $\theta_0\ge 1$ such that for all $f\in F$ and all $i\in \cI$
\begin{equation*}
\norm{f-f^*}_{L^2_P}\le \theta_0\norm{f-f^*}_{L^1_{P_i}}\enspace.
\end{equation*}
\end{Assumption}
By Cauchy-Schwarz inequality, $\norm{f-f^*}_{L^1_{P_i}}\leq \norm{f-f^*}_{L^2_{P_i}}$ for all $f\in F$ and $i\in\cI$. Therefore, Assumptions~\ref{ass:Mom2F} and~\ref{ass:small-ball} together imply that all norms $L^2_P, L_{P_i}^2, L_{P_i}^1, i\in\cI$ are equivalent over $F-f^*$. Note also that Assumption~\ref{ass:small-ball} is related to the small ball property (cf. \cite{MR3431642,Shahar-COLT}) as shown by Proposition~\ref{prop:SBP=L2/L1} bellow. The small ball property has been recently used in Learning theory and signal processing. We refer to \cite{MR3431642,LM_compressed,shahar_general_loss,MR3364699,Shahar-ACM,RV_small_ball} for examples of distributions satisfying this assumption.
\begin{Proposition}\label{prop:SBP=L2/L1} Let $Z$ be a real-valued random variable.
\begin{enumerate}
\item If there exist $\kappa_0$ and $u_0$ such that $\bP(|Z|\geq \kappa_0\norm{Z}_2)\ge u_0$ then $\norm{Z}_2\le (u_0\kappa_0)^{-1}\norm{Z}_1$.
\item If there exists $\param{0}$ such that $\norm{Z}_2\le \param{0}\norm{Z}_1$, then for any $\kappa_0<\param{0}^{-1}$, $\bP(|Z|\geq \kappa_0\norm{Z}_2)\ge u_0$ where $u_0=(\param{0}^{-1}-\kappa_0)^2$.
\end{enumerate}
\end{Proposition}
\begin{proof} If $\bP(|Z|\geq \kappa_0\norm{Z}_2)\ge u_0$ then
\begin{equation*}
\norm{Z}_{1}\ge \int_{|z|\ge \kappa_0\norm{Z}_{2}}|z|P_Z(dz)\ge u_0\kappa_0\norm{Z}_{2}\enspace,
\end{equation*}where $P_Z$ denotes the distribution of $Z$. Conversely, if $\norm{Z}_2\leq \param{0} \norm{Z}_1$, the Paley-Zigmund's argument \cite[Proposition~3.3.1]{MR1666908} shows that, if $p=\bP\left(|Z| \geq \kappa_0 \norm{Z}_{2}\right)$,
\begin{align*}
\norm{Z}_{2}&\le \param{0}\norm{Z}_{1} = \param{0} \pa{\E[|Z|I(|Z| \leq \kappa_0 \norm{Z}_{2})] + \E[|Z|I(|Z| \geq \kappa_0 \norm{Z}_{2})]}\\
&\le \param{0}\norm{Z}_{2}\pa{\kappa_0 + \sqrt{p}}\enspace.
\end{align*}
As one can assume that $\norm{Z}_2\ne 0$, $p\geq (\param{0}^{-1}-\kappa_0)^2$.
\end{proof}
\subsection{Complexity parameters and the link function}\label{sec:Complexity}
This section defines the link function $r(\cdot)$ making the connections between norms that will be required in the extension of Le Cam's approach to a simultaneous control of two norms (one of the two being unknown). For any $\rho\ge 0$ and any $f\in E$, let
\begin{gather*}
B(f,\rho) = \{g\in E : \norm{f-g}\leq \rho\},\qquad S(f,\rho) = \{g\in E : \norm{g-f} = \rho\}
\enspace.
\end{gather*}
\begin{Definition}\label{def:the-three-paraemters}
Let $(\eps_i)_{i\in \cI}$ be independent Rademacher random variables, independent from $(X_i,Y_i)_{i\in\cI}$ and let $\cJ=\{J\subset \cI, |J|\ge |\cI|/2\}$. For any $\gamma_Q,\gamma_M>0$ and $\rho>0$ let $F_{f^\star,\rho,r}=\{f \in F \cap B(f^\star,\rho) : \norm{f-f^\star}_{L^2_P} \leq r\}$,
\begin{align*}
\fQ^{\gamma_Q}_{f^\star,\rho}&=\left\{r>0: \forall J\in \cJ,\;\E \sup_{f \in F_{f^\star,\rho,r}} \left|\sum_{i\in J} \eps_i (f-f^\star)(X_i)\right| \leq \gamma_Q |J| r \right\}\enspace,\\
\fM^{\gamma_M}_{f^\star,\rho}&=\left\{r>0: \forall J\in \cJ,\;\E \sup_{f \in F_{f^\star,\rho,r}} \left|\sum_{i\in J} \eps_{i} (Y_i-f^\star(X_i)) (f-f^\star)(X_i)\right| \leq \gamma_M |J| r^2 \right\}
\end{align*}and the two fixed point functions
\begin{gather*}
r_Q(\rho,\gamma_Q)=
\sup_{f^\star\in F}\{\inf \fQ^{\gamma_Q}_{f^\star,\rho}\},\qquad r_M(\rho,\gamma_M) =
\sup_{f^\star \in F}\{\inf \fM^{\gamma_M}_{f^\star,\rho}\}\enspace.
\end{gather*}The \textbf{link function} is any continuous and non-decreasing function $r:\R_+\to \R_+$ such that for all $\rho>0$
\begin{equation}\label{eq:defr}
r(\rho) = r(\rho, \gamma_Q, \gamma_M)\geq \max(r_Q(\rho, \gamma_Q), r_M(\rho, \gamma_M)).
\end{equation}
\end{Definition}Note that if the function $\rho\to \max(r_Q(\rho, \gamma_Q), r_M(\rho, \gamma_M))$ is itself continuous and non-decreasing then it can be taken equal to $r(\cdot)$. In the next paragraph, we provide an explicit computation of the functions $r_Q(\cdot)$, $r_M(\cdot)$ and $r(\cdot)$ in the ``LASSO case''.
\paragraph*{Complexity parameters for the $\ell_1$-regularization}
One can derive $r_Q(\cdot)$ and $r_M(\cdot)$ from Gaussian mean widths defined
for any $V\subset\R^d$, by
\begin{equation}\label{eq:Gauss_mean_width}
\ell^*(V) = \E\cro{ \sup_{(v_j)\in V} \sum_{j=1}^d g_j v_j},\quad \text{where }\quad (g_1,\ldots,g_d)\sim\mathcal N_d(0,I_d)\enspace.
\end{equation}
The dual norm of the $\ell_1^d$-norm is $1$-unconditional with respect to the canonical basis of $\R^d$ \cite[Definition~1.4]{shahar_gafa_ln}. Therefore, \cite[Theorem~1.6]{shahar_gafa_ln} applies under the following assumption.
\begin{Assumption}\label{ass:shahar_theo16}
There exist constants $q_0>2$, $C_0$ and $L$ such that $\zeta\in L^{q_0}$, $X$ is isotropic ($\E \inr{X, t}^2 = \norm{t}_2^2$ for every $t\in\R^d$) and its coordinates have $C_0 \log d$ subgaussian moments: for every $1\leq j\leq d$ and every $1\leq p\leq C_0 \log d$, $\norm{\inr{X,e_j}}_{L^p}\le L\sqrt{p}\norm{\inr{X,e_j}}_{L^2}$.
\end{Assumption}
\noindent
Under Assumption~\ref{ass:shahar_theo16}, if $\sigma=\norm{\zeta}_{L^{q_0}}$, \cite[Theorem~1.6]{shahar_gafa_ln} shows that, for every $\rho>0$,
\begin{gather*}
\E \sup_{v\in \rho B_1^d \cap r B_2^d} \left|\sum_{i\in[N]}\eps_i \inr{v, X_i}\right|\leq c_2\sqrt{N}\ell^*(\rho B_1^d \cap r B_2^d)\enspace,\\
\E \sup_{v\in \rho B_1^d \cap r B_2^d} \left|\sum_{i\in[N]} \eps_i \zeta_i \inr{v, X_i}\right|\leq c_2\sigma\sqrt{N}\ell^*(\rho B_1^d \cap r B_2^d)\enspace.
\end{gather*}
Local Gaussian mean widths $\ell^*(\rho B_1^d \cap r B_2^d)$ are bounded from above in \cite[ Lemma~5.3]{LM_reg_comp} and computations of $r_M$ and $r_Q$ follow
\begin{align*}
r_M^2(\rho) &\lesssim_{L,q_0, \gamma_M}
\begin{cases}
\sigma^2\frac{ d}{N} & \mbox{ if } \rho^2 N \geq \sigma^2 d^2\\
\rho\sigma\sqrt{\frac{1}{N}\log\Big(\frac{e\sigma d}{\rho\sqrt{N}}\Big)} & \mbox{ otherwise}
\end{cases}
\enspace,\\
\notag r_Q^2(\rho) &
\begin{cases}
= 0 & \mbox{ if } N \gtrsim_{L, \gamma_Q} d \\
\lesssim_{L, \gamma_Q} \frac{\rho^2}{N}\log\Big(\frac{c(L, \gamma_Q)d}{N}\Big) & \mbox{ otherwise}
\end{cases}
\enspace.
\end{align*}
Therefore, a link function is explicitly given by
\begin{equation}\label{eq:r_function_LASSO}
r^2(\rho) \sim_{L,q_0, \gamma_Q, \gamma_M}
\begin{cases}
\max\left(\rho \sigma\sqrt{\frac{1}{N}\log\Big(\frac{e \sigma d}{\rho\sqrt{N}}\Big)}, \frac{\sigma^2 d}{N}\right) & \mbox{ if } N \gtrsim_L d\\
\max\left(\rho \sigma \sqrt{\frac{1}{N}\log\Big(\frac{e \sigma d}{\rho\sqrt{N}}\Big)},\frac{\rho^2}{N}\log\Big(\frac{d}{N}\Big)\right) & \mbox{ otherwise}
\end{cases}
\enspace.
\end{equation}
\subsection{The estimators}
\label{sec:estimators}
Let $(T_{K,\lambda}(g, f))_{f, g\in F}$ denote the family of tests defined in \eqref{eq:beat_on_Bk}.
For every function $f\in F$, let $\mathcal B_{K,\lambda}(f)=\{g\in F : T_{K,\lambda}(g,f)\ge 0\}$ denote the set of all functions $g\in F$ that beats $f$. As explained in Section~\ref{Sec:LFT}, these sets will be measured by two metrics.
First, let
\begin{gather*}
R_{K,\lambda}^{\text{reg}}(f)=\sup_{g\in \mathcal B_{K,\lambda}(f)}\left\{\|g-f\|\right\} \mbox{ and } \hat f_{K,\lambda}^{(1)}\in \arg\min_{f\in F} R_{K,\lambda}^{reg}(f)\enspace.
\end{gather*}
Next, let
\begin{gather*}
R_{K,\lambda}^{(2)}(f)=\sup_{g\in \mathcal B_{K,\lambda}(f)}\left\{\MOM{K}{|g-f|}\right\}.
\end{gather*}
Lemma~\ref{lem:Isometry} below proves that, with large probability, $\MOM{K}{|f-g|}$ and $\norm{f-g}_{L^2_P}$ are isomorphic distances. The second criterion is then given by
\[
C^{(2)}_{K,\lambda}(f)=\inf\left\{\rho\ge 0 : R_{K,\lambda}^{\text{reg}}(f)\le \rho \text{ and }R_{K,\lambda}^{(2)}(f)\le 85\param{r} r(\rho)\right\}\enspace,
\]
where $r(\cdot)$ is a link function as defined in Definition~\ref{def:the-three-paraemters}. That is a continuous and non-decreasing function such that for all $\rho>0$, $r(\rho)\geq \max(r_M(\rho,\gamma_M), r_Q(\rho,\gamma_Q))$ where the choice of $\gamma_Q$ and $\gamma_M$ is given in Theorem~\ref{theo:basic-combining-loss-and-reg} below.
The associated estimator is then given by
\[
\hat f_{K,\lambda}^{(2)}\in \argmin_{f\in F}C^{(2)}_{K,\lambda}(f)\enspace.
\]
\subsection{The sparsity equation}
\label{sub:the_sparsity_equation}
By \eqref{GenRiskBound}, estimation rates for $\hat f_{K,\lambda}^{(2)}$ will be derived from upper bounds on $C_{K,\lambda}^{(2)}(f^*)$.
To get these, our strategy is to show that $T_{K,\lambda}(f^*,f)> 0$ for all $f$ such that $\|f-f^*\|$ or $\|f-f^*\|_{L^2_P}$ is large.
Recall that the quadratic / multiplier decomposition of the excess quadratic risk:
\begin{equation}\label{eq:SE1}
T_{K,\lambda}(f^*,f)=\text{MOM}_K[(f-f^*)^2-2\zeta(f-f^*)]+\lambda(\|f\|-\|f^*\|)\enspace.
\end{equation}
Let $f\in F$ and $\rho=\norm{f-f^*}$. When $\rho$ is large and $\norm{f-f^*}_{L^2_P}$ is small, $T_{K,\lambda}(f^*,f)> 0$ thanks to the regularization term $\lambda(\|f\|-\|f^*\|)$ in \eqref{eq:SE1} because the quadratic term $(f-f^*)^2$ is likely to be small.
We will therefore derive a lower bound on the regularization term when the subdifferential of $\norm{\cdot}$ is ``large'' in the following sense.
First, we recall that the subdifferential of $\norm{\cdot}$ in $f\in F$ is the set
\begin{equation*}
(\partial\norm{\cdot})_f = \{z^*\in E^*: \norm{f+h}\geq \norm{f}+z^*(h) \mbox{ for every } h\in E \}\enspace,
\end{equation*}
where $(E^*, \norm{\cdot}^*)$ is the dual normed space of $(E,\norm{\cdot})$ (and $E$ is the linear space containing $F$ onto which $\norm{\cdot}$ is defined).
For all $\rho>0$, let $H_\rho$ denote the set
\[
H_{\rho} = \{f \in F : \norm{f-f^*}=\rho, \; \|f-f^*\|_{L^2_P} \leq r(\rho)\}
\]where $r(\cdot)$ is the \textit{link function} from Definition~\ref{def:the-three-paraemters}. Let $\Gamma_{f^*}(\rho)$ denote the union of all subdifferentials of $\norm{\cdot}$ at functions ``close" to $f^*$
\[
\Gamma_{f^*}(\rho)=\bigcup_{f\in B(f^*, \rho/20)}(\partial \norm{\cdot})_f\enspace.
\]
Intuitively, every norm is associated with a notion of ``sparsity'' if one agrees to say that a non-zero function $f^{**}$ is \textit{sparse} w.r.t. the norm $\norm{\cdot}$ when the subdifferential of this norm at $f^{**}$ is a ``large subset'' of the dual sphere (i.e. the sphere of $(E^*, \norm{\cdot}^*)$). Sparse functions $f^{**}$ are useful in our context because a large lower bound on $\norm{f}-\norm{f^{**}}$ (and so for $\norm{f}-\norm{f^{**}}$ when $\norm{f^{**}-f^*}$ is small enough) can be derived when the vector $f-f^{**}$ is in the right direction. This intuition are formalized in the sparsity equation. More precisely, let
\begin{equation*}
\forall \rho>0,\qquad \Delta(\rho) = \inf_{f \in H_{\rho}} \sup_{z^* \in \Gamma_{f^*}(\rho)} z^*(f-f^*)\enspace.
\end{equation*}
$\Delta(\rho)$ is a uniform lower bound on $\|f\|-\|f^{**}\|$ if $f^{**}\in B(f^*, \rho/20)$. Thus, $\|f\|-\|f^*\|\gtrsim \rho$, if $\sup_{f^{**}\in \Gamma_{f^*}(\rho)}(\|f\|-\|f^{**}\|)\gtrsim \rho$ or if the following \emph{sparsity equation} of \cite{LM_reg_comp} holds.
\begin{Definition}\label{def:sparsity_equation}
A radius $\rho>0$ satisfies the \textbf{sparsity equation} if $\Delta(\rho) \geq 4\rho/5$.
\end{Definition}
\noindent
If $\rho^*$ satisfies the sparsity equation, so do all $ \rho\geq \rho^*$. Therefore, one can define
\begin{equation}\label{eq:SE}
\rho^* = \inf\left(\rho>0: \Delta(\rho) \geq \frac{4\rho}{5}\right).
\end{equation}
\paragraph*{The sparsity equation in $\ell_1^d$-regularization}
The equation has been solved in this example in \cite[Lemma~4.2]{LM_reg_comp},
recall this result.
\begin{Lemma} \label{lem:sparse_equa_LASSO}
If there exists $v\in\R^d$ such that $v \in t^*+(\rho/20)B_1^d$ and $|{\rm supp}(v)| \leq c\rho^2/r^2(\rho)$ then
\begin{equation*}
\Delta(\rho)=\inf_{h \in \rho S_1^{d-1}\cap r(\rho)B_2^{d}} \sup_{g \in \Gamma_{t^*}(\rho)} \inr{h, g-t^*}\geq \frac{4\rho}{5}\enspace.
\end{equation*}
where $S_{1}^{d-1}$ is the unit sphere of the $\ell_1^d$-norm and $B_2^{d}$ is the unit Euclidean ball in $\R^d$.
\end{Lemma}
\noindent
If $N\gtrsim s \log(ed/s)$ and if there exists a $s$-sparse vector in $t^*+(\rho/20)B_1^d$, Lemma~\ref{lem:sparse_equa_LASSO} and the choice of $r(\cdot)$ in \eqref{eq:r_function_LASSO} imply that for $\sigma = \norm{\zeta}_{L^{q_0}}$,
\begin{equation*}
\rho^* \sim_{L, q_0} \sigma s \sqrt{\frac{1}{N}\log\left(\frac{ed}{s}\right)} \mbox{ and } r^2(\rho^*)\sim \frac{\sigma^2 s}{N}\log\left(\frac{ed}{s}\right)
\end{equation*}then $\rho^*$ satisfies the sparsity equation and $r^2(\rho^*)$ is the rate of convergence of the LASSO (cf. \cite{LM_reg_comp}).
\section{Main results}\label{sub:main_result}
\subsection{Performances of the estimators}
\noindent
Theorem \ref{theo:basic-combining-loss-and-reg} gathers estimation error bounds satisfied by the estimators $ {\hat f}_{K,\lambda}^{(j)}$ for $j=1, 2$ defined in Section~\ref{sec:estimators}.
\begin{Theorem}\label{theo:basic-combining-loss-and-reg} Grant Assumptions \ref{ass:Mom2F},~\ref{ass:margin} and \ref{ass:small-ball} and let $r_Q$, $r_M$ anr $r$ denote the functions introduced in Definition~\ref{def:the-three-paraemters} for
\begin{equation*}
\gamma_Q = \min\left(\frac{1}{661\theta_0}, \frac{1}{1764\theta_r}\right), \gamma_M = \frac{\eps}{ 168} \mbox{ and } \eps = \frac{3}{331\theta_0^2}.
\end{equation*} Let $\rho^*$ be defined in \eqref{eq:SE} and let $K^*$ denote the smallest integer such that
\[
K^*\ge \max\left( \frac{8K_o}{7}, \frac{N \eps^2 r^2(\rho^*)}{336\theta_m^2}\right)\enspace.
\]
For all $K\ge 1$, let $\rho_K$ be a solution of $r^2(\rho_K)= [16 \theta_m^2/(\eps^2\alpha)]\sqrt{K/N}$. Assume that for every $i\in \cI$, $K\in[K^*, N]$ and $f\in F\cap B(f^*,\rho_K)$,
\begin{equation}\label{eq:robust_theo_basic}
2(P_i-P) \zeta (f-f^*)\leq \eps\max\left(\frac{16 \theta_m^2}{\eps^2\alpha}\frac{K}{N}, r^2_M(\rho_K, \gamma_M), \norm{f-f^*}^2_{L^2_P}\right)\enspace.
\end{equation}
For all $K\in[K^*,N/(84\param{r}^2\theta_0^2)]$, on an event $\Omega_1(K)$ such that $\bP(\Omega_1(K))\ge 1-4\exp(-K/1008)$,
the estimators $ {\hat f}_{K,\lambda}^{(j)}$ for $j=1, 2$ defined in Section~\ref{sec:estimators} satisfy
\begin{equation*}
\norm{\est^{(1)}-f^*}\leq \rho_K\enspace,
\end{equation*}and
\begin{equation*}
\norm{\est^{(2)}-f^*}\leq \rho_K,\quad\norm{\est^{(2)}-f^*}_{L^2_P}\leq 340 \theta_0\theta_r r(\rho_K)
\end{equation*}when the regularization parameter satisfy
\[
\frac{20\eps}{7}\frac{r^2(\rho_K)}{\rho_K}<\lambda<\frac{10}{331\theta_0^2}\frac{r^2(\rho_K)}{\rho_K}\enspace.
\]
\end{Theorem}
To the best of our knowledge, Theorem~\ref{theo:basic-combining-loss-and-reg} provides the first statistical performance of an estimator operating in such a ``nasty'' environment: the dataset may be corrupted by complete outliers, the informative data may be heavy-tailed and their distribution $P_i$ for $i\in\cI$ is only asked to have a $L^2$ and $L^1$ geometry over $F-f^*$ equivalent to that of $P$.
The most surprising thing is that the rate we obtain for $K=K^*$ in Theorem~\ref{theo:basic-combining-loss-and-reg}, i.e. $r(\rho_{K^*})$ when the number of outliers $K_o$ is less than $N r^2(\rho^*)$ is the minimax rate we would have gotten in a very good i.i.d. subgaussian framework with independent noise.
This means that the quality of a dataset does not have to be as good as it is classically assumed in the literature to make estimation possible: all we need is that a large fraction of the data should be independent (even though we believe that some ``weak dependence'' could also be introduced) and distributed according to distributions inducing $L^1$ and $L^2$ geometries equivalent to the $L^2_P$ one.
In Theorem~\ref{theo:basic-combining-loss-and-reg}, $K$ can be as small as the infimum between the number of outliers and $N$ times the minimax rate of convergence. Henceforth, if the optimal rate is known, as in \cite{LugosiMendelson2016}, Theorem~\ref{theo:basic-combining-loss-and-reg} shows that Le Cam's champion of the median of means tournament with $K=K^*$ reaches the same performances as any champion in this paper.
Theorem~\ref{theo:basic-combining-loss-and-reg} is thus an extension of \cite{LugosiMendelson2016} to a non-i.d. corrupted setting for Le Cam's champion. Moreover, our control improves theirs if the upper bound on the radius of $f^*$ used in \cite{LugosiMendelson2016} is pessimistic (cf. Example~\ref{Ex:LugMen} in Section~\ref{sec:Example}).
Assumption~\ref{ass:Mom2F} is automatically satisfied in the i.i.d. case and so is Assumption~\eqref{eq:robust_theo_basic}.
Theorem~\ref{theo:basic-combining-loss-and-reg} goes beyond this i.i.d. setup, relaxing
the i.d. assumptions into proximity assumptions between $L_{P_i}^2$ and $L^2_P$ geometries, for informative data.
\paragraph*{Risk bounds in $\ell_1^d$ regularization}Let us now
compute explicit values of $\rho_K$ and $\lambda\sim r^2(\rho_K)/\rho_K$ in the $\ell_1^d$-regularization case. Let $K\in[N]$ and $\sigma = \norm{\zeta}_{L^{q_0}}$. The equation $K=c r(\rho_K)^2N$ is solved by
\begin{equation}\label{eq:LASSO_choice_rho_K}
\rho_K\sim_{L,q_0} \frac{K}{\sigma} \sqrt{\frac{1}{N}\log^{-1}\left(\frac{\sigma^2 d}{K}\right)}
\end{equation} for the $r(\cdot)$ function defined in \eqref{eq:r_function_LASSO}.
%
Therefore, \begin{equation}\label{eq:reg_param_lasso}
\lambda \sim \frac{r^2(\rho_K)}{\rho_K} \sim_{L,q_0} \sigma \sqrt{\frac{1}{N}\log\left(\frac{e\sigma d}{\rho_K\sqrt{N}}\right)} \sim_{L,q_0} \sigma \sqrt{\frac{1}{N}\log\left(\frac{e\sigma^2 d}{K}\right)}\enspace.
\end{equation}
The regularization parameter depends on the ``level of noise'' $\sigma$, the $L^{q_0}$-norm of $\zeta$. This parameter is unknown in practice. Nevertheless, it can be estimated and replaced by this estimator in the regularization parameter as in \cite[Sections~5.4 and 5.6.2]{MR3307991}.
\subsection{Adaptive choice of $K$ by Lepski's method}
The main drawback of Theorem~\ref{theo:basic-combining-loss-and-reg} is that optimal rates are only achieved when $K\approx K^*$. Since $K^*$ is unknown, it cannot be used in general. This issue is tackled in this section by Lepski's method.
Let $K_1=K^*$ and $K_2=N/(84\theta_0^2\param{r}^2)$ be defined as in Theorem~\ref{theo:basic-combining-loss-and-reg}. For any integer $K\in[K_1,K_2]$, let $\rho_K$ and $\lambda$ be defined as in Theorem~\ref{theo:basic-combining-loss-and-reg} and for $j=1, 2$ denote by $\hat f_K^{(j)} =\est^{(j)}$ for this choice of $\lambda$.
These estimators are the building blocks of the following confidence sets.
For all $f\in F$, let
\begin{equation*}
\hat B^{(2)}_K(f)=\left\{g\in F : \MOM{K}{|g-f|}\le 28900\theta_r^2\theta_0 r(\rho_K)\right\}\enspace.
\end{equation*}
Now, let
\begin{equation*}
R_K^{(1)}=B({\hat f}_{K}^{(1)},\rho_K),\quad R_K^{(2)}=B({\hat f}_{K}^{(2)},\rho_K)\cap \hat B^{(2)}_K(\hat f_K^{(2)})
\end{equation*}
and for every $j=1, 2$, let
\[
\hat{K}^{(j)}=\inf\left\{K\in[K_2]: \bigcap_{J=K}^{K_2}R_J^{(j)}\ne \emptyset\right\}\enspace.
\]
Finally, define adaptive (to $K$) estimators via Lepski's method: for $j=1, 2$, $
{\hat f}_{LE}^{(j)} \in \bigcap_{J=\hat{K}^{(j)}}^{K_2}R_J^{(j)}$.
\begin{Theorem}\label{theo:main_baby_Lepski}
Grant assumptions and notations of Theorem~\ref{theo:basic-combining-loss-and-reg}. There exist absolute constants $(\cabs{i})_{1\le i\le 2}$ such that the estimators $ {\hat f}_{LE}^{(j)}$ for $j=1, 2$
satisfy for every $K\in[ K^*, N/(84\theta_0^2\param{r}^2)]$, with probability at least $1-\cabs{1}\exp\left(-\cabs{2}K\right)$,
\begin{equation*}
\norm{ {\hat f}_{LE}^{(1)}-f^*}\leq 2\rho_{K}\enspace,
\end{equation*}and
\begin{equation*}
\norm{ {\hat f}_{LE}^{(2)}-f^*}\leq 2\rho_{K},\quad\norm{ {\hat f}_{LE}^{(2)}-f^*}_{L^2_P}\leq 680 \theta_r\theta_0 r(2\rho_K)\enspace.
\end{equation*}
In particular, for $K=K^*$, if the following regularity assumption holds: there exists an absolute constant $c_3$ such that for all $\rho>0$, $r(2\rho)\leq c_3 r(\rho)$ then with probability at least
\[
1-c_1\exp\left(-c_4 N \max\left(\frac{K_o}{N}, \frac{r^2(\rho^*)}{\theta_0^4\theta_m^2}\right)\right)
\]
then,
\begin{equation*}
\norm{ {\hat f}_{LE}^{(2)}-f^*}_{L^2_P}\leq c_5 \max\left(\theta_0^4\theta_m^2\frac{K_o}{N}, r^2(\rho^*)\right)\enspace.
\end{equation*}
\end{Theorem}
Recall an optimality result from \cite{LM13}. Assume that all $(X_i, Y_i), i\in[N]$ are distributed according to $(X,Y^{f^*})$, where $f^*\in F$, $Y^{f^*}=f^*(X) + \zeta$ and $\zeta$ is a centered Gaussian variable with variance $\sigma$ independent of $X$. Assume that $F$ is $L$-subgaussian : for every $f\in F$ and $p\ge 2$, $\norm{f}_{L^p}\leq L\sqrt{p} \norm{f}_{L^2}$. Then, \cite[Theorem~A${}^{\prime}$]{LM13} proves that if $\tilde f_N$ is an estimator such that for every $f^*\in F$ and every $r>0$, with probability at least $1-c_0\exp(- \sigma^{-1}r^2 N/c_0)$, $\norm{\tilde f_N-f^*}_{L^2_P}\leq \zeta_N$, then necessarily
\begin{equation}\label{eq:minimax_rate_LM13}
\zeta_N\gtrsim \min\left(r, {\rm diam}(F, L^2_P)\right).
\end{equation}
When $Y^{f^*}=f^*(X) + \zeta$, $c\sim 1/\param{m}\sim 1/\sigma$. Applying this result to $r=r(\rho_K)$ for some given $K\geq K^*$ shows no procedure can estimate $f^*$ in $L^2_P$ uniformly over $F$ with confidence at least $1-c_0\exp(-K/c_0)$ at a rate better than $r(\rho_K)$ (we implicitly assumed that $r(\rho_K)\leq {\rm diam}(F, L^2_P)$ since $r(\rho_K)$ can obviously be replaced by $r(\rho_K)\wedge {\rm diam}(F, L^2_P)$ in all results). Moreover, this rate is minimax since \cite[Theorem~A]{LM13} also shows that the ERM over $\rho_K B$, $\hat f^{ERM}_N \in\argmin_{f\in \rho_{K}B} P_N\ell_f$, satisfies $\norm{\hat f^{ERM}_N - f^*}_{L^2}\lesssim r(\rho_K)$ with probability at least $1-c_0\exp(-\sigma^{-1} r^2(\rho_{K})N/c_0)$ when $\sigma\gtrsim r_Q(\rho_{K})$.
Theorem~\ref{theo:main_baby_Lepski} shows that $\hat f_{LE}$ achieves the same rate of convergence with the same exponentially high confidence as a minimax estimator does in the Gaussian regression model (with independent noise).
These rates are achieved here under very weak stochastic assumptions allowing the presence of outliers, without assuming that the regression function lies in $F$ or that the data are i.i.d.. Compared to \cite{LugosiMendelson2016}, using a Lepski method, we don't have to \emph{choose} the integer $K$ in advance, we let the data decide the best choice and automatically get an estimator with the correct minimax rate of convergence. Moreover, the regularization parameter is chosen adaptively, which yields to exact minimax rates and, since this minimax rate is not required to build the estimators, these are naturally adaptive.
\paragraph*{Adaptive results in $\ell_1^d$ regularization}
The following result follows from Theorem~\ref{theo:main_baby_Lepski} together with the computation of $\rho^*$, $r_Q$, $r_M$ and $r$ from the previous sections. This is a slight extension of Theorem~\ref{theo:mom_lasso} to the case where the oracle $t^*$ is not exactly sparse but close to a sparse vector.
\begin{Theorem}\label{theo:mom_lasso_sharp}
Assume that $X$ is isotropic and
\begin{enumerate}
\item[o)] there exist $s\in[N]$ such that $N\geq c_1 s \log(ed/s)$ and $v\in\R^d$ such that $\norm{t^*-v}_1\leq \sigma s \sqrt{\log\left(ed/s\right)/N}/20$ and $|{\rm supp}(v)|\leq s$.
\item[i')] $|\cI|\geq N/2$ and $|\cO|\leq c_1 s \log(ed/s)$,
\item[ii)] $ \zeta=Y-\inr{X, t^*} \in L_{q_0}$ for some $q_0>2$
\item[iii')] for every $1\leq p\leq C_0 \log(ed)$, $\norm{\inr{X,e_j}}_{L_p}\leq L \sqrt{p}\norm{\inr{X,e_j}}_{L_2}$ where $(e_j)_{j\in[d]}$ is the canonical basis of $\R^d$ and $C_0$ is some absolute constant,
\item [iv')] there exists $\theta_0$ such that $\norm{\inr{X, t}}_{L^1}\leq \theta_0\norm{\inr{X, t}}_{L^2}$, for all $t\in \R^d$,
\item [v)] there exists $\theta_m$ such that ${\rm var}(\zeta\inr{X, t})\leq \param{m}^2 \norm{t}_2^2$, for all $t\in\R^d$.
\end{enumerate}
The MOM-LASSO estimator $\hat{t}_{LE}$ such that $\hat f_{LE}=\inr{\hat{t}_{LE}, \cdot}$ satisfies, with probability at least $1-c_2 \exp(-c_3 s \log(ed/s))$, for every $1\leq p\leq 2$,
\begin{equation*}
\norm{\hat{t}_{LE} - t^*}_p\leq c_4(L, \theta_m)\norm{\zeta}_{L_{q_0}}s^{1/p}\sqrt{\frac{1}{N}\log\left(\frac{ed}{s}\right)}\enspace,
\end{equation*}
\end{Theorem}
In particular, Theorem~\ref{theo:mom_lasso_sharp} shows that, for our estimator contrary to the one in \cite{LugosiMendelson2016}, the sparsity parameter $s$ does not have to be known in advance in the LASSO case.
\proof
It follows from Theorem~\ref{theo:main_baby_Lepski}, the computation of $r(\rho_K)$ from \eqref{eq:r_function_LASSO} and $\rho_K$ in \eqref{eq:LASSO_choice_rho_K} that with probability at least $1-c_0\exp(- cr(\rho_K)^2N/\overline{C})$, $\norm{\hat{t}_{LE} - t^*}_1\leq \rho_{K^*}$ and $\norm{\hat t_{LE} - t^*}_2\lesssim r(\rho_K)$.
The result follows since $\rho_{K^*}\sim\rho^* \sim_{L, q_0} \sigma s \sqrt{\frac{1}{N}\log\left(\frac{ed}{s}\right)}$
and $\norm{v}_p\leq \norm{v}_1^{-1+2/p}\norm{v}_2^{2-2/p}$ for all $v\in\R^d$ and $1\leq p\leq2$.
{\mbox{}\nolinebreak\hfill\rule{2mm}{2mm}\par\medbreak}
\section{Proofs}\label{sec:Proofs}
In all the proof section, we denote by $\bP$ the distribution of $(X_1,\ldots,X_N)$ and $\E$ the corresponding expectation. For any non-empty subset $B\subset [N]$ and any $f:\cX\to\R$ for which it makes sense, let $\overline P_Bf=\frac1{|B|}\sum_{i\in B}P_if$. For any $f\in L^2_P$ and $r\ge 0$, let
\[
B_2(f,r)=\{g\in L^2_P : \norm{f-g}_{L^2_P}\le r\},\quad S_2(f,r)=\{g\in L^2_P : \norm{f-g}_{L^2_P}= r\}\enspace.
\]We consider the set of indices of blocks $B_k$ containing only informative data:
\begin{equation*}
\cK = \left\{k\in[K]: B_k\subset \cI\right\}.
\end{equation*}
\subsection{Lower Bound on the quadratic process}
\begin{Lemma}\label{lem:UBQP} Grant Assumptions~\ref{ass:Mom2F} and~\ref{ass:small-ball}. Fix $\eta\in (0,1)$, $\rho>0$ and let $\alpha,\gamma_Q,\gamma,x\in (0,1)$ be such that $\gamma\left(1-\alpha-x-32\theta_0 \gamma_Q\right) \ge 1-\eta$. Let $K\in[ K_o/(1-\gamma),N\alpha/(2\theta_0\param{r})^2]$.
There exists an event $\Omega_Q(K, \rho)$ such that $\bP\left(\Omega_Q(K, \rho)\right)\geq 1-\exp(-K\gamma x^2/2)$ on which for all $f\in B(f^*,\rho)$ if $\norm{f-f^*}_{L^2_P}\ge r_Q(\rho,\gamma_Q)$ then
\begin{equation*}
\notag Q_{\eta,K}(|f-f^*|)\ge \frac1{4\theta_0}\norm{f-f^*}_{L^2_P}\mbox{ and }
Q_{\eta,K}((f-f^*)^2)\ge \frac1{(4\theta_0)^2}\norm{f-f^*}_{L^2_P}^2\enspace.
\end{equation*}
\end{Lemma}
\begin{proof}
For all $f\in F-\{f^*\}$, let $n_f=(f-f^*)/\norm{f-f^*}_{L^2_P}$. For $i\in \cI$, $P_i|n_f|\ge \theta_0^{-1}$ by Assumption~\ref{ass:small-ball} and $P_in_f^2\le \param{r}^2$ by Assumption~\ref{ass:Mom2F}. By Markov's inequality, for all $k\in \cK$,
\[
\bP\left(|P_{B_k}|n_f|-\overline P_{B_k}|n_f||>\frac{\param{r}}{\sqrt{\alpha|B_k|}}\right)\le \alpha
\]
and so
\[
\bP\left(P_{B_k}|n_f|\ge \frac1{\theta_0}-\frac{\param{r}}{\sqrt{\alpha|B_k|}}\right)\ge 1-\alpha\enspace.
\]
Since $K\le[\alpha/(2\theta_0\param{r})]^2N$ then $|B_k|=N/K\geq [\alpha/(2\theta_0\param{r})]^2$ and so we have
\begin{equation}\label{eq:markov_1}
\bP\left(P_{B_k}|n_f|\ge \frac1{2\theta_0}\right)\ge 1-\alpha\enspace.
\end{equation}
Let $\phi$ denote the function defined by $\phi(t)=(t-1)I(1\le t\le 2)+I(t\ge 2)$ for all $t\in\R_+$ and, for all $f\in F-\{f^*\}$, let $Z(f)=\sum_{k\in[K]}I(4\theta_0P_{B_k}|n_f|\ge 1)$. Since $I(t\ge 1)\ge \phi(t)$ for any $t\geq0$ then $Z(f)\ge \sum_{k\in\cK}\phi\left(4\theta_0P_{B_k}|n_f|\right)$. Since $\phi(t)\ge I(t\ge 2)$ for all $t\geq0$, it follows from \eqref{eq:markov_1} that
\begin{align*}
\E\left[\sum_{k\in \cK}\phi\left(4\theta_0P_{B_k}|n_f|\right)\right]\ge \sum_{k\in \cK}\bP\left(4\theta_0P_{B_k}|n_f|\ge 2\right)\ge |\cK|(1-\alpha)\enspace.
\end{align*}
Therefore, for all $f\in F$, we have
\begin{align*}
Z(f)\ge |\cK|(1-\alpha)+\sum_{k\in \cK}\left(\phi\left(4\theta_0P_{B_k}|n_f|\right)-\E\left[\phi\left(4\theta_0P_{B_k}|n_f|\right)\right]\right)\enspace.
\end{align*}
Let $\cF=\{f\in B(f^*, \rho) : \norm{f-f^*}_{L^2_P}\geq r_Q(\rho,\gamma_Q)\}$. By the bounded difference inequality (cf. \cite[Lemma~1.2]{MR1036755} or \cite[Theorem~6.2]{BouLugMass13}, there exists an event $\Omega(x)$ such that $\bP(\Omega(x))\ge 1-\exp(-x^2|\cK|/2)$, on which
\begin{multline*}
\sup_{f\in \cF
} \left|\sum_{k\in \cK}\left(\phi\left(4\theta_0P_{B_k}|n_f|\right)-\E\left[\phi\left(4\theta_0P_{B_k}|n_f|\right)\right]\right)\right|\\
\le \E\sup_{f\in \cF
} \left|\sum_{k\in \cK}\left(\phi\left(4\theta_0P_{B_k}|n_f|\right)-\E\left[\phi\left(4\theta_0P_{B_k}|n_f|\right)\right]\right)\right|+|\cK|x\enspace.
\end{multline*}
By the Gin{\'e}-Zynn symmetrization argument \cite[Lemma~11.4]{BouLugMass13},
\begin{align*}
\E\sup_{f\in \cF } \left|\sum_{k\in \cK}\left(\phi\left(4\theta_0P_{B_k}|n_f|\right)-\E\left[\phi\left(4\theta_0P_{B_k}|n_f|\right)\right]\right)\right|\le2\E\sup_{f\in \cF } \left|\sum_{k\in \cK}\epsilon_k\phi\left(4\theta_0P_{B_k}|n_f|\right)\right|
\end{align*}where $(\eps_k)_{k\in\cK}$ are independent Rademacher variables independent of the data. Moreover, $\phi$ is 1-Lipschitz and $\phi(0)=0$. By the contraction principle (cf. \cite[Theorem~4.12]{LT:91} or \cite[Theorem 11.6]{BouLugMass13}),
\[
\E\sup_{f\in \cF
} \left|\sum_{k\in \cK}\epsilon_k\phi\left(4\theta_0P_{B_k}|n_f|\right)\right| \le 4\theta_0\E\sup_{f\in \cF
} \left|\sum_{k\in \cK}\epsilon_kP_{B_k}|n_f|\right| \enspace.
\]
Applying again the symmetrization and contraction principles, we get,
\[
\E\sup_{f\in \cF} \left|\sum_{k\in \cK}\epsilon_kP_{B_k}|n_f|\right|\le \frac{4K}N\E\sup_{f\in \cF} \left|\sum_{i\in\cup_{k\in \cK}B_k}\epsilon_in_f(X_i)\right|\enspace.
\]
It follows from the convexity of $F$ that for all $f\in \cF$, $r_{Q}(\rho,\gamma_Q)n_f\in F-f^*$ and it also belongs to the $L^2_P$ sphere of radius $r_Q(\rho,\gamma_Q)$. Therefore, by definition of $r_Q:=r_Q(\rho,\gamma_Q)$ and for $J = \cup_{k\in \cK}B_k$,
\begin{align*}
\E\sup_{f\in \cF} &\left|\sum_{i\in J}\epsilon_in_f(X_i)\right|=\frac{1}{r_Q}\E\sup_{f\in F\cap S_2(f^*,r_Q)} \left|\sum_{i\in J}\epsilon_i(f-f^*)(X_i)\right| \le \gamma_Q\frac{|\cK|N}{K}\enspace.
\end{align*}
In conclusion, on $\Omega(x)$, all $f\in \cF$ is such that
\begin{align*}
Z(f)\ge |\cK|\left(1-\alpha-x-32\theta_0\gamma_Q \right)\ge (1-\eta)K\enspace.
\end{align*}
In other words, on $\Omega(x)$, for all $f\in \cF$, there exist at least $(1-\eta)K$ blocks $B_k$ such that $P_{B_k}|n_f|\ge (4\theta_0)^{-1}$. For any of these blocks $B_k$, $P_{B_k}n_f^2\ge (P_{B_k}|n_f|)^2$, hence, on $\Omega(x)$, $Q_{\eta,K}[|n_f|]\ge (4\theta_0)^{-1}$ and $Q_{\eta,K}[n_f^2]\ge (4\theta_0)^{-2}$.
\end{proof}
\subsection{Upper Bound on the multiplier process}
\begin{Lemma}\label{lem:proc_multiplicatif} Grant Assumption~\ref{ass:margin}. Fix $\eta\in (0,1)$, $\rho\in(0,+\infty]$, and let $\alpha,\gamma_M,\gamma,x$ and $\eps$ be positive absolute constants such that $\gamma\left(1-\alpha-x-8 \gamma_M/\eps\right) \ge 1-\eta$.
Let $K\in[K_o/(1-\gamma),N]$. There exists an event $\Omega_M(K, \rho)$ such that $\bP(\Omega_M(K, \rho))\ge 1-\exp(-\gamma K x^2/2)$ and on $\Omega_M(K, \rho)$, for all $f\in B(f^*, \rho)$ there is at least $(1-\eta)K$ blocks $B_k$ with $k\in\cK$ such that
\begin{align*}
\left|2(P_{B_k}-\overline P_{B_k})(\zeta(f-f^*))\right| \leq \eps\max\left(\frac{16 \theta_m^2}{\eps^2\alpha}\frac{K}{N}, r^2_M(\rho, \gamma_M), \norm{f-f^*}^2_{L^2_P}\right)\enspace.
\end{align*}
\end{Lemma}
\begin{proof}
For all $k\in [K]$ and $f\in F$, define $W_k(f)=2(P_{B_k}-\overline P_{B_k})\left(\zeta(f-f^*)\right)$ and
\begin{equation*}
\gamma_k(f)= \eps\max\left(\frac{16 \theta_m^2}{\eps^2\alpha}\frac{K}{N}, r^2_M(\rho, \gamma_M), \norm{f-f^*}^2_{L^2_P}\right)\enspace.
\end{equation*}
Let $f\in F$ and $k\in \mathcal K$. It follows from Markov's inequality and Assumption~\ref{ass:margin} that
\begin{align}\label{eq:multi_1}
\nonumber \bP&\left[2\Big|W_k(f)\Big|\ge \gamma_k(f)\right]\leq \frac{4\E \left[\Big(2(P_{B_k}-\overline P_{B_k})(\zeta (f-f^*))\Big)^2\right]}{ \frac{16\param{m}^2}{\alpha}\norm{f-f^*}_{L^2_P}^2\frac{K}{N}}\\
& \leq \frac{\alpha \sum_{i\in B_k}{\rm var}_{P_i}(\zeta (f-f^*))}{|B_k|^2 \param{m}^2\norm{f-f^*}_{L^2_P}^2\frac{K}N} \leq \frac{\alpha \param{m}^2 \norm{f-f^*}_{L^2_P}^2}{|B_k| \param{m}^2\norm{f-f^*}_{L^2_P}^2\frac{K}N} =\alpha \enspace.
\end{align}
Denote $J=\cup_{k\in\mathcal K}B_k$ and remark that $J\in \cJ$ as defined in Definition~\ref{def:the-three-paraemters}. Let $r_M := r_M(\rho,\gamma_M)$ for simplicity. We have
\begin{align*}
\E&\sup_{f\in B(f^*, \rho)}\sum_{k\in\mathcal K}\epsilon_k\frac{W_k(f)}{\gamma_k(f)}
\leq 2\E \sup_{f\in B(f^*, \rho)}\left|\sum_{k\in\mathcal K} \frac{\eps_k(P_{B_k}-\overline P_{B_k})(\zeta (f-f^*))}{\eps\max(r_M^2, \norm{f-f^*}^2_{L^2_P})}\right|\\
&\leq \frac{2}{\epsilon r_M^2}
\E \left[\sup_{f\in B(f^*, \rho)\setminus B_2(f^*, r_M)}\left|\sum_{k\in\mathcal K} \eps_k(P_{B_k}-\overline P_{B_k})\left(\zeta r_M\frac{f-f^*}{\norm{f-f^*}_{L^2_P}}\right)\right|\right.\\
&\qquad\qquad\qquad\;\left.\vee\sup_{f\in B(f^*, \rho)\cap B_2(f^*, r_M)}\left|\sum_{k\in\mathcal K} \eps_k(P_{B_k}-\overline P_{B_k})\left(\zeta (f-f^*)\right)\right|\right]\\
&\leq \frac{2}{\epsilon r_M^2}
\E \sup_{f\in B(f^*, \rho)\cap B_2(f^*, r_M)}\left|\sum_{k\in\mathcal K} \eps_k(P_{B_k}-\overline P_{B_k})\left(\zeta (f-f^*)\right)\right|\enspace,
\end{align*}where in the last but one inequality we used that $F$ is convex and the same argument as in the proof of Lemma~\ref{lem:UBQP}. Moreover, since the random variables $((\zeta_i(f-f^*)(X_i) -P_i\zeta (f-f^*)):i\in\cI)$ are centered and independent, the symmetrization argument applies and, by definition of $r_M$,
\begin{align}\label{eq:multi-2}
\nonumber \E\sup_{f\in B(f^*, \rho)}\sum_{k\in\mathcal K}\epsilon_k\frac{W_k(f)}{\gamma_k(f)}
&\leq \frac{4 K}{\epsilon r_M^2N} \E \sup_{f\in B(f^*, \rho) \cap B_2(f^*, r_M)}\left|\sum_{i\in J} \eps_{i}\zeta_i (f-f^*)(X_i)\right|\\
&\le \frac{4 K}{\epsilon N}\gamma_M|\cK|\frac NK=\frac{4\gamma_M}{\epsilon}|\cK|\enspace.
\end{align}
Now, let $\psi(t) = (2t-1)I(1/2\leq t \leq 1)+I(t\ge 1)$ for all $t\geq0$ and note that $\psi$ is $2$-Lipschitz, $\psi(0)=0$ and satisfies $I(t\ge 1)\le \psi(t)\le I(t\ge 1/2)$ for all $t\geq0$. Therefore, all $f\in B(f^*, \rho)$ satisfies
\begin{align*}
\sum_{k\in \cK}I&\left(|W_k(f)| < \gamma_k(f)\right)\\
&=|\cK|-\sum_{k\in \cK} I\left(\frac{|W_k(f)|}{\gamma_k(f)}\ge 1\right)\\
&\ge|\cK|-\sum_{k\in \cK}\psi\left(\frac{|W_k(f)|}{\gamma_k(f)}\right)\\
&=|\cK|-\sum_{k\in\cK}\bE \psi\left(\frac{|W_k(f)|}{\gamma_k(f)}\right)-\sum_{k\in \cK}\left[\psi\left(\frac{|W_k(f)|}{\gamma_k(f)}\right)-\bE \psi\left(\frac{|W_k(f)|}{\gamma_k(f)}\right)\right]\\
&\ge |\cK|-\sum_{k\in\cK}\bP\left(\frac{|W_k(f)|}{\gamma_k(f)}\ge \frac12\right)-\sum_{k\in \cK}\left[\psi\left(\frac{|W_k(f)|}{\gamma_k(f)}\right)-\bP\psi\left(\frac{|W_k(f)|}{\gamma_k(f)}\right)\right]\\
&\ge(1-\alpha)|\cK|-\sup_{f\in B(f^*, \rho)}\left|\sum_{k\in \cK}\left[\psi\left(\frac{|W_k(f)|}{\gamma_k(f)}\right)-\bE \psi\left(\frac{|W_k(f)|}{\gamma_k(f)}\right)\right]\right|
\end{align*}where we used \eqref{eq:multi_1} in the last inequality.
The bounded difference inequality ensures that, for all $x>0$, there exists an event $\Omega(x)$ satisfying $\bP(\Omega(x))\ge 1-\exp(-x^2|\cK|/2)$ on which
\begin{multline*}
\sup_{f\in B(f^*, \rho)} \left|\sum_{k\in \cK}\left[\psi\left(\frac{|W_k(f)|}{\gamma_k(f)}\right)-\bE\psi\left(\frac{|W_k(f)|}{\gamma_k(f)}\right)\right]\right|\\
\le \bE \sup_{f\in B(f^*, \rho)}\left|\sum_{k\in \cK}\left[\psi\left(\frac{|W_k(f)|}{\gamma_k(f)}\right)-\bE\psi\left(\frac{|W_k(f)|}{\gamma_k(f)}\right)\right]\right|+|\cK|x\enspace.
\end{multline*}
Furthermore, it follows from the symmetrization argument that
\begin{multline*}
\bE \sup_{f\in B(f^*, \rho)}\left|\sum_{k\in \cK}\left[\psi\left(\frac{|W_k(f)|}{\gamma_k(f)}\right)-\bE\psi\left(\frac{|W_k(f)|}{\gamma_k(f)}\right)\right]\right|\\
\le 2\bE\sup_{f\in B(f^*, \rho)}\left|\sum_{k\in \cK}\epsilon_k\psi\left(\frac{|W_k(f)|}{\gamma_k(f)}\right)\right|
\end{multline*}
and, from the contraction principle and \eqref{eq:multi-2}, that
\[
\bE\sup_{f\in B(f^*, \rho)}\left|\sum_{k\in \cK}\epsilon_k\psi\left(\frac{|W_k(f)|}{\gamma_k(f)}\right)\right|\le 2\bE \sup_{f\in B(f^*, \rho)}\left|\sum_{k\in \cK}\epsilon_k\frac{|W_k(f)|}{\gamma_k(f)}\right| \le \frac{8\gamma_M}{\eps}|\cK|\enspace.
\]
In conclusion, on $\Omega(x)$, for all $f\in B(f^*, \rho)$,
\begin{align*}
\sum_{k\in \cK}I\left(|W_k(f)|< \gamma_k(f)\right)&\ge \left(1-\alpha-x-8 \gamma_M/\eps\right)|\cK|\\
&\geq K \gamma \left(1-\alpha-x-8 \gamma_M/\eps\right)\geq (1-\eta)K\enspace.
\end{align*}
\end{proof}
\subsection{An isometry property of $\MOM{K}{\cdot}$ processes}
Besides the controls of the quadratic and multiplier MOM processes presented in Lemmas~\ref{lem:UBQP} and~\ref{lem:proc_multiplicatif} respectively, the estimation error bounds for the MOM estimators rely on the following isometry property of the MOM processus $f\in F\to \MOM{K}{|f-f^*|}$.
\begin{Lemma}\label{lem:Isometry}[Isometry property of the $\MOM{K}{\cdot}$ process]
Grant Assumptions~\ref{ass:Mom2F} and~\ref{ass:small-ball}. Fix $\eta\in (0,1)$, $\rho>0$ and let $\alpha,\gamma_Q, \gamma,x$ denote absolute constants in $(0,1)$ such that $\gamma\left(1-\alpha-x- 4 \theta_r\gamma_Q/\alpha\right)\ge 1-\eta$. Let $K\in [K_o/(1-\gamma), N\alpha/(2\theta_0 \theta_r)^2]$. There exists an event $\Omega_{iso}(K, \rho)\subset\Omega_Q(K, \rho)$ such that $\bP(\Omega_{iso}(K, \rho))\ge1-2\exp\left(-\gamma x^2 K/2\right)$ and on the event $\Omega_{iso}(K, \rho)$, for all $f\in B(f^*, \rho)$,
\begin{equation*}
Q_{1-\eta,K}{|f-f^*|}\le \theta_r \norm{f-f^*}_{L^2_P} + \frac{4\theta_r}{\alpha}\max\left(r_Q(\rho, \gamma_Q), \norm{f-f^*}_{L^2_P}\right)
\end{equation*} and if $\norm{f-f^*}_{L^2_P}\geq r_Q(\rho,\gamma_Q)$ then $Q_{\eta,K}{|f-f^*|}\ge (1/(4\theta_0))\norm{f-f^*}_{L^2_P}$.
In particular, for $\eta=1/2$, on the event $\Omega_{iso}(K, \rho)$, for all $f\in B(f^*, \rho)$, if $\norm{f-f^*}_{L^2_P}\geq r_Q(\rho,\gamma_Q)$, then
\begin{equation}\label{eq:iso_MOMK}
\frac{1}{4\theta_0}\norm{f-f^*}_{L^2_P} \leq \MOM{K}{|f-f^*|} \leq \theta_r \left(1+\frac{4}{\alpha}\right)\norm{f-f^*}_{L^2_P}.
\end{equation}
\end{Lemma}
\begin{proof}
It follows from Lemma~\ref{lem:UBQP} that on the event $\Omega_Q(K, \rho)$ for all $f\in B(f^*, \rho)$, if $\norm{f-f^*}_{L^2_P}\geq r_Q(\rho, \gamma_Q)$ then $Q_{\eta,K}{|f-f^*|}\ge (1/(4\theta_0)\norm{f-f^*}_{L^2_P}$. This yields the ``lower bound'' result in \eqref{eq:iso_MOMK}.
For the upper bound of the isomorphic result, we essentially repeat the proof of Lemma~\ref{lem:proc_multiplicatif}. Let us just highlight the main differences. We will use the same notation as in the proof of Lemma~\ref{lem:proc_multiplicatif} except that for all $f\in F$, we define
\begin{equation*}
W_k(f) = (P_{B_k} - \overline{P}_{B_k})|f-f^*| \mbox{ and } \gamma_k(f) = \frac{4\theta_r}{\alpha} \max\left(r_Q(\rho, \gamma_Q), \norm{f-f^*}_{L^2_P}\right).
\end{equation*}
It follows from Chebyshev's inequality and Assumption~\ref{ass:Mom2F} that
\begin{align*}
\bP\left[2 |W_k(f)|\geq \gamma_k(f)\right]\leq \frac{4 \overline{P}_{B_k}|f-f^*|}{\gamma_k(f)}\leq \frac{4 \theta_r \norm{f-f^*}_{L^2_P}}{\gamma_k(f)}\leq \alpha.
\end{align*}Moreover, by convexity of $F$, we have, for $r_Q:=r_Q(\rho, \gamma_Q)$,
\begin{align*}
(\star) &: =\E \sup_{f\in B(f^*, \rho)} \sum_{k\in\cK} \eps_k \frac{W_k(f)}{\gamma_k(f)}\\
& \leq \frac{4\theta_r}{\alpha r_Q} \E \sup_{f\in B(f^*, \rho)\cap S_2(f^*, r_Q)} \left|\sum_{k\in\cK} \eps_k (P_{B_k} - \overline{P}_{B_k})|f-f^*|\right|
\end{align*}and then using a symmetrization argument, we obtain that
\begin{align*}
(\star) \leq \frac{4\theta_r K}{\alpha r_Q N} \E \sup_{f\in B(f^*, \rho)\cap S_2(f^*, r_Q)} \left|\sum_{i\in J} \eps_i (f-f^*)(X_i)\right|\leq \frac{4 \theta_r \gamma_Q |\cK| }{\alpha }.
\end{align*}Finally, using the same argument as in the proof of Lemma~\ref{lem:proc_multiplicatif}, for all $x>0$ there exists an event $\Omega(x)$ such that $\bP(\Omega(x))\geq 1-\exp(-x^2|\cK|/2)$, on which for all $f\in B(f^*, \rho)$,
\begin{equation*}
\sum_{k\in\cK} I(|W_k(f)|\leq \gamma_k(f))\geq |\cK|(1-\alpha- x -4\theta_r \gamma_Q/\alpha)\geq (1-\eta)|\cK|.
\end{equation*}In particular, on the event $\Omega(x)$, for all $f\in B(f^*, \rho)$ there are more than $(1-\eta)K$ blocks $B_k$ for which, $P_{B_k}|f-f^*|\leq \overline{P}_{B_k}|f-f^*| + \gamma_k(f)$. Now, the result follows from Assumption~\ref{ass:Mom2F} since $\overline{P}_{B_k}|f-f^*|\leq \theta_r\norm{f-f^*}_{L^2_P}$.
\end{proof}
\subsection{Conclusion to the proof of Theorem~\ref{theo:basic-combining-loss-and-reg}}\label{sub:EndOfProof}
The proof relies on the following proposition.
\begin{Proposition}\label{prop:Useful}
Grant conditions of Theorem~\ref{theo:basic-combining-loss-and-reg}. Let $\gamma_Q=1/(661\theta_0)$, $\gamma_M=\eps/168$ for some $\eps < 7/(662 \theta_0^2)$ and the regularization parameter be such that
\begin{equation*}
\frac{20\eps r^2(\rho_K)}{7 \rho_K}< \lambda< \frac{10 r^2(\rho_K)}{331\theta_0^2 \rho_K}.
\end{equation*}
The event $\Omega_0(K) = \Omega_Q(K, \rho_K)\cap \Omega_M(K, \rho_K)$ is such that $\bP(\Omega_0(K))\ge 1-2\exp\left(-K/1008\right)$ and on $\Omega_0(K)$ for all $f\in F$ if $\|f-f^*\|_{L_P^2}\ge r(\rho_K)$ or $\|f-f^*\|\ge \rho_K$ then
\begin{equation*}
\text{MOM}_K\left[\ell_f-\ell_{f^*}\right]+\lambda(\|f\|-\|f^*\|)> 0\enspace.
\end{equation*}
\end{Proposition}
\begin{proof}
Using \eqref{prop:Cone}, \eqref{prop:Opposes} and \eqref{prop:Sum} together with the quadratic / multiplier decomposition of the excess quadratic loss yields that for all $f\in F$,
\begin{align}\label{Obj0}
\notag\text{MOM}_K\left[\ell_f-\ell_{f^*}\right] &= \text{MOM}_K\left[(f-f^*)^2-2\zeta(f-f^*)\right]\\
&\ge Q_{1/4,K}[(f-f^*)^2]-2Q_{3/4,K}[\zeta(f-f^*)]\enspace.
\end{align}
Note that $\gamma(1-\alpha - x - 32\theta_0 \gamma_Q) \geq 1-\eta$ when one chooses
\begin{equation}\label{eq:choice_constants}
\eta = \frac{1}{4}, \gamma = \frac{7}{8}, \alpha = \frac{1}{21}, x = \frac{1}{21}, \gamma_Q = \frac{1}{661 \theta_0}, \gamma_M = \frac{\eps}{168} \mbox{ and } \eps \leq \frac{1}{64 \theta_0^2}.
\end{equation}For this choice of constants, Lemma~\ref{lem:UBQP} applies and for $\rho=\rho_K$ we get that there exists an event $\Omega_Q(K, \rho_K)$ with probability larger than $1-\exp(-K/1008)$ and on that event, for all $f\in B(f^*,\rho_K)$, if $\norm{f-f^*}_{L^2_P}\geq r_Q(\rho_K,\gamma_Q)$ then
\begin{equation}\label{eq:quad_final_proof}
Q_{1/4,K}[(f-f^*)^2]\ge \frac1{(4\theta_0)^2}\norm{f-f^*}^2_{L^2_P}\enspace.
\end{equation}
Moreover, for the choice of parameters as in \eqref{eq:choice_constants}, we also have $\gamma(1-\alpha-x-8\gamma_M/\eps)\geq 1-\eta$, hence Lemma~\ref{lem:proc_multiplicatif} applies and for $\rho=\rho_K$ we get that there exists an event $\Omega_M(K, \rho_K)$ with probability larger than $1-\exp(-K/1008)$ and on that event, for all $f\in B(f^*,\rho_K)$ there are more than $3K/4$ blocks $B_k$ with $k\in \cK$ such that
\begin{equation*}
|2(P_{B_k}-\overline P_{B_k})\zeta(f-f^*)|\le \eps\max\left(\frac{16 \theta_m^2}{\eps^2\alpha}\frac{K}{N}, r^2_M(\rho_K, \gamma_M), \norm{f-f^*}^2_{L^2_P}\right)\enspace.
\end{equation*}
Combining the last result with Assumpion~\eqref{eq:robust_theo_basic}, it follows that on the event $\Omega_M(K, \rho_K)$, for all $f\in B(f^*,\rho_K)$,
\begin{equation}\label{eq:multi_final_proof}
2Q_{3/4,K}[\zeta(f-f^*)]\le 2\eps\max\left(\frac{16 \theta_m^2}{\eps^2\alpha}\frac{K}{N}, r^2_M(\rho_K, \gamma_M), \norm{f-f^*}^2_{L^2_P}\right)\enspace.
\end{equation}
Let us now prove that on the event $\Omega_M(K, \rho_K)\cap \Omega_Q(K, \rho_K)$, one has for all $f\in B(f^*,\rho_K)$,
\begin{equation}\label{Obj3}
\text{MOM}_K\left[(f-f^*)^2-2\zeta(f-f^*)\right]\ge -2\eps r^2(\rho_K) \enspace.
\end{equation}
Assume that $\Omega_M(K, \rho_K)\cap \Omega_Q(K, \rho_K)$ holds and let $f\in B(f^*,\rho_K)$. First assume that $\norm{f-f^*}_{L^2_P}\geq r^2(\rho_K)$. Then, it follows from \eqref{Obj0}, \eqref{eq:quad_final_proof} and \eqref{eq:multi_final_proof}, the choice of $\eps$ in \eqref{eq:choice_constants} and the definition of $\rho_K$ that
\begin{align}
\label{Obj2}&\text{MOM}_K\left[(f-f^*)^2-2\zeta(f-f^*)\right]\ge\pa{\frac1{(4\theta_0)^2}-2\epsilon}\norm{f-f^*}^2_{L^2_P} \ge\frac{\norm{f-f^*}^2_{L^2_P}}{32\theta_0^2}.
\end{align}Now, if $\norm{f-f^*}_{L^2_P}\leq r^2(\rho_K)$ then it follows from \eqref{Obj0}, \eqref{eq:multi_final_proof} and the definition of $\rho_K$ that
\begin{equation*}
\text{MOM}_K\left[(f-f^*)^2-2\zeta(f-f^*)\right]\ge -2\eps r^2(\rho_K)
\end{equation*}and \eqref{Obj3} follows.
\paragraph{Conclusion of the proof when the regularization distance is small (i.e. $\|f-f^*\|\leq \rho_K$) and the $L^2_P$-distance is large (i.e. $\norm{f-f^*}_{L_P^2}\ge r(\rho_K)$)} Let $f\in F$ be such that $\|f-f^*\|\le \rho_K$ and $\norm{f-f^*}_{L_P^2}\ge r(\rho_K)$. It follows from the triangular inequality that $\|f\|-\|f^*\|\ge -\|f-f^*\|\ge -\rho_K$. Combining this together with \eqref{Obj2}, it follows that
\begin{equation*}
\text{MOM}_K\left[\ell_f-\ell_{f^*}\right]+\lambda(\|f\|-\|f^*\|)\ge \frac{\norm{f-f^*}^2_{L^2_P}}{32\theta_0^2} - \lambda \rho_K\geq\frac{r^2(\rho_K)}{32\theta_0^2} - \lambda \rho_K>0
\end{equation*} when $\lambda < r^2(\rho_K)/(32\theta_0^2 \rho_K)$.
\paragraph{Conclusion of the proof when the regularization distance is large (i.e. $\|f-f^*\|\geq \rho_K$): the homogeneity argument}
\begin{Lemma}\label{lem:Hom1}
For all $f\in F$ such that $\|f-f^*\|\ge \rho_K$
\begin{equation*}
\norm{f}-\norm{f^*}\geq\sup_{z^*\in\Gamma_{f^*}(\rho_K)}z^*(f-f^*)-\frac{\rho_K}{10}\enspace.
\end{equation*}
\end{Lemma}
\begin{proof}
For every $f^{**}\in F^* +(\rho_K/20)B$ and every $z^*\in(\partial\norm{\cdot})_{f^{**}}$,
\begin{align*}
\norm{f}&-\norm{f^*}\geq \norm{f} - \norm{f^{**}} -\norm{f^{**}-f^*}\geq z^*(f-f^{**})-\frac{\rho_K}{20} \\
&= z^*(f-f^*)-z^*(f^{**}-f^*)-\frac{\rho_K}{20}\geq z^*(f-f^*)-\frac{\rho_K}{10}\enspace.
\end{align*}
\end{proof}
\begin{Lemma}\label{lem:Hom2}
Assume that, for all $f\in F\cap S(f^*,\rho_K)$,
\begin{equation}\label{Obj4}
\MOM{K}{(f-f^*)^2-2\zeta(f-f^*)}+\lambda\sup_{z^*\in\Gamma_{f^*}(\rho_K)}z^*(f-f^*)> \lambda\frac{\rho_K}{10}\enspace.
\end{equation}
Then \eqref{Obj4} holds for all $f\in F$ such that $\norm{f-f^*}\ge \rho_K$.
\end{Lemma}
\begin{proof}
Let $f\in F$ be such that $\norm{f-f^*}\ge \rho_K$. Define $g=f^*+\rho_K\frac{f-f^*}{\|f-f^*\|}$ and remark that $\norm{g-f^*}_{L^2_P} = \rho_K$ and that, by convexity of $F$, $g\in F$. It follows from \eqref{Obj4} that for $\kappa=\|f-f^*\|/\rho_K\ge 1$, one has
\begin{align*}
&\MOM{K}{(f-f^*)^2-2\zeta(f-f^*)}+\lambda\sup_{z^*\in\Gamma_{f^*}(\rho_K)}z^*(f-f^*)\\
&=\MOM{K}{\kappa^2(g-f^*)^2-2\kappa\zeta(g-f^*)}+\lambda\kappa\sup_{z^*\in\Gamma_{f^*}(\rho_K)}z^*(g-f^*)\\
&\ge \kappa\left(\MOM{K}{(g-f^*)^2-2\zeta(g-f^*)}+\lambda\sup_{z^*\in\Gamma_{f^*}(\rho_K)}z^*(g-f^*)\right)\\
&> \kappa \lambda\frac{\rho_K}{10}\geq \lambda\frac{\rho_K}{10}\enspace.
\end{align*}
\end{proof}
Let $f\in F$ be such that $\norm{f-f^*}\ge \rho_K$. By Lemma~\ref{lem:Hom1},
\begin{align*}
&\MOM{K}{(f-f^*)^2-2\zeta(f-f^*)}+\lambda(\|f\|-\|f^*\|)\\
&\ge \MOM{K}{(f-f^*)^2-2\zeta(f-f^*)}+\lambda\sup_{z^*\in\Gamma_{f^*}(\rho_K)}z^*(f-f^*)-\lambda\frac{\rho_K}{10}\enspace.
\end{align*}
Therefore, it will follow from Lemma~\ref{lem:Hom2} that
\[
\MOM{K}{(f-f^*)^2-2\zeta(f-f^*)}+\lambda(\|f\|-\|f^*\|)> 0
\]
if we can prove that for all $g\in F$ such that $\|g-f^*\|=\rho_K$ one has
\begin{equation}\label{eq:TODO}
\MOM{K}{(g-f^*)^2-2\zeta(g-f^*)}+\lambda\sup_{z^*\in\Gamma_{f^*}(\rho_K)}z^*(g-f^*)>\lambda\frac{\rho_K}{10}\enspace.
\end{equation}
Let us now prove that \eqref{eq:TODO} holds. Let $g\in F$ be such that $\|g-f^*\|=\rho_K$. First assume that $\norm{g-f^*}_{L^2_P}\leq r(\rho_K)$ so that $g\in H_{\rho_K}$. By definition $\sup_{z^*\in\Gamma_{f^*}(\rho_K)}z^*(g-f^*)\ge \Delta(\rho_K)$ and, since $\rho_K\ge \rho^*$, $\rho_K$ satisfies the sparsity equation and thus, $\sup_{z^*\in\Gamma_{f^*}(\rho_K)}z^*(g-f^*)\ge 4\rho_K/5$. Therefore, thanks to \eqref{Obj3}, when $\lambda> 20\eps r^2(\rho_K)/(7 \rho_K)$, one has
\begin{align*}
\MOM{K}{(g-f^*)^2 -2\zeta(g-f^*)}&+\lambda\sup_{z^*\in\Gamma_{f^*}(\rho_K)}z^*(g-f^*)\\
&\ge -2 \eps r^2(\rho_K)+\lambda\frac{4}{5}\rho_K> \lambda\frac{\rho_K}{10}\enspace.
\end{align*}
Finally assume that $\|g-f^*\|_{L^2_P}\ge r(\rho_K)$. Since $\sup_{z^*\in\Gamma_{f^*}(\rho_K)}z^*(f-f^*)\ge -\|f-f^*\|=-\rho_K$, it follows from \eqref{Obj2} that
\begin{align*}
&\MOM{K}{(g-f^*)^2-2\zeta(g-f^*)}+\lambda\sup_{z^*\in\Gamma_{f^*}(\rho_K)}z^*(g-f^*)\\
&\ge\frac1{32\param{0}^2}\norm{g-f^*}^2_{L^2_P}-\lambda \rho_K \geq\frac{r^2(\rho_K)}{32\param{0}^2}-\lambda \rho_K > \lambda\frac{\rho_K}{10}
\end{align*}when $\lambda< 10 r^2(\rho_K)/(331\theta_0^2 \rho_K)$.
\end{proof}
\paragraph{End of the proof of Theorem~\ref{theo:basic-combining-loss-and-reg}}
On the event $\Omega_0(K)$ of Proposition~\ref{prop:Useful}, $\cB_{K,\lambda}(f^*)$ is included in the ball $B(f^*,\rho_K)$, therefore, by definition of $\est^{(1)}$ (cf. \eqref{GenRiskBound}),
\[
\norm{\est^{(1)}-f^*}\leq C_{K,\lambda}^{(1)}(f^*)\le \rho_K\enspace.
\]
Again, by Proposition~\ref{prop:Useful}, on the same event $\Omega_0(K)$, $\cB_{K,\lambda}(f^*)\subset B(f^*, \rho_K)\cap B_2(f^*, r(\rho_K))$, hence, on $\Omega_0(K)\cap\Omega_{iso}(K)$, where $\Omega_{iso}(K)$ is an event defined in Lemma~\ref{lem:Isometry}, for all $f\in \cB_{K,\lambda}(f^*)$,
\[
\MOM{K}{|f-f^*|}\le 85\param{r}\norm{f-f^*}_{L^2_P}\le 85\param{r} r(\rho_K)
\]where $\alpha=1/21$ according to \eqref{eq:choice_constants}. Therefore, $C_{K,\lambda}^{(2)}(f^*)\le \rho_K$, which implies that $\norm{\est^{(2)}-f^*}\le \rho_K$ (cf. \eqref{GenRiskBound}) and that $C_{K,\lambda}^{(2)}(\est^{(2)})\le \rho_K$ and therefore, by Lemma~\ref{lem:Isometry}, on $\Omega_0(K)\cap\Omega_{iso}(K)$, either $\norm{\est^{(2)}-f^*}_{L^2_P}\leq r_Q(\rho_K, \gamma_K)$ and so $\norm{\est^{(2)}-f^*}_{L^2_P}\leq 340\param{0}\param{r}r(\rho_K)$ or $\norm{\est^{(2)}-f^*}_{L^2_P}\geq r_Q(\rho_K, \gamma_K)$ and so
\[
\norm{\est^{(2)}-f^*}_{L^2_P}\le 4\param{0}\MOM{K}{|\est^{(2)}-f^*|}\le 340\param{0}\param{r}r(\rho_K)\enspace.
\]
\subsection{Conclusion to the proof of Theorem~\ref{theo:main_baby_Lepski}}
First, it follows from Theorem~\ref{theo:basic-combining-loss-and-reg} that for all $K \in[K_1, K_2]$, with probability at least $1-\cabs{0}\exp(-\cabs{1}K)$, for both $j=1, 2$, $f^*\in \cap_{J=K}^{K_2} R^{(j)}_K$, so $\widehat K^{(j)}\le K$, which implies that both $f^*$ and $\ESTI{\text{LE}}{j}$ belong to $B(\ESTI{K,\lambda}{j},\rho_K)$, therefore, $\norm{f^*-\ESTI{\text{LE}}{j}}\le 2\rho_K$.
\noindent
Second, for the $L^2_P$-estimation error bound of $\ESTI{\text{LE}}{2}$, denote by $r_J=340\param{r}\param{0}r(\rho_J)$ the bound on the $L^2_P$ risk of the estimator $\hat f^{(2)}_J$ obtained in Theorem~\ref{theo:basic-combining-loss-and-reg}. Let $K\in [K_1,K_2]$. It follows from Lemma~\ref{lem:Isometry} for $\rho=2\rho_J, J\geq K$ that there exists absolute constants $\cabs{1},\cabs{2}$ and an event $\Omega_{iso}$ such that $\bP(\Omega_{iso})\geq 1-\cabs{1}\exp(-\cabs{2}K)$ and, on the event $\Omega_{iso}$, for all $J\ge K$, $\eta\in \{1/4,1/2,3/4\}$ and $f\in B(f^*,2\rho_J)$,
\[
\text{if }\norm{f-f^*}_{L^2_P}\ge r_Q(2\rho_J,\gamma_Q),\qquad Q_{\eta,J}(|f-f^*|)
\begin{cases}
\ge \frac{1}{4\param{0}}\norm{f-f^*}_{L^2_P}\\
\le 85\param{r}\norm{f-f^*}_{L^2_P}
\end{cases}
\enspace.
\]
Let $\Omega$ be the event defined as the following intersection:
\begin{equation*}
\Omega = \bigcap_{J=K}^{K_2}\left\{\norm{\ESTI{J}{2}-f^*}\le\rho_J \mbox{ and }\norm{\ESTI{J}{2}-f^*}_{L^2_P}\le r_J\right\} \bigcap \Omega(K) \bigcap \Omega_{iso}\enspace.
\end{equation*} It follows from Theorem~\ref{theo:basic-combining-loss-and-reg} that $\bP(\Omega)\ge 1-c_3 \exp(-c_4 K)$. Moreover, on $\Omega$, for all $J\ge K$,
\[
Q_{3/4,J}\pa{|f^*-\ESTI{J}{2}|}
\le 85\param{r}r_J\enspace.
\]
So, in particular, $f^*\in\cap_{J=K}^{K_2}\left\{f\in B(\ESTI{J}{2},\rho_J) : \MOM{J}{|f-\ESTI{J}{2}|}\le 85\param{r}r_J\right\}$. By definition of $\hat K^{(2)}$, this implies that $\hat K^{(2)}\le K$ on $\Omega$. Therefore, on $\Omega$,
\[
\hat f_{LE}^{(2)} \in \cap_{J=K}^{K_2}\left\{f\in B(f^*,2\rho_J) : \MOM{J}{|f-\ESTI{J}{2}|}\le 85\param{r}r_J\right\}\enspace.
\]
In particular,
\[
\MOM{K}{|\hat f_{LE}^{(2)}-\ESTI{K}{2}|}\le 85\param{r}r_K\enspace.
\]
Now on $\Omega_{iso}$, one has for all $f\in B(f^*,2\rho_K)$, if $\norm{f-f^*}_{L^2_P}\geq r_Q(2\rho_K,\gamma_Q)$ then
\[
Q_{1/4,J}[|f-f^*|]\ge \frac{1}{4\param{0}}\norm{f-f^*}_{L^2_P}\enspace.
\]
Therefore on $\Omega_{iso}$, one has either $\norm{\hat f_{LE}^{(2)}-f^*}_{L^2_P}\le r_Q(2\rho_K,\gamma_Q)$ or $\norm{\hat f_{LE}^{(2)}-f^*}_{L^2_P}\ge r_Q(2\rho_K,\gamma_Q)$ and in the latter case,
\begin{align*}
\norm{\hat f_{LE}^{(2)}-f^*}_{L^2_P}&\le4\param{0}Q_{1/4,K}[|\hat f_{LE}^{(2)}-f^*|]\\
&\le 4\param{0}\left(\MOM{K}{\big|\hat f_{LE}^{(2)}-\ESTI{K}{2}\big|}+Q_{3/4,K}(|\ESTI{K}{2}-f^*|)\right)\\
&\le 680\param{0} \param{r} r_K\enspace.
\end{align*}
{\mbox{}\nolinebreak\hfill\rule{2mm}{2mm}\par\medbreak}
\bibliographystyle{elsarticle-harv}
| {
"timestamp": "2017-07-19T02:07:20",
"yymm": "1701",
"arxiv_id": "1701.01961",
"language": "en",
"url": "https://arxiv.org/abs/1701.01961",
"abstract": "We obtain estimation error rates for estimators obtained by aggregation of regularized median-of-means tests, following a construction of Le Cam. The results hold with exponentially large probability -- as in the gaussian framework with independent noise- under only weak moments assumptions on data and without assuming independence between noise and design. Any norm may be used for regularization. When it has some sparsity inducing power we recover sparse rates of convergence.The procedure is robust since a large part of data may be corrupted, these outliers have nothing to do with the oracle we want to reconstruct. Our general risk bound is of order \\begin{equation*} \\max\\left(\\mbox{minimax rate in the i.i.d. setup}, \\frac{\\text{number of outliers}}{\\text{number of observations}}\\right) \\enspace. \\end{equation*}In particular, the number of outliers may be as large as (number of data) $\\times$(minimax rate) without affecting this rate. The other data do not have to be identically distributed but should only have equivalent $L^1$ and $L^2$ moments.For example, the minimax rate $s \\log(ed/s)/N$ of recovery of a $s$-sparse vector in $\\mathbb{R}^d$ is achieved with exponentially large probability by a median-of-means version of the LASSO when the noise has $q_0$ moments for some $q_0>2$, the entries of the design matrix should have $C_0\\log(ed)$ moments and the dataset can be corrupted up to $C_1 s \\log(ed/s)$ outliers.",
"subjects": "Statistics Theory (math.ST)",
"title": "Learning from MOM's principles: Le Cam's approach",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9770226300899744,
"lm_q2_score": 0.7248702880639792,
"lm_q1q2_score": 0.7082146753183464
} |
https://arxiv.org/abs/1108.5393 | New bounds on the maximum number of points on genus-4 curves over small finite fields | For prime powers q<100, we compute new upper and lower bounds on N_q(4), the maximal number of points on a genus-4 curve over a finite field with q elements. We determine the exact value of N_q(4) for 17 prime powers q for which the value was previously unknown. | \section{Introduction}
\label{S:intro}
For every prime power $q$ and integer $g\ge 0$, let $N_q(g)$ denote
the maximal number of points on a curve of genus~$g$ over~${\mathbf{F}}_q$. In
this paper we compute new upper and lower bounds on $N_q(4)$ for prime
powers $q$ less than $100$, and we find the exact value of $N_q(4)$ for
$17$ prime powers $q$ for which the value was previously unknown.
Weil's famous `Riemann Hypothesis' for curves over finite
fields~\cites{Weil1940,Weil1941,Weil1945,Weil1946} gives the
fundamental result that
\[ N_q(g) \le q + 1 + 2g\sqrt{q}, \]
which generalizes Hasse's theorem for the number of points on an
elliptic curve over a finite field~\cite{Hasse1936}. Weil's bound was
improved significantly in the case where $g$ is large with respect to
$q$ by Manin~\cite{Manin1981}, Ihara~\cite{Ihara1981}, and
Drinfel{\cprime}d\ and Vl\u adu\c t~\cite{DrinfeldVladut1983}, and for fixed $q$ we
have the Drinfel{\cprime}d-Vl\u adu\c t\ bound
\[
N_q(g)\leq (\sqrt{q}-1+o(1)) g \qquad\text{as $g \rightarrow \infty$.}
\]
When $q$ is a square this bound is asymptotically optimal, in the sense
that
\[ \limsup N_q(g)/g = \sqrt{q} - 1.\]
The value of this lim sup is not known for any nonsquare~$q$.
For any particular choice of $q$ and $g$, it can be a difficult
computational problem to determine the actual value of $N_q(g)$ and to
find a curve that attains this number of points. When $g$ is less than
$(q - \sqrt{q})/2$, the best upper bound known is often Serre's
improvement to the Weil bound~\cite{Serre1983a}:
\[ N_q(g) \le q + 1 + g \lfloor 2\sqrt{q}\rfloor. \]
But for every fixed genus $g>2$, nothing is known about the proportion
of prime powers for which this bound is attained.
For $g=1$ and $g=2$, the exact value of $N_q(g)$ can be calculated.
For $g=1$ this is due essentially to Deuring~\cite{Deuring1941}
(see~\cite{Waterhouse1969}*{Thm.~4.1, p.~536}), and for $g=2$ to
Serre~\cites{Serre1983a,Serre1983b,Serre1984} (see
also~\cite{HoweNartRitzenthaler2009}). But even for $g=3$ difficulties
arise, as is explained in~\cite{LRZ}.
The online tables available at \url{http://manypoints.org} collect the
best known upper and lower bounds on $N_q(g)$ for $g\le 50$ and for
various values of $q$: the primes less than $100$, the prime powers
$p^i$ for $p<20$ and $i\le 5$, and the powers of $2$ up to~$2^7$.
Despite the lack of an explicit formula for $N_q(3)$, for all of the
prime powers $q$ listed in these tables the value of $N_q(3)$ has been
calculated. This leaves $N_q(4)$ as the next challenge. In this paper,
we give new upper and lower bounds for $N_q(4)$ for $22$ of the $23$
prime powers $q<100$ for which the exact value had not been previously
computed; we find the exact value of $N_q(4)$ for $17$ of these prime
powers.
Our results are given in Table~\ref{T:results}. The entries in the
`old' column show the results that were listed on
\href{http://manypoints.org}{\texttt{manypoints.org}} on
1 August~2011. Note that the entries in the `new' column show that
there are now only six values of $q$ less than $100$ for which the
exact value of $N_q(4)$ is not known.
\begin{table}
\renewcommand{\arraystretch}{1.25}
\begin{center}
\begin{tabular}{|r|r|r|r|r|r|r|r|r|r|r|}
\cline{1-3}\cline{5-7}\cline{9-11}
$q$ & old & new &\qquad & $q$ & old & new & \qquad & $q$ & old & new \\
\cline{1-3}\cline{5-7}\cline{9-11}
$2$ & $8$ & $8$ && $23$ & --\,$58$ & $57$ && $59$ & --\,$118$ & $116$ \\
$3$ & $12$ & $12$ && $25$ & $66$ & $66$ && $61$ & --\,$122$ & $118$\,--\,$119$ \\
$4$ & $15$ & $15$ && $27$ & $64$ & $64$ && $64$ & $129$ & $129$ \\
$5$ & $18$ & $18$ && $29$ & --\,$70$ & $67$\,--\,$68$ && $67$ & --\,$132$ & $129$ \\
$7$ & $24$ & $24$ && $31$ & --\,$73$ & $72$ && $71$ & $132$\,--\,$136$ & $134$ \\
$8$ & $25$ & $25$ && $32$ & $71$\,--\,$72$ & $71$\,--\,$72$ && $73$ & --\,$139$ & $138$ \\
$9$ & $30$ & $30$ && $37$ & --\,$84$ & $82$ && $79$ & --\,$148$ & $148$ \\
$11$ & $33$\,--\,$34$ & $33$ && $41$ & --\,$90$ & $88$ && $81$ & $154$ & $154$ \\
$13$ & --\,$39$ & $38$ && $43$ & --\,$93$ & $92$ && $83$ & --\,$154$ & $152$ \\
$16$ & $45$ & $45$ && $47$ & --\,$100$ & $98$ && $89$ & --\,$162$ & $160$\,--\,$162$ \\
$17$ & --\,$48$ & $46$ && $49$ & --\,$106$ & $102$\,--\,$106$ && $97$ & --\,$174$ & $174$ \\
\cline{9-11}
$19$ & --\,$52$ & $48$\,--\,$50$ && $53$ & --\,$110$ & $108$ & \multicolumn{4}{}{} \\
\cline{1-3}\cline{5-7}
\end{tabular}
\end{center}
\vskip0.5em
\caption{Old and new ranges for $N_q(4)$, for $q<100$.}
\label{T:results}
\end{table}
We obtain our new upper bounds on $N_q(4)$ by using the results
of~\cite{HoweLauter2012}. For each~$q$, we use the computer programs
associated with that paper to obtain restrictions on genus-$4$ curves
over ${\mathbf{F}}_q$ with many points. Sometimes we learn that a curve with a
certain number of points must be a double cover of one of several
elliptic curves; in Section~\ref{S:covers} we show how to enumerate
such double covers. Sometimes we learn that a curve with a certain
number of points must correspond to a Hermitian lattice over a
quadratic order; we discuss these cases in Sections~\ref{S:maximal}
and~\ref{S:nonmaximal}. In two cases we find that the Jacobian of a
curve with a certain number of points must have complex multiplication
by ${\mathbf{Z}}[\zeta_5]$; our analysis of Hermitian forms over this ring in
Section~\ref{S:zeta} shows that such curves do not exist. Our new
lower bounds come from explicit examples of curves, which we present
in Section~\ref{S:lower}.
We have implemented all of our calculations in the computer algebra
package Magma~\cite{magma}. As we mentioned above, we discover
properties of genus-$4$ curves over ${\mathbf{F}}_q$ with a given number of
points by using the programs associated with the
paper~\cite{HoweLauter2012}. These programs are found in the package
\texttt{IsogenyClasses.magma}, which is available on the author's
website: Go to the bibliography page
\centerline{\href{http://alumni.caltech.edu/~however/biblio.html}
{\texttt{http://alumni.caltech.edu/{\lowtilde}however/biblio.html}}
}
\noindent
and follow the link associated with the paper~\cite{HoweLauter2012}.
The programs we use to enumerate double covers are also available
online, by starting at the URL given above and following the link
associated with the present paper.
\section{Restrictions on genus-$4$ curves with many points}
\label{S:results}
In this section we present the results we obtained from running the
programs in \texttt{IsogenyClasses.magma}.
Table~\ref{T:upper} lists the $(q,N)$ pairs that we will have to
eliminate in order to prove our new upper bounds; for each $q$ and~$N$,
the table shows what \texttt{IsogenyClasses.magma} tells us about
genus-$4$ curves over ${\mathbf{F}}_q$ with $N$ points. An entry of
``None exist'' means that \texttt{IsogenyClasses.magma} shows that no
genus-$4$ curve over ${\mathbf{F}}_q$ has exactly $N$ points. Entries of the
form ``Double cover of elliptic curve with trace $t$'' mean that any
genus-$4$ curve over ${\mathbf{F}}_q$ with $N$ points must be a double cover of
an elliptic curve over ${\mathbf{F}}_q$ with trace $t$. An entry of
``Hermitian module over $R$'', where $R$ is an order in an imaginary
quadratic field, means that any genus-$4$ curve over ${\mathbf{F}}_q$ with $N$
points must have a Jacobian that is isogenous to the fourth power of
an ordinary elliptic curve over ${\mathbf{F}}_q$ whose Frobenius endomorphism
generates the order $R$. An entry of
``Hermitian module over ${\mathbf{Z}}[\zeta_5]$'' means that any genus-$4$ curve
over ${\mathbf{F}}_q$ with $N$ points must have a Jacobian that is isogenous to
the square of an ordinary abelian surface with complex multiplication
by the ring of integers of the $5$th cyclotomic field.
\begin{table}
\renewcommand{\arraystretch}{1.25}
\begin{center}
\begin{tabular}{|r|r|l|}
\hline
$q$ & $N$ & Properties of a genus-$4$ curve over ${\mathbf{F}}_q$ with $N$ points\\
\hline
$11$ & $34$ & Hermitian module over ${\mathbf{Z}}[\zeta_5]$ \\
$13$ & $39$ & Double cover of elliptic curve with trace $-7$ \\
$17$ & $48$ & Double cover of elliptic curve with trace $-8$ \\
$17$ & $47$ & None exist \\
$19$ & $52$ & Hermitian module over ${\mathbf{Z}}[\sqrt{-3}]$ \\
$19$ & $51$ & None exist \\
$23$ & $58$ & Double cover of elliptic curve with trace $-9$ \\
$29$ & $70$ & Hermitian module over ${\mathbf{Z}}[2i]$ \\
$29$ & $69$ & None exist \\
$31$ & $73$ & Double cover of elliptic curve with trace $-11$ \\
$37$ & $84$ & Double cover of elliptic curve with trace $-12$ \\
$37$ & $83$ & None exist \\
$41$ & $90$ & Hermitian module over ${\mathbf{Z}}[\sqrt{-5}]$ \\
$41$ & $89$ & None exist \\
$43$ & $93$ & Double cover of elliptic curve with trace $-13$ \\
$47$ & $100$ & Hermitian module over ${\mathbf{Z}}[\alpha]$ \\
$47$ & $99$ & None exist \\
$53$ & $110$ & Hermitian module over ${\mathbf{Z}}[2i]$ \\
$53$ & $109$ & None exist \\
$59$ & $118$ & Double cover of elliptic curve with trace $-15$ \\
$59$ & $117$ & Double cover of elliptic curve with trace $-15$ \\
$61$ & $122$ & Hermitian module over ${\mathbf{Z}}[\alpha]$ \\
$61$ & $121$ & None exist \\
$61$ & $120$ & Double cover of elliptic curve with trace $-13$, or \\
& & Hermitian module over ${\mathbf{Z}}[\zeta_5]$ \\
$67$ & $132$ & Hermitian module over ${\mathbf{Z}}[\sqrt{-3}]$ \\
$67$ & $131$ & None exist \\
$67$ & $130$ & Double cover of elliptic curve with trace $-14$ \\
$71$ & $136$ & Hermitian module over ${\mathbf{Z}}[\sqrt{-7}]$ \\
$71$ & $135$ & None exist \\
$73$ & $139$ & Double cover of elliptic curve with trace $-17$ \\
$83$ & $154$ & Double cover of elliptic curve with trace $-18$ \\
$83$ & $153$ & Double cover of elliptic curve with trace $-18$ \\
\hline
\end{tabular}
\end{center}
\vskip0.5em
\caption{What \texttt{IsogenyClasses.magma} tells us about genus-$4$
curves over ${\mathbf{F}}_q$ with $N$ points. Here $\zeta_5$ is a
primitive $5$th root of unity, $i = \sqrt{-1}$, and $\alpha$
satisfies $\alpha^2 + \alpha + 5 = 0$.}
\label{T:upper}
\end{table}
\section{Double covers of elliptic curves}
\label{S:covers}
Table~\ref{T:upper} shows that there are a number of pairs $(q,N)$ such
that a genus-$4$ curve over ${\mathbf{F}}_q$ with $N$ points must be a double
cover of an elliptic curve with a certain trace~$t$. Thus, one way to
show that there are no genus-$4$ curves over ${\mathbf{F}}_q$ with $N$ points
would be to enumerate all of the elliptic curve over ${\mathbf{F}}_q$ of
trace~$t$, enumerate all of the genus-$4$ double covers of these
curves, count the number of points on the double covers, and verify
that none of them has~$N$ points. This is the strategy we use for the
appropriate entries in Table~\ref{T:upper}.
In this section we explain more details of this procedure. The actual
Magma programs we use can be found at the URL mentioned in the
introduction; follow the link associated with this paper, and then
download the file \texttt{Genus4.magma}.
Our first comment is that if $C$ is a double cover of an elliptic curve
$E$ over a finite field $k$, and if $\sigma$ is an automorphism of~$k$,
then $C^\sigma$ is a double cover of~$E^\sigma$. Since $C^\sigma$ and
$C$ have the same number of points, it will suffice for us to enumerate
all of the double covers of a set of representatives of the trace-$t$
elliptic curves up to automorphisms of~$k$.
The finite fields we must investigate are small enough that we use a
completely naive method of finding representatives for these elliptic
curves. We will only be working with fields of characteristic larger
than $3$, so every elliptic curve can be written in the form
$y^2 = x^3 + ax + b$, where $a$ and $b$ are elements of $k$ with
$4a^3 + 27b^2\ne 0$. The curve determined by one such pair $(a,b)$ is
isomorphic to a Galois conjugate of the curve determined by another
pair $(a',b')$ if and only if $(a',b') = (a^\sigma u^4, b^\sigma u^6)$
for some $u\in k^*$ and some automorphism $\sigma$ of~$k$. The
function \texttt{ECs} takes as input a finite field $k$ and a
trace~$t$, explicitly computes the orbits of the set of $(a,b)$ pairs
under the combined action of $k^*$ and $\Aut k$, and computes the trace
of one representative elliptic curve from each orbit. It returns those
representatives that have the desired trace~$t$.
Next, given an $E$ of trace~$t$, we must enumerate its genus-$4$ double
covers~$C$. We use the same general idea as
in~\cite{HoweLauter2003}*{\S6.1}; our description of the method is
adapted from the version given there.
The function field of such a $C$ is obtained from that of $E$ by
adjoining a root of $z^2 = f$, where $f$ is a function on $E$. By the
Riemann-Hurwitz formula, in order for $C$ to have genus $4$ the divisor
of $f$ must be of the form
\[ P_1 + \cdots + P_6 + 2D, \]
where the $P_i$ are distinct geometric points on $E$ such that the
divisor $P_1 + \cdots + P_6$ is $k$-rational, and where $D$ is a
divisor of degree~$-3$. There is a function $g$ on $E$ such that
\[ D + \divisor g = Q - 4\infty, \]
where $\infty$ is the infinite point on~$E$ and where $Q$ is a rational
point on~$E$. Replacing $f$ with $fg^2$ gives an isomorphic double
cover of $E$. Thus, we may assume that $C$ is given by adjoining a root
of $z^2 = f$, where $f$ is a function on $E$ whose divisor is of the
form
\begin{equation}
\label{EQ:gooddiv}
P_1 + \cdots + P_6 + 2Q - 8\infty,
\text{\quad where the $P_i$ are distinct.
}
\end{equation}
We can also change the map $C\to E$ by following it with a translation
map on~$E$. Translating $E$ by a rational point $R$ has the effect of
replacing $f$ with a function whose divisor is
\[ (P_1 + R) + \cdots + (P_6 + R) + 2(Q + R) - 8R \]
(where the sums in parentheses take place in the algebraic group~$E$).
By modifying this new $f$ by the square of a function we can get the
divisor of $f$ to be
\[ (P_1 + R) + \cdots + (P_6 + R) + 2(Q - 3R) - 8\infty. \]
We see that we only need to consider $Q$ that represent distinct
classes of $E(k)$ modulo $3E(k)$. Note that every class other than the
class of the identity element contains a representative that is not a
$2$-torsion point, so we may assume that $Q$ does not have order~$2$.
And finally, we note that we need only look at one representative from
each orbit of $\Aut E$ acting on~$E(k)/3E(k)$.
Next, given an $E$ and a $Q\in E(k)$ not of order~$2$, we must
enumerate all functions on $E$ with divisors of the
form~\eqref{EQ:gooddiv}. Suppose first that $Q\ne\infty$. By
translating co\"ordinates on $E$, we may assume that $E$ is given by
an equation $y^2 = x^3 + r x^2 + s x + t^2$ and that $Q$ is the point
$(0,t)$. Since $Q$ is not a $2$-torsion point we have $t\neq 0$, and
the tangent line to $E$ at $Q$ is given by $y = mx + t$, where
$m = s/(2t)$. Then there are two cases to consider: functions for
which none of the $P_i$ is equal to $\infty$, and functions for which
one of the $P_i$ is equal to~$\infty$.
If $f$ is a function on $E$ whose divisor has the desired form and for
which no $P_i$ is $\infty$, then $f$ has degree~$8$ and lies in the
Riemann-Roch space ${\mathscr L}(8\infty-2Q)$. We check that this space is
spanned by the functions
\[ \{ x^4,\ x^2(y-t), \ x^3, \ x(y-t), \ x^2, \ y - mx - t \}. \]
We can then run through all of the $f$'s in the $k$-span of these
functions, considering only those linear combinations where the
coefficient of $x^4$ is nonzero (so that the function actually has
degree~$8$). In fact, since $z^2 = f$ and $z^2 = d^2 f$ give
isomorphic extensions of $E$, we can restrict our attention to linear
combinations where the coefficient of $x^4$ is either $1$ or a fixed
nonsquare value.
For a given linear combination $f$, we can easily compute the number of
points on the extension $z^2 = f$ under the assumption that the divisor
of $f$ is of the form~\eqref{EQ:gooddiv}. (We get either $2$ or $0$
points over $\infty$ depending on the leading term of the Laurent
expansion of $f$ at $\infty$, and likewise for the points over $Q$;
for the other points $P$ of $E(k)$, we get either $0$, $1$, or $2$
points over $P$ depending on whether $f(P)$ is a nonsquare, zero, or a
nonzero square.) If the number we calculate is larger than our previous
best count, we can then check whether the divisor of $f$ is in fact of
the form~\eqref{EQ:gooddiv}.
The case where one of the $P_i$ is equal to $\infty$ is very similar;
the difference is that we now look in the Riemann-Roch space
${\mathscr L}(7\infty-2Q)$, and that now the extension $z^2=f$ always has
exactly one point lying over $\infty$.
The function \texttt{double\us{}covers\us{}genus\us{}4} in the file
\texttt{Genus4.magma} takes as input an elliptic curve $E$ over a
finite field, and runs the algorithm sketched above to find the largest
number of points on a genus-$4$ double cover of~$E$. The function
\texttt{double\us{}covers\us{}given\us{}trace} takes as input a prime
power $q$ and a trace $t$, and finds the maximal number of points on a
genus-$4$ curve over ${\mathbf{F}}_q$ that is a double cover of some elliptic
curve over ${\mathbf{F}}_q$ of trace~$t$.
For each pair $(q,N)$ whose associated entry in Table~\ref{T:upper}
says that a genus-$4$ curve over ${\mathbf{F}}_q$ with $N$ points must be a
double cover of an elliptic curve of trace $t$, we ran our program with
input $(q,t)$. For each pair, we found that the maximal number of
points on a genus-$4$ double cover of an elliptic curve with trace $t$
is less than $N$. Thus, for these pairs $(q,N)$, we find that
$N_q(4) < N$.
\section{Hermitian forms over maximal quadratic orders}
\label{S:maximal}
The programs in \texttt{IsogenyClasses.magma} show that for several of
the $(q,N)$ pairs listed in Table~\ref{T:upper}, every genus-$4$ curve
over ${\mathbf{F}}_q$ having $N$ points must have Jacobian isogenous to~$E^4$,
where $E$ is an ordinary elliptic curve over ${\mathbf{F}}_q$ whose Frobenius
endomorphism generates a specific imaginary quadratic order~$R$. The
study of abelian varieties isogenous to $E^4$ is simplified by the use
of Serre's `Hermitian modules'~\cite{LauterSerre2002}*{Appendix} or
Deligne's equivalence of categories~\cite{Deligne1969} between ordinary
abelian varieties and modules over a certain ring, combined with the
description of polarizations on these modules provided
in~\cite{Howe1995}.
In this section we analyze the three cases from Table~\ref{T:upper}
where the quadratic order in question is maximal; these are the cases
$(q,N) = (41,90)$, $(q,N) = (47,100)$, and $(q,N) = (61,122)$, where
the corresponding quadratic order ${\mathscr O}$ has discriminant $-20$,
$-19$, and $-19$, respectively. As is explained
in ~\cite{LauterSerre2002}*{Appendix}, a principally-polarized abelian
variety isogenous to $E^4$ corresponds to a principally-polarized
Hermitian ${\mathscr O}$-module of rank~$4$. Schiemann~\cite{Schiemann1998}
has computed all such principally-polarized Hermitian modules (up to
isomorphism) for the quadratic orders we are concerned with; the lists
can be found at
\centerline{\url{http://www.math.uni-sb.de/ag/schulze/Hermitian-lattices/}}
\noindent
as well as on the author's web site, mentioned in the introduction.
Schiemann calculated the automorphism groups of these polarized
Hermitian modules; these groups are isomorphic to the automorphism
groups of the corresponding polarized abelian four-folds. Schiemann's
tables show that for all of the Hermitian modules we must consider, the
automorphism groups have order divisible by~$4$. By Torelli's
theorem~\cite{Milne1986}*{Thm.~12.1, p.~202}, if the polarized Jacobian
of a curve $C$ has an automorphism group of order divisible by~$4$,
then either
\begin{itemize}
\item $C$ has an automorphism $\iota$ of order~$2$ such that the
quotient curve $C_0=C/\langle\iota\rangle$ has positive genus, or
\item $C$ is a hyperelliptic curve with an automorphism of order~$4$
whose square is the hyperelliptic involution.
\end{itemize}
For the $(q,N)$ pairs we have to consider, we find that if $C$ is a
genus-$4$ curve over ${\mathbf{F}}_q$ with $N$ points, then either $C$ is a
double cover of an elliptic curve isogenous to~$E$, or $C$ is a double
cover of a genus-$2$ curve whose Jacobian is isogenous to~$E^2$, or $C$
is a hyperelliptic curve with an automorphism whose square is the
hyperelliptic involution.
We have already seen how to enumerate the genus-$4$ double covers of an
elliptic curve; running \texttt{double\us{}covers\us{}given\us{}trace}
for the $(q,t)$ pairs coming from our $(q,N)$ pairs, we find no
genus-$4$ curves over ${\mathbf{F}}_q$ with $N$ points.
In Section~\ref{SS:genus2} below we will show how to enumerate the
genus-$4$ double covers of genus-$2$ curves, and in
Section~\ref{SS:hyperelliptic} we will show how to enumerate the genus-$4$
hyperelliptic curves with an automorphism whose square is the hyperelliptic involution.
For our $(q,N)$ pairs, we find no genus-$4$ curves over ${\mathbf{F}}_q$ having $N$ points.
\subsection{Double covers of genus-$2$ curves}
\label{SS:genus2}
Suppose $\varphi:C\to C_0$ is a degree-$2$ map from a genus-$4$ curve
to a genus-$2$ curve over a finite field $k$ of characteristic greater
than~$2$. The function field of $C$ is obtained from that of $C_0$ by
adjoining a root of $z^2 = f$, where $f$ is a function on $C_0$. By the
Riemann-Hurwitz formula, in order for $C$ to have genus $4$ the divisor
of $f$ must be of the form
\[ P_1 + P_2 + 2D, \]
where the $P_i$ are distinct geometric points on $E$ such that the
divisor $P_1 + P_2$ is $k$-rational, and where $D$ is a divisor of
degree~$-1$. Let $\infty$ be any rational point on~$C_0$. The
Riemann-Roch theorem shows that the Riemann-Roch space
${\mathscr L}(D + 4\infty)$ has dimension $2$; it follows that it must contain
a function $g$ such that $\divisor g + D + 3\infty$ is effective. Then
\[\divisor fg^2 = P_1 + P_2 + 2D' - 6\infty\]
for some effective divisor $D'$ of degree~$2$. Replacing $f$
with~$fg^2$, we find that $C$ can be obtained from $C_0$ by adjoining a
root of $z^2 = f$, with $\divisor f = P_1 + P_2 + 2D' - 6\infty$ for
some effective $D'$ of degree~$2$.
Thus, to enumerate all genus-$4$ curves that are double covers
of~$C_0$, we can simply enumerate all effective degree-$2$
divisors~$D'$, compute the Riemann-Roch space ${\mathscr L}(6\infty - 2D')$,
loop through the function $f$ in this space (up to squares in~$k$) that
are not squares in $\overline{k}(C_0)$, and consider the curves $z^2 = f$. It
is easy to count the number of points on a curve defined by such an
extension. We implement this algorithm in the function
\texttt{double\us{}covers\us{}genus\us{}4}, the same function we used
for doubles covers of elliptic curves; when the input to the function
is a curve of genus~$2$, the function runs through the procedure
sketched above, and outputs the largest number of points it finds.
For the case $(q,N) = (41,90)$ there are three possibilities for the
base curve~$C_0$; this can be determined by brute force (by enumerating
all genus-$2$ curves over~$k$) or by theory, by noting that Schiemann's
tables show exactly three unimodular rank-$2$ Hermitian ${\mathscr O}$-modules
that are not products of two rank-$1$ modules. The three curves are
\begin{align*}
y^2 &= x^6 + 7 x^4 + 8 x^2 - 7,\\
y^2 &= x^6 + 7 x^4 + 3 x^2 + 7, \quad\textup{and}\\
y^2 &= x^6 - 3 x^4 - 3 x^2 + 1.
\end{align*}
Running \texttt{double\us{}covers\us{}genus\us{}4} on these curves, we
find no genus-$4$ curves over ${\mathbf{F}}_{41}$ with $90$ points.
Similarly, we find one possible $C_0$ for each of the other two $(q,N)$
pairs we must consider.
For~${\mathbf{F}}_{47}$ and ${\mathbf{F}}_{61}$, these curves are
\[y^2 = x^6 + 7 x^4 - 9 x^2 - 6 \quad\textup{and}\quad
y^2 = x^6 + 10 x^4 - 11 x^2 - 1,\]
respectively.
Running \texttt{double\us{}covers\us{}genus\us{}4} on these curves, we
find no genus-$4$ curves over ${\mathbf{F}}_{47}$ with $100$ points, and no
genus-$4$ curves over ${\mathbf{F}}_{61}$ with $122$ points.
\subsection{Hyperelliptic curves with automorphisms of order $4$}
\label{SS:hyperelliptic}
A hyperelliptic curve over ${\mathbf{F}}_q$ can have at most $2(q+1)$ rational
points, so the only one of our three $(q,N)$ pairs that we need be
concerned with is $(q,N) = (61,122)$.
Suppose $C$ is a genus-$4$ hyperelliptic curve over a finite field $k$
with an automorphism $\alpha$ of order~$4$ whose square is the
hyperelliptic involution. (In fact,~\cite{GuralnickHowe2009}*{Thm.~1.1}
shows that \emph{every} order-$4$ automorphism of a genus-$4$
hyperelliptic curve has square equal to the hyperelliptic involution.)
We can choose a parameter for ${\mathbf{P}}^1$ so that the automorphism of
${\mathbf{P}}^1$ induced by $\alpha$ is $x \mapsto a/x$ for some~$a\in k^*$.
Writing $C$ as $y^2 = f$ for some separable polynomial $f\in k[x]$ of
degree $9$ or~$10$, we see that the automorphism $\alpha$ must have the
form $(x,y) \mapsto (a/x, b y / x^5)$, and since $\alpha^2$ is the
hyperelliptic involution we must have $b^2 = -a^5$. Replacing $x$ with
$a^2 x / b$, we compute that $\alpha$ is given by
$(x,y) \mapsto (-1/x, y / x^5)$.
We find that $f$ must be a linear combination of the following
polynomials:
\[ \{ x^{10} + 1 , \
x^9 - x , \
x^8 + x^2, \
x^7 - x^3, \
x^6 + x^4 \}.\]
For the one pair $(q,N)$ we must consider, we can examine all linear
combinations of these polynomials (up to squares in~${\mathbf{F}}_q$), and
compute the number of points on the associated hyperelliptic curves.
We find no curve over ${\mathbf{F}}_{61}$ having $122$ points.
The function \texttt{hyperelliptic} in the file \texttt{Genus4.magma}
implements this algorithm.
\section{Hermitian forms over nonmaximal quadratic orders}
\label{S:nonmaximal}
For five of the entries $(q,N)$ in Table~\ref{T:upper}, we see that the
Jacobian of a genus-$4$ curve over ${\mathbf{F}}_q$ with $N$ points must be
isogenous to~$E^4$, where $E$ lies in an isogeny class of ordinary
elliptic curves over ${\mathbf{F}}_q$ all of whose elements have complex
multiplication by a non-maximal order $R$ in a quadratic field~$K$;
here $R$ is the ring generated over ${\mathbf{Z}}$ by the Frobenius
endomorphism of~$E$. The five entries are $(19, 52)$, $(29, 70)$,
$(53, 110)$, $(67, 132)$, and $(71, 136)$, and the rings $R$ are the
orders of conductor~$2$ inside the maximal orders of, respectively,
${\mathbf{Q}}(\sqrt{-3})$, ${\mathbf{Q}}(i)$, ${\mathbf{Q}}(i)$, ${\mathbf{Q}}(\sqrt{-3})$, and
${\mathbf{Q}}(\sqrt{-7})$. The discriminants of the rings $R$ are,
respectively, $-12, -16, -16, -12,$ and~$-28$. We will show that for
each of these pairs $(q,N)$, a genus-$4$ curve over ${\mathbf{F}}_q$ having $N$
points must be a double cover of a curve of genus $1$ or~$2$.
\begin{proposition}
\label{P:nonmaximal}
Let $t$ be the trace of Frobenius for an isogeny class ${\mathscr C}$ of
ordinary elliptic curves over a finite field~${\mathbf{F}}_q$ such that
$\Delta := t^2 - 4q$ lies in $\{-12,-16,-28\}.$ Suppose $C$ is a
genus-$4$ curve over ${\mathbf{F}}_q$ whose Jacobian is isogenous to the fourth
power of an elliptic curve in ${\mathscr C}$.
\begin{enumerate}
\item If $\Delta=-12$ or $\Delta=-16$ then $C$ is a double cover of a
curve in~${\mathscr C}$.
\item If $\Delta=-28$ then either $C$ is a double cover of a curve
in~${\mathscr C}$, or $C$ is a double cover of a genus-$2$ curve whose
Jacobian is isogenous to the square of a curve in~${\mathscr C}$.
\end{enumerate}
\end{proposition}
For our five $(q,N)$ pairs, the function
\texttt{double\us{}covers\us{}given\us{}trace} finds no genus-$4$ curve
with $N$ points. For the pair $(q,N) = (71, 136)$ we must also run
\texttt{double\us{}covers\us{}genus\us{}4} on three genus-$2$ curves:
\begin{align*}
y^2 &= x^6 - 29 x^4 - 29 x^2 + 1, \\
y^2 &= x^6 - 13 x^4 - 13 x^2 + 1, \quad\textup{and}\\
y^2 &= x^6 - 12 x^4 - 15 x^2 - 1.
\end{align*}
Again, we find no genus-$4$ curve with $N$ points.
The proof of Proposition~\ref{P:nonmaximal} relies on several lemmas.
To begin, we show that for each of our five cases, there are only four
abelian varieties to consider.
\begin{lemma}
\label{L:products}
Let $t$ be the trace of Frobenius for an isogeny class ${\mathscr C}$ of
ordinary elliptic curves over a finite field~${\mathbf{F}}_q$ such that
$\Delta := t^2 - 4q$ lies in $\{-12,-16,-28\}.$ Then ${\mathscr C}$ contains
exactly two elliptic curves, one of them with endomorphism ring equal
to the maximal order ${\mathscr O}$ in $K:={\mathbf{Q}}(\sqrt{\Delta})$, and one with
endomorphism ring equal to the order $R$ of conductor $2$ in ${\mathscr O}$.
Let $E$ and $F$ be these two elliptic curves, where $E$ has the larger
endomorphism ring. Then every abelian variety over ${\mathbf{F}}_q$ isogenous
to $E^n$ is isomorphic to $E^i \times F^{n-i}$ for some $i$, and the
varieties arising from distinct values of $i$ are not isomorphic to one
another.
\end{lemma}
\begin{proof}
By Deligne's equivalence of categories~\cite{Deligne1969}, the elliptic
curves in ${\mathscr C}$ correspond to the isomorphism classes of nonzero
finitely-generated $R$-modules in~$K$. The endomorphism ring of such a
module is either $R$ or ${\mathscr O}$, and since both $R$ and ${\mathscr O}$ have
class number $1$, there is one elliptic curve $E$ with
$\End E \cong {\mathscr O}$ and one elliptic curve $F$ with $\End F \cong R$.
Likewise, Deligne's theorem shows that the varieties isogenous to $E^n$
correspond to rank-$n$ modules over $R$. By using a result of Borevich
and Faddeev~\cite{BorevichFaddeev1965}, we see that every such module
is isomorphic to a sum of copies of ${\mathscr O}$ plus a sum of copies
of~$R$, and that two such modules are isomorphic if and only if they
have the same number of each type of summand. The second statement of
the lemma follows upon applying Deligne's equivalence of categories.
\end{proof}
Next, we show that for each of our cases, any curve with Jacobian
isomorphic to $E^i \times F^{4-i}$ with $i>0$ must be a double cover
of~$E$.
\begin{lemma}
\label{L:pullback}
Let $t$ be the trace of Frobenius for an isogeny class of ordinary
elliptic curves over a finite field~${\mathbf{F}}_q$ such that
$\Delta := t^2 - 4q$ lies in $\{-12,-16,-28\},$ and let $E$ and $F$ be
as in Lemma~\textup{\ref{L:products}}. If $C$ is a genus-$4$ curve
over ${\mathbf{F}}_q$ whose Jacobian is isomorphic to $E^i\times F^{4-i}$ for
some $i>0$, then $C$ is a double cover of~$E$.
\end{lemma}
\begin{proof}
Our proof is computational, and uses the function
\texttt{pullback\us{}bound} from the package
\texttt{IsogenyClasses.magma}. Given a fundamental discriminant $D$
with $-11\le D < 0$, an integer $n$, and a dimension~$d$, this function
will return an integer $B$ such that for every $d\times d$ Hermitian
matrix of determinant $n$ with entries in the quadratic order ${\mathscr O}_D$
of discriminant $D$, the Hermitian form on ${\mathscr O}_D^d$ determined by
this matrix will have a nonzero vector of length at most~$B$. As is
explained in~\cite{HoweLauter2012}*{\S4}, this translates into a
statement about polarizations on $E^d$, where $E$ has complex
multiplication by ${\mathscr O}_D$; namely, for every degree-$n^2$
polarization $\lambda$ on $E^d$, there is an embedding
$\varphi\,{:}\, E\to E^d$ such that $\varphi^*\lambda$ is a polarization
on $E$ of degree at most~$B^2$.
Now suppose $C$ is a curve with Jacobian isomorphic to
$E^i\times F^{4-i}$, with $i>0$, and let $\lambda$ be the canonical
principal polarization on $\Jac C$. There is an isogeny of degree
$2^{4-i}$ from $E^4$ to $\Jac C$, and pulling back $\lambda$ via this
isogeny gives a polarization $\mu$ of degree $4^{4-i}$ on $E^4$. The
function \texttt{pullback\us{}bound} tells us that we can pull back
$\mu$ to a polarization of degree $1$ or $4$ on $E$, and
by~\cite{HoweLauter2012}*{Lem.~4.3} this gives us a map of degree $1$
or $2$ from $C$ to $E$. A map of degree~$1$ is clearly impossible, so
we find that $C$ is a double cover of~$E$.
\end{proof}
We see that to prove Proposition~\ref{P:nonmaximal} we need only
consider curves whose Jacobians are isomorphic to~$F^4$; in other
words, we need only consider principal polarizations on~$F^4$. For
$\Delta = -12$ and $\Delta = -16$ we will show that every such
polarization can be pulled back to either $E$ or $F$ to get a
polarization of degree $1$ or $4$. For $\Delta = -28$ we will show
that if such a polarization cannot be pulled back to a polarization of
degree $1$ or $4$ on $E$ or~$F$, and if the polarized variety is the
Jacobian of a curve, then the curve has an involution that makes it a
double cover of a genus-$2$ curve.
Let $\varphi$ be a degree-$2$ isogeny from $E$ to $F$, and let
$\Phi\,{:}\, E^4 \to F^4$ be the degree-$16$ product isogeny
$\varphi\times\varphi\times\varphi\times\varphi$. Denote the kernel of
$\Phi$ by $G$. Note that the smallest ${\mathscr O}$-stable subgroup of $E^4$
that contains $G$ is $E^4[2]$.
Suppose $\lambda$ is a principal polarization on $F^4$, and let
$\mu = \Phi^*\lambda$ be the pullback of $\lambda$ to $E^4$. Then $\mu$
is a polarization of degree $16^2$ on $E^4$, and since $\ker \mu$ is
stable under the action of ${\mathscr O}$ and contains $G$, we must have
$\ker\mu = E^4[2]$. That means that $\mu$ must be twice a principal
polarization.
Thus, to enumerate the possible principal polarizations $\lambda$
on~$F^4$, we can enumerate the principal polarizations on $E^4$ up to
isomorphism, multiply each of these by $2$ to get a polarization $\mu$
of degree $16^2$ on $E^4$, enumerate the subgroups $G$ of $E^4[2]$ that
generate all of $E^4[2]$ as an ${\mathscr O}$-module and that are isotropic
with respect to the Weil pairing on $E^4[2]$ determined by $\mu$, and
then compute the matrix in $M_4(R)$ that represents the polarization we
get by pushing $\mu$ down through an isogeny with kernel $G$. The
Magma programs we use to do this can be found at the URL mentioned in
the introduction; follow the link associated with this paper, and then
download the file \texttt{NonMaximalOrders.magma}.
\subsection{Proof of Proposition~\ref{P:nonmaximal} when $\Delta=-12$}
\label{SS:Delta12}
Schiemann's tables~\cite{Schiemann1998} show that up to isomorphism the
only principal polarization on $E^4$ is the product polarization, so
our only $\mu$ is represented by the diagonal matrix with $2$'s on the
diagonal. Embedding $E$ into $E^4$ along one of the factors, we find
that~$\mu$, and hence~$\lambda$, can be pulled back to a polarization
of degree $4$ on~$E$.
\subsection{Proof of Proposition~\ref{P:nonmaximal} when $\Delta=-16$}
\label{SS:Delta16}
Schiemann's tables show two principal polarizations on $E^4$, the
product polarization and the polarization given by the Hermitian matrix
\[
P = \left[ \
\begin{matrix}
2 & 1 & 0 & 0 \\
1 & 2 & 1-i & 1-i \\
0 & 1+i & 2 & 1 \\
0 & 1+i & 1 & 2
\end{matrix}
\ \right].
\]
Twice the product polarization can be pulled back to give a
polarization of degree $4$ on $E$, so we need only consider the
polarization~$2P$.
A computer calculation shows that there are $1024$ subgroups of
$E^4[2]$ that are maximal isotropic with respect to the Weil pairing
determined by $P$ and that generate all of $E^4[2]$ as an
${\mathscr O}$-module. For each such subgroup $G$, we compute the
polarization on $F^4$ obtained by pushing the polarization $2P$ down
through the isogeny with kernel $G$; this polarization can be
represented by a Hermitian matrix in $M_4(R)$, which gives a Hermitian
form on $R^4$. We can then compute the short vectors in this Hermitian
lattice. We find that for each group $G$, the Hermitian lattice has a
short vector of length $2$. This means that the polarization on $F^4$
can be pulled back to a polarization of degree $4$ on $F$.
\subsection{Proof of Proposition~\ref{P:nonmaximal} when $\Delta=-28$}
\label{SS:Delta28}
Schiemann's tables show that there are three principal polarizations
on $E^4$: the product polarization and the polarizations given by the
Hermitian matrices
\begin{align*}
P_1 &= \left[ \
\begin{matrix}
1 & 0 & 0 & 0 \\
0 & 2 & \frac{1 + \sqrt{-7}}{2} & -1\phantom{-} \\
0 & \frac{1 - \sqrt{-7}}{2} & 2 & \frac{1 + \sqrt{-7}}{2} \\
0 & -1\phantom{-} & \frac{1 - \sqrt{-7}}{2} & 2
\end{matrix}
\ \right]
\\
\intertext{and}
P_2 &= \left[ \
\begin{matrix}
2 & 1 & 0 & \frac{1 + \sqrt{-7}}{2} \\
1 & 2 & \frac{1 + \sqrt{-7}}{2} & \frac{1 + \sqrt{-7}}{2} \\
0 & \frac{1 - \sqrt{-7}}{2} & 2 & 1 \\
\frac{1 - \sqrt{-7}}{2} & \frac{1 - \sqrt{-7}}{2} & 1 & 2 \\
\end{matrix}
\ \right].
\end{align*}
There is a $2$ on the diagonal of $2P_1$, so this polarization can be
pulled back to a degree-$4$ polarization on $E$. Likewise, twice the
product polarization can be pulled back to a degree-$4$ polarization
on~$E$. That leaves us to consider the polarization~$2P_2$.
A computer calculation shows that there are $448$ subgroups of $E^4[2]$
that are maximal isotropic with respect to the Weil pairing determined
by $P_2$ and that generate all of $E^4[2]$ as an ${\mathscr O}$-module. Of
these subgroups, $256$ give rise to principal polarizations on $F^4$
whose associated Hermitian forms have short vectors of length~$2$. The
other $192$ subgroups give principal polarizations on $F^4$ that are
isomorphic to the polarizations defined by one of the following three
Hermitian matrices:
\begin{align*}
Q_1 &=\left[ \
\begin{matrix}
4 & 2 & 2 & -\sqrt{-7}\phantom{-} \\
2 & 4 & -\sqrt{-7}\phantom{-} & 2 \\
2 & \phantom{-}\sqrt{-7}\phantom{-} & 4 & -2\phantom{-} \\
\phantom{-}\sqrt{-7}\phantom{-} & 2 & -2\phantom{-} & 4 \\
\end{matrix}\ \right]\displaybreak[0]\\
Q_2 &= \left[ \
\begin{matrix}
3 & 1 & 1 & -1 - \sqrt{-7} \\
1 & 3 & -1 - \sqrt{-7} & 1 \\
1 & -1 + \sqrt{-7} & 4 & -2\phantom{-} \\
-1 + \sqrt{-7} & 1 & -2\phantom{-} & 4 \\
\end{matrix}\ \right]\displaybreak[0]\\
Q_3 &= \left[ \
\begin{matrix}
3 & 1 & 0 & -\sqrt{-7}\phantom{-} \\
1 & 3 & -\sqrt{-7}\phantom{-} & 0 \\
0 & \phantom{-}\sqrt{-7}\phantom{-} & 3 & -1\phantom{-} \\
\phantom{-}\sqrt{-7}\phantom{-} & 0 & -1\phantom{-} & 3 \\
\end{matrix}\ \right] .
\end{align*}
We check that for each of these three polarizations $Q$ there is an
involution of the polarized variety $(F^4,Q)$ that fixes a
$2$-dimensional subvariety of~$F^4$; such an involution can be
represented by a matrix $A$ such that $A^2 = I$ and $A^* Q A = Q$, and
such that $A-I$ has rank~$2$. For each $Q$ we note that the matrix
\[
A = \left[ \
\begin{matrix}
0 & 1 & 0 & 0 \\
1 & 0 & 0 & 0 \\
0 & 0 & 0 & 1 \\
0 & 0 & 1 & 0 \\
\end{matrix}\ \right]
\]
has these properties. It follows from Torelli's
theorem~\cite{Milne1986}*{Thm.~12.1, p.~202} that if $(F^4,Q)$ is the
Jacobian of a curve $C$, then $C$ has an involution $\alpha$ such that
the quotient of $C$ by $\alpha$ is a genus-$2$ curve whose Jacobian is
necessarily isogenous to~$F^2$.
This completes the proof of Proposition~\ref{P:nonmaximal}.\qed
\section{Hermitian forms over ${\mathbf{Z}}[\zeta_5]$}
\label{S:zeta}
Let ${\mathscr O} = {\mathbf{Z}}[\zeta_5]$. As Table~\ref{T:upper} indicates, the
programs in \texttt{IsogenyClasses.magma} show that if $C$ is a
genus-$4$ curve over ${\mathbf{F}}_{11}$ with $34$ points, the center of the
endomorphism ring of the Jacobian of $C$ is isomorphic to ${\mathscr O}$. (At
the end of this section we will explain how to derive this statement
from the output of the programs.) Likewise, if $C$ is a genus-$4$ curve
over ${\mathbf{F}}_{61}$ with $120$ points that is not a double cover of an
elliptic curve, then the center of the endomorphism ring of its
Jacobian is isomorphic to ${\mathscr O}$. In this section, we show that no
ordinary genus-$4$ curve over a finite field can have ${\mathscr O}$ as the
center of the endomorphism ring of its Jacobian.
\begin{proposition}
\label{P:zeta5}
Let $k$ be a finite field. There is no genus-$4$ curve over $k$ with
ordinary Jacobian $J$ such that the center of $\End J$ is isomorphic
to~${\mathbf{Z}}[\zeta_5]$.
\end{proposition}
\begin{proof}
Suppose, to obtain a contradiction, that such a curve $C$ exists. Let
$K = {\mathbf{Q}}(\zeta_5)$, let $\pi$ be the Frobenius endomorphism of~$J$, let
$\overline{\pi}$ be the Verschiebung endomorphism of $J$, and let $R$ be the
ring ${\mathbf{Z}}[\pi,\overline{\pi}]$. Since $R$ lies in the center of $\End J$ we
may view $R$ as a subring of ${\mathscr O}$, and since $\pi$ generates the
center of $(\End J)\otimes{\mathbf{Q}}$, we see that $R$ is an order in~${\mathscr O}$.
Let $M$ be the Deligne module associated with $J$
(see~\cite{Howe1995}), so that $M$ is a finitely-generated $R$-module
that is isomorphic to a submodule of~$K^2$. Since $\End J$ contains
${\mathscr O}$, we see that $M$ is in fact an ${\mathscr O}$-module, and since
${\mathscr O}$ has class number $1$ we have $M \cong {\mathscr O}\oplus{\mathscr O}$.
Translating this statement back from Deligne modules to ordinary
abelian varieties, we see that $J\cong A\times A$ for a $2$-dimensional
abelian variety with $\End A \cong {\mathscr O}$.
The field $K$ is ramified over its real subfield at a finite prime, so
by~\cite{Howe1995}*{Cor.~11.4, p.~2391} the variety $A$ has a principal
polarization $\lambda$. Let $\mu$ be the canonical polarization
on~$J$, viewed as a polarization on $A\times A$. Then $\mu$ is equal
to the product polarization $\lambda\times\lambda$ preceded by an
endomorphism $P$ of $A\times A$; viewing $P$ as an element of
$M_2({\mathscr O})$, we find that $P$ is a unimodular Hermitian matrix that is
totally positive (meaning that all of the roots of its minimal
polynomial are totally positive algebraic numbers). But
Lemma~\ref{L:zeta5} below says that all totally positive rank-$2$
unimodular Hermitian lattices over ${\mathscr O}$ are decomposable, so $\mu$
is isomorphic to the product polarization $\lambda\times\lambda$.
This contradicts the fact that the polarization on a Jacobian is never
a product.
\end{proof}
\begin{lemma}
\label{L:zeta5}
Let ${\mathscr O} = {\mathbf{Z}}[\zeta_5]$. Suppose $P$ is a totally positive
unimodular Hermitian matrix in $M_2({\mathscr O})$. Then there is an
invertible $C\in M_2({\mathscr O})$ such that $P = C^*C$, where $C^*$ is
the conjugate transpose of~$C$.
\end{lemma}
\begin{proof}
Our proof follows the lines set out in~\cite{HoweLauter2003}*{\S8}.
Let $K = {\mathbf{Q}}(\zeta_5)$, let $\varphi$ be a fundamental unit of the
maximal real subfield $K^+ = {\mathbf{Q}}(\sqrt{5})$ of $K$ with
$\Tr_{K^+/{\mathbf{Q}}} \varphi = 1$, and let $\phi$ be the real number
$(1 + \sqrt{5})/2$. Let $\psi_1$ and $\psi_2$ be embeddings of $K$
into the complex numbers ${\mathbf{C}}$ with $\psi_1(\varphi) =\phi$ and
$\psi_2(\varphi) = 1/\phi$. If $z$ is a complex number, we let $|z|$
be its magnitude and we let $||z||$ be its norm, so that
$||z|| = |z|^2 = z \overline{z}$.
Let $q$ be the quadratic form on ${\mathscr O}$ that sends $x$ to the trace
from $K$ to ${\mathbf{Q}}$ of~$x\overline{x}$, so that
$q(x) = 2||\psi_1(x)|| + 2||\psi_2(x)||$. Let $\Lambda$ be the lattice
$({\mathscr O},q)$. Using Magma, we compute the Voronoi cell for this
lattice, and we find that the covering radius of the lattice is~$2$;
that is, every element of $\Lambda\otimes{\mathbf{Q}}$ differs from a lattice
point by an element $x$ with $q(x)\le 2$. This means that every
element of $K$ differs from an element of ${\mathscr O}$ by an element $x$
satisfying
\[ ||\psi_1(x)|| + ||\psi_2(x)|| \le 1 . \]
In particular, we see that this $x$ also satisfies
\[ N_{K/{\mathbf{Q}}}(x) \le 1/4 \text{\quad and \quad} ||\psi_i(x)|| \le 1
\text{\quad for $i = 1,2$.} \]
In turn, this statement about the lattice $\Lambda$ gives us a
Euclidean algorithm on ${\mathscr O}$: Given elements $n$ and $d$ of~${\mathscr O}$,
there are elements $q$ and $r$ of ${\mathscr O}$ such that $n = qd + r$ with
\[
N(r) \le 1/4 \text{\quad and \quad }
||\psi_i(r)|| \le ||\psi_i(n)|| \text{\quad for $i = 1,2 $}.
\]
Write our totally positive unimodular Hermitian matrix $P$ as
\[
P =
\left[\ \begin{matrix}
\alpha & \overline{\beta} \\
\beta & \gamma
\end{matrix}\ \right]
\]
where $\alpha$ and $\gamma$ are totally real and where $\alpha$ and
$\alpha\gamma - \beta\overline{\beta}$ are totally positive. We will
repeatedly choose invertible matrices $C$ and replace $P$ with $C^*PC$
in order to reduce the size of the norm of the upper left element
of~$P$.
The determinant of $P$ is a totally positive unit in $K^+$, and so is
an even power of $\varphi$. By modifying $P$ by a matrix $C$ of the
form
\[
\left[\ \begin{matrix}
\varphi^i & 0 \\
0 & 1
\end{matrix}\ \right]
\]
we may assume that $P$ has determinant~$1$. Then by modifying $P$ by a
power of the matrix
\[
\left[\ \begin{matrix}
\varphi & 0 \\
0 & \varphi^{-1}
\end{matrix}\ \right]
\]
we can ensure that
\[
\frac{1}{\phi^2} \le \frac{\psi_1(\alpha) }{ \psi_2(\alpha)} \le \phi^2 .
\]
Another way of expressing this is to say that
\begin{equation}
\label{EQ:alphabound}
\frac{1}{\phi^2} \le \frac{\psi_i(\alpha)^2 }{\Norm_{K^+/{\mathbf{Q}}}(\alpha)}
\le \phi^2 \quad\text{for $i = 1, 2$.}
\end{equation}
Applying our Euclidean algorithm to $\beta$ and $\alpha$, we find that
$\beta = q \alpha + r$ for a $q$ and an~$r$ with
$||\psi_i(r)|| \le ||\psi_i(\alpha)||$ and with
$\Norm_{K/{\mathbf{Q}}}(r) \le (1/4) \Norm_{K/{\mathbf{Q}}}(\alpha)$. If we set
\[
C=
\left[\ \begin{matrix}
1 & -\overline{q} \\
0 & 1
\end{matrix}\ \right]
\]
then
\[
C^* P C =
\left[\ \begin{matrix}
\alpha & \overline{r}\\
r & \gamma'
\end{matrix}\ \right]
\]
for some integer $\gamma'$ in $K^+$. Replace $\beta$ with $r$ and
$\gamma$ with~$\gamma'$, so that now we have
\begin{equation}
\label{EQ:betabound}
||\psi_i(\beta)|| \le ||\psi_i(\alpha)|| \quad\text{for $i=1,2$}
\end{equation}
and
\begin{equation}
\label{EQ:betabound2}
\Norm_{K/{\mathbf{Q}}}(\beta) \le (1/4) \Norm_{K/{\mathbf{Q}}}(\alpha).
\end{equation}
Let $B = \beta \overline{\beta}$, so that $B$ is an integer in $K^+$. Note
that we have $\alpha\gamma - B = 1$, so
\[
\psi_i(\alpha) \psi_i(\gamma) = 1 + \psi_i(B) \quad \text{for $i = 1,2 $}
\]
and therefore
\begin{equation}
\label{EQ:gammabound}
\psi_i(\gamma) / \psi_i(\alpha) =
1/\psi_i(\alpha)^2 + \psi_i(B)/\psi_i(\alpha)^2 \text{\quad for $i = 1,2$.}
\end{equation}
Now let
\begin{align*}
b_1 &= \psi_1(B) / \psi_1(\alpha^2) \\
b_2 &= \psi_2(B) / \psi_2(\alpha^2) \\
c_1 &= 1/\psi_1(\alpha^2)\\
c_2 &= 1/\psi_2(\alpha^2)
\end{align*}
so that equation~(\ref{EQ:gammabound}) becomes
\[
\psi_1(\gamma) / \psi_1(\alpha) = b_1 + c_1 \text{\qquad and\qquad }
\psi_2(\gamma) / \psi_2(\alpha) = b_2 + c_2.
\]
Multiplying these last two equalities gives
\begin{equation}
\label{EQ:gammaalphabound}
\Norm_{K^+/{\mathbf{Q}}}(\gamma/\alpha) = b_1 b_2 + b_1 c_2 + b_2 c_1 + c_1 c_2.
\end{equation}
Note that
\begin{equation}
\label{EQ:bbbound}
b_1 b_2 = \frac{\Norm_{K^+/{\mathbf{Q}}}(B)} {\Norm_{K^+/{\mathbf{Q}}}(\alpha)^2}
= \frac{\Norm_{K/{\mathbf{Q}}}(\beta)}{\Norm_{K/{\mathbf{Q}}}(\alpha)}
\le \frac{1}{4}
\end{equation}
(where the final inequality comes from~(\ref{EQ:betabound2})) and
\begin{equation}
\label{EQ:ccbound}
c_1 c_2 = \frac{1}{\Norm_{K^+/{\mathbf{Q}}}(\alpha^2)}.
\end{equation}
Furthermore, from inequality~(\ref{EQ:alphabound}) we see that
\begin{equation}
\label{EQ:cbound}
c_1 \le \frac{\phi^2}{\Norm_{K^+/{\mathbf{Q}}}(\alpha)} \qquad\text{and}\qquad
c_2 \le \frac{\phi^2}{\Norm_{K^+/{\mathbf{Q}}}(\alpha)},
\end{equation}
and from inequality~(\ref{EQ:betabound}) we see that
\begin{equation}
\label{EQ:bbound}
b_1 = \frac{||\psi_1(\beta)||}{||\psi_1(\alpha)||} \le 1
\qquad\text{and}\qquad
b_2 = \frac{||\psi_2(\beta)||}{||\psi_2(\alpha)||} \le 1.
\end{equation}
If we view $b_1$, $b_2$, $c_1$, and $c_2$ as non-negative real
variables subject only to the conditions expressed in
relations~(\ref{EQ:bbbound}), (\ref{EQ:ccbound}), (\ref{EQ:cbound}),
and~(\ref{EQ:bbound}), and if we maximize $b_1c_1 + b_2c_2$ subject to
these conditions, we find that the maximum value occurs when $b_1 = 1$
and $c_1 = \phi^2/\Norm_{K^+/{\mathbf{Q}}}(\alpha)$. Thus we have
\begin{equation}
\label{EQ:crosstermbound}
b_1c_1 + b_2c_2 \le 1\cdot \frac{\phi^2}{\Norm_{K^+/{\mathbf{Q}}}(\alpha)}
+ \frac{1}{4} \cdot \frac{(1/\phi^2)}{\Norm_{K^+/{\mathbf{Q}}}(\alpha)}
\le \frac{2.72}{\Norm_{K^+/{\mathbf{Q}}}(\alpha)}.
\end{equation}
Let $\epsilon = 1/\Norm_{K^+/{\mathbf{Q}}}(\alpha)$. Then by combining the
relations~(\ref{EQ:gammaalphabound}), (\ref{EQ:bbbound}),
(\ref{EQ:ccbound}) and~(\ref{EQ:crosstermbound}) we find that
\[
\Norm_{K^+/{\mathbf{Q}}}(\gamma/\alpha) \le \epsilon^2 + 2.72 \, \epsilon + 1/4.
\]
If $\Norm_{K^+/{\mathbf{Q}}}(\alpha) \ge 4$ then $\epsilon \le 1/4$ and
$\Norm_{K^+/{\mathbf{Q}}}(\gamma/\alpha) < 1$. Then we can modify $P$ by
\[
\left[\ \begin{matrix}
0 & 1 \\
1 & 0 \end{matrix}\ \right]
\]
to exchange $\alpha$ and $\gamma$, and this decreases the norm of the
upper left element of $P$.
We repeat this procedure until we reach the point where
$\Norm_{K^+/{\mathbf{Q}}}(\alpha) \le 3$. The only totally positive integer of
$K^+$ with norm less than $4$ is $1$, so $\alpha=1$ and we can reduce
$\beta$ to be~$0$. Then we find $\gamma = 1$, so that we have reduced
$P$ to the identity matrix.
\end{proof}
Let us turn to the specific pairs $(q,N)$ that we must consider. For
$q=11$ and $N=34$, the function \texttt{isogeny\us{}classes} in the
package \texttt{IsogenyClasses.magma} shows that a genus-$4$ curve $C$
over ${\mathbf{F}}_{11}$ with $34$ points must have real Weil polynomial equal
to $(x^2 + 11x + 29)^2$; that is, the characteristic polynomial of
Frobenius plus Verschiebung is the square of this polynomial. It
follows that the characteristic polynomial of Frobenius is
\[ (x^4 + 11 x^3 + 51 x^2 + 121 x + 121)^2, \]
which means that the Jacobian of $C$ is isogenous to the square of an
abelian surface $A$ with characteristic polynomial
$x^4 + 11 x^3 + 51 x^2 + 121 x + 121$. Since the middle coefficient of
this characteristic polynomial is coprime to $q = 11$, we see that $A$
is ordinary. Furthermore, the polynomial has a root $\pi$ in
${\mathbf{Q}}(\zeta_5)$, namely $\pi = \zeta_5^2 + 2 \zeta_5 - 2$. One checks
that $\pi$ and its complex conjugate generate the ring ${\mathbf{Z}}[\zeta_5]$,
so the center of $\End J$, which contains Frobenius and Verschiebung,
must equal ${\mathbf{Z}}[\zeta_5]$. As we have seen, this is impossible.
For $q = 61$ and $N=120$, the function \texttt{isogeny\us{}classes}
tells us that if a genus-$4$ curve $C$ over ${\mathbf{F}}_{61}$ with $120$
points is not a double cover of an elliptic curve of trace $-13$, then
it must have real Weil polynomial equal to $(x^2 + 29 x + 209)^2$. As
above, we see that this implies that $C$ is ordinary and the center of
the endomorphism ring of its Jacobian is ${\mathbf{Z}}[\zeta_5]$, which is
impossible.
\begin{remark}
\label{R:simpler}
For our particular cases $(q,N) = (11, 34)$ and $(q,N) = (61,120)$,
there is also a more computational approach to showing that there is no
genus-$4$ curve $C$ over ${\mathbf{F}}_q$ with $N$ points. As we have seen, we
may assume that there is a fifth root of unity in the center of the
endomorphism ring of the Jacobian of $C$, and it follows that $C$ has
an automorphism of order~$5$. Since the finite fields we are concerned
with contain the fifth roots of unity, this shows that $C$ is a
degree-$5$ Kummer extension of another curve, and by using the
Riemann-Hurwitz formula we see that this second curve must be the
projective line. It is not hard to enumerate such Kummer covers; doing
so, we find no curves with $N$ points.
\end{remark}
\section{Lower bounds from explicit examples}
\label{S:lower}
In this section we prove the new lower bounds for $N_q(4)$ given in
Table~\ref{T:results} by providing examples of genus-$4$ curves with
many points. Each line of Table~\ref{T:lower} gives a prime power~$q$,
an integer~$N$, and the equations for a genus-$4$ curve over ${\mathbf{F}}_q$
having $N$ points.
\begin{table}
\renewcommand{\arraystretch}{1.25}
\begin{center}
\begin{tabular}{|r|r|lll|}
\hline
$q$ & $N$ & \multicolumn{3}{l|}{Equations for a genus-$4$ curve over ${\mathbf{F}}_q$ with $N$ points} \\
\hline
$13$ & $38$ & $y^2 = x^3 + 4$ && $z^2 = x^3 + x^2 - 4 x - 3$ \\
$17$ & $46$ & $y^2 = x^3 + x + 8$ && $z^2 = x^3 - 5 x^2 - 2 x - 8$ \\
$19$ & $48$ & $y^2 = x^3 + 8$ && $z^2 = x^3 - 9 x^2 - 5$ \\
$23$ & $57$ & $y^2 = x^3 - 6 x^2 - 3 x - 7$ && $z^2 = x y + 2 x^3 - 11 x^2 + 7 x + 1$ \\
$29$ & $67$ & $y^2 = x^3 - 3 x + 18$ && $z^2 = y + 7 x^3 + 6 x^2 + 2 x - 10$ \\
$31$ & $72$ & $y^2 = x^3 + x + 10$ && $z^2 = x^3 + 7 x^2 - 13 x - 13$ \\
$37$ & $82$ & $y^2 = x^3 + 2 x$ && $z^2 = x^3 + x^2 - 10 x - 13$ \\
$41$ & $88$ & $y^2 = x^3 + 6 x + 5$ && $z^2 = 3 x^3 - 17 x^2 - 11 x$ \\
$43$ & $92$ & $y^2 = x^3 + 2 x + 1$ && $z^2 = x^3 - 6 x^2 + 11 x$ \\
$47$ & $98$ & $y^2 = x^5 - 6 x^3 + 8 x^2 - 5 x + 12$ && $z^2 = y + 6 x^3 + 6 x^2 - x - 3$ \\
$49$ & $102$ & $y^2 = x^3 + x$ && $z^2 = x^3 + x^2 + x + 4$ \\
$53$ & $108$ & $y^2 = x^3 + 4 x + 10$ && $z^2 = x^3 + 12 x^2 + 17 x + 9$ \\
$59$ & $116$ & $y^2 = x^3 + 2 x + 22$ && $z^2 = 2 x^3 + x^2 - x + 9$ \\
$61$ & $118$ & $y^2 = x^3 + 4$ && $z^2 = x^3 + 23 x^2 + 25 x + 36$ \\
$67$ & $129$ & $y^2 = x^3 + 25$ && $z^2 = x y - 8 x^3 - 27 x + 4$ \\
$71$ & $134$ & $y^2 = x^3 + x + 9$ && $z^2 = x^3 + 9 x^2 + 24 x - 9$ \\
$73$ & $138$ & $y^2 = x^3 + 3 x + 11$ && $z^2 = x^3 + 34 x^2 + 18 x + 40$ \\
$79$ & $148$ & $y^2 = x^3 + x + 6$ && $z^3 = y + 33 x + 2$ \\
$83$ & $152$ & $y^2 = x^3 + 2 x + 19$ && $z^2 = x^3 + 38 x^2 - 6 x + 39$ \\
$89$ & $160$ & $y^2 = x^3 + 3 x$ && $z^2 = x^3 + 13 x^2 - 22 x + 28$ \\
$97$ & $174$ & $y^2 = x^3 + 5 x + 26$ && $z^3 = y + 37 x + 16$ \\
\hline
\end{tabular}
\end{center}
\vskip0.5em
\caption{Genus-$4$ curves over small finite fields with many points.
Note that for $q=79$ and $q=97$ the exponent on $z$ is $3$, not~$2$.}
\label{T:lower}
\end{table}
Almost all of the examples in Table~\ref{T:lower} were obtained by
running our program \texttt{double\us{}cover\us{}given\us{}trace}. The
exceptions are the example for $q=47$, which was found by running
\texttt{double\us{}cover\us{}genus\us{}4} on a carefully chosen
genus-$2$ curve, and the examples for $q=79$ and $q=97$, which were
found during a search of degree-$3$ Kummer covers of elliptic curves.
While searching for such Kummer covers, we also found a particularly
nice example for $q = 67$: the curve $y^6 = x^3 + x - 6$ attains
$N_{67}(4)$.
The function \texttt{check\us{}examples} in the file
\texttt{Genus4.magma} verifies that all of these examples do have the
number of points claimed.
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| {
"timestamp": "2012-03-12T01:00:28",
"yymm": "1108",
"arxiv_id": "1108.5393",
"language": "en",
"url": "https://arxiv.org/abs/1108.5393",
"abstract": "For prime powers q<100, we compute new upper and lower bounds on N_q(4), the maximal number of points on a genus-4 curve over a finite field with q elements. We determine the exact value of N_q(4) for 17 prime powers q for which the value was previously unknown.",
"subjects": "Algebraic Geometry (math.AG); Number Theory (math.NT)",
"title": "New bounds on the maximum number of points on genus-4 curves over small finite fields",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.977022627413796,
"lm_q2_score": 0.7248702880639791,
"lm_q1q2_score": 0.708214673378464
} |
https://arxiv.org/abs/2209.04176 | Modules in which pure submodule is essential in a direct summand | In this paper, we study the class of modules have the property that every pure submodule is essential in a direct summand. These modules are termed as pure extending modules which is a proper generalisation of extending modules. Examples and counterexamples are given. We study some properties of pure extending modules and characterize regular ring, semisimple ring, local ring and PDS ring in terms of pure extending modules. | \section{Introduction}\label{sec1}
Utumi\cite{YU} observed $C_1$ condition on a ring which is satisfied if the ring is self injective. Later, similarly $C_1$ condition on a module $M$ is defined as, every submodule of a module $M$ is essential in a direct summand of $M$. A module satisfies $C_1$ condition known as $C_1$ module or extending module. As we know that every submodule of a module $M$ need not be pure submodule ( Infact no submodule is pure submodule of $\mathbb{Z}$ as $\mathbb{Z}$ module). Motivated by the above facts, the objective of this paper is to extend the theory of extending modules to pure extending modules by using the concept of purity. Pure extending modules are those modules in which every pure submodule is essential in a direct summand. So pure extending modules is a proper generalisation of extending modules i.e. the class of pure extending modules is larger than the class of extending modules.
\\
Notion of pure submodules defined by some authors in different aspects which are listed below:\\
$(i)$ P.M. Cohn \cite{PMC} called a submodule $P$ of a right $R$-module $M$ to be a pure submodule, if for each left $R$-module $N$, the sequence $0\rightarrow P\otimes N\rightarrow M\otimes N$ is exact whenever the sequence $0\rightarrow P\rightarrow M$ is exact . \\
$(ii)$ According to Anderson and Fuller \cite{AF}, a submodule $P$ of a module $M$ is said to be pure if $\mathcal{I}P=P\cap \mathcal{I}M$ for every ideal $\mathcal{I}$ of $R$. \\
$(iii)$ In \cite{PR}, Ribenboim defined a pure submodule $P$ of a module $M$ if for every $r\in R$, $rP=rM \cap P$. In particular, $P$ is known as $RD$ (relatively divisible) pure submodule of $M$.\\
In above definitions, $(i)\Rightarrow (ii) \Rightarrow (iii)$ but converse need not be true, while in case of $M$ to be flat module all are equivalent. A right $R$-module $M$ is called flat if whenever $0 \rightarrow N_1 \rightarrow N_2$ is exact for left $R$-modules $N_1$ and $N_2$ then $ 0 \rightarrow M \otimes N_1 \rightarrow M\otimes N_2$ is also exact.
A module $M$ is said to be pure $C_2$ module if every pure submodule of $M$ that is isomorphic to a direct summand of $M$ is itself a direct summand of $M$\cite{SK}. An $R$-module $M$ is said to be pure $C_3$ module if $K$ and $L$ are disjoint direct summands of $M$ then $K\oplus L$ is a pure submodule of $M$ iff $K\oplus L$ is a direct summand of $M$ \cite{LM}. \\
In section 2, after defining the notion of pure extending modules, several examples and counter examples are given to distinguish this class of modules with the various classes of modules. We show that pure extending module is a proper generalisation of extending module by giving counter examples of pure extending module that are not extending module. We provide a sufficient condition under which pure extending module implies extending module. We prove that a direct summand and a pure submodule of pure extending module are pure extending. Let $P$ and $Q$ be $R$-modules, then a module $P$ is $Q$-pure injective if for every pure submodule $L$ of $Q$, $f\in Hom_R(L, P)$ can be extended to $g\in Hom_R(Q,P)$ such that $goh=f$ where $h\in Hom_R(L,Q)$. Further, $P$ is said to be a quasi-pure-injective if $P$ is a $P$-pure-injective, while $P$ is called a pure-injective if it is $Q$-pure injective for every $R$-module $Q$ (\cite{FH},\cite{AH},\cite{RW}).Here, we show that pure quasi injective module implies pure extending module.
In general the following chain holds,
$\xymatrix
{Injective \ar[d] \ar[r] & Quasi-injective \ar[d] \ar[r] & Extending \ar[d] \\
Pure-injective \ar[r] & Pure\: Quasi-injective \ar[r] & Pure \: extending}$
but converse of the above chain need not be true (see \cite{FH}, \cite{TY},\cite{RW} and Example \ref{abc}).\
Also, we show that direct sum of pure extending modules is pure extending (see Proposition \ref{dss}).We call a module $M$, $RD$-pure extending if every $RD$-pure submodule of $M$ is essential in a direct summand of $M$. As we have seen above that not every $RD$-pure submodule is pure submodule. We show that every $RD$-pure extending module is extending module and converse need not true.\\
In section 3, we charaterize regular ring, semisimple ring, local ring and PDS ring in terms of pure extending modules. Every pure extending module is flat over regular ring (see Proposition \ref{fe1}). We find the equivalent conditions of pure $C_i$ for $i=1,2,3$ to be projective modules over semisimple ring (see Proposition \ref{fe2}). A module $C$ is called cotorsion if $Ext_R ^1(F,C)=0$ for any flat module $F$ \cite{EE}. We show that flat cotorsion module is pure extending module ( see Proposition \ref{cm}) .\\
Throughout this article, we consider all rings to be associative with unity and all modules are right unital unless otherwise specified. The notations $N\leq M$, $N\leq ^{\oplus }M$ and $N\leq ^{e} M$ will denote $N$ is a submodule of $M$, $N$ is a direct summand of $M$ and $N$ is an essential submodule of $M$ respectively. A regular ring means to be von Neumann regular ring. $E(M)$ and $PE(M)$ denote the injective hull and pure injective hull of a module $M$ respectively. For undefined terms and notions, please refer to \cite{AF}.
\section{Pure Extending Modules}
In this section, we study extending modules in terms of purity by taking pure submodules. Now we introduce pure extending modules.
\begin{definition}
An $R$-Module $M$ is called pure extending ( or pure $C_1$), if every pure submodule of $M$ is essential in a direct summand of $M$.
\end{definition}
\begin{example}
\begin{enumerate}
\item [(i)] Since, only pure submodules of $\mathbb{Z}$-module $\mathbb{Z}$ are $\{0\}$ and $\mathbb{Z}$ itself. Therefore $\mathbb{Z}$ as a $\mathbb{Z}$-module is a pure extending.
\item[(ii)] Any pure injective module $M$ is pure extending module (since every pure injective module is pure quasi injective and by proposition \ref{prop1}).
\item [(iii)] Semisimple modules and injective modules are pure extending.
\item [(iv)] Any finitely generated module over a Noetherian ring is pure extending module. Since its pure submodules are just direct summands [Corollary 4.91,\cite{TY}]. In particular, every finitely generated $\mathbb{Z}$-module is pure extending.
\end{enumerate}
\end{example}
We observed that every $R$-module (in particular $\mathbb{Z}$-module) need not be pure extending.
\begin{example}
Consider a $\mathbb{Z}$-module $M$ such that $M=\Pi_{p\in P}\mathbb{Z}/<p>$ where $p$ varies through all primes.
Let $P=\bigoplus_{p\in P} \mathbb{Z}/<p>$ be pure submodule which is not essential in a direct summand.
\end{example}
\begin{proposition} \label{psm}
Pure submodule of pure extending module is pure extending.
\end{proposition}
\begin{proof}
Let $M$ be a pure extending module and $P$ be a pure submodule of $M$. Consider $K$ be a pure submodule of $P$. Thus from [Proposition 7.2,\cite{FH}], $K$ is pure submodule of $M$. So there exists a direct summand $L$ of $M$ such that $K\leq ^e L$ which implies $K\leq ^e L\cap P$. Since $L\leq^\oplus M$, so $M=L\bigoplus L'$ for some submodule $L'$ of $M$. Now, $M\cap P=(L\bigoplus L')\cap P$ which implies $P=(L\cap P)\bigoplus (L'\cap P)$. Hence $P$ is pure extending.
\end{proof}
\begin{remark} Submodule of a pure extending module need not be pure extending.\\
Let $N$ be a module which is not a pure extending and $PE(N)$ be pure injective hull of $N$. Then $PE(N)$ be pure injective module which implies $PE(N)$ is pure extending while $N\leq{PE(N)}$ is not pure extending. While over regular ring, every submodule of a pure extending module is pure extending.
\end{remark}
\begin{proposition} \label{DS1.1}
Direct summand of a pure extending module is pure extending.
\end{proposition}
\begin{proof}
Let $N$ be a direct summand such that $M=N\oplus N'$ of a pure extending module $M$ and $P$ be a pure submodule of $N$. Since $P\leq N \leq^{\oplus} M$, $P$ be a pure submodule of $M$. As $M$ is pure extending module, therefore $P$ is essential in a direct summand of $M$. Since $P\leq N$ and $P\cap N' = 0 $ then $P$ is essential in a direct summand of $N$.
\end{proof}
The following examples justifies that the class of pure extending modules is a proper generalisation of extending.
\begin{example} \label{abc}
Consider $M= \mathbb{Z}_{p}\oplus \mathbb{Z}_{p^3}$ as $\mathbb{Z}$ module, where $p$ be any prime. In particular for $p=2$, $P=\mathbb{Z}_{2}\oplus \mathbb{Z}_{8}$ as $\mathbb{Z}$ module. As its each pure submodule is direct summand [Corollary 4.91,\cite {TY}], so $P$ is pure extending whereas $P$ is not extending. In fact, its submodule $\mathbb{Z}(1+2\mathbb{Z},2+8\mathbb{Z})$ is not essential in any direct summand of $P$.
\end{example}
\begin{example}
Let $R={\begin{pmatrix}
\mathbb{Z} & \mathbb{Z}\\0 & \mathbb{Z} \end{pmatrix}}$, then $R_R$ is finitely generated and noetherian module so it is pure extending module whereas it is not extending module [Example 6.2,\cite{CAW}].
\end{example}
In the following proposition, we give a sufficient condition for the pure extending modules to be extending modules.
\begin{proposition} A ring $R$ is regular iff every pure extending right (resp.,left ) $R$-module is extending module.
\end{proposition}
\begin{proof}
Let $M$ be pure extending module then every pure submodule of $M$ is essential in a direct summand $M$. As $R$ is a von-Neumann regular ring so every submodule of an $R$-module $M$ is pure. Hence $M$ is a extending module.\\
Conversely, every pure extending right (resp., left ) $R$-module is extending module, that implies every submodule of $R$-module $M$ is pure. Hence, $R$ be a von-Neumann regular ring.
\end{proof}
\begin{proposition}
Let $M$ be a module which is fully invariant in its pure injective hull, then $M$ is pure extending module.
\end{proposition}
\begin{proof} Proof follows from [ lemma 3.1,\cite{AH}].
\end{proof}
\begin{proposition}
Let $R$ and $S$ be rings for which there is a Morita equivalence $F:$ Mod-$R\rightarrow$ Mod-$S$ and let $A\in$ Mod-$R$. Then $A$ is pure extending iff $F(A)$ is pure extending.
\end{proposition}
\begin{proof}
It follows from the morita invarient property of pure submodule.
\end{proof}
\begin{definition}
A submodule $K$ of a module $M$ is said to be pure essential in $M$ if $K$ is pure in $M$ and for any non zero submodule $N$ of $M$ either $K\cap N\neq 0$ or $(K\oplus N)/N$ is not pure in $M/N$ \cite{AH}.\\
\end{definition}
\begin{proposition}
If every submodule of a module $M$ is pure essential in $M$, then $M$ is pure extending.
\end{proposition}
\begin{proof}
Let $N$ be a pure submodule of $M$. Since every submodule is pure essential in $M$, $N$ is essential in $M$. Hence $M$ is pure extending module.
\end{proof}
Converse of the above statement need not true in general. Now, we provide the sufficient condition when it holds true.
\begin{proposition}
If $M$ is a pure simple pure extending module then every submodule of $M$ is pure essential submodule.
\end{proposition}
\begin{proof}
If $M$ be a pure simple module then it has no non proper non trivial pure submodule. Hence every submodule of $M$ is pure essential.
\end{proof}
\begin{corollary}
If $M$ is an indecomposable pure extending module then every submodule of $M$ is pure essential submodule.
\end{corollary}
\begin{proposition}
\label{prop1}
In any pure quasi injective module $M$, Every pure submodule of $M$ is essential in a direct summand of $M$ .
\end{proposition}
\begin{proof}
Let $N$ be a pure submodule of $M$ and write $PE(M)=PE(N)\oplus L$.
The pure quasi injectivity of $M$ implies $M\cap PE(M) = M \cap PE(N) \oplus M\cap L$ and $N{\le}^{e} M\cap PE(N)$ it implies pure extending.
\end{proof}
\begin{proposition}
Let $M$ be a $\mathbb{Z}$-module. If $M$ satisfies any one of the following conditions, then $M$ is pure extending module.
\begin{enumerate}
\item[(i)] $M$ is finitely generated.
\item[(ii)] $M$ is divisible.
\end{enumerate}
\end{proposition}
\begin{proof}
$(i)$ Since every pure submodule of a finite generated module over a noetherian ring is a direct summand (by Corollary 4.91,\cite{TY}). Hence, $M$ is pure extending.\\
$(ii)$ Let $M$ be a divisible module, this implies that it is an extending $\mathbb{Z}$ module. Hence $M$ is pure extending module.
\end{proof}
\begin{proposition}
Every pure split module is pure extending.
\end{proposition}
\begin{proof}
Let $M$ be a pure split $R$-module then every pure submodule of $M$ is direct summand. This implies $M$ is a pure extending module.
\end{proof}
\begin{proposition} \label{thm3}
Let $M$ be a module and $M=M_1\oplus M_2$ be direct sum decomposition . If $N$ be a pure submodule of $M$ then $N=N_ 1 \oplus N_2$, where $N_i$ is pure submodule of $M_i$ for $i=1,2$.
\end{proposition}
\begin{proof}
Let $N$ be a pure submodule of $M$ such that $N=N_1+N_2$. $N_1\cap N_2\leq N_1\leq M_1, N_1\cap N_2\leq N_2\leq M_2$ , which implies $N_1\cap N_2 \leq M_1 \cap M_2$, so $N_1\cap N_2 =0$. Hence, $N=N_1 \oplus N_2$. Since $N_i {\leq}^{\oplus} N$ then $N_i$ is pure submodule of $N$. $N_i$ is pure in $N$ and $N$ is pure in $M_i$ then $N_i$ is pure in $M_i$ for $i=1,2$.
\end{proof}
In the next proposition, we show when direct sum of pure extending modules is pure extending module.
\begin{proposition} \label{dss}
Let $M=\bigoplus_{i\in \Lambda} M_i$, where $\Lambda$ be an arbitrary index set. $M$ is pure extending iff for each $i\in \Lambda$, $M_i$ is pure extending module.
\end{proposition}
\begin{proof}
Let $M$ be a pure extending module. So, by proposition \ref{DS1.1} $M_i$ is pure extending module for each $i\in \Lambda$.\\
Conversely, Let $N$ be a pure submodule of $M$. So by proposition (\ref{thm3}), $N=\bigoplus _{i\in \Lambda} (N_i)$ such that every $N_i$ is a pure submodule of $M_i$ for each $i\in \Lambda$ . Since $M_i$ is pure extending, there exists $X_i \leq ^{\oplus} M_i$ such that $N_i \leq ^{e} X_i$. Therefore $\bigoplus _{i\in \Lambda}(N_i)\leq ^{e} \bigoplus_{i\in \Lambda} X_i\leq ^{\oplus} M$. Hence $M$ is pure extending module.
\end{proof}
We say an $R$-Module $M$ is $RD$-pure extending if every $RD$-pure submodule is essential in a direct summand of $M$.
\begin{lemma}\label{RDS1.1}
Every $RD$-pure extending module is pure extending.
\end{lemma}
\begin{proof}
Let $P$ be a pure submodule of $M$. Since every pure submodule is $RD$-pure and $M$ is $RD$-pure extending module. Therefore $P$ is essential in a direct summand of $M$. Hence the module $M$ is pure extending.
\end{proof}
\begin{example}
Let $M$ be an $R$ module such that every pure submodule is essential in a direct summand it implies $M$ is pure extending. In [page no.-158]\cite{TY}, there is an example which justify that every $RD$-pure submodule need not be pure submodule. So it may be possible that there exist $RD$-pure submodule of $M$ which are not essential in a direct summand of $M$. Hence $M$ need not be $RD$-pure extending.
\end{example}
\begin{proposition}
\begin{enumerate}
\item Direct summand of $RD$-pure extending module is $RD$-pure extending.
\item $RD$-pure submodule of $RD$-pure extending module is $RD$-pure extending.
\end{enumerate}
\end{proposition}
\begin{proof}
The proof is similar to proposition \ref{psm} and \ref{DS1.1}.
\end{proof}
\begin{proposition}
A flat $R$-module $M$ is pure extending iff $M$ is $RD$-pure extending.
\end{proposition}
\begin{proof}
The proof follows from lemma \ref{RDS1.1} and [Corollary 11.21,\cite{CF}].
\end{proof}
\begin{corollary}
A free (projective) $R$-module $M$ is pure extending iff $M$ is $RD$-pure extending.
\end{corollary}
\begin{corollary}
A faithful multiplicative $R$-module $M$ is pure extending iff $M$ is $RD$-pure extending.
\end{corollary}
\begin{proof}
The proof follows by the fact that every faithful multiplicative module is flat.
\end{proof}
\section{Characterization of rings using Pure extending modules}
In the next proposition we characterize the regular ring.
\begin{proposition}\label{fe1} For a ring $R$, the following conditions are equivalent:
\begin{enumerate}
\item $R$ is a regular ring.
\item Every pure extending $R$-module is flat.
\end{enumerate}
\end{proposition}
\begin{proof}
$(1)\Rightarrow (2)$ It is clear from [Theorem 4.21,\cite{TY}]\\
$(2)\Rightarrow (1)$ Let $M$ be an right $R$-module and $PE(M)$ be a pure injective hull of $M$. Then $0 \rightarrow M \rightarrow PE(M) \rightarrow PE(M)/M \rightarrow 0 $ is a pure exact sequence. From hypothesis, $PE(M)$ is a flat module so by [Corollary 4.86,\cite{TY}], $PE(M)/M$ is flat. Therefore $M$ is a flat which implies $R$ is regular ring [Corollary 4.21,\cite{TY}].
\end{proof}
In the next proposition we characterize the semisimple ring.
\begin{proposition}\label{fe2}
For a ring $R$, the following conditions are equivalent:
\begin{enumerate}
\item $R$ is a right semisimple ring.
\item Every pure $C_3$ $R$-module is projective.
\item Every pure $C_2$ $R$-module is projective
\item Every quasi pure injective $R$-module is projective.
\item Every pure injective $R$-module is projective.
\item Every pure extending $R$-module is projective.
\end{enumerate}
\end{proposition}
\begin{proof}
$(1)\Rightarrow (2)$ Let $R$ be a right semisimple ring then every $R$-module $M$ is projective [Proposition 20.7,\cite{RW}]. So $(2)$ holds.\\
$(2)\Rightarrow (3)$ Every pure $C_2$ module is pure $C_3$, so (3) holds.\\
$(3)\Rightarrow (4)$ Every quasi pure Injective right $R$- module is pure $C_2$, so (4) holds.\\
$(4)\Rightarrow (5)$ Every pure injective right $R$-module is quasi pure injective. so (5) holds.\\
$(5)\Rightarrow (1)$ Every pure injective right $R$-module is projective it implies every injective is projective. Therefore $(1)$ holds by [Proposition 20.7,\cite{RW}] \\
$(1) \Leftrightarrow (4)$ It follows from [Proposition 6,\cite{SK}].\\
$(1)\Rightarrow (6)$ Let $R$ be a semisimple ring it implies every $R$-module is injective (in particular pure extending) and projective [Proposition 20.7,\cite{RW}], so (6) holds.\\
$(6)\Rightarrow (1)$ By hypothesis, every pure extending module is projective it implies injective is projective. Hence, $R$ is a right semisimple ring \cite[Proposition 20.7]{RW}.
\end{proof}
\begin{proposition}
Let $R$ be a local ring then the module $R_R$ is pure extending.
\end{proposition}
\begin{proof}
By [Theorem 3,\cite{FDJ}], every local ring is pure simple it implies $R_R$ and 0 are its only pure submodules. Hence, $R_R$ is pure extending module.
\end{proof}
A ring $R$ is called left PDS if pure submodule of left $R$-module are direct summand and ring $R$ is said to be a PDS if it is both left and right PDS \cite{FH}.
\begin{proposition}
For a PDS ring $R$, every $R$-module is pure extending.
\end{proposition}
\begin{proof}
Let $M$ be an $R$-module. By hypothesis, $R$ is PDS ring which implies every pure submodule is essential in a direct summand. Hence $M$ be pure extending module.
\end{proof}
\begin{proposition}\label{cm}
Every flat cotorsion module is pure extending.
\end{proposition}
\begin{proof}
Let $M$ be a flat and cotorsion module then by [Proposition 3.2,\cite{LM}], $M$ is quasi pure injective. Therefore $M$ is a pure extending module.
\end{proof}
\begin{corollary}
If $R$ is a right cotorsion ring then $R_R$ is pure extending $R$-module.
\end{corollary}
| {
"timestamp": "2022-09-12T02:09:05",
"yymm": "2209",
"arxiv_id": "2209.04176",
"language": "en",
"url": "https://arxiv.org/abs/2209.04176",
"abstract": "In this paper, we study the class of modules have the property that every pure submodule is essential in a direct summand. These modules are termed as pure extending modules which is a proper generalisation of extending modules. Examples and counterexamples are given. We study some properties of pure extending modules and characterize regular ring, semisimple ring, local ring and PDS ring in terms of pure extending modules.",
"subjects": "Commutative Algebra (math.AC); Rings and Algebras (math.RA)",
"title": "Modules in which pure submodule is essential in a direct summand",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9770226274137959,
"lm_q2_score": 0.7248702880639791,
"lm_q1q2_score": 0.708214673378464
} |
https://arxiv.org/abs/1404.2133 | Convergence of a Moran model to Eigen's quasispecies model | We prove that a Moran model converges in probability to Eigen's quasispecies model in the infinite population limit. | \section{Introduction}
The concept of quasispecies was proposed by Manfred Eigen
in order to explain how a population of macromolecules behaves
when subject to an evolutionary process with selection and mutation.
In his celebrated paper~\cite{Eigen1},
Eigen models the evolution of a population of macromolecules
via a system of differential equations,
which arises from the laws of chemical kinetics.
Selection is performed according to a fitness landscape,
and mutations occur in the course of reproductions,
independently at each locus with rate $q$.
On the sharp peak landscape
---all but one sequence, the master sequence,
have the same fitness and the master sequence has higher fitness than the rest---
Eigen discovered that an error threshold phenomenon takes place:
there exists a critical mutation rate $q^*$
such that if $q>q^*$ then at equilibrium the population is totally random,
while if $q<q^*$ then at equilibrium the population forms a quasispecies,
i.e., it contains a positive fraction of the master sequence along with a cloud of mutants
that closely resemble the master sequence.
The concepts of error threshold and quasispecies might not only be relevant in molecular genetics,
but also in several other areas of biology, namely population genetics or virology~\cite{Domingo}.
Nevertheless, in Eigen's model the dynamics of the concentrations of the different genotypes
is driven by a system of differential equations,
which is a major drawback for the viability of the model in settings more complex than the molecular level~\cite{Wilke}.
A finite and stochastic version of Eigen's quasispecies model would be much more suitable
to expand the quasispecies theory to other areas~\cite{EMS,Schuster1,Wilke}.
The issue of designing a finite population version of the quasispecies model
has been tackled by several authors.
Different approaches have been considered in the literature:
Alves and Fontanari~\cite{AF}
propose a finite population model
and they study the dependence of the error threshold on the population size,
a similar approach is taken by McCaskill~\cite{McCaskill},
Park, Mu\~noz, Deem~\cite{PMD}
and Saakian, Deem, Hu~\cite{SDH},
who all suggest different kinds of finite population models.
In~\cite{NS}, Nowak and Schuster derive the error threshold for finite populations
using a birth and death chain.
More recently, in~\cite{CerfM,CerfWF},
Cerf shows that the error threshold and quasispecies concepts
arise for both the Moran model and the classical Wright--Fisher model
in the appropriate asymptotic regimes.
Some other authors propose stochastic models that converge to Eigen's model
in the infinite population limit,
this is the approach taken by Demetrius, Schuster, Sigmund~\cite{DSS},
who use branching processes,
Dixit, Srivastava, Vishnoi~\cite{DSV}
or Musso~\cite{Musso}.
Showing convergence of a finite population model
to Eigen's model is in general a delicate matter;
to our knowledge, all the works that have been done in this direction
prove that some stochastic process converges to Eigen's model in expectation.
As pointed out in~\cite{DSS}, convergence in expectation can be misleading sometimes,
mainly due to the fact that variation might increase as expectation converges,
leading to a poor understanding of the asymptotic behaviour of the stochastic process.
In the current work we consider the Moran model studied in~\cite{CerfM,CD}
driving the evolution of a finite population subject to selection and mutation effects.
This Moran model is shown to converge to Eigen's quasispecies model in the infinite population limit,
independently of the fitness landscape:
on any finite time interval, we prove convergence in probability
for the supremum norm.
The interest of our result not only lies on the type of convergence,
but also on the choice of the model:
the Moran model is possibly one of the simplest models
for which such a result can be expected.
The result is proven by means of a theorem due to Kurtz~\cite{Kurtz},
which gives sufficient conditions for the convergence of a sequence
of Markov processes to a deterministic trajectory,
characterised by a system of differential equations.
The article is organised as follows:
first we briefly introduce Eigen's quasispecies model
and the Moran model.
We state the main result in section~\ref{Main}.
In section~\ref{Convmark} we adapt Kurtz's theorem,
which originally deals with continuous state space Markov chains,
to the discrete state space setting.
Finally, in section~\ref{Proof},
we apply Kurtz's theorem in order to prove the main result.
\section{The Eigen and Moran models}\label{Models}
We present here Eigen's quasispecies model
and a discrete Moran model.
Consider a set of $N$ different genotypes,
labelled from 1 to $N$.
Both Eigen's model and the Moran model
describe the evolution of a population
of individuals having genotypes $1,\dots,N$.
In both models the evolution of the population
is driven by two main forces:
selection and mutation.
The selection and mutation mechanisms depend only on the genotypes,
and are common to both models.
Selection is performed with a fitness landscape
$(f_i)_{1\leq i\leq N}$,
$f_i$ being the reproduction rate of an individual having genotype $i$.
The mutation scheme is encoded in a mutation matrix
$(Q_{ij})_{1\leq i,j\leq N}$,
$Q_{ij}$ being the probability that an individual having genotype $i$
mutates into an individual having genotype $j$.
The mutation matrix is assumed to be stochastic,
i.e., its entries are non--negative and the rows add up to 1.
\textbf{Eigen's model.}
Eigen originally formulated the quasispecies model
to explain the evolution of a population of macromolecules.
The evolution of the concentration of the different genotypes
is driven by a system of differential equations,
obtained from the theory of chemical kinetics.
Let us denote by $\mathcal{S}_N$ the unit simplex, i.e.,
$$\mathcal{S}_N\,=\,
\lbrace\,
x\in\mathbb{R}^N:
x_i\geq 0,\ \, 1\leq i\leq N\quad
\text{and}\quad
x_1+\cdots+x_N=1
\,\rbrace\,.$$
An element $x\in\mathcal{S}_N$ represents a population
in which the concentration of the individuals having the $i$--th genotype
is $x_i$, for $1\leq i\leq N$.
Let $x^0\in\mathcal{S}_N$ be the starting population
and let us denote by $x(t)$ the population at time $t>0$.
Eigen's model describes the dynamics of $x(t)$
thorough the following system of differential equations:
$$
(*)\quad
x_i'(t)\,=\,\displaystyle\sum_{k=1}^N f_k Q_{ki} x_k(t)
-x_i(t)\sum_{k=1}^N f_k x_k(t)\,,\quad 1\leq i\leq N\,
$$
with initial condition $x(0)=x^0$.
The first term in the differential equation
accounts for the replication rate and mutations towards the $i$--th genotype,
while the second term helps to keep the total concentration constant.
A recent review on Eigen's quasispecies model can be found in~\cite{Schuster2}.
\textbf{The Moran model.}
Moran models aim at describing the evolution of a finite population.
The dynamics of the population is stochastic,
the evolution is described by a Markov chain.
Loosely speaking,
the Moran model evolves as follows:
at each step of time,
an individual is selected from the current population
according to its fitness,
this individual then produces an offspring,
which is subject to mutations.
Finally,
an individual chosen uniformly at random from the population is replaced by the offspring.
The state space of the Moran process
will be the set $\cP^m_{N}$ of the ordered partitions of the integer $m$
in at most $N$ parts:
$$\cP^m_{N}\,=\,\lbrace\,
z\in\mathbb{N}^N:
z_1+\cdots+z_N=m
\,\rbrace\,.$$
An element $z\in\cP^m_{N}$
represents a population in which $z_i$ individuals
have the genotype $i$, for $1\leq i\leq N$.
The only allowed changes at each time step
consist in replacing an individual from the current population by a new one.
If we denote by $(e_i)_{1\leq i\leq N}$
the canonical basis of $\mathbb{R}^N$,
the only allowed changes in a population are of the form
$$z\,\longrightarrow\,z-e_i+e_j\qquad 1\leq i,j\leq N\,.$$
Let $\lambda$ be a constant such that
$\lambda\,\geq\,\max\lbrace\,f_i:1\leq i\leq N\,\rbrace$.
The Moran process is the Markov chain
$(Z_n)_{n\geq 0}$
having state space $\cP^m_{N}$
and transition matrix $p$ given by:
for all $z\in\cP^m_{\ell +1}$ and $i,j\in\lbrace\,1,\dots,N\,\rbrace$ such that $i\neq j$,
$$p(z,z-e_i+e_j)\,=\,
\frac{z_i}{m}\times\frac{1}{\lambda m}\sum_{k=1}^N f_kQ_{kj}z_k\,.$$
The other non--diagonal coefficients of the transition matrix are null,
the diagonal coefficients are arranged so that the matrix is stochastic,
i.e., the entries are non--negative and the rows add up to 1.
\section{Main result}\label{Main}
Our aim is to show that Eigen's quasispecies model
arises as the infinite population limit of the Moran model.
More precisely, we will prove the following result:
\begin{theorem}\label{main}
Let $(Z_n)_{n\geq0}$
be the Moran process described above.
Suppose that we have the convergence of the initial conditions
towards $x^0$:
$$\lim_{m\to\infty}\frac{1}{m}Z_0=x^0\,,$$
and let
$x(t)$ be the solution of the system of differential equations $(*)$
with initial condition $x(0)=x^0$.
Then, for every $\delta,T>0$, we have
$$\lim_{m\to\infty}\,
P\bigg(
\sup_{0\leq t\leq T}\bigg|
\frac{1}{m}Z_{\lfloor\lambda mt\rfloor}-x(t)
\bigg|>\delta
\bigg)\,=\,0\,.$$
\end{theorem}
This result is an immediate consequence of theorem~4.7 in~\cite{Kurtz}.
In order to prove the result, we proceed in two steps.
We state first theorem~4.7 in~\cite{Kurtz},
and we show next that all the hypotheses needed to apply the theorem
are fulfilled in our particular setting.
\section{Convergence of a family of Markov chains}\label{Convmark}
Let $d\geq 1$ and let $E$ be a subset of $\mathbb{R}^d$.
Let
$\big(
(X^m_n)_{n\geq 0},
m\geq 1
\big)$
be a sequence of discrete time Markov chains
with state spaces $E_m\subset E$
and transition matrices $(p^m(x,y))_{x,y\in E_m}$.
Let $F:\mathbb{R}^d\to\mathbb{R}^d$ and consider
the system of differential equations
$$x_i'(t)\,=\,F_i(x(t))\,,\qquad 1\leq i\leq N\,.$$
Theorem~$4.7$ in~\cite{Kurtz}
gives a series of sufficient conditions
under which the sequence of Markov chains $(X^m)_{m\geq 1}$
converges to a solution of the above system of differential equations.
The original statement of theorem~4.7 in~\cite{Kurtz}
is written for the more general setting
of continuous state space Markov chains.
We modify just the notation in~\cite{Kurtz}
in order to state the result in a way which is more suited to our particular setting.
Theorem~4.7 in~\cite{Kurtz} can be applied if the following
set of conditions is satisfied.
There exist sequences of positive numbers
$(\alpha_m)_{m\geq1}$ and $(\varepsilon_m)_{m\geq1}$
such that
\begin{enumerate}
\vspace*{-15 pt}
\item $\displaystyle\lim_{m\to\infty}\alpha_m=\infty\ $
and
$\ \displaystyle\lim_{m\to\infty}\varepsilon_m=0$.
\item $\displaystyle\sup_{m\geq1}\,\sup_{x\in E_m}\,\alpha_m\sum_{y\in E_m}|y-x|p^m(x,y)<\infty$.
\vspace*{-5 pt}
\item $\displaystyle\lim_{m\to\infty}\,\sup_{x\in E_m}\,\alpha_m
\sum
{y\in E_m:|y-x|>\varepsilon_m}
|y-x|p^m(x,y)=0$.
\hspace*{-30 pt}Define, for $m\geq 1$,
$F^m(x)=\displaystyle\alpha_m\sum_{y\in E_m}(y-x)p^m(x,y)$.
\vspace*{-5 pt}
\item $\displaystyle\lim_{m\to\infty}\,\sup_{x\in E_m}\,|F^m(x)-F(x)|=0$.
\item There exists a constant $M$ such that
$$\forall x,y\in E\,,\qquad
|F(x)-F(y)|\,\leq\, M|x-y|\,.$$
\end{enumerate}
\begin{theorem}[Kurtz]\label{Kurtz}
Suppose that conditions 1--5 are satisfied.
Suppose further that we have the convergence of the initial conditions
$$\lim_{m\to\infty}X^m_0=x^0\,.$$
Then, for every $\delta,T>0$, we have
$$\lim_{m\to\infty}\,P\bigg(
\sup_{0\leq t\leq T}\big|
X^m_{\lfloor \alpha_m t\rfloor}-x(t)
\big|>\delta
\bigg)\,=\,0\,.$$
\end{theorem}
\section{Proof of theorem~\ref{main}}\label{Proof}
Let $(Z_n)_{n\geq 0}$
be the Moran process
defined in section~\ref{Models}.
Our aim is to apply theorem~\ref{Kurtz}
to the sequence of Markov chains
$\big( (Z_n/m)_{n\geq0},m\geq 1 \big)$.
We only need to find the appropriate sequences
$(\alpha_m)_{m\geq 1}$ and $(\varepsilon_m)_{m\geq 1}$
and verify that conditions~1--5 are satisfied in our setting.
For $m\geq 1$, let
$\alpha_m=\lambda m$ and $\varepsilon_m=2/m$.
The sequences $(\alpha_m)_{m\geq 1}$ and $(\varepsilon_m)_{m\geq 1}$
obviously verify condition~1.
As for condition~2,
we have, for $z\in\cP^m_{N}$
$$\sum_{z'\in\cP^m_{N}}\Big|
\frac{z}{m}-\frac{z'}{m}
\Big|p(z,z')\,=\,
\sum_{i,j=1}^N \Big|
-\frac{e_i}{m}+\frac{e_j}{m}
\Big|p(z,z-e_i+e_j)\,\leq\,\frac{\sqrt{2}}{m}\,.$$
Thus,
$$\sup_{m\geq 1}\,\sup_{z\in\cP^m_{N}}\,\alpha_m\sum_{z'\in\cP^m_{N}}
\Big|\frac{z'}{m}-\frac{z}{m}\Big|p(z,z')\,\leq\, \lambda\sqrt{2}\,<\,\infty\,,$$
as required for condition~2.
Since $p(z,z')>0$
if and only if
$|z-z'|\leq \sqrt{2}$,
for all $m\geq 1$ and $z\in\cP^m_{N}$, we have
$$\sum_{z'\in\cP^m_{N}:|z'-z|>m\varepsilon_m}
\Big|
\frac{z'}{m}-\frac{z}{m}
\Big|p(z,z')\,=\,0\,,$$
and condition~3 is also satisfied.
Let $F:\mathbb{R}^N\to\mathbb{R}^N$
be the function defined by
$$\forall i\in\lbrace\,1,\dots,N\,\rbrace\,,
\quad\forall x\in\mathbb{R}^N\,,\qquad
F_i(x)\,=\,\sum_{j=1}^N f_jQ_{ji}x_j-x_i\sum_{j=1}^N f_jx_j\,.$$
Since all the partial derivatives of $F$
are bounded on the simplex $\mathcal{S}_N$,
F is a Lipschitz function on $\mathcal{S}_N$,
i.e., condition~5 holds.
Finally,
let us compute,
for $m\geq 1$ and $x\in\cP^m_{N}/m$,
the value of $F^m(x)$.
By definition,
$$F^m(x)\,=\,
\lambda m\sum_{z\in\cP^m_{N}}\Big(
\frac{z}{m}-x
\Big)p(mx,z)
\,=\,\lambda\sum_{i,j=1}^N (-e_i+e_j)p(mx,mx-e_i+e_j)\,.
$$
Thus,
for $i\in\lbrace\,1,\dots,N\,\rbrace$
we have
\begin{align*}
F^m_i(x)\,&=\,
\lambda\sum_{k:k\neq i}p(mx,mx-e_k+e_i)
-\lambda\sum_{k:k\neq i}p(mx,mx-e_i+e_k)\\
&=\,\sum_{k:k\neq i}x_k\sum_{j=1}^N f_jQ_{ji}x_j
-\sum_{k:k\neq i}x_i\sum_{j=1}^N f_jQ_{jk}x_j\,.
\end{align*}
Since $x_1+\cdots+x_N=1$ and for all $i\in\lbrace\,1,\dots,N\,\rbrace$,
$Q_{i1}+\cdots+Q_{iN}=1$,
$$F^m_i(x)=
(1-x_i)\sum_{j=1}^N f_jQ_{ji}x_j-x_i\sum_{j=1}^N f_j(1-Q_{ji})x_j
=\sum_{j=1}^N f_jQ_{ji}x_j-x_i\sum_{j=1}^N f_jx_j.
$
Thus,
the function $F^m$ coincides with the function $F$
on the set $\cP^m_{N}/m$, which readily implies condition~4.
Since all five conditions are satisfied,
we can apply theorem~\ref{Kurtz}
to the sequence of Markov chains $\big(
(Z_n/m)_{n\geq 0}, m\geq 1
\big)$
and we obtain the desired result.
\bibliographystyle{plain}
| {
"timestamp": "2014-04-09T02:09:46",
"yymm": "1404",
"arxiv_id": "1404.2133",
"language": "en",
"url": "https://arxiv.org/abs/1404.2133",
"abstract": "We prove that a Moran model converges in probability to Eigen's quasispecies model in the infinite population limit.",
"subjects": "Populations and Evolution (q-bio.PE); Probability (math.PR)",
"title": "Convergence of a Moran model to Eigen's quasispecies model",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9770226347732859,
"lm_q2_score": 0.724870282120402,
"lm_q1q2_score": 0.7082146729061303
} |
https://arxiv.org/abs/2104.00577 | Vertex and edge metric dimensions of unicyclic graphs | In a graph G, the cardinality of the smallest ordered set of vertices that distinguishes every element of V (G) (resp. E(G)) is called the vertex (resp. edge) metric dimension of G. In [16] it was shown that both vertex and edge metric dimension of a unicyclic graph G always take values from just two explicitly given consecutive integers that are derived from the structure of the graph. A natural problem that arises is to determine under what conditions these dimensions take each of the two possible values. In this paper for each of these two metric dimensions we characterize three graph configurations and prove that it takes the greater of the two possible values if and only if the graph contains at least one of these configurations. One of these configurations is the same for both dimensions, while the other two are specific for each of them. This enables us to establish the exact value of the metric dimensions for a unicyclic graph and also to characterize when each of these two dimensions is greater than the other one. | \section{Introduction}
In this paper we consider only simple and connected graphs unless explicitly
stated otherwise. The distance between a pair of vertices $u$ and $v$ in a
graph $G$ is denoted by $d(u,v)$. We say that a vertex $s$ from $G$
\emph{distinguishes} or \emph{resolves} a pair of vertices $u$ and $v$ from
$G$ if $d(s,u)\not =d(s,v).$ A set of vertices $S\subseteq V(G)$ is called a
\emph{vertex metric generator,} if every pair of vertices in $G$ is
distinguished by at least one vertex from $S.$ The cardinality of a smallest
vertex generator in $G$ is called the \emph{vertex metric dimension} of $G,$
and it is denoted by $\mathrm{dim}(G).$ For this variant of metric dimension
the prefix "vertex" is sometimes omitted, so we say only metric generator and
metric dimension.
The concept of metric generator was introduced in \cite{SlaterVertex} under
the name of locating set and the problem of uniquely identifying the location
of an intruder in a network by its distances to the locating devices was
considered. Independently, the same concept was studied in \cite{HararyVertex}
under the name of resolving set. The complexity of approximating the metric
dimension of a graph was studied in \cite{KhullerVertex}, applications of
metric dimension in digital geometry in \cite{MelterVertex}, the comparison of
metric dimension of graphs and line graphs in \cite{KleinVertex}, how the
metric dimension can be affected by the addition of a single vertex in
\cite{BuczkowskiVertex}, vertices contained in all metric generators in
\cite{HakanenYero}, and the behaviour of metric dimension with respect to
various graph operations in \cite{ChartrandVertex, SaputroVertex}.
Recently, the concept of metric dimension was extended from distinguishing
vertices to distinguishing edges. Similarly as above, a vertex $s\in V(G)$
\emph{distinguishes} two edges $e,f\in E(G)$ if $d(s,e)\neq d(s,f)$, where
$d(e,s)=d(uv,s)=\min\{d(u,s),d(v,s)\}$. The authors of \cite{TratnikEdge}
noticed that there are graphs in which none of the smallest metric generators
distinguishes all pairs of edges, so they were motivated to introduce a notion
of an \emph{edge metric generator} as any set $S\subseteq V(G)$ which
distinguishes all pairs of edges, the \emph{edge metric dimension} (denoted by
$\mathrm{edim}(G)$) as the cardinality of a smallest edge metric generator,
and then study its relation with the vertex metric dimension. They presented
families of graphs for which $\mathrm{dim}(G)<\mathrm{edim}(G)$, or
$\mathrm{dim}(G)=\mathrm{edim}(G)$, or $\mathrm{dim}(G)>\mathrm{edim}(G)$,
also they established that determining the edge metric dimension of a graph is NP-hard.
This new variant of metric dimension immediately attracted a lot of interest.
The behaviour of edge metric dimension on several graph operations was studied
in \cite{ZubrilinaEdge}, the edge metric dimension of convex polytopes and its
related graphs in \cite{ZhangGaoEdge}, graphs with the maximum edge metric
dimension in \cite{ZhuEdge}, pattern avoidance in graphs with bounded metric
dimension or edge metric dimension in \cite{GenesonEdge}, an approximation
algorithm for the edge metric dimension problem in
\cite{HuangApproximationEdge}, comparison of metric dimensions in
\cite{KelHypercubes, Knor}, bounds on vertex and edge metric dimension of
graphs with edge disjoint cycles in \cite{SedSkreBounds}.
Recently the mixed metric dimension was also introduced \cite{KelencMixed},
where a mixed metric generator of a graph $G$ is defined as any set $S$ which
distinguishes all pairs from $V(G)\cup E(G),$ the size of a smallest such set
is the \emph{mixed metric dimension} of $G$, and it is denoted by
$\mathrm{mdim}(G)$. The paper \cite{KelencMixed} contains lower and upper
bounds for various graph classes. The mixed metric dimension was further
studied in \cite{SedSkrekMixed} for graphs with edge disjoint cycles, these
results were further generalized to graphs with prescribed cyclomatic number
in \cite{SedSkreTheta}. For a wider and systematic introduction of the topic
of these three variants of metric dimension, we recommend the PhD thesis of
Kelenc \cite{KelPhD}.
The focus of this paper is on the result from \cite{SedSkreBounds}, where it
was was established that both $\mathrm{dim}(G)$ and $\mathrm{edim}(G)$ of a
unicyclic graph $G$ can take its value from only two consecutive integers
which can be determined from the structure of the graph. In this paper, we go
further and characterize when these two metric dimensions take each of the two
possible values, a research direction which is proposed in \cite{Knor}. This
promptly resolves which of the following three situations $\mathrm{dim}%
(G)<\mathrm{edim}(G)$, or $\mathrm{dim}(G)=\mathrm{edim}(G)$, or
$\mathrm{dim}(G)>\mathrm{edim}(G)$ holds for a unicyclic graph $G$.
\section{Preliminaries}
Throughout the paper we will use the following notation. The cycle in a
unicyclic graph $G$ is denoted by $C=v_{0}v_{1}\cdots v_{g-1}$, where $g$ is
the length of $C$ (i.e. $g=\left\vert V(C)\right\vert $). Additionally, each
edge $v_{i}v_{i+1}$ of $C$ is denoted by $e_{i}.$ The connected component of
$G-E(C)$ containing vertex $v_{i}$ is denoted by $T_{v_{i}}$. A \emph{thread}
in a graph $G$ is a path $u_{1}u_{2}\cdots u_{k}$ in which $u_{k}$ is of
degree $1$, all other vertices of the thread are of degree $2$ and the vertex
$u_{1}$ is a neighbour of a vertex $v\in V(G)$ with $\mathrm{\deg}(v)\geq3$.
For a vertex $v$ from a unicyclic graph $G$ we say that it is a
\emph{branching} vertex if $v\not \in V(C)$ and $\deg(v)\geq3$ or $v\in V(C)$
and $\deg(v)\geq4$. We say that a vertex $v_{i}\in V(C)$ is
\emph{branch-active} if $T_{v_{i}}$ contains a branching vertex. Let us denote
by $b(C)$ the number of all branch-active vertices on $C$. As the cycle $C$ is
the only cycle in a unicyclic graph, we may use notation $b(G)$ instead of
$b(C)$ as well.
Notice that every branching vertex $v$ which belongs to $T_{v_{i}}$ has at
least two neighbours which are not distinguished by any vertex from outside of
$T_{v_{i}},$ even more - it may not be distinguished by some vertices from
$T_{v_{i}}$, see Figure \ref{FigureBranching}.a) for illustration. Similarly
holds for a pair of edges incident to branching vertices $v$ and $v_{k.}$ We
say that a set $S\subseteq V(G)$ of a (unicyclic) graph $G$ is
\emph{branch-resolving} if for every $v\in V(G)$ of degree at least $3$, the
set $S$ contains a vertex from all threads hanging at $v$ except possibly from
one such thread, see Figure \ref{FigureBranching}.b).
\begin{figure}[h]
\begin{center}
$%
\begin{array}
[c]{cccc}%
\text{a)} &
\text{\raisebox{-1\height}{\includegraphics[scale=0.5]{Figure12.pdf}}} &
\text{b)} &
\text{\raisebox{-1\height}{\includegraphics[scale=0.5]{Figure13.pdf}}}%
\end{array}
$
\end{center}
\caption{Both figures show the same graph for which: a) Two branching vertices
are $v$ and $v_{k},$ and each of them has a pair of neighbours which cannot be
distinguished by a vertex outside of $T_{v_{0}}$ and $T_{v_{k}}$ respectively.
b) $S=\{s_{1},s_{2}\}$ is an example of a smallest branch-resolving and the
set $\{v_{0},v_{k}\}$ is both the set of $S$-active vertices, and the set of
branch-active vertices.}%
\label{FigureBranching}%
\end{figure}
Let us denote by $\ell(v)$ the number of all threads attached to a vertex $v$
of degree $\geq3$ and let
\[
L(G)=\sum(\ell(v)-1),
\]
where the sum runs over all vertices $v$ of degree $\geq3$ of $G$ with
$l(v)>1.$ Note that for every branch-resolving set $S$ we have $\left\vert
S\right\vert \geq L(G)$ where equality holds if $S$ is a branch-resolving sets
of minimum cardinality.
For a set of vertices $S\subseteq V(G)$ we say that $v_{i}$ from $C$ is
$S$\emph{-active} if $T_{v_{i}}$ contains a vertex from $S$. The number of
$S$-active vertices on $C$ is denoted by $a_{S}(C)$. Note that for a
branch-resolving set $S$ it holds that $a_{S}(C)\geq b(C)$ with equality
holding for a smallest branch-resolving set $S$ in $G$. A smallest
branch-resolving set usually is not unique, but all smallest branch-resolving
sets have the same set of $S$-active vertices on the cycle which equals the
set of branch-active vertices on the cycle (see again Figure
\ref{FigureBranching}.b)). Finally, we say that a thread hanging at a vertex
$v$ of degree $\geq3$ is $S$\emph{-free} if it does not contain a vertex from
$S.$ Let us remark, as $v$ is not a vertex of the thread, if $v\in S,$ it does
not prevent the thread to be $S$-free.
The following two properties of branch-resolving sets were shown in
\cite{SedSkreBounds}.
\begin{lemma}
\label{Lemma_a(S)vj2} Let $S$ be a metric generator or an edge metric
generator of a unicyclic graph $G$. Then $S$ is a branch-resolving set with
$a_{S}(C)\geq2$.
\end{lemma}
\begin{lemma}
\label{Lemma_SameComponent} Let $G$ be a unicyclic graph and $S\subseteq V(G)$
a branch-resolving set with $a_{S}(C)\geq2$. Then, any two vertices (also any
two edges) from a same connected component of $G-E(C)$ are distinguished by
$S$.
\end{lemma}
We say that a set $S\subseteq V(G)$ is \emph{biactive} if $a_{S}(C)\geq2.$
Thus, according to Lemma \ref{Lemma_a(S)vj2}, if the set $S$ is not
branch-resolving or if it is not biactive, then $S$ certainly is not a vertex
nor an edge metric generator. The problem of non-distinguished pairs of
vertices (resp. edges), if $S$ is not branch-resolving, is already illustrated
by Figure \ref{FigureBranching}.a). Let us now consider when $S$ is not
biactive. If $a_{S}(C)=0$ then $S=\phi$ and $S$ cannot be a metric generator,
on the other hand if $a_{S}(C)=1$ then the pair of vertices (resp. edges) from
$C$ which are adjacent (resp. incident) to the only $S$-active vertex on $C$
are not distinguished by $S$, see Figure \ref{FigureSactive}.a) for illustration.
\begin{figure}[h]
\begin{center}
$%
\begin{array}
[c]{cccc}%
\text{a)} &
\text{\raisebox{-1\height}{\includegraphics[scale=0.5]{Figure14.pdf}}} &
\text{b)} &
\text{\raisebox{-1\height}{\includegraphics[scale=0.5]{Figure15.pdf}}}%
\end{array}
$
\end{center}
\caption{An example of a branch-resolving set $S$ with: a) $a_{S}(C)=1,$ b)
$a_{S}(C)=2,$ which is not a vertex metric generator. A pair of
non-distinguished vertices is marked in the figure.}%
\label{FigureSactive}%
\end{figure}
Yet, a biactive branch-resolving set $S$ may or may not be a vertex (edge)
metric generator, this depends also of the position of vertices on the cycle
which have an $S$-free thread attached. For example, the set $S$ from Figure
\ref{FigureBranching}.b) is a biactive branch-resolving set which is a vertex
(edge) metric generator, and the set $S$ from Figure \ref{FigureSactive}.b) is
also a biactive branch-resolving set but it is not a vertex (edge) metric
generator. Notice that even if we add to $S$ all vertices of $T_{v_{i}},$ for
$0\leq i\leq k,$ the set $S$ would still not be a vertex (edge) metric generator.
By the above, we need to introduce a configuration of $S$-active vertices on
the cycle $C$ which will suffice for a biactive branch-resolving set $S$ to
become a vertex (edge) metric generator. For this, let $v_{i}$, $v_{j}$, and
$v_{k}$ be three vertices of the cycle $C$. We say that $v_{i}$, $v_{j}$, and
$v_{k}$ form a \emph{geodesic triple} of vertices on $C$, if
\[
d(v_{i},v_{j})+d(v_{j},v_{k})+d(v_{i},v_{k})=|V(C)|.
\]
Observe that for any two vertices of $C$, we can easily choose a third one
such that they form a geodesic triple. It is also easy to observe that a
geodesic triple of vertices distinguishes vertices from $C.$ Moreover the
following property of geodesic triples was shown in \cite{SedSkreBounds}.
\begin{lemma}
\label{Lemma_GeodTrip} Let $G$ be a unicyclic graph, and let $S$ be a
branch-resolving set of $G$ with $a_{S}(C)\geq3$ and with three $S$-active
vertices on $C$ forming a geodesic triple. Then, $S$ is both a metric
generator and an edge metric generator of $G$.
\end{lemma}
\noindent By the all above, for a set of vertices $S\subseteq V(G)$ we can
conclude the following:
\begin{enumerate}
\item[S1.] If the set $S$ is not a biactive branch-resolving set, then $S$
cannot be a vertex (resp. edge) metric generator.
\item[S2.] If the set $S$ is a biactive branch-resolving set, then either:
\begin{enumerate}
\item $S$ is a vertex (resp. edge) metric generator by itself; or
\item $S$ is not a vertex (resp. edge) metric generator by itself, and it
suffices to introduce precisely one more vertex to $S$ in order to become a
vertex (resp. edge) metric generator according to Lemma \ref{Lemma_GeodTrip}.
\end{enumerate}
\end{enumerate}
In this paper we will establish necessary and sufficient condition under which
that one additional vertex must be introduced to a smallest biactive
branch-resolving set $S$ to become a vertex (resp. edge) metric generator. In
order to do so, we will first consider unicyclic graphs with $b(G)\geq2$ and
identify three structural configurations $\mathcal{A}$, $\mathcal{B}$, and
$\mathcal{C}$ (resp. $\mathcal{A}$, $\mathcal{D}$, and $\mathcal{E}$) in such
a graph which are the only obstacle for a smallest biactive branch-resolving
set to be a vertex (resp. edge) metric generator. Since these configurations
in graphs with $b(G)<2$ depend on the set $S,$ this approach is further
extended by introducing a more general property of a graph which enables us to
derive results which encapsulate also graphs with $b(G)<2,$ namely
$\mathcal{ABC}$-positivity and $\mathcal{ABC}$-negativity (resp.
$\mathcal{ADE}$-positivity and $\mathcal{ADE}$-negativity).
For characterization of the smallest biactive branch-resolving sets $S$ that
need to be introduced an additional vertex in order to become a metric
generator, the position of $S$-active vertices on the cycle $C$ matters. In
order to be able to deal with them, we introduce the following labelling of
the cycle.
\begin{definition}
Let $G$ be a unicyclic graph with the cycle $C$ of length $g$ and let $S$ be a
biactive branch-resolving set in $G$. We say that $C=v_{0}v_{1}\cdots v_{g-1}$
is \emph{canonically }labelled with respect to $S$ if $v_{0}$ is $S$-active
and $k=\max\{i:v_{i}$ is $S$-active$\}$ is as small as possible.
\end{definition}
\noindent Notice that when there is no geodesic triple of $S$-active vertices,
the canonical labelling implies $k\leq g/2$. In particular, if $a_{S}(C)=2$,
then $k\leq g/2$. Throughout the paper we will assume that the cycle $C$ is
canonically labelled with respect to the given biactive branch-resolving set
$S$, unless explicitly stated otherwise.
\section{Vertex metric dimension}
Regarding S2 we want to characterize when a smallest biactive branch-resolving
set is a vertex metric generator by itself, and when an additional vertex must
be added to such a set to become a vertex metric generator. For this, we
introduce three structural configurations $\mathcal{A}$, $\mathcal{B}$, and
$\mathcal{C}$ of the graph $G$ with respect to $S$ which will be crucial for
the characterization.
\begin{definition}
Let $G$ be a unicyclic graph, and let $S$ be a biactive branch-resolving set
in $G$. We say that the graph $G$ with respect to $S$ \emph{contains} configurations:
\begin{description}
\item {$\mathcal{A}$}. If $a_{S}(C)=2$, $g$ is even, and $k=g/2$;
\item {$\mathcal{B}$}. If $k\leq\left\lfloor g/2\right\rfloor -1$ and there is
an $S$-free thread hanging at a vertex $v_{i}$ for some $i\in\lbrack
k,\left\lfloor g/2\right\rfloor -1]\cup\lbrack\left\lceil g/2\right\rceil
+k+1,g-1]\cup\{0\}$;
\item {$\mathcal{C}$}. If $a_{S}(C)=2$, $g$ is even, $k\leq g/2$ and there is
an $S$-free thread of the length $\geq g/2-k$ hanging at a vertex $v_{i}$ for
some $i\in\lbrack0,k]$.
\end{description}
\end{definition}
Notice that configuration $\mathcal{C}$ with $k=g/2$ is also configuration
$\mathcal{A}$, and configuration $\mathcal{C}$ with $i\in\{0,k\}$ and $k\leq
g/2-1\ $is also configuration $\mathcal{B}$.
\begin{figure}[h]
\begin{center}
$%
\begin{array}
[c]{cccc}%
\text{a)} &
\text{\raisebox{-1\height}{\includegraphics[scale=0.5]{Figure07.pdf}}} &
\text{b)} &
\text{\raisebox{-1\height}{\includegraphics[scale=0.5]{Figure05.pdf}}}\\
\text{c)} &
\text{\raisebox{-1\height}{\includegraphics[scale=0.5]{Figure06.pdf}}} &
\text{d)} &
\text{\raisebox{-1\height}{\includegraphics[scale=0.5]{Figure08.pdf}}}%
\end{array}
$
\end{center}
\caption{In all examples we consider a branch-resolving set $S,$ where $v_{0}$
and $v_{k}$ are the only two $S$-active vertices on $C.$ Configuration
$\mathcal{A}$ with respect to $S$ is illustrated by a). Configuration
$\mathcal{B}$ is shown: b) for even cycle, c) for odd cycle. Finally,
configuration $\mathcal{C}$ is illustrated by d). In every graph a pair of
vertices is marked which is not distinguished by $S.$}%
\label{Figure_YesProblem}%
\end{figure}
The above configurations are illustrated by Figure \ref{Figure_YesProblem}. In
every graph from Figure \ref{Figure_YesProblem}, a pair of vertices is marked
which is not distinguished by $S.$ In the next theorem we will show that this
holds in general, i.e. that every set $S$ for which the cycle $C$ has one of
these configurations is not a vertex metric generator, and otherwise $S$ is a
vertex metric generator.
\begin{lemma}
\label{Lm_vertexGeneratorNec}Let $G$ be a unicyclic graph and let $S$ be a
biactive branch-resolving set in $G$. If $G$ contains configuration
$\mathcal{A}$, $\mathcal{B}$, or $\mathcal{C}$ with respect to $S,$ then $S$
is not a vertex metric generator in $G.$
\end{lemma}
\begin{proof}
Let us assume that $G$ contains configuration $\mathcal{A}$, $\mathcal{B}$, or
$\mathcal{C}$ with respect to $S$ and it is sufficient to find a pair of
vertices $x,x^{\prime}\in V(G)$ which are not distinguished by $S$.
If $G$ contains configuration $\mathcal{A}$ then $x=v_{1}$ and $x^{\prime
}=v_{g-1}$ are not distinguished by $S$.
Next, if $G$ contains configuration $\mathcal{B}$, let $v_{i}$ be a vertex
from $C$ with an $S$-free thread hanging at it, where $i\in\lbrack
k,\left\lfloor g/2\right\rfloor -1]\cup\lbrack\left\lceil g/2\right\rceil
+k+1,g]\cup\{0\}$. Let $w$ be the neighbour of $v_{i}$ which belongs to the
thread hanging at $v_{i}$. If $i\in\lbrack k,\left\lfloor g/2\right\rfloor
-1]$ then $x=w$ and $x^{\prime}=v_{i+1}$ are not distinguished by $S$. And if
$i\in\lbrack\left\lceil g/2\right\rceil +k+1,g-1]\cup\{0\},$ then $x=w$ and
$x^{\prime}=v_{i-1}$ are not distinguished by $S$.
Finally, suppose that $G$ contains configuration $\mathcal{C}$. Let $v_{i}$
with $i\in\lbrack0,k]$ be a vertex on the cycle $C$ with an $S$-free thread of
the length $\geq g/2-k$ hanging at it. Let $x$ be the vertex on that thread
such that $d(x,v_{i})=g/2-k$, let $j=2k-i+d(x,v_{i})=g/2+k-i$ and let
$x^{\prime}=v_{j}$. We argue that $x$ and $x^{\prime}$ are not distinguished
by $S$. Note that $i\in\lbrack0,k]$ implies $j\in\left[ g/2,g/2+k\right] $,
therefore $d(v_{j},v_{k})=j-k$ and $d(v_{j},v_{0})=g-j$. Now a simple
calculation yields that $x$ and $x^{\prime}$ are not distinguished by
$\{v_{0},v_{k}\},$ and therefore they are not distinguished by $S$.
\end{proof}
In the above lemma we have shown that configurations $\mathcal{A}$,
$\mathcal{B}$, and $\mathcal{C}$ are the obstacles for $S$ to be a vertex
metric generator in $G.$ Let us now prove that these configurations are the
only such.
\begin{lemma}
\label{Lm_vertexGeneratorSuf}Let $G$ be a unicyclic graph and let $S$ be a
biactive branch-resolving set in $G$. If $G$ does not contain any of the
configurations $\mathcal{A}$, $\mathcal{B}$, and $\mathcal{C}$ with respect to
$S,$ then the set $S$ is a vertex metric generator in $G$.
\end{lemma}
\begin{proof}
Let us assume that $G$ does not contain any of the configurations
$\mathcal{A}$, $\mathcal{B}$, and $\mathcal{C}$ with respect to $S$ and let us
suppose the contrary to the claim, i.e. $S$ is not a vertex metric generator.
By Lemma \ref{Lemma_GeodTrip}, there is no geodesic triple of $S$-active
vertices on $C$, and hence the canonical labelling of $G$ implies $k\leq g/2$.
Suppose first that $k=g/2$, which implies that $g$ is even. If $a_{S}(C)\geq
3$, then $k=g/2$ implies that the third $S$-active vertex on $C$ together with
$v_{0}$ and $v_{k}$ forms a geodesic triple of $S$-active vertices on $C,$
which is a contradiction. On the other hand, if $a_{S}(C)=2$, then $k=g/2$
implies that the graph $G$ contains the configuration $\mathcal{A}$ which is
again a contradiction.
Suppose now that $k<g/2$. As we assumed that $S$ is not a vertex metric
generator, there must exist a pair of vertices $x$ and $x^{\prime}$ in $G$
which is not distinguished by $S$. Let us assume that $x\in V(T_{v_{i}})$ and
$x^{\prime}\in V(T_{v_{j}})$. If $i=j$, then $x$ and $x^{\prime}$ would be
distinguished by $S$ according to Lemma \ref{Lemma_SameComponent}. Therefore,
without loss of generality, we may assume $i<j$. Now, we consider the
following five cases regarding $i$ and $j$ in order to conclude the proof.
\medskip\noindent\textbf{Case 1:} $i,j\in\left[ 0,k\right] $. If $x$ and
$x^{\prime}$ are not distinguished by $S\cap V(T_{v_{0}})$, then $i<j$ implies
$d(x,v_{i})>d(x^{\prime},v_{j})$ which further implies $d(x,s)>d(x^{\prime
},s)$ for every $s\in S\cap V(T_{v_{k}})$. Therefore, $S$ distinguishes $x$
and $x^{\prime}$ which is a contradiction.
\medskip\noindent\textbf{Case 2:} $i,j\in\left[ k+1,g-1\right] $. Since we
do not have a configuration $\mathcal{B}$, it must be $\left\lfloor
g/2\right\rfloor \leq i<j\leq\left\lceil g/2\right\rceil +k$. But then notice
the following. If $d(v_{j},x^{\prime})<d(v_{i},x)$ then $v_{0}$ distinguish
these two vertices as $x^{\prime}$ is closer to $v_{0}$ than $x$. And,
similarly if $d(v_{j},x^{\prime})>d(v_{i},x)$ then $v_{k}$ distinguishes these
two vertices as $x$ is closer. So we infer $d(v_{j},x^{\prime})=d(v_{i},x)$.
Then $v_{0}$ does not distinguish $x$ and $x^{\prime}$ only if $g$ is odd and
$v_{i}$ and $v_{j}$ are the antipodals of $v_{0}$. But then $v_{i}$ and
$v_{j}$ cannot be antipodals of $v_{k}$, and so $v_{k}$ distinguishes them.
\medskip\noindent\textbf{Case 3:} $i\in\left[ 1,k-1\right] $ \textit{and}
$j\in\left[ k+1,g-1\right] $. If $j\leq\left\lfloor g/2\right\rfloor ,$ then
the fact that $S\cap V(T_{v_{k}})$ does not distinguish $x$ and $x^{\prime}$
implies $d(x,v_{i})<d(x^{\prime},v_{i}),$ so $x$ and $x^{\prime}$ are
distinguished by $S\cap V(T_{v_{0}}).$ The similar argument holds if
$j\geq\left\lceil g/2\right\rceil +k.$ Therefore, we may assume that
$j\in\lbrack\left\lfloor g/2\right\rfloor +1,\left\lceil g/2\right\rceil
+k-1],$ which implies that $d(v_{j},v_{0})+d(v_{j},v_{k})=g-k.$ Now, the facts
that $S\cap V(T_{v_{0}})$ and $S\cap V(T_{v_{k}})$ do not distinguish $x$ and
$x^{\prime}$ imply
\begin{align*}
d(x,v_{i})+d(v_{i},v_{0}) & =d(x^{\prime},v_{j})+d(v_{j},v_{0})\\
d(x,v_{i})+d(v_{i},v_{k}) & =d(x^{\prime},v_{j})+d(v_{j},v_{k}),
\end{align*}
respectively. Summing these two equalities further implies $d(x,v_{i}%
)-d(x^{\prime},v_{j})=g/2-k$. The fact $k<g/2$ implies $g/2-k>0$, so plugging
this expression in the above equalities we obtain $d(v_{j},v_{0}%
)>d(v_{i},v_{0})$ and $d(v_{j},v_{k})>d(v_{i},v_{k})$. Now, in the case when
$a_{S}(C)\geq3$, there is $l\in\left( 0,k\right) $ such that $v_{l}$ is
$S$-active, but then for $l\in\left( 0,i\right] $ the fact that $S\cap
V(T_{v_{0}})$ does not distinguish $x$ and $x^{\prime}$ implies $S\cap
V(T_{v_{l}})$ distinguishes them which is a contradiction, and for
$l\in\left[ i,k\right) $ when $S\cap V(T_{v_{k}})$ instead of $S\cap
V(T_{v_{0}})$ a similar argument holds. So, we may assume that $a_{S}(C)=2$.
But in this case the fact $d(x,v_{i})-d(x^{\prime},v_{j})=g/2-k>0$ implies
$g/2-k$ is an integer so $g$ must be even and also it implies that there is a
sufficiently long $S$-free thread hanging at $v_{i}$ for $G$ to contain
configuration $\mathcal{C}$ which is a contradiction.
\medskip\noindent\textbf{Case 4:} $i=k$ \textit{and} $j>k$. The fact that
$S\cap V(T_{v_{k}})$ does not distinguish $x$ and $x^{\prime}$ implies
$d(x,v_{k})>d(x^{\prime},v_{j})$, which further implies $x\not =v_{k}$. Notice
that $x$ does not belong to an $S$-free thread hanging at $v_{k}$ as that
would mean $G$ contains configuration $\mathcal{B}$ in all cases except when
$g$ is odd and $k=\left\lfloor g/2\right\rfloor ,$ but in that case
$d(x,v_{k})>d(x^{\prime},v_{j})$ implies $d(x,v_{0})>d(x^{\prime},v_{0}),$ so
$x$ and $x^{\prime}$ are distinguished by $S\cap V(T_{v_{0}}),$ which is a
contradiction. Since $x$ does not belong to an $S$-free thread, we conclude
there must exist a vertex $s\in S\cap V(T_{v_{k}})$ distinct from $v_{k}$ such
that the shortest path $P$ from $x$ to $s$ does not contain $v_{k}.$
Let $v$ be the vertex on the path $P$ which is closest to $v_{k}$. Since $x$
and $x^{\prime}$ are not distinguished by $S\cap V(T_{v_{k}})$, by definition
we have $d(x,s)=d(x^{\prime},s),$ which implies%
\begin{equation}
d(x,v)=d(x^{\prime},v_{k})+d(v_{k},v) \label{Eq1}%
\end{equation}
and hence
\begin{equation}
d(x,v_{k})=d(x^{\prime},v_{k})+2d(v_{k},v). \label{Eq2}%
\end{equation}
The equality (\ref{Eq2}) implies $d(x,v_{k})>d(x^{\prime},v_{k})$. If
$j\leq\left\lfloor g/2\right\rfloor $, then the shortest path from both $x$
and $x^{\prime}$ to $v_{0}$ leads through $v_{k}$, so $d(x,v_{k})>d(x^{\prime
},v_{k})$ would imply that $x$ and $x^{\prime}$ are distinguished by $S\cap
V(T_{v_{0}})$, a contradiction.
Suppose therefore that $j>\left\lfloor g/2\right\rfloor $. In this case a
shortest path from $x^{\prime}$ to $v_{0}$ does not lead through $v_{k},$ so
we have $d(x^{\prime},v_{0})=d(x^{\prime},v_{j})+g-j$. Also, equality
(\ref{Eq1}) implies
\begin{align*}
d(x,v_{0}) & =d(x,v)+d(v,v_{k})+k\\
& =d(x^{\prime},v_{k})+2d(v,v_{k})+k\\
& =d(x^{\prime},v_{j})+j+2d(v,v_{k}).
\end{align*}
Subtracting these expressions for $d(x,v_{0})$ and $d(x^{\prime},v_{0})$ we
obtain%
\[
d(x^{\prime},v_{0})-d(x,v_{0})=g-2j-2d(v,v_{k}),
\]
where the fact that $d(v_{k},v)\not =0$ further implies $d(x^{\prime}%
,v_{0})-d(x,v_{0})\leq g-2j-2$. Note that $j>\left\lfloor g/2\right\rfloor $
implies $g-2j-2<0$, so we conclude that $S\cap V(T_{v_{0}})$ distinguishes $x$
and $x^{\prime}$ which is a contradiction.
\medskip\noindent\textbf{Case 5:} $i=0$ \textit{and} $j>k$. This case is
analogous to Case 4.
\bigskip By the above analysis, we have shown that any pair of vertices from
$G$ is distinguished by $S,$ so $S$ is a vertex metric generator, which
concludes the proof.
\end{proof}
The last two lemmas give us the necessary and sufficient condition for a
biactive branch-resolving set of vertices $S$ to be a vertex metric generator.
In order to establish the exact value of the vertex metric dimension for a
unicyclic graph $G$ we have to find a smallest set $S$ which meets the
condition from Lemmas \ref{Lm_vertexGeneratorNec} and
\ref{Lm_vertexGeneratorSuf}.
Notice that a branch-resolving set $S$ activates all branch-active vertices in
$G,$ so if $b(G)\geq2$ then every branch-resolving set is biactive. Therefore,
for a smallest branch-resolving set $S$ in a unicyclic graph with $b(G)\geq2$,
the set of $S$-active vertices is fixed by the structure of $G$ and coincides
with the set of branch-active vertices. Since the presence of configurations
$\mathcal{A}$, $\mathcal{B}$, and $\mathcal{C}$ by definition depends on the
position of $S$-active vertices, we can observe the following.
\begin{remark}
\label{Obs_b2_notdependS}If a unicyclic graph $G$ contains at least two
branch-active vertices on $C$, i.e. $b(G)\geq2$, then the graph $G$ contains
configuration $\mathcal{A}$, $\mathcal{B}$, or $\mathcal{C}$ either with
respect to all smallest biactive branch-resolving sets or to none of them.
\end{remark}
The above observation allows us to omit the set $S$ in the definition of
containment of configurations $\mathcal{A}$, $\mathcal{B}$, and $\mathcal{C}$
in a unicyclic graph $G$. We can simply say "$G$ contains a configuration"
instead of "$G$ contains a configuration with respect to $S$". Now, for
unicyclic graphs with at least two branch-active vertices we can state and
prove the following theorem which gives the exact value of the vertex metric dimension.
\begin{theorem}
\label{Tm_vDim_bc2}Let $G$ be a unicyclic graph with at least two
branch-active vertices, i.e. $b(G)\geq2$. Then%
\[
\mathrm{dim}(G)=L(G)+\Delta,
\]
where $\Delta=0$ if the graph $G$ does not contain any of the configurations
$\mathcal{A}$, $\mathcal{B}$, $\mathcal{C}$, and $\Delta=1$ otherwise.
\end{theorem}
\begin{proof}
Let $S$ be a smallest branch-resolving set in $G.$ Since $G$ has at least two
branch-active vertices on $C$, i.e. $b(G)\geq2,$ it follows that $S$ is
biactive, and so $a_{S}(C)=b(G).$ If the graph $G$ does not contain any of the
configurations $\mathcal{A}$, $\mathcal{B}$, and $\mathcal{C}$, then Lemma
\ref{Lm_vertexGeneratorSuf} implies that $S$ is a vertex metric generator.
Therefore, $\mathrm{dim}(G)=\left\vert S\right\vert =L(G).$
On the other hand, if $G$ contains any of the configurations $\mathcal{A}$,
$\mathcal{B}$, or $\mathcal{C}$, then $S$ is not a vertex metric generator
according to Lemma \ref{Lm_vertexGeneratorNec}. Let $v$ be a vertex from $C$
which forms a geodesic triple with two branch-active vertices on $C,$ and let
$S^{\prime}=S\cup\{v\}.$ Notice that according to Lemma \ref{Lemma_GeodTrip},
the set $S^{\prime}$ is a vertex metric generator, therefore $\mathrm{dim}%
(G)=\left\vert S^{\prime}\right\vert =L(G)+1.$
\end{proof}
\begin{figure}[h]
\begin{center}
$%
\begin{array}
[c]{cccc}%
\text{a)} &
\text{\raisebox{-1\height}{\includegraphics[scale=0.5]{Figure18.pdf}}} &
\text{b)} &
\text{\raisebox{-1\height}{\includegraphics[scale=0.5]{Figure17.pdf}}}\\
\text{c)} &
\text{\raisebox{-1\height}{\includegraphics[scale=0.5]{Figure19.pdf}}} &
\text{d)} &
\text{\raisebox{-1\height}{\includegraphics[scale=0.5]{Figure20.pdf}}}%
\end{array}
$
\end{center}
\caption{In all four examples we consider the same graph~$G$ without
branch-active vertices on $C$ and four different smallest biactive
branch-resolving sets $S=\{v_{0},v_{k}\}$ chosen so that the graph $G$ with
respect to $S$ contains: a) configuration $\mathcal{A}$, b) configuration
$\mathcal{B}$, c) configuration $\mathcal{C}$, d) none of the configurations
$\mathcal{A}$, $\mathcal{B}$, and $\mathcal{C}$. In the first three examples
the set $S$ is not a vertex metric generator, so a pair of vertices
non-distinguished by $S$ is marked in $G.$}%
\label{Fig_b01}%
\end{figure}
So far we have determined the exact value of unicyclic graphs $G$ with
$b(G)\geq2$ and our characterization depends on the presence of configurations
$\mathcal{A}$, $\mathcal{B}$, and $\mathcal{C}$ in the graph $G.$ Now we want
to deal with unicyclic graphs $G$ with $b(G)<2.$ The problem with such graphs
is that different biactive sets $S$ impose different $S$-active vertices on
$C,$ and we derive presence of different configurations (see Figure
\ref{Fig_b01}).
We conclude that we cannot speak of presence/absence of a particular
configuration in a graph, unless we have a biactive branch-resolving set $S$.
So, we introduce the following definition to unify both cases $b(G)\geq2$ and
$b(G)<2.$
\begin{definition}
Let $G$ be a unicyclic graph. We say that $G$ is $\mathcal{ABC}$%
\emph{-negative}, if there exists a smallest biactive branch-resolving set $S$
such that $G$ does not contain any of the configurations $\mathcal{A},$
$\mathcal{B},$ and $\mathcal{C}$ with respect to $S.$ Otherwise we say that
$G$ is $\mathcal{ABC}$\emph{-positive}.
\end{definition}
Notice that the above definition extends the case when $G$ contains two or
more branch-active vertices, i.e. $b(G)\geq2$, in which case $G$ will be
$\mathcal{ABC}$-negative if it does not contain any of the configurations
$\mathcal{A},$ $\mathcal{B},$ $\mathcal{C}$, and $G$ is $\mathcal{ABC}%
$-positive if it contains at least one of the configurations $\mathcal{A},$
$\mathcal{B},$ $\mathcal{C}.$
As an example of $\mathcal{ABC}$-negative and $\mathcal{ABC}$-positive graphs
$G$ with $b(G)<2$, we can consider corona product graphs $C_{n}\odot K_{1}$
which are obtained from the cycle $C_{n}$ by appending a leaf to every vertex
in $C_{n}.$ We leave to the reader to verify that if $n$ is odd, then corona
product $C_{n}\odot K_{1}$ is $\mathcal{ABC}$-negative, and for even $n\geq6$
it is $\mathcal{ABC}$-positive (see Figure \ref{Fig_corona}). Corona product
graphs $C_{n}\odot K_{1}$ are examples of unicyclic graphs with $b(G)=0,$ but
if we replace one leaf in $C_{n}\odot K_{1}$ by any acyclic structure, we
obtain a graph with $b(G)=1$ and the same reasoning holds.
\begin{figure}[h]
\begin{center}
$%
\begin{array}
[c]{cccc}%
\text{a)} &
\text{\raisebox{-1\height}{\includegraphics[scale=0.5]{Figure22.pdf}}} &
\text{b)} &
\text{\raisebox{-1\height}{\includegraphics[scale=0.5]{Figure23.pdf}}}%
\end{array}
$
\end{center}
\caption{Figure shows corona product graphs: a) $C_{9}\odot K_{1}$ which is
$\mathcal{ABC}$-negative and $S=\{s_{1},s_{2}\}$ is a vertex metric generator,
b) $C_{10}\odot K_{1}$ which is $\mathcal{ABC}$-positive with $S=\{s_{1}%
,s_{2}\}$ being such that it avoids configurations $\mathcal{A}$ and
$\mathcal{B}$, but does not avoid configuration $\mathcal{C}.$}%
\label{Fig_corona}%
\end{figure}
Now, we can state a more general version of Theorem \ref{Tm_vDim_bc2} which
encapsulates unicyclic graphs with less than two branch-active vertices.
\begin{theorem}
\label{Tm_dim}Let $G$ be a unicyclic graph. Then%
\[
\mathrm{dim}(G)=L(G)+\max\{0,2-b(G)\}+\Delta,
\]
where $\Delta=0$ if the graph $G$ is $\mathcal{ABC}$-negative, and $\Delta=1$
if $G$ is $\mathcal{ABC}$-positive.
\end{theorem}
\begin{proof}
Assume first that $G$ is $\mathcal{ABC}$-negative. This implies that there is
a smallest biactive branch-resolving set $S$ such that $G$ does not contain
any of the configurations $\mathcal{A},$ $\mathcal{B},$ and $\mathcal{C}$ with
respect to $S.$ Then, Lemma \ref{Lm_vertexGeneratorSuf} implies that $S$ is a
vertex metric generator, so $\mathrm{dim}(G)=\left\vert S\right\vert
=L(G)+\max\{0,2-b(G)\}.$
Assume now that $G$ is $\mathcal{ABC}$-positive and let $S$ be a smallest
biactive branch-resolving set in $G.$ Definition of $\mathcal{ABC}$-positivity
implies that $G$ contains at least one of the configurations $\mathcal{A}$,
$\mathcal{B}$, or $\mathcal{C}$ with respect to $S,$ so $S$ is not a vertex
metric generator according to Lemma \ref{Lm_edgeGeneratorNec}. But then, let
$v$ be a vertex from $C$ which forms a geodesic triple with two $S$-active
vertices on $C,$ and let $S^{\prime}=S\cup\{v\}.$ Now, Lemma
\ref{Lemma_GeodTrip} implies that $S^{\prime}$ is a metric generator, so
$\mathrm{dim}(G)=\left\vert S^{\prime}\right\vert =L(G)+\max\{0,2-b(G)\}+1.$
\end{proof}
\section{Edge metric dimension}
Now we want to apply a similar study for the edge metric dimension of
unicyclic graphs. Similarly as for the vertex metric dimension, Lemma
\ref{Lemma_a(S)vj2} implies that a set $S\subseteq V(G)$ which is not a
branch-resolving set or for which $a_{S}(C)<2$ cannot be an edge metric
generator, and Lemma \ref{Lemma_GeodTrip} implies that a branch-resolving set
$S$ with a geodesic triple of $S$-active vertices on the cycle $C$ certainly
is an edge metric generator. Therefore, it remains to consider
branch-resolving sets $S$ with $a_{S}(C)\geq2,$ but without a geodesic triple
of $S$-active vertices. Such a set may or may not be an edge metric generator,
and we want to establish necessary and sufficient conditions for it to be an
edge metric generator.
Let us consider the graphs from Figure \ref{Figure_YesProblem}.\ Notice that
in the graph $G$ from a), which contains configuration $\mathcal{A}$, there is
a pair of edges incident to $v_{0}$ which is not distinguished by $S$.
Similarly holds for graphs in b) and c) which contain configuration
$\mathcal{B}.$ Moreover, in the example c) where the cycle is odd there will
be a pair of undistinguished edges even if the $S$-free thread is hanging at
$v_{i}$ for $i=\left\lceil g/2\right\rceil -1$ or $i=\left\lfloor
g/2\right\rfloor +k+1.$ So, in the case of edge metric dimension configuration
$\mathcal{B}$ will have to be extended to a new configuration $\mathcal{D}.$
Finally, in the graph $G$ from d) there is no pair of undistinguished edges,
so configuration $\mathcal{C}$ is not an obstacle for $S$ to be an edge metric
generator, but there is a counterpart configuration for the edge metric
dimension which will be configuration $\mathcal{E}$.
\begin{definition}
Let $G$ be a unicyclic graph and let $S$ be a biactive branch-resolving set
$S$ in $G$. We say that the graph $G$ with respect to $S$ \emph{contains} configuration:
\begin{description}
\item $\mathcal{D}$. If $k\leq\left\lceil g/2\right\rceil -1$ and there is an
$S$-free thread hanging at a vertex $v_{i}$ for some $i\in\lbrack
k,\left\lceil g/2\right\rceil -1]\cup\lbrack\left\lfloor g/2\right\rfloor
+k+1,g-1]\cup\{0\}$;
\item {$\mathcal{E}$}. If $a_{S}(C)=2$ and there is an $S$-free thread of the
length $\geq\left\lfloor g/2\right\rfloor -k+1$ hanging at a vertex $v_{i}$
with $i\in\lbrack0,k].$ Moreover, if $g$ is even, an $S$-free thread must be
hanging at the vertex $v_{j}$ with $j=g/2+k-i$.
\end{description}
\end{definition}
Notice that if $G$ contains configuration $\mathcal{B},$ then it certainly
contains configuration $\mathcal{D}$, but the oposite does not hold. As for
configuration $\mathcal{E}$, notice that for odd $g$ we encounter
configuration $\mathcal{E}$ just with a thread hanging at $v_{i}$ (as $j$ is
not integer anyway). Configuration $\mathcal{E}$ is illustrated by Figure
\ref{Figure_YesProblemEdge}, where a pair of undistinguished edges is marked
and we will show that the same holds generally, i.e. that configuration
$\mathcal{E}$ is an obstacle for set $S$ to be an edge metric generator.
\begin{figure}[h]
\begin{center}
$%
\begin{array}
[c]{cccc}%
\text{a)} &
\text{\raisebox{-1\height}{\includegraphics[scale=0.5]{Figure09.pdf}}} &
\text{b)} &
\text{\raisebox{-1\height}{\includegraphics[scale=0.5]{Figure10.pdf}}}%
\end{array}
$
\end{center}
\caption{In both examples we consider a branch-resolving set $S,$ where
$v_{0}$ and $v_{k}$ are the only two $S$-active vertices on $C.$ Configuration
$\mathcal{E}$ with respect to $S$ is shown: a) for an even cycle, b) for an
odd cycle. In both graphs a pair of edges is marked which is not distinguished
by $S.$}%
\label{Figure_YesProblemEdge}%
\end{figure}
\begin{lemma}
\label{Lm_edgeGeneratorNec}Let $G$ be a unicyclic graph and let $S$ be a
biactive branch-resolving set in $G$. If the graph $G$ contains configuration
$\mathcal{A}$, $\mathcal{D}$, or $\mathcal{E}$ with respect to $S,$ then the
set $S$ is not an edge metric generator in $G$.
\end{lemma}
\begin{proof}
Let us assume that $G$ contains configuration $\mathcal{A}$, $\mathcal{D}$, or
$\mathcal{E}$ with respect to $S$, and it is sufficient to find a pair of
edges $x,x^{\prime}\in E(G)$ which are not distinguished by $S$.
If $G$ contains configuration $\mathcal{A}$, then edges $v_{0}v_{1}$ and
$v_{0}v_{g-1}$ are not distinguished by $S$.
Next, if $G$ contains configuration $\mathcal{D}$, let $v_{i}$ be the
"problematic" vertex on $C$ with an $S$-free thread hanging at it and let $w$
be the neighbour of $v_{i}$ on that thread. Then either pair of edges $wv_{i}$
and $v_{i}v_{i+1}$ or the pair $wv_{i}$ and $v_{i}v_{i-1}$ are not
distinguished by $S$.
Finally, assume that $G$ contains configuration $\mathcal{E}$. By definition
this implies that $a_{S}(C)=2$ and there is an $S$-free thread hanging at
$v_{i}$ for $i\in\lbrack0,k]$ of the length $\geq\left\lfloor g/2\right\rfloor
-k+1$ and also an $S$-free thread hanging at $v_{j}$ for $j=g/2+k-i$ (if $g$
is even). Let $e$ be an edge in $T_{v_{i}}$ such that $d(e,v_{i})=\left\lfloor
g/2\right\rfloor -k$, note that such an edge must exist due to the fact that
the thread attached to $v_{i}$ is of length $\geq\left\lfloor g/2\right\rfloor
-k+1$. If $g$ is even, then $v_{j}$ has an $S$-free thread attached, so let
$e^{\prime}$ be the first edge on that thread (i.e. $e^{\prime}$ is incident
to $v_{j}$). Then $e$ and $e^{\prime}$ are not distinguished by $S$. If $g$ is
odd, let $e^{\prime}=v_{\left\lfloor j\right\rfloor }v_{\left\lfloor
j\right\rfloor +1}$. Then again $e$ and $e^{\prime}$ are not distinguished by
$S$.
\end{proof}
So far we have shown that configurations $\mathcal{A}$, $\mathcal{D}$, and
$\mathcal{E}$ are indeed the obstacle for $S$ to be an edge metric generator.
Next, we show that these are the only obstacles.
\begin{lemma}
\label{Lm_edgeGeneratorSuf}Let $G$ be a unicyclic graph, and let $S$ be a
biactive branch-resolving set in $G$. If the graph $G$ does not contain any of
the configurations $\mathcal{A}$, $\mathcal{D}$, and $\mathcal{E}$ with
respect to $S,$ then the set $S$ is an edge metric generator in $G$.
\end{lemma}
\begin{proof}
Suppose that the graph $G$ does not contain any of the configurations
$\mathcal{A}$, $\mathcal{D},$ and $\mathcal{E}$ with respect to $S.$ Let us
suppose the contrary to the claim, i.e. that $S$ is not an edge metric
generator. Let $e=xy$ and $e^{\prime}=x^{\prime}y^{\prime}$ be two edges in
$G$ which are not distinguished by $S$. If $e$ (resp. $e^{\prime}$) is not an
edge of $C,$ then assume that $x$ (resp. $x^{\prime}$) is closer to $C$ than
$y$ (resp. $y^{\prime}$). Also, let $G_{1}=G/\{e,e^{\prime}\}$ and let the
vertex of $G_{1}$ obtained by contracting $e$ (resp. $e^{\prime}$) be denoted
by $x$ (resp. $x^{\prime}$). Denote by $d_{1}(u,v)$ the distance of vertices
$u$ and $v$ in $G_{1}$. The length of the cycle in $G_{1}$ will be denoted by
$g_{1}$. Now we consider the following three cases.
\medskip\noindent\textbf{Case 1:} $e,e^{\prime}\in E(C)$. Edges $e$ and
$e^{\prime}$ are not distinguished by $S$ only if $g$ is even and $a_{S}%
(C)=2$, where the only two $S$-active vertices are an antipodal pair, which
implies $k=g/2$. Hence, we infer that $G$ contains configuration $\mathcal{A}$
which is a contradiction.
\medskip\noindent\textbf{Case 2:} $e,e^{\prime}\not \in E(C)$. Let $e\in
E(T_{v_{i}})$ and $e^{\prime}\in E(T_{v_{j}})$. Lemma
\ref{Lemma_SameComponent} then implies $i\not =j$, where without loss of
generality we may assume $i<j$. Let $G_{y},G_{y^{\prime}}$ and $G_{x}$ be the
connected components of $G-\{e,e^{\prime}\}$ that contains vertices $y$,
$y^{\prime}$ and $x$ respectively. If there is a vertex $s\in S\cap
V(G_{y}\cup G_{y^{\prime}})$, then obviously $s$ distinguishes $e$ and
$e^{\prime}$. So, let us assume $S\subseteq V(G_{x})$ which implies
$d(s,e)=d(s,x)=d_{1}(s,x)$ and $d(s,e^{\prime})=d(s,x^{\prime})=d_{1}%
(s,x^{\prime})$. Hence, $e$ and $e^{\prime}$ are distinguished by $S$ in $G$
if and only if $x$ and $x^{\prime}$ are distinguished by $S$ in $G_{1}$.
As we assumed that $e$ and $e^{\prime}$ are not distinguished by $S$ in $G,$
Lemma \ref{Lm_vertexGeneratorSuf} implies that $G_{1}$ contains configurations
$\mathcal{A},$ $\mathcal{B}$ or $\mathcal{C}$. If $G_{1}$ contains
configuration $\mathcal{A}$ (resp. $\mathcal{B}$), then $G$ obviously contains
configuration $\mathcal{A}$ (resp. $\mathcal{D}$), which is contradiction. On
the other hand, if $x$ and $x^{\prime}$ are not distinguished by $S$ in
$G_{1}$ due to configuration $\mathcal{C},$ then $g=g_{1}$ is even and
$a_{S}(C)=2$. Also, from $i<j$ we infer that $x$ belongs to an $S$-free thread
hanging at a vertex $v_{i}$ for some $i\in\lbrack0,k]$ and $d(x,v_{i})\geq
g/2-k.$ Moreover, since $x$ and $x^{\prime}$ are not distinguished by $S$,
then $x^{\prime}$ must belong to $T_{v_{j}}$ for $j=g/2+k-i.$ Given the fact
that in $G$ there is an edge $e$ and $e^{\prime}$ attached to vertices $x$ and
$x^{\prime}$ respectively, this implies that $G$ contains configuration
$\mathcal{E}$, a contradiction.
\medskip\noindent\textbf{Case 3:} $e\in E(C)$ \textit{and} $e^{\prime
}\not \in E(C)$. Suppose $e=e_{i}=v_{i}v_{i+1}$ and $e^{\prime}\in E(T_{v_{j}%
})$. Let $G_{x^{\prime}}$ and $G_{y^{\prime}}$ be the connected components of
$G-e^{\prime}$ containing vertices $x^{\prime}$ and $y^{\prime}$ respectively.
If there is a vertex $s\in S\cap V(G_{y^{\prime}})$, then $e$ and $e^{\prime}$
would be distinguished by $s$. Therefore assume $S\subseteq V(G_{x^{\prime}}%
)$. If there is a vertex $s\in S$ such that the shortest path connecting
vertices $x^{\prime}$ and $s$ contains $e$, then $e$ and $e^{\prime}$ would
obviously be distinguished by $S$. Therefore, assume that no path from
$x^{\prime}$ to vertices from $S$ contains the edge $e$.
If $e$ and $e^{\prime}$ are incident, say $x^{\prime}=v_{i},$ then $e$ and
$e^{\prime}$ are not distinguished by $S$ only if $i\in\lbrack k,\left\lceil
g/2\right\rceil -1].$ Since $e^{\prime}\not \in E(C),$ this implies that $G$
contains configuration $\mathcal{D}$. So, let us assume that $e$ and
$e^{\prime}$ are not incident which implies $x\not =x^{\prime}$ in $G_{1}.$
Since no path from $x^{\prime}$ to vertices from $S$ contains the edge $e,$ we
conclude that $d(e,s)=d_{1}(x,s)$ and $d(e^{\prime},s)=d_{1}(x^{\prime},s)$
for every $s\in S$. As we assumed that $e$ and $e^{\prime}$ are not
distinguished by $S$ in $G,$ it follows that $x$ and $x^{\prime}$ are not
distinguished by $S$ in $G_{1},$ so according to Lemma
\ref{Lm_vertexGeneratorSuf} the graph $G_{1}$ contains configuration
$\mathcal{A},$ $\mathcal{B},$ or $\mathcal{C}.$ Notice that in this case
$g_{1}=g-1$.
Similarly as in the previous case, if $G_{1}$ contains configuration
$\mathcal{B}$, then $G$ contains configuration $\mathcal{D}$, which is
contradiction. If $G_{1}$ contains configuration $\mathcal{A},$ then $g_{1}$
is even, so $g$ is odd and $k=\left\lfloor g/2\right\rfloor $. Also, as $e\in
E(C)$ and $e^{\prime}\not \in E(C)$, it must hold $i\in\lbrack k,g-1]$ and
$j\in\lbrack0,k].$ Since $x^{\prime}\in T_{v_{j}}$ has an edge $e^{\prime
}\not \in E(C)$ attached to it, this implies there is an $S$-free thread
hanging at $v_{j}$ of the length $\geq1=\left\lfloor g/2\right\rfloor -k+1.$
Therefore, $G$ contains configuration $\mathcal{E}$ on odd cycle, a contradiction.
Finally, let us assume $x$ and $x^{\prime}$ are not distinguished by $S$ in
$G_{1}$ due to configuration $\mathcal{C}.$ This implies $g_{1}$ is even and
$x^{\prime}$ belongs to an $S$-free thread hanging at $v_{j}$ for $j\in
\lbrack0,k]$ and $d(x^{\prime},v_{j})\geq g/2-k.$ Since there is an edge
$e^{\prime}$ hanging at $x^{\prime}$ in $G,$ this implies $G$ contains
configuration $\mathcal{E}$ on odd cycle.
Altogether, we conclude that any two distinct edges are distinguished by $S,$
which implies that $S$ is an edge metric generator.
\end{proof}
In the previous two lemmas we have established the necessary and sufficient
condition for a set of vertices to be an edge metric generator, so now we can
proceed with determining the exact value of the edge metric dimension of a
unicyclic graph $G.$ Notice that configuration $\mathcal{D}$ and $\mathcal{E}%
$, similarly as configurations $\mathcal{A}$, $\mathcal{B}$, and $\mathcal{C}%
$, depend only on the position of $S$-active vertices on $C.$ Therefore,
Observation \ref{Obs_b2_notdependS} holds also for configurations
$\mathcal{D}$ and $\mathcal{E}$ and we can say that unicyclic graphs with
$b(G)\geq2$ contain configuration $\mathcal{D}$ or $\mathcal{E}$ without
explicitely stating the set $S$. So, for unicyclic graphs with $b(G)\geq2$ we
can state the theorem which gives the edge metric dimension as follows.
\begin{theorem}
\label{Tm_eDim_bc2}Let $G$ be a unicyclic graph with at least two
branch-active vertices. Then%
\[
\mathrm{dim}(G)=L(G)+\Delta_{e},
\]
where $\Delta_{e}=0$ if the graph $G$ does not contain any of the
configurations $\mathcal{A}$, $\mathcal{D}$, or $\mathcal{E},$ and $\Delta
_{e}=1$ otherwise.
\end{theorem}
\begin{proof}
The proof is analogous to the proof of Theorem \ref{Tm_vDim_bc2}, using
configurations $\mathcal{D}$ and $\mathcal{E}$ instead of configurations
$\mathcal{B}$ and $\mathcal{C}$ respectively. Also, Lemmas
\ref{Lm_edgeGeneratorNec} and \ref{Lm_edgeGeneratorSuf} need to be applied
instead of Lemmas \ref{Lm_vertexGeneratorNec} and \ref{Lm_vertexGeneratorSuf}, respectively.
\end{proof}
Finally, if a unicyclic graph $G$ contains less than two branch-active
vertices on $C,$ then a smallest branch-resolving set $S$ is not biactive and
needs to be introduced $2-b(G)$ vertices to become biactive, the consequence
of which is that different smallest \emph{biactive} branch-resolving sets $S$
may have different set of $S$-active vertices on $C.$ Since the presence of
configurations $\mathcal{D}$ and $\mathcal{E}$ depends on the position of
$S$-active vertices on $C,$ this implies that a unicyclic graph $G$ with
$b(G)<2$ does contain configuration $\mathcal{D}$ or $\mathcal{E}$ with
respect to one smallest biactive branch-resolving set, but not with respect to
another. Since we need a smallest biactive branch-resolving set such that $G$
does not contain any of the configurations $\mathcal{A}$, $\mathcal{D}$, and
$\mathcal{E}$ with respect to $S,$ we introduce the following definition.
\begin{definition}
We say that a unicyclic graph $G$ is $\mathcal{ADE}$\emph{-negative}, if there
is a smallest biactive branch-resolving set $S$ such that $G$ does not contain
any of the configurations $\mathcal{A}$, $\mathcal{D}$, and $\mathcal{E}$ with
respect to $S.$ Otherwise, we say that $G$ is $\mathcal{ADE}$\emph{-positive}.
\end{definition}
Again, a unicyclic graph $G$ with $b(G)\geq2$ is $\mathcal{ADE}$-negative if
it does not contain any of the configurations $\mathcal{A}$, $\mathcal{D}$,
and $\mathcal{E}$, otherwise it is $\mathcal{ADE}$-positive. In case of
unicyclic graphs with $b(G)<2$, we can again consider corona product graphs
$C_{n}\odot K_{1}$ as an example (see Figure \ref{Fig_corona}). Notice that in
this case it is opposite to the situation with vertex metric dimension, now a
corona graph $C_{n}\odot K_{1}$ with odd $n\geq7$ is $\mathcal{ADE}$-positive
as the set $S=\{s_{1},s_{2}\}$ shown in Figure \ref{Fig_corona}.a) does not
avoid configuration $\mathcal{E}$. On the other hand, a graph $C_{n}\odot
K_{1}$ with even $n$ is $\mathcal{ADE}$-negative (the set $S=\{s_{1},s_{2}\}$
shown in Figure \ref{Fig_corona}.b) avoids all three configurations and is
therefore an edge metric generator).
\begin{theorem}
\label{Tm_edim}Let $G$ be a unicyclic graph. Then%
\[
\mathrm{edim}(G)=L(G)+\max\{0,2-b(G)\}+\Delta_{e}%
\]
where $\Delta_{e}=0$ if the graph $G$ is $\mathcal{ADE}$-negative, and
$\Delta_{e}=1$ if $G$ is $\mathcal{ADE}$-positive.
\end{theorem}
\begin{proof}
Analogous to the proof of Theorem \ref{Tm_dim}.
\end{proof}
\section{Difference between metric dimensions}
Now, we can use the results proven in the previous two sections, and so we
answer a proposal from \cite{SedSkreBounds}. Namely, in \cite{SedSkreBounds}
it was shown that for a unicyclic graph $G$ it holds that $\left\vert
\mathrm{dim}(G)-\mathrm{edim}(G)\right\vert \leq1$ and the problem of
determining whether the difference $\mathrm{dim}(G)-\mathrm{edim}(G)$ is $-1$,
$0$ and $1$ was posed as a natural question. This can be easily answered for
unicyclic graphs $G$ with $b(G)\geq2$ using Theorems \ref{Tm_vDim_bc2} and
\ref{Tm_eDim_bc2}.
\begin{theorem}
\label{Tm_difference_bc2}Let $G$ be a unicyclic graph with $b(G)\geq2$. It
holds that%
\[
\mathrm{dim}(G)-\mathrm{edim}(G)=\left\{
\begin{array}
[c]{rl}%
1 & \text{if }G\text{ contains configuration }\mathcal{C},\text{ but none of
}\mathcal{A}\text{, }\mathcal{D}\text{, and }\mathcal{E}\text{,}\\
-1 & \text{if }G\text{ contains configuration }\mathcal{D}\text{ or
}\mathcal{E},\text{ but none of }\mathcal{A}\text{, }\mathcal{B}\text{, and
}\mathcal{C}\text{,}\\
0 & \text{otherwise.}%
\end{array}
\right.
\]
\end{theorem}
\begin{proof}
According to Theorems \ref{Tm_dim} and \ref{Tm_edim} both $\mathrm{dim}(G)$
and $\mathrm{edim}(G)$ take their value from
\[
L(G)+\max\{2-b(G),0\}\quad\hbox{ and }\quad L(G)+\max\{2-b(G),0\}+1.
\]
Moreover, $\mathrm{dim}(G)$ (resp. $\mathrm{edim}(G)$) will take the greater
value of the two, if $G$ contains configuration $\mathcal{A}$, $\mathcal{B}$,
or $\mathcal{C}$ (resp. configuration $\mathcal{A}$, $\mathcal{D}$, or
$\mathcal{E}$). The observation that $G$ cannot contain configuration
$\mathcal{B}$ without containing configuration $\mathcal{D}$ concludes the proof.
\end{proof}
A more general version of Theorem \ref{Tm_difference_bc2}, which encapsulates
also unicyclic graphs $G$ with $b(G)<2,$ can be established using Theorems
\ref{Tm_dim} and \ref{Tm_edim}.
\begin{theorem}
\label{Tm_difference}Let $G$ be a unicyclic graph. It holds that%
\[
\mathrm{dim}(G)-\mathrm{edim}(G)=\left\{
\begin{array}
[c]{rl}%
1 & \text{if }G\text{ is }\mathcal{ABC}\text{-positive and }\mathcal{ADE}%
\text{-negative,}\\
-1 & \text{if }G\text{ is }\mathcal{ABC}\text{-negative and }\mathcal{ADE}%
\text{-positive,}\\
0 & \text{otherwise.}%
\end{array}
\right.
\]
\end{theorem}
\begin{proof}
It goes similarly as the proof of Theorem \ref{Tm_difference_bc2}.
\end{proof}
An example of graphs $G$ with $b(G)<2$ which are at the same time
$\mathcal{ABC}$-positive and $\mathcal{ADE}$-negative, is the family of corona
graphs $C_{n}\odot K_{1}$ with $n\geq6$ even (see Figure \ref{Fig_corona}).
Therefore, for such graphs we have $\mathrm{dim}(C_{n}\odot K_{1}%
)-\mathrm{edim}(C_{n}\odot K_{1})=3-2=1.$ On the other hand, if $n\geq7$ is
odd, then the corona graphs $C_{n}\odot K_{1}$ can serve as an example of
unicyclic graphs $G$ with $b(G)<2$ which are $\mathcal{ABC}$-negative and
$\mathcal{ADE}$-positive. For them we derive $\mathrm{dim}(C_{n}\odot
K_{1})-\mathrm{edim}(C_{n}\odot K_{1})=2-3=-1$.
It is of interest to determine the classes of graphs for which $\mathrm{dim}%
(G)$ is smaller, equal or bigger from $\mathrm{edim}(G)$. Several such
families of graphs were presented in \cite{TratnikEdge}. We can now make the
same distinction in the class of unicyclic graphs.
\begin{corollary}
\label{Kor_evenOdd}Let $G$ be a unicyclic graph with its cycle $C$. If $C$ is
odd then $\mathrm{dim}(G)\leq\mathrm{edim}(G)$ and the inequality is strict if
$G$ is $\mathcal{ABC}$-negative and $\mathcal{ADE}$-positive. If $C$ is even
then $\mathrm{dim}(G)\geq\mathrm{edim}(G)$ and the inequality is strict if $G$
is $\mathcal{ABC}$-positive and $\mathcal{ADE}$-negative.
\end{corollary}
\begin{proof}
Let $C$ be the cycle in $G$ and assume first that $C$ is of odd length.
According to Theorem \ref{Tm_difference}, the inequality $\mathrm{dim}%
(G)\leq\mathrm{edim}(G)$ will hold if every $G$ which is $\mathcal{ABC}%
$-positive is also $\mathcal{ADE}$-positive. But that is the obvious
consequence of the fact that configuration $\mathcal{B}$ is also configuration
$\mathcal{D},$ i.e. graph $G$ which contains configuration $\mathcal{B}$
certainly contains configuration $\mathcal{D}$ with respect to the same set
$S,$ and the fact that graph on odd cycle cannot contain configuration
$\mathcal{C}$.
Assume now that $C$ is of even length. Again, Theorem \ref{Tm_difference}
implies that the inequality $\mathrm{dim}(G)\geq\mathrm{edim}(G)$ holds if
every $G$ which is $\mathcal{ABC}$-negative is also $\mathcal{ADE}$-negative.
Since configuration $\mathcal{B}$ is also $\mathcal{D}$ and on even cycle
every graph which contains $\mathcal{E}$ also contains $\mathcal{C},$ the
claim follows.
The claim on strictness is the direct consequence of Theorem
\ref{Tm_difference}.
\end{proof}
Regarding Corollary \ref{Kor_evenOdd}, let us mention that in
\cite{KelHypercubes}, it was shown that for bipartite graphs $\mathrm{dim}%
(G)\geq\mathrm{edim}(G).$ Our result extends to all unicylic graphs and
characterizes when the equality holds.
\section{Concluding remarks}
By our previous work, both the vertex and the edge metric dimensions of a
unicyclic graph $G$ takes their value in
\[
L(G)+\max\{2-b(G),0\}\quad\hbox{ and }\quad L(G)+\max\{2-b(G),0\}+1.
\]
In this paper, we first characterize for unicyclic graphs $G$ with $b(G)\geq2$
when the above two values are encountered depending on the presence of
configurations $\mathcal{A}$, $\mathcal{B}$, $\mathcal{C}$, $\mathcal{D}$, and
$\mathcal{E}$. Afterwards, the approach was extended to unicyclic graphs $G$
with $b(G)<2$ by extending the concept of containment of configuration to
$\mathcal{X}$-positivity and $\mathcal{X}$-negativity for $\mathcal{X}%
\in\{\mathcal{ABC},\mathcal{ADE}\}$.
One may try to characterize unicyclic graphs $G$ with $b(G)<2$ and that are
$\mathcal{X}$-positive and $\mathcal{Y}$-negative for distinct $\mathcal{X}%
,\mathcal{Y}\in\{\mathcal{ABC},\mathcal{ADE}\}$, as this may need different
approaches and this paper is already lengthy, we decided not to conduct this
direction of research here. Here we state it explicitly as a problem.
\begin{problem}
Characterize which unicyclic graphs $G$ with $b(G)<2$ are $\mathcal{X}%
$-positive and $\mathcal{Y}$-negative for distinct $\mathcal{X},\mathcal{Y}%
\in\{\mathcal{ABC},\mathcal{ADE}\}$.
\end{problem}
\bigskip
\bigskip\noindent\textbf{Acknowledgements.}~~The authors acknowledge partial
support of the Slovenian research agency ARRS program \ P1--0383 and ARRS
project J1-1692 and also Project KK.01.1.1.02.0027, a project co-financed by
the Croatian Government and the European Union through the European Regional
Development Fund - the Competitiveness and Cohesion Operational Programme.
| {
"timestamp": "2021-04-02T02:34:32",
"yymm": "2104",
"arxiv_id": "2104.00577",
"language": "en",
"url": "https://arxiv.org/abs/2104.00577",
"abstract": "In a graph G, the cardinality of the smallest ordered set of vertices that distinguishes every element of V (G) (resp. E(G)) is called the vertex (resp. edge) metric dimension of G. In [16] it was shown that both vertex and edge metric dimension of a unicyclic graph G always take values from just two explicitly given consecutive integers that are derived from the structure of the graph. A natural problem that arises is to determine under what conditions these dimensions take each of the two possible values. In this paper for each of these two metric dimensions we characterize three graph configurations and prove that it takes the greater of the two possible values if and only if the graph contains at least one of these configurations. One of these configurations is the same for both dimensions, while the other two are specific for each of them. This enables us to establish the exact value of the metric dimensions for a unicyclic graph and also to characterize when each of these two dimensions is greater than the other one.",
"subjects": "Combinatorics (math.CO)",
"title": "Vertex and edge metric dimensions of unicyclic graphs",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9770226347732859,
"lm_q2_score": 0.724870282120402,
"lm_q1q2_score": 0.7082146729061303
} |
https://arxiv.org/abs/1301.0743 | Covering monolithic groups with proper subgroups | Given a finite non-cyclic group $G$, call $\sigma(G)$ the smallest number of proper subgroups of $G$ needed to cover $G$. Lucchini and Detomi conjectured that if a nonabelian group $G$ is such that $\sigma(G) < \sigma(G/N)$ for every non-trivial normal subgroup $N$ of $G$ then $G$ is \textit{monolithic}, meaning that it admits a unique minimal normal subgroup. In this paper we show how this conjecture can be attacked by the direct study of monolithic groups. | \section{Covering nilpotent groups}
In this section we will compute the covering number of nilpotent groups in order to get the reader familiarized with the methods.
Let $p$ be a prime. Observe that the group $C_p \times C_p$ admits exactly $p+1$ proper subgroups, and all these subgroups are cyclic of order $p$ and index $p$. Let us visualize this in the subgroup lattice: $$\xymatrix{& C_p \times C_p & & \\ \bullet \ar@{-}[ur] & \bullet \ar@{-}[u] & \cdots & \bullet \ar@{-}[ull] \\ & \{1\} \ar@{-}[ul] \ar@{-}[u] \ar@{-}[urr] & & }$$Therefore there is a unique cover of $C_p \times C_p$, it is the one consisting of all of its non-trivial proper subgroups. We obtain that $\sigma(C_p \times C_p) = p+1$.
\
The following result (which generalizes the equality $\sigma(C_p \times C_p) = p+1$) is a direct consequence of Theorem \ref{tom}. However, we will prove it in detail.
\begin{prop} \label{nilpotent}
Let $G$ be a finite nilpotent group. Then $\sigma(G) = p+1$ where $p$ is the smallest prime divisor of $|G|$ such that the Sylow $p$-subgroup of $G$ is not cyclic.
\end{prop}
Let us first observe that if $G$ is any finite group and $\Phi(G)$ is the Frattini subgroup of $G$ (i.e. the intersection of the maximal subgroups of $G$) then $\sigma(G) = \sigma(G/\Phi(G))$. Indeed, in any minimal cover of $G$ consisting of maximal subgroups its members all contain the Frattini subgroup.
Now suppose $G$ is a non-cyclic $p$-group. It is well known that $G/\Phi(G) \cong {C_p}^d$ where $d$ is the smallest size of a subset of $G$ generating $G$. Therefore $\sigma(G) = \sigma({C_p}^d)$. The covering number of ${C_p}^d$ can be easily computed using the following basic lemma.
\begin{lemma}[Minimal Index Lower Bound] \label{minind}
Let $\mathcal{H}$ be a minimal cover of a finite group $T$. Then $$\min \{|T:H|\ :\ H \in \mathcal{H}\} < \sigma(T).$$
\end{lemma}
\begin{proof}
Write $\mathcal{H} = \{H_1,\ldots,H_k\}$, $k=\sigma(T)$, $\beta_i := |T:H_i|$ with $\beta_1 \leq \cdots \leq \beta_k$. Since the union $H_1 \cup \cdots \cup H_k$ is not disjoint (because $1 \in H_i$ for $i=1,\ldots,k$), we have $$|T| = |\bigcup_{i=1}^k H_i| < \sum_{i=1}^k |H_i| = \sum_{i=1}^k |T|/\beta_i \leq k|T|/\beta_1.$$It follows that $\beta_1 < k = \sigma(T)$.
\end{proof}
Lemma \ref{minind} implies that $\sigma({C_p}^d) > p$. On the other hand, since $d > 1$ (because $G$ is non-cyclic), ${C_p}^d$ projects onto ${C_p}^2 = C_p \times C_p$, therefore $p < \sigma({C_p}^d) \leq \sigma(C_p \times C_p) = p+1$. We deduce that $\sigma(G) = \sigma({C_p}^d) = p+1$. Since any finite nilpotent group is the direct product of its Sylow subgroups, Proposition \ref{nilpotent} follows from the following lemma.
\begin{lemma}
Let $A,B$ be two finite groups of coprime order. Then $$\sigma(A \times B) = \min \{\sigma(A),\sigma(B)\}.$$
\end{lemma}
\begin{proof}
Let $\pi_A: A \times B \to A$, $\pi_B: A \times B \to B$ be the canonical projections. Let $\mathcal{H}$ be a minimal cover of $A \times B$ consisting of maximal subgroups, and let $$\Omega_A := \{H \in \mathcal{H}\ :\ \pi_B(H)=B\}, \hspace{1cm} \Omega_B := \{H \in \mathcal{H}\ :\ \pi_A(H)=A\}.$$Since $|A|,|B|$ are coprime, any subgroup of $A \times B$ is of the form $C \times D$ with $C \leq A$ and $D \leq B$. It follows that $\mathcal{H} = \Omega_A \cup \Omega_B$. Let $$O_A := A - \bigcup_{C \times B \in \Omega_A} C, \hspace{1cm} O_B := B - \bigcup_{A \times D \in \Omega_B} D.$$Since $\mathcal{H}$ covers $A \times B$, it covers $O_A \times O_B$, so $O_A \times O_B = \emptyset$. Hence, either $O_A = \emptyset$, implying $\Omega_B = \emptyset$ by minimality of $\mathcal{H}$ and $\sigma(A \times B) = \sigma(A)$, or $O_B = \emptyset$, implying $\Omega_A = \emptyset$ by minimality of $\mathcal{H}$ and $\sigma(A \times B) = \sigma(B)$.
\end{proof}
\section{Direct products of groups}
The very first case to consider when dealing with Conjecture \ref{mainconj} is the direct product case. In a joint work with A. Lucchini we deal with this case. We prove
\begin{teor}[Lucchini A., Garonzi M. 2010 \cite{garluc}]
Let $\mathcal{M}$ be a minimal cover of a direct product $G=H_1 \times H_2$ of two groups.
Then one of the following holds:
\begin{enumerate}
\item $\mathcal{M}=\{X\times H_2\mid X\in \mathcal{X}\}$ where $\mathcal{X}$ is a minimal cover of $H_1.$ In this case $\sigma(G)=\sigma(H_1).$
\item $\mathcal{M}=\{H_1\times X\mid X\in \mathcal{X}\}$ where $\mathcal{X}$ is a minimal cover of $H_2.$ In this case $\sigma(G)=\sigma(H_2).$
\item There exist $N_1\trianglelefteq H_1,$ $N_2\trianglelefteq H_2$ with $H_1/N_1\cong H_2/N_2\cong C_p$
and $\mathcal{M}$ consists of the maximal subgroups of $H_1\times H_2$ containing $N_1\times N_2.$ In this case $\sigma(G)=p+1.$
\end{enumerate}
\end{teor}
We will now give the idea of how the proof goes when $H_1$ and $H_2$ are isomorphic non-abelian simple groups. This does not cover all the ideas of the proof but it covers quite well those used when $H_1$ and $H_2$ do not have common abelian factor groups.
\
Let $S$ be a non-abelian simple group. We want to prove that $\sigma(S \times S) = \sigma(S)$. Note that since $S$ is a quotient of $S \times S$, $\sigma(S \times S) \leq \sigma(S)$.
\begin{enumerate}
\item We know that the maximal subgroups of $S \times S$ are of the following three types: $$(1)\ K \times S, \hspace{1cm} (2)\ S \times K, \hspace{1cm} (3)\ \Delta_{\varphi}:=\{(x,\varphi(x))\ |\ x \in S\},$$where $K$ is a maximal subgroup of $S$ and $\varphi \in \text{Aut}(S)$.
\item Let $\mathcal{M} = \mathcal{M}_1 \cup \mathcal{M}_2 \cup \mathcal{M}_3$ be a minimal cover of $S \times S$, where $\mathcal{M}_i$ consists of subgroups of type $(i)$.
\item Let $\Omega := S \times S - \bigcup_{M \in \mathcal{M}_1 \cup \mathcal{M}_2} M = \Omega_1 \times \Omega_2$, where $\Omega_1 = S - \bigcup_{K \times S \in \mathcal{M}_1} K$ and $\Omega_2 = S - \bigcup_{S \times K \in \mathcal{M}_2} K$.
\item We claim that it is enough to prove that $\Omega = \emptyset$. Indeed if this is the case then either $\Omega_1 = \emptyset$, in which case $\bigcup_{K \times S \in \mathcal{M}_1} K = S$ and $\mathcal{M} = \mathcal{M}_1$ by minimality of $\mathcal{M}$, or $\Omega_2 = \emptyset$, in which case $\bigcup_{S \times K \in \mathcal{M}_2} K = S$, and $\mathcal{M} = \mathcal{M}_2$ by minimality of $\mathcal{M}$. In both cases we obtain $\sigma(S \times S) \geq \sigma(S)$ and hence $\sigma(S \times S) = \sigma(S)$. \\ Suppose by contradiction $\Omega \neq \emptyset$, i.e. $\Omega_1 \neq \emptyset \neq \Omega_2$, and let $\omega \in \Omega_1$.
\item The family $$\{ K<S\ |\ S \times K \in \mathcal{M}_2\} \cup \{\langle \varphi(\omega) \rangle\ |\ \Delta_{\varphi} \in \mathcal{M}_3\}$$ is a cover of $S$ of size $|\mathcal{M}_2|+|\mathcal{M}_3|$ (it consists of proper subgroups being $S$ non-abelian). Indeed, if $b \in S$ is such that $b \not \in K$ for any $K < S$ such that $S \times K \in \mathcal{M}_2$ then $(\omega,b) \in S \times S - \Omega_1 \times \Omega_2$ hence, being $\mathcal{M}$ a cover for $S \times S$, $(\omega,b) \in \Delta_{\varphi}$ for some $\varphi \in \Aut(S)$ such that $\Delta_{\varphi} \in \mathcal{M}_3$, and we conclude that $b = \varphi(\omega) \in \langle \varphi(\omega) \rangle$.
\item It follows that $$|\mathcal{M}_1| + |\mathcal{M}_2| + |\mathcal{M}_3| = |\mathcal{M}| = \sigma(S \times S) \leq \sigma(S) \leq |\mathcal{M}_2| + |\mathcal{M}_3|.$$ This implies that $\mathcal{M}_1=\emptyset$. Analogously $\mathcal{M}_2=\emptyset$. So $\mathcal{M}=\mathcal{M}_3$.
\item Observe that since $S$ is covered by its non-trivial cyclic subgroups, $\sigma(S) < |S|$. Since each member of $\mathcal{M}_3 = \mathcal{M}$ has index $|S|$, by the Minimal Index Lower Bound (Lemma \ref{minind}) $$|S| < \sigma(S \times S) \leq \sigma(S) < |S|,$$ a contradiction.
\end{enumerate}
\section{Sigma star}
Recall that a group $G$ is called ``primitive'' if it admits a core-free maximal subgroup, that is, a maximal subgroup $M$ such that $\bigcap_{g \in G} gMg^{-1} = \{1\}$. A primitive group has always at most two minimal normal subgroup, and if they are two, they are non-abelian.
Recall that a $G$-group is a group $A$ endowed with a homomorphism $f: G \to \text{Aut}(A)$. If $a \in A$ and $g \in G$, the element $f(g)(a)$ is usually denoted $a^g$ if no ambiguity is possible.
\begin{defi}
Let $G$ be a group, and let $A,B$ be two $G$-groups.
\begin{itemize}
\item $A,B$ are said to be $G$-isomorphic \index{$G$-isomorphic $G$-groups} (written $A \cong_G B$) if there exists an isomorphism $\varphi:A \to B$ such that $a^{\varphi g} = a^{g \varphi}$ for every $g \in G$.
\item $A,B$ are said to be $G$-equivalent \index{$G$-equivalent $G$-groups} (written $A \sim_G B$) if there exist isomorphisms $$\xymatrix{\varphi:A \ar[r] & B},\ \xymatrix{\Phi:G \ltimes A \ar[r] & G \ltimes B}$$ such that the following diagram commutes: $$\xymatrix{\{1\} \ar[r] & A \ar[r] \ar[d]^-{\varphi} & G \ltimes A \ar[r] \ar[d]^-{\Phi} & G \ar[r] \ar@{=}[d] & \{1\} \\ \{1\} \ar[r] & B \ar[r] & G \ltimes B \ar[r] & G \ar[r] & \{1\}}$$
\end{itemize}
\end{defi}
Let $N$ be a minimal normal subgroup of a group $G$. The conjugation action of $G$ on $N$ gives $N$ the structure of $G$-group. Define $I_G(N)$ to be the set of elements of $G$ which induce by conjugation an inner automorphism of $N$ and define $R_G(N)$ to be the intersection of the normal subgroups $K$ of $G$ contained in $I_G(N)$ with the property that $I_G(N)/K$ is non-Frattini (i.e. not contained in the Frattini subgroup of $G/K$) and $G$-equivalent to $N$.
Recall that the ``socle'' of a group $G$, denoted $\soc(G)$, is the subgroup of $G$ generated by the minimal normal subgroups of $G$. $\soc(G)$ is always a direct product of some minimal normal subgroups of $G$. $G$ is said to be ``monolithic'' if it admits a unique minimal normal subgroup, i.e. if $\soc(G)$ is a minimal normal subgroup of $G$.
\begin{teor}[Lucchini, Detomi \cite{cubo} Corollary 14] \label{struttura}
Let $H$ be a non-abelian $\sigma$-elementary group and let $N_1,\ldots,N_{\ell}$ be minimal normal subgroups of $H$ such that $\soc(H) = N_1 \times \cdots \times N_{\ell}$. Let $X_i := G/R_H(N_i)$ for $i=1,\ldots,\ell$. Then $X_i$ is a primitive monolithic group with socle isomorphic to $N_i$ for $i=1,\ldots,\ell$ ($X_i$ will be called ``\textit{the primitive monolithic group associated to} $N_i$'') and $H$ is a subdirect product of $X_1, \ldots, X_{\ell}$: the canonical homomorphism $$H \to X_1 \times \ldots \times X_{\ell}$$is injective.
\end{teor}
\begin{defi}[Sigma star] \label{star}
Let $X$ be a primitive monolithic group, and let $N$ be its unique minimal normal subgroup. If $\Omega$ is an arbitrary union of cosets of $N$ in $X$ define $\sigma_{\Omega}(X)$ to be the smallest number of supplements of $N$ in $X$ needed to cover $\Omega$. If $\Omega = \{Nx\}$ we will write $\sigma_{Nx}(X)$ instead of $\sigma_{\{Nx\}}(X)$. Define $$\sigma^{\ast}(X) := \min \{\sigma_{\Omega}(X)\ |\ \Omega = \bigcup_i N \omega_i,\ \langle \Omega \rangle = X \}.$$
\end{defi}
\begin{prop}[Lucchini, Detomi \cite{cubo} Proposition 16] \label{sigmastar}
Let $H$ be a non-abelian $\sigma$-elementary group with socle $N_1 \times \cdots \times N_{\ell}$, $$H \leq_{\text{subd}} X_1 \times \ldots \times X_{\ell}$$as in Theorem \ref{struttura}. For $i=1,\ldots,\ell$ let $\ell_{X_i}(N_i)$ be the smallest primitivity degree of $X_i$, i.e. the smallest index of a proper supplement of $N_i$ in $X_i$. Then $\ell_{X_i}(N_i) \leq \sigma^{\ast}(X_i)$ for $i=1,\ldots,\ell$ and $$\sum_{i=1}^{\ell} \ell_{X_i}(N_i) \leq \sum_{i=1}^{\ell} \sigma^{\ast}(X_i) \leq \sigma(H).$$
\end{prop}
\begin{prop}[\cite{cubo}, Proposition 10] \label{abmns}
Let $G$ be a finite group. If $V$ is a complemented normal abelian subgroup of $G$ and $V \cap Z(G) = \{1\}$ then $\sigma(G) \leq 2|V|-1$.
\end{prop}
\begin{proof}
Let $H$ be a complement of $V$ in $G$. The idea is to show that $G$ is covered by the family $\{H^v\ |\ v \in V\} \cup \{C_H(v)V\ |\ 1 \neq v \in V\}$. We omit the details.
\end{proof}
\section{Small covering numbers}
The content of this section is included in my Ph.D. thesis.
\begin{lemma} \label{spancoprime}
Let $N$ be a normal subgroup of a group $X$. If a set of subgroups of $X$ covers a coset $yN$ of $N$ in $X$, then it also covers every coset $y^{\alpha}N$ with $\alpha$ prime to $|y|$.
\end{lemma}
\begin{proof}
Let $s$ be an integer such that $s \alpha \equiv 1 \mod |y|$. As $s$ is prime to $|y|$, by Dirichlet's theorem there exist infinitely many primes in the arithmetic progression $\{s+|y|n\ |\ n \in \mathbb{N}\}$; we choose a prime $p > |X|$ in $\{s + |y|n\ |\ n \in \mathbb{N}\}$. Clearly, $y^p=y^s$. As $p$ is prime to $|X|$, there exists an integer $r$ such that $pr \equiv 1 \mod |X|$. Hence, if $yN \subseteq \cup_{i \in I} M_i$, for every $g \in y^{\alpha}N$ we have that $g^p \in (y^{\alpha})^p N = (y^{\alpha})^s N = yN \subseteq \cup_{i \in I} M_i$ and therefore also $g=(g^p)^r$ belongs to $\cup_{i \in I}M_i$.
\end{proof}
\begin{prop} \label{56}
Let $H$ be a non-abelian $\sigma$-elementary group such that $\sigma(H) \leq 55$. Then $H$ is primitive and monolithic.
\end{prop}
\begin{proof}
We will use the notations of Theorem \ref{struttura}.
It is proven in \cite{cubo} that any non-abelian $\sigma$-elementary group has at most one abelian minimal normal subgroup. Therefore we may assume that there exists a non-abelian minimal normal subgroup $N$ of $H$. Let $G$ be the primitive monolithic group associated to $N$. If $G$ has a primitivity degree at most $27$ then either $\ell_G(N) \geq 10$ and $G/N \in \{C_2 \times C_2,\Sym(3),D_8\}$ (by inspection) - contradicting the inequality $\ell_G(N) \leq \sigma(H) \leq \sigma(G)$ (being $\sigma(C_2 \times C_2) = \sigma(D_8) = 3$ and $\sigma(S_3)=4$) - or $G/N$ is cyclic of prime-power order. Assume the latter case holds. Then $G/N$ admits only one maximal subgroup. In other words, a subset of $G$ generates $G$ modulo $N$ if and only if it contains an element $g \in G$ such that $G/N = \langle gN \rangle$. Thus Lemma \ref{spancoprime} implies that $\sigma(G) \leq \sigma^{\ast}(G)+1$, so that $$\sigma^{\ast}(X_1) + \sigma^{\ast}(X_2) \leq \sigma(H) \leq \sigma(X_1) \leq \sigma^{\ast}(X_1) + 1.$$In particular $\ell_{X_2}(N_2) \leq \sigma^{\ast}(X_2) \leq 1$, and this is a contradiction ($\ell_{X_2}(N_2)$ is the index of a proper subgroup of $X_2$).
Therefore we may assume that $\ell_G(N) \geq 28$ whenever $N$ is a non-abelian minimal normal subgroup of $G$. Suppose $H$ has at least two minimal normal subgroups $N_1=N, N_2$. If $N_2$ is non-abelian then by assumption $\ell_{X_2}(N_2) \geq 28$ and Proposition \ref{sigmastar} implies $56 \leq \ell_{X_1}(N_1) + \ell_{X_2}(N_2) \leq \sigma(H)$, a contradiction. Hence $N_2$ is abelian. We have $\ell_{X_2}(N_2) = |N_2|$ and by Proposition \ref{sigmastar} and Proposition \ref{abmns} $$28 + |N_2| \leq \ell_{X_1}(N_1) + \ell_{X_2}(N_2) \leq \sigma(H) \leq \sigma(X_2) < 2 |N_2|,$$ therefore $\sigma(H)-28 \geq |N_2| > \frac{1}{2} \sigma(H)$, and this implies $\sigma(H) > 56$, a contradiction.
\end{proof}
Proposition \ref{56} allows us to list the $\sigma$-elementary groups with small covering number. Indeed, if $H$ is a $\sigma$-elementary group such that $\sigma(H) \leq 55$ then $H$ is a primitive monolithic group with a primitivity degree at most $55$ (cf. Proposition \ref{sigmastar}). Since there are only finitely many groups of a given primitivity degree, we are reduced to look at a finite list of groups. By giving bounds to their covering numbers we can list the $\sigma$-elementary groups $G$ with $\sigma(G) \leq 25$. The explicit bounds can be found in \cite{gar}.
\begin{table}
\label{table25}
\begin{tabular}{|c|c|}
\hline $\sigma$ & $\text{Groups}$ \\ \hline $3$ & $C_2 \times C_2$ \\ \hline $4$ & $C_3 \times C_3,\Sym(3)$ \\ \hline $5$ & $\Alt(4)$ \\ \hline $6$ & $C_5 \times C_5,D_{10},AGL(1,5)$ \\ \hline $7$ & $\emptyset$ \\ \hline $8$ & $C_7 \times C_7,D_{14},7:3,AGL(1,7)$ \\ \hline $9$ & $AGL(1,8)$ \\ \hline $10$ & $3^2:4,AGL(1,9),\Alt(5)$ \\ \hline $11$ & $\emptyset$ \\ \hline $12$ & $C_{11} \times C_{11},11:5,D_{22},AGL(1,11)$ \\ \hline $13$ & $\Sym(6)$ \\ \hline $14$ & $C_{13} \times C_{13},D_{26},13:3,13:4,13:6,AGL(1,13)$ \\ \hline $15$ & $SL(3,2)$ \\ \hline $16$ & $\Sym(5),\Alt(6)$ \\ \hline $17$ & $2^4:5,AGL(1,16)$ \\ \hline $18$ & $C_{17} \times C_{17},D_{34},17:4,17:8,AGL(1,17)$ \\ \hline $19$ & $\emptyset$ \\ \hline $20$ & $C_{19} \times C_{19},AGL(1,19),D_{38},19:3,19:6,19:9$ \\ \hline $21$ & $\emptyset$ \\ \hline $22$ & $\emptyset$ \\ \hline $23$ & $M_{11}$ \\ \hline $24$ & $C_{23} \times C_{23},D_{46},23:11,AGL(1,23)$ \\ \hline $25$ & $\emptyset$ \\ \hline
\end{tabular}
\caption{The list of $\sigma$-elementary groups $G$ with $3 \leq \sigma(G) \leq 25$.}
\end{table}
In general, the following fact holds.
\begin{prop}
For every fixed positive integer $n$, the set of $\sigma$-elementary groups $H$ with $\sigma(H) = n$ is finite, bounded by a function of $n$.
\end{prop}
\begin{proof}
We will use the notations of Theorem \ref{struttura}. Let $H$ be a $\sigma$-elementary group, and write $\soc(H) = N_1 \times \ldots \times N_{\ell}$. Let $X_1,\ldots,X_{\ell}$ be the primitive monolithic groups associated to $N_1,\ldots,N_{\ell}$ respectively. $H$ embeds in $X_1 \times \ldots \times X_{\ell}$, so in order to conclude it suffices to bound the number of possibilities for $\ell$ and each $X_i$ in terms of $\sigma(H)$. By Proposition \ref{sigmastar} $$\ell \leq \sum_{i=1}^{\ell} \ell_{X_i}(N_i) \leq \sum_{i=1}^{\ell} \sigma^{\ast}(X_i) \leq \sigma(H).$$Since there are finitely many primitive groups with a given primitivity degree, the result follows.
\end{proof}
\section{Considering some monolithic groups}
The content of this section is included in my Ph.D. thesis.
Proposition \ref{56} holds also for $56$, but for this number a quite different argument is needed. This is interesting because of the following result, which is \cite[Theorem 2]{gar2}. Here $A_5 \wr C_2$ denotes the wreath product of $A_5$ with $C_2$, i.e. the semidirect product $(A_5 \times A_5) \rtimes C_2$ with the action of $C_2 = \langle \varepsilon \rangle$ on $A_5 \times A_5$ given by $(x,y)^{\varepsilon} = (y,x)$.
\begin{teor}[\cite{gar2} Theorem 2] \label{a5wrc2}
$\sigma(A_5 \wr C_2) = 1 + 4 \cdot 5 + 6 \cdot 6 = 57$.
\end{teor}
A minimal cover of $G = A_5 \wr C_2$ is given by its socle, $\soc(G) = A_5 \times A_5$, together with the subgroups of the form $N_G(M \times M^a)$ where $a \in A_5$ and $M$ is either the stabilizer of $j \in \{1,2,3,4,5\}-\{i\}$ (for some $i \in \{1,2,3,4,5\}$) in $A_5$ or the normalizer of a Sylow $5$-subgroup of $A_5$.
The lower bounds for the covering number will be obtained by using the following tool, introduced by Mar\'oti in \cite{maroti}.
\begin{defi}[Definite unbeatability] \label{du}
\label{d1} Let $X$ be a group. Let $\mathcal{H}$ be a set
of proper subgroups of $X$, and let $\Pi \subseteq X$. Suppose
that the following four conditions hold for $\mathcal{H}$ and
$\Pi$.
\begin{enumerate}
\item $\Pi \cap H \neq \emptyset$ for every $H \in \mathcal{H}$;
\item $\Pi \subseteq \bigcup_{H \in \mathcal{H}} H$;
\item $\Pi \cap H_{1} \cap H_{2} = \emptyset$ for every distinct
pair of subgroups $H_{1}$ and $H_{2}$ of $\mathcal{H}$;
\item $|\Pi \cap K| \leq |\Pi \cap H|$ for every $H \in
\mathcal{H}$ and $K < X$ with $K \not \in \mathcal{H}$.
\end{enumerate}
Then $\mathcal{H}$ is said to be definitely unbeatable on $\Pi$.
\end{defi}
For $\Pi \subseteq X$ let $\sigma_X(\Pi)$ be the least cardinality
of a family of proper subgroups of $X$ whose union contains $\Pi$.
The following lemma is straightforward.
\begin{lemma}
\label{l6} If $\mathcal{H}$ is definitely unbeatable on $\Pi$ then
$\sigma_X(\Pi)=|\mathcal{H}|$.
\end{lemma}
It follows that if $\mathcal{H}$ is definitely unbeatable on $\Pi$
then $|\mathcal{H}| = \sigma_X(\Pi) \leq \sigma(X)$.
Let us give \cite[Theorem 3.1]{maroti} as an example. Let $n \geq 11$ be an odd integer, and let $X := \Sym(n)$ be the symmetric group on $n$ letters. Let $\mathcal{H}$ be the family of subgroups of $\Sym(n)$ consisting of the alternating group $\Alt(n)$ and the intransitive maximal subgroups of $\Sym(n)$. Let $\Pi$ be the subset of $\Sym(n)$ consisting of the permutations which are product of at most two disjoint cycles. Then $\mathcal{H}$ is a cover of $\Sym(n)$ which is definitely unbeatable on $\Pi$, therefore $\sigma(\Sym(n)) = |\mathcal{H}| = 2^{n-1}$.
This example was rivisited and generalized by Mar\'oti and me (cf. \cite{margar}, \cite{gar2}) and the results summarized in Theorems \ref{t1} and \ref{t2} below were obtained.
Let us fix some notations we will often use.
\begin{nota} \label{mono}
Let $G$ be a monolithic group with socle $N = \soc(G) = S_1 \times \cdots \times S_m$, where $S_1, \ldots ,S_m$ are pairwise isomorphic non-abelian simple groups. $X := N_G(S_1)/C_G(S_1)$ is an almost-simple group with socle $S := S_1 C_G(S_1)/C_G(S_1) \cong S_1$. The minimal normal subgroups of $S^m = S_1 \times \ldots \times S_m$ are precisely its factors, $S_1,\ldots,S_m$. Since automorphisms send minimal normal subgroups to minimal normal subgroups, it follows that $G$ acts on the $m$ factors of $N$. Let $\rho: G \to \Sym(m)$ be the homomorphism induced by the conjugation action of $G$ on the set $\{S_1, \ldots,S_m\}$. $K := \rho(G)$ is a transitive permutation group of degree $m$. By \cite[Remark 1.1.40.13]{spagn} $G$ embeds in the wreath product $X \wr K$. Let $L$ be the subgroup of $X$ generated by the following set: $$S \cup \{x_1 \cdots x_m\ |\ \exists k \in K:\ (x_1, \ldots, x_m) k \in G\}.$$Let $T$ be a normal subgroup of $X$ containing $S$ and contained in $L$ with the property that $L/T$ has prime order if $L \neq S$, and $T=L$ if $L=S$.
\end{nota}
Let $G$ be a primitive monolithic group with non-abelian socle $N$, and write $N=S^m$ with $S$ a non-abelian simple group. The covers of $G$ we often look at consist of some subgroups of $G$ containing $N$ and subgroups of the form $N_G(M \times M^{a_2} \times \cdots \times M^{a_m})$ with $M < S$, which will be called ``\textit{product type subgroups}''.
In the following if $n$ is a positive integer we denote by $\omega(n)$ the number of prime divisors of $n$. Suppose that $G/N$ is cyclic. The covers of $G$ we consider consist of all the $\omega(|G/N|)$ maximal subgroups of $G$ containing $N$ and some product type subgroups $N_G((S \cap M) \times (S \cap M)^{a_2} \times \cdots \times (S \cap M)^{a_m})$ where $a_1=1,a_2,\ldots,a_m \in S$ and $M$ varies in a family of maximal subgroups of $X$ supplementing $S$ which covers a coset $xS$ of $S$ in $X$ which generates the cyclic group $X/S$. This is how we obtain upper bounds for $\sigma(G)$ (the size of a cover of $G$ is an upper bound for $\sigma(G)$).
\begin{teor}[Mar\'oti A., Garonzi M. 2010 \cite{margar}] \label{t1}
Let $G$ be a monolithic group with non-abelian socle, and let us use Notations \ref{mono}. Suppose that $G/N$ is cyclic and that $X = S = \Alt(n)$. Then the following holds.
\begin{enumerate}
\item If $12 < n \equiv 2 \mod(4)$ then $$\sigma(G) = \omega(m) + \sum_{i=1,\ i\ \text{odd}}^{(n/2)-2} \binom{n}{i}^m + \frac{1}{2^m} \binom{n}{n/2}^m.$$
\item If $12 < n \not \equiv 2 \mod(4)$ then $$\omega(m) + \frac{1}{2} \sum_{i=1,\ i\ \text{odd}}^n \binom{n}{i}^m \leq \sigma(G).$$
\item Suppose $n$ has a prime divisor at most $\sqrt[3]{n}$. Then $$\sigma(G) \sim \omega(m) + \min_{\mathcal{M}} \sum_{M \in \mathcal{M}} |S:M|^{m-1}\ \text{as}\ n \to \infty.$$
\end{enumerate}
\end{teor}
\begin{teor}[Garonzi M. 2011 \cite{gar2}] \label{t2}
Let $G$ be a monolithic group with non-abelian socle, and let us use Notations \ref{mono}. Suppose that $G/N$ is cyclic and that $X = \Sym(n)$. Then the following holds.
\begin{enumerate}
\item Suppose that $n \geq 7$ is odd and $(n,m) \neq (9,1)$. Then $$\sigma(G) = \omega(2m) + \sum_{i=1}^{(n-1)/2} \binom{n}{i}^m.$$
\item Suppose that $n \geq 8$ is even. Then $$\left( \frac{1}{2} \binom{n}{n/2} \right)^m \leq \sigma(G) \leq \omega(2m) + \left( \frac{1}{2} \binom{n}{n/2} \right)^m + \sum_{i=1}^{[n/3]} \binom{n}{i}^m.$$In particular $\sigma(G) \sim \left( \frac{1}{2} \binom{n}{n/2} \right)^m$ as $n \to \infty$.
\end{enumerate}
\end{teor}
\section{Attacking the conjecture}
The content of this section is included in my Ph.D. thesis.
The following result provides a first partial reduction to monolithic groups.
\begin{prop} \label{starmin}
Let $H$ be a non-abelian $\sigma$-elementary group, let $N_1,\ldots,N_{\ell}$ be minimal normal subgroups of $H$ such that $\soc(H) = N_1 \times \cdots \times N_{\ell}$ and let $X_1,\ldots,X_{\ell}$ be the primitive monolithic groups associated to $N_1,\ldots,N_{\ell}$ respectively. Then at most one of $N_1,\ldots,N_{\ell}$ is abelian. Suppose that $N_1$ is non-abelian and that $\sigma^{\ast}(X_1) \leq \sigma^{\ast}(X_j)$ whenever $j \in \{1,\ldots,\ell\}$ and $N_j$ is non-abelian. If $\sigma(X_1) < 2 \sigma^{\ast}(X_1)$ then $H \cong X_1$, i.e. $H$ is monolithic.
\end{prop}
\begin{proof}
By Proposition \ref{sigmastar} $$\sigma^{\ast}(X_1) + \sum_{j=2}^{\ell} \sigma^{\ast}(X_j) \leq \sigma(H) \leq \sigma(X_1) < 2 \sigma^{\ast}(X_1).$$It follows that $\sum_{j=2}^{\ell} \sigma^{\ast}(X_j) < \sigma^{\ast}(X_1)$ hence, by the minimality hypothesis on $X_1$, $N_2,\ldots,N_{\ell}$ are abelian. In \cite[Corollary 14]{cubo} it is proved that any non-abelian $\sigma$-elementary group has at most one abelian minimal normal subgroup, thus $\ell = 2$. Since $N_2$ is abelian $\ell_{X_2}(N_2)=|N_2|$, and by Proposition \ref{sigmastar}
\begin{eqnarray}
\min \{2 \sigma^{\ast}(X_1), 2|N_2|\} & \leq & \sigma^{\ast}(X_1) + |N_2| = \sigma^{\ast}(X_1) + \ell_{X_2}(N_2) \leq \nonumber \\ & \leq & \sigma(H) \leq \min \{\sigma(X_1),\sigma(X_2)\}. \nonumber
\end{eqnarray}
Now by hypothesis $\sigma(X_1) < 2 \sigma^{\ast}(X_1)$, and $\sigma(X_2) < 2|N_2|$ by Proposition \ref{abmns}. This leads to a contradiction.
\end{proof}
In order to prove an inequality like $\sigma(G) < 2 \sigma^{\ast}(G)$ for $G$ a primitive monolithic group we first need some way to get as much general as possible upper bounds for $\sigma(G)$.
\begin{teor} \label{sopra}
Let $G$ be a monolithic group with non-abelian socle, and let us use Notations \ref{mono}. Assume that $X/S$ is abelian. Let $\mathcal{M}$ be a set of maximal subgroups of $X$ supplementing $S$ and such that $\bigcup_{M \in \mathcal{M}} M$ contains a coset $xS \in L$ with the property that $\langle x,T \rangle = L$.
Then $\sigma(G) \leq 2^{m-1} + \sum_{M \in \mathcal{M}} |S:S \cap M|^{m-1}$.
\end{teor}
Unfortunately the hypothesis ``$X/S$ abelian'' does not seem easy to bypass.
\begin{proof}
If $L \neq T$ define $$R := \{(x_1, \ldots ,x_m)k \in G\ |\ x_1 \cdots x_m \in T\}.$$Since $X/S$ is abelian, $R$ is a proper subgroup of $G$.
Let $\delta \in K$ be an $m$-cycle, $1 = a_1, a_2, \ldots ,a_m \in X$ and $M \in \mathcal{M}$. An element $(x_1, \ldots ,x_m) \delta \in X \wr K$ normalizes $(M \cap S) \times (M \cap S)^{a_2} \times \cdots \times (M \cap S)^{a_m}$ if and only if $$(M \cap S)^{a_{\delta^{-1}(1)} x_{\delta^{-1}(1)}} \times (M \cap S)^{a_{\delta^{-1}(2)} x_{\delta^{-1}(2)}} \times \cdots \times (M \cap S)^{a_{\delta^{-1}(m)} x_{\delta^{-1}(m)}} =$$ $$= (M \cap S) \times (M \cap S)^{a_2} \times \cdots \times (M \cap S)^{a_m},$$if and only if
\begin{equation} \label{prodeq}
a_{\delta^{-1}(1)} x_{\delta^{-1}(1)} a_1^{-1},\ a_{\delta^{-1}(2)} x_{\delta^{-1}(2)} a_2^{-1}, \ldots, a_{\delta^{-1}(m)} x_{\delta^{-1}(m)} a_m^{-1} \in N_X(M \cap S) = M.
\end{equation}
If $x_1 x_{\delta(1)} \cdots x_{\delta^{m-1}(1)} \in M$ then there exist $a_2, \ldots ,a_m \in X$ such that (\ref{prodeq}) is true. Since $M$ supplements $S$ in $X$, $a_2, \ldots ,a_m$ can be chosen in $S$. Therefore every element $(x_1, \ldots, x_m) \delta \in G$ such that $\delta$ is an $m$-cycle and $x_1 x_{\delta(1)} \cdots x_{\delta^{m-1}(1)} \in xS$ belongs to a subgroup of $G$ of the form $N_G((M \cap S) \times (M \cap S)^{a_2} \times \cdots \times (M \cap S)^{a_m})$ where $M \in \mathcal{M}$ and $a_2, \ldots, a_m \in S$. It follows that $G$ is covered by these subgroups together with $R$ (if $L \neq T$) and the pre-images through $\rho$ of $2^{m-1}-1$ maximal intransitive subgroups of $K$ (corresponding to the subsets of $\{1,\ldots,m\}$ of size from $1$ to $[m/2]$).
\end{proof}
Recall the structure of maximal subgroups of primitive monolithic groups.
\begin{defi}[\cite{spagn}, Definition 1.1.37]
Let $G = \prod_{i=1}^n S_i$ be a direct product of groups. A subgroup $H$ of $G$ is said to be ``full diagonal'' if each projection $\pi_i: H \to S_i$ is an isomorphism.
\end{defi}
What follows is part of the O'Nan-Scott theorem (reference: \cite[Remark 1.1.40]{spagn}). Let $G$ be a primitive monolithic group with non-abelian socle $N=S^m$. Let $H$ be a maximal subgroup of $G$ such that $N \not \subseteq H$, i.e. $HN=G$, i.e. $H$ supplements $N$. Suppose $N \cap H \neq \{1\}$, i.e. $H$ does not complement $N$. Since $N$ is the unique minimal normal subgroup of $G$ and $H$ is a maximal subgroup of $G$ not containing $N$, $H=N_G(N \cap H)$. In the following let $X := N_G(S_1)/C_G(S_1)$ (it is an almost simple group with socle $S_1C_G(S_1)/C_G(S_1) \cong S$). There are two possibilities for the intersection $N \cap H$:
\begin{enumerate}
\item \textbf{Product type}. Suppose the projections $H \to S_i$ are not surjective. Then there exists a subgroup $M$ of $S$ such that $N_X(M)$ supplements $S$ in $X$ and elements $a_2,\ldots,a_m \in S$ such that $$H \cap N = M \times M^{a_2} \times \ldots \times M^{a_m}.$$In this case $|H \cap N| = |M|^m$.
\item \textbf{Diagonal type}. Suppose the projections $H \to S_i$ are surjective. Then there exists an $H$-invariant partition $\Delta$ of $\{1,\ldots,m\}$ into blocks for the action of $H$ on $\{1,\ldots,m\}$ such that $$H \cap N = \prod_{D \in \Delta} (H \cap N)^{\pi_D}$$and for each $D \in \Delta$ the projection $(H \cap N)^{\pi_D}$ is a full diagonal subgroup of $\prod_{i \in D}S_i$. In this case $|H \cap N| \leq |S|^{m/r}$ where $r$ is the smallest prime divisor of $m$.
\end{enumerate}
We now prove a crucial lemma which we will need in the proof of the main theorem. Let $G$ be a monolithic group with non-abelian socle, and let us use Notations \ref{mono}.
Let $\mathcal{Z}$ be the set of pairs $(z,w)$ in $X \times X$ such that $\langle z^a, w^b \rangle \supseteq S$ for every $a,b \in S$. By \cite{kls}, $\mathcal{Z} \cap (S \times S) \neq \emptyset$. Let $k$ be a non-$m$-cycle in $K$, let $O_1 = (i_1, \ldots ,i_r)$, $O_2 = (j_1, \ldots ,j_s)$ be two cycles in the cyclic decomposition of $k$, and for $\rho^{-1}(k) \ni h = (x_1, \ldots ,x_m) k$, with $x_1,\ldots,x_m \in X$, let $h_{O_1} := x_{i_1} \cdots x_{i_r}$ and $h_{O_2} := x_{j_1} \cdots x_{j_s}$.
\begin{lemma} \label{nonciclo}
Let $\mathcal{E}_k := \{(h_{O_1},h_{O_2})\ |\ h \in \rho^{-1}(k)\} \cap \mathcal{Z}$. Let $r$ be the smallest prime divisor of $m$. If $g \in \rho^{-1}(k)$ then $\sigma_{Ng}(G) \geq |\mathcal{E}_k| \cdot |S|^{m-m/r-2}$.
\end{lemma}
\begin{proof}
Let $$\mathfrak{X} := \{h \in Ng\ |\ (h_{O_1}, h_{O_2}) \in \mathcal{E}_k \}.$$ Note that if $h \in Ng$, $\theta$, $\varphi \in X$ are such that $h_{O_1} \equiv \theta \mod S$ and $h_{O_2} \equiv \varphi \mod S$ then there exists $t \in N$ such that $(th)_{O_1} = \theta$, $(th)_{O_2} = \varphi$. This implies that $|\mathfrak{X}| \geq |\mathcal{E}_k| \cdot |S|^{m-2}$. It is easy to show that if $a_2, \ldots ,a_m \in S$ and $h \in \rho^{-1}(k) \cap N_G(M \times M^{a_2} \times \cdots \times M^{a_m})$ then $h_{O_1} \in N_X(M)^{a_{i_1}}$, $h_{O_2} \in N_X(M)^{a_{j_1}}$. By the definition of $\mathcal{E}_k$, we deduce that $\mathfrak{X} \cap H = \emptyset$ whenever $H$ is a supplement of $N$ of product type. Since the maximal subgroups of $G$ complementing $N$ intersect $Ng$ in at most one point, this implies that in order to cover $\mathfrak{X}$ with supplements of $N$ we need at least $|\mathcal{E}_k| \cdot |S|^{m-2}/|S|^{m/r}$ of them.
\end{proof}
We are ready to state the main theorem.
\begin{teor}
Let $H$ be a $\sigma$-elementary non-abelian group. We will use the notations of Theorem \ref{struttura}. Let $N = N_1$ be a non-abelian minimal normal subgroup of $H$ and let $G := H/R_H(N) = X_1$ be the primitive monolithic group associated to $N$. Assume that $\min \{\sigma^{\ast}(X_i)\ :\ i=1,\ldots,h\} = \sigma^{\ast}(G)$. Let us use Notations \ref{mono}, and let $r$ be the smallest prime divisor of $m$. Suppose that $X/S$ is abelian. Let $E_{nc} := \min \{|\mathcal{E}_k|\ |\ k \in K\ \text{non-}m\text{-cycle}\}$ ($\mathcal{E}_k$ is as in Lemma \ref{nonciclo}). Suppose that whenever $x \in X$ is such that $\langle x,S \rangle = T$ there exist families $\mathcal{M},\mathcal{J}$ of maximal subgroups of $X$ supplementing $S$ such that:
\begin{enumerate}
\item $xS \subseteq \bigcup_{M \in \mathcal{M} \cup \mathcal{J}} M$;
\item $\sum_{M \in \mathcal{M} \cup \mathcal{J}} |S:S \cap M|^{m-1} < E_{nc} \cdot |S|^{m-m/r-2}$;
\item $\sigma_{Ny}(Y) \geq \sum_{M \in \mathcal{M}} |S:S \cap M|^{m-1}$ (notation is as in Definition \ref{star}) whenever $Y$ is a primitive monolithic group with socle $N$ and $y \in Y$ is such that $\langle N,y \rangle = Y$.
\item $\sum_{M \in \mathcal{J}} |S:S \cap M|^{m-1} + 2^{m-1} < \sum_{M \in \mathcal{M}} |S:S \cap M|^{m-1}$;
\end{enumerate}
Then $H \cong G$, in other words $H$ is monolithic.
\end{teor}
\begin{proof}
By Lemma \ref{starmin}, it is enough to show that $\sigma(G) < 2 \sigma^{\ast}(G)$. Let us do that. Since (1) holds, we may apply Theorem \ref{sopra} and obtain that $\sigma(G) \leq 2^{m-1} + \sum_{M \in \mathcal{M} \cup \mathcal{J}} |S:S \cap M|^{m-1}$. Fix a set $\Omega$ of cosets of $N$ in $G$ such that $\sigma^{\ast}(G) = \sigma_{\Omega}(G)$. Clearly, if $gN \in \Omega$ then $\sigma^{\ast}(G) \geq \sigma_{Ng}(G) = \sigma_{Ng}( \langle N,g \rangle)$. By (2) and Lemma \ref{nonciclo}, $\rho(g)$ is an $m$-cycle, therefore $\langle N,g \rangle$ is a primitive monolithic group hence by (3) $$\sigma^{\ast}(G) \geq \sigma_{Ng} (G) = \sigma_{Ng} (\langle N,g \rangle) \geq \sum_{M \in \mathcal{M}} |S:S \cap M|^{m-1}.$$Therefore by Theorem \ref{sopra} and (4), $$\sigma(G) \leq 2^{m-1} + \sum_{M \in \mathcal{M} \cup \mathcal{J}} |S:S \cap M|^{m-1} < 2 \sum_{M \in \mathcal{M}} |S:S \cap M|^{m-1} \leq 2 \sigma^{\ast}(G).$$Therefore $\sigma(G) < 2 \sigma^{\ast}(G)$.
\end{proof}
Fulfilling condition (3) requires the type of results listed in Theorems \ref{t1} and \ref{t2} (indeed, note that in Condition (3) the quotient $Y/N$ is cyclic). In my Ph.D. thesis I give several examples of applications of this result, and in particular I prove the following.
\begin{cor}
Let $H$ be a non-abelian $\sigma$-elementary group. If all the minimal subnormal subgroups of $H$ are alternating groups of even degree larger than $30$ then $H$ is monolithic.
\end{cor}
| {
"timestamp": "2013-01-07T02:01:47",
"yymm": "1301",
"arxiv_id": "1301.0743",
"language": "en",
"url": "https://arxiv.org/abs/1301.0743",
"abstract": "Given a finite non-cyclic group $G$, call $\\sigma(G)$ the smallest number of proper subgroups of $G$ needed to cover $G$. Lucchini and Detomi conjectured that if a nonabelian group $G$ is such that $\\sigma(G) < \\sigma(G/N)$ for every non-trivial normal subgroup $N$ of $G$ then $G$ is \\textit{monolithic}, meaning that it admits a unique minimal normal subgroup. In this paper we show how this conjecture can be attacked by the direct study of monolithic groups.",
"subjects": "Group Theory (math.GR)",
"title": "Covering monolithic groups with proper subgroups",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9770226267447514,
"lm_q2_score": 0.7248702880639791,
"lm_q1q2_score": 0.7082146728934935
} |
https://arxiv.org/abs/1304.7717 | The Randomized Dependence Coefficient | We introduce the Randomized Dependence Coefficient (RDC), a measure of non-linear dependence between random variables of arbitrary dimension based on the Hirschfeld-Gebelein-Rényi Maximum Correlation Coefficient. RDC is defined in terms of correlation of random non-linear copula projections; it is invariant with respect to marginal distribution transformations, has low computational cost and is easy to implement: just five lines of R code, included at the end of the paper. | \section{Introduction}\label{sec:intro}
Measuring statistical dependence between random variables is a fundamental
problem in statistics. Commonly used measures of dependence, Pearson's rho,
Spearman's rank or Kendall's tau are computationally efficient and
theoretically well understood, but consider only a limited class of association
patterns, like linear or monotonically increasing functions. The development of
non-linear dependence measures is challenging because of the radically larger
amount of possible association patterns.
Despite these difficulties, many non-linear statistical dependence measures
have been developed recently. Examples include the Alternating Conditional
Expectations or \emph{backfitting algorithm} (ACE) \cite{Breiman85,Hastie86},
Kernel Canonical Correlation Analysis (KCCA) \cite{Bach02}, (Copula) Maximum
Mean Discrepancy (MMD, CMMD in their HSIC formulations)
\cite{Gretton05,Gretton12,Poczos12}, Distance or Brownian Correlation (dCor)
\cite{Szekely07,Szekely10} and the Maximal Information Coefficient (MIC)
\cite{Reshef11}. However, these methods exhibit high computational demands (at
least quadratic costs in the number of samples for KCCA, MMD, CMMD, dCor or
MIC), are limited to measuring dependencies between scalar random variables
(ACE, MIC), show poor performance under the existence of additive noise (MIC)
or can be difficult to implement (ACE, MIC).
This paper develops the \emph{Randomized Dependence Coefficient} (RDC), an
estimator of the Hirschfeld-Gebelein-R\'enyi Maximum Correlation Coefficient
(HGR) addressing the issues listed above. RDC defines dependence between two
random variables as the largest canonical correlation between random non-linear
projections of their respective empirical copula-transformations. RDC is
invariant to monotonically increasing transformations, operates on random
variables of arbitrary dimension, and has computational cost of $O(n\log n )$
with respect to the sample size. Moreover, it is easy to implement: just five
lines of R code, included in Appendix \ref{sec:code}.
The following Section reviews the classic work of Alfr\'ed R\'enyi
\cite{Renyi59}, who proposed seven desirable fundamental properties of
dependence measures, proved to be satisfied by the
Hirschfeld-Gebelein-R\'enyi's Maximum Correlation Coefficient (HGR). Section
\ref{sec:rdc} introduces the Randomized Dependence Coefficient as an estimator
designed in the spirit of HGR, since HGR itself is computationally intractable.
Properties of RDC and its relationship to other non-linear dependence measures
are analysed in Section \ref{sec:rdc_prop}. Section \ref{sec:exps} validates
the empirical performance of RDC on a series of numerical experiments on both
synthetic and real-world data.
\section{Hirschfeld-Gebelein-R\'enyi's Maximum Correlation
Coefficient}\label{sec:renyi}
In 1959 \cite{Renyi59}, Alfr\'ed R\'enyi argued that a measure of dependence
$\rho^* : \mathcal{X} \times \mathcal{Y} \rightarrow [0,1]$ between random
variables $X\in\mathcal{X}$ and $Y\in\mathcal{Y}$ should satisfy seven
fundamental properties:
\begin{enumerate}
\item $\rho^*(X,Y)$ is defined for any pair of non-constant random variables
$X$ and $Y$.
\item $\rho^*(X,Y) = \rho^*(Y,X)$
\item $0 \leq \rho^*(X,Y) \leq 1$
\item $\rho^*(X,Y) = 0$ iff $X$ and $Y$ are statistically independent.
\item For bijective Borel-measurable functions $f,g : \mathbb{R}
\rightarrow \mathbb{R}$, $\rho^*(X,Y) = \rho^*(f(X),g(Y))$.
\item $\rho^*(X,Y) = 1$ if for Borel-measurable functions $f$ or $g$, $Y =
f(X)$ or $X = g(Y)$.
\item If $(X,Y) \sim \mathcal{N}(\bm \mu, \bm \Sigma)$, then $\rho^*(X,Y) =
|\rho(X,Y)|$, where $\rho$ is the correlation coefficient.
\end{enumerate}
R\'enyi also showed the \emph{Hirschfeld-Gebelein-R\'enyi Maximum
Correlation Coefficient} (HGR) \cite{Gebelein41,Renyi59} to satisfy
all these properties. HGR was defined by Gebelein in 1941
\cite{Gebelein41} as the supremum of Pearson's correlation coefficient
$\rho$ over all Borel-measurable functions $f,g$ of finite variance:
\begin{equation}\label{eq:hgr}
\text{hgr}(X,Y) = \sup_{f,g} \rho(f(X),g(Y)),
\end{equation}
Since the supremum in \eqref{eq:hgr} is over an infinite-dimensional
space, HGR is not computable. It is an abstract concept, not a
practical dependence measure. In the following we propose a scalable
estimator with the same structure as HGR: the Randomized Dependence
Coefficient.
\section{Randomized Dependence Coefficient} \label{sec:rdc}
The \emph{Randomized Dependence Coefficient} (RDC) measures the dependence
between random samples $\bm X \in \mathbb{R}^{p\times n}$ and $\bm Y \in
\mathbb{R}^{q\times n}$ as the largest canonical correlation between $k$
randomly chosen non-linear projections of their copula transformations. Before
Section~\ref{sec:formal-definition-or} defines this concept formally, we
describe the three necessary steps to construct the RDC statistic:
copula-transformation of each of the two random samples
(Section~\ref{sec:estim-copula-transf}), projection of the copulas through $k$
randomly chosen non-linear maps (Section~\ref{sec:gener-rand-non}) and
computation of the largest canonical correlation between the two sets of
non-linear random projections (Section~\ref{sec:comp-canon-corr}). Figure
\ref{fig:rdcsteps} offers a sketch of this process.
\begin{figure}[h!]
\includegraphics[width=\textwidth]{pipeline.pdf}
\caption{RDC computation for a simple set of samples
$\{(x_i,y_i)\}_{i=1}^{100}$ drawn from a noisy circular pattern: The samples
are used to estimate the copula, then mapped with randomly drawn non-linear
functions. The RDC is the largest canonical correlation between these
non-linear projections.}
\label{fig:rdcsteps}
\end{figure}
\subsection{Estimation of Copula-Transformations}
\label{sec:estim-copula-transf}
To achieve invariance with respect to transformations on marginal distributions
(such as shifts or rescalings), we operate on the \emph{empirical copula
transformation} of the data \cite{Nelsen06,Poczos12}. Consider a random vector
$\bm X = (X_1, \ldots, X_d)$ with continuous marginal cumulative distribution
functions (cdfs) $P_i$, $1 \leq i \leq d$. Then the vector $\bm U =
(U_1,\ldots,U_d) := \bm P(\bm X) = (P_1(X_1),\ldots,P_d(X_d))$, known as the
\emph{copula transformation}, has uniform marginals:
\begin{theorem}
(Probability Integral Transform \cite{Nelsen06}) For a random variable $X$
with cdf $P$, the random variable $U := P(X)$ is uniformly distributed on
$[0,1]$.
\end{theorem}
The random variables $U_1, \ldots, U_d$ are known as the observation ranks of
$X_1, \ldots, X_d$. Crucially, $\bm U$ preserves the dependence structure of
the original random vector $\bm X$, but ignores each of its $d$ marginal forms
\cite{Nelsen06}. The joint distribution of $\bm U$ is known as the copula of
$\bm X$:
\begin{theorem}
(Sklar \cite{Sklar59}) Let the random vector $\bm X = (X_1, \ldots, X_d)$
have continuous marginal cdfs $P_i$, $1 \leq i \leq d$. Then, the joint
cumulative distribution of $\bm X$ is uniquely expressed as:
\begin{equation}
P(X_1, \ldots, X_d) = C(P_1(X_1), \ldots, P_d(X_d)),
\end{equation}
where the distribution $C$ is known as the copula of $\bm X$.
\end{theorem}
A practical estimator of the univariate cdfs $P_1, \ldots, P_d$ is the
\emph{empirical cdf}:
\begin{equation}
P_n(x) := \frac{1}{n} \sum_{i=1}^n \mathbb{I}(X_i \leq x),
\end{equation}
which gives rise to the \emph{empirical copula transformations} of a
multivariate sample:
\begin{equation}
\bm P_n(\bm x) = [{P}_{n,1}(x_1), \ldots, {P}_{n,d}(x_d)].
\end{equation}
The Massart-Dvoretzky-Kiefer-Wolfowitz inequality \cite{Massart90} can be used
to show that empirical copula transformations converge fast to the true
transformation as the sample size increases:
\begin{theorem}
(Convergence of the empirical copula, \cite[Lemma 7]{Poczos12}) Let $\bm X_1,
\ldots, \bm X_n$ be an i.i.d. sample from a probability distribution over
$\mathbb{R}^d$ with marginal cdf's $P_1, \ldots, P_d$. Let $\bm P(\bm X)$ be
the copula transformation and $\bm P_n(\bm X)$ the empirical copula
transformation. Then, for any $\epsilon > 0$:
\begin{equation}
\Pr \left[ \sup_{\bm x \in \mathbb{R}^d} \| \bm P(\bm x) - \bm P_n(\bm x)
\|_2 > \epsilon \right] \leq 2 d \exp \left( -\frac{2 m
\epsilon^2}{d}\right).
\end{equation}
\end{theorem}
Computing $\bm P_n(\bm X)$ involves sorting the marginals of $\bm X \in
\mathbb{R}^{d\times n}$, thus $O(d n \log (n))$ operations.
\subsection{Generation of Random Non-Linear Projections}
\label{sec:gener-rand-non}
The second step of the RDC computation is to augment the empirical copula
transformations with non-linear projections, so that linear methods can
subsequently be used to capture non-linear dependencies on the original data.
This is a classic idea also used in other areas, particularly in regression. In
an elegant result, Rahimi and Brecht \cite{Rahimi08} proved that linear
regression on random, non-linear projections of the original feature space can
generate high-performance regressors:
\begin{theorem}\label{thm:rahimi}
(Rahimi-Brecht)
Let $p$ be a distribution on $\Omega$ and $|\phi(\bm x; \bm w)| \leq 1$.
Let $\mathcal{F} = \left\lbrace \left. f(\bm x) = \int_\Omega
\alpha(\bm w) \phi(\bm x; \bm w) \mathrm{d}\bm w \right|
|\alpha(\bm w)| \leq C p(\bm w)\right\rbrace$.
Draw $\bm w_1, \ldots, \bm w_k$ iid from $p$.
Further let $\delta > 0$, and $c$ be some $L$-Lipschitz loss function,
and consider data $\{\bm x_i, y_i\}_{i=1}^n$ drawn iid from some
arbitrary $P(\bm X,Y)$. The $\alpha_1, \ldots, \alpha_k$ for which
${f}_k(\bm x) = \sum_{i=1}^k \alpha_i \phi(\bm x; \bm w_i)$ minimizes
the empirical risk $c(f_k(\bm x),y)$ has a distance from the $c$-optimal
estimator in $\mathcal{F}$ bounded by
\begin{equation}
\mathbb{E}_P[c({f}_k(\bm x),y)] - \min_{f\in \mathcal{F}}
\mathbb{E}_P[c(f(\bm x),y)] \leq O\left(\left( \frac{1}{\sqrt{n}} +
\frac{1}{\sqrt{k}} \right) LC\sqrt{\log \frac{1}{\delta}}\right)
\end{equation}
with probability at least $1-2\delta$.
\end{theorem}
Intuitively, Theorem \ref{thm:rahimi} states that randomly selecting $\bm w_i$
in $\sum_{i=1}^k \alpha_i \phi(\bm x; \bm w_i)$ instead of optimising them
causes only bounded error.
The choice of the non-linearities $\phi : \mathbb{R} \rightarrow \mathbb{R}$ is
the main, unavoidable assumption in RDC. This choice is a well-known problem
common to all non-linear regression methods and has been studied extensively in
the theory of regression as the selection of reproducing kernel Hilbert space
\cite[\textsection3.13]{learningwithkernels}. The choice of the family (space)
of features, and of probability distributions over it, is unlimited. The only
way to favour one such family and distribution over another is to use prior
assumptions about which kind of distributions the method will typically have to
analyse.
Features popular in parts of the literature are sigmoids, parabolas, radial
basis functions, complex sinusoids, sines or cosines. In our experiments, we
will use sine and cosine projections, $\phi(\bm w^T \bm x +b) := (\cos(\bm w^T
\bm x + b), \sin(\bm w^T \bm x + b))$. Arguments favouring this choice are
that shift-invariant kernels are approximated with these features when using
the appropriate random parameter sampling distribution
\cite{Rahimi08},\cite[p.~208]{gihman4s:_theor_stoch_proces}
\cite[p.~24]{stein99:_inter_spatial_data}, and that functions with absolutely
integrable Fourier transforms are approximated with $L_2$ error below
$O(1/\sqrt{k})$ by $k$ of these features \cite{Jones92}.
Let the random parameters $\bm w_i \sim \mathcal{N}(\bm 0, s\bm I)$, $b_i \sim
\mathcal{U}[-\pi,\pi]$. Choosing $\bm w_i$ to be Normal is analogous to the use
of the Gaussian kernel for MMD, CMMD or KCCA \cite{Rahimi08}.
Tuning $s$ is analogous to selecting the kernel width, that is, to regularize the
non-linearity of the random projections.
Given a data collection $\bm X = (\bm x_1, \ldots, \bm x_n)$, we will
denote by
\begin{equation}
\bm \Phi(\bm X; k,s) :=
\left(
\begin{array}{ccc}
\phi(\bm w_1^T \bm x_1+b_1) & \cdots & \phi(\bm w_k^T \bm x_1+b_k)\\
\vdots & \vdots & \vdots \\
\phi(\bm w_1^T \bm x_n+b_1) & \cdots & \phi(\bm w_k^T \bm x_n +b_k)
\end{array}
\right)^T
\end{equation}
the $k-$th order random non-linear projection from $\bm X \in \mathbb{R}^{d
\times n}$ to $\bm \Phi^{k,s}_{\bm X} := \bm \Phi(\bm X; k, s) \in
\mathbb{R}^{2k \times n}$. The computational complexity of computing $\bm
\Phi^{k,s}_{\bm X}$ using naive matrix multiplications is $O(kdn)$. However,
recent techniques \cite{Le13} allow computing these feature expansions within a
computational cost of $O(k\log(d)n)$ using $O(k)$ storage.
\subsection{Computation of Canonical Correlations} \label{sec:comp-canon-corr}
The final step of RDC is to compute the linear combinations of the augmented
empirical copula transformations that have maximal correlation. Canonical
Correlation Analysis (CCA, \cite{Haerdle07}) is the calculation of pairs of
basis vectors $(\bm \alpha, \bm \beta)$ such that the projections $\bm \alpha^T
\bm X$ and $\bm \beta^T \bm Y$ of two random samples $\bm X \in
\mathbb{R}^{p\times n}$ and $\bm Y \in \mathbb{R}^{q\times n}$ are maximally
correlated. The correlations between the projected (or canonical) random
samples are referred to as canonical correlations. There exist up to
$\max(\text{rank}(\bm X), \text{rank}(\bm Y))$ of them. Canonical correlations
$\rho^2$ are the solutions to the eigenproblem:
\begin{align}
\left(
\begin{array}{cc}
\bm 0 & \bm C_{xx}^{-1}\bm C_{xy}\\
\bm C_{yy}^{-1}\bm C_{yx} & \bm 0
\end{array}
\right)
\left(
\begin{array}{c}
\bm \alpha\\
\bm \beta
\end{array}
\right)
=
\rho^2
\left(
\begin{array}{c}
\bm \alpha\\
\bm \beta
\end{array}
\right)
,\label{eq:eigenset}
\end{align}
where $\bm C_{xy} = \text{cov}(\bm X, \bm Y)$ and the matrices $\bm C_{xx}$ and
$\bm C_{yy}$ are assumed to be invertible. Therefore, the largest canonical
correlation $\rho_1$ between $\bm X$ and $\bm Y$ is the supremum of the
correlation coefficients over their linear projections, that is:
$ \rho_1(\bm X, \bm Y) = \sup_{\bm \alpha, \bm \beta} \rho(\bm \alpha^T \bm X,
\bm \beta^T \bm Y).
$
When $p,q \ll n$, the cost of CCA is dominated by the estimation of the matrices $\bm
C_{xx}$, $\bm C_{yy}$ and $\bm C_{xy}$, hence being $O((p+q)^2n)$ for two
random variables of dimensions $p$ and $q$, respectively.
\subsection{Formal Definition or RDC}
\label{sec:formal-definition-or}
Given the random samples $\bm X \in \mathbb{R}^{p\times n}$ and $\bm Y \in
\mathbb{R}^{q\times n}$ and the parameters $k \in \mathbb{N}_+$ and $s \in
\mathbb{R}_+$, the Randomized Dependence Coefficient between $\bm X$ and $\bm
Y$ is defined as:
\begin{equation}\label{eq:rdc}
\text{rdc}(\bm X, \bm Y; k,s) :=
\sup_{\bm \alpha, \bm \beta}\rho\left(\bm \alpha^T \bm \Phi_{\bm P(\bm X)}^{k,s}, \bm \beta^T
\bm \Phi_{\bm P(\bm Y)}^{k,s}\right).
\end{equation}
\section{Properties of RDC}\label{sec:rdc_prop}
\paragraph{Computational complexity:} In the typical setup (very large $n$,
large $p$ and $q$, small $k$) the computational complexity of RDC is dominated
by the calculation of the copula-transformations. Hence, we achieve a cost in
terms of the sample size of $O((p+q) n \log n + kn\log(pq) + k^2n) \approx O(n
\log n)$.
\paragraph{Ease of implementation:} An implementation of RDC in R is included
in the Appendix \ref{sec:code}.
\paragraph{Relationship to the HGR coefficient:} \label{sec:relat-hgr}
It is tempting to wonder whether RDC is a consistent, or even an efficient
estimator of the HGR coefficient. However, a simple experiment shows
that it is not desirable to approximate HGR exactly on finite datasets:
Consider $p(X,Y)=\mathcal{N}(x;0,1)\mathcal{N}(y;0,1)$ which is independent,
thus, by both R\'enyi's 4th and 7th properties, has $\mathrm{hgr}(X,Y)=0$.
However, for finitely many $N$ samples from $p(X,Y)$, almost surely, values in
both $X$ and $Y$ are pairwise different and separated by a finite difference.
So there exist continuous (thus Borel measurable) functions $f(X)$ and $g(Y)$
mapping both $X$ and $Y$ to the sorting ranks of $Y$, i.e.
$f(x_i)=g(y_i)\;\forall (x_i,y_i)\in(\bm X,\bm Y)$. Therefore, the
finite-sample version of Equation \eqref{eq:hgr} is constant and equal to ``1''
for continuous random variables. Meaningful measures of dependence from finite
samples thus must rely on some form of regularization. RDC achieves this by
approximating the space of Borel measurable functions with the restricted
function class $\mathcal{F}$ from Theorem \ref{thm:rahimi}:
Assume the optimal transformations $f$ and $g$ (Equation 1) to belong to the
Reproducing Kernel Hilbert Space $\mathcal{F}$ (Theorem 4), with associated
shift-invariant, positive semi-definite kernel function $k(\bm x, \bm x') =
\langle \bm \phi(\bm x), \bm \phi(\bm x')\rangle_\mathcal{F} \leq 1$. Then, with
probability greater than $1-2\delta$:
\begin{equation}
\label{eq:err_total}
\mathrm{hgr}(\bm X, \bm Y; \mathcal{F}) -
\mathrm{rdc}(\bm X, \bm Y; k)
= O\left(\left(\frac{\|\bm
m\|_F}{\sqrt{n}}+\frac{LC}{\sqrt{k}}\right)
\sqrt{\log\frac{1}{\delta}}\right),\end{equation}
where $\bm m := \bm \alpha \bm \alpha^T +\bm \beta \bm \beta^T$ and $n$, $k$
denote the sample size and number of random features. The bound
(\ref{eq:err_total}) is the sum of two errors. The error $O(1/\sqrt{n})$ is due
to the convergence of CCA's largest eigenvalue in the finite sample size
regime. This result \cite[Theorem 6]{Hardoon09} is originally obtained by
posing CCA as a least squares regression on the product space induced by the
feature map $\bm \psi(\bm x, \bm y) = [\bm \phi(\bm x)\bm \phi(\bm x)^T, \bm
\phi(\bm y) \phi(\bm y)^T, \sqrt{2}\bm \phi(\bm x) \phi(\bm y)^T]^T$. Because of
approximating $\bm \psi$ with $k$ random features, an additional error
$O(1/\sqrt{k})$ is introduced in the least squares regression \cite[Lemma
3]{Rahimi08}. Therefore, an equivalence between RDC and KCCA is established if
RDC uses an infinite number of sine/cosine features, the random sampling distribution is
set to the inverse Fourier transform of the shift-invariant kernel used by KCCA and
the copula-transformations are discarded.
However, when $k \geq n$ regularization
is needed to avoid spurious perfect correlations, as discussed above.
\paragraph{Relationship to other estimators:}\label{sec:comparison} Table
\ref{table:comparison} summarizes several state-of-the-art dependence measures
showing, for each measure, whether it allows for general non-linear dependence
estimation, handles multidimensional random variables, is invariant with
respect to changes in marginal distributions, returns a statistic in $[0,1]$,
satisfy R\'enyi's properties (Section \ref{sec:renyi}), and how many parameters
it requires. As parameters, we here count the kernel function for kernel
methods, the basis function and number of random features for RDC, the stopping
tolerance for ACE and the search-grid size for MIC, respectively. Finally, the
table lists computational complexities with respect to sample size.
\begin{table}[h!]
\caption{Comparison between non-linear dependence measures.}
\vskip 0.3 cm
\resizebox{\textwidth}{!}
{
\begin{tabular}{lccccccl}
\hline
\head{1.5cm}{\textbf{Name of Coeff.}} &
\head{ .9cm}{\textbf{Non-Linear}} &
\head{1.2cm}{\textbf{Vector Inputs}} &
\head{1.6cm}{\textbf{Marginal Invariant}} &
\head{1.9cm}{\textbf{Renyi's Properties}} &
\head{1.1cm}{Coeff. \textbf{$\in [0,1]$}} &
\head{1cm}{\# \textbf{Par.}} &
\head{1cm}{\textbf{Comp. Cost}}\\\hline\hline
Pearson's $\rho$ & $\times$ & $\times$ & $\times$ & $\times$ & $\checkmark$ & 0 & $n$ \\ \hline
Spearman's $\rho$ & $\times$ & $\times$ & $\checkmark$ & $\times$ & $\checkmark$ & 0 & $n \log n$ \\ \hline
Kendall's $\tau$ & $\times$ & $\times$ & $\checkmark$ & $\times$ & $\checkmark$ & 0 & $n \log n$ \\ \hline
CCA & $\times$ & $\checkmark$ & $\times$ & $\times$ & $\checkmark$ & 0 & $n$ \\ \hline
KCCA \cite{Bach02} & $\checkmark$ & $\checkmark$ & $\times$ & $\times$ & $\checkmark$ & 1 & $n^3$ \\ \hline
ACE \cite{Breiman85} & $\checkmark$ & $\times$ & $\times$ & $\checkmark$ & $\checkmark$ & 1 & $n$ \\ \hline
MIC \cite{Reshef11} & $\checkmark$ & $\times$ & $\times$ & $\times$ & $\checkmark$ & 1 & $2^n$ \\ \hline
dCor \cite{Szekely07} & $\checkmark$ & $\checkmark$ & $\times$ & $\times$ & $\checkmark$ & 1 & $n^2$ \\ \hline
MMD \cite{Gretton12} & $\checkmark$ & $\checkmark$ & $\times$ & $\times$ & $\times$ & 1 & $n^2$ \\ \hline
CMMD \cite{Poczos12} & $\checkmark$ & $\checkmark$ & $\checkmark$ & $\times$ & $\times$ & 1 & $n^2$ \\ \hline
\textbf{RDC} & $\checkmark$ & $\checkmark$ & $\checkmark$ & $\checkmark$ & $\checkmark$ & 2 & $n \log n$ \\ \hline\hline
\end{tabular}
}
\label{table:comparison}
\end{table}
\paragraph{Testing for independence with RDC:} Consider the hypothesis ``the
two sets of non-linear projections are mutually uncorrelated''. Under
normality assumptions (or large sample sizes), Bartlett's approximation
\cite{Mardia79} can be used to show:
\begin{equation}
\left(\frac{2k+3}{2}-n\right) \log \prod_{i=1}^k (1-\rho_i^2) \sim
\chi^2_{k^2},
\end{equation}
where $\rho_1, \ldots, \rho_k$ are the canonical correlations between the two
sets of non-linear projections $\bm \Phi_{\bm P(\bm X)}^{k,s}$ and $\bm \Phi_{\bm P(\bm Y)}^{k,s}$. Alternatively, non-parametric asymptotic
distributions can be obtained from the spectrum of the inner products of the
non-linear random projection matrices \cite[Theorem 3]{Zhang12}.
\section{Experimental Results}\label{sec:exps}
We performed numerical experiments on both synthetic and real-world data to
validate the empirical performance of RDC versus the non-linear dependence
measures listed in Table \ref{table:comparison}. In some experiments, we don't compare against to
KCCA due its prohibitive running times (see Table \ref{fig:times}).
\paragraph{Parameter selection:} The number of random features for RDC was set
to $k=10$ symmetrically for both random samples, since no significant
improvements were observed for larger values. However, this parameter can be
set to the largest value that fits within the available computational
budget. The random sampling parameters $(s_{\bm X}, s_{\bm Y})$ were set
independently for each of the two random samples, equal to their squared
euclidean distance empirical median \cite{Gretton12}. Competing kernel methods
make use of Gaussian RBF kernels of the form $k(\bm x, \bm x'; s_{\bm X}) =
exp(-\| \bm x- \bm x'\|^2/s_{\bm X})$ for the random variable $\bm X$ and
analogously for the random variable $\bm Y$. For the MIC statistic, the
search-grid size is set to $B(n) = n^{0.6}$, as recommended by the authors
\cite{Reshef11}. The stopping tolerance for ACE is set to $\epsilon = 0.01$,
the default value in the R package
\texttt{acepack}\footnote{\url{http://cran.r-project.org/web/packages/acepack/index.html}}.
\subsection{Synthetic Data}
\paragraph{Resistance to additive noise:}
We define the \emph{power} of a dependence measure as its ability to discern
between dependent and independent samples that share equal marginal forms. In
the spirit of Simon and
Tibshirani\footnote{\url{http://www-stat.stanford.edu/~tibs/reshef/comment.pdf}},
we conducted experiments to estimate the power of RDC as a measure of
non-linear dependence. We chose 8 bivariate association patterns, depicted inside little
boxes in Figure \ref{fig:power}. For each of the 8 association patterns, 500
repetitions of 500 samples were generated, in which the input variable was
uniformly distributed on the unit interval. Next, we regenerated the input
variable randomly, to generate independent versions of each sample with equal
marginals. Figure \ref{fig:power} shows the power for the discussed non-linear dependence
measures as the variance of some zero-mean Gaussian additive noise increases
from $1/30$ to $3$. RDC shows worse performance in the linear association
pattern due to noise overfitting and in the step-function due to the smoothness
prior induced by the use of sine/cosine basis functions, but has good
performance in non-functional association patterns (such as the circle and the
mixture of sinusoidal waves).
\paragraph{Running times:}
Table \ref{fig:times} summarizes running times (in seconds) for the considered
non-linear dependence measures on scalar, uniformly distributed, independent
samples of sizes $\{10^3, \ldots, 10^6\}$ when averaging over 100 runs. Single
runs above ten minutes were cancelled (empty cells in table).
In this comparison, Pearson's $\rho$, ACE, dCor and
MIC are using compiled C code, while RDC, along with MMD, CMMD and KCCA are
implemented as interpreted R code.
\begin{table}[h!]
\caption{Average running times (in seconds) for dependence measures on
versus sample sizes.}
\vskip 0.3 cm
\resizebox{\textwidth}{!}{
\begin{tabular}{l|cccccccc}
\hline
\bf sample size& \bf Pearson's $\rho$ & \bf RDC & \bf ACE & \bf
dCor & \bf MMD & \bf CMMD & \bf MIC & \bf KCCA\\\hline\hline
1,000 & 0.0001 & 0.0047 & 0.0080 & 0.3417 & 0.3103 & 0.3501 & 1.0983 & 166.29 \\\hline
10,000 & 0.0002 & 0.0557 & 0.0782 & 59.587 & 27.630 & 29.522 & --- & --- \\\hline
100,000 & 0.0071 & 0.3991 & 0.5101 & --- & --- & --- & --- & --- \\\hline
1,000,000 & 0.0914 & 4.6253 & 5.3830 & --- & --- & --- & --- & --- \\\hline
\hline
\end{tabular}
}
\label{fig:times}
\end{table}
\paragraph{Value of statistic in $[0,1]$:} Figure \ref{fig:pairs} shows RDC,
ACE, dCor, MIC, Pearson's $\rho$, Spearman's rank and Kendall's $\tau$
dependence estimates for 14 different associations of two scalar random
variables. RDC scores values close to one on all the proposed dependent
associations, whilst scoring values close to zero for the independent
association, depicted last. When the associations are Gaussian (first row),
RDC scores values close to the Pearson's correlation coefficient, as suggested
in the seventh property of R\'enyi (Section \ref{sec:renyi}).
\subsection{Feature Selection in Real-World Data} We performed greedy feature
selection via dependence maximization \cite{Song12} on eight real-world
datasets. More specifically, we attempted to construct the subset of features
$\mathcal{G} \subset \mathcal{X}$ that minimizes the normalized mean squared
regression error (NMSE) of a Gaussian process regressor. We do so by selecting
the feature $x^{(i)}$ maximizing dependence between the feature set
$\mathcal{G}_{i} = \{\mathcal{G}_{i-1} , x^{(i)}\}$ and the target variable $y$
at each iteration $i \in \{1, \ldots 10\}$, such that $\mathcal{G}_0 = \{
\emptyset \}$ and $x^{(i)} \notin \mathcal{G}_{i-1}$.
We considered 12 heterogeneous datasets, obtained from the UCI dataset
repository\footnote{\url{http://www.ics.uci.edu/~mlearn}}, the Gaussian process
web site Data\footnote{\url{http://www.gaussianprocess.org/gpml/data/}} and the
Machine Learning data set repostitory\footnote{\url{http://www.mldata.org}}.
Random training/test partitions are computed to be disjoint and equal sized.
Since $\mathcal{G}$ can be multi-dimensional, we compare RDC to the non-linear
methods dCor, MMD and CMMD. Given their quadratic computational demands, dCor,
MMD and CMMD use up to $1,000$ points when measuring dependence; this
constraint only applied on the \texttt{sarcos} and \texttt{calcensus} datasets.
Results are averages of $20$ random training/test partitions.
\begin{figure}[h!]
\includegraphics[width=\textwidth]{real.pdf}
\caption{Feature selection experiments on real-world datasets.}
\label{fig:featsel}
\end{figure}
Figure \ref{fig:featsel} summarizes the results for all datasets and algorithms
as the number of selected features increases. RDC performs best in most
datasets, with much lower running time than its
contenders.
\section{Conclusion}
\label{sec:conclusion}
We have presented the randomized dependence coefficient, a lightweight
non-linear measure of dependence between multivariate random samples.
Constructed as a finite-dimensional estimator in the spirit of the
Hirschfeld-Gebelein-R\'enyi maximum correlation coefficient, RDC performs well
empirically, is scalable to very large datasets, and is easy to adapt to concrete
problems.
\newpage
\clearpage
\begin{figure}[t!]
\includegraphics[width=\textwidth]{power.pdf}
\caption{Power of discussed measures on several bivariate association patterns
as noise increases. Insets show the noise-free form of each association
pattern.}
\label{fig:power}
\end{figure}
\begin{figure}[h!]
\centering
\includegraphics[width=\textwidth]{pairs.pdf}
\vskip -.5 cm
\caption{RDC, ACE, dCor, MINE, Pearson's $\rho$, Spearman's rank and Kendall's
$\tau$ estimates (numbers in tables above plots, in that order) for several
bivariate association patterns.}
\label{fig:pairs}
\end{figure}
| {
"timestamp": "2013-06-04T02:03:27",
"yymm": "1304",
"arxiv_id": "1304.7717",
"language": "en",
"url": "https://arxiv.org/abs/1304.7717",
"abstract": "We introduce the Randomized Dependence Coefficient (RDC), a measure of non-linear dependence between random variables of arbitrary dimension based on the Hirschfeld-Gebelein-Rényi Maximum Correlation Coefficient. RDC is defined in terms of correlation of random non-linear copula projections; it is invariant with respect to marginal distribution transformations, has low computational cost and is easy to implement: just five lines of R code, included at the end of the paper.",
"subjects": "Machine Learning (stat.ML)",
"title": "The Randomized Dependence Coefficient",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.977022625406662,
"lm_q2_score": 0.7248702880639791,
"lm_q1q2_score": 0.7082146719235523
} |
https://arxiv.org/abs/1709.09325 | Self-Similar Tilings of Fractal Blow-Ups | New tilings of certain subsets of $\mathbb{R}^{M}$ are studied, tilings associated with fractal blow-ups of certain similitude iterated function systems (IFS). For each such IFS with attractor satisfying the open set condition, our construction produces a usually infinite family of tilings that satisfy the following properties: (1) the prototile set is finite; (2) the tilings are repetitive (quasiperiodic); (3) each family contains self-similartilings, usually infinitely many; and (4) when the IFS is rigid in an appropriate sense, the tiling has no non-trivial symmetry; in particular the tiling is non-periodic. | \section{Introduction}
\label{sec:intro}
The subject of this paper is a new type of tiling of certain subsets $D$ of
$\mathbb{R}^{M}$. Such a domain $D$ is a fractal blow-up (as defined in
Section~\ref{defsec}) of certain similitude iterated function systems (IFSs);
see also \cite{manifold, strichartz}. For an important class of such tilings
it is the case that $D=\mathbb{R}^{M}$, as exemplified by the tiling of
Figure~\ref{fig:b} (on the right ) that is based on the \textquotedblleft
golden b" tile (on the left). We are also interested, however, in situations
where $D$ has non-integer Hausdorff dimension. The left panel in
Figure~\ref{sidebyside} shows the domain $D$, the right panel a tiling of $D$.
These examples are explored in Section~\ref{exsec}. In this work, tiles may be
fractals; pairs of distinct tiles in a tiling are required to be
non-overlapping, i.e., they intersect on a set whose Hausdorff dimension is
lower than that of the individual tiles. \begin{figure}[tbh]
\includegraphics[width=3cm, keepaspectratio]{GB.png} \hskip 10mm
\includegraphics[width=8cm, keepaspectratio]{EX2a.png} \caption{Golden b and
golden b tiling.}%
\label{fig:b}%
\end{figure}%
\begin{figure}[ptb]%
\centering
\includegraphics[
height=1.6475in,
width=5.2477in
]%
{sidebyside.png}%
\caption{The left image shows part of an infinite fractal blow-up $D$; the
right image shows part of a tiling of $D$ using a finite set of prototiles.
See Section \ref{exsec}.}%
\label{sidebyside}%
\end{figure}
These tilings come in families, one family for each similitude IFS whose
functions $f_{1},f_{2}\dots,f_{N}$ have scaling ratios that are integer powers
$s^{a_{1}},s^{a_{2}},\dots,s^{a_{N}}$ of a single real number $s$ and whose
attractor is non-overlapping. Each such family contains, in general, an
uncountable number of tilings. Each family has a finite set of prototiles.
The paper is organized as follows. Sections \ref{tilingsec} and \ref{defsec}
provide background and definitions relevant to tilings and to iterated
function systems. The construction of our tilings is given in Section
\ref{defsec}. The main theorems are stated precisely in Section \ref{defsec}
and proved in subsequent sections. Results appear in Section \ref{realabsec}
that define and discuss the relative and absolute addresses of tiles. These
concepts, useful towards understanding the relationships between different
tilings, are illustrated in Section \ref{exsec}. Also in Section~\ref{exsec}
are examples of tilings of $\mathbb{R}^{2}$ and of a quadrant of
$\mathbb{R}^{2}$. The Ammann (the golden b) tilings and related fractal
tilings are also discussed in that section, as is a blow-up of a Cantor set.
A subset $P$ of a tiling $T$ is called a \textit{patch} of $T$ if it is
contained in a ball of finite radius. A tiling $T$ is \textit{quasiperiodic}
(also called repetitive) if, for any patch $P$, there is a number $R>0$ such
that any disk of radius $R$ centered at a point contained in a tile of $T$
contains an isometric copy of $P$. Two tilings are \textit{locally isomorphic}
if any patch in either tiling also appears in the other tiling. A tiling $T$
is \textit{self-similar} if there is a similitude $\psi$ such that $\psi(t)$
is a union of tiles in $T$ for all $t\in T$. Such a map $\psi$ is called a
\textit{self-similarity}.
Let $\mathcal{F}$ be a similitude IFS whose functions have scaling ratios
$s^{a_{1}},s^{a_{2}},\dots,s^{a_{N}}$ as defined above. Let $[N]^{\ast}$ be
the set of finite words over the alphabet $[N]:=\{1,2,\dots,N\}$ and
$[N]^{\infty}$ be the set of infinite words over the alphabet $[N]$. For a
fixed IFS $\mathcal{F}$, our results show that:
\begin{enumerate}
\item For each $\theta\in[N]^{*}$, our construction yields a bounded tiling,
and for each $\theta\in[N]^{\infty}$, our construction yields an unbounded
tiling. In the latter case, the tiling, denoted $\pi(\theta)$, almost always
covers ${\mathbb{R}}^{M}$ when the attractor of the IFS has nonempty interior.
\item The mapping $\theta\mapsto\pi(\theta)$ is continuous with respect to the
standard topologies on the domain and range of $\pi$.
\item Under quite general conditions, the mapping $\theta\mapsto\pi(\theta)$
is injective.
\item For each such tiling, the prototile set is $\{sA, s^{2}A,\dots,
s^{a_{\max}} A\}$, where $A$ is the attractor of the IFS and $a_{\max} =
\max\{a_{1}, a_{2}, \dots, a_{N}\}$.
\item The constructed tilings, in the unbounded case, are repetitive
(quasiperiodic) and any two such tilings are locally isomorphic.
\item For all $\theta\in[N]^{\infty}$, if $\theta$ is eventually periodic,
then $\pi(\theta)$ is self-similar.
\item If $\mathcal{F}$ is strongly rigid, then how isometric copies of a pair
bounded tilings can overlap is extremely restricted: if the two tilings are
such that their overlap is a subset of each, then one tiling must be contained
in the other.
\item If $\mathcal{F}$ is strongly rigid, then the constructed tilings have no
non-identity symmetry. In particular, they are non-periodic.
\end{enumerate}
The concept of a rigid and a strongly rigid IFS is discussed in
Sections~\ref{strongsec}.
A special case of our construction (polygonal tilings, no fractals) appears in
\cite{polygon}, in which we took a more recreational approach, devoid of
proofs. Other references to related material are \cite{anderson, sadun}. This
work extends, but is markedly different from \cite{tilings}.
\section{\label{tilingsec}Tilings, Similitudes and Tiling Spaces}
Given a natural number $M$, this paper is concerned with certain tilings of
strict subsets of Euclidean space $\mathbb{R}^{M}$ and of $\mathbb{R}^{M}$
itself. A \textit{tile} is a perfect (i.e. no isolated points) compact
nonempty subset of $\mathbb{R}^{M}$. Fix a Hausdorff dimension $0<D_{H}\leq
M$. A \textit{tiling} in $\mathbb{R}^{M}$ is a set of tiles, each of Hausdorff
dimension $D_{H}$, such that every distinct pair is non-overlapping. Two tiles
are \textit{non-overlapping} if their intersection is of Hausdorff dimension
strictly less than $D_{H}$. The \textit{support} of a tiling is the union of
its tiles. We say that a tiling tiles its support. Some examples are presented
in Section~\ref{exsec}.
A \textit{similitude} is an affine transformation $f:{\mathbb{R}}%
^{M}\rightarrow{\mathbb{R}}^{M}$ of the form $f(x)=s\,O(x)+q$, where $O$ is an
orthogonal transformation and $q\in\mathbb{R}^{M}$ is the translational part
of $f(x)$. The real number $s>0$, a measure of the expansion or contraction of
the similitude, is called its \textit{scaling} \textit{ratio}. An
\textit{isometry} is a similitude of unit scaling ratio and we say that two
sets are isometric if they are related by an isometry. We write $\mathcal{E}$
to denote the group of isometries on $\mathbb{R}^{M}$.
The \textit{prototile set} $\mathcal{P}$ of a tiling $T$ is a minimal set of
tiles such that every tile in $T$ is an isometric copy of a tile in
$\mathcal{P}$. The tilings constructed in this paper have a finite prototile set.
Given a tiling $T$ we define $\partial T$ to be the union of the set of
boundaries of all of the tiles in $T$ and we let $\rho:\mathbb{R}%
^{M}\rightarrow\mathbb{S}^{M}$ be the usual $M$-dimensional stereographic
projection to the $M$-sphere, obtained by positioning $\mathbb{S}^{M}$ tangent
to $\mathbb{R}^{M}$ at the origin. We define the distance between tilings $T$
and $T^{\prime}$ to be%
\[
d_{\tau}(T,T^{\prime})=h(\overline{\rho(\partial T)},\overline{\rho(\partial
T^{\prime})})
\]
where the bar denotes closure and $h$ is the Hausdorff distance with respect
to the round metric on $\mathbb{S}^{M}$. Let $\mathbb{K(R}^{M})$ be the set of
nonempty compact subsets of $\mathbb{R}^{M}$. It is well known that $d_{\tau}$
provides a metric on the space $\mathbb{K}({\mathbb{R}}^{M})$ and that
$(\mathbb{K}({\mathbb{R}}^{M}),d_{\tau})$ is a compact metric space.
This paper examines spaces consisting, for example, of $\pi(\theta)$ indexed
by $\theta\in\left[ N\right] ^{\ast}$ with metric $d_{\tau}$. Although we
are aware of the large literature on tiling spaces, we do not explore the
larger spaces obtained by taking the closure of orbits of our tilings under
groups of isometries as in, for example, \cite{anderson, sadun}. We focus on
the relationship between the addressing structures associated with IFS theory
and the particular families of tilings constructed here.
\section{\label{defsec} Definition and Properties of IFS Tilings}
Let $\mathbb{N=\{}1,2,\cdots\}$ and $\mathbb{N}_{0}=\{0,1,2,\cdots\}$. For
$N\in\mathbb{N}$, let $[N]=\{1,2,\cdots,N\}$. Let $[N]^{\ast}=\cup
_{k\in\mathbb{N}_{0}}[N]^{k}$, where $[N]^{0}$ is the empty string, denoted
$\varnothing$.
See \cite{hutchinson} for formal background on iterated function systems
(IFSs). Here we are concerned with IFSs of a special form: let $\mathcal{F}%
=\{{\mathbb{R}}^{M};f_{1},f_{2},\cdots,f_{N}\}$, with $N\geq2$, be an IFS of
contractive similitudes where the scaling factor of $f_{n}$ is $s^{a_{n}}$
with $0<s<1$ where $a_{n}\in\mathbb{N}$. There is no loss of generality in
assuming that the greatest common divisor is one: $\gcd\{a_{1},a_{2}%
,\cdots,a_{N}\}=1$. That is, for $x\in{\mathbb{R}}^{M}$, the function
$f_{n}:\mathbb{R}^{M}\rightarrow\mathbb{R}^{M}$ is defined by
\[
f_{n}(x)=s^{a_{n}}O_{n}(x)+q_{n}%
\]
where $O_{n}$ is an orthogonal transformation and $q_{n}\in{\mathbb{R}}^{M}$.
It is convenient to define%
\[
a_{\max}=\max\{a_{i}:i=1,2,\dots,N\}.
\]
The \textit{attractor} $A$ of $\mathcal{F}$ is the unique solution in
$\mathbb{K(R}^{M})$ to the equation
\[
A=\bigcup\limits_{i\in\lbrack N]}f_{i}(A)\text{.}%
\]
It is assumed throughout that $A$ obeys the open set condition (OSC) with
respect to $\mathcal{F}$. As a consequence, the intersection of each pair of
distinct tiles in the tilings that we construct either have empty intersection
or intersect on a relatively small set. More precisely, the OSC implies that
the Hausdorff dimension of $A$ is strictly greater than the Hausdorff
dimension of the \textit{set of overlap} $\mathcal{O=\cup}_{i\neq j}%
f_{i}(A)\cap f_{j}(A)$. Similitudes applied to subsets of the set of overlap
comprise the sets of points at which tiles may meet. See \cite[p.481]{bandt}
for a discussion concerning measures of attractors compared to measures of the
set of overlap.
In what follows, the space $[N]^{\ast}\cup\lbrack N]^{\infty}$ is equipped
with a metric $d_{[N]^{\ast}\cup\lbrack N]^{\infty}}$ such that it becomes
compact. First, define the \textquotedblleft length" $\left\vert
\theta\right\vert $ of $\theta\in\lbrack N]^{\ast}\cup\lbrack N]^{\infty}$ as
follows. For $\theta=\theta_{1}\theta_{2}\cdots\theta_{k}\in\lbrack N]^{\ast}$
define $\left\vert \theta\right\vert =k$, and for $\theta\in\lbrack
N]^{\infty}$ define $\left\vert \theta\right\vert =\infty$. Now define
$d_{[N]^{\ast}\cup\lbrack N]^{\infty}}(\theta,\omega)=0$ if $\theta=\omega,$
and
\[
d_{[N]^{\ast}\cup\lbrack N]^{\infty}}(\theta,\omega)=2^{-\mathcal{N(}%
\theta,\omega)}%
\]
if $\theta\neq\omega$, where $\mathcal{N(}\theta,\omega)$ is the index of the
first disagreement between $\theta$ and $\omega$ (and $\theta$ and $\omega$
are understood to disagree at index $k$ if either $|\theta|<k$ or $|\omega|<k$
). It is routine to prove that $([N]^{\ast}\cup\lbrack N]^{\infty
},d_{[N]^{\ast}\cup\lbrack N]^{\infty}})$ is a compact metric space.
A point $\theta\in\lbrack N]^{\infty}$ is \textit{eventually periodic} if
there exists $m\in\mathbb{N}_{0}$ and $n\in\mathbb{N}$ such that $\theta
_{m+i}=\theta_{m+n+i}$ for all $i\geq1$. In this case we write $\theta
=\theta_{1}\theta_{2}\cdots\theta_{m}\overline{\theta_{m+1}\theta_{m+2}%
\cdots\theta_{m+n}}$.
For $\theta=\theta_{1}\theta_{2}\cdots\theta_{k}\in\lbrack N]^{\ast}$, the
following simplifying notation will be used:
\[
\begin{aligned} f_{\theta} &= f_{\theta_{1}} f_{\theta_{2}}\cdots f_{\theta_k} \\
f_{-\theta} &=f_{\theta_{1}}^{-1}f_{\theta_{2}}^{-1}\cdots f_{\theta_k}^{-1}=(f_{\theta_k\theta_{k-1}\cdots\theta_{1}})^{-1},
\end{aligned}
\]
with the convention that $f_{\theta}$ and $f_{-\theta}$ are the identity
function $id$ if $\theta=\varnothing$. Likewise, for all $\theta\in\lbrack
N]^{\infty}$ and $k\in\mathbb{N}_{0}$ define $\theta|k=\theta_{1}\theta
_{2}\cdots\theta_{k}$, and
\[
f_{-\theta|k}=f_{\theta_{1}}^{-1}f_{\theta_{2}}^{-1}\cdots f_{\theta_{k}}%
^{-1}=(f_{\theta_{k}\theta_{k-1}\cdots\theta_{1}})^{-1},
\]
with the convention that $f_{-\theta|0}=id$.
For $\sigma=\sigma_{1}\sigma_{2}\cdots\sigma_{k}\in\lbrack N]^{\ast}$ and with
$\left\{ a_{1},\dots,a_{N}\right\} $ the scaling powers defined above, let
\[
e(\sigma)=a_{\sigma_{1}}+a_{\sigma_{2}}+\cdots+a_{\sigma_{k}}\qquad
\text{and}\qquad e^{-}(\sigma)=a_{\sigma_{1}}+a_{\sigma_{2}}+\cdots
+a_{\sigma_{k-1}},
\]
with the conventions $e(\varnothing)=e^{-}(\varnothing)=0$. Let
\[
\Omega_{k}:=\{\sigma\in\lbrack N]^{\ast}:e(\sigma)>k\geq e^{-}(\sigma)\}
\]
for all $k\in\mathbb{N}_{0}$, and note that $\Omega_{0}=[N]$. We also write,
in some places, $\sigma^{-}=\sigma_{1}\sigma_{2}\cdots\sigma_{k-1}$ so that
\[
e^{-}(\sigma)=e(\sigma^{-}).
\]
\begin{definition}
\label{defONE} A mapping $\mathbb{\pi}$ from $[N]^{\ast}\cup\lbrack
N]^{\infty}$ to collections of subsets of $\mathbb{R}^{M}$ is defined as
follows. For $\theta\in\lbrack N]^{\ast}$
\[
\mathbb{\pi}(\theta):=\{f_{_{-\theta}}f_{\sigma}(A):\sigma\in\Omega
_{e(\theta)}\},
\]
and for $\theta\in\lbrack N]^{\infty}$
\[
\mathbb{\pi}(\theta):=\bigcup\limits_{k\in\mathbb{N}_{0}} \pi(\theta|k).
\]
Let $\mathbb{T}$ be the image of $\pi$, i.e.
\[
\mathbb{T=\{\pi}(\theta):\theta\in\lbrack N]^{\ast}\cup\lbrack N]^{\infty}\}.
\]
\end{definition}
It is consequence of Theorem~\ref{theoremONE}, stated below, that the elements
of $\mathbb{T}$ are tilings. We refer to $\mathbb{\pi}(\theta)$ as an
\textit{IFS tiling}, but usually drop the term \textquotedblleft IFS". It is a
consequence of the proof of Theorem \ref{theoremONE}, given in Section
\ref{ProofofONE}, that the support of $\mathbb{\pi}(\theta)$ is what is
sometimes referred to as a \textit{fractal blow-up} \cite{manifold,
strichartz}. More exactly, if $F_{k}:=f_{_{-\theta|k}}(A)$, then
\[
\text{support}\,(\mathbb{\pi}(\theta))=\bigcup\limits_{k\in\mathbb{N}_{0}%
}F_{k}.
\]
Thus the support of $\mathbb{\pi}(\theta)$ is the limit of an increasing union
of sets $F_{0}\subseteq F_{1}\subseteq F_{2}\subseteq\cdots$, each similar to
$A$.
The theorems of this paper are summarized in the rest of this section. The
first two theorems, as well as a proposition in Section \ref{realabsec},
reveal general information about the tilings in $\mathbb{T}$ without the
rigidity condition that is assumed in the second two theorems. The proof of
the following theorem appears in Section~\ref{ProofofONE}.
\begin{theorem}
\label{theoremONE} Each set $\pi(\theta)$ in $\mathbb{T}$ is a tiling of a
subset of $\mathbb{R}^{M}$, the subset being bounded when $\theta\in[N]^{*}$
and unbounded when $\theta\in\lbrack N]^{\infty}$. For all $\theta\in\lbrack
N]^{\infty}$ the sequence of tilings $\left\{ \pi(\theta|k)\right\}
_{k=0}^{\infty}$ is nested according to%
\begin{equation}
\{f_{i}(A):i\in\lbrack N]\}=\pi(\varnothing)\subset\pi(\theta|1)\subset
\pi(\theta|2)\subset\pi(\theta|3)\subset\cdots\text{ .} \label{eqthmONE}%
\end{equation}
For all $\theta\in\lbrack N]^{\infty}$, the prototile set for $\mathbb{\pi
}(\theta)$ is $\{s^{i}A:i=1,2,\cdots,a_{\max}\}$.\textit{ }Furthermore
\[
\pi:[N]^{\ast}\cup\lbrack N]^{\infty}\rightarrow\mathbb{T}%
\]
is a continuous map from the compact metric space $[N]^{\ast}\cup\lbrack
N]^{\infty}$ into the space $(\mathbb{K}({\mathbb{R}}^{M}),d_{\tau})$.
\end{theorem}
The proof of the following theorem is given in Section \ref{proofofTHREE}.
\begin{theorem}
\label{theoremTHREE}
\begin{enumerate}
\item Each tiling in $\mathbb{T}$ is quasiperiodic and each pair of such
tilings in $\mathbb{T}$ are locally isomorphic.
\item If $\theta$ is eventually periodic, then $\pi(\theta)$ is self-similar.
In fact, if $\theta=\alpha\overline{\beta}$ for some $\alpha,\beta\in\left[
N\right] ^{\ast}$ then $f_{-\alpha}f_{-\beta}\left( f_{-\alpha}\right)
^{-1}$ is a self-similarity of $\pi(\theta)$.
\end{enumerate}
\end{theorem}
In Section \ref{strongsec} the concept of \textit{rigidity} of an IFS is
defined. We postpone the definition because additional notation is required.
There are numerous examples of rigid $\mathcal{F}$, including the golden b IFS
in Section \ref{exsec}. The following theorem is proved in Section
\ref{strongsec}.
\begin{theorem}
\label{intersectthm} Let $\mathcal{F}$ be strongly rigid. If $\theta
,\theta^{\prime}\in\lbrack N]^{\ast}$and $E\in\mathcal{E}$ are such that
$\pi(\theta)\cap E\pi(\theta^{\prime})$ is a nonempty common tiling, then
either $\pi(\theta)\subset E\pi(\theta^{\prime})$ or $E\pi(\theta^{\prime
})\subset\pi(\theta)$. If $e(\theta)=e(\theta^{\prime}),$ then $E\pi
(\theta^{\prime})=\pi(\theta).$
\end{theorem}
A \textit{symmetry} of a tiling is an isometry that takes tiles to tiles. A
tiling is \textit{periodic} if there exists a translational symmetry;
otherwise the tiling is \textit{non-periodic}. For example, any tiling of a
quadrant of $\mathbb{R}^{2}$ by congruent squares is periodic. The proof of
the following theorem is given in Section \ref{proofofTWO}.
\begin{theorem}
\label{theoremTWO}If $\mathcal{F}$ is strongly rigid, then there does not
exist any non-identity isometry $E\in\mathcal{E}$ and $\theta\in\lbrack
N]^{\infty}$ such that $E\pi(\theta)\subset\pi(\theta)$.
\end{theorem}
The following theorem is proved in Section~\ref{invertsec}.
\begin{theorem}
\label{1to1thm}If $\pi(i)\cap\pi(j)$ does not tile $\left( support\text{ }%
\pi(i)\right) \cap\left( support\text{ }\pi(j)\right) $ for all $i\neq j$,
then $\pi:[N]^{\ast}\cup\lbrack N]^{\infty}\rightarrow\mathbb{T}$ is one-to-one.
\end{theorem}
\section{Structure of $\{\Omega_{k}\}$ and Symbolic IFS Tilings}
The results in this section, which will be applied later, relate to a symbolic
version of the theory in this paper. The next two lemmas provide recursions
for the sequence $\Omega_{k}:=\{\sigma\in\lbrack N]^{\ast}:e(\sigma)>k\geq
e^{-}(\sigma)\}$. In this section the square union symbol $\bigsqcup$ denotes
a disjoint union.
\begin{lemma}
\label{lemma1} For all $k\geq a_{\max}$
\begin{equation}
\Omega_{k}={\bigsqcup_{i=1}^{N}i\, \Omega_{k-a_{i}}}. \label{indexformula}%
\end{equation}
\end{lemma}
\begin{proof}
For all $k\in\mathbb{N}_{0}$ we have%
\begin{align*}
i\,\Omega_{k} & =\{i\sigma:\sigma\in\lbrack N]^{\ast},e(\sigma) > k \geq
e^{-}(\sigma)\}\\
& =\{\omega:\omega\in\lbrack N]^{\ast},e(\omega) > k+a_{i} \geq e^{-}%
(\omega),\omega_{1}=i\}\\
& =\Omega_{k+a_{i}}\cap i[N]^{\ast}\text{.}%
\end{align*}
It follows that
\[
i\,\Omega_{k-a_{i}}=\Omega_{k}\cap i[N]^{\ast}%
\]
for all $k\geq a_{i}$, from which it follows that $\Omega_{k}={\bigsqcup
_{i=1}^{N}i\Omega_{k-a_{i}}}$ for all $k\geq a_{\max}$.
\end{proof}
\begin{lemma}
\label{lemstruc2} With $\Omega_{k}^{^{\prime}} :=\{\omega\in\lbrack N]^{\ast
}:e(\omega)=k+1\}$, we have $\Omega_{k}^{^{\prime}}\subset\Omega_{k}$ and
\[
\Omega_{k+1}=\{\Omega_{k}\backslash\Omega_{k}^{\prime}\} \, \bigsqcup\,
\left\{ {\bigsqcup_{i=1}^{N}\Omega_{k}^{^{\prime}}i}\right\} .
\]
\end{lemma}
\begin{proof}
(i) We first show that $\{\Omega_{k}\backslash\Omega_{k}^{\prime}\} \,
\bigsqcup\, \left\{ {\bigsqcup_{i=1}^{N}\Omega_{k}^{^{\prime}}i}\right\}
\subset\Omega_{k+1}$.
Suppose $\theta\in\Omega_{k}\backslash\Omega_{k}^{\prime}$. Then $e^{-}%
(\theta)\leq k<e(\theta)$ and $e(\theta)\neq k+1$. Hence $e^{-}(\theta)\leq
k+1<e(\theta)$ and so $\theta\in\Omega_{k+1}$.
Suppose ${\theta\in\Omega_{k}^{^{\prime}}i}$ for some $i\in\lbrack N].$ Then
${\theta=\theta}^{-}{i}$ where ${\theta}^{-}\in{\Omega_{k}^{^{\prime}}}$,
$e^{-}(\theta)=e({\theta}^{-})=k+1$ and $e(\theta)=e({\theta}^{-}%
{i)=k+1+a}_{i}.$ Hence $e\left( \theta\right) >k+1=e^{-}(\theta)$. Hence
$e^{-}(\theta)\leq k+1<e\left( \theta\right) $. Hence $\theta\in\Omega
_{k+1}$.
(ii) We next show that $\Omega_{k+1}\subset\{\Omega_{k}\backslash\Omega
_{k}^{\prime}\}\, \bigsqcup\, \left\{ {\bigsqcup_{i=1}^{N}\Omega
_{k}^{^{\prime}}i}\right\} $.
Let $\theta\in\Omega_{k+1}.$ Then $e^{-}(\theta)=e(\theta^{-})\leq
k+1<e(\theta)$.
If $e(\theta^{-})=k+1$, then $\theta\in{\Omega_{k}^{^{\prime}}}\theta
_{\left\vert \theta\right\vert }\subset\{\Omega_{k}\backslash\Omega
_{k}^{\prime}\} \, \bigsqcup\, \left\{ {\bigsqcup_{i=1}^{N}\Omega
_{k}^{^{\prime}}i}\right\} $.
If $e(\theta^{-})\neq k+1$, then $e(\theta^{-})<k+1$. So $e(\theta^{-})\leq
k<k+1<e(\theta)$; so $\theta\in\Omega_{k}\backslash\Omega_{k}^{\prime}%
\subset\{\Omega_{k}\backslash\Omega_{k}^{\prime}\} \, \bigsqcup\, \left\{
{\bigsqcup_{i=1}^{N}\Omega_{k}^{^{\prime}}i}\right\} $.
\end{proof}
For all $\theta\in\lbrack N]^{\ast},$ define $c(\theta)=\{\omega\in\lbrack
N]^{\infty}:\omega_{1}\omega_{2}\cdots\omega_{\left\vert \theta\right\vert
}=\theta\}$. (Such sets are sometimes called \textit{cylinder sets}.) With the
metric on $[N]^{\infty}$ defined to be $d_{0}(\theta,\omega)=2^{-\min
\{k:\theta_{k}\neq\omega_{k}\}}$ for $\theta\neq\omega$, the diameter of
$c(\theta)$ is $2^{-(\left\vert \theta\right\vert +1)}$. The following lemma
tells us how $\{c(\theta):\theta\in\Omega_{k}\}$ may be considered as a tiling
of the symbolic space $[N]^{\infty}$.
\begin{lemma}
\label{lemstruc3} For each $k\in\mathbb{N}_{0}$ the collection of sets
$\{c(\theta):\theta\in\Omega_{k}\}$ form a partition of $[N]^{\infty}$, each
part of which has diameter belonging to $\{s^{k+1},s^{k+2},\dots s^{k+a_{\max
}}\}$ where $s=1/2$. That is,
\[
\left[ N\right] ^{\infty}=\bigsqcup\limits_{\theta\in\Omega_{k}}c(\theta)
\]
for all $k\in\mathbb{N}_{0}$.
\end{lemma}
\begin{proof}
Assume that $\omega\in[N]^{\infty}$. There is a unique $j$ such that $\omega|j
\in\Omega_{k}$. Letting $\theta= w|j$ we have $\omega\in c(\theta)
\subset[N]^{\infty}$. Therefore $[N]^{\infty} = \bigcup_{\theta\in\Omega_{k}}
c(\theta)$.
Assume that $\theta,\theta^{\prime}\in\Omega_{k}$. If $\omega\in c(\theta)\cap
c(\theta^{\prime})$, then by the definition of cylinder set either
$\theta=\theta^{\prime}$ or $|\theta|\neq|\theta^{\prime}|$. However, if
$|\theta|\neq|\theta^{\prime}|$, then $\omega\big ||\theta|=\theta\in
\Omega_{k}$ and $\omega\big ||\theta^{\prime}|=\theta^{\prime}\in\Omega_{k}$,
which would contradict the uniqueness of $j$. Therefore $[N]^{\infty
}=\bigsqcup_{\theta\in\Omega_{k}}c(\theta)$.
\end{proof}
\section{A Canonical Sequence of Self-similar Tilings}
To facilitate the proofs of the theorems stated in Section~\ref{defsec},
another family of tilings is introduced, tilings isometric to those that are
the subject of this paper. Let
\[
A_{k}=s^{-k}A
\]
for all $k\in\mathbb{N\cup\{}-1,-2,\dots,-a_{\max}\}$, and define, for all
$k\in\mathbb{N}$, a sequence of tilings $T_{k}$ of $A_{k}$ by
\[
T_{k}=\{ s^{-k} f_{\sigma}(A):\sigma\in\Omega_{k}\}.
\]
The following lemma says, in particular, that $T_{k}$ is a non-overlapping
union of copies of $T_{k-a_{i}}$ for $i\in\lbrack N]$ when $k\geq a_{\max}$,
and $T_{k}$ may be expressed as a non-overlapping union of copies of
$T_{k-e(\omega)}$ for $\omega\in\Omega_{{l}}$ when $k$ is somewhat larger than
$l\in\mathbb{N}_{0}$. In this section the square union notation $\bigsqcup$
denotes a non-overlapping union.
\begin{lemma}
\label{lemma02} For all $k\in\mathbb{N}_{0}$ the support of $T_{k}$ is $A_{k}%
$. For all $\theta\in\lbrack N]^{\ast}$,
\[
\pi(\theta)=E_{\theta}T_{e(\theta)}%
\]
where $E_{\theta}$ is the isometry $f_{-\theta}s^{e(\theta)}$. Also
\begin{equation}
T_{k}={\bigsqcup_{i=1}^{N}}E_{k,i}T_{k-a_{i}} \label{Tkformula}%
\end{equation}
for all $k\geq a_{\max}$, where each of the mappings $E_{k,i}=s^{-k}\circ
f_{i}\circ s^{k-a_{i}}$ is an isometry. More generally,
\begin{equation}
T_{k}={\bigsqcup_{\omega\in\Omega_{l}}}E_{k,\omega}T_{k-e(\omega)},
\label{Tkformula2}%
\end{equation}
for all $k\geq l+a_{\max}$ and for all $l\in\mathbb{N}_{0}$, where each of the
mappings $E_{k,\omega} =s^{-k}\circ f_{\omega}\circ s^{k-e(\omega)}$ is an isometry.
\end{lemma}
\begin{proof}
It is well-known that if $\mathcal{P}$ is a partition of $[N]^{\infty},$ then
$A=%
{\textstyle\bigcup_{\omega\in\mathcal{P}}}
\phi(\omega)$ where $\phi:[N]^{\infty}\rightarrow A$ is the usual (continuous)
coding map defined by $\phi(\omega)=\lim_{k\rightarrow\infty}f_{\omega|k}(x)$
for any fixed $x\in A$. By Lemma \ref{lemstruc3} we can choose $\mathcal{P=}%
\{c(\theta):\theta\in\Omega_{k}\}$. Hence, the support of $T_{k}$ is
\begin{align*}
s^{-k}\{%
{\textstyle\bigcup}
\{f_{\sigma}(A) :\sigma\in\Omega_{k}\}\} & =s^{-k}\{%
{\textstyle\bigcup}
\{\phi(\omega):\omega\in\{c(\theta):\theta\in\Omega_{k}\}\}\}\\
& =s^{-k}A\text{.}%
\end{align*}
The expression $\pi(\theta)=E_{\theta}T_{e(\theta)}$ where $E_{\theta}$ is the
isometry $f_{-\theta}s^{e(\theta)}$ follows from the definitions of
$\pi(\theta)$ and $T_{k}$ on taking $k=e(\theta)$.
Equation (\ref{Tkformula}) follows from Lemma \ref{lemma1} according to these
steps.
\begin{align*}
T_{k} & = \{ s^{-k} f_{\sigma}(A):\sigma\in\Omega_{k}\}\text{ (by
definition)}\\
& =s^{-k}\{f_{\sigma}(A):\sigma\in{\bigsqcup_{i=1}^{N}i\Omega_{k-a_{i}}%
\}}\text{ (by Lemma \ref{lemma1})}\\
& =s^{-k}{\bigsqcup_{i=1}^{N}}\{f_{i\sigma}(A):\sigma\in\Omega{_{k-a_{i}}%
\}}\text{ (identity)}\\
& =s^{-k}{\bigsqcup_{i=1}^{N}}f_{i}(\{f_{\sigma}(A):\sigma\in\Omega
{_{k-a_{i}}\})}\text{ (identity)}\\
& ={\bigsqcup_{i=1}^{N}}E_{k,i}T_{k-a_{i}}\text{ (by definition)}%
\end{align*}
The function $E_{k,i}=s^{-k}\circ f_{i}\circ s^{k-a_{i}}$ is an isometry
because it is a composition of three similitudes, of scaling ratios $s^{-k}$,
$s^{a_{i}},$ and $s^{k-a_{i}}$. The proof of the last assertion is immediate:
tiles meet at images under similitudes of the set of overlap $\mathcal{O=\cup
}_{i\neq j}f_{i}(A)\cap f_{j}(A)$.
Equation (\ref{Tkformula2}) can be proved by induction on $l,$ starting from
Equation (\ref{Tkformula}) and using Lemma \ref{lemstruc2}.
\end{proof}
The following definition, formalizing the notion of an ``isometric combination
of tilings", will be used later, but it is convenient to place it here.
\begin{definition}
Let $\{U_{i}:i\in\mathcal{I\}}$ be a collection of tilings. An
\textbf{isometric combination of the set of tilings} $\{U_{i}:i\in
\mathcal{I\}}$ is a tiling $V$ that can be written in the form
\[
V={\bigsqcup_{i=1}^{K}}E^{(i)}U^{(i)}%
\]
for some $K\in\mathbb{N}$, where $E^{(i)}\in\mathcal{E}$, $U^{(i)}\in
\{U_{i}:i\in\mathcal{I\}}$, for all $i\in\{1,2,\dots,K\}.$
\end{definition}
For example, Lemma \ref{lemma02} tells us that any $T_{k}$ can be written as
an isometric combination of any set of tilings of the form $\{T_{j,}%
T_{j+1},\dots,T_{j+a_{\max}-1}\}$ when $k\geqslant j.$
\begin{proposition}
\label{lemmass} The sequence $\left\{ T_{k}\right\} $ of tilings is
self-similar in the following sense. Each of the sets in the magnified tiling
$s^{-1}T_{k}$ is a union of tiles in $T_{k+1}$.
\end{proposition}
\begin{proof}
This follows at once from Lemma \ref{lemstruc2}. The tiling $T_{k+1}$ is
obtained from $T_{k}$ by applying the similitude $s^{-1}$ and then splitting
those resulting sets that are isometric to $A$. By splitting we mean we
replace $EA$ by $\{Ef_{1}(A),$ $Ef_{2}(A),\dots, Ef_{N}(A)\}$, see Section
\ref{strongsec}.
\end{proof}
\section{Theorem \ref{theoremONE}: Existence and Continuity of
Tilings\label{ProofofONE}}
Let
\[
A_{-\theta|k}:=f_{-\theta|k}A
\]
for all $\theta\in\lbrack N]^{\infty}$. It is immediate from Definition
\ref{defONE} that the support of the tiling $\pi(\theta|k)$ is $A_{-\theta|k}$
and that $\pi(\theta|k)$ is isometric to the tiling $T_{e(k)}$ of $A_{e(k)}$.
We use this fact repeatedly in the rest of this paper.
\begin{flushleft}
\textbf{Theorem~\ref{theoremONE}.} Each set $\pi(\theta)$ in $\mathbb{T}$ is a
tiling of a subset of $\mathbb{R}^{M}$, the subset being bounded when
$\theta\in[N]^{*}$ and unbounded when $\theta\in\lbrack N]^{\infty}$. For all
$\theta\in\lbrack N]^{\infty}$ the sequence of tilings $\left\{ \pi
(\theta|k)\right\} _{k=0}^{\infty}$ is nested according to%
\begin{equation}
\{f_{i}(A):i\in\lbrack N]\}=\pi(\varnothing)\subset\pi(\theta|1)\subset
\pi(\theta|2)\subset\pi(\theta|3)\subset\cdots\text{ .}%
\end{equation}
For all $\theta\in\lbrack N]^{\infty}$, the prototile set for $\mathbb{\pi
}(\theta)$ is $\{s^{i}A:i=1,2,\cdots,a_{\max}\}$.\textit{ }Furthermore
\[
\pi:[N]^{\ast}\cup\lbrack N]^{\infty}\rightarrow\mathbb{T}%
\]
is a continuous map from the compact metric space $[N]^{\ast}\cup\lbrack
N]^{\infty}$ into the space $(\mathbb{K}({\mathbb{R}}^{M}),d_{\tau})$.
\end{flushleft}
\begin{proof}
Using Lemma \ref{lemma02}, for $\theta=\theta_{1}\theta_{2}\cdots\theta_{l}%
\in\lbrack N]^{\ast}$ and $\theta^{-}=\theta_{1}\theta_{2}\cdots\theta_{l-1}%
$,
\begin{align*}
\mathbb{\pi}(\theta) & =E_{\theta}T_{e(\theta)}={\bigsqcup_{i=1}^{N}%
}E_{\theta}E_{e(\theta),i}T_{k-a_{i}}\\
& \supset E_{\theta}E_{e(\theta),\theta_{l}}T_{k-a_{\theta_{l}}}%
=E_{\theta^{-}}T_{e(\theta^{-})}=\mathbb{\pi}(\theta^{-})\text{.}%
\end{align*}
It follows that $\{\pi(\theta|k)\}$ is an increasing sequence of tilings for
all $\theta\in\lbrack N]^{\infty}$, as in Equation (\ref{eqthmONE}), and so
converges to a well-defined limit. Since the maps in the IFS are strict
contractions, their inverses are expansive, whence $\pi(\theta)$ is unbounded
for all $\theta\in\lbrack N]^{\infty}$.
The fact that the tiles here are indeed tiles as we defined them at the start
of this paper follows from three readily checked observations. (i) The tiles
are nonempty perfect compact sets because they are isometric to the attractor,
that is not a singleton, of an IFS of similitudes. (ii) There are only
finitely many tiles that intersect any ball of finite radius. (iii) Any two
tiles can meet only on a set that is contained in the image under a similitude
of the set of overlap.
Next we prove that there are exactly $a_{\max}$ distinct tiles, up to
isometry, in any tiling $\pi(\theta)$ for $\theta\in\lbrack N]^{\infty}$. The
tiles of $\pi(\theta)$ take the form $\{f_{_{-\theta|k}}f_{\sigma}%
(A):\sigma\in\Omega_{e(\theta|k)}\}$ for some $k\in\mathbb{N}$. The mappings
here are similitudes whose scaling factors are $\{s^{e(\sigma)-e(\theta
|k)}:e(\sigma)-e(\theta|k)>0\geq e(\sigma)-e(\theta|k)-a_{|\sigma|}\},$ namely
$\{s^{m}:m>0\geq m-a_{|\sigma|}\}$ for which the possible values are at most
all of $\{1,2,\dots,a_{\max}\}$. That all of these values occur for large
enough $k$ follows from $\gcd\{a_{i}:i=1,2,\dots, N\}=1$.
Next we prove that $\pi:[N]^{\ast}\cup\lbrack N]^{\infty}\rightarrow
\mathbb{T}$ is a continuous map from the compact metric space $[N]^{\ast}%
\cup\lbrack N]^{\infty}$ onto the space $(\mathbb{T},d_{T}).$ The map
$\pi|_{[N]^{\ast}}:[N]^{\ast}\rightarrow\mathbb{T}$ is continuous on the
discrete part of the space $([N]^{\ast},d_{[N]^{\ast}\cup\lbrack N]^{\infty}%
})$ because each point $\theta\in\lbrack N]^{\ast}$ possesses an open
neighborhood that contains no other points of $[N]^{\ast}\cup\lbrack
N]^{\infty}$. To show that $\pi$ is continuous at points of $[N]^{\infty}$ we
follow a similar method to the one in \cite{anderson}. Let $\varepsilon>0$ be
given and let $B(R)$ be the open ball of radius $R$ centered at the origin.
Choose $R$ so large that $h(\rho(\overline{B(R)}),\mathbb{S}^{M})<\varepsilon
$. This implies that if two tilings differ only where they intersect the
complement of $\overline{B(R)}$, then their distance $d_{\tau}$ apart is less
than $\varepsilon$. But geometrical consideration of the way in which
\textit{support(}$\pi(\theta_{1}\theta_{2}\theta_{3}..\theta_{k}))$ grows with
increasing $k$ shows that we can choose $K$ so large that \textit{support(}%
$\pi(\theta_{1}\theta_{2}\theta_{3}..\theta_{k}))\cap\overline{B(R)}$ is
constant for all $k\geq K$. It follows that%
\[
h(\rho(\pi(\theta_{1}\theta_{2}..\theta_{k})),\rho(\pi(\theta_{1}\theta
_{2}..\theta_{l})))\leq\varepsilon
\]
and as a consequence
\[
h(\rho(\partial\pi(\theta_{1}\theta_{2}..\theta_{k})),\rho(\partial\pi
(\theta_{1}\theta_{2}..\theta_{l})))\leq\varepsilon
\]
for all $k,l\geq K$. It follows that $h(\rho(\pi(\theta)),\rho(\pi
(\omega)))\leq\varepsilon)$ whenever $\theta_{1}\theta_{2}..\theta_{K}%
=\omega_{1}\omega_{2}..\omega_{K}$. It follows that $\pi$ is continuous.
\end{proof}
\section{\label{proofofTHREE}Theorem \ref{theoremTHREE}: When Do all Tilings
Repeat the Same Patterns?}
\begin{flushleft}
\textbf{Theorem 2.}
\end{flushleft}
\begin{enumerate}
\item Each unbounded tiling in $\mathbb{T}$ is quasiperiodic and all tilings
in $\mathbb{T}$ have the local isomorphism property.
\item If $\theta$ is eventually periodic, then $\pi(\theta)$ is self-similar.
In fact, if $\theta=\alpha\overline{\beta}$ for some $\alpha,\beta\in\left[
N\right] ^{\ast},$ then $f_{-\alpha}f_{-\beta}\left( f_{-\alpha}\right)
^{-1}$ is a self-similarity of $\pi(\theta)$.
\end{enumerate}
\begin{proof}
(1) First we prove quasiperiodicity. This is related to the self-similarity of
the sequence of tilings $\left\{ T_{k}\right\} $ mentioned in Proposition
\ref{lemmass}.
Let $\theta\in\left[ N\right] ^{\infty}$ be given and let $P$ be a patch in
$\pi(\theta)$. There is a $K_{1}\in\mathbb{N}$ such that $P$ is contained in
$\pi(\theta|K_{1})$. Hence an isometric copy of $P$ is contained in $T_{K_{2}%
}$ where $K_{2}=e(\theta|K_{1})$. Now choose $K_{3}\in\mathbb{N}$ so that an
isometric copy of $T_{K_{2}}$ is contained in each $T_{k}$ with $k\geq K_{3}.$
That this is possible follows from the recursion (\ref{Tkformula2}) of Lemma
\ref{lemma02} and \textit{gcd}$\{a_{i}\}=1$. In particular, $T_{K_{2}}\subset
T_{K_{3}+i}$ for all $i\in\{1,2,...,a_{\max}\}$.
Now let $K_{4}=K_{3}+a_{\max}$. Then, for all $k\geq K_{4}$, the tiling
$T_{k}$ is an isometric combination of $\{T_{K_{3}+i}:$\textit{ }%
$i=1,2,...,a_{\max}\}$, and each of these tilings contains a copy of
$T_{K_{2}}$ and in particular a copy of $P$.
Let $D=\max\{\left\Vert x-y\right\Vert :x,y\in A\}$ be the diameter of $A$.
The support of $T_{k}$ is $s^{-k}A$ which has diameter $s^{-k}D.$ Hence
\textit{support}$(T_{k})\subset$ $B(x,2s^{-k}D)$, the ball centered at $x$ of
radius $2s^{-k}D$, for all $x\in$ \textit{support}$(T_{k})$. It follows that
if $x\in$\textit{support}$\pi(\theta^{\prime})$ for any $\theta^{\prime}%
\in\left[ N\right] ^{\infty}$, then $B(x,2s^{-K_{4}}D)$ contains a copy of
\textit{support}$(T_{K_{2}})$ and hence a copy of $P$. Therefore all unbounded
tilings in $\mathbb{T}$ are quasiperiodic.
In \cite{Rad} Radin and Wolff define a tiling to have the local ismorphism
property if for every patch $P$ in the tiling there is some distance $d(P)$
such that every sphere of diameter $d(P)$ in the tiling contains an isometric
copy of $P$. Above, we have proved a stronger property of tilings, as defined
here, of fractal blow-ups. Given $P,$ there is a distance $d(P)$ such that
each sphere of diameter $d(P),$ centered at any point belonging to the support
of any unbounded tiling in $\mathbb{T}$, contains a copy of $P$.
(2) Let $\theta=\alpha\overline{\beta}=\alpha_{1}\alpha_{2}\cdots\alpha
_{l}\beta_{1}\beta_{2}\cdots\beta_{m}\beta_{1}\beta_{2}\cdots\beta_{m}%
\beta_{1}\beta_{2}\cdots\beta_{m}\cdots$. We have the equivalent increasing
unions
\[
\pi(\theta)=%
{\textstyle\bigcup\limits_{k\in\mathbb{N}}}
E_{\theta|k}T_{e(\theta|k)}=%
{\textstyle\bigcup\limits_{j\in\mathbb{N}}}
E_{\theta|(l+jm)}T_{e(\theta|(l+jm))}=%
{\textstyle\bigcup\limits_{j\in\mathbb{N}}}
E_{\theta|(l+jm+m)}T_{e(\theta|(l+jm+m))}%
\]
where, for all $k$,
\[
E_{\theta|k}=f_{-\theta|k}s^{e(\theta|k)}\text{.}%
\]
We can write
\[
\pi(\theta)=%
{\textstyle\bigcup\limits_{j\in\mathbb{N}}}
E_{\theta|(l+jm)}T_{e(\theta|(l+jm))}=f_{-\alpha}%
{\textstyle\bigcup\limits_{j\in\mathbb{N}}}
f_{-\beta}^{j}s^{e(\theta|(l+jm))}T_{e(\theta|(l+jm))},
\]
and also
\[
\pi(\theta)=%
{\textstyle\bigcup\limits_{j\in\mathbb{N}}}
E_{\theta|(l+jm+m)}T_{e(\theta|(l+jm+m))}=f_{-\alpha}f_{-\beta}%
{\textstyle\bigcup\limits_{j\in\mathbb{N}}}
f_{-\beta}^{j}s^{e(\theta|(l+jm+m))}T_{e(\theta|(l+jm+m))}\text{.}%
\]
Here $f_{-\beta}^{j}s^{e(\theta|(l+jm+m))}T_{e(\theta|(l+jm+m))}$ is a
refinement of $f_{-\beta}^{j}s^{e(\theta|(l+jm))}T_{e(\theta|(l+jm))}$. It
follows that $\left( f_{-\alpha}f_{-\beta}\right) ^{-1}\pi(\theta)$ is a
refinement of $\left( f_{-\alpha}\right) ^{-1}\pi(\theta)$, from which it
follows that $\left( f_{-\alpha}\right) \left( f_{-\alpha}f_{-\beta
}\right) ^{-1}\pi(\theta)$ is a refinement of $\pi(\theta)$. Therefore, every
set in $\left( f_{-\alpha}f_{-\beta}\right) \left( f_{-\alpha}\right)
^{-1}\pi(\theta)$ is a union of tiles in $\pi(\theta)$.
\end{proof}
\section{\label{realabsec} Relative and Absolute Addresses}
In order to understand how different tilings relate to one another, the
notions of relative and absolute addresses of tiles are introduced. Given an
IFS $\mathcal{F}$, the \textit{set of absolute addresses} is defined to be:
\[
\mathbb{A}:=\{\theta.\omega:\theta\in\lbrack N]^{\ast},\,\omega\in
\Omega_{e(\theta)},\,\theta_{\left\vert \theta\right\vert }\neq\omega_{1}\}.
\]
Define $\widehat{\pi}:\mathbb{A\rightarrow\{}t\in T:T\in\mathbb{T\}}$ by
\[
\widehat{\pi}(\theta.\omega)=f_{-\theta}.f_{\omega}(A).
\]
We say that $\theta.\omega$ is an \textit{absolute address} of the tile
$f_{-\theta}.f_{\omega}(A)$. It follows from Definition \ref{defONE} that the
map $\widehat{\pi}$ is surjective: every tile of $\mathbb{\{}t\in
T:T\in\mathbb{T\}}$ possesses at least one address. The condition
$\theta_{\left\vert \theta\right\vert }\neq\omega_{1}$ is imposed to make
cancellation unnecessary.
The \textit{set of relative addresses} is associated with the tiling $T_{k}$
of $A_{k}=s^{-k}A$ and is defined to be $\{.\omega:\omega\in\Omega_{k}\}$.
\begin{proposition}
\label{lembij}There is a bijection between the set of relative addresses
$\{.\omega:\omega\in\Omega_{k}\}$ and the tiles of $T_{k}$, for all
$k\in\mathbb{N}_{0}$.
\end{proposition}
\begin{proof}
This follows from the non-overlapping union
\[
A=%
{\textstyle\bigsqcup\limits_{\omega\in\Omega_{k}}}
f_{\omega}(A)\text{.}%
\]
This expression follows immediately from Lemma \ref{lemstruc3}; see the start
of the proof of Lemma \ref{lemma02}.
\end{proof}
Accordingly, we say that $.\omega$, or equivalently $\varnothing.\omega,$
where $\omega\in\Omega_{k}$, is \textit{the relative address} of the tile
$s^{-k}f_{\omega}(A)$ in the tiling $T_{k}$ of $A_{k}$. Note that a tile of
$T_{k}$ may share the same relative address as a different tile of $T_{l}$ for
$l\neq k$.
Define the \textit{set of labelled tiles} of $T_{k}$ to be%
\[
\mathcal{A}_{k}=\{(.\omega,s^{-k}f_{\omega}(A)):\omega\in\Omega_{k}\}
\]
for all $k\in\mathbb{N}_{0}$. A key point about relative addresses is that the
set of labelled tiles of $T_{k}$ for $k\in\mathbb{N}$ can be computed
recursively. Define
\[
\mathcal{A}_{k}^{^{\prime}}=\{(\omega,s^{-k}f_{\omega}(A))\in\mathcal{A}%
_{k}:e(\omega)=k+1\}\subset\mathcal{A}_{k}\text{.}%
\]
An example of the following inductive construction is illustrated in\ Figure
\ref{l-maps}, and some corresponding tilings $\pi(\theta)$ labelled by
absolute addresses are illustrated in Figure \ref{absolute}.
\begin{lemma}
\label{branchlem}For all $k\in\mathbb{N}_{0}$ we have%
\[
\mathcal{A}_{k+1}=\mathcal{L(A}_{k}\backslash\mathcal{A}_{k}^{^{\prime}}%
)\cup\mathcal{M(A}_{k}^{^{\prime}})
\]
where
\begin{align*}
\mathcal{L}(\omega,s^{-k}f_{\omega}(A)) & =(\omega,s^{-k-1}f_{\omega}(A)),\\
\mathcal{M}(\omega,s^{-k}f_{\omega}(A)) & =\big \{(\omega i,s^{-k-1}%
f_{\omega i}(A)):i\in\lbrack N]\big \}\text{.}%
\end{align*}
\end{lemma}
\begin{proof}
This follows immediately from Lemma \ref{lemstruc2}.
\end{proof}
\section{\label{strongsec}Strong Rigidity, Definition of ``Amalgamation and
Shrinking" Operation $\alpha$ on Tilings, and Proof of Theorem
\ref{intersectthm}.}
We begin this key section by introducing an operation, called
\textquotedblleft amalgamation and shrinking", that maps certain tilings into
tilings. This leads to the main result of this section, Theorem
\ref{intersectthm}, which, in turn, leads to Theorem \ref{theoremTWO}.
\begin{definition}
\label{rigiddef} Let $T_{0}=\{f_{i}(A):i\in\left[ N\right] \}$. The IFS
$\mathcal{F}$ is said to be \textbf{rigid} if (i) there exists no non-identity
isometry $E\in\mathcal{E}$ such that $T_{0}\cap ET_{0}$ is non-empty and tiles
$A\cap ET$, and (ii) there exists no non-identity isometry $E\in\mathcal{E}$
such that $A=EA$.
\end{definition}
\begin{definition}
Define $\mathbb{T}^{\prime}$ to be the set of all tilings using the set of
prototiles $\left\{ s^{i}A:i=1,2,...,a_{\max}\right\} $. Any tile that is
isometric to $s^{a_{\max}}A$ is called a \textbf{small tile}, and any tile
that is isometric to $sA$ is called a \textbf{large tile}. We say that a
tiling $P\in\mathbb{T}^{\prime}$ comprises a set of \textbf{partners }if
$P=ET_{0}$ for some $E\in\mathcal{E}$. Define $\mathbb{T^{\prime\prime}\subset
T}^{\prime}$ to be the set of all tilings in $\mathbb{T}^{\prime}$ such that,
given any $Q\in\mathbb{T^{\prime\prime}}$ and any small tile $t\in Q$, there
is a set of partners of $t$, call it $P(t)$, such that $P(t)\subset Q$. Given
any $Q\in\mathbb{T^{\prime\prime}}$ we define $Q^{\prime}$ to be the union of
all sets of partners in $Q$.
\end{definition}
\begin{definition}
Let $\mathcal{F}$ be a rigid IFS. The amalgamation and shrinking operation
$\alpha:\mathbb{T^{\prime\prime}\rightarrow T}^{\prime}$ is defined by
\[
\alpha Q=\{st:t\in Q\backslash Q^{\prime}\}\cup%
{\displaystyle\bigsqcup_{\{E\in\mathcal{E}:ET_{0}\subset Q^{\prime}\}}}
sEA\text{.}%
\]
\end{definition}
\begin{lemma}
\label{inverselemma} If $\mathcal{F}$ is rigid, the function $\alpha
:\mathbb{T^{\prime\prime}\rightarrow T}^{\prime}$is well-defined and
bijective; in particular, $\alpha^{-1}:\mathbb{T}^{\prime}\rightarrow
\mathbb{T^{\prime\prime}}$ is well defined by
\[
\alpha^{-1}(Q)=\{\alpha_{Q}^{-1}(q):q\in Q\}
\]
where
\[
\alpha_{Q}^{-1}(q)=\left\{
\begin{array}
[c]{c}%
s^{-1}q\text{ if }q\in Q\text{ is not a large tile}\\
s^{-1}ET_{0}\text{ if }Eq\text{ is a large tile, some }E\in\mathcal{E}%
\end{array}
\right.
\]
\end{lemma}
\begin{proof}
Because $\mathcal{F}$ is rigid, there can be no ambiguity with regard to which
sets of tiles in a tiling are partners, nor with regard to which tiles are the
partners of a given small tile. Hence $\alpha:\mathbb{T^{\prime\prime
}\rightarrow T}^{\prime}$ is well defined. Given any $T^{\prime}\in
\mathbb{T}^{\prime}$ we can find a unique $Q\in\mathbb{T^{\prime\prime}}$ such
that $\alpha(Q)=T^{\prime},$ namely $\alpha^{-1}(Q)$ as defined in the lemma.
\end{proof}
\begin{lemma}
\label{alphaTlem}Let $\mathcal{F}$ be rigid and $k\in\mathbb{N}$. Then
(i) $T_{k}\in\mathbb{T^{\prime\prime}}$;
(ii) $\alpha T_{k}=T_{k-1}$ and $\alpha^{-1}T_{k-1}=T_{k}$.
\end{lemma}
\begin{proof}
As described in Lemma \ref{branchlem}$,$ $T_{k}$ can constructed in a
well-defined manner, starting from from $T_{k-1}$, by scaling and splitting,
that is, by applying $\alpha^{-1}$. Conversely $T_{k-1}$ can be constructed
from $T_{k}$ by applying $\alpha$. Statements (i) and (ii) are consequences.
\end{proof}
\begin{lemma}
\label{srinterlem} If $\mathcal{F}$ is rigid, $L,M\in\mathbb{T^{\prime\prime}%
}$, and $L\cap M$ tiles support($L)\,\cap\,$support($M),$ then $L\, \cap\,
M\in\mathbb{T^{\prime\prime}}$. Moreover,
\[
\alpha(L\cap M)=\alpha(L)\cap\alpha(M),
\]
and $\alpha(L\cap M)$ tiles support$\, \alpha(L)\, \cap\, $support$\,
\alpha(M)$.
\end{lemma}
\begin{proof}
Since $L,M\in\mathbb{T^{\prime\prime}\subset T}^{\prime}$ lie in the range of
$\alpha^{-1},$ we can find unique $L^{\prime},M^{\prime}\in\mathbb{T}^{\prime
}$ such that
\[
L=\alpha^{-1}L^{\prime}\text{ and }M=\alpha^{-1}M^{\prime}.
\]
Note that $\alpha^{-1}(T^{\prime})=\left\{ \alpha^{-1}(t):t\in T^{\prime
}\right\} $ for all $T^{\prime}\in\mathbb{T}^{\prime}$, which implies that
$\alpha^{-1}$ commutes both with unions of disjoint tilings and also with
intersections of tilings whose intersections tile the intersections of their
supports. It follows that $L\cap M\in\mathbb{T^{\prime\prime}}$,
\begin{align*}
\alpha(L\cap M) & =\alpha(\alpha^{-1}L^{\prime}\cap\alpha^{-1}M^{\prime})\\
& =\alpha(\alpha^{-1}(L^{\prime}\cap M^{\prime}))\\
& =L^{\prime}\cap M^{\prime}\\
& =\alpha\left( L\right) \cap\alpha\left( M\right) \text{,}%
\end{align*}
and support $\alpha(L\cap M)=$\, support $\, \alpha\left( L\right) \cap
\,$support $\alpha\left( M\right) $.
\end{proof}
\begin{definition}
\label{strongdef}$\mathcal{F}$ is \textbf{strongly rigid} if $\mathcal{F}$ is
rigid and whenever $i,j\in\{0,1,2,\dots,a_{\max}-1\},E\in\mathcal{E}$, and
$T_{i}\cap ET_{j}$ tiles $A_{i}\cap EA_{j}$, either $T_{i}\subset ET_{j}$ or
$T_{i}\supset ET_{j}.$
\end{definition}
Section \ref{exsec} contain a few examples of strongly rigid IFSs.
\begin{lemma}
\label{intersectlemma} Let $\mathcal{F}$ be strongly rigid, $k,l\in
\mathbb{N}_{0}$, and $E\in\mathcal{E}$.
(i) If $ET_{k}\cap T_{k}$ is nonempty and tiles $EA_{k}\cap A_{k},$ then
$E=id$.
(ii) If $EA_{k}\cap A_{k+l}$ is nonempty and $ET_{k}\cap T_{k+l}$ tiles
$EA_{k}\cap A_{k+l}$, then $ET_{k}\subset T_{k+l}$.
\end{lemma}
\begin{proof}
Suppose $ET_{k}\cap T_{l}\neq\varnothing$ and t.i.s. (tiles intersection of
supports). Without loss of generality assume $k\leq l$, for if not, then apply
$E^{-1}$, then redefine $E^{-1}$ as $E$.
Both $ET_{k}$ and $T_{l}$ lie in the domain of $\alpha^{k}$, so we can apply
Lemma \ref{srinterlem} $k$ times, yielding
\begin{align}
\alpha^{k}(ET_{k}\cap T_{l}) & =s^{k}Es^{-k}T_{0}\cap T_{l-k} \label{aboveq}%
\\
& :=\widetilde{E}T_{0}\cap T_{l-k}\neq\varnothing,\nonumber
\end{align}
where $\widetilde{E}T_{0}\cap T_{l-k}$ t.i.s. Now observe that by Lemma
\ref{lemma02} we can write, for all $k^{\prime}\geq l^{\prime}+a_{\max}$,%
\[
T_{k^{\prime}}={\bigsqcup_{\omega\in\Omega_{l^{\prime}}}}E_{k^{\prime},\omega
}T_{k^{\prime}-e(\omega)}\left( =\left\{ E_{k^{\prime},\omega}T_{k^{\prime
}-e(\omega)}:\omega\in\Omega_{l^{\prime}}\right\} \right) ,
\]
where $E_{k^{\prime},\omega}\in\mathcal{E}$ for all $k^{\prime},\omega$.
Choosing $l^{\prime}=k^{\prime}-a_{\max}$ and noting that, for $\omega
\in\Omega_{l^{\prime}}$, we have $e(\omega)\in\{l^{\prime}+1,\dots,l^{\prime
}+a_{\max}\},$ and for $\omega\in\Omega_{k^{\prime}-a_{\max}}$ we have
$e(\omega)\in\{k^{\prime}-a_{\max}+1,\dots,k^{\prime}\}.$ Therefore
$k^{\prime}-e(\omega)\in\{0,1,\dots,a_{\max}-1\}$ and we obtain the explicit
representation%
\[
T_{k^{\prime}}={\bigsqcup_{\omega\in\Omega_{k^{\prime}-a_{\max}}}}%
E_{k^{\prime},\omega}T_{k^{\prime}-e(\omega)}%
\]
which is an isometric combination of $\{T_{0},T_{1},\dots,T_{a_{\max}-1}\}$.
In particular, we can always reexpress $T_{l-k}$ in (\ref{aboveq}) as
isometric combination of $\{T_{0},T_{1},\dots,T_{a_{\max}-1}\}$ and so there
is some $E^{\prime}$ and some $T_{m}\in\{T_{0},T_{1},\dots,T_{a_{\max}-1}\}$
such that
\[
\widetilde{E}T_{0}\cap E^{\prime}T_{m}\neq\varnothing\text{ and t.i.s.}%
\]
By the strong rigidity assumption, this implies $\widetilde{E}T_{0}\subset
E^{\prime}T_{m}$, which in turn implies
\[
\widetilde{E}T_{0}\subset T_{l-k}%
\]
and t.i.s. Now apply $\alpha^{-k}$ to both sides of this last equation to
obtain the conclusions of the lemma.
\end{proof}
\begin{flushleft}
\textbf{Theorem \ref{intersectthm}.} Let $\mathcal{F}$ be strongly rigid. If
$\theta,\theta^{\prime}\in\lbrack N]^{\ast}$and $E\in\mathcal{E}$ are such
that $\pi(\theta)\cap E\pi(\theta^{\prime})$ is not empty and tiles
$A_{-\theta}\cap EA_{-\theta^{\prime}}$, then either $\pi(\theta)\subset
E\pi(\theta^{\prime})$ or $E\pi(\theta^{\prime})\subset\pi(\theta)$. In this
situation, if $e(\theta)=e(\theta^{\prime}),$ then $E\pi(\theta^{\prime}%
)=\pi(\theta).$
\end{flushleft}
\begin{proof}
This follows from Lemma \ref{intersectlemma}. If $\theta,\theta^{\prime}%
\in\lbrack N]^{\ast}$and $E\in\mathcal{E}$ are such that $\pi(\theta)\cap
E\pi(\theta^{\prime})$ is not empty and tiles $A_{-\theta}\cap EA_{-\theta
^{\prime}}$, then $\theta,\theta^{\prime}\in\lbrack N]^{\ast}$and
$E\in\mathcal{E}$ are such that $E_{\theta}T_{e(\theta)}\cap EE_{\theta
^{\prime}}T_{e(\theta^{\prime})}$ is not empty and tiles $E_{\theta
}A_{e(\theta)}\cap EE_{\theta^{\prime}}A_{e(\theta^{\prime})}$, where
$E_{\theta}=f_{-\theta}s^{e(\theta)}$ and $E_{\theta^{\prime}}=f_{-\theta
^{\prime}}s^{e(\theta^{\prime})}$ are isometries$.$ Assume, without loss of
generality, that $e(\theta)\leq e(\theta^{\prime})$ and apply $E_{\theta
^{\prime}}^{-1} E^{-1}$ to obtain that $\theta,\theta^{\prime}\in\lbrack
N]^{\ast}$ and $E^{\prime}=E_{\theta^{\prime}}^{-1}E^{-1} E_{\theta}%
\in\mathcal{E}$ are such that $E^{\prime}T_{e(\theta)}\cap T_{e(\theta
^{\prime})}$ is not empty and tiles $E^{\prime}A_{e(\theta)}\cap
A_{e(\theta^{\prime})}.$ By Lemma \ref{intersectlemma} it follows that
$E^{\prime}T_{e(\theta)}\subset T_{e(\theta^{\prime})}$, i.e. $E_{\theta
^{\prime}}^{-1}E^{-1}E_{\theta}T_{e(\theta)}\subset T_{e(\theta^{\prime})},$
i.e. $\pi(\theta)\subset E\pi(\theta^{\prime}).$ If also $e(\theta^{\prime
})\leq e(\theta)$ (i.e. $e(\theta^{\prime})=e(\theta)$), then also
$E\pi(\theta^{\prime})\subset\pi(\theta)$. Therefore $E\pi(\theta^{\prime
})=\pi(\theta).$
\end{proof}
\section{\label{proofofTWO}Theorem \ref{theoremTWO}: When is a Tiling
Non-Periodic?}
\begin{flushleft}
\textbf{Theorem \ref{theoremTWO}.} \textit{If }$F$\textit{ is strongly rigid,
then there does not exist any non-identity isometry }$E\in\mathcal{E}$\textit{
and }$\theta\in\lbrack N]^{\infty}$\textit{ such that }$E\pi(\theta)\subset
\pi(\theta)$\textit{.}
\end{flushleft}
\begin{proof}
Suppose there exists an isometry $E$ such that $E\pi(\theta)=\pi(\theta).$
Then we can choose $K\in\mathbb{N}_{0}$ so large that $E\pi(\theta|K)\cap
\pi(\theta|K)\neq\varnothing$ and $E\pi(\theta|K)\cap\pi(\theta|K)$ tiles
$EA_{-\theta|K}\cap A_{-\theta|K}.$ By Theorem \ref{intersectthm} it follows
that%
\[
E\pi(\theta|K)=\pi(\theta|K)
\]
This implies
\[
EE_{\theta}T_{e\left( \theta|K\right) }=E_{\theta}T_{e\left( \theta
|K\right) }%
\]
whence, because $E_{\theta}T_{e\left( \theta|K\right) }$ is in the domain of
$\alpha^{e\left( \theta|K\right) }$ and $\alpha^{e\left( \theta|K\right)
}T_{e\left( \theta|K\right) }=T_{0},$ we have by Lemma \ref{alphaTlem}
\begin{align*}
\alpha^{e\left( \theta|K\right) }E & E_{\theta}T_{e\left( \theta
|K\right) } =\alpha^{e\left( \theta|K\right) }E_{\theta}T_{e\left(
\theta|K\right) }\\
& \Longrightarrow s^{e\left( \theta|K\right) }EE_{\theta}s^{-e\left(
\theta|K\right) }\alpha^{e\left( \theta|K\right) }T_{e\left(
\theta|K\right) }=s^{e\left( \theta|K\right) }E_{\theta}s^{-e\left(
\theta|K\right) }\alpha^{e\left( \theta|K\right) }T_{e\left(
\theta|K\right) }\\
& \Longrightarrow s^{e\left( \theta|K\right) }EE_{\theta}s^{-e\left(
\theta|K\right) }T_{0}=s^{e\left( \theta|K\right) }E_{\theta}s^{-e\left(
\theta|K\right) }T_{0}\\
& \Longrightarrow s^{e\left( \theta|K\right) }EE_{\theta}s^{-e\left(
\theta|K\right) }=s^{e\left( \theta|K\right) }E_{\theta}s^{-e\left(
\theta|K\right) }\text{ (using rigidity)}\\
& \Longrightarrow E=id\text{.}%
\end{align*}
\end{proof}
It follows that if $\mathcal{F}$ is strongly rigid, then $\pi(\theta)$ is
non-periodic for all $\theta$.
\section{\label{invertsec}When is $\pi:[N]^{\ast}\cup\lbrack N]^{\infty
}\rightarrow\mathbb{T}$ invertible?}
\begin{lemma}
\label{noninjlem}For all $\mathcal{F}$ the restricted mapping $\pi|_{\left[
N\right] ^{\ast}\text{.}}:[N]^{\ast}\rightarrow\mathbb{T}$ is injective.
\end{lemma}
\begin{proof}
To simplify notation, write $\pi=\pi|_{\left[ N\right] ^{\ast}\text{.}}$ We
show how to calculate $\theta$ given $\pi\left( \theta\right) $ for
$\theta\in\left[ N\right] ^{\ast}.$ By Lemma~\ref{lemma02} we have
$\pi(\theta)=E_{\theta}T_{e(\theta)}$, where $E$ is the isometry $f_{-\theta
}s^{e(\theta)}$. Given $\pi(\theta)$, we can calculate
\[
e(\theta)=\frac{\ln\left\vert A\right\vert -\ln\left\vert \pi(\theta
)\right\vert }{\ln s},
\]
where $\left\vert U\right\vert $ denotes the diameter of the set $U$.
We next show that if $E_{\theta}=E_{\theta^{\prime}}$ for some $\theta
\neq\theta^{\prime}$ with $e(\theta)=e(\theta^{\prime})$, then $\pi
(\theta)\neq\pi(\theta^{\prime})$. To do this, suppose that $E_{\theta
}=E_{\theta^{\prime}}$. This implies that $f_{-\theta}=f_{-\theta^{\prime}}$
which implies
\[
\left( f_{-\theta^{\prime}}\right) ^{-1}f_{-\theta}=id\text{,}%
\]
which is not possible when $\theta\neq\theta^{\prime},$ as we prove next. The
similitude $\left( f_{-\theta^{\prime}}\right) ^{-1}f_{-\theta}$ maps
$\left( f_{-\theta}\right) ^{-1}(A)\subset A$ to $\left( f_{-\theta
^{\prime}}\right) ^{-1}(A)\subset A$, and these two subsets of\ $A$ are
distinct for all $\theta,\theta^{\prime}\in\left[ N\right] ^{\ast}$with
$\theta\neq\theta^{\prime}$, as we prove next.
Let $\omega,\omega^{\prime}$ denote the two strings $\theta,\theta^{\prime}$
written in inverse order, so that $\theta\neq\theta^{\prime}$ is equivalent to
$\omega\neq\omega^{\prime}$. First suppose $\left\vert \omega\right\vert
=\left\vert \omega^{\prime}\right\vert =m$ for some $m\in\mathbb{N}.$ Then
use
\[
A=%
{\displaystyle\bigsqcup\limits_{\omega\in\left[ N\right] ^{m}}}
f_{\omega}(A),
\]
which tells us that $f_{\omega}(A)$ and $f_{\omega^{\prime}}(A)$ are disjoint.
Since $\left( f_{-\theta^{\prime}}\right) ^{-1}f_{-\theta}$ maps
\linebreak$\left( f_{-\theta}\right) ^{-1}(A)=f_{\omega}(A)$ to the distinct
set $\left( f_{-\theta^{\prime}}\right) ^{-1}(A)=f_{\omega^{\prime}}(A)$, we
must have $\left( f_{-\theta^{\prime}}\right) ^{-1}f_{-\theta}\neq id$.
Now suppose $\left\vert \omega\right\vert =m<\left\vert \omega^{\prime
}\right\vert =m^{\prime}.$ If both strings $\omega$ and $\omega^{\prime}$
agree through the first $m$ places, then $f_{\omega}(A)$ is a strict subset of
$f_{\omega^{\prime}}^{-1}(A)$ and again we cannot have $\left( f_{-\theta
^{\prime}}\right) ^{-1}f_{-\theta}=id$. If both strings $\omega$ and
$\omega^{\prime}$ do not agree through the first $m$ places, then let $p<m$ be
the index of their first disagreement. Then we find that $f_{\omega}(A)\ $is a
subset of $f_{\omega|p}(A)$, while $f_{\omega^{\prime}}(A)$ is a subset of the
set $f_{\omega^{\prime}|p}(A)$, which is disjoint from $f_{\omega|p}(A)$.
Since $\left( f_{-\theta^{\prime}}\right) ^{-1}f_{-\theta}$ maps $f_{\omega
}(A)$ to $f_{\omega^{\prime}}(A)$, we again have that $\left( f_{-\theta
^{\prime}}\right) ^{-1}f_{-\theta}\neq id$.
\end{proof}
We are going to need a key property of certain shifts maps on tilings, defined
in the next lemma.
\begin{lemma}
The mappings $S_{i}:\{\pi(\theta):\theta\in\lbrack N]^{l}\cup\lbrack
N]^{\infty},l\geq a_{i}\}\rightarrow\mathbb{T}^{\prime}$ for $i\in\lbrack N]$
are well-defined by%
\[
S_{i}=f_{i}s^{-a_{i}}\alpha^{a_{i}}\text{.}%
\]
It is true that%
\[
S_{\theta_{1}}\pi(\theta)=\pi(S\theta)
\]
for all $\theta\in\lbrack N]^{l}\cup\lbrack N]^{\infty}$ where $l\geq
a_{\theta_{1}}$.
\end{lemma}
\begin{proof}
We only consider the case $\theta\in\lbrack N]^{\infty}$. The case $\theta
\in\lbrack N]^{l}$ is treated similarly. A detailed calculation, outlined
next, is needed. The key idea is that $\pi\left( \theta\right) $ is broken
up into a countable union of disjoint tilings, each of which belongs to the
domain of $\alpha^{k}$ for all $k\leq K$ for any $K\in\mathbb{N}$. For all
$K\in\mathbb{N}$ we have:%
\[
\pi\left( \theta\right) =E_{\theta|K}T_{e\left( \theta|K\right) }%
{\textstyle\bigsqcup}
{\textstyle\bigsqcup_{k=K}^{\infty}}
E_{\theta|k+1}T_{e\left( \theta|k+1\right) }\backslash E_{\theta
|k}T_{e\left( \theta|k\right) }\text{.}%
\]
The tilings on the r.h.s. are indeed disjoint, and each set belongs to the
domain of $\alpha^{e\left( \theta|K\right) }$, so we can use Lemma
\ref{srinterlem} applied countably many times to yield%
\[
S_{\theta_{1}}\pi\left( \theta\right) =S_{\theta_{1}}\left( E_{\theta
|K}T_{e\left( \theta|K\right) }\right)
{\textstyle\bigsqcup_{k=K}^{\infty}}
S_{\theta_{1}}\left( E_{\theta|k+1}T_{e\left( \theta|k+1\right) }\right)
\backslash S_{\theta_{1}}\left( E_{\theta|k}T_{e\left( \theta|k\right)
}\right) \text{.}%
\]
Evaluating, we obtain successively
\begin{align*}
S_{\theta_{1}}\pi\left( \theta\right) & =f_{\theta_{1}}s^{-a_{\theta_{1}}%
}\alpha^{a_{\theta_{1}}}\left( E_{\theta|K}T_{e\left( \theta|K\right)
}\right)
{\textstyle\bigsqcup_{k=K}^{\infty}}
f_{\theta_{1}}s^{-a_{\theta_{1}}}\alpha^{a_{\theta_{1}}}\left( E_{\theta
|k+1}T_{e\left( \theta|k+1\right) }\right) \backslash f_{\theta_{1}%
}s^{-a_{\theta_{1}}}\alpha^{a_{\theta_{1}}}\left( E_{\theta|k}T_{e\left(
\theta|k\right) }\right) ,\\
S_{\theta_{1}}\pi\left( \theta\right) & =f_{\theta_{1}}E_{\theta
|K}s^{-a_{\theta_{1}}}\alpha^{a_{\theta_{1}}}T_{e\left( \theta|K\right) }%
{\textstyle\bigsqcup_{k=K}^{\infty}}
f_{\theta_{1}}E_{\theta|k+1}s^{-a_{\theta_{1}}}\alpha^{a_{\theta_{1}}%
}T_{e\left( \theta|k+1\right) }\backslash f_{\theta_{1}}E_{\theta
|k+1}s^{-a_{\theta_{1}}}\alpha^{a_{\theta_{1}}}T_{e\left( \theta|k\right)
},\\
S_{\theta_{1}}\pi\left( \theta\right) & =f_{\theta_{1}}E_{\theta
|K}s^{-a_{\theta_{1}}}T_{e\left( S\theta|K-1\right) }%
{\textstyle\bigsqcup_{k=K}^{\infty}}
f_{\theta_{1}}E_{\theta|k+1}s^{-a_{\theta_{1}}}T_{e\left( S\theta|k\right)
}\backslash f_{\theta_{1}}E_{\theta|k}s^{-a_{\theta_{1}}}T_{e\left(
S\theta|k-1\right) },\\
S_{\theta_{1}}\pi\left( \theta\right) & =E_{S\theta|\left( K-1\right)
}T_{e\left( S\theta|K-1\right) }%
{\textstyle\bigsqcup_{k=K}^{\infty}}
E_{S\theta|k}T_{e\left( S\theta|k-1\right) }\backslash E_{S\theta
|k-1}T_{e\left( S\theta|k-1\right) }=\pi\left( S\theta\right) .
\end{align*}
\end{proof}
\begin{flushleft}
\textbf{Theorem \ref{1to1thm}.} If $\pi(i)\cap\pi(j)$ does not tile $\left(
support\text{ }\pi(i)\right) \cap\left( support\text{ }\pi(j)\right) $ for
all $i\neq j$, then $\pi:[N]^{\ast}\cup\lbrack N]^{\infty}\rightarrow
\mathbb{T}$ is one-to-one.
\end{flushleft}
\begin{proof}
The map $\pi$ is one-to-one on $[N]^{\ast}$ by Lemma \ref{noninjlem}, so we
restrict attention to points in $[N]^{\infty}$. If $\theta$ and $\theta
^{\prime}$ are such that $\theta_{1}=i$ and $\theta_{1}^{\prime}=j$, then the
result is immediate because $\pi(\theta)$ contains $\pi(i)$ and $\pi
(\theta^{\prime})$ contains $\pi(j)$. If $\theta$ and $\theta^{\prime}$ agree
through their first $K$ terms with $K\geq1$ and $\theta_{K+1}\neq$
$\theta_{K+1}^{\prime}$, then $\pi(S^{K}\theta)\neq\pi(S^{K}\theta^{\prime})$.
Now apply $S_{\theta_{1}}^{-1}S_{\theta_{2}}^{-1}...S_{\theta_{K}}^{-1}$ to
obtain $\pi(\theta)\neq\pi(\theta^{\prime})$. (We can do this last step
because $S_{i}^{-1}=\left( f_{i}s^{-a_{i}}\alpha^{a_{i}}\right) ^{-1}%
=\alpha^{-a_{i}}s^{a_{i}}f_{i}^{-1}$ has as its domain all of $\mathbb{T}%
^{\prime}$ and maps $\mathbb{T}^{\prime}$ into $\mathbb{T}^{\prime}$.)
\end{proof}
\section{\label{exsec}Examples}
\subsection{Golden b tilings}
A \textit{golden b} $G\subset{\mathbb{R}}^{2}$ is illustrated in Figure
\ref{golden-01}. This hexagon is the only rectilinear polygon that can be
tiled by a pair of differently scaled copies of itself \cite{S, Sch}.
Throughout this subsection the IFS is
\[
\mathcal{F}=\{\mathbb{R}^{2};f_{1},f_{2}\}
\]
where
\[
f_{1}(x,y)=%
\begin{pmatrix}
0 & s\\
-s & 0
\end{pmatrix}%
\begin{pmatrix}
x\\
y
\end{pmatrix}
+%
\begin{pmatrix}
0\\
s
\end{pmatrix}
,\quad f_{2}(x,y)=%
\begin{pmatrix}
-s^{2} & 0\\
0 & s^{2}%
\end{pmatrix}%
\begin{pmatrix}
x\\
y
\end{pmatrix}
+%
\begin{pmatrix}
1\\
0
\end{pmatrix}
,
\]
where the scaling ratios $s$ and $s^{2}$ obey $s^{4}+s^{2}=1$, which tells us
that $s^{-2}=\alpha^{-2}$ is the golden mean. The attractor of $\mathcal{F}$
is $A=G$. It is the union of two prototiles $f_{1}(G)$ and $f_{2}(G)$. Copies
of these prototiles are labelled $L$ and $S$. In this example, note that
$e(\theta)=\theta_{1}+\theta_{2}+\cdots+\theta_{\left\vert \theta\right\vert
}$ for $\theta\in\lbrack2]^{\ast}$.
\begin{figure}[h]
\centering
\includegraphics[width=5cm, keepaspectratio]{golden01.png}\caption{A golden b
is a union of two tiles, a small one and its partner, a large one. The
vertices of this golden b are located at $(0,0)$ $(1,0)$ $(1,\alpha^{3})$
$(\alpha^{2},\alpha^{3})$ $(\alpha^{2},\alpha)$ $(0,\alpha)$ in
counterclockwise order, starting at the lower left corner, where $\alpha^{-2}$
is the golden mean. This picture also represents a tiling $T_{0}%
=\pi(\varnothing)$. }%
\label{golden-01}%
\end{figure}
The figures in this section illustrate some earlier concepts in the context of
the golden b. Using some of these figures, it is easy to check that
$\mathcal{F}$ is strongly rigid, so the tilings $\pi(\theta)$ have all of the
properties ascribed to them by the theorems in the earlier sections.
\begin{figure}[ptb]
\centering
\includegraphics[width=8cm, keepaspectratio]{GS-C.png} \caption{Structures of
$A_{\theta_{1}\theta_{2}\cdots\theta_{k}1}$ and $A_{\theta_{1}\theta_{2}%
\cdots\theta_{k}2}$ relative to $A_{\theta_{1}\theta_{2}\cdots\theta_{k}}$.}%
\label{construction}%
\end{figure}
\begin{figure}[ptb]
\centering
\vskip 7mm \includegraphics[width=8cm, keepaspectratio]{GS-fixedagain.png}
\caption{Some of the sets $A_{\theta_{1}\theta_{2}\theta_{3}..\theta_{k}}$ and
the corresponding tilings $\pi(\theta_{1}\theta_{2}\theta_{3}..\theta_{k})$.
The recursive organization is such that $\pi(\varnothing)\subset\pi(\theta
_{1})\subset\pi(\theta_{1}\theta_{2})\subset\cdots$ regardless of the choice
$\theta_{1}\theta_{2}\theta_{3}..\in\{1,2\}^{\infty}$. }%
\label{img_1015x}%
\end{figure}
\begin{figure}[ptb]
\centering
\includegraphics[width=8cm, keepaspectratio
]{L-maps.png}\caption{Relative addresses, the addresses of the tiles that
comprise the tilings $T_{0},T_{1},T_{2},T_{3}$ of $A_{0},A_{1},A_{2}%
,A_{3}\text{.}$}%
\label{l-maps}%
\end{figure}
\begin{figure}[ptb]
\centering
\vskip 9mm \includegraphics[width=8cm, keepaspectratio]{GS-absolute.png}
\caption{Absolute addresses associated with the golden b.}%
\label{absolute}%
\end{figure}
\begin{figure}[ptb]
\centering
\vskip 9mm \includegraphics[width=8cm, keepaspectratio]{IMG_1013.png}
\caption{The boundaries of the tilings $\pi(\emptyset)$, $\pi(1)$, $\pi(2)$,
with the parts of the boundaries of the tiles in $\pi(1)$ that are not parts
of the boundaries of tiles in $\pi(2)$ superimposed in red on the rightmost
image.}%
\label{img1013}%
\end{figure}
The relationships between $A_{\theta_{1}\theta_{2}\cdots\theta_{k}1}$ and
$A_{\theta_{1}\theta_{2}\cdots\theta_{k}2}$ relative to $A_{\theta_{1}%
\theta_{2}\cdots\theta_{k}}$ are illustrated in Figure \ref{construction}.
Figure \ref{img_1015x} illustrates some of the sets $A_{\theta_{1}\theta
_{2}\theta_{3}..\theta_{k}}$ and the corresponding tilings $\pi(\theta
_{1}\theta_{2}\theta_{3}..\theta_{k})$.
In Section \ref{realabsec}, procedures were described by which the relative
addresses of tiles in $T(\theta|k)$ and the absolute addresses of tiles in
$\pi(\theta|k)$ may be calculated recursively. Relative addresses for some
golden b tilings are illustrated in Figure \ref{l-maps}. Figure \ref{absolute}
illustrates absolute addresses for some golden b tilings.
The map $\pi:[2]^{\ast}\cup\lbrack2]^{\infty}\rightarrow\mathbb{T}$ is 1-1 by
Theorem \ref{1to1thm}, because $\pi(1)\cup\pi(2)$ does not tile the
interesection of the supports of $\pi(1)$ and $\pi(2),$ as illustrated in
Figure \ref{img1013}.
We note that $\pi(\overline{12})$ and $\pi(\overline{21})$ are aperiodic
tilings of the upper right quadrant of $\mathbb{R}^{2}$.
\subsection{Fractal tilings with non-integer dimension}
The left hand image in Figure \ref{gold8map}, shows the attractor of the IFS
represented by the different coloured regions, there being 8 maps, and
provides an example of a strongly rigid IFS. The right hand image represents
the attractor of the same IFS minus one of the maps, also strongly rigid, but
in this case the dimensions of the tiles is less than two and greater than
one. Figure \ref{sidebyside} (in Section~\ref{sec:intro}) illustrates a part
of a fractal blow up of a different but related 7 map IFS, also strongly
rigid, and the corresponding tiling.%
\begin{figure}[ptb]%
\centering
\includegraphics[
height=2.5728in,
width=5.2477in
]%
{gold8map2.png}%
\caption{See text.}%
\label{gold8map}%
\end{figure}
Figure \ref{beetile} left shows a tiling associated with the IFS $\mathcal{F}$
represented on the left in Figure \ref{gold8map}, while the tiling on the
right is another example of a fractal tiling, obtained by dropping one of the
maps of $\mathcal{F}$.%
\begin{figure}[ptb]%
\centering
\includegraphics[
height=2.5728in,
width=5.2468in
]%
{beetiling.png}%
\caption{See text.}%
\label{beetile}%
\end{figure}
\subsection{Tilings derived from Cantor sets}
Our results apply to the case where $\mathcal{F}=\{\mathbb{R}^{M}%
;f_{i}(x)=s^{a_{i}}O_{i}+q_{i},i\in\lbrack N]\}$ where $\{O_{i},q_{i}%
:i\in\lbrack N]\}$ is fixed in a general position, the $a_{i}$s are positive
integers, and $s$ is chosen small enough to ensure that the attractor is a
Cantor set. In this situation the set of overlap is empty and it is to be
expected that $\mathcal{F}$ is strongly rigid, in which case all tilings (by a
finite set of prototiles, each a Cantor set) will be non-periodic. We can then
take $s$ to be the supremum of value such that the set of overlap is nonempty,
to yield interesting ``just touching" tilings.
| {
"timestamp": "2017-09-28T02:04:06",
"yymm": "1709",
"arxiv_id": "1709.09325",
"language": "en",
"url": "https://arxiv.org/abs/1709.09325",
"abstract": "New tilings of certain subsets of $\\mathbb{R}^{M}$ are studied, tilings associated with fractal blow-ups of certain similitude iterated function systems (IFS). For each such IFS with attractor satisfying the open set condition, our construction produces a usually infinite family of tilings that satisfy the following properties: (1) the prototile set is finite; (2) the tilings are repetitive (quasiperiodic); (3) each family contains self-similartilings, usually infinitely many; and (4) when the IFS is rigid in an appropriate sense, the tiling has no non-trivial symmetry; in particular the tiling is non-periodic.",
"subjects": "Dynamical Systems (math.DS); Metric Geometry (math.MG)",
"title": "Self-Similar Tilings of Fractal Blow-Ups",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9770226247376174,
"lm_q2_score": 0.7248702880639791,
"lm_q1q2_score": 0.7082146714385816
} |
https://arxiv.org/abs/1908.08479 | Iterative Hard Thresholding for Low CP-rank Tensor Models | Recovery of low-rank matrices from a small number of linear measurements is now well-known to be possible under various model assumptions on the measurements. Such results demonstrate robustness and are backed with provable theoretical guarantees. However, extensions to tensor recovery have only recently began to be studied and developed, despite an abundance of practical tensor applications. Recently, a tensor variant of the Iterative Hard Thresholding method was proposed and theoretical results were obtained that guarantee exact recovery of tensors with low Tucker rank. In this paper, we utilize the same tensor version of the Restricted Isometry Property (RIP) to extend these results for tensors with low CANDECOMP/PARAFAC (CP) rank. In doing so, we leverage recent results on efficient approximations of CP decompositions that remove the need for challenging assumptions in prior works. We complement our theoretical findings with empirical results that showcase the potential of the approach. | \section{Introduction}
The field of compressive sensing \cite{RefWorks:45,RefWorks:373} has lead to a rich corpus of results showcasing that intrinsically low-dimensional objects living in large ambient dimensional space can be recovered from small numbers of linear measurements. As a complement to the so-called sparse vector recovery problem is the low-rank matrix recovery problem. Motivated by applications in signal processing (see e.g. \cite{ahmed2014compressive,zhang2013hyperspectral,gross2010quantum,candes2011robust}) and data science (see e.g. \cite{basri2003lambertian,candes2009exact}), the latter asks for a low-rank matrix to be recovered from a small number of linear measurements or observations. Formally, one considers a matrix $\boldsymbol{X}\in\mathbb{R}^{n_1\times n_2}$ of (nearly) rank $r \ll \min(n_1, n_2)$ along with a linear operator $\mathcal{A} : \mathbb{R}^{n_1\times n_2} \rightarrow \mathbb{R}^m$ with $m \ll n_1n_2$, and the goal is to recover $\boldsymbol{X}$ from the measurements $\bm{y} = \mathcal{A}(\boldsymbol{X})$. A common approach is to consider the relaxation of the NP-hard rank minimization \cite{eldar2012uniqueness}, leading to the so-called nuclear-norm minimization problem \cite{candes2009exact,recht2010guaranteed},
\begin{equation}\label{eq:NNM}
\hat{\boldsymbol{X}} = \argmin_{\boldsymbol{X}\in\mathbb{R}^{n_1\times n_2}} \|\boldsymbol{X}\|_* \quad\text{subject to}\quad \mathcal{A}(\boldsymbol{X}) = \bm{y},
\end{equation}
where the nuclear-norm is the L1 norm of the singular values: $\|\boldsymbol{X}\|_* := \sum_i \sigma_i(\boldsymbol{X}) = \operatorname*{trace}(\sqrt{\boldsymbol{X}^*\boldsymbol{X}})$. It has been shown that \eqref{eq:NNM} yields exact recovery of $\boldsymbol{X}$ (or approximate when the measurements contain noise) when the measurement operator $\mathcal{A}$ obeys some assumptions such as incoherence, restricted isometry, or is constructed from some random models \cite{candes2010tight,candes2009exact,recht2010guaranteed}. Examples include when $\mathcal{A}$ is constructed by taking matrix inner products with matrices containing i.i.d. (sub)-Gaussian entries or when $\mathcal{A}$ views entries of the matrix $\boldsymbol{X}$ selected uniformly at random. In these and most other cases, the number of measurements required is on the order of $m\approx r\max(n_1, n_2)$.
As in vector sparse recovery, an alternative to optimization based programs like \eqref{eq:NNM} is to use iterative methods that produce estimates that converge to the solution $\boldsymbol{X}$. Relevant to this paper is the Iterative Hard Thresholding (IHT) method \cite{PaperIHT,blumensath2010normalized,tanner2013normalized}, that can be described succinctly by the update
\begin{equation}\label{eq:IHT}
\boldsymbol{X}^{j+1} = \mathcal{H}_r\left(\boldsymbol{X}^j + \mathcal{A}^*(\bm{y} - \mathcal{A}(\boldsymbol{X}^j))\right),
\end{equation}
where $\boldsymbol{X}^0$ is chosen either as the zero vector/matrix or randomly. Here, $\mathcal{A}^*$ denotes the adjoint of the operator $\mathcal{A}$, and the function $\mathcal{H}_r$ is a thresholding operator. In the vector sparse recovery setting, $\mathcal{H}_r$ simply keeps the $r$ largest in magnitude entries of its input and sets the rest to zero, thereby returning the closest $r$-sparse vector to its input. In the matrix case, it returns the closest rank-$r$ matrix to its input. To guarantee recovery via the IHT method, one may consider the restricted isometry property (RIP) \cite{RefWorks:48}, which asks that the operator $\mathcal{A}$ roughly preserves the geometry of sparse/low-rank vectors/matrices:
$$
(1-\delta_r) \|\boldsymbol{X}\|_F^2 \leq \|\mathcal{A}(\boldsymbol{X})\|_2^2 \leq (1+\delta_r)\|\boldsymbol{X}\|_F^2 \quad\text{for all $r$-sparse/rank-$r$ matrices $\boldsymbol{X}$},
$$
where $0 < \delta_r < 1$ is a controlled constant that may depend on $r$ and $\| \cdot \|_F$ denotes the Frobenius norm. For example, when the operator $\mathcal{A}$ satisfies the RIP for $3r$-sparse vectors with constant $\delta_{3r} < 1/\sqrt{32}$, after a suitable number of iterations, IHT exactly recovers any $r$-sparse vector $\bm{x}$ from the measurements $\bm{y} = \mathcal{A}(\bm{x})$. Moreover, the result is robust and shows that when the measurements $\bm{y}$ are corrupted by noise $\bm{y} = \mathcal{A}\bm{x} + \bm{e}$ and the vector $\bm{x}$ is not exactly sparse but well-approximated by its $r$-sparse representation $\bm{x}_r$, that IHT still produces an accurate estimate to $\bm{x}$ with error proportional to $\|\bm{e}\|_2 + \frac{1}{\sqrt{r}}\|\bm{x}-\bm{x}_r\|_1 + \|\bm{x}-\bm{x}_r\|_2$. See \cite{blumensath2010normalized} for details.
\subsection{Extension to tensor recovery}
Extending IHT from sparse vector recovery to low-rank matrix recovery is somewhat natural. Indeed, a matrix is low-rank if and only if its vector of singular values is sparse. Extensions to the tensor setting, however, yield some non-trivial challenges. Nonetheless, there are many applications that motivate the use of tensors, ranging from video and longitudinal imaging \cite{liu2012tensor,bengua2017efficient} to machine learning \cite{romera2013multilinear,vasilescu2005multilinear} and differential equations \cite{beck2000multiconfiguration,lubich2008quantum}.
We will write a $d$-order tensor as $\boldsymbol{X}\in\mathbb{R}^{n_1\times n_2\times\ldots\times n_d}$, where $n_i$ denotes the dimension in the $i$th mode. Unlike the matrix case, for order $d\geq 3$ tensors, there is not one unique notion of rank. In fact, many notions of rank along with their corresponding decompositions have now been defined and studied, including Tucker rank and higher order SVD \cite{tucker1966some,de2000multilinear}, CP-rank \cite{carroll1970analysis,harshman1970foundations}, and tubal rank \cite{kilmer2011factorization, zhang2014novel}. We refer the reader to \cite{kolda2009tensor} for a nice review of tensors and these various concepts. Succinctly, the Tucker format relies on \textit{unfoldings} of the tensor whereas the CP format relies on rank-1 tensor factors. For example, an order $d$ tensor $\boldsymbol{X}$ can be \textit{matricized} in $d$ ways by unfolding it along each of the $d$ modes \cite{kolda2009tensor}. One can then consider a notion of rank for this tensor as a $d$-tuple $(r_1, r_2, \ldots, r_d)$ where $r_i$ is the rank of the $i$th unfolding. Such a notion is attractive since rank is well-defined for matrices. The CP format on the other hand avoids the need to matricize or unfold the tensor. For the purposes of this work, we will focus on CP-decompositions and CP-rank.
For vectors $\bm{x}$ and $\bm{z}$ denote by $\bm{x}\otimes\bm{z}$ their outer product and for any integer $r$, let $ [r] = \{1,2,...,r \}$. Then one can build a tensor in $\mathbb{R}^{n_1\times n_2\times\ldots\times n_d}$ by taking the outer product $\bm{x}_1\otimes\bm{x}_2\ldots\otimes \bm{x}_d$ where $\bm{x}_i \in \mathbb{R}^{n_i}$ and $\bm{x}_1\otimes\bm{x}_2\ldots\otimes \bm{x}_d$ is a rank-$1$ tensor. This leads to the notation of a rank-$r$ tensor by considering vectors $\bm{x}_{ij}\in\mathbb{R}^{n_j}$ for $i\in[r]$ and $j\in[d]$ and considering the sum of $r$ rank-$1$ tensors:
$$
\boldsymbol{X} = \sum_{i=1}^r \bm{x}_{i1}\otimes\bm{x}_{i2}\otimes\ldots\otimes\bm{x}_{id}.
$$
When $\boldsymbol{X}$ can be written via this decomposition, $\boldsymbol{X}$ is at most rank $r$. The smallest number of rank-$1$ tensors that can be used to express a tensor $\boldsymbol{X}$ is then defined to be the rank of the tensor, and a decomposition using that number of rank-$1$ tensors is its rank decomposition. Note that often one may also ask that the vectors $\bm{x}_{ij}$ have unit norm, and aggregate the magnitude information into constants $\lambda_i$ so that
$$
\boldsymbol{X} = \sum_{i=1}^r \lambda_i\bm{x}_{i1}\otimes\bm{x}_{i2}\otimes\ldots\otimes\bm{x}_{id}.
$$
Note that there are many differences between matrix rank and tensor rank.
For example, the rank of a real-valued tensor may be different when considered over $\mathbb{R}$ versus $\mathbb{C}$ (i.e. if one allows the factor vectors above to be complex-valued or restricted to the reals).
Throughout this paper, we will consider real-valued tensors in $\mathbb{R}$, but our analysis extends to the complex case as well. The CP-rank and CP-decompositions can be viewed as natural extensions of the matrix rank and SVD, and are well motivated by applications such as topic modeling, psychometrics, signal processing, linguistics and many others \cite{carroll1970analysis,harshman1970foundations,anandkumar2014tensor}.
Unfortunately, not only is rank-minimization of tensors NP-Hard, but even the relaxation to the nuclear norm minimization using any of these notions of rank is also NP-Hard \cite{hillar2013most}. Therefore, it is even more crucial to consider other types of methods for tensor recovery. Fortunately, many iterative methods have natural extensions to the tensor setting.
Here, we will focus on the extension of the IHT method \eqref{eq:IHT} to the tensor setting, as put forth in \cite{rauhut2017low}. The authors prove accuracy of the tensor variant under a tensor RIP assumption. Likewise, we will consider measurements of the form $\bm{y} = \mathcal{A}(\boldsymbol{X})$, where $\mathcal{A} : \mathbb{R}^{n_1\times\ldots\times n_d}\rightarrow \mathbb{R}^m$ is a linear operator. The tensor IHT method (TIHT) of \cite{rauhut2017low} is summarized as in Algorithm \ref{alg:TIHT}.
\begin{algorithm}[H]
\caption{Tensor Iterative Hard Thresholding (TIHT)}\label{alg:TIHT}
\begin{algorithmic}[1]
\State\textbf{Input:} operator $ \mathcal{H}_r$, rank $r$, measurements $\bm{y}$, number of iterations $T$
\State\textbf{Output:} $\hat{\boldsymbol{X}}=\boldsymbol{X}^{T}$.
\State\textbf{Initialize:} $\boldsymbol{X}^1=\mathbf{0}$
\For {$j=0,2,\ldots,T-1$}
\State $\mathbf{W}^j = \boldsymbol{X}^j + \mathcal{A}^*(\bm{y} - \mathcal{A}(\boldsymbol{X}^j))$
\State $\boldsymbol{X}^{j+1} = \mathcal{H}_r(\mathbf{W}^j)$
\EndFor
\end{algorithmic}
\end{algorithm}
Note that the algorithm depends on the ``thresholding'' operator\footnote{We note that this operator need not really be a true ``thresholding'' operator, but use this nomenclature since the method is derived from the classical iterative hard thresholding method.} as input. In the vector case this operator performs simple thresholding, thus the name of the algorithm. In the matrix case it typically performs a low-rank projection via the SVD. In the tensor case, there are several options for this operator. In \cite{rauhut2017low}, the authors ask that $\mathcal{H}_r$ computes an approximation to the closest rank-$r$ tensor to its input. Such an approximation is necessary since computing the best rank-$r$ tensor may be ill-defined or NP-Hard depending on the notion of rank used. For $d$-order tensors, the approximation in \cite{rauhut2017low} is asked to satisfy
$$
\|\boldsymbol{X} - \mathcal{H}_r(\boldsymbol{X})\|_F \leq C\sqrt{d}\|\boldsymbol{X} - \boldsymbol{X}_r\|_F,
$$
where $\boldsymbol{X}_r$ is the best rank-$r$ approximation to $\boldsymbol{X}$ using various notions of the Tucker rank. We will adapt this operator to the CP-rank and propose a valid approximation for our purposes later. As in \cite{rauhut2017low}, the recovery of low CP-rank tensors by TIHT will rely on a tensor variant of the RIP, defined below in Definition~\ref{cptrip}.
We will utilize the TRIP for an appropriate set $S_{r,R}$ corresponding to normalized tensors with low CP-rank. In \cite{rauhut2017low}, the TRIP is utilized for several other notions of Tucker rank, namely for tensors with low rank higher order SVD (HOSVD) decompositions, hierarchical Tucker (HT) decompositions, and tensor train (TT) decompositions. The authors then prove that various randomly constructed measurement maps $\mathcal{A}$ satisfy those variations of the TRIP with high probability. Indeed, letting the rank $r$ bound the rank entries $r_i$ of the appropriate Tucker $d$-tuple (see \cite{rauhut2017low} for details), the TRIP is satisfied when the number of measurements $m$ is on the order of $\delta_r^{-2}(r^d + dnr)\log(d)$ and $\delta_r^{-2}(dr^3 + dnr)\log(dr)$ or HOSVD and TT/HT decompositions, respectively. Such random constructions include those obtained by taking tensor inner products of $\boldsymbol{X}$ with tensors having i.i.d. (sub)-Gaussian entries. Under the TRIP assumption, the main result of \cite{rauhut2017low} shows that TIHT provides recovery of low Tucker rank tensors, as summarized by the following theorem.
\begin{theorem}[\cite{rauhut2017low}]
Let $\mathcal{A} : \mathbb{R}^{n_1\times n_2\times\ldots \times n_d} \rightarrow \mathbb{R}^m$ satisfy the TRIP (for rank$-r$ HOSVD or TT or HT tensors) with $\delta_{3r} \leq \delta < 1$, and run TIHT with noisy measurements $\bm{y} = \mathcal{A}\boldsymbol{X} +\bm{e}$. Assume that the following holds at each iteration of TIHT:
\begin{equation}\label{eq:verify}
\|\mathbf{W}^j - \boldsymbol{X}^{j+1}\|_F \leq (1+\varepsilon)\|\mathbf{W}^j - \boldsymbol{X}\|_F.
\end{equation}
Then the estimates produced by TIHT satisfy:
$$
\|\boldsymbol{X}^{j+1} - \boldsymbol{X}\|_F \leq c^j\|\boldsymbol{X}\|_F + C\|\bm{e}\|_2,
$$
where $0<c<1$ and $C$ denote constants that may depend on $\delta$.
As a consequence, after $T = C'\log\left(\|\boldsymbol{X}\|_F / \|\bm{e}\|_2\right)$ iterations, the estimate satisfies
$$
\|\boldsymbol{X}^T - \boldsymbol{X}\|_F \leq C''\|\bm{e}\|_2,
$$
where $C'$ and $C''$ denote constants that may depend on $\delta$.
\end{theorem}
As the authors themselves point out, the challenge with this result is that \eqref{eq:verify} may be challenging to verify.
\subsection{Contributions and Organization}
{The main contribution of this work is the extension and analysis of TIHT to low CP-rank tensors. Using recent work in low CP-rank tensor approximations, this work provides theoretical guarantees for the recovery of low CP-rank tensors without requiring assumptions on the hard thresholding operation $\mathcal{H}_r$. We also show that tensor measurement maps with properly normalized Gaussian random variables satisfy a CP-rank version of the tensor RIP (TRIP) with high probability. These contributions are then supported by synthetically generated as well as real world experiments on video data.}
The remainder of the paper is organized as follows. Section \ref{sec:main} contains our main results. Theorem \ref{mainthm} proves accurate recovery of TIHT for tensors with low CP-rank, under an appropriate TRIP assumption, without the need to verify an assumption like \eqref{eq:verify}. In Section \ref{sec:maintrip} we prove Theorem \ref{coverthm} showing that measurement maps satisfying our TRIP assumption can be obtained by random constructions. Section \ref{sec:exps} showcases numerical results for real and synthetic tensors, and we conclude in Section \ref{sec:conclude}.
\section{Main Results for TIHT for CP-rank}\label{sec:main}
First, let us formally define the set of tensors for which we will prove accurate recovery using TIHT. Given some $R>0$, let us define the set of tensors:
\begin{equation}\label{SrR}
S_{r,R} := \{ \boldsymbol{X} \in \mathbb{R}^{n_1\times n_2\times\ldots\times n_d} : \|\boldsymbol{X}\|_F = 1, \boldsymbol{X} = \sum_{i=1}^r \bm{x}_{i1}\otimes \bm{x}_{i2}\otimes\ldots\otimes \bm{x}_{id}, x_{ij}\in\mathbb{R}^{n_j}, \|\bm{x}_{ij}\|_2 \leq R\}.
\end{equation}
In other words, $S_{r,R}$ is the set of all CP-rank $r$ tensors with bounded factors. Such tensors are not unusual and have been used to provide theoretical guarantees in previous works~\cite{song2019relative,bhaskara2014uniqueness}.
We first define an analog of the tensor RIP (TRIP) for low CP-rank tensors.
\begin{definition}\label{cptrip}
The measurement operator $\mathcal{A} : \mathbb{R}^{n_1\times n_2\times\ldots \times n_d} \rightarrow \mathbb{R}^m$ satisfies the TRIP adapted to $S_{r,R}$ with parameter $\delta_r>0$ when
$$
(1-\delta_r)\|\boldsymbol{X}\|_F^2 \leq \|\mathcal{A}(\boldsymbol{X})\|_2^2 \leq (1+\delta_r)\|\boldsymbol{X}\|_F^2
$$
for all $\boldsymbol{X}\in S_{r,R}$, defined in \eqref{SrR}.
\end{definition}
We will utilize the method and result from \cite{song2019relative} that guarantees the following.
\begin{theorem}[\cite{song2019relative}, Theorem 1.2]\label{woodruff}
Let $\mathbf{W}$ be an arbitrary order-$d$ tensor, $\varepsilon,\alpha>0$, and positive integer $r$, and set
$$
\gamma := \min_{\hat{\mathbf{W}} : \text{rank}(\hat{\mathbf{W}})=r} \|\hat{\mathbf{W}} - \mathbf{W}\|_F.
$$
Suppose there is a rank-$r$ tensor $\hat{\mathbf{W}}$ satisfying $\|\hat{\mathbf{W}} - \mathbf{W}\|_F^2 \leq \gamma^2 + 2^{-n^\alpha}$ and whose CP factors have norms bounded by $2^{n^\alpha}$. Then there is an efficient algorithm that outputs a rank-$r$ tensor estimate $\tilde{\mathbf{W}}$ such that
$$
\|\mathbf{W} - \tilde{\mathbf{W}}\|_F^2 \leq (1+\varepsilon)\gamma^2 + 2^{-n^\alpha}.
$$
We will write this method as $\mathcal{H}_r(\mathbf{W}) = \tilde{\mathbf{W}}$.
\end{theorem}
Our variant of the TIHT method will utilize this result. It can be summarized by the update steps, initialized with $\boldsymbol{X}^0 = \mathbf{0}$:
\begin{align}
\mathbf{W}^j &= \boldsymbol{X}^j + \mathcal{A}^*(\bm{y} - \mathcal{A}(\boldsymbol{X}^j)) \label{eq:cpTIHT_GD}\\
\boldsymbol{X}^{j+1} &= \mathcal{H}_r(\mathbf{W}^j) \quad\text{as in Theorem \ref{woodruff}.\label{eq:cpTIHT_HT}}
\end{align}
In our context, Theorem \ref{woodruff} means the following.
\begin{corollary}\label{woodcor}
Let $\alpha, \varepsilon >0 $, and let $\boldsymbol{X}$ be an arbitrary CP-rank $r$ tensor with bounded factors; in particular let $\boldsymbol{X}\in S_{r, 2^{n^\alpha}}$. Assume $\mathcal{A}$ is a measurement operator satisfying the TRIP with parameter $\delta = \delta_{3r} < \frac{1}{2}2^{-n^\alpha}$.
Assume measurements $\bm{y} = \mathcal{A}(\boldsymbol{X}) + \bm{z}$ with bounded noise $\|\bm{z}\|_2 \leq \frac{2^{-0.5n^\alpha}}{2\|\mathcal{A}\|_{2\rightarrow 2}}$. Using the notation in~\eqref{eq:cpTIHT_GD} and~\eqref{eq:cpTIHT_HT}, we have
$$
\|\mathbf{W}^0 - \boldsymbol{X}^{1}\|_F^2 \leq (1+\varepsilon)\|\mathbf{W}^0 - \boldsymbol{X}\|_F^2 + 2^{-n^\alpha},
$$
and $\boldsymbol{X}^1$ is of CP-rank $r$ with factors bounded by $2^{n^\alpha}$.
In addition, if $\|\boldsymbol{X}^{j} - \boldsymbol{X}\|_F \leq \|\boldsymbol{X}^{0} - \boldsymbol{X}\|_F$, then we have that
$$
\|\mathbf{W}^{j} - \boldsymbol{X}^{j+1}\|_F^2 \leq (1+\varepsilon)\|\mathbf{W}^j - \boldsymbol{X}\|_F^2 + 2^{-n^\alpha},
$$
and $\boldsymbol{X}^{j+1}$ is of CP-rank $r$ with factors bounded by $2^{n^\alpha}$.
\end{corollary}
\begin{proof}
To apply Theorem \ref{woodruff}, we will verify there is a rank-$r$ tensor $\hat{\mathbf{W}}$ with bounded factors that satisfies $\|\hat{\mathbf{W}} - \mathbf{W}^0\|_F^2 \leq 2^{-n^\alpha} \leq \gamma^2 + 2^{-n^\alpha}$. Our choice for $\hat{\mathbf{W}}$ is precisely the tensor $\boldsymbol{X}$. Indeed, we have
\begin{align*}
\|\mathbf{W}^0 - \boldsymbol{X}\|_F &= \|\mathcal{A}^*\mathcal{A}(\boldsymbol{X}) + \mathcal{A}^*\bm{z} - \boldsymbol{X}\|_F\\
&\leq \|(\mathcal{A}^*\mathcal{A} - \mathcal{I})(\boldsymbol{X})\|_F + \|\mathcal{A}^*\bm{z}\|_F\\
&\leq \delta\|\boldsymbol{X}\|_F + \|\mathcal{A}\|_{2\rightarrow 2}\|\bm{z}\|_2\\
&\leq \frac{1}{2}2^{-n^\alpha} + \frac{1}{2}2^{-0.5n^\alpha}.
\end{align*}
Thus, $\|\mathbf{W}^0 - \boldsymbol{X}\|_F^2 \leq (\frac{1}{2}2^{-n^\alpha} + \frac{1}{2}2^{-0.5n^\alpha})^2 \leq 2^{-n^\alpha}$, so by Theorem \ref{woodruff}, the output $\boldsymbol{X}^1$ satisfies
$$
\|\mathbf{W}^0 - \boldsymbol{X}^{1}\|_F^2 \leq (1+\varepsilon)\min_{\hat{\mathbf{W}} : \text{rank}(\hat{\mathbf{W}})=r} \|\hat{\mathbf{W}} - \mathbf{W}^1\|_F^2 + 2^{-n^\alpha} \leq (1+\varepsilon)\|\mathbf{W}^0 - \boldsymbol{X}\|_F^2 + 2^{-n^\alpha}.
$$
To prove the second part, we proceed in the same way. Namely, we have
\begin{align*}
\|\mathbf{W}^j - \boldsymbol{X}\|_F &= \|\boldsymbol{X}^j + \mathcal{A}^*\mathcal{A}(\boldsymbol{X}) + \mathcal{A}^*\bm{z} - \mathcal{A}^*\mathcal{A}\boldsymbol{X}^j - \boldsymbol{X}\|_F\\
&\leq \|(\mathcal{A}^*\mathcal{A} - \mathcal{I})(\boldsymbol{X} - \boldsymbol{X}^j)\|_F + \|\mathcal{A}^*\bm{z}\|_F\\
&\leq \delta\|\boldsymbol{X} - \boldsymbol{X}^j\|_F + \|\mathcal{A}\|_{2\rightarrow 2}\|\bm{z}\|_2\\
&\leq \delta\|\boldsymbol{X} - \boldsymbol{X}^0\|_F + \|\mathcal{A}\|_{2\rightarrow 2}\|\bm{z}\|_2\\
&= \delta\|\boldsymbol{X}\|_F + \|\mathcal{A}\|_{2\rightarrow 2}\|\bm{z}\|_2\\
&\leq \frac{1}{2}2^{-n^\alpha} + \frac{1}{2}2^{-0.5n^\alpha}.
\end{align*}
Thus, $\|\mathbf{W}^j - \boldsymbol{X}\|_F^2 \leq (\frac{1}{2}2^{-n^\alpha} + \frac{1}{2}2^{-0.5n^\alpha})^2 \leq 2^{-n^\alpha}$, so by Theorem \ref{woodruff}, the output $\boldsymbol{X}^{j+1}$ satisfies
$$
\|\mathbf{W}^j - \boldsymbol{X}^{j+1}\|_F^2 \leq (1+\varepsilon)\min_{\hat{\mathbf{W}} : \text{rank}(\hat{\mathbf{W}})=r} \|\hat{\mathbf{W}} - \mathbf{W}^j\|_F^2 + 2^{-n^\alpha} \leq (1+\varepsilon)\|\mathbf{W}^j - \boldsymbol{X}\|_F^2 + 2^{-n^\alpha}.
$$
\end{proof}
Our goal will be to prove that the TIHT variant described in~\eqref{eq:cpTIHT_GD}-\eqref{eq:cpTIHT_HT} provides accurate recovery of tensors in $S_{r,R}$ \eqref{SrR}, provided that the measurement operator $\mathcal{A}$ satisfies the CP-rank analog of the TRIP.
We now proceed with our main theorems.
\begin{theorem}[TIHT with bounded low CP-rank]\label{mainthm}
Let $\boldsymbol{X}\in S_{r,2^{n^\alpha}}$.
Consider the TIHT method described in~\eqref{eq:cpTIHT_GD}-\eqref{eq:cpTIHT_HT}, assume $\mathcal{A}$ satisfies the TRIP with parameter $\delta=\delta_{3r} \leq \frac{1}{2}2^{-n^\alpha}$ as in Definition~\ref{cptrip}, and run TIHT with noisy measurements $\bm{y} = \mathcal{A}(\boldsymbol{X}) + \bm{z}$ where the noise is bounded $\|\bm{z}\|_2 \leq \frac{2^{-n^\alpha}}{2\|\mathcal{A}\|_{2\rightarrow 2}}$. Then TIHT has iterates that satisfy
\begin{equation}
\|\boldsymbol{X}^{j+1} - \boldsymbol{X} \|_F \leq (2 \delta)^j \|\boldsymbol{X}^{0} - \boldsymbol{X} \|_F + \frac{2 \sqrt{1+\delta}}{1 - 2 \delta}\|\bm{z} \|_2 + \frac{(1+\epsilon) 2^{-0.5n^\alpha}}{1 - 2 \delta}.
\end{equation}
As a consequence, recovery error on the order of the upper bound of the noise, $2^{-n^\alpha}$, is achieved after roughly $\lceil \log_{1/2\delta}(\|\boldsymbol{X}^0 - \boldsymbol{X}\|_F / \|\bm{z}\|_2) \rceil $ iterations.
\end{theorem}
\begin{proof}
The proof follows that of Theorem 1 in \cite{rauhut2017low} with some crucial modifications. In particular, instead of requiring assumption (31) in the proof the aforementioned theorem, we have Corollary~\ref{woodcor}. As a direct result, we also have a nice upper bound on $\|\mathbf{W}^j - \boldsymbol{X} \|_F$ whereas Theorem 1 in \cite{rauhut2017low} requires additional computation for upper bounding this term.
Starting from Corollary~\ref{woodcor}, we have that
\begin{equation*}
2^{-n^\alpha} + (1+\epsilon) \|\mathbf{W}^j - \boldsymbol{X} \|^2_F \geq \|\mathbf{W}^{j} - \boldsymbol{X}^{j+1} \|^2_F.
\end{equation*}
Adding and subtracting $\boldsymbol{X}$ to the right hand side, rearranging terms, and substituting the value of $\mathbf{W}^j$ from~\eqref{eq:cpTIHT_GD}, we can write:
\begin{align*}
\|\boldsymbol{X}^{j+1} - \boldsymbol{X} \|^2_F &\leq 2 \langle \boldsymbol{X}^j - \boldsymbol{X}, \boldsymbol{X}^{j+1} - \boldsymbol{X} \rangle - 2 \langle \mathcal{A} ( \boldsymbol{X}^j - \boldsymbol{X}), \mathcal{A}(\boldsymbol{X}^{j+1} - \boldsymbol{X}) \rangle\\
&\quad\quad +2 \langle \bm{z}, \mathcal{A}(\boldsymbol{X}^{j+1} - \boldsymbol{X}) \rangle + (2\epsilon + \epsilon^2) \| \mathbf{W}^j - \boldsymbol{X} \|_F^2 + 2^{-n^\alpha}.
\end{align*}
Using the fact that $rank(\boldsymbol{X}^{j+1} - \boldsymbol{X}) \leq 2r < 3r$, we invoke TRIP and the Cauchy-Schwarz inequality on the third term and the upper bound $\| \mathbf{W}^j - \boldsymbol{X}\|_F^2 \leq 2^{-n^\alpha}$ shown in Corollary~\ref{woodcor} on the fourth term in the summation to obtain
\begin{align*}
\|\boldsymbol{X}^{j+1} - \boldsymbol{X} \|^2_F &\leq 2 \langle \boldsymbol{X}^j - \boldsymbol{X}, \boldsymbol{X}^{j+1} - \boldsymbol{X} \rangle - 2 \langle \mathcal{A} ( \boldsymbol{X}^j - \boldsymbol{X}), \mathcal{A}(\boldsymbol{X}^{j+1} - \boldsymbol{X}) \rangle\\ &\quad\quad +2 \sqrt{1+\delta_{3r} } \| \boldsymbol{X}^{j+1} - \boldsymbol{X} \|_F\|\bm{z}\|_2 + (1+\epsilon)^2 2^{-n^\alpha}.
\end{align*}
Let $\mathcal{Q}^j:\mathbb{R}^{n_1 \times n_2 \times \dots \times n_d } \rightarrow \mathbb{R}^{U^j}$ by the orthogonal projection operator into the subspace spanned by $\boldsymbol{X}^{j+1}$, $\boldsymbol{X}^j$, and $\boldsymbol{X}$. Additionally denote $\mathcal{A}_\mathcal{Q}^j(\mathbf{Z}):=\mathcal{A} (\mathcal{Q}^j(\mathbf{Z}))$ for all $\mathbf{Z} \in \mathbb{R}^{n_1\times n_2 \times \dots n_d}$. Using this notation, we can rewrite the above inequality as:
\begin{align*}
\|\boldsymbol{X}^{j+1} - \boldsymbol{X} \|^2_F &\leq 2 \langle \boldsymbol{X}^j - \boldsymbol{X}, \boldsymbol{X}^{j+1} - \boldsymbol{X} \rangle - 2 \langle \mathcal{A}_\mathcal{Q}^j ( \boldsymbol{X}^j - \boldsymbol{X}), \mathcal{A}_\mathcal{Q}^j (\boldsymbol{X}^{j+1} - \boldsymbol{X}) \rangle\\ &\quad\quad +2 \sqrt{1+\delta_{3r} } \| \boldsymbol{X}^{j+1} - \boldsymbol{X} \|_F\|\bm{z}\|_2 + (1+\epsilon)^2 2^{-n^\alpha} \\
&\leq 2 \| \mathcal{I} - {\mathcal{A}_\mathcal{Q}^j}^*\mathcal{A}_\mathcal{Q}^j \|_{2 \rightarrow 2} \|\boldsymbol{X}^{j+1} - \boldsymbol{X} \|_F \|\boldsymbol{X}^{j} - \boldsymbol{X} \|_F \\ &\quad\quad + 2 \sqrt{1+\delta_{3r} } \| \boldsymbol{X}^{j+1} - \boldsymbol{X} \|_F\|\bm{z}\|_2 + (1+\epsilon)^2 2^{-n^\alpha},
\end{align*}
where the final inequality uses simplification to combine the first two terms and the Cauchy-Schwarz inequality.
Let $\beta$, $\gamma \in [0,1]$ such that $\beta + \gamma = 1$ then we can write:
\begin{align*}
(1-\beta-\gamma)\|\boldsymbol{X}^{j+1} - \boldsymbol{X} \|^2_F & \leq 2 \| \mathcal{I} - {\mathcal{A}_\mathcal{Q}^j}^*\mathcal{A}_\mathcal{Q}^j \|_{2 \rightarrow 2} \|\boldsymbol{X}^{j+1} - \boldsymbol{X} \|_F \|\boldsymbol{X}^{j} - \boldsymbol{X} \|_F \\
\beta \|\boldsymbol{X}^{j+1} - \boldsymbol{X} \|^2_F &\leq 2 \sqrt{1+\delta_{3r} } \| \boldsymbol{X}^{j+1} - \boldsymbol{X} \|_F\|\bm{z}\|_2 \\
\gamma \|\boldsymbol{X}^{j+1} - \boldsymbol{X} \|^2_F &\leq (1+\epsilon)^2 2^{-n^\alpha}.
\end{align*}
Dividing the first two inequalities by $\| \boldsymbol{X}^{j+1} - \boldsymbol{X}\|_F$, square rooting both sides of the last inequality, and taking the sum of all three inequalities results in:
\begin{equation}
\|\boldsymbol{X}^{j+1} - \boldsymbol{X} \|_F \leq f(\beta) \left( 2 \| \mathcal{I} - {\mathcal{A}_\mathcal{Q}^j}^*\mathcal{A}_\mathcal{Q}^j \|_{2 \rightarrow 2} \|\boldsymbol{X}^{j} - \boldsymbol{X} \|_F + 2 \sqrt{1+\delta_{3r} }\|\bm{z} \|_2 + (1+\epsilon) 2^{-0.5n^\alpha}\right),
\end{equation}
where $f(\beta) = \left(1 - \beta + \sqrt{\beta} \right)$. Noting that $0 \leq f(\beta) \leq 1$ for $\beta \in [0,1]$, we can drop the $f(\beta)$ term proceeding inequalities. With the bound from the proof of \cite{rauhut2017low}[Theorem 1], we have that $\| \mathcal{I} - {\mathcal{A}_\mathcal{Q}^j}^*\mathcal{A}_\mathcal{Q}^j \|_{2 \rightarrow 2} \leq \delta_{3r}$ leading to the inequality:
\begin{equation*}
\|\boldsymbol{X}^{j+1} - \boldsymbol{X} \|_F \leq 2 \delta_{3r} \|\boldsymbol{X}^{j} - \boldsymbol{X} \|_F + 2 \sqrt{1+\delta_{3r}}\|\bm{z} \|_2 + (1+\epsilon) 2^{-0.5n^\alpha}.
\end{equation*}
Finally, iterating the upper bound leads to the desired result:
\begin{equation*}
\|\boldsymbol{X}^{j+1} - \boldsymbol{X} \|_F \leq (2 \delta_{3r})^j \|\boldsymbol{X}^{0} - \boldsymbol{X} \|_F + \frac{2 \sqrt{1+\delta_{3r}}}{1 - 2 \delta_{3r}} \|\bm{z} \|_2+ \frac{(1+\epsilon) 2^{-0.5n^\alpha}}{1 - 2 \delta_{3r}}.
\end{equation*}
\end{proof}
Note that the TIHT method and main theorem have been modified in two crucial ways. First, the ``thresholding'' operator $\mathcal{H}_r$ has been replaced by the output guaranteed by Theorem \ref{woodruff}. This allows us to obtain an efficient thresholding step at the price of outputting a tensor of slightly higher rank. This higher rank output in turn requires a stricter assumption on the operator $\mathcal{A}$, namely that the TRIP is satisfied with that higher rank. However, we typically assume $d$ is small and bounded, so the increase is not severe. That leads to the second modification, which is that the TRIP is defined for low CP-rank matrices rather than other types of low rankness. Thus, what remains to be proved is for what types of measurement operators $\mathcal{A}$ this TRIP holds.
\subsection{Measurement maps satisfying our TRIP}\label{sec:maintrip}
The following theorem shows that Gaussian measurement maps satisfy the desired TRIP with high probability.
\begin{theorem}\label{coverthm}
Let $\mathcal{A} : \mathbb{R}^{n_1\times n_2\times\ldots \times n_d} \rightarrow \mathbb{R}^m$ be represented by a tensor in $\mathbb{R}^{n_1\times n_2\times\ldots \times n_d\times m}$ whose entries are properly normalized, i.i.d. Gaussian random variables.\footnote{This can easily be extended to mean zero $L$-subgaussian random variables, as in \cite{rauhut2017low}, where the constant $C$ in \eqref{eq:mtrip} depends on the subgaussian parameter $L$.}
Then $\mathcal{A}$ satisfies the TRIP as in Definition \ref{cptrip} with parameter $\delta_r$ as long as
\begin{equation}
m \geq C\delta^{-2}\cdot \max\left\{\log(\varepsilon^{-1}), ~r\log(drR^d)\sum_{i=1}^d n_i \right\}.
\label{eq:mtrip}
\end{equation}
\end{theorem}
\begin{proof}
We proceed again as in \cite{rauhut2017low}, with the main change being the construction of the covering of the set $S_{r,R}$. To that end, we first obtain a bound on the covering number of this set, defined to the minimal cardinality of an $\epsilon$-net $\mathcal{X}$ such that for any point $\boldsymbol{X}\in S_{r,R}$, there is a point $\hat{\boldsymbol{X}}\in\mathcal{X}$ such that $\|\boldsymbol{X} - \hat{\boldsymbol{X}}\|_F \leq \epsilon$. We denote this covering number as $\mathcal{N}(S_{r,R},\epsilon)$. See e.g. \cite{rogers1964packing} for more details on covering numbers.
\begin{lemma}\label{coverlem}
For any $\epsilon > 0$, the covering number of $S_{r,R}$ satisfies
$$
\mathcal{N}(S_{r,R},\epsilon) \leq \left(\frac{3drR^d}{\epsilon}\right)^{r\sum_{i=1}^d n_i}.
$$
\end{lemma}
\begin{proof}
For simplicity, we will prove the result for $d=3$ and $n_1 = n_2 = n_3 := n$, and the general case follows similarly. Denote by $B_2^n$ the unit ball in $\mathbb{R}^n$, namely $B_2^n = \{\bm{x}\in\mathbb{R}^n : \|\bm{x}\|_2 \leq 1\}$.
First, we begin by obtaining an $\epsilon_1$-net $\mathcal{X}_1$ of the ball of radius $R$, namely $RB_2^n$ ($\epsilon_1$ will be chosen later). Classical results~\cite{vershynin2010introduction}
utilizing volumetric estimates show that such an $\epsilon_1$-net exists with cardinality at most $(3R/\epsilon_1)^n$. Create the set of tensors
$$\mathcal{X}_2 := \left\{\sum_{i=1}^r \bm{x}_{i1}'\otimes \bm{x}_{i2}'\otimes \bm{x}_{i3}' : \bm{x}_{ij}'\in\mathcal{X}_1 \right\} \subset\mathbb{R}^{n^3},$$
and note that that its cardinality is at most
\begin{equation}\label{cover}
|\mathcal{X}_2| \leq |\mathcal{X}_1|^{3r} \leq (3R/\epsilon_1)^{3nr}.
\end{equation}
Now let $\boldsymbol{X}\in S_{r,R}$ be given. Thus, $\boldsymbol{X}$ can be written as $\boldsymbol{X} = \sum_{i=1}^r \bm{x}_{i1}\otimes \bm{x}_{i2}\otimes \bm{x}_{i3}$ for $\bm{x}_{ij}\in RB_2^n$. For each $1\leq i \leq r$ and $1\leq j \leq 3$, choose $\bm{x}_{ij}'\in \mathcal{X}_1$ so that $\|\bm{x}_{ij} - \bm{x}_{ij}'\|_2 \leq \epsilon_1$, and note that $\boldsymbol{X}' := \sum_{i=1}^r \bm{x}_{i1}'\otimes \bm{x}_{i2}'\otimes \bm{x}_{i3}' \in \mathcal{X}_2$. First, we want to bound a single term $\|\bm{x}_{i1}\otimes \bm{x}_{i2}\otimes \bm{x}_{i3} - \bm{x}_{i1}'\otimes \bm{x}_{i2}'\otimes \bm{x}_{i3}'\|_F$. For notational convenience, let us drop the subscript $i$, and write $\bm{x}_{j,a}$ to denote the $a$th entry of the vector $\bm{x}_j := \bm{x}_{ij}$. In addition, let us define the gradually modified tensors $\boldsymbol{Y} = \bm{x}_1\otimes \bm{x}_2\otimes \bm{x}_3$, $\boldsymbol{Y}' = \bm{x}_1'\otimes \bm{x}_2\otimes \bm{x}_3$, $\boldsymbol{Y}'' = \bm{x}_1'\otimes \bm{x}_2'\otimes \bm{x}_3$, and $\boldsymbol{Y}''' = \bm{x}_1'\otimes \bm{x}_2'\otimes \bm{x}_3'$. Our goal is thus to bound $\|\boldsymbol{Y} - \boldsymbol{Y}'''\|_F$. To that end, we have
\begin{align*}
\|\boldsymbol{Y} - \boldsymbol{Y}'\|_F &=
\|\bm{x}_1\otimes \bm{x}_2\otimes \bm{x}_3 - \bm{x}_1'\otimes \bm{x}_2\otimes \bm{x}_3\|_F \\
&= \left( \sum_{a,b,c=1}^n (\bm{x}_{1,a}\bm{x}_{2,b}\bm{x}_{3,c} - \bm{x}_{1,a}'\bm{x}_{2,b}\bm{x}_{3,c})^2\right)^{1/2}\\
&= \left( \sum_{b,c=1}^n\bm{x}_{2,b}^2\bm{x}_{3,c}^2\sum_{a=1}^n (\bm{x}_{1,a} - \bm{x}_{1,a}')^2\right)^{1/2}\\
&= \left( \sum_{b,c=1}^n\bm{x}_{2,b}^2\bm{x}_{3,c}^2\|\bm{x}_{1} - \bm{x}_{1}'\|^2\right)^{1/2}\\
&\leq \epsilon_1\left( \sum_{b,c=1}^n\bm{x}_{2,b}^2\bm{x}_{3,c}^2\right)^{1/2}\\
&= \epsilon_1 \|\bm{x}_2\|_2 \|\bm{x}_3\|_2\\
&\leq \epsilon_1 R^2.
\end{align*}
Similarly, we bound $\|\boldsymbol{Y}' - \boldsymbol{Y}''\|_F \leq \epsilon_1 R^2$ and $\|\boldsymbol{Y}'' - \boldsymbol{Y}'''\|_F \leq \epsilon_1 R^2$. Thus,
\begin{align*}
\|\boldsymbol{Y} - \boldsymbol{Y}'''\|_F &\leq \|\boldsymbol{Y} -\boldsymbol{Y}'\|_F + \|\boldsymbol{Y}' - \boldsymbol{Y}''\|_F + \|\boldsymbol{Y}'' - \boldsymbol{Y}'''\|_F\\
&\leq 3\epsilon_1 R^2
\end{align*}
Finally,
\begin{align*}
\|\boldsymbol{X} - \boldsymbol{X}'\|_F &= \|\sum_{i=1}^r \bm{x}_{i1}\otimes \bm{x}_{i2}\otimes\ldots\otimes \bm{x}_{id} - \sum_{i=1}^r \bm{x}_{i1}'\otimes \bm{x}_{i2}'\otimes\ldots\otimes \bm{x}_{id}'\|_F\\
&\leq \sum_{i=1}^r \|\bm{x}_{i1}\otimes \bm{x}_{i2}\otimes\ldots\otimes \bm{x}_{id} - \bm{x}_{i1}'\otimes \bm{x}_{i2}'\otimes\ldots\otimes \bm{x}_{id}'\|_F\\
&\leq 3r\epsilon_1 R^2.
\end{align*}
Thus, $\mathcal{X}_2$ is a $(3r\epsilon_1 R^2)$-net for $S_{r,R}$. Since our goal is to obtain an $\epsilon$-net, we choose $\epsilon_1 = \epsilon / (3rR^2)$. By \eqref{cover}, this yields a covering number of
$$
|\mathcal{X}_2| \leq \left(\frac{9rR^3}{\epsilon}\right)^{3nr}.
$$
Note that in the case of arbitrary $d$-order tensors, the same proof yields
$$
|\mathcal{X}_2| \leq \left(\frac{3drR^d}{\epsilon}\right)^{r\sum_{i=1}^d n_i}.
$$
\end{proof}
With Lemma \ref{coverlem} in tow, we may now use more classical results from concentration inequalities (see e.g. the proof of Theorem 2 of \cite{rauhut2017low} for a complete proof in the tensor case) that show the number of measurements for a sub-Gaussian operator to satisfy the RIP with parameter $\delta$ over a space $S$ scales like $\log\mathcal{N}(S, \epsilon)$. This completes our proof.
\end{proof}
Since Corollary \ref{woodcor} and Theorem \ref{mainthm} require the TRIP with parameter $\delta_{3r} < \frac{1}{2}2^{-n^\alpha}$, we will apply Theorem \ref{coverthm} with $R = \frac{1}{2}2^{-n^\alpha}$ (and $r$ replaced by $3r$). Theorem \ref{coverthm} then immediately yields the following. Note there is of course a tradeoff in choosing $\alpha$ small or large. Larger $\alpha$ allows for larger noise tolerance and a weaker TRIP requirement, but incurs additional runtime cost in utilizing Corollary \ref{woodcor}.
\begin{corollary}
A Gaussian measurement operator $\mathcal{A}$ will satisfy the TRIP assumptions of Theorem \ref{mainthm} and Corollary \ref{woodcor} so long as
$$
m \geq C'\delta^{-2}\cdot \max\left(\log(\varepsilon^{-1}, r\log(dr2^{-dn^\alpha})\sum_{i=1}^d n_i \right).
$$
\end{corollary}
\section{Numerical Results}\label{sec:exps}
Here we present some synthetic and real experiments that showcase the performance of TIHT when applied to low CP-rank tensors. All experiments are done on order-3 tensors, so $d=3$. We utilize Gaussian measurements as motivated by Theorem \ref{coverthm}. To be precise, for a tensor $\boldsymbol{X}\in\mathbb{R}^{n_1\times n_2\times n_3}$, we construct for specified number of measurements $m$ a matrix $\mathcal{A}\in\mathbb{R}^{n_1 n_2 n_3\times m}$ whose entries are i.i.d. Gaussian with mean zero and variance $1/m$. We then compute the measurements $\bm{y} = \mathcal{A}\boldsymbol{X}$ by applying the matrix $\mathcal{A}$ to the vectorized tensor $\boldsymbol{X}$. Note there are of course other natural ways of applying such Gaussian operators and we do not seek optimality in terms of computation here. In addition, the ``thresholding'' step $\mathcal{H}_r$ of \eqref{eq:cpTIHT_HT} is performed using the \texttt{cpd} function from the Tensorlab package \cite{vervliet2016tensorlab}, which we note may be implemented differently than the method guaranteed by Theorem \ref{woodruff}. We measure the relative recovery error as $\|\boldsymbol{X}-{\boldsymbol{X}^j}\|_F / \|\boldsymbol{X}\|_F$ for a given tensor $\boldsymbol{X}$. We refer to the measurement rate as the percentage of the total number of pixels; a rate of $C\%$ corresponds to number of measurements $m = \frac{C}{100}n_1 n_2 n_3$. The rank reported is the rank used in the TIHT method \eqref{eq:cpTIHT_HT}. We present recovery results for this model here.
\subsection{Synthetic data}
For our first set of experiments, we create synthetic low-rank tensors and test the performance of TIHT with varying parameters $r$, $m$ and $\|\bm{z}\|_2$, corresponding to the tensor rank, number of measurements, and the noise level, respectively. All tensors are created with i.i.d. standard normal entries, and the noise is also Gaussian, normalized as described. Figure \ref{fig:synth} displays results showcasing the effect of varying the number of measurements as well as running TIHT on tensors of varying ranks. The left and center plots of this figure show that as expected, lower numbers of measurements lead to slower convergence rates in both the noiseless and noisy case. In the noisy case (center), we also see different levels of the so-called ``convergence horizon.'' We see the same effect in the right plot, showing that lower rank tensors exhibit faster convergence, again as expected. We repeat these experiments but for the tensor completion setting, where instead of taking Gaussian measurements we simply observe randomly chosen entries of the tensor. The results are displayed in Figure \ref{fig:synth2}, where we observe similar behavior.
\begin{figure}[h!]
\includegraphics[width=2.1in]{{fig1a}.png}$\;$\includegraphics[width=2.1in]{{fig1b}.png}$\;$\includegraphics[width=2.1in]{{fig1c}.png}
\caption{Gaussian measurements. Relative recovery error as a function of iteration. Left: Various measurement rates with $r=2$, $n_1=n_2=n_3=10$, no noise $\bm{z}=0$. Center: Various measurement rates with $r=2$, $n_1=n_2=n_3=10$, noise level $\|\bm{z}\|_2=0.01$. Right: Various ranks $r$ with $n_1=n_2=n_3=10$, noise level $\|\bm{z}\|_2=0.01$.}\label{fig:synth}
\end{figure}
\begin{figure}[h!]
\includegraphics[width=2.1in]{{fig2a}.png}$\;$\includegraphics[width=2.1in]{{fig2b}.png}$\;$\includegraphics[width=2.1in]{{fig2c}.png}
\caption{Tensor completion. Relative recovery error as a function of iteration. Left: Various measurement rates with $r=2$, $n_1=n_2=n_3=10$, no noise $\bm{z}=0$. Center: Various measurement rates with $r=2$, $n_1=n_2=n_3=10$, noise level $\|\bm{z}\|_2=0.01$. Right: Various ranks $r$ with $n_1=n_2=n_3=10$, noise level $\|\bm{z}\|_2=0.01$.}\label{fig:synth2}
\end{figure}
\subsection{Real data}
There is an abundance of real applications that involve tensors. Here, we consider only two, that are appealing for presentation reasons, and for the sole purpose of verifying our theoretical findings.
Namely, we will consider tensors arising from color images and from grayscale video. For $n_1\times n_2$ color (RGB) images, we view the image as a tensor in $\mathbb{R}^{n_1\times n_2\times 3}$ where the third mode represents the three color channels. Similarly, we use a $n_1\times n_2\times n_3$ tensor representation for a video with $n_3$ frames, each of size $n_1\times n_2$. These experiments are meant to showcase that TIHT can indeed be reasonably applied to such real applications.
Figure \ref{fig:kopen} shows the relative recovery error as a function of TIHT iterations for the RGB images shown, which are already low-rank (by construction). Figure \ref{fig:truekopen} shows the same result for a real image which is not made to be low-rank. We compute its approximate distance to its nearest rank-15 tensor using the TensorLab \texttt{cpd} function to be $0.1$. Note that we use small image sizes for sake of computation (the image is actually a patch out of a larger image), which causes the lack of sharpness in the visualization of the original images as displayed. We see in Figure~\ref{fig:truekopen} that the relative error reaches around the optimal, namely the relative distance to its low-rank representation. Thus, this error can be viewed as reaching the noise floor. Figure \ref{fig:candle} shows the same information for the videos whose frames are displayed, and we see again that the error decays until approximately the noise floor. Note that for computational reasons we consider only small image sizes, since the measurement maps themselves grow quite large quickly. Of course, other types of maps may reduce this problem, although that is not the focus of this paper.
\begin{figure}[h!]
\includegraphics[width=2.1in]{fig3a}$\;$\includegraphics[width=2.1in]{fig3b.png}$\;$\includegraphics[width=2.1in]{fig3c.png}
\caption{Color image of size $60\times 60\times 3$, $60\%$ measurement rate, rank $r=15$. Left: Low rank original image. Center: Relative reconstruction error per iteration. Right: Reconstructed image after $200$ iterations. }\label{fig:kopen}
\end{figure}
\begin{figure}[h!]
\includegraphics[width=2.1in]{fig4a}$\;$\includegraphics[width=2.1in]{fig4b.png}$\;$\includegraphics[width=2.1in]{fig4c}
\caption{Color image of size $60\times 60\times 3$, $60\%$ measurement rate. The true image has relative error $0.09$ to a rank-15 image, and rank $r=15$ is used in the algorithm. Left: Original image. Center: Relative reconstruction error per iteration. Right: Reconstructed image after $200$ iterations.}\label{fig:truekopen}
\end{figure}
\begin{figure}[h!]
\includegraphics[width=2.1in]{fig5a}$\;$\includegraphics[width=2.1in]{fig5b.png}$\;$\includegraphics[width=2.1in]{fig5c}
\caption{Candle video of size $30\times 30\times 10$, $80\%$ measurement rate, rank $r=15$. Left: Original image corresponding to the first frame. Center: Relative reconstruction error per iteration. Right: Reconstructed first frame after $130$ iterations. }\label{fig:candle}
\end{figure}
\section{Conclusion}\label{sec:conclude}
This work presents an Iterative Hard Thresholding approach to low CP-rank tensor approximation from linear measurements. We show that the proposed algorithm converges to a low-rank tensor when a linear operator satisfies an RIP-type condition for low rank tensors (TRIP). In addition, we prove that Gaussian measurements satisfy the TRIP condition for low CP-rank signals. Our numerical experiments not only verify our theoretical findings but also highlight the potential of the proposed method. Future directions for this work include extensions to stochastic iterative hard thresholding~\cite{nguyen2017linear} and utilizing different types of tensor measurement maps~\cite{georgieva2019greedy}.
\section*{Acknowledgements}
This material is based upon work supported by the National Security Agency under Grant No. H98230-19-1-0119, The Lyda Hill Foundation, The McGovern Foundation, and Microsoft Research, while the authors were in residence at the Mathematical Sciences Research Institute in Berkeley, California, during the summer of 2019. In addition, Needell was funded by NSF CAREER DMS $\#1348721$ and NSF BIGDATA DMS $\#1740325$. Li was supported by the NSF grants CCF-1409258, CCF-1704204, and the DARPA Lagrange Program under ONR/SPAWAR contract N660011824020. Qin is supported by the NSF grant DMS-1941197.
\bibliographystyle{ieeetr}
| {
"timestamp": "2019-08-23T02:14:31",
"yymm": "1908",
"arxiv_id": "1908.08479",
"language": "en",
"url": "https://arxiv.org/abs/1908.08479",
"abstract": "Recovery of low-rank matrices from a small number of linear measurements is now well-known to be possible under various model assumptions on the measurements. Such results demonstrate robustness and are backed with provable theoretical guarantees. However, extensions to tensor recovery have only recently began to be studied and developed, despite an abundance of practical tensor applications. Recently, a tensor variant of the Iterative Hard Thresholding method was proposed and theoretical results were obtained that guarantee exact recovery of tensors with low Tucker rank. In this paper, we utilize the same tensor version of the Restricted Isometry Property (RIP) to extend these results for tensors with low CANDECOMP/PARAFAC (CP) rank. In doing so, we leverage recent results on efficient approximations of CP decompositions that remove the need for challenging assumptions in prior works. We complement our theoretical findings with empirical results that showcase the potential of the approach.",
"subjects": "Numerical Analysis (math.NA); Machine Learning (stat.ML)",
"title": "Iterative Hard Thresholding for Low CP-rank Tensor Models",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9770226320971079,
"lm_q2_score": 0.724870282120402,
"lm_q1q2_score": 0.7082146709662485
} |
https://arxiv.org/abs/1505.04262 | Hold-in, pull-in, and lock-in ranges of PLL circuits: rigorous mathematical definitions and limitations of classical theory | The terms hold-in, pull-in (capture), and lock-in ranges are widely used by engineers for the concepts of frequency deviation ranges within which PLL-based circuits can achieve lock under various additional conditions. Usually only non-strict definitions are given for these concepts in engineering literature. After many years of their usage, F.~Gardner in the 2nd edition of his well-known work, Phaselock Techniques, wrote "There is no natural way to define exactly any unique lock-in frequency" and "despite its vague reality, lock-in range is a useful concept." Recently these observations have led to the following advice given in a handbook on synchronization and communications "We recommend that you check these definitions carefully before using them." In this survey it is shown that, from a mathematical point of view, in some cases the hold-in and pull-in "ranges" may not be the intervals of values but a union of intervals and thus their widely used definitions require clarification. Rigorous mathematical definitions for the hold-in, pull-in, and lock-in ranges are given. An effective solution for the problem on the unique definition of the lock-in frequency, posed by Gardner, is suggested. | \section{Introduction}
\IEEEPARstart{T}{he} phase-locked loop based circuits (PLL)
are widely used in various applications. A PLL is essentially a nonlinear
control system and its nonlinear analysis is a challenging task.
Much engineering writing is devoted to the study
of PLL-based circuits and the various characteristics for their stability
(see, e.g. a rather comprehensive bibliography of pioneering works in \cite{LindseyT-1973}).
An important engineering characteristic of PLL
is a set of parameters' values for which a PLL achieves lock.
In the classical books on PLLs \cite{Gardner-1966,Viterbi-1966,ShahgildyanL-1966},
published in 1966, such concepts as hold-in, pull-in, lock-in,
and other frequency ranges for which PLL can achieve lock,
were introduced. They are widely used nowadays
(see, e.g. contemporary engineering literature
\cite{Gardner-2005-book,Best-2007,Kroupa-2012} and other publications).
Usually in engineering literature only non-strict definitions are given
for these concepts.
F.~Gardner in 1979\footnote{A year later, in 1980, F.Gardner was elected IEEE Fellow
``for contributions to the understanding and applications of phase lock loops''.}
in the 2nd edition of his well-known work, \emph{Phaselock Techniques},
formulated the following problem \cite[p.70]{Gardner-1979-book}
(see also the 3rd edition \cite[p.187-188]{Gardner-2005-book}):
``{\it{There is no natural way to define exactly any unique lock-in frequency}}''.
The lack of rigorous explanations led
to the paradox:
``{\it{despite its vague reality, lock-in range is a useful concept}}''
\cite[p.70]{Gardner-1979-book}.
Many years of using definitions based on the above concepts
has led to the advice given in a handbook on synchronization and
communications, namely to check the definitions carefully
before using them \cite[p.49]{KiharaOE-2002}.
\smallskip
In this paper it is shown that, from a mathematical point of view,
in some cases the hold-in and pull-in ``ranges''
may be not intervals of values but a union of intervals,
and thus their widely used definitions require clarification.
Next, rigorous mathematical definitions for the hold-in, pull-in, and lock-in ranges are given.
In addition we suggest an effective solution
for the problem of the unique definition of the lock-in frequency,
posed by Gardner.
\section{Classical nonlinear mathematical models
of PLL-based circuits in a signal's phase space}
In classical engineering publications various analog PLL-based circuits
are represented in a \emph{signal's phase space}
(also named \emph{frequency-domain} \cite[p.338]{Davis-2011})
by the block diagram shown in Fig.~\ref{costas-pll-sim-model}.
\begin{figure}[H]
\centering
\includegraphics[width=0.73\linewidth]{non-linear-tfs.pdf}
\caption{
PLL-based circuit in a signal's phase space.
}
\label{costas-pll-sim-model}
\end{figure}
Considering the corresponding mathematical model:
the Phase Detector (PD) is a nonlinear block;
the phases $\theta_{1,2}(t)$ of
the input (reference) and VCO signals
are PD block inputs and the output is
a function $\varphi(\theta_\Delta(t)) = \varphi(\theta_1(t)-\theta_2(t))$
named a \emph{phase detector characteristic},
where
\begin{equation}
\label{theta_delta_def}
\begin{aligned}
& \theta_\Delta(t) = \theta_1(t) - \theta_2(t),
\end{aligned}
\end{equation}
named the phase error.
The relationship between the input $\varphi(\theta_\Delta(t))$
and the output $g(t)$ of the linear filter (Loop filter) is as follows:
\begin{equation}\label{loop-filter}
\begin{aligned}
& \dot x = A x + b \varphi(\theta_\Delta(t)),
\ g(t) = c^*x + h\varphi(\theta_\Delta(t)),
\end{aligned}
\end{equation}
where $A$ is a constant matrix,
$x(t) \in \mathbb{R}^n$ the filter state,
$x(0)$ the initial state of filter,
$b$ and $c$ constant vectors, and h a number.
The filter transfer function has the form:\footnote{
In the control theory the transfer function is often defined with the opposite sign
(see, e.g. \cite{LeonovK-2014}): $H(s) = c^*(A-sI)^{-1}b-h.$
}
\begin{equation}
H(s) = -c^*(A-sI)^{-1}b+h.
\end{equation}
A lead-lag filter \cite{Best-2007} (usually $H(0)=-c^*A^{-1}b+h=1$,
but $H(0)$ can also be any nonzero value
when an active lead-lag filter is used), or a PI filter ($H(0)$ is infinite)
is usually used as the filter.
The solution of \eqref{loop-filter} with initial data $x(0)$
(the filter output for the initial state $x(0)$) is as follows:
\begin{equation}\label{loop-filter-int}
\begin{array}{c}
g(t,x_0) = \alpha_0(t,x(0)) +
\int\limits_0^t
\gamma(t - \tau)\varphi(\theta_\Delta(\tau))
{\rm d}\tau
+ h \varphi(\theta_\Delta(t)),
\end{array}
\end{equation}
where $\gamma(t - \tau)=c^*e^{A(t-\tau)}b$ is the impulse response function of the filter
and $\alpha_0(t,x(0))= c^*e^{At}x(0)$ the zero input response
(natural response, i.e. when the input of the filter is zero).
The control signal $g(t)$ adjusts the VCO frequency to
the frequency of the input signal:
\begin{equation} \label{vco first}
\dot\theta_2(t) = \omega_2(t) = \omega_2^{\text{free}} + Lg(t),
\end{equation}
where $\omega_2^{\text{free}}$ is the VCO free-running frequency
(i.e. for $g(t)\equiv 0$) and $L$ the VCO gain.
Nonlinear VCO models can be similarly considered, see, e.g. \cite{Margaris-2004,Suarez-2009}.
The frequency of the input signal (reference frequency) is usually assumed
to be constant:
\begin{equation}\label{omega1-const}
\dot\theta_1(t) = \omega_1(t) \equiv \omega_1.
\end{equation}
The difference between the reference frequency and the VCO free-running frequency
is denoted as $\omega_\Delta^{\text{free}}$:
\begin{equation}
\label{omega_delta_def}
\begin{aligned}
& \omega_\Delta^{\text{free}} \equiv \omega_1 - \omega_2^{\text{free}}.
\end{aligned}
\end{equation}
By combining equations \eqref{theta_delta_def}, \eqref{loop-filter}, and \eqref{vco first}--\eqref{omega_delta_def}
a \emph{nonlinear mathematical model in the signal's phase space} is obtained
(i.e. in the state space: the filter's state $x$ and the difference between the signal's phases $\theta_\Delta$):
\begin{equation}\label{final_system}
\begin{aligned}
& \dot{x} = A x + b \varphi(\theta_{\Delta}), \\
& \dot\theta_{\Delta} = \omega_{\Delta}^{\text{free}}
- Lc^*x - Lh\varphi(\theta_{\Delta}).
\end{aligned}
\end{equation}
Nowadays nonlinear model \eqref{final_system} is widely used
(see, e.g. \cite{Abramovitch-2002,Abramovitch-2004,Best-2007})
to study acquisition processes of various circuits.
The model can be obtained from the corresponding model in \emph{the signal space}
(called also \emph{time-domain} \cite[p.329]{Davis-2011})
by averaging under certain conditions \cite{KrylovB-1947,KudrewiczW-2007,LeonovKYY-2012-TCASII,LeonovK-2014,LeonovKYY-2015-SIGPRO},
a rigorous consideration of which is often omitted
(see, e.g. classical books \cite[p.12,15-17]{Viterbi-1966}, \cite[p.7]{Gardner-1966})
while their violation may lead to unreliable results
(see, e.g. \cite{KuznetsovKLNYY-2015-ISCAS,BestKKLYY-2015-ACC}).
Usually the PD characteristic is an odd function
(e.g. a PD realization such as a multiplier,
JK-flipflop, EXOR, PFD,
and other elements \cite{Best-2007}).
Note that the PD characteristic $\varphi(\theta_{\Delta})$ depends
on the waveforms of the considered
signals \cite{LeonovKYY-2012-TCASII,LeonovKYY-2015-SIGPRO}).
For the classical PLL with sinusoidal signals
and a two-phase PLL we have $\varphi(\theta_\Delta)=\frac{1}{2}\sin(\theta_\Delta)$,
for the classical BPSK Costas loop with ideal low-pass filters and a two-phase Costas loop
we have $\varphi(\theta_\Delta)=\frac{1}{8}\sin(2\theta_\Delta)$.
Classical PD characteristics are bounded piecewise smooth $2\pi$ periodic
functions\footnote{
If $\varphi(\theta_\Delta(t))$ has another period
(e.g. $\pi$ for the Costas loop models),
it has to be considered in the further discussion instead of $2\pi$.
}:
\[
\varphi(\theta_\Delta(t)+2\pi k)=\varphi(\theta_\Delta(t)),
\quad \forall k=0,1,2...
\]
Thus, it is convenient to assume that $\theta_\Delta$ mod $2\pi$
is a cyclic variable,
and the analysis is restricted to the range of $\theta_\Delta(0) \in [-\pi, \pi)$.
For the case of an odd PD characteristic\footnote{
There are examples of non odd PD characteristics, where \eqref{odd-change}
does not hold true (see, e.g.
BPSK Costas loop with sawtooth signals
\cite{LeonovKYY-2012-TCASII,LeonovKYY-2015-SIGPRO} and others).
}, system \eqref{omega_delta_def} is not changed by the transformation
\begin{equation}\label{odd-change}
\big(\omega_{\Delta}^{\text{free}},x(t),\theta_{\Delta}(t)) \rightarrow
\big(-\omega_{\Delta}^{\text{free}},-x(t),-\theta_{\Delta}(t)).
\end{equation}
Property \eqref{odd-change} allows the analysis of system \eqref{final_system}
with only $\omega_\Delta^{\text{free}}>0$
and introduces the concept of \emph{frequency deviation}
\begin{center}{$|\omega_\Delta^{\text{free}}| = |\omega_1 - \omega_2^{\text{free}}|$.}\end{center}
\section{Locked state}
The locked states (also called steady states) of
the model in the signal's phase space must satisfy the following conditions:
\begin{itemize}
\item the phase error $\theta_\Delta$ is constant, the frequency error $\dot \theta_\Delta$ is zero;
\item the model in a locked state approaches the same locked state
after small perturbations (of the VCO phase, input signal phase, and filter state).
\end{itemize}
The locally asymptotically stable equilibrium (stationary) points
of model \eqref{final_system}:
\begin{equation}
\label{eq-points-def}
\begin{aligned}
& \theta_\Delta(t) \equiv \theta_{eq} + 2\pi k,\quad x(t) \equiv x_{eq},
\end{aligned}
\end{equation}
are locked states, i.e. satisfy the above conditions\footnote{
It can be proved that if the filter is controllable and observable,
then only equilibria satisfy locked state conditions, i.e.
the filter state $x(t)$ must be constant in the locked state \cite{LeonovK-2014}.}.
Considering the case of a nonsingular matrix $A$
(i.e. the transfer function of the filter does not have zero poles),
the equilibria of \eqref{final_system} (stationary points)
are given by the equations
\begin{equation}\label{zeros1}
\begin{aligned}
& \varphi(\theta_{eq}) = \frac{\omega_{\Delta}^{\text{free}}}{L(c^*A^{-1}b - h)}
= \frac{\omega_{\Delta}^{\text{free}}}{LH(0)}, \\
& x_{eq} = -A^{-1}b \varphi(\theta_{eq}) =
- A^{-1}b\frac{\omega_{\Delta}^{\text{free}}}{L(c^*A^{-1}b - h)}.
\end{aligned}
\end{equation}
Thus, the equilibria can be considered as
a multiple-valued function of variable $\omega_\Delta^{\text{free}}$:
$\big(x_{eq}(\omega_{\Delta}^{\text{free}}),\theta_{eq}(\omega_{\Delta}^{\text{free}})\big)$.
From the boundedness of the PD characteristic $\varphi(\theta_{eq})$
it follows that there are no equilibria
for sufficiently large $|\omega_{\Delta}^{\text{free}}|$.
\section{Engineering definitions of stability ranges}
The widely used engineering assumption (see Viterbi's pioneering writing
\cite[p.15]{Viterbi-1966}) is that
the zero input response of filter $\alpha_0(t,x_0)$ does not affect the synchronization of the loop.
This assumption allows the filter state $x(t)$ to be excluded
from the consideration and
a \emph{simplified mathematical model of PLL-based circuit in the signal's phase space}
to be obtained from \eqref{loop-filter-int} and \eqref{vco first}
(see, e.g. \cite[p.17, eq.2.20]{Viterbi-1966} for $h=0$
and \cite[p.41, eq.4-26]{Gardner-1966} for $\gamma \equiv 0$):
\begin{equation}\label{mathmodel-class-simple}
\dot\theta_{\Delta}\!=\!
\omega_{\Delta}^{\text{free}}-L\!\!\!\int_0^t\!\!\!\!\!\gamma(t - \tau)
\varphi(\theta_{\Delta}(\tau)){\rm d}\tau
-Lh \varphi(\theta_{\Delta}(t)).
\end{equation}
For an example of this one-dimensional integro-differential equation
the following intervals (\!\!\cite{Gardner-1966,Viterbi-1966}) are defined:
the hold-in range includes $|\omega_{\Delta}^{\text{free}}|$
such that model \eqref{mathmodel-class-simple}
has an equilibrium $\theta_{\Delta}(t) \equiv \theta_{eq}$,
which is locally stable
(local stability, i.e. for some initial phase error $\theta_{\Delta}(0)$);
the pull-in range includes $|\omega_{\Delta}^{\text{free}}|$
such that any solution of model \eqref{mathmodel-class-simple}
is attracted to one of the equilibria $\theta_{eq}$
(global stability, i.e. for any initial phase error $\theta_{\Delta}(0)$).
Thus, the block diagram of the loop in Fig.~\ref{costas-pll-sim-model}
is usually considered without initial data $x(0)$ and $\theta_{\Delta}(0)$
(see, e.g. \cite[p.17, fig.2.3]{Viterbi-1966}).
Viterbi \cite{Viterbi-1966} explains the above assumption for the stable matrix $A$,
but considers also various filters with marginally stable matrixes
(e.g. a filter -- perfect integrator, where $A=0$).
At the same time, even for a stable matrix $A$,
the initial filter state $x(0)$ and $\alpha_0(t,x_0)$
may affect the acquisition process and stability ranges
(see, e.g. corresponding examples for the classical PLL
\cite{KuznetsovKLNYY-2015-ISCAS} and Costas loops
\cite{KudryashovaKKLSYY-2014-ICINCO,KuznetsovKLNYY-2014-ICUMT-QPSK,KuznetsovKLSYY-2014-ICUMT-BPSK,BestKKLYY-2015-ACC}).
While the above assumption allows
introduction of the above one-dimensional stability sets,
defined only by $|\omega_{\Delta}^{\text{free}}|$,
for rigorous study the multi-dimensional stability domains have to considered,
taking into account $x(0)$,
and their relationships with the classical engineering ranges
have to be explained.
In \cite[p.187]{Gardner-2005-book} it is noted that the consideration of all state variables is of utmost importance
in the study of cycle slips and the \emph{lock-in} concept.
\smallskip
\section{Rigorous definitions of stability sets}
The rigorous mathematical definitions
of the hold-in, pull-in, and lock-in sets are now given
for the nonlinear mathematical model of PLL-based circuits in the signal's phase space
\eqref{final_system} and corresponding nontrivial examples are considered.
\subsection{Local stability and hold-in set}
We now consider the linearization\footnote{
Here it is assumed that
the PD characteristic $\varphi(\theta_\Delta)$ is smooth at the point $\theta_\Delta=\theta_{eq}$.
However, there are PLL-based circuits with nonsmooth or discontinuous PD characteristics
(see, e.g. the sawtooth PD characteristic for PLL \cite{Gardner-2005-book},
the model of QPSK Costas loop \cite{BestKLYY-2014-IJAC},
and some others \cite{biggio2014accurate,biggio2015efficient,bizzarri2012periodic}).
In such a case care has to be taken of the definition of solutions,
the linearization of the model
and the analysis of possible sliding solutions (see, e.g. \cite{GeligLY-1978}).}
of system \eqref{final_system} along
an equilibrium $(x_{eq},\theta_{eq})$.
Taking into account \eqref{zeros1} and $\varphi'(\theta) := d \varphi(\theta)/ d \theta$,
the linearized system is as follows:
\begin{equation}\label{linearized_system_2}
\begin{aligned}
\left(
\begin{array}{c}
\dot{x} \\
\dot\theta_\Delta \\
\end{array}
\right)
=
\left(
\begin{array}{cc}
A & b\varphi'(\theta_{eq}) \\
-Lc^* & -Lh\varphi'(\theta_{eq})
\end{array}
\right)
\left(
\begin{array}{c}
x-x_{eq} \\
\theta_\Delta - \theta_{eq} \\
\end{array}
\right)
\end{aligned}
\end{equation}
The characteristic polynomial of linear system \eqref{final_system}
can be written (using the Schur complement, e.g. \cite{LeonovK-2014}) in the following form:
$\chi(s) = \big(-Lh\varphi'(\theta_{eq}) - s +Lc^*(A-sI)^{-1}b\varphi'(\theta_{eq})\big)\det(A-sI)$,
or can be expressed in terms of the filter's transfer function $H(s)=\frac{a(s)}{d(s)}$, where $a(s)$ and $d(s)$ are polynomials:
\begin{equation}\label{tfdenumerator}
\begin{aligned}
\chi(s) = -\big(sd(s)+a(s)L \varphi'(\theta_{eq})\big).
\end{aligned}
\end{equation}
The characteristic polynomial corresponds to the denominator of the closed loop transfer function\footnote{
Consideration of linearized model \eqref{linearized_system_2} allows to avoid
the rigorous discussion of initial states $(x(0),\theta_\Delta(0))$
related to the Laplace transformation and transfer functions \cite{LeonovK-2014}.
}.
To study the local stability of equilibria \eqref{zeros1},
it is necessary to check whether all the roots of the characteristic polynomial \eqref{tfdenumerator}
for the linearization of model \eqref{final_system} along the equilibria
(i.e. the poles of the closed loop transfer function)
have negative real parts.
For this purpose, at the stage of \emph{pre-design analysis}
when all parameters of the loop can be chosen precisely,
the Routh-Hurwitz criterion and its analogs
(see, e.g. Kharitonov's generalization \cite{Kharitonov-1978}
for interval polynomials) can be effectively applied.
At the stage of \emph{post-design analysis}
when only the input and VCO output are considered and the parameters
are known only approximately,
various frequency characteristics of the loop
(see, e.g. Nyquist and Bode plots)
and the continuation principle can be used (see, e.g, \cite{Gardner-2005-book,Best-2007}).
If the PD characteristic
is an odd function and hence $\varphi'(\theta_{eq})$ is an even function,
from \eqref{odd-change} we conclude that
1) there are symmetric equilibria:
$\big( x_{eq}(\omega_\Delta^{\text{free}}), \theta_{eq}(\omega_\Delta^{\text{free}}) \big)
=\big( -x_{eq}(-\omega_\Delta^{\text{free}}), -\theta_{eq}(-\omega_\Delta^{\text{free}}) \big)$,
2) these symmetric equilibria are simultaneously stable or unstable.
The same holds true for nonstationary trajectories.
\begin{definition}\label{def-hold}
A set of all frequency deviations
$|\omega_{\Delta}^{\text{free}}|$
such that the mathematical model of the loop
in the signal's phase space has a locally asymptotically stable equilibrium
is called a hold-in set $\Omega_{\text{hold-in}}$.
\end{definition}
Thus, a value of frequency deviation belongs to the hold-in set
if the loop re-achieves locked state after small perturbations
of the filter's state, the phases and frequencies of VCO and the input signals.
This effect is also called \emph{steady-state stability}.
In addition, for a frequency deviation within the hold-in set,
\begin{figure}[ht]
\centering
\includegraphics[width=0.4\textwidth]{hold-in.pdf}
\caption{
Phase portrait for $\omega_{\Delta}^{\text{free}}$ from the hold-in range.
The system's evolving state over time traces a trajectory $(x(t),\theta_\Delta(t))$.
Trajectories can not intersect.
Unstable equilibrium points, such as saddles --- black dots,
locally asymptotically stable equilibria --- green dots,
any of which has its own basin of attraction (shaded domain)
bounded by stable saddle separatrices (black trajectories going to the saddles).
There are initial states and corresponding trajectories (see, e.g. dashed trajectory),
which are not attracted to an equilibrium.
}
\label{hold-in-phase-portrait}
\end{figure}
the loop in a locked state tracks
small changes in input frequency,
i.e. achieves a new locked state (\emph{tracking process}).
In the literature the following explanations of the hold-in range
(sometimes also called a \emph{lock range} \cite[p.507]{PedersonM-2008-book}, \cite[p.10-2]{BakshiG-2009-book},
a \emph{synchronization range} \cite{Blanchard-1976}, a \emph{tracking range} \cite[p.49]{KiharaOE-2002})
can be found:
``{\it{The hold-in range is obtained by calculating the frequency where
the phase error is at its maximum}}''\cite[p.171]{brendel2013millimeter},
``{\it{The maximum frequency difference
before losing lock of the PLL system is called the hold-in range}}''\cite[p.258]{Kroupa-2012}.
The following example shows that these explanations may not be correct,
because for high-order filters
the hold-in ``range'' may have holes.
The following example shows that the hold-in set may not include
$\omega_\Delta^{\text{free}} = 0$.
\begin{example}[the hold-in set does not contain $\omega_\Delta^{\text{free}} = 0$]
Consider the classical PLL with the sinusoidal PD characteristic
$\varphi(\theta_\Delta) = \frac{1}{2}\sin(\theta_\Delta)$, VCO input gain $L=8$,
and the filter transfer function
\begin{equation}
H(s) = \frac{a(s)}{d(s)}= \frac{1+0.5s}{1+0.5s+0.5s^2}.
\end{equation}
From \eqref{zeros1} the following equation for equilibria is obtained:
\begin{equation}
\label{eq-points-1}
\begin{aligned}
& \frac{1}{2}\sin(\theta_{eq}) = \frac{1}{8}\omega_\Delta^{\text{free}}.
\end{aligned}
\end{equation}
\begin{figure}[ht]
\centering
\includegraphics[width=0.4\textwidth]{no-equilibria-phase-portrait.pdf}
\caption{
Phase portrait for $\omega_{\Delta}^{\text{free}}$ outside the hold-in range:
there are no locally stable equilibria.}
\label{hold-in-phase-portrait-noeq}
\end{figure}
Applying the Routh-Hurwitz stability criterion\footnote{
For a third-order polynomial $\chi(s) = a_3s^3 + a_2s^2 + a_1s + a_0$,
all the roots have negative real parts and the corresponding linear system is asymptotically stable
if $a_{1,2,3} > 0$ and $a_2a_1 > a_3a_0$.
For
$\chi(s) = a_4s^4 + a_3s^3 + a_2s^2 + a_1s + a_0$,
all the coefficients must satisfy $a_{1,2,3,4} > 0$, and
$a_3a_2 > a_4a_1$ and $a_3a_2a_1 > a_4a_1^2 + a_3^2a_0$.
}
to the denominator of the closed loop transfer function \eqref{tfdenumerator}
\begin{equation}
\label{pol1}
\begin{aligned}
& s^3 + s^2 + s (2+4\cos(\theta_{eq})) + 8\cos(\theta_{eq}),
\end{aligned}
\end{equation}
the following conditions are obtained:
\begin{equation}
\begin{aligned}
& \cos(\theta_{eq}) >0, \ (2+4\cos(\theta_{eq})) > 0,\\
& (2+4\cos(\theta_{eq})) > 8\cos(\theta_{eq}).
\end{aligned}
\end{equation}
Then $0< \cos(\theta_{eq}) < \frac{1}{2}$, and
for the locked state the steady-state phase error (i.e. corresponding to an equilibrium)
is obtained
\begin{equation}
\label{theta-eqs-1}
\begin{aligned}
& \theta_{eq} \in (-\frac{\pi}{2},-\frac{\pi}{3})\cup
(\frac{\pi}{3},\frac{\pi}{2}).
\end{aligned}
\end{equation}
Taking into account \eqref{eq-points-1}, \eqref{theta-eqs-1},
one obtains the hold-in set
\begin{equation}
\begin{aligned}
& |\omega_\Delta^{\text{free}}| \in (2\sqrt{3},4).
\end{aligned}
\end{equation}
\end{example}
\smallskip
The next example shows that the hold-in set may not actually be a range
(i.e., an interval)
but a union of intervals, one of which may include
$\omega_\Delta^{\text{free}} = 0$.
\begin{example}[the hold-in set is a union of disjoint
intervals, one of which contains $\omega_\Delta^{\text{free}} = 0$]
\label{interval-union-example}
Consider the classical PLL with the sinusoidal PD characteristic
$\varphi(\theta_\Delta) = \frac{1}{2}\sin(\theta_\Delta)$, the VCO input gain $L=80$,
and the filter transfer function
\begin{equation}
H(s) = \frac{1+0.25s+0.5s^2}{1+2s+2s^2+2s^3}.
\end{equation}
From \eqref{zeros1} the following equation for the equilibria is obtained:
\begin{equation}
\label{eq-points-omega-2}
\begin{aligned}
& \frac{1}{2}\sin(\theta_{eq}) = \frac{1}{80}\omega_\Delta^{\text{free}}.
\end{aligned}
\end{equation}
An equilibrium is asymptotically stable if and only if all the roots
of polynomial \eqref{tfdenumerator}:
\begin{equation}\label{pol2}
\begin{aligned}
& s(1+2s+2s^2+2s^3) + K(1+0.25s+0.5s^2) =\\
& 2s^4 + 2s^3 + s^2(2+0.5K) + s(1+0.25K) + K,\\
& K = L\varphi'(\theta_{eq})= 40\cos(\theta_{eq})
\end{aligned}
\end{equation}
have negative real parts.
Using the Routh-Hurwitz criterion, we obtain
\begin{equation}
\begin{aligned}
& 2+0.5K >0,\ 1+0.25K > 0,\ K > 0,\\
& 2(2+0.5K) > 2(1+0.25K),\\
& 2(2+0.5K)(1+0.25K) > 2(1+0.25K)^2+2^2K.
\end{aligned}
\end{equation}
From these inequalities we have
\begin{equation}
\label{theta-eqs-2}
\begin{aligned}
& K = 40\cos(\theta_{eq}) \in (0,12 - 8\sqrt{2})\cup(12+8\sqrt{2},\infty),\\
& \theta_{eq} \in (-\frac{\pi}{2},-1.5536) \cup (-0.9486,0.9486) \cup (1.5536,\frac{\pi}{2}).
\end{aligned}
\end{equation}
Note that for other values of $\theta_{eq}$
at least one root of the polynomial \eqref{pol2}
has a positive real part, making the corresponding equilibrium unstable.
Combining \eqref{eq-points-omega-2} and \eqref{theta-eqs-2},
we obtain the hold-in set
\begin{equation}
\begin{aligned}
& |\omega_\Delta^{\text{free}}| \in [0,32.5) \cup (39.9942,40).
\end{aligned}
\end{equation}
Note that in this case, for the values of the VCO input gain $L > 24+16\sqrt{2}$
the hold-in set is always a union of disjoint intervals.
For $L=80$ the simulation results of transition process
in Simulink model\,\footnote{Following
the above classical consideration,
the filter is often represented in MatLab Simulink as the block \emph{Transfer Fcn}
with zero initial state
(see, e.g. \cite{BrigatiFMPP-2001,NicolleTMOJ-2007,Zucchelli-2007,KoivoE-2009,KaaldLHS-2009}).
It is also related to the fact that the transfer function (from $\varphi$ to $g$)
of linear system \eqref{loop-filter}
is defined by the Laplace transformation for zero initial data $x(0) \equiv 0$.
In Fig.~\ref{simulink-model} we use the block \emph{Transfer Fcn (with initial states)}
to take into account the initial filter state $x(0)$;
the initial phase error $\theta_\Delta(0)$
can be taken into account by the property \emph{initial data} of the \emph{Intergator} blocks.
Note that the corresponding initial states in SPICE
(e.g. capacitor's initial charge)
are zero by default
but can be changed manually \cite{BianchiKLYY-2015}.}
in Fig.~\ref{simulink-model} are shown in Figs.~\ref{sim-1}--\ref{sim-3}
for the initial data $(x(0)=(0;0;0.9990), \theta_\Delta(0) = 1.5585)$
and various $\omega_\Delta^{\text{free}}$.
\begin{figure}[ht]
\centering
\includegraphics[width=0.48\textwidth]{simulink-model.pdf}
\caption{MatLab Simulink: the signal's phase space model of the classical PLL}
\label{simulink-model}
\end{figure}
\end{example}
Related discussion on the frequency responses of loop
with high-order filters can be found in \cite[p.34-38, 52-56]{Gardner-2005-book}.
\begin{remark}
For the first order filters, the set $\Omega_{\text{hold-in}}$
is an interval $|\omega_\Delta^{\text{free}}| < \omega_h$.
For higher order filters, the set $\Omega_{\text{hold-in}}$
may be more complex.
Thus, from an engineering point of view,
it is reasonable to require that
$\omega_\Delta^{\text{free}} = 0$ belongs to the hold-in set
and to define a hold-in range as
the largest interval $[0,\omega_{h})$ from the hold-in set
\[
[0,\omega_{h}) \subset \Omega_{\text{hold-in}}
\]
such that a certain stable equilibrium varies continuously
when $\omega_\Delta^{\text{free}}$ is changed within the range\footnote{
In general (when the stable equilibria coexist and some of them may appear or disappear),
the stable equilibria can be considered as
a multiple-valued function of variable $\omega_\Delta^{\text{free}}$,
in which case the existence of its continuous singlevalue branch
for $|\omega_\Delta^{\text{free}}| \in [0,\omega_{h})$ is required.}.
Here $\omega_{h}$ is called
a \emph{hold-in frequency} (see \cite[p.38]{Gardner-1966}).
\end{remark}
\begin{remark}
In the general case when there is no symmetry with respect to
$\omega_{\Delta}^{\text{free}}$ (see \eqref{odd-change})
the hold-in set need not be symmetric
and the set $\omega_{\Delta}^{\text{free}} \in \Omega_{\text{hold-in}}$
must be considered in Definition~\ref{def-hold}.
\end{remark}
\begin{figure}[ht]
\centering
\includegraphics[width=0.24\textwidth]{eq-3-phases.pdf}
\includegraphics[width=0.24\textwidth]{eq-3-filters.pdf}
\caption{$\omega_\Delta^{\text{free}} = 3$: stable locked state exists.}
\label{sim-1}
\end{figure}
\begin{figure}[ht]
\centering
\includegraphics[width=0.24\textwidth]{eq-35-phases.pdf}
\includegraphics[width=0.24\textwidth]{eq-35-filters.pdf}
\caption{$\omega_\Delta^{\text{free}} = 35$: there are no locked states (see also Fig.~\ref{hold-in-phase-portrait-noeq}).}
\label{sim-2}
\end{figure}
\begin{figure}[ht]
\centering
\includegraphics[width=0.24\textwidth]{eq-39-phases.pdf}
\includegraphics[width=0.24\textwidth]{eq-39-filters.pdf}
\caption{$\omega_\Delta^{\text{free}} = 39.997$: stable locked state exists.}
\label{sim-3}
\end{figure}
\subsection{Global stability (stability in the large) and pull-in set}
Assume that the loop power supply is initially switched off and then at $t = 0$ the power is switched on,
and assume that the initial frequency difference is sufficiently large.
The loop may not lock within one beat note,
but the VCO frequency will be slowly tuned
toward the reference frequency (\emph{acquisition process}).
This effect is also called a \emph{transient stability}.
The pull-in range is used to name such frequency deviations
that make the acquisition process possible
(see, e.g. explanations in
\cite[p.40]{Gardner-1966}, \cite[p.61]{Best-2007}).
To define a \emph{pull-in range}
(called also a \emph{capture range} \cite{Talbot-2012-book},
an \emph{acquisition range} \cite[p.253]{Blanchard-1976})
rigorously, consider first an important definition from stability theory.
\begin{definition}
If for a certain $\omega_{\Delta}^{\text{free}}$
any trajectory of system \eqref{final_system} tends to an equilibrium,
then the system with such $\omega_{\Delta}^{\text{free}}$
is called globally asymptotically stable
(see Fig.~\ref{fig-pullin}).
\end{definition}
\begin{figure}[ht]
\centering
\includegraphics[width=0.36\textwidth]{pull-in-phase-portrait.pdf}
\caption{ Phase portrait for $\omega_{\Delta}^{\text{free}}$
from the pull-in range:
any trajectory is attracted to an equilibrium
(equilibria: green--stable and black--unstable circles);
for a sufficiently large initial state of the filter,
cycle slipping is possible (see, e.g. dashed trajectory).}
\label{fig-pullin}
\end{figure}
We now consider a possible rigorous definition.
\begin{definition} \label{def-pull}
A set of all frequency deviations $|\omega_\Delta^{\text{free}}|$
such that the mathematical model of the loop in the signal's phase space
is globally asymptotically stable
is called a pull-in set $\Omega_{\text{pull-in}}$.
\end{definition}
\begin{remark}
In the general case when there is no symmetry with respect to
$\omega_{\Delta}^{\text{free}}$
the set $\omega_{\Delta}^{\text{free}} \in \Omega_{\text{pull-in}}$
has to be considered in Definition~\ref{def-pull}.
\end{remark}
\begin{remark}
The pull-in set is a subset of the hold-in set:
$\Omega_{\text{pull-in}} \subset \Omega_{\text{hold-in}}$,
and need not be an interval.
From an engineering point of view,
it is reasonable to require that
$\omega_\Delta^{\text{free}} = 0$ belongs to the pull-in set
and to define a pull-in range as
the largest interval $[0,\omega_{p})$ from the pull-in set:
\[
[0,\omega_{p}) \subset \Omega_{\text{pull-in}},
\]
where $\omega_{p}$ is called a \emph{pull-in frequency} (see \cite[p.40]{Gardner-1966}).
\end{remark}
\begin{remark}
If all possible states of the filter are bounded:
\[
x \in X_{\text{real}}\ (\text{e.g.\ } X_{\text{real}} =
\{x: c_{\text{min}} <|x| < c_{\text{max}}\}),
\]
by the design of the circuit
(e.g. capacitors have limited maximum and minimum charges,
the VCO frequency is limited etc.),
then in the definition of pull-in set
it is reasonable to require that only
solutions with $x(0) \in X_{\text{real}}$
tend to the stationary set.
Trajectories, with initial data outside of
the domain defined by $x(0) \in X_{\text{real}}$
(here the initial phase error $\theta_\Delta(0)$ can take any value),
need not tend to the stationary set.
\end{remark}
For the model without filter (i.e. $H(s)=const$)
the pull-in set coincides with the hold-in set.
The pull-in set of PLL-based circuits with first-order filters
can be estimated using phase plane analysis methods \cite{Tricomi-1933,AndronovVKh-1937},
but in general its rigorous study is a challenging task
\cite{Viterbi-1966,Stensby-1997,Margaris-2004,KudrewiczW-2007,PinheiroP-2014}.
For the case of the passive lead-lag filter
$H(s) = \frac{1+s \tau_2}{1+s(\tau_1 + \tau_2)}$,
a recent work \cite[p.123]{Margaris-2004} notes that
``{\it the determination of the width of the capture range together
with the interpretation of the capture effect in the second order type-I loops
have always been an attractive theoretical problem.
This problem has not yet been provided with a satisfactory solution}''.
At the same time in \cite{Shakhtarin-1969,Belyustina-1970-eng,LeonovK-2013-IJBC,LeonovK-2014}
it is shown that the basin of attraction of the stationary set may be bounded
(e.g. by a semistable periodic trajectory, which
may appear as the result of collision of unstable and stable periodic solutions),
and corresponding analytical estimations and bifurcation diagram are given.
Note that in this case a numerical simulation may give wrong estimates
and should be used very carefully.
For example, in \cite{BianchiKLYY-2015}
the SIMetrics SPICE model for a two-phase PLL with a lead-lag filter
gives two essentially different results of simulation
with default ``auto'' sampling step (acquires lock) and minimum sampling step set to $1m$
(does not acquire lock --- such behaviour agrees with the theoretical analysis).
The same problems
are also observed in MatLab Simulink \cite{KuznetsovLYY-2014-IFAC,KuznetsovKLNYY-2015-ISCAS,BestKKLYY-2015-ACC},
see, e.g. Fig.~\ref{pll_hidden}.
These examples demonstrate the difficulties of numerical search of so-called \emph{hidden oscillations}
\cite{LeonovK-2013-IJBC,KuznetsovL-2014-IFACWC,LeonovKM-2015-EPJST},
whose basin of attraction does not overlap with the neighborhood of an equilibrium point,
and thus may be difficult to find numerically\footnote{In \cite{LauvdalMF-1997}
the crash of aircraft YF-22 Boeing in April 1992,
caused by the difficulties of rigorous analysis and design of nonlinear control systems with saturation,
is discussed and the conclusion is made
that since stability in simulations does not imply
stability of the physical control system,
stronger theoretical understanding is required
(see, e.g. similar problem with the simulation of PLL in Fig.~\ref{pll_hidden}).
These difficulties in part are related to well-known Aizerman's and Kalman's conjectures on
the global stability of nonlinear control systems,
which are valid from the standpoint of
simplified analysis by the linearization, harmonic balance, and describing function methods
(note that all these methods are also widely used to the analysis of nonlinear oscillators used in VCO \cite{Margaris-2004,Suarez-2009}).
However the counterexamples (multistable high-order nonlinear systems where
the only equilibrium, which is stable, coexists with a hidden periodic oscillation) can be constructed
to these conjectures \cite{LeonovK-2013-IJBC,HeathCS-2015}.}.
In this case the observation of one or another stable solution may depend on the initial data
and integration step.
\begin{figure}[h]
\centering
\includegraphics[width=0.72\linewidth]{simulink_hidden.pdf}
\caption{
Simulation of two-phase PLL described by Fig.~\ref{simulink-model} or model \eqref{final_system} \cite{BianchiKLYY-2015}:
$\tau_1\!=\!0.0448$, $\tau_2\!=\!0.0185$,
$A\!=\!-\frac{1}{\tau_1+\tau_2}$, $b\!=\!1 - \frac{\tau_2}{\tau_1+\tau_2}$,
$c\!=\!\frac{1}{\tau_1+\tau_2}$, $h\!=\!\frac{\tau_2}{\tau_1+\tau_2}$; $\varphi(\theta_\Delta)=\frac{1}{2}\sin(\theta_\Delta)$;
$\omega_1\!=\!10000$, $\omega_2^{\text{free}}\!=\!10000 - 178.9$, $L\!=\!500$.
Filter output $g(t)$ for the initial data $x_0\!=\!0.1318, \theta_{\Delta}(0)\!=\!0$
obtained for default ``auto'' relative tolerance (red) --- acquires lock,
relative tolerance set to ``1e-3''(green) --- does not acquire lock.}
\label{pll_hidden}
\end{figure}
\noindent S.~Goldman, who has worked at Texas Instruments over 20 years, notes that
PLLs are used as pipe cleaners for breaking simulation tools \cite[p.XIII]{Goldman-2007-book}.
While PLL-based circuits are nonlinear control systems and
for their nonlocal analysis it is essential to apply
the classical stability criteria, which are developed in control theory,
however their direct application to analysis of the PLL-based models is often impossible,
because such criteria are usually not adapted for
the cylindrical phase space\footnote{For example,
in the classical Krasovskii--LaSalle principle on global stability
the Lyapunov function has to be radially unbounded
(e.g. $V(x,\theta_\Delta) \to +\infty$ as $||(x,\theta_\Delta)|| \to +\infty$).
While for the application of this principle to the analysis of phase synchronization systems
there are usually used Lyapunov functions periodic in $\theta_\Delta$
(e.g. $V(x,\theta_\Delta)$ in Remark~\ref{remarkPI} is bounded
for any $||(0,\theta_\Delta)|| \to +\infty$),
and the discussion of this gap is often omitted
(see, e.g. patent \cite{Abramovitch-2004} and works \cite{Bakaev-1963,Abramovitch-1990,Abramovitch-2003}).
Rigorous discussion can be found, e.g. in \cite{GeligLY-1978,LeonovK-2014}.
};
in the tutorial {\it Phase Locked Loops: a Control Centric Tutorial}
\cite{Abramovitch-2002},
presented at {\it the American Control Conference 2002}, it was said that
``{\it The general theory of PLLs and ideas on how to make
them even more useful seems
to cross into the controls literature only rarely}''.
At the same time the corresponding modifications of classical stability criteria
for the nonlinear analysis of control systems in cylindrical phase space
were well developed in the second half of the 20th century,
see, e.g. \cite{GeligLY-1978,LeonovRS-1992,LeonovPS-1996,LeonovBSh-1996}.
A comprehensive discussion and the current state of the art can be found in \cite{LeonovK-2014}.
One reason why these works have remained
almost unnoticed by the contemporary engineering community
may be that they were written in
the language of control theory and the theory of dynamical systems,
and, thus, may not be well adapted to the terms and objects used
in the engineering practice of phase-locked loops.
Another possible reason, as noted in \cite[p.1]{Tranter-2010-book},
is that the nonlinear analysis techniques are well beyond the scope
of most undergraduate courses in communication theory and circuits design.
Note that for the application of various stability criteria
it is often necessary to represent system \eqref{final_system} in the Lur'e form:
\begin{equation}\label{Lurie}
\begin{aligned} &
\begin{aligned}
\left(\!\!\!
\begin{array}{c}
\dot{\bar{x}} \\
\dot\theta_\Delta\\
\end{array}
\!\!\!\right)
=\left(\!\!\!
\begin{array}{cc}
A & 0 \\
-Lc^* & 0 \\
\end{array}
\!\!\!\right)
\left(\!\!\!
\begin{array}{c}
\bar{x} \\
\theta_\Delta\\
\end{array}
\!\!\!\right)
+
\left(\!\!\!
\begin{array}{c}
b \\
-Lh \\
\end{array}
\!\!\!\right)
\bar{\varphi}(\theta_\Delta)
\end{aligned}, \\
\end{aligned}
\end{equation}
where
\[
\begin{aligned}
& \bar{x} = x-x_{eq} = x+A^{-1}b\varphi(\theta_{eq}),\
\bar{\varphi}(\theta_\Delta) = \varphi(\theta_\Delta)-\varphi(\theta_{eq}), \\
& \varphi(\theta_{eq}) = \omega_{\Delta}^{\text{free}}L^{-1}(c^*A^{-1}b - h)^{-1}.
\end{aligned}
\]
See also discussion of some nonlinear methods for the analysis of PLL-based models
in recent books \cite{SuarezQ-2003,Margaris-2004,KudrewiczW-2007,Suarez-2009}.
\subsection{Cycle slips and lock-in range}
Let us rigorously define \emph{cycle slipping} in the phase space of system \eqref{final_system}.
\begin{definition}\label{def-cs}
If
\begin{equation}\label{eq-cs}
\begin{aligned}
& \limsup\limits_{t\to+\infty} |\theta_\Delta(0) - \theta_\Delta(t)| > 2\pi,
\end{aligned}
\end{equation}
it is then said that cycle slipping occurs
(see, e.g. dashed trajectory in Fig.~\ref{fig-pullin}).
\end{definition}
Here, sometimes, instead of the limit of the difference,
the maximum of the difference is considered
(see, e.g. \cite[p.131]{Stensby-1997}).
\noindent{\bf Definition \ref{def-cs}'}
{\it
If
\begin{equation}\label{eq-cs-sup}
\begin{aligned}
& \sup\limits_{t>0} |\theta_\Delta(0) - \theta_\Delta(t)| > 2\pi,
\end{aligned}
\end{equation}
it is then said that cycle slipping has occurred.
}
Note that, in general, Definition~\ref{def-cs}' need not mean that finally
(after acquisition) condition \eqref{eq-cs} can not be fulfilled.
Sometimes, the number of cycle slips is of interest.
\begin{definition}
If
\begin{equation}
\begin{aligned}
& 2k\pi < \limsup\limits_{t\to\infty}
|\theta_\Delta(0) - \theta_\Delta(t)| < 2(k+1)\pi,
\end{aligned}
\end{equation}
it is then said that $k$ cycle slips occurred.
\end{definition}
A numerical study of cycle slipping in classical PLL can be found in
\cite{AscheidM-1982}.
Analytical tools for estimating the number of cycle slips
depending on the parameters of the loop
can be found, e.g. in \cite{ErshovaL-1983,LeonovRS-1992,LeonovK-2014}.
The concepts of \emph{lock-in frequency} and \emph{lock-in range}
(called also a \emph{lock range}\cite[p.256]{Yeo-2010-book}, a \emph{seize range} \cite[p.138]{Egan-2007-book}),
were intended to describe the set of frequency deviations for which
the loop can acquire lock within one beat without cycle slipping.
In \cite[p.40]{Gardner-1966} the following definition was introduced:
``{\it{If, for some reason, the frequency difference between input and VCO
is less than the loop bandwidth, the loop will lock up almost instantaneously
without slipping cycles. The maximum frequency difference for which
this fast acquisition is possible is called the lock-in frequency}}''.
However, in general, even for zero frequency deviation ($\omega_\Delta^{\text{free}}=0$)
and a sufficiently large initial state of filter ($x(0)$),
cycle slipping may take place (see, e.g. dashed trajectory in Fig.~\ref{lockin3def},\,left).
Thus, considering of all state variables is of utmost importance
for the cycle slip analysis and, therefore,
the concept \emph{lock-in frequency}
lacks rigor for classical simplified model \eqref{mathmodel-class-simple}
because it does not take into account the initial state of the filter.
The above definition of the lock-in frequency and corresponding definition of the lock-in range
were subsequently in various engineering publications
(see, e.g.
\cite[p.34-35]{Best-1984},\cite[p.161]{Wolaver-1991},\cite[p.612]{HsiehH-1996},\cite[p.532]{Irwin-1997},\cite[p.25]{CraninckxS-1998-book},
\cite[p.49]{KiharaOE-2002},\cite[p.4]{Abramovitch-2002},\cite[p.24]{DeMuerS-2003-book},\cite[p.749]{Dyer-2004-book},\cite[p.56]{Shu-2005},
\cite[p.112]{Goldman-2007-book},\cite[p.61]{Best-2007},\cite[p.138]{Egan-2007-book},\cite[p.576]{Baker-2011},\cite[p.258]{Kroupa-2012}).
\begin{figure*}[ht]
\centering
\includegraphics[width=\textwidth]{lockin_omega_delta_zero.pdf}
\caption{
Phase portraits for the classical PLL with the following parameters:
$H(s)= \frac{1+s\tau_2}{1+s(\tau_1 + \tau_2)}$,
$\tau_1 = 4.48\cdot10^{-2}$,
$\tau_2 = 1.85\cdot10^{-2}$,
$L=250$, $\varphi(\theta_\Delta)= \frac{1}{2}\sin(\theta_\Delta)$, and various frequency deviations.
Black color is for the system with positive $\omega_\Delta^{\text{free}}=|\widetilde{\omega}|$.
Red is for the system with negative $\omega_\Delta^{\text{free}}=-|\widetilde{\omega}|$.
Equilibria (dots), separatrices pass in and out of the saddles,
local lock-in domains are shaded
(upper black horizontal lines is for $\omega_\Delta^{\text{free}}>0$,
lower red vertical lines is for $\omega_\Delta^{\text{free}}<0$).
Left subfig: $\omega_\Delta^{\text{free}} = 0$;
middle subfig: $\omega_\Delta^{\text{free}} = \pm 65$;
right subfig: $\omega_\Delta^{\text{free}} = \pm 68$.
}. \label{lockin3def}
\end{figure*}
The loop model \eqref{final_system}
has a subdomain of the phase space, where trajectories do not slip cycles
(called a lock-in domain), for each value of $\omega_\Delta^{\text{free}}$.
The lock-in domain is the union of local lock-in domains,
each of which corresponds to one of the equilibria and has
its own shape
(see, e.g. shaded domain in Fig.~\ref{lockin3def},\,left defined by corresponding separatrices).
The shape of the lock-in domain significantly varies depending
on $\omega_{\Delta}^{\text{free}}$.
In \cite[p.50]{Viterbi-1966}) a lock-in domain is called a \emph{frequency lock}.
Some writers (e.g. \cite[p.132]{Stensby-1997},\cite[p.355]{Meyer-2004-book})
use the concept \emph{lock-in range} to denote a \emph{lock-in domain}.
In general, taking into account nonuniform behavior of the lock-in domain shape,
Gardner wrote {\it ``There is no natural way to define exactly any unique lock-in frequency''}
\cite[p.70]{Gardner-1979-book}, \cite[p.188]{Gardner-2005-book}.
Below we demonstrate how to overcome these problems and rigorously define
a unique lock-in frequency and range.
We now consider a specific $\omega_{\Delta}^{\text{free}}$
and denote by $D_{\text{lock-in}}(\omega_{\Delta}^{\text{free}})$
the corresponding lock-in domain.
Such a domain exists for any
$|\omega_{\Delta}^{\text{free}}| \in \Omega_{\text{hold-in}}$
because at least the equilibria are contained in this domain.
For a set $\omega_\Delta^{\text{free}} \in \Omega$
we consider the intersection of corresponding lock-in domains
(see, e.g. the intersections of local lock-in domains for various $\omega_{\Delta}^{\text{free}}=\pm|\widetilde{\omega}|$
in Fig.~\ref{lockin3def} --- domains shaded both by red vertical and black horizontal lines):
\[
{\rm D}_{\text{lock-in}}(\Omega) = \bigcap\limits_{\omega_\Delta^{\text{free}} \in \Omega} {\rm D}_{\text{lock-in}}(\omega_{\Delta}^{\text{free}}).
\]
\begin{definition} \label{def-lock}
A lock-in range is the largest interval $[0,\omega_l)$
such that for any $|\omega_\Delta^{\text{free}}| \in [0,\omega_l)$
the mathematical model of the loop in the signal's phase space
is globally asymptotically stable (i.e. $[0,\omega_l) \subset [0,\omega_p)$)
and the following domain
\[
{\rm D}_{\text{lock-in}}\big((-\omega_l,\omega_l)\big) =
\bigcap\limits_{|\omega_\Delta^{\text{free}}| < \omega_l} {\rm D}_{\text{lock-in}}(\omega_{\Delta}^{\text{free}}).
\]
contains all corresponding equilibria:
\[
\big( x_{eq}(\omega_{\Delta}^{\text{free}}),
\theta_{eq}(\omega_{\Delta}^{\text{free}})\big)
\in {\rm D}_{\text{lock-in}}\big((-\omega_l,\omega_l)\big).
\]
\end{definition}
We call such domain ${\rm D}_{\text{lock-in}}={\rm D}_{\text{lock-in}}\big((-\omega_l,\omega_l)\big)$
\emph{a uniform lock-in domain} (uniform with respect to $(-\omega_l,\omega_l)$),
$\omega_{l}$ is called a \emph{lock-in frequency} (see \cite[p.40]{Gardner-1966}).
\smallskip
Various additional requirements may be imposed on the shape of the uniform lock-in domain
${\rm D}_{\text{lock-in}}$, e.g. it has to contain the line defined
by $x \equiv 0$ (see, e.g. \cite[p.258]{Kroupa-2012}) or the band defined by $|x| < c_{\text{max}}$.
If instead of global stability in the definition of the pull-in set
we consider stability in
the domain defined by $X_{\text{real}}$,
then we require that the intersection ${\rm D}_{\text{lock-in}} \bigcap X_{\text{real}}$
contains all corresponding equilibria.
\begin{remark}
In the general case when there is no symmetry with respect to
$\omega_{\Delta}^{\text{free}}$
we have to consider a unsymmetrical interval containing zero
in Definition~\ref{def-lock}.
\end{remark}
\noindent Similarly, we can define an extension of the lock-in range: $\Omega_{\text{lock-in}} \supset [0, \omega_l)$,
called a lock-in set (however, in general, such an extension may be not unique).
In other words, the definition implies that
\emph{if the loop is in a locked state,
then after an abrupt change of $\omega_{\Delta}^{\text{free}}$
within a lock-in range $[0, \omega_l)$,
the corresponding acquisition process in the loop leads,
if it is not interrupted, to a new locked state
without cycle slipping}.
Finally, our definitions give
\(
\Omega_{\text{lock-in}} \subset \Omega_{\text{pull-in}}
\subset \Omega_{\text{hold-in}},
\)
\[
[0,\omega_l) \subset [0,\omega_p)
\subset [0,\omega_h)
\]
which is in agreement with the classical consideration (see, e.g. \cite[p.34]{Best-1984},\cite[p.612]{HsiehH-1996},\cite[p.61]{Best-2007},\cite[p.138]{Egan-2007-book},\cite[p.258]{Kroupa-2012}).
\subsection{Approximations of the lock-in range of the classical PLL}
For the case of the classical odd PD characteristic (see Fig.~\ref{lockin3def}),
taking into account that equilibria are proportional
to the frequency deviation (see \eqref{zeros1})
and using the symmetry
$\big(x_{eq}(\omega_{l}), \theta_{eq}(\omega_{l})\big) =
- \big(x_{eq}(-\omega_{l}), \theta_{eq}(-\omega_{l})\big)$,
we can effectively determine $\omega_{l}$.
For that, we have to increase the frequency deviation
$|\omega_{\Delta}^{\text{free}}|$ step by step
and at each step, after the loop achieves a locked state,
to change $\omega_{\Delta}^{\text{free}}=\widetilde{\omega}$ abruptly to $\omega_{\Delta}^{\text{free}}=-\widetilde{\omega}$
and to check if the loop can achieve a new locked state without cycle
slipping. If so, then the considered value $|\omega_{\Delta}^{\text{free}}|$
belongs to $\Omega_{\text{lock-in}}$.
If $\omega_{\Delta}^{\text{free}}\!=\!0$ belongs to $\Omega_{\text{pull-in}}$,
then it is clear that $0$ belongs to
$\Omega_{\text{lock-in}}$ (see Fig.~\ref{lockin3def},\,left).
The limit value $\omega_{l}$ is defined by the case in Fig.~\ref{lockin3def},\,middle.
At the next step
when a value $|\omega_{\Delta}^{\text{free}}|=|\widetilde{\omega}| > \omega_{l}$ is considered,
for $\omega_{\Delta}^{\text{free}}=-|\widetilde{\omega}|$
the \newline
trajectory from the initial point, corresponding to a stable equilibrium
for $\omega_{\Delta}^{\text{free}}\!=\!|\widetilde{\omega}|$
(see Fig.~\ref{lockin3def},\,right: red trajectory outgoing from a black dot),
is attracted to an equilibrium
only after cycle slipping.
In other words \cite{KuznetsovLYY-2015-IFAC-Ranges}, for this case:
{\it
The lock-in range is a subset of the pull-in range
such that for each corresponding frequency deviation
the lock-in domain
(i.e. a domain of the loop states, where fast acquisition without cycle slipping is possible)
contains both symmetric locked states
(i.e. locked states for
\begin{figure}[ht]
\centering
\includegraphics[width=0.45\textwidth]{horizontal_lines_lockind.pdf}
\caption{
Phase portrait. Separatrices, equilibria
and corresponding local lock-in domains (shaded):
upper black is for $\omega_\Delta^{\text{free}} = 61.5$,
lower red is for $\omega_\Delta^{\text{free}} = -61,5$.
The uniform lock-in domain is approximated by the band between
two blue horizontal lines: $|x|\leq 0.0110$.
}. \label{lockindefband}
\end{figure}
the positive and negative value of the difference between the reference frequency and the VCO free-running frequency).
}
In Fig.~\ref{lockin3def},\,middle the set ${\rm D}_{\text{lock-in}}$:
contains all equilibria $x_{eq}(\omega_{\Delta}^{\text{free}})$ for
$0 \leq |\omega_{\Delta}^{\text{free}}| < \omega_{l}$.
However for some non-equilibrium initial states
from the band defined by $\{x: |x|< |x_{eq}(\omega_{l})|\}$
(phase error $\theta_{\Delta}$ takes all possible values),
cycle slipping can take place.
For example, see the points to the left and to the right of the black equilibrium states
(i.e. for $\omega_{\Delta}^{\text{free}}=|\omega_{l}|>0$),
lying above the red separatrix (i.e. for $\omega_{\Delta}^{\text{free}}=-|\omega_{l}|<0$),
correspond to the red trajectories
(i.e. for $\omega_{\Delta}^{\text{free}}=-|\omega_{l}|<0$),
which are attracted to an equilibrium only after cycle slipping.
To approximate the ${\rm D}_{\text{lock-in}}$ by a band,
$\omega_{l}$ can be slightly decreased to cut the above points.
In Fig.~\ref{lockindefband}
the band defined by $X_{\text{lock-in}}=\{x: |x|< |x_{eq}(\widetilde\omega_{l})|,\ \widetilde\omega_{l}< \omega_{l} \}$
is contained in ${\rm D}_{\text{lock-in}}$
and for any initial state from the band
the corresponding acquisition process in the loop leads, if it is not interrupted,
to lock up without cycle slipping.
Such a construction is more laborious and
requires rigorous analysis of the phase space or exhaustive simulation.
\begin{remark}
If we define (see, e.g. \cite[p.92]{PurkayasthaS-2015})
cycle slipping by the interval of maximum length $2\pi$
instead of $4\pi$ in Definition~\ref{def-cs}:
i.e. $\limsup_{t\to\infty} |\theta_\Delta(0) - \theta_\Delta(t)| > \pi$,
then for any $|\omega_\Delta^{\text{free}}| > 0$
a distance between neighboring unstable and stable equilibria
and a phase deviation of the corresponding unstable saddle separatrix
may exceed $\pi$ (see, e.g. Fig.~\ref{lockindefband}).
Thus, the lock-in range may contain only $|\omega_\Delta^{\text{free}}| = 0$.
\end{remark}
\smallskip
\begin{remark}\label{remarkPI}
If the filter -- perfect integrator can be implemented in considered architecture,
the loop can be designed with the first order PI filter having
the transfer function $H(s) = \frac{1+s\tau_2}{s\tau_1}$.
Equations of the loop in this case become
\begin{figure}[ht]
\centering
\includegraphics[width=0.45\textwidth]{phase_portrait_pi.pdf}
\caption{
Phase portraits for the classical PLL with the following parameters:
$H(s)=\frac{1+0.0225s}{0.0633s}$, $L=250$, and $\omega_\Delta^{\text{free}} = \pm 47$.
Separatrices, equilibria
and corresponding local lock-in domains (shaded):
upper black is for $\omega_\Delta^{\text{free}} = 47$,
lower red is for $\omega_\Delta^{\text{free}} = -47$.
The uniform lock-in domain is approximated by the band between
the two blue horizontal lines: $|x|\leq 0.0119$.
} \label{lockindefband_pi}
\end{figure}
\begin{equation}\label{pi-system}
\dot{x} = \frac{1}{\tau_1} \varphi(\theta_\Delta), \ \
\dot\theta_\Delta = \omega_\Delta^{\text{free}}
- Lx - L\frac{\tau_2}{\tau_1}\varphi(\theta_\Delta),
\end{equation}
or equivalently
\begin{equation}
\label{pi_equation}
\ddot\theta_\Delta = -L\frac{1}{\tau_1} \varphi(\theta_\Delta)
-L\frac{\tau_2}{\tau_1}\varphi'(\theta_\Delta)\dot\theta_\Delta.
\end{equation}
Here the equilibria are defined from the equations
\[
\varphi(\theta_{eq}) = 0, \
\ x_{eq} = {\omega_{\Delta}^{\text{free}} L^{-1}}.
\]
Because model \eqref{pi_equation} does not depend explicitly on $\omega_{\Delta}^{\text{free}}$,
the hold-in and pull-in ranges are either infinite or empty.
Note, that the parameter $\omega_{\Delta}^{\text{free}}$ shifts the phase plane vertically
(in the variable $x$) without distorting trajectories,
which simplifies the analysis of the uniform lock-in domain and range (see Fig.~\ref{lockindefband_pi}).
If the transfer function $H(s)$ of a high order filter
has the term $s^r$ with $r\in\mathbb{N}$ in the denominator,
then instead of equilibria we have a stationary linear manifold:
$
\varphi(\theta_{eq}) = 0, \ c_1 x_{eq}^1 +\ldots+ c_r x_{eq}^r
= \frac{-\omega_\Delta^{\text{free}}}{L}.
$
For the classical PLL with the filter's transfer function
$H(s) = \frac{\beta+\alpha s}{s}$
it can be analytically proved that the pull-in range is theoretically infinite.
Some needed explanations are given by Viterbi \cite{Viterbi-1966}
using phase plane analysis.
But, even in such a simple case, rigorous phase plane analysis
is a complex task
(e.g. \cite{AlexandrovKLNS-2015-IFAC},
the proof of the nonexistence of heteroclinic and first-order cycles is omitted in \cite{Viterbi-1966}).
The rigorous analytical proof can be effectively achieved
by considering a special Lyapunov function \cite{Bakaev-1963,LeonovK-2014,AlexandrovKLNS-2015-IFAC}: $V(x,\theta_\Delta)=\frac{1}{2}\big(x-\frac{\omega_\Delta^{\text{free}}}{L}\big)^2+\frac{2\beta}{L}\sin^2{\frac{\theta_\Delta}{2}} \geq 0$
and $\dot V(x,\theta_\Delta) = - h\beta\sin^2{\theta_\Delta} \leq 0$.
Here it is important that for any $\omega_\Delta^{\text{free}}$
the set $\dot V(x,\theta_\Delta)\equiv0$ does not contain the whole trajectories of system \eqref{pi-system} except for equilibria.
\end{remark}
\subsection{Initial and free-running frequencies of VCO}
Note that in the above Definitions~\ref{def-hold}, \ref{def-pull},
and \ref{def-lock} the hold-in, pull-in, and lock-in sets
are defined by the frequency deviation,
i.e. by the absolute value of the difference between
VCO free-running frequency (in the open loop)
and the input signal's frequency:
$|\omega_{\Delta}^{\text{free}}| = |\omega_1-\omega_2^{\text{free}}|$.
The VCO free-running frequency $\omega_2^{\text{free}}$
is different from the VCO initial frequency $\omega_2(0)$:
\( \omega_2(0) = \omega_2^{\text{free}} + g(0), \)
where $g(0) = c^{*}x(0)+h\varphi(\theta_{\Delta}(0))$
is the initial control signal,
depending on the initial states of the filter $x(0)$
and the initial phase difference $\theta_{\Delta}(0)$.
It is interesting that for simplified model \eqref{mathmodel-class-simple}
with $h=0$ (see eq.~2.20 in the classic reference \cite{Viterbi-1966})
the absolute value of the initial difference between frequencies
$|\dot \theta_{\Delta}(0)| = |\omega_{\Delta}(0)|=|\omega_1-\omega_2(0)|$ is equal
to the frequency deviation $|\omega_{\Delta}^{\text{free}}| = |\omega_1-\omega_2^{\text{free}}|$.
Following such simplified consideration
in engineering literature the concept of an ``{\it initial frequency difference}''
can be found to be in use instead of the concept of a ``{\it frequency deviation}'':
see, e.g. \cite[p.44]{Gardner-1966}
``{\it If the initial frequency difference (between VCO and input)
is within the pull-in range, the VCO frequency will slowly change
in a direction to reduce the difference}'',
\cite[p.1792]{Chen-2002-book} ``{\it The maximum frequency difference
between the input and the output that the PLL can lock
within one single beat note is called the lock-in range of the PLL}'',
\cite[p.49]{KiharaOE-2002} ``{\it Whether the PLL can get synchronized
at all or not depends on the initial frequency difference between
the input signal and the output of the controlled oscillator.}''
In general, the change of $\omega_2^{\text{free}}$ to $\omega_2(0)$
may lead to wrong results in the above definitions of ranges
because in the case of $x(0) \neq 0$, $h \neq 0$
or non-odd function $\varphi(\theta_{\Delta})$
for the same values of $\omega_2(0)$
the loop can achieve synchronization or not
depending on the filter's initial state $x(0)$,
the initial phase difference $\theta_{\Delta}(0)$, and $\omega_2^{\text{free}}$.
See the corresponding example.
\begin{example}\label{change}
Consider the behavior of model \eqref{final_system}
for the sinusoidal signals
(i.e. $\varphi(\theta_{\Delta}) = {1 \over 2}\sin(2\theta_{\Delta})$)
and the fixed parameters:
$\omega_\Delta = 100, H(s) = \frac{(1+s \tau_2)}{1+s(\tau_1 + \tau_2)},
\tau_1 = 0.0448, \tau_2 = 0.0185,
L = 250$.
In Fig.~\ref{small omega vatiation sim3}
the phase portrait of system \eqref{final_system} is shown.
The blue dash line consists of points for which the initial frequency difference
is zero: $\omega_\Delta(0)=\dot\theta_\Delta(0)=0$.
Despite the fact that the initial frequency differences of all trajectories outgoing
from the blue line are the same (equal to $0$),
the green trajectory tends to a locked state
while the magenta trajectory can not achieve this.
\begin{figure}[h]
\centering
\includegraphics[width=0.45\textwidth]{intial_freq_example.pdf}
\caption{
Phase portrait for $\omega_{\Delta}^{\text{free}} = 100$.
Blue dash curve corresponds to the set defined by $\dot\theta_\Delta(0)=0$.
Initial points of the green (upper) and magenta (lower) trajectories correspond
to the same initial frequency difference $\omega_\Delta(0)=0$.}
\label{small omega vatiation sim3}
\end{figure}
\end{example}
\section{\uppercase{Conclusions}}
This survey discussed a disorder and inconsistency in the definitions of ranges
currently used.
An attempt is made to discuss and fill some of the gaps identified
between mathematical control theory, the theory of dynamical systems
and the engineering practice of phase-locked loops.
Rigorous mathematical definitions for
the hold-in, pull-in, and lock-in ranges are suggested.
The problem of unique lock-in frequency definition,
posed by Gardner \cite{Gardner-1979-book},
is solved and an effective way to determine
the unique lock-in frequency is suggested.
\section*{\uppercase{Acknowledgements}}
This work was supported by the Russian Scientific Foundation (project 14-21-00041)
and Saint-Petersburg State University.
The authors would like to thank Roland~E.~Best,
the founder of the Best Engineering Company, Oberwil, Switzerland
and the author of the bestseller on PLL-based circuits \cite{Best-2007}
for valuable discussion.
\newcommand{\noopsort}[1]{} \newcommand{\printfirst}[2]{#1}
\newcommand{\singleletter}[1]{#1} \newcommand{\switchargs}[2]{#2#1}
| {
"timestamp": "2015-09-08T02:09:39",
"yymm": "1505",
"arxiv_id": "1505.04262",
"language": "en",
"url": "https://arxiv.org/abs/1505.04262",
"abstract": "The terms hold-in, pull-in (capture), and lock-in ranges are widely used by engineers for the concepts of frequency deviation ranges within which PLL-based circuits can achieve lock under various additional conditions. Usually only non-strict definitions are given for these concepts in engineering literature. After many years of their usage, F.~Gardner in the 2nd edition of his well-known work, Phaselock Techniques, wrote \"There is no natural way to define exactly any unique lock-in frequency\" and \"despite its vague reality, lock-in range is a useful concept.\" Recently these observations have led to the following advice given in a handbook on synchronization and communications \"We recommend that you check these definitions carefully before using them.\" In this survey it is shown that, from a mathematical point of view, in some cases the hold-in and pull-in \"ranges\" may not be the intervals of values but a union of intervals and thus their widely used definitions require clarification. Rigorous mathematical definitions for the hold-in, pull-in, and lock-in ranges are given. An effective solution for the problem on the unique definition of the lock-in frequency, posed by Gardner, is suggested.",
"subjects": "Dynamical Systems (math.DS)",
"title": "Hold-in, pull-in, and lock-in ranges of PLL circuits: rigorous mathematical definitions and limitations of classical theory",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9770226320971079,
"lm_q2_score": 0.724870282120402,
"lm_q1q2_score": 0.7082146709662485
} |
https://arxiv.org/abs/2210.11436 | Revisiting Le Cam's Equation: Exact Minimax Rates over Convex Density Classes | We study the classical problem of deriving minimax rates for density estimation over convex density classes. Building on the pioneering work of Le Cam (1973), Birge (1983, 1986), Wong and Shen (1995), Yang and Barron (1999), we determine the exact (up to constants) minimax rate over any convex density class. This work thus extends these known results by demonstrating that the local metric entropy of the density class always captures the minimax optimal rates under such settings. Our bounds provide a unifying perspective across both parametric and nonparametric convex density classes, under weaker assumptions on the richness of the density class than previously considered. Our proposed `multistage sieve' MLE applies to any such convex density class. We further demonstrate that this estimator is also adaptive to the true underlying density of interest. We apply our risk bounds to rederive known minimax rates including bounded total variation, and Holder density classes. We further illustrate the utility of the result by deriving upper bounds for less studied classes, e.g., convex mixture of densities. |
\section{Introduction}
It is well known that (global) metric entropy often times determines the minimax
rates for density estimation. Specifically, the following equation sometimes
informally referred to as the `Le Cam equation' is used to heuristically
determine the minimax rate of convergence
\begin{align*}
\log{\mgloc{\cF}{\varepsilon}} \asymp n \varepsilon^2,
\end{align*}
where $n$ is the sample size, $\log{\mgloc{\cF}{\varepsilon}}$ is the
\emph{global} metric entropy of the density set $\cF$ at a Hellinger distance
$\varepsilon$ (see Definition \ref{nrmk:cF-compact-packing-numbers}), and
$\varepsilon^2$ determines the order of the minimax rate. In this paper we
complement these known results, by establishing that \emph{local} metric entropy
\emph{always} determines the minimax rate for convex density classes, where the
densities are assumed to be (uniformly) bounded from above and below.
In detail, under the setting of density estimation just described, we suggest a
small revision to the Le Cam equation: namely, change the global entropy to
local entropy, and the Hellinger metric to the $L_2$-metric. Furthermore, the
same result holds when the convex density class contains densities only
(uniformly) bounded from above, and a single density which is bounded from
below. Unlike previous known results, our result unites minimax density
estimation under both parametric and nonparametric convex density classes. A
further contribution is that our proposed `multistage sieve' maximum likelihood
estimator (MLE) achieves these bounds regardless of the density class (as long
as it is convex).
We will now formally describe the setting we consider. To that end, we first
define a general class of bounded densities, \ie, $\cF_{B}^{[\alpha, \beta]}$. Later, we
will assume that the true density of interest belongs to a known convex subset
of this general ambient density class.
\begin{restatable}[Ambient density class
$\cF_{B}^{[\alpha, \beta]}$]{ndefn}{densityclassFalphabeta}\label{ndefn:density-class-F-alpha-beta}
Given constants $0 < \alpha < \beta < \infty$, for some fixed dimension
$p \in \NN$, and a common known (Borel measurable) compact support set
$B \subseteq \RR^{p}$ (with positive measure), we then define the class
of density functions, $\cF_{B}^{[\alpha, \beta]}$, as follows:
\begin{equation}\label{neqn:density-class-F-alpha-beta}
\cF_{B}^{[\alpha, \beta]}
\defined
\thesetb{f \colon B \to [\alpha, \beta]}{\int_{B}{f} \dlett{\mu} = 1, \textnormal{$f$ measurable}},
\end{equation}
where $\mu$ is the dominating finite measure on $B$. We always take $\mu$ to
be a (normalized) probability measure on $B$.
\end{restatable}
Furthermore, we can endow $\cF_{B}^{[\alpha, \beta]}$ with the $L_{2}$-metric. That is, for
any two densities $f, g \in \cF_{B}^{[\alpha, \beta]}$, we denote the $L_{2}$-metric between
them to be
\begin{equation}\label{neqn:L2-metric-density-defn}
\normb{f - g}_{2}
\defined
\parens{\int_{B} (f - g)^{2} \dlett{\mu}}^{\frac{1}{2}}.
\end{equation}
\begin{restatable}{nrmk}{densityclasscfalphabeta}\label{nrmk:density-class-F-alpha-beta}
Qualitatively, we have that $\cF_{B}^{[\alpha, \beta]}$ is the class of all densities
that are uniformly $\alpha$-lower bounded and $\beta$-upper bounded, on a
common compact support $B \subseteq \RR^{p}$. Furthermore,
\Cref{ndefn:density-class-F-alpha-beta} implies that $\cF_{B}^{[\alpha, \beta]}$ forms a
convex set, and that the metric space $(\cF_{B}^{[\alpha, \beta]}, \normb{\cdot}_{2})$
is complete, bounded, but may not be totally bounded\footnote{These
fundamental (and additional) analytic properties of $\cF_{B}^{[\alpha, \beta]}$ are
formally justified in \Cref{app:prop-of-class-F-alpha-beta}.}.
\end{restatable}
In this paper we will focus on the scenario where it is known that the true
density $f \in \cF \subset \cF_{B}^{[\alpha, \beta]}$, where $\cF$ is a known convex set.
The set $\cF$ represents our knowledge on the true density, before observing any
data. With these mathematical preliminaries, we formalize our core density
estimation problem of interest as follows.
\begin{tcolorbox}[breakable]
\begin{itquote}
\textbf{Core problem:}
Suppose that we observe $n$ observations $X \defined (X_1,
\ldots, X_n)^{\top} \distiid f$, for some (fixed but unknown) $f \in
\cF$. Here $\cF \subset \cF_{B}^{[\alpha, \beta]}$ is a convex set, which is
known to the observer. Can we propose a universal estimator for $f$,
and derive the exact (up to constants) squared $L_{2}$-minimax rate
of estimation, in expectation?
\end{itquote}
\end{tcolorbox}
For convenience, we can illustrate the generating process for a univariate
example of our density estimation problem of interest in
\Cref{fig:generating-process}. It will serve as a useful conceptual guide to
later help visualize our proposed estimator over such general convex class of
densities $\cF$.
\iftoggle{showfigs}{%
\begin{figure}[!ht]
\centering
\includegraphics[scale=0.75]{figures/generating-process-02.pdf}
\caption{Illustrative example of the generating process for a univariate density $f \in \cF \subset \cF_{B}^{[\alpha, \beta]}$.}
\label{fig:generating-process}
\end{figure}
}{%
}
Now, without further ado, we will informally state our main result as a direct
affirmative answer to our core question of interest. Namely, there does exist a
likelihood-based estimator (one can think of it as a multistage sieve MLE), \ie,
$\nu^{*}(X)$, which achieves the following rate of estimation error
\begin{equation}\label{neqn:intro-main-result-bound}
\sup_{f \in \cF} \EE \normb{\nu^{*}(X) - f}_{2}^{2} \lesssim \varepsilon^{*2} \wedge \operatorname{diam}_{2}(\cF)^2.
\end{equation}
Here $\varepsilon^* \defined \sup \thesetc{\varepsilon}{n \varepsilon^{2} \leq
\log{\mlocc{\cF}{\varepsilon}{c}}}$, with
$\log{\mlocc{\cF}{\varepsilon}{c}}$ being the $L_{2}$-\emph{local}
metric entropy of $\cF$ (see Definition \ref{ndefn:local-entropy}). The
quantity $\operatorname{diam}_{2}(\cF)$, refers to the $L_{2}$-diameter of $\cF$,
which is finite by the boundedness of $\cF_{B}^{[\alpha, \beta]}$ in our setting. In
addition, the rate above is minimax optimal, as there is a matching (up
to constants) lower bound.
\begin{restatable}{nrmk}{relaxingdensityclassFalphabeta}\label{nrmk:relaxing-density-class-F-alpha-beta}
We will later see that we can largely relax the $\alpha$-lower boundedness
condition on $\cF$. That is, the results we are about to derive can be
readily generalized to convex subsets $\cF \subset \cF_{B}^{[0, \beta]}$. This is so
as long as the class $\cF$ contains a \emph{single} density which is bounded
away from $0$.
\end{restatable}
Next, we turn our attention to reviewing some relevant literature.
\subsection{Relevant Literature}\label{sec:relevant-literature}
\vspace{4pt}
\nit \underline{\textbf{Classical work}}
\vspace{4pt}
As noted, density estimation is a classical statistical estimation problem with
a rich history. Lively accounts of the key references, particularly for
nonparametric density estimation as relevant to our setting, are already covered
in \citet[Section~1]{yang1999information} and
\citet[Section~6.1]{bilodeau21density}. We similarly begin with a brief
panoramic overview of these references in regard to minimax risk bounds for
density estimation, before comparing and contrasting the results from the most
relevant references to our work.
In terms of minimax lower bounds on density estimation,
\citet{boyd1978lowerbddsnonparamest} prove a fundamental $n^{-1}$ rate in the
mean integrated $p^{\textnormal{th}}$ power error (with $p \geq 1$), for
\emph{any} arbitrary density estimator. Such generalized lower bounds on density
estimation were also further studied in \citet{devroye1983arbslowratesconv}. In
the case of density estimation over classes with more assumed structure (\eg,
smoothness, or regularity assumptions) minimax lower bounds have been developed
based on hypothesis testing approaches coupled with information-theoretic
techniques. We now provide brief highlights of such key works in this direction.
In \citet{bretagnolle1979estrisqueminimax}, the authors derive sharp lower
bounds for Sobolev smooth densities in $\RR^{d}$ ($d \in \NN$), with risk
measured with respect to a power of the $L_{q}$-metric ($q \geq 1$). In
\citet{birge1986estimatinghellingerfacts}, sharp risk bounds for more general
classes of such smooth families were provided using metric entropy based
methods, with an emphasis on the Hellinger loss. The work of
\citet{efroimovich1982estsquareintdensityrv} provided precise (asymptotic)
analysis for an ellipsoidal class of densities in the $L_{2}$-metric. Across a
wide-ranging series of related and collaborative efforts
\citet{hasminski1978lowerboundunifmetric, ibragimov1977estimationparwhitenoise,
ibragimov1978capacitycommsmoothsig,ibragimov1980estdistdensity} used Fano's
inequality type arguments to establish lower bounds over a variety of density
estimation settings. These range from deriving lower bounds on nonparametric
density estimation in the uniform metric, to minimax risk bounds for the
Gaussian white noise model, for example. The authors also develop metric entropy
based techniques in \cite{hasminski1990denskolmogorovappr} to derive minimax
lower bounds for a wide variety of density classes defined on $\RR^{d}$ ($d \in
\NN$), in $L_{q}$-loss ($q \geq 1$). Numerous applications of optimal lower
bounds using both Assouad's and Fano's lemma arguments for densities on a
compact support, are demonstrated in
\citep[Section~29.3]{yu1997assouadfanolecamminimaxlb}. Later
\citet{yang1999information} demonstrated that \emph{global} metric entropy
bounds capture minimax risk for sufficiently rich density classes over a common
compact support. Classical reference texts on minimax lower bound techniques
with an emphasis on nonparametric density estimation include
\citet{devroye1987coursedensityest,devroye1985nonparamdensityestl1view,
lecam1986asympmethodsdectheory}. More modern such references include
\citet{tsybakov2009introduction} and \citet[Chapter~15]{wainwright2019high}. The
latter in particular, also incorporates metric entropy based lower bound
techniques.
In addition, there is a large body of work in deriving upper bounds for specific
density estimators using metric entropy methods. This includes
\citet{yatracos1985ratesconvmindistent, barron1991mincompdensityest}, who employ
the minimum distance principle to derive density estimators and their metric
entropy-based upper bounds in the Hellinger and $L_{1}$-metric, respectively. In
a similar spirit to \citet{birge1983approximation,
birge1986estimatinghellingerfacts}, \citet{vandegeer1993hellingerconsnpmle} is
also concerned with density estimation using Hellinger loss. However, its focus
is to use techniques from empirical process theory in order to specifically
establish the Hellinger consistency of the nonparametric MLE, over convex
density classes. Upper bounds for density estimation based on the `sieve' MLE
technique is studied in \citet{wong1995probability}. Recall that a `sieve'
estimator effectively estimates the parameter of interest via an optimization
procedure (\eg, maximum likelihood) over a constrained subset of the parameter
space \citep[Chapter~8]{grenander1981abstractinference}. In
\citet{birge1993ratesconvmincontrast} the authors study `minimum contrast
estimators' (MCEs), which include the MLE, least squares estimators (LSEs) \etc,
and apply them to density estimation. This is further developed in
\cite{birge1994mincontrastsieve} where they analyze convergence of MCEs using
sieve-based approaches.
\vspace{4pt}
\nit \underline{\textbf{Comparison to our work}}
\vspace{4pt}
By stating our main result early in the introduction, we now turn to contrasting
it with the most relevant results in the literature. These include both the
aforementioned classical references, and more recent work on convex density
estimation, which have most directly inspired our efforts in this work.
First we would like to comment on the closely related landmark papers
\citep{lecam1973convergence,
birge1983approximation,birge1986estimatinghellingerfacts}. These works consider
very abstract settings and show upper bounds based on Hellinger ball testing.
Although widely believed that they do, whether these results lead to bounds that
are minimax optimal is unclear. Moreover, their estimator is quite involved and
non-constructive. In contrast, in this paper we offer a simple to state,
\emph{constructive} multistage sieve MLE type of estimator, which is provably
minimax optimal over any convex density class $\cF$. A crucial difference is
that we metrize the space $\cF$ with the $L_{2}$-metric as we mentioned above.
Even though in our instance the two distances are equivalent, in contrast to the
Hellinger distance, the $\varepsilon$-\emph{local} metric entropy of the convex
density class in the $L_{2}$-metric can be shown to be monotonic in
$\varepsilon$. This key observation enables us to match the upper and lower
bounds exactly.
Next, we will compare our work with the celebrated paper of
\cite{yang1999information}, who inspect a very similar problem.
\cite{yang1999information} demonstrate a lower and upper bound which need not
match in general but do match under certain sufficient conditions. Notably their
bounds involve only quantities depending on the global entropies of the set
$\cF$ (which is also assumed to be convex for some results of
\cite{yang1999information}). This is convenient as often times global metric
entropy is easier to work with compared to local metric entropy, however under
\cite{yang1999information}'s sufficient condition it can be seen that the two
notions are equivalent. Hence, our work can be thought of as removing the
sufficient condition requirement from \cite{yang1999information} and also
unifying parametric and nonparametric density estimation problems (over convex
classes) for which one typically needs to use different tools to obtain the
accurate rates. Finally we would like to mention \cite{wong1995probability}. In
that paper the authors propose a sieve MLE estimator and demonstrate that it is
\emph{nearly} minimax optimal under certain conditions. Our estimator is not the
same as the one considered by \cite{wong1995probability}, and we can provably
match the minimax rate over whatever be the convex set $\cF$. A notable
difference is that \cite{wong1995probability} work with the Hellinger metric and
KL divergence, which although equivalent to $L_{2}$-metric in our problem, are
actually less practical in terms of matching the bounds exactly as we explained
above. We will now turn our attention to reviewing some further relevant
literature.
\vspace{4pt}
\nit \underline{\textbf{Recent work}}
\vspace{4pt}
Our estimator and proof techniques thereof, are inspired by the recent work of
\cite{neykov2022minimax} on the Gaussian sequence model. We would like to stress
on the fact that the sequence model is a very distinct problem from density
estimation. In particular, our underlying metric space of interest is $(\cF,
\normb{\cdot}_{2})$, as compared to\footnote{Note that $\normb{\cdot}_{2}$ here
is the Euclidean metric on $\RR^{n}$.} $(\RR^{n}, \normb{\cdot}_{2})$ for the
sequence model. Both of these metric spaces differ \emph{vastly} from each other
in their underlying geometric structure. Furthermore, unlike our setting, the
sequence model contains additional Gaussian information on the underlying
generating process, which can be directly exploited for estimation purposes. As
such, given that \cite{neykov2022minimax} provides a guiding template for our
analysis, some resulting structural similarities to their work are to be
expected. However, all corresponding proofs, and estimators thereof, have to be
non-trivially adapted to our nonparametric density estimation setting. A notable
example of such required modifications, is that our estimator presented in this
paper does not use proximity in Euclidean norm, but is a likelihood-based
estimator.
We additionally note that density estimation in both abstract and more concrete
settings, continues to be an active area of research. It is not feasible to
detail such a large and growing body of references. However, we provide a
selective overview of some interesting recent directions in density estimation,
to simply indicate the diversity of the research efforts thereof. For example,
\citet{cleanthous2020kernwaveletdensityest, baldi2009adaptdensityestspherical}
study convergence properties of density estimators using wavelet-based methods.
The papers
\citet{goldenshluger2014onadaptivedensityestrd,efromovich2008adaptiveestoracle,
rigollet2006adaptiveestblockstein,rigollet2007linearconvaggdensityest,
samarov2007aggdensityest,birge2014modselectdensityest} study adaptive minimax
density estimation on $\RR^{d}$ ($d \geq 1$) under $L_{p}$-loss ($p \geq 1$).
Here, `adaptive' refers to the fact that the density class is defined by an
unknown tuning hyperparameter, which must be explicitly accounted for during the
estimation process. Recently \citet{wang2022minimaxdensityoptimaltransport} used
techniques from optimal transport to study the convergence properties of various
nonparametric density estimators. Interestingly, \citet{bilodeau21density}
applied empirical (metric) entropy methods to establish minimax optimal rates in
the adjacent setting of \emph{conditional} density estimation. Although these
works do not directly study our core problem of interest, we note that they
represent new and important perspectives on classical minimax density
estimation, and related problems.
\subsection{Notation}\label{sec:notation}
We outline some commonly used notation here. We use $a \vee b$ and $a \wedge b$
for the $\max$ and $\min$ of two numbers $\theset{a, b}$, respectively.
Throughout the paper $\normb{\cdot}_{2}$ denotes the $L_{2}$-metric in $\cF$.
Constants may change values from line to line. For an integer $m\in \NN$, we use
the shorthand $[m] \defined \{1, \ldots, m\}$. We use $B_2(\theta,r)$ to denote
a closed $L_{2}$-ball centered at the point $\theta$ with radius $r$. We use
$\lesssim$ and $\gtrsim$ to mean $\leq$ and $\geq$ up to absolute (positive)
constant factors, and for two sequences $a_n$ and $b_n$ we write $a_n \asymp
b_n$ if both $a_n \lesssim b_n$ and $a_n \gtrsim b_n$ hold. Throughout the paper
we use $\log$ to denote the natural logarithm, or we specify the base explicitly
otherwise. Our use of $\theset{\alpha, \beta}$ is \emph{only} used to refer to
the constants in \Cref{ndefn:density-class-F-alpha-beta}, of $\cF_{B}^{[\alpha, \beta]}$
(and thus $\cF$). We will introduce additional section-specific notation as
needed.
\subsection{Organization}\label{sec:organization-paper} The rest of this paper
is organized as follows. In \Cref{sec:minimax-upper-and-lower-bounds} we prove
risk bounds for our underlying setting. We first establish the key toplogical
equivalence between the $L_{2}$-metric and the Kullback-Leibler divergence in
$\cF_{B}^{[\alpha, \beta]}$. We then proceed to derive minimax lower bounds for our setting
in \Cref{sec:lower-bound}, introducing additional relevant mathematical
background as needed, \eg, local metric entropy. In \Cref{sec:upper-bound} we
define our likelihood-based estimator, and provide intuition behind its
construction. We then derive its (matching) minimax risk upper bound. In
\Cref{sec:examples}, we apply our results to specific examples of commonly used
convex density classes. We then conclude in \Cref{sec:discussion} by summarizing
our results, and discuss some future research directions.
\section{Minimax Lower and Upper
Bounds}\label{sec:minimax-upper-and-lower-bounds}
Before establishing our main results, we establish a key technical lemma which
drives much of the geometric arguments in our analysis to follow. Note that for
any two densities $f, g \in \cF_{B}^{[\alpha, \beta]}$, the $\mathsf{KL}$-divergence between
them is defined to be
\begin{equation}\label{neqn:kldiv-density-defn}
\kldiva{f}{g}
\defined
\int_{B} f \log\parens{\frac{f}{g}} \dlett{\mu}
\defines
\EE_{f}\log\parens{\frac{f(X)}{g(X)}},
\end{equation}
where $X \sim f$ in \eqref{neqn:kldiv-density-defn}.
\begin{restatable}{nrmk}{kldivdensitywelldefined}\label{nrmk:kldiv-density-well-defined}
We observe that $\kldiva{f}{g}$ is well-defined in
\eqref{neqn:kldiv-density-defn} for our setting, since $\inf_{x \in B} g(x)
\geq \alpha > 0$, by \Cref{ndefn:density-class-F-alpha-beta}. We further
emphasize that $\mathsf{KL}$-divergence is not valid metric in general,
since it is not symmetric in its arguments.
\end{restatable}
The crucial fact in the risk bounds we will soon derive, is the `topological
equivalence' of the $L_{2}$-metric and $\mathsf{KL}$-divergence, on the density
class $\cF_{B}^{[\alpha, \beta]}$. Since it is hard to find a concrete reference for this
folklore fact, we formalize this equivalence for our setting in
\Cref{nlem:equiv-kl-euc-metric}.
\begin{restatable}[$\mathsf{KL}$-$L_{2}$ equivalence on
$\cF_{B}^{[\alpha, \beta]}$]{nlem}{equivkleucmetric}\label{nlem:equiv-kl-euc-metric}
For each pair of densities $f, g \in \cF_{B}^{[\alpha, \beta]}$, the following
relationship holds:
\begin{equation}\label{neqn:nlem:equiv-kl-euc-metric-01}
c(\alpha,\beta) \normb{f-g}_{2}^{2}
\leq \kldiva{f}{g}
\leq (1 / \alpha) \normb{f - g}_{2}^{2},
\end{equation}
where we denote $c(\alpha,\beta) \defined \frac{h(\beta / \alpha)}{\beta} >
0$. Here $h : (0, \infty) \to \RR$ is defined to be
\begin{equation}\label{neqn:nlem:equiv-kl-euc-metric-02}
h(\gamma)
\defined
\begin{cases}
\frac{\gamma - 1 - \log{\gamma}}{(\gamma - 1)^{2}} & \text{if $\gamma \in (0, \infty) \setminus \theset{1}$} \\
\frac{1}{2} = \lim_{x \to 1} \frac{x - 1 - \log{x}}{(x - 1)^{2}}
& \text{if $\gamma = 1$},
\end{cases}
\end{equation}
and is positive over its entire support. It is also easily seen that on
$\cF_{B}^{[\alpha, \beta]}$, $d_{\mathsf{KL}}$ (and hence the $L_{2}$-metric) is also
equivalent to the Hellinger metric. Furthermore, these properties are also
inherited by $\cF \subset \cF_{B}^{[\alpha, \beta]}$, which is our density class of
interest.
\end{restatable}
\begin{restatable}{nrmk}{klemelaequivkleucmetric}\label{nrmk:equiv-kl-euc-metric}
We note that both the upper and lower bounds in
\eqref{neqn:nlem:equiv-kl-euc-metric-01} are stated without proof and
without tracking constants in \citet[Lemma~11.6]{klemela2009smoothing}. We
formally prove this claim in \Cref{app:mathematical-preliminaries}.
Importantly, the validity of \eqref{neqn:nlem:equiv-kl-euc-metric-01} relies
on the assumption of the boundedness of the densities, which holds in our
setting.
\end{restatable}
\subsection{Minimax Lower Bound}\label{sec:lower-bound} We will first establish
a lower bound. For completeness, we need to introduce some additional relevant
background and notation. We start by stating Fano's inequality for our convex
density class, $\cF$ \citep[see][Lemma~2.10]{tsybakov2009introduction}.
\begin{restatable}[Fano's inequality for
$\cF$]{nlem}{fanoinequality}\label{nlem:fano-inequality} Let
$\theset{f^1, \ldots, f^m} \subset \cF$ be a collection of
$\varepsilon$-separated densities (\ie $\normb{f^{i} - f^{j}}_{2} >
\varepsilon$ for $i \neq j$), in the $L_{2}$-metric. Suppose $J$ is
uniformly distributed over the index set $[m]$, and $(X_{i} | J = j)
\distiid f^j$ for each $i \in [n]$. Then
\begin{align*}
\inf_{\widehat{\nu}} \sup_{f} \EE \|\widehat{\nu}(X) - f\|_{2}^{2}
\geq
\frac{\varepsilon^2}{4}\bigg(1 - \frac{n I(X_1; J) + \log 2}{\log m}\bigg).
\end{align*}
\end{restatable}
In the above $I(X_1;J) \defined \frac{1}{m}\sum_{j = 1}^{m} \kldiva{f^{j}}{\bar
f}$, where $\bar f = \frac{1}{m}\sum_{j = 1}^m f^j$ is the mutual
information between $X_1$ and the randomly sampled index $J$. Further,
the infimum is taken over all measurable functions of the data. Next, we
define the important notion of a packing set for $\cF$ \citep[see
Section~5.2][\eg, for more details]{wainwright2019high}.
\begin{restatable}[Packing sets and packing numbers of $\cF$ in the
$L_{2}$-metric]{ndefn}{packingsets}\label{ndefn:packing-sets} Given any
$\varepsilon > 0$, an $\varepsilon$-packing set of $\cF$ in the
$L_{2}$-metric, is a set $\theset{f^1, \ldots, f^m} \subset \cF$ of
$\varepsilon$-separated densities (\ie, $\normb{f^{i} - f^{j}}_{2} >
\varepsilon$ for $i \neq j$) in the $L_{2}$-metric. The corresponding
$\varepsilon$-packing number, denoted by $M(\varepsilon, \cF)$, is the
cardinality of the largest (maximal) $\varepsilon$-packing of $\cF$. We
refer to $\log{\mgloc{\cF}{\varepsilon}} \defined \log{M(\varepsilon,
\cF)}$ as the \emph{global metric entropy} of $\cF$.
\end{restatable}
\begin{restatable}{nrmk}{cFcompactpackingnumbers}\label{nrmk:cF-compact-packing-numbers}
Note that we are not assuming here that $\cF$ is totally bounded, hence some
(or perhaps all) of the packing numbers may be infinite; this however does
not cause a problem in what follows. Henceforth, all packing sets (or
packing numbers) of $\cF$, will be assumed to be with reference to the
$L_{2}$-metric, unless stated otherwise. We will use the standard fact that
a $\varepsilon$-maximal packing of $\cF$, is also a $\varepsilon$-covering
set of $\cF$.
\end{restatable}
We will now define the notion of \emph{local metric entropy}, which will play a
key role in the development of our risk bounds.
\begin{restatable}[Local metric entropy of
$\cF$]{ndefn}{localentropy}\label{ndefn:local-entropy} Let $c > 0$ be
fixed, and $\theta \in \cF$ be an arbitrary point. Consider the
set\footnote{Observe that this set may also fail to be totally bounded,
since while the ball $B_2(\theta,\varepsilon)$ is a bounded set, it is
not totally bounded.} $\cF \cap B_{2}(\theta, \varepsilon)$. Let
$M(\varepsilon/c, \cF \cap B_{2}(\theta, \varepsilon))$ denote the
$\varepsilon/c$-packing number of $\cF \cap B_{2}(\theta, \varepsilon)$,
in the $L_{2}$-metric. Let
\begin{align*}
\mlocc{\cF}{\varepsilon}{c}
\defined
\sup_{\theta \in \cF} M(\varepsilon/c, \cF \cap B_{2}(\theta, \varepsilon))
\defines
\sup_{\theta \in \cF} \mgloc{\cF \cap B_{2}(\theta, \varepsilon)}{\varepsilon/c}.
\end{align*}
We refer to $\log{\mlocc{\cF}{\varepsilon}{c}}$ as the \emph{local metric
entropy} of $\cF$.
\end{restatable}
We show the following minimax lower bound for our convex density estimation
setting over $\cF$. It is a direct consequence of Fano's inequality per
\Cref{nlem:fano-inequality}.
\begin{restatable}[Minimax lower
bound]{nlem}{minimaxlowerbound}\label{nlem:minimax-lower-bound} Let $c >
0$ be fixed, and independent of the data samples $X$. Then
the minimax rate satisfies
\begin{align*}
\inf_{\widehat{\nu}} \sup_{f \in \cF} \EE_f \normb{\widehat{\nu}(X) - f}_{2}^{2} \geq \frac{\varepsilon^2}{8c^2},
\end{align*}
if $\varepsilon$ satisfies $\log{\mlocc{\cF}{\varepsilon}{c}} > 2 n
\varepsilon^{2} / \alpha + 2\log 2$.
\end{restatable}
\subsection{Upper Bound}\label{sec:upper-bound}
We now turn our attention to the upper bound. We note that our universal
estimator over $\cF$, will be a likelihood-based estimator for $f$. As such for
any two densities $g, g^{\prime} \in \cF$, we will routinely work with the
\emph{log-likelihood difference} for the $n$ observed samples $X
\defined (X_{1}, \ldots, X_{n})^{\top} \distiid f \in \cF$. We will denote this
by
\begin{equation}\label{neqn:log-likelihood-g-g-prime}
\psi(g, g^{\prime}, X)
\defined
\log{\parens{\prod_{i = 1}^{n} \frac{g(X_{i})}{g^{\prime}(X_{i})}}}
=
\sum_{i = 1}^{n} \log\parens{\frac{g(X_{i})}{g^{\prime}(X_{i})}}
=
\sum_{i = 1}^{n} \log g(X_{i}) - \sum_{i = 1}^{n} \log g^{\prime}(X_{i}).
\end{equation}
\begin{restatable}{nrmk}{loglikelihoodwelldefined}\label{nrmk:log-likelihood-well-defined}
We note that the log-likelihood difference $\psi(g, g^{\prime},
X)$ in \eqref{neqn:log-likelihood-g-g-prime}, is well-defined.
This follows since for each $i \in [n]$, the individual random variables
$\log g(X_{i})/g^{\prime}(X_{i})$ are well-defined (as $\alpha > 0$), and
bounded. That is, $-\infty < \log \alpha/\beta \leq \log
g(X_{i})/g^{\prime}(X_{i}) \leq \log \beta/\alpha < \infty$, for each $i \in
[n]$.
\end{restatable}
We will use the log-likelihood difference to help us decide which of the two
densities is ``more'' correct, given the observed data samples $X$.
Given this, we will first need a concentration result on the density
log-likelihood difference. We do this by establishing the following lemma.
\begin{restatable}[Log-likelihood difference concentration in
$\cF$]{nlem}{loglikelihoodhoeffding}\label{nlem:log-likelihood-bernstein}
Let $\delta > 0$ be arbitrary, and let $X \defined (X_{1},
\ldots, X_{n})^{\top} \distiid f \in \cF$, be the $n$ observed samples.
Suppose we are trying to distinguish between two densities $g,
g^{\prime} \in \cF$. Let $\psi(g, g^{\prime}, X)$ denote
their log-likelihood difference per
\eqref{neqn:log-likelihood-g-g-prime}. We then have
\begin{equation}\label{neqn:log-likelihood-bernstein-01}
\sup_{\substack{g, g^{\prime} \colon \normb{g - g^{\prime}}_{2} \geq C\delta, \\
\normb{g^{\prime}-f}_{2} \leq \delta}}\PP(\psi(g, g^{\prime}, X) > 0)
\leq
\exp\parens{-n L(\alpha, \beta, C) \delta^{2}}
\end{equation}
where
\begin{align}
C
& >
1 + \sqrt{1 / (\alpha c(\alpha,\beta))}
\label{neqn:log-likelihood-bernstein-02a} \\
L(\alpha, \beta, C)
& \defined
\frac{\parens{\sqrt{c(\alpha,\beta)} (C-1) - \sqrt{1 / \alpha}}^{2}
}{2\braces{ 2 K(\alpha, \beta) +\frac{1}{3} \log \beta / \alpha}},
\label{neqn:log-likelihood-bernstein-02b}
\end{align}
with $K(\alpha, \beta) \defined \beta / (\alpha^{2} c(\alpha, \beta))$, and
$c(\alpha, \beta)$ is as defined in \Cref{nlem:equiv-kl-euc-metric}. In the
above $\PP$ is taken with respect to the true density function $f$, \ie,
$\PP = \PP_f$.
\end{restatable}
From \Cref{nlem:log-likelihood-bernstein}, we derive a key concentration result
concerning a packing set in $\cF$, as summarized in
\Cref{nlem:critical-log-likelihood-concentration}. The relevance of such a
result will become clearer later, when we introduce our sieve-based MLE for $f$.
Our sieve estimator will be constructed using packing sets of $\cF$, thus
\Cref{nlem:critical-log-likelihood-concentration} will be an important tool to
enable us to handle the concentration properties of our estimator.
\begin{restatable}[Maximum likelihood concentration in
$\cF$]{nlem}{criticalloglikelihoodconcentration}\label{nlem:critical-log-likelihood-concentration}
Let $\delta > 0$ be arbitrary, and let $X \defined (X_{1},
\ldots, X_{n})^{\top} \distiid f \in \cF$, be the $n$ observed samples.
Suppose further that we have a maximal $\delta$-packing set of
$\cF^{\prime} \subset \cF$, \ie, $\theset{g_1, \ldots, g_m} \subset
\cF^{\prime}$ such that $\normb{g_i- g_{j}}_{2} > \delta$ for all $i
\neq j$, and it is known that $f \in \cF^{\prime}$. Now let $j^{*} \in
[m]$, denote the index of a density whose likelihood is the largest. We
then have
\begin{equation*}
\PP(\|g_{j^*}-f\|_2 > (C + 1) \delta) \leq
m \exp\parens{-n L(\alpha, \beta, C) \delta^{2}},
\end{equation*}
where $C$ is assumed to satisfy \eqref{neqn:log-likelihood-bernstein-02a},
and $L(\alpha, \beta, C)$ is defined as per
\eqref{neqn:log-likelihood-bernstein-02b}.
\end{restatable}
Next we establish that the map $\varepsilon \mapsto \log
\mlocc{\cF}{\varepsilon}{c}$ is non-increasing. This lemma is made possible
by the fact that the set $\cF$ is convex by assumption, and that we are
using the $L_{2}$-metric. This monotonicity property of the
$\varepsilon$-local metric entropy in the $L_{2}$-metric is a
\emph{critical} technical ingredient used in the proofs establishing our
upper bound.
\begin{restatable}[Monotonicity of local metric
entropy]{nlem}{localmetricentmonotone}\label{nlem:local-metric-ent-monotone}
The map $\varepsilon \mapsto \log{\mlocc{\cF}{\varepsilon}{c}}$ is
non-increasing.
\end{restatable}
We now turn our attention to describing our proposed likelihood-based estimator,
\ie, $\nu^{*}(X)$, of $f \in \cF$. In the discussion that follows we
let $d \defined \operatorname{diam}_{\operatorname{2}}(\cF)$, which is finite by the
boundedness of $\cF$. The estimator is directly inspired by a recent
construction used in \citet{neykov2022minimax}, who applied it to the Gaussian
sequence model. However, there the underlying space used is $(\RR^{n},
\normb{\cdot}_{2})$, whereas in our case it is $(\cF, \normb{\cdot}_{2})$, which
has a \emph{vastly} different underlying geometric structure. Importantly since
we are performing density estimation, our proposed estimator uses a
fundamentally different \emph{log-likelihood}-based selection criteria, compared
to the \emph{projection}-based sequence model estimator in
\citet{neykov2022minimax}. Although our estimator can also be described
constructively, it is not intended to be practically computable.
\vspace{8pt}
\begin{mycolorbox}{blue}{\textbf{Construction of the multistage sieve MLE,
$\nu^{*}(X)$, of $f \in \cF$.}}
\benum[wide=0pt, label={\color{blue} \textbf{Step \arabic*}}, align=left, start=1]
\item \label{itm:likelihood-based-estimator-01} \textbf{Initialize inputs.}
\vspace{1mm}
\newline
\nit
Let $X \defined (X_{1}, \ldots, X_{n})^{\top}$ denote our $n$
observed \iid data samples. Fix some sufficiently large $c > 0$, and then
define $C$ such that $c \defined 2(C + 1)$. Importantly, the constant $c$
should be set \emph{without} looking at the data samples, \ie,
\emph{independently} of $X$.
\item \label{itm:likelihood-based-estimator-02} \textbf{Construct a maximal
packing set tree of depth $\overline{J}$ \emph{before} seeing the data.}
\vspace{1mm}
\newline
\nit
Construct a tree of packing sets of depth $\overline{J} \in \NN$, which is
\emph{independent} of the data samples $X$. Here, $\overline{J}$
is as defined in \Cref{nthm:upper-bound-rate-finite-iterations}. The
explicit construction of such a packing set tree proceeds as follows. First,
fix any arbitrary point $\Upsilon_{1} \in \cF$, which is the root node, \ie,
the first level of the packing set tree. In the case where $\overline{J} =
1$, the tree construction stops at this single root node.
Assuming the (more interesting) case where $\overline{J} > 1$, we then let
$d \defined \operatorname{diam}{(\cF)}$, and construct a maximal $\frac{d}{2(C +
1)}$-packing set of $B_{2}(\Upsilon_{1}, d) \cap \cF = \cF$. Denote this
packing set by $P_{\Upsilon_{1}} \defined \theset{m_{1}, m_{2}, m_{3},
\ldots, m_{\absb{P_{\Upsilon_{1}}}}}$. The set $P_{\Upsilon_{1}}$ forms the
children (densities) of our root node, that is the second level of the
tree\footnote{By \emph{convention}, the children forming the packing set
densities are arbitrarily indexed in an increasing alphanumeric manner, from
left child node to right child node.}. Now, for \emph{each} density in
$P_{\Upsilon_{1}}$, we again construct a maximal packing set as follows. For
example, taking the density $m_{3} \in P_{\Upsilon_{1}}$, we construct a
maximal $\frac{d}{4(C + 1)}$-packing set of $B_{2}(m_{3}, d/2) \cap \cF$,
which we denote as $P_{m_{3}} \defined \theset{m_{3,1}, m_{3,2}, m_{3,3},
\ldots, m_{3,\absb{P_{m_{3}}}}}$. Here, the (finite) packing set $P_{m_{3}}$
again forms the children of the node density $m_{3}$. Iterating this process
over each density in $P_{\Upsilon_{1}}$, forms the complete second level of
the tree.
Now we can further iterate this process over each density in the second
level of the tree to construct the third level of the tree. For example,
taking the density $m_{3, 3}$, we construct a maximal $\frac{d}{8(C +
1)}$-packing set of $B_{2}(m_{3,3}, d/4) \cap \cF$, which we denote as
$P_{m_{3, 3}} \defined \theset{m_{3,3,1}, m_{3,3,2}, m_{3,3,3}, \ldots,
m_{3,3\absb{P_{m_{3,3}}}}}$, which forms the children of node $m_{3, 3}$.
This process is iterated so that for the $k^{\textnormal{th}}$-level of the
tree, we construct $\frac{d}{2^{k}(C + 1)}$-packing sets, with closed balls
$B_{2}(\cdot, d/2^{k - 1}) \cap \cF$. In particular the packing set tree is
extended for each depth level $k \in \theset{2, 3, \ldots, \overline{J}-1}$.
This process results in a maximal packing set tree of depth $\overline{J}$,
as claimed.
\item \textbf{Build a finite sequence of densities by traversing our packing
set tree.}
\label{itm:likelihood-based-estimator-03}
\vspace{1mm}
\newline
\nit
Now, \emph{after} observing our data sample $X$, we construct a
\emph{finite} sequence of densities, \ie, $\Upsilon \defined
\theseqb{\Upsilon_{k}}{k = 1}{\overline{J}}$, using our packing set tree
construction in \ref{itm:likelihood-based-estimator-02}. First, we
initialize the first term of our sequence to $\Upsilon_{1}$, \ie, the root
node already chosen in \ref{itm:likelihood-based-estimator-02}. If
$\overline{J} = 1$, then the sequence $\Upsilon \defined (\Upsilon_{1})$.
Otherwise, if $\overline{J} > 1$, we traverse down one level of our packing
set tree, and assign $\Upsilon_{2}$ to be the density from
$P_{\Upsilon_{1}}$ which maximizes the log-likelihood \emph{given} the data.
That is, set $\Upsilon_{2} \defined \argmax_{\nu \in P_{\Upsilon_{1}}}
\sum_{i = 1}^{n} \log \nu(X_{i})$. Since $P_{\Upsilon_{1}}$ is a finite set,
this will be exhausted for each such iteration in finitely many steps.
Moreover, we note that when assigning $\Upsilon_{2}$, there may be ties in
children densities who all simultaneously maximize the log-likelihood. To
break ties, by \emph{convention}, we always select the \emph{left-most}
child from our packing set tree\footnote{This selection rule thus
effectively assigns the child density maximizing log-likelihood with the
\emph{smallest} such alphanumeric index.}. Once the $\Upsilon_{2}$ is
assigned from our packing set tree, once again assign $\Upsilon_{3}$ from
its children by again maximizing the log-likelihood. Keep iterating in this
manner for each index\footnote{Note that $k$ here refers to index of the
$k^{\textnormal{th}}$-term our sequence $\Upsilon$.} $k \in \theset{2, 3,
\ldots, \overline{J}}$, and construct the finite, \ie, \emph{terminating}
sequence $\Upsilon$.
\item \textbf{Output estimator as the $\overline{J}^{\textnormal{th}}$-term
of the sequence.}
\label{itm:likelihood-based-estimator-04}
\vspace{1mm}
\newline
\nit
Finally, we note that the finite sequence $\Upsilon \defined
\theseqb{\Upsilon_{k}}{k = 1}{\overline{J}}$ satisfies\footnote{We will
formally justify this in the appendix in
\Cref{nlem:upsilon-is-cauchy-sequence}.} $\normb{\Upsilon_{J} -
\Upsilon_{J^{\prime}}}_{2}
\leq
\frac{d}{2^{J^{\prime} - 2}}$, for each pair of positive integers
$J^{\prime} < J$. Our multistage sieve MLE, \ie, $\nu^{*}(X)$,
can be taken as the final term of this sequence. That is
$\nu^{*}(X) \defined \Upsilon_{\overline{J}}$. The estimator
$\nu^{*}(X)$ is readily understood by comparing\footnote{We note
that in \Cref{fig:packing-set-tree-construction} if $\overline{J} = 1$, the
estimator would just output $\Upsilon_{1}$. In the case where $\overline{J}
> 1$, the maximal packing sets for each level of the tree are illustrated on
the left, and the corresponding constructed tree level is shown on the
right. In this instance the finite sequence of $\overline{J}$ densities is
given by $\Upsilon = \parens{\Upsilon_{1}, m_{3}, m_{3, 3}, m_{3, 3, 2},
\ldots, m_{3, 3, 2, \ldots, 5}}$. The estimator then takes the
$\overline{J}^{\textnormal{th}}$-term of $\Upsilon$, \ie,
$\nu^{*}(X)=m_{\underbrace{3,3,2, \ldots,
5}_{(\bar{J}-1)-\text{terms}}}$.} \Cref{fig:packing-set-tree-construction}
with the qualitative description in
\ref{itm:likelihood-based-estimator-01}-\ref{itm:likelihood-based-estimator-04}.
\eenum
\end{mycolorbox}
\iftoggle{showfigs}{%
\begin{figure}[!ht]
\centering
\includegraphics[scale=0.72]{packing-set-tree-06.pdf}
\caption{Maximal packing set tree construction in
\ref{itm:likelihood-based-estimator-02}.}
\label{fig:packing-set-tree-construction}
\end{figure}
}{%
}
\begin{restatable}{nrmk}{packingsettreeconstructionscale}\label{nrmk:packing-set-tree-construction-scale}
We emphasize that \Cref{fig:packing-set-tree-construction} is not drawn to
any precise scale. In reality the $L_2$-balls should be much ``wider'' than
the set $\cF$ (and $\cF_{B}^{[\alpha, \beta]}$). This is because they do not impose that
their elements are proper densities, unlike the elements of the set $\cF$
(and $\cF_{B}^{[\alpha, \beta]}$) which are non-negative and integrate to $1$. It is
intended to be useful conceptual guide to understanding the construction of
our multistage sieve MLE.
\end{restatable}
We observe that our proposed estimator $\nu^{*}(X)$ can be thought of
as an ``multistage sieve MLE'' in the spirit of \citet{wong1995probability}.
Broadly speaking a `sieve' MLE effectively takes the MLE over a strategically
constrained subset of the parameter space, \ie, $\cF$ in our setting \citep[see
Chapter~8][\eg, for more details]{grenander1981abstractinference}. Specifically,
as we traverse the down the finite-depth maximal packing set tree, each group of
children densities along with the MLE selection rule can be thought of as a
``sieve''. We note that the sieve MLE proposed in \citet{wong1995probability} is
a construction which is also not practically computable for general density
classes $\cF$.
\begin{restatable}[An \emph{online} finite packing set tree
construction]{nrmk}{packingsettreeonline}\label{nrmk:packing-set-tree-online}
We note that the finite-depth maximal packing set tree described in
\ref{itm:likelihood-based-estimator-02}, can be replaced with a
conceptually simpler \emph{online} finite-depth maximal packing set tree
construction. This proceeds as follows. Once again, as per
\ref{itm:likelihood-based-estimator-02}, we can initialize $\Upsilon_{1}
\in \cF$ to be the root node independently of the data. We then
construct the second level of our packing set tree, \ie,
$P_{\Upsilon_{1}} \defined \theset{m_{1}, m_{2}, m_{3}, \ldots,
m_{\absb{P_{\Upsilon_{1}}}}}$, as the previously described maximal
packing set. This first level is constructed without looking at the data
samples $X$. This time however, we can traverse down the
first level of the tree and set $\Upsilon_{2} \defined \argmax_{\nu \in
P_{\Upsilon_{1}}} \sum_{i = 1}^{n} \log \nu(X_{i})$, \ie, by using the
data samples $X$. Given $\Upsilon_{2}$ selected in this data
driven manner, we can construct the second level of the tree as the
children of $\Upsilon_{2}$, \ie, the maximal packing set
$P_{\Upsilon_{2}}$ \emph{without} using the data samples. We can then
set $\Upsilon_{3} \defined \argmax_{\nu \in P_{\Upsilon_{2}}} \sum_{i =
1}^{n} \log \nu(X_{i})$, once again using the data. We can thus repeat
this recursive process for $\overline{J}$ iterations, whereby the
maximal packing set of children of each parent node are constructed
without seeing the data. The specific child node is selected after
seeing the data, \emph{and then} the estimator can traverse to one of
these children. This does not require the all possible children of all
possible parent nodes of the maximal packing set tree to be constructed
up front as described in \ref{itm:likelihood-based-estimator-02}.
Instead, we only construct the children as required in a simple
\emph{sequential} manner.
\end{restatable}
We next show that our multistage sieve MLE is a measurable function of the data
with respect to the Borel $\sigma$-field on $\cF$ in $L_{2}$-metric topology.
This is important, because all upper bound risk rates in expectation for
$\nu^{*}(X)$ that follow, are with respect to the $L_{2}$-metric
topology on $\cF$.
\begin{restatable}[Measurability of
$\nu^{*}(X)$]{nprop}{estimatormeasurability}\label{nprop:nu-star-estimator-measurability}
The multistage sieve MLE, \ie, $\nu^{*}(X)$, is a measurable
function of the data with respect to the Borel $\sigma$-field on $\cF$
in the $L_{2}$-metric topology.
\end{restatable}
With the measurability of $\nu^{*}(X)$ established, the main theorem
establishing the performance of $\nu^{*}(X)$ is
\Cref{nthm:upper-bound-rate-finite-iterations} below.
\begin{restatable}[Upper bound rate for the multistage sieve MLE
$\nu^{*}(X)$]{nthm}{upperboundratefiniteiters}\label{nthm:upper-bound-rate-finite-iterations}
Let, $\nu^{*}(X) = \Upsilon_{\overline{J}}$ be the output of
the multistage sieve MLE which is run for $\overline{J} \in \NN$ steps.
Here $\overline{J}$ is defined as the maximal integer $J \in \NN$, such
that $\varepsilon_J \defined \frac{\sqrt{L(\alpha, \beta, c/2 - 1)}
d}{2^{(J-2)}c}$ satisfies\footnote{Observe that by the definition of
$\varepsilon_{\overline J}$ and \eqref{upper:bound:suff:cond} we have
that all packing sets used in the construction of the estimator must be
finite, even though we are not assuming that the set $\cF$ is totally
bounded.}
\begin{align}\label{upper:bound:suff:cond}
n \varepsilon_J^2
>
2 \log \mlocc{\cF}{\varepsilon_J
\frac{c}{\sqrt{L(\alpha, \beta, c / 2 - 1)}}}{c}
\vee \log 2,
\end{align}
or $\overline{J} = 1$ if no such $J$ exists. Then
\begin{align*}
\EE \normb{\nu^{*}(X) - f}^{2}_{2}
\leq
\bar C \varepsilon^{*2},
\end{align*}
for some universal constant $\bar C$, and where $\varepsilon^* \defined
\varepsilon_{\overline{J}}$. We remind the reader that $c \defined 2(C +
1)$ is the constant from the definition of local metric entropy, which
is assumed to be sufficiently large. Here $C$ is assumed to satisfy
\eqref{neqn:log-likelihood-bernstein-02a}, and $L(\alpha, \beta, C)$ is
defined as per \eqref{neqn:log-likelihood-bernstein-02b}.
\end{restatable}
We will now formally illustrate that the above estimator achieves the minimax
rate. The precise expression of the rate is quantified in the following result.
\begin{restatable}[Minimax
rate]{nthm}{sharpminimaxrate}\label{nthm:sharp-minimax-rate} Define
$\varepsilon^* \defined \sup \thesetc{\varepsilon}{n \varepsilon^{2}
\leq \log{\mlocc{\cF}{\varepsilon}{c}}}$, where $c$ in the definition of
local metric entropy is a sufficiently large absolute constant. Then
the minimax rate is given by $\varepsilon^{*2} \wedge d^2$ up to
absolute constant factors.
\end{restatable}
\begin{restatable}[Extending results to loss functions in
$\mathsf{KL}$-divergence and the Hellinger
metric]{nrmk}{extendkldivhellinger}\label{nthm:extend-kldiv-hellinger}
Recall that by \Cref{nlem:equiv-kl-euc-metric} we have the ``topological
equivalence'' of the $\mathsf{KL}$-divergence and squared Hellinger
metric with the squared $L_{2}$-metric on $\cF$. This means that we can
readily extend our minimax risk bounds in \Cref{nthm:sharp-minimax-rate}
to loss functions measured via $\mathsf{KL}$-divergence and the squared
Hellinger metric. The important consideration is that
\eqref{upper:bound:suff:cond} is still solved (in both cases) using the
local metric entropy of $\cF$ using the squared $L_{2}$-metric. Note
that for the $\mathsf{KL}$-divergence to be well-defined, we require
that all densities are strictly positively lower bounded over the common
compact support.
\end{restatable}
We now argue that the minimax rate for a class $\cF \subset \cF_{B}^{[0, \beta]}$ which
is convex and not necessarily lower bounded by $\alpha > 0$ is given by the same
equation, as long as there exists a single density in $f_{\alpha} \in \cF$ which
is $\alpha$-lower bounded. The argument used to establish this claim essentially
the same as used in \citet[Lemma~1]{yang1999information}, which we formalize for
our setting in \Cref{nprop:extend-zero-bounded-densities}. For completeness, we
provide all details for our setting in the Appendix.
\begin{restatable}[Extending results to
$\cF_{B}^{[0, \beta]}$]{nprop}{extendzeroboundeddensities}\label{nprop:extend-zero-bounded-densities}
Let $\cF \subset \cF_{B}^{[0, \beta]}$ be a convex class of densities, with at
least one $f_{\alpha} \in \cF$ that is $\alpha$-lower bounded, with
$\alpha > 0$. Then the minimax rate in the squared $L_{2}$-metric is
$\varepsilon^{*2} \wedge d^{2}$, where $\varepsilon^* \defined \sup
\thesetc{\varepsilon}{n \varepsilon^{2} \leq
\log{\mlocc{\cF}{\varepsilon}{c}}}$.
\end{restatable}
\section{Examples}\label{sec:examples}
We will now apply our work to derive risk bounds for density estimation (under
the squared $L_{2}$-metric) for various examples of convex density classes
$\cF$. To that end, per \Cref{nprop:extend-zero-bounded-densities} our risk
bounds only require us to establish that the stated class $\cF$ is indeed
convex, and importantly that there exists at least one density $f_{\alpha} \in
\cF$ that is positively bounded away from 0 over the entire support $B$. In
order to establish the latter fact we can usually take $f_{\alpha} \sim
\distUnif{[B]}$, and check that it lies in our density class $\cF$, and by
suitably expanding our ambient space\footnote{We reiterate that our use of
$\theset{\alpha, \beta}$ in this section (and throughout the paper) is
\emph{only} used to refer to the constants in
\Cref{ndefn:density-class-F-alpha-beta}, of $\cF_{B}^{[\alpha, \beta]}$ and thus $\cF$.}
$\cF_{B}^{[\alpha, \beta]}$. We will also use the following key fact relating
$L_{2}$-\emph{local} and $L_{2}$-\emph{global} metric entropies.
\begin{equation}\label{neqn:local-global-entropy-bounds}
\log{\mgloc{\cF}{\varepsilon / c}} - \log{\mgloc{\cF}{\varepsilon}}
\leq
\log{\mlocc{\cF}{\varepsilon}{c}}
\leq
\log{\mgloc{\cF}{\varepsilon / c}}
\end{equation}
Here, \eqref{neqn:local-global-entropy-bounds} follows directly from
\citet[Lemma~2]{yang1999information}, where it is only proved for the case $c =
2$. However, their proof directly extends to the more general case for each $c >
0$, which is required for our setting. For the various examples of $\cF$ that
follow below, we will show that a stronger sufficient condition on global
entropy is satisfied, namely
\begin{equation}\label{neqn:local-global-entropy-bounds-suff-condn}
\log{\mgloc{\cF}{\varepsilon / c}} - \log{\mgloc{\cF}{\varepsilon}}
\asymp
\log{\mgloc{\cF}{\varepsilon / c}},
\end{equation}
provided we take $c$ to be sufficiently large enough, which is within our
control to do, per our packing set tree construction. In short,
\eqref{neqn:local-global-entropy-bounds-suff-condn} will enable us to bound the
local metric entropy via \eqref{neqn:local-global-entropy-bounds}. To illustrate
this, we initially consider two examples from
\citet[see][Section~6]{yang1999information}. We begin with the class $\cF
\defined \mathsf{Lip}_{\gamma, q}(\Psi)$, \ie, the $(\gamma, q, \Psi)$-Lipschitz density class
defined as per \eqref{neqn:lipschitz-density-class-01}. As noted in
\citet[Section~6.4]{yang1999information}, with fixed constants $\max{\theset{1 /
q - 1 / 2, 0}} < \gamma \leq 1$, and $1 \leq q \leq \infty$, the
$\varepsilon$-\emph{global} metric entropy of $\mathsf{Lip}_{\gamma, q}(\Psi)$ is of the order
$\varepsilon^{-1 / \gamma}$ per \citet{birman1980quantsobolevimbedding}.
\begin{restatable}[Lipschitz density class
$\cF$]{nexa}{exalipschitzdensityclass}\label{nexa:lipschitz-density-class}
Let $1 < \Psi < \beta < \infty$, $\max{\theset{1 / q - 1 / 2, 0}} <
\gamma \leq 1$, and $1 \leq q \leq \infty$ be fixed constants, and $B
\defined [0, 1]$. Now, let $\cF \defined \mathsf{Lip}_{\gamma, q}(\Psi)$ denote the space
of $(\gamma, q, \Psi)$-Lipschitz densities with total variation at most
$\beta$. That is,
\begin{equation}\label{neqn:lipschitz-density-class-01}
\mathsf{Lip}_{\gamma, q}(\Psi)
\defined
\thesetb{f \colon B \to [0, \Psi]}{%
\normb{f(x + h) - f(x)}_{q} \leq \Psi h^{\gamma},
\normb{f}_{q} \leq \Psi,
\int_{B} f \dlett{\mu} = 1,
f \text{ measurable}},
\end{equation}
and $\normb{f}_{q} \defined \parens{\int_{B} \absa{f(x)}^{q} \dlett{\mu}}^{1
/ q}$. Note that in \eqref{neqn:lipschitz-density-class-01} we have
that $x \in B$, and only consider $h > 0$ such that $x + h \in B$,
so that the predicate of $\mathsf{Lip}_{\gamma, q}(\Psi)$ is well-defined. Then
$\mathsf{Lip}_{\gamma, q}(\Psi)$ is a convex density class, there exists a density
$f_{\alpha} \in \mathsf{Lip}_{\gamma, q}(\Psi)$ that is strictly positively bounded
away from 0, and the minimax rate (in the squared $L_{2}$-metric)
for estimating $f \in \mathsf{Lip}_{\gamma, q}(\Psi)$ is of the order $n^{- \frac{2
\gamma}{2 \gamma + 1}}$.
\end{restatable}
Another well studied density estimation problem is the case where $\cF \defined
\mathsf{BV}_{\zeta}$ is total bounded variation at most $\zeta$, defined as per
\eqref{neqn:bdd-total-variation-density-class-01}. Importantly we note that
the $\varepsilon$-\emph{global} $L_{2}$-metric entropy of this well studied
function class is of the order $\varepsilon^{-1}$ \citep[see
Section~6.4][\eg, for more details]{yang1999information}.
\begin{restatable}[Bounded total variation density class
$\cF$]{nexa}{exabddtotvardensityclass}\label{nexa:bdd-total-variation-density-class}
Let $1 < \zeta < \beta < \infty$ be a fixed constant, and $B \defined
[0, 1]$. Now, let $\cF \defined \mathsf{BV}_{\zeta}$ denote the space of
univariate densities with total variation at most $\beta$. That is,
\begin{equation}\label{neqn:bdd-total-variation-density-class-01}
\mathsf{BV}_{\zeta}
\defined
\thesetb{f \colon B \to [0, \zeta]}{%
\normb{f}_{\infty} \leq \zeta,
V(f) \leq \zeta,
\int_{B} f \dlett{\mu} = 1,
f \text{ measurable}},
\end{equation}
where we define the total variation of $f$, \ie, $V(f)$ as
\begin{equation}\label{neqn:bdd-total-variation-density-class-02}
V(f)
\defined
\sup_{\thesetb{x_{1}, \ldots, x_{m}}{0 \leq x_1 < \cdots < x_m \leq 1, m \in \NN}}
\sum_{i=1}^{m - 1} \absb{f\parens{x_{i+1}}-f\parens{x_{i}}},
\end{equation}
and $\normb{f}_{\infty} \defined \sup_{x \in B} \absa{f(x)}$. Then the
minimax rate (in the squared $L_{2}$-metric) for estimating $f \in
\mathsf{BV}_{\zeta}$ is of the order $n^{- 2 / 3}$.
\end{restatable}
Another interesting example illustrating the use case of our bounds is that
where $\cF \defined \mathsf{Quad}_{\gamma}$, forms the density class of
$\gamma$-quadratic functionals defined as per
\eqref{neqn:quad-functional-density-class-01}. Importantly we note that the
$\varepsilon$-\emph{global} $L_{2}$-metric entropy of this well studied function
class is of the order $\varepsilon^{-1 / 4}$ \citep[see Example~15.8 and
Example~15.22][\eg, for more details]{wainwright2019high}.
\begin{restatable}[Quadratic functional density class
$\cF$]{nexa}{quadfunctionaldensityclass}\label{nexa:quad-functional-density-class}
Let $0 < \alpha < 1 < \beta < \infty$, and $\gamma > 1$ be fixed
constants, with $B \defined [0, 1]$. Now, let $\cF \defined
\mathsf{Quad}_{\gamma}$ denote the space of univariate quadratic functional
densities. That is,
\begin{equation}\label{neqn:quad-functional-density-class-01}
\mathsf{Quad}_{\gamma}
\defined
\thesetb{f \colon B \to [\alpha, \beta]}{%
\normb{f^{\prime \prime}}_{\infty} \leq \gamma,
\int_{B} f \dlett{\mu} = 1,
f \text{ measurable}}.
\end{equation}
Then $\mathsf{Quad}_{\gamma}$ is a convex density class, there exists a density
$f_{\alpha} \in \mathsf{Quad}_{\gamma}$ that is strictly positively bounded away
from 0, and the minimax rate (in the squared $L_{2}$-metric) for estimating
$f \in \mathsf{Quad}_{\gamma}$ is of the order $n^{- 4 / 5}$.
\end{restatable}
We now turn our attention to an interesting example, which demonstrates that our
results can yield useful bounds in cases where $L_{2}$-\emph{global} metric
entropy of $\cF$ may be unknown (or difficult to compute), but the
$L_{2}$-\emph{local} metric entropy can be controlled.
\begin{restatable}[Convex mixture density class
$\cF$]{nexa}{exaconvexmixturedensityclass}\label{nexa:convex-mixture-density-class}
Let $\cF \defined \mathsf{Conv}_{k}$ where
\begin{equation}\label{neqn:convex-mixture-density-class-01}
\mathsf{Conv}_{k}
\defined
\thesetb{\sum_{i = 1}^k \alpha_{i} f_{i}}{\sum_{i = 1}^k \alpha_{i} = 1,
\alpha_{i} \geq 0,
f_{i} \in \cF_{B}^{[\alpha, \beta]}},
\end{equation}
for some fixed $k \in \NN$ and $f_i \in \cF_{B}^{[\alpha, \beta]}$ for each $i \in [k]$.
Further, let $\bfG = \parens{\bfG_{ij}}_{i, j \in [k]}$ denote the Gram
matrix with $\bfG_{ij} \defined \int_B f_{i} f_{j} \mu(\dlett{x})$, which we
assume is positive definite, \ie, $\bfG \succ \bfzero$. Then the minimax
rate for estimating $f \in \mathsf{Conv}_{k}$ is bounded from above by
$\sqrt{\frac{k}{n}}$ up to absolute constant factors.
\end{restatable}
\section{Discussion}\label{sec:discussion}
In this paper we derived exact minimax rates for density estimation over convex
density classes. Our work builds on seminal research of
\citet{lecam1973convergence,birge1983approximation,yang1999information,wong1995probability}.
More directly, we non-trivially adapted the techniques of
\cite{neykov2022minimax}, who used it for deriving exact rates for the Gaussian
sequence model. Our results demonstrate that the $L_{2}$-\emph{local} metric
entropy \emph{always} determines that minimax rate under squared $L_{2}$-loss in
this setting. We thus provide a unifying perspective across parametric
\emph{and} nonparametric convex density classes, and under weaker assumptions
than those used by \citet{yang1999information}.
An important open question that we would like to think further about is whether
there exists a computationally tractable estimator which is also minimax optimal
in our setting. We can also consider applying our techniques to the
nonparametric regression setting (with Gaussian noise) where $f$ is a uniformly
bounded regression function of interest. We leave these exciting directions for
future work. Finally, we hope that this research stimulates further activity in
approximating $L_{2}$-\emph{local} metric entropy for various convex density
classes.
\section{Acknowledgments}\label{sec:acknowledgments}
We thank Wanshan Li of Carnegie Mellon University for providing insightful
feedback during this work. All figures in this paper were drawn using the
\texttt{Mathcha}\footnote{\url{https://www.mathcha.io/editor}} editor.
\section{Preliminary}\label{app:mathematical-preliminaries}
We begin with some basic mathematical preliminaries for our work.
\subsection{Properties of
\texorpdfstring{$\cF_{B}^{[\alpha, \beta]}$}{$\cF_{B}^{[\alpha, \beta]}$}}\label{app:prop-of-class-F-alpha-beta}
Here we provide some basic analytic properties of our core density class
$\cF_{B}^{[\alpha, \beta]}$, as per \Cref{ndefn:density-class-F-alpha-beta}. Many of
these facts will be used, sometimes implicitly, in our proofs. We hope that
by documenting them rigorously, they provide the reader with a much richer
understanding of the geometry of this broader density class. This may also
be a useful reference for researchers in working in similar density
estimation settings. We provide suitable references where the properties
follow from standard real analysis theory.
\begin{restatable}[Convexity of
$\cF_{B}^{[\alpha, \beta]}$]{nlem}{convexitycfalphabeta}\label{nlem:convexity-of-class-F-alpha-beta}
The density class $\cF_{B}^{[\alpha, \beta]}$, forms a convex set, in the
$L_{2}$-metric.
\end{restatable}
\bprfof{\Cref{nlem:convexity-of-class-F-alpha-beta}} In order to show the
convexity of $\cF_{B}^{[\alpha, \beta]}$, Let $f, g \in \cF_{B}^{[\alpha, \beta]}$, and let $\kappa \in
[0, 1]$ be arbitrary. Then for each $x \in B$, we observe that
\begin{align}
(\kappa f + (1 - \kappa) g)(x)
\defined \kappa f(x) + (1 - \kappa) g(x)
& \geq \kappa \alpha + (1 - \kappa) \alpha
\geq \alpha
\label{neqn:prop-of-class-F-alpha-beta-01} \\
(\kappa f + (1 - \kappa) g)(x)
\defined \kappa f(x) + (1 - \kappa) g(x)
& \leq \kappa \beta + (1 - \kappa) \beta
\leq \beta
\label{neqn:prop-of-class-F-alpha-beta-02}
\end{align}
From \eqref{neqn:prop-of-class-F-alpha-beta-01} and
\eqref{neqn:prop-of-class-F-alpha-beta-02}, it follows that $\kappa f + (1 -
\kappa) g \colon B \to [\alpha, \beta]$. Moreover, since $\int_{B} f \dlett{\mu}
= \int_{B} g \dlett{\mu} = 1$, we have
\begin{equation}
\int_{B} (\kappa f + (1 - \kappa) g) \dlett{\mu}
= \kappa \int_{B} f \dlett{\mu} +
(1 - \kappa) \int_{B} g \dlett{\mu}
= 1.
\end{equation}
Since $f, g$ are measurable functions, then so is their convex combination, \ie,
$\kappa f + (1 - \kappa) g$. Combining the above we have shown that $\kappa f +
(1 - \kappa) g \in \cF_{B}^{[\alpha, \beta]}$, which proves the convexity of
$\cF_{B}^{[\alpha, \beta]}$, as required.
\eprfof
\begin{restatable}[Boundedness of
$\cF_{B}^{[\alpha, \beta]}$]{nlem}{boundednesscfalphabeta}\label{nlem:boundedness-of-class-F-alpha-beta}
The density class $\cF_{B}^{[\alpha, \beta]}$, is bounded, in the $L_{2}$-metric.
\end{restatable}
\bprfof{\Cref{nlem:boundedness-of-class-F-alpha-beta}} We now show that
$\cF_{B}^{[\alpha, \beta]}$ is bounded in the $L_{2}$-metric. To see this observe that for
any $f, g \in \cF_{B}^{[\alpha, \beta]}$:
\begin{equation}
\normb{f - g}_{2}^{2}
\defined \int_{B} (f-g)^{2} \dlett{\mu}
\leq \int_{B} \absa{f-g} 2 \beta \dlett{\mu}
\leq 2 \beta \parens{\int_{B} \absa{f} \dlett{\mu} + \int_{B} \absa{g} \dlett{\mu}}
= 4 \beta.
\end{equation}
It follows that $\operatorname{diam}_{2}\parens{\cF_{B}^{[\alpha, \beta]}} \defined \sup\thesetb{\normb{f
- g}_{2}}{f, g \in \cF_{B}^{[\alpha, \beta]}} \leq 2 \sqrt{\beta} < \infty$, as
required.
\eprfof
\begin{restatable}[$\cF_{B}^{[\alpha, \beta]}$ lies in
$L^{2}(B)$]{nlem}{subsetlebtwospacecfalphabeta}\label{nlem:subset-l2-space-class-F-alpha-beta}
The density class $\cF_{B}^{[\alpha, \beta]}$, satisfies $\cF_{B}^{[\alpha, \beta]} \subset
L^{2}(B)$, where
\begin{equation}\label{neqn:l2-space-on-set-b-01}
L^{2}(B)
\defined
\thesetb{f \colon B \to \RR}{\int_{B}{f}^{2} \dlett{\mu} < \infty, \textnormal{$f$ measurable}}.
\end{equation}
As such $(\cF_{B}^{[\alpha, \beta]}, \normb{\cdot}_{2})$ is an induced metric subspace
of $(L^{2}(B), \normb{\cdot}_{2})$.
\end{restatable}
\bprfof{\Cref{nlem:subset-l2-space-class-F-alpha-beta}} Let $f \in
\cF_{B}^{[\alpha, \beta]}$ be arbitrary. We then observe that
\begin{equation}
\int_{B}{f}^{2} \dlett{\mu}
\leq \int_{B}f {\beta} \dlett{\mu}
= \beta
< \infty,
\end{equation}
since $f \leq \beta$ by definition of $\cF_{B}^{[\alpha, \beta]}$. Given that $f \colon B
\to [\alpha, \beta] \subset \RR$, we have that $f \in L^{2}(B)$, \ie,
$\cF_{B}^{[\alpha, \beta]} \subset L^{2}(B)$ as required.
\eprfof
\begin{restatable}[Completeness and separability of
$L^{2}(B)$]{nlem}{completenesslebtwospace}\label{nlem:completeness-separability-l2-space}
The metric space $(L^{2}(B), \normb{\cdot}_{2})$, with $L^{2}(B)$
defined as per \Cref{neqn:l2-space-on-set-b-01}, is complete and
separable.
\end{restatable}
\bprfof{\Cref{nlem:completeness-separability-l2-space}} We note that
completeness of $(L^{2}(B), \normb{\cdot}_{2})$ follows directly from
\citet[Theorem~4.8]{brezis2011funcanalysis}, and separability follows from
\citet[Theorem~4.13]{brezis2011funcanalysis}.
\eprfof
\begin{restatable}[Completeness and separability of $(\cF_{B}^{[\alpha, \beta]},
\normb{\cdot}_{2})$]{nlem}{completenesscfalphabeta}\label{nlem:completeness-of-class-F-alpha-beta}
The metric space $(\cF_{B}^{[\alpha, \beta]}, \normb{\cdot}_{2})$ is complete
and separable.
\end{restatable}
\bprfof{\Cref{nlem:boundedness-of-class-F-alpha-beta}} Firstly we note that
$(\cF_{B}^{[\alpha, \beta]}, \normb{\cdot}_{2})$ is an induced metric subspace of
$(L^{2}(B), \normb{\cdot}_{2})$ per
\Cref{nlem:subset-l2-space-class-F-alpha-beta}. Now separability of
$(\cF_{B}^{[\alpha, \beta]}, \normb{\cdot}_{2})$ follows, since it is inherited from
$(L^{2}(B), \normb{\cdot}_{2})$ by applying
\citet[Proposition~2.3.16]{shirali2006metricspaces}. We now show the
completeness of $(\cF_{B}^{[\alpha, \beta]}, \normb{\cdot}_{2})$. Take an arbitrary Cauchy
sequence $\theseqb{f_k}{k = 1}{\infty}$ in $\cF_{B}^{[\alpha, \beta]}$. Since $(L^{2}(B),
\normb{\cdot}_{2})$ is complete per
\Cref{nlem:completeness-separability-l2-space}, it follows that the $L_{2}$
limit of $\theseqb{f_k}{k = 1}{\infty}$ exists in $L^{2}(B)$. Let $f$ be that
limit, \ie, $\lim_{k \to \infty} f_{k} \defines f \in L^{2}(B)$. We will show
that $f \in \cF_{B}^{[\alpha, \beta]}$. First let us show that it is a density, \ie, it
integrates to 1. By Cauchy-Schwartz $\int_{B} |f_k(x) - f(x)| \mu(\dlett{x})
\leq \sqrt{\int_{B} \absb{f_k(x) - f(x)}^2 \mu(\dlett{x})} \rightarrow 0$ so
that $\int f(x) \mu(\dlett{x}) = 1$. Next consider the function $f^{\prime} = (f
\wedge \alpha) \vee \beta$. Since for any $x \in [\alpha ,\beta]$ and any $y$ we
have $|x - y| \geq |x - (y \wedge \alpha) \vee \beta|$ then that implies (since
$f_k \in \cF_{B}^{[\alpha, \beta]}$) that $\int_{B} |f_k(x) - f^{\prime}(x)|^2
\mu(\dlett{x}) \leq \int_{B} |f_k(x) - f(x)|^2 \mu(\dlett{x}) \rightarrow 0$ and
so $f^{\prime}$ must also be a limit of $f_{k}$. Since the limits are unique (up
to considering equivalence classes modulo sets of measure $0$ with respect to
$\mu$) then $f = f^{\prime}$ and hence it belongs to $\cF_{B}^{[\alpha, \beta]}$.
\eprfof
The following lemma shows that in general $\cF_{B}^{[\alpha, \beta]}$ is not totally
bounded. We consider a restricted case of $B \defined [0, 1]$, to construct a
suitable counterexample.
\begin{restatable}[Non total boundedness of $(\cFalphabetasupp{[0, 1]},
\normb{\cdot}_{2})$]{nlem}{nontotbddlebtwospace}\label{nlem:non-total-bdd-l2-zero-one}
Suppose $\beta \geq 2 - \alpha$. Then the metric space
$(\cFalphabetasupp{[0, 1]}, \normb{\cdot}_{2})$ is not totally
bounded, and hence not compact.
\end{restatable}
\bprfof{\Cref{nlem:non-total-bdd-l2-zero-one}} We note that per
\citet[Theorem~5.1.12]{shirali2006metricspaces} a metric space is totally
bounded if and only if every sequence contains a Cauchy subsequence. We will use
this characterization to construct a counterexample to demonstrate that
$(\cFalphabetasupp{[0, 1]}, \normb{\cdot}_{2})$ is not totally bounded. In
particular, we will define a sequence in $(\cFalphabetasupp{[0, 1]},
\normb{\cdot}_{2})$ which can't contain \emph{any} Cauchy subsequence.
Specifically we consider the sequence of functions $(1, \{x\mapsto \sin(2\pi j
x)\}_{j \in \NN})$. These functions are orthonormal in $L^2([0, 1])$.
Construct the sequence of functions $f_{j}(x) = 1 + (1-\alpha)\sin(2\pi j
x)$ for $j \in \NN$. By the orthogonality of $1$ and $\sin(2\pi j x)$ we
have that $\int_0^1 f_{j}(x) dx = 1$. Furthermore, $\alpha \leq f_{j}(x)
\leq 2 - \alpha$ for all $x \in [0,1]$, hence since $\beta \geq 2 -\alpha$
we have $f_{j}(x) \in \mathcal{F}^{[\alpha,\beta]}_{[0,1]}$. Take any two $j
\neq k \in \NN$, and consider
\begin{align*}
\|f_{j} - f_k\|_2^2 = (1-\alpha)^2\|\sin(2\pi j x) - \sin(2\pi k x)\|_2^2 = 2(1-\alpha)^2 > 0.
\end{align*}
This shows that there cannot be a Cauchy subsequence and hence the set is not
totally bounded.
\eprfof
\subsection{Elementary inequalities}\label{app:elementary-inequalities}
We will state and prove \Cref{nlem:elem-log-inequality}, which will provide the
key fact to will assist us in the proof of the lower bound in
\Cref{nlem:equiv-kl-euc-metric}.
\begin{restatable}[Elementary $\log$
inequality]{nlem}{elemlogineq}\label{nlem:elem-log-inequality} For each
$\gamma > 0$, and for any $x \in (0, \gamma]$, the following
relationship holds:
\begin{equation}\label{neqn:nlem:elem-log-inequality-01}
\log{x}
\leq
(x - 1) - h(\gamma) (x - 1)^{2}.
\end{equation}
Here $h : (0, \infty) \to \RR$ is defined as in
\eqref{neqn:nlem:equiv-kl-euc-metric-02}, and is positive over its entire
support.
\end{restatable}
\bprfof{\Cref{nlem:elem-log-inequality}} We first argue that $h(x) > 0$ for $x
\in (0,\infty)$. This is by the elementary inequality $\log(x + 1) \leq x$
for all $x \geq -1$. Next, it suffices to show that the map $x \mapsto h(x)$
is decreasing for $x > 0$ where $h$ is defined in
\eqref{neqn:nlem:equiv-kl-euc-metric-02}. This is because
\eqref{neqn:nlem:elem-log-inequality-01} holds for $x = 1$, and if $ x \neq
1$ it is equivalent to
\begin{align*}
h(\gamma)
\leq
\frac{(x-1) - \log x}{(x-1)^2},
\end{align*}
for $x \leq \gamma$. It is simple to verify that
\begin{align*}
h'(x) = \frac{-x^2 + 2x \log x + 1}{(x-1)^3 x}.
\end{align*}
We will show that the above function is negative on $(0,\infty)$ which will
complete the proof. First we will evaluate it at $x = 1$. By a triple
application of L'H\^opital's rule it is simple to verify that $\frac{d}{dx}h(x)
\vert_{x = 1} = -\frac{1}{3} < 0$. Thus, it remains to show that for $x\neq 1$,
\begin{align*}
(-x^2 + 2x \log x + 1)(x-1) < 0.
\end{align*}
Now let $f(x) \defined \frac{x^{2}-1}{2 x} - \log{x}$. We want to show that
$f(x) > 0$, for each $x > 1$ and $f(x) < 0$ for $x < 1$. First observe that
$f(1) = 0$. Moreover, we have that
\begin{equation}
f^{\prime}(x) = \frac{(x - 1)^{2}}{2 x^{2}} > 0.
\end{equation}
That is, $f(x)$ is \emph{strictly} increasing, which implies that $f(x) > 0$ for
each $x \in (1, \infty)$, and $f(x) < 0$ for each $x < 1$ as required.
\eprfof
\clearpage
\section{Main proofs}\label{appendix:main-proofs}
\subsection{Proof of
\texorpdfstring{\Cref{nlem:equiv-kl-euc-metric}}{\autoref{nlem:equiv-kl-euc-metric}}}\label{app:nlem:equiv-kl-euc-metric}
\equivkleucmetric*
\bprfof{\Cref{nlem:equiv-kl-euc-metric}} We will prove the upper and lower bound
in turn.\\
\newline
\textbf{(Upper bound in \eqref{neqn:nlem:equiv-kl-euc-metric-01}):} We seek to
show that $\kldiva{f}{g} \leq \frac{1}{\alpha} \normb{f - g}_{2}^{2}$. First,
for any two densities $f, g \in \cF$, we define the $\chi^{2}$-divergence
between $f$ and $g$, as follows:
\begin{equation}\label{neqn:chi-sq-divergence-density}
\chisqdiva{f}{g}
\defined \int_{B} \frac{(f - g)^{2}}{g} \dlett{\mu}.
\end{equation}
Per \Cref{nrmk:kldiv-density-well-defined}, we note that $\chi^{2}(f || g)$ in
\eqref{neqn:chi-sq-divergence-density} is similarly well-defined. We then have:
\begin{align}
\kldiva{f}{g}
& \leq \chisqdiva{f}{g}
\tag{per \citet[Theorem~5]{gibbs2002chooseboundprobmetrics}} \\
& \defined \int_{B} \frac{(f - g)^{2}}{g} \dlett{\mu}
\tag{using \eqref{neqn:chi-sq-divergence-density}} \\
& \leq \frac{1}{\alpha} \int_{B} (f - g)^{2} \dlett{\mu}
\tag{since $\inf_{x \in B} g(x) \geq \alpha > 0$} \\
& \defines \frac{1}{\alpha} \normb{f - g}_{2}^{2}.
\tag{by definition}
\end{align}
As required. \qedbsquare \\
\newline
\noindent\textbf{(Lower bound in \eqref{neqn:nlem:equiv-kl-euc-metric-01}):} We
seek to show that $\kldiva{f}{g} \geq c(\alpha,\beta) \normb{f - g}_{2}^{2}$. We
proceed as follows: First observe that for any $f, g \in \cF$, we have that $0 <
\frac{g}{f} \leq \frac{\beta}{\alpha} < \infty$
\begin{align}
\kldiva{f}{g}
& \defined
\int_{B} f \log\parens{\frac{f}{g}} \dlett{\mu}
\tag{per \eqref{neqn:kldiv-density-defn}} \\
& =
\int_{B} - f \log\parens{\frac{g}{f}} \dlett{\mu}
\tag{since $\inf_{x \in B} f(x) \geq \alpha > 0$} \\
& \geq
\int_{B} - f \parens{\parens{\frac{g}{f} - 1} - h(\beta / \alpha)\parens{\frac{g}{f} - 1}^{2}} \dlett{\mu}
\tag{using \Cref{nlem:elem-log-inequality}, with $C = \frac{\beta}{\alpha}$ and $x = \frac{g}{f}$} \\
& =
\int_{B} (f - g) \dlett{\mu} +
h(\beta / \alpha) \int_{B} \frac{(g - f)^{2}}{f} \dlett{\mu}
\nonumber \\
& \geq \frac{h(\beta / \alpha)}{\beta} \int_{B} (g - f)^{2} \dlett{\mu}
\tag{since $\int_{B} (f - g) \dlett{\mu} = 0$, and $0 < \sup_{x \in B} f(x) \leq \beta$} \\
& \defines \frac{h(\beta / \alpha)}{\beta} \normb{f - g}_{2}^{2}
\nonumber \\
& \defines c(\alpha, \beta) \normb{f - g}_{2}^{2},
\end{align}
where we define $c(\alpha, \beta) \defined \frac{h(\beta / \alpha)}{\beta} > 0$.
This proves the lower bound in \eqref{neqn:nlem:equiv-kl-euc-metric-01}, as
required. \qedbsquare
We now show the following equivalence between the Hellinger, \ie,
$d_{\mathsf{H}}$-metric, and the $L_{2}$ metric in $\cF_{B}^{[\alpha, \beta]}$.
\begin{equation}\label{neqn:equiv-hellinger-l2-metric-01}
(1 / 4 \beta) \normb{f-g}_{2}^{2}
\leq
\hellmet{f}{g}^{2}
\leq
(1 / \alpha) \normb{f - g}_{2}^{2}.
\end{equation}
To prove the upper bound in \eqref{neqn:equiv-hellinger-l2-metric-01}, we note
that
\begin{align}
\hellmet{f}{g}^{2}
& \leq
\kldiva{f}{g}
\tag{from \citet{gibbs2002chooseboundprobmetrics}} \\
& \leq
(1 / \alpha) \normb{f - g}_{2}^{2},
\tag{per \eqref{neqn:nlem:equiv-kl-euc-metric-01}}
\end{align}
as required. In order to prove the lower bound we observe that
\begin{align}
\normb{f - g}_{2}^{2}
& =
\int_{B} (f - g)^{2} \dlett{\mu}
\nonumber \\
& =
\int_{B} (\sqrt{f} + \sqrt{g})^{2} (\sqrt{f} - \sqrt{g})^{2} \dlett{\mu}
\nonumber \\
& \leq
4 \beta \int_{B} (\sqrt{f} - \sqrt{g})^{2} \dlett{\mu}
\tag{since $f, g \leq \beta$.} \\
& \defines
4 \beta \hellmet{f}{g}^{2},
\tag{by definition}
\end{align}
which implies the required lower bound in
\eqref{neqn:equiv-hellinger-l2-metric-01}. We have thus established the required
upper and lower bounds in both \eqref{neqn:nlem:equiv-kl-euc-metric-01} and
\eqref{neqn:equiv-hellinger-l2-metric-01}.
Finally, we note that $(\cF, \normb{\cdot}_{2})$ is metric space, since it is
the restriction of the metric space $(\cF_{B}^{[\alpha, \beta]}, \normb{\cdot}_{2})$. And so
the bounds \eqref{neqn:nlem:equiv-kl-euc-metric-01} and
\eqref{neqn:equiv-hellinger-l2-metric-01}, are also inherited by $\cF \subset
\cF_{B}^{[\alpha, \beta]}$.
\eprfof
\subsection{Proof of
\texorpdfstring{\Cref{nlem:minimax-lower-bound}}{\autoref{nlem:minimax-lower-bound}}}\label{app:nlem:minimax-lower-bound}
\minimaxlowerbound*
\bprfof{\Cref{nlem:minimax-lower-bound}} Let $c > 0$ be fixed, and $\theta \in
\cF$ be an arbitrary point. Consider maximal packing the set $\theset{f^{1},
\ldots, f^{m}} \subset \cF \cap B_{2}(\theta, \varepsilon)$ at a
$L_{2}$-``distance'' at least $\varepsilon/c$. Here $B_{2}(\theta,
\varepsilon)$ denotes a closed $L_{2}$-ball around the point $\theta$, with
radius $\varepsilon$. Suppose it has $m$ elements. Then we know that
\begin{equation*}
I(X;J)
\leq
\frac{1}{m}\sum_{j = 1}^{m} \kldiva{f^{j}}{\theta}
\leq
\max_{j \in [m]} \kldiva{f^{j}}{\theta}
\leq
\max_{j \in [m]} (1 / \alpha) \normb{f^{j} - \theta}_{2}^{2}
\leq
\varepsilon^{2} / \alpha.
\end{equation*}
Here the final two inequalities follow by applying
\eqref{neqn:nlem:equiv-kl-euc-metric-01}, and using the fact that
$\theset{f^{1}, \ldots, f^{m}} \subset \cF \cap B_{2}(\theta, \varepsilon)$,
respectively. Hence, if the packing number satisfies $\log m \geq 2 n
\varepsilon^{2} / \alpha + 2\log 2$ we will have a lower bound proportional to
$\varepsilon^2$ (it will be $\varepsilon^2/(8c^2)$). By taking the supremum over
$\theta$, we conclude that if $\log{\mlocc{\cF}{\varepsilon}{c}} > 2 n
\varepsilon^{2} / \alpha + 2\log 2$ we have a lower bound proportional to
$\varepsilon^2$.
\eprfof
\subsection{Proof of
\texorpdfstring{\Cref{nlem:log-likelihood-bernstein}}{\autoref{nlem:log-likelihood-bernstein}}}\label{app:nlem:log-likelihood-bernstein}
\loglikelihoodhoeffding*
\bprfof{\Cref{nlem:log-likelihood-bernstein}} We first observe per
\Cref{nrmk:log-likelihood-well-defined} that the log-likelihood, $\psi(g,
g^{\prime}, X)$, is well-defined. Next the mean of these variables, for each $i
\in [n]$, is
\begin{align}
\EE_f \brackets{\log \frac{g(X_{i})}{g^{\prime}(X_{i})}}
& = \EE_f \brackets{\log\parens{\frac{f(X_{i})}{g^{\prime}(X_{i})} \Big/ \frac{f(X_{i})}{g(X_{i})}}}
\tag{which is well-defined by \Cref{nrmk:log-likelihood-well-defined}.} \\
& = \EE_f \brackets{\log \frac{f(X_{i})}{g^{\prime}(X_{i})}} - \EE_f \brackets{\log \frac{f(X_{i})}{g(X_{i})}}
\nonumber \\
& = \kldiva{f}{g^{\prime}} - \kldiva{f}{g}.
\label{neqn:log-likelihood-bernstein-03}
\end{align}
Where the last line follows by definition using \eqref{neqn:kldiv-density-defn}.
We then have
\begin{align}
\PP( \psi(g,g^{\prime},X) > 0)
& = \PP\parens{\frac{1}{n}\sum_{i = 1}^{n} \log\frac{g(X_{i})}{g^{\prime}(X_{i})} > 0}
\tag{using \eqref{neqn:log-likelihood-g-g-prime}} \\
& = \PP\parens{\frac{1}{n}\sum_{i = 1}^{n} \log\frac{g(X_{i})}{g^{\prime}(X_{i})}
- \EE_f \log \frac{g(X_{1})}{g^{\prime}(X_{1})}
> \EE_f \log \frac{g^{\prime}(X_{1})}{g(X_{1})}}
\nonumber \\
& = \PP\parens{\frac{1}{n}\sum_{i} \log \frac{g(X_{i})}{g^{\prime}(X_{i})} - \EE_f \log \frac{g(X_{1})}{g^{\prime}(X_{1})} > \kldiva{f}{g} - \kldiva{f}{g^{\prime}}}
\tag{using \eqref{neqn:log-likelihood-bernstein-03}} \\
& \leq
\exp \parens{-\frac{n^2 t^{2}}{2\braces{\sum_{i=1}^{n} \EE\brackets{Y_{i}^{2}}+\frac{1}{3} n \kappa t}}}
\nonumber \\
& =
\exp \parens{-\frac{n^2 t^{2}}{2\braces{n \EE\brackets{Y_{1}^{2}}+\frac{1}{3} n \kappa t}}}
\tag{since $Y_{i}$ are \iid} \\
& =
\exp \parens{-\frac{n t^{2}}{2\braces{ \EE\brackets{Y_{1}^{2}}+\frac{1}{3} \kappa t}}}
\label{neqn:log-likelihood-bernstein-03b}
\end{align}
where $\kappa \defined \log \beta/\alpha$, $t \defined \kldiva{f}{g} -
\kldiva{f}{g^{\prime}}$, and $Y_{i} \defined \log
\frac{g(X_{i})}{g^{\prime}(X_{i})} - \EE_f \log
\frac{g(X_{1})}{g^{\prime}(X_{1})}$. This follows by the boundedness of
$\log g(X_{i})/g^{\prime}(X_{i})$, and then by applying Bernstein's
inequality, provided that $t > 0$. In order to check this final positivity
condition, we first note that there exists a $C > 0$ such that $\normb{g -
g^{\prime}}_{2} \geq C\delta$, and $\normb{g^{\prime}-f}_{2} \leq \delta$
both hold. We then have
\begin{equation}\label{neqn:log-likelihood-bernstein-04}
\normb{f - g}_{2}
\geq (C - 1) \delta.
\end{equation}
To see this we observe that by assumption, and the triangle inequality
respectively that $C \delta \leq \normb{g - g^{\prime}} \leq \normb{g - f} +
\normb{f - g^{\prime}}$. Then using $\normb{f - g^{\prime}} \leq \delta$ by
assumption and re-arranging, we obtain \eqref{neqn:log-likelihood-bernstein-04}
as required. As a result we obtain the following two inequalities
\begin{align}
\sqrt{\kldiva{f}{g}}
& \geq \sqrt{c(\alpha,\beta)} \|f-g\|_2
\geq
\sqrt{c(\alpha,\beta)} (C-1) \delta
\label{neqn:log-likelihood-bernstein-05} \\
\sqrt{\kldiva{f}{g^{\prime}}}
& \leq
\sqrt{1 / \alpha}\|f-g^{\prime}\|_2 \leq \sqrt{1 / \alpha}\delta
\label{neqn:log-likelihood-bernstein-06},
\end{align}
where $C > 0$ is defined to be a constant satisfying $c(\alpha,\beta) (C-1)^{2}
> 1 / \alpha$, \ie,
\begin{equation}\label{neqn:crit-cond-bernstein}
C > 1 + \sqrt{1 / (\alpha c(\alpha,\beta))}.
\end{equation}
Under the condition specified by \eqref{neqn:crit-cond-bernstein}, and by
squaring and subtracting \eqref{neqn:log-likelihood-bernstein-06} from
\eqref{neqn:log-likelihood-bernstein-05}, we obtain
\begin{equation}\label{neqn:log-likelihood-bernstein-07}
t
\defined
\kldiva{f}{g} - \kldiva{f}{g^{\prime}}
\geq
(c(\alpha,\beta)(C-1)^{2} - 1 / \alpha)\delta^2
> 0
\end{equation}
Now we show that $\EE_{f}\parens{Y_{1}^{2}} \lesssim \kldiva{f}{g} +
\kldiva{f}{g^{\prime}}$. To see this
\begin{align}
\EE_{f}\parens{Y_{1}^{2}}
& \leq
\EE_f \brackets{\parens{\log \frac{g(X_{1})}{g^{\prime}(X_{1})}}^{2}}
\nonumber \\
& =
\EE_f \brackets{\log\parens{\frac{f(X_{1})}{g^{\prime}(X_{1})} \Big/ \frac{f(X_{1})}{g(X_{1})}}}
\tag{which is well-defined by \Cref{nrmk:log-likelihood-well-defined}.} \\
& =
\EE_f \brackets{\parens{\log \frac{f(X_{1})}{g^{\prime}(X_{1})} -
\log \frac{f(X_{1})}{g(X_{1})}}^{2}}
\nonumber \\
& \leq
2\ubrace{\EE_f \brackets{\parens{\log \frac{f(X_{1})}{g(X_{1})}}^{2}}}{\defines A}{} +
2\ubrace{\EE_f \brackets{\parens{\log \frac{f(X_{1})}{g^{\prime}(X_{1})}}^{2}}}{\defines B}{}
\tag{using $(a - b)^{2} \leq 2(a^{2} + b^{2})$, for $a, b \geq 0$.}
\end{align}
We now bound the $A$ term above, with $B$ handled similarly. We observe that:
\begin{align}
A
& \defined
\EE_f \brackets{\parens{\log \frac{f(X_{1})}{g(X_{1})}}^{2}}
\tag{by definition} \\
& =
\int f \parens{\log \frac{f}{g}}^{2} \dlett{\mu}
\nonumber \\
& =
\int_{f \leq g} f \parens{\log \frac{g}{f}}^{2} \dlett{\mu} +
\int_{g < f} f \parens{\log \frac{f}{g}}^{2} \dlett{\mu}.
\label{neqn:bernstein-bdd-second-moment-01}
\end{align}
Now using $\log x \leq x - 1$, for each $x \in \RR_{>0}$, we have that
\begin{equation}\label{neqn:bernstein-bdd-second-moment-02}
\parens{\log \frac{g}{f}}^{2}
\leq
\parens{\frac{g - f}{f}}^{2}
\text{ and }
\parens{\log \frac{f}{g}}^{2}
\leq
\parens{\frac{f - g}{g}}^{2},
\end{equation}
which hold for $f \leq g$ (\ie, $\frac{g}{f} \geq 1$), and $g < f$ (\ie,
$\frac{f}{g} > 1$), respectively. Now we have:
\begin{align}
A
& \leq
\int_{f \leq g} \frac{(g - f)^{2}}{f} \dlett{\mu} +
\int_{g < f} \frac{(f - g)^{2} f}{g^{2}} \dlett{\mu}
\tag{using \eqref{neqn:bernstein-bdd-second-moment-01} and \eqref{neqn:bernstein-bdd-second-moment-02}.} \\
& \leq
(1 / \alpha) \int_{f \leq g} (g - f)^{2} \dlett{\mu} +
(\beta / \alpha^{2}) \int_{g < f} (f - g)^{2} \dlett{\mu}
\tag{since $0 < \alpha < f, g \leq \beta$} \\
& \leq
(\beta / \alpha^{2}) \normb{f - g}_{2}^{2}
\tag{since $\beta / \alpha^{2} \geq 1 / \alpha$.} \\
& \leq
K(\alpha, \beta) \kldiva{f}{g},
\label{neqn:bernstein-bdd-second-moment-03}
\end{align}
where $K(\alpha, \beta) \defined \beta / (\alpha^{2} c(\alpha, \beta))$, where
$c(\alpha, \beta)$ is as defined in \Cref{nlem:equiv-kl-euc-metric}. By a
similar argument, we also have that
\begin{equation}\label{neqn:bernstein-bdd-second-moment-04}
B
\leq
K(\alpha, \beta) \kldiva{f}{g^{\prime}}.
\end{equation}
Let $z \defined \kldiva{f}{g} + \kldiva{f}{g^{\prime}}$. Then using
\eqref{neqn:bernstein-bdd-second-moment-03} and
\eqref{neqn:bernstein-bdd-second-moment-04}, we obtain
\begin{equation}\label{neqn:bernstein-bdd-second-moment-05}
\EE_{f}\parens{Y_{1}^{2}}
\leq 2 K(\alpha, \beta) [\kldiva{f}{g} + \kldiva{f}{g^{\prime}}]
\defines 2 z K(\alpha, \beta)
\end{equation}
Now we use the basic inequality $a + b \leq \parens{\sqrt{a} + \sqrt{b}}^{2}
\leq 2 (a + b)$, to obtain
\begin{equation}
z
\leq
\parens{\sqrt{\kldiva{f}{g}} + \sqrt{\kldiva{f}{g^{\prime}}}}^{2}
\leq
2 z.
\end{equation}
Now, $t^{2} \defined \parens{\kldiva{f}{g} - \kldiva{f}{g^{\prime}}}^{2} =
\parens{\sqrt{\kldiva{f}{g}} - \sqrt{\kldiva{f}{g^{\prime}}}}^{2}
\parens{\sqrt{\kldiva{f}{g}} + \sqrt{\kldiva{f}{g^{\prime}}}}^{2}$, we have:
\begin{equation}
\parens{\sqrt{\kldiva{f}{g}} - \sqrt{\kldiva{f}{g^{\prime}}}}^{2}
z
\leq
t^{2}
\leq
2 \parens{\sqrt{\kldiva{f}{g}} - \sqrt{\kldiva{f}{g^{\prime}}}}^{2}
z.
\end{equation}
We then conclude using
\eqref{neqn:log-likelihood-bernstein-03b},\eqref{neqn:log-likelihood-bernstein-07},
that
\begin{align}
\PP( \psi(g,g^{\prime},X) > 0)
& \leq
\exp \parens{-\frac{n t^{2}}{2\braces{ \EE\brackets{Y_{1}^{2}}+\frac{1}{3} \kappa t}}}
\tag{per \eqref{neqn:log-likelihood-bernstein-03b}} \\
& \leq
\exp \parens{-\frac{n \parens{\sqrt{\kldiva{f}{g}} - \sqrt{\kldiva{f}{g^{\prime}}}}^{2}
z}{2\braces{ 2 z K(\alpha, \beta) +\frac{1}{3} \kappa z}}}
\tag{since $t \leq z$ and \eqref{neqn:bernstein-bdd-second-moment-05}} \\
& =
\exp \parens{-\frac{n \parens{\sqrt{\kldiva{f}{g}} - \sqrt{\kldiva{f}{g^{\prime}}}}^{2}
}{2\braces{ 2 K(\alpha, \beta) + \frac{1}{3} \kappa}}}
\nonumber \\
& \leq \exp \parens{-\frac{n \parens{\sqrt{c(\alpha,\beta)} (C-1) - \sqrt{1 / \alpha}}^{2}\delta^2
}{2\braces{ 2 K(\alpha, \beta) +\frac{1}{3} \kappa}}}
\tag{by subtracting
\eqref{neqn:log-likelihood-bernstein-06} from
\eqref{neqn:log-likelihood-bernstein-05}} \\
& \defines \exp\parens{-n L(\alpha, \beta, C) \delta^{2}},
\nonumber
\end{align}
whenever condition \eqref{neqn:crit-cond-bernstein} holds, and $L(\alpha, \beta,
C) \defined \frac{ \parens{\sqrt{c(\alpha,\beta)} (C-1) - \sqrt{1 /
\alpha}}^{2} }{2\braces{ 2 K(\alpha, \beta) +\frac{1}{3} \log \beta /
\alpha}}$. Now, taking the supremum over all $g, g^{\prime} \colon \normb{g
- g^{\prime}}_{2} \geq C\delta, \normb{g^{\prime}-f}_{2} \leq \delta$, the
required result follows.
\eprfof
\subsection{Proof of
\texorpdfstring{\Cref{nlem:critical-log-likelihood-concentration}}{\autoref{nlem:critical-log-likelihood-concentration}}}\label{app:nlem:critical-log-likelihood-concentration}
Recall that \Cref{nlem:critical-log-likelihood-concentration} is concerning
a packing set. Suppose we have a maximal packing set of $\cF^{\prime}
\subset \cF$, \ie, $\theset{g_1, \ldots, g_m} \subset \cF^{\prime} \subset
\cF$ such that $\normb{g_i- g_{j}}_{2} > \delta$ for all $i \neq j$, and it
is known that $f \in \cF^{\prime}$. We then obtain a key concentration
result as per \Cref{nlem:critical-log-likelihood-concentration}.
\criticalloglikelihoodconcentration*
\bprfof{\Cref{nlem:critical-log-likelihood-concentration}} We first define the
intermediate \emph{thresholding} random variables
\begin{align*}
T_{k} \defined
\begin{cases}
\max_{j \in [m]} \normb{g_{j}- g_{k}}_{2}
& , \mbox{s.t. }
\sum_{i = 1}^{n} \log g_{j}(X_{i}) \geq \sum_{i = 1}^{n} \log g_{k}(X_{i}), \normb{g_{j}- g_{k}}_{2} > C\delta \\
0 & , \mbox{otherwise},
\end{cases}
\end{align*}
for each $k \in [m]$. Without loss of generality suppose that $\normb{g_{k} -
f}_{2} \leq \delta$. Next
\begin{align*}
\PP(\|g_{j^*}-f\|_2 > (C + 1)\delta)
& \leq
\mathbb{P}(j^* \in \{j: \normb{g_{j} - g_{k}}_{2} > C\delta\})
\\
& \leq
\PP(T_{k} > 0).
\end{align*}
On the other hand
\begin{align*}
\PP(T_{k} > 0)
& =
\PP\parens{\exists j \in [m] \colon
\sum_{i = 1}^{n} \log g_{j}(X_{i})
\geq
\sum_{i = 1}^{n} \log g_{k}(X_{i}),
\normb{g_{j}- g_{k}}_{2}
> C \delta} \\
& =
\PP\parens{\bigcup_{j = 1}^{m}
\braces{\sum_{i = 1}^{n} \log g_{j}(X_{i})
\geq
\sum_{i = 1}^{n} \log g_{k}(X_{i}),
\normb{g_{j}- g_{k}}_{2}
> C \delta}} \\
& \leq
m \exp\parens{-n L(\alpha, \beta, C) \delta^{2}},
\tag{using union bound and \Cref{nlem:log-likelihood-bernstein}.}
\end{align*}
where $C$ is assumed to satisfy \eqref{neqn:log-likelihood-bernstein-02a}, and
$L(\alpha, \beta, C)$ is defined as per
\eqref{neqn:log-likelihood-bernstein-02b}.
\eprfof
\subsection{Proof of
\texorpdfstring{\Cref{nlem:local-metric-ent-monotone}}{\autoref{nlem:local-metric-ent-monotone}}}\label{app:nlem:local-metric-ent-monotone}
\localmetricentmonotone*
\bprfof{\Cref{nlem:local-metric-ent-monotone}} It suffices to show that if $g_1,
\ldots, g_m \in \cF \cap B_{2}(\theta, \varepsilon)$ is a maximal packing
set at a distance $\varepsilon/c$, then we can pack $ B_{2}(\theta,
\varepsilon') \cap \cF$ at a distance $\varepsilon'/c$ with at least $m$
points where $\varepsilon' < \varepsilon$. Consider the points $\theta (1-
\varepsilon'/\varepsilon) + \varepsilon'/\varepsilon g_{j}$. These points
clearly are densities since $\theta, g_j \in \cF$. We will show that these
points are an $\varepsilon'/c$ packing of $B_{2}(\theta, \varepsilon') \cap
\cF$. First let us convince ourselves that the points belong to the set. We
have
\begin{align*}
\|\theta (1- \varepsilon'/\varepsilon) + \varepsilon'/\varepsilon g_{j} - \theta\|_2 = \varepsilon'/\varepsilon \|g_{j}- \theta\|_2 \leq\varepsilon',
\end{align*}
and using the fact that $\cF$ is convex (by assumption) grants the conclusion.
Next
\begin{align*}
\|\theta (1-\varepsilon'/\varepsilon) + \varepsilon'/\varepsilon g_{j}- \theta (1-
\varepsilon'/\varepsilon) - \varepsilon'/\varepsilon g_k\|_2 =
\varepsilon'/\varepsilon \|g_{j}- g_k\|_2 > \varepsilon'/c,
\end{align*}
which completes the proof.
\eprfof
\subsection{Proof of
\texorpdfstring{\Cref{nprop:nu-star-estimator-measurability}}{\autoref{nprop:nu-star-estimator-measurability}}}\label{app:nprop:nu-star-estimator-measurability}
\estimatormeasurability*
\bprfof{\Cref{nprop:nu-star-estimator-measurability}} Recall our multistage
sieve estimator $\nu^{*}(X) \defined \Upsilon_{\overline{J}}$, where
$X \defined (X_{1}, \ldots, X_{n})^{\top}$ is a fixed data sample.
Here $\Upsilon_{\overline{J}}$ denotes the last term of the \emph{finite}
sequence $\Upsilon \defined \theseqb{\Upsilon_{k}}{k = 1}{\overline{J}}$ as
described in \Cref{sec:upper-bound}.
In order to show the measurability of $\nu^{*}(X)$ we need to
formalize our setting. We note that our estimator $\nu^{*} \colon B^{n} \to
\cF$, is more precisely a map from the measurable space $(B^{n}, \sigma(B^{n}))$
to the measurable space $(\cF, \sigma(\cF))$. Here $\sigma(B^{n})$ and
$\sigma(\cF)$ denote the Borel $\sigma$-field with respect to the Euclidean and
$L_{2}$-metric topologies on $B^{n}$ and $\cF$, respectively.
Our proof strategy will be to proceed by induction on $k \in [\overline{J}]$
over the sequence $\Upsilon$. We will show that each
$k^{\textnormal{th}}$-indexed map in $\Upsilon$, \ie, $\Upsilon_{k}$, is Borel
measureable, which in turn will imply the measureability of the
$\nu^{*}(X)$. Following our (maximal) packing set construction as
described in \Cref{sec:upper-bound} and
\Cref{fig:packing-set-tree-construction}, we need to consider the case where the
traversal down the tree is not necessarily unique at each level, \ie, there may
be collisions (ties) in the packing set children nodes, where the likelihood is
equal. We do always ensure a unique path down the maximal packing set tree, by
selecting the smallest alphanumerically indexed children node at each level.
However, our measurability proof must account for this selection rule
explicitly.
In order to proceed by induction, we consider the base case for $k = 1$, \ie,
$\Upsilon_{1} \in \cF$. Importantly, we note that $\Upsilon_{1}$ is chosen
arbitrarily from $\cF$ independently of the data samples, $X$. Let $A
\in \sigma(\cF)$ be any Borel set. Since all samples $X \in B^{n}$
are mapped to $\Upsilon_{1}$ in our setting, then $\Upsilon_{1}^{-1}[A] = B^{n}$
if $\Upsilon_{1} \in A \in \sigma(\cF)$, or $\Upsilon_{1}^{-1}[A] =
\varnothing$, otherwise. In either case we have $\varnothing, B^{n} \in
\sigma(B^{n})$, which shows that $\Upsilon_{1}$ is Borel measurable. Now
consider the event $\theset{\Upsilon_{2} = m_{s}} \defined \thesetb{(X_{1},
\ldots, X_{n})^{\top} \in B^{n}}{\Upsilon_{2}(X_{1}, \ldots, X_{n}) = m_{s}}
\subset B^{n}$, for some index $s \in \NN$. Then we have
\begin{align}
\theset{\Upsilon_{2} = m_{s}}
& \defined
\thesetb{(X_{1}, \ldots, X_{n})^{\top} \in B^{n}}{\Upsilon_{2}(X_{1}, \ldots, X_{n}) = m_{s}}
\label{neqn:upsilon2-measurability-01}
\\
& =
\bigcap_{g \in P_{\Upsilon_{1}}}
\thesetb{(X_{1}, \ldots, X_{n})^{\top} \in B^{n}}{\sum_{i = 1}^{n}
\log (m_{s}(X_{i})) \geq \sum_{i = 1}^{n} \log (g(X_{i}))} \; \bigcap
\nonumber \\
& \bigcap_{j = 1}^{s - 1}
\thesetb{(X_{1}, \ldots, X_{n})^{\top} \in B^{n}}{\sum_{i = 1}^{n}
\log (m_{s}(X_{i})) > \sum_{i = 1}^{n} \log (m_{j}(X_{i}))}.
\label{neqn:upsilon2-measurability-02}
\end{align}
In \eqref{neqn:upsilon2-measurability-02}, we observe that $\theset{\Upsilon_{2}
= m_{s}} \subset B^{n}$ is represented as the intersection of 2 separate
(finite) set intersections. Note that the second intersection set
\emph{explicitly} accounts for our alphanumerical index selection rule
in the children densities of $P_{\Upsilon_{1}}$. Consider the first
finite intersection term. Here, each $g \in P_{\Upsilon_{1}} \subset
\cF$ are Borel measurable by \eqref{neqn:density-class-F-alpha-beta}. We
note that the $\log$ and the addition (\ie, ``$+_{\RR}$'') functions are
both continuous and measurable, and therefore, so is their composition.
Thus the resulting \emph{finite} sum, $\sum_{i = 1}^{n} \log f(X_{i})$,
is a measurable function, for any density $f \in \cF$ (which is always
measurable). As such the $\Upsilon_{2}$ is measurable since all these
inequalities give rise to measurable sets and when one intersects them
(they are finitely many) one obtains another measurable set. Once again,
let $A \in \sigma(\cF)$ be any Borel set. Then such an $A$ contains
either no such densities $m_{s}$, or at most finitely many (since the
number of children of our maximal packing set tree is always finite). If
no such $m_{s} \in A$, then $\Upsilon_{2}^{-1}[A] = \varnothing \in
\sigma(B^{n})$. Thus $\Upsilon_{2}$ is indeed Borel measurable in this
case. In the case where there exist finitely many such $m_{s} \in A$, it
follows that
\begin{equation}\label{neqn:upsilon2-measurability-03}
\Upsilon_{2}^{-1}[A]
=
\bigcup_{\thesetb{s}{m_{s} \in A}} \theset{\Upsilon_{2} = m_{s}}
\defines
\bigcup_{\thesetb{s}{m_{s} \in A}} \thesetb{(X_{1}, \ldots, X_{n})^{\top} \in B^{n}}{\Upsilon_{2}(X_{1}, \ldots, X_{n}) = m_{s}}
\end{equation}
In \eqref{neqn:upsilon2-measurability-03} we note that $\Upsilon_{2}^{-1}[A]$
represents a finite union of Borel measurable sets as per
\eqref{neqn:upsilon2-measurability-02}, which is again Borel measurable. That
is, we have shown that $\Upsilon_{2}^{-1}[A] \in \sigma(B^{n})$, which indeed
implies the Borel measurability of $\Upsilon_{2}$, as required.
Similarly, consider the event $\theset{\Upsilon_{3} \defines m_{s, t}} \subset
B^{n}$, for some $t \in \NN$ and $s \in \NN$ taken as per
\eqref{neqn:upsilon2-measurability-01}. Here, the indexed density $m_{s, t}$
signifies that $\Upsilon_{3}$ is derived from the children of the packing
set of $\Upsilon_{2} \defines m_{s}$, as denoted by $P_{m_{s}}$ in our work.
Once again we can write this $\Upsilon_{3}$ as
\begin{align}
\theset{\Upsilon_{3} = m_{s, t}}
& \defined
\thesetb{(X_{1}, \ldots, X_{n})^{\top} \in B^{n}}{\Upsilon_{3}(X_{1}, \ldots, X_{n}) = m_{s, t}}
\nonumber \\
& =
\bigcap_{g \in P_{m_{s}}}
\thesetb{(X_{1}, \ldots, X_{n})^{\top} \in B^{n}}{\sum_{i = 1}^{n}
\log (m_{s, t}(X_{i})) \geq \sum_{i = 1}^{n} \log (g(X_{i}))} \; \bigcap
\nonumber \\
& \bigcap_{j = 1}^{t - 1}
\thesetb{(X_{1}, \ldots, X_{n})^{\top} \in B^{n}}{\sum_{i = 1}^{n}
\log (m_{s, t}(X_{i})) > \sum_{i = 1}^{n} \log (m_{s, j}(X_{i}))}
\; \bigcap \nonumber \\
& \bigcap \; \theset{\Upsilon_{2} = m_{s}}.
\end{align}
By a similar argument to the measurability of $\Upsilon_{2}$ it follows that
$\Upsilon_{3}$ is also measurable. As such, given the recursive construction of
the finite sequence $\Upsilon \defined \theseqb{\Upsilon_{k}}{k =
1}{\overline{J}}$ via our maximal packing set tree traversal, this pattern
inductively repeats for each $k \in \theset{4, \ldots, \overline{J}}$. Since
$\nu^{*}(X) \defined \Upsilon_{\overline{J}}$, this implies the
measurability of $\nu^{*}(X)$, as required.
\eprfof
\subsection{Proof of
\texorpdfstring{\Cref{nthm:upper-bound-rate-finite-iterations}}{\autoref{nthm:upper-bound-rate-finite-iterations}}}\label{app:nthm:upper-bound-rate}
We begin with a useful result, which will enable us to construct upper bounds
for estimator $\nu^{*}(X)$.
\begin{restatable}{nlem}{upsiloniscauchy}\label{nlem:upsilon-is-cauchy-sequence}
The finite sequence $\Upsilon \defined \theseqb{\Upsilon_{k}}{k =
1}{\overline{J}}$, as defined in the construction of our estimator
$\nu^{*}(X)$, satisfies
\begin{equation}
\normb{\Upsilon_{J} - \Upsilon_{J^{\prime}}}_{2}
\leq
\frac{d}{2^{J^{\prime} - 2}},
\end{equation}
for each pair of positive integers $J^{\prime} < J$.
\end{restatable}
\bprfof{\Cref{nlem:upsilon-is-cauchy-sequence}} Let $\Upsilon_{J^{\prime}},
\Upsilon_{J} \in \Upsilon$, for any positive integers $J > J^{\prime} \geq
1$. We then have
\begin{equation}\label{neqn:upsilon-is-cauchy-sequence-01}
\normb{\Upsilon_{J} - \Upsilon_{J^{\prime}}}_{2}
\leq
\sum_{i = J^{\prime}}^{J - 1}
\normb{\Upsilon_{i + 1} - \Upsilon_{i}}_{2}
\leq
\sum_{i = J^{\prime}}^{J - 1} \frac{d}{2^{i-1}}
\leq
\frac{d}{2^{J^{\prime} - 2}}.
\end{equation}
As required.
\eprfof
\begin{restatable}[Telescoping sum of conditional
probabilities]{nlem}{telescopingsumscondprobs}\label{nlem:telescoping-sum-cond-prob}
Let $n \geq 2$ be a fixed integer, and $\theset{A_{1}, A_{2}, \ldots,
A_{n}}$ denote events on a common probability space, with
$\PP(\stcomp{A}_{j}) > 0$ for each $j \geq 1$. We then have
\begin{equation}\label{neqn:telescoping-sum-cond-prob-01}
\PP(A_{n})
\leq
\sum_{j = n}^{2}\PP(A_{j} \mid \stcomp{A}_{j -1}) + \PP(A_{1}).
\end{equation}
\end{restatable}
\bprfof{\Cref{nlem:telescoping-sum-cond-prob}} We will prove this by induction
on $n \geq 2$. We check the induction base case for $n = 2$. We first observe
that
\begin{equation}\label{neqn:telescoping-sum-cond-prob-02}
A_{2}
\subseteq
A_{1} \cup A_{2}
=
(A_{2} \cap \stcomp{A}_{1})
\sqcup
A_{1},
\end{equation}
where the latter set is a \emph{disjoint} union. It then follows that
\begin{align}
\PP(A_{2})
& \leq
\PP\parens{A_{2} \cap \stcomp{A}_{1}} +
\PP(A_{1})
\tag{by monotonicity of $\PP$ applied to \eqref{neqn:telescoping-sum-cond-prob-02}} \\
& \leq
\frac{\PP\parens{A_{2} \cap \stcomp{A}_{1}}}{\PP(\stcomp{A}_{1})} +
\PP(A_{1})
\tag{since $\PP(\stcomp{A}_{1}) \in (0, 1]$, by assumption} \\
& \defines
\PP(A_{2} \mid \stcomp{A}_{1}) + \PP(A_{1}),
\nonumber
\end{align}
which proves the base case for $n = 2$. Now, by induction assume the result is
true for each integer $n = k > 2$. We then have for $n = k + 1$ that:
\begin{align}
\PP(A_{k + 1})
& \leq
\PP(A_{k + 1} \mid \stcomp{A}_{k}) + \PP(A_{k})
\tag{using induction base case} \\
& \leq
\PP(A_{k + 1} \mid \stcomp{A}_{k}) +
\sum_{j = k}^{2}\PP(A_{j} \mid \stcomp{A}_{j -1}) + \PP(A_{1})
\tag{using induction hypothesis} \\
& =
\sum_{j = k + 1}^{2}\PP(A_{j} \mid \stcomp{A}_{j -1}) + \PP(A_{1}),
\end{align}
as required. So the result is true for $n = k + 1$, and thus by induction holds
for each integer $n \geq 2$.
\eprfof
\upperboundratefiniteiters*
\bprfof{\Cref{nthm:upper-bound-rate-finite-iterations}} Combining the results of
Lemma \ref{nlem:critical-log-likelihood-concentration} (with $c \defined 2(C+1)$
where $c$ is the constant from the definition of local packing entropy) and
Lemma \ref{nlem:local-metric-ent-monotone} we conclude that for each $j \in
\theset{2, \ldots, J}$ we have
\begin{align}
& \mathbb{P}\parens{\normb{f - \Upsilon_{j}}_{2}
>
\frac{d}{2^{j-1}}
\,\middle|\,
\normb{f - \Upsilon_{j-1}}_{2}
\leq
\frac{d}{2^{j-2}},
\Upsilon_{j - 1}}
\nonumber
\\
& \leq
\absb{P_{\Upsilon_{j-1}}}
\exp\parens{-\frac{n L(\alpha, \beta, C) d^2}{2^{2(j-1)}(C+1)^2}}
\label{neqn:telescoping-bound-01} \\
& \leq
\mlocc{\cF}{\frac{d}{2^{J - 2}}}{c}
\exp\parens{-\frac{n L(\alpha, \beta, C) d^2}{2^{2(j-1)}(C+1)^2}}
\label{neqn:telescoping-bound-02}
\end{align}
where $P_{\Upsilon_{j}}$ are the maximal packing sets described in the
construction of $\nu^{*}(X)$. Crucially, we observe that the $\rhs$
of \eqref{neqn:telescoping-bound-02} does not depend on the conditioned random
variables, \ie, $\Upsilon_{j - 1}$, for each $j \in \theset{2, \ldots, J}$ hence
we can drop $\Upsilon_{j - 1}$ from the conditioning. Now let denote $A_{j}
\defined \theset{\normb{f - \Upsilon_{j}}_{2} > \frac{d}{2^{j-1}}}$, for each
integer $j \geq 1$. Then we can proceed by working with the unconditional events
$A_{j}$ in \eqref{neqn:telescoping-bound-02}.
Moreover, we then have that $\stcomp{A}_{j - 1} \defined \theset{\normb{f -
\Upsilon_{j - 1}}_{2}
\leq
\frac{d}{2^{j-2}}}$ for each integer $j \geq 2$. In particular
$\PP(\stcomp{A}_{1}) = \theset{\normb{f - \Upsilon_{1}}_{2} \leq d} = 1$, since
$f, \Upsilon_{1} \in \cF$, so indeed $\normb{f - \Upsilon_{1}}_{2} \leq
\operatorname{diam}_{2}{(\cF)} \defines d$ almost surely. By aligning our notation directly
with \Cref{nlem:telescoping-sum-cond-prob}, we can apply the telescoping bound
to $\PP(A_{j})$ as follows
\begin{align}
\PP(A_{J})
& \defined
\mathbb{P}\parens{\normb{f - \Upsilon_{J}}_{2}
>
\frac{d}{2^{J-1}}}
\tag{by definition} \\
& \leq
\mlocc{\cF}{\frac{d}{2^{J - 2}}}{c}
\sum_{j = 1}^{J-1}\exp\parens{-\frac{n L(\alpha, \beta, C) d^2}{2^{2j}(C+1)^2}}
\tag{per \eqref{neqn:telescoping-bound-02}} \\
& \leq
\mlocc{\cF}{\frac{d}{2^{J - 2}}}{c}
a (1 + a^{4-1} + a^{16-1} + \ldots)\mathbbm{1}(J > 1)
\label{neqn:telescoping-bound-03} \\
& \leq
\mlocc{\cF}{\frac{d}{2^{J - 2}}}{c}
a (1 + a + a^{2} + \ldots)\mathbbm{1}(J > 1)
\nonumber \\
& \leq
\mlocc{\cF}{\frac{d}{2^{J - 2}}}{c}
\frac{a}{1-a} \mathbbm{1}(J > 1),
\label{neqn:telescoping-bound-04}
\end{align}
where for brevity in \eqref{neqn:telescoping-bound-03} we denote
\begin{align*}
a
\defined
\exp\parens{-\frac{n L(\alpha, \beta, C) d^2}{2^{2(J-1)}(C+1)^2}}.
\end{align*}
Since $C$ is assumed to satisfy \eqref{neqn:log-likelihood-bernstein-02a}, and
$L(\alpha, \beta, C)$ is defined as per
\eqref{neqn:log-likelihood-bernstein-02b}, it follows that $a < 1$. Note here
that the above bound \eqref{neqn:telescoping-bound-04} holds, provided that
$\PP(\stcomp{A}_{j}) > 0$ for $j < J$ as required by
\Cref{nlem:telescoping-sum-cond-prob}. Suppose that the $\rhs$ of
\eqref{neqn:telescoping-bound-04} is strictly smaller than $1$. In that case for
all $j$, $\PP(\stcomp{A}_{j}) > 0$ since bound \eqref{neqn:telescoping-bound-04}
holds inductively for all $\PP(A_j)$ for $j \leq J$. On the other hand, if the
$\rhs$ of \eqref{neqn:telescoping-bound-04} is $\geq 1$ then
\eqref{neqn:telescoping-bound-04} trivially holds. In both cases we conclude
that \eqref{neqn:telescoping-bound-04} holds.
If one sets $\varepsilon_J \defined \frac{\sqrt{L(\alpha, \beta, C)}
d}{2^{(J-1)}(C+1)}$, we have that if
\begin{align*}
n \varepsilon_J^2 > 2 \log
\mlocc{\cF}{\varepsilon_J \frac{2 (C+1)}{\sqrt{L(\alpha, \beta, C)}}}{c} = 2
\log \mlocc{\cF}{\frac{d}{2^{J - 2}}}{c},
\end{align*} and $a \defined \exp(-n \varepsilon_J^2) <
1/2 \iff n \varepsilon_J^2 > \log 2$, the above probability in
\eqref{neqn:telescoping-bound-04} will be bounded from above by $2 \exp(-n
\varepsilon_J^2 / 2)$. This condition is implied when
\begin{align}\label{suff:condition:epsJ}
n \varepsilon_J^2
>
2 \log \mlocc{\cF}{\varepsilon_J
\frac{2 (C+1)}{\sqrt{L(\alpha, \beta, C)}}}{c}
\vee \log 2.
\end{align}
We now have
\begin{align}\label{mu:upislon:ineq}
\normb{\nu^*_{\overline{J}} - f}_2
\leq
\normb{\Upsilon_{\overline{J}} - \Upsilon_J}_2 +
\normb{\Upsilon_J - f}_2
\leq
3 \varepsilon_J \frac{C+1}{\sqrt{L(\alpha, \beta, C)}},
\end{align} with probability at least $1 - 2\exp(-n\varepsilon_J^2 / 2)$ which holds for all $J$ satisfying \eqref{suff:condition:epsJ} (including $\overline{J}$). Here we want to clarify that the last inequality in \eqref{mu:upislon:ineq} follows from the fact that $\|\Upsilon_{\overline{J}} - \Upsilon_J\|_2 \leq d/2^{J-2}$, as seen when we verified that $\Upsilon$ forms a Cauchy sequence in \Cref{nlem:upsilon-is-cauchy-sequence} (and since $\overline{J} \geq J$).
Let $J^*$ be selected as the maximum integer $J$ such that
\eqref{suff:condition:epsJ} holds, or otherwise if such $J$ does not exist $J^*
= 1$, i.e. $J^* \equiv \overline{J}$. Let $\eta = 3\frac{C+1}{\sqrt{L(\alpha,
\beta, C)}}$, $\underline{C} = 2$ and $C' = 1 / 2$. We have established that the
following bound holds
\begin{equation*}
\mathbb{P}(\normb{f - \nu^*_{\overline{J}}}_{2}
>
\eta \varepsilon_J)
\leq
\underline C \exp(-C'n\varepsilon_J^2) \mathbbm{1}(J > 1)
\leq
\underline C \exp(-C'n\varepsilon_J^2) \mathbbm{1}(J^* > 1),
\end{equation*}
for all $1 \leq J \leq J^*$, where this bound also holds in the case when $J^* =
1$ by exception. Observe that we can extend this bound to all $J \in
\mathbb{Z}$ and $J \leq J^*$, since for $J < 1$ we have $\eta \varepsilon_J
\geq 6 d$ and so
\begin{equation*}
\mathbb{P}(\normb{f - \nu^*_{\overline{J}}}_{2}
>
\eta \varepsilon_J)
\leq
0
\leq
\underline C \exp(-C'n\varepsilon_J^2) \mathbbm{1}(J^* > 1).
\end{equation*}
We conclude that
\begin{equation*}
\mathbb{P}(\normb{f - \nu^*_{\overline{J}}}_{2}
>
\eta \varepsilon_J)
\leq
0
\leq
\underline C \exp(-C'n\varepsilon_J^2) \mathbbm{1}(J^* > 1),
\end{equation*}
for any $J \leq J^*$. Now for any $\varepsilon_{J-1} > x \geq \varepsilon_{J}$
for $J \leq J^*$ we have that
\begin{align*}
\mathbb{P}(\normb{f - \nu^*_{\overline{J}}}_{2}
>
2\eta x)
& \leq
\mathbb{P}(\normb{f - \nu^*_{\overline{J}}}_{2}
>
\eta \varepsilon_{J - 1}) \\
& \leq
\underline C \exp(-C'n\varepsilon_{J-1}^2) \mathbbm{1}(J^* > 1) \\
& \leq
\underline C \exp(-C'nx^2)\mathbbm{1}(J^* > 1),
\end{align*}
where the last inequality follows due to the fact that the map $x \mapsto
\underline C \exp(-C'nx^2)$ is monotonically decreasing for positive reals.
We will now integrate the tail bound:
\begin{equation}\label{important:prob:bound}
\mathbb{P}(\normb{f - \nu^*_{\overline{J}}}_{2}
>
2 \eta x)
\leq
\underline C \exp(-C'nx^2) \mathbbm{1}(J^* > 1),
\end{equation}
which holds true for $x \geq \varepsilon^*$, where $\varepsilon_J =
\frac{\sqrt{L(\alpha, \beta, C)} d}{2^{(J-1)}(C+1)}$, always (since even if
$J^* = 1$ by exception, this bound is still valid). We then have
\begin{align*}
\EE \normb{f - \nu^*_{\overline{J}}}_{2}^{2}
& =
\int_{0}^{\infty} 2 x
\mathbb{P}(\normb{f - \nu^*_{\overline{J}}}_{2} > x) \dlett{x} \\
& \leq
C''' \varepsilon^{*2} + \int_{2\eta\varepsilon^*}^{\infty} 2 x \underline C \exp(-C''nx^2) \mathbbm{1}(J^* > 1) \dlett{x} \\
& =
C''' \varepsilon^{*2} + C^{''''}n^{-1}\exp(-C'''''n\varepsilon^{*2})\mathbbm{1}(J^* > 1).
\end{align*}
Now $n\varepsilon^{*2}$ is bigger than a constant (\ie, $\log 2$) otherwise $J^*
= 1$. Hence, the above is smaller than $\bar C \varepsilon^{*2}$ for some
absolute constant $\bar C$.
\eprfof
\subsection{Proof of
\texorpdfstring{\Cref{nthm:sharp-minimax-rate}}{\autoref{nthm:sharp-minimax-rate}}}\label{app:nthm:sharp-minimax-rate}
\sharpminimaxrate*
\bprfof{\Cref{nthm:sharp-minimax-rate}} First suppose that $\varepsilon^*$
satisfies $n\varepsilon^{*2} > 4 \log 2$. Then for $\delta^* :=
\varepsilon^*/\sqrt{4(1 / \alpha \vee 1)}$ we have $\log
\mlocc{\cF}{\delta^*}{c}
\geq
\log \mlocc{\cF}{\varepsilon^*}{c}
\geq
n\varepsilon^{*2}/2 + n \varepsilon^{*2}/2 > 2n \delta^{*2} / \alpha + 2
\log 2$ and so this implies the sufficient condition for the lower bound per
\Cref{nlem:minimax-lower-bound}. Let $\eta \defined \frac{c}{\sqrt{L(\alpha,
\beta, c / 2 - 1)}} \wedge 1$. For a constant $C$ such that $C \eta > 1$, we
have
\begin{align*}
C^2 n \varepsilon^{*2}
& \geq
1/\eta^2
\log \mlocc{\cF}{C \eta \varepsilon^*}{c}
\geq
\log \mlocc{\cF}{C \eta \varepsilon^*}{c} \\
& \geq
\log \mlocc{\cF}{C \varepsilon^* \frac{c}{\sqrt{L(\alpha, \beta, c / 2 - 1)}}}{c}
\end{align*}
Setting $\delta \defined C \varepsilon^*$ we obtain that
\begin{align*}
n\delta^2
\geq
\log \mlocc{\cF}{\delta \frac{c}{\sqrt{L(\alpha, \beta, c / 2 - 1)}}}{c}.
\end{align*}
In addition since $C > 1$, $\delta$ satisfies \eqref{upper:bound:suff:cond}
(taking into account that $n \varepsilon^{*2} > 4 \log 2$, which implies $n
\delta^2 \geq 4 \log 2 C^2 > \log 2$). We note that the map $0 < x \mapsto n x^2
- \log \mlocc{\cF}{x \frac{c}{\sqrt{L(\alpha, \beta, c / 2 - 1)}}}{c} \vee \log
2$ is non-decreasing by \Cref{nlem:local-metric-ent-monotone}. Now, with
$\varepsilon_{J^*}$ defined as per
\Cref{nthm:upper-bound-rate-finite-iterations}, this implies that $\delta \geq
\varepsilon_{J^*}/2$. This shows that the rate in this case is of the order
$\varepsilon^{*2}$.
Next, suppose that $\varepsilon^{*}$ defined by $\sup \{\varepsilon: n
\varepsilon^{2} \leq \log{\mlocc{\cF}{\varepsilon}{c}}\}$ satisfies $n
\varepsilon^{*2} \leq 4\log 2$. For $2\varepsilon^*$, we have $16 \log 2
\geq 4\varepsilon^{*2}n \geq \log{\mlocc{\cF}{2 \varepsilon^{*}}{c}}$. If
$c$ in the definition of local packing is large enough, we could put points
in the diameter of the ball with radius $2 \varepsilon^*$ such that the
packing set has more than $\exp(16\log 2)$ many points. But that implies
that the set $\cF$ is entirely inside a ball of radius $\sqrt{16\log 2}
n^{-1/2}$ (as $\varepsilon^{*2} \leq (4\log 2) n^{-1} $). To see the latter,
one can take the midpoint of the line segment connecting the endpoints of a
diameter of $\cF$ and position a ball of radius $2\varepsilon^*$ there. In
such a case, for the lower bound, we could pick $\varepsilon$ to be
proportional to the diameter of the set (with a small proportionality
constant). That will ensure that $\varepsilon \sqrt{n}$ is upper bounded by
some constant (as $2\sqrt{(16 \log 2)}n^{-1/2}$ is bigger than the
diameter), and at the same time $\log{\mlocc{\cF}{\varepsilon}{c}}$ can be
made bigger than a constant (provided that $c$ in the definition of a local
packing is large enough) -- by taking $\theta$ (where $\theta$ is the center
of the localized set $B_2(\theta, \varepsilon) \cap \cF$) to be the midpoint
of a diameter of the set $K$ and then placing equispaced points on the
diameter. Hence, the diameter of the set is a lower bound (up to constant
factors) in this case, which is of course always an upper bound too (up to
constant factors). So we conclude that either for $\varepsilon^{*}$ defined
by $\sup \thesetc{\varepsilon}{ \varepsilon^{2} n \leq
\log{\mlocc{\cF}{\varepsilon}{c}}}$ satisfies $\varepsilon^{*2}n > 4\log 2$
or the lower and upper bounds are of the order of the diameter of the set.
In summary the rate is given by the $\varepsilon^{*2} \wedge d^2$. This is
true since in the second case, $4\varepsilon^*$ is bigger than the diameter
of the set.
\eprfof
\subsection{Proof of
\texorpdfstring{\Cref{nprop:extend-zero-bounded-densities}}{\autoref{nprop:extend-zero-bounded-densities}}}\label{app:nprop:extend-zero-bounded-densities}
\extendzeroboundeddensities*
\bprfof{\Cref{nprop:extend-zero-bounded-densities}} We argue this as follows.
Let $f_{\alpha} \in \cF$, which is lower bounded by some $\alpha > 0$. Now
consider the following set of $f_{\alpha}$-mixture densities, \ie, $\cF^{\prime}
= \thesetc{(1/2) f_{\alpha} + (1/2) f}{f \in \cF} \subset \cF$. By construction,
all densities in $\cF^{\prime}$ are thus lower bounded by $\alpha/2$, \ie
$\cF^{\prime} \subset \cF_B^{[\alpha/2,\beta]}$. Moreover, $\cF^{\prime}$ forms
a convex density class. Hence, the minimax rate would be given by
$\varepsilon^{2} \wedge \operatorname{diam}_2(\cF^{\prime})^2$ where
$\varepsilon = \sup \thesetc{\varepsilon}{n \varepsilon^{2} \leq
\log{\mlocc{\cF^{\prime}}{\varepsilon}{c}}}$. We can artificially create
variables from the class $\cF^{\prime}$ by randomizing $X_{i}$ as follows
\begin{align*}
Z_{i} = \begin{cases}
T_{i} \distiid f_{\alpha} & \mbox{ with probability } $1/2$, \\
X_{i} & \mbox{ with probability } $1/2$.
\end{cases}
\end{align*}
Then let $\hat f$ be our estimator of $(1/2) f_{\alpha} + (1/2)f$. We know:
\begin{align*}
\EE_Z \normb{\hat f - ((1/2) f_{\alpha} + (1/2) f)}_{2}^{2} \lesssim \varepsilon^{2} \wedge \operatorname{diam}(\cF^{\prime})^2,
\end{align*}
so that
\begin{align*}
\EE_{X, T, V} \|(2\hat f-f_{\alpha}) - f\|_{2}^{2} \lesssim 4 \varepsilon^{2} \wedge \operatorname{diam}(\cF^{\prime})^2,
\end{align*}
where $T = (T_1,\ldots, T_n)$ and $V = (V_1,\ldots, V_n)$ are the values of the
coin flips in the definition of $Z_i$. Hence, $\EE_{T,V}2 \hat f-f_{\alpha}$
achieves the same rate for $f$ since by Jensen's inequality
\begin{align*}
\EE_Y \|\EE_{T, V} (2\hat f-f_{\alpha}) - f\|_{2}^{2} \leq \EE_{Y, T, V} \|(2\hat f-f_{\alpha}) - f\|_{2}^{2} \lesssim 4 \varepsilon^{2} \wedge \operatorname{diam}(\cF^{\prime})^2.
\end{align*}
Moreover, note that since $\hat f \in \cF^{\prime}$ for each of value of $T,V$
we have $\EE_{T,V} 2\hat f-f_{\alpha} \in \cF$. Thus, the upper bound is the
same for the two sets. On the other hand since $\cF^{\prime} \subset \cF$ the
lower bound is also of the same rate. Finally, observe that $
\log{\mlocc{\cF^{\prime}}{\varepsilon}{c}} = \log{\mlocc{\cF}{2
\varepsilon}{c}}$ so that the order of $\varepsilon^* = \sup
\thesetc{\varepsilon}{n \varepsilon^{2} \leq
\log{\mlocc{\cF^{\prime}}{\varepsilon}{c}}}$ is the same as that of the equation
$\varepsilon^* = \sup \thesetc{\varepsilon}{n \varepsilon^{2} \leq
\log{\mlocc{\cF}{\varepsilon}{c}}}$. In addition, it is also clear that
$2\operatorname{diam}_{2}(\cF^{\prime}) = \operatorname{diam}_{2}(\cF)$.
\eprfof
\subsection{Proof justification for
\texorpdfstring{\Cref{nexa:lipschitz-density-class}}{\autoref{nexa:lipschitz-density-class}}}\label{app:nexa:lipschitz-density-class}
Before proving
\Cref{nexa:lipschitz-density-class,nexa:bdd-total-variation-density-class,nexa:quad-functional-density-class},
we first prove a useful lemma. This lemma will provide a sufficient condition to
ensure that $L_{2}$-\emph{local} and $L_{2}$-\emph{global} metric entropies are
of the same order for various forms of the density class $\cF$, as specified in
our chosen examples.
\begin{restatable}[Asymptotic order global metric
entropy]{nlem}{localglobalentropyequivcF}\label{nlem:local-global-ent-cF}
Let $\cF \subset \cF_{B}^{[\alpha, \beta]}$, such that for any fixed $\eta > 0$, we
have $0 < \varepsilon \mapsto \log{\mgloc{\cF}{\varepsilon}} \asymp
\varepsilon^{- 1 / \eta}$. Then there exists a $c > 0$, such that the
following holds
\begin{equation}\label{neqn:local-global-ent-cF-01}
\log{\mgloc{\cF}{\varepsilon / c}} - \log{\mgloc{\cF}{\varepsilon}}
\asymp
\log{\mgloc{\cF}{\varepsilon / c}}
\end{equation}
\end{restatable}
\bprfof{\Cref{nlem:local-global-ent-cF}} We firstly note that
\eqref{neqn:local-global-ent-cF-01} has the following equivalence
\begin{align}
& \log{\mgloc{\cF}{\varepsilon / c}} - \log{\mgloc{\cF}{\varepsilon}}
\asymp
\log{\mgloc{\cF}{\varepsilon / c}}
\nonumber \\
\iff
\exists 0 < k_{1} < k_{2} \text{ s.t. }
k_{1} \log{\mgloc{\cF}{\varepsilon / c}}
\leq
& \log{\mgloc{\cF}{\varepsilon / c}} - \log{\mgloc{\cF}{\varepsilon}}
\leq
k_{2} \log{\mgloc{\cF}{\varepsilon / c}}
\label{neqn:local-global-ent-cF-02}
\end{align}
In general, for \Cref{neqn:local-global-ent-cF-02} we observe that since
$\log{\mgloc{\cF}{\varepsilon / c}} > 0$, it follows that
$\log{\mgloc{\cF}{\varepsilon / c}} - \log{\mgloc{\cF}{\varepsilon}} \leq
\log{\mgloc{\cF}{\varepsilon / c}}$. So taking $k_{2} = 1$ will always suffice
to ensure \eqref{neqn:local-global-ent-cF-02} holds. It remains to check that we
can also find a $k_{1} \in (0, 1)$ such that \eqref{neqn:local-global-ent-cF-02}
also holds. In our case, since $\log{\mgloc{\cF}{\varepsilon}} \asymp
\varepsilon^{- 1 / \eta}$ by assumption, we have that
$\log{\mgloc{\cF}{\varepsilon / c}} \geq C_{1} (\varepsilon / c)^{- 1 / \eta}$
and $\log{\mgloc{\cF}{\varepsilon}} \leq C_{2} \varepsilon^{- 1 / \eta}$ for
some universal constants $C_{1}, C_{2} > 0$. It then follows
\begin{align}
\frac{\log{\mgloc{\cF}{\varepsilon / c}} - \log{\mgloc{\cF}{\varepsilon}}}{\log{\mgloc{\cF}{\varepsilon / c}}}
& \geq 1 - \parens{\frac{C_{2}}{C_{1}}}c^{- \frac{1}{\eta}}
\nonumber \\
& \geq
k_{1}
\tag{as required, for $k_{1} \in (0, 1)$} \\
& \defines
1 - \delta
\tag{for some $\delta \in (0, 1)$, since $k_{1} \in (0, 1)$} \\
\text{if }
c
& \geq
\parens{\frac{C_{2}}{C_{1} \delta}}^{\eta}.
\label{neqn:local-global-ent-cF-03}
\end{align}
That is, there exists such a $k_{1} \in (0, 1)$, if we choose $c \geq
\parens{\frac{C_{2}}{C_{1} \delta}}^{\eta}$, for each $\eta > 0$. So indeed
\eqref{neqn:local-global-ent-cF-01} holds, for the specified class $\cF$, as
required.
\eprfof
\exalipschitzdensityclass*
\bprfof{\Cref{nexa:lipschitz-density-class}} In order to establish the minimax
rate for $\mathsf{Lip}_{\gamma, q}(\Psi)$, we need to show that $\mathsf{Lip}_{\gamma, q}(\Psi)$ is a convex
density class, and that there exists a density $f_{\alpha} \in \mathsf{Lip}_{\gamma, q}(\Psi)$
that is strictly positively bounded away from 0. We can then apply
\Cref{nprop:extend-zero-bounded-densities}. We first verify that $\mathsf{Lip}_{\gamma, q}(\Psi)$
here is a convex density class. To that end, let $f, g \in \mathsf{Lip}_{\gamma, q}(\Psi)$, and
let $\kappa \in [0, 1]$, be arbitrary. Then for each $x \in B \defined [0, 1]$,
we observe that
\begin{align}
(\kappa f + (1 - \kappa) g)(x)
\defined \kappa f(x) + (1 - \kappa) g(x)
& \geq \kappa (0) + (1 - \kappa) (0)
=
0
\label{neqn:lipschitz-density-class-convex-01} \\
(\kappa f + (1 - \kappa) g)(x)
\defined \kappa f(x) + (1 - \kappa) g(x)
& \leq
\kappa \Psi + (1 - \kappa) \Psi
=
\Psi
\label{neqn:lipschitz-density-class-convex-02}
\end{align}
From \eqref{neqn:lipschitz-density-class-convex-01} and
\eqref{neqn:lipschitz-density-class-convex-02}, it follows that
\begin{equation}\label{neqn:lipschitz-density-class-02}
\kappa f + (1 - \kappa) g \colon B \to [0, \Psi].
\end{equation} Moreover, since $\int_{B} f \dlett{\mu}
= \int_{B} g \dlett{\mu} = 1$, we have
\begin{equation}\label{neqn:lipschitz-density-class-03}
\int_{B} (\kappa f + (1 - \kappa) g) \dlett{\mu}
= \kappa \int_{B} f \dlett{\mu} +
(1 - \kappa) \int_{B} g \dlett{\mu}
= 1.
\end{equation}
Since $f, g \in \mathsf{Lip}_{\gamma, q}(\Psi)$, we have both $\normb{f}_{q}, \normb{g}_{q} \leq
\Psi$. Then by the triangle inequality it follows
\begin{equation}\label{neqn:lipschitz-density-class-04}
\normb{\kappa f + (1 - \kappa) g}_{q}
\leq
\normb{\kappa f}_{q} +
\normb{(1 - \kappa) g}_{q}
\leq
\kappa \Psi + (1 - \kappa) \Psi
=
\Psi.
\end{equation}
Since $f, g$ are measurable functions, then so is their convex combination, \ie,
$\kappa f + (1 - \kappa) g$. Now we observe
\begin{align}
& \normb{(\kappa f + (1 - \kappa) g)(x + h)
- (\kappa f + (1 - \kappa) g)(x)}_{q}
\nonumber \\
& =
\normb{\kappa (f(x + h) - f(x))
+ (1 - \kappa) (g(x + h) - g(x))}_{q}
\nonumber \\
& \leq
\normb{\kappa (f(x + h) - f(x))}_{q} +
\normb{(1 - \kappa) (g(x + h) - g(x))}_{q}
\tag{by the triangle inequality.} \\
& \leq
\kappa h^{\gamma} + (1 - \kappa) h^{\gamma}
\tag{since $f, g \in \mathsf{Lip}_{\gamma, q}(\Psi)$} \\
& =
h^{\gamma}
\label{neqn:lipschitz-density-class-05},
\end{align}
as required. Combining
\eqref{neqn:lipschitz-density-class-02},\eqref{neqn:lipschitz-density-class-03},\eqref{neqn:lipschitz-density-class-04},
and \eqref{neqn:lipschitz-density-class-05} we have shown that $\kappa f + (1 -
\kappa) g \in \mathsf{Lip}_{\gamma, q}(\Psi)$. This proves the convexity of $\mathsf{Lip}_{\gamma, q}(\Psi)$, as
required.
Now let $f_{\alpha} \sim \distUnif{[B]}$, \ie $f_{\alpha}(x) \defined \Indb{[0,
1]}{x}$. Therefore, $\normb{f_{\alpha}(x + h) - f(x)}_{q} = 0
\leq \Psi h^{\gamma}$, for each $x \in B$, and $h > 0$ such that
$f_{\alpha}(x + h)$ is defined. Moreover,
$\normb{f_{\alpha}(x)}_{q} = 1 < \Psi$, by assumption, for each
$1 \leq q \leq \infty$. Now we have that $\int_{B} f_{\alpha}
\dlett{\mu} = \int_{B} \Indb{[0, 1]}{x} \dlett{\mu}(x) = 1$, and
$f$ is measurable since it is a simple function. So indeed we
have found $f_{\alpha} \in \mathsf{Lip}_{\gamma, q}(\Psi)$, such that it is
$\alpha$-lower bounded (with $\alpha = 1$).
We now proceed to check that $L_{2}$-\emph{global} metric entropy is of the same
order as the $L_{2}$-\emph{local} metric entropy for $\mathsf{Lip}_{\gamma, q}(\Psi)$. That is,
we want to check that \eqref{neqn:local-global-ent-cF-01} holds. Here $\cF =
\mathsf{Lip}_{\gamma, q}(\Psi)$, with $0 < \varepsilon \mapsto \log{\mgloc{\cF}{\varepsilon}}
\asymp \varepsilon^{- 1 / \gamma}$. Thus, we can apply
\Cref{nlem:local-global-ent-cF} with $\eta \defined \gamma \in (0, 1]$ to
conclude that indeed \eqref{neqn:local-global-ent-cF-01} holds, as required.
Since we have checked all the sufficient conditions in order to apply
\Cref{nprop:extend-zero-bounded-densities} for $\mathsf{Lip}_{\gamma, q}(\Psi)$, we can obtain
the minimax rate of density estimation by solving
\begin{equation}
n \varepsilon^{2}
\asymp
\varepsilon^{- \frac{1}{\gamma}}
\iff
\varepsilon
\asymp
n^{-\frac{\gamma}{2 \gamma + 1}}
\iff
\varepsilon^{2}
\asymp
n^{-\frac{2 \gamma}{2 \gamma + 1}}.
\end{equation}
So the minimax rate is (up to constants) the order of $n^{-\frac{2 \gamma}{2
\gamma + 1}}$ as required.
\eprfof
\subsection{Proof justification for
\texorpdfstring{\Cref{nexa:bdd-total-variation-density-class}}{\autoref{nexa:bdd-total-variation-density-class}}}\label{app:nexa:bdd-total-variation-density-class}
\exabddtotvardensityclass*
\bprfof{\Cref{nexa:bdd-total-variation-density-class}} In order to establish the
minimax rate for $\mathsf{BV}_{\zeta}$, we need to show that $\mathsf{BV}_{\zeta}$ is a convex
density class, and that there exists a density $f_{\alpha} \in \mathsf{BV}_{\zeta}$
that is strictly positively bounded away from 0. We can then apply
\Cref{nprop:extend-zero-bounded-densities}. We first verify that $\mathsf{BV}_{\zeta}$
here is a convex density class. To that end, let $f, g \in \mathsf{BV}_{\zeta}$, and
let $\kappa \in [0, 1]$, be arbitrary. Then for each $x \in B \defined [0, 1]$,
it follows by an identical argument to
\eqref{neqn:lipschitz-density-class-convex-01} and
\eqref{neqn:lipschitz-density-class-convex-02} that
\begin{equation}\label{neqn:bdd-total-variation-convex-03}
\kappa f + (1 - \kappa) g \colon B \to [0, \zeta].
\end{equation} Moreover, since $\int_{B} f \dlett{\mu}
= \int_{B} g \dlett{\mu} = 1$, we have
\begin{equation}\label{neqn:bdd-total-variation-convex-04}
\int_{B} (\kappa f + (1 - \kappa) g) \dlett{\mu}
= \kappa \int_{B} f \dlett{\mu} +
(1 - \kappa) \int_{B} g \dlett{\mu}
= 1.
\end{equation}
Since $f, g \in \mathsf{BV}_{\zeta}$, we have both $\normb{f}_{\infty},
\normb{g}_{\infty} \leq \zeta$. Then by the triangle inequality it follows
\begin{equation}\label{neqn:bdd-total-variation-convex-05}
\normb{\kappa f + (1 - \kappa) g}_{\infty}
\leq
\normb{\kappa f}_{\infty} +
\normb{(1 - \kappa) g}_{\infty}
\leq
\kappa \zeta + (1 - \kappa) \zeta
=
\zeta.
\end{equation}
Since $f, g$ are measurable functions, then so is their convex combination, \ie,
$\kappa f + (1 - \kappa) g$. Finally, fix any $m \in \NN$, and let $a \leq x_{1}
< \cdots < x_{m} \leq b$ be any fixed partition of $B$. Now we observe
\begin{align}
& \sum_{i=1}^{m - 1} \absb{(\kappa f + (1 - \kappa) g)\parens{x_{i+1}}-(\kappa f + (1 - \kappa) g)\parens{x_{i}}}
\nonumber \\
& =
\sum_{i=1}^{m - 1} \absb{\kappa (f(x_{i+1}) - f(x_{i}))
+ (1 - \kappa) (g(x_{i+1}) - g(x_{i}))}
\nonumber \\
& \leq
\kappa \sum_{i=1}^{m - 1} \absb{f(x_{i+1}) - f(x_{i})} +
(1 - \kappa) \sum_{i=1}^{m - 1} \absb{g(x_{i+1}) - g(x_{i})}
\tag{by the triangle inequality.} \\
& \leq
\kappa V(f) + (1 - \kappa) V(g)
\nonumber \\
& \leq
\kappa (\zeta) + (1 - \kappa) (\zeta)
\tag{since $V(f), V(g) \leq \zeta$, by definition of $\mathsf{BV}_{\zeta}$.} \\
& =
\zeta.
\nonumber
\end{align}
Taking the supremum over all $m \in \NN$ and all partitions of length $m$ of $B$
of the $\lhs$ sum we obtain:
\begin{equation}\label{neqn:bdd-total-variation-convex-06}
V(\kappa f + (1 - \kappa) g)
\leq
\zeta,
\end{equation}
as required. Combining
\eqref{neqn:bdd-total-variation-convex-03},\eqref{neqn:bdd-total-variation-convex-04},\eqref{neqn:bdd-total-variation-convex-05},
and \eqref{neqn:bdd-total-variation-convex-06} we have shown that $\kappa f + (1
- \kappa) g \in \mathsf{BV}_{\zeta}$. This proves the convexity of $\mathsf{BV}_{\zeta}$, as
required.
Similar to the proof of \Cref{nexa:lipschitz-density-class}, we let $f_{\alpha}
\sim \distUnif{[B]}$, \ie $f_{\alpha}(x) \defined \Indb{[0, 1]}{x}$.
Therefore, $\normb{f}_{\infty} = 1 \leq \zeta$ by assumption. Also, $V(f) =
0 < \zeta$, by assumption. Now we have that $\int_{B} f_{\alpha} \dlett{\mu}
= \int_{B} \Indb{[0, 1]}{x} \dlett{\mu}(x) = 1$, and $f$ is measurable since
it is a simple function. So indeed we have found $f_{\alpha} \in
\mathsf{BV}_{\zeta}$, such that it is $\alpha$-lower bounded (with $\alpha = 1$).
We now proceed to check that $L_{2}$-\emph{global} metric entropy is of the same
order as the $L_{2}$-\emph{local} metric entropy for $\mathsf{BV}_{\zeta}$. That is, we
want to check that \eqref{neqn:local-global-ent-cF-01} holds. Here $\cF =
\mathsf{BV}_{\zeta}$, with $0 < \varepsilon \mapsto \log{\mgloc{\cF}{\varepsilon}}
\asymp \varepsilon^{- 1}$. Thus, we can apply \Cref{nlem:local-global-ent-cF}
with $\eta \defined 1$ to conclude that indeed
\eqref{neqn:local-global-ent-cF-01} holds, as required.
Since we have checked all the sufficient conditions in order to apply
\Cref{nprop:extend-zero-bounded-densities} for $\mathsf{BV}_{\zeta}$, we can obtain the
minimax rate of density estimation by solving
\begin{equation}
n \varepsilon^{2}
\asymp
\varepsilon^{- 1}
\iff
\varepsilon
\asymp
n^{-\frac{1}{3}}
\iff
\varepsilon^{2}
\asymp
n^{- \frac{2}{3}}.
\end{equation}
So the minimax rate is (up to constants) the order of $n^{- \frac{2}{3}}$ as
required.
\eprfof
\subsection{Proof justification for
\texorpdfstring{\Cref{nexa:quad-functional-density-class}}{\autoref{nexa:quad-functional-density-class}}}\label{app:nexa:quad-functional-density-class}
\quadfunctionaldensityclass*
\bprfof{\Cref{nexa:quad-functional-density-class}} In order to establish the
minimax rate for $\mathsf{Quad}_{\gamma}$, we need to show that $\mathsf{Quad}_{\gamma}$ is a
convex density class. We can then apply
\Cref{nprop:extend-zero-bounded-densities}. We first verify that
$\mathsf{Quad}_{\gamma}$ here is a convex density class. To that end, let $f, g \in
\mathsf{Quad}_{\gamma}$, and let $\kappa \in [0, 1]$, be arbitrary. Then for each $x \in
B \defined [0, 1]$, it follows by an identical argument to
\eqref{neqn:lipschitz-density-class-convex-01} and
\eqref{neqn:lipschitz-density-class-convex-02} that
\begin{equation}\label{neqn:quad-functional-density-class-02}
\kappa f + (1 - \kappa) g \colon B \to [0, \beta].
\end{equation} Moreover, since $\int_{B} f \dlett{\mu}
= \int_{B} g \dlett{\mu} = 1$, we have
\begin{equation}\label{neqn:quad-functional-density-class-03}
\int_{B} (\kappa f + (1 - \kappa) g) \dlett{\mu}
= \kappa \int_{B} f \dlett{\mu} +
(1 - \kappa) \int_{B} g \dlett{\mu}
= 1.
\end{equation}
Since $f, g$ are measurable functions, then so is their convex combination, \ie,
$\kappa f + (1 - \kappa) g$. Now we observe
\begin{align}
\normb{(\kappa f + (1 - \kappa) g)^{\prime \prime}}_{\infty}
\nonumber
& =
\normb{\kappa f^{\prime \prime} + (1 - \kappa) g^{\prime \prime}}_{\infty}
\tag{by linearity of $2^\textnormal{nd}$ derivative.} \\
& \leq
\normb{\kappa f^{\prime \prime}}_{\infty} +
\normb{(1 - \kappa) g^{\prime \prime}}_{\infty}
\tag{by the triangle inequality.} \\
& =
\kappa \normb{f^{\prime \prime}}_{\infty} +
(1 - \kappa) \normb{g^{\prime \prime}}_{\infty}
\nonumber \\
& \leq
\kappa \gamma + (1 - \kappa) \gamma
\tag{since $f, g \in \mathsf{Quad}_{\gamma}$} \\
& =
\gamma
\label{neqn:quad-functional-density-class-04},
\end{align}
as required. Combining
\eqref{neqn:quad-functional-density-class-02},\eqref{neqn:quad-functional-density-class-03},
and \eqref{neqn:quad-functional-density-class-04} we have shown that $\kappa f +
(1 - \kappa) g \in \mathsf{Quad}_{\gamma}$. This proves the convexity of
$\mathsf{Quad}_{\gamma}$, as required.
Similar to the proof of \Cref{nexa:lipschitz-density-class}, we let $f_{\alpha}
\sim \distUnif{[B]}$, \ie $f_{\alpha}(x) \defined \Indb{[0, 1]}{x}$. Since
$\normb{f^{\prime \prime}}_{\infty} = 0 \leq \gamma$. Here, for the boundary
points of $B \defined [0, 1]$, we are careful to take all derivatives of
$f_{\alpha}(x)$ at $x = 0$ from the right, and all derivatives from the left
at $x = 1$. Now we have that $\int_{B} f_{\alpha} \dlett{\mu} = \int_{B}
\Indb{[0, 1]}{x} \dlett{\mu}(x) = 1$, and $f$ is measurable since it is a
simple function. So indeed we have found $f_{\alpha} \in \mathsf{Quad}_{\gamma}$,
such that it is $\alpha$-lower bounded (with $\alpha = 1$).
We now proceed to check that $L_{2}$-\emph{global} metric entropy is of the same
order as the $L_{2}$-\emph{local} metric entropy for $\mathsf{Quad}_{\gamma}$. That is,
we want to check that \eqref{neqn:local-global-ent-cF-01} holds. Here $\cF =
\mathsf{Quad}_{\gamma}$, with $0 < \varepsilon \mapsto \log{\mgloc{\cF}{\varepsilon}}
\asymp \varepsilon^{- 1 / 4}$. Thus, we can apply
\Cref{nlem:local-global-ent-cF} with $\eta \defined 4$ to conclude that indeed
\eqref{neqn:local-global-ent-cF-01} holds, as required.
Since we have checked all the sufficient conditions in order to apply
\Cref{nprop:extend-zero-bounded-densities} for $\mathsf{Quad}_{\gamma}$, we can obtain
the minimax rate of density estimation by solving
\begin{equation}
n \varepsilon^{2}
\asymp
\varepsilon^{- \frac{1}{2}}
\iff
\varepsilon
\asymp
n^{-\frac{2}{5}}
\iff
\varepsilon^{2}
\asymp
n^{- \frac{4}{5}}.
\end{equation}
So the minimax rate is (up to constants) the order of $n^{- \frac{4}{5}}$ as
required.
\eprfof
\subsection{Proof justification for
\texorpdfstring{\Cref{nexa:convex-mixture-density-class}}{\autoref{nexa:convex-mixture-density-class}}}\label{app:nexa:convex-mixture-density-class}
\exaconvexmixturedensityclass*
\bprfof{\Cref{nexa:convex-mixture-density-class}} Let $\bfG =
\parens{\bfG_{ij}}_{i, j \in [k]}$ denote the Gram matrix $\bfG_{ij}
\defined \int_B f_{i} f_{j} \mu(\dlett{x})$. Then it is simple to see that
for some point $\theta \in \cF$ which can be represented as the convex
combination $\theta = \sum_{i \in [k]} \alpha_{i} f_i$, the packing set
should consist of functions $g_{i} = \sum_{j \in [k]} \beta_{ij} f_{j}$
satisfying both
\begin{align*}
(\alpha - \beta_i){ ^\mathrm{\scriptscriptstyle T} }} \def\v{{\varepsilon} \bfG (\alpha - \beta_i) \leq \varepsilon^2, \\
(\beta_{i} - \beta_{j}){ ^\mathrm{\scriptscriptstyle T} }} \def\v{{\varepsilon} \bfG (\beta_{i} - \beta_{j}) > \varepsilon^2/c^2, \mbox{ for } i \neq j,
\end{align*}
where $\beta_i$ are vectors from the $k$-dimensional unit simplex, \ie, $\sum_{j
\in [k]}\beta_{ij}= 1$, $\beta_{ij} \geq 0$. Now suppose that $\bfG
\succ 0$. Then upon substituting $\alpha' = \sqrt{\bfG} \alpha$,
$\beta_i' = \sqrt{\bfG}\beta_{i}$ and dropping the simplex requirements
on the $\beta$ we obtain the set
\begin{align*}
\|\alpha ' - \beta_i'\| \leq \varepsilon \\
\|\beta_i' - \beta_j'\| > \varepsilon/c,
\end{align*}
which is like packing the unit sphere at a distance $1/c$. Hence, the log
cardinality of such a packing is always $\lesssim k$ \cite[see Chapter
5]{wainwright2019high}. If $k$ is not allowed to scale with $n$, we conclude
therefore that the minimax rate is upper bounded by $n^{-1/2}$ which is the
parametric rate as we would expect. If $k$ is allowed to scale with $n$ the rate
is smaller than $\sqrt{\frac{k}{n}}$.
\eprfof
| {
"timestamp": "2022-10-21T02:18:10",
"yymm": "2210",
"arxiv_id": "2210.11436",
"language": "en",
"url": "https://arxiv.org/abs/2210.11436",
"abstract": "We study the classical problem of deriving minimax rates for density estimation over convex density classes. Building on the pioneering work of Le Cam (1973), Birge (1983, 1986), Wong and Shen (1995), Yang and Barron (1999), we determine the exact (up to constants) minimax rate over any convex density class. This work thus extends these known results by demonstrating that the local metric entropy of the density class always captures the minimax optimal rates under such settings. Our bounds provide a unifying perspective across both parametric and nonparametric convex density classes, under weaker assumptions on the richness of the density class than previously considered. Our proposed `multistage sieve' MLE applies to any such convex density class. We further demonstrate that this estimator is also adaptive to the true underlying density of interest. We apply our risk bounds to rederive known minimax rates including bounded total variation, and Holder density classes. We further illustrate the utility of the result by deriving upper bounds for less studied classes, e.g., convex mixture of densities.",
"subjects": "Statistics Theory (math.ST); Machine Learning (stat.ML)",
"title": "Revisiting Le Cam's Equation: Exact Minimax Rates over Convex Density Classes",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9770226294209298,
"lm_q2_score": 0.724870282120402,
"lm_q1q2_score": 0.7082146690263664
} |
https://arxiv.org/abs/1109.3436 | The monotone secant conjecture in the real Schubert calculus | The monotone secant conjecture posits a rich class of polynomial systems, all of whose solutions are real. These systems come from the Schubert calculus on flag manifolds, and the monotone secant conjecture is a compelling generalization of the Shapiro conjecture for Grassmannians (Theorem of Mukhin, Tarasov, and Varchenko). We present some theoretical evidence for this conjecture, as well as computational evidence obtained by 1.9 teraHertz-years of computing, and we discuss some of the phenomena we observed in our data. | \section{Introduction}
A system of real polynomial equations with finitely many solutions has some, but likely
not all, of its solutions real.
In fact, sometimes the structure of the equations leads to upper bounds~\cite{BBS, Kh91}
ensuring that not all solutions can be real.
The Shapiro Conjecture and the Monotone Secant Conjecture posit a family of systems of
polynomial equations with the extreme property of having all their solutions be real.
The Shapiro Conjecture asserts that a zero-dimensional intersection of Schubert
subvarieties of a flag manifold consists only of real points provided that the Schubert
varieties are given by flags tangent to (osculating) a real rational normal curve.
Eremenko and Gabrielov gave a proof in the special case the Grassmannian of
codimension-two planes~\cite{EG10, EG_02}.
The general case for Grassmannians was established by Mukhin, Tarasov, and
Varchenko~\cite{MTV_Annals,MTV_JAMS}.
A complete story of this conjecture and its proof can be found in~\cite{So_FRSC}.
The Shapiro conjecture is false for non-Grassmannian flag manifolds, but in a
very interesting manner.
This failure was first noticed in~\cite{So_Shap} and systematic computer experimentation
suggested a correction, the Monotone Conjecture~\cite{RSSS, So_fulton},
that appears to be valid for flag manifolds.
Eremenko, Gabrielov, Shapiro, and
Vainshtein~\cite{EGSV} proved a result that implied the Monotone Conjecture for some
manifolds of two-step flags.
This result concerned codimension-two subspaces that meet flags
which are secant to the rational normal curve along disjoint intervals.
This suggested the Secant Conjecture, which
asserts that an intersection of Schubert varieties in a Grassmannian is transverse with all
points real, provided that the Schubert varieties are defined by flags secant to a
rational normal curve along disjoint intervals.
This was posed and evidence was presented for its validity in~\cite{FRSC_Sec}.
The Monotone Secant Conjecture is a common extension of both the
Monotone Conjecture and the Secant Conjecture.
We give here the simplest open instance, expressed as a system of
polynomial equations in local coordinates for the variety of flags $E_2\subset E_3$ in
${\mathbb{C}}^5$, where $\dim E_i=i$. Let $x_1,\ldots,x_8$ be indeterminates and consider the
polynomials
\begin{equation}\label{Eq:F235}
f(s,t,u;x)\ :=\ \det\left(\begin{matrix}
1&0&x_1&x_2&x_3\\
0&1&x_4&x_5&x_6\\\hline
1&s&s^2&s^3&s^4\rule{0pt}{12pt}\\
1&t&t^2&t^3&t^4\\
1&u&u^2&u^3&u^4
\end{matrix}\right)\ ,
\hspace{10pt
g(v,w;x)\ :=\ \det\left(\begin{matrix}
1&0&x_1&x_2&x_3\\
0&1&x_4&x_5&x_6\\
0&0&1&x_7&x_8\\\hline
1&v&v^2&v^3&v^4\rule{0pt}{12pt}\\
1&w&w^2&w^3&w^4\\
\end{matrix}\right),
\end{equation}
which depend upon parameters $s,t,u$ and $v,w$ respectively.
\begin{conj}\label{C:first}
Let $\Blue{s_1<t_1<u_1<\dotsb< s_4< t_4}\,
<\, \G{u_4<v_1<w_1<\dotsb< v_4\,<\, w_4}$ be real numbers.
Then the system of polynomial equations
\begin{eqnarray}\label{Eq:polysys}
\B{f(s_1,t_1,u_1; x)\ = \ f(s_2,t_2,u_2; x)\ =\ f(s_3,t_3,u_3; x)\ =\ f(s_4,t_4,u_4; x)} &=& 0 \\
\nonumber\G{g(v_1,w_1; x) \ = \ g(v_2,w_2; x) \ = \ g(v_3,w_3; x) \ = \ g(v_4,w_4; x)} &=& 0
\end{eqnarray}
has twelve solutions, and all of them are real.
\end{conj}
Geometrically, the equation $f(s,t,u;x)=0$ is the condition that a general 2-plane
$E_2$ (spanned by the first two rows of the matrix) meets the 3-plane which is secant
to the rational normal curve $\gamma\colon y\mapsto (1,y,y^2,y^3,y^4)$ at the points
$\gamma(s),\gamma(t),\gamma(u)$.
Similarly, the equation $g(v,w;x)=0$ is the condition that a general 3-plane $E_3$
meets the 2-plane secant to $\gamma$ at the points $\gamma(v), \gamma(w)$.
The monotonicity hypothesis is that the four 3-planes given by $s_i,t_i, u_i$ are
secant along intervals $\B{[s_i,u_i]}$ which are pairwise disjoint and
occur before the intervals $\G{[v_i, w_i]}$ where the 2-planes
are secant.
If the order of the intervals $\B{[s_4,u_4]}$ and $\G{[v_1, w_1]}$ is switched, the
evaluation is no longer monotone.
We tested $1,000,000$ instances of Conjecture~\ref{C:first}, finding only real solutions.
In contrast, we tested $7,000,000$ with the monotonicity condition relaxed, finding
instances in which not all solutions were real.
We formulate the Monotone Secant Conjecture, explain its relation to the other
conjectures, and present overwhelming computational evidence that supports its validity.
These data are obtained in an experiment we are conducting on a supercomputer at
Texas A\&M University whose day job is Calculus instruction.
Our data can be viewed online; these data and the code can be accessed from our
website~\cite{FRSC}.
The design and execution of this kind of large-scale experiment was described
in~\cite{Exp-FRSC}.
This paper is organized as follows. In Section \ref{Sec:fourlines} we illustrate the rich geometry behind Schubert problems and we make use of the classical problem of four lines to depict the Monotone Secant Conjecture. Section \ref{Sec:background} provides a primer on flag
manifolds, states the Shapiro, Secant, and Monotone conjectures and there we state in detail the Monotone Secant Conjecture.
In Section \ref{Sec:results} we discuss the results collected from the observations of our
data. and we give a brief guide to our data.
Lastly, in Section \ref{Sec:method} we describe the methods we used to test the conjecture
\section{The problem of four lines}\label{Sec:fourlines}
The classical problem of four lines asks for the
finitely many lines $\M{m}$ that meet four given general lines $\G{\ell_1},\, \B{\ell_2},\,
\Red{\ell_3},\, \ell_4$ in (projective) three-space.
Three general lines
$\G{\ell_1},\,\B{\ell_2},\,\Red{\ell_3}$ lie in one ruling of a doubly-ruled
quadric surface $\Brown{Q}$, with the other ruling consisting of all lines that meet the
first three.
The line $\ell_4$ meets $\Brown{Q}$ in two points, and through each of these points there
is a line in the second ruling.
These two lines, $\M{m_1},\, \M{m_2}$, are the solutions to this problem.
\[
\begin{picture}(310,178)(0,4)
\put(0,0){\includegraphics[height=190pt]{4lines.eps}}
\put(266,148){\G{$\ell_1$}} \put(258,3){\Red{$\ell_3$}}
\put(288,85){\Blue{$\ell_2$}} \put(0,125){$\ell_4$}
\put(10, 95){\M{$m_1$}} \put(68, 0){\M{$m_2$}}
\put(146,132){\Brown{$Q$}}
\end{picture}
\]
If the lines $\G{\ell_1},\, \Blue{\ell_2},\, \Red{\ell_3},\, \ell_4$ are real,
then so is \Brown{$Q$}, but the intersection of \Brown{$Q$} with $\ell_4$ may consist
either of two real points or of a complex conjugate pair of points.
In the first case, the problem of four lines has two real solutions, while in the second,
it has no real solutions.
The Shapiro Conjecture asserts that if the four given lines are tangent to a rational
normal curve, then both solutions are real.
We illustrate this.
Set $\gamma(t):=(6t^2-1,\, \frac{7}{2}t^3+\frac{3}{2}t,\, -\frac{1}{2}t^3+\frac{3}{2}t )$,
a rational normal curve $\DeCo{\gamma}\colon{\mathbb{R}}\to{\mathbb{R}}^3$.
Write $\ell(t)$ for the line tangent at the point $\gamma(t)$.
Our given lines will be $\Red{\ell(-1)}, \Red{\ell(0)}, \Red{\ell(1)}$, and
$\G{\ell(s)}$ for $\G{s}\in(0,1)$.
The first three lines lie on the quadric \Brown{$Q$} defined by $x^2 - y^2 + z^2 = 1$.
The line $\G{\ell(s)}$ meets the quadric in two real points, as illustrated in Figure
\ref{F:throat},
\begin{figure}[htb]
\[
%
\begin{picture}(303, 106)(-22,4)
\put(0,0){\includegraphics[height=110pt]{shapiro.eps}}
\put(-23,24){\Red{$\ell(1)$}}
\put(250,30){\Red{$\ell(-1)$}}
\put(202,115){\Red{$\ell(0)$}}
\put(104,115){\M{$m_1$}} \put(250,16){\M{$m_2$}}
\put(250,95){\Brown{$Q$}}
\put(155,58.5){\Color{0 0.482353 0.482353 0.356863}{$\gamma(s)$}}
\put(-12,2){\B{$\gamma$}} \put(-23,83){\G{$\ell(s)$}}
\end{picture}
\]
\caption{Four tangent lines give two real solutions.}
\label{F:throat}
\end{figure}
giving two real solutions to this instance of the problem of four lines.
For the problem of four lines, the Secant Conjecture replaces the four tangent lines by
four lines that are secant to $\gamma$.
Suppose that the four are close to four tangents in that the intervals along
$\gamma$ given by their points of secancy are disjoint.
Figure~\ref{F:secant_four} shows three lines secant to
$\gamma$ along disjoint intervals and the quadric \Brown{$Q$} that they lie upon.
\begin{figure}[htb]
\[
\begin{picture}(320,110)(0,7)
\put(0,7){\includegraphics[height=119pt,viewport=5 78 430 240,clip]{secant.eps}}
\thicklines
\put(119,49){$\gamma$}\put(128,51){\vector(2,-1){25}}
\put(250,0){\vector(0,1){82.3}}
\put(246,-10){$I$}
\put(320,90){\Brown{$Q$}}
\end{picture}
\]
\caption{The problem of four secant lines.}\vspace{-10pt}
\label{F:secant_four}
\end{figure}
Their intervals of secancy are disjoint from the indicated
interval $I$.
Any line secant along $I$ will meet \Brown{$Q$} in two points, giving two real
solutions to this instance of the Secant Conjecture.
Figure~\ref{F:secant_four} also illustrates the Monotone
Secant Conjecture.
Consider flags of a line lying on a plane, $m\subset M$.
We require that the line $m$ meets three fixed lines secant to $\gamma$ and
that the plane $M$ meets two points, $\gamma(s)$ and $\gamma(t)$, of $\gamma$.
Since the plane $M$ contains the two points $\gamma(s)$ and $\gamma(t)$, it contains
the secant line they span, $\ell(s,t)$.
But the line $m$ also lies in $M$, and therefore it must also meet the secant line
$\ell(s,t)$, in addition to the three original secant lines.
If the three original secant lines are the three lines in Figure~\ref{F:secant_four} and
the points $\gamma(s),\gamma(t)$ lie in the interval $I$, then there will be two
real lines $m$ meeting all four secant lines, and for each line $m$, the plane $M$ is the span
of $m$ and $\ell(s,t)$.
\begin{figure}[htb]
\[
\begin{picture}(310,110)(0,15)
\put(0,7){\includegraphics[height=119pt,viewport=5 78 430 240,clip]{not_monotone.eps}}
\thicklines
\put(119,49){$\gamma$}\put(128,51){\vector(2,-1){25}}
\put(119,109){\vector(1,0){134}}
\put(90,106){$\ell(s,t)$}
\put(320,90){\Brown{$Q$}}
\end{picture}
\]
\caption{A non-monotone evaluation.}
\label{F:not_monotone}
\end{figure}
If the points $\gamma(s)$ and $\gamma(t)$ are chosen as in
Figure~\ref{F:not_monotone}, so that the secant line $\ell(s,t)$
does not meet the quadric \Brown{$Q$}, then the solutions will not be real.
Thus the positions of the points $\gamma(s),\gamma(t)$ relative to the other intervals of
secancy affect whether or not the solutions are real.
The schematic in Figure~\ref{F:schematic}
illustrates the relative positions of the secancies along $\gamma$ (which is homeomorphic
to the circle).
\begin{figure}[htb]
\[
\begin{picture}(80,90)(0,-14)
\put(0,0){\includegraphics{NC_int.eps}}
\put(22,-4){$s$} \put(52,-4){$t$}
\put(0,-17){All solutions real}
\end{picture}
\qquad\quad
\begin{picture}(84,90)(-2,-14)
\put(0,0){\includegraphics{CR_int.eps}}
\put(-2,20){$s$} \put(77,20){$t$}
\put(-7,-17){Not all solutions real}
\end{picture}
\]
\caption{Schematic for the secancies.}
\label{F:schematic}
\end{figure}
The idea behind the Monotone Secant Conjecture is to attach to each interval the dimension
of that part of the flag, 1 for $m$ and 2 for $M$, which it affects.
Then the schematic on the left has labels $1,1,1,2,2$, reading clockwise, starting just past
the point $s$, while the schematic on the right reads $1,1,2,1,2$.
In the first, the labels increase monotonically, while in the second, they do not.
\section{Background}\label{Sec:background}
We develop the background for the statement of the Monotone Secant Conjecture, defining
flag varieties and their Schubert problems.
Fix positive intgers $\alpha:=(a_1 < \cdots < a_k)$ and $n$ with $a_k<n$.
A \demph{flag $E_{\bullet}$ of type $\alpha$} is a sequence of subspaces
\[
E_{\bullet}\ \colon\ \{0\}\subset E_{a_1}\ \subset\ E_{a_2}\ \subset\
\dotsb\ \subset\ E_{a_k}\ \subset\ {\mathbb{C}}^n\,,
\qquad\mbox{where\ }\dim(E_{a_i})=a_i\,.
\]
The set of all such flags forms the
\demph{flag manifold $\mathbb{F}\ell(\alpha;n)$}, which has dimension
$\dim(\alpha):=\sum_{i=1}^k(n-a_i)(a_i-a_{i-1})$, where $a_0:=0$.
When $\alpha=(a)$ is a singleton, $\mathbb{F}\ell(\alpha;n)$ is the \demph{Grassmannian}
of $a$-planes in ${\mathbb{C}}^n$, written ${\rm Gr}(a,n)$.
Flags of type $1<2<\cdots<n-1$ in ${\mathbb{C}}^n$ are \demph{complete}.
The positions of flags of type $\alpha$ relative to a fixed complete flag $F_{\bullet}$ stratify
$\mathbb{F}\ell(\alpha;n)$ into cells whose closures are \demph{Schubert varieties}.
These positions are indexed by certain permutations.
The \demph{descent set $\delta(\sigma)$} of a permutation $\sigma\in S_n$ is the set of
numbers $i$ such that $\sigma(i)>\sigma(i{+}1)$.
Given a permutation $\sigma\in S_n$ with descent set a subset of $\alpha$,
set $r_\sigma(i,j):=\ |\{ l\leq i \mid j+\sigma(l)>n\}|$.
Then the Schubert variety $X_\sigma F_{\bullet}$ is
\[
X_\sigma F_{\bullet} \ =\
\{ E_{\bullet}\in\mathbb{F}\ell(\alpha;n) \mid
\dim E_{a_i}\cap F_j \geq r_\sigma(a_i,j), \ i=1,\ldots, k,\ j=1,\ldots,n \}.
\]
Flags $E_{\bullet}$ in $X_\sigma F_{\bullet}$ have position $\sigma$ relative to $F_{\bullet}$.
A permutation $\sigma$ with descent set contained in $\alpha$ is a \demph{Schubert
condition} on flags of type $\alpha$.
The Schubert variety $X_\sigma F_{\bullet}$ is irreducible with codimension
$\ell(\sigma):= |\{i<j \mid \sigma(i)>\sigma(j)\}|$.
A \demph{Schubert problem} for $\mathbb{F}\ell(\alpha;n)$ is a list of Schubert conditions
$(\sigma_1,\ldots, \sigma_m)$ for $\mathbb{F}\ell(\alpha;n)$ satisfying
$\ell(\sigma_1)+\cdots + \ell(\sigma_m) = \dim(\alpha)$.
Given a Schubert problem $(\sigma_1, \ldots, \sigma_m)$ for $\mathbb{F}\ell(\alpha;n)$ and complete flags
$F_{\bullet}^1,\dots, F_{\bullet}^m$, consider the intersection
\begin{equation}\label{eq:SchubIntersect}
X_{\sigma_1}F_{\bullet}^1\cap\cdots\cap X_{\sigma_m}F_{\bullet}^m\,.
\end{equation}
When the flags are in general position, this intersection is transverse and
zero-dimensional~\cite{Kleiman}, and it consists of all flags $E_{\bullet}\in\mathbb{F}\ell(\alpha;n)$
having position $\sigma_i$ relative to $F_{\bullet}^i$, for each
$i=1,\ldots, m$.
Such a flag $E_{\bullet}$ is a \demph{solution} to the Schubert problem.
The degree of a zero-dimensional intersection (\ref{eq:SchubIntersect}) is independent of
the choice of the flags and we call this number $d(\sigma_1,\dots, \sigma_m)$ the
\demph{degree} of the Schubert problem.
When the intersection is transverse, the number of solutions to a Schubert problem equals its degree.
When the flags $F_{\bullet}^1,\dots, F_{\bullet}^m$ are real, the solutions to the Schubert
problem need not be real.
The Monotone Secant Conjecture posits a method to select the flags $F_{\bullet}$ so that all
solutions are real, for a certain class of Schubert problems.
Let $\gamma\colon{\mathbb{R}}\to{\mathbb{R}}^n$ be a rational normal curve, which is affinely equivalent to
the moment curve $\gamma(t):=(1,t,t^2,\ldots,t^{n-1})$.
A flag $F_{\bullet}$ is \demph{secant along an interval $I$} of $\gamma$ if every subspace in
the flag is spanned by its intersection with $I$.
A list of flags $F_{\bullet}^1,\dotsc,F_{\bullet}^m$ secant to $\gamma$ is \demph{disjoint} if the
intervals of secancy are pairwise disjoint.
Disjoint flags are naturally ordered by order in which their intervals of secancy lie
within ${\mathbb{R}}$.
A permutation $\sigma$ is \demph{Grassmannian} of \demph{type $\delta(\sigma):=a_i$} if
its only descent is at position $a_i$.
A \demph{Grassmannian Schubert problem} is one that involves only Grassmannian Schubert
conditions.
A list of disjoint secant flags $F_{\bullet}^1,\dotsc,F_{\bullet}^m$ is \demph{monotone} with respect to
a Grassmannian Schubert problem $(\sigma_1,\dotsc,\sigma_m)$ if the function $F_{\bullet}^i
\mapsto \delta(\sigma_i)$ is monotone; in other words, if
\[
\delta(\sigma_i)\, <\, \delta(\sigma_j)\
\Longrightarrow\ F^i < F^j\,,\qquad\mbox{for all } i,j\,.
\]
\begin{monsecconj}\label{C:monosec}
For any Grassmannian Schubert problem $(\sigma_1,\ldots, \sigma_m)$ on the flag manifold
$\mathbb{F}\ell(\alpha;n)$ and any disjoint secant flags $F_{\bullet}^1,\ldots,F_{\bullet}^m$ that are monotone with
repsect to the Schubert problem, the intersection
\[
X_{\sigma_1}F_{\bullet}^1\cap X_{\sigma_2}F_{\bullet}^2\cap
\dotsb \cap X_{\sigma_m}F_{\bullet}^m
\]
is transverse with all points real.
\end{monsecconj}
Conjecture~\ref{C:first} is the Monotone Secant Conjecture for a
Schubert problem on $\mathbb{F}\ell(2,3;5)$ involving the Schubert conditions
$\sigma:=(1\,3\,2\,4\,5)$ and $\tau:=(1\,2\,4\,3\,5)$.
Then $\delta(\sigma)=2$, $\delta(\tau)=3$, and $\ell(\sigma)=\ell(\tau)=1$,
so that $(\sigma,\sigma,\sigma,\sigma,\tau,\tau,\tau,\tau)=(\sigma^4,\tau^4)$ is a
Schubert problem for $\mathbb{F}\ell(2,3;5)$, as $\dim(\mathbb{F}\ell(2,3;5))=8$.
The corresponding Schubert varieties are
\begin{eqnarray*}
X_{\sigma}F_{\bullet}\ =\ \{ E_{\bullet}\in\mathbb{F}\ell(2,3;5) \mid \dim E_2\cap F_3 \geq 1 \}\,,\\
X_{\tau}F_{\bullet}\ =\ \{ E_{\bullet}\in \mathbb{F}\ell(2,3;5) \mid \dim E_3 \cap F_2 \geq 1 \}\,,
\end{eqnarray*}
that is, the set of flags $E_{\bullet}$ whose $2$-plane $E_2$ meets a fixed $3$-plane $F_3$
non-trivially, and the set of $E_{\bullet}$ where $E_3$ meets a fixed $2$-plane $F_2$
non-trivially, respectively.
For $s,t,u,v,w\in{\mathbb{R}}$, let $F_3(s,t,u)$ be the linear span of $\gamma(s), \gamma(t)$, and
$\gamma(u)$ and $F_2(v,w)$ be the linear span of $\gamma(v)$ and $\gamma(w)$; these are a
secant 3-plane and a secant 2-plane to the rational normal curve, respectively.
The condition $f(s,t,u;x)=0$ of Conjecture~\ref{C:first} implies that $E_{\bullet}\in
X_\sigmaF_{\bullet}(s,t,u)$, where we ignore the larger subspaces in the flag $F_{\bullet}(s,t,u)$.
Similarly, the condition $g(v,w;x)=0$ implies that $E_{\bullet}\in X_\tauF_{\bullet}(v,w)$.
Lastly, the condition on the ordering of the points $s_i,t_i,u_i,v_i,w_i$ in
Conjecture~\ref{C:first} implies that the flags $F_{\bullet}(s_i,t_i,u_i)$ and $F_{\bullet}(v_i,w_i)$
are disjoint secant flags that are monotone with respect to this Schubert problem.
Three conjectures that have driven progress in enumerative real algebraic
geometry are specializations of the Monotone Secant Conjecture.
Observe that in the Grassmannian ${\rm Gr}(a;n)$, any list of disjoint secant flags
$F_{\bullet}^1,\dotsc,F_{\bullet}^m$ is monotone with respect to any Schubert problem
$(\sigma_1,\dotsc,\sigma_m)$, as all the conditions have the same descent.
In this way, the Monotone Secant Conjecture reduces to the Secant Conjecture, when the
flag manifold is a Grassmannian.
\begin{sconj}\label{C:sec}
For any Schubert problem $(\sigma_1, \dots, \sigma_m)$ on a Grassmannian ${\rm Gr}(a;n)$ and any
disjoint secant flags $F_{\bullet}^1,\ldots, F_{\bullet}^m$, the intersection
\[
X_{\sigma_1}F_{\bullet}^1\cap X_{\sigma_2}F_{\bullet}^2\cap \cdots \cap X_{\sigma_m}F_{\bullet}^m
\]
is transverse with all points real.
\end{sconj}
We studied this conjecture in a large-scale experiment whose results are described
in~\cite{FRSC_Sec}, solving 1,855,810,000 instances of 703 Schubert problems on 13
different Grassmannians, verifying the Secant Conjecture in each of the 448,381,157
instances checked.
This took 1.065 terahertz years of computing.
The limit of any family of flags whose intervals of secancy shrink to a point $\gamma(t)$
is the \demph{osculating flag $F_{\bullet}(t)$}.
This is the flag whose $j$-dimensional
subspace is the span of the first $j$ derivatives $\gamma(t),\
\gamma'(t),\ldots,\gamma^{(j-1)}(t)$ of $\gamma$ at $t$.
In this way, the limit of the Monotone Secant Conjecture, as the secant flags become
osculating flags, is a similar conjecture where we replace monotone secant flags by
monotone osculating flags.
\begin{monconj}\label{C:monotone}
For any Schubert problem $(\sigma_1,\ldots, \sigma_m)$ on the flag manifold
$\mathbb{F}\ell(\alpha;n)$ and any flags $F_{\bullet}^1,\ldots, F_{\bullet}^m$ osculating a rational normal
curve $\gamma$ at points that are monotone with respect to the Schubert problem, the
intersection
\[
X_{\sigma_1}F_{\bullet}^1\cap X_{\sigma_2}F_{\bullet}^2\cap \cdots\cap X_{\sigma_m}F_{\bullet}^m
\]
is transverse with all points real.
\end{monconj}
Ruffo, et al.~\cite{RSSS} formulated and studied this conjecture, establishing special
cases and giving substantial experimental evidence in support of it.
The specialization of the Monotone Secant Conjecture that both restricts to the
Grassmannian and to osculating flags is the Shapiro Conjecture which
was posed around 1995 by Boris
Shapiro and Michael Shapiro, studied in~\cite{So_Shap}, and for which proofs were given by
Eremenko and Gabrielov for ${\rm Gr}(n{-}2;n)$~\cite{EG_02} and in complete generality by
Mukhin, Tarasov, and Varchenko~\cite{MTV_Annals,MTV_JAMS}.
\begin{Shapiroconj}\label{ShapiroConj}
For any Schubert problem $(\sigma_1, \ldots, \sigma_m)$ in ${\rm Gr}(a;n)$ and any distinct
real numbers $t_1,\ldots, t_m$, the intersection
\[
X_{\sigma_1}F_{\bullet}(t_1)\cap X_{\sigma_2}F_{\bullet}(t_2)\cap\cdots\cap X_{\sigma_m}F_{\bullet}(t_m)
\]
is transverse with all points real.
\end{Shapiroconj}
\section{Results}\label{Sec:results}
The Secant Conjecture (like the Shapiro Conjecture before it) cannot hold for flag
manifolds. The monotonicity condition seems to correct this failure in both conjectures.
Here, we give more details on the relation of the Monotone Conjecture to the Monotone
Secant Conjecture, and then discuss some of our data in an ongoing experiment testing both conjectures.
\subsection{The Monotone Conjecture is the limit of the Monotone Secant Conjecture}\label{Sec:LimitMono}
The osculating plane $F_i(s)$ is the unique $i$-dimensional subspace having maximal order
of contact with the rational normal curve $\gamma$ at the point $\gamma(s)$, and therefore
it is a limit of secant flags.
\begin{proposition}
Let $\{s_1^{(j)}\ldots, \ldots, s_i^{(j)}\}$ for $j=1,2,\ldots$ be a sequence of lists of
$i$ distinct complex numbers with the property that for each $p=1,\ldots, i$, we have
$$
\lim_{j\to \infty} s_p^{(j)} = s,
$$
for some number $s$. Then,
$$
\lim_{j\to \infty} \mbox{\rm span}\{ \gamma(s_1^{(j)}), \ldots, \gamma(s_i^{(j)}) \} = F_i(s).
$$
\end{proposition}
As explained in the provious section, the Monotone Conjecture is implied by the Monotone
Secant Conjecture by this proposition.
There is a partial converse which follows from a standard limiting argument.
\begin{theorem}\label{T:limit}
Let $(\sigma_1,\ldots, \sigma_m)$ be a Schubert problem on $\mathbb{F}\ell(a;n)$ for which the
Monotone Conjecture holds.
Then, for any distinct real numbers that are monotone with respect to $(\sigma_1,\ldots,
\sigma_m)$, there exists a number $\epsilon>0$ such that, if for each $i=1,\ldots,m$,
$F_{\bullet}^i$ is a flag secant to $\gamma$ along an interval of length $\epsilon$ containing
$t_i$, then the intersection
$$
X_{\sigma_1}F_{\bullet}^1\cap X_{\sigma_2}F_{\bullet}^2\cap \cdots \cap X_{\sigma_m}F_{\bullet}^m
$$
is transverse with all points real.
\end{theorem}
\subsection{Experimental evidence for the Monotone Secant Conjecture}\label{Sec:ExpEvidence}
While its relation to existing conjectures led to positing the Monotone Secant Conjecture, we believe the immense weight of empirical evidence is the strongest support for it. Our ongoing experiment is testing this conjecture and related notions
for many computable Schubert problems.
As of 4 February 2011, we have solved 4,090,490,116 instances of 775 Schubert problems.
About 4.5\% of these (176,809,563) were instances of the Monotone Secant Conjecture, and
in every case, it was verified by symbolic computation.
Other computations tested the Monotone conjecture for comparison.
The remaining instances involved disjoint secant flags, but
with the monotonicity condition violated.
Table \ref{T:X1^4Y1^4=12} shows the data we obtained for the Schubert problem
$(\Blue{\sigma^4},\G{\tau^4})$ with 12 solutions on the Flag manifold $\mathbb{F}\ell(2,3;5)$
introduced in Conjecture \ref{C:first}.
\begin{table}[htb]
\noindent{\small
\noindent\begin{tabular}{r|r||r|r|r|r|r|r|c||r|}
\multicolumn{10}{c}{Real Solutions}\\
\cline{2-10}
\multirow{10}{*}{\begin{sideways}Necklace\end{sideways}} &
\textbackslash & 0&2&4&6&8&10&12 &Total\\\cline{2-10}\noalign{\smallskip}\cline{2-10}
&\Blue{2222}\G{3333} &&&&&&&1000000 & 1000000\\\cline{2-10}
&\Blue{222}\G{33}\Blue{2}\G{33} &&&6& 68210& 181738& 415395& 334651 & 1000000\\\cline{2-10}
& \Blue{2}\Blue{2}\G{3}\Blue{2}\Blue{2}\G{3}\G{3}\G{3} &&& 70& 134436& 357068& 322668& 185758& 1000000\\\cline{2-10}
& \Blue{2}\Blue{2}\G{3}\G{3}\Blue{2}\Blue{2}\G{3}\G{3} &&& 147& 267567& 399979& 216682& 115625& 1000000\\\cline{2-10}
& \Blue{2}\Blue{2}\G{3}\Blue{2}\G{3}\G{3}\Blue{2}\G{3} && 354& 23116& 100299& 313296& 374515& 188420& 1000000\\\cline{2-10}
& \Blue{2}\Blue{2}\G{3}\Blue{2}\G{3}\Blue{2}\G{3}\G{3} && 11148& 316401& 419371& 186548& 54634& 11898& 1000000\\\cline{2-10}
& \Blue{2}\Blue{2}\Blue{2}\G{3}\Blue{2}\G{3}\G{3}\G{3} && 31172& 95108& 153468& 336276& 249805& 134171& 1000000\\\cline{2-10}
& \Blue{2}\G{3}\Blue{2}\G{3}\Blue{2}\G{3}\Blue{2}\G{3} & 295403& 284925& 276937& 99691& 34520& 7807& 717& 1000000\\\cline{2-10}\noalign{\smallskip}\cline{2-10}
&Total & 295403& 327599& 711785& 1243042& 1809425& 1641506& 1971240 & 8000000 \\\cline{2-10}
\end{tabular}
}\vspace{5pt}
\caption{Necklaces vs. real solutions for $(\Blue{\sigma^4},\G{\tau^4})$ in $\mathbb{F}\ell(2,3;5)$.}\vspace{-20pt}
\label{T:X1^4Y1^4=12}
\end{table}
We computed 8,000,000 instances of this problem, all involving flags that were secant to the rational normal curve along disjoint intervals. This took 15.058 gigahertz-years. The columns are indexed by even integers numbers from 0 to 12, indicating the number of real solutions. The rows are indexed by \demph{necklaces}, which are sequences $\{\delta(\sigma_1),\ldots, \delta(\sigma_m)\}$, where $\delta(\sigma_i)$ denotes the unique descent of the Grassmannian condition $\sigma_i$, as described in Section \ref{Sec:background}. Therefore, in our example, a \Blue{2} represents the condition on the two-plane $E_2$ given by the permutation $\Blue{\sigma} =\Blue{(1\, 3\, 2\, 4\, 5)}$, and similarly a \G{3} represents the condition on $E_3$ given by the permutation $\G{\tau} =\G{(1\, 2\, 4\, 3\, 5)}$.
In Table \ref{T:X1^4Y1^4=12}, the first row labeled with $\Blue{2222}\G{3333}$ represents
tests of the Monotone Secant conjecture, since the only entries are in the column for 12
real solutions, the Monotone Secant conjecture was verified in 1,000,000 instances.
This is the only row with only real solutions.
Compare this to the 8,000,000 instances of the same Schubert problem, but with osculating
flags.
These data are presented in Table \ref{T:2X1^4Y1^4=12}. This computation took 67.460
gigahertz-days.
\begin{table}[htb]
\noindent{\small
\noindent\begin{tabular}{r|r||r|r|r|r|r|r|c||r|}
\multicolumn{10}{c}{Real Solutions}\\
\cline{2-10}
\multirow{10}{*}{\begin{sideways}Necklace\end{sideways}} &
\textbackslash & 0&2&4&6&8&10&12 &Total\\\cline{2-10}\noalign{\smallskip}\cline{2-10}
&\Blue{2222}\G{3333}
&&&&&&&1000000 & 1000000\\\cline{2-10}
&\Blue{222}\G{33}\Blue{2}\G{33}
&&& 514& 123534& 290754& 291572& 293626 & 1000000\\\cline{2-10}
& \Blue{2}\Blue{2}\G{3}\Blue{2}\Blue{2}\G{3}\G{3}\G{3}
&&& 765& 132416& 310881& 291640& 264298& 1000000\\\cline{2-10}
& \Blue{2}\Blue{2}\G{3}\G{3}\Blue{2}\Blue{2}\G{3}\G{3}
&&& 4467& 108818& 430805& 251237& 204673& 1000000\\\cline{2-10}
& \Blue{2}\Blue{2}\G{3}\Blue{2}\G{3}\G{3}\Blue{2}\G{3}
&&\begin{picture}(1,1)\put(-30,-2){\Dandelion{\rule{35pt}{9pt}}}\end{picture}
& 59935& 201234& 333260& 274979& 130592& 1000000\\\cline{2-10}
& \Blue{2}\Blue{2}\G{3}\Blue{2}\G{3}\Blue{2}\G{3}\G{3}
&& 3697& 127290& 215573& 332693& 210303& 110444& 1000000\\\cline{2-10}
& \Blue{2}\Blue{2}\Blue{2}\G{3}\Blue{2}\G{3}\G{3}\G{3}
&& 12857& 68514& 113824& 207927& 212245& 384633& 1000000\\\cline{2-10}
& \Blue{2}\G{3}\Blue{2}\G{3}\Blue{2}\G{3}\Blue{2}\G{3}
& 24493& 62798& 279201& 198460& 258211& 121806& 55031& 1000000\\\cline{2-10}\noalign{\smallskip}\cline{2-10}
&Total & 24493& 79352& 540686& 1093859& 2164531& 1653782& 2443297& 8000000 \\\cline{2-10}
\end{tabular}
}\vspace{5pt}
\caption{Necklaces vs. real solutions for $(\Blue{\sigma^4},\G{\tau^4})$ in $\mathbb{F}\ell(2,3;5)$.}\vspace{-20pt}
\label{T:2X1^4Y1^4=12}
\end{table}
Both tables are similar with nearly identical ``inner borders'', except for the shaded box
in Table~\ref{T:2X1^4Y1^4=12}.
\subsection{Lower bounds and inner borders}\label{Sec:inner}
Probably the most enigmatic phenomenon that we observe in our data is the presence of an
``inner border" for may geometric problems, as we have pointed out in example of
Table~\ref{T:X1^4Y1^4=12}.
That is, for some necklaces (besides the monotone ones), there appears to be a lower bound
on the number of real solutions.
We do not understand this phenomenon, even conjecturally.
Another common phenomenon is that for many problems, there are always at least some
solutions real, for any necklace.
(Note that the last rows of Tables~\ref{T:X1^4Y1^4=12} and~\ref{T:2X1^4Y1^4=12} had
instanecs with no real solutions).
Table~\ref{T14:W3X4=21_MSC} displays an example of this for
a Schubert problem on $\mathbb{F}\ell(2,3;6)$ involving three conditions
$\Purple{W}:=(1\,3\,2\,4\,5\,6)$ and five involving $\Blue{X}:=(1\,2\,4\,3\,5\,6)$ that
\vspace{-5pt}
\begin{table}[htb]
{\small
\[
\begin{tabular}{r|r||r|r|r|r|r|r|r|r|r|r|r|r|}
\multicolumn{14}{c}{Real Solutions}\\
\cline{2-14}
\multirow{7}{*}{\begin{sideways}Necklace\end{sideways}} &
\textbackslash &1&3&5&7&9&11&13&15&17&19&21&Total\\\cline{2-14}\noalign{\smallskip}\cline{2-14}
& \Purple{W}\W\Purple{W}\Blue{X}\X\Blue{X}\X\Blue{X}&&&&&&&&&&&80000&80000\\\cline{2-14}
& \Purple{W}\W\Blue{X}\Purple{W}\Blue{X}\X\Blue{X}\X&&&&&&&921&16549&26267&14475&21788&80000\\\cline{2-14}
& \Purple{W}\W\Blue{X}\X\Purple{W}\Blue{X}\X\Blue{X}&&&&&&39&1208&24559&39013&13947&1234&80000\\\cline{2-14}
& \Purple{W}\Blue{X}\Purple{W}\Blue{X}\X\Purple{W}\Blue{X}\X&&&&&&612&9544&43256&23583&2927&78&80000\\\cline{2-14}
& \Purple{W}\Blue{X}\Purple{W}\Blue{X}\Purple{W}\Blue{X}\X\Blue{X}&&&&&&3244&19887&31931&13688&3632&7618&80000\\\cline{2-14}
&Total&&&&&&3895&31560&116295&102551&34981&110718&400000\\\cline{2-14}
\end{tabular}
\]
}\caption{Enumerative Problem $W^3X^5= 21$ on $\mathbb{F}\ell(2, 3 ; 6)$}\vspace{-20pt}
\label{T14:W3X4=21_MSC}
\end{table}
has 21 solutions.
Very prominently, it appears that at least 11 of the solutions will always be real.
These lower bounds and inner borders were observed in the computations studying the
Monotone Conjecture~\cite{RSSS}.
Eremenko and Gabrielov established lower bounds for the Wronski map~\cite{EG01} in
Schubert calculus for the Grassmannian, and more recently, Azar and
Gabrielov~\cite{AzarGab} established lower bounds for some instances of the Monotone
Conjecture which were observed in~\cite{RSSS}.
\section{Method}\label{Sec:method}
Our experimentation was possible as instances of Schubert problems are simple to model on a computer. The procedure we use may be semi-automated and run on supercomputers. We will not describe how this automation is done, for that is the subject of the paper~\cite{Exp-FRSC}; instead, we explain here the computations we are performing.
For a Schubert condition $\sigma_i$, a fixed flag instantiating $\sigma_i$ is secant to the rational normal curve $\gamma$ along some $t_i$ points. Thus, for a Schubert problem $(\sigma_1, \ldots, \sigma_m)$, we need $t:= t_1+\cdots+ t_m$ points of $\gamma$ for the given Schubert problem. We construct secant flags by choosing $t$ points on $\gamma$. With the flags selected, we formulate the Schubert problem as a system of equations by a choice of local coordinates, whose common zeroes represent the solutions to the Schubert problem in the local coordinates. This was illustrated in the Introduction when Conjecture \ref{C:first} was presented. We direct the reader to ~\cite{Fu97,RSSS, So_Shap} for details. We then eliminate all but one variable from the equations, obtaining an \demph{eliminant}. If the eliminant is square-free and has degree equals to the expected number of complex solutions (this is easily verified;) then, the Shape Lemma concludes that the number of real roots of the eliminant equals the number of real solutions to the Schubert problem.
Instead of solving the Schubert problem, we may determine its number of real solutions in specific instances with software tools using implementations of Sturm sequences. If the software is reliably implemented, which we believe, then this computation provides a proof that the given instance has the computed number of real solutions to the original Schubert problem.
For each Schubert problem, we perform these steps thousands to millions of times, starting by selecting $t$ points in $\gamma$ and constructing flags secant to the rational normal curve and distributed with respect to a necklace. For every Schubert problem we have a list of necklaces with the first always being the necklace that creates an instance of the Monotone Secant conjecture, and the rest being necklaces where the monotonicity is broken.
Our experiment is not only testing the Monotone Secant Conjecture, but is also testing the Monotone Conjecture by just taking a point of tangency to $\gamma$ instead of an interval of secancy. With this, we are not only extending those computations made by Ruffo, et al.~\cite{RSSS}, but also comparing the results with those from the Montone Secant Conjecture in order to understand both conjectures deeply.
\bibliographystyle{amsplain}
| {
"timestamp": "2011-09-16T02:03:59",
"yymm": "1109",
"arxiv_id": "1109.3436",
"language": "en",
"url": "https://arxiv.org/abs/1109.3436",
"abstract": "The monotone secant conjecture posits a rich class of polynomial systems, all of whose solutions are real. These systems come from the Schubert calculus on flag manifolds, and the monotone secant conjecture is a compelling generalization of the Shapiro conjecture for Grassmannians (Theorem of Mukhin, Tarasov, and Varchenko). We present some theoretical evidence for this conjecture, as well as computational evidence obtained by 1.9 teraHertz-years of computing, and we discuss some of the phenomena we observed in our data.",
"subjects": "Algebraic Geometry (math.AG)",
"title": "The monotone secant conjecture in the real Schubert calculus",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9770226294209299,
"lm_q2_score": 0.724870282120402,
"lm_q1q2_score": 0.7082146690263664
} |
https://arxiv.org/abs/1310.1709 | General inner approximation of vector-valued functions | This paper addresses the problem of evaluating a subset of the range of a vector-valued function. It is based on a work by Gold- sztejn and Jaulin which provides methods based on interval analysis to address this problem when the dimension of the domain and co-domain of the function are equal. This paper extends this result to vector-valued functions with domain and co-domain of different dimensions. This ex- tension requires the knowledge of the rank of the Jacobian function on the whole domain. This leads to the sub-problem of extracting an in- terval sub-matrix of maximum rank from a given interval matrix. Three different techniques leading to approximate solutions of this extraction are proposed and compared. |
\section{Introduction}
Computing the values a function can take over some domain is generally of great interest in the analysis of numerical programs as in abstract interpretation~\cite{Cousot1977}, in robust control of dynamic systems~\cite{jaulin1996guaranteed}, or in global optimization~\cite{neumaier2004complete}.
Computing the image of a domain by a function (also called direct image or \emph{range}) exactly is intractable in general.
The classical solution is then to compute an outer approximation of this range, which can unfortunately be very pessimistic.
This outer approximation may introduce many values which do not belong to the range.
Providing, in addition, an inner approximation can be helpful to state the quality of the outer approximation.
For scalar-valued functions, an inner approximation can be evaluated using modal intervals~\cite{HerreroSVJ05} (using Kaucher arithmetic~\cite{kaucher1980interval}) or twin arithmetic~\cite{nesterov1997}.
When $f$ maps $\ensuremath{\mathbb{R}}^n$ to $\ensuremath{\mathbb{R}}^n$, modal intervals~\cite{goldsztejn2005} can also be used in the linear case.
For the non-linear case, set inversion~\cite{jaulin2001applied} can be used when $f$ is \emph{globally} invertible (when an inverse function $f^{-1}$ can be produced).
In the more general case of $f$ being \emph{locally} invertible, the method described by Goldsztejn and Jaulin in \cite{Goldsztejn2010} can be applied.
This technique requires however the inverse of the Jacobian of $f$.
Thus it can only be applied for functions from $\ensuremath{\mathbb{R}}^n$ to $\ensuremath{\mathbb{R}}^n$ of constant rank $n$.
This paper proposes a generalization of the method in~\cite{Goldsztejn2010} to deal with functions $f$ from $\mathcal{D} \subseteq \ensuremath{\mathbb{R}}^m$ to $\ensuremath{\mathbb{R}}^n$, with $m \neq n$, with rank $r$.
It describes a method to compute an inner approximation for at most $r$ components of $f$.
As in \cite{Goldsztejn2010}, the evaluation of the Jacobian of the function on a given subset of its domain is needed.
There, the identification of the components that can be used to compute an inner approximation has to be done by extracting the sub-matrix of full rank in its Jacobian.
Checking regularity of interval matrices is a NP-hard problem~\cite{poljak1993checking}, so is the problem of extracting an interval sub-matrix of full rank.
To our knowledge, no necessary and sufficient condition for checking regularity can be used to address this problem (a list of necessary and sufficient conditions for an interval matrix to be regular can be found in~\cite{rohn1989systems}).
This paper is organized as follows: Section~\ref{sec:gold} recalls the main result of \cite{Goldsztejn2010} on the computation of an inner approximation of the range of vector-valued functions with domain and co-domain of the same dimension.
Section~\ref{sec:extension} describes the extension of this result to functions with domain and co-domain of different dimensions.
Section~\ref{sec:algo} addresses algorithms for computing and inner approximation and describes how sub-matrices of full rank can be extracted from a given interval matrix using different techniques.
The computation of an inner approximation of the range of functions is illustrated on examples in Section~\ref{sec:app}.
\subsection*{\textbf{Notations}}
${\bf x} = \inter{\lb{x},\ub{x}} \triangleq \{x \in \ensuremath{\mathbb{R}} : \lb{x} \leqslant x \leqslant \ub{x}\}$ is an \emph{interval} where $\lb{x}$ and $\ub{x}$ are respectively its lower and its upper bound.
$\mathbb{IR} \triangleq \{\inter{\lb{x},\ub{x}} : \lb{x}, \ub{x} \in \ensuremath{\mathbb{R}}, \lb{x} \leqslant \ub{x}\}$ represents the set of intervals.
A box is the Cartesian product of $n$ intervals in $\mathbb{IR}^n$.
For an interval ${\bf x} = \inter{\lb{x}, \ub{x}}$, the \emph{width} is $\wid{{\bf x}} \triangleq \ub{x} - \lb{x}$, the \emph{midpoint} is $\midpoint{{\bf x}} \triangleq \frac{1}{2}(\ub{x} + \lb{x})$, the \emph{interior} is $\interior({\bf x}) \triangleq \{x \in \ensuremath{\mathbb{R}} | \lb{x} < x < \ub{x} \}$, and the \emph{boundary} is denoted by $\partial {\bf x}$.
The \emph{magnitude} is denoted $|{\bf x}| \triangleq \max\{|\lb{x}|, |\ub{x}|\}$ and the \emph{mignitude} is $\langle {\bf x} \rangle \triangleq \min\{|\lb{x}|, |\ub{x}|\}$ if $0 \notin {\bf x}$ and $\langle {\bf x} \rangle = 0$ otherwise.
The width of an interval vector ${\bf x} \in \mathbb{IR}^n$ is $\max_{1 \leqslant i \leqslant n}(\wid{{\bf x}_i})$.
The core of interval analysis is its fundamental theorem (see, \textit{e.g.}, \cite{moore1966} or \cite{neumaier1990}) asserting that an evaluation of an expression using intervals gives an outer approximation of the range of this expression over the considered intervals.
An interval function is an inclusion function denoted here ${\bf f}$: $f({\bf x}) = \{ f(x): x \in {\bf x} \} \subseteq {\bf f}({\bf x})$ for ${\bf x}$ included in the domain of $f$.
For an interval square matrix ${\bf A} \in \mathbb{IR}^{n \times n}$, $\diag {\bf A} \in \mathbb{IR}^{n \times n}$ is the diagonal interval matrix whose diagonal entries are $(\diag {\bf A})_{ii} = {\bf A}_{ii}$, $1 \leqslant i \leqslant n$, and 0 elsewhere.
$\offdiag {\bf A} \in \mathbb{IR}^{n \times n}$ is the interval matrix with null diagonal and with off-diagonal entries such that $(\offdiag {\bf A})_{ij} = {\bf A}_{ij}$.
For a vector-valued function $\ensuremath{f:\mathcal{D} \subseteq\R^m\rightarrow\R^n}$ and $x \in \mathcal{D}$, $f_{\num{i}{j}}(x)\triangleq (f_i(x), f_{i+1}(x), \dots, f_j(x))^T$ for $i \leqslant j$.
For the Jacobian $J^f$ of $f$ and $x \in \mathcal{D}$,
\begin{equation*}
J^{f_\num{i}{j}, x_\num{k}{\ell}}(x) \triangleq
\left(
\begin{array}{cccc}
\frac{\partial f_i}{\partial x_k}(x) & \frac{\partial f_i}{\partial x_{k+1}}(x) & \dots & \frac{\partial f_i}{\partial x_\ell}(x)\\
\frac{\partial f_{i+1}}{\partial x_k}(x) & \frac{\partial f_{i+1}}{\partial x_{k+1}}(x) & \dots & \frac{\partial f_{i+1}}{\partial x_\ell}(x)\\
\vdots & \vdots & \vdots\\
\frac{\partial f_j}{\partial x_k}(x) & \frac{\partial f_j}{\partial x_{k+1}}(x) & \dots & \frac{\partial f_j}{\partial x_\ell}(x)
\end{array}
\right)
\end{equation*}
is the restriction of the Jacobian of $f$ for $j-i$ components of $f$ and $\ell-k$ components of $x$.
$I_k \in \ensuremath{\mathbb{R}}^{k\times k}$ is the identity matrix of dimension $k$.
The null matrix with $k$ rows and $\ell$ columns is denoted $0_{k\times \ell}$ and the null vector of $k$ entries is denoted $0_k \triangleq(0, \dots, 0)^T$.
\section{Inner approximation for functions with domain and co-domain of the same dimension}
\label{sec:gold}
This section recalls the main result of \cite{Goldsztejn2010} to evaluate an inner approximation of the range of a function with domain and co-domain of the same dimension.
\begin{corollary}
\label{cor:gold}
Let ${\bf x} \in \mathbb{IR}^n$ and $f : {\bf x} \rightarrow \ensuremath{\mathbb{R}}^n$ be a continuous function continuously differentiable in $\interior({\bf x})$.
Consider ${\bf y} \in \mathbb{IR}^n$ and $\tilde{x} \in {\bf x}$ such that $f(\tilde{x}) \in {\bf y}$.
Consider also an interval matrix ${\bf J} \in \mathbb{IR}^{n\times n}$ such that $f'(x) \in {\bf J}$ for all $x \in {\bf x}$.
Assume that $0 \notin {\bf J}_{ii}$ for all $i \in \inter{1,\dots,n}$.
Let
\begin{equation}
H({\bf J}, \tilde{x}, {\bf x}, {\bf y}) = \tilde{x} + (\diag^{-1} {\bf J})\Big( {\bf y} - f(\tilde{x}) - (\offdiag {\bf J})({\bf x} - \tilde{x}) \Big)
\label{eq:hf}
\end{equation}
If $H({\bf J}, \tilde{x}, {\bf x}, {\bf y}) \subseteq \interior({\bf x})$ then ${\bf y} \subseteq \mathop{\mathrm{range}}(f, {\bf x})$.
\end{corollary}
This corollary provides an efficient test for a box ${\bf y}$ to be a subset of the range of a vector-valued function.
It can be used to compute an inner approximation of functions $f$ from $\ensuremath{\mathbb{R}}^n$ to $\ensuremath{\mathbb{R}}^n$, see Section~\ref{sec:algo}.
The restriction on $f$ having same dimension of domain and co-domain comes from the matrix inversion of $\diag {\bf J}$ in \eqref{eq:hf}.
\begin{figure}
\centering
\def0.52\columnwidth} \input{plot_05.pdf_tex{\columnwidth}
\input{explications.pdf_tex}
\caption{Sets and functions involved in Corollary~\ref{cor:gold} for inner approximation.}
\label{fig:exp}
\end{figure}
Figure~\ref{fig:exp} illustrates the computation in Corollary~\ref{cor:gold}.
The left part of Figure~\ref{fig:exp} represents the domain ${\bf x}$ and the right part the co-domain of $f$.
The set-valued map $f_S$ is defined from $\mathcal{P}(\ensuremath{\mathbb{R}}^n)$ (power set of $\ensuremath{\mathbb{R}}^n$) to $\mathcal{P}(\ensuremath{\mathbb{R}}^n)$ and returns the set $\{ f(x) : x \in \mathcal{D} \}$ for a given set $\mathcal{D}$, see~\cite{aubin2008set}.
From a given box $\tilde{{\bf x}} \subset {\bf x}$, one wants to know if the box $\tilde{{\bf y}}$ computed by an inclusion function of $f$ over $\tilde{{\bf x}}$ belongs to the range of $f$ or equivalently if $\tilde{{\bf y}} = {\bf f}(\tilde{{\bf x}})$ is a subset of $f_S({\bf x})$.
If $\tilde{{\bf x}}$ is too large compared to ${\bf x}$, one might have $\tilde{{\bf y}} = {\bf f}(\tilde{{\bf x}}) \nsubseteq f_S({\bf x})$.
To prove that $\tilde{{\bf y}} \subset f_S({\bf x})$, it is sufficient to prove that $f^{-1}_S(\tilde{{\bf y}}) \subset {{\bf x}}$.
The function $H({\bf J}, \tilde{x}, {\bf x}, {\bf y})$ in \eqref{eq:hf} can be seen as an inclusion function for $f^{-1}_S\circ f_S({\bf x})$.
\section{Extension for functions with domain and co-domain of different dimensions}
\label{sec:extension}
Corollary~\ref{cor:gold} only applies for functions having the same dimension for domain and co-domain.
It also needs that the determinant of the Jacobian is different from 0.
Consider now the case of a function $f$ with domain and co-domain of different dimensions.
In what follows, assume that $\ensuremath{f:\mathcal{D} \subseteq\R^m\rightarrow\R^n}$ is a $C^1$ function of rank greater than or equal to $r$ on $\mathcal{D}$.
It is assumed that there exist $r$ components $(x_{i_1},\dots,x_{i_r})$ of $x = (x_1, \dots, x_m) \in \mathcal{D} \subseteq\ensuremath{\mathbb{R}}^m$ and $r$ components $(f_{j_1}(x), \dots, f_{j_r}(x))$ such that
\begin{equation}
\label{eq:detRank}
\forall x\in\mathcal{D} \subseteq\ensuremath{\mathbb{R}}^m,\
\det
\left(
\frac{\partial f_{j_k}}{\partial x_{i_\ell}}(x)
\right)_{1\leqslant k, \ell \leqslant r}
\neq 0
\end{equation}
Hereafter, without loss of generality, $f$ is considered after the permutation of the $r$ coordinates $(x_{i_1},\dots,x_{i_r})$ and the $r$ coordinates $(f_{j_1}(x), \dots, f_{j_r}(x))$ (this permutation is discussed later in Section~\ref{sub:perm}).
It means that the Jacobian of $f$ has an $r \times r$ sub-matrix on the upper left such that
\begin{equation}
\forall x\in\mathcal{D} \subseteq\ensuremath{\mathbb{R}}^m,\ \det
\left(
\frac{\partial f_j}{\partial x_{k}}(x)
\right)_{1\leqslant j,k \leqslant r}
\neq 0
\label{eq:inv}
\end{equation}
Theorem~3.1 in \cite{Goldsztejn2010} provides sufficient conditions for a box ${\bf y}$ to be included in the range of a function.
Theorem~\ref{theo:th} bellow generalizes this characterization by providing sufficient conditions for a box ${\bf y}_1 \in \mathbb{IR}^r$ to be inside the projection on the first $r$ components of the image of $\ensuremath{f:\mathcal{D} \subseteq\R^m\rightarrow\R^n}$ when $f$ verifies \eqref{eq:inv}.
\begin{theorem}
\label{theo:th}
Let $f: \mathcal{D} \subseteq \ensuremath{\mathbb{R}}^m \rightarrow \ensuremath{\mathbb{R}}^n$ be a $C^1$ function that verifies \eqref{eq:inv}, ${\bf u} \subset \mathcal{D}$ a box in $\mathbb{IR}^m$ and ${\bf y}_1 \in \mathbb{IR}^r$.
Assume that the two following conditions are satisfied
\begin{enumerate}\renewcommand{\labelenumi}{(\roman{enumi})}
\item ${\bf y}_1 \cap {f_{1:r}}(\partial {\bf u}) = \emptyset$;\label{cond:1}
\item ${f_{1:r}}(\tilde{u}) \in {\bf y}_1$ for some $\tilde{u} \in {\bf u}$,\label{cond:2}
\end{enumerate}
then ${\bf y}_1 \subseteq {f_{1:r}}({\bf u})$.
\end{theorem}
Before starting the proof the next result is needed.
\begin{lemma}
\label{lem:1}
Let $f: \mathcal{D} \rightarrow \ensuremath{\mathbb{R}}^n$ be a $C^1$ function satisfying \eqref{eq:inv} and let $E$ be a compact such that $E \subset \mathcal{D}$.
Then one has $\partial( f_\num{1}{r}(E) ) \subseteq f_\num{1}{r}(\partial E)$.
\end{lemma}
\begin{proof}
Consider any $y_1 \in \partial( f_\num{1}{r}(E) )$.
As $f$ is continuous, ${f_{1:r}}$ is continuous as well.
Then the image of $E$, compact, by ${f_{1:r}}$ is also compact.
In particular it is closed so $\partial({f_{1:r}}(E) )$ is included in ${f_{1:r}}(E)$.
So there exists $x \in E$ such that $y_1 = f_\num{1}{r}(x)$.
Now suppose that $x \in \interior E$.
We now prove that this leads to a contradiction.
As $x \in \interior E$, there exists $U$ open of $E$ with $x \in U$.
Because of \eqref{eq:inv}, ${f_{1:r}}$ is a submersion and as submersions are open maps (see \cite{tu2010introduction}), $V = f_{1:r}(U)$ is open in $\ensuremath{\mathbb{R}}^r$.
We have $y_1 \in V$ then $y_1 \in \interior f_\num{1}{r}(E)$ which contradicts $y_1 \in \partial( f_\num{1}{r}(E) )$.
As a conclusion we have $x \in \partial E$ and eventually, $\partial( f_\num{1}{r}(E) ) \subseteq f_\num{1}{r}(\partial E)$.
\end{proof}
\begin{proof}[Theorem~\ref{theo:th}]
${\bf u}$ is a compact of $\mathcal{D}$ and $f$ is a $C^1$ function that verifies \eqref{eq:inv} so we have, from Lemma~\ref{lem:1}, $\partial( {f_{1:r}}({\bf u}) ) \subseteq {f_{1:r}}(\partial {\bf u})$.
So ${\bf y}_1 \cap \partial({f_{1:r}}({\bf u})) \subseteq {\bf y}_1 \cap {f_{1:r}}(\partial {\bf u}) =\emptyset$
therefore
\begin{equation}
\label{eq:empty}
{\bf y}_1 \cap \partial( f_\num{1}{r}({\bf u}) ) = \emptyset.
\end{equation}
The set ${f_{1:r}}({\bf u})$ is compact because ${\bf u}$ is compact and ${f_{1:r}}$ is continuous.
Let $\tilde{u}$ and $y_1 = {f_{1:r}}(\tilde{u}) \in {\bf y}_1$ be given in ($ii$).
As the intersection of ${\bf y}_1$ and $\partial {f_{1:r}}({\bf u})$ is empty by \eqref{eq:empty}, $y_1 \in \interior {f_{1:r}}({\bf u})$.
Consider any $z \in {\bf y}_1$ and suppose that $z \notin {f_{1:r}}({\bf u})$.
Since ${\bf y}_1$ is path connected, there exists a path included in ${\bf y}_1$ between $y_1$ and $z$ such that, by Lemma~A.1. in \cite{Goldsztejn2010}, this path intersects $\partial {f_{1:r}}({\bf u})$ which is not possible from \eqref{eq:empty}.
Therefore $z \in {f_{1:r}}({\bf u})$ which concludes the proof.
\end{proof}
Theorem~\ref{theo:th} is a generalization of Theorem~3.1 in \cite{Goldsztejn2010} for a function satisfying \eqref{eq:inv}.
In Theorem~3.1 in \cite{Goldsztejn2010}, the set $\Sigma = \{ x \in \interior{{\bf x}}\ |\ \det f'(x) = 0 \}$ can be extended for ${f_{1:r}}$ by $\Sigma_2 = \{ x \in \interior{{\bf x}}\ |\ \text{rank}({f_{1:r}}(x)) < r\}$.
Due to \eqref{eq:inv}, one has $\Sigma_2 = \emptyset$.
In what follows, Corollary~\ref{cor:rang} of Theorem~\ref{theo:th}, which extends the inclusion test of Corollary~\ref{cor:gold}, is introduced.
\begin{corollary}
\label{cor:rang}
Let $f:\mathcal{D} \subseteq\ensuremath{\mathbb{R}}^m\rightarrow\ensuremath{\mathbb{R}}^n$ be a $C^1$ function that satisfies \eqref{eq:inv} and ${\bf u} = ({\bf u}_1, {\bf u}_2)\in \mathbb{IR}^r\times\mathbb{IR}^{m-r}$.
Consider ${\bf y}_1 \in \mathbb{IR}^r$, $\tilde{u} = (\tilde{u}_1, \tilde{u}_2) \in ({\bf u}_1, {\bf u}_2)$ such that ${f_{1:r}}(\tilde{u}) \in {\bf y}_1$ and ${\bf J}^{f_{1:r}} = \left( {\bf J}^{{f_{1:r}}, u_1}\ {\bf J}^{{f_{1:r}}, u_2} \right)\in \mathbb{IR}^{r\times m}$ an interval matrix containing $J^{f_{1:r}}$ the Jacobian of ${f_{1:r}}$ on $({\bf u}_1, {\bf u}_2)$ such that $0 \notin \left({\bf J}^{{f_{1:r}}, u_1}\right)_{ii}$ for $1 \leqslant i \leqslant r$.
Let
\begin{eqnarray*}
&&H_{f_{1:r}}({\bf J}^{f_{1:r}}, \tilde{u}, {\bf u}, {\bf y}_1) = \tilde{u}_1 + (\diag^{-1} {\bf J}^{{f_{1:r}}, u_1}) \times\\
&&\Big({\bf y}_1 - {f_{1:r}}(\tilde{u}) - (\offdiag {\bf J}^{{f_{1:r}}, u_1})({\bf u}_1 - \tilde{u}_1) - {\bf J}^{{f_{1:r}}, u_2}({\bf u}_2 - \tilde{u}_2)\Big).
\end{eqnarray*}
If
\begin{equation}
H_{f_{1:r}}({\bf J}^{f_{1:r}}, \tilde{u}, {\bf u}, {\bf y}_1) \subseteq \interior({\bf u}_1),
\label{eq:rang}
\end{equation}
then
\begin{equation*}
{\bf y}_1\subseteq {f_{1:r}}({\bf u}).
\end{equation*}
\end{corollary}
\begin{proof}
It is sufficient to prove that if \eqref{eq:rang} is satisfied, the conditions of Theorem~\ref{theo:th} are satisfied too.
($i$) Let $u = (u_1, u_2) \in \partial {\bf u}$.
Since $u \in {\bf u}$, the mean value theorem applied to $f_{1:r}$ (see~\cite{neumaier1990}) shows that
\begin{equation}
{f_{1:r}}(u) \in {f_{1:r}}(\tilde{u}) + {\bf J}^{{f_{1:r}}} ({\bf u} - \tilde{u})
\end{equation}
Let us show that ${f_{1:r}}(\tilde{u}) + {\bf J}^{{f_{1:r}}} ({\bf u} - \tilde{u}) \cap {\bf y}_1 \neq \emptyset$ which implies $(i)$ false contradicts \eqref{eq:rang}.
Assume that there exists $J \in {\bf J}^{{f_{1:r}}}$, $J = (J_1 J_2)$ with $J_1 \in \ensuremath{\mathbb{R}}^{r \times r}$ and $J_2 \in \ensuremath{\mathbb{R}}^{r \times m - r}$; $u = (u_1, u_2)^T$, $\tilde{u} = (\tilde{u}_1, \tilde{u}_2)^T$, and $y_1 \in {\bf y}_1$ such that
\begin{eqnarray}
\label{eq:mv}
y_1 &=& {f_{1:r}}(\tilde{u}) + J(u - \tilde{u})\nonumber\\
&=& {f_{1:r}}(\tilde{u}) + J_1(u_1 - \tilde{u}_1) + J_2(u_2 - \tilde{u}_2)
\label{eq:temp}
\end{eqnarray}
By splitting $J_1$ in $\diag J_1 + \offdiag J_1$ in~\eqref{eq:temp}, we obtain:
\begin{equation*}
y_1 - {f_{1:r}}(\tilde{u}) - J_2(u_2 - \tilde{u}_2) = (\diag J_1)(u_1 - \tilde{u}_1) + (\offdiag J_1)(u_1 - \tilde{u}_1)
\end{equation*}
\begin{equation*}
\tilde{u}_1 + (\diag^{-1} J_1)\left(y_1 - {f_{1:r}}(\tilde{u}) - (\offdiag J_1)(u_1 - \tilde{u}_1) - J_2(u_2 - \tilde{u}_2)\right) = u_1
\end{equation*}
As $u \in {\bf u}$, $y_1 \in {\bf y}_1$, $\diag^{-1} J_1 \in \diag^{-1} {\bf J}_1$, $\offdiag J_1 \in \offdiag {\bf J}_1$, and $J_2 \in {\bf J}_2$, one gets
\begin{equation}
u_1 \in H_{f_{1:r}}(({\bf J}_1\ {\bf J}_2), \tilde{u}, {\bf u}, {\bf y}_1)
\end{equation}
and $u_1 \in \partial {\bf u}_1$ which contradicts \eqref{eq:rang}.
Then \eqref{eq:rang} implies $(i)$.
$(ii)$ By hypothesis, ${f_{1:r}}(\tilde{u}) \in {\bf y}_1$.
\end{proof}
For a function $\ensuremath{f:\mathcal{D} \subseteq\R^m\rightarrow\R^n}$, Corollary~\ref{cor:rang} gives a test for a box to belong to the image of $r$ components of $f$.
It can only be performed if the Jacobian for $r$ components of the function evaluated over the considered box is of full rank $r$.
When the rank of $f$ equals the dimension of the co-domain, $f$ is a \emph{submersion}~\cite{arnold1985}, Corollary~\ref{cor:rang} can be used to compute an inner approximation of the entire range of $f$.
\begin{example}
Let $f : \mathcal{D} \subset \ensuremath{\mathbb{R}} \rightarrow \ensuremath{\mathbb{R}}^3$ be a function that satisfies \eqref{eq:inv} for $r = 1$.
\begin{eqnarray}
f : && {\bf x} \subset \ensuremath{\mathbb{R}} \rightarrow \ensuremath{\mathbb{R}}^3\nonumber\\
&& x \mapsto
\left(
\begin{array}{c}
\sin 2x\\
\sin x\\
\frac{x}{2}
\end{array}
\right)
\label{eq:ex1}
\end{eqnarray}
The box ${\bf x} = \inter{0, \pi} \subset \mathcal{D}$ is considered as the domain on which $f$ is studied.
The function is of constant rank 1 then using Corollary~\ref{cor:rang}, one is able to compute an inner approximation of the range of a single component of $f$ e.g. $f_1({\bf x})$, $f_2({\bf x})$ or $f_3({\bf x})$.
\begin{figure}
\centering
\def0.52\columnwidth} \input{plot_05.pdf_tex{.75\columnwidth}
\input{ex_param1.pdf_tex}
\caption{Example for $f: ({\bf x}) \subseteq \ensuremath{\mathbb{R}} \mapsto \ensuremath{\mathbb{R}}^3$ of constant rank 1.}
\label{fig:gen}
\end{figure}
There is of course no proper box of dimension 2 or 3 included in the range of $f$.
Figure~\ref{fig:gen} represents the range of the function defined in \eqref{eq:ex1}.
\end{example}
\section{Algorithms}
\label{sec:algo}
\subsection{When domain and co-domain have the same dimension}
Algorithm~1 in \cite{Goldsztejn2010} computes an inner approximation for functions with domain and co-domain of the same dimension using Corollary~\ref{cor:gold} and a bisection algorithm.
The method is as follows.
For a given box $\tilde{{\bf x}}$ included in the initial domain ${\bf x}$, a box $\tilde{{\bf y}}$ such that $f({\bf x}) = \{ f(x) : x \in \tilde{{\bf x}} \} \subseteq \tilde{{\bf y}}$ is computed using the interval extension ${\bf f}$ of $f$.
If the hypotheses of Corollary~\ref{cor:gold} are satisfied, $\tilde{{\bf y}}$ is part of an inner approximation of the range of $f$.
If they are not satisfied, $\tilde{{\bf x}}$ is partitioned into two smaller boxes $\tilde{{\bf x}}'$ and $\tilde{{\bf x}}''$ that are treated like $\tilde{{\bf x}}$ was.
If the box $\tilde{{\bf x}} = (\tilde{{\bf x}}_1, \dots, \tilde{{\bf x}}_m)$ is deemed too small to be further bisected (i.e. when $\wid{{\bf x}} < \varepsilon$ where $\varepsilon$ is a user-defined parameter), then the iterations stop for this box.
This is described in Algorithm~\ref{algo:bis}.
It uses the function Inner described in Algorithm~\ref{algo:inner}, to decide if a box belongs to the range of a function.
\input{algo_gold}
Algorithm~\ref{algo:inner} decides for a given box $\tilde{{\bf x}} \subset {\bf x}$ whether ${\bf f}(\tilde{{\bf x}})$ belongs to the range of $f$ over ${\bf x}$.
The parameters $\tau$ and $\mu$ are used for the domain inflation (see Section~5.2 in \cite{Goldsztejn2010}) and $C$ is used to precondition the interval matrix ${\bf J}^f$ (see Section~4 in \cite{Goldsztejn2010}).
\input{algo_inner}
\begin{example}
Let $f(x) = Ax$ with $A =
\left(
\begin{array}{rc}
1 & 1\\
-1 & 1
\end{array}
\right)\text{,}
$
and an initial domain ${\bf x} = \left( \inter{-2, 2}, \inter{-2, 2} \right)$.
The aim is to compute an inner approximation of the set $\{ f(x) : x \in {\bf x} \}$.
Of course, in this too simple case, direct methods would be applicable since $A$ is an invertible matrix, but this is intended to exemplify the method.
\begin{figure}
\centering
\includegraphics[scale=0.45]{function41.png}
\caption{Inner approximation of the range of $f(x) = Ax$ when $x \in \inter{-2, 2}^2$ : $\mathcal{L}_{\texttt{Boundary}}$ (in black) and $\mathcal{L}_{\texttt{Inside}}$ are evaluated using Algorithm~\ref{algo:bis} with $\varepsilon = 10^{-3}$}
\label{fig:ax}
\end{figure}
Figure~\ref{fig:ax} shows the result obtained using Algorithm~\ref{algo:bis}.
Since bisections occur in the domain, the result consists of a set of overlapping boxes, obtained by an inclusion function computing outer approximations.
Dark areas in Figure~\ref{fig:ax} indicate many overlapping boxes.
\end{example}
\subsection{When the domain and co-domain have different dimensions}
We now extend the method in \cite{Goldsztejn2010} to compute an inner approximation of the projection on $r$ components of $f$.
Algorithm~\ref{algo:bis} is used unchanged, except for the inner inclusion test Inner in Line~7 which is now implemented by Algorithm~\ref{algo:inner2} instead of Algorithm~\ref{algo:inner}.
\input{algo_inner2}
When using Algorithm~\ref{algo:inner2}, the vector-valued function $f$ is assumed to satisfy \eqref{eq:inv}.
The main difference with the method in \cite{Goldsztejn2010} is in the construction of the variables needed in the application of Corollary~\ref{cor:rang}: in Algorithm~\ref{algo:inner2}, Lines 7--10 are dedicated to the definition of the vectors $(u_1, u_2)$, the interval vectors $({\bf u}_1, {\bf u}_2)$ and the interval matrices $({\bf J}^{{f_{1:r}}, u_1}$, ${\bf J}^{{f_{1:r}}, u_2})$ from Corollary~\ref{cor:rang}.
First, the $r$ components must be separated from the others to obtain $(u_1, u_2)$.
In Line~7, we construct from a vector in $\mathcal{D} \subseteq\ensuremath{\mathbb{R}}^m$, the initial domain, a vector in $\ensuremath{\mathbb{R}}^r\times \ensuremath{\mathbb{R}}^{m - r}$.
\begin{equation*}
(x_1, \dots, x_m)\mapsto ( (x_1, \dots, x_r), (x_{r+1}, \dots, x_{m}) )
\end{equation*}
Lines~8 and 9 construct the same information as at Line~7 but for ${\bf x} \in \mathbb{IR}^m$, an interval vector instead of a vector in $\ensuremath{\mathbb{R}}^m$, to get $({\bf u}_1, {\bf u}_2) \in \mathbb{IR}^r \times \mathbb{IR}^{m - r}$.
In Line~10, the pair of interval matrices $({\bf J}^{{f_{1:r}}, u_1}$, ${\bf J}^{{f_{1:r}}, u_2})) \in \ensuremath{\mathbb{R}}^{r \times r} \times \ensuremath{\mathbb{R}}^{r\times m - r}$ are obtained from an interval matrix ${\bf J}^{f_{1:r}} \in \ensuremath{\mathbb{R}}^{r \times n}$.
\paragraph{Preconditioning}
In~\cite{Goldsztejn2010}, the function has the same dimension for domain and co-domain and the Jacobian is then a square matrix.
This interval square matrix which is an outer approximation of the Jacobian has to be preconditioned in order to apply the test in Corollary~\ref{cor:gold} with an H-matrix (see Definition~\ref{def:hmat} and Section~4 in \cite{Goldsztejn2010}).
Here we need also to extend this preconditioning operation.
In practice, the preconditioning matrix is computed as follows:
For a given box ${\bf u} \in \mathbb{IR}^m$, $J^{f_{1:r}}(\tilde{u})$, the Jacobian of the $r$ first components of $f$, is computed for $\tilde{u} = \midpoint{{\bf u}}$ and supplemented with the $(m-r)$ last lines of the identity matrix $I_{m}$ to obtain an $m \times m $ matrix
\begin{equation*}
D_{\tilde{u}} =
\left(
\begin{array}{c|c}
\multicolumn{2}{c}{J^{f_{1:r}}(\tilde{u})}\\
0_{m-r \times r} & I_{m-r}
\end{array}
\right).
\end{equation*}
The inverse of $D_{\tilde{u}}$ is computed and its $r$ first columns are extracted to be the preconditioning matrix $C$.
Decomposing $C$ into $(C_1, C_2)^T$ with $C_1 \in \ensuremath{\mathbb{R}}^{r \times r}$ and $C_2 \in \ensuremath{\mathbb{R}}^{m-r \times r}$, the test in Corollary~\ref{cor:rang} becomes
\begin{equation}
H_{f_{1:r}}(C{\bf J}^{f_{1:r}}, \tilde{u}, {\bf u}, C{\bf y}_1) \subseteq \interior({\bf u}_1).
\label{eq:rang2}
\end{equation}
\subsection{Extracting the sub-matrix of maximum rank from an interval matrix}
\label{sub:perm}
The use of Algorithm~\ref{algo:inner2} requires that the rank $r$ of the Jacobian of $f$ is known and that $f$ satisfies \eqref{eq:inv}.
The Jacobian matrix is an interval matrix containing the Jacobian of $f$ over some box.
In the general case, we thus need to extract an interval sub-matrix of constant rank from the Jacobian of $f$.
In this section, we first define the rank of an interval matrix.
Then, we propose different methods to extract sub-matrices of full rank from a given interval matrix.
Some results on the evaluation of the eigenvalues of an interval matrix are well documented (see, e.g., \cite{rohn1993interval}) but are not tractable for our problem.
The extraction of an $r\times r$ sub-matrix of full rank is also not tractable.
Thus, we chose to rely on three more tractable - though more approximate - methods aiming at extracting a sub-matrix of high rank from a given interval matrix.
\begin{definition}[Regular interval square matrix~\cite{rohn1989systems}]\ \\
Let ${\bf A} \in \mathbb{IR}^{n \times n}$ be an interval matrix.
${\bf A}$ is regular if and only if for all matrix $A \in {\bf A}$, $A$ is not singular.
\end{definition}
\begin{definition}[Rank of an interval matrix~\cite{ishida1996reasoning}]\ \\
Let ${\bf A} \in \mathbb{IR}^{n \times m}$ be an interval matrix.
${\bf A}$ is of constant rank $r$ if and only if the largest regular interval square sub-matrix ${\bf A}_0$ of ${\bf A}$, is of dimension $r$.
\label{def:rankinter}
\end{definition}
Definition~\ref{def:rankinter} means that for all $A \in {\bf A} \in \mathbb{IR}^{n\times m}$, the rank of $A$ is larger than or equal to $r$.
To extract a regular interval square matrix of dimension equal to the rank of ${\bf A}$, three techniques are proposed in what follows.
\subsubsection{Building strictly dominant interval sub-matrices}
\label{sec:sdd}
This first method relies on the Levy-Desplanques theorem on strictly dominant matrices as a simple test for non-singularity.
We uses this test to formulate the extraction of sub-matrices of full rank as a linear programming problem.
\begin{definition}[Strictly diagonally dominant matrix~\cite{golub1996}]\ \\
Let $A = \left(a_{ij}\right)_{1\leqslant i,j \leqslant m}\in\ensuremath{\mathbb{R}}^{m\times m}$ be a square matrix.
$A$ is a strictly diagonally dominant matrix if and only if
\begin{equation}
\forall i\in \{1,\dots,m\}, |a_{ii}|> \sum_{\substack{i=1\\i\neq j}}^m |a_{ij}|.
\end{equation}
\end{definition}
This definition can be extended to interval matrices, using magnitude and mignitude instead of the absolute value:
\begin{definition}[Strictly diagonally dominant interval matrix~\cite{neumaier1990}]\ \\
Let ${\bf A} = \left({\bf a}_{ij}\right)_{1\leqslant i,j \leqslant m}\in \mathbb{IR}^{m\times m}$ be a square interval matrix.
${\bf A}$ is a strictly diagonally dominant interval matrix if and only if
\begin{equation}
\forall i\in \{1,\dots,m\}, \langle{\bf a}_{ii}\rangle> \sum_{\substack{i=1\\i\neq j}}^m |{\bf a}_{ij}|
\label{eq:diag}
\end{equation}
\label{def:sdd}
\end{definition}
\begin{theorem}[Levy-Desplanques theorem~\cite{levy1881,taussky1949recurring}]
\label{theo:levy}
A strictly diagonally dominant (interval) matrix is regular.
\end{theorem}
Consider ${\bf A} \in \mathbb{IR}^{n\times m}$, an interval matrix.
We introduce the decision variables $x_{ij}$ with $1 \leqslant i \leqslant n$ and $1 \leqslant j \leqslant m$.
The boolean $x_{ij}$ equals 1 if the component ${\bf a}_{ij}$ of ${\bf A}$ is picked to be an element of the diagonal of the rank $k$ sub-matrix of ${\bf A}$ and 0 otherwise.
The $x_{ij}$s are obtained as solutions of the following constrained optimization problem
\begin{equation}
\begin{aligned}
\max\ & f(x) = \sum_{i=1}^n\sum_{j=1}^m x_{ij} &\\
s.t. &
\left\{
\begin{array}{lr}
\displaystyle{\sum_{i=1}^n x_{ij}} \leqslant 1 & j = 1,\dots,m\\
\displaystyle{\sum_{j=1}^m x_{ij}} \leqslant 1 & i = 1,\dots,n\\
\displaystyle{\sum_{k = 1}^n x_{kl} \ \langle {\bf a}_{kl} \rangle > \sum_{i = 1}^n \sum_{j = 1}^m x_{ij} (1 - x_{il}) |{\bf a}_{il}|} & \begin{array}{c}k = 1,\dots,n\\l = 1,\dots,m\end{array}\\
x_{ij}\in \{0,1\} &
\end{array}
\right.
\end{aligned}
\label{eq:lp}
\end{equation}
The objective is to maximize the size of a square regular sub-matrix of ${\bf A}$.
The two first constraints ensure that at most one component on each row and column of the interval matrix ${\bf A}$ is taken (it corresponds to the problem of placing towers in a possibly not square chess board).
The last constraint corresponds to Theorem~\ref{theo:levy}.
Figure~\ref{fig:solpl} shows an example of solution provided by the constrained optimization problem \eqref{eq:lp}.
\begin{figure}
\centering
\begin{tabular}{|c|c|c|c|c|}
\hline
\;\;\; & $\bigstar$ & $\blacklozenge$ & $\blacklozenge$ & \;\;\; \\
\hline
\ & $\blacklozenge$ & $\blacklozenge$ & $\bigstar$ & \\
\hline
\ & & & & \\
\hline
& $\blacklozenge$ & $\bigstar$ & $\blacklozenge$ & \\
\hline
\end{tabular}
\caption{Example of result provided by the method using strictly dominance (see Section~\ref{sec:sdd}) and the one using H-matrices (see Section~\ref{sec:hmat}) on an interval matrix ${\bf A} \in \mathbb{IR}^{4 \times 5}$: $\bigstar$ represents components of the matrix that have been chosen ($x_{ij} = 1$) for the diagonal and $\blacklozenge$ represents the non-diagonal entries of the sub-matrix. Empty boxes represent components that are not part of the sub-matrix.}
\label{fig:solpl}
\end{figure}
A component of the interval matrix is picked if and only if it satisfies \eqref{eq:diag} and then leads to a strictly diagonally dominant interval matrix.
The last constraints in \eqref{eq:lp} are quadratic and have to be turned into linear constraints for efficiency reasons since linear programming techniques are generally fast.
A given ${\bf a}_{kl}$ is chosen if the sum of all the other ${\bf a}_{il}$ for $i = 1, \dots, m$ for which there exists an ${\bf a}_{ij}$ that is part of the diagonal of the extracted sub-matrix is lower.
Equivalently,
\begin{equation}
x_{kl} = 1 \Rightarrow \langle {\bf a}_{kl}\rangle > \sum_{\substack{i=1\\i\ne k}}^n\sum_{j=1}^m x_{ij}|{\bf a}_{il}|.
\end{equation}
Using the so called Big-M relaxation (see, e.g., \cite{hooker2011integrated}), this constraint can be rewritten as follows.
\begin{equation}
\sum_{\substack{i=1\\i\ne k}}^n\sum_{j=1}^m x_{ij}|{\bf a}_{il}| \leqslant M + (\langle{\bf a}_{kl}\rangle - \mu - M)x_{kl}
\label{eq:constraint}
\end{equation}
with $M$ chosen to be larger than $\sum_{\substack{i=1\\i\ne k}}^n \sum_{j=1}^m x_{ij}|{\bf a}_{il}|$ in order to deactivate the constraint when $x_{kl} = 0$ and $\mu$ as small as possible to approximate the strict inequality but not too small to avoid introduction of numerical instability.
Using~\eqref{eq:constraint} in~\eqref{eq:lp}, the constrained optimization problem~\eqref{eq:lp} becomes
\begin{equation}
\begin{aligned}
\max\ & f(x) = \sum_{i=1}^n\sum_{j=1}^m x_{ij} &\\
s.t. &
\left\{
\begin{array}{lr}
\displaystyle{\sum_{i=1}^n x_{ij}} \leqslant 1 & j = 1,\dots,m\\
\displaystyle{\sum_{j=1}^m x_{ij}} \leqslant 1 & i = 1,\dots,n\\
\displaystyle{\sum_{\substack{i=1\\i\ne k}}^n\sum_{j=1}^m} x_{ij}|{\bf a}_{il}| \leqslant M + (\langle{\bf a}_{kl}\rangle+ \mu - M)x_{kl} & \begin{array}{c}k = 1,\dots,n\\l = 1,\dots,m\end{array}\\
x_{ij}\in \{0,1\} &
\end{array}
\right.
\end{aligned}
\label{eq:lpsdd}
\end{equation}
Using a linear programming solver on~\eqref{eq:lpsdd}, a strictly dominant interval matrix can be extracted from ${\bf A}$.
The property for an interval matrix ${\bf A}$ to be a strictly dominant interval matrix is on the rows of ${\bf A}$.
This definition can apply also for ${\bf A}^T$ the transpose of ${\bf A}$, this is why the linear program is solved for both ${\bf A}$ and ${\bf A}^T$ to obtain the best result.
\subsubsection{Building H-sub-matrices}
\label{sec:hmat}
A second method is now investigated.
It uses a generalization of strictly dominant interval matrices, i.e., the notion of H-matrices~\cite{neumaier1990}.
Basic results on H-matrices are first provided before showing the slight changes in the constraint~\eqref{eq:constraint} that have to be done in order to detect H-sub-matrices in an interval matrix.
\begin{definition}[Comparison Matrix~\cite{neumaier1990}]\ \\
Let ${\bf A} \in \ensuremath{\mathbb{R}}^{m \times m}$ be a square interval matrix.
The comparison matrix $\langle {\bf A} \rangle$ is built as follows
\begin{equation*}
\langle {\bf A} \rangle_{ij} =
\begin{cases}
\langle {\bf A}_{ij} \rangle \text{ if } i = j\\
-|{\bf A}_{ij}| \text{ otherwise}
\end{cases}\text{ with } i, j = 1, \dots, m.
\end{equation*}
\end{definition}
\begin{definition}[H-matrix~\cite{neumaier1990}]\ \\
Let ${\bf A} \in \ensuremath{\mathbb{R}}^{m \times m}$ be a square interval matrix.
${\bf A}$ is an H-matrix if and only if there exists $u > 0_{m}$ such that $\langle {\bf A} \rangle u > 0_m$.
\label{def:hmat}
\end{definition}
\begin{theorem}[\cite{neumaier1990}]
Every H-matrix is regular.
\end{theorem}
\begin{remark}
The notion of H-matrices generalizes the one of strictly dominant interval matrices since a strictly diagonally dominant interval matrix is a particular case of an H-matrix by fixing $u = (\underbrace{1, \dots, 1}_{m})^T$ in Definition~\ref{def:hmat}.
\label{rmk:h}
\end{remark}
From Remark~\ref{rmk:h}, only slight changes have to be done in order to detect an H-matrix instead of a strictly diagonally one.
The constrained optimization Problem~\eqref{eq:lpsdd} is transformed into
\begin{equation}
\begin{aligned}
\max\ & f(x) = \sum_{i=1}^n\sum_{j=1}^m x_{ij} &\\
s.t. &
\left\{
\begin{array}{lr}
\displaystyle{\sum_{i=1}^n x_{ij}} \leqslant 1 & j = 1,\dots,m\\
\displaystyle{\sum_{j=1}^m x_{ij}} \leqslant 1 & i = 1,\dots,n\\
\displaystyle{\sum_{\substack{i=1\\i\ne k}}^n\sum_{j=1}^m} x_{ij}|{\bf a}_{il}|u_{ij} \leqslant M + (\langle{\bf a}_{kl}\rangle u_{kl} - \mu - M)x_{kl} & \begin{array}{c}k = 1,\dots,n\\l = 1,\dots,m\end{array}\\
x_{ij}\in \{0,1\} &\\
u_{ij}> 0 &
\end{array}
\right.
\end{aligned}
\label{eq:lp2bis}
\end{equation}
Figure~\ref{fig:solpl} shows an example of solution provided by the constrained optimization Problem \eqref{eq:lp2bis}.
In \eqref{eq:lp2bis}, the last constraint requires a matrix of variables $U = (u_{ij})_{\substack{1\leq i \leq n\\1 \leq j \leq m}}$ to be introduced.
It corresponds to the vector $u$ in Definition~\ref{def:hmat}.
This means we now have to solve a quadratic problem that could be tackled using SDP solvers.
In order to solve \eqref{eq:lp2bis} as efficiently as possible, i.e., by using linear programming techniques, we thus chose a particular $u$ before solving~\eqref{eq:lp2bis}.
All components of $u$ are chosen to be the inverse of the mignitude of the diagonal entries of the considered sub-matrix (as recommended in~\cite{neumaier1990}): $u = (\langle {\bf a}_{ij} \rangle^{-1})_{1\leq i \leq n; 1 \leq j \leq m}$.
The linear program is then
\begin{equation}
\begin{aligned}
\max\ & f(x) = \sum_{i=1}^n\sum_{j=1}^m x_{ij} &\\
s.t. &
\left\{
\begin{array}{lr}
\displaystyle{\sum_{i=1}^n x_{ij}} \leqslant 1 & j = 1,\dots,m\\
\displaystyle{\sum_{j=1}^m x_{ij}} \leqslant 1 & i = 1,\dots,n\\
\displaystyle{\sum_{\substack{i=1\\i\ne k}}^n\sum_{j=1}^m} x_{ij}|{\bf a}_{il}|\langle {\bf a}_{ij} \rangle^{-1} \leqslant M + (\underbrace{\langle{\bf a}_{kl}\rangle \langle {\bf a}_{kl} \rangle^{-1}}_{=1} - \mu - M)x_{kl} & \begin{array}{c}k = 1,\dots,n\\l = 1,\dots,m\end{array}\\
x_{ij}\in \{0,1\} &
\end{array}
\right.
\end{aligned}
\label{eq:lp2}
\end{equation}
\begin{remark}
Note that $\langle {\bf a}_{kl} \rangle$ has to be different from 0 because of the division that occurs in the constraint.
\end{remark}
As in the method using strictly diagonally dominant matrices, the linear program~\eqref{eq:lp2} can be solved using a linear programming solver.
\subsubsection{Combinatorial method}
A random search for an interval sub-matrix of maximum rank is performed.
It could rely on the two previous conditions (Definition~\ref{def:sdd} or Definition~\ref{def:hmat}) to determine whether a matrix is of full rank.
However, since no linear programming formulation has to be consider, one may use a more sophisticated test for full rank verification.
We use for that a result provided in~\cite{rohn1989systems}.
\begin{theorem}[Corollary 5.1 in \cite{rohn1989systems}]
Let ${\bf A} \in \mathbb{IR}^{m \times m}$ be an square interval matrix.
Let $\Delta$ be a matrix such that ${\bf A} = \midpoint{{\bf A}} + \inter{-\Delta, \Delta}$.
Let $D = |\text{\emph{mid}}({\bf A})^{-1}|\Delta$.
If the spectral radius $\rho(D) < 1$ then ${\bf A}$ is regular.
\label{thm:regular}
\end{theorem}
We combine this criterion derived from Theorem~\ref{thm:regular} by extracting randomly chosen components of an interval matrix and testing whether the resulting sub-matrix is regular.
This process is described in Algorithm~\ref{algoCombi}.
\begin{algorithm2e}
\label{algoCombi}
\caption{Extraction of regular interval sub-matrix}
\KwIn{${\bf A} \in \ensuremath{\mathbb{R}}^{n,m}$}
\KwOut{${\bf B} \in \mathbb{IR}^{k,k}$ a regular interval matrix}
\If{$n = m$ and ${\bf A}$ is regular}
{
\Return{{\bf A}}
}
\For{$k = \min(n, m)$ \emph{\textbf{downto}} 1}
{
\For{$i = 1$ \emph{\textbf{to}} MAX\_ITERATION}
{
${\bf B} \leftarrow$ extraction$({\bf A}, k)$\tcp{Extraction takes randomly $k$ components of ${\bf A}$ to be in the diagonal of ${\bf B}$ and the other components are deducted from this diagonal.}
\If{${\bf B}$ is regular}
{
\Return{{\bf B}}
}
}
}
\end{algorithm2e}
\subsubsection{Experiments on the sub-matrix extraction}
In this section, some results on extracting an interval square sub-matrix of maximum rank from a given interval matrix are now described for the three methods that have been previously described.
Two types of experiments have been performed depending on how the considered interval matrix has been produced.
The linear programs for the first two methods have been solved using the GLPK interface for C++~\cite{makhorin2006glpk}.
All experiments have been done on a 2.3 Ghz Intel core i5 processor based laptop with 8 GBytes memory.
In all experiments, the constant for the linear programs~\eqref{eq:lpsdd} and~\eqref{eq:lp2} are $M = \sum_{i=1}^m \sum_{j=1}^n \ub{{\bf a}}_{ij}$ and $\mu = 10^{-2}$.
For the random extraction, Algorithm~\ref{algoCombi} has been used with a fixed MAX\_ITERATION equal to 500.
Results have been averaged over 200 realizations.
First experiments have been done on an interval matrix generated randomly but containing a strictly dominant interval sub-matrix with a fixed dimension.
The matrix is constructed as follows: the size $m$ of an interval square matrix ${\bf A}$ is chosen.
For each component ${\bf a}_{ij}$ of ${\bf A}$, ${\bf a}_{ij} = \inter{\lb{{\bf a}}_{ij}, \ub{{\bf a}}_{ij}}$ with $1 \leqslant i, j \leqslant m$, $\lb{{\bf a}}_{ij}$ is a (pseudo) random number in $\inter{0,9}$ and $\ub{{\bf a}}_{ij}$ is equal to $\lb{{\bf a}}_{ij} + 1$ for all $i, j = 1, \dots, m$.
An a priori rank $r$ is chosen.
Then $r$ coordinates $(i, j)$ are randomly picked and for each of these pairs, the associated interval ${\bf a}_{ij} = \inter{\lb{{\bf a}}_{ij}, \ub{{\bf a}}_{ij}}$ is taken as
\begin{equation*}
\lb{{\bf a}}_{ij} = 1 + \sum_{k = 1}^m \ub{{\bf a}}_{kj}\text{ and }\ub{{\bf a}}_{ij} = \lb{{\bf a}}_{ij} + 1
\end{equation*}
Using this construction, there is in the resulting interval matrix ${\bf A}$ an $r \times r$ interval sub-matrix of ${\bf A}$ and $r$ is a lower bound of the actual rank of ${\bf A}$.
\begin{figure}
\begin{scriptsize}
\centering
\def0.52\columnwidth} \input{plot_05.pdf_tex{.8\columnwidth}
\input{Timesi.pdf_tex}
\caption{Computing time for the LP solver and the random extraction for a full rank interval matrix of size $i\times i$.}
\label{fig:times1}
\end{scriptsize}
\end{figure}
Figure~\ref{fig:times1} depicts a first experiment showing the average execution time as a function of the dimension of the considered interval matrix for the three methods.
The constructed matrices here are square, have a dimension from 2 to 8 and are strictly dominant interval matrices ($r = m$).
This experiment shows the exponential increase of the computing time needed while the dimension of the initial matrix for the methods using an LP solver and the apparently better behaviour of the combinatorial method (with a fixed number of iterations).
\begin{figure}
\begin{scriptsize}
\centering
\def0.52\columnwidth} \input{plot_05.pdf_tex{.8\columnwidth}
\input{Times8.pdf_tex}
\caption{Average computing time (in seconds) over iterations of the LP solver for the search of H-sub-matrices and strictly diagonally dominant sub-matrices, and for the random sub-matrix extraction for an interval matrix of size $8\times 8$ as a function of the minimum expected rank.}
\label{fig:times2}
\end{scriptsize}
\end{figure}
Figure~\ref{fig:times2} shows average computing times of the LP solver for the search of H-sub-matrices and strictly diagonally dominant sub-matrices, and for the random sub-matrix extraction for an interval square matrix of dimension 8.
In this case the matrix created randomly is constructed with a known minimum rank $r$ with $2 \leqslant r \leqslant 8$ that is the size of the interval H sub-matrix created.
Using the LP solver, the execution time is more or less constant.
We stress here the fact that the method using random extraction becomes faster for larger ranks because they are detected very early with Algorithm~\ref{algoCombi} and because the matrix can contain components equal to 0, the method using H-matrices in the case where $u$ is fixed as presented can no longer be applied since a division by 0 can occur in the linear program \eqref{eq:lp2}.
Finally two last experiments are provided: they use a different technique to construct the interval matrix.
To test the efficiency of the proposed algorithms, matrices with known rank $r$ are built.
For that purpose, a triangular matrix $A$ (with $(A)_{ii} \neq 0,\ 1 \leqslant i \leqslant r$) is first created.
It is the upper left part of a matrix $J \in \ensuremath{\mathbb{R}}^{n\times n}$ with $n \geqslant r$.
The $(n-r)$ remaining columns of $J$ are created as linear combinations of the $r$ first columns.
Then a sequence of rotations are applied to the matrix $J$ (here we applied $n+1$ rotations).
An angle $\theta_i$ is chosen randomly with $0 \leqslant \theta_i \leqslant \pi$ to compose the rotation matrix.
A pair $(k,l)$ with $k \neq l$ and $1 \leqslant k, l \leqslant n$ is chosen randomly and the rotation is applied for coordinates $(k, l)$ and $(l, k)$.
The matrix $A$ constructed in this way is not interval.
Figure~\ref{fig:ranks} shows the results for a first experiment using this matrix construction which has the advantage to let us know the expected rank of the results.
Using Theorem~\ref{theo:levy} as a criteria for sub-matrix extraction generally leads to a sub-matrix of dimension less than the rank of the matrix that is worse than the combinatorial method.
\begin{figure}
\begin{scriptsize}
\centering
\def0.52\columnwidth} \input{plot_05.pdf_tex{0.8\columnwidth}
\input{ranksVarMat.pdf_tex}
\caption{Execution from a matrix $J \in \ensuremath{\mathbb{R}}^{8\times 8}$ of constant rank $i = 2, \dots, 8$. Results are average values over 200 computations.}
\label{fig:ranks}
\end{scriptsize}
\end{figure}
The matrix construction can be used to show the impact of the interval width of the components of the matrix on the method efficiency.
For the last experiment, the same matrix construction than the previous one is used except that the components of the matrix constructed are thickened to obtain an interval matrix (the interval $\inter{-0.25, 0.25}$ is added to each component of the matrix for one experiment and $\inter{-0.5, 0.5}$ for another).
Results are shown in Figure~\ref{fig:ranks2}.
\begin{figure}
\begin{scriptsize}
\centering
\def0.52\columnwidth} \input{plot_05.pdf_tex{0.8\columnwidth}
\input{ranksVarMat025.pdf_tex}
\def0.52\columnwidth} \input{plot_05.pdf_tex{0.8\columnwidth}
\input{ranksVarMat05.pdf_tex}
\caption{Estimate of the rank of an interval matrix ${\bf A} \in \mathbb{IR}^{8\times 8}$ of constant rank $i = 1, \dots, 8$. Upper picture: $\inter{-0.25, 0.25}$ is added to each component, lower picture: $\inter{-0.5, 0.5}$ is added to each component Results are average values over 200 evaluations. Standard deviation is provided.}
\label{fig:ranks2}
\end{scriptsize}
\end{figure}
These experiments show that the combinatorial method is better than the method extracting strictly diagonally dominant sub-matrices.
This is due to Theorem~\ref{thm:regular} which can detect a bigger subset of regular matrices than just strictly diagonally dominant ones.
\subsubsection*{Limitations}
As previously mentioned, this method cannot guarantee to obtain the sub-matrix of maximum rank.
For a given function $f : \mathcal{D} \subseteq \ensuremath{\mathbb{R}}^m \rightarrow \ensuremath{\mathbb{R}}^n$ of constant rank $r$, only $p \leqslant r$ components can be detected from the LP program \eqref{eq:lp}.
An inner approximation will be obtained only for these $p$ components of $f$.
\section{Computation of an inner approximation of the range of a vector-valued function}
\label{sec:app}
This section shows some results of the computation of inner approximations of immersions and submersions.
The functions considered in these examples satisfy \eqref{eq:inv} with $r$ known.
\subsection{Immersion}
\label{subsec:imm}
Consider the problem of finding the range of the function
\begin{align}
f:\ensuremath{\mathbb{R}}^2 & \rightarrow \ensuremath{\mathbb{R}}^3\nonumber\\
(u,v) & \mapsto
\left\lbrace
\begin{array}{l}
f_1(u,v) = \cos(u)\cos(v)\\
f_2(u,v) = \sin(u)\cos(v)\\
f_3(u,v) = \sin(v)
\end{array}
\right.
\label{ex:immersion}
\end{align}
over the box $({\bf u}, {\bf v}) = (\left[\frac{3\pi}{2}+\tau,2\pi-\tau\right];\left[\tau,\frac{\pi}{2}-\tau\right]),\ \tau>0$.
$f$ is of constant rank 2 in $({\bf u}, {\bf v})$.
The rank is equal to the dimension of the domain of $f$, it is then an \emph{immersion}.
Corollary~\ref{cor:rang} can be used to get an inner approximation of the range of two components of $f$.
Here we compute the range of the two first components, but the two last or the first and the last components could also be considered.
\begin{figure}
\begin{scriptsize}
\centering
\def0.52\columnwidth} \input{plot_05.pdf_tex{0.6\columnwidth}
\input{ex_function1.pdf_tex}
\caption{Example 1: Range of the immersion defined in (\ref{ex:immersion}) for the initial domain $({\bf u}, {\bf v}) = (\left[\frac{3\pi}{2}+\tau,2\pi-\tau\right];\left[\tau,\frac{\pi}{2}-\tau\right]),\ \tau>0$.}
\label{fig:boule}
\end{scriptsize}
\end{figure}
Figure~\ref{fig:boule} represents the image of $({\bf u}, {\bf v})$ by $f$ which has no volume in its co-domain $\ensuremath{\mathbb{R}}^3$.
Results of the computation of an inner approximation of this range are shown in Figure~\ref{fig:result} for different values of $\varepsilon$ in Algorithm~\ref{algo:bis}.
The smaller the $\varepsilon$, the more accurate the inner approximation and the longer the computing time.
In Figure~\ref{fig:result}, empty boxes in gray represent boxes Algorithm~\ref{algo:bis} was unable to prove to be in the range.
Black boxes all belong to the range.
\begin{figure}
\begin{tiny}
\centering
\begin{tabular}{ccc}
\def0.52\columnwidth} \input{plot_05.pdf_tex{0.3\columnwidth} \input{plot_exemple510.pdf_tex}&
\def0.52\columnwidth} \input{plot_05.pdf_tex{0.3\columnwidth} \input{plot_exemple506.pdf_tex}&
\def0.52\columnwidth} \input{plot_05.pdf_tex{0.3\columnwidth} \input{plot_exemple502.pdf_tex}
\end{tabular}
\caption{Example 1: Results of the computation of an inner approximation for different values of parameter $\varepsilon$ (0.1, 0.06, and 0.02).}
\label{fig:result}
\end{tiny}
\end{figure}
The left part of Figure~\ref{fig:result} is for $\varepsilon = 0.1$, the middle part for $\varepsilon = 0.06$ and the last part for $\varepsilon = 0.02$.
The computing times are respectively 0.026 s, 0.10 s, and 0.64 s.
\subsection{Submersion}
Consider now the computation of an inner approximation of the range of the function
\begin{align}
f: ({\bf x}, {\bf y}, {\bf z}) \subseteq \ensuremath{\mathbb{R}}^3 & \rightarrow \ensuremath{\mathbb{R}}^2\nonumber\\
(x,y,z) & \mapsto
\left\lbrace
\begin{array}{l}
(x+r\cos(z))\cos(y)\\
(x+r\sin(z))\sin(y)
\end{array}
\right.
\label{ex:sub}
\end{align}
with $({\bf x}, {\bf y}, {\bf z}) = (\inter{2, 4.5}, \inter{0, 2\pi - \tau}, \inter{0, 2 \pi - \tau})$, $\tau = 10^{-3}$.
Figure~\ref{fig:donut} represents the range of $f$ and Figure~\ref{fig:result2} represents different computations of an inner approximation according to the parameter $\varepsilon$ in Algorithm~\ref{algo:bis}.
On Figure~\ref{fig:result2}a $\varepsilon = 0.5$ and it took 0.18s to get the result.
For Figure~\ref{fig:result2}b, $\varepsilon = 0.3$ and computation time is 6.25s.
In Figure~\ref{fig:result2}c, it took 145.53s with $\varepsilon = 0.1$ to get these results.
Finally Figure~\ref{fig:result2}~d is for $\varepsilon = 0.05$ and the computing time is 838.67s.
The time needed for computation is longer for this experiment than for the previous one on immersion.
It is due to the fact that the Jacobian of $f$ is not of full rank in the entire domain $(\inter{2, 4.5}, \inter{0, 2\pi - \tau}, \inter{0, 2 \pi - \tau})$.
It is why an area (cf. Figure~\ref{fig:result2}d remains out of the range of $f$.
\begin{figure}
\begin{scriptsize}
\centering
\def0.52\columnwidth} \input{plot_05.pdf_tex{0.3\columnwidth}
\input{donut.pdf_tex}
\caption{Example 2: Range of the submersion defined in (\ref{ex:sub}) for the initial domain $({\bf x}, {\bf y}, {\bf z}) = (\inter{2, 4.5}, \inter{0, 2\pi - \tau}, \inter{0, 2 \pi - \tau})$, $\tau = 10^{-3}$.}
\label{fig:donut}
\end{scriptsize}
\end{figure}
\begin{figure}
\begin{scriptsize}
\centering
\begin{tabular}{cc}
\textbf{a)}\def0.52\columnwidth} \input{plot_05.pdf_tex{0.4\columnwidth} \input{plot_5.pdf_tex}&
\textbf{b)}\def0.52\columnwidth} \input{plot_05.pdf_tex{0.4\columnwidth} \input{plot_3.pdf_tex}\\
\textbf{c)}\def0.52\columnwidth} \input{plot_05.pdf_tex{0.4\columnwidth} \input{plot_1.pdf_tex}&
\textbf{d)}\def0.52\columnwidth} \input{plot_05.pdf_tex{0.52\columnwidth} \input{plot_05.pdf_tex}
\end{tabular}
\caption{Example 2: Results of computation of an inner approximation according to the parameter $\varepsilon$ in Algorithm~\ref{algo:bis} ($\varepsilon = 0.5$, 0.3, 0.1 and 0.05).}
\label{fig:result2}
\end{scriptsize}
\end{figure}
\section{Conclusion}
Goldsztejn and Jaulin in~\cite{Goldsztejn2010} proposed a way to compute an inner approximation of the range of a vector-valued function.
This paper provides an algorithm to evaluate inner approximations of the range of vector-valued functions without restriction on the dimension of its domain and co-domain.
Using the proposed algorithm, one is able, for functions from $\ensuremath{\mathbb{R}}^m$ to $\ensuremath{\mathbb{R}}^n$ to evaluate an inner approximation of the projection of the range of $f$ on at most $r$ components, where $r$ is the rang of the Jacobian matrix of $f$.
In the general case, this rank $r$ is unknown a priori, it is thus necessary to develop several techniques to extract a sub-matrix of maximal rank from a given interval matrix.
The restriction of this method is providing an inner approximation of at most $r$ components of the function if this function has a constant rank $r$.
\bibliographystyle{plain}
| {
"timestamp": "2013-10-08T02:10:24",
"yymm": "1310",
"arxiv_id": "1310.1709",
"language": "en",
"url": "https://arxiv.org/abs/1310.1709",
"abstract": "This paper addresses the problem of evaluating a subset of the range of a vector-valued function. It is based on a work by Gold- sztejn and Jaulin which provides methods based on interval analysis to address this problem when the dimension of the domain and co-domain of the function are equal. This paper extends this result to vector-valued functions with domain and co-domain of different dimensions. This ex- tension requires the knowledge of the rank of the Jacobian function on the whole domain. This leads to the sub-problem of extracting an in- terval sub-matrix of maximum rank from a given interval matrix. Three different techniques leading to approximate solutions of this extraction are proposed and compared.",
"subjects": "Numerical Analysis (math.NA)",
"title": "General inner approximation of vector-valued functions",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9770226287518853,
"lm_q2_score": 0.724870282120402,
"lm_q1q2_score": 0.7082146685413959
} |
https://arxiv.org/abs/1411.2689 | Avoiding the Global Sort: A Faster Contour Tree Algorithm | We revisit the classical problem of computing the \emph{contour tree} of a scalar field $f:\mathbb{M} \to \mathbb{R}$, where $\mathbb{M}$ is a triangulated simplicial mesh in $\mathbb{R}^d$. The contour tree is a fundamental topological structure that tracks the evolution of level sets of $f$ and has numerous applications in data analysis and visualization.All existing algorithms begin with a global sort of at least all critical values of $f$, which can require (roughly) $\Omega(n\log n)$ time. Existing lower bounds show that there are pathological instances where this sort is required. We present the first algorithm whose time complexity depends on the contour tree structure, and avoids the global sort for non-pathological inputs. If $C$ denotes the set of critical points in $\mathbb{M}$, the running time is roughly $O(\sum_{v \in C} \log \ell_v)$, where $\ell_v$ is the depth of $v$ in the contour tree. This matches all existing upper bounds, but is a significant improvement when the contour tree is short and fat. Specifically, our approach ensures that any comparison made is between nodes in the same descending path in the contour tree, allowing us to argue strong optimality properties of our algorithm.Our algorithm requires several novel ideas: partitioning $\mathbb{M}$ in well-behaved portions, a local growing procedure to iteratively build contour trees, and the use of heavy path decompositions for the time complexity analysis. | \section{Introduction}
Geometric data is often represented as a function $f: \mathbb{R}^d \to \mathbb{R}$. Typically, a finite representation is given
by considering $f$ to be piecewise linear over some triangulated mesh (i.e.\ simplicial complex) $\mathbb{M}$ in $\mathbb{R}^d$.
\emph{Contour trees} are a topological structure used to represent and visualize
the function $f$.
It is convenient to think of $f$ as a manifold sitting in $\mathbb{R}^{d+1}$,
with the last coordinate (i.e.\ height) given by $f$. Imagine sweeping the hyperplane $x_{d+1} = h$
with $h$ going from $+\infty$ to $-\infty$. At every instance, the intersection of this plane
with $f$ gives a set of connected components, the \emph{contours} at height $h$. As the sweeping
proceeds various events occur: new contours are created or destroyed, contours merge into each other or
split into new components, contours acquire or lose handles.
The contour tree is a concise representation of all these events.
Throughout we follow the definition of contour trees from \cite{kobps-ctsssit-97} which includes all changes in topology.
For $d>2$, some subsequent works, such as \cite{csa-cctad-00}, only include changes in the number of components.
If $f$ is smooth, all points where the gradient of $f$ is zero are \emph{critical points}.
These points are the ``events" where the contour topology changes and form the vertices of the contour tree.
An edge of the contour tree connects two critical points if one event immediately ``follows"
the other as the sweep plane makes its pass. (We provide formal definitions later.)
\Fig{relative} and \Fig{sorting}
show examples of simplicial complexes, with heights and their contour trees.
Think of the contour tree edges as pointing downwards. Leaves are either maxima or minima,
and internal nodes are either ``joins" or ``splits".
Consider $f: \mathbb{M} \to \mathbb{R}$, where $\mathbb{M}$ is a triangulated mesh with $n$ vertices, $N$ faces in total, and $t \leq n$ critical points.
(We assume that $f:\mathbb{M} \to \mathbb{R}$ a linear interpolant over distinct valued vertices, where the contour tree
$T$ has maximum degree $3$. The degree assumption simplifies the presentation, and is commonly made~\cite{kobps-ctsssit-97}.)
A fundamental result in this area is the algorithm of Carr, Snoeyink, and Axen to compute contour trees,
which runs in $O(n\log n + N\alpha(N))$ time~\cite{csa-cctad-00} (where $\alpha(\cdot)$ denotes the inverse
Ackermann function). In practical applications, $N$ is typically $\Theta(n)$
(certainly true for $d=2$). The most expensive operation is an initial sort of all the vertex
heights. Chiang \textit{et~al.}\xspace build on this approach to get a faster algorithm that only sorts the critical vertices,
yielding a running time of $O(t\log t + N)$~\cite{cllr-sooscctmp-05}.
Common applications for contour trees involve turbulent combustion or noisy data, where the number of critical points is likely to be $\Omega(n)$.
There is a worst-case
lower bound of $\Omega(t\log t)$ by Chiang \textit{et~al.}\xspace \cite{cllr-sooscctmp-05}, based on a construction of Bajaj \textit{et~al.}\xspace \cite{BaKr+98}.
All previous algorithms begin by sorting (at least) the critical points. Can we beat this sorting bound for certain instances,
and can we characterize which inputs are hard? Intuitively, points that are incomparable in the contour tree
do not need to be compared. Look at \Fig{relative} to see such an example. All previous algorithms waste time
sorting all the maxima. Also consider the surface of \Fig{sorting}. The final contour tree is basically two binary
trees joined at their roots, and we do not need the entire sorted order of critical points
to construct the contour tree.
\begin{figure}[h]\centering
\includegraphics[width=.26\linewidth]{figs/spikey1}%
\hspace{.35in}
\includegraphics[width=.26\linewidth]{figs/spikey2}%
\hspace{.4in}
\includegraphics[width=.215\linewidth]{figs/spikeyTree}
\caption{Two surfaces with different orderings of the maxima, but the same contour tree.}
\label{fig:relative}
\end{figure}
\begin{figure}[h]\centering
\includegraphics[width=.35\linewidth,height=.15\linewidth]{figs/balanced4}
\hspace{.3in}
\includegraphics[width=.35\linewidth,height=.15\linewidth]{figs/trees}
\caption{On left, a surface with a balanced contour tree, but whose join and split trees have long tails.
On right (from left to right), the contour, join and split trees.}
\label{fig:sorting}
\end{figure}
Our main result gives an affirmative answer. Remember that we can consider
the contour tree as directed from top to bottom. For any node $v$ in the tree, let $\ell_v$ denote the length of the
longest directed path passing through $v$.
\begin{theorem} \label{thm:main-corr} Consider a simplicial complex $f:\mathbb{M} \to \mathbb{R}$, described as above, and denote
the contour tree by $T$ with vertex set (the critical points) $C(T)$. There exists an algorithm to compute the contour tree $T$
in $O(\sum_{v \in C(T)} \log \ell_v + t\alpha(t) + N)$ time. Moreover, this algorithm only
compares function values at pairs of points that are ancestor-descendant in $T$.
\end{theorem}
Essentially, the ``run time per critical point" is the height/depth of the point in the contour tree.
This bound immediately yields a run time of $O(t\log D + t\alpha(t) + N)$,
where $D$ is the diameter of the contour tree.
This is a significant improvement for short and fat contour trees. For example, if the tree is balanced,
then we get a bound of $O(t\log\log t)$.
Even if $T$ contains a long path of length $O(t/\log t)$, but is otherwise short, we get the improved bound of $O(t\log\log t)$.
\subsection{A refined bound with optimality properties}\label{sec:more-refined}
\Thm{main-corr} is a direct corollary of a stronger but more cumbersome theorem.
\begin{definition}
\label{def:path}
For a contour tree $T$, a \emph{leaf path} is any path in $T$ containing a leaf,
which is also monotonic in the height values of its vertices.
Then a \emph{path decomposition}, $P(T)$, is a partition of the vertices of $T$ into a set of vertex disjoint leaf paths.
\end{definition}
\begin{theorem}
\label{thm:main-alg}
There is a deterministic algorithm to compute the contour tree, $T$, whose running time is $O(\sum_{p \in P(T)} |p|\log |p| + t\alpha(t) + N)$,
where $P(T)$ is a specific path decomposition (constructed implicitly by the algorithm).
The number of comparisons made is $O(\sum_{p \in P(T)} |p|\log |p| + N)$.
In particular, any comparisons made are only between ancestors and descendants in the contour tree.
\end{theorem}
Note that \Thm{main-corr} is a direct corollary of this statement. For any $v$, $\ell_v$ is at most
the length of the path in $P(T)$ that contains $v$.
This bound is strictly stronger,
since for any balanced contour tree, the run time bound of \Thm{main-alg} is $O(t\alpha(t) + N)$, and $O(t)$
comparisons are made.
The bound of \Thm{main-alg} may seem artificial, since it actually depends on the $P(T)$ that
is implicitly constructed by the algorithm. Nonetheless, we prove that the algorithm of \Thm{main-alg}
has strong optimality properties. For convenience, fix some value of $t$, and consider the set
of terrains ($d=2$) with $t$ critical points. The bound of \Thm{main-alg} takes values ranging
from $t$ to $t\log t$. Consider some $C \in [t,t\log t]$, and consider the set of terrains
where the algorithm makes $C$ comparisons. Then \emph{any algorithm} must make roughly $C$
comparisons in the worst-case over this set. (All further details are in \InSoCGVer{Appendix~}\Sec{lb}.)
\begin{theorem}
\label{thm:main-lb}
There exists some absolute constant $\alpha$ such that the following holds.
For sufficiently large $t$ and any $C\in [t, t\log t]$, consider the set ${\bf F}_C$ of terrains with $t$ critical points such that
the number of comparisons made by the algorithm of \Thm{main-alg} on these terrains is in $[C,\alpha C]$.
Any algebraic decision tree that correctly computes
the contour tree on all of ${\bf F}_C$ has a worst case running time of $\Omega(C)$.
\end{theorem}
\subsection{Previous Work}
Contour trees were first used to study terrain maps by Boyell and Ruston, and Freeman and Morse~\cite{BoRu63,FrMo67}.
Contour trees have been applied in analysis of fluid mixing, combustion simulations,
and studying chemical systems~\cite{LaBe+06,BrWe+10,BeWe+11,BrWe+11,MaGr+11}. Carr's thesis~\cite{c-tmi-04} gives various
applications of contour trees for data visualization and is an excellent reference for contour tree definitions and algorithms.
The first formal result was an $O(N\log N)$ time algorithm for functions over 2D
meshes and an $O(N^2)$ algorithm for higher dimensions, by van Kreveld \textit{et~al.}\xspace \cite{kobps-ctsssit-97}.
Tarasov and Vyalya \cite{tv-cct-98} improved the running time to $O(N\log N)$ for the 3D case.
The influential paper of Carr \textit{et~al.}\xspace \cite{csa-cctad-00} improved the running time for all dimensions to $O(n\log n + N\alpha(N))$.
Pascucci and Cole-McLaughlin \cite{pc-ectls-02} provided an $O(n+t\log n)$ time algorithm for
$3$-dimensional structured meshes. Chiang \textit{et~al.}\xspace \cite{cllr-sooscctmp-05} provide an unconditional $O(N+t\log t)$ algorithm.
Contour trees are a special case of Reeb graphs, a general topological representation for real-valued functions
on any manifold. Algorithms for computing Reeb graphs
is an active topic of research~\cite{sk-crgacs-91,cehnp-lrbm-03,PaScBr07,DoNa09,HaWaWe10,Pa12}, where
two results explicitly reduce to computing contour trees~\cite{TiGySi09,DoNa13}.
\section{Contour tree basics} \label{sec:basics}
We detail the basic definitions about contour trees, following the terminology of Chapter 6 of Carr's thesis \cite{c-tmi-04}.
All our assumptions and definitions
are standard for results in this area, though there is some variability in notation.
The input is a continuous piecewise-linear function $f:\mathbb{M} \to \mathbb{R}$, where $\mathbb{M}$ is a simply connected and fully triangulated simplicial complex in $\mathbb{R}^d$,
except for specially designated \emph{boundary facets}. So $f$ is explicitly defined only on the vertices of $\mathbb{M}$,
and all other values are obtained by linear interpolation.
We assume that the boundary values satisfy a special property. This is mainly for convenience
in presentation.
\begin{definition} \label{def:bound} The function $f$ is \emph{boundary critical} if the following holds.
Consider a boundary facet $F$. All vertices of $F$ have the same function value. Furthermore, all
neighbors of vertices in $F$, which are not also in $F$ itself,
either have all function values strictly greater than or all function values strictly less than the function value at $F$.
\end{definition}
This is convenient, as we can now assume that $f$ is defined on $\mathbb{R}^d$. Any point inside a boundary facet
has a well-defined height, including the infinite facet, which is required to be a boundary facet. However,
we allow for other boundary facets, to capture the resulting surface pieces after our algorithm makes a horizontal cut.
We think of the dimension $d$, as constant, and assume that $\mathbb{M}$ is represented in a data structure that allows constant-time access to neighboring simplices
in $\mathbb{M}$ (e.g.~\cite{BoMa12}). (This is analogous to a doubly connected edge list, but for higher dimensions.)
Observe that $f:\mathbb{M} \rightarrow \mathbb{R}$ can be thought of as a $d$-dimensional simplicial complex living in $\mathbb{R}^{d+1}$,
where $f(x)$ is the ``height" of a point $x \in \mathbb{M}$, which is encoded in the representation of $\mathbb{M}$.
Specifically, rather than writing our input as $(\mathbb{M},f)$, we abuse notation and typically just write $\mathbb{M}$ to denote the lifted complex.
\begin{definition} \label{def:level} The \emph{level set} at value $h$ is the set $\{x| f(x) = h\}$.
A \emph{contour} is a connected component of a level set. An \emph{$h$-contour} is a contour where $f$-values are $h$.
\end{definition}
Note that a contour that does not contain a boundary is itself a simplicial complex of one dimension lower, and is represented (in our algorithms) as such.
We let $\delta$ and $\varepsilon$ denote infinitesimals. Let $B_\varepsilon(x)$ denote a ball of radius $\varepsilon$ around $x$, and let
$f|B_\varepsilon(x)$ be the restriction of $f$ to $B_\varepsilon(x)$.
\begin{definition} \label{def:deg} The \emph{Morse up-degree} of $x$ is the number of $(f(x) + \delta)$-contours of $f|B_\varepsilon(x)$
as $\delta, \varepsilon \rightarrow 0^+$. The \emph{Morse down-degree} is the number of $(f(x) - \delta)$-contours of $f|B_\varepsilon(x)$
as $\delta, \varepsilon \rightarrow 0^+$.
A \emph{regular} point has both Morse up-degree and down-degree $1$. A \emph{maximum} has Morse up-degree $0$, while
a \emph{minimum} has Morse down-degree $0$. A \emph{Morse Join} has Morse up-degree strictly greater than $1$,
while a \emph{Morse Split} has Morse down-degree strictly greater than $1$. Non-regular points are called \emph{critical}.
\end{definition}
The set of critical points is denoted by ${\cal V}(f)$.
Because $f$ is piecewise-linear, all critical points are vertices in $\mathbb{M}$.
A value $h$ is called \emph{critical}, if $f(v) = h$, for some $v \in {\cal V}(f)$.
A contour is called \emph{critical}, if it contains a critical point, and it is called \emph{regular} otherwise.
The critical points are exactly where the topology of level sets change.
By assuming that our manifold is boundary critical, the vertices on a given boundary are either collectively all maxima or all minima.
We abuse notation and refer to this entire set of vertices as a maximum or minimum.
\begin{definition} \label{def:equiv} Two regular contours $\psi$ and $\psi'$ are \emph{equivalent}
if there exists an $f$-monotone path $p$ connecting a point in $\psi$ to $\psi'$,
such that no $x \in p$ belongs to a critical contour.
\end{definition}
This equivalence relation gives a set of \emph{contour classes}.
Every such class maps to intervals
of the form $(f(x_i),f(x_j))$, where $x_i, x_j$ are critical points. Such a class is said
to be created at $x_i$ and destroyed at $x_j$.
\begin{definition} \label{def:tree} The \emph{contour tree} is the graph on vertex set ${\cal V}= {\cal V}(f)$, where
edges are formed as follows. For every contour class that is created at $v_i$ and destroyed $v_j$,
there is an edge $(v_i,v_j)$. (Conventionally, edges are directed from higher to lower function value.)
\end{definition}
We denote the contour tree of $\mathbb{M}$
by $\cC(\mathbb{M})$. The corresponding node and edge sets are denoted as ${\cal V}(\cdot)$ and ${\cal E}(\cdot)$.
It is not immediately obvious that this graph is a tree, but alternate definitions of the contour tree
in~\cite{csa-cctad-00} imply this is a tree.
Since this tree has height values associated with the vertices, we can talk about up-degrees and down-degrees in $\cC(\mathbb{M})$.
Similar to \cite{kobps-ctsssit-97} (among others), multi-saddles are treated as a set of ordinary saddles, which can be realized
via vertex unfolding (which can increase surface complexity if multi-saddle degrees are allowed to be super-constant).
Therefore, to simplify the presentation, for the remainder of the paper up and down-degrees are at most $2$, and total degree is at most $3$.
\InSoCGVer{See \Sec{techmarks} for further technical remarks on the above definitions.}
\newcommand{\technicalRemarks}{
Note that if one intersects $\mathbb{M}$ with a given ball $B$, then a single contour in $\mathbb{M}$ might be split into more than one contour in the intersection.
In particular, two $(f(x)+\delta)$-contours of $f|_{B_\varepsilon(x)}$, given by \Def{deg}, might actually be the same contour in $\mathbb{M}$.
Alternatively, one can define the up-degree (as opposed to \emph{Morse} up-degree) as the number of $(f(x)+\delta)$-contours (in the full $\mathbb{M}$)
that intersect $B_\varepsilon(x)$, a potentially smaller number. This up-degree
is exactly the up-degree of $x$ in $\cC(\mathbb{M})$. (Analogously, for down-degree.)
When the Morse up-degree is $2$ but the up-degree is $1$, the topology of the level set changes but not by the number of connected components changing.
For example, when $d=3$ this is equivalent to the contour gaining a handle.
When $d=2$, this distinction is not necessary, since any point with Morse degree strictly greater than $1$ will have degree strictly greater than $1$ in $\cC(\mathbb{M})$.
As Carr points out in Chapter 6 of his thesis, the term contour tree can be used for a family of related structures.
Every vertex in $\mathbb{M}$ is associated with an edge in $\cC(\mathbb{M})$, and sometimes the
vertex is explicitly placed in $\cC(\mathbb{M})$ (by subdividing the respective edge). This is referred
to as augmenting the contour tree, and it is common to augment $\cC(\mathbb{M})$ with all vertices.
Alternatively, one can smooth out all vertices of up-degree and down-degree $1$ to get the
unaugmented contour tree. (For $d=2$, there are no such vertices in $\cC(\mathbb{M})$.)
The contour tree of \Def{tree} is the typical definition in all results on output-sensitive
contour trees, and is the smallest tree that contains all the topological changes
of level sets. \Thm{main-alg} is applicable for any augmentation of $\cC(\mathbb{M})$ with a predefined
set of vertices, though we will not delve into these aspects in this paper.
}
\InNotSoCGVer{
\subsection{Some technical remarks}
\technicalRemarks
}
\newcommand{\newintuitionSection}{
\section{A tour of the new contour tree algorithm} \label{sec:approach}
We provide a short, high-level description of the main ideas of subsequent sections.
A longer more detailed version of this high-level description can be found in the full version \CiteFullVer.
\myparagraph{Cutting $\mathbb{M}$ into extremum dominant pieces:} We define a simplicial complex
endowed with a height to be \emph{extremum dominant} if there exists only a single minimum, or only a single maximum.
(When formally defined in \Sec{rain}, extremum dominant complexes will allow additional trivial minima or maxima which are a small complication resulting from the following cutting procedure.)
We first cut $\mathbb{M}$ into disjoint extremum dominant pieces in linear time.
Take an arbitrary maximum $x$ and imagine torrential rain at the maximum. The water flows down, wetting any point that has a non-ascending path
from $x$. We end up with two portions, the wet part of $\mathbb{M}$ and the dry part. This is similar
to \emph{watershed} algorithms used for image segmentation~\cite{RoMe00}. The wet part is obviously connected and
can be shown to be extremum dominant. We can cut along the interface of the wet and dry, remove
the wet part, and recurse on the dry parts. This is carefully done to ensure that the total complexity
of the pieces is linear.
We prove a simple contour surgery theorem that builds the contour tree of $\mathbb{M}$
from the contour trees of the various pieces created.
Using ideas from~\cite{csa-cctad-00}, we prove that the contour tree of an extremum dominant complex is (basically) just the \emph{join tree},
which unlike the contour tree tracks superlevel sets rather than level sets.
Consider sweeping down the hyperplane $x_{d+1} = h$ and taking the connected components of the portion of $\mathbb{M}$ above height $h$. For a terrain, these superlevel
sets are a collection of ``mounds". As we sweep downwards, these mounds keep joining each other,
until finally, we end up with all of $\mathbb{M}$. The join tree tracks exactly these events.
\myparagraph{Join trees from painted mountaintops:} Our main result is a faster algorithm for join trees.
The key idea is \emph{paint spilling}. Start with each maximum having a large can of paint, with distinct
colors for each maximum. In arbitrary order, we spill paint from each maximum, wait till it flows down,
then spill from the next, etc. Paint is viscous, and only flows down edges. Furthermore, our paints do not mix, so each edge receives a unique color, decided by the first
paint to reach it.
Our algorithm incrementally builds the join tree from the leaves (maxima) to the root.
Refer to the left part of \Fig{colors}.
Consider two sibling leaves $\ell_1, \ell_2$ and their common parent $v$. The leaves are maxima,
and $v$ is a join that ``merges" $\ell_1, \ell_2$.
In that case, there are ``mounds" corresponding to $\ell_1$ and $\ell_2$ that merge
at a valley $v$. Suppose this was the entire input, and $\ell_1$ was colored blue and $\ell_2$ was colored red.
Both mounds are colored completely blue or red, while $v$ is touched by both colors.
So this indicates that $v$ joins the blue maximum and red maximum in the join tree and is a critical point.
To process $v$, we ``merge" the colors red and blue into a new color, purple.
In terms of the join tree, this
is equivalent to removing leaves $\ell_1$ and $\ell_2$, and making $v$ a new leaf.
Of course, things are more complicated when there are other mounds. There may be a yellow mound,
corresponding to $\ell_3$ that joins with the blue mound higher up at some vertex $u$ (see the right part of \Fig{colors}). In the join tree, $\ell_1$ and $\ell_3$
are sibling leaves, and $\ell_2$ is a sibling of some ancestor of these leaves. So we cannot
merge red and blue, until yellow and blue merge.
A critical insight is the use of \emph{binomial heaps}~\cite{Vu78} to handle all the priorities and the merging.
This ensures that all comparisons are only made between points that are ancestor-descendant in the join tree.
There are numerous details to handle to prove the final run time bound, but this should provide the reader
some intuition behind our algorithm.
\begin{figure}[h]\centering
\includegraphics[width=.25\linewidth]{figs/redBlue}%
\hspace{.07in}
\includegraphics[width=.25\linewidth]{figs/redBlue2}%
\hspace{.07in}
\includegraphics[width=.03\linewidth]{figs/redBlueTree}
\hspace{.09in}
\includegraphics[width=.3\linewidth]{figs/mound}%
\hspace{.07in}
\includegraphics[width=.065\linewidth]{figs/moundTree}
\caption{On the left, red and blue merge to make purple, followed by the contour tree with initial colors. On the right, additional maxima and the resulting contour tree.}
\label{fig:colors}
\end{figure}
}
\newcommand{\intuitionSection}{
\section{A tour of the new contour tree algorithm} \label{sec:approach}
\InNotSoCGVer{
Our final algorithm is quite technical and has numerous moving parts. However, for the $d=2$ case,
where the input is just a triangulated terrain, the main ideas of the parts of the algorithm
can be explained clearly. Therefore, here we first provide a high level view of the entire result.
}
\InSoCGVer{
For the $d=2$ case, where the input is just a triangulated terrain, the main ideas of the parts of the algorithm
can be explained clearly. Therefore, here we provide a high level view of the entire result.
}
In the interest of presentation, the definitions and theorem statements in this section will
slightly differ from those in the main body. They may also differ from the original definitions proposed
in earlier work.
\myparagraph{Do not globally sort:} The starting point for this work is \Fig{relative}. We have two terrains
with exactly the same contour tree, but different orderings of (heights of) the critical points.
Turning it around, we cannot deduce the full height ordering of critical points from the contour tree.
Sorting all critical points is computationally unnecessary for constructing the contour tree.
In \Fig{sorting}, the contour tree consists of two balanced binary trees, one of the joins,
another of the splits. Again, it is not necessary to know the relative ordering between the mounds
on the left (or among the depressions on the right) to compute the contour tree. Yet some ordering
information is necessary: on the left, the little valleys are higher than the big central valley,
and this is reflected in the contour tree. Leaf paths in the contour tree have points
in sorted order, but incomparable points in the tree are unconstrained.
How do we sort exactly what is required, without knowing the contour tree in advance?
\subsection{Breaking $\mathbb{M}$ into simpler pieces} \label{sec:break}
Let us begin with the algorithm of Carr, Snoeyink, and Axen~\cite{csa-cctad-00}. The key insight is to build
two different trees, called the join and split trees, and then merge them together into the contour tree.
Consider sweeping down the hyperplane $x_{d+1} = h$ and taking the \emph{superlevel} sets. These
are the connected components of the portion of $\mathbb{M}$ above height $h$. For a terrain, the superlevel
sets are a collection of ``mounds". As we sweep downwards, these mounds keep joining each other,
until finally, we end up with all of $\mathbb{M}$. The join tree tracks exactly these events.
Formally, let $\mathbb{M}^+_v$ denote the simplicial complex induced on the subset of
vertices which are higher than $v$.
\begin{definition}
\label{def:int-criticalJoin}
The \emph{join tree} ${\cal J}(\mathbb{M})$ is built on the set ${\cal V}$ of all critical points.
The directed edge $(u,v)$ is present when $u$ is the smallest valued vertex in ${\cal V}$ in a connected component of $\mathbb{M}^+_v$
and $v$ is adjacent (in $\mathbb{M}$) to a vertex in this component.
\end{definition}
Refer to \Fig{sorting} for the join tree of a terrain. Note that nothing happens at splits, but these
are still put as vertices in the join tree. They simply form a long path. The split tree
is obtained by simply inverting this procedure, sweeping upwards and tracking sublevel sets.
A major insight of~\cite{csa-cctad-00} is an ingeniously simple linear time procedure to construct
the contour tree from the join and split trees. So the bottleneck is computing these trees. Observe
in \Fig{sorting} that the split vertices form a long path in the join tree (and vice versa). Therefore, constructing
these trees forces a global sort of the splits, an unnecessary computation for the contour tree. Unfortunately, in general (i.e.\ unlike \Fig{sorting})
the heights of joins and splits may be interleaved in a complex manner, and hence the final merging of~\cite{csa-cctad-00}
to get the contour tree requires having the split vertices in the join tree. Without
this, it is not clear how to get a consistent view of both joins and splits, required for the contour tree.
Our aim is to break $\mathbb{M}$ into smaller pieces, where this unnecessary computation can be avoided.
\myparagraph{Contour surgery:} We first need a divide-and-conquer lemma. Any contour $\phi$ can be associated
with an edge $e$ of the contour tree. Suppose we ``cut" $\mathbb{M}$ along this contour. We prove that $\mathbb{M}$
is split into two disconnected pieces, such the contour trees of these pieces is obtained
by simply cutting $e$ in $\cC(\mathbb{M})$. Alternatively, the contour trees of these pieces can
be glued together to get $\cC(\mathbb{M})$. This is not particularly surprising, and is fairly easy to prove
with the right definitions. The idea of loop surgery has been used to reduce Reeb graphs to contour trees~\cite{TiGySi09,DoNa13}.
Nonetheless, our theorem appears to be new and works for all dimensions.
\begin{figure}[h]\centering
\includegraphics[width=.172\linewidth]{figs/tent2}%
\hspace{1.2in}
\includegraphics[width=.172\linewidth]{figs/tent1}
\caption{On left, downward rain spilling only (each shade of gray represents a piece created by each different spilling), producing a grid.
Note we are assuming raining was done in sorted order of the maxima (i.e.\ lowest to highest).
On right, flipping the direction of rain spilling.}
\label{fig:order}
\end{figure}
\myparagraph{Cutting $\mathbb{M}$ into extremum dominant pieces:} We define a simplicial complex
endowed with a height to be \emph{minimum dominant} if there exists only a single minimum.
(Our real definition is more complicated, and involves simplicial complexes
that allow additional ``trivial'' minima.)
In such a complex, there exists a non-ascending path from any point to this unique minimum.
Analogously, we can define maximum dominant complexes,
and both are collectively called extremum dominant.
We will cut $\mathbb{M}$ into disjoint extremum dominant pieces, in linear time.
One way to think of our procedure is a meteorological analogy. Take an arbitrary maximum $x$,
and imagine torrential rain at the maximum. The water flows down, wetting any point that has a non-ascending path
from $x$. We end up with two portions, the wet part of $\mathbb{M}$ and the dry part. This is similar
to \emph{watershed} algorithms used for image segmentation~\cite{RoMe00}. The wet part is obviously connected,
while there may be numerous disconnected dry parts. The interface between the dry and wet parts
is a set of contours\footnote{Technically, they are not contours, but rather the limits of sequences of contours.},
given by the ``water line". The wet part is clearly maximum dominant, since all
wet points have a non-descending path to $x$. So we can simply cut along the interface
contours to get the wet maximum dominant piece $\mathbb{M}'$. By our contour surgery theorem,
we are left with a set of disconnected dry parts, and we can recur this procedure on them.
But here's the catch. Every time we cut $\mathbb{M}$ along a contour, we potentially increase
the complexity of $\mathbb{M}$. Water flows in the interior of faces, and the interface will naturally
cut some faces. Each cut introduces new vertices, and a poor choice of repeated raining leads
to a large increase in complexity. Consider the left of \Fig{order}. Each raining produces a single
wet and dry piece, and each cut introduces many new vertices. If we wanted to partition this terrain into
maximum dominant simplicial complexes, the final complexity would be forbiddingly large.
A simple trick saves the day. Unlike reality, we can choose rain to flow solely downwards or solely upwards.
Apply the procedure above to get a single wet maximum dominant $\mathbb{M}'$ and a set of dry pieces.
Observe that a single dry piece $\mathbb{N}$ is boundary critical with the newly introduced boundary $\phi$ (the wet-dry interface) behaving as a minimum.
So we can rain upwards from this minimum, and get a \emph{minimum dominant} portion $\mathbb{N}'$.
This ensures that the new interface (after applying the procedure on $\mathbb{N}$) does not
cut any face previously cut by $\phi$.
For each of the new dry pieces, the newly introduced boundary is now a maximum. So we rain downwards from there.
More formally, we alternate between raining upwards and downwards as we go down the recursion tree.
We can prove that an original face of $\mathbb{M}$ is cut at most once, so the final complexity can be bounded.
In \Fig{order}, regardless of the choice of the starting maximum, this procedure would yield (at most) two pieces,
one maximum dominant, and one minimum dominant.
Using the contour surgery theorem previously discussed, we can build the contour tree of $\mathbb{M}$
from the contour trees of the various pieces created. All in all, we prove the following theorem.
\begin{theorem} \label{thm:int-rain} There is an $O(N)$ time procedure that cuts $\mathbb{M}$ into extremum dominant simplicial complexes
$\mathbb{M}_1, \mathbb{M}_2, \ldots$. Furthermore, given the set of contour trees $\{{\cal C}(\mathbb{M}_i)\}$, ${\cal C}(\mathbb{M})$ can be constructed in $O(N)$ time.
\end{theorem}
\myparagraph{Extremum dominance simplifies contour trees:} We will focus on minimum dominant simplicial complexes $\mathbb{M}$. By \Thm{int-rain}, it suffices
to design an algorithm for contour trees on such inputs.
For the $d=2$ case, it helps to visualize such an input as a terrain with no minima,
except at a unique boundary face (think of a large boundary triangle that is the boundary).
All the saddles in such a terrain are necessarily joins, and there can be no splits.
In \Fig{sorting}, the portion on the left is minimum dominant in exactly this fashion, albeit in one dimension lower.
More formally, $\mathbb{M}^-_v$ is connected for all $v$, so there are no splits.
We can prove that the split tree is just a path, and the contour tree is exactly
the join tree. The formal argument is a little involved, and we employ the merging procedure
of~\cite{csa-cctad-00} to get a proof. We demonstrate that the merging procedure will
actually just output the join tree, so we do not need to actually compute the split tree.
(The real definition of minimum dominant is a little more complicated,
so the contour tree is more than just the join tree. But computationally, it suffices
to construct the join tree.)
We stress the importance of this step for our approach. Given the algorithm of~\cite{csa-cctad-00},
one may think that it suffices to design faster algorithms for join trees. But this
cannot give the sort of optimality we hope for. Again, consider \Fig{sorting}. Any algorithm
to construct the true join tree must construct the path of splits, which implies
sorting all of them. It is absolutely necessary to cut $\mathbb{M}$ into pieces where the cost
of building the join tree can be related to that of building ${\cal C}(\mathbb{M})$.
\subsection{Join trees from painted mountaintops} \label{sec:join-paint}
Arguably, everything up to this point is a preamble for the main result: a faster
algorithm for join trees. Our algorithm does not require the initial input to be extremum
dominant. This is only required to relate the join trees, of the resulting subcomplexes of Theorem~\ref{thm:int-rain}, to the contour tree of the initial input $\mathbb{M}$.
For clarity, we use $\mathbb{N}$ to denote the input here. Note that in \Def{int-criticalJoin},
the join tree is defined purely combinatorially in terms of the 1-skeleton (the underlying graph) of $\mathbb{N}$.
The join tree ${\cal J}(\mathbb{N})$ is a rooted tree with the dominant minimum at the root, and we direct edges downwards (towards the root). So it makes sense to talk
of comparable vs incomparable vertices.
We arrive at the main challenge: how to sort only the comparable critical points, without
constructing the join tree? The join tree algorithm of~\cite{csa-cctad-00} is a typical event-based
computational geometry algorithm. We have to step away from this viewpoint to avoid the global sort.
The key idea is \emph{paint spilling}. Start with each maximum having a large can of paint, with distinct
colors for each maximum. In arbitrary order, we spill paint from each maximum, wait till it flows down,
then spill from the next, etc. Paint is viscous, and only flows down edges. \emph{It does not
paint the interior of higher dimensional faces.} That is, this process is restricted to the 1-skeleton of $\mathbb{N}$.
Furthermore, our paints do not mix, so each edge receives a unique color, decided by the first
paint to reach it. In the following, $[n]$ denotes the set $\{1, \ldots, n\}$, for any natural number $n$.
\begin{definition} \label{def:paint1} \InSoCGVer{(Restatement of \Def{paint2}.)} Let the 1-skeleton of $\mathbb{N}$ have edge set $E$ and maxima $X$.
A \emph{painting} of $\mathbb{N}$ is a map $\chi:X \cup E \to [|X|]$ with the following property.
Consider an edge $e$. There exists a descending path from some maximum $x$ to $e$
consisting of edges in $E$, such that all edges along this path have the same color as $x$.
An \emph{initial} painting has the additional property that the restriction $\chi:X \to [|X|]$ is a bijection.
\end{definition}
Note that a painting colors edges, and not vertices (except for maxima). Our definition also does not require the timing aspect
of iterating over colors, though that is one way of painting $\mathbb{N}$. We begin with an initial painting, since all maximum
colors are distinct. A few comments on paint vs water. The interface between two regions of different color
is \emph{not} a contour, and so we cannot apply the divide-and-conquer
approach of contour surgery. On the other hand, painting does not cut $\mathbb{N}$, so there is no increase in complexity.
Clearly, an initial painting can be constructed in $O(N)$ time. This is the tradeoff between water and paint.
Water allows for an easy divide-and-conquer, at the cost of more complexity in the input. For an extremum
dominant input, using water to divide the input $\mathbb{N}$ raises the complexity too much.
Our algorithm incrementally builds ${\cal J}(\mathbb{M})$ from the leaves (maxima) to the root (dominant minimum).
We say that vertex $v$ is \emph{touched} by color $c$, if there is a $c$-colored edge with lower endpoint $v$.
Let us focus on an initial painting, where the colors have 1-1 correspondence with the maxima. Refer to the left part of \Fig{colors}.
Consider two sibling leaves $\ell_1, \ell_2$ and their common parent $v$. The leaves are maxima,
and $v$ is a join that ``merges" $\ell_1, \ell_2$.
In that case, there are ``mounds" corresponding to $\ell_1$ and $\ell_2$ that merge
at a valley $v$. Suppose this was the entire input, and $\ell_1$ was colored blue and $\ell_2$ was colored red.
Both mounds are colored completely blue or red, while $v$ is touched by both colors.
So this indicates that $v$ joins the blue maximum and red maximum in ${\cal J}(\mathbb{M})$.
This is precisely how we hope to exploit the information in the painting. We prove later that
when some join $v$ has all incident edges with exactly two colors, the corresponding maxima (of those colors) are exactly
the children of $v$ in ${\cal J}(\mathbb{M})$. To proceed further, we ``merge" the colors red and blue into a new color, purple. In other words,
we replace all red and blue edges by purple edges. This indicates that the red and blue maxima have been
handled. Imagine flattening the red and blue mounds until reaching $v$, so that the former join $v$
is now a new maximum, from which purple paint is poured. In terms of ${\cal J}(\mathbb{M})$, this
is equivalent to removing leaves $\ell_1$ and $\ell_2$, and making $v$ a new leaf.
Alternatively, ${\cal J}(\mathbb{M})$ has been constructed up to $v$, and it remains to determine $v$'s parent.
The merging of the colors is not explicitly performed as that would be too expensive; instead we maintain a union-find
data structure for that.
Of course, things are more complicated when there are other mounds. There may be a yellow mound,
corresponding to $\ell_3$ that joins with the blue mound higher up at some vertex $u$ (see the right part of \Fig{colors}). In ${\cal J}(\mathbb{M})$, $\ell_1$ and $\ell_3$
are sibling leaves, and $\ell_2$ is a sibling of some ancestor of these leaves. So we cannot
merge red and blue, until yellow and blue merge. Naturally, we use priority queues to handle this issue.
We know that $u$ must also be touched by blue. So all critical vertices touched by blue
are put into a priority queue keyed by height, and vertices are handled in that order.
\begin{figure}[h]\centering
\includegraphics[width=.25\linewidth]{figs/redBlue}%
\hspace{.07in}
\includegraphics[width=.25\linewidth]{figs/redBlue2}%
\hspace{.07in}
\includegraphics[width=.03\linewidth]{figs/redBlueTree}
\hspace{.09in}
\includegraphics[width=.3\linewidth]{figs/mound}%
\hspace{.07in}
\includegraphics[width=.065\linewidth]{figs/moundTree}
\caption{On the left, red and blue merge to make purple, followed by the contour tree with initial colors. On the right, additional maxima and the resulting contour tree.}
\label{fig:colors}
\end{figure}
What happens when finally blue and red join at $v$? We merge the two colors, but now have
blue and red queues of critical vertices, which also need to be merged to get a consistent painting.
This necessitates using a priority queue with efficient merges.
Specifically, we use \emph{binomial heaps}~\cite{Vu78}, as they provide logarithmic time merges and deletes (though other similar heaps work).
We stress that the feasibility of the entire approach hinges on the use of such an efficient heap structure.
In this discussion, we ignored an annoying problem. Vertices
may actually be touched by numerous colors, not just one or two as assumed above. A simple solution
would be to insert vertices into heaps corresponding to all colors touching it. But there could
be super-constant numbers of copies of a vertex, and handling all these copies would lead
to extra overhead. We show that it suffices to simply put each vertex $v$ into at most two heaps, one for each
``side" of a possible join. We are guaranteed that when $v$ needs to be processed, all edges have
at most $2$ colors, because of all the color merges that previously occurred.
\myparagraph{The running time analysis:}
All the non-heap operations can be easily bounded by $O(t\alpha(t) + N)$ (the $t\alpha(t)$ is from the union-find data structure
for colors). It is not hard to argue that at all times, any heap always contains a subset of a leaf to root path.
This observation suffices to get a running time bound which is the analogue of \Thm{main-corr}, but for join trees.
Each heap deletion and merge can be charged to a vertex in the join tree (where each vertex gets charged only a constant number of times).
Let $d_v$ denote the distance to the root for a vertex $v$ in the join tree, then since a heap's elements are on a single leaf to root
path, the size of $v$'s heap at the time an associated heap operation is made is at most $d_v$.
Therefore, the total cost (of the heap operations) is at most $O(\sum_v \log d_v)$. This immediately proves a bound of $O(t\log D)$, where
$D$ is the maximum distance to the root, an improvement over previous work.
However, this bound is non-optimal. For example, for a balanced binary tree, this gives a bound of $O(t\log\log t)$, however, by using
an analysis involving path decompositions we can get an $O(t)$ bound. Imagine walking from some leaf
towards the root. Each vertex on this path can have at most two colors (the ones getting merged), however, as we get closer to the root
the competition for which two colors a vertex gets assigned to grows, as the number of descendant leaf colors grows. This means that for some vertices,
$v$, the size of the heap for an associated heap operation will be significantly smaller than $d_v$.
The intuition is that the paint spilling from the maxmima in the simplicial complex, corresponds to paint spilling from the leaves in
the join tree, which decomposes the join tree into a set of colored paths. Unfortunately, the situation is more complex since
while a given color class is confined to a single leaf to root path, it may \emph{not} appear contiguously on this path, as the right part of \Fig{colors} shows.
Specifically, in this figure the far left saddle (labeled $i$) is hit by blue paint. However, there is another
saddle on the far right (labeled $j$) which is not hit by blue paint. Since this far right saddle is slightly
higher than the far left one, it will merge into the component containing the blue mound (and also the yellow and red mounds)
before the far left one. Hence, the vertices initially touched by blue are not contiguous in the join tree.
This non-contiguous complication along with the fact that heap size keep changing as color classes merge, causes the analysis to be technically challenging.
We employ a variant of \emph{heavy path decompositions}, first used by Sleator and Tarjan for analyzing link/cut trees~\cite{st-dsdt-83}.
The final analysis charges expensive heap operations to long paths in the decomposition, resulting in the bound stated in \Thm{main-alg}.
\subsection{The lower bound}
Consider a contour tree $T$ and the path decomposition $P(T)$ used to bound the running time.
Denoting $\mathop{cost}(P(T)) = \sum_{p \in P(T)} |p|\log |p|$, we construct a set of $\prod_{p \in P(T)} |p|!$
functions on a fixed domain such that each function has a distinct (labeled) contour tree. By a simple
entropy argument, any algebraic decision tree that correctly computes the contour tree on all instances
requires worst case $\Omega(\mathop{cost}(P(T)))$ time. We prove that our algorithm makes $\Theta(\mathop{cost}(P(T)))$
comparisons on all these instances.
We have a fairly simple construction that works for terrains. In $P(T)$, consider the path
$p$ that involves the root. The base of the construction is a conical ``tent", and there
will be $|p|$ triangular faces that will each have a saddle. The heights of these saddles
can be varied arbitrarily, and that will give $|p|!$ different choices. Each of these saddles
will be connected to a recursive construction involving other paths in $P(T)$. Effectively,
one can think of tiny tents that are sticking out of each face of the main tent. The contour
trees of these tiny tents attach to a main branch of length $|p|$. Working out the details,
we get $\prod_{p \in P(T)} |p|!$ terrains each with a distinct contour tree.
}
\InSoCGVer{
\newintuitionSection
}
\InNotSoCGVer{
\intuitionSection
}
\section{Divide and conquer through contour surgery} \label{sec:surgery}
{\bf The cutting operation:} We define a ``cut" operation on $f:\mathbb{M} \rightarrow \mathbb{R}$ that cuts along a regular contour to create
a new simplicial complex with an added boundary. Given a contour $\phi$, roughly speaking, this constructs the simplicial complex $\mathbb{M} \setminus \phi$.
We will always enforce the condition that $\phi$ never passes through a vertex of $\mathbb{M}$.
Again, we use $\varepsilon$ for an infinitesimally small value. We denote $\phi^+$ (resp. $\phi^-$) to be
the contour at value $f(\phi) + \varepsilon$ (resp. $f(\phi) - \varepsilon$), which is at distance $\varepsilon$
from $\phi$.
An $h$-contour is achieved by intersecting $\mathbb{M}$ with the hyperplane $x_{d+1} = h$ and taking a connected component. (Think of the $d+1$-dimension
as height.) Given some point $x$ on an $h$-contour $\phi$, we can walk along $\mathbb{M}$ from $x$ to determine $\phi$.
We can ``cut" along $\phi$ to get a new (possibly) disconnected simplicial complex $\mathbb{M}'$. This is achieved
by splitting every face $F$ that $\phi$ intersects into an ``upper" face and ``lower" face. Algorithmically,
we cut $F$ with $\phi^+$ and take everything above $\phi^+$ in $F$ to make the upper face. Analogously, we cut with $\phi^-$ to get the lower face.
The faces are then triangulated to ensure that they are all simplices.
\Ben{Reviewer wanted clarification on how triangulation is done. Maybe can use bottom-vertex triangulation of Clarkson 88, discussed in Matousek book?}
This creates the two new boundaries $\phi^+$ and $\phi^-$, and we maintain the property of constant $f$-value at a boundary.
Note that by assumption $\phi$ cannot cut a boundary face, and moreover all non-boundary faces
have constant size. Therefore, this process takes time linear in $|\phi|$, the number of faces $\phi$ intersects.
This new simplicial complex is denoted by ${\tt cut}(\phi,\mathbb{M})$.
We now describe a high-level approach to construct $\cC(\mathbb{M})$ using this cutting procedure.
\Ben{Surgery should be rewritten to take instead take as input the output of cut, i.e. two complexes and a contour.
This is how it is eventually used in Claim\ref{clm:rain-reeb}. Currently it does nothing really.}
\medskip
\fbox{
\begin{minipage}{0.9\textwidth}
{\bf ${\tt surgery}(\mathbb{M},\phi)$}
\smallskip
\begin{asparaenum}
\item Let $\mathbb{M}' = {\tt cut}(\mathbb{M},\phi)$.
\item Construct $\cC(\mathbb{M}')$ and let $A, B$ be the nodes corresponding to the new boundaries created
in $\mathbb{M}'$. (One is a minimum and the other is maximum.)
\item Since $A, B$ are leaves, they each have unique neighbors $A'$ and $B'$, respectively. Insert
edge $(A',B')$ and delete $A, B$ to obtain $\cC(\mathbb{M})$.
\end{asparaenum}
\end{minipage}}
\medskip
\InSoCGVer{The following theorems are intuitively obvious, and are proven in \Sec{proofOfSurgery}.}
\begin{theorem} \label{thm:surgery} For any regular contour $\phi$, the output of ${\tt surgery}(\mathbb{M},\phi)$ is $\cC(\mathbb{M})$.
\end{theorem}
\newcommand{\surgeryBody}{
To prove Theorem~\ref{thm:surgery}, we require a theorem from \cite{c-tmi-04} (Theorems 6.6) that map paths in ${\cal C}(\mathbb{M})$ to $\mathbb{M}$.
\begin{theorem} \label{thm:carr-path} For every path $P$ in $\mathbb{M}$, there exists a path $Q$ in the contour tree corresponding
to the contours passing through points in $P$.
For every path $Q$ in the contour tree, there exists at least one path $P$ in $\mathbb{M}$ through points present
in contours involving $Q$.
In particular, for every monotone path $P$ in $\mathbb{M}$, there exists a monotone path $Q$ in the contour tree to which $P$
maps, and vice versa.
\end{theorem}
Theorem~\ref{thm:surgery} is a direct consequence of the following lemma.
\begin{lemma} \label{lem:cut} Consider a regular contour $\phi$ contained in a contour class (of an edge of $\cC(\mathbb{M}))$
$(u,v)$ and let $\mathbb{M}' = {\tt cut}(\mathbb{M},\phi)$. Then ${\cal V}(\cC(\mathbb{M}')) = \{\phi^+,\phi^-\} \cup {\cal V}(\mathbb{M})$
and ${\cal E}(\cC(\mathbb{M}')) = \{(u,\phi^+),(\phi^-,v)\} \cup ({\cal E}(\mathbb{M}) \setminus (u,v))$.
\end{lemma}
\begin{proof} First observe that since $\phi$ is a regular contour, the vertex set in the complex $\mathbb{M}'$ is the same
as the vertex set in $\mathbb{M}$, except with the addition of the newly created vertices on $\phi^+$ and $\phi^-$.
Moreover, ${\tt cut}(\mathbb{M},\phi)$ does not affect the local neighborhood of any vertex in $\mathbb{M}$.
Therefore since a vertex being critical is a local condition, with the exception of new boundary vertices, the
critical vertices in $\mathbb{M}$ and $\mathbb{M}'$ are the same. Finally, the new vertices on $\phi^+$ and $\phi^-$
collectively behave as a minimum and maximum, respectively, and so ${\cal V}(\cC(\mathbb{M}')) = \{\phi^+,\phi^-\} \cup {\cal V}(\mathbb{M})$.
Now consider the edge sets of the contour trees. Any contour class in $\mathbb{M}'$ (i.e.\ edge in ${\cal C}(\mathbb{M}')$) that does not involve $\phi^+$ or $\phi^-$ is also
a contour class in $\mathbb{M}$. Furthermore, a maximal contour class satisfying these properties is also
maximal in $\mathbb{M}$. So all edges of ${\cal C}(\mathbb{M}')$ that do not involve $\phi^+$ or $\phi^-$ are edges of ${\cal C}(\mathbb{M})$.
Analogously, every edge of ${\cal C}(\mathbb{M})$ not involving $\phi$ is an edge of ${\cal C}(\mathbb{M}')$.
Consider the contour class corresponding to edge $(u,v)$ of ${\cal C}(\mathbb{M})$. There is a natural ordering
of the contours by function value, ranging from $f(u)$ to $f(v)$. All contours in this class ``above" $\phi$
form a maximal contour class in $\mathbb{M}'$, represented by edge $(u,\phi^+)$. Analogously, there is another
contour class represented by edge $(\phi^-,v)$. We have now accounted for all contours in ${\cal C}(\mathbb{M}')$,
completing the proof.
\end{proof}
A useful corollary of this lemma shows that a contour actually splits the simplicial complex into
two disconnected complexes.
}
\InNotSoCGVer{
\surgeryBody
}
\begin{theorem} \label{thm:jordan} ${\tt cut}(\mathbb{M},\phi)$ consists of two
disconnected simplicial complexes.
\end{theorem}
\newcommand{\proofofDisconnected}{
\begin{proof} Denote (as in \Lem{cut}) the edge containing $\phi$ to be $(u,v)$. Suppose for contradiction that there is a path between vertices $u$ and $v$
in $\mathbb{M}' = {\tt cut}(\mathbb{M},\phi)$. By \Thm{carr-path}, there is a path in ${\cal C}(\mathbb{M}')$ between $u$ and $v$. Since $\phi^+$ and $\phi^-$
are leaves in ${\cal C}(\mathbb{M}')$, this path cannot use their incident edges. Therefore by \Lem{cut},
all the edges of this path are in ${\cal E}({\cal C}(\mathbb{M})) \setminus (u,v)$. So we get a cycle in ${\cal C}(\mathbb{M})$, a contradiction.
To show that there are exactly two connected components in ${\tt cut}(\mathbb{M},\phi)$, it suffices
to see that ${\cal C}(\mathbb{M}')$ has two connected components (by \Lem{cut}) and then applying \Thm{carr-path}.
\end{proof}
}
\InNotSoCGVer{\proofofDisconnected}
\section{Raining to partition $\mathbb{M}$} \label{sec:rain}
In this section, we describe a linear time procedure that partitions $\mathbb{M}$ into special
\emph{extremum dominant} simplicial complexes.
\begin{definition} \label{def:dom} A simplicial complex is \emph{minimum dominant} if there exists
a minimum $x$ such that every non-minimal \emph{vertex} in the manifold has a non-ascending path to $x$.
Analogously define \emph{maximum dominant}.
\end{definition}
The first aspect of the partitioning is ``raining''. Start at some point $x \in \mathbb{M}$ and imagine rain at $x$.
The water will flow downwards along non-ascending paths and ``wet'' all the points encountered. Note that this procedure considers all points
of the manifold, not just vertices.
\begin{definition} \label{def:wet} Fix $x \in \mathbb{M}$. The set of points $y \in \mathbb{M}$ such that there is a non-ascending path from $x$ to $y$
is denoted by ${\tt wet}(x,\mathbb{M})$ (which in turn is represented as a simplicial complex). A point $z$ is at the \emph{interface} of ${\tt wet}(x,\mathbb{M})$ if every neighborhood of $z$
has non-trivial intersection with ${\tt wet}(x,\mathbb{M})$ (i.e.\ the intersection is neither empty nor the entire neighborhood).
\end{definition}
The following claim gives a description of the interface.
\begin{claim} \label{clm:inter} For any $x$, each component of the interface of ${\tt wet}(x,\mathbb{M})$ contains a join vertex.
\end{claim}
\begin{proof} If $p \in {\tt wet}(x,\mathbb{M})$, all the points in any contour containing $p$ are also in ${\tt wet}(x,\mathbb{M})$.
(Follow the non-ascending path from $x$ to $p$ and then walk along the contour.) The converse is also true,
so ${\tt wet}(x,\mathbb{M})$ contains entire contours.
Let $\varepsilon, \delta$ be sufficiently small as usual. Fix some $y$ at the interface.
Note that $y \in {\tt wet}(x,\mathbb{M})$. (Otherwise, $B_\varepsilon(y)$ is dry.)
The points in $B_\varepsilon(y)$ that lie below $y$ have a descending path from $y$ and hence must be wet.
There must also be a dry point in $B_\varepsilon(y)$ that is above $y$, and hence,
there exists a dry, regular $(f(y)+\delta)$-contour $\phi$ intersecting $B_\varepsilon(y)$.
Let $\Gamma_y$ be the contour containing $y$. Suppose for contradiction
that $\forall p \in \Gamma_y$, $p$ has up-degree $1$ (see \Def{deg}). Consider the non-ascending path from $x$ to $y$ and let $z$
be the first point of $\Gamma_y$ encountered. There exists a wet, regular $(f(y) + \delta)$-contour $\psi$
intersecting $B_\varepsilon(z)$. Now, walk from $z$ to $y$ along $\Gamma_y$. If all points $w$ in this walk
have up-degree $1$, then $\psi$ is the unique $(f(y)+\delta)$-contour
intersecting $B_\varepsilon(w)$. This would imply that $\phi = \psi$, contradicting the fact that $\psi$ is wet
and $\phi$ is dry.
\end{proof}
Note that ${\tt wet}(x,\mathbb{M})$ (and its interface) can be computed in time linear in the size of the wet simplicial complex.
We perform a non-ascending search from $x$. Any face $F$ of $\mathbb{M}$ encountered is partially (if not entirely) in ${\tt wet}(x,\mathbb{M})$.
The wet portion is determined by cutting $F$ along the interface. Since each component of the interface is a contour, this is equivalent
to locally cutting $F$ by a hyperplane. All these operations can be performed to output ${\tt wet}(x,\mathbb{M})$ in time linear in $|{\tt wet}(x,\mathbb{M})|$.
We define a simple {\tt lift}{} operation on the interface components. Consider such a component $\phi$ containing
a join vertex $y$. Take any dry increasing edge incident to $y$, and pick the point $z$ on this edge at height
$f(y) + \delta$ (where $\delta$ is an infinitesimal, but larger than the value $\varepsilon$ used in the definition of ${\tt cut}$).
Let ${\tt lift}(\phi)$ be the unique contour through the regular point $z$. Note that ${\tt lift}(\phi)$ is dry.
The following claim follows directly from \Thm{jordan}.
\begin{claim} \label{clm:cut-int} Let $\phi$ be a connected component of the interface. Then ${\tt cut}(\mathbb{M},{\tt lift}(\phi))$
results in two disjoint simplicial complexes, one consisting entirely of dry points.
\end{claim}
\InNotSoCGVer{
\begin{proof} By \Thm{jordan}, ${\tt cut}(\mathbb{M},{\tt lift}(\phi))$ results in two disjoint simplicial complexes. Let $\mathbb{N}$ be the complex containing
the point $x$ (the argument in ${\tt wet}(x,\mathbb{M})$), and let $\mathbb{N}'$ be the other complex.
Any path from $x$ to $\mathbb{N}'$ must intersect ${\tt lift}(\phi)$, which is dry. Hence $\mathbb{N}'$ is dry.
\end{proof}
}
We describe the main partitioning procedure that cuts a simplicial complex $\mathbb{N}$ into extremum
dominant complexes. It takes an additional input of a maximum $x$. To initialize,
we begin with $\mathbb{N}$ set to $\mathbb{M}$ and $x$ as an arbitrary maximum.
When
we start, rain flows downwards. In each recursive call, the direction of rain is \emph{switched} to the
opposite direction. This is crucial to ensure a linear running time.
The switching is easily implemented by inverting a complex $\mathbb{N}'$, achieved by negating the height values.
We can now let rain flow downwards, as it usually does in our world.
\medskip
\fbox{
\begin{minipage}{0.9\textwidth}
{\bf ${\tt rain}(x,\mathbb{N})$}
\smallskip
\begin{compactenum}
\item Determine interface of ${\tt wet}(x,\mathbb{N})$.
\item If the interface is empty, simply output $\mathbb{N}$. Otherwise, denote the connected components by $\phi_1, \phi_2, \ldots, \phi_k$ and set $\phi'_i = {\tt lift}(\phi_i)$.
\item Initialize $\mathbb{N}_1 = \mathbb{N}$.
\item For $i$ from $1$ to $k$:
\begin{compactenum}
\item Construct ${\tt cut}(\mathbb{N}_i,\phi'_i)$, consisting of dry complex $\mathbb{L}_i$ and remainder $\mathbb{N}_{i+1}$.
\item Let the newly created boundary of $\mathbb{L}_i$ be $B_i$. Invert $\mathbb{L}_i$ so that $B_i$ is a maximum. Recursively
call ${\tt rain}(B_i,\mathbb{L}_i)$.
\end{compactenum}
\item Output $\mathbb{N}_{k+1}$ together with any complexes output by recursive calls.
\end{compactenum}
\end{minipage}}
\medskip
For convenience, denote the total output of ${\tt rain}(x,\mathbb{M})$ by $\mathbb{M}_1, \mathbb{M}_2, \ldots, \mathbb{M}_r$.
\begin{lemma} \label{lem:rain-1} Each output $\mathbb{M}_i$ is extremum dominant.
\end{lemma}
\begin{proof} Consider a call to ${\tt rain}(x,\mathbb{N})$. If the interface is empty, then all of $\mathbb{N}$
is in ${\tt wet}(x,\mathbb{N})$, so $\mathbb{N}$ is trivially extremum dominant. So suppose the interface
is non-empty and consists of $\phi_1, \phi_2, \ldots, \phi_k$ (as denoted in the procedure).
By repeated applications of \Clm{cut-int}, $\mathbb{N}_{k+1}$ contains ${\tt wet}(x,\mathbb{M})$.
Consider ${\tt wet}(x,\mathbb{N}_{k+1})$. The interface must exactly be $\phi_1, \phi_2, \ldots, \phi_k$.
So the only dry vertices are those in the boundaries $B_1, B_2, \ldots, B_k$. But these
boundaries are maxima.
\end{proof}
As ${\tt rain}(x,\mathbb{M})$ proceeds, new faces/simplices are created because of repeated cutting.
The key to the running time of ${\tt rain}(x,\mathbb{M})$ is bounding the number of newly created faces, for which we have the following lemma.
\begin{lemma}\label{lem:new-verts}
A face $F \in \mathbb{M}$ is cut\footnote{Technically what we are calling a single cut is done with two hyperplanes.} at most once during ${\tt rain}(x,\mathbb{M})$.
\end{lemma}
\begin{proof} Notation here follows the pseudocode of ${\tt rain}$.
First, by \Thm{jordan}, all the pieces on which ${\tt rain}$ is invoked are disjoint.
Second, all recursive calls are made on dry complexes.
Consider the first time that $F$ is cut, say, during the call to ${\tt rain}(x,\mathbb{N})$.
Specifically, say this happens when ${\tt cut}(\mathbb{N}_i,\phi'_i)$ is constructed.
${\tt cut}(\mathbb{N}_i,\phi'_i)$ will cut $F$ with two horizontal cutting planes, one $\varepsilon$ above $\phi'_i$
and one $\varepsilon$ below $\phi'_i$. This breaks $F$ into lower and upper portions which are then triangulated
(there is also a discarded middle portion). The lower portion, which is adjacent to $\phi_i$, gets included
in $\mathbb{N}_{k+1}$, the complex containing the wet points, and hence does not participate in any later recursive calls.
The upper portion (call it $U$) is in $\mathbb{L}_i$. Note that the lower
boundary of $U$ is in the boundary $B_i$. Since a recursive call is made to ${\tt rain}(B_i,\mathbb{L}_i)$
(and $\mathbb{L}_i$ is inverted), $U$ becomes wet. Hence $U$, and correspondingly $F$, will not be subsequently cut.
\end{proof}
The following are direct consequences of \Lem{new-verts} and the ${\tt surgery}$ procedure.
\begin{theorem} \label{thm:rain-time} The total running time of ${\tt rain}(x,\mathbb{M})$ is $O(|\mathbb{M}|)$.
\end{theorem}
\InNotSoCGVer{
\begin{proof} The only non-trivial operations performed are ${\tt wet}$ and ${\tt cut}$.
Since ${\tt cut}$ is a linear time procedure, Lemma~\ref{lem:new-verts} implies the total time for all calls to ${\tt cut}$ is $O(|\mathbb{M}|)$.
As for the ${\tt wet}$ procedure, observe that Lemma~\ref{lem:new-verts} additionally implies there are only $O(|\mathbb{M}|)$ new faces created by ${\tt rain}$.
Therefore, since ${\tt wet}$ is also a linear time procedure, and no face is ever wet twice, the total time for all calls to ${\tt wet}$ is $O(|\mathbb{M}|)$.
\end{proof}
}
\begin{claim} \label{clm:rain-reeb} Given $\cC(\mathbb{M}_1), \cC(\mathbb{M}_2), \ldots, \cC(\mathbb{M}_r)$,
$\cC(\mathbb{M})$ can be constructed in $O(|\mathbb{M}|)$ time.
\end{claim}
\InNotSoCGVer{
\begin{proof} Consider the tree of recursive calls in ${\tt rain}(x,\mathbb{M})$, with each node
labeled with some $\mathbb{M}_i$.
Walk through this tree in a leaf first ordering. Each time we visit a node we connect its contour tree to the
contour tree of its children in the tree using the ${\tt surgery}$ procedure.
Each ${\tt surgery}$ call takes constant time, and the total time is the size of the recursion tree.
\end{proof}
}
\section{Contour trees of extremum dominant manifolds} \label{sec:extreme}
The previous section allows us to restrict attention to extremum dominant manifolds.
We will orient so that the extremum in question is always a \emph{minimum}.
We will fix such a simplicial complex $\mathbb{M}$, with the dominant minimum $m^*$.
For vertex $v$, we use $\mathbb{M}^+_v$ to denote the simplicial complex obtained by only
keeping vertices $u$ such that $f(u) > f(v)$. Analogously, define $\mathbb{M}^-_v$. Note that $\mathbb{M}^+_v$ may contain
numerous connected components.
\InNotSoCGVer{
The main theorem of this section asserts that contour trees of minimum dominant manifolds have a simple description.
The exact statement will require some definitions and notation.
We require the notions of \emph{join} and \emph{split} trees, as given by~\cite{csa-cctad-00}.
Conventionally, all edges are directed from higher to lower function value.
}
\InSoCGVer{
The main theorem of this section asserts that contour trees of minimum dominant manifolds have a simple description.
Intuitively, the cutting procedure of \Sec{rain} introduced ``stumps'' where we made cuts, which
is why we allowed for non-dominant minima and maxima in the definition of extremum dominant manifolds.
The punchline is that, ignoring these stumps, the contour tree of an extremum dominant manifold is equivalent to its \emph{join} (or \emph{split}) tree, defined in~\cite{csa-cctad-00}.
Hence the critical join tree below is just the join tree without these stumps.
Additional definitions and proofs are given in \Sec{jstedm}.
}
\newcommand{\joinTreeSplitTreeMerge}{
\begin{definition} \label{def:join} The \emph{join tree} ${\cal J}(\mathbb{M})$ of $\mathbb{M}$ is built on vertex set ${\cal V}(\mathbb{M})$.
The directed edge $(u,v)$ is present when $u$ is the smallest valued vertex in a connected component of $\mathbb{M}^+_v$
\emph{and} $v$ is adjacent to a vertex in this component (in $\mathbb{M}$). The \emph{split tree} ${\cal S}(\mathbb{M})$ is obtained
by looking at $\mathbb{M}^-_v$ (or alternatively, by taking the join tree of the inversion of $\mathbb{M}$).
\end{definition}
Some basic facts about these trees.
All outdegrees in ${\cal J}(\mathbb{M})$ are at most $1$, all indegree $2$ vertices are joins, all leaves are maxima,
and the global minimum is the root. All indegrees in ${\cal S}(\mathbb{M})$ are at most $1$, all outdegree $2$
vertices are splits, all leaves are minima, and the global maximum is the root.
As these trees are rooted, we can use ancestor-descendant terminology.
Specifically, for two adjacent vertices $u$ and $v$, $u$ is the parent of $v$ if $u$ is closer to the root
(i.e.\ each node can have at most one parent, but can have two children).
The key observation is that ${\cal S}(\mathbb{M})$ is trivial for a minimum dominant $\mathbb{M}$.
\begin{lemma} \label{lem:split} ${\cal S}(\mathbb{M})$ consists of:
\begin{asparaitem}
\item A single path (in sorted order) with all vertices except non-dominant minima.
\item Each non-dominant minimum is attached to a unique split (which is adjacent to it).
\end{asparaitem}
\end{lemma}
\begin{proof} It suffices to prove that each split $v$ has one child that is just a leaf, which
is a non-dominant minimum. Specifically, any minimum is a leaf in ${\cal S}(\mathbb{M})$ and thereby attached to a split,
which implies that if we removed all non-dominant minima, we must end up with a path, as asserted above.
Consider a split $v$. For sufficiently small $\varepsilon, \delta$, there are exactly two $(f(v) - \delta)$-contours
$\phi$ and $\psi$ intersecting $B_\varepsilon(v)$. Both of these are regular contours. There must be a non-ascending
path from $v$ to the dominant minimum $m^*$. Consider the first edge (necessarily decreasing from $v$)
on this path. It must intersect one of the $(f(v) - \delta)$-contours, say $\phi$. By \Thm{jordan}, ${\tt cut}(\mathbb{M},\phi)$ has
two connected components, with one (call it $\mathbb{L}$) having $\phi^-$ as a boundary maximum. This complex
contains $m^*$ as the non-ascending path intersects $\phi$ only once. Let the other
component be called $\mathbb{M}'$.
Consider ${\tt cut}(\mathbb{M}',\psi)$ with connected component $\mathbb{N}$ having $\psi^-$ as a boundary. $\mathbb{N}$
does not contain $m^*$, so any path from the interior of $\mathbb{N}$ to $m^*$ must intersect the boundary $\psi^-$.
But the latter is a maximum in $\mathbb{N}$, so there can be no non-ascending path from the interior to $m^*$.
Since $\mathbb{M}$ is overall minimum dominant, the interior of $\mathbb{N}$ can only contain a single vertex $w$, a non-dominant
minimum.
The split $v$ has two children in ${\cal S}(\mathbb{M})$, one in $\mathbb{N}$ and one in $\mathbb{L}$. The child in $\mathbb{N}$ can only
be the non-dominant minimum $w$, which is a leaf.
\end{proof}
It is convenient to denote the non-dominant minima as $m_1, m_2, \ldots, m_k$
and the corresponding splits (as given by the lemma above) as $s_1, s_2, \ldots, s_k$.
Using the above lemma we can now prove that computing the contour tree for a minimum dominant manifold
amounts to computing its join tree. Specifically, to prove our main theorem, we rely on the correctness of the
merging procedure from~\cite{csa-cctad-00} that constructs the contour tree from the join and split trees.
It actually constructs the \emph{augmented contour tree} ${\cal A}(\mathbb{M})$, which is
obtained by replacing each edge in the contour tree with a path of all regular vertices
(sorted by height) whose corresponding contour belongs to the equivalence class of that edge.
Consider a tree $T$ with a vertex $v$ of in and out degree at most $1$.
\emph{Erasing} $v$ from $T$ is the following operation: if $v$ is a leaf, just delete $v$. Otherwise,
delete $v$ and connect its neighbors by an edge (i.e.\ smooth $v$ out). This tree is denoted by $T \ominus v$.
\medskip
\fbox{
\begin{minipage}{0.9\textwidth}
{\bf ${\tt merge}({\cal J}(\mathbb{M}),{\cal S}(\mathbb{M}))$}
\smallskip
\begin{compactenum}
\item Set ${\cal J} = {\cal J}(\mathbb{M})$ and ${\cal S} = {\cal S}(\mathbb{M})$.
\item Denote $v$ as a \emph{candidate} if the sum of its indegree in ${\cal J}$ and outdegree in ${\cal S}$ is $1$.
\item Add all candidates to queue.
\item While candidate queue is non-empty:
\begin{compactenum}
\item Let $v$ be head of queue. If $v$ is leaf in ${\cal J}$, consider its edge in ${\cal J}$. Otherwise
consider its edge in ${\cal S}$. In either case, denote the edge by $(v,w)$.
\item Insert $(v,w)$ in ${\cal A}(\mathbb{M})$.
\item Set ${\cal J} = {\cal J} \ominus v$ and ${\cal S} = {\cal S} \ominus v$. Enqueue any new candidates.
\end{compactenum}
\item Smooth out all regular vertices in ${\cal A}(\mathbb{M})$ to get ${\cal C}(\mathbb{M})$.
\end{compactenum}
\end{minipage}}
\medskip
}
\InNotSoCGVer{\joinTreeSplitTreeMerge}
\begin{definition}
\label{def:criticalJoin}
The \emph{critical join tree} $\cJ_C(\mathbb{M})$ is built on the set $V'$ of all
critical points other than the non-dominant minima.
The directed edge $(u,v)$ is present when $u$ is the smallest valued vertex in $V'$ in a connected component of $\mathbb{M}^+_v$
and $v$ is adjacent (in $\mathbb{M}$) to a vertex in this component. \InSoCGVer{The \emph{join tree}, ${\cal J}(\mathbb{M})$, is defined analogously, but instead on ${\cal V}(\mathbb{M})$.}
\end{definition}
\InSoCGVer{Each non-dominant minimum, $m_i$, connects to the contour tree at a corresponding split $s_i$. We have the following (see \Sec{jstedm} for more details).}
\begin{theorem} \label{thm:contour-tree} Let $\mathbb{M}$ have a dominant minimum.
The contour tree ${\cal C}(\mathbb{M})$ consists of all edges $\{(s_i, m_i)\}$ and $\cJ_C(\mathbb{M})$.
\end{theorem}
\newcommand{\proofofContourJoinEquiv}{
\begin{proof} We first show that ${\cal A}(\mathbb{M})$ is ${\cal J}(\mathbb{M}) \ominus \{m_i\}$ with edges $\{(s_i,m_i)\}$.
We have flexibility in choosing the order of processing in ${\tt merge}$. We first put the non-dominant
maxima $m_1, \ldots, m_k$ into the queue. As these are processed, the edges $\{(s_i,m_i)\}$ are inserted
into ${\cal A}(\mathbb{M})$. Once all the $m_i$'s are erased, ${\cal S}$ becomes a path, so all outdegrees are at most $1$.
The join tree is now ${\cal J}(\mathbb{M}) \ominus \{m_i\}$. We can now process ${\cal J}$ leaf by leaf, and all
edges of ${\cal J}$ are inserted into ${\cal A}(\mathbb{M})$.
Note that ${\cal C}(\mathbb{M})$ is obtained by smoothing out all regular points from ${\cal A}(\mathbb{M})$. Similarly, smoothing out regular
points from ${\cal J}(\mathbb{M}) \ominus \{m_i\}$ yields the edges of $\cJ_C(\mathbb{M})$.
\end{proof}
}
\InNotSoCGVer{\proofofContourJoinEquiv}
\begin{Remark}\label{rem:joinAndCriticalJoin}
The above theorem, combined with the previous sections, implies that in order to get an efficient contour tree algorithm,
it suffices to have an efficient algorithm for computing $\cJ_C(\mathbb{M})$.
Due to minor technicalities, it is easier to phrase the following section instead in terms of computing ${\cal J}(\mathbb{M})$ efficiently.
Note however that for minimum dominant complexes output by ${\tt rain}$, converting between $\cJ_C$ and ${\cal J}$ is trivial, as ${\cal J}$ is just $\cJ_C$
with each non-dominant minimum $m_i$ augmented along the edge leaving $s_i$.
\end{Remark}
\section{Painting to compute contour trees}
\label{sec:paint}
The main algorithmic contribution is a new algorithm for computing join trees of any triangulated simplicial
complex $\mathbb{M}$.
{\bf Painting:} The central tool is a notion
of \emph{painting} $\mathbb{M}$. Initially associate a color with each maximum. Imagine there being a large
can of paint of a distinct color at each maximum $x$. We will spill different paint from each maximum and watch it flow down.
This is analogous to the raining of \Sec{rain}, but paint is a much more viscous liquid.
\emph{So paint only flows down edges, and it does not color the interior of higher dimensional faces.} Furthermore, paints
do not mix, so every edge of $\mathbb{M}$ gets a unique color. This process (and indeed the entire algorithm)
works purely on the 1-skeleton of $\mathbb{M}$, which is just a graph.
\InNotSoCGVer{We now restate \Def{paint1}.}
\begin{definition} \label{def:paint2} Let the 1-skeleton of $\mathbb{M}$ have edge set $E$ and maxima $X$.
A \emph{painting} of $\mathbb{M}$ is a map $\chi:X \cup E \to [|X|]$ with the following property.
Consider an edge $e$. There exists a descending path from some maximum $x$ to $e$
consisting of edges in $E$, such that all edges along this path have the same color as $x$.
An \emph{initial} painting has the additional property that the restriction $\chi:X \to [|X|]$ is a bijection.
\end{definition}
\begin{definition} \label{def:color-set} Fix a painting $\chi$ and vertex $v$.
\begin{asparaitem}
\item An \emph{up-star} of $v$ is the set of edges that all connected to a fixed component of $\mathbb{M}^+_v$.
\item A vertex $v$ is \emph{touched by color $c$} if $v$ is incident to a $c$-colored
edge with $v$ at the lower endpoint. For $v$, $col(v)$ is the set of colors that touch $v$.
\item A color $c \in col(v)$ \emph{fully touches} $v$ if all edges in an up-star are colored $c$.
\item For any maximum $x\in X$, we say that $x$ is both touched and fully touched by $\chi(x)$.
\end{asparaitem}
\end{definition}
\subsection{The data structures} \label{sec:struct}
\noindent
{\bf The binomial heaps $T(c)$:} For each color $c$, $T(c)$ is a subset of vertices touched by $c$,
This is stored as a \emph{binomial max-heap} keyed by vertex heights. Abusing notation, $T(c)$ refers
both to the set and the data structure used to store it.
\medskip
\noindent
{\bf The union-find data structure on colors:} We will repeatedly perform unions
of classes of colors, and this will be maintained as a standard union-find data structure.
For any color $c$, $rep(c)$ denotes the representative of its class.
\medskip
\noindent
{\bf The stack $K$:} This consists of non-extremal critical points, with monotonically increasing
heights as we go from the base to the head.
\ignore{
Each point $x \in K$ has an associated subset of $col(x)$, denoted $mcol(x)$.
Both $mcol(x)$ and its complement are stored as hash table. So lookups, inserts, and deletes
are in these sets are all constant time operations. The stack is guaranteed to satisfy
the following invariants.
\begin{asparaitem}
\item For every $x \in K$: For every $c \in mcol(x)$, $x$ is the highest element
in $T(c)$. Furthermore, $c = rep(c)$.
\item Consider $x, y \in K$ such that $y$ was pushed on $x$. There exists $c \in col(x) \setminus
mcol(x)$ such that $x$ is not highest in $T(c)$ but $y$ is highest in $T(c)$.
\end{asparaitem}
}
\medskip
\noindent
{\bf Attachment vertex $att(c)$:} For each color $c$, we maintain a critical point $att(c)$ of this color.
We will maintain the guarantee that the portion of the contour tree above (and including) $att(c)$ has already been constructed.
\subsection{The algorithm} \label{sec:algo}
We formally describe the algorithm below.
We require a technical definition of \emph{ripe} vertices.
\begin{definition} \label{def:ripe} A vertex $v$ is \emph{ripe} if: for all $c \in col(v)$, $v$
is present in $T(rep(c))$ and is also the highest vertex in this heap.
\end{definition}
\medskip
\fbox{
\begin{minipage}{0.9\textwidth}
{\bf ${\tt init}(\mathbb{M})$}
\smallskip
\begin{compactenum}
\item Construct an initial painting of $\mathbb{M}$ using a descending BFS from maxima that does not explore previously colored edges.
\item Determine all critical points in $\mathbb{M}$. For each $v$, look at $(f(v) \pm \delta)$-contours in $f|_{B_\varepsilon(v)}$
to determine the up and down degrees.
\item Mark each critical $v$ as unprocessed.
\item For each critical $v$ and each up-star, pick an arbitrary color $c$ touching $v$. Insert $v$ into $T(c)$.
\item Initialize $rep(c) = c$ and set $att(c)$ to be the unique maximum colored $c$.
\item Initialize $K$ to be an empty stack.
\end{compactenum}
\end{minipage}}
\medskip
\fbox{
\begin{minipage}{0.9\textwidth}
{\bf ${\tt build}(\mathbb{M})$}
\smallskip
\begin{compactenum}
\item Run ${\tt init}(\mathbb{M})$.
\item While there are unprocessed critical points:
\begin{compactenum}
\item Run ${\tt update}(K)$. Pop $K$ to get $h$.
\item Let $cur(h) = \{rep(c) | c \in col(h)\}$.
\item For all $c' \in cur(h)$:
\begin{compactenum}
\item Add edge $(att(c'),h)$ to ${\cal J}(\mathbb{M})$.
\item Delete $h$ from $T(c')$.
\end{compactenum}
\item Merge heaps $\{T(c') | c' \in cur(h)\}$.
\item Take union of $cur(h)$ and denote resulting color by $\widehat{c}$.
\item Set $att(\widehat{c}) = h$ and mark $h$ as processed.
\end{compactenum}
\end{compactenum}
\end{minipage}}
\medskip
\fbox{
\begin{minipage}{0.9\textwidth}
{\bf ${\tt update}(K)$}
\smallskip
\begin{compactenum}
\item If $K$ is empty, push arbitrary unprocessed critical point $v$.
\item Let $h$ be the head of $K$.
\item While $h$ is not ripe:
\begin{compactenum}
\item Find $c \in col(h)$ such that $h$ is not the highest in $T(rep(c))$.
\item Push the highest of $T(rep(c))$ onto $K$, and update head $h$.
\end{compactenum}
\end{compactenum}
\end{minipage}}
\bigskip
A few simple facts:
\begin{compactitem}
\item At all times, the colors form a valid painting.
\item Each vertex is present in at most $2$ heaps. After processing, it is removed from all heaps.
\item After $v$ is processed, all edges incident to $v$ have the same (representative) color.
\item Vertices on the stack are in increasing height order.
\end{compactitem}
\begin{observation}
\label{obs:twoQueues}
Each unprocessed vertex is always in exactly one queue of the colors in each of its up-stars. Specifically,
for a given up-star of a vertex $v$, ${\tt init}(\mathbb{M})$ puts $v$ into the queue of exactly one of the colors of the up-star, say $c$. As time goes on this queue
may merge with other queues, but while $v$ remains unprocessed, it is only ever (and always) in the queue of $rep(c)$,
since $v$ is never added to a new queue and is not removed until it is processed.
In particular, finding the queues of a vertex in ${\tt update}(K)$ requires at most two union find operations (assuming each vertex records its two colors from ${\tt init}(\mathbb{M})$).
\end{observation}
\medskip
\ignore{
Suppose $mcol(h) = \{c\}$ (so $h$ is split).
\begin{compactenum}
\item Connect $h$ (in $\cC(\mathbb{M})$) to $att(c)$ and the unique minimum corresponding to split $h$.
\item Delete $h$ from $T(c)$, set $att(c) = h$.
\item End procedure.
\end{compactenum}
\item Let $mcol(h) = \{c_1, c_2\}$ (so $h$ is merge).
\item Connect $h$ in $\cC(\mathbb{M})$ to $att(c_1)$ and $att(c_2)$.
\item Delete all copies of $h$ from $T(c_1)$ and $T(c_2)$.
\item Perform union of colors $c_1$ and $c_2$ (denote merged color as $c$). Merge heaps
to get $T(c) = T(c_1) \cup T(c_2)$.
\item Set $att(c) = h$.
We state the primary invariant below.
First, some notation. For any $v$, let $\psi^-_v$ denote the $(f(v)-\delta)$-contour intersecting
$B_\varepsilon(v)$. If there are two such contours (so $v$ is a split), choose the one that contains
the dominant minimum. The \emph{palette} $P$ is the set of colors currently used,
which is $\{T(rep(c)) | c \in |X|\}$.
Observe that ${\tt build}(\mathbb{M})$ loops over all unprocessed critical point.
The invariant is true at the starting point of each such iteration.
\medskip
\textbf{Invariant:}
\begin{compactenum}
\item For every $c \in P$, the subtree of ${\cal J}(\mathbb{M})$ rooted at $att(c)$ has been constructed.
\item Fix $c \in P$. Consider the set $S$ of maxima of $\mathbb{M}$ in the subtree of ${\cal J}(\mathbb{M})$ rooted at $att(c)$.
Then $rep(c) = \bigcup_{s \in S} \chi(s)$.
\item Let $\Psi = \{\psi^-_{att(c)} | c \in P\}$. The coloring given by $rep(\cdot)$
is a valid painting of ${\tt cut}(\mathbb{M},\Psi)$, where $\psi^-_{att(c)}$ has color $c$.
\end{compactenum}
\medskip
It is easy to see that the invariant is true at the very beginning of the algorithm.
Each $att(c)$ is simply the maximum colored with $c$, and we have a valid painting of $\mathbb{M}$.
}
\subsection{Proving correctness} \label{sec:correct}
\ignore{
\begin{claim} \label{clm:process} Assume the invariant. Any vertex with a non-increasing path to some $att(c)$
for $c \in P$ has been processed.
\end{claim}
\begin{proof} This is a direct consequence of \Thm{carr-mono}, which relates monotone paths in $\mathbb{M}$ to ${\cal C}(\mathbb{M})$.
Since the subtree of ${\cal J}(\mathbb{M})$ (which is basically the subtree of ${\cal C}(\mathbb{M})$) rooted at $att(c)$
has been found, all vertices in this subtree must be processed. These are all the vertices with non-increasing paths
to $att(c)$ in ${\cal C}(\mathbb{M})$, which by \Thm{carr-mono} is the same as those in ${\cal C}(\mathbb{M})$.
\end{proof}
}
Our main workhorse is the following technical lemma. In the following, the current color of an edge, $e$, is the value of $rep(\chi(e))$,
where $\chi(e)$ is the color of $e$ from the initial painting.
\begin{lemma} \label{lem:full} Suppose vertex $v$ is connected to a component $\mathbb{P}$ of $\mathbb{M}^+_v$
by an edge $e$ which is currently colored $c$. Either all edges in $\mathbb{P}$ are currently colored $c$, or there
exists a critical vertex $w \in \mathbb{P}$ fully touched by $c$ and touched by another color.
\end{lemma}
\begin{proof} Since $e$ has color $c$,
there must exist vertices in $\mathbb{P}$ touched by $c$. Consider the highest
vertex $w$ in $\mathbb{P}$ that is touched by $c$ and some other color. If no such vertex exists,
this means all edges incident to a vertex touched by $c$ are colored $c$. By walking through
$\mathbb{P}$, we deduce that all edges are colored $c$.
So assume $w$ exists. Take the $(f(w)+\delta)$-contour $\phi$ that intersects $B_\varepsilon(w)$
and intersects some $c$-colored edge incident to $w$. Note that all edges intersecting $\phi$ are also colored $c$,
since $w$ is the highest vertex to be touched by $c$ and some other color. (Take the path of $c$-colored
edges from the maximum to $w$. For any point on this path, the contour passing through this point must
be colored $c$.) Hence, $c$ fully touches $w$.
But $w$ is touched by another color, and the corresponding edge cannot intersect $\phi$. So $w$
must have up-degree $2$ and is critical.
\end{proof}
\begin{corollary}
\label{cor:terminate}
Each time ${\tt update}(K)$ is called, it terminates with a ripe vertex on top of the stack.
\end{corollary}
\begin{proof}
${\tt update}(K)$ is only called if there are unprocessed vertices remaining, and so by the time we reach step 3 in
${\tt update}(K)$, the stack has some unprocessed vertex $h$ on it.
If $h$ is ripe, then we are done, so suppose otherwise.
Let $\mathbb{P}$ be one of the components of $\mathbb{M}^+_h$. By construction, $h$ was put in the heap of some initial
adjacent color $c$. Therefore, $h$ must be in the current heap of $rep(c)$ (see \Obs{twoQueues}).
Now by \Lem{full}, either all edges in $\mathbb{P}$ are colored $rep(c)$ or
there is some vertex $w$ fully touched by $rep(c)$ and some other color.
The former case implies that if there are any unprocessed vertices in $\mathbb{P}$ then they are all in $T(rep(c))$,
implying that $h$ is not the highest vertex and a new higher up unprocessed vertex will be put on the stack for the next iteration of the while loop.
Otherwise, all the vertices in $\mathbb{P}$ have been processed.
However, it cannot be the case that all vertices in all components of $\mathbb{M}^+_h$ have already been processed,
since this would imply that $h$ was ripe, and so one can apply the same argument to the other non-fully processed component.
Now consider the latter case, where we have a non-monochromatic vertex $w$.
In this case $w$ cannot have been processed (since after being processed it is touched only by one color),
and so it must be in $T(rep(c))$ since it must be in some heap of a color in each up-star (and one up-star is entirely colored $rep(c)$).
As $w$ lies above $h$ in $\mathbb{M}$, this implies $h$ is not on the top of this heap.
\end{proof}
\begin{claim} \label{clm:upstar} Consider a ripe vertex $v$ and take the up-star connecting
to some component of $\mathbb{M}^+_v$. All edges in this component and the up-star have the same color.
\end{claim}
\begin{proof} Let $c$ be the color of some edge in this up-star.
By ripeness, $v$ is the highest in $T(rep(c))$.
Denote the component of $\mathbb{M}^+_v$ by $\mathbb{P}$.
By \Lem{full}, either all edges in $\mathbb{P}$ are colored $rep(c)$ or there exists critical vertex $w \in \mathbb{P}$
fully touched by $rep(c)$ and another color. In the latter case, $w$ has not been processed,
so $w \in T(rep(c))$ (contradiction to ripeness). Therefore, all edges in $\mathbb{P}$ are colored $rep(c)$.
\end{proof}
\begin{claim} \label{clm:process} The partial output on the processed vertices is exactly
the restriction of ${\cal J}(\mathbb{M})$ to these vertices.
\end{claim}
\begin{proof} More generally, we prove the following: all outputs on processed vertices
are edges of ${\cal J}(\mathbb{M})$ and for any current color $c$, $att(c)$ is the lowest processed vertex
of that color. We prove this by induction on the processing order. The base case is trivially
true, as initially the processed vertices and attachments of the color classes are the set of maxima.
For the induction step, consider the situation
when $v$ is being processed.
Since $v$ is being processed, we know by \Cor{terminate} that it is ripe. Take any up-star of $v$, and the corresponding component $\mathbb{P}$
of $\mathbb{M}^+_v$ that it connects to. By \Clm{upstar}, all edges in $\mathbb{P}$ and the up-star
have the same color (say $c$). If some critical vertex in $\mathbb{P}$ is not processed,
it must be in $T(c)$, which violates the ripeness of $v$.
Thus, all critical vertices in $\mathbb{P}$ have been processed, and so by the induction hypothesis, the restriction of ${\cal J}(\mathbb{M})$ to $\mathbb{P}$ has been correctly computed.
Additionally, since all critical vertices in $\mathbb{P}$ have processed, they all have the same color $c$ of the lowest critical vertex in $\mathbb{P}$.
Thus by the strengthened induction hypothesis, this lowest critical vertex is $att(c)$.
If there is another component of $\mathbb{M}^+_v$, the same argument implies
the lowest critical vertex in this component is $att(c')$ (where $c'$ is the color of edges in the respective component).
Now by the definition of ${\cal J}(\mathbb{M})$, the critical vertex $v$ connects to the lowest critical vertex in each component of $\mathbb{M}^+_v$,
and so by the above $v$ should connect to $att(c)$ and $att(c')$, which is precisely what $v$ is connected to by ${\tt build}(\mathbb{M})$.
Moreover, ${\tt build}$ merges the colors $c$ and $c'$ and correctly sets $v$ to be the attachment,
as $v$ is the lowest processed vertex of this merged color (as by induction $att(c)$ and $att(c')$ were the lowest vertices before merging colors).
\end{proof}
\begin{theorem}
\label{thm:correct}
Given an input complex $\mathbb{M}$, ${\tt build}(\mathbb{M})$ terminates and outputs ${\cal J}(\mathbb{M})$.
\end{theorem}
\begin{proof}
First observe that each vertex can be processed at most once by ${\tt build}(\mathbb{M})$. By \Cor{terminate}, we know that as long as there
is an unprocessed vertex, ${\tt update}(K)$ will be called and will terminate with a ripe vertex which is ready to be processed.
Therefore, eventually all vertices will be processed, and so by \Clm{process} the algorithm will terminate having
computed ${\cal J}(\mathbb{M})$.
\end{proof}
\InNotSoCGVer{
\subsection{Running Time}
\label{sec:runTime}
We now bound the running time of the algorithm of \Sec{algo}.
In subsequent sections, through a sophisticated charging argument, this bound is then related to matching upper
and lower bounds in terms of path decompositions.
Therefore, it will be useful to set up some terminology that can
be used consistently in both places. Specifically, the path decomposition bounds will be purely combinatorial
statements on colored rooted trees, and so the terminology is of this form.
Any tree $T$ considered in following will be a rooted binary tree\footnote{Note
that technically the trees considered should have a leaf vertex hanging below the root
in order to represent the global minimum of the complex. This vertex is
(safely) ignored to simplify presentation.} where the height of a vertex is its distance
from the root $r$ (i.e.\ conceptually $T$ will be a join tree with $r$ at the bottom).
As such, the children of a vertex $v\in T$ are the adjacent vertices of larger height
(and $v$ is the parent of such vertices). Then the subtree rooted at $v$, denoted $T_v$ consists of the graph induced on all
vertices which are descendants of $v$ (including $v$ itself). For two vertices $v$ and $w$ in $T$ let $d(v,w)$ denote the
length of the path between $v$ and $w$.
We use $A(v)$ to denote the set of ancestors of $v$.
For a set of nodes $U$, $A(U) = \bigcup_{u \in U} A(u)$.
\begin{definition}
\label{def:leafAssign}
A \emph{leaf assignment} $\chi$ of a tree $T$ assigns \emph{two} distinct leaves to each internal vertex $v$,
one from the left child and one from the right child subtree of $v$ (naturally if $v$ has only one child it is assigned
only one color).
\end{definition}
For a vertex $v\in T$, we use $H_v$ to denote the \emph{heap} at $v$.
Formally, $H_v = \{u | u \in A(v), \chi(u) \cap L(T_v) \neq \emptyset\}$, where $L(T_v)$ is the set of leaves of $T_v$.
In words, $H_v$ is the set of ancestors of $v$ which are colored by some leaf in $T_v$.
\begin{definition}
\label{def:initialColoring}
Note that the subroutine ${\tt init}(\mathbb{M})$ from \Sec{algo} naturally defines a leaf assignment to ${\cal J}(\mathbb{M})$
according to the priority queue for each up-star we put a given vertex in. Call this the \emph{initial coloring}
of the vertices in ${\cal J}(\mathbb{M})$. Note also that this initial coloring defines the $H_v$ values for all $v\in {\cal J}(\mathbb{M})$.
\end{definition}
The following lemma should justify these technical definitions.
\begin{lemma}
\label{lem:runTimeUpper}
Let $\mathbb{M}$ be a simplicial complex with $t$ critical points. For every vertex in ${\cal J}(\mathbb{M})$,
let $H_v$ be defined by the initial coloring of $\mathbb{M}$.
The running time of ${\tt build}(\mathbb{M})$ is $O(N+t\alpha(t) + \sum_{v \in {\cal J}(\mathbb{M})} \log |H_v|)$.
\end{lemma}
}
\newcommand{\proofofRunTime}{
\begin{proof}
First we look at the initialization procedure ${\tt init}(\mathbb{M})$. This procedure runs in $O(N)$ time.
Indeed, the painting procedure consists of several BFS's but as each vertex is only explored by one of the BFS's, it is linear time overall.
Determining the critical points is a local computation on the neighborhood of each vertex as so is linear (i.e.\ each edge is viewed at most twice).
Finally, each vertex is inserted into at most two heaps and so initializing the heaps takes linear time in the number of vertices.
Now consider the union-find operations performed by ${\tt build}$ and ${\tt update}$.
Initially the union find data structure has a singleton component for each leaf (and no new components are ever created),
and so each union-find operation takes $O(\alpha(t))$ time.
For ${\tt update}$, by \Obs{twoQueues}, each iteration of the while loop requires a constant number of finds (and no unions).
Specifically, if a vertex is found to be ripe (and hence processed next) then these can be charged to that vertex.
If a vertex is not ripe, then these can be charged to the vertex put on the stack.
As each vertex is put on the stack or processed at most once, ${\tt update}$ performs $O(t)$ finds overall.
Finally, ${\tt build}(\mathbb{M})$ performs one union and at most two finds for each vertex. Therefore the total number of union find
operations is $O(t)$.
For the remaining operations, observe that for every iteration of the loop in ${\tt update}$, a vertex is pushed onto the stack and each
vertex can only be pushed onto the stack once (since the only way it leaves the stack is by being processed).
Therefore the total running time due to ${\tt update}$ is linear (ignoring the find operations).
What remains is the time it takes to process a vertex $v$ in ${\tt build}(\mathbb{M})$.
In order to process a vertex there are a few constant time operations, union-find operations, and queue operations.
Therefore the only thing left to bound are the queue operations.
Let $v$ be a vertex in ${\cal J}(\mathbb{M})$, and let $c_1$ and $c_2$ be its children (the same argument holds if $v$ has only one child).
At the time $v$ is processed, the colors and queues of all vertices in a given component of $\mathbb{M}^+_v$ have merged together.
In particular, when $v$ is processed we know it is ripe and so all vertices above $v$ in each component of $\mathbb{M}^+_v$ have been processed, implying
these merged queues are the queues of the current colors of $c_1$ and $c_2$. Again since $v$ is ripe, it must be on the top of these queues and so
the only vertices left in these queues are those in $H_{c_1}$ and $H_{c_2}$.
Now when $v$ is handled, three queue operations are performed. Specifically, $v$ is removed from the queues of $c_1$ and $c_2$, and then the queues are
are merged together. By the above arguments the sizes of the queues for each of these operations are $H_{c_1}$, $H_{c_2}$, and $H_v$, respectively.
As merging and deleting takes logarithmic time in the heap size for binomial heaps, the claim now follows.
\end{proof}
}
\InNotSoCGVer{
\proofofRunTime
}
\InSoCGVer{
\subsection{Upper Bounds for Running Time}\label{sec:runTime}
The algorithm ${\tt build}(\mathbb{M})$ processes vertices in ${\cal J}(\mathbb{M})$ one at a time. The main processing cost
comes from priority queue operations. The cost of these operations is a function of the size of the queue which is in turn a function of the choices made by the subroutine ${\tt init}(\mathbb{M})$.
\begin{definition}\label{def:initialColoring}
A \emph{leaf assignment} $\chi$ of a binary tree $T$ assigns \emph{two} distinct leaves to each internal vertex $v$,
one from the left and one from the right subtree of $v$ (or only one leaf if $v$ has only one child).
The subroutine ${\tt init}(\mathbb{M})$ naturally defines a leaf assignment to ${\cal J}(\mathbb{M})$ (which
is a rooted binary tree) according to the priority queue for each up-star we put a given vertex in.
Call this the \emph{initial coloring} of the vertices in ${\cal J}(\mathbb{M})$, and denote it by $\chi$.
For a vertex $v$ in ${\cal J}(\mathbb{M})$, let $L(v)$ denote the set of leaves of the subtree rooted at $v$, and let $A(v)$ denote the set of
ancestors of $v$, i.e.\ the vertices on the $v$ to root path. For a vertex $v\in {\cal J}(\mathbb{M})$, and an initial coloring $\chi$,
we use $H_v$ to denote the \emph{heap} at $v$. Formally, $H_v = \{u | u \in A(v), \chi(u) \cap L(v) \neq \emptyset\}$, i.e.\
the set of ancestors colored by some leaf in $L(v)$.
\end{definition}
Given the above technical definition, the proof of the following lemma is straightforward, though long and so has been moved to \Sec{runTimeProof}.
\begin{lemma}\label{lem:runTimeUpper}
Let $\mathbb{M}$ be a simplicial complex with $t$ critical points.
The running time of ${\tt build}(\mathbb{M})$ is $O(N+t\alpha(t) + \sum_{v \in {\cal J}(\mathbb{M})} \log |H_v|)$, where $H_v$ is defined by an initial coloring.
\end{lemma}
}
\InSoCGVer{Our main result, } \Thm{main-corr}, is an easy corollary of the above lemma. Specifically, consider a critical point $v$ of the initial input
complex. By \Thm{jordan} this vertex appears in exactly one of the pieces output by ${\tt rain}$. As in the \Thm{main-corr} statement, let $\ell_v$
denote the length of the longest directed path passing through $v$ in the contour tree of the input complex, and let $\ell_v'$
denote the longest directed path passing through $v$ in the join tree of the piece containing $v$.
By \Thm{surgery}, ignoring non-dominant extrema introduced from cutting (whose cost can be charged to a corresponding saddle), the join tree on each piece output by ${\tt rain}$ is isomorphic to some
connected subgraph of the contour tree of the input complex, and hence $\ell_v'\leq \ell_v$.
Moreover, $|H_v|$ only counts vertices in a $v$ to root path and so trivially $|H_v|\leq \ell_v'$, implying \Thm{main-corr}.
Note that there is fair amount of slack in this argument as $|H_v|$ may be significantly smaller than $\ell_v'$.
This slack allows for the more refined upper and lower bounds mentioned in \Sec{more-refined}. Quantifying this slack however is quite challenging,
and requires a significantly more sophisticated analysis involving path decompositions, which is the subject of \Sec{pathDecomp} and \Sec{lb}.
\newcommand{\leafAssignPathDecomp}{
\section{Leaf assignments and path decompositions}
\label{sec:pathDecomp}
In this section, we set up a framework to analyze the time taken to compute a join tree ${\cal J}(\mathbb{M})$ (see \InSoCGVer{\Def{criticalJoin}}\InNotSoCGVer{\Def{join}}).
We adopt all notation already defined in \Sec{runTime}. From here forward we will often assume binary trees are full binary trees
(this assumption simplifies the presentation but is not necessary).
Let $\chi$ be some fixed leaf assignment to a rooted binary tree $T$, which in turn fixes all the heaps $H_v$.
We choose a special path decomposition that is best defined as a subset of edges in $T$ such that each internal
vertex has degree at most $2$. This naturally gives a path decomposition.
For each internal vertex $v\in T$, add the edge from $v$ to $\arg \max_{v_l, v_r} \{|H_{v_l}|, |H_{v_r}|\}$
where $v_l$ and $v_r$ are the children of $v$ (if $|H_{v_l}|=|H_{v_r}|$ then pick one arbitrarily).
This is called the \emph{maximum} path decomposition, denoted by $P_{\max}(T)$.
Our main goal in this section is to prove the following theorem. We use $|p|$ to denote the number of vertices in $p$.
\begin{theorem} \label{thm:runtime} $\sum_{v \in T} \log |H_v| = O(\sum_{p\in P_{\max}(T)} |p| \log |p|)$.
\end{theorem}
We conclude this section in \Sec{implications} by showing that proving this theorem implies our main result \Thm{main-alg}.
\subsection{Shrubs, tall paths, and short paths}
\label{sec:shrubs}
The paths in $P(T)$ naturally define a tree\footnote{Please excuse the
overloading of the term 'tree', it is the most natural term to use here.} of their own. Specifically, in the original
tree $T$ contract each path down to its root. Call the resulting tree the \emph{shrub} of $T$ corresponding to
the path decomposition $P(T)$. Abusing notation, we simply use $P(T)$ to denote the shrub.
As a result, we use terms like `parent', `child', `sibling', etc. for paths as well.
The shrub gives a handle on the heaps of a path.
We use $b(p)$ to denote the \emph{base} of the path, which is vertex in $p$ closest
to root of $T$. We use $\ell(p)$ to denote the leaf in $p$.
We use $H_p$ to denote the $H_{b(p)}$.
\begin{lemma}
\label{lem:adjacent}
Let $p$ be any path in $P(T)$ and let $\{q_1, \dots q_k\}$ be the children on $p$.
Then $H_{\ell(p)} + \sum_{i=1}^k |H_{q_i}| \leq |H_p|+2|p|$.
\end{lemma}
\begin{proof} For convenience, denote $H_i = H_{q_i}$ and $H_0 = H_{\ell(p)}$.
Consider $v \in \bigcup_{i} H_i$ that lies below $b(p)$ in $T$.
Note that such a vertex has only one of its two colors in $L(b(p))$.
Since the colors tracked by $H_i$ and $H_j$ for $i\neq j$ are disjoint,
such a vertex can appear in only one of the $H_i$'s.
On the other hand, a vertex $u\in p$ can appear in more than one $H_i$,
but since any vertex has exactly two colors it can appear in at most two
such heaps. Hence, $\sum_i |H_i| \leq |H_p| + 2|p|$.
\ignore{
First observe that for all $i$, $r_{i}$ lies above $r_p$ on some root to leaf path.
Let $H_i'$ be the subset of $H_i$ that lies below $r_p$. Observe that $H_i'\subseteq H_p$, as $L(T_{r_i})\subseteq L(T_{r_p})$.
Now for any $i\neq j$, $H_i \cap H_j = \emptyset$, as $r_{i}$ and $r_{j}$ are not on the same root to leaf path (i.e.\ $L(T_{r_i}) \cap L(T_{r_j}) = \emptyset$).
In particular, $(H_i\setminus H_i') \cap (H_j \setminus H_j') = \emptyset$, and so $\sum_{i=0}^k |H_i\setminus H_i'| \leq |p|$
as $H_i\setminus H_i'$ is precisely the subset of $H_i$ that lies on $p$.
Moreover, for any $i\neq j$, $ H_i' \cap H_j' = \emptyset$, and so since $H_i'\subseteq H_p$ we have $\sum_{i=0}^k |H_i'|\leq |H_p|$.
Putting these together gives
\[
\sum_{i=0}^k |H_i| = \sum_{i=0}^k |H_i'| + \sum_{i=0}^k |H_i\setminus H_i'| \leq |H_p|+|p|.
\]
}
\end{proof}
We wish to prove $\sum_{v\in T} \log |H_v| = O(\sum_{p\in P} |p| \log |p|)$.
The simplest approach is to prove $\forall p\in P$, $\sum_{v\in p} \log |H_v|= O(|p|\log|p|)$.
This is unfortunately not true, which is why we divide paths into two categories.
\begin{definition}
For $p\in P(T)$, $p$ is \emph{short} if $|p| < \sqrt{|H_{p}|}/100$, and \emph{tall} otherwise.
\end{definition}
\ignore{
\begin{observation}
\label{obs:decrease}
Let $v$ be a vertex in $T$ and let $w$ be its parent. Then $|H_w| \geq |H_v| -1$, as $L(T_v) \subseteq L(T_w)$ and the path from
$w$ to the root has one less vertex than the path from $v$ to the root.
In particular, we have the following more general property.
Let $v$ and $u$ be any two vertices in the same root to leaf path of $T$, such that $v$ is a descendant of $u$.
Then $|H_v| \leq |H_u| + d(u,v)$.
\end{observation}
The following can be thought of as a generalization of the above observation, and will be useful in later sections.
}
The following lemma demonstrates that tall paths can ``pay'' for themselves.
\begin{lemma}
\label{lem:pathBounds} If $p$ is tall, $\sum_{v\in p} \log |H_v| = O(|p| \log |p|)$.
If $p$ is short, $\sum_{v\in p} \log |H_v| = O(|H_p| \log |H_p|)$.
\end{lemma}
\begin{proof} \ignore{
By $\Obs{decrease}$ we know that for any vertex $v\in p$, $|H_v|\leq |H_{p}| + |p|$ (as $v$ is a descendant of $r_p$ along $p$).
}
For $v \in p$, $|H_v|\leq |H_{p}| + |p|$ (as $v$ is a descendant of $b(p)$ along $p$).
Hence, $\sum_{v\in p} \log |H_v| \leq \sum_{v\in p} \log(|H_{p}| + |p|) = |p| \log (|H_{p}| + |p|)$.
If $p$ is a tall path, then $|p| \log (|H_{p}| + |p|) = O(|p| \log |p|)$. If $p$ is short, then
$|p| \log (|H_{p}| + |p|) = O(|p| \log |H_{p}| )$. For short paths, $|p| = O(|H_p|)$.
\end{proof}
\ignore{
For a short path $p$ we can think of the quantity $|p|(\log |H_p| - \log |p|)$ as the excess of $p$, i.e.\ what was not ``paid for'' locally by $p$.
Our eventual goal will be to show that by choosing the right path decomposition, this excess
can be recharged to tall paths. In the next couple sections we introduce the appropriate path decomposition and prove
some useful facts about it. Then given these facts, in \Sec{} we are able to make the recharging argument.
}
There are some short paths that we can also ``pay" for. Consider any short path $p$ in the shrub. We will refer to the
\emph{tall support chain} of $p$ as the tall ancestors of $p$ in the shrub which have a path to $p$ which does not use any
short path (i.e.\ it is a chain of paths adjacent to $p$).
\begin{definition} \label{def:support} A short path $p$ is \emph{supported} if
at least $|H_p|/100$ vertices $v$ in $H_p$ lie in paths in the tall support chain of $p$.
\end{definition}
Let ${\cal L}$ be the set of short paths, ${\cal L}'$ be the set of supported short paths, and ${\cal H}$ be the set of tall paths
given by $P_{\max}(T)$.
We now construct the shrub of unsupported short paths. Consider $p \in {\cal L} \setminus {\cal L}'$,
and traverse the chain of ancestors from $p$. Eventually, we must reach another short path $q$.
(If not, we have reached the root $r$ of $P_{\max}(T)$. Hence, $p$ is supported.)
Insert edge from $p$ to $q$, so $q$ is the parent of $p$ in ${\cal U}$. This construction leads
to the shrub forest of ${\cal L} \setminus {\cal L}'$, where all the roots are supported short paths,
and the remaining nodes are the unsupported short paths.
Most of the work goes into proving the following technical lemma.
\begin{lemma} \label{lem:cu-root} Let ${\cal U}$ denote a connected component (shrub) in the shrub forest of ${\cal L} \setminus {\cal L}'$ and let $r$
be the root of ${\cal U}$. (i) For any $v \in p$ such that $p \in {\cal U}$,
$|H_v| = O(|H_r|)$. (ii) $\sum_{p \in {\cal U}} |p| = O(|H_r|)$.
\end{lemma}
We split the remaining argument into two subsections. We first prove \Thm{runtime} from \Lem{cu-root},
which involves routine calculations. Then we prove \Lem{cu-root}, where the interesting work happens.
\subsection{Proving \Thm{runtime}} \label{sec:thm-runtime}
We split the summation into tall, short, and unsupported short paths.
\begin{eqnarray*}
\sum_{p \in {\cal L}} \sum_{v \in p} \log |H_v| & = & \sum_{p \in {\cal L} \setminus {\cal L}'} \sum_{v \in p} \log |H_v| + \sum_{p \in {\cal L}'} \sum_{v \in p} \log |H_v| + \sum_{p \in {\cal H}} \sum_{v \in p} \log |H_v|
\end{eqnarray*}
The last term can be bounded by $O(\sum_{p \in P_{\max}(T)} |p|\log |p|)$, by \Lem{pathBounds}.
The second term can be bounded by $O(\sum_{p \in {\cal L}'} |H_p| \log |H_p|)$, by \Lem{pathBounds} again.
The following claim shows that this in turn is at most the last term.
\begin{claim} \label{clm:above} $\sum_{p \in {\cal L}'} |H_p| \log |H_p| = O(\sum_{q \in {\cal H}} \sum_{v \in q} \log |H_v|)$.
\end{claim}
\begin{proof}
Pick $p \in {\cal L}'$.
As we traverse the tall support chain of $p$, there
are at least $|H_p|/100$ vertices of $H_p$ that lie in these paths. These are encountered in
a fixed order. Let $H'_p$ be the first $|H_p|/200$ of these vertices. When $v \in H'_p$
is encountered, there are $|H_p|/200$ vertices of $H_p$ not yet encountered. Hence,
$|H_v| \geq |H_p|/200$. Hence, $|H_p|\log |H_p| = O(\sum_{v \in H'_p} \log |H_v|)$.
Since all the vertices lie in tall paths,
we can write this as $O(\sum_{q \in {\cal H}} \sum_{v \in H'_p \cap q} \log |H_v|)$.
Summing over all $p$, the expression is $\sum_{q \in {\cal H}} \sum_{p \in {\cal L}'} \sum_{v \in H'_p \cap q} \log |H_v|$.
Consider any $v \in H'_p$. Let $S$ be the set of paths $\widetilde{p}\in {\cal L}'$ such that
$v\in H'_{\widetilde{p}}$. We now show $|S|\leq 2$ (i.e.\ it contains at most one path other than $p$).
First observe that any two paths in $S$ must be unrelated (i.e.\ $S$ is an anti-chain),
since paths which have an ancestor-descendant relationship have disjoint tall support chains.
However, any vertex $v$ receives exactly one color from each of its two subtrees (in $T$), and therefore $|S|\leq 2$ since any two
paths which share descendant leaves in $T$ (i.e.\ their heaps are tracking the same color) must have an ancestor-descendant relationship.
In other words, any $\log |H_v|$ appears at most twice in the above triple summation.
Hence, we can bound it by $O(\sum_{q \in {\cal H}} \sum_{v \in q} \log |H_v|)$.
\end{proof}
The first term (unsupported short paths) can be charged to the second term (supported short paths).
This is where the critical \Lem{cu-root} plays a role.
\begin{claim} \label{clm:first} $\sum_{p \in {\cal L} \setminus {\cal L}'} \sum_{v \in p} \log |H_v| = O(\sum_{p \in {\cal L}'} |H_p| \log |H_p|)$.
\end{claim}
\begin{proof} Let ${\cal U}$ denote a connected component of the shrub forest.
We have $\sum_{p \in {\cal L} \setminus {\cal L}'} \sum_{v \in p} \log |H_v| \allowbreak \leq \sum_{{\cal U}} \sum_{p \in {\cal U}} \sum_{v \in p} \log |H_v|$.
By \Lem{cu-root}, $|H_v| = O(|H_r|)$, where $r$ is the root of ${\cal U}$.
Furthermore, $\sum_{p \in {\cal U}} |p| = O(|H_r|)$.
We have $\sum_{p \in {\cal U}} \sum_{v \in p} \log |H_v| = O((\log |H_r|) \sum_{p \in {\cal U}} |p|) = O(|H_r|\log |H_r|)$.
We sum this over all ${\cal U}$ in the shrub forest, and note that roots in the shrub forest are supported short paths.
\end{proof}
\ignore{
Now, for the critical piece of the approach. Focus on $P_{\max}(T)$. Let $L$ be the set of short paths,
and $\hat{L} \subseteq L$ be the set of paths such that no ancestor of $p \in L'$
is short. Obviously, $\hat{L}$ is an anti-chain in $P(T)$.
\begin{lemma} \label{lem:short} For the path decomposition $P_{\max}(T)$, $\sum_{p \in {\cal L}} \sum_{v \in p} \log |H_v| = O(\sum_{p \in {\cal L}'} |H_p| \log |H_p|)$.
\end{lemma}
With this lemma, the proof of \Thm{runtime} is straightforward.
\begin{proof} (of \Thm{runtime}) We focus on $P_{\max}(T)$ and split into tall and short paths.
We have $\sum_{v \in T} \log |H_v| =$ $\sum_{p \in {\cal L}} \sum_{v \in p} \log |H_v| + $ $\sum_{p \in {\cal H}} \sum_{v \in p} \log |H_v|$.
For the first term, we apply \Lem{short} to bound by $O(\sum_{p \in {\cal L}'} |H_p| \log |H_p|)$.
Then we apply \Clm{above} to bound by $O(\sum_{p \in {\cal H}} |p|\log |p|)$.
The latter is also bounded by the same expression, by \Lem{pathBounds}. By definition,
$\sum_{p \in {\cal H}} |p|\log |p| \leq \mathop{cost}(P_{\max}(T))$.
\end{proof}
The main challenge is proving \Lem{short}, which we defer to the next section.
\begin{definition}
Let $\chi(T)$ be a valid coloring of a binary tree $T$. We define a set of disjoint paths over the vertices of
$T$ as follows.
The corresponding collection of paths that all such edges correspond to is a path decomposition of $T$, which
we refer to as a \emph{maximum} path decomposition, and is denoted by $\mathcal{P}(T)$.
\end{definition}
From the previous section we now have a lower bound of $\Omega(f_{path}(P))$ for computing the merge tree in terms of any path decomposition $P$.
We now show that the running time of our algorithm for computing the merge tree is upper bounded by this expression for a specific path decomposition,
which we will call the maximum path decomposition (defined below).
Specifically, for a given valid coloring $\chi$ of a tree $T$, we will prove that for the maximum path decomposition $f_{alg}(\chi(T)) = O(f_{path}(P(T)))$.
Throughout this section, for a path $p\in P(T)$, we use the notation $H_p$ to refer to the heap at the root of $p$, i.e.\ $H_p = H_{r_p}$.
}
\subsection{Proving \Lem{cu-root}: the root is everything in ${\cal U}$} \label{sec:weight}
\Lem{cu-root} asserts the root $r$ in ${\cal U}$ pretty much encompasses all sizes and heaps in ${\cal U}$.
We will work with the \emph{reduced} heap $\widetilde{H}_p$. This is the subset of vertices of $H_p$ that
do not appear on the tall support chain of $p$.
By definition, for any unsupported short path
(hence, any non-root $p \in {\cal U}$), $|\widetilde{H}_p| \geq 99|H_p|/100$.
We begin with a key property, which is where the construction of $P_{\max}(T)$ enters the picture.
\begin{lemma}
\label{lem:geometric}
Let $q$ be the child of some path $p$ in ${\cal U}$, then $|H_p|\geq \frac{3}{2}|H_q|$.
Moreover, if $p\neq r({\cal U})$, then $|\widetilde{H}_p|\geq \frac{3}{2}|\widetilde{H}_q|$.
\end{lemma}
\begin{proof} Let $h(q)$ denote the tall path that is a child of $p$ in $P_{\max}(T)$,
and an ancestor of $q$. If no such tall path exists, then by construction $p$ is the parent
of $q$ in $P_{\max}(T)$, and the following argument will go through by setting $h(q)=q$.
The chain of ancestors from $q$ to $h(q)$ consists only of tall paths.
Since $q$ is unsupported, these paths contain at most $|H_q|/100$ vertices of $H_q$.
Thus, $|H_{h(q)}| \geq 99|H_q|/100$.
Consider the base
of $h(q)$, which is a node $w$ in $T$. Let $v$ denote the sibling of $w$ in $T$.
Their parent is called $u$. Note that both $u$ and $v$ are nodes in the path $p$.
Now, the decomposition $P_{\max}(T)$ put $u$ and $v$ in the same path $p$. This
implies $|H_v| \geq |H_w|$. Since $|H_u| \geq |H_v| + |H_w| - 2$,
$|H_u| \geq 2|H_w| - 2$. Let $b$ be the base of $p$.
We have $|H_p| = |H_b| \geq |H_u| - |p| \geq 2|H_w| - |p| - 2$. Since $p$ is a short path, $|p| < \sqrt{|H_p|}/100$.
Applying this bound, we get $|H_p| \geq (2-\delta)|H_w|$ (for a small constant $\delta > 0$).
Since $w$ is the base of $h(q)$, $H_w = H_{h(q)}$. We apply the bound $|H_{h(q)}| \geq 99|H_q|/100$
to get $|H_p| \geq 197|H_q|/100$, implying the first part of the lemma. For the second part, observe that
if $p\neq r({\cal U})$, then $p$ is unsupported and so $|\widetilde{H}_p| \geq 99|H_p|/100$, and therefore the second part follows since $|H_q| \geq |\widetilde{H}_q|$.
\end{proof}
This immediately proves part (i) of \Lem{cu-root}. Part (ii) requires much more work.
We define a \emph{residue} $R_p$ for each $p \in {\cal U}$. Suppose $p$ has children
$q_1, q_2, \ldots, q_k$ in ${\cal U}$. Then $R_p = |\widetilde{H}_p| - \sum_i |\widetilde{H}_{q_i}|$.
By definition, $|\widetilde{H}_p| = \sum_{q \in {\cal U}_p} R_p$.
Note that $R_p$ can be negative.
Now, define $R^+_p = \max(R_p,0)$, and set $W_p = \sum_{q \in {\cal U}_p} R^+_p$.
Observe that $W_p \geq |\widetilde{H}_p|$. We also get an approximate converse.
\begin{claim} \label{clm:loss} For any path $p\in {\cal U}$, $|\widetilde{H}_p| \geq W_p - 2\sum_{q \in {\cal U}_p} |q|$.
\end{claim}
\begin{proof} We write $W_p - |\widetilde{H}_p| = \sum_{q \in {\cal U}_p} R^+_q - R_q$ $= -\sum_{q \in {\cal U}_p: R_q < 0} R_q$.
Consider $q \in {\cal U}_p$ and denote the children in ${\cal U}_p$ by $q'_1, q'_2, \ldots$.
Note that $R_q$ is negative exactly when $|\widetilde{H}_q| < \sum_i |\widetilde{H}_{q'_i}|$.
Traverse $P_{\max}(T)$ from $q'_i$ to $q$.
Other than $q$, all other nodes encountered are in the tall support chain of $q'_i$ and hence
do not affect its reduced heap.
The vertices of $\widetilde{H}_{q'_i}$ that are deleted
are exactly those present in the path $q$.
Any vertex in $q$ can be deleted from at most two of the reduced heaps (of the children of $q$ in ${\cal U}_p$), since
theses reduced heaps do not have an ancestor-descendant relationship.
Therefore when $R_q$ is negative, it is at most by $2|q|$.
We sum over all $q$ to complete the proof.
\end{proof}
\ignore{Let $D_q = H_q \setminus \bigcup_i H_{q'_i}$. In other words, as we traverse ${\cal U}$,
$D_q$ is the set of vertices ``added" at $q$. Suppose we traverse $P_{\max}(T)$ from $q'_i$
to $q$. All vertices of $H_{q'_i}$ that lie on these paths are removed.
and traverse $P_{\max}(T)$ up to $p$. Suppose we encounter
the paths $q_0 = q, q_1, q_2, \ldots, q_k = p$. The vertices in the residue $R_q$ that
are removed from $H_p$ are exactly the vertices that lie in these paths.
We split this contribution into tall and short paths, observing that the latter
are all unsupported. Hence, we can (crudely) upper bound the number of vertices
removed by $\sum_{q \in {\cal U}_p} |q| + \sum_{q \in {\cal U}_p} \sum_{h \in {\cal H}} |h \cap q|$.
Each vertex can be removed from at most two distinct $R_q$'s, completing the proof.
}
The main challenge of the entire proof is bounding the sum of path lengths, which is done next.
We stress that the all the previous work is mostly the setup for this claim.
\begin{claim} \label{clm:charge} Fix any path $p \in {\cal U}\setminus\{r({\cal U})\}$. Suppose for any $q,q' \in {\cal U}_p$
where $q$ is a parent of $q'$ in ${\cal U}_p$, $W_q \geq (4/3)W_{q'}$. Then $\sum_{q \in {\cal U}_p} |q| \leq W_p/20$.
\end{claim}
\begin{proof} Since $q$ is an unsupported short path, $|q| < \sqrt{|H_q|}/100 \leq \sqrt{|\widetilde{H}_q|}/99 \leq \sqrt{|W_q|}/99$.
We prove that $\sum_{q \in {\cal U}_p} \sqrt{|W_q|}/99 \leq W_p/20$
by a charge redistribution scheme. Assume that each $q \in {\cal U}_p^-$
starts with $\sqrt{W_q}/99$ units of charge. We redistribute this charge over all
nodes in ${\cal U}_q$, and then calculate the total charge. For $q \in {\cal U}_p$, spread
its charge to all nodes in ${\cal U}_q$ proportional to $R^+$ values. In other words,
give $(\sqrt{W_q}/99)\cdot(R^+_{q'}/W_q)$ units of charge to each $q' \in {\cal U}_q$.
After the redistribution, let us compute the charge deposited at $q$. Every ancestor in ${\cal U}_p^-$
$q = a_0, a_1, a_2, \ldots, a_k$ contributes to the charge at $q$. The charge is expressed in the following
equation. We use the assumption that $W_{a_i} \geq (4/3) W_{a_{i-1}}$ and hence $W_{a_i} \geq (4/3)^i W_{a_0} \geq (4/3)^i$,
as $a_0$ is an unsupported short path and hence $W_{a_0}\geq 1$.
$$ ({R^+_q}/99)\sum_{a_i} 1/\sqrt{W_{a_i}} \leq ({R^+_q}/99)\sum_{a_i} (3/4)^{i/2} \leq R^+_q/20 $$
The total charge is $\sum_{q \in {\cal U}_p} R^+_p/20 = W_p/20$.
\end{proof}
\begin{corollary} \label{cor:chargeCor}
Let $r$ be the root of ${\cal U}$, and suppose that for any paths $q,q' \in {\cal U}\setminus \{r\}$, where $q$ is a parent of $q'$ in ${\cal U}$, $W_q \geq (4/3)W_{q'}$.
Then $\sum_{p \in {\cal U}} |p| \leq W_r/20+|r|$.
\end{corollary}
\begin{proof}
Let $c_1, \dots, c_m$ be the children of $r$ in ${\cal U}$. By definition, $W_r = \sum_i W_{c_i}+R^+_r \geq \sum_i W_{c_i}$.
By \Clm{charge}, for each $c_i$ we have $W_{c_i}/20\geq \sum_{p \in {\cal U}_{c_i}} |p|$. Combining these to facts yields the claim.
\end{proof}
We wrap it all up by proving part (ii) of \Lem{cu-root}.
\begin{claim} $\sum_{p \in {\cal U}} |p| \leq |H_{r({\cal U})}|/10$.
\end{claim}
\begin{proof} We use $r$ for $r({\cal U})$.
Suppose $W_q \geq (4/3)W_{q'}$ (for any choice in ${\cal U}\setminus\{r\}$ of $q$ parent of $q'$),
then by \Cor{chargeCor}, $\sum_{p \in {\cal U}} |p| \leq W_{r}/20+|r|$.
By \Clm{loss}, $|\widetilde{H}_r| \geq W_r - 2\sum_{p \in {\cal U}} |p|$, and so combining these inequalities gives,
\[
\sum_{p \in {\cal U}} |p|\leq \frac{10}{9} \pth{|\widetilde{H}_r|/20+|r|} \leq
\frac{10}{9}\pth{|H_r|/20+\sqrt{|H_r|}/100} \leq |H_r|/10.
\]
We now prove that for any $q$ parent of $q'$ (other than $r$), $W_q \geq (4/3)W_{q'}$.
Suppose not. Let $p, p'$ be the counterexample furthest from the root,
where $p$ is the parent of $p'$.
Note that for $q$ and child $q'$ in ${\cal U}_{p'}$, $W_q \geq (4/3)W_{q'}$.
We will apply \Clm{charge} for ${\cal U}_{p'}$ to deduce that $\sum_{q \in {\cal U}_{p'}} |q| \leq W_{p'}/20$.
Combining this with \Clm{loss} gives, $|\widetilde{H}_{p'}| \geq 19W_{p'}/20$. By \Lem{geometric}, $|\widetilde{H}_p| \geq (3/2)|\widetilde{H}_{p'}|$.
Noting that $W_p \geq |\widetilde{H}_p|$, we deduce that $W_p \geq (4/3)W_{p'}$.
Hence, $p, p'$ is not a counterexample, and more generally, there is no counterexample.
That completes the whole proof.
\end{proof}
\subsection{Our Main Result}
\label{sec:implications}
We now show that \Thm{runtime} allows us to upper bound the running time for our join tree and contour
tree algorithms in terms of path decompositions.
\begin{theorem}
Let $f:\mathbb{M} \to \mathbb{R}$ be the linear interpolant over distinct valued vertices, where the join tree ${\cal J}(\mathbb{M})$ has maximum degree $3$.
There is an algorithm to compute the join tree whose running time is $O(\sum_{p \in P_{\max}({\cal J})} |p|\log |p| + t\alpha(t) + N)$.
\end{theorem}
\begin{proof}
By \Thm{correct} we know that ${\tt build}(\mathbb{M})$ correctly outputs ${\cal J}(\mathbb{M})$, and by \Lem{runTimeUpper}
we know this takes $O(\sum_{v \in {\cal J}(\mathbb{M})} \log |H_v| + t\alpha(t) + N)$ time, where the $H_v$ values are determined as in \Def{initialColoring}.
Therefore by \Thm{runtime}, ${\tt build}(\mathbb{M})$ takes $O(\sum_{p \in P_{\max}({\cal J})} |p|\log |p| + t\alpha(t) + N)$
time to correctly compute ${\cal J}(\mathbb{M})$.
\end{proof}
This result for join trees easily implies our main result, \Thm{main-alg}, which we now restate and prove.
\begin{theorem}
Let $f:\mathbb{M} \to \mathbb{R}$ be the linear interpolant over distinct valued vertices, where the contour tree ${\cal C}={\cal C}(\mathbb{M})$ has maximum degree $3$.
There is an algorithm to compute ${\cal C}$ whose running time is $O(\sum_{p \in P({\cal C})} |p|\log |p| + t\alpha(t) + N)$,
where $P(T)$ is a specific path decomposition (constructed implicitly by the algorithm).
\end{theorem}
\begin{proof}
First, lets review the various pieces of our algorithm.
On a given input simplicial complex, we first break it into extremum dominant pieces using ${\tt rain}(\mathbb{M})$ (and in $O(|\mathbb{M}|)$ time by \Thm{rain-time}).
Specifically, \Lem{rain-1} proves that the output of ${\tt rain}(\mathbb{M})$ is a set of extremum dominant pieces,
$\mathbb{M}_1, \dots, \mathbb{M}_k$, and \Clm{rain-reeb} shows that given the contour trees, ${\cal C}(\mathbb{M}_1), \dots, {\cal C}(\mathbb{M}_k)$,
the full contour tree, ${\cal C}(\mathbb{M})$, can be constructed (in $O(|\mathbb{M}|)$ time).
Now one of the key observations was that for extremum dominant manifolds, computing the contour tree is roughly the same as
computing the join tree. Specifically, \Thm{contour-tree} implies that given $\cJ_C(\mathbb{M}_i)$ ,
we can obtain ${\cal C}(\mathbb{M}_i)$ by simply sticking on the non-dominant minima at their respective splits (which can easily be done in linear time).
Remark~\ref{rem:joinAndCriticalJoin} implies that $\cJ_C(\mathbb{M}_i)$ is trivially obtained from the ${\cal J}(\mathbb{M}_i)$, and
by the above theorem we know ${\cal J}(\mathbb{M}_i)$ can be computed in $O(\sum_{p \in P_{\max}({\cal J}(\mathbb{M}_i))} |p|\log |p| + t_i\alpha(t_i) + N_i)$
(where $t_i$ and $N_i$ are the number of critical points and faces when restricted to $\mathbb{M}_i$).
At this point we can now see what the path decomposition referenced in theorem statement should be. It is just the union of all the maximum
path decomposition across the extremum dominant pieces, $P_{\max}({\cal C}(\mathbb{M})) = \cup_{i=1}^k P_{\max}({\cal J}(\mathbb{M}_i))$.
Since all procedures besides computing the join trees take linear time in the size of the input complex, we can therefore
compute the contour tree in time
\InSoCGVer{
\begin{align*}
\hspace{.8in} O\pth{N+ \sum_{i=1}^k \pth{\sum_{p \in P_{\max}({\cal J}(\mathbb{M}_i))} |p|\log |p|} + t_i\alpha(t_i) + N_i}& \\
=\break O\pth{\pth{\sum_{p \in P_{\max}({\cal C}(\mathbb{M}))} |p|\log |p|} + t\alpha(t) + N}&
\end{align*}
}
\InNotSoCGVer{
\[
O\pth{N+ \sum_{i=1}^k \pth{\sum_{p \in P_{\max}({\cal J}(\mathbb{M}_i))} |p|\log |p|} + t_i\alpha(t_i) + N_i}
=\break O\pth{\pth{\sum_{p \in P_{\max}({\cal C}(\mathbb{M}))} |p|\log |p|} + t\alpha(t) + N}
\]
}
\end{proof}
}
\InNotSoCGVer{
\leafAssignPathDecomp
}
\newcommand{\lowerBoundbyPathDecomp}{
\section{Lower Bound by Path Decomposition}
\label{sec:lb}
We first prove a lower bound for join trees, and then generalize to contour trees.
Note that the form of the theorem statements in this section differ from \Thm{main-lb},
as they are stated directly in terms of path decompositions. \Thm{main-lb}
is an immediate corollary of the final theorem of this section, \Thm{path-main-lb}.
\subsection{Join Trees}
We focus on terrains, so $d=2$.
Consider any path decomposition $P$ of a valid join tree (i.e.\ any rooted binary tree).
When we say ``compute the join tree", we require the join tree to
be labeled with the corresponding vertices of the terrain.
\begin{figure}[h]\centering
\includegraphics[width=.18\linewidth]{figs/diamond}\hspace{2cm}
\includegraphics[width=.3\linewidth]{figs/cones}
\caption{Left: angled view of a tent / Right: a parent and child tent put together}
\label{fig:diamond}
\end{figure}
\begin{lemma}
\label{lem:pathDecomp}
Fix any path decomposition $P$. There is a family of terrains, ${\bf F}_P$, all with the same triangulation,
such that $|{\bf F}_P| = \Pi_{p_i\in P} (|p_i|-1)!$, and no two terrains in ${\bf F}_P$ define the same join tree.
\end{lemma}
\begin{proof}
We describe the basic building block of these terrains, which corresponds to a fixed path $p\in P$.
Informally, a \emph{tent} is an upside down cone with $m$ triangular faces (see \Fig{diamond}).
Construct a slightly tilted cycle
of length $m$ with the two antipodal points at heights $1$ and $0$. These are called the anchor and trap of the tent, respectively.
The remaining $m-2$ vertices are evenly spread around the cycle and heights decrease monotonically when going from the anchor to the trap.
Next, create an apex vertex at some appropriately large height, and add an edge to each vertex in the cycle.
Now we describe how to attach two different tents. In this process, we glue the base of a scaled down ``child'' tent on to a triangular cone face of the larger ``parent'' tent (see \Fig{diamond}).
Specifically, the anchor of the child tent is attached directly to a face of the parent tent at some height $h$. The remainder of the base of the child cone is then
extended down (at a slight angle) until it hits the face of the parent.
The full terrain is obtained by repeatedly gluing tents. For each path $p_i\in P$, we create a tent of size $|p_i|+1$.
The two faces adjacent to the anchor are always empty, and the remaining faces are for gluing on other tents.
(Note that tents have size $|p_i|+1$ since $|p_i|-1$ faces represent the joins of $p_i$, the apex represents the leaf,
and we need two empty faces next to the anchor.)
Now we glue together tents of different paths in the same way the paths are connected in the shrub $P_{\mathcal{S}}$ (see \Sec{shrubs}).
Specially, for two paths $p,q\in P$ where
$p$ is the parent of $q$ in $P_{\mathcal{S}}$, we glue $q$ onto a face of the tent for $p$ as described above.
(Naturally for this construction to work, tents for a given path will be scaled down relative to the size of the tent of their parent).
By varying the heights of the gluing, we get the family of terrains.
Observe now that the only saddle points in this construction are the anchor points. Moreover, the only maxima are the apexes of the
tents. We create a global boundary minimum by setting the vertices at the base of the tent representing the root of $P_{\mathcal{S}}$ all to the same height (and there are no other minima).
Therefore, the saddles on a given tent will appear contiguously on a root to leaf path in the join tree of the terrain, where the leaf corresponds to the maximum of the tent
(since all these saddles have a direct line of sight to this apex). In particular,
this implies that, regardless of the heights assigned to the anchors, the join tree has a path decomposition whose corresponding shrub is equivalent to $P_{\mathcal{S}}$.
There is a valid instance of this described construction for any relative ordering of the heights of the saddles on a given tent.
In particular, there are $(|p_i|-1)!$ possible orderings of the heights of the saddles on the tent for $p_i$, and hence $\Pi_{p_i\in P} (|p_i|-1)!$ possible
terrains we can build. Each one of these functions will result in a different (labeled) join tree. All saddles on a given tent will appear in sorted order in the join tree.
So, any permutation of the heights on a given tent corresponds to a permutation of the vertices along a path in $P$.
\end{proof}
Two path decompositions $P_1$ and $P_2$ (of potentially different complexes and/or height functions) are equivalent
if: there is a 1-1 correspondence between the sizes of the constituent paths, and the shrubs are isomorphic.
\begin{lemma}
\label{lem:cost}
For all $\mathbb{M} \in {\bf F}_P$, the total number of heap operations performed by ${\tt build}(\mathbb{M})$ is $O(\sum_{p\in P} |p|\log|p|)$.
\end{lemma}
\begin{proof}
The primary ``non-determinism" of the algorithm is the initial painting constructed by ${\tt init}(\mathbb{M})$.
We show that regardless of how paint spilling is done, the number of heap operations is bounded as above.
Consider an arbitrary order of the initial paint spilling over the surface.
Consider any join on a face of some tent, which is the anchor point of some connecting child tent.
The join has two up-stars, each of which has exactly one edge. Each edge connects to a maximum
and must be colored by that maximum.
Hence, the two colors touching this join (according to \Def{initialColoring})
are the colors of the apexes of the child and parent tent.
Take any join $v$, with two children $w_1$ and $w_2$. Suppose $w_1$ and $v$ belong to the same path
in the decomposition. The key is that any color from a maximum in the subtree at $w_2$ cannot touch
any ancestor of $v$. This subtree is exactly the join tree of the child tent attached at $v$.
The base of this tent is completely contained in a face of the parent tent. So all colors
from the child ``drain off" to the base of the parent, and do not touch any joins on the parent tent.
Hence, $|H_v|$ is at most the size of the path in $P$ containing $v$. By \Lem{runTimeUpper}, the total number of heap operations
is at most $\sum_v \log |H_v|$, completing the proof.
\end{proof}
The following is the equivalent of \Thm{main-lb} for join trees, and immediately follows from
the previous lemmas.
\begin{theorem}\label{thm:joinLB}
Consider a rooted tree $T$ and an arbitrary path decomposition $P$ of $T$.
There is a family ${\bf F}_P$ of terrains such that any algebraic decision
tree computing the join tree\footnote{Note that for the referenced family of terrains, the join tree and contour tree are equivalent}
(on ${\bf F}_P$) requires $\Omega(\sum_{p \in P} |p|\log |p|)$ time.
Furthermore, our algorithm makes $O(\sum_{p \in P} |p|\log |p|)$ comparisons on all these instances.
\end{theorem}
\begin{proof}
The proof is a basic entropy argument. Any algebraic decision tree that is correct on all of ${\bf F}_P$
must distinguish all inputs in this family. By Stirling's approximation, the depth of this tree is $\Omega(\sum_{p_i\in P} |p_i|\log|p_i|)$.
\Lem{cost} completes the proof.
\end{proof}
\subsection{Contour Trees}
We first generalize previous terms to the case of contour trees.
In this section $T$ will denote an arbitrary contour tree with every internal vertex of degree $3$.
For simplicity we now restrict our attention to path decompositions
consistent with the raining procedure described in \Sec{rain}
(more general decompositions can work, but it is not needed for our purposes).
\begin{definition}
\label{def:path2} A path decomposition, $P(T)$, is called \emph{rain consistent} if its paths can be obtained as follows.
Perform an downward BFS from an arbitrary maximum $v$ in $T$, and mark all vertices encountered.
Now recursively run a directional BFS from all vertices adjacent to the current marked set.
Specifically, for each BFS run, make it an downward BFS if it is at an odd height in the recursion tree and upward otherwise.
This procedure partitions the vertex set into disjoint rooted subtrees of $T$, based on which BFS marked a vertex.
For each such subtree, now take any partition of the vertices into leaf paths.\footnote{Note that the subtree
of the initial vertex is rooted at a maximum. For simplicity we require that the path this vertex belongs to
also contains a minimum.}
\end{definition}
The following is analogous to \Lem{pathDecomp}, and in particular uses it as a subroutine.
\begin{lemma}
\label{lem:pathDecomp2}
Let $P$ be any rain consistent path decomposition of some contour tree.
There is a family of terrains, ${\bf F}_P$, all with the same triangulation,
such that the size of ${\bf F}_P$ is $\Pi_{p_i\in P} (|p_i|-1)!$, and no two terrains in ${\bf F}_P$ define the same contour tree.
\end{lemma}
\begin{proof}
As $P$ is rain consistent, the paths can be partitioned into sets $P_1, \dots, P_k$, where $P_i$ is the set of all paths with
vertices from a given BFS, as described in \Def{path2}. Specifically, let $T_i$ be the subtree of $T$ corresponding to $P_i$ and
let $r_i$ be the root vertex of this subtree.
Note that the $P_i$ sets naturally define a tree where $P_i$ is the parent of $P_j$ if $r_i$ (i.e.\ the root of $T_i$) is adjacent to a vertex in $P_j$.
As the set $P_i$ is a path decomposition of a rooted binary tree $T_i$, the terrain construction of \Lem{pathDecomp} for $P_i$ is well defined.
Actually the only difference is that here the rooted tree is not a full binary tree, and so some of the (non-achor adjacent) faces of the constructed tents will be blank.
Specifically, these blank faces correspond to the adjacent children of $P_i$, and they tell us how to connect the terrains of the different $P_i$'s.
So for each $P_i$ construct a terrain as described in \Lem{pathDecomp}. Now each $T_i$ is (roughly speaking) a join or a split tree, depending on
whether the BFS which produced it was an upward or downward BFS, respectively. As the construction in \Lem{pathDecomp} was for join trees, each terrain
we constructed for a $P_i$ which came from a split tree, must be flipped upside down.
Now we must described how to glue the terrains together.
\begin{figure}[h]\centering
\includegraphics[width=.28\linewidth]{figs/cones2}
\caption{A child tent attached to a parent tent with opposite orientation.}
\label{fig:diamond2}
\end{figure}
By construction, the tents corresponding to the paths in $P_i$ are connected
into a tree structure (i.e.\ corresponding to the shrub of $P_i$). Therefore the bottoms of all these tents are covered except for the one corresponding
to the path containing the root $r_i$.
If $r_i$ corresponds to the initial maximum that the rain consistent path decomposition was defined from, then this will be flat and corresponds to the global outer face.
Otherwise, $P_i$ has some parent $P_j$ in which case we connect the bottom of the tent for $r_i$ to a free face of a tent in the construction for $P_j$, specifically, the face corresponding to the
vertex in $T$ which $r_i$ is adjacent to. This gluing is done in the same manner as in \Lem{pathDecomp}, attaching the anchor for the root of $P_i$ directly the corresponding face of $P_j$, except that now $P_i$ and $P_j$ have opposite orientations. See \Fig{diamond2}.
Just as in \Lem{pathDecomp} we now have one fixed terrain structure, such that each different relative ordering of the heights of the join and split vertices
on each tent produces a surface with a distinct contour tree. The specific bound on the size of ${\bf F}_P$, defining these distinct contour trees, follows by applying the
bound from \Lem{pathDecomp} to each $P_i$.
\end{proof}
\begin{lemma}
For all $\mathbb{M} \in {\bf F}_P$, the number of heap operations is $\Theta(\sum_{p\in P} |p|\log |p|)$
\end{lemma}
\begin{proof}
This lemma follows immediately from \Lem{cost}. The heap operations can be partitioned into the
operations performed in each $P_i$.
Apply \Lem{cost} to each of the $P_i$ separately and take the sum.
\end{proof}
We now restate \Thm{main-lb}, which follows immediately from an entropy argument, analogous to \Thm{joinLB}.
\begin{theorem}\label{thm:path-main-lb}
Consider any rain consistent path decomposition $P$. There exists a family ${\bf F}_P$ of terrains ($d=2$) with the following properties.
Any contour tree algorithm makes $\Omega(\sum_{p \in P} |p|\log |p|)$ comparisons in the worst case over ${\bf F}_P$.
Furthermore, for any terrain in ${\bf F}_P$, our algorithm makes $O(\sum_{p \in P} |p|\log |p|)$ comparisons.
\end{theorem}
\begin{Remark}
Note that for the terrains described in this section, the number of critical points is within a constant factor of
the total number of vertices. In particular, for this family of terrains, all previous algorithms required $\Omega(n\log n)$ time.
\end{Remark}
}
\InNotSoCGVer{
\lowerBoundbyPathDecomp
}
\myparagraph{Acknowledgements.}
We thank Hsien-Chih Chang, Jeff Erickson, and Yusu Wang for numerous useful discussions. This work is
supported by the Laboratory Directed Research and Development
(LDRD) program of Sandia National Laboratories. Sandia National
Laboratories is a multi-program laboratory managed and operated
by Sandia Corporation, a wholly owned subsidiary of Lockheed
Martin Corporation, for the U.S. Department of Energy's National
Nuclear Security Administration under contract DE-AC04-94AL85000.
\InNotSoCGVer{
\bibliographystyle{alpha}
}
\InSoCGVer{
\bibliographystyle{plain}
}
| {
"timestamp": "2015-12-11T02:12:49",
"yymm": "1411",
"arxiv_id": "1411.2689",
"language": "en",
"url": "https://arxiv.org/abs/1411.2689",
"abstract": "We revisit the classical problem of computing the \\emph{contour tree} of a scalar field $f:\\mathbb{M} \\to \\mathbb{R}$, where $\\mathbb{M}$ is a triangulated simplicial mesh in $\\mathbb{R}^d$. The contour tree is a fundamental topological structure that tracks the evolution of level sets of $f$ and has numerous applications in data analysis and visualization.All existing algorithms begin with a global sort of at least all critical values of $f$, which can require (roughly) $\\Omega(n\\log n)$ time. Existing lower bounds show that there are pathological instances where this sort is required. We present the first algorithm whose time complexity depends on the contour tree structure, and avoids the global sort for non-pathological inputs. If $C$ denotes the set of critical points in $\\mathbb{M}$, the running time is roughly $O(\\sum_{v \\in C} \\log \\ell_v)$, where $\\ell_v$ is the depth of $v$ in the contour tree. This matches all existing upper bounds, but is a significant improvement when the contour tree is short and fat. Specifically, our approach ensures that any comparison made is between nodes in the same descending path in the contour tree, allowing us to argue strong optimality properties of our algorithm.Our algorithm requires several novel ideas: partitioning $\\mathbb{M}$ in well-behaved portions, a local growing procedure to iteratively build contour trees, and the use of heavy path decompositions for the time complexity analysis.",
"subjects": "Computational Geometry (cs.CG)",
"title": "Avoiding the Global Sort: A Faster Contour Tree Algorithm",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9770226280828407,
"lm_q2_score": 0.7248702821204019,
"lm_q1q2_score": 0.7082146680564253
} |
https://arxiv.org/abs/1705.08574 | A Gromov hyperbolic metric vs the hyperbolic and other related metrics | We mainly consider two metrics: a Gromov hyperbolic metric and a scale invariant Cassinian metric. We compare these two metrics and obtain their relationship with certain well-known hyperbolic-type metrics, leading to several inclusion relations between the associated metric balls. | \section{Introduction}
Comparison of hyperbolic-type metrics, defined over proper subdomains of $ {\mathbb{R}^n} $, is a significant part of geometric function theory as it reveals the geometric property of the domain. For example, uniform
domains are defined by comparing the quasihyperbolic metric \cite{GP76} and the distance ratio metric \cite{GO79}.
In recent years, many authors have contributed to
the study of hyperbolic-type metrics. Some of the familiar hyperbolic-type metrics are
the Apollonian metric \cite{Bea98,BI08,Has03,Ibr03}, the half-Apollonian metric \cite{HL04}, the Seittenranta metric \cite{Sei99}, the Cassinian metric \cite{Ibr09,IMSZ,KMS,HKVZ},
the triangular ratio metric \cite{CHKV15}, etc. These metrics are also referred as the {\em relative metrics} since they are defined in proper subdomains of the Euclidean space $ {\mathbb{R}^n} $, $n\ge 2$, relative to domain boundaries.
A general form of some of these relative metrics has been considered by P. H\"ast\"o in \cite[Lemma~6.1]{Has02}, in a different context.
Very recently, a scale invariant version of the Cassinian metric has been studied
by Ibragimov in \cite{Ibr16} which is defined by
$$\tilde{\tau}_D(x,y)=\log\left(1+\sup_{p\in \partial D}\frac{|x-y|}{\sqrt{|x-p||p-y|}}\right), \quad x,y\in D\subsetneq {\mathbb{R}^n} .
$$
The interesting part of this metric is that many
properties in arbitrary domains are revealed in the setting of once-punctured spaces.
For example, $\tilde{\tau}_D$ is a metric in an arbitrary domain $D\subsetneq {\mathbb{R}^n} $ if it is a metric
on once-punctured spaces. The $\tilde{\tau}_D$-metric is comparable with the Vuorinen's distance ratio metric \cite{Vuo85} in
arbitrary domains $D\subsetneq {\mathbb{R}^n} $ if they are comparable in the punctured spaces (see \cite{Ibr16}).
It is appropriate here to recall that the $\tilde{\tau}_D$-metric satisfies the domain monotonicity property, i.e. if $D_1,D_2\subsetneq {\mathbb{R}^n} $ with $D_1\subset D_2$, then
$\tilde{\tau}_{D_2}(x,y)\le \tilde{\tau}_{D_1}(x,y)$ for all $x,y\in D_1$ \cite[p.~2]{Ibr16}.
Gromov in 1987 introduced the notion of an abstract hyperbolic space \cite{Gro87}.
Let $(D,d)$ be
a metric space and $x,y,z\in D$. The {\em Gromov product}\index{Gromov product} of $x$ and $y$ with respect to $z$
is defined by the formula
$$(x|y)_z=\frac{1}{2} \left[d(x,z)+d(y,z)-d(x,y)\right].
$$
The metric space $(D,d)$ is said to be {\em Gromov hyperbolic}\index{Gromov hyperbolic} if there exists $\beta \ge 0$ such that
$$(x|y)_w\ge (x|z)_w \wedge (z|y)_w -\beta
$$
for all $x,y,z,w\in D$. We also say that $D$ is $\beta$-hyperbolic. Equivalently,
the metric space $(D,d)$ is called Gromov hyperbolic if and only if there exist a constant $\beta>0$ such that
$$d(x,z)+d(y,w)\le (d(x,w)+d(y,z))\vee (d(x,y)+d(z,w))+2\beta
$$
for all points $x,y,z,w \in D$. Note that Gromov hyperbolicity is preserved under
quasi-isometries\index{quasi-isometry}. That means it is preserved under the mappings $f:(D,d_1)\to (f(D),d_2)$ satisfying
$$\frac{1}{\lambda} d_1(x,y)-k\le d_2(f(x),f(y))\le \lambda d_1(x,y)+k, \quad x,y\in D,
$$
where $\lambda \ge 1$ and $k\ge 0$. Literature on Gromov hyperbolicity are available in
\cite{BS00,Gro87,Has05,Ibr11,Ibr11-I,Vai05}.
One natural question was to investigate {\em whether a metric space is hyperbolic in the
sense of Gromov or not}?
Ibragimov in \cite{Ibr11} introduced a metric, $u_Z$, which hyperbolizes
(in the sense of Gromov) the locally compact non-complete metric space $(Z,d)$ without changing its quasiconformal geometry, by
$$u_Z(x,y)=2\log \frac{d(x,y)+\max\{{\rm dist(x, \partial Z)},{\rm dist(y, \partial Z)}\}}{\sqrt{{\rm dist(x, \partial Z)}\,{\rm dist(y, \partial Z)}}}, \quad x,y\in Z.
$$
For a domain $D\subsetneq {\mathbb{R}^n} $ equipped with the Euclidean metric, the {\em $u_D$-metric} is defined by
$$u_D(x,y)=2\log \frac{|x-y|+\max\{{\rm dist}(x,\partial D),{\rm dist}(y,\partial D)\}}
{\sqrt{{\rm dist}(x,\partial D)\,{\rm dist}(y,\partial D)}}, \quad x,y\in D.
$$
Note that the $u_D$-metric does not satisfy the domain monotonicity property and
it coincides with the Vuorinen's distance ratio metric in punctured spaces $ {\mathbb{R}^n} \setminus\{p\}$, for $p\in {\mathbb{R}^n} $.
Though comparisons of the $u_D$-metric with some hyperbolic-type metrics are studied in \cite{Ibr11},
in this paper, we further compare the $u_D$-metric with the hyperbolic metric and other related metrics which were not
considered in \cite{Ibr11}.
It is well known that the geometric structure of a metric space can be viewed from the geometric structure of the fundamental element, namely, the metric balls. Hence it is reasonable to study metric balls and their inclusion relations
with other metric balls by fixing the centre common to each pair of metric balls. In this
regard we study inclusion relations associated with the metric balls.
The paper is organized as follows:
Section~\ref{prel} is devoted to the preliminaries for the upcoming sections.
In Section~\ref{u-rho-bn}, we compare the $u_D$-metric with the hyperbolic metric of the
unit ball and upper half space. In Section~\ref{u-tildetau}, we compare the $u_D$-metric
with the $\tilde{\tau}_D$-metric. Comparisons of the $u_D$-metric and the $\tilde{\tau}_D$-metric with other metrics are discussed in Section~\ref{u-tildetau-other}.
\section{Preliminaries}\label{prel}
Throughout the paper, $D$ denotes an arbitrary, proper subdomain of the Euclidean space $\mathbb{R}^n$.
Symbolically, we write $D\subsetneq \mathbb{R}^n$.
Given $x\in\mathbb{R}^n$ and $r>0$, the open ball centered at $x$ and of radius $r$ is denoted by $B(x,r)=\{y\in\mathbb{R}^n\colon\, |x-y|<r\}$.
Set $\mathbb{B}^n=B(0,1)$. We denote by $D_p$ and $D_{p,q}$, the punctured spaces $ {\mathbb{R}^n} \setminus\{p\}$ and $ {\mathbb{R}^n} \setminus\{p,q\}$ respectively.
For a given $x\in D$, we set $d(x):={\rm dist}(x,\partial D)$.
For real numbers $r$ and $s$, we set $r\vee s=\max\{r,\, s\}$ and $r\wedge s=\min\{r,\, s\}$.
The {\it distance ratio metric}, $\tilde{j}_D$, is defined by
$$
\tilde{j}_D(x,y)=\log\left(1+\frac{|x-y|}{d(x)\wedge d(y)}\right), \quad x,y\in D.
$$
The above form of the metric $\tilde{j}_D$, which was first considered in \cite{Vuo85},
is a slight modification of the original distance ratio metric, $j_D$, of Gehring and Osgood \cite{GO79}, defined by
$$j_D(x,y)=\frac{1}{2}\log \left(1+\frac{|x-y|}{d(x)}\right) \left(1+\frac{|x-y|}{d(y)}\right),\quad x,y\in D.
$$
The $\tilde{j}_D$-metric has been widely studied in the literature; see, for instance, \cite{Vuo88}.
These two distance ratio metrics are related by:
$\tilde{j}_D(x,y)/2 \le j_D(x,y)\le \tilde{j}_D(x,y)$; see \cite{Sei99,Has05}.
The {\it{hyperbolic metric}}, $\rho_{\mathbb{B}^n}$, of the unit ball $\mathbb{B}^n$ is given by
$$
\rho_{ {\mathbb{B}^n} }(x,y)=\inf_\gamma\int_\gamma \frac{2|{\rm d}z|}{1-|z|^2},
$$
where the infimum is taken over all rectifiable curves $\gamma\subset \mathbb{B}^n$
joining $x$ and $y$.
Now,
we define the $d$-metric ball (metric ball with respect to the metric $d$) as follows: let $(D,d)$
be a metric space. Then the set
$$B_d(x,R)=\{z\in D:\,d(x,z)<R\}
$$
is called the {\em $d$-metric ball} of the domain $D$.
\section{Comparison of the $u_D$-metric with the hyperbolic metric}\label{u-rho-bn}
This section is devoted to the comparison of the $u_D$-metric and the $\rho_{D}$-metric
when $D= {\mathbb{B}^n} $ and $D=\mathbb{H}^n:=\{(x_1,x_2,\ldots,x_{n}):x_n>0\}$.
The hyperbolic metric of the unit ball $ {\mathbb{B}^n} $, $\rho_{ {\mathbb{B}^n} }$, can be computed by the following formula
(see \cite[p.~40]{Bea95}).
\begin{equation}\label{def-rho}
\sinh\left(\frac{\rho_{ {\mathbb{B}^n} }(x,y)}{2}\right)=\frac{|x-y|}{\sqrt{(1-|x|^2)(1-|y|^2)}}.
\end{equation}
We first establish the relationship between the $u_ {\mathbb{B}^n} $-metric and the $\rho_ {\mathbb{B}^n} $-metric.
In this setting, the following lemma is useful which yields a relationship between
the $u_D$-metric and the $j_D$-metric in arbitrary subdomains of $ {\mathbb{R}^n} $.
\begin{lemma}\label{lem-u-j}
Let $D\subsetneq {\mathbb{R}^n} $ be arbitrary. Then for $x,y\in D$ we have
$$2j_D(x,y)\le u_D(x,y)\le 4 j_D(x,y).
$$
The first inequality becomes equality when $d(x)=d(y)$.
\end{lemma}
\begin{proof}
The first inequality is proved in \cite[Theorem~3.1]{Ibr11}. From the definitions
of the $j_D$-metric and the $u_D$-metric, it follows that $2j_D(x,y)=u_D(x,y)$ whenever $d(x)=d(y)$.
Now, we shall prove the second inequality. Without loss of generality we assume that $d(x)\ge d(y)$ for $x,y\in D\subsetneq {\mathbb{R}^n} $. To show our claim, it is enough to prove that
$$\frac{|x-y|+d(x)}{\sqrt{d(x)d(y)}}\le \left(1+\frac{|x-y|}{d(x)}\right)\left(1+\frac{|x-y|}{d(y)}\right),
$$
or, equivalently,
$$d(x)d(y)\le (d(y)+|x-y|)^2
$$
which is true by the triangle inequality. The proof is complete.
\end{proof}
\begin{theorem}\label{thm-rho-u}
For all $x,y\in {\mathbb{B}^n} $ we have
$$\frac{1}{2}\rho_{ {\mathbb{B}^n} }(x,y)\le u_{ {\mathbb{B}^n} }(x,y)\le 4\rho_{ {\mathbb{B}^n} }(x,y).
$$
\end{theorem}
\begin{proof}
From \cite[p.~151]{AVV} and from \cite[p.~29]{Vuo88} we have respectively the following two inequalities:
\begin{equation}\label{jrho}
\tilde{j}_{ {\mathbb{B}^n} }(x,y)\le \rho_{ {\mathbb{B}^n} }(x,y)\le 2\tilde{j}_{ {\mathbb{B}^n} }(x,y)
\mbox{ and } \frac{1}{2}\tilde{j}_D(x,y)
\le j_D(x,y)\le \tilde{j}_D(x,y).
\end{equation}
Now the proof of our theorem follows from Lemma~\ref{lem-u-j}.
\end{proof}
Observe that for the choice of points $x,y\in {\mathbb{B}^n} $ with $y=-x$,
\[u_{ {\mathbb{B}^n} }(x,-x)=2\log \left(\frac{1+|x|}{1-|x|}\right)=2 \rho_{ {\mathbb{B}^n} }(0,x)=\rho_{ {\mathbb{B}^n} }(x,-x).
\]
This observation leads to the following conjecture.
\begin{conjecture}
For $x,y\in {\mathbb{B}^n} $ we have $\rho_{ {\mathbb{B}^n} }(x,y)\le u_{ {\mathbb{B}^n} }(x,y)\le 2\rho_{ {\mathbb{B}^n} }(x,y)$.
\end{conjecture}
As a consequence of Theorem~\ref{thm-rho-u} we have the following inclusion relation.
\begin{corollary}
Let $x\in {\mathbb{B}^n} $ and $t>0$. Then
$$B_{\rho_{ {\mathbb{B}^n} }}(x,r)\subseteq B_{u_{ {\mathbb{B}^n} }}(x,t)\subseteq B_{\rho_{ {\mathbb{B}^n} }}(x,R),
$$
where $r=t/4$ and $R=2t$.
\end{corollary}
\begin{proof}
The proof follows directly from Theorem~\ref{thm-rho-u}.
\end{proof}
Next theorem shows that the $u_{ {\mathbb{B}^n} }$-metric and the $\rho_{ {\mathbb{B}^n} }$-metric are satisfying
the quasi-isometry property.
\begin{theorem}
For all $x,y\in {\mathbb{B}^n} $, we have
$$
\rho_{\mathbb{B}^n}(x,y)-2\log 2\le u_{\mathbb{B}^n}(x,y)\le 2\rho_{\mathbb{B}^n}(x,y)+2\log 2.
$$
\end{theorem}
\begin{proof}
The right hand side inequality easily follows from \eqref{jrho} and \cite[Theorem~3.1]{Ibr11}.
For the left hand side inequality we assume that $x,y\in {\mathbb{B}^n} $ with $|x|\le |y|$.
It is clear that $(1-|x|)(1-|y|)\le (1-|x|^2)(1-|y|^2)$. Now with the help of the formula \eqref{def-rho} we have
\begin{eqnarray*}
u_{ {\mathbb{B}^n} } (x,y) &=& 2\log \left(\frac{|x-y|+1-|x|}{\sqrt{(1-|x|)(1-|y|)}}\right)
\ge 2\log \left(1+\frac{|x-y|}{\sqrt{(1-|x|^2)(1-|y|^2)}}\right)\\
&=& 2\log \left(1+\sinh\left(\frac{\rho_{ {\mathbb{B}^n} }(x,y)}{2}\right)\right)
\ge \rho_{\mathbb{B}^n}(x,y)-2\log 2,\\
\end{eqnarray*}
where the first inequality follows from the fact that $(1-|x|)/(1-|y|)\ge 1$ and the second inequality
follows from the fact that $1+\sinh(r)\ge e^r/2, r\ge 0$. The proof is complete.
\end{proof}
Now, we compare the $u_{\mathbb{H}^n}$-metric with the $\rho_{\mathbb{H}^{n}}$-metric. Note that for $x,y\in \mathbb{H}^{n}$, the $\rho_{\mathbb{H}^{n}}$-metric can be computed by the formula (see \cite[p.~35]{Bea95})
\begin{equation}\label{rho-hn}
2\sinh\left(\frac{\rho_{\mathbb{H}^{n}}(x,y)}{2}\right)=\frac{|x-y|}{\sqrt{x_n y_n}}.
\end{equation}
\begin{theorem}
For $x,y\in \mathbb{H}^n$ we have
$$\rho_{\mathbb{H}^n}(x,y)\le u_{\mathbb{H}^n}(x,y).
$$
The inequality is sharp.
\end{theorem}
\begin{proof}
Suppose that $x,y\in \mathbb{H}^n$. Without loss of generality we assume that $x_n\ge y_n$. Now,
\begin{eqnarray*}
u_{\mathbb{H}^n}=2\log\left(\frac{|x-y|+x_n}{\sqrt{x_n y_n}}\right) \ge 2\log\left(2\sinh\left(\frac{\rho_{\mathbb{H}^n}(x,y)}{2}\right)+1\right)\ge \rho_{\mathbb{H}^n}(x,y),
\end{eqnarray*}
where the first inequality follows from \eqref{rho-hn} and the hypothesis. However, the second inequality follows from the fact that
$1+2\sinh (r)=1+e^r-e^{-r}\ge e^r$ for $r\ge 0$. For sharpness, consider the points $x=te_2$ and $y=(1/t)e_2$ with $t>1$. Then
$$\rho_{\mathbb{H}^n}(te_2,(1/t)e_2)=2\sinh^{-1} \left(\frac{t^2-1}{2t}\right)=2\log t \mbox{ and }
u_{\mathbb{H}^n}(te_2,(1/t)e_2)=2\log\left(\frac{2t^2-1}{t}\right).
$$
Now taking the limits as $t\to \infty$ we get
$$\lim_{t\to \infty} \frac{\rho_{\mathbb{H}^n}(te_2,(1/t)e_2)}{u_{\mathbb{H}^n}(te_2,(1/t)e_2)}=\lim_{t\to
\infty} \frac{\log t}{\log\left(\displaystyle\frac{2t^2-1}{t}\right)}=1.
$$
Hence completing the proof.
\end{proof}
\section{Comparison of the $u_D$-metric and the $\tilde{\tau}_D$-metric}\label{u-tildetau}
We begin this section with the proof of the comparisons stated in Table~\ref{T1}.
\begin{center}
\begin{table}[h!]
\begin{tabular}{ | c | c | c | }
\hline
& Comparison with $\tilde{\tau}_D$ & Comparison with $u_D$\\[1mm]
\hline
&&\\[-3mm]
& $\frac{1}{2}j_D\le \tilde{\tau}_D\le j_D$ & $2j_D\le u_D\le 4j_D$\\[1mm]
$j_D$ & \cite[Theorems~5.1,5.4]{Ibr16} & [Lemma~\ref{lem-u-j}]\\[1mm]
&(Right hand side inequality is sharp)&(Left hand side inequality is sharp)\\
\hline
&&\\[-3mm]
& $\frac{1}{2}\tilde{j}_D\le \tilde{\tau}_D\le \tilde{j}_D$ & $\tilde{j}_D\le u_D\le 2\tilde{j}_D$\\[1mm]
$\tilde{j}_D$ & \cite[Lemma~4.1, Theorem~4.3]{Ibr16}& [Theorem~\ref{lem-tildej-u}]\\[1mm]
&(Both the inequalities are sharp)&(Left hand side inequality is sharp)\\
\hline
&&\\[-3mm]
& & $2\tilde{\tau}_D\le u_D\le 4\tilde{\tau}_D$\\[1mm]
$\tilde{\tau}_D$ & - & [Theorem~\ref{tau-u-g}]\\[1mm]
&& (Both the inequalities are sharp)\\[1mm]
\hline
\end{tabular}
\vspace*{0.5cm}
\caption{Comparison of $\tilde{\tau}_D$-metric and $u_D$-metric with other hyperbolic-type metrics}\label{T1}
\end{table}
\end{center}
First, we compare the $u_D$-metric with the $\tilde{\tau}_D$-metric in arbitrary domains $D\subsetneq {\mathbb{R}^n} $. Ibragimov in \cite{Ibr16}, proved that the $\tilde{\tau}_D$-metric is Gromov-hyperbolic ($\eta$-hyperbolic) with the constant $\eta=\log 3$ by comparing this with the $\tilde{j}_D$-metric.
Comparison of the $u_D$-metric and the $\tilde{\tau}_D$-metric leads to an improvement in the constant of Gromov hyperbolicity of the $\tilde{\tau}_D$-metric from $\log 3$ to $\log 2$. In light of \cite[Theorem~3.1]{Ibr11} and \cite[Theorem~3.7]{Ibr16},
it can easily be seen that
$$u_D(x,y)\leq 4\tilde\tau_{D}(x,y)+2\log 2.
$$
holds for $x,y\in D\subsetneq {\mathbb{R}^n} $; however, this has been improved in Theorem~\ref{tau-u-g}.
Next theorem proves the comparison of both the $\tilde{\tau}_D$-metric and $u_D$-metric in the other way, i.e., the $\tilde{\tau}_D$-metric as a lower bound to the $u_D$-metric.
\begin{theorem}\label{tildetau-u}
Let $D\subsetneq {\mathbb{R}^n} $ be any domain with $\partial D\neq \emptyset$ and $x,y\in D$. Then
$$ 2\tilde{\tau}_D(x,y)\le u_D(x,y).
$$
Equality holds whenever $d(x)=|x-p|=|y-p|=d(y)$ for some $p\in \partial D$. Moreover, there exists no
constant $k\ge 0$ such that
$$ u_D(x,y)\le 2\tilde{\tau}_D(x,y)+k
$$
for all $x,y\in D$ unless $D$ is a once-punctured space. If $D$ is a once-punctured space, then
$$ u_D(x,y)\le 2\tilde{\tau}_D(x,y)+ 2\log 2
$$
and the inequality is sharp.
\end{theorem}
\begin{proof}
Without loss of generality we assume that $d(x)\ge d(y)$. For $x,y\in D$, the relation
$|x-p||y-p|\ge d(x)d(y)$ clearly holds for all $p\in \partial D$. Then we have
\begin{eqnarray*}
u_D(x,y) &=& 2 \log \frac{|x-y|+d(x)}{\sqrt{d(x)d(y)}}
\ge 2\log \left(1+\frac{|x-y|}{\sqrt{d(x)d(y)}}\right)\\
&\ge &2 \log\left(1+\sup_{p\in \partial D}\frac{|x-y|}{\sqrt{|x-p||p-y|}}\right)=2 \tilde{\tau}_D(x,y),\\
\end{eqnarray*}
where the first inequality follows from the fact that $d(x)/d(y)\ge 1$. It is clear that if $d(x)=|x-p|=|y-p|=d(y)$ for some $p\in \partial D$, then both the above inequalities turn into an equality and hence the sharpness part is proved.
To prove the second part, suppose that $D$ has more than one boundary point and $k\ge 0$
such that $u_D(x,y)\le 2\tilde{\tau}_D(x,y)+k$ for all $x,y\in D$. Since $u_D$ is $\delta$-hyperbolic
in $D$ and $2\tilde{\tau}_D(x,y)\le u_D(x,y)\le 2\tilde{\tau}_D(x,y)+k$, we conclude that $\tilde{\tau}_D$
is $\delta$-hyperbolic in $D$, contradicting \cite[Remark~4.4]{Ibr16}.
The translation invariance of the $\tilde{\tau}_D$-metric and the $u_D$-metric
allows us to take the punctured space to be $D_0$ without any loss to
generality. Again we assume that $|x|\ge |y|$. To show the third part, it is sufficient to
show
$$ \frac{|x-y|+|x|}{\sqrt{|x||y|}}\le 2 \left(1+\frac{|x-y|}{\sqrt{|x||y|}}\right),
$$
or, equivalently,
$$ \frac{|x|-|x-y|}{\sqrt{|x||y|}}\le 2.
$$
The hypothesis $|x|\ge |y|$ along with the triangle inequality yields
$$ \frac{|x|-|x-y|}{\sqrt{|x||y|}}\le \frac{|y|}{\sqrt{|x||y|}}\le 2.
$$
To prove the sharpness, let $y=e_1$ and $x=te_1, t>1$. Then
$$\lim_{t\to \infty} u_D(x,y)-2\tilde{\tau}_D(x,y)=\lim_{t\to \infty} 2\log \frac{2t-1}{t+\sqrt{t}-1}=2\log 2.
$$
Hence the proof is complete.
\end{proof}
The following inclusion relation holds true between the $u_D$-metric ball and the $\tilde{\tau}_D$-metric ball.
\begin{corollary}
Let $D\subsetneq {\mathbb{R}^n} $ be any arbitrary domain and $x\in D$. Then
$$B_{u_D}(x,r)\subseteq B_{\tilde{\tau}_D}(x,t),
$$
where $r=2t$ and the inclusion is sharp.
\end{corollary}
\begin{proof}
Suppose that $y\in B_{u_D}(x,2t)$. Then $u_D(x,y)<2t$. Now it follows from Theorem~\ref{tildetau-u} that $\tilde{\tau}_D(x,y)<t$ and hence the proof is complete. For the sharpness, consider the domain $D= {\mathbb{R}^n} \setminus\{0\}$ and $x\in D$. Choose the point $y=\partial B_{u_D}(x,2t)\cap \partial B(0,|x|)$. Now,
$u_D(x,y)=2t$ implies $\tilde{\tau}_D(x,y)=t$. This proves our corollary.
\end{proof}
Next, we aim to prove the other way of comparison. That is, to find a constant $k$ such that
$u_D\le k \tilde{\tau}_D$. First, we prove this result in once-punctured spaces and then we extend this to arbitrary proper subdomains of $ {\mathbb{R}^n} $. Next result shows that in punctured spaces the constant $k=4$.
\begin{lemma}\label{u-tau-d0}
Let $x,y\in D_0$. Then
$$u_{D_0}(x,y)\le 4\tilde{\tau}_{D_0}(x,y).
$$
\end{lemma}
\begin{proof}
Without loss of generality, we assume that $|x|\ge |y|$. To prove the required inequality, it suffices to show that
$$\frac{|x-y|+|x|}{\sqrt{|x||y|}}\le \left(1+\frac{|x-y|}{\sqrt{|x||y|}}\right)^2
$$
or, equivalently,
$$ \frac{|x|-|x-y|}{\sqrt{|x||y|}}\le 1+ \frac{|x-y|^2}{|x||y|}.
$$
This holds true, because
$$\frac{|x|-|x-y|}{\sqrt{|x||y|}}\le \frac{|y|}{\sqrt{|x||y|}}\le 1 \le 1+ \frac{|x-y|^2}{|x||y|},
$$
completing the proof of our lemma.
\end{proof}
We now prove that the conclusion of Lemma~\ref{u-tau-d0} still holds if we replace the once-punctured space by twice-punctured spaces.
\begin{lemma}\label{u-tau}
Let $x,y\in D_{p,q}$. Then
$$u_{D_{p,q}}(x,y)\le 4\tilde{\tau}_{D_{p,q}}(x,y).
$$
\end{lemma}
\begin{proof}
Suppose that $x,y\in D_{p,q}$. If $d(x)=|x-p|$ and $d(y)=|y-p|$ (or $d(x)=|x-q|$ and $d(y)=|y-q|$), then
the proof follows from Theorem~\ref{u-tau-d0}. Hence, without loss of generality, we assume
$d(x)=|x-p|,d(y)=|y-q|$, and $|x-p|\ge |y-q|$.
Note that $\tilde{\tau}_{D_{p,q}}(x,y)=\tilde{\tau}_{D_{p}}(x,y)\vee \tilde{\tau}_{D_{q}}(x,y)$.
Hence, to prove our claim, it is enough to establish the inequality $u_{D_{p,q}}(x,y)\le 4 \tilde{\tau}_{D_q}(x,y)$. That is to prove the inequality
$$\frac{|x-y|+|x-p|}{\sqrt{|x-p||y-q|}}\le \left(1+\frac{|x-y|}{\sqrt{|x-q||y-q|}}\right)^2.
$$
Let $|x-y|=a|y-q|$, where $a>0$. From the assumption we know that
\begin{equation}\label{proof-1}|x-p|\leq |x-q|\leq |x-y|+|y-q|.\end{equation}
Now
\begin{eqnarray}\label{proof-2}\frac{|x-y|+|x-p|}{\sqrt{|x-p||y-q|}}&=&\frac{|x-y|}{\sqrt{|x-p||y-q|}}+\frac{|x-p|}{\sqrt{|x-p||y-q|}}\\ \nonumber&\leq&
\frac{|x-y|}{|y-q|}+\sqrt{\frac{|x-p|}{|y-q|}}\\ \nonumber&\leq& \frac{|x-y|}{|y-q|}+\sqrt{\frac{|x-y|}{|y-q|}+1}\\ \nonumber&\leq&
a+\sqrt{a+1}.\end{eqnarray}
We obtain from (\ref{proof-1}) that \begin{equation}\label{proof-3}\frac{2|x-y|}{\sqrt{|x-q||y-q|}}\geq \frac{2|x-y|}{\sqrt{1+a}|y-q|}=\frac{2a}{\sqrt{1+a}}.\end{equation}
Next we divide the proof into two cases.
\begin{ca} $a<1$.\end{ca}
It follows from
(\ref{proof-2}) and (\ref{proof-3}) that
\begin{eqnarray}
\bigg(1+\frac{|x-y|}{\sqrt{|x-q||y-q|}}\bigg)^2-\frac{|x-y|+|x-p|}{\sqrt{|x-p||y-q|}}
\nonumber &> & \frac{2|x-y|}{\sqrt{|x-q||y-q|}}+1-\frac{|x-y|+|x-p|}{\sqrt{|x-p||y-q|}}\\
\nonumber&\geq&\frac{2a}{\sqrt{1+a}}+1-a-\sqrt{a+1}>0
\end{eqnarray}
\begin{ca} $a\ge 1$.\end{ca}
From (\ref{proof-1}) we have
\begin{equation} \label{proof-4}
\frac{|x-y|^2}{|x-q||y-q|}\geq \frac{|x-y|^2}{(|x-y|+|y-q|)|y-q|}=\frac{a^2}{1+a}.
\end{equation}
Then it follows from (\ref{proof-2}), (\ref{proof-3}) and \eqref{proof-4} that
\begin{eqnarray} \bigg(1+\frac{|x-y|}{\sqrt{|x-q||y-q|}}\bigg)^2-\frac{|x-y|+|x-p|}{\sqrt{|x-p||y-q|}}\nonumber &\geq & \frac{2a}{\sqrt{1+a}}+\frac{a^2}{1+a}+1-a-\sqrt{a+1}\\ \nonumber
&\ge & 0.
\end{eqnarray}
This completes the proof of our lemma.
\end{proof}
Combining Lemma~\ref{tildetau-u} and Lemma~\ref{u-tau} we obtain
\begin{theorem}\label{tau-u-g}
Let $x,y\in D\subsetneq {\mathbb{R}^n} $. Then
$$2 \tilde{\tau}_D(x,y) \le u_D(x,y)\le 4\tilde{\tau}_D(x,y).
$$
Both the inequalities are sharp.
\end{theorem}
\begin{proof}
The first inequality is proved in Theorem~\ref{tildetau-u}. Now, we prove the second inequality.
Suppose that $p,q\in \partial D$ such that $d(x)=|x-p|$ and $d(y)=|y-q|$.
Clearly, $D\subset D_{p,q}$ and $u_D(x,y)=u_{D_{p,q}}(x,y)$. Now,
$$u_D(x,y)=u_{D_{p,q}}(x,y)\le 4 \tilde{\tau}_{D_{p,q}}(x,y)\le 4\tilde{\tau}_{D}(x,y),
$$
where the first inequality follows from Lemma~\ref{u-tau} and the second inequality follows from
the monotone property of $\tilde{\tau}_D$ \cite[p.~2]{Ibr16}.
The sharpness of the first inequality is given in Theorem~\ref{tildetau-u}. For the sharpness of the second inequality we consider the unit ball $ {\mathbb{B}^n} $. Choose the points $x$ and $y$ such that $y=-x$. Now,
$$u_{ {\mathbb{B}^n} }(x,-x)=2\log\left(\frac{1+|x|}{1-|x|}\right) \mbox{ and }
\tilde{\tau}_{ {\mathbb{B}^n} }(x,-x)=\log\left(1+\frac{2|x|}{\sqrt{1-|x|^2}}\right).
$$
It follows that
$$\lim_{|x|\to 1} \frac{u_{ {\mathbb{B}^n} }(x,-x)}{4\tilde{\tau}_{ {\mathbb{B}^n} }(x,-x)}=\lim_{|x|\to 1} \frac{(2|x|+\sqrt{1-|x|^2})^2}{(1+|x|)^2}=1.
$$
Hence the proof is complete.
\end{proof}
An immediate consequence of Theorem~\ref{tau-u-g} is the following inclusion relation.
\begin{corollary}\label{u-tau-inclusion}
Let $x\in D\subsetneq {\mathbb{R}^n} $ and $t>0$. Then we have
$$B_{u_D}(x,r)\subseteq B_{\tilde{\tau}_D}(x,t)\subseteq B_{u_D}(x,R),
$$
where $r=2t$ and $R=4t$. The radii $r$ and $R$ are the best possible.
\end{corollary}
\begin{proof}
Let $y\in B_{u_D}(x,r)$, $r=2t$ and $R=4t$. Then by Theorem~\ref{tau-u-g} we have $\tilde{\tau}_D(x,y)<t$. So, $B_{u_D}(x,r)\subseteq B_{\tilde{\tau}_D}(x,t)$. Conversely, if $y\in B_{\tilde{\tau}_D}(x,t)$, then also by Theorem~\ref{tau-u-g} we have $y\in B_{u_D}(x,R)$. So, $B_{\tilde{\tau}_D}(x,t)\subseteq B_{u_D}(x,R)$ and hence the inclusion follows. Next, we need to prove the sharpness part.
First we consider the domain $D=D_0$ and let $x\in D_0$. Now choose $y\in B(0,|x|)\cap \partial B_{\tilde{\tau}_{D_0}}(x,t)$. Then
$$\tilde{\tau}_{D_0}(x,y)=t=\log\left(1+\frac{|x-y|}{|x|}\right)=\frac{u_{D_0}(x,y)}{2},
$$
which proves the sharpness of the first inclusion.
Secondly, consider $D= {\mathbb{B}^n} $ and let $x\in {\mathbb{B}^n} $ be arbitrary.
Choose $y\in {\mathbb{B}^n} $ such that $x$ and $y$ lie on a diameter of $ {\mathbb{B}^n} $ with $0$ lying
in-between and $|y|\le |x|$. Now,
$$u_{ {\mathbb{B}^n} }(x,y)=2\log\left(\frac{|x-y+1-|y||}{\sqrt{(1-|x|)(1-|y|)}}\right) \mbox{ and }
\tilde{\tau}_{ {\mathbb{B}^n} }(x,y)=\log\left(1+\frac{|x-y|}{\sqrt{(1-|x|)(1+|y|)}}\right).
$$
It follows that
\begin{eqnarray*}
\lim_{x\to e_1} \frac{u_{ {\mathbb{B}^n} }(x,y)}{\tilde{\tau}_{ {\mathbb{B}^n} }(x,y)} &=&
\lim_{x\to e_1} \frac{(|x-y|+1-|y|)(1-|x|)(1+|y|)}{\sqrt{(1-|x|)(1-|y|)}(|x-y|+\sqrt{(1-|x|)(1+|y|)})^2}\\
&=&\left\{
\begin{array}{ll}
1, & \mbox{ if } y=-x,\\
0, & \mbox{otherwise}.\\
\end{array}\right.\\
\end{eqnarray*}
Hence we conclude that
for each $x\in {\mathbb{B}^n} $ with $|x|\to 1$ and $t>0$, there exist $y=-x$ such that $y\in \partial B_{\tilde{\tau}_D}(x,t)$ and $u_D(x,y)=4t$. This proves the sharpness of the second inclusion
relation and hence the proof is complete.
\end{proof}
Recall that
\begin{equation}\label{j-tau}
\frac{1}{2} \tilde{j}_D(x,y)\le \tilde{\tau}_D(x,y)\le \tilde{j}_D(x,y)
\end{equation}
holds true for
$D\subsetneq {\mathbb{R}^n} $ (see \cite[Theorem~4.2, 4.3]{Ibr16}). Both the inequalities are sharp.
The proof of the sharpness part of the left hand side inequality is done by the method of contradiction in \cite{Ibr16}.
Here we give a precise example to prove the sharpness part of the left hand side inequality.
Consider the unit ball $ {\mathbb{B}^n} $ and $x,y\in {\mathbb{B}^n} $ with $y=-x$. Now we see that
$$\tilde{\tau}_{ {\mathbb{B}^n} }(x,-x)=\log\left(1+\frac{2|x|}{\sqrt{1-|x|^2}}\right) \mbox{ and } \tilde{j}_{ {\mathbb{B}^n} }(x,-x)=\log\left(
1+\frac{2|x|}{1-|x|}\right).
$$
It follows that
\begin{equation}\label{j-tau-eqn}
\lim_{|x|\to 1}\frac{\tilde{j}_{ {\mathbb{B}^n} }(x,-x)}{2\tilde{\tau}_{ {\mathbb{B}^n} }(x,-x)}=\lim_{|x|\to 1} \frac{2|x|+\sqrt{1-|x|^2}}{2}=1.
\end{equation}
Now, we establish inclusion relation between the $\tilde{j}_D$-metric and the $\tilde{\tau}_D$-metric balls.
\begin{theorem}
Let $D\subsetneq {\mathbb{R}^n} $ and $x\in D$ and $t>0$. Then the following inclusion property holds true:
$$B_{\tilde{j}_D}(x,r)\subseteq B_{\tilde{\tau}_D}(x,t)\subseteq B_{\tilde{j}_D}(x,R).
$$
Here $r=t$ and $R=2t$. The radii $r$ and $R$ are the best possible.
\end{theorem}
\begin{proof}
The proof follows from (\ref{j-tau}). To show that the radius $r$ is the best possible, consider the domain
$D=D_0= {\mathbb{R}^n} \setminus\{0\}$ and $x\in D$. Choose $y\in \partial B(0,|x|)\cap \partial B_{\tilde{\tau}_D}(x,t)$.
Now clearly, $\tilde{j}_D(x,y)=\tilde{\tau}_D(x,y)=t$.
To show $R$ is the best possible, consider the domain
$D= {\mathbb{B}^n} $. With the similar argument given in the proof of Corollary~\ref{u-tau-inclusion}, for the second
inclusion property, we can show that for each $x\in {\mathbb{B}^n} $ with $|x|\to 1$ and $t>0$, there exist $y(=-x)$ such that $y\in \partial B_{\tilde{\tau}_{ {\mathbb{B}^n} }}(x,t)$ and $\tilde{j}_{ {\mathbb{B}^n} }(x,y)=2t$. This completes the proof of our theorem.
\end{proof}
By Theorem~\ref{tau-u-g} and \eqref{j-tau} we have
$$\tilde{j}_D(x,y)\le 2\tilde{\tau}_D(x,y)\le u_D(x,y)
$$
and also
$$u_D(x,y)\le 4\tilde{\tau}_D(x,y)\le 4\tilde{j}_D(x,y).
$$
Hence we have the following relationship between the $\tilde{j}_D$-metric and the $u_D$-metric.
\begin{theorem}\label{lem-tildej-u}
For $D\subsetneq {\mathbb{R}^n} $ we have
$$\tilde{j}_D(x,y)\le u_D(x,y)\le 4 \tilde{j}_D(x,y).
$$
The first inequality is sharp.
\end{theorem}
\begin{proof}
For the sharpness part, consider the domain $D= {\mathbb{R}^n} \setminus\{-e_1,e_1\}$. Choose $x=0$ and $y=te_2, t>1$. Then $\tilde{j}_D(0,te_2)=\log(1+t)$ and
$$ u_D(0,te_2)=2\log\left(\frac{t+\sqrt{1+t^2}}{(1+t^2)^{1/4}}\right)=\log\left(\frac{1+2t^2+2t\sqrt{1+t^2}}{\sqrt{1+t^2}}\right).
$$
Now we see that
\begin{eqnarray*}
\lim_{t\to \infty} \frac{\tilde{j}_D(0,te_2)}{u_D(0,te_2)} &=&\lim_{t\to \infty} \frac{\log(1+t)}{\log\left(\displaystyle\frac{1+2t^2+2t\sqrt{1+t^2}}{\sqrt{1+t^2}}\right)}\\
&=& \lim_{t\to \infty} \displaystyle \frac{(1+t^2)(1+2t^2+2t\sqrt{1+t^2})}{(1+t)(4t(1+t^2)+2(1+t^2)^{3/2}-t-2t^3)}=1.
\end{eqnarray*}
This completes the proof of our theorem.
\end{proof}
\begin{remark}
The constant $4$ in the right hand side inequality of Lemma~\ref{lem-tildej-u} can't be replaced by $2$ due to the fact that
$$u_D(x,y)\le 2\tilde{j}_D(x,y)\iff |x-y|^2\ge d(x)d(y)
$$
for every $x,y\in D$, which is not true in general.
\end{remark}
As an immediate consequence of Theorem~\ref{lem-tildej-u} we have the following inclusion relation.
\begin{corollary}
Let $D\subsetneq {\mathbb{R}^n} $, $x\in D$, and $t>0$. Then
$$B_{u_D}(x,r)\subseteq B_{\tilde{j}_D}(x,t)\subset B_{u_D}(x,R),
$$
where $r=t$ and $R=4t$. The radius $r$ is best possible.
\end{corollary}
\begin{proof}
Proof follows from Theorem~\ref{lem-tildej-u}.
\end{proof}
The next result shows that the $\tilde{\tau}_D$-metric balls can be written as the intersection of $\tilde{\tau}$-metric balls in punctured spaces over the boundary points of $D$.
\begin{proposition}
Let $D\subsetneq {\mathbb{R}^n} $, $x\in D$, and $r>0$. Then
$$B_{\tilde{\tau}_D}(x,r)=\cap_{p\in \partial D} B_{\tilde{\tau}_{ {\mathbb{R}^n} \setminus\{p\}}}(x,r).
$$
\end{proposition}
\begin{proof}
Suppose that $y\in \cap_{p\in \partial D} B_{\tilde{\tau}_{ {\mathbb{R}^n} \setminus\{p\}}}(x,r)$. Then $\tilde{\tau}_{ {\mathbb{R}^n} \setminus\{p\}}(x,y)<r$ for all $p\in \partial D$. In particular,
$$\tilde{\tau}_D(x,y)=\sup_{p\in \partial D} \tilde{\tau}_{ {\mathbb{R}^n} \setminus\{p\}}(x,y)<r.
$$
So, $\cap_{p\in \partial D} B_{\tilde{\tau}_{ {\mathbb{R}^n} \setminus\{p\}}}(x,r)\subseteq B_{\tilde{\tau}_D}(x,r)$.
Conversely, suppose that $y\in B_{\tilde{\tau}_D}(x,r)$ and let $p\in \partial D$. Then
$$B_{\tilde{\tau}_D}(x,r)\subseteq B_{\tilde{\tau}_{ {\mathbb{R}^n} \setminus\{p\}}}(x,r)
$$
by the monotone property of the $\tilde{\tau}_D$-metric.
Hence, $ B_{\tilde{\tau}_D}(x,r) \subseteq \cap_{p\in \partial D} B_{\tilde{\tau}_{ {\mathbb{R}^n} \setminus\{p\}}}(x,r)$
and the proof is complete.
\end{proof}
\section{Comparison with other related metrics}\label{u-tildetau-other}
In this section, we consider the Cassinian \cite{Ibr09}, the Seittenranta \cite{Sei99}, the triangular ratio \cite{CHKV15} and the half-Apollonian \cite{HL04} metrics, and compare them with the $\tilde{\tau}_D$-metric and the $u_D$-metric.
Main results of this section are stated in Table~\ref{T2}.
\begin{center}
\begin{table}[H]
\begin{tabular}{ | c | c | c | }
\hline
& Comparison with $\tilde{\tau}_D$ & Comparison with $u_D$\\[1mm]
\hline
&&\\[-3mm]
& $\cfrac{1}{4}\delta_D(x,y)\le \tilde{\tau}_D(x,y)\le \delta_D(x,y)$ & $\cfrac{\delta_D}{2}\le u_D\le 4j_D$\\[1mm]
$\delta_D$ & [Theorem~\ref{delta-tau}] & [Corollary~\ref{delta-u}]\\[1mm]
&(Right hand side inequality is sharp)&\\[1mm]
\hline
&&\\[-3mm]
$c_D$ & $c_D(x,y)\le \cfrac{e^{4\tilde{\tau}_D(x,y)}-1}{d(y)}$ & $c_D(x,y)\le \cfrac{e^{2u_D(x,y)}-1}{d(y)}$\\[1mm]
& [Corollary~\ref{cor-cD-tau-u}] & [Corollary~\ref{cor-cD-tau-u}]\\[1mm]
\hline
&&\\[-3mm]
& $(\log 3)s_D\le \tilde{\tau}_D$ & $(\log 9)s_D\le u_D$\\[1mm]
$s_D$ & [Theorem~\ref{tau-sD}] & [Corollary~\ref{cor-sD-u}]\\[1mm]
&The inequality is sharp&\\
\hline
&&\\[-3mm]
& $\frac{1}{2}\eta_D\le \tilde{\tau}_D\le \log(2+e^{\eta_D})$ & $\eta_D\le u_D\le 4\log(2+e^{\eta_D})$\\[1mm]
$\eta_D$ & \cite{IS} & [Lemma~\ref{eta-u}]\\[1mm]
&(Both inequalities are sharp)& \\[1mm]
\hline
\end{tabular}
\vspace*{0.5cm}
\caption{Comparisons with other hyperbolic-type metrics}\label{T2}
\end{table}
\end{center}
We begin by comparing the Cassinian metric with the Seittenranta metric.
The {\it Cassinian metric}, $c_D$, of the domain $D\subsetneq {\mathbb{R}^n} $ is defined as
$$
c_D (x,y)=\sup_{p\in \partial D} \frac{|x-y|}{|x-p||p-y|} .
$$
This metric was first introduced and studied in \cite{Ibr09} and subsequently studied in \cite{IMSZ,HKVZ,KMS}.
Geometrically, the $c_D$-metric can be defined by taking the maximal Cassinian oval in $D$ with foci at
$x$ and $y$ (see, \cite{Ibr09}).
Clearly, the supremum in the definition is attained at some point $p \in \partial D$.
The {\it{Seittenranta metric}}, $\delta_D$, introduced in \cite{Sei99}, is defined by
$$\delta_D(x,y)=\log(1+m_D(x,y))
$$
where
$$m_D(x,y)=\sup_{a,b\in \partial D} \frac{|x-y||a-b|}{|x-a||y-b|}.
$$
Note that the quantity $m_D(x,y)$ does not define a metric. The Seittenranta metric is M\"obius invariant and coincides with the hyperbolic metric of the unit ball $ {\mathbb{B}^n} $.
The $c_D$-metric and the $\delta_D$-metric are exponentially related, which is stated in the following
theorem.
\begin{theorem}\label{c-delta}
Let $D\subsetneq \overline{ {\mathbb{R}^n} }$ be any domain. Then
$$c_D(x,y)\le \frac{e^{\delta_D(x,y)}-1}{d(y)}.
$$
The inequality is sharp.
\end{theorem}
\begin{proof}
Let $p\in \partial D$ such that
$$c_D(x,y)=\frac{|x-y|}{|x-p||p-y|}.
$$
Choose $q\in \partial D$ such that $|p-q|\ge |y-q|$. Now,
\begin{eqnarray*}
c_D(x,y) &=& \frac{|x-y|}{|x-p||p-y|}\\
&=& \frac{|x-y||p-q|}{|x-p||y-q|}.\frac{|y-q|}{|p-q||p-y|}\\
&\le & \frac{m_D(x,y)}{d(y)}.
\end{eqnarray*}
Hence we get
$$\delta_D(x,y)=\log(1+m_D(x,y))\ge \log(1+d(y).c_D(x,y))
$$
and the proof is complete. For the sharpness, we consider the punctured space $D_p$. Let $x,y\in D_p$ with
$|x-p|\le |y-p|$. It is clear that $\delta_{D_p}(x,y)=\tilde{j}_{D_p}(x,y)=\log(1+|x-y|/|x-p|)$ and hence the sharpness
follows.
\end{proof}
An immediate corollary to Theorem~\ref{c-delta} is the following inclusion relation.
\begin{corollary}
Let $D\subsetneq {\mathbb{R}^n} $, $x\in D$, and $t>0$. Then
$$B_{\delta_D}(x,t)\subseteq B_{c_D}(x,R),
$$
where $R=(e^t-1)/d(y)$. The inclusion is sharp.
\end{corollary}
\begin{proof}
If $\delta_D(x,y)<y$, then by Theorem~\ref{c-delta}, we have $c_D(x,y)<(e^t-1)/d(y)$. For the sharpness,
choose a point $y\in \partial B_{\delta_D}(x,t)\cap L$, where $L$ is the line passing
through $0$ and $x$ with $|x|<|y|$. Then
$$\delta_D(x,y)=t=\log\left(1+\frac{|x-y|}{|x|}\right).
$$
Now,
$$c_D(x,y)=\frac{|x-y|}{|x||y|}=\frac{1}{|y|}(e^t-1)
$$
and hence the proof is complete.
\end{proof}
Again the $\delta_D$-metric is bilipschitz equivalent to the $\tilde{j}_D$-metric. Indeed, we have
\begin{equation}\label{tildej-delta}
\tilde{j}_D\le \delta_D\le 2\tilde{j}_D,
\end{equation}
see, for instance \cite[p.~525]{Sei99}. Hence \eqref{j-tau} along with \eqref{tildej-delta} yield the
following inequality between the $\delta_D$-metric and the $\tilde{\tau}_D$-metric.
\begin{theorem}\label{delta-tau}
Let $x,y\in D\subsetneq {\mathbb{R}^n} $. Then the following holds true:
$$\frac{1}{4}\delta_D(x,y)\le \tilde{\tau}_D(x,y)\le \delta_D(x,y).
$$
The second inequality is sharp.
\end{theorem}
\begin{proof}
The proof of the inequality follows directly from \eqref{j-tau} and \eqref{tildej-delta}.
For the sharpness of the second inequality, consider the domain $D_0$ and choose $x,y\in D_0$ with $y=-x$.
Then $\tilde{\tau}_{D_0}(x,-x)=\log 3=\delta_{D_0}(x,-x)$.
\end{proof}
The following inclusion relation holds true.
\begin{corollary}
Let $x\in D\subsetneq {\mathbb{R}^n} $ and $t>0$. Then
$$B_{\delta_D}(x,r)\subseteq B_{\tilde{\tau}_D}(x,t)\subseteq B_{\delta_D}(x,R),
$$
where $r=t$ and $R=4t$.
\end{corollary}
\begin{proof}
Proof follows from Theorem~\ref{delta-tau}.
\end{proof}
Theorem~\ref{tau-u-g} and Theorem~\ref{delta-tau} together yield the following:
\begin{corollary}\label{delta-u}
Let $x,y\in D\subsetneq {\mathbb{R}^n} $. Then we have
$$\frac{1}{2}\delta_D(x,y)\le u_D(x,y)\le 4\delta_D(x,y).
$$
\end{corollary}
Corollary~\ref{delta-u} leads to the following inclusion relation.
\begin{corollary}
Let $x\in D\subsetneq {\mathbb{R}^n} $ and $t>0$. Then
$$B_{\delta_D}(x,r)\subseteq B_{u_D}(x,t)\subseteq B_{\delta_D}(x,R),
$$
where $r=t/4$ and $R=2t$.
\end{corollary}
Hence, as a consequence to Theorem~\ref{c-delta}, we have
\begin{corollary}\label{cor-cD-tau-u}
Let $x,y\in D\subsetneq {\mathbb{R}^n} $. Then we have
$$c_D(x,y)\le \frac{e^{4\tilde{\tau}_D(x,y)}-1}{d(y)} \mbox{ and } c_D(x,y)\le \frac{e^{2u_D(x,y)}-1}{d(y)}.
$$
\end{corollary}
\begin{proof}
The first inequality follows from Theorem~\ref{c-delta} and Lemma~\ref{delta-tau}, whereas the second inequality
follows from Theorem~\ref{c-delta} and Corollary~\ref{delta-u}.
\end{proof}
Now, we compare the $\tilde{\tau}_D$-metric with
the triangular ratio metric, $s_D$, defined in a proper subdomain $D$ of $ {\mathbb{R}^n} $ by
$$s_D(x,y)=\sup_{p\in \partial D} \frac{|x-y|}{|x-p|+|p-y|}, \quad x,y\in D.
$$
Geometrically, the triangular ratio metric can be viewed by taking the maximal ellipse in $D$ with foci at $x$ and $y$ in the similar fashion as in the geometric definition the Apollonian metric \cite{Has03}. For more details on $s_D(x,y)$ we refer \cite{CHKV15}.
\begin{theorem}\label{tau-sD}
Let $D\subsetneq {\mathbb{R}^n} $ and $x,y\in D$. Then
$$\tilde{\tau}_D(x,y)\ge (\log 3) s_D(x,y).
$$
The inequality is sharp.
\end{theorem}
\begin{proof}
From AM-GM inequality, it follows that
$$\frac{1}{\sqrt{|x-p||y-p|}}\ge \frac{2}{|x-p|+|y-p|}.
$$
Now,
\begin{eqnarray*}
\tilde{\tau}_D(x,y) &\ge & \log\left(1+\frac{|x-y|}{\sqrt{|x-p||y-p|}}\right)\\
&\ge & \log\left(1+\frac{2|x-y|}{|x-p|+|y-p|}\right)\\
&\ge & \frac{|x-y|}{|x-p|+|y-p|} \log 3
\end{eqnarray*}
holds for all $p\in \partial D$. Here the last inequality follows from the well known Bernoulli's
inequality:
$$\log(1+ax)\ge a \log(1+x)\quad \mbox{ for } a\in (0,1), x>0.
$$
In particular, we have $\tilde{\tau}_D(x,y)\ge (\log 3) s_D(x,y)$. To prove the sharpness,
consider the domain $D_0$ and $x,y\in D_0$ with $y=-x$. Then
$\tilde{\tau}_D(x,-x)=\log 3$ and $s_D(x,-x)=1$.
\end{proof}
\begin{remark}
Combining Theorem~\ref{tau-sD} and \cite[Theorem~4.3]{Ibr16}, one can obtain Theorem~3.3 of \cite{CHKV15}.
\end{remark}
Theorem~\ref{tau-sD} leads to the following inclusion relation.
\begin{corollary}
Let $x\in D\subsetneq {\mathbb{R}^n} $ and $t>0$. Then
$$B_{\tilde{\tau}_D}(x,t)\subseteq B_{s_D}(x,R),
$$
where $R=t/\log 3$. The inclusion is sharp.
\end{corollary}
\begin{proof}
It follows from Theorem~\ref{tau-sD} that for $\tilde{\tau}_D(x,y)<t$,
$s_D(x,y)<t/\log 3$. Hence, $B_{\tilde{\tau}_D}(x,t)\subseteq B_{s_D}(x,R)$ with $R=t/\log 3$.
To prove the sharpness part, consider the domain $D=D_0$ and $t=\log 3$. Then we have $R=1$ and
$$\tilde{\tau}_{D_0}(x,y)=\log 3 \iff |x-y|=2\sqrt{|x||y|}
$$
and hence
$$s_{D_0}(x,y)=\frac{2\sqrt{|x||y|}}{|x|+|y|}.
$$
To show $s_{D_0}(x,y)=1$, we need to choose points $x$ and $y$ such that $|x|=|y|$. This implies $x$ and $y$ are co-linear. i.e. $y=-x$. From the definition of the $\tilde{\tau}_D$ metric it is clear that the point
$-x$ lies on the sphere $\partial B_{\tilde{\tau}_{D_0}}(x,\log 3)$. Now, for any $x\in D_0$, choose
$y\in \partial B_{\tilde{\tau}_{D_0}}(x,\log 3)\cap L$, where $L$ is the line passing through $0$ and $x$
with $|x|=|y|$. Then
$$\tilde{\tau}_{D_0}(x,y)=\log 3 \iff s_{D_0}(x,y)=1.
$$
Hence, the proof is complete.
\end{proof}
Another consequence of Theorem~\ref{tau-sD} leads to the following corollary.
\begin{corollary}\label{cor-sD-u}
Let $D\subsetneq {\mathbb{R}^n} $. Then for all $x,y\in D$ we have
$$s_D(x,y)\le \frac{1}{\log 9} u_D(x,y).
$$
\end{corollary}
\begin{proof}
The proof follows from Theorem~\ref{tau-u-g} and Theorem~\ref{tau-sD}.
\end{proof}
Next, we discuss the inclusion properties associated with the $\tilde{\tau}_D$-metric and the half-Apollonian metric balls. The half-Apollonian metric, $\eta_D$, of a domain $D\subsetneq {\mathbb{R}^n} $ is defined by
$$\eta_D(x,y)=\sup_{p\in \partial D} \left|\log \frac{|x-p|}{|y-p|}\right|, \quad x,y\in D.
$$
The $\eta_D$-metric was introduced by H\"ast\"o and Linden in \cite{HL04}.
It is now appropriate to recall the following result by Seittenranta.
\begin{lemma}\cite[Theorem~3.11]{Sei99}\label{sei3.11}
Let $D\subset \overline{ {\mathbb{R}^n} }$ be an open set with ${\rm card }\,\, \partial D\ge 2$. Then
$$\alpha_D(x,y)\le \delta_D(x,y)\le \log (e^{\alpha_D}+2)
$$
The inequalities give the best possible bounds for $\delta_D$ expressed in terms of
$\alpha_D$ only.
\end{lemma}
Here the quantity $\alpha_D$ represents the Apollonian metric, introduced by Beardon in \cite{Bea98}.
As a special case of Lemma~\ref{sei3.11}, the following result holds true, which is proved in \cite{IS}.
\begin{lemma}\label{tau-eta}
Let $D\subsetneq {\mathbb{R}^n} $ and $x,y\in D$. Then
$$ \frac{1}{2} \eta_D(x,y) \le \tilde{\tau}_D(x,y) \le \log(2+e^{\eta_D(x,y)}).
$$
Both the inequalities are sharp.
\end{lemma}
Lemma~\ref{tau-eta} obtains the following inclusion property.
\begin{corollary}
Let $D\subsetneq {\mathbb{R}^n} $ and $x\in D$ and $t>0$. Then the following inclusion property holds true:
$$B_{\eta_D}(x,r)\subseteq B_{\tilde{\tau}_D}(x,t)\subseteq B_{\eta_D}(x,R).
$$
Here $r=\log(e^t-2)$ and $R=2t$. The radii $r$ and $R$ are best possible.
\end{corollary}
\begin{proof}
Let $y\in B_{\tilde{\tau}_D}(x,t)$, i.e. $\tilde{\tau}_D(x,y)<t$.
From the left hand side inequality of
Theorem~\ref{tau-eta}, we have $\eta_D(x,y)<2t(=R)$. On the other hand, if $\eta_D(x,y)<\log(e^t-2)(=r)$, then
from the right hand side inequality of Theorem~\ref{tau-eta}, we have $\tilde{\tau}_D(x,y)<t$. With the similar argument
given in the proof for the sharpness part of second inclusion relation in Corollary~\ref{u-tau-inclusion}, we
can show that the radius $R$ is the best possible in the punctured space $D_0$ with $t=\log 3$.
\end{proof}
Comparison of Theorem~\ref{tau-u-g} and Lemma~\ref{tau-eta} together lead to the following relation
between the $\eta_D$-metric and the $\tilde{\tau}_D$-metric.
\begin{lemma}\label{eta-u}
Let $D\subsetneq {\mathbb{R}^n} $ and $x,y\in D$. Then
$$\eta_D(x,y)\le u_D(x,y)\le 4\log(2+e^{\eta_D(x,y)}).
$$
\end{lemma}
Subsequently, we have the following trivial inclusion relation.
\begin{corollary}
Let $x\in D\subsetneq {\mathbb{R}^n} $ and $t>0$. Then
$$B_{\eta_D}(x,r)\subseteq B_{u_D}(x,t)\subseteq B_{\eta_D}(x,R),
$$
where $r=\log(e^{t/4}-2)$ and $R=t$.
\end{corollary}
\bigskip
\begin{acknowledgement}
The second author would like to thank Zair Ibragimov for bringing the interesting paper \cite{Ibr11} to his attention and for
useful discussion on this topic when the author visited him during June 2016.
The authors would also like to thank Manzi Huang for her valuable comments, specially
for nice conversation in the proof of Lemma~\ref{u-tau}. The research was partially
supported by NBHM, DAE $($Grant No: $2/48 (12)/2016/${\rm NBHM (R.P.)/R \& D II}/$13613)$.
\end{acknowledgement}
| {
"timestamp": "2017-05-25T02:02:58",
"yymm": "1705",
"arxiv_id": "1705.08574",
"language": "en",
"url": "https://arxiv.org/abs/1705.08574",
"abstract": "We mainly consider two metrics: a Gromov hyperbolic metric and a scale invariant Cassinian metric. We compare these two metrics and obtain their relationship with certain well-known hyperbolic-type metrics, leading to several inclusion relations between the associated metric balls.",
"subjects": "Metric Geometry (math.MG)",
"title": "A Gromov hyperbolic metric vs the hyperbolic and other related metrics",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9770226354423304,
"lm_q2_score": 0.7248702761768248,
"lm_q1q2_score": 0.7082146675840912
} |
https://arxiv.org/abs/1310.7696 | Delaunay stability via perturbations | We present an algorithm that takes as input a finite point set in Euclidean space, and performs a perturbation that guarantees that the Delaunay triangulation of the resulting perturbed point set has quantifiable stability with respect to the metric and the point positions. There is also a guarantee on the quality of the simplices: they cannot be too flat. The algorithm provides an alternative tool to the weighting or refinement methods to remove poorly shaped simplices in Delaunay triangulations of arbitrary dimension, but in addition it provides a guarantee of stability for the resulting triangulation. |
\section{Algorithm}
\label{sec:algorithm}
In this section we present the algorithm. We start, in
\Secref{sec:main.result}, by announcing the guarantees of the
algorithm as our main theorem.
\subsection{Main result}
\label{sec:main.result}
The goal and primary contribution of this paper is the presentation of
the perturbation \Algref{alg1}, and the demonstration of its
guarantees.
In our analysis we employ three positive parameters, $\delta_0$,
$\Gamma_0$, and $\rho_0} %{\eta_0$, which are logically distinct. The
parameter $\delta_0$ specifies the protection that will be guaranteed
for the Delaunay $m$-simplices in $\rdelsmhull{\ppts}$, and $\Gamma_0$ is a
bound on the quality of these simplices. The analysis places an upper
bound on $\delta_0$ with respect to $\Gamma_0$, and so for the
statement of our results, and the description of \Algref{alg1}, it is
convenient to combine the parameters by setting $\delta_0$ to be equal
to this upper bound:
\begin{equation*}
\delta_0 = \Gamma_0^{m+1}.
\end{equation*}
Our primary interest is in $\delta_0$, but it is more convenient to
express the results in terms of $\Gamma_0$. The analysis also places
an upper bound on $\Gamma_0$ with respect to the parameter $\rho_0} %{\eta_0$
that governs the amount of perturbation the input points may be
subjected to. We fix $\Gamma_0$ with respect to this upper
bound, and let $\rho_0} %{\eta_0$ be the only free parameter for the
algorithm.
The following theorem is demonstrated in \Secref{sec:analysis} and is
stated in full generality as \Thmref{thm:raw.main}:
\begin{thm}[Main result]
\label{thm-main-theorem-of-the-paper}
Taking as input a $\mueps$-net\ $\mathsf{P} \subset \reel^m$, where
$\sparseconst$ and $\epsilon$ are known, and a positive parameter
$\rho_0} %{\eta_0 \leq \frac{\sparseconst}{4}$,
\Algref{alg1} produces a $\pmueps$-net\
$\pts'$ that is a $\rho_0} %{\eta_0\epsilon$-perturbation of $\mathsf{P}$ such that all
the Delaunay $m$-simplices in $\rdelsmhull{\ppts}$ are $\Gamma_0$-good and
$\delta$-protected, with
\begin{equation*}
\Gamma_0 = \frac{\rho_0} %{\eta_0}{C},
\quad \text{and} \quad
\delta = \Gamma_0^{m+1}\sparseconst' \samconst',
\end{equation*}
where $C = \left( \frac{2}{\sparseconst} \right)^{3m^2 + 5m + 17}$,
and $\sparseconst' = \frac{\sparseconst-2\rho_0} %{\eta_0}{1+ \rho_0} %{\eta_0}$, and
$\samconst' = (1+\rho_0} %{\eta_0)\epsilon$.
The expected time complexity is
\begin{equation*}
\bigo{m}(\card{\mathsf{P}})^2 + \left(\frac{2}{\sparseconst}\right)^{\bigo{m^2}}\card{\mathsf{P}},
\end{equation*}
where the constant in the big-$O$ notation is an absolute constant.
\end{thm}
Although we require knowledge of two sampling parameters, $\sparseconst$, and $\epsilon$, in practice one is easily deduced from the other by finding the minimum distance between two points in $\mathsf{P}$, and using the relation $\distEm{p}{q} \geq \sparseconst \epsilon$.
We recall that by itself $\delta_0 = \Gamma_0^{m+1}$ guarantees a lower
thickness bound proportional to $\delta_0^2 = \Gamma_0^{2m+2}$ on the
Delaunay $m$-simplices \cite[Theorem 3.11]{boissonnat2013stab1}, but
this is much smaller than the $\Gamma_0^m$ thickness guaranteed by
\Thmref{thm-main-theorem-of-the-paper}. If we were to set $\delta_0 =
0$ we would have a ``sliver exudation'' algorithm which would not
guarantee any $\delta$-genericity, but
$\Gamma_0$ would only increase by a factor of two.
\subsection{Algorithm overview}
We present an algorithm that will perturb an input $\mueps$-net\ $\mathsf{P}$ to
obtain a $\pmueps$-net\ $\pts'$ which contains no forbidden configurations} %{$\dg$-configuration. The algorithm
takes as input a finite $\mueps$-net\ $\mathsf{P} = \{p_1, \ldots, p_{n} \}
\subset \reel^m$. The output is obtained after $n$ iterations, such that
at the $i^{\text{th}}$ iteration a perturbation $\mathsf{P}_i = \{p'_1,
\ldots, p'_i, p_{i+1}, \ldots, p_n \}$ is produced by perturbing the
point $p_i \mapsto p'_i$ in a way that ensures that there are no
forbidden configuration} %{$\dg$-configuration s incident to $p'_i$ in $\mathsf{P}_i$. Thus we have a sequence of
perturbations
\begin{equation*}
\mathsf{P} = \mathsf{P}_0 \to \mathsf{P}_1 \to \cdots \to \mathsf{P}_n,
\end{equation*}
such that for all $i \in [1, \ldots, n]$, $\mathsf{P}_i$ is a perturbation
of $\mathsf{P}$ as well as of $\mathsf{P}_{i-1}$, and $\mathsf{P}_{i-1} \setminus
\{p_i\} = \mathsf{P}_i \setminus \{p'_i\}$. Thus all the sets $\mathsf{P}_i$ are
$\pmueps$-net s.
At the $i^{th}$ iteration of the algorithm, all the points $p_{1}$ to
$p_{i-1}$, have already been perturbed, and the points $p_{i}$ to
$p_{n}$ have not yet been perturbed. Using a uniform distribution, we
pick a random point $x \in B(p_{i}, \rho_0} %{\eta_0\epsilon)$.
\begin{de}
\label{def:good.perturbation}
We say that $x$ is a \defn{good perturbation} of $p_i$ if for all
simplices $\sigma \in \mathsf{P}_{i-1} \setminus \{ p_i \}$, the
simplex $\splxjoin{x}{\sigma}$ is not a forbidden configuration} %{$\dg$-configuration.
\end{de}
If $x$ is a good perturbation of $p_i$, we let $p'_i = x$ and go on to
the next iteration, otherwise we choose a new random point from
$\ballEm{p_i}{\rho_0} %{\eta_0\epsilon}$. The algorithm for determining if
$x$ is a good perturbation is discussed in
\Secref{sec:good.perturbations}, and the existence of good
perturbations is established in \Secref{sec:analysis}. The essential
ingredient is the $\hoopbnd$-hoop\ property, and especially the symmetric nature
of this property.
The algorithm is shown in pseudocode in Algorithm~\ref{alg1}. Since a
good perturbation $p \mapsto p'$ ensures that there are no forbidden configuration} %{$\dg$-configuration s
incident to $p'$ in the current point set, and in particular that no
new forbidden configuration} %{$\dg$-configuration s are created, the output of the algorithm cannot
contain any forbidden configurations} %{$\dg$-configuration:
\begin{lem}
\label{lem:no.forbidden.output}
After the $i^{\text{th}}$ iteration of the algorithm, there are no
forbidden configuration} %{$\dg$-configuration s in $\cpltcplx{\mathsf{P}_i}$ incident to $p'_j \in \mathsf{P}_i$
for any $j \in [1, \ldots, i]$. In particular, when the $n^{\text{th}}$
iteration is completed, $\mathsf{P}_n$ contains no forbidden configurations} %{$\dg$-configuration.
\end{lem}
\begin{proof}
By the definition of a good perturbation, there is no forbidden configuration} %{$\dg$-configuration\
incident to $p_1 \in \mathsf{P}_1$ after the first iteration has
completed. Assume that at the $i^{\text{th}}$ iteration there are no
forbidden configuration} %{$\dg$-configuration s in $\mathsf{P}_{i-1}$ incident to any $p'_j \in \mathsf{P}_{i-1}$ for
all $j < i$. At the completion of the $i^{\text{th}}$ iteration
$\mathsf{P}_{i-1} \setminus \{p_i\} = \mathsf{P}_i \setminus \{p'_i\}$, so if
there is a forbidden configuration} %{$\dg$-configuration\ $\tau \subset \mathsf{P}_i$ that includes a $p'_j$
with $j < i$, then $\tau$ must also include $p'_i$, since
otherwise we would have $\tau \subset \mathsf{P}_{i-1}$. But this
contradicts the fact that $p'_i$ was chosen to be a good
perturbation of $p_i$, thus establishing the claim.
\end{proof}
\begin{algorithm}
\caption{Randomized perturbation algorithm}
\label{alg1}
\begin{algorithmic}
\STATE{\rm Input:}\quad $\mueps$-net\ $\mathsf{P}_0 = \{p_{1}, \dots, p_{n}\}
\subset \reel^m$ and $\rho_0} %{\eta_0$
\FOR{$i = 1$ \TO $n$}
\STATE ${\rm Flag} \gets 0$
\STATE $x \gets p_i$
\WHILE{${\rm Flag} \neq 1$}
\IF{\texttt{good\_perturbation}$(x,p_i,\mathsf{P}_{i-1})$}
\STATE $p'_{i} \gets x$
\STATE $\mathsf{P}_i \gets ( \mathsf{P}_{i-1} \setminus \{p_i\} ) \cup \{p'_i\}$
\STATE ${\rm Flag} \gets 1$
\ELSE
\STATE // \texttt{random\_point}$(B(p_{i}, \rho_0} %{\eta_0\epsilon))$
outputs a point from the uniform distribution on $B(p_{i},
\rho_0} %{\eta_0\epsilon)$
\STATE $x \gets$ \texttt{random\_point}$(B(p_{i},
\rho_0} %{\eta_0\epsilon))$
\ENDIF
\ENDWHILE
\ENDFOR
\STATE // $\mathsf{P}_{n} = \{ p_{1}', \, \dots, \, p_{n}'\}$, a
$\delta$-generic $\pmueps$-net, as described in
\Thmref{thm-main-theorem-of-the-paper}
\STATE{\rm Output:}\quad $\mathsf{P}_{n}$
\end{algorithmic}
\end{algorithm}
\subsection{Implementation of good perturbations}
\label{sec:good.perturbations}
The geometric computations of the algorithm occur in the
\texttt{good\_perturbation} procedure, which is outlined in
\Algref{alg:good.perturbation}.
The check for a good perturbation is a local operation.
We first establish a bound on the number of possible distinct
forbidden configuration} %{$\dg$-configuration s incident to $p'$ in a perturbation $\pts'$ of
$\mathsf{P}$. The first step is to bound the radius of a ball centred on $p$
that contains all such forbidden configurations} %{$\dg$-configuration:
\begin{lem}
\label{lem:bound.forbid.rad}
Suppose $\pts'$ is a perturbation of $\mathsf{P}$, and $\tau \subset
\pts'$ is a forbidden configuration} %{$\dg$-configuration, with $\delta_0 \leq \frac{2}{5}$. If $p \in
\mathsf{P}$ and $p \mapsto p' \in \tau$, then all the vertices of
$\tau$ originate from elements of $\mathsf{P}$ contained in the ball
$\ballEm{p}{r}$, with $r = (3 + \frac{\sparseconst}{2})\epsilon$.
\end{lem}
\begin{proof}
Suppose $q' \in \tau$ originates from $q \in \mathsf{P}$. Then, using
Property~\ref{hyp:diam.bnd} and the perturbation
bound~\eqref{eq:pertbnd}, the triangle inequality yields
\begin{equation*}
\begin{split}
\distEm{p}{q}
&\leq \longedge{\tau} + \distEm{p}{p'} + \distEm{q}{q'}\\
&< \frac{5}{2}(1 + \frac{1}{2}\delta_0 \sparseconst)\epsilon + 2\rho_0} %{\eta_0
\epsilon\\
&\leq (3 + \frac{1}{2}\sparseconst)\epsilon.
\end{split}
\end{equation*}
\end{proof}
We exploit \Lemref{lem:bound.forbid.rad} to define the local
structures in which we check for forbidden configurations} %{$\dg$-configuration. For any
point $p \in \mathsf{P}$, let
\begin{equation*}
\mathcal{N}_{p} = \ballEm{p}{(3 + \frac{\sparseconst}{2})\epsilon}
\cap \mathsf{P} \setminus \{p\},
\end{equation*}
and define $\mathcal{S}_p$ to be the $m$-skeleton
of the complete complex on $\mathcal{N}_p$. In other words,
$\mathcal{S}_p$ consists of all $j$-simplices with vertices in
$\mathcal{N}_p$ and $j \leq m$.
We let $\mathcal{S}_{p_i}(\mathsf{P}_{i-1})$ denote the simplices in
$\mathsf{P}_{i-1}$ that correspond to simplices in $\mathcal{S}_{p_i}$. If
$\sigma' \in \mathsf{P}_{i-1} \setminus \{p_i\}$ is such that it forms a
forbidden configuration} %{$\dg$-configuration\ with $x \in \ballEm{p_i}{\rho_0} %{\eta_0 \epsilon}$, then
$\sigma'$ belongs to $\mathcal{S}_{p_i}(\mathsf{P}_{i-1})$.
\begin{algorithm}[ht]
\caption{\texttt{good\_perturbation}$(x,p, \pts')$}
\label{alg:good.perturbation}
\begin{algorithmic}[1]
\STATE // Test if $x$ is a good perturbation of $p$ in $\pts'$.
\STATE // $\mathcal{S}_p(\pts')$ is defined in
\Secref{sec:good.perturbations}, and $\alpha_0$
is defined by Property~\ref{hyp:clean.hoop.bnd} of
\Thmref{thm:prop.forbid.cfg}.
\STATE compute $\mathcal{S}_p(\pts')$
\FOR {each $\sigma \in \mathcal{S}_p(\pts')$}
\IF {$\circrad{\sigma} < \infty$} \label{lin:radbnd}
\IF
$\abs{\distEm{x}{\circcentre{\sigma}} - \circrad{\sigma}} \leq
\alpha_0 2\epsilon$} \label{lin:hoopbnd}
\RETURN \FALSE
\ENDIF
\ENDIF
\ENDFOR
\RETURN \TRUE
\end{algorithmic}
\end{algorithm}
\Algref{alg:good.perturbation} reveals that Algorithm~\ref{alg1} uses
two geometric predicates: (1) a distance comparison (to compute
$\mathcal{S}_p(\pts')$), and (2) the in-sphere tests implicit in
Line~\ref{lin:hoopbnd} of \Algref{alg:good.perturbation}. The
complexity of the algorithm will be discussed in
\Secref{ssec-algorithm-complexity}.
\begin{remk}
\label{rem:bad.good.facet}
We observe that \texttt{good\_perturbation} does not explicitly
exploit Property~\ref{hyp:good.facets} of forbidden configuration} %{$\dg$-configuration s. Also,
Property~\ref{hyp:clean.facet.rad.bnd} is only really used for the
bound on the right hand side of the inequality of
Line~\ref{lin:hoopbnd}. The volumetric analysis presented in
\Secref{sec:analysis} counts all simplices $\sigma$ that could be a
facet of a simplex with diameter bounded by
Property~\ref{hyp:diam.bnd}, without consideration of the
circumradius or thickness of $\sigma$. However, Properties
\ref{hyp:good.facets} and \ref{hyp:clean.facet.rad.bnd} may be
important in applications, and Line~\ref{lin:radbnd} serves as a
reminder that they may be taken into account.
\end{remk}
\section{Analysis of the algorithm}
\label{sec:analysis}
In this section we will prove
\Thmref{thm-main-theorem-of-the-paper}. We begin with a calculation of
the number of simplices contained in the local complexes
$\mathcal{S}_p(\pts')$. Then in \Secref{sec:good.perts.exist}, following a standard practice in the analysis of perturbation algorithms~\cite{edelsbrunner2000smoothing,halperin2004controlled}, we
perform the volume calculations that show the existence of good
perturbations, and the probability of finding one with a random
point. Then in \Secref{ssec-algorithm-complexity} we analyse the
complexity and precision required by the algorithm.
\begin{lem}
\label{lem-analysis-size-of-the-np-complex}
Let $\mathsf{P} \subset \reel^m$ be a $\mueps$-net. For all $p \in \mathsf{P}$, we have
$\card{\mathcal{N}_{p}} \leq E_{1} \stackrel{{\rm def}}{=}
\left( \frac{8}{\sparseconst} \right)^m$, and
\begin{equation*}
\card{\mathcal{S}_{p}} < E
\stackrel{{\rm def}}{=}
2\left( \frac{8}{\sparseconst} \right)^{m^2 +m}.
\end{equation*}
\end{lem}
\begin{proof}
In order to bound $\card{\mathcal{N}_p}$ we will use a packing
argument in the ball $\ballEm{p}{(3 +
\frac{\sparseconst}{2})\epsilon}$ described in
\Lemref{lem:bound.forbid.rad}. We extend the radius by the packing
radius $r = \frac{\sparseconst \epsilon}{2}$ of $\mathsf{P}$. Thus let $R
= (3 + \sparseconst)\epsilon$. It follows then that for any $p \in
\mathsf{P}$
\begin{equation*}
\card{\mathcal{N}_{p}} \leq \left(\frac{R}{r}\right)^{m} =
\left( \frac{2}{\sparseconst} \left( 3 + \sparseconst \right) \right)^m
\leq \left( \frac{8}{\sparseconst} \right)^m = E_1.
\end{equation*}
This implies that for all $p\in \mathsf{P}$,
\begin{equation*}
\card{\mathcal {S}_{p}} \leq \sum_{j=1}^{m+1} E_{1}^{j} <
2E_1^{m+1} \leq 2\left( \frac{8}{\sparseconst} \right)^{m^2 +m} = E.
\end{equation*}
\end{proof}
\subsection{Existence of good perturbations}
\label{sec:good.perts.exist}
Recall that for any simplex $\sigma$ with $\circrad{\sigma} < \infty$
the circumsphere $\circsphere{\sigma}$ is contained in the diametric
sphere $\diasphere{\sigma}$. Thus if $\dist{x}{\diasphere{\sigma}} >
\alpha_0 \circrad{\sigma}$, then $\dist{x}{\circsphere{\sigma}} >
\alpha_0 \circrad{\sigma}$, and $\tau = \splxjoin{x}{\sigma}$ cannot
have the $\hoopbnd$-hoop\ property. As discussed below, it is convenient to use
$\diasphere{\sigma}$ instead of $\circsphere{\sigma}$, and there is
little cost since these objects coincide when $\sigma$ is an
$m$-simplex, and this dominates the calculation we are about to
describe.
The \texttt{good\_perturbation} procedure uses this sufficient
criterion to filter for good perturbations. The probability of
successfully finding a good perturbation by choosing a random point is
based on a volume calculation. Specifically, exploiting Properties
\ref{hyp:clean.hoop.bnd} and \ref{hyp:clean.facet.rad.bnd} of
forbidden configuration} %{$\dg$-configuration s described in \Thmref{thm:prop.forbid.cfg}, we define the
\defn{forbidden volume} $F_p(\sigma)$ for $p$ contributed by $\sigma$
as the volume occupied in the perturbation ball
$\ballEm{p}{\rho}$ for $p$ consisting of those points that are
within a distance $\alpha_0 2 \epsilon$ from $\diasphere{\sigma}$, as
depicted in \Figref{fig:forbidden.volume}.
\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.6\linewidth]{imgs/pert_ball_shell}
\end{center}
\caption[Forbidden volume]{The \defn{forbidden volume} $F_p(\sigma)$
that a simplex $\sigma$ removes from the perturbation ball
$\ballEm{p}{\rho}$ constitutes the points in
$\ballEm{p}{\rho}$ that are within a distance $\alpha_0
2\epsilon$ from $\diasphere{\sigma}$, as suggested by Properties
\ref{hyp:clean.hoop.bnd} and \ref{hyp:clean.facet.rad.bnd} of
\Thmref{thm:prop.forbid.cfg}.}
\label{fig:forbidden.volume}
\end{figure}
We let $\ballvol{j}$ denote the volume of a $j$-dimensional Euclidean
unit ball. The following lemma yields a bound on the forbidden volumes
$F_p(\sigma)$:
\begin{lem}[Forbidden volume]
\label{lem:shell.vol.bnd}
If $S^{m-1}$ is a sphere of radius $R$ in $\reel^m$, then for any $p
\in \reel^m$, and $\rho < R - \beta$, the volume
$F_p(\rho, \beta, S^{m-1})$ of points contained in
$\ballEm{p}{\rho}$, and within a distance $\beta$ from
$S^{m-1}$ is bounded by
\begin{equation*}
F_p(\rho,\beta, S^{m-1}) \leq
\ballvol{m-1}(\frac{\pi}{2}\rho)^{m-1} 2\beta.
\end{equation*}
\end{lem}
\begin{proof}
Consider an $(m-1)$-sphere $S$, concentric with $S^{m-1}$ and with
radius $\tilde{R}$ with $R - \beta \leq \tilde{R} \leq R +
\beta$. The intersection of $\ballEm{p}{\rho}$ with $S$
will be a geodesic ball $\mathcal{B} \subset S$. Since $\rho <
\tilde{R}$, the geodesic radius of $\mathcal{B}$, say
$r=\tilde{R}\theta$, is subtended by an angle $\theta$ that is less
than $\pi/2 $, and $\frac{2}{\pi}\theta \leq \sin \theta \leq
\rho/\tilde{R}$. It follows that $r \leq
\frac{\pi}{2}\rho$, independent of $R$ or $\tilde{R}$.
Since the volume of a geodesic ball in an $(m-1)$-sphere is smaller
than a Euclidean $(m-1)$-dimensional ball of the same
radius~\cite[Theorem III.4.2]{chavel2006}, we have
\begin{equation*}
\vol(\mathcal{B}) \leq \ballvol{m-1}
(\frac{\pi}{2}\rho)^{m-1},
\end{equation*}
and the stated bound follows.
\end{proof}
\begin{remk}
If $\sigma \in \mathcal{S}_p(\pts')$ is a $j$-simplex, with $j \leq
m$, then it is also the face of many $m$-simplices in
$\mathcal{S}_p(\pts')$. Thus if $\dist{x}{\diasphere{\sigma}} \leq
\alpha_0 2 \epsilon$, then we will also have
$\dist{x}{\circsphere{\tau}} \leq \alpha_0 2 \epsilon$ for any
$m$-simplex $\tau$ such that $\sigma \leq \tau$. Thus the
\texttt{good\_perturbation} \Algref{alg:good.perturbation} only
really needs to consider the $m$-simplices in
$\mathcal{S}_p(\pts')$. This would save a factor of two in the
estimate of $\card{\mathcal{S}_p}$, but if we wish to exploit
Property~\ref{hyp:good.facets} of \Thmref{thm:prop.forbid.cfg}, as
must be done in the context of finite precision, then all the lower
dimensional simplices must also be taken into consideration. Indeed,
if $\sigma$ is $\Gamma_0$-good and has a small circumradius, we
cannot assume that it is the face of an $m$-simplex with these
properties.
\end{remk}
We now prove that at the $i$-th iteration of the algorithm there
exists a $p_{i}' \in B(p_{i}, \rho_0} %{\eta_0\epsilon)$ that is a good
perturbation of $p_{i}$. We also establish an upper bound on the
expected number of times we have to pick random points from $B(p_{i},
\rho_0} %{\eta_0\epsilon)$ in order to get a good perturbation. In the
description of the algorithm we let $\rho_0} %{\eta_0$ determine $\delta_0$
and $\Gamma_0$, but here we keep all three as separate parameters,
subject to constraint inequalities.
\begin{lem}[Existence of good perturbations]
\label{lem:good.perturbation.existence}
If
\begin{equation}
\label{eq:dno.and.gammaz.bnd}
\delta_0 \leq \Gamma_0^{m+1}, \quad \text{ and } \quad
\Gamma_0 < \frac{\rho_0} %{\eta_0}{K},
\end{equation}
where $K = \frac{\ballvol{m-1}}{\ballvol{m}} \left(
\frac{8}{\sparseconst} \right)^{m^2} \left(
\frac{16}{\sparseconst} \right)^{m+4}$, then at the
$i^{\text{th}}$ iteration of the algorithm there exists a good
perturbation $p_{i}'$ of $p_{i}$ such that no forbidden configuration} %{$\dg$-configuration\ is incident
to $p_{i}'$ in $\mathsf{P}_i$, and the expected number of times we have to
pick random points from $B(p_{i}, \rho_0} %{\eta_0\epsilon)$ to get a good
perturbation of $p_{i}$ is less than
\begin{equation*}
T = \frac{1}{1-\gamma},
\end{equation*}
where
\begin{equation*}
\gamma = \frac{K \Gamma_0}{\rho_0} %{\eta_0}.
\end{equation*}
\end{lem}
\begin{proof}
We exploit \Thmref{thm:prop.forbid.cfg}. Say that $x$ is a bad
perturbation of $p \in \pts'$ if there is a $\sigma \in
\mathcal{S}_p(\pts')$ such that $\distEm{x}{\diasphere{\sigma}} \leq
\alpha_0 2\epsilon$, with $\alpha_0$ defined by
Property~\ref{hyp:clean.hoop.bnd}. Let $F_p(\sigma) :=
F_p(\rho_0} %{\eta_0\epsilon, \alpha_0 2\epsilon, \diasphere{\sigma})$
denote the volume in $\ballEm{p}{\rho_0} %{\eta_0\epsilon}$ that represents
bad perturbations with respect to $\sigma$. Then
\Lemref{lem:shell.vol.bnd} implies
\begin{equation*}
F_p(\sigma) \leq
\ballvol{m-1}(\frac{\pi}{2})^{m-1} \rho_0} %{\eta_0^{m-1}\epsilon^{m-1}
\alpha_0 4\epsilon.
\end{equation*}
Using $E$ defined in \Lemref{lem-analysis-size-of-the-np-complex},
we obtain a bound on $F_p$, the total volume of the bad
perturbations in $\ballEm{p}{\rho_0} %{\eta_0\epsilon}$:
\begin{align*}
F_p &\leq E F_p(\sigma) &\\
&\leq 8\left( \frac{8}{\sparseconst} \right)^{m^2 +m}
\left(\frac{\pi}{2} \right)^{m-1}
\alpha_0 \rho_0} %{\eta_0^{m-1}\ballvol{m-1}\epsilon^m &\\
&\leq 16\left( \frac{8}{\sparseconst} \right)^{m^2 +m}
\left(\frac{\pi}{2} \right)^{m-1}
\left(\frac{16}{\sparseconst} \right)^3 \Gamma_0
\rho_0} %{\eta_0^{m-1}\ballvol{m-1}\epsilon^m &
\text{by Property~\ref{hyp:clean.hoop.bnd}} \\
&\leq \left( \frac{8}{\sparseconst} \right)^{m^2}
\left( \frac{16}{\sparseconst} \right)^{m + 4}
\Gamma_0 \rho_0} %{\eta_0^{m-1}\ballvol{m-1}\epsilon^m &\\
\end{align*}
Therefore, the volume of the set of good perturbations of $p$ in
$\ballEm{p}{\rho_0} %{\eta_0 \epsilon}$ is greater than
\begin{equation*}
\ballvol{m} \rho_0} %{\eta_0^{m}\epsilon^{m} - K\ballvol{m} \rho_0} %{\eta_0^{m-1} \Gamma_0
\epsilon^{m},
\end{equation*}
and it follows that the probability of getting a good perturbation
of $p$ by a picking random point from $B(p, \rho_0} %{\eta_0\epsilon)$ is
greater than $1-\gamma$, where $\gamma = \frac{K
\Gamma_0}{\rho_0} %{\eta_0}$. Therefore the expected number of trials
required to get a good perturbation is not greater than
\begin{equation*}
\sum_{i=0}^{\infty} (i+1)\gamma^{i}(1-\gamma) = \frac{1}{1-\gamma}.
\end{equation*}
\end{proof}
\subsection{Complexity of the algorithm}
\label{ssec-algorithm-complexity}
Lemmas \ref{lem-analysis-size-of-the-np-complex} and
\ref{lem:good.perturbation.existence} lead directly to bounds on the
asymptotic properties of the algorithm:
\begin{lem}
\label{lem:time.space.complexity}
The expected time complexity of \Algref{alg1} is
\begin{equation*}
\bigo{m}(\card{\mathsf{P}})^2 +
(1-\gamma)^{-1} \left( \frac{2}{\sparseconst} \right)^{\bigo{m^2}}
\card{\mathsf{P}}.
\end{equation*}
The space complexity required to run the algorithm is
\begin{equation*}
\left(\frac{2}{\sparseconst} \right)^{\bigo{m}} \card{\mathsf{P}}
+ \left( \frac{2}{\sparseconst} \right)^{\bigo{m^2}}.
\end{equation*}
\end{lem}
\begin{proof}
The sets $\mathcal{N}_p$ can be computed by a na\"ive algorithm in
$\bigo{m}(\card{\mathsf{P}})^2$ time, while being stored in $\left(
\frac{2}{\sparseconst} \right)^{\bigo{m}} \card{\mathsf{P}}$ space,
which is also sufficient to store the input and output point sets.
The algorithm visits each point once, and it computes and stores the
set $\mathcal{S}_p(\pts')$ which has size $\left(
\frac{2}{\sparseconst} \right)^{\bigo{m^2}}$. The
\texttt{good\_perturbation} procedure
(\Algref{alg:good.perturbation}) evaluates
$\abs{\distEm{x}{\circcentre{\sigma}} - \circrad{\sigma}} \leq
2\alpha_0\epsilon$ for every simplex $\sigma \in
\mathcal{S}_p(\pts')$. This computation can be performed via
determinant evaluations in $\bigo{m^3}$ time, so the time required
to run the \texttt{good\_perturbation} algorithm is $\left(
\frac{2}{\sparseconst} \right)^{\bigo{m^2}}$. The expected number
of times it must be run on each point is $(1-\gamma)^{-1}$, and this
yields the stated bound.
\end{proof}
\subsection{Summary of guarantees}
\Lemref{lem:no.forbidden.output} and
\Lemref{lem:good.perturbation.existence} guarantee that \Algref{alg1}
terminates with $\mathsf{P}_n$ which contains no forbidden configuration} %{$\dg$-configuration s and is a
perturbation of $\mathsf{P}$. \Lemref{lem:time.space.complexity}
establishes the complexity bound. Since
Condition~\eqref{eq:dno.and.gammaz.bnd} demanded by
\Lemref{lem:good.perturbation.existence} implies
Condition~\eqref{eq:dnobnd} required for
\Thmref{thm:no.hoops.implies.protection}, the main result
is established:
\begin{thm}[Main result]
\label{thm:raw.main}
\Algref{alg1} takes as input a $\mueps$-net\ $\mathsf{P} \subset \reel^m$ and
positive parameters $\rho_0} %{\eta_0 \leq \frac{\sparseconst}{4}$ and
$\Gamma_0$, with
\begin{equation}
\label{eq:raw.gamma.z.bnd}
\Gamma_0 < \frac{\rho_0} %{\eta_0}{K},
\end{equation}
where
\begin{equation}
\label{eq:raw.K}
K = \frac{\ballvol{m-1}}{\ballvol{m}} \left(\frac{8}{\sparseconst}
\right)^{m^2}\left(\frac{16}{\sparseconst} \right)^{m+4},
\end{equation}
and $\ballvol{j}$ is the volume of the $j$-dimensional unit ball.
By sequentially perturbing the points, it produces a $\pmueps$-net\
$\pts'$ that is a $\delta$-generic, $\rho_0} %{\eta_0
\epsilon$-perturbation of $\mathsf{P}$ and such that all the Delaunay
$m$-simplices in $\rdelsmhull{\ppts}$ are $\Gamma_0$-good and
\begin{equation*}
\delta = \Gamma_0^{m+1}\sparseconst' \samconst',
\end{equation*}
where $\sparseconst'$ and $\samconst'$ are defined in
\Lemref{lem:perturb.Delone}.
The expected time complexity is less than
\begin{equation*}
\bigo{m}(\card{\mathsf{P}})^2 + (1-\gamma)^{-1} \left(
\frac{2}{\sparseconst} \right)^{O(m^{2})} \card{\mathsf{P}},
\end{equation*}
where the constant in the big-$O$ notation is an absolute constant
and
\begin{equation*}
\gamma = \frac{K \Gamma_0}{\rho_0} %{\eta_0}.
\end{equation*}
\end{thm}
\Thmref{thm-main-theorem-of-the-paper} is a restatement of this
result, simplified by setting $\Gamma_0 = \frac{\rho_0} %{\eta_0}{2K}$, and
by also observing that
\begin{equation}
\label{eq:bound.sphere.vol.ratio}
\frac{\ballvol{m-1}}{\ballvol{m}} \leq 2^m.
\end{equation}
Indeed, $\frac{\ballvol{m-1}}{\ballvol{m}}$ is a slowly growing function
of $m$, and the crude bound~\eqref{eq:bound.sphere.vol.ratio} can be
obtained from an elementary calculation using the
expression~\cite[Eq. (18), p. 9]{conway1988sphere} for $\log_2 V_m$.
The constant $K$ involved in the bound on $\Gamma_0$ has been
computed explicitly, and cannot easily be reduced significantly. This
means that \Eqnref{eq:dno.and.gammaz.bnd}
yields a $2^{-\bigo{m^3}}$ bound on $\delta_0$, which results in very
small numbers, even in low dimensions.
Two of the powers of $m$ in the exponent come
from the consideration of all $m$-simplices in the neighbourhood of a
point (\Lemref{lem-analysis-size-of-the-np-complex}), and the other
comes from the dimension-gradated thickness bound introduced in the
\Defref{def:flake} of a flake. Analyses of traditional sliver
exudation algorithms suffer from similar tiny bounds, but in practice
these bounds appear to be pessimistic.
\section{Background}
\label{sec-background-definition}
We work in $m$-dimensional Euclidean space $\reel^m$, where distances are
determined by the standard norm, $\norm{\cdot}$. The distance between
a point $p$ and a set $\X \subset \reel^m$, is the infimum of the
distances between $p$ and the points of $\X$, and is denoted
$\distEm{p}{\X}$. We refer to the distance between two points $a$ and
$b$ as $\norm{b-a}$ or $\distEm{a}{b}$ as convenient. A ball
$\ballEm{c}{r} = \{ x \, | \, \distEm{x}{c}< r \}$ is open, and
$\cballEm{c}{r}$ is its topological closure. Generally, we denote the
topological closure of a set $\X$ by $\close{\X}$, the interior by
$\intr{\X}$, and the boundary by $\bdry{\X}$. The convex hull is
denoted $\convhull{\X}$, and the affine hull is $\affhull{\X}$.
The cardinality of a finite set $\mathsf{P}$ is $\card{\mathsf{P}}$.
\subsection{Sampling parameters}
The structures of interest will be built from a finite set $\mathsf{P}
\subset \reel^m$, which we consider to be a set of \defn{sample
points}. If $D \subset \reel^m$, then
$\mathsf{P}$ is \defn{$\epsilon$-dense} for $D$ if $\distEm{x}{\mathsf{P}} <
\epsilon$ for all $x \in D$.
We say that $\epsilon$ is a \defn{sampling radius} for $D$ satisfied
by $\mathsf{P}$. If no domain $D$ is specified, we say
$\mathsf{P}$ is $\epsilon$-dense if $\distEm{x}{\mathsf{P} \cup
\bdry{\convhull{\mathsf{P}}}} < \epsilon$ for all $x \in
\convhull{\mathsf{P}}$. Equivalently, $\mathsf{P}$ is $\epsilon$-dense if it
satisfies a sampling radius $\epsilon$ for
\begin{equation}
\label{eq:contracted.hull}
D_\epsilon(\mathsf{P}) = \{ x \in \convhull{\mathsf{P}} \, | \,
\distEm{x}{\bdry{\convhull{\mathsf{P}}}} \geq \epsilon \}.
\end{equation}
A convenience of this definition is expressed in
\Lemref{lem:perturb.Delone} below.
The set $\mathsf{P}$ is \defn{$\sparsity$-separated} if $\distEm{p}{q} \geq
\sparsity$ for all $p,q \in \mathsf{P}$. We usually assume that $\sparsity =
\sparseconst \epsilon$ for some positive $\sparseconst \leq 1$. Such
a set is said to be a \defn{$\mueps$-net}, and if $\sparseconst = 1$, then
$\mathsf{P}$ is an \defn{$\epsilon$-net}. If $\mathsf{P}$ is a $\mueps$-net\ for $D$,
then the open balls of radius $\epsilon$ centred at the points of
$\mathsf{P}$ cover $D$, and the likewise centred open balls of radius
$\frac{\sparseconst \epsilon}{2}$ are pairwise disjoint. The sampling
radius is sometimes called a \defn{covering radius}, and
$\frac{\sparseconst \epsilon}{2}$ is a \defn{packing radius} for
$\mathsf{P}$. This consistent use of open balls to describe packing and
covering radii yields the strict and non strict inequalities in our
definitions of density and separation. The density and separation parameters are used extensively in the computational geometry literature on sampling and mesh generation, while the equivalent terminology of covering radius and packing radius is favoured in the crystalography and sphere packing literature. There is no standard notation for point sets described by these parameters. In our notation $\sparseconst$ is a dimensionless quantity that gives some measure of the quality of $\mathsf{P}$, while $\epsilon$ is a distance and is just an indication of scale.
We work with $\mueps$-net s, but this should not be viewed as a significant constraint on the point sets considered. Indeed \emph{any} finite set of distinct points is a $\mueps$-net\ for a large enough $\epsilon$ and a small enough $\sparseconst$. Thus $\epsilon$ and $\sparseconst$ are simply parameters that describe the point set. However, the parameter $\sparseconst$ has a direct bearing on the output guarantees of the algorithm. Our main result, \Thmref{thm-main-theorem-of-the-paper}, reveals that the expected running time of the algorithm, as well as the stability properties of the Delaunay triangulation of the output points, both depend on $\sparseconst$. Also, our results only begin to become interesting when $D_{\epsilon}(\mathsf{P})$ defined in \Eqnref{eq:contracted.hull} is non-empty; as explained in \Secref{sec:Delaunay.complexes}, the stability claims (\Thmref{thm:thick.eucl.stability}) about Delaunay simplices only apply to simplices that are not too close to the boundary of the convex hull.
\subsection{Perturbations}
Our algorithm will return a perturbation of a given $\mueps$-net. Here we
define perturbations in our context, and observe that a perturbed
$\mueps$-net\ is itself a $\pmueps$-net.
\begin{de}[Perturbation]
\label{def:perturbation}
A \defn{$\rho$-perturbation} of a $\mueps$-net\ $\mathsf{P} \subset \reel^m$
is a bijective application $\zeta} % perturbation function: P \to \tilde{P: \mathsf{P} \to \pts' \subset \reel^m$ such
that $\distEm{\zeta} % perturbation function: P \to \tilde{P(p)}{p} \leq \rho$ for all $p \in
\mathsf{P}$, and $\rho < \frac{\sparseconst \epsilon}{2}$.
For convenience, we will demand a stronger bound on $\rho$ and
omit the explicit qualification: unless otherwise specified, a
\defn{perturbation} will always refer to a
$\rho$-perturbation, with $\rho = \rho_0} %{\eta_0 \epsilon$
for some
\begin{equation}
\label{eq:pertbnd}
\rho_0} %{\eta_0 \leq \frac{\sparseconst}{4}.
\end{equation}
We also refer to $\pts'$ itself as a perturbation of $\mathsf{P}$.
We generally use $p'$ to denote the point $\zeta} % perturbation function: P \to \tilde{P(p) \in \pts'$, and
similarly, for any point $q' \in \pts'$ we understand $q$ to be its
preimage in $\mathsf{P}$.
\end{de}
Given a perturbation constrained by \Eqnref{eq:pertbnd}, we do not expect a close relationship between the associated Delaunay complexes (defined in \Secref{sec:Delaunay.complexes}), but we can at least relate the sampling parameters of the two point sets:
\begin{lem}
\label{lem:perturb.Delone}
If $\mathsf{P} \subset \reel^m$ is a $\mueps$-net, and $\pts'$ is a $\rho_0} %{\eta_0\epsilon$-perturbation of
$\mathsf{P}$, with $\rho_0} %{\eta_0 \leq \frac{\sparseconst}{4}$, then $\pts'$ is a $\pmueps$-net, where
\begin{itemize}
\item $\samconst' = (1 + \rho_0} %{\eta_0)\epsilon \leq \frac{5}{4}\epsilon$,
and
\item $\sparseconst' = \frac{\sparseconst - 2\rho_0} %{\eta_0}{1 + \rho_0} %{\eta_0}
\geq \frac{2}{5} \sparseconst$.
\end{itemize}
\end{lem}
\begin{proof}
\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.5\linewidth]{imgs/conv_hull_pert}
\end{center}
\caption[Perturbed convex hull]{\Lemref{lem:perturb.Delone}: $\bdry{\convhull{P}}$ and $\bdry{\convhull{P'}}$ must be close.}
\label{fig:conv.hull.pert}
\end{figure}
The only non-trivial assertion is the density bound.
We will show that
\begin{equation*}
D_{\psamconst}(\ppts) \subseteq D_{\samconst}(\pts).
\end{equation*}
It follows that for any $x \in D_{\samconst'}(\pts')$, we have
$\distEm{x}{\pts'} \leq \distEm{x}{\mathsf{P}} + \rho_0} %{\eta_0\epsilon < (1 +
\rho_0} %{\eta_0) \epsilon = \samconst'$.
We first observe that for any $y \in \convhull{\mathsf{P}}$, we have
\begin{equation}
\label{eq:bnd.y.convppts}
\distEm{y}{\convhull{\pts'}} \leq \rho_0} %{\eta_0 \epsilon.
\end{equation}
To see this, we use Carath\'eodory's Theorem to write $y =
\sum_{i=0}^m \lambda_i p_i$, where $p_i \in \mathsf{P}$ and the
$\lambda_i$ are non-negative barycentric coordinates: $\sum_{i=0}^m
\lambda_i = 1$. It follows that the point $y^* = \sum_{i=0}^m
\lambda_i p'_i$ lies in $\convhull{\pts'}$, and $\norm{y^* - y} \leq
\sum_{i=0}^m \lambda_i \norm{p'_i - p_i} \leq \rho_0} %{\eta_0
\epsilon$. Similarly, we have that if $z \in \convhull{\pts'}$,
then
\begin{equation}
\label{eq:bnd.z.convpts}
\distEm{z}{\convhull{\mathsf{P}}} \leq \rho_0} %{\eta_0 \epsilon.
\end{equation}
This implies that if $y \in \bdry{\convhull{\mathsf{P}}}$, then
$\distEm{y}{\bdry{\convhull{\pts'}}} \leq \rho_0} %{\eta_0
\epsilon$. Indeed, assume that $y \in \convhull{\pts'}$, since
otherwise the assertion is an immediate consequence of
\Eqnref{eq:bnd.y.convppts}. To reach a contradiction, assume
$\distEm{y}{\bdry{\convhull{\pts'}}} = R > \rho_0} %{\eta_0 \epsilon$. Then
$\close{B} = \cballEm{y}{R} \subseteq \convhull{\pts'}$. Let $H$ be
a hyperplane through $y$ and supporting $\convhull{\mathsf{P}}$, and let
$z \in \bdry{\close{B}}$ lie on a line through $y$ and orthogonal to
$H$ and in the open half-space that doesn't contain
$\convhull{\mathsf{P}}$, as shown in \Figref{fig:conv.hull.pert}. Then
$\dist{z}{\convhull{\mathsf{P}}} = R > \rho_0} %{\eta_0 \epsilon$, contradicting
\Eqnref{eq:bnd.z.convpts}.
Suppose $x \in D_{\psamconst}(\ppts)$. Let $y \in
\bdry{\convhull{\mathsf{P}}}$ be such that $\distEm{x}{y} =
\distEm{x}{\bdry{\convhull{\mathsf{P}}}}$, and let $z \in
\bdry{\convhull{\pts'}}$ satisfy $\distEm{y}{z} =
\distEm{y}{\bdry{\convhull{\pts'}}}$. Then
\begin{equation*}
\begin{split}
\samconst' &\leq \distEm{x}{z} \leq \distEm{x}{y} +
\distEm{y}{z}\\
&= \distEm{x}{\bdry{\convhull{\mathsf{P}}}} +
\distEm{y}{\bdry{\convhull{\pts'}}}\\
&\leq \distEm{x}{\bdry{\convhull{\mathsf{P}}}} + \rho_0} %{\eta_0 \epsilon,
\end{split}
\end{equation*}
and we obtain $\distEm{x}{\bdry{\convhull{\mathsf{P}}}} \geq \samconst' -
\rho_0} %{\eta_0\epsilon = \epsilon$. Hence $x \in D_{\samconst}(\pts)$.
\end{proof}
\subsection{Simplices}
Although our problem setting is geometric in nature, it is convenient
to work with the framework of abstract simplices and complexes. A
\defn{simplex} $\sigma$ is a non-empty finite set. The
\defn{dimension} of $\sigma$ is given by $\dim{\sigma} = \card{\sigma}
- 1$, and a $j$-simplex refers to a simplex of dimension $j$. The
dimension of a simplex is sometimes indicated with a superscript:
$\sigma^j$. The elements of $\sigma$ are called the \defn{vertices}
of $\sigma$. We do not distinguish between a $0$-simplex and its
vertex. If a simplex $\sigma$ is a subset of $\tau$, we say it is a
\defn{face} of $\tau$, and we write $\sigma \leq \tau$. A
$1$-dimensional face is called an \defn{edge}. If $\sigma$ is a proper
subset of $\tau$, we say it is a \defn{proper face} and we write
$\sigma < \tau$. A \defn{facet} of $\tau$ is a face $\sigma$ with
$\dim{\sigma} = \dim{\tau} - 1$.
For any vertex $p \in \sigma$, the \defn{face opposite} $p$ is the
face determined by the other vertices of $\sigma$, and is denoted
$\opface{p}{\splxs}$. If $\sigma$ is a $j$-simplex, and $p$ is not a vertex of
$\sigma$, we may construct a $(j+1)$-simplex $\tau =
\splxjoin{p}{\sigma}$, called the \defn{join} of $p$ and $\sigma$. It
is the simplex defined by $p$ and the vertices of $\sigma$, i.e.,
$\sigma = \opface{p}{\splxt}$.
We will be considering simplices whose vertices are points in $\reel^m$,
and this endows the simplices with geometric properties, but we do not
require the vertices to be affinely independent. If $\sigma \subset
\reel^m$ and $x \in \sigma$, then $x$ is a vertex of $\sigma$.
The \defn{length} of an edge is the distance between its vertices. The
\defn{diameter} of a simplex $\sigma$ is its longest edge length, and
is denoted $\longedge{\sigma}$. The shortest edge length is
denoted $\shortedge{\sigma}$. If $\sigma$ is a $0$-simplex, we define
$\shortedge{\sigma} = \longedge{\sigma} = 0$.
The \defn{altitude} of $p$ in $\sigma$ is $\splxalt{p}{\sigma} =
\distEm{p}{\affhull{\opface{p}{\splxs}}}$. A poorly-shaped simplex can be
characterized by the existence of a relatively small altitude. The
\defn{thickness} of a $j$-simplex $\sigma$ is the dimensionless
quantity
\begin{equation*}
\thickness{\sigma} =
\begin{cases}
1& \text{if $j=0$} \\
\min_{p \in \sigma} \frac{\splxalt{p}{\sigma}}{j
\longedge{\sigma}}& \text{otherwise.}
\end{cases}
\end{equation*}
We say that $\sigma$ is $\Upsilon_0$-thick, if $\thickness{\sigma} \geq
\Upsilon_0$. If $\sigma$ is $\Upsilon_0$-thick, then so are
all of its faces. Indeed if $\sigma^j \leq \sigma$, then the smallest
altitude in $\sigma^j$ cannot be smaller than that of $\sigma$, and also
$\longedge{\sigma^j} \leq \longedge{\sigma}$.
A \defn{circumscribing ball} for a simplex $\sigma$ is any
$m$-dimensional ball that contains the vertices of $\sigma$ on its
boundary. If $\thickness{\sigma} = 0$, we say that $\sigma$ is
\defn{degenerate}, and such a simplex may not admit any circumscribing
ball. If $\sigma$ admits a circumscribing ball, then it has a
\defn{circumcentre}, $\circcentre{\sigma}$, which is the centre of the
unique smallest circumscribing ball for $\sigma$. The radius of this
ball is the \defn{circumradius} of $\sigma$, denoted
$\circrad{\sigma}$. A degenerate simplex $\sigma$ may or may not have
a circumcentre and circumradius; we write $\circrad{\sigma} < \infty$
to indicate that it does. In this case we can also define the
\defn{diametric sphere} as the boundary of the smallest circumscribing
ball: $\diasphere{\sigma} =
\bdry{\ballEm{\circcentre{\sigma}}{\circrad{\sigma}}}$, and the
\defn{circumsphere}: $\circsphere{\sigma} = \diasphere{\sigma} \cap
\affhull{\sigma}$. Observe that if $\sigma \leq \tau$, then
$\circsphere{\sigma} \subseteq \circsphere{\tau}$. If $\dim \sigma =
m$, then $\circsphere{\sigma} = \diasphere{\sigma}$.
\subsection{Complexes}
An \defn{abstract simplicial complex} (we will just say
\defn{complex}) is a set $\mathcal{K}$ of simplices such that if
$\sigma \in \mathcal{K}$, then all the faces of $\sigma$ are also
members of $\mathcal{K}$. The union of the vertices of all the
simplices of $\mathcal{K}$ is the \defn{vertex set} of
$\mathcal{K}$. We say that $\mathcal{K}$ is a \defn{complex on $\mathsf{P}$}
if $\mathsf{P}$ includes the vertex set of $\mathcal{K}$. Our complexes are
finite and the number of simplices in a complex $\mathcal{K}$ is
denoted $\card{\mathcal{K}}$. The \defn{complete complex} on $\mathsf{P}$,
denoted $\cpltcplx{\mathsf{P}}$, is set of all simplices that have vertices
in $\mathsf{P}$. If we let $\pwrset{\mathsf{P}}$ denote the set of subsets of
$\mathsf{P}$, then $\cpltcplx{\mathsf{P}} = \pwrset{\mathsf{P}} \setminus \emptyset$. A
complex $\mathcal{K}$ is the complete complex on $\mathsf{P}$ if and only if
$\mathsf{P}$ is the vertex set of $\mathcal{K}$ and $\mathsf{P} \in \mathcal{K}$.
A subset $\mathcal{L} \subseteq \mathcal{K}$ is a \defn{subcomplex} of
$\mathcal{K}$ if it is also a complex. If $\mathcal{K}$ is a complex
on $\mathsf{P}$, and $\mathcal{K}'$ is a complex on $\pts'$, then a map
$\zeta} % perturbation function: P \to \tilde{P: \mathsf{P} \to \pts'$ induces a \defn{simplicial map} $\mathcal{K}
\to \mathcal{K}'$ if for every $\sigma \in \mathcal{K}$,
$\zeta} % perturbation function: P \to \tilde{P(\sigma) \in \mathcal{K}'$. Thus the image of the simplicial map is a
subcomplex of $\mathcal{K}'$. We denote the simplicial map with
the same symbol, $\zeta} % perturbation function: P \to \tilde{P$. If $\zeta} % perturbation function: P \to \tilde{P$ is injective on
$\mathsf{P}$, and $\zeta} % perturbation function: P \to \tilde{P(\mathcal{K}) = \mathcal{K}'$, then $\zeta} % perturbation function: P \to \tilde{P$
is an \defn{isomorphism}.
Although we prefer to work with abstract simplices and complexes, the
underlying motivation for this work is centred in the concept of a
\defn{triangulation}, which demands traditional geometric simplicial
complexes for its definition. A \defn{geometric realisation} of a
complex $\mathcal{K}$ with vertex set $\mathsf{P}$, is a topological
space $\carrier{\mathcal{K}} \subset \amb$ such that there is a
bijection $g: \mathsf{P} \to \tilde{\pts} \subset \carrier{\mathcal{K}}$ with the
property that $\bigcup_{\sigma \in \mathcal{K}} \convhull{g(\sigma)} =
\carrier{\mathcal{K}}$, and if $\tau, \tau' \in \mathcal{K}$, then
$\convhull{g(\tau)} \cap \convhull{g(\tau')} = \X$, where either
$\X = \emptyset$, or $\X = \convhull{g(\sigma)}$
with $\sigma = (\tau \cap \tau') \in \mathcal{K}$.
If $\mathcal{K}$ is a complex on $\mathsf{P} \subset \reel^m$, we say that
$\mathcal{K}$ is \defn{embedded} if the inclusion map $\incl: \mathsf{P}
\hookrightarrow \reel^m$ yields a geometric realisation of
$\mathcal{K}$. A \defn{triangulation of a connected set $\X \subset
\reel^m$} is an embedded complex $\mathcal{K}$ on $\mathsf{P} \subset \X$
such that $\carrier{\mathcal{K}} = \X$. A \defn{triangulation of
$\mathsf{P} \subset \reel^m$} is a triangulation of $\convhull{\mathsf{P}}$.
\subsection{Delaunay complexes}
\label{sec:Delaunay.complexes}
Our definition of the Delaunay complex is equivalent to defining it as
the nerve of the Voronoi diagram, however we do not exploit
the Voronoi diagram in this work.
An \defn{empty ball} is one that contains no point from $\mathsf{P}$.
\begin{de}[Delaunay complex]
\label{def:Delaunay.complex}
A \defn{Delaunay ball} is a maximal empty ball. Specifically, $B =
\ballEm{x}{r}$ is a Delaunay ball if any empty ball centred at $x$
is contained in $B$. A simplex $\sigma$ is a \defn{Delaunay
simplex} if there exists some Delaunay ball $B$ such that the
vertices of $\sigma$ belong to $\bdry{B}\cap \mathsf{P}$. The
\defn{Delaunay complex} is the set of Delaunay simplices, and is
denoted $\delof{\pts}$.
\end{de}
If $\X \subset \reel^m$, then the \defn{Delaunay complex of $\mathsf{P}$
restricted to $\X$} is the subcomplex of $\delof{\pts}$ consisting of those
simplices that have a Delaunay ball centred in $\X$. We are interested
in the case where $\X = D_{\samconst}(\pts)$ for a finite $\epsilon$-dense
sample set $\mathsf{P}$. We denote the Delaunay complex of $\mathsf{P}$ restricted
to $D_{\samconst}(\pts)$ by $\rdelsmhull{\pts}$. Our interest in this subcomplex is due
to the following observation that is an immediate consequence of the
definitions. If the radius of a Delaunay ball $\sigma$ exceeds $\epsilon$, then the centre of that ball is at a distance of more than $\epsilon$ from any point in $\mathsf{P}$. Thus we have:
\begin{lem}
\label{lem:rdel.small.circrad}
If $\mathsf{P}$ is $\epsilon$-dense, then
every simplex $\sigma \in \rdelsmhull{\pts}$ has a Delaunay ball with radius
less than $\epsilon$, and in particular $\circrad{\sigma} <
\epsilon$.
\end{lem}
A Delaunay simplex $\sigma$ is \defn{$\delta$-protected} if it has a
Delaunay ball $B$ such that $\distEm{q}{\bdry{B}} > \delta$ for all
$q \in \mathsf{P} \setminus \sigma$. We say that $B$ is a
$\delta$-protected Delaunay ball for $\sigma$.
We say that $\sigma$ is \defn{protected} to mean that it is
$\delta$-protected for some unspecified $\delta > 0$.
A $\mueps$-net\ $\mathsf{P} \subset \reel^m$ is \defn{$\delta$-generic} if all the Delaunay $m$-simplices in $\rdelsmhull{\pts}$ are $\delta$-pro\-tec\-ted. The set $\mathsf{P}$ is simply \defn{generic} if it is $\delta$-generic for some unspecified $\delta > 0$. If $\mathsf{P}$ is generic, then $\rdelsmhull{\pts}$ is embedded \cite[Lemmas 3.5]{boissonnat2013stab1}, and with an abuse of language we call $\rdelsmhull{\pts}$ the \defn{restricted Delaunay triangulation} of $\mathsf{P}$. (We are abusing the language because in general $\rdelsmhull{\pts}$ coincides with neither $\convhull{P}$ nor $D_{\epsilon}$.)
If $\mathsf{P}$ is a $\delta$-generic $\mueps$-net, then the Delaunay
triangulation exhibits stability with respect to small perturbations
of the points or of the metric~\cite{boissonnat2013stab1}. This gives us motivation to demonstrate that $\delta$-generic point sets can be produced algorithmically, which is the primary contribution of the current work.
We will present an algorithm that, when given a $\mueps$-net, and a small
positive parameter $\Gamma_0 < 1$, will generate a $\delta$-generic
$\pmueps$-net\ $\pts'$ such that all the $m$ simplices in $\rdelsmhull{\ppts}$ are
$\Gamma_0^m$-thick. As an example in this context, the stability
with respect to the sample positions \cite[Theorem
4.14]{boissonnat2013stab1}, can be stated as:
\begin{thm}[Delaunay stability]
\label{thm:thick.eucl.stability}
Suppose $\pts' \subset \reel^m$ is a $\pmueps$-net, and all the $m$-simplices
in $\rdelsmhull{\ppts}$ are $\Gamma_0^m$-thick and $\delta$-protected, where
$\delta = \delta_0 \sparseconst' \samconst'$, with $0 \leq \delta_0
\leq 1$. If $\zeta} % perturbation function: P \to \tilde{P: \pts' \to \tilde{\pts}$ is a $\rho$-perturbation
of $\pts'$ with
\begin{equation*}
\rho \leq \frac{\Gamma_0^m \sparseconst'^2
\delta_0}{18}\samconst',
\end{equation*}
then $\zeta} % perturbation function: P \to \tilde{P: \rdelsmhull{\ppts} \to \mathcal{K} \subseteq \delof{\tilde{\pts}}$ is a
simplicial isomorphism onto an embedded subcomplex $\mathcal{K}$ of
$\delof{\tilde{\pts}}$.
\end{thm}
\section{Conclusions}
\label{sec:conclusions}
We have demonstrated an algorithm that will produce a $\delta$-generic
$\pmueps$-net\ $\pts'$ that is a perturbation of a given $\mueps$-net\
$\mathsf{P}$. The Delaunay triangulation of $\pts'$ is then quantifiably
stable with respect to changes in the metric or the points themselves.
Although our exposition assumes a finite set $\mathsf{P}$, it is worth
observing that the analysis requires only local finiteness (the
intersection of $\mathsf{P}$ with any compact set is a finite set), and the
algorithm extends trivially to the case of a periodic set $\tilde{\pts}
\subset \reel^m$. For example, we may have $\tilde{\pts} = \tilde{\pts} + v$ for any
$v \in \mathbb{Z}^m$, and $\tilde{\pts}$ is $\epsilon$-dense with respect to all
of $\reel^m$. In this framework we require that $\epsilon < 1/2$, and we
may view $\tilde{\pts}$ as a finite set $\mathsf{P}$ in the standard flat torus
${\mathbb{T}}^{m} = \reel^m / \mathbb{Z}^m$. This has the advantage of avoiding boundary
considerations. It is also closer in spirit to the primary motivating
application of this work, which is the construction of Delaunay
triangulations of compact manifolds.
Funke et al.~\cite{funke2005} hinted at a much simpler analysis for
arguing that a perturbation of points in $\reel^m$, for arbitrary $m$,
has a good probability of being $\delta$-generic, with
$\Gamma_0$-good simplicies. For a given point $p$, one simply
calculates the volumes of $\delta$-thick shells around the diametric
spheres of the nearby $m$-simplices (i.e., take $\beta=\delta$ in
\Figref{fig:forbidden.volume}), and one also accounts for the volumes
of ``slabs'' (i.e., the affine hull of each nearby $j$-simplex
thickened by an offset proportional to $\Gamma_0^j$). The probability
that the perturbed point $p'$ violates the protection of a Delaunay
ball, or becomes the vertex of a $\Gamma_0$-bad simplex, can thus be
made as small as required by appropriately reducing the size of
$\delta$ and $\Gamma_0$, or by increasing the perturbation parameter
$\rho_0} %{\eta_0$.
The problem with this simplified analysis is that although the
probability calculated for a given point depends only on points in a
neighbourhood (assuming a sampling density), these probabilities are
not independent. Conceptually, all the points must be perturbed at
once, and the probability of success is proportional to the total
number of points. Funke et al.~\cite[Section 4.3]{funke2005} mentioned
this limitation of their analysis.
In this paper we have shown that the hoop property provides a way to
circumvent this difficulty and obtain a $\delta$-generic $\pts'$,
where $\delta/\epsilon$ is only ultimately constrained by the separation
parameter $\sparseconst$, via Equations \ref{eq:pertbnd} and
\ref{eq:dno.and.gammaz.bnd}, and not by the sampling density or total
number of sample points. This is essential for our intended
application to meshing non-flat manifolds, which we have developed in other work~\cite{boissonnat2013manmesh.inria}. Building on the algorithm presented here, we give a constructive demonstration of the existence of Delaunay triangulations on compact abstract Riemannian manifolds.
Thus we are already exploiting the theoretical benefits of the algorithm. The obstruction to a practical implementation is the computation required to verify that a perturbation is good. We are currently exploring an approach that avoids this problem by using only combinatorial tests and a result of Moser and Tardos~\cite{moser2010}.
\section{\texorpdfstring{forbidden configuration} %{$\dg$-configuration s}{Forbidden configurations}}
\section{Forbidden configurations}
\label{sec:forbidden}
Our goal is to produce a point set whose Delaunay triangulation has
nice properties. In this section we identify specific configurations
of points whose existence in a $\pmueps$-net\ $\pts'$ implies that $\pts'$
does not meet the requirements of \Thmref{thm:thick.eucl.stability}.
These configurations are a particular family of thin simplices that we
call \defn{forbidden configuration} %{$\dg$-configuration s}.
For a $\mueps$-net\ the Delaunay triangles automatically enjoy a lower bound
on their thickness due to the bounds on their circumradius and
shortest edge (as verified by a calculation similar to the one in Lemma 3.13 of the Delaunay stability paper~\cite{boissonnat2013stab1}). However, higher dimensional Delaunay simplices may have
arbitrarily small thickness. The problem simplices in three
dimensional Delaunay triangulations have their vertices all near ``the
equator'' of their circumsphere, and were dubbed
\defn{slivers}~\cite{cheng2000}. They were characterised as simplices
that had an upper bound on both their thickness and the ratio of their
circumradius to shortest edge length.
The essential property of slivers, that is exploited by many
algorithms that seek to remove them, is the fact that every vertex
lies close to the circumcircle of its opposing facet. This property is
a consequence of the defining characteristics of a sliver, and it is
demonstrated in a ``Torus Lemma'' \cite{edelsbrunner2000smoothing}.
The Torus Lemma is important because it places a bound on the volume
of possible positions of a fourth vertex that would make a sliver when
joined with a fixed set of three vertices.
The concept of a sliver has been extended to higher dimensions in
various works, and likewise there is a higher dimensional analogue of
the Torus Lemma \cite{li2003}. In our current context, we will be
considering unwanted simplices that are not subjected to an upper
bound on their circumradius, because they are not Delaunay
simplices. For this reason, we introduce \defn{flakes} in
\Secref{sec:flakes}. Flakes have one of the important properties of
slivers: there is an upper bound on all of the altitudes, but flakes
are not subjected to a circumradius bound.
A flake that appears in the Delaunay complex of a $\mueps$-net\ is
necessarily a sliver in the traditional sense, but the Torus Lemma
does not apply to flakes in general. In \Secref{sec:delta.gen.forbid}
we introduce the \defn{forbidden configuration} %{$\dg$-configuration s}, a subfamily of flakes that may be
considered to be a generalisation of slivers. In
\Secref{sec:hoop.property} we show that forbidden configuration} %{$\dg$-configuration s will exhibit
the important property embodied in the Torus Lemma. We call this
property the \defn{hoop property}, and the Hoop \Lemref{lem:hoop} is
our extension of the Torus Lemma to the current context.
\subsection{Flakes}
\label{sec:flakes}
In dimensions higher than three, a simple upper bound on the thickness
of a simplex is not sufficient to bound \emph{all} of the altitudes of
the simplex. In order to obtain an effective bound on all of the
altitudes, a small upper bound on the thickness needs to be coupled
with a relatively larger lower bound on the thickness of the
facets. For this reason we introduce a thickness requirement that is
gradated with the dimension. We exploit a positive real parameter
$\Gamma_0$, which is no larger than one. In the following definition, $\Gamma_0^j$ means $\Gamma_0$ raised to the $j^{\text{th}}$ power.
\begin{de}[$\Gamma_0$-good simplices and $\Gamma_0$-flakes]
\label{def:flake}
\label{def:good.simplex}
A simplex $\sigma$ is \defn{$\Gamma_0$-good} if for all $j$ with
$0\leq j \leq \dim \sigma$, we have $\thickness{\sigma^j} \geq
\Gamma_0^j$ for all $j$-simplices $\sigma^j \leq \sigma$. A simplex
is \defn{$\Gamma_0$-bad} if it is not $\Gamma_0$-good. A
\defn{$\Gamma_0$-flake} is a $\Gamma_0$-bad simplex in which all
the proper faces are $\Gamma_0$-good.
\end{de}
Observe that a flake must have dimension at least $2$, since
$\thickness{\sigma^j} = 1$ for $j < 2$. Also, since a flake may be
degenerate, but its facets cannot, the dimension of a flake can be as
high as $m+1$, but no higher.
Earlier definitions of slivers in higher dimensions
\cite{li2003,cheng2005} correspond to flakes together with the
additional requirement that the circumradius to shortest edge ratio be
bounded. The dimension-gradated requirement on simplex quality
(altitude bound) is implicitly present in these earlier works.
Ensuring that all simplices in a complex $\mathcal{K}$ are
$\Gamma_0$-good is the same as ensuring that there are no flakes in
$\mathcal{K}$. Indeed, if $\sigma$ is $\Gamma_0$-bad, then it has a
$j$-face $\sigma^j \leq \sigma$ that is not $\Gamma_0^j$-thick. By
considering such a face with minimal dimension we arrive at the
following important observation:
\begin{lem}
\label{lem:bad.has.flake}
A simplex is $\Gamma_0$-bad if and only if it has a face that is a
$\Gamma_0$-flake.
\end{lem}
We obtain an upper bound on the altitudes of a $\Gamma_0$-flake
through a consideration of dihedral angles. In particular, we observe
the following general relationship between simplex altitudes:
\newcommand{\sigma_{pq}}{\sigma_{pq}}
\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.5\linewidth]{imgs/dihedral}
\end{center}
\caption[Dihedral angle]{The sine of the dihedral angle $\theta$ between the facets $\opface{q}{\splxs} = \asimplex{p,u,v}$, and $\opface{p}{\splxs} = \asimplex{q,u,v}$ of $\sigma = \asimplex{p,q,u,v}$ is given by $\frac{\splxalt{p}{\sigma}}{\splxalt{p}{\opface{q}{\splxs}}}$, i.e., the ratio of the altitude of $p$ in $\sigma$ to the altitude of $p$ in $\opface{q}{\splxs}$. The point $p_*$ is the orthogonal projection of $p$ into the affine hull of $\sigma_{pq} = \asimplex{u,v}$.
}
\label{fig:dihedral}
\end{figure}
\begin{lem}
\label{lem:alt.ratios}
If $\sigma$ is a $j$-simplex with $j \geq 2$, then for any two
vertices $p,q \in \sigma$, the dihedral angle between $\opface{p}{\splxs}$ and
$\opface{q}{\splxs}$ defines an equality between ratios of altitudes:
\begin{equation*}
\sin \angleop{\affhull{\opface{p}{\splxs}}}{\affhull{\opface{q}{\splxs}}} =
\frac{\splxalt{p}{\sigma}}{\splxalt{p}{\opface{q}{\splxs}}}
=
\frac{\splxalt{q}{\sigma}}{\splxalt{q}{\opface{p}{\splxs}}}.
\end{equation*}
\end{lem}
\begin{proof}
An example of the assertion is depicted in \Figref{fig:dihedral}.
Let $\sigma_{pq} = \opface{p}{\splxs} \cap \opface{q}{\splxs}$, and let $p_*$ be the projection
of $p$ into $\affhull{\sigma_{pq}}$. Taking $p_*$ as the origin, we see
that $\frac{p-p_*}{\splxalt{p}{\opface{q}{\splxs}}}$ has the maximal distance
to $\affhull{\opface{p}{\splxs}}$ out of all the unit vectors in
$\affhull{\opface{q}{\splxs}}$, and this distance is
$\frac{\splxalt{p}{\sigma}}{\splxalt{p}{\opface{q}{\splxs}}}$. By definition
this is the sine of the angle between $\affhull{\opface{p}{\splxs}}$ and
$\affhull{\opface{q}{\splxs}}$. A symmetric argument is carried out with $q$ to
obtain the result.
\end{proof}
The usefulness of the definition of flakes lies in the following observation:
\begin{lem}[Flakes have small altitude]
\label{lem:flake.alt.bnd}
If $\tau$ is a $\Gamma_0$-flake, then for any vertex $p \in
\tau$,
\begin{equation*}
\splxalt{p}{\tau} < \frac{ 2\longedge{\tau}^{2}
\Gamma_0 }{ \shortedge{\tau} }.
\end{equation*}
\end{lem}
\begin{proof}
Recalling \Lemref{lem:alt.ratios} we have
\begin{equation*}
\splxalt{p}{\tau} = \frac{\splxalt{q}{\tau}
\splxalt{p}{\opface{q}{\splxt}} } {\splxalt{q}{\opface{p}{\splxt}}},
\end{equation*}
and taking $q$ to be a vertex with minimal altitude, we have
\begin{equation*}
\splxalt{q}{\tau} = k \thickness{\tau} \longedge{\tau}
< k \Gamma_0^k \longedge{\tau},
\end{equation*}
and
\begin{equation*}
\begin{split}
\splxalt{q}{\opface{p}{\splxt}}
&\geq (k-1) \thickness{\opface{p}{\splxt}} \longedge{\opface{p}{\splxt}}\\
&\geq (k-1) \Gamma_0^{k-1} \longedge{\opface{p}{\splxt}}\\
&\geq (k-1) \Gamma_0^{k-1} \shortedge{\tau},
\end{split}
\end{equation*}
and
\begin{equation*}
\splxalt{p}{\opface{q}{\splxt}} \leq \longedge{\opface{q}{\splxt}} \leq \longedge{\tau},
\end{equation*}
and since $k \leq 2(k-1)$, the bound is obtained.
\end{proof}
\subsection{Properties of $\delta$-generic point sets}
\label{sec:delta.gen.forbid}
\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.5\linewidth]{imgs/gen_sliver}
\end{center}
\caption[Forbidden configuration]{ A forbidden configuration is a
flake $\tau$ that has a vertex $p$ that lies within a distance
$\delta$ from a small circumscribing ball of the opposing facet
$\opface{p}{\splxt}$.}
\label{fig:forbidden.config}
\end{figure}
In order to ensure a $\delta$-generic point set $\pts'$, we need to
consider simplices that may not appear in any Delaunay
triangulation. Specifically, we do not have a circumradius bound on
the problem configurations. This makes their description more
complicated than the traditional definition of a sliver. As
schematically depicted in \Figref{fig:forbidden.config}, we have the
following characterisation of the configurations that we need to
avoid:
\begin{de}[Forbidden configuration
\label{def:forbidden.config}
Let $\mathsf{P}' \subset \reel^m$ be a $\pmueps$-net. A $(k+1)$-simplex $\tau
\subseteq \mathsf{P}'$, is a \defn{forbidden configuration} %{$\dg$-configuration} in $\pts'$ if it is a
$\Gamma_0$-flake, with $k\leq m$, and there exists a $p \in \tau$
such that $\opface{p}{\tau}$ has a circumscribing ball $B =
\ballEm{C}{R}$ with $R < \samconst'$, and $\abs{\distEm{p}{C} - R}
\leq \delta$, where $\delta = \delta_0 \sparseconst' \samconst'$. We
say that the forbidden configuration} %{$\dg$-configuration\ is \defn{certified} by $p$ and $B$.
\end{de}
We remark that the definition of a forbidden configuration} %{$\dg$-configuration\ depends on two
parameters, $\Gamma_0$, and $\delta_0$, as well as on the parameters
which we associate with the sample set $\pts'$, namely
$\sparseconst'$, and $\samconst'$.
In order to guarantee that the $\pmueps$-net\ $\pts'$ is $\delta$-generic,
with $\delta = \delta_0 \sparseconst' \samconst'$, it is sufficient to
ensure that there is no forbidden configuration} %{$\dg$-configuration\ with vertices in $\pts'$:
\begin{lem}
\label{lem:non.protection.implies.forbidden}
Suppose $\pts' \subset \reel^m$ is a $\pmueps$-net. If there exists an
$m$-simplex $\sigma^m \in \rdelsmhull{\ppts}$ which is not
$\delta$-protected, with $\delta = \delta_0 \sparseconst'
\samconst'$, then $\cpltcplx{\pts'}$ contains a forbidden configuration} %{$\dg$-configuration. Likewise,
if any $\sigma^m \in \rdelsmhull{\ppts}$ is not $\Gamma_0$-good, then
$\cpltcplx{\pts'}$ contains a forbidden configuration} %{$\dg$-configuration.
\end{lem}
\begin{proof}
Suppose $\sigma^m \in \rdelsmhull{\ppts}$ is not $\delta$ protected. Then
there exists a $p \in \pts' \setminus \sigma^m$ such that $0 \leq
\distEm{p}{\circcentre{\sigma^m}} - \circrad{\sigma^m} \leq \delta$. The
$(m+1)$-simplex $\tilde{\tau} = \splxjoin{p}{\sigma^m}$ is
necessarily degenerate, therefore, by \Lemref{lem:bad.has.flake},
there is a $\Gamma_0$-flake $\tau \leq \tilde{\tau}$. If $p$ belongs
to $\tau$, then $\tau$ is necessarily a forbidden configuration} %{$\dg$-configuration\ certified by
$p$ and $B = \ballEm{\circcentre{\sigma^m}}{\circrad{\sigma^m}}$,
because $\delta \leq \delta_0 \sparseconst' \samconst'$. If $p$ does
not belong to $\tau$, then it is a forbidden configuration} %{$\dg$-configuration\ certified by any one
of its vertices and $B$.
A similar argument reveals a forbidden configuration} %{$\dg$-configuration\ if $\sigma^m$ is not
$\Gamma_0$-good.
\end{proof}
\subsection{The Hoop property}
\label{sec:hoop.property}
We characterise the property of forbidden configuration} %{$\dg$-configuration s that is important for
algorithmic purposes as follows:
\begin{de}[Hoop property]
\label{def:hoop}
A simplex $\tau \subset \reel^m$ has the \defn{$\hoopbnd$-hoop\ property} if
there is a constant $\alpha_0 > 0$ such that for every $p \in
\tau$, the opposing facet has a circumcentre and
\begin{equation*}
\distEm{p}{\circsphere{\opface{p}{\tau}}} \leq \alpha_0
\circrad{\opface{p}{\tau}} < \infty.
\end{equation*}
\end{de}
\subsubsection{The Hoop Lemma}
\label{sec:hoop.lem}
We emphasise that the \emph{symmetric} nature of the hoop property is essential for our purposes. The hoop property says that \emph{every} vertex is close to the circumsphere of the opposing facet. We obtain this bound in two steps. First we exploit the thickness of the facets to show that forbidden configuration} %{$\dg$-configuration s have a natural symmetry characterised by the fact that \emph{every} vertex lies close to some small circumscribing sphere of its opposing facet:
\begin{lem}[Symmetry of forbidden configurations} %{$\dg$-configuration]
\label{lem:close.to.some.sphere}
Suppose $\tau = \splxjoin{q}{\sigma}$ is a $(k+1)$-simplex
certified by $q$ and $\ballEm{C}{R}$ as a forbidden configuration} %{$\dg$-configuration\ in a
$\pmueps$-net. If $\delta_0 \leq \frac{1}{4}$, then for any $p \in
\tau$ there exists a ball $B = \ballEm{C_p}{R_p}$ circumscribing
$\opface{p}{\splxt}$ and such that
\begin{equation*}
R_p \leq \left(1 + \frac{3\delta_0}{\sparseconst' \Gamma_0^k}
\right) R,
\end{equation*}
and
\begin{equation*}
\distEm{p}{\bdry{B}} \leq \left( \frac{6\delta_0}{\sparseconst'^2
\Gamma_0^k} \right) \shortedge{\opface{p}{\splxt}}.
\end{equation*}
\end{lem}
\begin{proof}
The idea is that $C$ is ``almost'' a circumcentre for $\opface{p}{\splxt}$ in that the distances between $C$ and the vertices of $\opface{p}{\splxt}$ are all very close. Since $\opface{p}{\splxt}$ is thick, we can exploit a result \cite[Lemma 4.3]{boissonnat2013stab1} that says that $\opface{p}{\splxt}$ must have a circumscribing ball with a centre near $C$. The bounds then follow from a consideration of the triangle inequality, and the fact that $\opface{p}{\splxt}$ and $\sigma$ must have a vertex in common.
We observe that for any $u,v \in \opface{p}{\splxt}$ we have
\begin{equation*}
\big| \distEm{u}{C} - \distEm{v}{C} \big| \leq \delta_0
\shortedge{\sigma}.
\end{equation*}
It follows then, from \cite[Lemma 4.3]{boissonnat2013stab1}, that
there is a circumscribing ball $B = \ballEm{C_p}{R_p}$ for $\opface{p}{\splxt}$
with
\begin{equation*}
\distEm{C_p}{C} \leq \frac{ (R + \delta_0
\shortedge{\sigma} ) \delta_0 \shortedge{\sigma}}
{\thickness{\opface{p}{\splxt}}\longedge{\opface{p}{\splxt}} }.
\end{equation*}
Since $\tau$ is a $\Gamma_0$-flake, $\thickness{\opface{p}{\splxt}} \geq
\Gamma_0^k$. Thus, using
$\frac{\shortedge{\sigma}}{\longedge{\opface{p}{\splxt}}} \leq
\frac{2}{\sparseconst'}$, and $R \leq
\frac{1}{\sparseconst'}\shortedge{\opface{p}{\splxt}}$ and $\delta_0 <
\frac{1}{4}$, we find
\begin{equation*}
\distEm{C_p}{C} \leq \frac{ 2(R + 2\delta_0 R) \delta_0}
{\sparseconst' \Gamma_0^k}
\leq \frac{ 3\delta_0 R}
{\sparseconst' \Gamma_0^k} \leq
\frac{ 3 \delta_0 \shortedge{\opface{p}{\splxt}} }
{\sparseconst'^2 \Gamma_0^k}.
\end{equation*}
We have $k \geq 1$, since $\tau$ is a flake, so $\sigma$ and $\opface{p}{\splxt}$
must share a common vertex. Thus the bounds follow from the triangle
inequality.
\end{proof}
In the next step we arrive at the $\hoopbnd$-hoop\ property by exploiting the
altitude bound on every vertex that is guaranteed by
\Lemref{lem:flake.alt.bnd} because a forbidden configuration} %{$\dg$-configuration\ is a
$\Gamma_0$-flake. The Symmetry \Lemref{lem:close.to.some.sphere}
allows us to exploit an argument similar to the traditional
demonstration of the torus lemma. The full proof is described in
\Secref{sec:hoop.lem.proof}. We arrive at the following Hoop Lemma, which is
a restatement of \Lemref{lem:shell.lem}:
\begin{lem}[Hoop Lemma]
\label{lem:hoop}
If
\begin{equation*}
\delta_0 \leq \frac{\sparseconst'^2 \Gamma_0^m}{6},
\end{equation*}
then a forbidden configuration} %{$\dg$-configuration\ $\tau$ in a $\pmueps$-net\ has the $\hoopbnd$-hoop\ property
with
\begin{equation*}
\alpha_0 = \left(\frac{6}{\sparseconst'} \right)^3
\left( \Gamma_0 + \frac{\delta_{0}}{\Gamma_0^m} \right).
\end{equation*}
Furthermore,
the facets of $\tau$ are subject to a circumradius bound:
\begin{equation*}
\circrad{\opface{p}{\splxt}} < \left(1 + \frac{3 \delta_0}{\sparseconst'
\Gamma_0^m} \right) \samconst',
\end{equation*}
for all $p \in \tau$.
\end{lem}
The definition of forbidden configuration} %{$\dg$-configuration s is cumbersome, but the Hoop
\Lemref{lem:hoop} provides us with a symmetric property of forbidden configuration} %{$\dg$-configuration s
that is easy to exploit. In particular, when we perturb a point $p \mapsto p'$, then for any nearby simplex $\sigma$, we are able to check whether $\tau = \splxjoin{p'}{\sigma}$ is a forbidden configuration simply by examining the distance between $p'$, and the circumsphere for $\sigma$; we do not have to check this for all the vertices of $\tau$.
\subsubsection{The perturbation setting}
Although we have described forbidden configuration} %{$\dg$-configuration s and the Hoop Lemma in terms of
a $\pmueps$-net\ $\pts'$, rather than a $\mueps$-net\ $\mathsf{P}$, the notation is
simply a convenience for our current purposes. Until now we have not
supposed that $\pts'$ was a perturbation of a $\mueps$-net. We now review
the results in this setting.
If we constrain $\Gamma_0$ and constrain $\delta_0$ relative to
$\Gamma_0$, we observe that, for a forbidden configuration} %{$\dg$-configuration\ that appears in a
perturbed point set, the properties expressed in the Hoop
\Lemref{lem:hoop}
can be simplified and, by using \Lemref{lem:perturb.Delone}, they can
be expressed in terms of the parameters of the original $\mueps$-net:
\begin{lem}[Hoop Lemma for perturbed points]
\label{lem:clean.bad.hoop}
Suppose $\pts'$ is a perturbation of the $\mueps$-net\ $\mathsf{P}$, and $\tau
\subset \pts'$ is a forbidden configuration} %{$\dg$-configuration.
If
\begin{equation*}
\delta_0 \leq \Gamma_0^{m+1} \quad \text{and} \quad \Gamma_0
\leq \frac{2\sparseconst^2 }{75},
\end{equation*}
then $\tau$ has the $\hoopbnd$-hoop\ property, with
\begin{equation*}
\alpha_0 = 2 \left( \frac{16}{\sparseconst} \right)^3 \Gamma_0.
\end{equation*}
Also, for all $p \in \tau$,
\begin{equation*}
\circrad{\opface{p}{\splxt}} < 2\epsilon.
\end{equation*}
\end{lem}
For convenience, we restate the consequences of
\Lemref{lem:non.protection.implies.forbidden} in terms of the
algorithmically convenient property guaranteed by
\Lemref{lem:clean.bad.hoop}, together with a couple of other
properties that are a direct consequence of
\Defref{def:forbidden.config}. In particular, if $\tau$ is a
forbidden configuration} %{$\dg$-configuration, then it follows directly from \Defref{def:forbidden.config}
that
\begin{equation*}
\longedge{\tau} < (2 + \delta_0\sparseconst')\samconst'.
\end{equation*}
From this observation, and \Lemref{lem:perturb.Delone},
we obtain the diameter bound \ref{hyp:diam.bnd} below.
\begin{thm}[Properties of forbidden configurations]
\label{thm:no.hoops.implies.protection}
\label{thm:prop.forbid.cfg}
Suppose that $\mathsf{P} \subset \reel^m$ is a $\mueps$-net\ and that $\pts'$ is a
perturbation of $\mathsf{P}$ such that there is no simplex $\tau \subset
\pts'$ that satisfies \emph{all} of the following properties:
\begin{enumerate}[label={$\mathcal{P}$\arabic*}]
\item \label{hyp:clean.hoop.bnd}
Simplex $\tau$ has the $\hoopbnd$-hoop\ property, with
$\alpha_0 = 2 \left( \frac{16}{\sparseconst} \right)^3 \Gamma_0$.
\item \label{hyp:clean.facet.rad.bnd}
For all $p \in \tau$,
$\circrad{\opface{p}{\splxt}} < 2\epsilon$.
\item \label{hyp:diam.bnd}
$\longedge{\tau} < \frac{5}{2}(1 + \frac{1}{2}\delta_0\sparseconst)\epsilon$.
\item \label{hyp:good.facets}
Every facet of $\tau$ is $\Gamma_0$-good.
\end{enumerate}
If
\begin{equation}
\label{eq:dnobnd}
\delta_0 \leq \Gamma_0^{m+1} \quad \text{and} \quad \Gamma_0
\leq \frac{2\sparseconst^2 }{75},
\end{equation}
then $\pts'$ contains no forbidden configuration} %{$\dg$-configuration s, and thus all the $m$-simplices
in $\rdelsmhull{\ppts}$ are $\Gamma_0$-good and $\delta$-protected, with
$\delta = \delta_0 \sparseconst' \samconst'$.
\end{thm}
In order to eliminate forbidden configurations, we only need to ensure that any one of the four properties of \Thmref{thm:prop.forbid.cfg} cannot occur in any simplex. As discussed in \Remref{rem:bad.good.facet} below, the algorithm does not exploit \ref{hyp:good.facets}, and only partially exploits \ref{hyp:clean.facet.rad.bnd}.
\section{Introduction}
\label{sec:intro}
The main contribution of this paper is to provide a proof that, for a
quantifiable $\delta$, a $\delta$-generic point set may be obtained as
a perturbation of an existing point set. In Euclidean space $\reel^m$, a
discrete point set $\mathsf{P}$ is said to be \defn{$\delta$-generic} if
every Delaunay $m$-simplex has no other sample points within a
distance of $\delta$ from its circumsphere.
The Delaunay triangulation of such a point set is stable with respect
to small perturbations of either the points or of the metric
\cite{boissonnat2013stab1}. This makes $\delta$-generic sets important
in various contexts. The original motivation for this work is the
desire to establish a general framework for Delaunay triangulations on
Riemannian manifolds.
The stability issue with geometric structures also arises in the
context of robust computation, where a high precision may be demanded
to resolve near degenerate configurations. Halperin and
Shelton~\cite{halperin1998} developed a general technique of
controlled perturbation in this setting. Funke et al.~\cite{funke2005}
presented a controlled perturbation algorithm for computing planar
Delaunay triangulations, which may be extended to higher dimensions.
Their algorithm can also be seen as seeking to produce a
$\delta$-generic point set, and in this respect, although the
motivation and context are different, our algorithm also shares some
properties with theirs. However, in their approach all the points are
perturbed simultaneously with a probability of success that decreases
with the total size of the input point set. This makes the approach
unworkable for our desired application of triangulating general
manifolds.
By contrast, in the algorithm we present here each point is perturbed
in turn and is never subsequently visited after a successful
perturbation is found for that point. The probability of success is
independent of the total number of points or even the local
sampling density. We discuss the difference between our algorithm and
the approach of Funke et al.~\cite{funke2005} in more detail when we
conclude in \Secref{sec:conclusions}.
A well known issue with higher dimensional Delaunay triangulations is
the presence of poorly shaped (flat) ``sliver'' simplices. This
creates poorly conditioned systems in numerical applications, and
technical problems in geometric applications such as meshing
submanifolds. In fact, the issue is related to the above mentioned
problems with computing the Delaunay triangulation itself; the
existence of slivers is an indication that the point set is close to a
degenerate configuration \cite{boissonnat2013stab1}.
Existing work on removing slivers from high dimensional Euclidean
Delaunay triangulations has been based on two main techniques. The
first approach involves weighting the points to obtain a weighted
Delaunay triangulation with no slivers \cite{cheng2000}. This
technique was employed in the first work on reconstructing a
submanifold of arbitrary dimension in Euclidean space
\cite{cheng2005}, as well as in more recent work which avoids the
exponential cost of constructing a Delaunay triangulation of the
ambient space \cite{boissonnat2014tancplx.dcg}. The other approach is to
refine the point set \cite{li2003}. This technique was used for
constructing anisotropic triangulations based on locally defined
Riemannian metrics \cite{boissonnat2011aniso.tr}, and also for meshing
submanifolds in Euclidean space \cite{boissonnat2010meshing}.
The algorithm presented here provides a third approach, and it
guarantees a Delaunay triangulation that is stable in addition to
being sliver free. The perturbation approach enjoys the best aspects
of the other two methods. If the sample set is sufficiently dense,
there is no need to add more sample points. We also have the benefit
of using the standard metric, rather than squared distances where the
triangle inequality no longer applies. This latter aspect of the
weighting paradigm becomes awkward when considering perturbations of
the metric.
In spirit our algorithm is an extension of the algorithm presented by
Edelsbrunner et al. \cite{edelsbrunner2000smoothing} for creating a
sliver free Delaunay triangulation in $\reel^3$. We extend this work
in two ways: We extend it into higher dimensions, and we also extend
it to provide $\delta$-genericity. It is this latter aspect that
embodies our primary technical contribution. In our context the
concept of sliver, and the existing extensions to higher dimensions,
were inadequate; we need to eliminate simplices that do not belong to
a Delaunay triangulation, and have no upper bound on their
circumradius. The heart of the reason for this need to consider non-Delaunay simplices is that a violation of $\delta$-genericity is witnessed by a set $\tau$ of $m+2$ points, where $p \in \tau$ is within a distance $\delta$ of the circumsphere of the Delaunay simplex $\sigma = \tau \setminus \{p\}$. This simplex $\tau$ is not a Delaunay simplex in general, but either it, or one of its faces, represents a problem that we need to eliminate.
Our algorithm perturbs each point at most once. The correctness demonstration for this approach relies heavily on the Hoop Lemma~\ref{lem:hoop}, which says that the simplices that need to be eliminated have the property that every vertex lies close to the circumsphere of its opposing facet.
The algorithm itself is characterised by its simplicity. It is much
simpler than the refinement or weighting schemes. In essence, at each
iteration we perturb a point $p \mapsto p'$ in such a way as to ensure
that $p'$ does not lie too close to the circumsphere of any nearby
$m$-simplex in the current point set $\pts' \setminus \{p'\}$. It is
not immediately obvious that this should result in a $\delta$-generic
point set: if $p'$ is not ``too close'' to the circumsphere of an $m$-simplex $\sigma$ in the current point set we need to be ensured that the distance from $p'$ to the circumsphere of $\sigma$ remains greater than $\delta$ even after the vertices of $\sigma$ itself have been perturbed. The analysis reveals that we can get this ensurance, even though the algorithm never explicitly considers the circumspheres of simplices containing the point that is being perturbed.
\subsection{Proof of the Hoop Lemma}
\label{sec:hoop.lem.proof}
In this appendix we demonstrate the Hoop Lemma~\ref{lem:hoop}, which
can be stated in full detail as:
\begin{lem}[Hoop Lemma]
\label{lem:shell.lem}
Let $\tau$ be a $(k+1)$-dimensional forbidden configuration} %{$\dg$-configuration\ in a $\pmueps$-net. If
\begin{equation*}
\delta_0 \leq \frac{\sparseconst'^2 \Gamma_0^k}{6},
\end{equation*}
then for any $p \in \tau$
\begin{equation*}
\distEm{p}{\circsphere{\opface{p}{\splxt}}} \leq
\left( \frac{84}{\sparseconst'^3} \frac{\delta_0}{\Gamma_0^k}
+ \frac{216}{\sparseconst'^3} \Gamma_0 \right)
\circrad{\opface{p}{\splxt}},
\end{equation*}
and
\begin{equation*}
\circrad{\opface{p}{\splxt}} < \left(1 + \frac{3\delta_0}{\sparseconst'
\Gamma_0^k} \right) \samconst'.
\end{equation*}
\end{lem}
Recall that \Lemref{lem:close.to.some.sphere} demonstrated that any
vertex in a forbidden configuration} %{$\dg$-configuration\ lies close to a circumscribing sphere for its
opposing face. We now use the fact that a forbidden configuration} %{$\dg$-configuration\ is a flake to
bound the distance from a vertex to the circumsphere of its opposing
face. We employ the following characterisation of the altitudes of a
triangle:
\begin{lem}[Triangle altitude bound]
\label{lem:tri.alt.bnd}
For any non-degenerate triangle $\zeta = \simplex{\tilde{p},u,v}$, we have
\begin{equation*}
\splxalt{\tilde{p}}{\zeta} = \frac{\norm{\tilde{p}-v}\norm{\tilde{p}-u}}{2\circrad{\zeta}}.
\end{equation*}
\end{lem}
\begin{proof}
Let $\alpha = \angle \tilde{p} uv$ and observe that
\begin{equation*}
\sin \alpha = \frac{\norm{\tilde{p}-v}}{2\circrad{\zeta}}.
\end{equation*}
Since $\splxalt{\tilde{p}}{\zeta} = \norm{\tilde{p}-u} \sin \alpha$, the result
follows.
\end{proof}
\begin{lem}[Distance to circumsphere]
\label{lem:dist.to.circumsphere}
Suppose $\tau$ is a $\Gamma_0$-flake with
$\longedge{\tau} \leq 3\samconst'$ and $\shortedge{\tau} \geq
\sparseconst' \samconst'$. If there exists a $p \in \tau$ and a
ball $B = \ballEm{C}{R}$ circumscribing $\opface{p}{\splxt}$, with $R <
\frac{3}{2}\samconst'$, and such that $\distEm{p}{\bdry{B}} \leq
\tilde{\delta}_0 \shortedge{\opface{p}{\splxt}}$ for some $\tilde{\delta}_0 \geq 0$, then
$\distEm{p}{\circsphere{\opface{p}{\splxt}}} \leq \alpha_0 \circrad{\opface{p}{\splxt}}$, with
\begin{equation*}
\alpha_0 = \frac{14}{\sparseconst'} \tilde{\delta}_0 +
\frac{216}{\sparseconst'^3} \Gamma_0.
\end{equation*}
\end{lem}
\begin{figure
\begin{center}
\includegraphics[width=.6\columnwidth]{imgs/bound_to_circsphere}
\end{center}
\caption{Diagram for \Lemref{lem:dist.to.circumsphere}. }
\label{fig:bound.to.Staup}
\end{figure}
\begin{proof}
We are given that $p$ lies close to a circumscribing sphere $\bdry{B}$ for $\opface{p}{\splxt}$. The fact that $\tau$ is a flake implies that $p$ must also lie close to the affine hull of $\opface{p}{\splxt}$. The result follows since $\circsphere{\opface{p}{\splxt}} = \bdry{B} \cap \affhull{\opface{p}{\splxt}}$. We quantify this by reducing the problem to two dimensions.
Consider the plane $Q$ defined by $p$, $C$, and
$\circcentre{\opface{p}{\splxt}}$; if two of these three points coincide, we
may choose $Q$ to be any plane which contains the three points. If
$p=C$, then we have $\distEm{p}{\circsphere{\opface{p}{\splxt}}} = R = \distEm{p}{\bdry{B}}
\leq \tilde{\delta}_0 \shortedge{\opface{p}{\splxt}} \leq \tilde{\delta}_0 2 \circrad{\opface{p}{\splxt}}$ which
immediately implies the result. Thus suppose $p \neq C$. Let $\tilde{p}$ be
the point of intersection of the ray from $C$ through $p$ with
$\bdry{B}$, let $u \in \circsphere{\opface{p}{\splxt}} \cap Q$ be the point closest to $\tilde{p}$,
and let $v \in \circsphere{\opface{p}{\splxt}} \cap Q$ be the farther point, as shown in
\Figref{fig:bound.to.Staup}. Then
\begin{equation}
\label{eq:bound.to.Staup}
\distEm{p}{u} \leq \distEm{p}{\tilde{p}} + \distEm{\tilde{p}}{u}.
\end{equation}
If $\tilde{p} = u \in \circsphere{\opface{p}{\splxt}}$, then the result follows immediately, so we
suppose these points to be distinct, and we consider the triangle
$\zeta = \simplex{\tilde{p},u,v}$. Since $\circrad{\zeta} = R$,
\Lemref{lem:tri.alt.bnd} yields
\begin{equation*}
\distEm{\tilde{p}}{u} = \frac{2 R
\splxalt{\tilde{p}}{\zeta}}{\distEm{\tilde{p}}{v}}.
\end{equation*}
Using our definition of $u$ we find
\begin{equation*}
\distEm{\tilde{p}}{v} \geq \frac{1}{2}\distEm{u}{v} =
\circrad{\opface{p}{\splxt}} \geq \frac{1}{2} \shortedge{\opface{p}{\splxt}}.
\end{equation*}
The altitude is bounded by
\begin{equation*}
\begin{split}
\splxalt{\tilde{p}}{\zeta}
&\leq \distEm{\tilde{p}}{p} +
\distEm{p}{\affhull{\seg{u}{v}}}\\
&= \distEm{\tilde{p}}{p} + \splxalt{p}{\tau}.
\end{split}
\end{equation*}
Indeed, if $p^*$ is the orthogonal projection of $p$ into
$\affhull{\opface{p}{\splxt}}$, then $\seg{p}{p^*}$ is parallel to
$\seg{C}{\circcentre{\opface{p}{\splxt}}}$, because $\affhull{\opface{p}{\splxt}}$ has
codimension one in $\affhull{\tau}$. It follows that $p^* \in
Q \cap \affhull{\opface{p}{\splxt}} = \affhull{\seg{u}{v}}$.
By \Lemref{lem:flake.alt.bnd}
and the fact that $\longedge{\tau} < 3\samconst'$, we have
\begin{equation*}
\splxalt{p}{\tau} \leq \frac{2\longedge{\tau}^2
\Gamma_0} { \shortedge{\tau}}
\leq \frac{6 \longedge{\tau}
\Gamma_0} { \sparseconst' }
\leq \frac{18 \Gamma_0 \shortedge{\opface{p}{\splxt}}} { \sparseconst'^2 }
\end{equation*}
Finally, recalling that $\distEm{p}{\tilde{p}} \leq \tilde{\delta}_0
\shortedge{\opface{p}{\splxt}}$, and $R < \frac{3}{2}\samconst'$,
we return to \Eqnref{eq:bound.to.Staup} and expand it using all of the
subsequent displayed observations:
\begin{equation}
\label{eq:final.bound.to.Sw}
\begin{split}
\distEm{p}{u}
&\leq \tilde{\delta}_0 \shortedge{\opface{p}{\splxt}}
+ \frac{2 R \splxalt{\tilde{p}}{\zeta}}{\distEm{\tilde{p}}{v}}\\
&\leq \tilde{\delta}_0 \shortedge{\opface{p}{\splxt}} + \frac{4R}{\shortedge{\opface{p}{\splxt}}}
\left(\tilde{\delta}_0 \shortedge{\opface{p}{\splxt}}
+ \frac{18 \Gamma_0}{\sparseconst'^2}
\shortedge{\opface{p}{\splxt}} \right)\\
&< \tilde{\delta}_0 2\circrad{\opface{p}{\splxt}} + \frac{12}{\sparseconst'}
\left(\tilde{\delta}_0 + \frac{18 \Gamma_0}{\sparseconst'^2} \right)
\circrad{\opface{p}{\splxt}}\\
&\leq \left( \frac{14}{\sparseconst'} \tilde{\delta}_0
+ \frac{216}{\sparseconst'^3} \Gamma_0 \right)
\circrad{\opface{p}{\splxt}}.
\end{split}
\end{equation}
\end{proof}
\noindent
\textbf{Proof of \Lemref{lem:shell.lem}.}
Using \Lemref{lem:close.to.some.sphere}, we apply
\Lemref{lem:dist.to.circumsphere} with
\begin{equation*}
\tilde{\delta}_0 = \frac{6\delta_0}{\sparseconst'^2 \Gamma_0^k}.
\end{equation*}
\hspace*{\fill}~$\square$
\section{Precision requirements}
\label{sec:precision}
We can estimate the order of magnitude of the precision requirements
associated with the algorithm by considering error tolerances without
specifically analysing the computation of the geometric predicate.
\Algref{alg1} needs to evaluate the geometric predicate
\begin{equation}
\label{eq:the.predicate}
\abs{\distEm{x}{\circcentre{\sigma}} - \circrad{\sigma}} - \alpha_0
2\epsilon \leq 0
\end{equation}
that appears in \Algref{alg:good.perturbation}. This is essentially an
``insphere'' predicate, but the sphere $\diasphere{\sigma}$ may be
defined by a simplex $\sigma$ whose dimension is less than $m$. If we
are to guarantee that the algorithm will terminate with given desired
effective values of $\delta_0$ and $\Gamma_0$, there is an upper
bound on the error that can be tolerated in the evaluation of
Predicate~\eqref{eq:the.predicate}. The magnitude of this error gives
us an estimate of the precision required to run the algorithm with
these parameters; the actual value of the required precision depends
on the details of how Predicate~\eqref{eq:the.predicate} is evaluated.
For simplicity we set $\sparseconst = 1$ and $\epsilon = 1$. Let
$e_{\alpha}$ be the error incurred in the evaluation of
Predicate~\eqref{eq:the.predicate}. This error can be bounded because
Properties \ref{hyp:good.facets} and
\ref{hyp:clean.facet.rad.bnd} of \Thmref{thm:prop.forbid.cfg} ensure
that Predicate~\eqref{eq:the.predicate} does not need to be evaluated
on arbitrarily thin simplices.
We consider $\alpha_0$ to be a parameter that we give to the
algorithm. Then in order to exploit our termination guarantees, we
need to consider the largest possible hoop parameter that could be
implicit in a computation. In other words, the hoop parameter
determines the volume of the bad perturbations within the perturbation
ball, see \Figref{fig:forbidden.volume}, and if the error in the
evaluation of Predicate~\eqref{eq:the.predicate} is too large we lose
the guarantee that a good perturbation will be found. We say the error
induces an implicit hoop parameter which we express as $\alpha_H$,
where
\begin{equation*}
2\alpha_H = 2\alpha_0 + e_{\alpha}.
\end{equation*}
Using Property~\ref{hyp:clean.hoop.bnd} of
\Thmref{thm:prop.forbid.cfg}, we write $\alpha_H = 2^{13}
\Gamma_H$, where $\Gamma_H$ is \emph{defined} by this equation,
and represents the largest flake parameter that could be implicit in a
calculation. For the goals of the current argument, we may assume that
$m \leq 5$, so that $\ballvol{m-1}/\ballvol{m} < 1$, and we can ignore
this factor in the definition of $K$ that appears in
\Eqnref{eq:raw.K}. Then, letting $\rho_0} %{\eta_0 = 2^{-2}$,
\Eqnref{eq:raw.gamma.z.bnd} demands
\begin{equation*}
\alpha_0 + \frac{e_{\alpha}}{2} = 2^{13} \Gamma_H < 2^{11} K^{-1} =
2^{-(3m^2 + 4m + 5)} \stackrel{\text{def}}{=} Z.
\end{equation*}
We observe that in order to guarantee termination of the algorithm we
require $e_{\alpha} < 2Z$. Similarly, we must choose $\alpha_0$ to be
less than $Z$. It is convenient to write $\alpha_0 = 2^{-j}Z$ and
$e_{\alpha} = 2^{-(k-1)}Z$ for positive integers $j$ and $k$. In order
to estimate the effective $\delta_0$ and $\Gamma_0$ that we can hope
to attain, we choose $j=1$. We then observe, again from
Property~\ref{hyp:clean.hoop.bnd}, that the flake parameter that we
can actually guarantee is given by $\Gamma_0 = 2^{-13}\alpha_L$,
where
\begin{equation*}
\alpha_L = \alpha_0 - \frac{e_{\alpha}}{2} = (2^{-1} - 2^{-k})Z.
\end{equation*}
Assuming $k \geq 2$, we have $\Gamma_0 \geq 2^{-15}Z$.
We can now give estimates for the largest magnitudes of the parameters
$\delta_0 = \Gamma_0^{m+1}$ that we could hope to attain. If $m=2$,
then $Z = 2^{-25}$ and we need $e_{\alpha} \leq 2^{-26}$ in order to
obtain $\Gamma_0 = 2^{-40}$ and $\delta_0 = 2^{-120}$. If $m=3$, then
$e_{\alpha} \leq 2^{-45}$ yields $\Gamma_0 = 2^{-59}$ and $\delta_0 =
2^{-236}$. These small values for $\delta_0$ do not indicate a
practical relevance for the perturbation algorithm.
However, the algorithm is formulated for arbitrary dimensions, and in
dimension 2 in particular, it does not make sense to consider flakes
and a parameter $\Gamma_0$. Indeed, in two dimensions the altitudes
of the facets of our $\hoopbnd$-hoop s will be already bounded by the sampling
parameters $\sparseconst$ and $\epsilon$.
Given a $\pmueps$-net\ $\pts'$, that is $\delta$-generic, the restricted
Delaunay triangulation $\rdelsmhull{\ppts}$ around points sufficently far from
the boundary of $\convhull{\pts'}$ can in principle be computed using
floating point arithmetic. We estimate the order of magnitude of the
precision required to evaluate the \texttt{insphere}$(x,\sigma^m)$
predicate, assuming that $\pts'$ is the output of \Algref{alg1} with
guaranteed $\delta_0$ and $\Gamma_0$ as discussed above. We can write
the insphere predicate as~\cite{funke2005}
\begin{equation}
\label{eq:insphere.orient}
\text{\texttt{insphere}}(x,\sigma^m) =
\text{\texttt{orient}}(\sigma^m)(\distEm{x}{\circcentre{\sigma^m}}^2 -
\circrad{\sigma^m}^2),
\end{equation}
where \texttt{insphere} and \texttt{orient} represent determinants
\cite{funke2005}. The \texttt{insphere} predicate is what needs to be
evaluated to compute the Delaunay triangulation; we will use the right
hand side of inequality~\eqref{eq:insphere.orient} to estimate the
magnitude of the error that can be tolerated while still guaranteeing
that \texttt{insphere} produces the correct sign.
Again assuming $\sparseconst = 1$ and $\epsilon = 1$, we have
$\sparseconst' \samconst' = \frac{1}{2}$ and putting $R =
\distEm{x}{\circcentre{\sigma^m}}$ we get
\begin{equation*}
\abs{R^2 - \circrad{\sigma^m}^2} \geq \circrad{\sigma^m} \abs{R -
\circrad{\sigma^m}} \geq \frac{\sparseconst' \samconst'}{2}
\delta_0\sparseconst' \samconst' = 2^{-3} \delta_0.
\end{equation*}
The \texttt{orient} predicate represents $m!$ times the signed volume
of $\sigma^m$. The absolute value of this can be expressed as a
telescopic product of altitudes of faces of progressively lower
dimension. If $\sigma^m$ is not $\Gamma_0$-good, then it cannot be a
Delaunay simplex. This could be tested by evaluating the magnitude of
\texttt{orient} for example. We assume then that $\sigma^m$ is
$\Gamma_0$-good. Then the altitude of a $j$-dimensional face
$\sigma^j \leq \sigma^m$ is bounded below by $j
\thickness{\sigma^j}\longedge{\sigma} \geq \frac{1}{2} j \Gamma_0^j
$. It follows then that
\begin{equation*}
\abs{ \text{\texttt{orient}}(\sigma^m) } \geq 2^{-m}
\Gamma_0^{\frac{m(m+1)}{2}}.
\end{equation*}
Let $e_S$ be the error incurred in the evaluation of
\texttt{insphere}. The preceding observations, together with
\Eqnref{eq:insphere.orient}, imply that \texttt{insphere} will
evaluate to the correct sign provided that
\begin{equation*}
e_S < 2^{-(m+3)} \Gamma_0^{\frac{m(m+1)}{2}} \delta_0
\leq 2^{-(m+3)} \Gamma_0^{\frac{1}{2}m^2 + \frac{3}{2}m + 1}.
\end{equation*}
Using the bound $\Gamma_0 = 2^{-15}Z$ found above, we find that we
require $e_S < 2^{\bigo{m^4}}$.
| {
"timestamp": "2015-05-08T02:01:40",
"yymm": "1310",
"arxiv_id": "1310.7696",
"language": "en",
"url": "https://arxiv.org/abs/1310.7696",
"abstract": "We present an algorithm that takes as input a finite point set in Euclidean space, and performs a perturbation that guarantees that the Delaunay triangulation of the resulting perturbed point set has quantifiable stability with respect to the metric and the point positions. There is also a guarantee on the quality of the simplices: they cannot be too flat. The algorithm provides an alternative tool to the weighting or refinement methods to remove poorly shaped simplices in Delaunay triangulations of arbitrary dimension, but in addition it provides a guarantee of stability for the resulting triangulation.",
"subjects": "Computational Geometry (cs.CG)",
"title": "Delaunay stability via perturbations",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9770226341042415,
"lm_q2_score": 0.7248702761768249,
"lm_q1q2_score": 0.7082146666141504
} |
https://arxiv.org/abs/0705.0762 | The bounded isometry conjecture for the Kodaira-Thurston manifold and 4-Torus | The purpose of this note is to study the bounded isometry conjecture proposed by Lalonde and Polterovich. In particular, we show that the conjecture holds for the Kodaira-Thurston manifold with the standard symplectic form and for the 4-torus with all linear symplectic forms. | \section{Introduction and main results}\label{introduction}
\smallskip
Let $(M, \omega)$ be a closed symplectic manifold. There is a
natural bi-invariant norm, called the Hofer norm $\rho$, defined on
the Hamiltonian diffeomorphism group ${\rm Ham}(M,\omega)$. That is,
$\rho(f)$ is the Hofer distance between the identity map $id$ and
$f$ for all $f \in {\rm Ham}(M,\omega)$, see Section \ref{hofernorm}
for details. Lalonde and Polterovich \cite{LP} have studied the full
symplectomorphism group ${\rm Symp}(M,\omega)$ within the framework
of Hofer's geometry. We first recall the notion of bounded and
unbounded symplectomorphisms. Namely, for each $\phi \in {\rm
Symp}(M,\omega)$, define
$$r(\phi):={\rm sup}\, \{\,\rho([\phi,f]) \mid f \in {\rm Ham}(M,\omega)\},$$
where $[\phi,f]:=\phi f \phi^{-1} f^{-1}$ is the commutator of
$\phi$ and $f$.
\smallskip
\begin{definition}\label{bounded}
An element $\phi \in {\rm Symp}(M,\omega)$ is bounded if
$r(\phi)<\infty$, and is unbounded if $r(\phi)= \infty$.
\end{definition}
Denote by $BI_0(M,\omega)$ the set of all bounded elements in the
identity component ${\rm Symp}_0(M,\omega)$ of ${\rm
Symp}(M,\omega)$. Since $\rho$ is bi-invariant, it follows from the
inequality $\rho([\phi,f]) \leqslant 2 \rho(\phi)$ that ${\rm
Ham}(M,\omega)$ is a subgroup of $BI_0(M,\omega)$. The converse is
the following conjecture in \cite{LP}.
\smallskip
\begin{guess}[Bounded isometry conjecture]\label{bic} For all symplectic manifolds $(M,\omega)$,
$BI_0(M,\omega)={\rm Ham}(M,\omega)$.
\end{guess}
This conjecture was proved in \cite{LP} for closed surfaces with
area form and for arbitrary products of closed surfaces of genus
greater than 0 with product symplectic form; Lalonde and Pestieau
\cite{LPe} confirmed it for product symplectic manifolds $M = N
\times W$ with $N$ being any product of closed surfaces and $W$
being any closed symplectic manifold of first real Betti number
equal to zero. In this note, we give a positive answer to this
conjecture for the Kodaira-Thurston manifold with the standard
symplectic form and for the 4-torus with all linear symplectic
forms.
\smallskip
\begin{thm}\label{bicforkt} The bounded isometry conjecture holds
for the Kodaira-Thurston manifold $M$ with the standard symplectic
form $\omega$.
\end{thm}
\begin{thm}\label{bicfortorus} The bounded isometry conjecture holds for the $4$-torus $(\mathbb
T^4, \omega)$ with any linear symplectic form $\omega:=\sum_{i <
j}\, a_{ij}\, dx_i \wedge dx_j$.
\end{thm}
{\noindent} \textbf{Organization of the paper:} We begin with some
preparations in Section \ref{fluxsubgroup}-\ref{admissiblelift}.
Then we prove Theorem \ref{bicforkt} in Section \ref{proofbicforkt}
and Theorem \ref{bicfortorus} in Section \ref{proofbicfortorus}. We
study the same conjecture for the Kodaira-Thurston manifold with all
linear symplectic forms in Section \ref{sec8}. While we are unable
to prove the conjecture in this case, some partial results are
provided and the difficulties are discussed.
\bigskip
{\noindent} \textbf{Acknowledgements:} This work is part of the author's
Ph.D. thesis, being carried out under the supervision of Dusa McDuff
at Stony Brook University. The author wishes to thank her for her
great guidance and continual support. He also thanks Xiaojun Chen,
Basak Gurel, Leonid Polterovich, Guanyu Shi, Yujen Shu, Dennis
Sullivan and Aleksey Zinger for useful comments and discussions.
\bigskip
\section{The flux subgroup}\label{fluxsubgroup}
\smallskip
The flux homomorphism is best defined on the universal cover
$\widetilde{\rm Symp}_0(M,\omega)$ of ${\rm Symp}_0(M,\omega)$,
$${\rm flux} : \widetilde{\rm Symp}_0(M,\omega) \to H^1(M, \mathbb R).$$
Let $\{\phi_t\} \in \widetilde{\rm Symp}_0(M,\omega)$, i.e. $\phi_t$
is a smooth isotopy in ${\rm Symp}_0(M,\omega)$. There exists a
unique family of vector fields $X_t$ which generates the flow
$\phi_t$, i.e.
$$\frac{d}{dt} \phi_t=X_t \circ \phi_t.$$
Define
$${\rm flux}(\{\phi_t\}):=\int_0^1 \iota(X_t)\omega \,dt.$$
In particular, if $\{\phi_t\}$ is the flow of the time-independent
symplecitc vector field $X$ on the time interval $0 \leqslant t
\leqslant 1$, then
\begin{equation} \label{eq:1}
\begin{split}
{\rm flux}(\{\phi_t\})=\iota(X)\omega.
\end{split}
\end{equation}
This fact will often be used in later calculations.
The flux subgroup $\Gamma:=\Gamma_\omega$ is the image
$${\rm flux}(\pi_1({\rm Symp}_0(M,\omega)) \subset H^1(M, \mathbb R)$$ of the
fundamental group of ${\rm Symp}_0(M,\omega)$ under the flux
homomorphism. Thus there is an induced map from ${\rm
Symp}_0(M,\omega)$, still denoted by ${\rm flux}$,
$${\rm flux} : {\rm Symp}_0(M,\omega) \to H^1(M, \mathbb
R)/\Gamma.$$ It is well known that this map is surjective, and its
kernel is equal to ${\rm Ham}(M,\omega)$. In other words, we have
the following exact sequence of groups
$$\begin{CD} 0 @>>> {\rm Ham}(M,\omega) @>>> {\rm Symp}_0(M,\omega)
@>{\rm flux}>>H^1(M, \mathbb R)/\Gamma @>>> 0. \end{CD}$$ We refer
to \cite{MS1} Chapter 10 for more details.
\smallskip
Since whether or not the flux is equal to 0 distinguishes a
Hamiltonian diffeomorphism from a nonHamiltonian symplectomorphism,
one main step in our applications is to understand the flux subgroup
$\Gamma$.
For this, we denote as in \cite{Ke} by $C(M)$ the space of
continuous maps from $M$ to $M$ with the compact open topology.
Given $p \in M$, we define the evaluation map $ev_c : C(M) \to M$ by
$ev_c(f)=f(p)$. Denote by $ev_s$ the restriction of $ev_c$ to ${\rm
Symp}_0(M,\omega)$. We will use the same notation for the induced
maps on the fundamental groups. By $\widetilde{ev}_s$ we denote the
homomorphism from $\pi_1({\rm Symp}_0(M,\omega))$ to $H_1(M, \mathbb
Z)$, which is the composition of $ev_s$ with the Hurewitz map from
$\pi_1(M)$ to $H_1(M,\mathbb Z)$.
The following commutative diagram due to Lalonde, McDuff and
Polterovich \cite{LMP2} plays a crucial role in the calculation of
the flux subgroup $\Gamma$.
\begin{lemma} [LMP] \label{commute}
Let $(M, \omega)$ be a closed symplectic manifold of dimension $2n$.
Then the following diagram commutes.
$$\begin{CD}
\pi_1({\rm Symp}_0(M,\omega)) @>\widetilde{ev}_s>>H_1(M, \mathbb Z) @>{\rm PD}>>H^{2n-1}(M, \mathbb Z) \\
@VidVV && @VV\centerdot(n-1)!{\rm vol}(M)V\\
\pi_1({\rm Symp}_0(M,\omega)) @>{\rm flux}>>H^1(M, \mathbb R) @>\wedge[\omega]^{n-1}>> H^{2n-1}(M, \mathbb R). \end{CD}$$
\end{lemma}\hfill$\Box$\medskip
\bigskip
\section{The Kodaira-Thurston manifold}\label{ktmanifold}
\smallskip
Let $G$ be the group $(\mathbb{Z}^4, \cdot)$ where $$(m_1, n_1, k_1,
\ell_1) \cdot (m_2, n_2, k_2, \ell_2)=(m_1+m_2, n_1+n_2, k_1+k_2+m_1
\ell_2, \ell_1+\ell_2).$$ $G$ acts on $\mathbb R^4$ via
$$G \to \mbox{Diff}(\mathbb R^4) : (m, n, k, \ell) \mapsto \rho_{m n k
\ell}$$ where
$$\rho_{m n k \ell}(s, t, x, y)=(s+m, t+n, x+k+m y, y+\ell).$$
Note that $\rho_{m n k \ell}$ preserves the symplectic form
$\omega=ds \wedge dt+dx\wedge dy$ on $\mathbb R^4$. Hence the
quotient $(M:=\mathbb R^4/G, \omega)$ is a closed symplectic
manifold, known as the Kodaira-Thurston manifold, see \cite{Th}. It
was the first known example of a closed symplectic manifold which
admits no k\"{a}hler structure, since its first betti number
$b_1=3$, see \cite{MS1} Example 3.8.
The manifold $M=\mathbb R^4/G$ can also be described as a torus
bundle over a torus, that is $M=\mathbb R^2 \times_{\mathbb{Z}^2}
\mathbb T^2$. Here $\mathbb{Z}^2$ acts on $\mathbb R^2$ in the usual
way, and it acts on $\mathbb T^2$ via
$$(m, n) \to A_{m n} : \left(
\begin{matrix}x\\y \end{matrix} \right)
\mapsto \left( \begin{matrix}1&m\\0&1
\end{matrix} \right) \left( \begin{matrix}x\\y
\end{matrix} \right).$$ Therefore $M=\mathbb R \times S^1 \times
\mathbb T^2/\sim$, where
$$(s, t, x, y) \sim (s+1, t, x+y, y).$$
\smallskip
Our first task is to understand the flux subgroup $\Gamma$ of the
Kodaira-Thurston manifold described above. In particular, we have
\begin{thm}\label{flux} The flux subgroup $\Gamma \subset H^1(M,\mathbb R)$ of
the Kodaira-Thurston manifold with the standard symplectic form
$\omega=ds \wedge dt+dx\wedge dy$ has rank $2$ over $\mathbb{Z}$.
Namely, $\Gamma=\mathbb{Z}\langle ds, dy\rangle $.
\end{thm}
To prove Theorem \ref{flux}, we need the following result on the
cohomology groups of the Kodaira-Thurston manifold.
\begin{lemma}\label{cohomology}
The cohomology groups of the Kodaira-Thurston manifold $M$ described
above are as follows: $H^1(M, \mathbb R)$ is of rank $3$, generated
by $ds, \, dt$ and $dy$; $H^2(M, \mathbb R)$ is of rank $4$,
generated by $\gamma \wedge ds,\, \gamma \wedge dy,\, ds \wedge dt$
and $dy \wedge dt$; and $H^3(M, \mathbb R)$ is of rank $3$,
generated by $\gamma \wedge dy \wedge dt,\, \gamma \wedge dy \wedge
ds$ and $\gamma \wedge ds \wedge dt$, where $\gamma=dx-s dy$.
\end{lemma}
\begin{proof}
This follows from an easy calculation.
\end{proof}
\smallskip
{\noindent} {\textbf{Proof of Theorem \ref{flux}.}} We use the commutative
diagram in Lemma \ref{commute}. For manifolds of dimension 4, the
diagram reads as
$$\begin{CD}
\pi_1({\rm Symp}_0(M,\omega)) @>\widetilde{ev}_s>>H_1(M, \mathbb Z) @>{\rm PD}>>H^3(M, \mathbb Z) \\
@VidVV && @VV\centerdot {\rm vol}(M)V\\
\pi_1({\rm Symp}_0(M,\omega)) @>{\rm flux}>>H^1(M, \mathbb R) @>\wedge[\omega]>> H^3(M, \mathbb R).
\end{CD}$$
\smallskip
Denote by $C_0(M)$ the identity component of $C(M)$. It was proved
in Gottlieb \cite{Go} (Theorem III.2) that for all aspherical
manifolds $M$,
$$ev_c : \pi_1(C_0(M))\cong Z(\pi_1(M))$$ is a group isomorphism,
where $Z(\pi_1(M))$ stands for the center of $\pi_1(M)$. For the
Kodaira-Thurston manifold $M=\mathbb R^4/G$, we have $\pi_1(M)=G$.
It is easy to check that
$Z(\pi_1(M))=\mathbb{Z}\langle\frac{\partial}{\partial t},
\frac{\partial}{\partial x}\rangle$, and the commutator group $[\,
\pi_1(M), \pi_1(M)]=\mathbb{Z}\langle\frac{\partial}{\partial
x}\rangle $, see Example 3.8 in \cite{MS1}. Thus the image of
$\widetilde{ev}_s$ in $H_1(M, \mathbb Z)$ is contained in
$$Z(\pi_1(M))/ [\, \pi_1(M),
\pi_1(M)]=\mathbb{Z}\langle\frac{\partial}{\partial t},
\frac{\partial}{\partial x}\rangle /
\mathbb{Z}\langle\frac{\partial}{\partial x}\rangle
=\mathbb{Z}\langle\frac{\partial}{\partial t}\rangle.$$
Note that $PD(\frac{\partial}{\partial t})=-dx \wedge dy \wedge
ds=-\gamma \wedge dy \wedge ds$, where $\gamma=dx-sdy$. Now look at
the map $\wedge \omega : H^1(M, \mathbb R) \to H^3(M, \mathbb R)$,
$$ds \mapsto ds \wedge \omega=\gamma \wedge dy \wedge ds \ne 0,$$
$$dt \mapsto dt \wedge \omega=\gamma \wedge dy \wedge dt \ne 0,$$
$$dy \mapsto dy \wedge \omega=dy \wedge ds \wedge dt=0.$$
{\noindent} Here we have used the fact that the 3-form $dy \wedge ds \wedge
dt = d (\gamma \wedge dt)$ is exact, so it vanishes on the
cohomology level. Since ${\rm vol}(M)=1$, we conclude from the above
commutative diagram that the flux subgroup $\Gamma \subset H^1(M,
\mathbb R)$ is contained in $\mathbb{Z}\langle ds, dy\rangle $. An
explicit construction shows that $\Gamma$ is actually equal to
$\mathbb{Z}\langle ds, dy\rangle $. Namely, we take two elements
$\{\phi_\theta\}$ and $\{\psi_\theta\}$ in $\pi_1({\rm
Symp}_0(M,\omega))$ such that
$$\phi_\theta(s, t, x, y)=(s, t-\theta, x, y), 0 \leqslant\theta \leqslant1, $$
$$\psi_\theta(s, t, x, y)=(s, t, x+\theta, y), 0 \leqslant\theta \leqslant1.$$
Using (\ref{eq:1}) in Section \ref{fluxsubgroup}, one can show that
${\rm flux}(\{\phi_\theta \})=ds$ and ${\rm flux}(\{\psi_\theta
\})=dy$. This completes the proof of Theorem \ref{flux}.\hfill$\Box$\medskip
\bigskip
\section{The Hofer norm}\label{hofernorm}
\smallskip
Let $(M,\omega)$ be a closed symplectic manifold of dimension $2n$.
Denote by $\mathcal{A}$ the space of all normalized smooth functions
on $M$ with respect to the volume form $\omega^n$, i.e.
$$\mathcal{A} := \{ F \in C^\infty (M) \mid \int_M F \,\omega^n
=0\}.$$ It is well known that $\mathcal{A}$ can be identified with
the space of Hamiltonian vector fields, which is the Lie
algebra\footnote{As a vector space, the Lie algebra is by definition
the tangent space to the Lie group at the identity. The tangent
spaces to the Lie group at other points are identified with the Lie
algebra with the help of right shifts of the group.} of the
$\infty$-dimensional Lie group ${\rm Ham}(M,\omega)$.
The $L_\infty$ norm on $\mathcal{A}$
$$||F||_\infty={\rm max}\,F-{\rm min}\,F$$
gives rise to the Hofer metric $d$ on ${\rm Ham}(M,\omega)$ in the
following way: we define the Hofer length of a smooth Hamiltonian
path $\alpha : [0,1] \to {\rm Ham}(M,\omega)$ as
$${\rm length}(\alpha):=\int_0^1 ||\dot{\alpha}(t)||_\infty dt=\int_0^1 ||F_t||_\infty dt,$$
where $F_t(x)=F(t,x)$ is the time-dependent Hamiltonian function
generating the path $\alpha$. The Hofer distance $d$ between two
Hamiltonian diffeomorphisms $f$ and $g$ is defined by
$$d(f,g):={\rm inf} \,\{\,{\rm length}(\alpha)\},$$
where the infimum is taken over all Hamiltonian paths $\alpha$
connecting $f$ and $g$. The Hofer norm $\rho(f)$ is the Hofer
distance between the identity map $id$ and $f$, i.e.
$$\rho(f):=d(id,f).$$
It is easy to check that $d$ is bi-invariant in the sense that
$$d(fh, gh)=d(hf, hg)=d(f, g)$$
for all $f,g,h \in {\rm Ham}(M,\omega)$. The fact that $d$ is
nondegenerate is highly nontrivial. This was proved by Hofer
\cite{Ho} for the case of $\mathbb R^{2n}$, then generalized by
Polterovich \cite{Po} to some larger class of symplectic manifolds,
and finally proved in the full generality by Lalonde and McDuff
\cite{LM} using the following energy-capacity inequality
$$e(S) \geqslant \frac{1}{2} {\rm capacity}(S)$$ for a subset $S$ of
$M$. Here the capacity of $S$ is equal to $\pi r^2$ when $S$ is a
symplectically embedded ball of radius $r$, and is defined in
general as the supremum of the capacities of all symplectically
embedded balls in $S$. The displacement energy $e(S)$ is defined to
be the infimum of the Hofer norms of all $f \in {\rm Ham}(M,\omega)$
such that $f(S) \cap S=\emptyset$.
Note that the energy-capacity inequality provides a lower bound for
the Hofer norm. Namely, we have
$$f(S) \cap S=\emptyset, \,\,{\rm capacity}(S)>c \Longrightarrow \rho(f) >c/2.$$
This fact will be crucial in our proof of Theorem \ref{bicforkt}.
Recall in Definition \ref{bounded} that an element $\phi \in {\rm
Symp}(M,\omega)$ is called unbounded if
$$r(\phi):={\rm sup}\,
\{\,\rho([\phi,f]) \mid f \in {\rm Ham}(M,\omega)\} = \infty.$$ Note
that all Hamiltonian diffeomorphisms are bounded since $r(g)
\leqslant 2 \rho(g)< \infty$ for all $g \in {\rm Ham}(M,\omega)$,
where $\rho(g)$ is the Hofer norm of $g$. According to Proposition
1.2.A in \cite{LP}, $r$ satisfies the triangle inequality $r(\phi
\psi) \leqslant r(\phi)+r(\psi)$. Since ${\rm Ham}(M,\omega)$ is the
kernel of the flux homomorphism, two symplectomorphisms $\phi$ and
$\psi$ have the same flux if and only if they differ by a
Hamiltonian diffeomorphism. Combining these facts, we have the
following
\bigskip
{\noindent} \textbf{Observation A.} \cite{LP} In order to prove
$BI_0(M,\omega)={\rm Ham}(M,\omega)$, it suffices to show that for
each nonzero value $v \in H^1(M, \mathbb R)/\Gamma$, there exists
some unbounded element $\phi \in {\rm Symp}_0(M,\omega)$ with ${\rm
flux}(\phi)=v$.
\bigskip
\section{The admissible lift}\label{admissiblelift}
\smallskip
To prove an element $\phi \in {\rm Symp}_0(M,\omega)$ is unbounded,
one has to show that $\rho([\,\phi, f])$ can be arbitrarily large by
choosing different $f \in {\rm Ham}(M,\omega)$. Hence the
energy-capacity inequality will not work directly for closed
manifolds since the capacity of the manifold itself is finite. To go
around this difficulty, we recall the notion of admissible lifts
which was first introduced by Lalonde and Polterovich \cite{LP}. We
shall point out that our definition is slightly different from
theirs, but the two definitions are equivalent.
Let $\pi : (\widetilde{M},\widetilde{\omega}) \to (M,\omega)$ be a
symplectic covering map, i.e. a covering map $\pi$ between two
symplectic manifolds such that $\widetilde{\omega}=\pi^* \omega$.
\begin{definition}
For every $g \in {\rm Ham}(M,\omega)$, assume $g$ is the time-1 map
of the Hamiltonian flow generated by time-dependent Hamiltonian
function $H_t$. An admissible lift $\widetilde{g} \in {\rm
Ham}(\widetilde{M},\widetilde{\omega})$ of $g$ with respect to $\pi$
is defined to be the time-1 map of the Hamiltonian flow generated by
$H_t \circ \pi$.
\end{definition}
\smallskip
\begin{lemma} [existence and uniqueness of admissible lifts]
For all $g \in {\rm Ham}(M,\omega)$, such an admissible lift
$\widetilde{g} \in {\rm Ham}(\widetilde{M},\widetilde{\omega})$
exists and is unique.
\end{lemma}
\begin{proof}
The existence follows from the definition. For the uniqueness, it
suffices to show that the admissible lift $\widetilde{g}$ of $g$ is
independent of the choice of the Hamiltonian function $H_t$.
Note that the choice of $H_t$ is equivalent to the choice of the
Hamiltonian isotopy $g_t$ connecting $id$ to $g$. For every point $p
\in M$, let $$\widetilde{ev}_p : \pi_1({\rm Ham}(M,\omega),id) \to
\pi_1(M,p)$$ be the map induced by the evaluation map $ev_p : {\rm
Ham}(M,\omega) \to M$ which takes $g$ to $g(p)$. It follows from
Floer theory that for all symplectic manifolds $(M,\omega)$, the
induced map $\widetilde{ev}_p$ is trivial, see Chapter 11 \cite{MS1}
for instance. This deep result implies that for any two different
paths $g_t^1$ and $g_t^2$ in ${\rm Ham}(M,\omega)$ connecting $id$
to $g$, $g_t^1(p)$ and $g_t^2(p)$ must be homotopic paths in $M$.
Therefore, for every point $\widetilde{p} \in \widetilde{M}$, the
image $\widetilde{g} (\widetilde{p})$ of $\widetilde{p}$ under
$\widetilde{g}$, being the endpoint of the lift of the path
$g_t(p)$, is independent of the choice of the Hamiltonian isotopy
$g_t$. This proves the uniqueness of admissible lifts.
\end{proof}
\smallskip
For our purposes, we consider the universal cover $\widetilde{M}$ of
$M$. Note that $\widetilde{M}$ is not necessarily compact, and the
admissible lift $\widetilde{g}$ of $g \in {\rm Ham}(M,\omega)$ is
not necessarily compactly supported in $\widetilde{M}$. Instead, it
belongs to ${\rm Ham}_b(\widetilde{M},\widetilde{\omega})$ of
time-$1$ maps of bounded Hamiltonians $\widetilde{M} \times [0, 1]
\to \mathbb R$. The Hofer norm is still well defined and the same
energy-capacity inequality still holds for this setting. This idea
is due to Lalonde and Polterovich \cite{LP}. We shall spell out some
details here for the sake of clarity.
Denote by $(N, \sigma)$ a noncompact symplectic manifold without
boundary. We do not often consider the group ${\rm Ham}(N, \sigma)$
of all Hamiltonian diffeomorphisms with arbitrary support. One
reason in our context is that it would not be possible to define the
Hofer norm on ${\rm Ham}(N, \sigma)$ using the $L_\infty$ norm on
the space $\mathcal A$ of all Hamiltonian functions with arbitrary
support, since not all elements in $\mathcal A$ have finite
$L_\infty$ norms.
One may consider the group ${\rm Ham}_c(N, \sigma)$ of Hamiltonian
diffeomorphisms with compact support. The Hofer norm $\rho$ is well
defined on ${\rm Ham}_c(N, \sigma)$, and the energy-capacity
inequality
$$e_c(S) \geqslant \frac{1}{2} {\rm capacity}(S)$$
is valid as usual, where
$$e_c(S):= {\rm inf}\,\{\, \rho(f)
\mid f \in {\rm Ham}_c(N, \sigma), \, f(S) \cap S=\emptyset\}.$$ As
we have already pointed out, however, this setting is not sufficient
for our purposes since the admissible lift is usually not compactly
supported. Hence we need to consider the larger group ${\rm
Ham}_b(N,\sigma)$ of Hamiltonian diffeomorphisms which are time-$1$
maps of bounded Hamiltonians $H: N \times [0, 1] \to \mathbb R$.
Note that one can not use an arbitrary bounded Hamiltonians $H$,
since the Hamiltonian flow generated by $H$ need not be integrable.
Instead, we only restrict to those bounded Hamiltonians whose flows
are integrable.
The Hofer norm can be defined on ${\rm Ham}_b(N,\sigma)$ exactly the
same way as in Section \ref{hofernorm}. For a subset $S$ of $N$,
define also the bounded displacement energy $e_b(S)$ as
$$e_b(S):= {\rm inf}\,\{\, \rho(f)
\mid f \in {\rm Ham}_b(N, \sigma), \, f(S) \cap S=\emptyset\}.$$
Note that ${\rm Ham}_c(N, \sigma) \subset {\rm Ham}_b(N, \sigma)$
implies $e_b(S) \leqslant e_c(S)$ for any subset $S \subset N$. In
fact, for any compact subset $S$, we have $$e_b(S) = e_c(S).$$ To
prove the other inequality, note that if $f \in {\rm Ham}_b(N,
\sigma)$ displaces a compact subset $S$ from itself, one can easily
construct some cut-off $f_{cut} \in {\rm Ham}_c(N, \sigma)$ of $f$
which still displaces $S$ from itself, and the Hofer norm satisfies
$\rho(f) \geqslant \rho(f_{cut})$. Taking the infimum implies
$e_b(S) \geqslant e_c(S)$.
The above argument implies that the energy-capacity inequality still
holds for the bounded displacement energy. That is
$$e_b(S) \geqslant \frac{1}{2} {\rm capacity}(S).$$
Now back to our discussion about the admissible lift. Note that the
admissible lift $\widetilde{g}$ of $g \in {\rm Ham}(M,\omega)$
belongs to ${\rm Ham}_b(\widetilde{M},\widetilde{\omega})$. And it
follows from the definition of the admissible lift that
$$\rho(g) \geqslant \rho(\widetilde{g})$$
for all $g \in {\rm Ham}(M,\omega)$ and the admissible lift
$\widetilde{g} \in {\rm Ham}_b(\widetilde{M},\widetilde{\omega})$ of
$g$. Here the two $\rho$'s are the Hofer norms on ${\rm
Ham}(M,\omega)$ and ${\rm Ham}_b(\widetilde{M},\widetilde{\omega})$
respectively. Combining the above discussions, we have
\bigskip
{\noindent} \textbf{Observation B.} \cite{LP} To construct $g \in {\rm
Ham}(M,\omega)$ of arbitrarily large Hofer norm, it suffices to make
sure that the unique admissible lift $\widetilde{g} \in {\rm
Ham}_b(\widetilde{M},\widetilde{\omega})$ of $g$ displaces from
itself a symplectic ball in $\widetilde{M}$ of arbitrarily large
capacity.
\bigskip
\section{Proof of Theorem \ref{bicforkt}}\label{proofbicforkt}
\smallskip
In this section, we prove Theorem \ref{bicforkt}. Recall that $(M,
\omega)$ is the Kodaira-Thurston manifold with the standard
symplectic form $\omega=ds \wedge dt+dx \wedge dy$. Recall also that
$H^1(M, \mathbb R)=\mathbb R\langle ds, dy, dt\rangle $ and the flux
subgroup $\Gamma=\mathbb{Z}\langle ds, dy\rangle $ by Lemma
\ref{cohomology} and Theorem \ref{flux}. In view of Observation A,
to prove $BI_0(M,\omega)={\rm Ham}(M,\omega)$, it suffices to show
that for every nonzero element $v \in H^1(M, \mathbb
R)/\Gamma=\mathbb R/\mathbb{Z} \langle ds, dy\rangle \oplus \mathbb
R\langle dt\rangle $, there exists some unbounded symplectomorphism
with flux equal to $v$. We begin with an explicit construction of
symplectomorphisms with given fluxes.
\begin{lemma}\label{phiabc}
Let $v$ be an element in $H^1(M, \mathbb R)/\Gamma=\mathbb
R/\mathbb{Z} \langle ds, dy\rangle \oplus \mathbb R\langle dt\rangle
$, say $v=\alpha ds+\beta dy+c dt$ where $\alpha, \beta \in \mathbb
R/\mathbb{Z}$ and $c \in \mathbb R$. Then there exists an element
$\phi_{\alpha \beta c} \in {\rm Symp}_0(M,\omega)$ with ${\rm
flux}(\phi_{\alpha \beta c})=v$. Namely,
$$\phi_{\alpha \beta c}(s, t, x, y) = (s+c, t-\alpha, x+\beta, y).$$
\end{lemma}
\begin{proof}
First $\phi_{\alpha \beta c}$ is well-defined. For instance, since
$(s, t, x, y)$ and $(s+1, t, x+y, y)$ represent the same point on
$M$, one has to show that
$$\phi_{\alpha \beta c}(s, t, x, y) \sim \phi_{\alpha \beta c}(s+1, t, x+y, y).$$
This is true since
$$\phi_{\alpha \beta c} (s, t, x, y)=(s+c, t-\alpha, x+\beta, y),$$
and $$\phi_{\alpha \beta c} (s+1, t, x+y, y)=(s+1+c, t-\alpha,
x+y+\beta, y).$$
{\noindent} It is easy to see that $\phi_{\alpha \beta c}$ preserves
$\omega$, and the obvious isotopy from $id$ to $\phi_{\alpha \beta
c}$ implies that $\phi_{\alpha \beta c} \in {\rm Symp}_0(M,\omega)$.
The calculation for ${\rm flux}(\phi_{\alpha \beta c})=v$ is
straightforward using (\ref{eq:1}) in Section \ref{fluxsubgroup}.
\end{proof}
\smallskip
The following theorem due to Lalonde and Polterovich \cite{LP} is an
important criteria for unbounded symplectomorphisms.
\smallskip
\begin{thm}[Theorem 1.4.A \cite{LP}]\label{displace}
Let $L \subset M$ be a closed Lagrangian submanifold admitting a
Riemannian metric with non-positive sectional curvature, and whose
inclusion in $M$ induces an injection on fundamental groups. Let
$\phi$ be an element in ${\rm Symp}_0(M,\omega)$ such that $\phi(L)
\cap L =\emptyset$. Then $\phi$ is unbounded.
\end{thm}
For the proof, one passes to the universal cover $\widetilde{M}$ of
$M$. The hypothesis implies that the lift of a neighbourhood $U$ of
$L$ has infinite capacity. One then constructs a Hamiltonian isotopy
$f_\tau$ supported in $U$ so that the admissible lift
$\widetilde{[\phi, f_\tau]}$ of the commutator $[\phi, f_\tau]$ will
displace a symplectic ball of arbitrarily large capacity as $\tau$
goes to infinity. This implies $\phi$ is unbounded according to
Observation B. See \cite{LP} for details.
\bigskip
{\noindent} {\textbf{Proof of Theorem \ref{bicforkt}.}} In view of
Observation A, it suffices to show that the symplectomorphisms
$\phi_{\alpha \beta c}$ constructed in Lemma \ref{phiabc} are
unbounded in all cases, as long as the flux $v=\alpha ds+\beta dy+c
dt$ does not vanish. We argue case by case. In the first two cases,
this is a direct consequence of Theorem \ref{displace}.
{\noindent} \textbf{Case 1.} $\alpha \ne 0 \in \mathbb R/ \mathbb{Z}$.
Let $L \subset M$ be the subset of $M$ defined by
$$L := \{(s,t,x,y) \in M \mid t=0,y=0\}.$$
It is easy to check that $L$ is a Lagrangian torus satisfying the
hypothesis of Theorem \ref{displace}, and $\phi_{\alpha \beta c}$
displaces $L$ from itself. Thus $\phi_{\alpha \beta c}$ is
unbounded.
\bigskip
{\noindent} \textbf{Case 2.} $\alpha = 0 \in \mathbb R/ \mathbb{Z}, \, \beta
\ne 0 \in \mathbb R/ \mathbb{Z}$ and $c=0 \in \mathbb R$.
In this case, $\phi_\beta := \phi_{\alpha \beta c}$ maps $(s, t, x,
y)$ to $(s, t, x+\beta, y)$. As in the first case, $\phi_\beta$
displaces from itself a Lagrangian torus $L$ of $M$ defined by
$$L := \{(s, t, x, y) \in M \mid s=0, x=0 \}.$$
We again get $\phi_\beta$ is unbounded in view of Theorem
\ref{displace}.
\bigskip
{\noindent} \textbf{Case 3.} $\alpha = 0 \in \mathbb R/ \mathbb{Z}$ and $c
\ne 0 \in \mathbb R$.
We write $\phi_{\beta c}$ for $\phi_{\alpha \beta c}$ in this case,
$$\phi_{\beta c} := \phi_{\alpha \beta c} : (s, t, x, y) \mapsto (s+c, t, x+\beta, y).$$
Consider two different situations, one of which is simple, while the
other is more complicated.
\bigskip
{\noindent} \textbf{3A.} $\alpha = 0 \in \mathbb R/ \mathbb{Z}$ and $c
\notin \mathbb{Z}$.
As in case 1 and 2, $\phi_{\beta c}$ is unbounded as it displaces
from itself
$$L := \{(s, t, x, y) \in M \mid s=0, x=0 \}.$$
\bigskip
{\noindent} \textbf{3B.} $\alpha = 0 \in \mathbb R/ \mathbb{Z}$ and $c \in
\mathbb{Z} \backslash \{0 \}$.
Note that $(s+c, t, x+\beta, y) \sim (s, t, x +\beta -c y, y)$. So
the map $\phi_{\beta c} : M \to M$ can also be expressed as
$$\phi_{\beta c}(s, t, x, y)=(s, t, x +\beta -c y, y).$$
In contrast to all previous cases where we used the same argument,
here we are facing a difficulty. The trouble is that in this case we
are unable to find a Lagrangian torus of $M$ which is disjoined from
itself by the map $\phi_{\beta c}$. Thus the above argument breaks
down.
To resolve this difficulty, we take $f_\tau$ to be the Hamiltonian
isotopy whose support is in the subset
$$U:=\{(s, t, x, y) \in M \mid |s|<\epsilon, |x|<\epsilon \}$$ of $M$.
We require $f_\tau$ to flow only along $y$ and $t$ direction in $U$
and its restriction to
$$V:=\{(s, t, x, y) \in M \mid |s|<\epsilon/2, |x|<\epsilon/2 \}$$
is defined by
$$f_\tau(s, t, x, y)=(s, t, x, y-\tau).$$
In the discussion below, $[f, g]:= f g f^{-1} g^{-1}$ stands for the
commutator of $f$ and $g$. Our goal is to show that the unique
admissible lift $\widetilde{[\phi_{\beta c}, f_\tau]}$ of
$[\phi_{\beta c}, f_\tau]$ still displaces from itself a subset of
$\mathbb R^4$ of arbitrarily large capacity when $\tau$ goes to
infinity. For this, we need the following
\begin{lemma}
Let $\phi \in {\rm Symp}_0(M,\omega)$, and $f_\tau$ be a Hamiltonian
isotopy of $M$. Let $\widetilde{\phi} :\widetilde{M} \to
\widetilde{M}$ be any lift of $\phi$, and $\widetilde{[\phi,
f_\tau]}$ and $\widetilde{f}_\tau$ be the unique admissible lift of
$[\phi, f_\tau]$ and $f_\tau$ respectively. Then
$$\widetilde{[\phi, f_\tau]} =
[\widetilde{\phi}, \widetilde{f}_\tau].$$
\end{lemma}
\begin{proof}
Note that $f_\tau$ is Hamiltonian implies $[\phi, f_\tau]$ is
Hamiltonian. So both admissible lifts $\widetilde{[\phi, f_\tau]}$
and $\widetilde{f}_\tau$ make sense. To simplify notation, denote
$$A_\tau:=\widetilde{[\phi, f_\tau]} \,\, \mbox{and} \,\,
B_\tau:=[\widetilde{\phi}, \widetilde{f}_\tau].$$ We want to show
$A_\tau=B_\tau$, which is equivalent to $A_\tau B_\tau^{-1}=id$.
Since $A_\tau$ and $B_\tau$ are both lifts of $[\phi, f_\tau]$,
$A_\tau B_\tau^{-1}$ is the deck transformation of the covering map
$\pi : \widetilde{M} \to M$. Now $A_0 B_0=id$, and $\tau \to A_\tau
B_\tau^{-1}$ is a continuously parametrized path into the discrete
set of all deck transformations. Thus $A_\tau B_\tau^{-1}=id \,$ for
all $\tau$.
\end{proof}
\bigskip
Now back to the proof of Theorem \ref{bicforkt}. To prove
$\phi_{\beta c}$ is unbounded, we need to show that the commutator
$[\phi_{\beta c}, f_\tau]$ has arbitrarily large Hofer norm when
$\tau$ goes to infinity. Let $V_0 \subset \mathbb R^4$ be the subset
of $\mathbb R^4$ defined by
$$V_0:=\{(s,t,x,y) \in \mathbb R^4 \mid |s|< \epsilon/2, t
\in \mathbb R, |x|<\epsilon/2, 0<y<\tau/2 \}.$$ Since $V_0$ has
arbitrarily large capacity as $\tau$ goes to infinity, according to
Observation B, it suffices to show that the admissible lift
$\widetilde{[\phi_{\beta c}, f_\tau]}$ of $[\phi_{\beta c}, f_\tau]$
displaces $V_0$ from itself.
For this, denote by $\widetilde{\phi}_{\beta c} : \mathbb R^4 \to
\mathbb R^4$ the preferred lift of the map $\phi_{\beta c}$ such
that
$$\widetilde{\phi}_{\beta c}(s, t, x, y)=(s, t, x +\beta -c y, y).$$
By the above lemma, it suffices to show that
$[\widetilde{\phi}_{\beta c}, \widetilde{f}_\tau] (V_0) \cap V_0 =
\emptyset$, which is equivalent to
$$\widetilde{\phi}_{\beta c}^{-1} \widetilde{f}_\tau^{-1}(V_0) \cap
\widetilde{f}_\tau^{-1} \widetilde{\phi}_{\beta c}^{-1}(V_0)=
\emptyset.$$ Note that the restriction of $\widetilde{f}_\tau$ to
$$\widetilde{V} :=\{(s,t,x,y) \in \mathbb R^4 \mid |s|< \epsilon/2, t
\in \mathbb R, |x|<\epsilon/2, y \in \mathbb R \}$$ is defined by
$$\widetilde{f}_\tau(s, t, x, y)=(s, t, x, y-\tau).$$ We have
$$\widetilde{f}_\tau^{-1}(V_0)= \{|s|<
\epsilon/2, t \in \mathbb R, |x|<\epsilon/2, \tau<y<3 \tau/2 \}.$$ Hence
$$\widetilde{\phi}_{\beta c}^{-1} \widetilde{f}_\tau^{-1}(V_0)=\{|s|< \epsilon/2, t \in \mathbb
R, |x +\beta -c y|<\epsilon/2, \tau<y<3 \tau/2 \}.$$ On the other hand,
$$\widetilde{\phi}_{\beta c}^{-1}(V_0)=\{|s|< \epsilon/2, t \in
\mathbb R, |x +\beta -c y|<\epsilon/2, 0<y<\tau/2 \}.$$
{\noindent} Note that in the set $\widetilde{\phi}_{\beta c}^{-1}
\widetilde{f}_\tau^{-1}(V_0)$ we have
$$|x|>|c y| -|\beta| -\epsilon/2>|c| \tau -|\beta| -\epsilon/2,$$
and in $\widetilde{\phi}_{\beta c}^{-1} (V_0)$ we have
$$|x|<|c y| +|\beta| +\epsilon/2<|c| \tau/2 +|\beta| +\epsilon/2.$$
Thus for sufficiently large $\tau$, these two sets do not share the
same values in $x$ coordinates. Since the flow
$\widetilde{f}_\tau^{-1}$ only changes the $y$ and $t$-coordinates
when restricted to $\widetilde{\phi}_{\beta c}^{-1}(V_0)$, we
conclude
$$\widetilde{\phi}_{\beta c}^{-1} \widetilde{f}_\tau^{-1}(V_0) \cap
\widetilde{f}_\tau^{-1} \widetilde{\phi}_{\beta c}^{-1}(V_0)=
\emptyset.$$ As we have already mentioned above, this implies
$\phi_{\beta c}$ is unbounded in case 3B, which completes the proof
of Theorem \ref{bicforkt}. \qed
\bigskip
\section{Proof of Theorem \ref{bicfortorus}}\label{proofbicfortorus}
\smallskip
We have already mentioned in Section \ref{introduction} that the
bounded isometry conjecture holds for the torus with the standard
symplectic form. In this section we prove Theorem \ref{bicfortorus}
which states that the conjecture holds for the 4-torus with any
linear symplectic form. We begin with a remark on the linear
symplectic form $\omega$ on $\mathbb T^4$.
\begin{rmk}\rm
The 2-form $\omega=\sum_{i < j}\, a_{ij}\, dx_i \wedge dx_j$ on
$\mathbb T^4$ is symplectic, i.e. nondegenerate if and only if
$a_{12}a_{34}-a_{13}a_{24}+a_{14}a_{23} \ne 0$.
\end{rmk}
For each $1 \leqslant i \leqslant 4$, let $\{\phi^i_\theta \} \in
\pi_1({\rm Symp}_0(\mathbb T^4,\omega))$ be the loop of rotations of
$\mathbb T^4$ along $x_i$ direction. Let $\xi_i \in H^1(\mathbb T^4,
\mathbb R)$ be the image of $\{\phi^i_\theta\}$ under the flux
homomorphism. Using (\ref{eq:1}) in Section \ref{fluxsubgroup}, one
easily gets
$$\xi_i := {\rm flux}(\{\phi^i_\theta\})=\displaystyle \sum_{j=1}^4
\,a_{ij} \, dx_j.$$ Here we take the convention that
$a_{ij}=-a_{ji}$. In particular, $a_{ii}=0$.
\begin{lemma}
For the $4$-torus with the linear symplectic form $\omega:=\sum_{i <
j}\, a_{ij} dx_i \wedge dx_j$, the flux subgroup $\Gamma \subset
H^1(\mathbb T^4, \mathbb R)$ is generated by the above $\xi_i's$
over $\mathbb Z$. That is, $\Gamma = \mathbb Z \langle \xi_1, \xi_2,
\xi_3, \xi_4 \rangle$.
\end{lemma}
\begin{proof}
According to Lemma \ref{commute}, we have the following commutative
diagram for the manifold $(\mathbb T^4, \omega)$.
$$\begin{CD}
\pi_1({\rm Symp}_0(\mathbb T^4,\omega)) @>\widetilde{ev}_s>>H_1(\mathbb T^4, \mathbb Z) @>{\rm PD}>>H^3(\mathbb T^4, \mathbb Z) \\
@VidVV && @VV\centerdot {\rm vol}(\mathbb T^4)V\\
\pi_1({\rm Symp}_0(\mathbb T^4,\omega)) @>{\rm flux}>>H^1(\mathbb T^4, \mathbb R) @>\wedge[\omega]>>H^3(\mathbb T^4, \mathbb R).
\end{CD}$$
\bigskip
{\noindent} Note that $\widetilde{ev}_s$ is surjective, and $\wedge[\omega]
: H^1(\mathbb T^4, \mathbb R) \to H^3(\mathbb T^4, \mathbb R)$ is an
isomorphism. Note also that ${\rm vol}(\mathbb
T^4)=a_{12}a_{34}-a_{13}a_{24}+a_{14}a_{23}$. It follows from a
similar argument as in the proof of Theorem \ref{flux} that $\xi_i
(1 \leqslant i \leqslant 4)$ span the flux subgroup $\Gamma$ over
$\mathbb Z$.
\end{proof}
Now let $\phi \in Symp_0(\mathbb T^4,\omega)$ such that
$$\phi(x_1, x_2, x_3, x_4)=(x_1+\alpha_1, x_2+\alpha_2, x_3+\alpha_3,
x_4+\alpha_4)$$ where $\alpha_i \in \mathbb R/\mathbb Z$ for $1
\leqslant i \leqslant 4$. Then
$${\rm flux}(\phi)= \displaystyle \sum_{i=1}^4 \, \alpha_i \,
\xi_i.$$
Recall that in view of Observation A in Section \ref{hofernorm}, to
prove Theorem \ref{bicfortorus}, it suffices to show $\phi$ is
unbounded as long as at least one $\alpha_i \in \mathbb R/\mathbb Z$
is nonzero. One may attempt to apply Theorem \ref{displace} by
showing $\phi$ disjoins some Lagrangian torus $L \subset \mathbb
T^4$ from itself. For a general symplectic form $\omega$, however,
there may not exist any such Lagrangian torus in $\mathbb T^4$.
Nevertheless, we can still prove $\phi$ is unbounded using the
following
\begin{lemma}\label{nocontractible}
Let $(M,\omega)$ be an aspherical symplectic manifold. Let $f_\tau
\in {\rm Ham}(M, \omega)$ be the flow generated by an autonomous
Hamiltonian which has no nonconstant contractible orbits. Then the
Hofer norm $\rho(f_\tau)$ goes to infinity as $\tau$ goes to
infinity.
\end{lemma}
This result can be found in Oh \cite{Oh}, Schwarz \cite{Sch} and
Kerman-Lalonde \cite{KL}. The main idea of the argument is that the
Hofer norm is bounded from below by the spectral norm, while the
spectral norm of such $f_\tau$ grows linearly with respect to
$\tau$.
\bigskip
{\noindent} {\bf Proof of Theorem \ref{bicfortorus}.} Let $\phi \in
Symp_0(\mathbb T^4,\omega)$ such that
$$\phi(x_1, x_2, x_3, x_4)=(x_1+\alpha_1, x_2+\alpha_2, x_3+\alpha_3,
x_4+\alpha_4).$$ As discussed above, it suffices to show $\phi$ is
unbounded when at least one $\alpha_i \in \mathbb R/\mathbb Z$ is
nonzero. Assume $\alpha_1 \ne 0$ without loss of generality. Thus
$\phi (U) \cap U = \emptyset$ where $U \subset \mathbb T^4$ is
defined by
$$U := \{(x_1, x_2, x_3, x_4) \in \mathbb T^4 \mid |x_1|<
\epsilon \}.$$ for sufficiently small $\epsilon$.
Let $H$ be a time-independent Hamiltonian function of $\mathbb T^4$
supported in $U$. Denote by $f_\tau$ the (autonomous) Hamiltonian
flow generated by $H$. Since $\phi (U) \cap U = \emptyset$, we know
that $[\phi, f_\tau]:=\phi f_\tau \phi^{-1} f_\tau^{-1}$ is also an
autonomous Hamiltonian flow supported in the union of two disjoint
sets $U \cup \phi (U)$. If we further require that $H$ depend only
on the first coordinate $x_1$, using the fact that $\omega$ is a
linear symplectic form, we conclude that $[\phi, f_\tau]$ has no
nonconstant contractible orbits. Thus it follows from Lemma
\ref{nocontractible} that the Hofer norm $\rho([\phi, f_\tau])$ goes
to infinity as $\tau$ goes to infinity. Hence $\phi$ is unbounded in
the sense of Definition \ref{bounded}. \qed
\bigskip
\section{The Kodaira-Thurston manifold with linear symplectic
forms}\label{sec8}
\smallskip
So far we have studied bounded isometries for the Kodaira-Thurston
manifold with the standard symplectic form and for the 4-torus with
all linear symplectic forms. In particular, we have shown that the
bounded isometry conjecture holds in both cases. In this section we
will study the same question for the Kodaira-Thurston manifold with
all linear symplectic forms.
\begin{question}\label{bicforktlinear}
Does the bounded isometry conjecture hold for the Kodaira-Thurston
manifold with all linear symplectic forms?
\end{question}
We expect the answer to be positive. Although we are not able to
give a complete proof yet at this time, we shall provide some
partial results below. We begin by describing the linear symplectic
forms on the Kodaira-Thurston manifold $M$. Recall that it follows
from Lemma \ref{cohomology} that $H^2(M, \mathbb R)$ is of rank $4$,
generated by $\gamma \wedge ds, \, \gamma \wedge dy, \, ds \wedge
dt, \, \mbox{and} \,dy \wedge dt$ where $\gamma=dx-s dy$. We
consider linear 2-forms
$$\omega_{a b e f}:=a \gamma \wedge ds + b\gamma \wedge d y + e ds \wedge dt + f dy \wedge dt.$$
Note that $\omega_{a b e f}$ is a symplectic form if and only if $b
e - a f \ne 0$. In particular, the standard symplectic form
corresponds to $b = e =1$ and $ a = f =0$. The following lemma on
the flux subgroup generalizes Theorem \ref{flux}.
\begin{lemma}\label{fluxlinear}
The flux subgroup $\Gamma \subset H^1(M,\mathbb R)$ of the
Kodaira-Thurston manifold with the linear symplectic form $\omega_{a
b e f}$ has rank $2$ over $\mathbb{Z}$. More precisely, we have
$\Gamma=\mathbb{Z}\langle e ds + f dy, \,a ds + b dy \rangle $.
\end{lemma}
\begin{proof}
The proof follows the same lines as that of Theorem \ref{flux}.
According to Lemma \ref{commute}, we have the following commutative
diagram.
$$\begin{CD}
\pi_1({\rm Symp}_0(M,\omega_{abef})) @>\widetilde{ev}_s>>H_1(M, \mathbb Z) @>{\rm PD}>>H^3(M, \mathbb Z) \\
@VidVV && @VV\centerdot {\rm vol}(M)V\\
\pi_1({\rm Symp}_0(M,\omega_{abef})) @>{\rm flux}>>H^1(M, \mathbb R) @>\wedge[\omega_{abef}]>> H^3(M, \mathbb R).
\end{CD}$$
\bigskip
As in the proof of Theorem \ref{flux}, the image of
$\widetilde{ev}_s$ in $H_1(M, \mathbb Z)$ is contained in
$\mathbb{Z}\langle\frac{\partial}{\partial t}\rangle$. Note that
$PD(\frac{\partial}{\partial t})=-\gamma \wedge dy \wedge ds$, where
$\gamma=dx-sdy$. Now look at the map $\wedge\omega_{abef} : H^1(M,
\mathbb R) \to H^3(M, \mathbb R)$,
\begin{equation}\nonumber
\begin{aligned}
ds \mapsto ds \wedge \omega_{abef} &=b \gamma \wedge dy \wedge ds - f dy \wedge ds \wedge dt =b \gamma \wedge dy \wedge ds,\\
dt \mapsto dt \wedge \omega_{abef} &=a \gamma \wedge ds \wedge dt + b \gamma \wedge dy \wedge dt,\\
dy \mapsto dy \wedge \omega_{abef} &= -a \gamma \wedge dy \wedge ds + e dy \wedge ds \wedge dt=-a \gamma \wedge dy \wedge ds.
\end{aligned}.
\end{equation}
\smallskip
{\noindent} Here we have used the fact that the 3-form $dy \wedge ds \wedge
dt = d (\gamma \wedge dt)$ is exact, so it vanishes on the
cohomology level. Since ${\rm vol}(M)=be-af \ne 0$, we conclude by
tracing the diagram that the flux subgroup $\Gamma \subset H^1(M,
\mathbb R)$ is contained in $\mathbb{Z}\langle e ds + f dy, a ds+ b
dy\rangle $. Note that the fact $be-af \ne 0$ implies that $e ds + f
dy$ and $a ds+ b dy$ are linearly independent. An explicit
construction shows that $\Gamma$ is actually equal to
$\mathbb{Z}\langle e ds + f dy, a ds + b dy \rangle$. Namely, we
take two elements $\{\phi_\theta\}$ and $\{\psi_\theta\}$ in
$\pi_1({\rm Symp}_0(M,\omega_{abef}))$ such that
$$\phi_\theta(s, t, x, y)=(s, t-\theta, x, y), 0 \leqslant\theta \leqslant1, $$
$$\psi_\theta(s, t, x, y)=(s, t, x+\theta, y), 0 \leqslant\theta \leqslant1.$$
A straightforward calculation using (\ref{eq:1}) in Section
\ref{fluxsubgroup} shows that ${\rm flux}(\{\phi_\theta \}) =e ds +
f dy$ and ${\rm flux}(\{\psi_\theta \})=a ds + b dy$.
\end{proof}
As in Lemma \ref{phiabc}, we explicitly construct below
symplectomorphisms with given fluxes.
\begin{lemma}\label{phiabclinear}
Let $v$ be an element in $$H^1(M, \mathbb R)/\Gamma=\mathbb
R/\mathbb{Z} \langle e ds + f dy, \, a ds + b dy \rangle \oplus
\mathbb R\langle dt\rangle,$$ say $$v=\alpha (e ds + f dy) +\beta (a
ds + b dy)+c (be - af) dt$$ where $\alpha, \beta \in \mathbb
R/\mathbb{Z}$ and $c \in \mathbb R$. Then there exists $\phi_{\alpha
\beta c} \in {\rm Symp}_0(M,\omega_{abef})$ with ${\rm
flux}(\phi_{\alpha \beta c})=v$. Namely,
$$\phi_{\alpha \beta c}(s, t, x, y) = (s+ b c, t-\alpha, x+\beta - a c s, y - a c).$$
\end{lemma}
\begin{proof}
First $\phi_{\alpha \beta c}$ is well-defined. For instance, since
$(s, t, x, y)$ and $(s+1, t, x+y, y)$ represent the same point in
$M$, one has to show that
$$\phi_{\alpha \beta c}(s, t, x, y) \sim \phi_{\alpha \beta c}(s+1, t, x+y, y).$$
This is true since
$$\phi_{\alpha \beta c}(s, t, x, y) = (s+ b c, t-\alpha, x+\beta - a c s, y - a c)$$
and $$\phi_{\alpha \beta c} (s+1, t, x+y, y)=(s+ 1 + b c, t-\alpha,
x+ y + \beta - a c (s+1), y - a c)$$ also represent the same point.
One can check that $\phi_{\alpha \beta c}^* \omega_{abef} =
\omega_{abef}$, and the obvious isotopy from $id$ to $\phi_{\alpha
\beta c}$ implies that $\phi_{\alpha \beta c} \in {\rm
Symp}_0(M,\omega_{abef})$.
It remains to show that ${\rm flux}(\phi_{\alpha \beta c})=v$. Note
that $\phi_{\alpha \beta c}$ is the time-1 map of the flow generated
by the time-independent symplectic vector field
$$X:= bc \frac{\partial}{\partial s}- \alpha
\frac{\partial}{\partial t} + (\beta - acs) \frac{\partial}{\partial
x} - ac \frac{\partial}{\partial y}.$$ Using (\ref{eq:1}) in Section
\ref{fluxsubgroup}, we have
\begin{equation}\nonumber
\begin{split}
{\rm flux} (\phi_{\alpha \beta c})&=\iota(X)\,\omega_{a b e f}\\
&= \iota (bc
\frac{\partial}{\partial s}- \alpha \frac{\partial}{\partial t}
+ (\beta - acs) \frac{\partial}{\partial x} - ac \frac{\partial}{\partial y})\, \omega_{a b e f}\\
&= -abc (dx - s dy) + bce dt + \alpha e ds + \alpha f dy \\
&+ a (\beta - acs) ds + b (\beta - acs) dy + a^2 c sds + abc dx - acf dt\\
&= \alpha (e ds + f dy) + \beta (a ds + b dy) + c(be - af) dt\\
&= v.
\end{split}
\end{equation}
\end{proof}
To answer Question \ref{bicforktlinear}, one has to check whether
$\phi_{\alpha \beta c}$ constructed in Lemma \ref{phiabclinear} is
always unbounded whenever its flux $v$ is nonzero in $H^1(M, \mathbb
R)/\Gamma$. This is in general a very hard question. In the
remaining of this section, we will give a proof for some known
cases. For the unknown cases, we will try to point out what
difficulty is involved.
\bigskip
{\noindent} {\bf Case 1}: $\alpha \ne 0 \in \mathbb R/\mathbb{Z}$. In this
case we will prove $\phi_{\alpha \beta c}$ is always unbounded. Note
that $\phi_{\alpha \beta c}(U) \cap U = \emptyset$ where $U \subset
M$ is defined by
$$U := \{(s, t, x, y) \in M \mid |t|<
\epsilon \}$$ for sufficiently small $\epsilon$. We will apply Lemma
\ref{nocontractible} as in the proof of Theorem \ref{bicfortorus}.
Recall that the only thing we need to do is to construct
time-independent Hamiltonian $H$ supported in $U$ whose flow has no
nonconstant contractible orbits. This follows from a tedious but
straightforward calculation which asserts that
$$\iota\,(X)\,\omega_{abef}=dt$$ where
$$X:=\frac{1}{be-af}(-as\frac{\partial}{\partial x}-a
\frac{\partial}{\partial y}+b \frac{\partial}{\partial s}).$$
{\noindent} Note that this is actually a special case of the construction in
Lemma \ref{phiabclinear}. And the fact that $X$ is a well defined
vector field on $M$ follows from the equivalence relation $(s,t,x,y)
\sim (s+1,t,x+y,y)$. Since $a$ and $b$ can not be both zero, if we
further require $H$ to depend only on the $t$-coordinates, we know
that the Hamiltonian flow generated by $H$ will have no nonconstant
contractible orbits. Therefore $\phi_{\alpha \beta c}$ is always
unbounded in this case.
\bigskip
{\noindent} {\bf Case 2}: $\alpha = 0 \in \mathbb R/\mathbb{Z}$ and $c \ne 0
\in \mathbb R$. First we assume $ac$ and $bc$ are not both integers.
Note that this is always the case when the ratio $a : b$ is
irrational. Under this assumption, $\phi_{\beta c}:=\phi_{\alpha
\beta c}$ is unbounded in view of Theorem \ref{displace} as it
disjoins a Lagrangian torus
$$L:=\{(s,t,x, y) \in M \mid s =0, \, y=0 \}.$$
If the ratio $a : b$ is rational, then there exists $c \ne 0$ such
that both $ac$ and $bc$ are integers. In this case, using the
equivalence relation $(s,t,x,y) \sim (s+1,t,x+y,y)$, we can write
the map
$$\phi_{\beta c}:(s, t, x, y) \mapsto (s+ b c, t, x+\beta - a c s, y - a c)$$
as
$$\phi_{\beta c}:(s, t, x, y) \mapsto (s, t, x+\beta - a c s-b c y, y).$$
It is natural to attempt the admissible lift argument as in Case 3B
of Theorem \ref{bicforkt} for the standard Kodaira-Thurston
manifold. One would try to construct a Hamiltonian isotopy
$\widetilde{f}_\tau$ on $\mathbb R^4$ supported in
$$\widetilde{U}:=\{(s, t, x, y) \in \mathbb R^4 \mid |es+fy|<\epsilon, |x|<\epsilon \}$$
which flows only along $s$ and $y$ directions, and whose restriction
to
$$\widetilde{V}:=\{(s, t, x, y) \in \mathbb R^4 \mid |es+fy|<\epsilon/2, |x|<\epsilon/2 \}$$
is defined by
$$\widetilde{f}_\tau(s, t, x, y)=(s+f \tau, t, x, y-e \tau).$$
Note that the above construction allows us to show that the lift
$$\widetilde{\phi}_{\beta c} :(s,
t, x, y) \mapsto (s, t, x+\beta - a c s-b c y, y)$$ of $\phi_{\beta
c}$ is unbounded on the universal cover level. For this, one would
argue as in Case 3B of Theorem \ref{bicforkt}, that the commutator
$[\widetilde{\phi}_{\beta c}, \widetilde{f}_\tau]$ displaces some
subset $V_0 \subset \mathbb R^4$ of arbitrarily large capacity with
respect to the symplectic form
$\widetilde{\omega}_{abef}:=\pi^*\omega_{abef}$. Namely,
$$V_0:=\{|es+fy|<\epsilon/2, \, t
\in \mathbb R,\, |x|<\epsilon/2, 0<as+by<|be-af|\tau/2 \}.$$ The problem
here is that $\widetilde{f}_\tau$ does not descend to a Hamiltonian
isotopy on $M$. Note that in proving $\phi_{\beta c}$ itself is
unbounded, it is crucial to have such a Hamiltonian isotopy on $M$,
not just on the universal cover $\mathbb R^4$. Hence this case is
still unsolved.
\bigskip
{\noindent} {\bf Case 3}: $\alpha = 0 \in \mathbb R/\mathbb{Z}$, $c = 0 \in
\mathbb R$ and $\beta \ne 0 \in \mathbb R/\mathbb{Z}$. In this case,
the map $\phi_\beta:= \phi_{\alpha \beta c}$ has the simple form
$$\phi_\beta :(s, t, x, y) \mapsto (s, t, x+\beta, y).$$
{\noindent} We do not know in general how to prove $\phi_\beta$ is unbounded
for this seemingly easy case. The difficulty in applying Theorem
\ref{displace} is that the obvious torus
$$L := \{(s, t, x, y) \in M \mid s=0, x=0 \}$$ displaced by
$\phi_\beta$ is not necessarily Lagrangian with respect to all
symplectic forms $\omega_{abef}$. If we assume $f=0$, then $L$ is
actually a Lagrangian torus, and $\phi_\beta$ will be unbounded in
view of Theorem \ref{displace}.
Note also that Lemma \ref{nocontractible} does not work here either
since our situation here is different from Case 1 above. The main
reason is that
$$U := \{(s, t, x, y) \in M \mid |x|< \epsilon \}$$
is not a well defined set in $M$. Thus one can no longer apply Lemma
\ref{nocontractible} by constructing a time-independent Hamiltonian
$H$ supported in $U$ whose flow has no nonconstant contractible
orbits.
\bigskip
| {
"timestamp": "2007-05-05T21:14:35",
"yymm": "0705",
"arxiv_id": "0705.0762",
"language": "en",
"url": "https://arxiv.org/abs/0705.0762",
"abstract": "The purpose of this note is to study the bounded isometry conjecture proposed by Lalonde and Polterovich. In particular, we show that the conjecture holds for the Kodaira-Thurston manifold with the standard symplectic form and for the 4-torus with all linear symplectic forms.",
"subjects": "Symplectic Geometry (math.SG)",
"title": "The bounded isometry conjecture for the Kodaira-Thurston manifold and 4-Torus",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9770226260757067,
"lm_q2_score": 0.724870282120402,
"lm_q1q2_score": 0.7082146666015136
} |
https://arxiv.org/abs/2205.12572 | The SO(3) and SE(3) Lie Algebras of Rigid Body Rotations and Motions and their Application to Discrete Integration, Gradient Descent Optimization, and State Estimation | Classical mathematical techniques such as discrete integration, gradient descent optimization, and state estimation (exemplified by the Runge-Kutta method, Gauss-Newton minimization, and extended Kalman filter or EKF, respectively), rely on linear algebra and hence are only applicable to state vectors belonging to Euclidean spaces when implemented as described in the literature. This document discusses how to modify these methods so they can be applied to non-Euclidean state vectors, such as those containing rotations and full motions of rigid bodies. To do so, this document provides an in-depth review of the concept of manifolds or Lie groups, together with their tangent spaces or Lie algebras, their exponential and logarithmic maps, the analysis of perturbations, the treatment of uncertainty and covariance, and in particular the definitions of the Jacobians required to employ the previously mentioned calculus methods. These concepts are particularized to the specific cases of the SO(3) and SE(3) Lie groups, known as the special orthogonal and special Euclidean groups of R3, which represent the rigid body rotations and motions, describing their various possible parameterizations as well as their advantages and disadvantages. | \section*{Abstract}
Common mathematical techniques such as discrete integration, gradient descent optimization, and state estimation (exemplified by the Runge-Kutta method, Gauss-Newton minimization, and extended Kalman filter or \hypertt{EKF}, respectively), rely on linear algebra and hence are only applicable to state vectors belonging to Euclidean spaces when implemented as described in the literature. This article describes how to modify these methods so they can be applied to non Euclidean state vectors, such as those containing rotations and full motions of rigid bodies.
To do so, this article provides an in-depth review of the \nm{\mathbb{SO}(3)} and \nm{\mathbb{SE}(3)} Lie groups, known as the special orthogonal and special Euclidean groups of \nm{\mathbb{R}^3}, which represent the rigid body rotations and motions, placing special emphasis on the different possible representations, their tangent spaces, the analysis of perturbations, and in particular the definitions of the jacobians required to employ the previously mentioned calculus methods.
\textbf{\emph{Keywords}}: Lie algebra, SO(3), SE(3), manifold, tangent space, state estimation, EKF, discrete integration, Runge-Kutta, gradient descent optimization, minimization, Gauss-Newton
\section*{Acronyms}
\begin{table}[ht]
\begin{tabular}{lp{6.4cm}p{0.1cm}lp{6.4cm}}
\hypertarget{CDF}{\texttt{CDF}} & Cumulative Distribution Functions & & \hypertarget{PDF}{\texttt{PDF}} & Probability Density Function \\
\hypertarget{ECEF}{\texttt{ECEF}} & Earth Centered Earth Fixed & & \hypertarget{PMF}{\texttt{PMF}} & Probability Mass Function \\
\hypertarget{EKF}{\texttt{EKF}} & Extended Kalman Filter & & \hypertarget{PSD}{\texttt{PSD}} & Power Spectral Density \\
\hypertarget{NED}{\texttt{NED}} & North East Down & & \hypertarget{ScLERP}{\texttt{ScLERP}} & Screw linear interpolation \\
\hypertarget{ODE}{\texttt{ODE}} & Ordinary Differential Equation & & \hypertarget{SLERP}{\texttt{SLERP}} & Spherical linear interpolation \\
\end{tabular}
\end{table}
\section{Introduction and Outline}\label{sec:Outline}
The widespread implementations of common calculus techniques such as discrete integration (exemplified by the Runge-Kutta method), gradient descent optimization (Gauss-Newton or Levenberg-Marquardt), and state estimation (extended Kalman filter or \hypertt{EKF}), are designed to work on state vectors belonging to linear Euclidean spaces \cite{Blanco2020}, and hence rely of linear algebra. There exist two possible approaches for those cases in which the state vector contains non Euclidean components, such as rigid body motions:
\begin{itemize}
\item The solution most commonly observed in the literature is to incorporate each component of the pose (position plus attitude) as an unconstrained real number into the state vector. The above techniques can then be employed without difficulties, but the resulting state vector does not comply with the constraints imposed by having some of its members being components of a rigid body pose, as these constraints have not been taken into account when integrating, optimizing, or estimating. The solutions hence need to be projected back into the space of valid rigid body poses, but this does not hide the fact that the whole process has been performed without respecting the motion constraints, which often has negative consequences for the accuracy, stability, and consistency of the solution \cite{Sola2018}.
\item The approach taken in some recent robotics literature, in particular in the field of motion estimation for navigation, consists on reformulating the above calculus techniques (integration, optimization, filtering) taking into account that some members of the state vector represent rigid body poses and hence can not be treated as Euclidean. By modeling these states properly, the quality of the solution can be improved. The use of Lie theory, with its manifolds and tangent spaces, enables the construction of a rigorous calculus corpus to handle uncertainties, derivatives, and integrals of non euclidean elements with precision and ease \cite{Sola2018}.
\end{itemize}
This article begins with a review of the concepts of random variables, stochastic processes, and white noise in section \ref{sec:Error_Random}, which are necessary for the section \ref{sec:Calculus_Euclidean} descriptions of the Runge-Kutta discrete integration, Gauss-Newton minimization, and \hypertt{EKF} when applied to Euclidean spaces. Section \ref{sec:Algebra} introduces the concepts of Lie groups and their tangent spaces or Lie algebras, and adapts the three calculus techniques so they can be applied when the state vector contains non Euclidean Lie group components. Sections \ref{sec:Rotate} and \ref{sec:Motion} particularize the generic concepts of section \ref{sec:Algebra} to the specific cases of both rigid body rotations as well as full rigid body motions.
\section{Random Variables, Stochastic Processes, and White Noise}\label{sec:Error_Random}
This section provides an introduction to the random variables and processes required to model those physical systems that can not be represented by deterministic models due to their inherent randomness, which results in the same set of parameter values and initial conditions leading to different outputs.
\subsection{Random Variables}\label{subsec:Error_RandomVariables}
Consider a random experiment (one in which the outcome is uncertain) with a sample space \nm{\Omega} (collection of possible elementary outcomes of the experiment), and let \nm{\omega} be a sample point belonging to \nm{\Omega}. A \emph{random variable} \nm{X\lrp{\omega}} (generally just \nm{X}) is a single valued real function that assigns a real number, called the value of \nm{X\lrp{\omega}}, to each sample point \nm{\omega \in \Omega} \cite{Ibe2005,Papoulis2002}. A random variable hence represents a map between the sample and real spaces \nm{\{X : \Omega \rightarrow \mathbb{R} \ | \ \omega \in \Omega \rightarrow X\lrp{\omega} \in \mathbb{R}\}}. The \emph{realization} of a random variable is the real variable obtained after a given experiment.
A random variable \nm{X} is completely described by its \emph{cumulative distribution function} (\hypertt{CDF}) \nm{F_X}, which represents the probability that the value of \nm{X} is less or equal than the function input \cite{Farrell2008}:
\neweq{F_X\lrp{x} = P\lrsb{X \leq x} \ \ \ \ \ \ \ -\infty < x < \infty}{eq:Error_rvar_CDF}
Random variables can also be described by the \hypertt{CDF} derivative, known as the \emph{probability mass function} (\hypertt{PMF}) \nm{p_X} in case of discrete random variables (those that can take at most a countable number of possible values) or as \emph{probability density function} (\hypertt{PDF}) \nm{f_X} for continuous ones (those that can take an uncountable number of possible values):
\begin{eqnarray}
\nm{F_X\lrp{x}} & = & \nm{\sum_{x_k \leq x} p_X\lrp{x_k}}\label{eq:Error_rvar_PMF} \\
\nm{F_X\lrp{x}} & = & \nm{\int_{- \infty}^x f_X\lrp{y} \, dy} \label{eq:Error_rvar_PDF}
\end{eqnarray}
The \emph{expected value} \nm{E\lrsb{X}} or \emph{mean} \nm{\mu_X} of a random variable \nm{X} is a function defined as its average value over a large number of experiments, and represents its central or typical value:
\neweq{E\lrsb{X} = \mu_X = \begin{dcases*}
\nm{\sum_k x_k \, p_X\lrp{x_k}} & when \nm{X} is discrete \\
\nm{\int_{- \infty}^{\infty} y \, f_X\lrp{y} \, dy} & when \nm{X} is continuous
\end{dcases*}} {eq:Error_rvar_Mean}
If a function acts on a random variable, then its output is also a random variable \cite{Simon2006}, and it is hence possible to compute the expected value of the output random variable\footnote{Note that the mean can be considered as the expected value of the \nm{f\lrp{X} = X} function.}. The \emph{variance} \nm{\sigma_X^2}, \nm{Var\lrp{X}}, or second central moment of a random variable \nm{X}, is the expected value of the squared deviation of \nm{X} from its mean, and measures the spread of its \hypertt{PMF} or \hypertt{PDF} about its expected value \cite{Ibe2005}, this is, how much the random variable is expected to deviate from its mean. The square root of the variance is called the \emph{standard deviation} \nm{\sigma_X}.
\neweq{Var\lrp{X} = \sigma_X^2 = E\lrsb{\lrp{X - \mu_X}^2} = \begin{dcases*}
\nm{\sum_k \lrp{x_k - \mu_X}^2 \, p_X\lrp{x_k}} & when \nm{X} is discrete \\
\nm{\int_{- \infty}^{\infty} \lrp{y - \mu_X}^2 \, f_X\lrp{y} \, dy} & when \nm{X} is continuous
\end{dcases*}} {eq:Error_rvar_Variance}
The variance and mean of a random variable are related by the following expression, where \nm{E\lrsb{X^2}} is the second moment of \nm{X}.
\neweq{\sigma_X^2 = E\lrsb{\lrp{X - \mu_X}^2} = E\lrsb{X^2} - 2 \, \mu_X \, E\lrsb{X} + \mu_X^2 = E\lrsb{X^2} - \mu_X^2 = \mu_{\small{X}\ds{^2}} - \mu_X^2}{eq:Error_rvar_VarianceMean}
The notation \nm{X \sim \lrp{\mu_X, \ \sigma_X^2}} means that the random variable X has \nm{\mu_X} mean and \nm{\sigma_X^2} variance. A \emph{normal} or \emph{Gaussian} random variable X of parameters \nm{\mu_X} and \nm{\sigma_X^2} \cite{Farrell2008}, represented as \nm{X \sim N\lrp{\mu_X, \ \sigma_X^2}}, is one whose \hypertt{PDF} responds to:
\neweq{f_X\lrp{x} = \dfrac{1}{\sqrt{2 \, \pi \, \sigma_X^2}} \, exp\lrp{- \, \dfrac{\lrp{x - \mu_X}^2}{2 \; \sigma_X^2}} \ \ \ \ \ \ \ -\infty < x < \infty}{eq:Error_Normal}
The expected value and variance of a normal random variable \nm{N\lrp{\mu_X, \ \sigma_X^2}} are \nm{\mu_X} and \nm{\sigma_X^2}, respectively. A normal random variable \nm{N\lrp{0, \, 1}} of zero mean and unit variance is called a \emph{standard normal random variable}. It is worth noting that any affine\footnote{In this contest affine means a function of the form \nm{y = a \, x + b}, while linear means \nm{y = a \, x}.} function of a Gaussian random variable results in a Gaussian random variable \cite{Simon2006}.
The \emph{discrete uniform distribution} assigns the same probability to each of its N possible values \cite{Ibe2005}. Represented by \nm{X \sim U\lrp{a, \ a + N - 1}}, its expected value is \nm{a + (N - 1) / 2} and its variance is \nm{(N^2 - 1) / 12}. Its \hypertt{PMF} responds to:
\neweq{p_X\lrp{x_k} = \begin{dcases*}
\nm{\frac{1}{N}} & \nm{x_k = a, \ a+1, \ldots, \ a+N-1}\\
\nm{0} & otherwise
\end{dcases*}} {eq:Error_uniform}
Consider now two random variables \nm{X} and \nm{Y} defined in the same sample space \nm{\Omega} with expected values \nm{\mu_X} and \nm{\mu_Y}, respectively, and variances \nm{\sigma_X^2} and \nm{\sigma_Y^2}. They are called \emph{independent} if their results do not depend on each other:
\neweq{F_{XY}\lrp{x, \, y} = P\lrsb{X \leq x, \, Y \leq y} = P\lrsb{X \leq x} \, P\lrsb{Y \leq y} = F_X\lrp{x} \, F_Y\lrp{y}}{eq:Error_rvar_indep}
The \emph{central limit theorem} states that the sum of independent random variables tends towards a Gaussian random variable, regardless of the \hypertt{CDF} of the individual random variables that contribute to the sum \cite{Ibe2005,Simon2006}. If a given random variable \nm{X} is realized many times, the \emph{law of large numbers} states that the average of the realizations is close to the random variable expected value \nm{\mu_X}, and tends to it as the numbers of realizations grows \cite{Ibe2005}.
\emph{Stochastic simulations}, also known as \emph{Monte Carlo simulations}, use randomness to solve complex problems that may have a deterministic nature \cite{Ripley1987, Sawilowsky2003} and that are based on multiple unknown parameters, many of which are difficult to obtain experimentally \cite{Shojaeefard2003}. They rely on defining the domain of possible inputs, randomly generating inputs from a probability distribution over the domain, performing a deterministic computation based on those inputs, and finally aggregating the results by means of a set of metrics.
The \emph{correlation} \nm{R_{XY}} and the \emph{covariance} \nm{C_{XY}} of the random variables X and Y are two measures of the linear correlation between both random variables \cite{Ibe2005}. They are defined as:
\begin{eqnarray}
\nm{R\lrp{X, \, Y} = R_{XY}} & = & \nm{E\lrsb{X \cdot Y} = \mu_{X \cdot Y}}\label{eq:Error_rvar_Correlation} \\
\nm{C\lrp{X, \, Y} = C_{XY}} & = & \nm{E\lrsb{\lrp{X - \mu_X} \cdot \lrp{Y - \mu_Y}} = E\lrsb{X \cdot Y} - \mu_X \, \mu_Y = R_{XY} - \mu_X \, \mu_Y}\label{eq:Error_rvar_Covariance}
\end{eqnarray}
Two random variables are \emph{uncorrelated} if their covariance is zero \nm{\lrp{C_{XY} = 0 \rightarrow R_{XY} = \mu_X \, \mu_Y}}. Independent random variables are always uncorrelated, but not the other way around, as two uncorrelated random variables may not necessarily be independent if there exists a non linear dependence between them. \emph{Orthogonal} random variables are those whose correlation is zero \nm{\lrp{R_{XY} = 0}}, so they may or may not also be uncorrelated. If they are, at least one of them is zero mean.
The expected value and variance of the sum and product of two random variables are of particular interest. Given two random variables \nm{X} and \nm{Y} with expected values \nm{\lrb{\mu_X, \mu_Y}} and variances \nm{\lrb{\sigma_X^2, \sigma_Y^2}}, its sum \nm{X + Y} and product \nm{X \cdot Y} are also random variables, as indicated above. The following expressions can be easily obtained by applying the equations above\footnote{These expressions can be further simplified when \nm{X} and \nm{Y} are uncorrelated.} \cite{Frishman1971}:
\begin{eqnarray}
\nm{\mu_{X+Y}} & = & \nm{E\lrsb{X + Y} = \mu_X + \mu_Y}\label{eq:Error_sum_mean} \\
\nm{\sigma_{X+Y}^2} & = & \nm{E\lrsb{\lrp{X + Y}^2} - \mu_{X+Y}^2 = \sigma_X^2 + 2 \, C_{XY} + \sigma_Y^2}\label{eq:Error_sum_variance} \\
\nm{\mu_{X \cdot Y}} & = & \nm{E\lrsb{X \cdot Y} = R_{XY} = C_{XY} + \mu_X \, \mu_Y}\label{eq:Error_product_mean} \\
\nm{\sigma_{X \cdot Y}^2} & = & \nm{E\lrsb{\lrp{X \cdot Y}^2} - \mu_{X \cdot Y}^2 = C_{{\small{X}}{\ds{^2}}{\small{Y}}{\ds{^2}}} - C_{XY}^2 - 2 \, C_{XY} \, \mu_X \, \mu_Y + \sigma_X^2 \, \sigma_Y^2 + \sigma_X^2 \, \mu_Y^2 + \mu_X^2 \, \sigma_Y^2} \label{eq:Error_product_variance}
\end{eqnarray}
\subsection{Random Vectors}\label{subsec:Error_RandomVectors}
A \emph{random vector} \nm{\vec X = \lrsb{X_1,\dotsc,X_n}^T} is a collection of random variables obtained from the same sample space \nm{\Omega} \cite{Ibe2005,Papoulis2002}. The random vector joint \hypertt{CDF}, \hypertt{PMF}, and \hypertt{PDF} are defined as follows:
\begin{eqnarray}
\nm{F_{X_1,\dotsc,X_n}\lrp{x_1,\dotsc,x_n}} & = & \nm{P\lrsb{\{X_1 \leq x_1\} \cap \ldots \cap \{X_n \leq x_n\}} = P\lrsb{X_1 \leq x_1,\dotsc,X_n \leq x_n}} \label{eq:Error_rvec_CDF} \\
\nm{F_{X_1,\dotsc,X_n}\lrp{x_1,\dotsc,x_n}} & = & \nm{\sum_{k_1 \leq x_1} \dots \sum_{k_n \leq x_n} p_{X_1,\dotsc,X_n}\lrp{k_1,\dotsc,k_n}} \label{eq:Error_rvec_PMF2_Joint} \\
\nm{F_{X_1,\dotsc,X_n}\lrp{x_1,\dotsc,x_n}} & = & \nm{\int_{- \infty}^{x_1} \dots \int_{- \infty}^{x_n} f_{X_1,\dotsc,X_n}\lrp{y_1,\dotsc,y_n}\, dy_n \ldots dy_1} \label{eq:Error_rvec_PDF_Joint}
\end{eqnarray}
When the components \nm{X_1,\dotsc,X_n} of the random vector \nm{\vec X} are independent from each other, its joint \hypertt{CDF}, \hypertt{PMF}, and \hypertt{PDF} are just the product of the respective functions of each of the random vector components \cite{Farrell2008}. The expected value \nm{E\lrsb{\vec X}} or mean \nm{\vec \mu_X} and the variance \nm{\vec \sigma_X^2} of a random vector \nm{\vec X} are defined as the vectors of those of its components:
\begin{eqnarray}
\nm{E\lrsb{\vec X}} & = & \nm{\vec \mu_X = \lrsb{\mu_{X1},\dotsc,\mu_{Xn}}^T} \label{eq:Error_rvec_mean} \\
\nm{Var\lrp{\vec X}} & = & \nm{\vec \sigma_X^2 = \lrsb{\sigma_{X1}^2,\dotsc,\sigma_{Xn}^2}^T} \label{eq:Error_rvec_var}
\end{eqnarray}
Given two random vectors \nm{\vec X \in \mathbb{R}^m} and \nm{\vec Y \in \mathbb{R}^n}, their \emph{correlation matrix} \nm{\vec R_{XY}} is defined so \nm{R_{ij} = R\lrp{X_i, \, Y_j}}, while their \emph{covariance matrix} \nm{\vec C_{XY}} verifies that \nm{C_{ij} = C\lrp{X_i, \, Y_j}} \cite{Farrell2008}:
\begin{eqnarray}
\nm{\vec R\lrp{\vec X, \, \vec Y} = \vec R_{XY}} & = & \nm{E\lrsb{\vec X \, \vec Y^T}}\label{eq:Error_rvec_CorrelationMatrix} \\
\nm{\vec C\lrp{\vec X, \, \vec Y} = \vec C_{XY}} & = & \nm{ E\lrsb{\lrp{\vec X - \vec \mu_X}\lrp{\vec Y - \vec \mu_Y}^T} = E\lrsb{\vec X \, \vec Y^T} - \vec \mu_X \, \vec \mu_Y^T = \vec R_{XY} - \vec \mu_X \, \vec \mu_Y^T}\label{eq:Error_rvec_CovarianceMatrix}
\end{eqnarray}
The \emph{autocorrelation} and \emph{autocovariance} matrices \nm{\vec R_{XX}} and \nm{\vec C_{XX}} of a random vector \nm{\vec X \in \mathbb{R}^m} are defined as the correlation and covariance matrices of that vector with itself. Both are square, symmetric \nm{\lrp{\vec R_{XX} = \vec R_{XX}^T, \, \vec C_{XX} = \vec C_{XX}^T}}, positive semidefinite \nm{\lrp{\vec z^T \, \vec R_{XX} \, \vec z \geq 0, \, \vec z^T \, \vec C_{XX} \, \vec z \geq 0, \forall \, \vec z \in \mathbb{R}^m}}, and their diagonals contain the second moments and variances of each of the random variables \nm{X_i} within \nm{\vec X}:
\begin{eqnarray}
\nm{\vec R\lrp{\vec X} = \vec R_{XX}} & = & \nm{E\lrsb{\vec X \, \vec X^T}}\label{eq:Error_rvec_autoCorrelationMatrix} \\
\nm{\vec C\lrp{\vec X} = \vec C_{XX}} & = & \nm{E\lrsb{\lrp{\vec X - \vec \mu_X}\lrp{\vec X - \vec \mu_X}^T} = E\lrsb{\vec X \, \vec X^T} - \vec \mu_X \, \vec \mu_X^T = \vec R_{XX} - \vec \mu_X \, \vec \mu_X^T}\label{eq:Error_rvec_autoCovarianceMatrix}
\end{eqnarray}
A normal or Gaussian random vector is that whose components are all normal random variables. As in the case of scalar random variables, an affine transformation of a Gaussian random vector results in a new Gaussian random vector.
\subsection{Stochastic Processes}\label{subsec:Error_StochasticProcesses}
A \emph{random process} or \emph{stochastic process} enlarges the concept of random vector (or random variable when the vector size is one) to include time. Given a sample vector \nm{\vec \omega} of the sample space \nm{\Omega} and a parameter t belonging to a parameter set \nm{\mathbb{T}} (generally time), a stochastic process assigns a real vector \nm{\{\vec X : \mathbb{T}, \Omega \rightarrow \mathbb{R}^m \ | \ t \in \mathbb{T}, \, \vec \omega \in \Omega \rightarrow \vec X\lrp{t, \, \vec \omega} \in \mathbb{R}^m\}} \cite{Ibe2005,Papoulis2002,Hoel1972}. If the sample vector \nm{\vec \omega} is fixed, the random process \nm{\vec X\lrp{t}} behaves as a function of time; on the other hand, if the time is fixed, the stochastic process \nm{\vec X\lrp{\vec \omega}} defaults to a random vector. A stochastic process is thus a family of random vectors (either discrete or continuous) indexed by a continuous parameter \nm{t \in \mathbb{T}}. If the parameter is discrete \nm{t \in \mathbb{Z}^+} \footnote{\nm{\mathbb{Z}^+} represents the set of positive integers.}, then the appropriate name is \emph{stochastic sequence} \cite{Farrell2008, Simon2006}.
Size one stochastic processes \nm{X\lrp{t, \, \omega}}, generally represented just by \nm{X\lrp{t}}, are completely described by their \hypertt{CDF}, while if \nm{\vec X\lrp{t}} is instead a random vector, it is represented by its joint \hypertt{CDF}:
\begin{eqnarray}
\nm{F_X\lrp{x; \, t}} & = & \nm{P\lrsb{X\lrp{t} \leq x}}\label{eq:Error_rpro_CDF_single} \\
\nm{F_X\lrp{x_1,\dotsc,x_n; \, t}} & = & \nm{P\lrsb{X_1\lrp{t} \leq x_1,\dotsc,X_n\lrp{t} \leq x_n}}\label {eq:Error_rpro_CDF}
\end{eqnarray}
The joint \hypertt{PMF} and \hypertt{PDF} are also defined in similar fashion. The \emph{ensemble average} or mean of a random process becomes a function of time:
\neweq{\vec \mu_X\lrp{t} = E\lrsb{\vec X\lrp{t}}} {eq:Error_rpro_Mean}
Note that the random process \nm{\vec X\lrp{t}} evaluated at different times comprises different random vectors of the same size. It is then possible to apply the concepts of autocorrelation and autocovariance introduced in section \ref{subsec:Error_RandomVectors} to any two of these random vectors, providing quantitative measures of the similarity of the random process at two different times, this is, measuring by how much a signal is similar to its time shifted version \cite{Ibe2005}. This results in the \emph{autocorrelation} \nm{\vec R_{XX}\lrp{t,\, t + \tau}} and the \emph{autocovariance} \nm{\vec C_{XX}\lrp{t,\, t + \tau}}:
\begin{eqnarray}
\nm{\vec R_{XX}\lrp{t, \, t + \tau}} & = & \nm{E\lrsb{\vec X\lrp{t} \, \vec X^T\lrp{t + \tau}}}\label{eq:Error_rpro_Autocorrelation} \\
\nm{\vec C_{XX}\lrp{t, \, t + \tau}} & = & \nm{E\lrsb{\big(\vec X\lrp{t} - \vec \mu_X\lrp{t}\big)\big(\vec X\lrp{t + \tau} - \vec \mu_X\lrp{t + \tau}\big)^T}} \nonumber \\
& = & \nm{E\lrsb{\vec X\lrp{t} \, \vec X^T\lrp{t + \tau}} - \vec \mu_X\lrp{t} \, \vec \mu_X^T\lrp{t + \tau} = \vec R_{XX}\lrp{t, \, t + \tau} - \vec \mu_X\lrp{t} \, \vec \mu_X^T\lrp{t + \tau}}\label{eq:Error_rpro_Autocovariance}
\end{eqnarray}
The autocovariance is zero when the two observations of \nm{\vec X} are independent, meaning that there is no coupling between \nm{\vec X\lrp{t}} and \nm{\vec X\lrp{t + \tau}}, and they are called uncorrelated. As with the random vectors, the reverse is not true, as two uncorrelated observations does not necessarily mean that they are independent.
A \emph{wide sense stationary process} is that in which the mean does not vary with time and the autocorrelation depends exclusively on the time difference\footnote{Strict sense stationary processes are those in which the complete \hypertt{CDF} is time invariant, not only its mean and autocorrelation. The definition is usually too restrictive for any practical use.}:
\begin{eqnarray}
\nm{\vec \mu_{X\sss{WSS}}\lrp{t}} & = & \nm{E\lrsb{\vec X_{\sss {WSS}}\lrp{t}} = \vec \mu_{X\sss{WSS}}}\label{eq:Error_rpro_Mean_Stationary} \\
\nm{\vec R_{XX\sss{WSS}}\lrp{t, \, t + \tau}} & = & \nm{E\lrsb{\vec X_{\sss{WSS}}\lrp{t} \, \vec X_{\sss{WSS}}^T\lrp{t + \tau}} = \vec R_{XX\sss{WSS}}\lrp{\tau}}\label{eq:Error_rpro_Autocorrelation_Stationary}
\end{eqnarray}
Consider now two stochastic processes \nm{\vec X\lrp{t}} and \nm{\vec Y\lrp{t}} defined in the same sample space \nm{\Omega}. The \emph{crosscorrelation} \nm{\vec R_{XY}\lrp{t,\, t + \tau}} and \emph{crosscovariance} \nm{\vec C_{XY}\lrp{t,\, t + \tau}} measure how similar two different processes (or signals) are when one is time shifted with respect to the other \cite{Ibe2005}:
\begin{eqnarray}
\nm{\vec R_{XY}\lrp{t, \, t + \tau}} & = & \nm{E\lrsb{\vec X\lrp{t} \, \vec Y^T\lrp{t + \tau}}} \label{eq:Error_rpro_Crosscorrelation} \\
\nm{\vec C_{XY}\lrp{t, \, t + \tau}} & = & \nm{E\lrsb{\big(\vec X\lrp{t} - \vec \mu_X\lrp{t}\big)\big(\vec Y\lrp{t + \tau} - \vec \mu_Y\lrp{t + \tau}\big)^T}} \nonumber \\
& = & \nm{E\lrsb{\vec X\lrp{t} \, \vec Y^T\lrp{t + \tau}} - \vec \mu_X\lrp{t} \, \vec \mu_Y^T\lrp{t + \tau} = \vec R_{XY}\lrp{t, \, t + \tau} - \vec \mu_X\lrp{t} \, \vec \mu_Y^T\lrp{t + \tau}}\label{eq:Error_rpro_Crosscovariance}
\end{eqnarray}
Two processes \nm{\vec X\lrp{t}} and \nm{\vec Y\lrp{t}} are orthogonal if their crosscorrelation is zero for all t and \nm{t + \tau}, while they are uncorrelated if their crosscovariance is zero. They are jointly wide sense stationary if their crosscorrelation is independent of the absolute time:
\neweq{\vec R_{XY\sss{WSS}}\lrp{t, \, t + \tau} = \vec R_{XY_{\sss{WSS}}}\lrp{\tau}} {eq:Error_rpro_Crosscorrelation_Stationary}
Consider also a stochastic process \nm{\vec X\lrp{t}} that has one realization \nm{\vec x\lrp{t}}. It is then possible to define the \emph{time average} \nm{A\big[\vec X\lrp{t}\big]} and the \emph{time autocorrelation} \nm{R\big[\vec X\lrp{t}, \, \tau\big]} for continuous processes as:
\begin{eqnarray}
\nm{A\big[\vec X\lrp{t}\big]} & = & \nm{\lim\limits_{T \to \infty} \frac{1}{2\, T} \int_{- T}^{T} \vec x\lrp{t} \, \mathrm{d}t} \label{eq:Error_rpro_TimeAverage} \\
\nm{R\big[\vec X\lrp{t}, \, \tau\big]} & = & \nm{A\lrsb{\vec X\lrp{t} \, \vec X^T\lrp{t + \tau}}}\label{eq:Error_rpro_TimaAutoCorrelation}
\end{eqnarray}
The discrete time definitions can be derived accordingly. Finally, an \emph{ergodic} process \cite{Simon2006} is a stationary random process for which
\begin{eqnarray}
\nm{A_{\sss{ERG}}\big[\vec X\lrp{t}\big]} & = & \nm{E_{\sss{ERG}}\lrsb{\vec X}} \label{eq:Error_rpro_Ergodic1} \\
\nm{R_{\sss{ERG}}\big[\vec X\lrp{t}, \, \tau\big]} & = & \nm{\vec R_{XX\sss{ERG}}\lrp{\tau}}\label{eq:Error_rpro_Ergodic2}
\end{eqnarray}
\subsection{White Noise}\label{subsec:Error_WhiteNoise}
If any two random vectors \nm{\vec X\lrp{t_1}} and \nm{\vec X\lrp{t_2}} taken from a stochastic process \nm{\vec X\lrp{t}} are independent for all \nm{t_1 \neq t_2}, then the random process \nm{\vec X\lrp{t}} is called \emph{white noise}. Otherwise, it is known as \emph{colored noise} \cite{Simon2006}.
The whiteness or color content of a stochastic process can be characterized by its \emph{power spectrum} or \emph{power spectral density} (\hypertt{PSD}) \nm{S_{XX}\lrp{\omega}}. For wide sense stationary processes, it is defined as the Fourier transform of its autocorrelation function \nm{R_{XX}\lrp{\tau}} \cite{Simon2006}:
\neweq{\vec S_{XX}\lrp{\omega} = \begin{dcases*}
\nm{\sum_{k = - \infty}^{\infty} \vec R_{XX}\lrp{k} \, exp\lrp{- i \, \omega \, k} \ \ \ \omega \in \lrsb{- \pi, \, \pi}} & when \nm{X} is discrete \\
\nm{\int_{- \infty}^{\infty} \vec R_{XX}\lrp{\tau} \, exp\lrp{- i \, \omega \, \tau} \, \mathrm{d}\tau} & when \nm{X} is continuous
\end{dcases*}} {eq:Error_whitenoise_Fourier}
The autocorrelation can be recovered by means of the inverse Fourier transform:
\neweq{\begin{dcases*}
\nm{\vec R_{XX}\lrp{k} = \dfrac{1}{2\pi}\int_{- \infty}^{\infty} \vec S_{XX}\lrp{\omega} \, exp\lrp{i \, \omega \, k} \, \mathrm{d}\omega} & when \nm{\vec X\lrp{t}} is discrete \\
\nm{\vec R_{XX}\lrp{\tau} = \dfrac{1}{2\pi}\int_{- \infty}^{\infty} \vec S_{XX}\lrp{\omega} \, exp\lrp{i \, \omega \, \tau} \, \mathrm{d}\omega} & when \nm{\vec X\lrp{t}} is continuous
\end{dcases*}} {eq:Error_whitenoise_Fourier_inverse}
In case of continuous time wide sense stationary stochastic processes\footnote{Similar expressions can be easily obtained for discrete time processes.}, the \emph{power} is defined as:
\neweq{\vec P_{XX} = \dfrac{1}{2\pi}\int_{- \infty}^{\infty} \vec S_{XX}\lrp{\omega} \, \mathrm{d}\omega}{eq:Error_whitenoise_Fourier_power}
In the case of two continuous time jointly wide sense stationary stochastic processes \nm{\vec X\lrp{t}} and \nm{\vec Y\lrp{t}}, the \emph{cross power spectrum} \nm{S_{XY}\lrp{\omega}} is defined as the Fourier transform of their crosscorrelation \nm{R_{XY}\lrp{\tau}} \cite{Simon2006}:
\begin{eqnarray}
\nm{\vec S_{XY}\lrp{\omega}} & = & \nm{\int_{- \infty}^{\infty} \vec R_{XY}\lrp{\tau} \, exp\lrp{- i \, \omega \, \tau} \, \mathrm{d}\tau}\label{eq:Error_whitenoise_Fourier_cross} \\
\nm{\vec R_{XY}\lrp{\tau}} & = & \nm{\dfrac{1}{2\pi}\int_{- \infty}^{\infty} \vec S_{XY}\lrp{\omega} \, exp\lrp{i \, \omega \, \tau} \, \mathrm{d}\omega}\label{eq:Error_whitenoise_Fourier_inverse_cross}
\end{eqnarray}
A white noise process \nm{\vec N\lrp{t}} (continuous time) or \nm{\vec N\lrp{k}} (discrete time) is one whose \hypertt{PSD} is constant for all frequencies, this is, a random process having equal power at all frequencies \cite{Farrell2008}. These processes do not have any correlation with themselves except at the present time \cite{Simon2006}. The definition for discrete time processes relies on the \emph{Kronecker delta function} \nm{\delta_k}\footnote{The Kronecker delta function \nm{\delta\lrp{k}} is valued 0 for all k except at \nm{k = 0}, where it is \nm{1}.}:
\begin{eqnarray}
\nm{\vec R_{NN}\lrp{k}} & = & \nm{\vec \sigma^2 \, \delta_k} \label{eq:Error_whitenoise_DiscreteWhiteNoise2} \\
\nm{\vec S_{NN}\lrp{\omega}} & = & \nm{\vec \sigma^2 = \vec R_{NN}\lrp{0} \ \ \ \forall \, \omega \in \lrsb{- \pi, \, \pi} } \label{eq:Error_whitenoise_DiscreteWhiteNoise}
\end{eqnarray}
while that for continuous time processes makes use of the \emph{impulse Dirac delta function} \nm{\delta\lrp{\tau}}\footnote{The Dirac delta function \nm{\delta\lrp{\tau}} is valued 0 everywhere except at \nm{\tau = 0}, where it is \nm{\infty}. Its integral over any space containing \nm{\tau = 0} is 1.}:
\begin{eqnarray}
\nm{\vec R_{NN}\lrp{\tau}} & = & \nm{\vec \sigma^2 \, \delta\lrp{\tau}} \label{eq:Error_whitenoise_ContinuousWhiteNoise2} \\
\nm{\vec S_{NN}\lrp{\omega}} & = & \nm{\vec \sigma^2 = \vec R_{NN}\lrp{0} \ \forall \omega \in \mathbb{R}} \label{eq:Error_whitenoise_ContinuousWhiteNoise}
\end{eqnarray}
\section{Calculus Methods in Euclidean Space}\label{sec:Calculus_Euclidean}
This section describes the most common approaches to three frequent calculus problems, such as discrete integration, optimization, and state estimation. The solutions, known as the Runge-Kutta integration method, the Gauss-Newton optimization, and the \hypertt{EKF}, are intended for state vectors in which all components can be considered Euclidean. Note that the main objective of this article is how to modify these three techniques so they can cope with non Euclidean state vectors.
\subsection{Discrete Integration in Euclidean Spaces}\label{subsec:euclidean_integration}
Let's consider an Euclidean space time varying state vector \nm{\vec x\lrp{t} \in \mathbb{R}^m} in which its value at a given discrete time \nm{\vec x_k = \vec x\lrp{t_k}} is known. The objective is to determine the state vector value at a later time \nm{\vec x_{k+1} = \vec x\lrp{t_{k+1}} = \vec x\lrp{t_k + \Delta t}} by relying on evaluations of the state vector derivative with time:
\neweq{\xvecdot\lrp{t} = f\big(\xvec\lrp{t}, \, t\big)} {eq:algebra_integration_derivative}
The initial value first order \emph{ordinary differential equation} (\hypertt{ODE}) problem can be solved with varying degrees of complexity and accuracy \cite{Press2002}, three of which are described below:
\begin{itemize}
\item \emph{Euler's method} is a first order approach that relies on evaluating the time derivative at \nm{t_k} and considering that its value does not change for the duration of the integration interval \nm{\Delta t}. Its error is proportional to the square of the integration interval:
\neweq{\vec x_{k+1} \approx \vec x_k + \Delta t \ \xvecdot(\vec x_k, \, t_k)}{eq:algebra_integration_euler}
\item \emph{Heun's method} is a second order approach that requires two evaluations of the time derivative. The constant gradient is estimated as the average between the time derivative evaluation at the initial state and that at the result of Euler's method, and results in an error proportional to the cube of the integration interval:
\begin{eqnarray}
\nm{\vec v_1} & = & \nm{\xvecdot(\vec x_k, \, t_k)} \label{eq:algebra_integration_heun_one} \\
\nm{\vec v_2} & = & \nm{\xvecdot(\vec x_k + \Delta t \ \vec v_1, t_k + \Delta t)} \label{eq:algebra_integration_heun_two} \\
\nm{\vec x_{k+1}} & \nm{\approx} & \nm{\vec x_k + \dfrac{\Delta t}{2} \ \lrsb{\vec v_1 + \vec v_2}} \label{eq:algebra_integration_heun}
\end{eqnarray}
\item The \nm{\vec 4^{th}} \emph{order Runge-Kutta method} is the defacto standard and relies on four evaluations of the state vector time derivative to obtain an error proportional to the fifth power of the integration interval:
\begin{eqnarray}
\nm{\vec v_1} & = & \nm{\xvecdot(\vec x_k, \, t_k)} \label{eq:algebra_integration_rk4th_one} \\
\nm{\vec v_2} & = & \nm{\xvecdot(\vec x_k + \dfrac{\Delta t \ \vec v_1}{2}, t_k + \dfrac{\Delta t}{2})} \label{eq:algebra_integration_rk4th_two} \\
\nm{\vec v_3} & = & \nm{\xvecdot(\vec x_k + \dfrac{\Delta t \ \vec v_2}{2}, t_k + \dfrac{\Delta t}{2})} \label{eq:algebra_integration_rk4th_three} \\
\nm{\vec v_4} & = & \nm{\xvecdot(\vec x_k + \Delta t \ \vec v_3, t_k + \Delta t)} \label{eq:algebra_integration_rk4th_four} \\
\nm{\vec x_{k+1}} & \nm{\approx} & \nm{\vec x_k + \Delta t \ \lrsb{\dfrac{\vec v_1}{6} + \dfrac{\vec v_2}{3} + \dfrac{\vec v_3}{3} + \dfrac{\vec v_4}{6}}} \label{eq:algebra_integration_rk4th}
\end{eqnarray}
\end{itemize}
\subsection{Gradient Descent Optimization in Euclidean Spaces}\label{subsec:euclidean_gradient_descent}
Let's consider an Euclidean vector \nm{\vec x \in \mathbb{R}^m}, a nonlinear map \nm{\lrb{\vec f: \mathbb{R}^m \rightarrow \mathbb{R}^n \ | \ \vec f\lrp{\vec x} \in \mathbb{R}^n, \forall \ \vec x \in \mathbb{R}^m}} for which it is also possible to evaluate its jacobian \nm{\lrb{\vec J: \mathbb{R}^m \rightarrow \mathbb{R}^{nxm} \ | \ \vec J\lrp{\vec x} = \partial{\vec f\lrp{\vec x}} / \partial{\vec x} \in \mathbb{R}^{nxm}, \forall \ \vec x \in \mathbb{R}^m}}, and an error or cost function \nm{\vec{\mathcal E}\lrp{\vec x} = \vec f\lrp{\vec x} - \vec f_T \ \in \mathbb{R}^n} containing the difference between the map \nm{\vec f} and a target result \nm{\vec f_T}.
Let's also consider that the objective is to determine an input vector \nm{\vec x = \vec x_0 + \Delta \vec x} in the vicinity of a known initial value \nm{\vec x_0}, for which the cost function norm \nm{\| \vec{\mathcal E}\lrp{\vec x} \| \in \mathbb{R}} holds a local minimum, this is, \nm{\| \vec{\mathcal E}\lrp{\vec x_0 + \Delta \vec x} \| < \| \vec{\mathcal E}\lrp{\vec x_0} \|, \, \forall \ \vec x_0 \in \mathbb{R}^m}. The \emph{Gauss-Newton} optimization method provides a solution to this problem that relies on iteratively advancing the solution per (\ref{eq:algebra_gradient_descent_iterative}) starting with \nm{\vec x_0}:
\neweq{\vec x_{k+1} \longleftarrow \vec x_k + \Delta \vec x_k}{eq:algebra_gradient_descent_iterative}
Adopting a lighter notation in which \nm{\vec f_k = \vec f\lrp{\vec x_k}}, \nm{\vec J_k = \vec J\lrp{\vec x_k}}, and \nm{\vec{\mathcal E}_k = \vec{\mathcal E} \lrp{\vec x_k}}, the process concludes when the step diminution of the cost function norm falls below a given threshold (\nm{\| \vec{\mathcal E}_k \| - \| \vec{\mathcal E}_{k+1} \| < \delta}). The Gauss-Newton method consists on linearizing each step by performing a first order Taylor expansion of the cost function before minimizing its norm by equaling its derivative with respect to \nm{\Delta \vec x_k} to zero \cite{Hartley2003}:
\begin{eqnarray}
\nm{\vec{\mathcal E}_{k+1}} & = & \nm{\vec f_{k+1} - \vec f_T \approx \vec f_k + \vec J_k \, \Delta \vec x_k - \vec f_T = \vec{\mathcal E}_k + \vec J_k \, \Delta \vec x_k} \label{eq:algebra_gradient_descent_taylor} \\
\nm{\| \vec{\mathcal E}_{k+1} \|} & = & \nm{\vec{\mathcal E}_{k+1}^T \, \vec{\mathcal E}_{k+1} = \vec{\mathcal E}_k^T \, \vec{\mathcal E}_k + \Delta \vec x_k^T \, \vec J_k^T \, \vec J_k \, \Delta \vec x_k + 2 \, \Delta \vec x_k^T \, \vec J_k^T \, \vec{\mathcal E}_k} \label{eq:algebra_gradient_descent_norm} \\
\nm{\pderpar{\| \vec{\mathcal E}_{k+1} \|}{\Delta \vec x_k}} & = & \nm{0 \ \longrightarrow \ 2 \, \vec J_k^T \, \vec J_k \, \Delta \vec x_k + 2 \, \vec J_k^T \, \vec{\mathcal E}_k = 0 \ \longrightarrow \ \Delta \vec x_k = - \big(\vec J_k^T \, \vec J_k\big)^{-1} \, \vec J_k^T \, \vec{\mathcal E}_k} \label{eq:algebra_gradient_descent_solution}
\end{eqnarray}
The Gauss-Newton algorithm is just one type of a more generic class of iterative minimization methods grouped under the name of \emph{gradient descent methods}. The \emph{Newton} method relies on minimizing (equaling its \nm{\Delta \vec x_k} derivative to zero) a second order Taylor expansion of the cost function norm \nm{N\lrp{\vec x} = \| \vec{\mathcal E}\lrp{\vec x} \| \ \in \mathbb{R}}, which requires the computation of both its gradient \nm{\lrb{\vec \nabla: \mathbb{R}^m \rightarrow \mathbb{R}^{1xm} \ | \ \vec \nabla\lrp{\vec x} = \partial{N\lrp{\vec x}} / \partial{\vec x} \in \mathbb{R}^{1xm}, \forall \ \vec x \in \mathbb{R}^m}} and its Hessian \nm{\lrb{\vec H: \mathbb{R}^m \rightarrow \mathbb{R}^{mxm} \ | \ \vec H\lrp{\vec x} = \partial^2{N\lrp{\vec x}} / \partial{\vec x^2} \in \mathbb{R}^{mxm}, \forall \ \vec x \in \mathbb{R}^m}} at each step, resulting in:
\neweq{\Delta \vec x_k = - \vec H_k^{-1} \ \vec \nabla_k^T}{eq:algebra_newton_solution}
As the Hessian \nm{\vec H} can be difficult or expensive to compute, there exist several approximations that reduce the computational cost of each step, such as the \emph{steepest descent} method, which replaces the Hessian with the product of a constant and the identity matrix \nm{\vec I_m \in \mathbb{R}^{mxm}}, and the \emph{diagonal approximation}, which sets to zero all \nm{\vec H} components outside its main diagonal. In this sense, the Gauss-Newton method is just another approximation that employs a first order simplification of the Hessian, as proven in \cite{Baker2004}.
The convergence of none of these methods is guaranteed. In general, both Gauss-Newton and Newton work better near the local minimum, where the quadratic approximation is good, but may diverge when the initial value is further away, where the steepest descent and diagonal approximation methods may be more robust. To ensure that the error gets smaller in each iteration, it may be convenient to advance with a smaller \nm{\Delta \vec x_k} step. The \emph{Levenberg-Marquardt} algorithm employs a varying ratio between the Gauss-Newton (or Newton) and diagonal approximations to the Hessian, moving towards the former when the error \nm{\| \vec{\mathcal E}_k \|} decreases, and towards the later while repeating the step if it increases.
\subsection{State Estimation in Euclidean Spaces}\label{subsec:SS}
\emph{State estimation} is the problem of determining the value of the state of a dynamic system based on a series of noisy equations that describe the evolution of the state with time, together with a series of noisy measurements or observations of variables that also depend on the state. \emph{State} or \emph{state vector} refers to those variables that provide a representation of the condition or status of the system at a given instant in time. Section \ref{subsubsec:SS_SampledDataSystems} discusses the equations that describe the system dynamics, this is, the state evolution with time, and what is the best possible state estimation that can be obtained from them. Section \ref{subsubsec:SS_SampledObservations} describes the measurement or observation equations, and also reaches the best possible state estimate from the information they contain. Both approaches are combined in section \ref{subsubsec:SS_EKF}, which describes the extended Kalman filter or \hypertt{EKF}, the most widely used non linear state estimation algorithm.
\subsubsection{Sampled Data Systems}\label{subsubsec:SS_SampledDataSystems}
A \emph{state space system} is a mathematical representation of a physical process in which the variables (both state and input) are related by first order differential equations (for continuous systems) or difference equations (for discrete ones). If the state of the system (the value of the state variables) is known at a given time, and so are all the present and future inputs (the evolution with time of the input variables), it is then possible to obtain the evolution with time of all the state variables.
A \emph{sampled data system} is one whose dynamics are described by continuous time differential equations, but whose inputs only change at discrete time instants. Additionally, it is only necessary to estimate the state variables, or to be precise its mean and covariance\footnote{Although the term covariance is traditionally employed in state estimation, it is in fact referring to the state random vector autocovariance provided by (\ref{eq:Error_rvec_autoCovarianceMatrix}), or to the state random process autocovariance given by (\ref{eq:Error_rpro_Autocovariance}), depending on context.}, at those same discrete time instants \cite{Simon2006}. A continuous time nonlinear state system can be written as
\neweq{\xvecdot\lrp{t} = f\big(\xvec\lrp{t}, \, \uvec\lrp{t}, \ \wvec\lrp{t}, \, t\big)} {eq:SS_cont_time_system}
where \nm{\xvec \in \mathbb{R}^m} is the state vector, \nm{\uvec \in \mathbb{R}^n} is the known \emph{control} or \emph{input vector}, and \nm{\wvec \in \mathbb{R}^p} is the \emph{process noise}. These three vectors may have different sizes. Let's also assume that the process noise \nm{\wvec\lrp{t}} can be modeled by a zero mean continuous time white noise random process\footnote{Note that the process noise does not need to be Gaussian.} of covariance \nm{\Qvec_c} (sections \ref{subsec:Error_RandomVariables} and \ref{subsec:Error_WhiteNoise}):
\begin{eqnarray}
\nm{\wvec\lrp{t}} & \nm{\sim} & \nm{\lrp{\vec 0, \, \Qvec_c}}\label{eq:SS_cont_time_system_noise1} \\
\nm{\vec R_{ww}\lrp{t, \, \tau}} & = & \nm{E\lrsb{\wvec\lrp{t} \, \wvec^T\lrp{\tau}} = \Qvec_c \, \delta\lrp{t - \tau}}\label{eq:SS_cont_time_system_noise2}
\end{eqnarray}
\textbf{Linearization of Continuous Time Systems}
The dynamics represented by (\ref{eq:SS_cont_time_system}) can be linearized by performing a Taylor expansion around an unknown nominal state \nm{\xvec_N\lrp{t}} and process noise \nm{\wvec_N\lrp{t}}\footnote{As the input vector \nm{\uvec\lrp{t}} is known, there is no need to expand around it.}, assuming without loss of generality that \nm{\wvec_N\lrp{t} = \vec 0}. If it is not, it can be written as the sum of a zero mean part and a known deterministic part, which can then be added to the control vector. The expansion is truncated so only the first order terms remain, introducing linearization errors; these are higher the more nonlinear that \nm{f\lrp{\xvec, \, \uvec, \ \wvec, \, t}} is with respect to \nm{\xvec} and \nm{\wvec}, and the farther away that \nm{\xvec\lrp{t}} lies from \nm{\xvec_N\lrp{t}} and \nm{\wvec\lrp{t}} from \nm{\wvec_N = \vec 0} \cite{Simon2006}.
\neweq{\xvecdot\lrp{t} \approx f\rvert_N + \pderpar{f}{\xvec}\Bigr\rvert_N \, \lrp{\xvec - \xvec_N} + \pderpar{f}{\wvec}\Bigr\rvert_N \, \wvec = \pderpar{f}{\xvec}\Bigr\rvert_N \, \xvec + \lrp{f\rvert_N - \pderpar{f}{\xvec}\Bigr\rvert_N \, \xvec_N} + \pderpar{f}{\wvec}\Bigr\rvert_N \, \wvec} {eq:SS_cont_time_system_taylor}
where \nm{\mid_N} stands for evaluation at \nm{\big(\xvec_N\lrp{t}, \, \uvec\lrp{t}, \, \vec 0, \, t\big)}. The state system is now continuous time but linear:
\begin{eqnarray}
\nm{\xvecdot\lrp{t}} & \nm{\approx} & \nm{\Avec\lrp{t} \, \xvec\lrp{t} + \Bvec\lrp{t} \, \utilde\lrp{t} + \wtilde\lrp{t}}\label{eq:SS_cont_time_system_linear} \\
\nm{\Avec\lrp{t}} & = & \nm{\pderpar{f}{\xvec}\big(\xvec_N\lrp{t}, \, \uvec\lrp{t}, \, \vec 0, \, t\big)}\label{eq:SS_cont_time_system_linear_system_matrix} \\
\nm{\Bvec\lrp{t}} & = & \nm{\Ivec}\label{eq:SS_cont_time_system_linear_other_matrix} \\
\nm{\Lvec\lrp{t}} & = & \nm{\pderpar{f}{\wvec}\big(\xvec_N\lrp{t}, \, \uvec\lrp{t}, \, \vec 0, \, t\big)}\label{eq:SS_cont_time_system_linear_input_matrix} \\
\nm{\utilde\lrp{t}} & = & \nm{f\big(\xvec_N\lrp{t}, \, \uvec\lrp{t}, \, \vec 0, \, t\big) - \Avec\lrp{t} \, \xvec_N\lrp{t}}\label{eq:SS_cont_time_system_linear_input_vector} \\
\nm{\wtilde\lrp{t}} & = & \nm{\Lvec\lrp{t} \, \wvec\lrp{t} \sim \lrp{\vec 0, \, \Lvec \, \Qvec_c \, \Lvec^T} = \lrp{\vec 0, \, \Qtilde_c\lrp{t}}}\label{eq:SS_cont_time_system_linear_noise1} \\
\nm{\vec R_{\widetilde{w}\widetilde{w}}\lrp{t, \, \tau}} & = & \nm{E\lrsb{\wtilde\lrp{t} \, \wtilde^T\lrp{\tau}} = \Qtilde_c\lrp{t} \, \delta\lrp{t - \tau}}\label{eq:SS_cont_time_system_linear_noise2}
\end{eqnarray}
The above linear state system is based on a unitary \emph{input matrix} \nm{\Bvec \in \mathbb{R}^{mxm}} and a \emph{system matrix} \nm{\Avec\lrp{t} \in \mathbb{R}^{mxm}} that is the jacobian of the non linear system with respect to the state vector evaluated at the unknown nominal state. It also employs modified input \nm{\utilde\lrp{t} \in \mathbb{R}^m} and process noise \nm{\wtilde\lrp{t} \in \mathbb{R}^m} vectors.
\textbf{Comparison of Integrated Continuous and Discrete White Noise Processes}
Before continuing, let's make a parenthesis to compare the behavior of an integrated continuous white noise process with that of a discrete one, as the result is essential to the discretization of the continuous time state system (\ref{eq:SS_cont_time_system_linear}). According to section \ref{subsec:Error_WhiteNoise}, a continuous zero mean white noise is defined by \nm{\wvec\lrp{t} \sim \lrp{\vec 0, \, \Qvec_c}} and \nm{E\lrsb{\wvec\lrp{t} \, \wvec^T\lrp{\tau}} = \Qvec_c \, \delta\lrp{t - \tau}}, while a zero mean discrete time white noise process responds to \nm{\wvec_k \sim \lrp{\vec 0, \, \Qvec_d}} and \nm{E\lrsb{\wvec_k \, \wvec_l^T} = \Qvec_d \, \delta_{k-l}}. Let's integrate the continuous white noise with \nm{\dot{\zvec}\lrp{t} = \wvec\lrp{t}, \ \zvec\lrp{0} = \vec 0} and analyze the variation with time of the mean \nm{\vec \mu_z\lrp{t}} and covariance \nm{\vec C_{zz}\lrp{t}} of the integrated noise \nm{\zvec\lrp{t}}:
\begin{eqnarray}
\nm{\vec \mu_z\lrp{t}} & = & \nm{E\lrsb{\zvec\lrp{t}} = E\lrsb{\int_0^t \wvec\lrp{\alpha} \, \mathrm{d}\alpha} = \int_0^t E\lrsb{\wvec\lrp{\alpha}} \, \mathrm{d}\alpha = \vec 0}\label{eq:SS_cont_integr_noise_mean} \\
\nm{\vec C_{zz}\lrp{t}} & = & \nm{E\lrsb{\zvec\lrp{t} \, \zvec^T\lrp{t}} - \vec \mu_z\lrp{t} \, \vec \mu_z^T\lrp{t} = E\lrsb{\int_0^t \wvec\lrp{\alpha} \, \mathrm{d}\alpha \, \int_0^t \wvec^T\lrp{\beta} \, \mathrm{d}\beta}} \nonumber \\
& = & \nm{\int_0^t \int_0^t E\lrsb{\wvec\lrp{\alpha} \, \wvec^T\lrp{\beta}} \, \mathrm{d}\alpha \, \mathrm{d}\beta = \Qvec_c \, \int_0^t \int_0^t \delta\lrp{\alpha - \beta} \, \mathrm{d}\alpha \, \mathrm{d}\beta = \Qvec_c \, \int_0^t \mathrm{d}\beta = \Qvec_c \, t}\label{eq:SS_cont_integr_noise_covariance}
\end{eqnarray}
This expression shows that the mean of an integrated continuous white noise is always zero, but its covariance grows linearly with time. Integrating now the difference equation \nm{\zvec_k = \zvec_{k-1} + \wvec_{k-1}, \ \zvec_0 = \vec 0}, let's also evaluate the variation with time of the mean \nm{\vec \mu_k} and covariance \nm{\vec C_{zz,k}} of the integrated noise \nm{\zvec_k}:
\begin{eqnarray}
\nm{\vec \mu_k} & = & \nm{E\lrsb{\zvec_k} = E\lrsb{\sum_{l=0}^{k-1} \, \wvec_l} = \sum_{l=0}^{k-1} \, E\lrsb{\wvec_l} = \vec 0}\label{eq:SS_discr_integr_noise_mean} \\
\nm{\vec C_{zz,k}} & = & \nm{E\lrsb{\zvec_k \, \zvec_k^T} - \vec \mu_k \, \vec \mu_k^T = E\lrsb{\sum_{l=0}^{k-1} \, \wvec_l \, \sum_{m=0}^{k-1} \, \wvec_m^T} = \sum_{l=0}^{k-1} \, \sum_{m=0}^{k-1} \, E\lrsb{\wvec_l \, \wvec_m^T}} \nonumber \\
& = & \nm{\Qvec_d \, \sum_{l=0}^{k-1} \, \sum_{m=0}^{k-1} \, \delta_{l-m} = \Qvec_d \, \sum_{l=0}^{k-1} \, 1 = \Qvec_d \, k}\label{eq:SS_discr_integr_noise_covariance}
\end{eqnarray}
The covariance of the integrated discrete white noise process also grows linearly with time. Considering a sampling period of \nm{\Deltat, \, t = k \cdot \Deltat}, a discrete zero mean white noise process can be considered equivalent \cite{Simon2006} to a continuous one if their covariances are related by:
\neweq{\Qvec_d = \Qvec_c \cdot \Deltat}{eq:SS_integr_white_noise_cov_equiv}
\textbf{Discretization of Linear Continuous Time Systems}
Returning to the main argument, and considering that the state vector needs to be known only at a series of discrete time points, it is possible to discretize the (\ref{eq:SS_cont_time_system_linear}) linear continuous time system if \nm{\Avec\lrp{t}}, \nm{\Bvec\lrp{t}}, and \nm{\utilde\lrp{t}} are considered constant during the integration interval, which starts at \nm{t_{k-1} = \lrp{k-1} \cdot \Deltat} and concludes at \nm{t_k = k \cdot \Deltat}. The introduced discretization errors are higher the farther away this assumption is from reality. Introducing (\ref{eq:SS_integr_white_noise_cov_equiv}), the state system is now discrete and linear \cite{Simon2006}:
\begin{eqnarray}
\nm{\xvec_k} & \nm{\approx} & \nm{\Fvec_{k-1} \, \xvec_{k-1} + \Gvec_{k-1} \, \utilde_{k-1} + \wtilde_{k-1}}\label{eq:SS_discr_time_system_linear} \\
\nm{\xvec_k} & = & \nm{\xvec\lrp{t_k} = \xvec \lrp{k \, \Deltat}}\label{eq:SS_discr_time_system_linear_x} \\
\nm{\Fvec_k} & = & \nm{exp\lrp{\Avec_k \, \Deltat} = exp\big(\Avec\lrp{k \, \Deltat} \, \Deltat\big)}\label{eq:SS_discr_time_system_linear_system_matrix} \\
\nm{\Gvec_k} & = & \nm{\Fvec_k \, \int_0^{\Deltat} \, exp\big(- \Avec\lrp{\tau} \ \tau\big) \, \mathrm{d}\tau \, \Bvec\lrp{k \, \Deltat}} \nonumber \\
& = & \nm{\Fvec_k \, \lrsb{\Ivec - exp\big(- \Avec\lrp{k \, \Deltat} \Deltat\big)} \Avec^{-1}\lrp{k \, \Deltat} \, \Bvec\lrp{k \, \Deltat}}\label{eq:SS_discr_time_system_linear_input_matrix} \\
\nm{\utilde_k} & = & \nm{\utilde\lrp{t_k} = \utilde \lrp{k \, \Deltat}}\label{eq:SS_discr_time_system_linear_u} \\
\nm{\wtilde_k} & = & \nm{\wtilde\lrp{k \, \Deltat} \sim \lrp{\vec 0, \, \Qtilde_c\lrp{k \cdot \Deltat} \cdot \Deltat} = \lrp{\vec 0, \, \Lvec_k \, \Qvec_c \, \Lvec_k^T \, \Deltat} = \lrp{\vec 0, \, \Qtilde_{d,k}}}\label{eq:SS_discr_time_system_linear_noise1} \\
\nm{\Rvec_{\widetilde{w}\widetilde{w},kj}} & = & \nm{E\lrsb{\wtilde_k \, \wtilde_j^T} = \Qtilde_{d,k} \, \delta_{k-j}}\label{eq:SS_discr_time_system_linear_noise2}
\end{eqnarray}
Note that both the \emph{system state transition matrix} \nm{\Fvec_k \in \mathbb{R}^{mxm}} and the \emph{input transition matrix} \nm{\Gvec_k \in \mathbb{R}^{mxm}} make use of the matrix exponential function, although computing the later is not required, as shown in section \ref{subsubsec:SS_EKF}.
\textbf{Mean and Covariance of State Vector}
It is possible to evaluate the mean \nm{\vec \mu_{x,k}} and covariance \nm{\vec C_{xx,k} = \Pvec_k} of the state vector given by (\ref{eq:SS_discr_time_system_linear}), which provide their variation with time\footnote{To compute the covariance, note that there is no correlation between \nm{\lrp{\xvec_{k-1} - \mu_{\xvec_{k-1}}}} and \nm{\wtilde_{k-1}}.}:
\begin{eqnarray}
\nm{\vec \mu_{x,k}} & = & \nm{E\lrsb{\xvec_k} = \Fvec_{k-1} \, \vec \mu_{x,k-1} + \Gvec_{k-1} \, \utilde_{k-1}}\label{eq:SS_discr_time_system_state_mean} \\
\nm{\vec C_{xx,k}} & = & \nm{\Pvec_k = E\lrsb{\lrp{\xvec_k - \vec \mu_{x,k}} \, \lrp{\xvec_k - \vec \mu_{x,k}}^T}} \nonumber \\
& = & \nm{E\lrsb{\Big(\Fvec_{k-1} \, \lrp{\xvec_{k-1} - \vec \mu_{x,k-1}} + \wtilde_{k-1}\Big) \Big(\Fvec_{k-1} \, \lrp{\xvec_{k-1} - \vec \mu_{x,k-1}} + \wtilde_{k-1}\Big)^T}} \nonumber \\
& = & \nm{\Fvec_{k-1} \, \vec C_{xx,k-1} \, \Fvec_{k-1}^T + \Qtilde_{d,k-1} = \Fvec_{k-1} \, \Pvec_{k-1} \, \Fvec_{k-1}^T + \Qtilde_{d,k-1}}\label{eq:SS_discr_time_system_state_cov}
\end{eqnarray}
Based on (\ref{eq:SS_discr_time_system_linear}), \nm{\xvec_k} is a linear combination of a series of known real vectors \nm{\utilde_0, \dots, \utilde_{k-1}} plus a series of independent random vectors \nm{\xvec_0, \, \wtilde_0, \dots, \wtilde_{k-1}}. According to the central limit theorem stated in section \ref{subsec:Error_RandomVariables}, \nm{\xvec_k \sim N\lrp{\vec \mu_{x,k}, \, \vec C_{xx,k}} = N\lrp{\vec \mu_{x,k}, \, \Pvec_k}} is a normal or Gaussian random vector completely characterized by its mean and covariance.
The summary of this section is that given a continuous time non linear state space system such as (\ref{eq:SS_cont_time_system}), it is possible, with some linearization and discretization errors, to transform it into an equivalent discrete time linear system (\ref{eq:SS_discr_time_system_linear}) that can be integrated to obtain the estimated value of the state vector \nm{\xvec\lrp{t}} at a series of discrete times \nm{t_k = k \, \Deltat} characterized by its mean \nm{\vec \mu_{x,k}} (\ref{eq:SS_discr_time_system_state_mean}) and covariance \nm{\vec C_{xx,k} = \Pvec_k} (\ref{eq:SS_discr_time_system_state_cov}). Without further assistance, (\ref{eq:SS_discr_time_system_state_cov}) shows that the uncertainty of the results grows with time because of the accumulation of the white noise present in the system \cite{Simon2006}. The next section shows how the addition of measurements can solve this problem.
\subsubsection{Sampled Observations}\label{subsubsec:SS_SampledObservations}
Given the sampled data system of section \ref{subsubsec:SS_SampledDataSystems}, let's imagine that there exist a series of sensors capable of measuring certain variables related to the state vector at the same time points at which the state system is discretized in section \ref{subsubsec:SS_SampledDataSystems}:
\neweq{\yvec_k = h\lrp{\xvec_k, \, \vvec_k, \, t_k}}{eq:SS_measur_nonlinear}
where \nm{\yvec_k = \yvec\lrp{t_k} \in \mathbb{R}^q} is the \emph{measurement} or \emph{observation vector} provided by the sensors, \nm{\xvec_k = \xvec\lrp{t_k} \in \mathbb{R}^m} is the state vector, \nm{t_k = t\lrp{k \, \Deltat}} is the discrete time at which the measurements are taken, and \nm{\vvec_k = \vvec\lrp{t_k} \in \mathbb{R}^q} is the \emph{measurement} or \emph{observation noise}, which can be modeled by a zero mean white noise random process\footnote{Note that the measurement noise does not need to be Gaussian.} of covariance \nm{\Rvec} (sections \ref{subsec:Error_RandomVariables} and \ref{subsec:Error_WhiteNoise}):
\begin{eqnarray}
\nm{\vvec_k} & \nm{\sim} & \nm{\lrp{\vec 0, \, \Rvec}}\label{eq:SS_measur_nonlinear_noise1} \\
\nm{\Rvec_{vv,kj}} & = & \nm{E\lrsb{\vvec_k \, \vvec_j^T} = \Rvec \, \delta_{k-j}}\label{eq:SS_measur_nonlinear_noise2}
\end{eqnarray}
Let's also assume that the measurement noise and the process noise of section \ref{subsubsec:SS_SampledDataSystems} are orthogonal:
\neweq{\Rvec_{vw,kj} = E\lrsb{\vvec_k \, \wvec_j^T} = \vec 0}{eq:SS_measur_nonlinear_noise3}
\textbf{Linearization of Observations}
The discrete observations represented by (\ref{eq:SS_measur_nonlinear}) can be linearized by performing a Taylor expansion around an unknown nominal state \nm{\xvec_{Nk} = \xvec_N\lrp{t_k}} and observation noise \nm{\vvec_{Nk} = \vvec_N\lrp{t_k}}, assuming without loss of generality that \nm{\vvec_{Nk} = \vec 0}. If it is not, it can be written as the sum of a zero mean part and a known deterministic part, which can then be included in the nonlinear function \nm{h}. The expansion is truncated so only the first order terms remain, introducing linearization errors; these are higher the more nonlinear that \nm{h\lrp{\xvec_k, \ \vvec_k, \, t_k}} is with respect to \nm{\xvec_k} and \nm{\vvec_k}, and the farther away that \nm{\xvec_k} is from \nm{\xvec_{Nk}} and \nm{\vvec_k} from \nm{\vvec_{Nk} = \vec 0} \cite{Simon2006}.
\neweq{\yvec_k \approx h\rvert_N + \pderpar{h}{\xvec_k}\Bigr\rvert_N \, \lrp{\xvec_k - \xvec_{Nk}} + \pderpar{h}{\vvec_k}\Bigr\rvert_N \, \vvec_k = \pderpar{h}{\xvec_k}\Bigr\rvert_N \, \xvec_k + \lrp{h\rvert_N - \pderpar{h}{\xvec_k}\Bigr\rvert_N \, \xvec_{Nk}} + \pderpar{h}{\vvec_k}\Bigr\rvert_N \, \vvec_k} {eq:SS_measur_taylor}
where \nm{\mid_N} stands for evaluation at \nm{\lrp{\xvec_{Nk}, \, \vec 0, \, t_k}}. The observations system is now discrete time and linear:
\begin{eqnarray}
\nm{\yvec_k} & \nm{\approx} & \nm{\Hvec_k \, \xvec_k + \zvec_k + \vtilde_k}\label{eq:SS_measur_linear} \\
\nm{\Hvec_k} & = & \nm{\Hvec\lrp{t_k} = \pderpar{h}{\xvec_k}\lrp{\xvec_{Nk}, \, \vec 0, \, t_k}}\label{eq:SS_measur_linear_output_matrix} \\
\nm{\zvec_k} & = & \nm{\zvec\lrp{t_k} = h\lrp{\xvec_{Nk}, \, \vec 0, \, t_k} - \Hvec_k \, \xvec_{Nk}}\label{eq:SS_measur_linear_extra_vector} \\
\nm{\Mvec_k} & = & \nm{\Mvec\lrp{t_k} = \pderpar{h}{\vvec_k}\lrp{\xvec_{Nk}, \, \vec 0, \, t_k}}\label{eq:SS_measur_linear_input_matrix} \\
\nm{\vtilde_k} & = & \nm{\vtilde\lrp{t_k} = \Mvec_k \, \vvec_k \sim \lrp{\vec 0, \, \Mvec_k \, \Rvec \, \Mvec_k^T} = \lrp{\vec 0, \, \Rtilde_k}}\label{eq:SS_measure_linear_noise1} \\
\nm{\Rvec_{\widetilde{v}\widetilde{v},kj}} & = & \nm{E\lrsb{\vtilde_k \, \vtilde_j^T} = \Rtilde_k \, \delta_{k-j}}\label{eq:SS_measure_linear_noise2}
\end{eqnarray}
The above observation system is based on an \emph{output matrix} \nm{\Hvec_k \in \mathbb{R}^{qxm}} that is the jacobian of the non linear system with respect to the state vector evaluated at the unknown nominal state, and a vector \nm{\zvec_k \in \mathbb{R}^q} that depends exclusively of the nominal state. It also employs a modified observation noise vector \nm{\vtilde_k \in \mathbb{R}^q}. It is worth noting that computation of \nm{\zvec_k} is not necessary to obtain the solution, as shown in section \ref{subsubsec:SS_EKF}.
\textbf{Constant State Vector Estimation based on Observations}
The objective of this section is to obtain the best possible estimate \nm{\xvecest_k} of a constant\footnote{Note that this is the only section where the state vector \nm{\xvec_k} is required to be constant, this is, \nm{\xvec = \xvec_0 = \xvec_k \ \forall \, k}.} state vector \nm{\xvec} based on the observations \nm{\yvec_k} provided by (\ref{eq:SS_measur_linear}) and the previous estimate \nm{\xvecest_{k-1}}. Let's use an expression like (\ref{eq:SS_measur_linear_estimate}), where \nm{\Kvec_k \in \mathbb{R}^{mxq}} is called the \emph{gain matrix} and \nm{\rvec_k \in \mathbb{R}^q} the \emph{innovations vector}:
\neweq{\xvecest_k = \xvecest_{k-1} + \Kvec_k \, \rvec_k = \xvecest_{k-1} + \Kvec_k \, \lrp{\yvec_k - \Hvec_k \, \xvecest_{k-1} - \zvec_k}}{eq:SS_measur_linear_estimate}
The \emph{estimation error} \nm{\vec \varepsilon_{x,k}} and its mean can then be computed based on (\ref{eq:SS_measur_linear_estimate}) and (\ref{eq:SS_measur_linear}):
\begin{eqnarray}
\nm{\vec \varepsilon_{x,k}} & = & \nm{\xvec - \xvecest_k = \lrp{\Ivec - \Kvec_k \, \Hvec_k} \, \vec \varepsilon_{x,k-1} - \Kvec_k \, \vtilde_k}\label{eq:SS_measur_linear_estimation_error} \\
\nm{\vec \mu_{\varepsilon x,k}} & = & \nm{E\lrsb{\vec \varepsilon_{x,k}} = \lrp{\Ivec - \Kvec_k \, \Hvec_k} \, \vec \mu_{\varepsilon x,k-1}}\label{eq:SS_measur_linear_estimation_error_mean}
\end{eqnarray}
As the linearized discrete noise \nm{\vtilde_k} is zero mean per (\ref{eq:SS_measure_linear_noise1}), (\ref{eq:SS_measur_linear_estimate}) is called an \emph{unbiased estimator} \cite{Simon2006}, because if the initial estimate \nm{\xvecest_0} is set equal to the expected value of the state vector \nm{\lrp{\xvecest_0 = \vec \mu_x \rightarrow \vec \mu_{\varepsilon x,0} = \vec 0}}, then \nm{\vec \mu_{\varepsilon x,k} = E\lrsb{\varepsilon_{x,k}} = \vec 0 \ \forall \, k}, this is, the expected value of \nm{\xvecest_k} is equal to \nm{\vec \mu_x = E\lrsb{\xvec}} for all \nm{t_k}. This is regardless of the value of the gain matrix \nm{\Kvec_k}.
Let's follow a similar process to compute the covariance of the estimation error \nm{\vec C_{xx,k} = \Pvec_k}. To do so, note that the observation noise is independent from the estimation error \nm{\lrp{E\lrsb{\vtilde_k \, \vec \varepsilon_{x,k-1}^T} = \vec 0}}:
\neweq{\vec C_{xx,k} = \Pvec_k = E\lrsb{\vec \varepsilon_{x,k} \, \vec \varepsilon_{x,k}^T} - \vec \mu_{\varepsilon x,k} \, \vec \mu_{\varepsilon x,k}^T = \lrp{\Ivec - \Kvec_k \, \Hvec_k}\, \Pvec_{k-1} \, \lrp{\Ivec - \Kvec_k \, \Hvec_k}^T + \Kvec_k \, \Rtilde_k \, \Kvec_k^T}{eq:SS_measur_linear_covariance}
This expression guarantees that \nm{\vec C_{xx,k} = \Pvec_k} is positive definite (as all covariance matrices) given that so are \nm{\Pvec_{k-1}} and \nm{\Rtilde_k}. Let's use the minimization of the sum of the variances of the estimation errors as the criterion to obtain the gain matrix \nm{\Kvec_k}, and hence fill up (\ref{eq:SS_measur_linear_estimate}) and (\ref{eq:SS_measur_linear_covariance}) to obtain the estimation of the state vector as well as the covariance of its error. That way, the estimation error is not only zero mean but it is also consistently as close as possible to zero \cite{Simon2006}.
\begin{eqnarray}
\nm{\vec J_k} & = & \nm{E\lrsb{\lrp{x_1 - \hat{x}_{1,k}}^2 + \ldots + \lrp{x_m - \hat{x}_{m,k}}^2} = E\lrsb{\varepsilon_{x1,k}^2 + \ldots + \varepsilon_{xm,k}^2}}\nonumber \\
& = & \nm{E\lrsb{\vec \varepsilon_{x,k}^T \, \vec \varepsilon_{x,k}} = E\lrsb{Tr\lrp{\vec \varepsilon_{x,k} \, \vec \varepsilon_{x,k}^T}} = Tr \, \vec C_{xx,k} = Tr \, \Pvec_k}\label{eq:SS_measur_linear_J} \\
\nm{\pderpar{\vec J_k}{\Kvec_k}} & = & \nm{2 \, \lrp{\Ivec - \Kvec_k \, \Hvec_k} \, \Pvec_{k-1} \, \lrp{- \Hvec_k^T} + 2 \, \Kvec_k \, \Rtilde_k}\label{eq:SS_measur_linear_Jderiv}
\end{eqnarray}
where \nm{Tr\lrp{}} stands for trace of a matrix, and some not so obvious matrix algebra properties have been employed. Setting the (\ref{eq:SS_measur_linear_Jderiv}) derivative to zero provides the optimum gain matrix:
\neweq{\Kvec_k = \Pvec_{k-1} \, \Hvec_k^T \lrp{\Hvec_k \, \Pvec_{k-1} \, \Hvec_k^T + \Rtilde_k}^{-1}}{eq:SS_measur_linear_optimal_gain}
\subsubsection{The Extended Kalman Filter}\label{subsubsec:SS_EKF}
Provided with discrete time and linear state dynamics (\ref{eq:SS_discr_time_system_linear}) and observations (\ref{eq:SS_measur_linear}), the goal of state estimation is to obtain the best possible estimate of the state vector \nm{\xvec_k = \xvec\lrp{t_k}} based on the knowledge of the system provided by the state dynamics and the availability of observations \cite{Simon2006}. At a given time \nm{t_k = k \, \Deltat}, the \emph{a priori estimation} \nm{\xvecest_k^-} is defined as the estimation of \nm{\xvec_k}, this is, the estimation of the state vector at time \nm{t_k}, making use of all measurements taken before \nm{t_k} but not including those at \nm{t_k}. The \emph{a posteriori estimation} \nm{\xvecest_k^+} is defined as the estimation of \nm{\xvec_k} that makes use of all measurements up and including \nm{t_k}. In the same way, it is possible to define the \emph{a priori} and \emph{a posteriori covariances} of the estimation error \nm{\Pvec_k^-} and \nm{\Pvec_k^+}:
\begin{eqnarray}
\nm{\Pvec_k^-} & = & \nm{E\lrsb{\lrp{\xvec_k - \xvecest_k^-} \, \lrp{\xvec_k - \xvecest_k^-}^T} - E\lrsb{\xvec_k - \xvecest_k^-} \, E\lrsb{\xvec_k - \xvecest_k^-}^T} \label{eq:SS_EKF_covariance_apriori_definition} \\
\nm{\Pvec_k^+} & = & \nm{E\lrsb{\lrp{\xvec_k - \xvecest_k^+} \, \lrp{\xvec_k - \xvecest_k^+}^T} - E\lrsb{\xvec_k - \xvecest_k^+} \, E\lrsb{\xvec_k - \xvecest_k^+}^T} \label{eq:SS_EKF_covariance_aposteriori_definition}
\end{eqnarray}
The process starts with an initial estimation of the state vector \nm{\xvecest_0^+} before any measurements are available (they start at \nm{k = 1}). Since no measurements are available, it is reasonable to form \nm{\xvecest_0^+} as the expected value of the initial state \nm{\xvec_0} \cite{Simon2006}:
\neweq{\xvecest_0^+ = \vec \mu_{x,0} = E\lrsb{\xvec_0} = E\big[\xvec\lrp{t_0}\big]}{eq:SS_EKF_x0_initial_state}
The covariance of the initial estimation error \nm{\Pvec_0^+} is also required, representing the uncertainty in the initial estimation \nm{\xvecest_0^+} \footnote{If the initial state is known with exactitude, use \nm{\Pvec_0^+ = 0}. Otherwise, use higher values the less confidence the user has in the accuracy of \nm{\xvecest_0^+}.}:
\neweq{\Pvec_0^+ = E\lrsb{\lrp{\xvec_0 - \xvecest_0^+} \, \lrp{\xvec_0 - \xvecest_0^+}^T} - E\lrsb{\xvec_0 - \xvecest_0^+} \, E\lrsb{\xvec_0 - \xvecest_0^+}^T = E\lrsb{\lrp{\xvec_0 - \vec \mu_{x,0}} \, \lrp{\xvec_0 - \vec \mu_{x,0}}^T}}{eq:SS_EKF_P0_initial_covariance}
\textbf{Time Update and Measurement Update Equations}
The next step is to propagate the state estimation without the use of any observations from \nm{\xvecest_0^+} to \nm{\xvecest_1^-}, with the objective of obtaining an estimation that coincides with the state vector mean, this is, \nm{\xvecest_1^- = \vec \mu_{x,1} = E\lrsb{\xvec_1}}. Recalling the evolution of the state vector expected value provided by (\ref{eq:SS_discr_time_system_state_mean}), and extending the same reasoning to all steps, it makes sense intuitively to propagate the state estimate the same way that the mean of the state propagates \cite{Simon2006}. Hence, the time propagation for the state estimate results in:
\neweq{\xvecest_k^- = \Fvec_{k-1} \, \xvecest_{k-1}^+ + \Gvec_{k-1} \, \utilde_{k-1}}{eq:SS_EKF_xest_minus_propagate}
A similar reasoning is employed for the propagation of the covariance of the estimation error in the absence of observations. Recalling the evolution of the state vector covariance provided by (\ref{eq:SS_discr_time_system_state_cov}) and extending the same reasoning to all steps, the time propagation of the covariance results in \cite{Simon2006}:
\neweq{\Pvec_k^- = \Fvec_{k-1} \, \Pvec_{k-1}^+ \, \Fvec_{k-1}^T + \Qtilde_{d,k-1}}{eq:SS_EKF_P_minus_propagate}
The above equations are called the \emph{time update equations}. Once the a priori estimation and error covariance have been computed, it is possible to update them with the information contained in the observation. This is done with the expressions derived in section \ref{subsubsec:SS_SampledObservations}, replacing \nm{\xvecest_{k-1}} with \nm{\xvecest_k^-}, \nm{\xvecest_k} with \nm{\xvecest_k^+}, \nm{\Pvec_{k-1}} with \nm{\Pvec_k^-}, and \nm{\Pvec_k} with \nm{\Pvec_k^+} \cite{Simon2006}. These are called the \emph{measurement update equations}:
\begin{eqnarray}
\nm{\Kvec_k} & = & \nm{\Pvec_k^- \, \Hvec_k^T \lrp{\Hvec_k \, \Pvec_k^- \, \Hvec_k^T + \Rtilde_k}^{-1}}\label{SS_EKF_kalman_gain} \\
\nm{\xvecest_k^+} & = & \nm{\xvecest_k^- + \Kvec_k \, \rvec_k = \xvecest_k^- + \Kvec_k \, \lrp{\yvec_k - \Hvec_k \, \xvecest_k^- - \zvec_k}}\label{eq:SS_EKF_xest_plus_propagate} \\
\nm{\Pvec_k^+} & = & \nm{\lrp{\Ivec - \Kvec_k \, \Hvec_k}\, \Pvec_k^- \, \lrp{\Ivec - \Kvec_k \, \Hvec_k}^T + \Kvec_k \, \Rtilde_k \, \Kvec_k^T}\label{eq:SS_EKF_P_plus_propagate}
\end{eqnarray}
\textbf{Introduction of the Nominal Trajectory}
The time and measurement update equations developed in the previous section provide the means to compute the variation with time of the estimated state vector \nm{\lrp{\xvecest_k^-, \, \xvecest_k^+}} as well as that of the covariance of the estimation errors \nm{\lrp{\Pvec_k^-, \, \Pvec_k^+}}. However, to do so, it is necessary to define what is the nominal point \nm{\xvec_{Nk} = \xvec_N\lrp{t_k}, \, \wvec_{Nk} = \wvec_N\lrp{t_k} = \vec 0, \, \vvec_{Nk} = \vvec_N\lrp{t_k} = \vec 0} around which the dynamics system is linearized in section \ref{subsubsec:SS_SampledDataSystems} and the observations in section \ref{subsubsec:SS_SampledObservations}.
The \emph{extended Kalman filter} (\hypertt{EKF}) provides a solution to this problem that is simple but not too intuitive. The \hypertt{EKF} considers its own a priori state estimate as the nominal trajectory, this is, the nonlinear state system and observations are linearized around the \hypertt{EKF} estimate, and simultaneously that same estimate depends on the linearized system \cite{Simon2006}:
\neweq{\xvec_{Nk} = \xvec_N\lrp{t_k} = \xvecest_k^-}{eq:SS_EKF_assumption}
This assumption can be introduced into the expressions for the observations output matrix \nm{\Hvec_k} and observations input vector \nm{\zvec_k} provided by (\ref{eq:SS_measur_linear_output_matrix}) and (\ref{eq:SS_measur_linear_extra_vector}), with the results introduced into the state estimation measurement update equation (\ref{eq:SS_EKF_xest_plus_propagate}):
\begin{eqnarray}
\nm{\xvecest_k^+} & = & \nm{\xvecest_k^- + \Kvec_k \, \lrp{\yvec_k - \pderpar{h}{\xvec_k}\lrp{\xvecest_k^-, \, \vec 0, \, t_k} \, \xvecest_k^- - h\lrp{\xvecest_k^-, \, \vec 0, \, t_k} + \pderpar{h}{\xvec_k}\lrp{\xvecest_k^-, \, \vec 0, \, t_k} \, \xvecest_k^-}}\nonumber \\
& = & \nm{\xvecest_k^- + \Kvec_k \, \big[\yvec_k - h\lrp{\xvecest_k^-, \, \vec 0, \, t_k}\big]}\label{eq:SS_EKF_xest_plus_propagate_bis}
\end{eqnarray}
Note that, in contrast with (\ref{eq:SS_EKF_xest_plus_propagate}), it is no longer necessary to compute the observations input vector \nm{\zvec_k}. In order to diminish the state system linearization errors described in section \ref{subsubsec:SS_SampledDataSystems}, it is also possible to replace the state estimate time update equation (\ref{eq:SS_EKF_xest_minus_propagate}) with a zeroth order forward integration of the continuous time state system (\ref{eq:SS_cont_time_system}), with the time derivative evaluated at \nm{\xvecest_{k-1}^+}:
\neweq{\xvecest_k^- = \xvecest_{k-1}^+ + \Deltat \cdot f \, \big(\xvecest_{k-1}^+, \, \uvec_{k-1}, \ \vec 0, \, t_{k-1}\big)} {eq:SS_xest_minus_propagate_bis}
An extra benefit of this approach compared with (\ref{eq:SS_EKF_xest_minus_propagate}) is that it is no longer necessary to perform the expensive computations required to evaluate \nm{\Gvec_{k-1}} (\ref{eq:SS_discr_time_system_linear_input_matrix}).
\textbf{EKF Summary}
Given a continuous time nonlinear state system (\ref{eq:SS_cont_time_system}) with process noise provided by (\ref{eq:SS_cont_time_system_noise1}) and (\ref{eq:SS_cont_time_system_noise2}), together with a series of discrete time nonlinear observations (\ref{eq:SS_measur_nonlinear}) with measurement noise given by (\ref{eq:SS_measur_nonlinear_noise1}) and (\ref{eq:SS_measur_nonlinear_noise2}), and considering no correlation between both noises (\ref{eq:SS_measur_nonlinear_noise3}), it is possible to compute estimations of the state at the same time points at which the observations are provided, in such a way that their errors (difference with respect to the true state) are zero mean and have a covariance that is also computed by means of the following equations:
\begin{eqnarray}
\nm{\xvecest_0^+} & = & \nm{\vec \mu_{x,0} = E\lrsb{\xvec_0}}\label{eq:SS_EKF_x0_initial_state_FINAL} \\
\nm{\Pvec_0^+} & = & \nm{E\lrsb{\lrp{\xvec_0 - \vec \mu_{x,0}} \, \lrp{\xvec_0 - \vec \mu_{x,0}}^T}}\label{eq:SS_EKF_P0_initial_covariance_FINAL} \\
\nm{\xvecest_k^-} & = & \nm{\xvecest_{k-1}^+ + \Deltat \cdot f \, \big(\xvecest_{k-1}^+, \, \uvec_{k-1}, \ \vec 0, \, t_{k-1}\big)}\label{eq:SS_xest_minus_propagate_bis_FINAL} \\
\nm{\Pvec_k^-} & = & \nm{\Fvec_{k-1} \, \Pvec_{k-1}^+ \, \Fvec_{k-1}^T + \Qtilde_{d,k-1}}\label{eq:SS_EKF_P_minus_propagate_FINAL} \\
\nm{\Kvec_k} & = & \nm{\Pvec_k^- \, \Hvec_k^T \lrp{\Hvec_k \, \Pvec_k^- \, \Hvec_k^T + \Rtilde_k}^{-1}}\label{eq:SS_EKF_kalman_gain_FINAL} \\
\nm{\xvecest_k^+} & = & \nm{\xvecest_k^- + \Kvec_k \, \lrsb{\yvec_k - h\lrp{\xvecest_k^-, \, \vec 0, \, t_k}}}\label{eq:SS_EKF_xest_plus_propagate_bis_FINAL} \\
\nm{\Pvec_k^+} & = & \nm{\lrp{\Ivec - \Kvec_k \, \Hvec_k}\, \Pvec_k^- \, \lrp{\Ivec - \Kvec_k \, \Hvec_k}^T + \Kvec_k \, \Rtilde_k \, \Kvec_k^T}\label{eq:SS_EKF_P_plus_propagate_FINAL}
\end{eqnarray}
In the discussion that follows, \nm{\xvecest_k} is employed to refer to both the a priori and a posteriori state vector estimations \nm{\lrp{\xvecest_k^-, \, \xvecest_k^+}}, and \nm{\vec \varepsilon_k = \xvec_k - \xvecest_k} for the state estimation errors. The above equations show that \nm{\xvecest_k} is a linear combination of a random vector \nm{\xvec_0^+} plus a series of random processes \nm{\utilde_k}, so it is itself a random process, and so is \nm{\vec \varepsilon_k}.
Let's leave aside for the time being the linearization and discretization errors of sections \ref{subsubsec:SS_SampledDataSystems} and \ref{subsubsec:SS_SampledDataSystems}, and focus on a problem composed by a discrete time linear state system (\ref{eq:SS_discr_time_system_linear}) and discrete time linear observations (\ref{eq:SS_measur_linear}). Provided with any user defined positive definite weighting matrix \nm{\Svec_k}, it can be proved that the solution provided verifies (\ref{eq:SS_EKF_KF_minimum}), this is, results in a state estimation that always minimizes the weighted sum of squared estimation errors \cite{Simon2006}, as long as the process and observations noises are Gaussian zero mean uncorrelated white noise processes. If they are not Gaussian, then \nm{\xvecest_k} provides the best linear (in the sense of the previous paragraph) solution to the (\ref{eq:SS_EKF_KF_minimum}) minimization, although there may be a better nonlinear solution.
\neweq{\xvecest_k = \argmin E\lrsb{\vec \varepsilon_k^T \, \Svec_k \, \vec \varepsilon_k}}{eq:SS_EKF_KF_minimum}
The errors induced by the discretization of the linear continuous time dynamics system in section \ref{subsubsec:SS_SampledDataSystems} generally do not result in significant errors as modern systems are capable of running the estimation algorithms at elevated frequencies. The system matrix \nm{\Avec\lrp{t}} and input vector \nm{\utilde\lrp{t}} in the continuous time system (\ref{eq:SS_cont_time_system_linear}) generally do not vary much during the integration interval, and hence the discretization errors are small. The linearization errors of sections \ref{subsubsec:SS_SampledDataSystems} and \ref{subsubsec:SS_SampledObservations} are a different story, and can induce the \hypertt{EKF} to provide unreliable estimates or even to diverge in case the nonlinearities are severe \cite{Simon2006}.
\section{Introduction to Lie Algebra}\label{sec:Algebra}
This section begins with some basic abstract and linear algebra concepts in sections \ref{subsec:algebra_structures} and \ref{subsec:algebra_points_and_vectors}, and then introduces Lie algebra in section \ref{subsec:algebra_lie}, followed by the derivation of some useful Lie jacobians in section \ref{subsec:algebra_lie_jacobians}. Its application to the discrete integration of states is discussed in section \ref{subsec:algebra_integration}, to gradient descent optimization in section \ref{subsec:algebra_gradient_descent}, and to state estimation in section \ref{subsec:algebra_SS}. The contents of this section are generic to any Lie group without making further mention to rigid bodies. It is only in sections \ref{sec:Rotate} and \ref{sec:Motion} where the Lie theory concepts are applied first to rotational motion and then to the more generic rigid body motion.
\subsection{Algebraic Structures, Maps, and Metric Spaces}\label{subsec:algebra_structures}
In algebra, a \emph{set} is a well defined collection of objects, named elements, while an \emph{operation} \nm{\ast} is a uniquely defined rule that assigns to each ordered pair of elements exactly a third element \nm{\lrb{\ast : \mathbb{A} \times \mathbb{B} \rightarrow \mathbb{C} \ | \ a \ast b = c \in \mathbb{C}, \forall \ a \in \mathbb{A}, \forall \ b \in \mathbb{B}}} \cite{Pinter1990}. Although an operation may involve up to three different sets \nm{\lrp{\mathbb{A}, \mathbb{B}, \mathbb{C}}}, often two or even the three of them coincide. A set \nm{\mathbb{A}} is a \emph{subset} of a set \nm{\mathbb{B}} if all elements of \nm{\mathbb{A}} are also elements of \nm{\mathbb{B}}. An \emph{algebraic structure} is a combination of a set and one or multiple operations that complies with certain axioms.
A set \nm{\mathbb{A}} has \emph{group} structure under operation \nm{\lrb{\ast: \mathbb{A} \times \mathbb{A} \rightarrow \mathbb{A}}} if it complies with the following four axioms \nm{\forall \ a, b, c \in \mathbb{A}} \cite{Pinter1990}:
\begin{enumerate}
\item Closure: \nm{a \ast b \in \mathbb{A}}
\item Associativity: \nm{\lrp{a \ast b} \ast c = a \ast \lrp{b \ast c}}
\item Identity: \nm{\exists \ e \in \mathbb{A} \ | \ e \ast a = a \ast e = a}
\item Inverse: \nm{\exists \ f \in \mathbb{A} \ | \ f \ast a = a \ast f = e}
\end{enumerate}
An \emph{abelian group} is that which in addition also complies with commutativity \nm{\lrb{a \ast b = b \ast a}}. A set \nm{\mathbb{A}} has \emph{ring} structure under two operations, usually named addition \nm{\lrb{+: \mathbb{A} \times \mathbb{A} \rightarrow \mathbb{A}}} and multiplication \nm{\lrb{\cdot: \mathbb{A} \times \mathbb{A} \rightarrow \mathbb{A}}}, if, in addition to being an abelian group under addition, complies with the following four axioms \nm{\forall \ a, b, c \in \mathbb{A}} \cite{Pinter1990}:
\begin{enumerate}
\item Closure of \nm{\cdot} : \nm{a \cdot b \in \mathbb{A}}
\item Associativity of \nm{\cdot} : \nm{\lrp{a \cdot b} \cdot c = a \cdot \lrp{b \cdot c}}
\item Distributivity of \nm{\cdot} with respect to \nm{+} : \nm{a \cdot \lrp{b + c} = a \cdot b + a \cdot c, \ \lrp{a + b} \cdot c = a \cdot c + b \cdot c}
\item Identity of \nm{\cdot} : \nm{\exists \ 1 \in \mathbb{A} \ | \ 1 \cdot a = a \cdot 1 = a}
\end{enumerate}
An \emph{abelian ring} is that which in addition also complies with commutativity over multiplication \nm{\lrb{a \cdot b = b \cdot a}}. Note that by convention, the identity and inverse of addition are denoted 0 and \nm{-a}, respectively, while those of multiplication are denoted 1 and \nm{a^{-1}}. A set \nm{\mathbb{A}} has \emph{field} structure under operations \nm{+} and \nm{\cdot} if \nm{\mathbb{A}} is an abelian group under \nm{+} and \nm{\mathbb{A} - \lrb{0}} (the set \nm{\mathbb{A}} without the additive identity 0) is an abelian group under \nm{\cdot} \cite{Pinter1990}. In an \emph{ordered field}, the implementation of the addition and multiplication operations enables determining if one element is greater, equal, or lower than a second element. The set of real numbers \nm{\mathbb{R}} endowed with the operations of addition \nm{+} and multiplication \nm{\cdot} forms an ordered field, known as the field of real numbers \nm{\langle \mathbb{R}, +, \cdot \rangle}, nearly always abbreviated to simply \nm{\mathbb{R}}.
A \emph{topological space} is an ordered pair \nm{\lrp{\mathbb{A}, \mathbb{\tau}}}, where \nm{\mathbb{A}} is a set and \nm{\mathbb{\tau}} is a collection of subsets of \nm{\mathbb{A}}, satisfying the following axioms \cite{Armstrong1983}:
\begin{enumerate}
\item The empty set and \nm{\mathbb{A}} itself belong to \nm{\mathbb{\tau}}.
\item Any arbitrary (finite or infinite) union of members of \nm{\mathbb{\tau}} still belongs to \nm{\mathbb{\tau}}.
\item The intersection of any finite number of members of \nm{\mathbb{\tau}} still belongs to \nm{\mathbb{\tau}}.
\end{enumerate}
The elements of \nm{\mathbb{\tau}} are called open sets and the collection \nm{\mathbb{\tau}} is called a topology on \nm{\mathbb{A}}. Topological spaces comprise the most general notion of a mathematical space; all other spaces defined below are specializations with extra structure or constraints.
A \emph{vector space} (\emph{linear space}) over a field \nm{\mathbb{F}} is a set \nm{\mathbb{V}} together with two operations, addition \nm{\lrb{+ : \mathbb{V} \times \mathbb{V} \rightarrow \mathbb{V}}} and scalar multiplication \nm{\lrb{\cdot : \mathbb{F} \times \mathbb{V} \rightarrow \mathbb{V}}} that, in addition of \nm{\mathbb{V}} being an abelian group under +, satisfies the following axioms \nm{\forall \ u, v \in \mathbb{V}} and \nm{\forall \ a, b \in \mathbb{F}} \cite{Shuster1993, Roman2005}. Elements of \nm{\mathbb{F}} are called scalars, while those of \nm{\mathbb{V}} vectors.
\begin{enumerate}
\item Closure of \nm{\cdot} : \nm{a \cdot u \in \mathbb{V}}
\item Compatibility of \nm{\cdot} with field \nm{\cdot} : \nm{a \cdot \lrp{b \cdot v} = \lrp{a \cdot b} \cdot v}
\item Identity of \nm{\cdot} : \nm{1 \cdot v = v}, where 1 denotes the field \nm{\cdot} identity.
\item Distributivity of \nm{\cdot} with respect to \nm{+} : \nm{a \cdot \lrp{u + v} = a \cdot u + a \cdot v}
\item Distributivity of \nm{\cdot} with respect to field \nm{+} : \nm{\lrp{a + b} \cdot v = a \cdot v + b \cdot v}
\end{enumerate}
A \emph{map} or \emph{morphism} is a rule that to every element in a set \nm{\mathbb{A}} assigns a unique element in a different set \nm{\mathbb{B}} \nm{\lrb{f : \mathbb{A} \rightarrow \mathbb{B} \ | \ f\lrp{a} = b \in \mathbb{B}, \forall \ a \in \mathbb{A}}} \cite{Pinter1990}. A map is \emph{injective} if each element in \nm{\mathbb{B}} is the image or map output of no more than one element of \nm{\mathbb{A}}, \emph{surjective} if each element in \nm{\mathbb{B}} is the image of at least one element of \nm{\mathbb{A}}, and \emph{bijective} is the map is simultaneously injective and surjective.
A \emph{homomorphism} is a structure preserving map between two algebraic structures of the same type (groups, rings, fields, vector spaces, etc.) \cite{Pinter1990}. Note that neither the sets nor the operations of the structures need to coincide, and that a homomorphism preserves every operation contained in the algebraic structures. In the case of groups, considering a homomorphism \nm{\lrb{f : \langle \mathbb{A}, \ + \rangle \rightarrow \langle \mathbb{B}, \ \cdot \rangle}}, it complies with all group axioms \nm{\forall \ a, b, c \in \mathbb{A}}:
\begin{enumerate}
\item Closure: \nm{f\lrp{a + b} = f\lrp{a} \cdot f\lrp{b}}
\item Associativity: \nm{f\big(\lrp{a + b} + c\big) = f\big(a + \lrp{b + c}\big) = \big(f\lrp{a} \cdot f\lrp{b}\big) \cdot f\lrp{c} = f\lrp{a} \cdot \big(f\lrp{b} \cdot f\lrp{c}\big)}
\item Identity: \nm{f\lrp{0} = 1 \rightarrow f\lrp{a} = f\lrp{a + 0} = f\lrp{0 + a} = f\lrp{a} \cdot f\lrp{0} = f\lrp{0} \cdot f\lrp{a} = f\lrp{a} \cdot 1 = 1 \cdot f\lrp{a}}
\item Inverse: \nm{f\lrp{-a} = f^{-1}\lrp{a} \rightarrow f\lrp{0} = f\lrp{a - a} = f\lrp{a} \cdot f\lrp{-a} = f\lrp{a} \cdot f^{-1}\lrp{a} = f^{-1}\lrp{a} \cdot f\lrp{a} = 1}
\end{enumerate}
Homomorphisms for other algebraic structures are defined similarly. A bijective homomorphism is known as an \emph{isomorphism} \cite{Pinter1990}. A \emph{metric} is an operation between two elements of the same set onto a field \nm{\lrb{d : \mathbb{V} \times \mathbb{V} \rightarrow \mathbb{F}}}. It defines the concept of distance between any two members of the set and complies with the following axioms \nm{\forall \ u, v, w \in \mathbb{V}}:
\begin{enumerate}
\item Identity of indiscernibles: \nm{d\lrp{u, v} = 0 \Leftrightarrow u = v}
\item Symmetry: \nm{d\lrp{u, v} = d\lrp{v, u}}
\item Subadditivity: \nm{d\lrp{u, w} \leq d\lrp{u, v} + d\lrp{v, w}}
\end{enumerate}
Based on these three axioms, it is straightforward to prove that \nm{d\lrp{u,v} \geq 0 \ \forall \ u, v \in \mathbb{V}}. The most common metric is the \emph{inner product} \nm{\lrb{\langle \cdot \, , \cdot \rangle: \mathbb{V} \times \mathbb{V} \rightarrow \mathbb{F}}}, usually associated to a vector space, which satisfies the following three axioms \nm{\forall \ u, v, w \in \mathbb{V}} and \nm{\forall \ a, b \in \mathbb{F}} \cite{Shuster1993}:
\begin{enumerate}
\item Commutativity: \nm{\langle u, v\rangle = \langle v, u\rangle}
\item Linearity with respect to \nm{+} and \nm{\cdot} : \nm{\langle u, a \cdot v + b \cdot w\rangle = a \cdot \langle u, v\rangle + b \cdot \langle u, w\rangle}
\item Positive definiteness: \nm{\langle v, v\rangle \geq 0} and \nm{\langle v, v\rangle = 0 \Leftrightarrow v = 0}
\end{enumerate}
A \emph{metric space} is a combination of a set with a metric on that same set \cite{Bryant1985}, while an \emph{inner product space} restricts the definition to the case of a vector space \nm{\mathbb{V}} over a field \nm{\mathbb{F}} endowed with an inner product metric over the same field \nm{\mathbb{F}} \cite{Jain1995}. An \emph{Euclidean space} is a finite-dimensional inner product space over the field of the real numbers \nm{\mathbb{R}} \cite{Artin1957}.
A \emph{group action} on a space is a group homomorphism of a given group \nm{\langle \mathbb{A}, \ast \rangle} into the group of transformations of the space \nm{\lrb{g\lrp{}: \mathbb{A} \times \mathbb{V} \rightarrow \mathbb{V} \ | \ g_a\lrp{u} = v \in \mathbb{V}, \forall \ a \in \mathbb{A}, \forall \ u \in \mathbb{V}}}, and needs to verify two axioms \nm{\forall \ a, b \in \mathbb{A}, \forall \ u \in \mathbb{V}}:
\begin{enumerate}
\item Identity: \nm{g_e\lrp{u} = u}, where e is the identity of \nm{\mathbb{G}}.
\item Compatibility: \nm{g_{a \ast b}\lrp{u} = g_a\big(g_b\lrp{u}\big)}
\end{enumerate}
Returning to the case of inner product spaces, two vectors are \emph{orthogonal} if their inner product is zero, while the length or \emph{norm} of a vector is \nm{\| v \| = \sqrt{\langle v, v\rangle}}. A unit vector is that whose norm is one. An inner product space can also be endowed with an additional operation, the \emph{cross product} \nm{\lrb{\times : \mathbb{V} \times \mathbb{V} \rightarrow \mathbb{V}}}, which complies with the following axioms \cite{Shuster1993}:
\begin{enumerate}
\item Anti commutativity: \nm{u \times v = - v \times u}
\item Compatibility with \nm{\cdot} : \nm{\lrp{a \cdot u} \times v = u \times \lrp{a \cdot v} = a \cdot \lrp{u \times v}}
\item Distributivity with + : \nm{\lrp{u + v} \times w = \lrp{u \times w} + \lrp{v \times w}}
\end{enumerate}
It can also be quickly derived that \nm{\langle u \times v , u\rangle = \langle u \times v , v\rangle = 0}.
\subsection{Points, Vectors, and Axes}\label{subsec:algebra_points_and_vectors}
Note that in the abstract discussion above it has not yet been defined what a vector is. This section focuses on the three-dimensional Euclidean space \nm{\mathbb{E}^3} \cite{Soatto2001}, which can be represented by a cartesian frame, where every \emph{point} \nm{\vec p \in \mathbb{E}^3} can be identified by its three coordinates \nm{\vec p = \lrsb{p_{\sss 1}, p_{\sss 2}, p_{\sss 3}}^T \in \mathbb{R}^3}. A \emph{vector} in \nm{\mathbb{E}^3} is defined by a pair of points \nm{\vec p, \vec q \in \mathbb{E}^3} with a directed arrow connecting \nm{\vec p} to \nm{\vec q}, where the vector \nm{\vec v = \lrsb{v_{\sss 1}, v_{\sss 2}, v_{\sss 3}}^T \in \mathbb{R}^3} is a triplet of numbers, each one being the difference between the corresponding coordinates of the two points \nm{\vec q} and \nm{\vec p} (\nm{\vec v = \vec q - \vec p \in \mathbb{R}^3}). Although they share notation, points and vectors are different geometric objects. A \emph{free vector} is one that does not depend on its starting or base point.
The set of all free vectors in \nm{\mathbb{R}^3} form an inner product space with cross product over the field of real numbers \nm{\mathbb{R}}, with both products defined as follows by making use of matrix notation:
\begin{eqnarray}
\nm{\langle \vec u , \vec v \rangle} & = & \nm{\vec u \cdot \vec v = {\vec u}^T \, \vec v = u_{\sss 1} \, v_{\sss 1} + u_{\sss 2} \, v_{\sss 2} + u_{\sss 3} \, v_{\sss 3}} \label{eq:SO3_inner_product} \\
\nm{\vec u \times \vec v} & = & \nm{\widehat{\vec u} \; \vec v = \begin{bmatrix} \nm{0} & \nm{- u_{\sss 3}} & \nm{+ u_{\sss 2}} \\ \nm{+ u_{\sss 3}} & \nm{0} & \nm{- u_{\sss 1}} \\ \nm{- u_{\sss 2}} & \nm{+ u_{\sss 1}} & \nm{0} \end{bmatrix} \begin{bmatrix} v_{\sss 1} \\ v_{\sss 2} \\ v_{\sss 3} \end{bmatrix} = \begin{bmatrix} \nm{u_{\sss 2} \, v_{\sss 3} - u_{\sss 3} \, v_{\sss 2}} \\ \nm{u_{\sss 3} \, v_{\sss 1} - u_{\sss 1} \, v_{\sss 3}} \\ \nm{u_{\sss 1} \, v_{\sss 2} - u_{\sss 2} \, v_{\sss 1}} \end{bmatrix} = - \vec v \times \vec u = - \widehat{\vec v} \; \vec u} \label{eq:SO3_cross_product}
\end{eqnarray}
where \nm{{\vec v}^T} is the transpose of \nm{\vec v} and \nm{\widehat{\vec v}} its skew-symmetric form\footnote{An skew-symmetric matrix is one whose negative equals its transpose.}. The inner product or euclidean metric can measure distances and angles, while the cross product defines orientation.
Any vector \nm{\vec v} in \nm{\mathbb{R}^3} can be written as \nm{\vec v = v_{\sss 1} \ \vec e_{\sss1} + v_{\sss 2} \ \vec e_{\sss 2} + v_{\sss 3} \ \vec e_{\sss 3}}, where \nm{\vec e_{\sss1}}, \nm{\vec e_{\sss 2}}, and \nm{\vec e_{\sss 3}} are the three linearly independent basis vectors and \nm{v_{\sss 1}, \ v_{\sss 2}, \ v_{\sss 3}} the coordinates or components of \nm{\vec v} with respect to that basis \cite{Strang2006}. The \emph{basis} is called orthogonal if \nm{{\vec e_i}^T \ \vec e_j = 0} when \nm{i \neq j}, orthonormal if additionally \nm{{\vec e_i}^T \ \vec e_j = 1} when \nm{i = j}, and right handed if additionally \nm{\epsilon_{ijk} = {\vec e_i}^T \ \widehat{\vec e}_j \ \vec e_k} is 1 for \nm{\epsilon_{123}}, \nm{\epsilon_{231}}, and \nm{\epsilon_{312}}, -1 for \nm{\epsilon_{132}}, \nm{\epsilon_{213}}, and \nm{\epsilon_{321}}, and 0 in all other cases \cite{Shuster1993}.
An \emph{axis} or \emph{line} is defined by its direction \nm{\vec n} (provided by a free vector) and a point \nm{\vec p} that it passes through. Its coordinates are \nm{\lrp{\vec n, \, \vec m} \in \mathbb{R}^6}, where \nm{ \vec m = \widehat{\vec p} \, \vec n} is called the \emph{moment} of the line. The coordinates \nm{\lrp{\vec n, \, \vec m}} are independent of \nm{\vec p}. The moment \nm{\vec m} is normal to the plane through the line and the origin with norm equal to the distance from the line to the origin. The point belonging to the line that is closest to the origin responds to \nm{\vec p_{\perp} = \widehat{\vec n} \, \vec m}. A line has four degrees of freedom and hence two redundancies, provided by \nm{\vec n} being a direction and hence a unit vector (\nm{\|\vec n\| = 1}) and \nm{\vec n} being orthogonal to \nm{\vec m} by definition (\nm{\vec n^T \, \vec m = 0}) \cite{Jia2013}.
It is important to remark that although this is the formal definition of an axis, it can also be informally considered that an axis is synonymous with just a direction \nm{\vec n} with two degrees of freedom (\nm{\|\vec n\| = 1}), passing through the origin \nm{\lrp{\vec p = \vec 0 \rightarrow \vec m = \vec 0}}. The reason is that in most occasions it is convenient to consider that a rigid body rotates about the origin of the reference frame representing it.
\subsection{Lie Groups and Lie Algebras}\label{subsec:algebra_lie}
A \emph{manifold} is a topological space that locally resembles Euclidean space near each element, so each element of an \emph{m} dimensional manifold has a neighborhood that is homeomorphic\footnote{A homeomorphism or topological isomorphism is a continuous function between topological spaces that has a continuous inverse function.} to the \emph{m} dimensional Euclidean space \cite{Gamelin1999}. Manifolds, which are embedded in spaces of higher dimension, are curved, smooth (hyper) surfaces with no edges or spikes; they are defined by the constraints imposed on the state \cite{Sola2018}, this is, the state vector is restricted to moving within the manifold.
A \emph{Lie group} \nm{\langle \mathcal{G}, \circ \rangle} is a smooth manifold whose elements satisfy the group axioms. They combine the local properties of smooth manifolds, enabling the use of calculus, with the global properties of groups, allowing the nonlinear composition of distant objects \cite{Sola2018}. Elements of \nm{\mathcal{G}} are denoted with \nm{\mathcal{X}}, the identity with \nm{\mathcal E}, and the inverse with \nm{\mathcal{X}^{-1}}. As in any other group, Lie groups are capable of transforming elements of other sets by means of their actions. In this sense, the group operation \nm{\lrb{\circ: \mathcal{G} \times \mathcal{G} \rightarrow \mathcal{G}}}, generally called \emph{composition}, can be considered as an action of the group on itself.
If \nm{\mathcal{X}\lrp{t}} is an element or point of the Lie group moving on the manifold, its derivative with time belongs to the space tangent to \nm{\mathcal{G}} at \nm{\mathcal{X}}, denoted by \nm{T_{\mathcal X}\mathcal G}. There exists a unique tangent space at each point, but the structure of such tangent spaces is the same everywhere \cite{Sola2018}. The \emph{tangent space} at a point is a real vector space of the same dimension as the manifold that intuitively contains all the possible directions in which one can tangentially pass through the point.
The \emph{Lie algebra} \nm{\mathfrak{m}} is defined as the tangent space at the identity \nm{\lrb{\mathfrak{m} = T_{\mathcal E}\mathcal G}}, and it is a vector space whose elements can be identified with vectors in \nm{\mathbb{R}^m}, with \emph{m} being the number of degrees of freedom of the Lie group \nm{\mathcal G} \cite{Sola2018}. Elements of the tangent space are usually denoted \nm{\vec v} when referring to velocities and \nm{\vec \tau = \vec v \cdot t} for more general elements. Lie algebras can be defined at any manifold point \nm{\mathcal{X}}, establishing local coordinates for \nm{T_{\mathcal X}\mathcal G}, and its elements are denoted by the \nm{<\cdot^{\wedge}>} symbol, such as \nm{\vec v^{\mathcal{X}\wedge} \in T_{\mathcal X}\mathcal G} or \nm{\vec \tau^{\mathcal{E}\wedge} \in T_{\mathcal E}\mathcal G}.
As all tangent spaces or Lie algebras have the same structure no matter the position of the element \nm{\mathcal X} within the Lie group \nm{\mathcal G}, it can be considered with no loss of generality\footnote{This is just a convention, and the opposite one is employed in some texts.} that all group actions at \nm{\mathcal X \in \mathcal G}, noted as \nm{g_{\mathcal X}()}, transform elements viewed in the \emph{local} or \emph{body} frame represented by \nm{\mathcal X} into the \emph{global} or \emph{space} frame represented by \nm{\mathcal E} \cite{Sola2018}. The opposite is true for the inverse operations noted as \nm{g_{\mathcal X}^{-1}() = g_{\mathcal{X}^{-1}}()}. When the group action is the composition \nm{\circ} itself, elements to the right of \nm{\mathcal X} belong to the body frame, while those on the left are viewed on the global frame.
\subsubsection{Lie Algebra Velocities, Hat and Vee Operators}\label{subsubsec:algebra_lie_velocities}
The structure of the Lie algebra can be obtained by time derivating the group inverse constraint \cite{Sola2018}. Every Lie group employed in this article (refer to sections \ref{sec:Rotate} and \ref{sec:Motion}) has a \nm{\circ} composition operator realized by some type of multiplication. If this is the case, the group inverse constraint responds to \nm{\mathcal X \circ \mathcal X^{-1} = \mathcal X^{-1} \circ \mathcal X = \mathcal E}, and its derivation with time leads to the Lie algebra \emph{velocities} viewed in either the body or local frames\footnote{Note that \nm{\mathcal X^{\dot{-}1}} represents the time derivative of the inverse, not the inverse of the time derivative.}:
\begin{eqnarray}
\nm{\dot{\mathcal X} \circ \mathcal X^{-1} + \mathcal X \circ \mathcal X^{\dot{-}1} = 0} & \nm{\rightarrow} & \nm{\vec v^{\mathcal{E}\wedge} = \dot{\mathcal X} \circ \mathcal X^{-1} = - \mathcal X \circ \mathcal X^{\dot{-}1}} \label{eq:algebra_vE} \\
\nm{\mathcal X^{-1} \circ \dot{\mathcal X} + \mathcal X^{\dot{-}1} \circ \mathcal X = 0} & \nm{\rightarrow} & \nm{\vec v^{\mathcal{X}\wedge} = \mathcal X^{-1} \circ \dot{\mathcal X} = - \mathcal X^{\dot{-}1} \circ \mathcal X} \label{eq:algebra_vX}
\end{eqnarray}
The \nm{\vec \tau^\wedge} or \nm{\vec v^\wedge} elements of the Lie algebra hence do not have trivial structures but can always be expressed as linear combinations of some base elements \nm{\vec e_i}, which are called the \emph{generators} of \nm{\mathfrak{m}} \cite{Sola2018}. It is generally more convenient to manipulate them as vectors \nm{\vec \tau \in \mathbb R^m}, as they can then be grouped together in larger state vectors and operated by means of linear algebra. The isomorphisms that linearly convert between them are called \emph{hat} \nm{\lrb{\cdot^\wedge: \mathbb{R}^m \rightarrow \mathfrak{m} \ | \ \vec \tau \rightarrow \vec \tau^\wedge}} and \emph{vee} \nm{\lrb{\cdot^\vee: \mathfrak{m} \rightarrow \mathbb{R}^m \ | \ \lrp{\vec \tau^\wedge}^\vee \rightarrow \vec \tau}}.
\subsubsection{Exponential and Logarithmic Maps, Plus and Minus Operators}\label{subsubsec:algebra_lie_exp_plus}
The \emph{exponential map} \nm{\lrb{exp\lrp{} : \mathfrak{m} \rightarrow \mathcal{G} \ | \ \mathcal{X} = exp\lrp{\vec \tau^\wedge}}} wraps the tangent element around the manifold following the geodesic or minimum distance line, effectively converting elements of the Lie algebra into those of the manifold or Lie group. The unwrapping or inverse operation is the \emph{logarithmic map} \nm{\lrb{log\lrp{} : \mathcal{G} \rightarrow \mathfrak{m} \ | \ \vec \tau^\wedge = log\lrp{\mathcal X}}}. The hat and vee operators can be incorporated into these maps, resulting in the \emph{capitalized maps} \nm{\lrb{Exp\lrp{} : \mathbb{R}^m \rightarrow \mathcal{G} \ | \ \mathcal{X} = Exp\lrp{\vec \tau}}} and \nm{\lrb{Log\lrp{} : \mathcal{G} \rightarrow \mathbb{R}^m \ | \ \vec \tau = Log\lrp{\mathcal X}}}. Note that the exponential map complies with the following properties \nm{\forall \ t \in \mathbb R} \cite{Sola2018}:
\begin{eqnarray}
\nm{exp\lrp{t \, \vec \tau^{\wedge}}} & = & \nm{exp\lrp{\vec \tau^\wedge}^t} \label{eq:algebra_power} \\
\nm{exp\lrp{\mathcal X \circ \vec \tau^{\wedge} \circ \mathcal{X}^{-1}}} & = & \nm{\mathcal X \circ exp\lrp{\vec \tau^{\wedge}} \circ \mathcal{X}^{-1}} \label{eq:algebra_exp}
\end{eqnarray}
The \emph{plus} and \emph{minus operators} enable operating with increments of the nonlinear manifold expressed in the corresponding linear tangent vector space \cite{Sola2018}. As the Lie group \nm{\mathcal G} is not abelian, there exist right \nm{\oplus} and \nm{\ominus} operators as well as left \nm{\boxplus} and \nm{\boxminus} ones. Note that the addition of (usually) small perturbations \nm{\Delta \vec \tau} to a given manifold \nm{\mathcal X} result in a perturbed manifold \nm{\mathcal Y}:
\begin{eqnarray}
\nm{\mathcal Y} & = & \nm{\mathcal X \oplus \Delta \vec \tau^{\mathcal X} = \mathcal X \circ Exp\lrp{\Delta \vec \tau^{\mathcal X}} \in \mathcal G} \label{eq:algebra_plus_right} \\
\nm{\Delta \vec \tau^{\mathcal X}} & = & \nm{\mathcal Y \ominus \mathcal X = Log\lrp{\mathcal X^{-1} \circ \mathcal Y} \in T_{\mathcal X}\mathcal G}\label{eq:algebra_minus_right} \\
\nm{\mathcal Y} & = & \nm\Delta {\vec \tau^{\mathcal E} \boxplus \mathcal X = Exp\lrp{\Delta \vec \tau^{\mathcal E}} \circ \mathcal X \in \mathcal G} \label{eq:algebra_plus_left} \\
\nm{\Delta \vec \tau^{\mathcal E}} & = & \nm{\mathcal Y \boxminus \mathcal X = Log\lrp{\mathcal Y \circ \mathcal X^{-1}} \in T_{\mathcal E}\mathcal G}\label{eq:algebra_minus_left}
\end{eqnarray}
\subsubsection{Adjoint Action}\label{subsubsec:algebra_lie_adjoint}
The vectors or elements of the tangent space at \nm{\mathcal{X}} can be transformed to the tangent space at the identity \nm{\mathcal E} by means of the \emph{adjoint}
\nm{\lrb{\vec{Ad}\lrp{}: \mathcal{G} \times \mathfrak{m} \rightarrow \mathfrak{m}}} \cite{Sola2018}. The adjoint is hence an action of the Lie group that operates on its own Lie algebra. The adjoint action can be obtained by the equivalence of the perturbed state \nm{\mathcal Y} in (\ref{eq:algebra_plus_right}, \ref{eq:algebra_plus_left}) by means of (\ref{eq:algebra_exp}):
\neweq{\vec \tau^{\mathcal{E}\wedge} = \vec{Ad}_{\mathcal X}\lrp{\vec \tau^{\mathcal{X}\wedge}} = \mathcal X \circ \vec \tau^{\mathcal{X}\wedge} \circ \mathcal{X}^{-1}} {eq:algebra_adjoint}
The adjoint action is a linear homomorphism, and hence complies with the following expressions \nm{\forall \ \mathcal{X}}, \nm{\mathcal{Y} \in \mathcal{G}}, \nm{\forall \ a, b \in \mathbb{R}}, \nm{\forall \ \vec \tau^{\mathcal{X}\wedge}, \vec \sigma^{\mathcal{X}\wedge} \in T_{\mathcal X}\mathcal G}, and \nm{\forall \ \vec \tau^{\mathcal{Y}\wedge} \in T_{\mathcal Y}\mathcal G}:
\begin{eqnarray}
\nm{\vec{Ad}_{\mathcal X}\lrp{a \, \vec \tau^{\mathcal{X}\wedge} + b \, \vec \sigma^{\mathcal{X}\wedge}}} & = & \nm{a \, \vec{Ad}_{\mathcal X}\lrp{\vec \tau^{\mathcal{X}\wedge}} + b \, \vec{Ad}_{\mathcal X}\lrp{\vec \sigma^{\mathcal{X}\wedge}}} \label{eq:algebra_adjoint_linear} \\
\nm{\vec{Ad}_{\mathcal X}\lrp{\vec{Ad}_{\mathcal Y}\lrp{\vec \tau^{\mathcal{Y}\wedge}}}} & = & \nm{\vec{Ad}_{\mathcal X \circ \mathcal Y}\lrp{\vec \tau^{\mathcal{Y}\wedge}}} \label{eq:algebra_adjoint_homo}
\end{eqnarray}
As the adjoint is a linear transform, it is always possible to obtain an equivalent matrix operator, the \emph{adjoint matrix} \nm{\lrb{\vec{Ad}: \mathcal{G} \times \mathbb{R}^m \rightarrow \mathbb{R}^m \ | \ \vec{Ad}_{\mathcal X} \cdot \vec \tau = \lrp{\mathcal X \, \vec \tau^{\wedge} \, \mathcal{X}^{-1}}^{\vee}, \ \vec{Ad}_{\mathcal X} \in \mathbb{R}^{mxm}}}, that maps the cartesian tangent space vectors instead of the Lie algebra elements \cite{Sola2017}. Both maps share the same symbols but are easily distinguished by context.
\neweq{\vec \tau^{\mathcal{E}} = \vec{Ad}_{\mathcal X} \cdot \vec \tau^{\mathcal{X}} = \lrp{\mathcal X \, \vec \tau^{{\mathcal X}\wedge} \, \mathcal{X}^{-1}}^{\vee} } {eq:algebra_adjoint_matrix}
The adjoint matrix complies with the following properties:
\begin{eqnarray}
\nm{\mathcal{X} \oplus \vec \tau^{\mathcal X}} & = & \nm{\lrp{\vec{Ad}_{\mathcal X} \cdot \vec \tau^{\mathcal{X}}} \boxplus \mathcal{X}} \label{eq:algebra_adjoint_matrix_general} \\
\nm{\vec{Ad}_{\mathcal X^{-1}}} & = & \nm{\vec{Ad}_{\mathcal X}^{-1}} \label{eq:algebra_adjoint_matrix_inverse} \\
\nm{\vec{Ad}_{\mathcal X \circ \mathcal Y}} & = & \nm{\vec{Ad}_{\mathcal X} \, \vec{Ad}_{\mathcal Y}} \label{eq:algebra_adjoint_matrix_product}
\end{eqnarray}
\subsubsection{Right and Left Lie Group Derivatives}\label{subsubsec:algebra_lie_derivatives}
Given a function \nm{\lrb{f: \mathcal{G} \rightarrow \mathcal {H} \ | \ \mathcal {Y} = f\lrp{\mathcal {X}} \in \mathcal {H}, \, \forall \mathcal {X} \in \mathcal {G}}} that maps together two manifolds or Lie groups of dimensions \emph{m} and \emph{n} respectively, it is possible to make use of the plus and minus operators to establish right and left derivatives (jacobians) that linearly map their respective Lie algebras or tangent spaces, either locally \nm{\lrp{T_{\mathcal X}\mathcal G \rightarrow T_{f\lrp{\mathcal X}}\mathcal H}} if employing the right \nm{\oplus} and \nm{\ominus} operators, or globally \nm{\lrp{T_{\mathcal E}\mathcal G \rightarrow T_{\mathcal E}\mathcal H}} when using the left \nm{\boxplus} and \nm{\boxminus} operators \cite{Sola2018}. As a tangent space can always be identified to a Euclidean space of the same dimension, this enables the application of the concepts of random vectors, stochastic processes, and their correlation (section \ref{sec:Error_Random}) to Lie algebras, leading to the section \ref{subsubsec:algebra_lie_covariance} expressions for Lie covariances. In addition, these derivatives constitute the basis for the construction of the Lie group jacobians in section \ref{subsec:algebra_lie_jacobians}, which in turn are indispensable for the establishment of rigorous solutions for the gradient descent minimization (optimization) and state estimation in Lie groups, as described in sections \ref{subsec:algebra_gradient_descent} and \ref{subsec:algebra_SS}, respectively.
The \emph{right jacobian} of \nm{f\lrp{\mathcal X}} is defined as the derivative of \nm{f\lrp{\mathcal X}} with respect to \nm{\mathcal X} when the increments are viewed in their respective local tangent spaces, this is, tangent respectively at \nm{\mathcal X \in \mathcal G} and \nm{f\lrp{\mathcal X} \in \mathcal H} \cite{Sola2018}. The \emph{left jacobian} of \nm{f\lrp{\mathcal X}} is defined similarly, but with the increments viewed in the global tangent spaces for \nm{\mathcal G} and \nm{\mathcal H} respectively:
\begin{eqnarray}
\nm{\vec J_{\ds{\oplus \; \mathcal X}}^{\ds{\oplus \; f(\mathcal X)}}} & = & \nm{\lim_{\Delta \vec \tau^{\mathcal X} \to \vec 0} \dfrac{f\lrp{\mathcal X \oplus \Delta \vec \tau^{\mathcal X}} \ominus f(\mathcal X)}{\Delta \vec \tau^{\mathcal X}} = \lim_{\Delta \vec \tau^{\mathcal X} \to \vec 0} \dfrac{Log\lrsb{f^{-1}(\mathcal X) \circ f\Big(\mathcal X \circ Exp\lrp{\Delta \vec \tau^{\mathcal X}}\Big)}}{\Delta \vec \tau^{\mathcal X}} \in \mathbb{R}^{nxm}} \label{eq:algebra_lie_derivative_right} \\
\nm{\vec J_{\ds{\boxplus \; \mathcal X}}^{\ds{\boxplus \; f(\mathcal X)}}} & = & \nm{\lim_{\Delta \vec \tau^{\mathcal E} \to \vec 0} \dfrac{f\lrp{\Delta \vec \tau^{\mathcal E} \boxplus \mathcal X} \boxminus f\lrp{\mathcal X}}{\Delta \vec \tau^{\mathcal E}} = \lim_{\Delta \vec \tau^{\mathcal E} \to \vec 0} \dfrac{Log\lrsb{f\Big(Exp\lrp{\Delta \vec \tau^{\mathcal E} \circ \mathcal X}\Big) \circ f^{-1}\lrp{\mathcal X}}}{\Delta \vec \tau^{\mathcal E}} \in \mathbb{R}^{nxm}} \label{eq:algebra_lie_derivative_left}
\end{eqnarray}
The following first order Taylor expansions can then be directly established:
\begin{eqnarray}
\nm{f\lrp{\mathcal X \oplus \Delta \vec \tau^{\mathcal X}}} & \nm{\approx} & \nm{f\lrp{\mathcal X} \oplus \lrsb{\vec J_{\ds{\oplus \; \mathcal X}}^{\ds{\oplus \; f\lrp{\mathcal X}}} \, \Delta \vec \tau^{\mathcal X}} = \mathcal Y \oplus \Delta \vec \tau^{\mathcal Y} \ \ \in \mathcal H} \label{eq:algebra_lie_derivative_right_taylor} \\
\nm{f\lrp{\Delta \vec \tau^{\mathcal E_{\mathcal G}} \boxplus \mathcal X}} & \nm{\approx} & \nm{\lrsb{\vec J_{\ds{\boxplus \; \mathcal X}}^{\ds{\boxplus \; f\lrp{\mathcal X}}} \, \Delta \vec \tau^{\mathcal E_{\mathcal G}}} \boxplus f\lrp{\mathcal X} = \Delta \vec \tau^{\mathcal E_{\mathcal H}} \boxplus \mathcal Y \ \ \in \mathcal H} \label{eq:algebra_lie_derivative_left_taylor}
\end{eqnarray}
Note that the \nm{\oplus} or \nm{\boxplus} symbols that appear as jacobian subindexes in (\ref{eq:algebra_lie_derivative_right}) and (\ref{eq:algebra_lie_derivative_left}) indicate that the domain \nm{\mathcal G} is indeed a Lie group, and should be replaced by a standard \nm{+} operator if this is not the case and the function \emph{f} domain is in fact a real or Euclidean space. If this is the case, the expressions \nm{\lrp{\mathcal X \oplus \Delta \vec \tau^{\mathcal X}}} and \nm{\lrp{\Delta \vec \tau^{\mathcal E_{\mathcal G}} \boxplus \mathcal X}} within the equations (\ref{eq:algebra_lie_derivative_right}) through (\ref{eq:algebra_lie_derivative_left_taylor}) shall both be replaced by \nm{\lrp{\mathcal X + \Delta \vec \tau}}.
Similarly, the \nm{\oplus} and \nm{\boxplus} symbols that appear as jacobian superindexes indicate that the function \emph{f} image or codomain \nm{\mathcal H} is also a Lie group, and should otherwise be replaced by \nm{+} if the codomain is a Euclidean space. If this is the case, the \nm{\ominus} and \nm{\boxminus} operators within (\ref{eq:algebra_lie_derivative_right}) and (\ref{eq:algebra_lie_derivative_left}) shall be replaced by the standard \nm{-} operator, and the \nm{f\lrp{\mathcal X} \oplus} and \nm{\boxplus \, f\lrp{\mathcal X}} expressions within (\ref{eq:algebra_lie_derivative_right_taylor}) and (\ref{eq:algebra_lie_derivative_left_taylor}) shall both be replaced by \nm{f\lrp{\mathcal X} +}.
Equations (\ref{eq:algebra_lie_derivative_right_taylor}) and (\ref{eq:algebra_lie_derivative_left_taylor}) lead to the following expressions for the function \emph{f} propagation of the tangent spaces:
\begin{eqnarray}
\nm{\Delta \vec \tau^{\mathcal Y}} & = & \nm{\Delta \vec \tau^{f\lrp{\mathcal X}} = \vec J_{\ds{\oplus \; \mathcal X}}^{\ds{\oplus \; f\lrp{\mathcal X}}} \, \Delta \vec \tau^{\mathcal X}} \label{eq:algebra_lie_derivative_right_equiv} \\
\nm{\Delta \vec \tau^{\mathcal E_{\mathcal H}}} & = & \nm{\vec J_{\ds{\boxplus \; \mathcal X}}^{\ds{\boxplus \; f\lrp{\mathcal X}}} \, \Delta \vec \tau^{\mathcal E_{\mathcal G}}} \label{eq:algebra_lie_derivative_left_equiv} \\
\end{eqnarray}
In addition, (\ref{eq:algebra_adjoint_matrix_general}) enables establishing a relationship between the right and left jacobians of \nm{f\lrp{\mathcal X}}:
\neweq{\vec J_{\ds{\boxplus \; \mathcal X}}^{\ds{\boxplus \; f\lrp{\mathcal X}}} = \vec{Ad}_{f\lrp{\mathcal X}} \ \vec J_{\ds{\oplus \; \mathcal X}}^{\ds{\oplus \; f\lrp{\mathcal X}}} \ \vec{Ad}_{\mathcal X}^{-1}} {eq:algebra_lie_derivative_relationship}
\subsubsection{Lie Groups Uncertainty and Covariance}\label{subsubsec:algebra_lie_covariance}
As the \nm{\oplus} and \nm{\ominus} operators can be employed to define perturbations in the local tangent space \nm{\lrp{T_{\vec \mu_{\mathcal X}}\mathcal G}} around a nominal or expected point \nm{E\lrsb{\mathcal X} = \vec \mu_{\mathcal X} \in \mathcal G}, it is possible to employ a \emph{local autocovariance} definition similar to the one used for Euclidean spaces (\ref{eq:Error_rvec_autoCovarianceMatrix}), leading to the definition of stochastic variables (vectors) on manifolds \nm{\mathcal X \sim \lrp{\vec \mu_{\mathcal X}, \ \vec C_{\mathcal X \mathcal X}^{\mathcal X}}} \cite{Sola2018}:
\begin{eqnarray}
\nm{\mathcal X} & = & \nm{\vec \mu_{\mathcal X} \oplus \Delta \vec \tau^{\mathcal X} \ \ \in \mathcal G} \label{eq:algebra_lie_covariance_right_plus} \\
\nm{\Delta \vec \tau^{\mathcal X}} & = & \nm{\mathcal X \ominus \vec \mu_{\mathcal X} \ \ \in T_{\vec \mu_{\mathcal X}}\mathcal G}\label{eq:algebra_lie_covariance_right_minus} \\
\nm{\vec C_{\mathcal X \mathcal X}^{\mathcal X}} & = & \nm{E\lrsb{\Delta \vec \tau^{\mathcal X} \, \Delta \vec \tau^{{\mathcal X,T}}} = E\lrsb{\lrp{\mathcal X \ominus \vec \mu_{\mathcal X}}\lrp{\mathcal X \ominus \vec \mu_{\mathcal X}}^T} \ \ \in \mathbb{R}^{mxm}}\label{eq:algebra_lie_covariance_right_def}
\end{eqnarray}
The different types of correlation matrices for stochastic processes introduced in section \ref{subsec:Error_StochasticProcesses} can also be defined accordingly. Note that although the notation of \nm{\vec C_{\mathcal X \mathcal X}^{\mathcal X}} refers to the covariance of the manifold or Lie group \nm{\mathcal X \in \mathcal G}, the definition in fact refers to the covariance of the nominal point local tangent space \nm{\Delta \vec \tau^{\mathcal X} \in T_{\vec \mu \mathcal X}\mathcal G}, with its dimension matching the number of degrees of freedom of the manifold.
A similar process employing the \nm{\boxplus} and \nm{\boxminus} operators leads to the \emph{global autocovariance}:
\begin{eqnarray}
\nm{\vec C_{\mathcal X \mathcal X}^{\mathcal E}} & = & \nm{E\lrsb{\Delta \vec \tau^{\mathcal E} \, \Delta \vec \tau^{{\mathcal E},T}} = E\lrsb{\lrp{\mathcal X \boxminus \vec \mu_{\mathcal X}}\lrp{\mathcal X \boxminus \vec \mu_{\mathcal X}}^T} \ \ \in \mathbb{R}^{mxm}}\label{eq:algebra_lie_covariance_left_def} \\
\nm{\vec C_{\mathcal X \mathcal X}^{\mathcal E}} & = & \nm{E\lrsb{\vec{Ad}_{\mathcal X} \, \Delta \vec \tau^{\mathcal X} \, \Delta \vec \tau^{{\mathcal X},T} \, \vec{Ad}_{\mathcal X}^T} = \vec{Ad}_{\mathcal X} \ \vec C_{\mathcal X \mathcal X}^{\mathcal X} \ \vec{Ad}_{\mathcal X}^T}\label{eq:algebra_lie_covariance_left_relationship} \\
\nm{\vec C_{\mathcal X \mathcal X}^{\mathcal X}} & = & \nm{E\lrsb{\vec{Ad}_{\mathcal X}^{-1} \, \Delta \vec \tau^{\mathcal E} \, \Delta \vec \tau^{{\mathcal E},T} \, \vec{Ad}_{\mathcal X}^{-T}} = \vec{Ad}_{\mathcal X}^{-1} \ \vec C_{\mathcal X \mathcal X}^{\mathcal E} \ \vec{Ad}_{\mathcal X}^{-T}}\label{eq:algebra_lie_covariance_right_relationship}
\end{eqnarray}
Given a function \nm{\lrb{f: \mathcal{G} \rightarrow \mathcal {H} \ | \ \mathcal {Y} = f\lrp{\mathcal {X}} \in \mathcal {H}, \, \forall \mathcal {X} \in \mathcal {G}}} as that introduced in section \ref{subsubsec:algebra_lie_derivatives}, which maps together two manifolds or Lie groups of dimensions \emph{m} and \emph{n} respectively, it is possible to propagate the covariances \nm{\vec C_{\mathcal X \mathcal X}^{\mathcal X}} and \nm{\vec C_{\mathcal X \mathcal X}^{\mathcal E}} from the domain manifold \nm{\mathcal G} to the image one \nm{\mathcal H}:
\begin{eqnarray}
\nm{\vec C_{\mathcal Y \mathcal Y}^{\mathcal Y}} & = & \nm{E\lrsb{\Delta \vec \tau^{\mathcal Y} \, \Delta \vec \tau^{\mathcal Y,T}} = E\lrsb{\Delta \vec \tau^{f\lrp{\mathcal X}} \, \Delta \vec \tau^{f\lrp{\mathcal X},T}}} \nonumber \\
& = & \nm{E\lrsb{\vec J_{\ds{\oplus \; \mathcal X}}^{\ds{\oplus \; f\lrp{\mathcal X}}} \, \Delta \vec \tau^{\mathcal X} \, \Delta \vec \tau^{{\mathcal X},T} \, \vec J_{\ds{\oplus \; \mathcal X}}^{{\ds{\oplus \; f\lrp{\mathcal X}}},T}} = \vec J_{\ds{\oplus \; \mathcal X}}^{\ds{\oplus \; f\lrp{\mathcal X}}} \ \vec C_{\mathcal X \mathcal X}^{\mathcal X} \ \vec J_{\ds{\oplus \; \mathcal X}}^{{\ds{\oplus \; f\lrp{\mathcal X}}},T} \ \ \ \ \in \mathbb{R}^{nxn}} \label{eq:algebra_lie_covariance_right_propagation} \\
\nm{\vec C_{\mathcal Y \mathcal Y}^{\mathcal E}} & = & \nm{E\lrsb{\Delta \vec \tau^{\mathcal E_{\mathcal H}} \, \Delta \vec \tau^{\mathcal E_{\mathcal H},T}} = \vec J_{\ds{\boxplus \; \mathcal X}}^{\ds{\boxplus \; f\lrp{\mathcal X}}} \ \vec C_{\mathcal X \mathcal X}^{\mathcal E} \ \vec J_{\ds{\boxplus \; \mathcal X}}^{{\ds{\boxplus \; f\lrp{\mathcal X}}},T} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \in \mathbb{R}^{nxn}} \label{eq:algebra_lie_covariance_left_propagation}
\end{eqnarray}
The establishment and propagation of covariance matrices of the proper dimensions is key for the application of state estimation techniques such as Kalman filtering when some of the state vector components belong to Lie manifolds and their tangent spaces, as described in section \ref{subsec:algebra_SS}.
\subsection{Euclidean and Lie Jacobians}\label{subsec:algebra_lie_jacobians}
The definition of the proper derivative matrices or jacobians is indispensable for all calculus techniques that rely on linearization, such as optimization by means of the gradient descent method (section \ref{subsec:algebra_gradient_descent}) or state estimation through Kalman filtering (section \ref{subsec:algebra_SS}). Given a function \nm{\lrb{f\lrp{\vec x}: \mathbb{R}^m \rightarrow \mathbb{R}^n}}, its (Euclidean) jacobian \nm{\vec J_{\ds{+ \; \vec x}}^{\ds{+ \; f(\vec x)}} \in \mathbb{R}^{nxm}} stacks the partial derivatives of each component of the output space with respect to those of the input space:
\neweq{J_{{\ds{+ \; \vec x}}, ij}^{\ds{+ \; f(\vec x)}} = \lim_{\Delta x_j\to 0} \dfrac{f_i\lrp{x_j + \Delta x_j} - f_i\lrp{x_j}}{\Delta x_j}}{eq:algebra_jacobian_euclidean}
This section relies on the right and left Lie group derivatives introduced in section \ref{subsubsec:algebra_lie_derivatives} to properly define jacobians when either the input or output spaces (or both) are not Euclidean but Lie groups. The various jacobians listed in table \ref{tab:algebra_lie_jacobians} have been obtained by means of the expressions that appear on section \ref{subsec:algebra_lie} together with the chain rule, and include instances in which both the domain and codomain of the \nm{f\lrp{\mathcal X}} function are either Euclidean or Lie groups. Some are generic, while others, which rely on group actions, depend on the specific set on which the action is applied and hence can only be established for a specific Lie group, such as rotational or rigid body motions (sections \ref{sec:Rotate} and \ref{sec:Motion}).
There are two constructions that appear repeatedly within table \ref{tab:algebra_lie_jacobians}. The first is the adjoint matrix (\ref{eq:algebra_adjoint_matrix}), which maps the local and global tangent spaces at a given point on a manifold or Lie group, while the second are the right and left jacobians of the capitalized exponential function, also known as simply the \emph{right jacobian} \nm{J_R\lrp{\vec \tau}} and the \emph{left jacobian} \nm{J_L\lrp{\vec \tau}}. These compare variations in the tangent space of the output \nm{Exp\lrp{\vec \tau}} map (locally for \nm{J_R} and globally for \nm{J_L}) with (Euclidean) variations in the input argument \nm{\vec \tau}, and are obtained in sections \ref{subsec:RigidBody_rotation_calculus_jacobians} and \ref{subsec:RigidBody_motion_calculus_jacobians} for the specific cases of rotational and rigid body motions, respectively.
\begin{eqnarray}
\nm{\vec{Ad}_{Exp\lrp{\vec \tau}}} & = & \nm{\vec J_L\lrp{\vec \tau} \, \vec J_R^{-1}\lrp{\vec \tau}} \label{eq:algebra_lie_jacobians_right_left1} \\
\nm{\vec J_R\lrp{- \vec \tau}} & = & \nm{\vec J_L\lrp{\vec \tau}} \label{eq:algebra_lie_jacobians_right_left2}
\end{eqnarray}
Being located in the tangent spaces, the right and left jacobians can be related by means of the adjoint, resulting in (\ref{eq:algebra_lie_jacobians_right_left1}). Expression (\ref{eq:algebra_lie_jacobians_right_left2}) in turn can be obtained by means of the chain rule.
\renewcommand{\arraystretch}{1.5}
\begin{center}
\begin{tabular}{lcp{0.2cm}rcll}
\hline
\textbf{Jacobian} & \textbf{Result} & & \multicolumn{3}{c}{\textbf{Taylor Expansion}} & \\
\hline
\nm{\vec J_{\ds{\oplus \; \mathcal X}}^{\ds{\oplus \; \mathcal X}^{-1}}} & \nm{- \vec{Ad}_{\mathcal X}} & & \nm{\lrp{\mathcal X \oplus \Delta \vec \tau}^{-1}} & \nm{\approx} & \nm{\mathcal X^{-1} \oplus \lrsb{\vec J_{\ds{\oplus \; \mathcal X}}^{\ds{\oplus \; \mathcal X}^{-1}} \, \Delta \vec \tau}} & \nm{\in \mathcal G} \\
\nm{\vec J_{\ds{\boxplus \; \mathcal X}}^{\ds{\boxplus \; \mathcal X}^{-1}}} & \nm{- \vec{Ad}_{\mathcal X}^{-1}} & & \nm{\lrp{\Delta \vec \tau \boxplus \mathcal X}^{-1}} & \nm{\approx} & \nm{\lrsb{\vec J_{\ds{\boxplus \; \mathcal X}}^{\ds{\boxplus \; \mathcal X}^{-1}} \, \Delta \vec \tau} \boxplus \mathcal X^{-1}} & \nm{\in \mathcal G} \\
\nm{\vec J_{\ds{\oplus \; \mathcal X}}^{\ds{\oplus \; \mathcal X \circ \mathcal Y}}} & \nm{\vec{Ad}_{\mathcal Y}^{-1}} & & \nm{\lrp{\mathcal X \oplus \Delta \vec \tau} \circ \mathcal Y} & \nm{\approx} & \nm{\lrp{\mathcal X \circ \mathcal Y} \oplus \lrsb{\vec J_{\ds{\oplus \; \mathcal X}}^{\ds{\oplus \; \mathcal X \circ \mathcal Y}} \, \Delta \vec \tau}} & \nm{\in \mathcal G} \\
\nm{\vec J_{\ds{\boxplus \; \mathcal X}}^{\ds{\boxplus \; \mathcal X \circ \mathcal Y}}} & \nm{\vec{I}_{mxm}} & & \nm{\lrp{\Delta \vec \tau \boxplus \mathcal X} \circ \mathcal Y} & \nm{\approx} & \nm{\lrsb{\vec J_{\ds{\boxplus \; \mathcal X}}^{\ds{\boxplus \; \mathcal X \circ \mathcal Y}} \, \Delta \vec \tau} \boxplus \lrp{\mathcal X \circ \mathcal Y}} & \nm{\in \mathcal G} \\
\nm{\vec J_{\ds{\oplus \; \mathcal Y}}^{\ds{\oplus \; \mathcal X \circ \mathcal Y}}} & \nm{\vec{I}_{mxm}} & & \nm{\mathcal X \circ \lrp{\mathcal Y \oplus \Delta \vec \tau}} & \nm{\approx} & \nm{\lrp{\mathcal X \circ \mathcal Y} \oplus \lrsb{\vec J_{\ds{\oplus \; \mathcal Y}}^{\ds{\oplus \; \mathcal X \circ \mathcal Y}} \, \Delta \vec \tau}} & \nm{\in \mathcal G} \\
\nm{\vec J_{\ds{\boxplus \; \mathcal Y}}^{\ds{\boxplus \; \mathcal X \circ \mathcal Y}}} & \nm{\vec{Ad}_{\mathcal X}} & & \nm{\mathcal X \circ \lrp{\Delta \vec \tau \boxplus \mathcal Y}} & \nm{\approx} & \nm{\lrsb{\vec J_{\ds{\boxplus \; \mathcal Y}}^{\ds{\boxplus \; \mathcal X \circ \mathcal Y}} \, \Delta \vec \tau} \boxplus \lrp{\mathcal X \circ \mathcal Y}} & \nm{\in \mathcal G} \\
\nm{\vec J_{\ds{+ \; \vec \tau}}^{\ds{\oplus \; Exp\lrp{\vec \tau}}}} & \nm{ J_R\lrp{\vec \tau}}\footnotemark & & \nm{Exp\lrp{\vec \tau + \Delta \vec \tau}} & \nm{\approx} & \nm{Exp\lrp{\vec \tau} \oplus \lrsb{J_R\lrp{\vec \tau} \, \Delta \vec \tau}} & \nm{\in \mathcal G} \\
\nm{J_R^{-1}\lrp{\vec \tau}} & & & \nm{\vec \tau + \vec J_R^{-1}\lrp{\vec \tau} \, \Delta \vec \tau} & \nm{\approx}\footnotemark & \nm{Log\Big(Exp\lrp{\vec \tau} \oplus \Delta \vec \tau\Big)} & \nm{\in \mathbb{R}^m} \\
\nm{\vec J_{\ds{+ \; \vec \tau}}^{\ds{\boxplus \; Exp\lrp{\vec \tau}}}} & \nm{J_L\lrp{\vec \tau}}\footnotemark & & \nm{Exp\lrp{\vec \tau + \Delta \vec \tau}} & \nm{\approx} & \nm{\lrsb{J_L\lrp{\vec \tau} \, \Delta \vec \tau} \boxplus Exp\lrp{\vec \tau}} & \nm{\in \mathcal G} \\
\nm{J_L^{-1}\lrp{\vec \tau}} & & & \nm{\vec \tau + \vec J_L^{-1}\lrp{\vec \tau} \, \Delta \vec \tau} & \nm{\approx}\footnotemark & \nm{Log\Big(\Delta \vec \tau \boxplus Exp\lrp{\vec \tau}\Big)} & \nm{\in \mathbb{R}^m} \\
\nm{\vec J_{\ds{\oplus \; \mathcal X}}^{\ds{+ \; Log\lrp{\mathcal X}}}} & \nm{J_R^{-1}\big(Log\lrp{\mathcal X}\big)} & & \nm{Log\lrp{\mathcal X \oplus \Delta \vec \tau}} & \nm{\approx} & \nm{Log\lrp{\mathcal X} + \lrsb{\vec J_{\ds{\oplus \; \mathcal X}}^{\ds{+ \; Log\lrp{\mathcal X}}} \, \Delta \vec \tau}} & \nm{\in \mathbb{R}^m} \\
\nm{\vec J_{\ds{\boxplus \; \mathcal X}}^{\ds{+ \; Log\lrp{\mathcal X}}}} & \nm{J_L^{-1}\big(Log\lrp{\mathcal X}\big)} & & \nm{Log\lrp{\Delta \vec \tau \boxplus \mathcal X}} & \nm{\approx} & \nm{Log\lrp{\mathcal X} + \lrsb{\vec J_{\ds{\boxplus \; \mathcal X}}^{\ds{+ \; Log\lrp{\mathcal X}}} \, \Delta \vec \tau}} & \nm{\in \mathbb{R}^m} \\
\nm{\vec J_{\ds{\oplus \; \mathcal X}}^{\ds{\oplus \; \mathcal X \oplus \vec \tau}}} & \nm{\vec{Ad}_{Exp\lrp{\vec \tau}}^{-1}} & & \nm{\lrp{\mathcal X \oplus \Delta \vec \tau} \oplus \vec \tau} & \nm{\approx} & \nm{\lrp{\mathcal X \oplus \vec \tau} \oplus \lrsb{\vec J_{\ds{\oplus \; \mathcal X}}^{\ds{\oplus \; \mathcal X \oplus \vec \tau}} \, \Delta \vec \tau}} & \nm{\in \mathcal G} \\
\nm{\vec J_{\ds{\boxplus \; \mathcal X}}^{\ds{\boxplus \; \vec \tau \boxplus \mathcal X}}} & \nm{\vec{Ad}_{Exp\lrp{\vec \tau}}} & & \nm{\vec \tau \boxplus \lrp{\Delta \vec \tau \boxplus \mathcal X}} & \nm{\approx} & \nm{\lrsb{\vec J_{\ds{\boxplus \; \mathcal X}}^{\ds{\boxplus \; \vec \tau \boxplus\mathcal X}} \, \Delta \vec \tau} \boxplus \lrp{\vec \tau \boxplus \mathcal X}} & \nm{\in \mathcal G} \\
\nm{\vec J_{\ds{+ \; \vec \tau}}^{\ds{\oplus \; \mathcal X \oplus \vec \tau}}} & \nm{J_R\lrp{\vec \tau}} & & \nm{\mathcal X \oplus \lrp{\vec \tau + \Delta \vec \tau}} & \nm{\approx} & \nm{\lrp{\mathcal X \oplus \vec \tau} \oplus \lrsb{\vec J_{\ds{+ \; \vec \tau}}^{\ds{\oplus \; \mathcal X \oplus \vec \tau}} \, \Delta \vec \tau}} & \nm{\in \mathcal G} \\
\nm{\vec J_{\ds{+ \; \vec \tau}}^{\ds{\boxplus \; \vec \tau \boxplus \mathcal X}}} & \nm{J_L\lrp{\vec \tau}} & & \nm{\lrp{\vec \tau + \Delta \vec \tau} \boxplus \mathcal X} & \nm{\approx} & \nm{\lrsb{\vec J_{\ds{+ \; \vec \tau}}^{\ds{\boxplus \; \vec \tau \boxplus \mathcal X}} \, \Delta \vec \tau} \boxplus \lrp{\vec \tau \boxplus \mathcal X}} & \nm{\in \mathcal G} \\
\nm{\vec J_{\ds{\oplus \; \mathcal X}}^{\ds{+ \; \mathcal Y \ominus \mathcal X}}} & \nm{- J_L^{-1}\lrp{\mathcal Y \ominus \mathcal X}} & & \nm{\mathcal Y \ominus \lrp{\mathcal X \oplus \Delta \vec \tau}} & \nm{\approx} & \nm{\lrp{\mathcal Y \ominus \mathcal X} + \lrsb{\vec J_{\ds{\oplus \; \mathcal X}}^{\ds{+ \; \mathcal Y \ominus \mathcal X}} \, \Delta \vec \tau}} & \nm{\in \mathbb{R}^m} \\
\nm{\vec J_{\ds{\boxplus \; \mathcal X}}^{\ds{+ \; \mathcal Y \boxminus \mathcal X}}} & \nm{- J_R^{-1}\lrp{\mathcal Y \boxminus \mathcal X}} & & \nm{\mathcal Y \boxminus \lrp{\Delta \vec \tau \boxplus \mathcal X}} & \nm{\approx} & \nm{\lrp{\mathcal Y \boxminus \mathcal X} + \lrsb{\vec J_{\ds{\boxplus \; \mathcal X}}^{\ds{+ \; \mathcal Y \boxminus \mathcal X}} \, \Delta \vec \tau}} & \nm{\in \mathbb{R}^m} \\
\nm{\vec J_{\ds{\oplus \; \mathcal Y}}^{\ds{+ \; \mathcal Y \ominus \mathcal X}}} & \nm{J_R^{-1}\lrp{\mathcal Y \ominus \mathcal X}} & & \nm{\lrp{\mathcal Y \oplus \Delta \vec \tau} \ominus \mathcal X} & \nm{\approx} & \nm{\lrp{\mathcal Y \ominus \mathcal X} + \lrsb{\vec J_{\ds{\oplus \; \mathcal Y}}^{\ds{+ \; \mathcal Y \ominus \mathcal X}} \, \Delta \vec \tau}} & \nm{\in \mathbb{R}^m} \\
\nm{\vec J_{\ds{\boxplus \; \mathcal Y}}^{\ds{+ \; \mathcal Y \boxminus \mathcal X}}} & \nm{J_L^{-1}\lrp{\mathcal Y \boxminus \mathcal X}} & & \nm{\lrp{\Delta \vec \tau \boxplus \mathcal Y} \boxminus \mathcal X} & \nm{\approx} & \nm{\lrp{\mathcal Y \boxminus \mathcal X} + \lrsb{\vec J_{\ds{\boxplus \; \mathcal Y}}^{\ds{+ \; \mathcal Y \boxminus \mathcal X}} \, \Delta \vec \tau}} & \nm{\in \mathbb{R}^m} \\
\nm{\vec J_{\ds{\oplus \; \mathcal X}}^{\ds{+ \; g_{\mathcal X}(\vec u)}}} & Tables & & \nm{g_{\mathcal X \oplus \Delta \vec \tau}\lrp{\vec u}} & \nm{\approx} & \nm{g_{\mathcal X}\lrp{\vec u} + \lrsb{\vec J_{\ds{\oplus \; \mathcal X}}^{\ds{+ \; g_{\mathcal X}(\vec u)}} \, \Delta \vec \tau}} & \nm{\in \mathbb{R}^u} \\
\nm{\vec J_{\ds{\boxplus \; \mathcal X}}^{\ds{+ \; g_{\mathcal X}(\vec u)}}} & \ref{tab:RigidBody_rotation_jacobians} & & \nm{g_{\Delta \vec \tau \boxplus \mathcal X}\lrp{\vec u}} & \nm{\approx} & \nm{g_{\mathcal X}\lrp{\vec u} + \lrsb{\vec J_{\ds{\boxplus \; \mathcal X}}^{\ds{+ \; g_{\mathcal X}(\vec u)}} \, \Delta \vec \tau}} & \nm{\in \mathbb{R}^u} \\
\nm{\vec J_{\ds{+ \; \vec u}}^{\ds{+ \; g_{\mathcal X}(\vec u)}}} & \& & & \nm{g_{\mathcal X}\lrp{\vec u + \Delta \vec u}} & \nm{\approx} & \nm{g_{\mathcal X}\lrp{\vec u} + \lrsb{\vec J_{\ds{+ \; \vec u}}^{\ds{+ \; g_{\mathcal X}(\vec u)}} \, \Delta \vec u}} & \nm{\in \mathbb{R}^u} \\
\nm{\vec J_{\ds{\oplus \; \mathcal X}}^{\ds{+ \; g_{\mathcal X}^{-1}(\vec u)}}} & \ref{tab:RigidBody_motion_jacobians} & & \nm{g_{\mathcal X \oplus \Delta \vec \tau}^{-1}\lrp{\vec u}} & \nm{\approx} & \nm{g_{\mathcal X}^{-1}\lrp{\vec u} + \lrsb{\vec J_{\ds{\oplus \; \mathcal X}}^{\ds{+ \; g_{\mathcal X}^{-1}(\vec u)}} \, \Delta \vec \tau}} & \nm{\in \mathbb{R}^u} \\
\hline
\end{tabular}
\end{center}
\addtocounter{footnote}{-3}
\footnotetext{Obtained in sections \ref{subsec:RigidBody_rotation_calculus_jacobians} and \ref{subsec:RigidBody_motion_calculus_jacobians} for rotational and rigid body motion, respectively.}
\addtocounter{footnote}{1}
\footnotetext{Obtained by replacing \nm{\Delta \vec \tau} by \nm{\vec J_R^{-1} \ \Delta \vec \tau} in the expression above.}
\addtocounter{footnote}{1}
\footnotetext{Obtained in sections \ref{subsec:RigidBody_rotation_calculus_jacobians} and \ref{subsec:RigidBody_motion_calculus_jacobians} for rotational and rigid body motion, respectively.}
\addtocounter{footnote}{1}
\footnotetext{Obtained by replacing \nm{\Delta \vec \tau} by \nm{\vec J_L^{-1} \ \Delta \vec \tau} in the expression above.}
\begin{center}
\begin{tabular}{lcp{0.2cm}rcll}
\hline
\textbf{Jacobian} & \textbf{Result} & & \multicolumn{3}{c}{\textbf{Taylor Expansion}} & \\
\hline
\nm{\vec J_{\ds{\boxplus \; \mathcal X}}^{\ds{+ \; g_{\mathcal X}^{-1}(\vec u)}}} & & & \nm{g_{\Delta \vec \tau \boxplus \mathcal X}^{-1}\lrp{\vec u}} & \nm{\approx} & \nm{g_{\mathcal X}^{-1}\lrp{\vec u} + \lrsb{\vec J_{\ds{\boxplus \; \mathcal X}}^{\ds{+ \; g_{\mathcal X}^{-1}(\vec u)}} \, \Delta \vec \tau}} & \nm{\in \mathbb{R}^u} \\
\nm{\vec J_{\ds{+ \; \vec u}}^{\ds{+ \; g_{\mathcal X}^{-1}(\vec u)}}} & & & \nm{g_{\mathcal X}^{-1}\lrp{\vec u + \Delta \vec u}} & \nm{\approx} & \nm{g_{\mathcal X}^{-1}\lrp{\vec u} + \lrsb{\vec J_{\ds{+ \; \vec u}}^{\ds{+ \; g_{\mathcal X}^{-1}(\vec u)}} \, \Delta \vec u}} & \nm{\in \mathbb{R}^u} \\
\nm{\vec J_{\ds{\oplus \; \mathcal X}}^{\ds{+ \; \vec{Ad}_{\mathcal X}(\vec v)}}} & & & \nm{\vec{Ad}_{\mathcal X \oplus \Delta \vec \tau}\lrp{\vec v}} & \nm{\approx} & \nm{\vec{Ad}_{\mathcal X}\lrp{\vec v} + \lrsb{\vec J_{\ds{\oplus \; \mathcal X}}^{\ds{+ \; \vec{Ad}_{\mathcal X}(\vec v)}} \, \Delta \vec \tau}} & \nm{\in \mathbb{R}^m} \\
\nm{\vec J_{\ds{\boxplus \; \mathcal X}}^{\ds{+ \; \vec{Ad}_{\mathcal X}(\vec v)}}} & Tables & & \nm{\vec{Ad}_{\Delta \vec \tau \boxplus \mathcal X}\lrp{\vec v}} & \nm{\approx} & \nm{\vec{Ad}_{\mathcal X}\lrp{\vec v} + \lrsb{\vec J_{\ds{\boxplus \; \mathcal X}}^{\ds{+ \; \vec{Ad}_{\mathcal X}(\vec v)}} \, \Delta \vec \tau}} & \nm{\in \mathbb{R}^m} \\
\nm{\vec J_{\ds{+ \; \vec v}}^{\ds{+ \; \vec{Ad}_{\mathcal X}(\vec v)}}} & \ref{tab:RigidBody_rotation_jacobians} & & \nm{\vec{Ad}_{\mathcal X}\lrp{\vec v + \Delta \vec v}} & \nm{\approx} & \nm{\vec{Ad}_{\mathcal X}\lrp{\vec v} + \lrsb{\vec J_{\ds{+ \; \vec v}}^{\ds{+ \; \vec{Ad}_{\mathcal X}(\vec v)}} \, \Delta \vec v}} & \nm{\in \mathbb{R}^m} \\
\nm{\vec J_{\ds{\oplus \; \mathcal X}}^{\ds{+ \; \vec{Ad}_{\mathcal X}^{-1}(\vec v)}}} & \& & & \nm{\vec{Ad}_{\mathcal X \oplus \Delta \vec \tau}^{-1}\lrp{\vec v}} & \nm{\approx} & \nm{\vec{Ad}_{\mathcal X}^{-1}\lrp{\vec v} + \lrsb{\vec J_{\ds{\oplus \; \mathcal X}}^{\ds{+ \; \vec{Ad}_{\mathcal X}^{-1}(\vec v)}} \, \Delta \vec \tau}} & \nm{\in \mathbb{R}^m} \\
\nm{\vec J_{\ds{\boxplus \; \mathcal X}}^{\ds{+ \; \vec{Ad}_{\mathcal X}^{-1}(\vec v)}}} & \ref{tab:RigidBody_motion_jacobians} & & \nm{\vec{Ad}_{\Delta \vec \tau \boxplus \mathcal X}^{-1}\lrp{\vec v}} & \nm{\approx} & \nm{\vec{Ad}_{\mathcal X}^{-1}\lrp{\vec v} + \lrsb{\vec J_{\ds{\boxplus \; \mathcal X}}^{\ds{+ \; \vec{Ad}_{\mathcal X}^{-1}(\vec v)}} \, \Delta \vec \tau}} & \nm{\in \mathbb{R}^m} \\
\nm{\vec J_{\ds{+ \; \vec v}}^{\ds{+ \; \vec{Ad}_{\mathcal X}^{-1}(\vec v)}}} & & & \nm{\vec{Ad}_{\mathcal X}^{-1}\lrp{\vec v + \Delta \vec v}} & \nm{\approx} & \nm{\vec{Ad}_{\mathcal X}^{-1}\lrp{\vec v} + \lrsb{\vec J_{\ds{+ \; \vec v}}^{\ds{+ \; \vec{Ad}_{\mathcal X}^{-1}(\vec v)}} \, \Delta \vec v}} & \nm{\in \mathbb{R}^m} \\
\nm{\vec J_{\ds{+ \; \vec \tau}}^{\ds{+ \; g_{Exp\lrp{\vec \tau}}(\vec u)}}} & & & \nm{g_{Exp\lrp{\vec \tau + \Delta \vec \tau}}\lrp{\vec u}} & \nm{\approx} & \nm{g_{Exp\lrp{\vec \tau}}\lrp{\vec u} + \lrsb{\vec J_{\ds{+ \; \vec \tau}}^{\ds{+ \; g_{Exp\lrp{\vec \tau}}(\vec u)}} \, \Delta \vec \tau}} & \nm{\in \mathbb{R}^u} \\
\nm{\vec J_{\ds{+ \; \vec \tau}}^{\ds{+ \; g_{Exp\lrp{\vec \tau}}^{-1}(\vec u)}}} & & & \nm{g_{Exp\lrp{\vec \tau + \Delta \vec \tau}}^{-1}\lrp{\vec u}} & \nm{\approx} & \nm{g_{Exp\lrp{\vec \tau}}^{-1}\lrp{\vec u} + \lrsb{\vec J_{\ds{+ \; \vec \tau}}^{\ds{+ \; g_{Exp\lrp{\vec \tau}}^{-1}(\vec u)}} \, \Delta \vec \tau}} & \nm{\in \mathbb{R}^u} \\
\hline
\end{tabular}
\end{center}
\captionof{table}{Lie jacobians} \label{tab:algebra_lie_jacobians}
\renewcommand{\arraystretch}{1.0}
\subsection{Discrete Integration in Lie Groups}\label{subsec:algebra_integration}
Following on the discrete integration in Euclidean spaces described in section \ref{subsec:euclidean_integration}, let's now consider that the state system is composed by a vector \nm{\vec y \in \mathbb{R}^n}, an element of a Lie group \nm{\mathcal X \in \mathcal G}, and a vector \nm{\vec v^{\mathcal X} \in \mathbb{R}^m} representing the velocity of \nm{\mathcal X} as it moves along its manifold, contained in the local or body tangent space \nm{T_{\mathcal X}\mathcal G}. Neither \nm{\mathcal X} nor its components are Euclidean, and hence the section \ref{subsec:euclidean_integration} integration schemes are not applicable. If treated so, the resulting element \nm{\mathcal X_{k+1}} would not be located on the manifold as it would not comply with the Lie group constraints, and it would be necessary to reproject it back to it, incurring in errors that although small for a single integration step may become significant when accumulated.
Let's group these states into a composite state vector made up by \emph{n} plus \emph{m} components of an Euclidean space plus an element of a Lie group. As in the Euclidean case, the initial composite vector value is known:
\begin{eqnarray}
\nm{\vec x} & = & \nm{\lrsb{\vec y, \, \vec v^{\mathcal X}, \, \mathcal X}^T} \label{eq:algebra_integration_comp} \\
\nm{\vec x_k} & = & \nm{\vec x\lrp{t_k} = \lrsb{\vec y_k, \, \vec v_k^{\mathcal X}, \, \mathcal X_k}^T} \label{eq:algebra_integration_comp_initial}
\end{eqnarray}
As in the Euclidean case, the objective is the determination of the composite state vector value at a time \nm{\vec x_{k+1} = \vec x\lrp{t_{k+1}} = \vec x\lrp{t_k + \Delta t}} by relying on evaluations of the \nm{\vec y} and \nm{\vec v^{\mathcal X}} time derivatives:
\begin{eqnarray}
\nm{\vec {\dot y}\lrp{t}} & = & \nm{f_y\big(\yvec\lrp{t}, \, \vec v^{\mathcal X}\lrp{t}, \, \mathcal X\lrp{t}, \, t\big)} \label{eq:algebra_integration_comp_x_deriv} \\
\nm{\vec {\dot v}^{\mathcal X}\lrp{t}} & = & \nm{f_v\big(\yvec\lrp{t}, \, \vec v^{\mathcal X}\lrp{t}, \, \mathcal X\lrp{t}, \, t\big)} \label{eq:algebra_integration_comp_v_deriv}
\end{eqnarray}
The solution consists on employing the Euclidean integration method of choice to obtain \nm{\vec y_{k+1}} and \nm{\vec v_{k+1}^{\mathcal X}}, but rely on the right plus operator and the capitalized exponential map of the Lie group to determine \nm{\mathcal X_{k+1}}. In case of Euler's method, the solution is the following:
\begin{eqnarray}
\nm{\vec y_{k+1}} & \nm{\approx} & \nm{\vec y_k + \Delta t \ \vec{\dot y}(\vec y_k, \, \vec v_k^{\mathcal X}, \, \mathcal X_k, \, t_k)} \label{eq:algebra_integration_comp_y_euler} \\
\nm{\vec v_{k+1}^{\mathcal X}} & \nm{\approx} & \nm{\vec v_k^{\mathcal X} + \Delta t \ \vec{\dot v}^{\mathcal X}(\vec y_k, \, \vec v_k^{\mathcal X}, \, \mathcal X_k, \, t_k)} \label{eq:algebra_integration_comp_v_euler} \\
\nm{\mathcal X_{k+1}} & \nm{\approx} & \nm{\mathcal X_k \oplus \lrsb{\Delta t \ \vec v_k^{\mathcal X}} = \mathcal X_k \circ Exp\lrp{\Delta t \ \vec v_k^{\mathcal X}}} \label{eq:algebra_integration_comp_X_euler} \\
\nm{\vec x_{k+1}} & = & \nm{\vec x\lrp{t_{k+1}} \approx \lrsb{\vec y_{k+1}, \, \vec v_{k+1}^{\mathcal X}, \, \mathcal X_{k+1}}^T} \label{eq:algebra_integration_comp_final}
\end{eqnarray}
In case the state vector velocity \nm{\vec v^{\mathcal E}} is that of the global or space tangent space \nm{T_{\mathcal E}\mathcal G}, it is necessary to modify (\ref{eq:algebra_integration_comp_X_euler}) to employ the left plus operator:
\neweq{\mathcal X_{k+1} \approx \lrsb{\Delta t \ \vec v_k^{\mathcal E}} \boxplus \mathcal X_k = Exp\lrp{\Delta t \ \vec v_k^{\mathcal E}} \circ \mathcal X_k} {eq:algebra_integration_comp_X_euler_left}
It is necessary to proceed in a similar manner if a different integration method is selected. In the case of Heun's method with local velocity \nm{\vec v^{\mathcal X}}, the modified integration scheme is the following:
\begin{eqnarray}
\nm{\vec {\dot y}_1} & = & \nm{\vec{\dot y}(\vec y_k, \, \vec v_k^{\mathcal X}, \, \mathcal X_k, \, t_k)} \label{eq:algebra_integration_comp_y_heun_one} \\
\nm{\vec {\dot y}_2} & = & \nm{\vec{\dot y}(\vec y_k + \Delta t \ \vec {\dot y}_1, \vec v_k^{\mathcal X} + \Delta t \ \vec {\dot v}_1^{\mathcal X}, \mathcal{X}_k \oplus \Delta t \ \vec v_k^{\mathcal X},t_k + \Delta t)} \label{eq:algebra_integration_comp_v_heun_one} \\
\nm{\vec {\dot v}_1^{\mathcal X}} & = & \nm{\vec{\dot v}^{\mathcal X}(\vec y_k, \, \vec v_k^{\mathcal X}, \, \mathcal X_k, \, t_k)} \label{eq:algebra_integration_comp_y_heun_two} \\
\nm{\vec {\dot v}_2^{\mathcal X}} & = & \nm{\vec{\dot v}^{\mathcal X}(\vec y_k + \Delta t \ \vec {\dot y}_1, \vec v_k^{\mathcal X} + \Delta t \ \vec {\dot v}_1^{\mathcal X}, \mathcal{X}_k \oplus \Delta t \ \vec v_k^{\mathcal X},t_k + \Delta t)} \label{eq:algebra_integration_comp_v_heun_two} \\
\nm{\vec y_{k+1}} & \nm{\approx} & \nm{\vec y_k + \dfrac{\Delta t}{2} \ \lrsb{\vec {\dot y}_1 + \vec {\dot y}_2}} \label{eq:algebra_integration_comp_y_heun} \\
\nm{\vec v_{k+1}^{\mathcal X}} & \nm{\approx} & \nm{\vec v_k^{\mathcal X} + \dfrac{\Delta t}{2} \ \lrsb{\vec {\dot v}_1^{\mathcal X} + \vec {\dot v}_2^{\mathcal X}}} \label{eq:algebra_integration_comp_v_heun} \\
\nm{\mathcal X_{k+1}} & \nm{\approx} & \nm{\mathcal X_k \oplus \lrsb{\dfrac{\Delta t}{2} \ \lrsb{\vec v_k^{\mathcal X} + \lrp{\vec v_k^{\mathcal X} + \Delta t \ \vec {\dot v}_1^{\mathcal X}}}} = \mathcal X_k \oplus \lrsb{\Delta t \ \vec v_k^{\mathcal X} + \dfrac{\Delta t^2}{2} \ \vec {\dot v}_1^{\mathcal X}}} \nonumber \\
& = & \nm{\mathcal X_k \circ Exp\lrp{\Delta t \ \vec v_k^{\mathcal X} + \dfrac{\Delta t^2}{2} \ \vec {\dot v}_1^{\mathcal X}}} \label{eq:algebra_integration_comp_X_heun}
\end{eqnarray}
In case of the \nm{4^{th}} order Runge-Kutta integration scheme, it results in the following:
\begin{eqnarray}
\nm{\vec {\dot y}_1} & = & \nm{\vec{\dot y}\lrp{\vec y_k, \, \vec v_k^{\mathcal X}, \, \mathcal X_k, \, t_k}} \label{eq:algebra_integration_comp_y_rk4th_one} \\
\nm{\vec {\dot v}_1^{\mathcal X}} & = & \nm{\vec{\dot v}^{\mathcal X}\lrp{\vec y_k, \, \vec v_k^{\mathcal X}, \, \mathcal X_k, \, t_k}} \label{eq:algebra_integration_comp_vv_rk4th_one} \\
\nm{\vec {\dot y}_2} & = & \nm{\vec{\dot y}\lrp{\vec y_k + \dfrac{\Delta t}{2} \ \vec {\dot y}_1, \vec v_k^{\mathcal X} + \dfrac{\Delta t}{2} \ \vec {\dot v}_1^{\mathcal X}, \mathcal{X}_k \oplus \dfrac{\Delta t}{2} \ \vec v_k^{\mathcal X},t_k + \dfrac{\Delta t}{2}}} \label{eq:algebra_integration_comp_y_rk4th_two} \\
\nm{\vec {\dot v}_2^{\mathcal X}} & = & \nm{\vec{\dot v}^{\mathcal X}\lrp{\vec y_k + \frac{\Delta t}{2} \ \vec {\dot y}_1, \vec v_k^{\mathcal X} + \dfrac{\Delta t}{2} \ \vec {\dot v}_1^{\mathcal X}, \mathcal{X}_k \oplus \dfrac{\Delta t}{2} \ \vec v_k^{\mathcal X},t_k + \dfrac{\Delta t}{2}}} \label{eq:algebra_integration_comp_vv_rk4th_two} \\
\nm{\vec {\dot y}_3} & = & \nm{\vec{\dot y}\lrp{\vec y_k + \dfrac{\Delta t}{2} \ \vec {\dot y}_2, \vec v_k^{\mathcal X} + \dfrac{\Delta t}{2} \ \vec {\dot v}_2^{\mathcal X}, \mathcal{X}_k \oplus \dfrac{\Delta t}{2} \ \lrsb{\vec v_k^{\mathcal X} + \dfrac{\Delta t}{2} \ \vec {\dot v}_1^{\mathcal X}},t_k + \dfrac{\Delta t}{2}}} \label{eq:algebra_integration_comp_y_rk4th_three} \\
\nm{\vec {\dot v}_3^{\mathcal X}} & = & \nm{\vec{\dot v}^{\mathcal X}\lrp{\vec y_k + \frac{\Delta t}{2} \ \vec {\dot y}_2, \vec v_k^{\mathcal X} + \dfrac{\Delta t}{2} \ \vec {\dot v}_2^{\mathcal X}, \mathcal{X}_k \oplus \dfrac{\Delta t}{2} \ \lrsb{\vec v_k^{\mathcal X} + \dfrac{\Delta t}{2} \ \vec {\dot v}_1^{\mathcal X}},t_k + \dfrac{\Delta t}{2}}} \label{eq:algebra_integration_comp_vv_rk4th_three} \\
\nm{\vec {\dot y}_4} & = & \nm{\vec{\dot y}\lrp{\vec y_k + \Delta t \ \vec {\dot y}_3, \vec v_k^{\mathcal X} + \Delta t \ \vec {\dot v}_3^{\mathcal X}, \mathcal{X}_k \oplus \Delta t \ \lrsb{\vec v_k^{\mathcal X} + \dfrac{\Delta t}{2} \ \vec {\dot v}_2^{\mathcal X}}, t_k + \Delta t}} \label{eq:algebra_integration_comp_y_rk4th_four} \\
\nm{\vec {\dot v}_4^{\mathcal X}} & = & \nm{\vec{\dot v}^{\mathcal X}\lrp{\vec y_k + \Delta t \ \vec {\dot y}_3, \vec v_k^{\mathcal X} + \Delta t \ \vec {\dot v}_3^{\mathcal X}, \mathcal{X}_k \oplus \Delta t \ \lrsb{\vec v_k^{\mathcal X} + \dfrac{\Delta t}{2} \ \vec {\dot v}_2^{\mathcal X}}, t_k + \Delta t}} \label{eq:algebra_integration_comp_vv_rk4th_four} \\
\nm{\vec y_{k+1}} & \nm{\approx} & \nm{\vec y_k + \Delta t \ \lrsb{\dfrac{\vec {\dot y}_1}{6} + \dfrac{\vec {\dot y}_2}{3} + \dfrac{\vec {\dot y}_3}{3} + \dfrac{\vec {\dot y}_4}{6}}} \label{eq:algebra_integration_comp_y_rk4th} \\
\nm{\vec v_{k+1}^{\mathcal X}} & \nm{\approx} & \nm{\vec v_k^{\mathcal X} + \Delta t \ \lrsb{\dfrac{\vec {\dot v}_1^{\mathcal X}}{6} + \dfrac{\vec {\dot v}_2^{\mathcal X}}{3} + \dfrac{\vec {\dot v}_3^{\mathcal X}}{3} + \dfrac{\vec {\dot v}_4^{\mathcal X}}{6}}} \label{eq:algebra_integration_comp_vv_rk4th} \\
\nm{\mathcal X_{k+1}} & \nm{\approx} & \nm{\mathcal X_k \oplus \lrsb{\dfrac{\Delta t}{6} \ \vec v_k^{\mathcal X} + \dfrac{\Delta t}{3} \lrp{\vec v_k^{\mathcal X} + \dfrac{\Delta t}{2} \ \vec {\dot v}_1^{\mathcal X}} + \dfrac{\Delta t}{3} \lrp{\vec v_k^{\mathcal X} + \dfrac{\Delta t}{2} \ \vec {\dot v}_2^{\mathcal X}} + \dfrac{\Delta t}{6} \lrp{\vec v_k^{\mathcal X} + \Delta t \ \vec {\dot v}_3^{\mathcal X}}}} \nonumber \\
& = & \nm{\mathcal X_k \oplus \lrsb{\Delta t \ \vec v_k^{\mathcal X} + \dfrac{\Delta t^2}{6} \ \vec {\dot v}_1^{\mathcal X} + \dfrac{\Delta t^2}{6} \ \vec {\dot v}_2^{\mathcal X} + \dfrac{\Delta t^2}{6} \ \vec {\dot v}_3^{\mathcal X}}} \nonumber \\
& = & \nm{\mathcal X_k \circ Exp\lrp{\Delta t \ \vec v_k^{\mathcal X} + \dfrac{\Delta t^2}{6} \ \vec {\dot v}_1^{\mathcal X} + \dfrac{\Delta t^2}{6} \ \vec {\dot v}_2^{\mathcal X} + \dfrac{\Delta t^2}{6} \ \vec {\dot v}_3^{\mathcal X}}} \label{eq:algebra_integration_comp_X_rk4th}
\end{eqnarray}
Similar modifications to that of (\ref{eq:algebra_integration_comp_X_euler_left}) are required if the tangent space velocity \nm{\vec v^{\mathcal E}} is viewed in the global space.
\subsection{Gradient Descent Optimization in Lie Groups}\label{subsec:algebra_gradient_descent}
The Gauss-Newton implementation of the gradient descent optimization method is described in section \ref{subsec:euclidean_gradient_descent} for the case of Euclidean spaces. Let's now consider a Lie group element \nm{\mathcal X \in \mathcal G} and a nonlinear map based on its tangent space \nm{\lrb{\vec f: \mathbb{R}^m \rightarrow \mathbb{R}^n \ | \ \vec f\lrp{\vec \tau} \in \mathbb{R}^n, \vec \tau = Log\lrp{\mathcal X} \in \mathbb{R}^m, \vec \tau^\wedge = log\lrp{\mathcal X} \in \mathfrak{m}, \forall \, \mathcal X \in \mathcal G}}. As in the Euclidean case, it is possible to evaluate its jacobian \nm{\lrb{\vec J: \mathbb{R}^m \rightarrow \mathbb{R}^{nxm} \ | \ \vec J\lrp{\vec \tau} = \partial{\vec f\lrp{\vec \tau}} / \partial{\vec \tau} \in \mathbb{R}^{nxm}, \forall \ \vec \tau \in \mathbb{R}^m}}, and the error function is defined as \nm{\vec{\mathcal E}\lrp{\vec \tau} = \vec{\mathcal E}\big(Log(\mathcal X)) = \vec f\lrp{\vec \tau} - \vec f_T}.
Given an initial state \nm{\mathcal X_0 = Exp\lrp{\vec \tau_0}}, the objective is to determine a Lie group element \nm{\mathcal X = Exp\lrp{\vec \tau} = \Delta \vec \tau^{\mathcal E} \boxplus \mathcal X_0 = \Delta \vec \tau^{\mathcal E} \circ Exp\lrp{\vec \tau_0}} in the vicinity of \nm{\mathcal X_0} for which the cost function norm \nm{\| \vec{\mathcal E}\lrp{\vec \tau} \| \in \mathbb{R}} holds a local minimum. As the \nm{\vec \tau^{{\mathcal E}\wedge}} perturbation belongs to the spatial tangent space \nm{T_{\mathcal E}\mathcal G}, all gradient descent methods advance the solution by means of (\ref{eq:algebra_gradient_descent_iterative_lie_left}):
\neweq{\mathcal X_{k+1} \longleftarrow \Delta \vec \tau_k^{\mathcal E} \boxplus \mathcal X_k = \Delta \vec \tau_k^{\mathcal E} \circ Exp\lrp{\vec \tau_k}}{eq:algebra_gradient_descent_iterative_lie_left}
A robust formulation is necessary to ensure that the transform vector \nm{\tau}, which is treated as Euclidean by the functions \nm{\vec{\mathcal E}} and \nm{\vec f}, is advanced as a member of the tangent space that adheres to the Lie group constraints, as guaranteed by the use of the left jacobian inverse \nm{\vec J_L^{-1}\lrp{\vec \tau}} (table \ref{tab:algebra_lie_jacobians}). The first step of the Gauss-Newton method hence consists of two first order Taylor expansions, one for the logarithmic map and a second for the function \nm{\vec f}:
\begin{eqnarray}
\nm{\vec{\mathcal E}_{k+1}} & = & \nm{\vec f_{k+1} - \vec f_T = \vec f\Big(Log\lrp{\Delta \vec \tau_k^{\mathcal E} \circ Exp\lrp{\vec \tau_k}}\Big) - \vec f_T \approx \vec f\Big(\vec \tau_k + \vec J_L^{-1}\Bigr\rvert_{\ds{Exp(\vec \tau_k)}} \ \Delta \vec \tau_k^{\mathcal E}\Big) - \vec f_T} \nonumber \\
& \nm{\approx} & \nm{\vec f_k + \vec J_k \, \vec J_L^{-1}\Bigr\rvert_{\ds{Exp(\vec \tau_k)}} \ \Delta \vec \tau_k^{\mathcal E} - \vec f_T = \vec{\mathcal E}_k + \vec J_k \, \vec J_{Lk}^{-1} \ \Delta \vec \tau_k^{\mathcal E}} \label{eq:algebra_gradient_descent_taylor_lie_left}
\end{eqnarray}
The remaining Gauss-Newton steps are modified accordingly:
\begin{eqnarray}
\nm{\| \vec{\mathcal E}_{k+1} \|} & = & \nm{\vec{\mathcal E}_{k+1}^T \, \vec{\mathcal E}_{k+1} = \vec{\mathcal E}_k^T \, \vec{\mathcal E}_k + \Delta \vec \tau_k^{{\mathcal E}T} \, \lrsb{\vec J_k \, \vec J_{Lk}^{-1}}^T \, \lrsb{\vec J_k \, \vec J_{Lk}^{-1}} \, \Delta \vec \tau_k^{\mathcal E} + 2 \, \Delta \vec \tau_k^{{\mathcal E}T} \, \lrsb{\vec J_k \, \vec J_{Lk}^{-1}}^T \, \vec{\mathcal E}_k} \label{eq:algebra_gradient_descent_norm_lie_left} \\
\nm{\pderpar{\| \vec{\mathcal E}_{k+1} \|}{\Delta \vec \tau_k^{\mathcal E}}} & = & \nm{0 \ \longrightarrow \ \Delta \vec \tau_k^{\mathcal E} = - \lrsb{\lrsb{\vec J_k \, \vec J_{Lk}^{-1}}^T \, \lrsb{\vec J_k \, \vec J_{Lk}^{-1}}}^{-1} \, \lrsb{\vec J_k \, \vec J_{Lk}^{-1}}^T \, \vec{\mathcal E}_k \ \longrightarrow} \nonumber \\
\nm{\Delta \vec \tau_k^{\mathcal E}} & = & \nm{- \lrsb{\vec J_{Lk}^{-T} \, \vec J_k^T \, \vec J_k \, \vec J_{Lk}^{-1}}^{-1} \, \vec J_{Lk}^{-T} \, \vec J_k^T \, \vec{\mathcal E}_k} \label{eq:algebra_gradient_descent_solution_lie_left}
\end{eqnarray}
If the perturbation \nm{\vec \tau^{{\mathcal X}\wedge}} instead belongs to the local tangent space \nm{T_{\mathcal X}\mathcal G}, it is necessary to employ the right jacobian instead of the left one, resulting in:
\begin{eqnarray}
\nm{\mathcal X_{k+1}} & \nm{\longleftarrow} & \nm{\mathcal X_k \oplus \Delta \vec \tau_k^{\mathcal X} = Exp\lrp{\vec \tau_k} \circ \Delta \vec \tau_k^{\mathcal X}} \label{eq:algebra_gradient_descent_iterative_lie_right} \\
\nm{\Delta \vec \tau_k^{\mathcal X}} & = & \nm{- \lrsb{\vec J_{Rk}^{-T} \, \vec J_k^T \, \vec J_k \, \vec J_{Rk}^{-1}}^{-1} \, \vec J_{Rk}^{-T} \, \vec J_k^T \, \vec{\mathcal E}_k} \label{eq:algebra_gradient_descent_solution_lie_right}
\end{eqnarray}
\subsection{State Estimation in Lie Groups}\label{subsec:algebra_SS}
The state estimation \hypertt{EKF} discussion of section \ref{subsec:SS} states that given a continuous time nonlinear Euclidean state system (\ref{eq:SS_cont_time_system}) with process noise provided by (\ref{eq:SS_cont_time_system_noise1}) and (\ref{eq:SS_cont_time_system_noise2}), together with a series of discrete time nonlinear observations (\ref{eq:SS_measur_nonlinear}) with measurement noise given by (\ref{eq:SS_measur_nonlinear_noise1}) and (\ref{eq:SS_measur_nonlinear_noise2}), and considering no correlation between both noises (\ref{eq:SS_measur_nonlinear_noise3}), it is possible to compute estimations of the Euclidean state and its covariance at the same time points at which the observations are provided, in such a way that the estimation errors (difference with respect to the true state) are zero mean. The estimations are obtained by means of (\ref{eq:SS_EKF_x0_initial_state_FINAL}) through (\ref{eq:SS_EKF_P_plus_propagate_FINAL}).
Instead of the Euclidean space time varying state vector \nm{\vec x\lrp{t} \in \mathbb{R}^m} considered in section \ref{subsec:SS}, let's now consider that the continuous time nonlinear state system is composed by a vector \nm{\vec z\lrp{t} \in \mathbb{R}^n}, an element of a Lie group \nm{\mathcal X\lrp{t} \in \mathcal G}, and a vector \nm{\vec v^{\mathcal X}\lrp{t} \in \mathbb{R}^m} representing the velocity of \nm{\mathcal X} as it moves along its manifold, contained in the local or body tangent space \nm{T_{\mathcal X}\mathcal G}. As \nm{\mathcal X} is not Euclidean, the direct application of the \hypertt{EKF} state estimation scheme of section \ref{subsec:SS} would result in the need to continuously reproject the estimated Lie group elements \nm{\mathcal X_k} back to the manifold as otherwise the estimated states would not comply with the Lie group constraints. The repeated deviations and reprojections from and to the manifold may result in a significant degradation in the estimation accuracy.
The most rigorous and precise way to adapt the \hypertt{EKF} scheme is to exclude the Lie group element \nm{\mathcal X \in \mathcal G} from the state system, replacing it by a local tangent space perturbation \nm{\Delta \vec \tau^{\mathcal X} \in T_{\mathcal X}\mathcal G}. Each filter step now consists on estimating the Lie group element \nm{\hat{\mathcal X}_k^+ = \hat{\mathcal X}^+\lrp{t_k} \in \mathcal G}, the state vector \nm{\xvecest_k^+ = \xvecest^+\lrp{t_k} = \lrsb{\Delta \hat{\vec \tau}_k^{{\mathcal X}+}, \, \hat{\vec v}_k^{{\mathcal X}+}, \hat{\vec z}_k^+}^T \in \mathbb{R}^{2 \, m + n}}, and its covariance \nm{\Pvec_k^+ = \Pvec^+\lrp{t_k} \in \mathbb{R}^{(2 \, m + n)x(2 \, m + n)}}, based on their values at \nm{t_{k-1} = \lrp{k-1} \, \Deltat}. For clarity purposes the Lie velocity \nm{\vec v^{\mathcal X}} and the state vector \nm{\vec z} can be grouped into a bigger Euclidean state vector \nm{\vec p = \lrsb{\vec v^{\mathcal X}, \, \vec z}^T \in \mathbb{R}^{m+n}}, so \nm{\xvecest_k^+ = \lrsb{\Delta \hat{\vec \tau}_k^{\mathcal X}, \, \hat{\vec p}_k}^T}.
Considering with no loss of generality that the local tangent state perturbation \nm{\Delta \vec \tau^{\mathcal X}} is located on the first positions of the combined state vector \nm{\xvec}, the definition of the covariance matrix is a combination of that of its Euclidean components (\ref{eq:SS_discr_time_system_state_cov}) and its local Lie counterparts (\ref{eq:algebra_lie_covariance_right_def}), with additional combined members:
\begin{eqnarray}
\nm{\Pvec_k} & = & \nm{\begin{bmatrix} \nm{\vec C_{{\mathcal {XX}},k}^{\mathcal X}} & \nm{\vec C_{{\mathcal X}p,k}^{\mathcal X}} \\ \nm{\vec C_{p{\mathcal X},k}^{\mathcal X}} & \nm{\vec C_{pp,k}} \end{bmatrix} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \in \mathbb{R}^{(2 \, m + n)x(2 \, m + n)}} \label{eq:algebra_SS_EKF_P_generic} \\
\nm{\vec C_{{\mathcal {XX}},k}^{\mathcal X}} & = & \nm{E\lrsb{\Delta \vec \tau_k^{\mathcal X} \, \Delta \vec \tau_k^{{\mathcal X,T}}} = E\lrsb{\lrp{\mathcal X_k \ominus \vec \mu_{{\mathcal X},k}} \, \lrp{\mathcal X_k \ominus \vec \mu_{{\mathcal X},k}}^T} \ \ \ \ \ \ \ \ \ \ \ \ \in \mathbb{R}^{mxm}} \label{eq:algebra_SS_EKF_P_lie} \\
\nm{\vec C_{{\mathcal X}p,k}^{\mathcal X}} & = & \nm{E\lrsb{\Delta \vec \tau_k^{\mathcal X} \, \lrp{\vec p_k - \vec \mu_{p,k}}^T} = E\lrsb{\lrp{\mathcal X_k \ominus \vec \mu_{{\mathcal X},k}} \, \lrp{\vec p_k - \vec \mu_{p,k}}^T} \ \ \ \ \in \mathbb{R}^{mx(m+n)}} \label{eq:algebra_SS_EKF_P_lieeuc} \\
\nm{\vec C_{p{\mathcal X},k}^{\mathcal X}} & = & \nm{E\lrsb{\lrp{\vec p_k - \vec \mu_{p,k}} \, \Delta \vec \tau_k^{{\mathcal X},T}} = E\lrsb{\lrp{\vec p_k - \vec \mu_{p,k}} \, \lrp{\mathcal X_k \ominus \vec \mu_{{\mathcal X},k}}^T} \ \ \ \in \mathbb{R}^{(m+n)xm}} \label{eq:algebra_SS_EKF_P_euclie} \\
\nm{\vec C_{pp,k}} & = & \nm{E\lrsb{\lrp{\vec p_k - \vec \mu_{p,k}} \, \lrp{\vec p_k - \vec \mu_{p,k}}^T} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \in \mathbb{R}^{(m+n)x(m+n)}} \label{eq:algebra_SS_EKF_P_euc}
\end{eqnarray}
The following paragraphs do not describe the full state estimation process, but only the changes with respect to the Euclidean case described in section \ref{subsec:SS}:
\begin{itemize}
\item \textbf{Initialization}. The Lie group element \nm{\hat{\mathcal X}_0^+}, Lie velocity \nm{\hat{\vec v}_0^{{\mathcal X}+}}, and Euclidean state vector \nm{\hat{\vec y}_0^+} are initialized with their expected values, while the local tangent space perturbation \nm{\Delta \hat{\vec \tau}_0^{{\mathcal X}+}} is initialized to zero. The covariance of the initial estimation error \nm{\Pvec_0^+} represents the uncertainty in the initial estimation \nm{\xvecest_0^+}.
\begin{eqnarray}
\nm{\hat{\mathcal X}_0^+} & = & \nm{\vec \mu_{\mathcal X,0} = E\lrsb{\mathcal X_0}} \label{eq:algebra_SS_EKF_man0_initial_manifold} \\
\nm{\xvecest_0^+} & = & \nm{\vec \mu_{x,0} = E\lrsb{\xvec_0} = E\Big[\lrsb{\vec{0}_m, \vec v_0^{\mathcal X}, \vec z_0}^T\Big]} \label{eq:algebra_SS_EKF_x0_initial_state} \\
\nm{\Pvec_0^+} & = & \nm{E\lrsb{\lrp{\xvec_0 - \vec \mu_{x,0}} \, \lrp{\xvec_0 - \vec \mu_{x,0}}^T}} \label{eq:algebra_SS_EKF_P0_initial_covariance} \\
\end{eqnarray}
\item \textbf{Time Update Equations}. The first \hypertt{EKF} step propagates the state estimation without the use of any observations, and is similar to that described in section \ref{subsec:SS}. The (\ref{eq:SS_cont_time_system}) continuous time non linear state system is however replaced by (\ref{eq:algebra_SS_cont_time_system}):
\begin{eqnarray}
\nm{\xvecdot\lrp{t}} & = & \nm{\lrsb{\Delta \vec{\dot \tau}^{\mathcal X}, \, \vec{\dot v}^{\mathcal X}, \, \vec {\dot z}}^T = f\big(\mathcal X\lrp{t} \oplus \Delta \vec \tau^{\mathcal X}\lrp{t}, \, \vec v^{\mathcal X}\lrp{t}, \, \vec z\lrp{t}, \, \uvec\lrp{t}, \ \wvec\lrp{t}, \, t\big)} \label{eq:algebra_SS_cont_time_system} \\
\nm{\Delta \vec{\dot \tau}^{\mathcal X}} & = & \nm{\vec v^{\mathcal X}} \label{eq:algebra_SS_cont_time_system_tau} \\
\nm{\vec{\dot p}} & = & \nm{f_p\big(\mathcal X\lrp{t} \oplus \Delta \vec \tau^{\mathcal X}\lrp{t}, \, \vec v^{\mathcal X}\lrp{t}, \, \vec z\lrp{t}, \, \uvec\lrp{t}, \ \wvec\lrp{t}, \, t\big)} \label{eq:algebra_SS_cont_time_system_y}
\end{eqnarray}
Linearization of the continuous time system results in the (\ref{eq:SS_cont_time_system_linear_system_matrix}) system matrix \nm{\Avec\lrp{t} \in \mathbb{R}^{(2\,m+n)x(2\,m+n)}}, where various blocks are likely to be based on the \nm{\oplus} and \nm{+} jacobians of table \ref{tab:algebra_lie_jacobians}. Its discretization by means of (\ref{eq:SS_discr_time_system_linear_system_matrix}) leads to the system state transition matrix \nm{\Fvec_k \in \mathbb{R}^{(2\,m+n)x(2\,m+n)}}. With no other differences with respect to the section \ref{subsec:SS} Euclidean process, the manifold element is left unchanged, while the a priori state vector and error covariance are obtained by means of (\ref{eq:algebra_SS_xest_minus_propagate}) and (\ref{eq:algebra_SS_P_minus_propagate}):
\begin{eqnarray}
\nm{\hat{\mathcal X}_k^-} & = & \nm{\hat{\mathcal X}_{k-1}^+} \label{eq:algebra_SS_man_minus_propagate} \\
\nm{\xvecest_k^-} & = & \nm{\xvecest_{k-1}^+ + \Deltat \cdot f \, \big(\hat{\mathcal X}_{k-1}^+ \oplus \Delta \hat{\vec \tau}_{k-1}^{{\mathcal X}+}, \, \hat{\vec v}_{k-1}^{{\mathcal X}+}, \, \hat{\vec z}_{k-1}^+, \, \uvec_{k-1}, \ \vec 0, \, t_{k-1}\big)}\label{eq:algebra_SS_xest_minus_propagate} \\
\nm{\Pvec_k^-} & = & \nm{\Fvec_{k-1} \, \Pvec_{k-1}^+ \, \Fvec_{k-1}^T + \Qtilde_{d,k-1}}\label{eq:algebra_SS_P_minus_propagate}
\end{eqnarray}
\item \textbf{Measurement Update Equations}. The second \hypertt{EKF} step updates the estimations by means of the observations, and is very similar to that described in section \ref{subsec:SS}. The (\ref{eq:SS_measur_nonlinear}) discrete time non linear observation system is however replaced by (\ref{eq:algebra_SS_measur_nonlinear}):
\neweq{\yvec_k = h\lrp{\mathcal X_k \oplus \Delta \vec \tau_k^{\mathcal X}, \, \vec v_k^{\mathcal X}, \, \vec z_k, \, \, \vvec_k, \, t_k}}{eq:algebra_SS_measur_nonlinear}
Its linearization results in the (\ref{eq:SS_measur_linear_output_matrix}) output matrix \nm{\Hvec_k \in \mathbb{R}^{qx(2\,m+n)}}, where it is also likely for various blocks to be based on the \nm{\oplus} and \nm{+} jacobians of table \ref{tab:algebra_lie_jacobians}. With no other differences with respect to the section \ref{subsec:SS} Euclidean process, the manifold element is again left unchanged, while the a posteriori state vector and error covariance are obtained by means of (\ref{eq:algebra_SS_EKF_xest_plus_propagate}) and (\ref{eq:algebra_SS_EKF_P_plus_propagate}):
\begin{eqnarray}
\nm{\Kvec_k} & = & \nm{\Pvec_k^- \, \Hvec_k^T \lrp{\Hvec_k \, \Pvec_k^- \, \Hvec_k^T + \Rtilde_k}^{-1}}\label{eq:algebra_SS_EKF_kalman_gain} \\
\nm{\hat{\mathcal X}_k^+} & = & \nm{\hat{\mathcal X}_k^-} \label{eq:algebra_SS_man_plus_propagate} \\
\nm{\xvecest_k^+} & = & \nm{\xvecest_k^- + \Kvec_k \, \lrsb{\yvec_k - h\lrp{\hat{\mathcal X}_k^- \oplus \Delta \hat{\vec \tau}_k^{{\mathcal X}-}, \, \hat{\vec v}_k^{{\mathcal X}-}, \, \hat{\vec z}_k^-, \, \vec 0, \, t_k}}} \label{eq:algebra_SS_EKF_xest_plus_propagate} \\
\nm{\Pvec_k^+} & = & \nm{\lrp{\Ivec - \Kvec_k \, \Hvec_k}\, \Pvec_k^- \, \lrp{\Ivec - \Kvec_k \, \Hvec_k}^T + \Kvec_k \, \Rtilde_k \, \Kvec_k^T}\label{eq:algebra_SS_EKF_P_plus_propagate}
\end{eqnarray}
\item \textbf{Reset Equations}. The third \hypertt{EKF} step, which is not necessary for purely Euclidean systems, resets the a posteriori estimation of the tangent space perturbation \nm{\Delta \hat{\vec \tau}_k^{{\mathcal X}+}} to zero while modifying the a posteriori estimations for the Lie group element \nm{\hat{\mathcal X}_k^+} and the error covariance \nm{\Pvec_k^+} accordingly\footnote{The a posteriori estimations for the Lie velocity \nm{\hat{\vec v}_k^{{\mathcal X}+}} and the Euclidean components \nm{\hat{\vec z}_k^+} are not modified in this step.}. Note that the accuracy of the linearizations present in the other two steps, which result in the \nm{\vec A\lrp{t}} and \nm{\vec H_k} system and output matrices, is based on the first order Taylor expansions present in table \ref{tab:algebra_lie_jacobians}, which are directly related to the size of the tangent space perturbations. Although it is not strictly necessary to execute this step in every \hypertt{EKF} cycle, the accuracy of the whole state estimation process is improved by maintaining the perturbations as small as possible, so it is recommended to never bypass the reset step.
Taking into account that the Lie group element is going to be updated per (\ref{eq:algebra_SS_man_reset}), the error covariance is propagated to the new Lie group element per (\ref{eq:algebra_lie_covariance_right_propagation}) as follows:
\begin{eqnarray}
\nm{\Pvec_k^+} & \nm{\longleftarrow} & \nm{\vec D \, \Pvec_k^+ \, \vec D^T} \label{eq:algebra_SS_P_reset} \\
\nm{\vec D} & = & \nm{\begin{bmatrix} \nm{\vec J_{\ds{\oplus \; \mathcal X}}^{\ds{\oplus \; \hat{\mathcal X}_k^+ \oplus \Delta \hat{\vec \tau}_k^{{\mathcal X}+}}}} & \nm{\vec{0}_{mx(m+n)}} \\ \nm{\vec{0}_{(m+n)xm}} & \nm{\vec {I}_{(m+n)x(m+n)}} \end{bmatrix} \ \ \ \ \in \mathbb{R}^{(2\,m+n)x(2\,m+n)}} \label{eq:algebra_SS_D_reset}
\end{eqnarray}
Once propagated, the Lie group element is updated with the local tangent space perturbation, which is itself reset to zero:
\begin{eqnarray}
\nm{\hat{\mathcal X}_k^+} & \nm{\longleftarrow} & \nm{\hat{\mathcal X}_k^+ \oplus \Delta \hat{\vec \tau}_k^{{\mathcal X}+}} \label{eq:algebra_SS_man_reset} \\
\nm{\Delta \hat{\vec \tau}_k^{{\mathcal X}+}} & \nm{\longleftarrow} & \nm{\vec{0}_m} \label{eq:algebra_SS_pertur_reset}
\end{eqnarray}
\end{itemize}
Although the above process has been described making use of a local tangent space perturbation \nm{\Delta \vec \tau^{\mathcal X} \in T_{\mathcal X}\mathcal G} and a vector \nm{\vec v^{\mathcal X} \in \mathbb{R}^m} also viewed in the local tangent space that represents the velocity of \nm{\mathcal X \in \mathcal G} as it moves along its manifold, there exists an equivalent formulation that employs perturbations and velocities viewed in the manifold global tangent space, this is, \nm{\Delta \vec \tau^{\mathcal E} \in T_{\mathcal E}\mathcal G} and \nm{\vec v^{\mathcal E} \in \mathbb{R}^m}. In this case, the filter estimates the Lie group element \nm{\hat{\mathcal X}_k^+ \in \mathcal G}, the state vector \nm{\xvecest_k^+ = \lrsb{\Delta \hat{\vec \tau}_k^{{\mathcal E}+}, \, \hat{\vec v}_k^{{\mathcal E}+}, \hat{\vec z}_k^+}^T \in \mathbb{R}^{2 \, m + n}}, and the error covariance \nm{\Pvec_k^+}:
\begin{eqnarray}
\nm{\Pvec_k} & = & \nm{\begin{bmatrix} \nm{\vec C_{{\mathcal {XX}},k}^{\mathcal E}} & \nm{\vec C_{{\mathcal X}p,k}^{\mathcal E}} \\ \nm{\vec C_{p{\mathcal X},k}^{\mathcal E}} & \nm{\vec C_{pp,k}} \end{bmatrix} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \in \mathbb{R}^{(2 \, m + n)x(2 \, m + n)}} \label{eq:algebra_SS_EKF_P_generic_global} \\
\nm{\vec C_{{\mathcal {XX}},k}^{\mathcal E}} & = & \nm{E\lrsb{\Delta \vec \tau_k^{\mathcal E} \, \Delta \vec \tau_k^{{\mathcal E,T}}} = E\lrsb{\lrp{\mathcal X_k \boxminus \vec \mu_{{\mathcal X},k}} \, \lrp{\mathcal X_k \boxminus \vec \mu_{{\mathcal X},k}}^T} \ \ \ \ \ \ \ \ \ \ \ \ \in \mathbb{R}^{mxm}} \label{eq:algebra_SS_EKF_P_lie_global} \\
\nm{\vec C_{{\mathcal X}p,k}^{\mathcal E}} & = & \nm{E\lrsb{\Delta \vec \tau_k^{\mathcal E} \, \lrp{\vec p_k - \vec \mu_{p,k}}^T} = E\lrsb{\lrp{\mathcal X_k \boxminus \vec \mu_{{\mathcal X},k}} \, \lrp{\vec p_k - \vec \mu_{p,k}}^T} \ \ \ \ \in \mathbb{R}^{mx(m+n)}} \label{eq:algebra_SS_EKF_P_lieeuc_global} \\
\nm{\vec C_{p{\mathcal X},k}^{\mathcal E}} & = & \nm{E\lrsb{\lrp{\vec p_k - \vec \mu_{p,k}} \, \Delta \vec \tau_k^{{\mathcal E},T}} = E\lrsb{\lrp{\vec p_k - \vec \mu_{p,k}} \, \lrp{\mathcal X_k \boxminus \vec \mu_{{\mathcal X},k}}^T} \ \ \ \in \mathbb{R}^{(m+n)xm}} \label{eq:algebra_SS_EKF_P_euclie_global}
\end{eqnarray}
The only additional changes are the use of (\nm{\Delta \vec \tau^{\mathcal E} \boxplus \mathcal X}) instead of (\nm{\mathcal X \oplus \Delta \vec \tau^{\mathcal X}}), the fact that the blocks of the \nm{\vec A\lrp{t}} and \nm{\vec H_k} matrices are now based on the \nm{\boxplus} and \nm{+} jacobians of table \ref{tab:algebra_lie_jacobians}, and the use of \nm{\vec J_{\ds{\boxplus \; \mathcal X}}^{\ds{\boxplus \; \vec \tau \boxplus \mathcal X}} \Bigr\rvert_{\ds{\Delta \hat{\vec \tau}_k^{{\mathcal E}+} \boxplus \hat{\mathcal X}_k^+}}} to propagate the covariance.
\section{Rotation of Rigid Bodies} \label{sec:Rotate}
A \emph{rigid body} is an object in which the distance between any two of its points is constant, as is the orientation between any two of its vectors (refer to section \ref{subsec:algebra_points_and_vectors} for the definition of points and vectors in \nm{\mathbb{R}^3}). Rotational rigid body motion is that in which one of its points, named the \emph{center of rotation} \nm{\vec O_{\sss {CR}}}, does not move. Rotations of rigid bodies do not comply with the axioms of an Euclidean space (section \ref{subsec:algebra_structures}) but with those of Lie groups (section \ref{subsec:algebra_lie}), and hence this section heavily relies on the concepts of Lie theory discussed in sections \ref{subsec:algebra_lie} and \ref{subsec:algebra_lie_jacobians}. Table \ref{tab:Rotate_lie_comparison} provides a comparison between the generic nomenclature employed in section \ref{sec:Algebra} and their rotation equivalents. The different representations discussed in this section are summarized in Table \ref{tab:Rotate_summary}.
\begin{center}
\begin{tabular}{lcclcc}
\hline
\textbf{Concept} & \textbf{Lie Theory} & \textbf{Rotation} & \textbf{Concept} & \textbf{Lie Theory} & \textbf{Rotation} \\
\hline
Lie group & \nm{\mathcal G} & \nm{\mathbb{SO}(3)} & Lie group elements & \nm{\mathcal X, \, \mathcal Y} & \nm{\mathcal R, \, \mathcal S} \\
Concatenation & \nm{\circ} & \nm{\circ} & Lie algebra & \nm{\mathfrak{m}} & \nm{\mathfrak{so}(3)} \\
Identity & \nm{\mathcal E} & \nm{\mathcal {I_R}} & Inverse & \nm{\mathcal X^{-1}} & \nm{\mathcal R^{-1}} \\
Velocity & \nm{\vec v} & \nm{\vec \omega} & Tangent element & \nm{\vec \tau} & \nm{\vec r} \\
Local frame & \nm{\mathcal X} & B & Global frame & \nm{\mathcal E} & N \\
Point action & \nm{g_{\mathcal X}()} & \nm{\vec g_{\mathcal R}(\vec p)} & Vector action & \nm{g_{\mathcal X}()} & \nm{\vec g_{\mathcal R*}(\vec v)} \\
Adjoint & \nm{\vec{Ad}_{\mathcal X}\lrp{\vec \tau^{\wedge}}} & \nm{\vec{Ad}_{\mathcal R}\lrp{\vec r^{\wedge}}} & Adjoint matrix & \nm{\vec{Ad}_{\mathcal X}\, \vec \tau} & \nm{\vec{Ad}_{\mathcal R} \, \vec r} \\
\hline
\end{tabular}
\end{center}
\captionof{table}{Comparison between generic Lie elements and those of rigid body rotations} \label{tab:Rotate_lie_comparison}
This section begins with an introduction to rotational motion in section \ref{subsec:RigidBody_bases}, followed by a description of the different rotation Lie group representations: the rotation matrix (section \ref{subsec:RigidBody_rotation_dcm}), the rotation vector (section \ref{subsec:RigidBody_rotation_rotv}), the unit quaternion (section \ref{subsec:RigidBody_rotation_rodrigues}), the half rotation vector (section \ref{subsec:RigidBody_rotation_halfrotv}), and the Euler angles (section \ref{subsec:RigidBody_rotation_euler}). Algebraic operations on rigid body rotations, such as powers, linear interpolation, and the plus and minus operators, are introduced in section \ref{subsec:RigidBody_rotation_algebra}. Section \ref{subsec:RigidBody_rotation_calculus_derivatives} presents the rotation time derivative that leads to the definition of the angular velocity in the tangent space. The velocity of the rigid body points is discussed in section \ref{subsec:RigidBody_rotation_velocity}, followed by the adjoint map in section \ref{subsec:RigidBody_rotation_adjoint}, which transforms elements of the tangent space while the rotation advances on its manifold, and by an analysis of uncertainty and covariances applied to rotation motion (section \ref{subsec:RigidBody_rotation_covariance}). An extensive analysis of the rotation jacobians is presented in section \ref{subsec:RigidBody_rotation_calculus_jacobians}. Sections \ref{subsec:RigidBody_rotation_integration}, \ref{subsec:RigidBody_rotation_gauss_newton}, and \ref{subsec:RigidBody_rotation_SS} apply the discrete integration of Lie groups, the Gauss-Newton optimization of Lie group functions, and the state estimation of Lie groups contained in sections \ref{subsec:algebra_integration}, \ref{subsec:algebra_gradient_descent}, and \ref{subsec:algebra_SS} to the case of rotations. Finally, the advantages and disadvantages of each rotation representation are discussed in section \ref{subsec:RigidBody_rotation_applications}.
\subsection{Special Orthogonal (Lie) Group}\label{subsec:RigidBody_bases}
A rigid body can be represented with a cartesian frame attached to any of its points (the origin), with the basis vectors \nm{\vec e_1}, \nm{\vec e_2}, and \nm{\vec e_3} being simply unit vectors along the main axes. It can be assumed with no loss of generality that the frame origin coincides with the center of rotation. Rigid body rotations can be combined and reversed, complying with the algebraic concept of group, but are not endowed with a metric, so they are not part of a metric or Euclidean space (section \ref{subsec:algebra_structures}). They do however comply with the axioms of a Lie group (section \ref{subsec:algebra_lie}), and hence the set of rigid body rotations together with the operation of rotation concatenation comprises \nm{\langle \mathbb{SO}(3), \circ \rangle}, known as the \emph{rotation group} or \emph{special orthogonal group} of \nm{\mathbb{R}^3} \cite{Sola2017}, where its elements are denoted by \nm{\mathcal R}, the identify rotation by \nm{\mathcal {I_R}}, and the inverse by \nm{\mathcal R^{-1}}. The rotation group has two main actions, which are the rotation of points \nm{\lrb{\vec g() : \mathbb{SO}(3) \times \mathbb{R}^3 \rightarrow \mathbb{R}^3 \ | \ \vec p \rightarrow \vec g_{\mathcal R}\lrp{\vec p}}} and that of vectors \nm{\lrb{\vec g_*() : \mathbb{SO}(3) \times \mathbb{R}^3 \rightarrow \mathbb{R}^3 \ | \ \vec v \rightarrow \vec g_{\mathcal R*}\lrp{\vec v}}}.
Based on the rigid body definition above, its motion corresponds to an \emph{orthogonal transformation}, this is, one that preserves the norm (maintaining distances between points as well as angles between vectors) and the cross product (maintaining orientation)\footnote{An orthogonal transformation can also be defined as one that preserves both the inner and cross products.} \cite{Soatto2001}. These are called \emph{orthogonality} and \emph{handedness} \cite{Sola2017}:
\begin{itemize}
\item Norm: \nm{\|\vec g_{\mathcal R*}\lrp{\vec v}\| = \|\vec v\|, \forall \, \vec v \in \mathbb{R}^3}
\item Cross product: \nm{\vec g_{\mathcal R*}\lrp{\vec u} \times \vec g_{\mathcal R*}\lrp{\vec v} = \vec g_{\mathcal R*}\lrp{\vec u \times \vec v}, \forall \, \vec u, \vec v \in \mathbb{R}^3}
\end{itemize}
Noting that a vector represents the difference between two points, \nm{\vec g_{\mathcal R*}\lrp{\vec v} = \vec g_{\mathcal R*}\lrp{\vec q - \vec p} = \vec g_{\mathcal R}\lrp{\vec q} - \vec g_{\mathcal R}\lrp{\vec p}}, and considering the possibility that one of the points may be the origin, \nm{\vec g_{\mathcal R}\lrp{\vec p} = \vec g_{\mathcal R}\lrp{\vec 0} = \vec 0}, results in the equivalence between the point and vector rotation maps:
\neweq{\vec g_{\mathcal R*}\lrp{\vec q - \vec 0} = \vec g_{\mathcal R*}\lrp{\vec q} = \vec g_{\mathcal R}\lrp{\vec q} - \vec g_{\mathcal R}\lrp{\vec 0} = \vec g_{\mathcal R}\lrp{\vec q} \rightarrow \vec g_{\mathcal R}() = \vec g_{\mathcal R*}()} {eq:SO3_equivalence}
\begin{center}
\begin{tabular}{lccc}
\hline
\textbf{Representation} & \textbf{Symbol} & \textbf{Structure} & \textbf{Space} \\
\hline
Rotation matrix & \nm{\vec R} & orthogonal 3x3 matrix & \nm{\mathbb{SO}(3)} \\
Angular velocity & \nm{\vec \omega^\wedge = \omegaskew} & skew-symmetric matrix & \nm{\mathfrak{so}(3)} \\
& \nm{\vec \omega} & free 3-vector & \\
Rotation vector & \nm{\vec r^\wedge = \rskew} & skew-symmetric matrix & \nm{\mathbb{SO}(3)} \& \nm{\mathfrak{so}(3)} \\
& \nm{\vec r = \vec \omega \, t = \vec n \, \phi} & free 3-vector & \\
Unit quaternion & \nm{\vec q} & unit quaternion & \nm{\mathbb{SO}(3)} \\
Half angular velocity & \nm{\vec \Omega^\wedge} & pure quaternion & \nm{\mathfrak{so}(3)} \\
& \nm{\vec \Omega = \vec \omega / 2} & free 3-vector & \\
Half rotation vector & \nm{\vec h^\wedge} & pure quaternion & \nm{\mathbb{SO}(3)} \& \nm{\mathfrak{so}(3)} \\
& \nm{\vec h = \vec \Omega \, t = \vec n \, \theta = \vec r / 2} & free 3-vector & \\
Euler angles & \nm{\vec \phi} & 3 angles & \nm{\mathbb{SO}(3)} \\
\hline
\end{tabular}
\end{center}
\captionof{table}{Summary of rotational motion representations} \label{tab:Rotate_summary}
The \nm{\mathbb{SO}(3)} analysis below adopts the convention introduced in section \ref{subsec:algebra_lie}, in which all actions, including concatenation \nm{\lrb{\circ : \mathbb{SO}\lrp{3} \times \mathbb{SO}\lrp{3} \rightarrow \mathbb{SO}\lrp{3}}}, transform elements viewed in the local or body frame \nm{F_{\sss B} = \{\OB, \vec b_1, \vec b_2, \vec b_3\}} into elements viewed in the global or spatial frame \nm{F_{\sss N} = \{\ON, \vec n_1, \vec n_2, \vec n_3\} = \{\vec g_{\mathcal R}\lrp{\OB}, \vec g_{\mathcal R*}\lrp{\vec b_1}, \vec g_{\mathcal R*}\lrp{\vec b_2}, \vec g_{\mathcal R*}\lrp{\vec b_3}\}}\footnote{The local and spatial bases are denoted B and N as they usually correspond to the body and \hypertt{NED} frames respectively.}:
\begin{eqnarray}
\nm{\pN} & = & \nm{\vec g_{\mathcal R_{NB}}\lrp{\pB}} \label{eq:Rotate_point_action} \\
\nm{\vN} & = & \nm{\vec g_{\mathcal R*_{NB}}\lrp{\vB}} \label{eq:Rotate_vector_action}
\end{eqnarray}
\subsection{Rotation Matrix}\label{subsec:RigidBody_rotation_dcm}
The three basis vectors of the output frame can be stacked side by side into a matrix \nm{\vec R = \RNB = \lrsb{\vec n_1 \ \vec n_2 \ \vec n_3} \in \mathbb{R}^{3x3}}, called the \emph{rotation matrix}. Since its columns form a right handed orthonormal basis, it complies with the orthogonality and handedness conditions, and it can be proven that the rotation matrix \nm{\vec R} is an special orthogonal matrix\footnote{Orthogonal means that the transpose equals the inverse, while special or proper means that the determinant is positive one.}. Rotation matrices hence represent rigid body rotations, and their space \nm{\mathbb{SO}\lrp{3} = \{\vec R \in \mathbb{R}^{3x3} \ | \ {\vec R}^T \vec R = \vec I_3, \ det\lrp{\vec R} = +1\}} has group structure under matrix multiplication \nm{\{\mathbb{R}^{3x3} \times \mathbb{R}^{3x3} \rightarrow \mathbb{R}^{3x3} \ | \ \vec R_a \, \vec R_b \in \mathbb{R}^{3x3}, \forall \ \vec R_a, \ \vec R_b \in \mathbb{R}^{3x3}\}} \cite{Pinter1990}. While having dimension nine, the special orthogonal group \nm{\mathbb{SO}\lrp{3}} defined by means of rotation matrices constitutes a three dimensional manifold to euclidean space \nm{\mathbb{E}^3}. Note that in this group the identity element is given by the identity matrix \nm{\lrp{\vec I = \vec I_3}}, and the inverse coincides with the transpose \nm{\lrp{\vec R^{-1} = \vec R^T}}.
\begin{center}
\begin{tabular}{lcclcc}
\hline
\textbf{Concept} & \nm{\mathbb{SO}^3} & \textbf{Rotation Matrix} & \textbf{Concept} & \nm{\mathbb{SO}^3} & \textbf{Rotation Matrix} \\
\hline
Lie group element & \nm{\mathcal R} & \nm{\vec R} & Concatenation & \nm{\circ} & Matrix product \\
Identity & \nm{\mathcal {I_R}} & \nm{\vec I_3} & Inverse & \nm{\mathcal R^{-1}} & \nm{\vec R^T} \\
Point rotation & \nm{\vec g_{\mathcal R}(\vec p)} & \nm{\vec R \, \vec p} & Vector rotation & \nm{\vec g_{\mathcal R*}(\vec v)} & \nm{\vec R \, \vec v} \\
\hline
\end{tabular}
\end{center}
\captionof{table}{Comparison between generic \nm{\mathbb{SO}(3)} and rotation matrix} \label{tab:Rotate_lie_dcm}
The rotation matrix \nm{\vec R} represents the actual coordinate transformation from the local to the global frame:
\neweq{\vec g_{\mathcal R*}\lrp{\vec v} = \vec R \; \vec v} {eq:SO3_dcm_transform}
The inverse rotation (from the global to the local frame), is simply the transpose:
\neweq{\vec R^{-1} = \vec R^T} {eq:SO3_dcm_inverse}
The concatenation of rotations is also straight forward as it coincides with matrix multiplication. Note that \nm{\mathbb{SO}\lrp{3}} as defined above is not an abelian group, so the order of the factors is important.
\neweq{\vec R_{\sss EB} = \vec R_{\sss EN} \, \vec R_{\sss NB}} {eq:SO3_dcm_concatenation}
\subsection{Rotation Vector as Tangent Space}\label{subsec:RigidBody_rotation_rotv}
As discussed in section \ref{subsubsec:algebra_lie_velocities}, the structure of the Lie algebra associated to \nm{\mathbb{SO}(3)} can be obtained by time derivating the Lie group inverse constraint, \nm{\vec R^T\lrp{t} \, \vec R\lrp{t} = \vec R\lrp{t} \, \vec R^T\lrp{t} = \vec I_3}, resulting in the following particularizations of (\ref{eq:algebra_vE}) and (\ref{eq:algebra_vX}):
\begin{eqnarray}
\nm{\vec \omega_{\sss {NB}}^{\sss N\wedge} = \wNBNskew} & = & \nm{\RNBdot \; \RNBtrans = - \RNB \; \RNBdottrans} \label{eq:SO3_dcm_omega_space} \\
\nm{\vec \omega_{\sss {NB}}^{\sss B\wedge} = \wNBBskew} & = & \nm{\RNBtrans \; \RNBdot = - \RNBdottrans \; \RNB} \label{eq:SO3_dcm_omega_body}
\end{eqnarray}
The Lie algebra velocity \nm{\vec v^\wedge} of \nm{\mathbb{SO}(3)} is known as the \emph{angular velocity} \nm{\vec \omega^\wedge}, and as shown in (\ref{eq:SO3_dcm_omega_space}) and (\ref{eq:SO3_dcm_omega_body}), has the structure of a skew-symmetric matrix because its negative coincides with its transpose, so it is generally denoted as \nm{\widehat{\vec \omega}}. An alternative definition of the angular velocity is presented in section \ref{subsec:RigidBody_rotation_calculus_derivatives}. Inverting the previous equations results in the rotation matrix time derivative, which is linear:
\neweq{\RNBdot = \wNBNskew \; \RNB = \RNB \; \wNBBskew} {eq:SO3_dcm_dot}
Notice that if \nm{\vec R\lrp{t_0} = \vec I_3}, then \nm{\vec{\dot R}\lrp{t_0} = \omegaskew \lrp{t_0}}, and hence the skew-symmetric matrix \nm{\omegaskew\lrp{t_0}} provides a first order approximation of the rotation matrix around the identity matrix \nm{\vec I_3}:
\neweq{\vec R\lrp{t_0 + \Deltat} \approx \vec I_3 + \omegaskew \lrp{t_0} \, \Deltat}{eq:SO3_rotv_taylor}
The \emph{space of skew-symmetric matrices} \nm{\mathfrak{so}\lrp{3} = \{\omegaskew \in \mathbb{R}^{3x3} \ | \ \vec \omega \in \mathbb{R}^3, \ - \omegaskew = \omegaskew^T\}} is hence the \emph{tangent space} of \nm{\mathbb{SO}\lrp{3}} at the identity \nm{\vec I_3} \cite{Soatto2001}, denoted as \nm{T_{\vec I_3}{\mathcal R}}. The \emph{hat} \nm{\lrb{\cdot^\wedge: \mathbb{R}^3 \rightarrow \mathfrak{so}(3) \ | \ \vec \omega \rightarrow \vec \omega^\wedge = \widehat{\vec \omega}}} and \emph{vee} \nm{\lrb{\cdot^\vee: \mathfrak{so}(3) \rightarrow \mathbb{R}^3 \ | \ \lrp{\widehat{\vec \omega}^\vee \rightarrow \vec \omega}}} operators convert the cartesian vector form of the angular velocity into its skew-symmetric form, and viceversa.
If \nm{\vec R\lrp{t_0} \neq \vec I_3}, the tangent space needs to be transported right multiplying by \nm{\RNB\lrp{t_0}} (in the case of space angular velocity), or left multiplying for the local based velocity:
\begin{eqnarray}
\nm{\RNB\lrp{t_0 + \Deltat}} & \nm{\approx} & \nm{\RNB\lrp{t_0} + \lrsb{\wNBNskew \lrp{t_0} \, \Deltat} \, \RNB\lrp{t_0} = \lrsb{\vec I_3 + \wNBNskew \lrp{t_0} \, \Deltat} \, \RNB\lrp{t_0} }\label{eq:SO3_rotv_taylor_space} \\
\nm{\RNB\lrp{t_0 + \Deltat}} & \nm{\approx} & \nm{\RNB\lrp{t_0} + \RNB\lrp{t_0} \, \lrsb{\wNBBskew \lrp{t_0} \, \Deltat} = \RNB\lrp{t_0} \, \lrsb{\vec I_3 + \wNBBskew \lrp{t_0} \, \Deltat}}\label{eq:SO3_rotv_taylor_body}
\end{eqnarray}
Note that the solution to the ordinary differential equation \nm{\vec{\dot x}\lrp{t} = \vec x\lrp{t} \, \omegaskew, \ \vec x\lrp{t} \in \mathbb{R}^3}, where \nm{\omegaskew} is constant, is \nm{\vec x\lrp{t} = \vec x\lrp{0} \, e^{\ds{\omegaskew t}}}. Based on it, assuming \nm{\vec R\lrp{0} = \vec I_3} as initial condition, and considering for the time being that \nm{\omegaskew} is constant,
\neweq{\vec R\lrp{t} = e^{\ds{\omegaskew t}} = \vec I_3 + \omegaskew t + \frac{\lrp{\omegaskew t}^2}{2!} + \dots + \frac{\lrp{\omegaskew t}^n}{n!} + \dots}{eq:SO3_rotv_exponential3}
which is indeed a rotation matrix as it complies with the \nm{\mathbb{SO}\lrp{3}} conditions of orthogonality and handedness \cite{Soatto2001}.
\begin{center}
\begin{tabular}{lcc}
\hline
\textbf{Concept} & \textbf{Lie Theory} & \nm{\mathbb{SO}^3} \\
\hline
Tangent space element & \nm{\vec \tau^\wedge} & \nm{\vec r^\wedge = \widehat{\vec r}} \\
Velocity element & \nm{\vec v^\wedge} & \nm{\vec \omega^\wedge = \widehat{\vec \omega}} \\
Structure & \nm{\wedge} & skew symmetric matrix \\
\hline
\end{tabular}
\end{center}
\captionof{table}{Comparison between generic \nm{\mathbb{SO}(3)} and rotation vector as tangent space} \label{tab:Rotate_lie_rotv}
Remembering that so far \nm{\omegaskew} is constant, (\ref{eq:SO3_rotv_exponential3}) means that any rotation \nm{\vec R\lrp{t} = e^{\ds{\omegaskew t}}} can be realized by maintaining a constant angular velocity \nm{\vec \omega} for a given time t. This is analogous to stating that any angular displacement \nm{\vec R\lrp{\phi} = e^{\ds{\nskew \phi}}} can be achieved by rotating an angle \nm{\phi} about a fixed unitary rotation axis \nm{\vec n}, which enables the definition of the \emph{rotation vector} \nm{\vec r}, also known as the \emph{exponential coordinates} of the \nm{\mathcal R} rotation, as
\neweq{\vec r = \vec \omega \, t = \vec n \, \phi \in \mathbb{R}^3}{eq:SO3_rotv_definition}
Note that the rotation vector \nm{\vec r} belongs to the tangent space as it is a multiple of the angular velocity \nm{\vec \omega \in \mathfrak{so}\lrp{3}}, and hence tends to coincide with it as time tends to zero. The \emph{exponential map} \nm{\lrb{exp\lrp{} : \mathfrak{so}(3) \rightarrow \mathbb{SO}(3) \ | \ \mathcal R = exp\lrp{\vec r^\wedge}}} and its capitalized form \nm{\lrb{Exp\lrp{} : \mathbb{R}^3 \rightarrow \mathbb{SO}(3) \ | \ \mathcal R = Exp\lrp{\vec r}}} wrap the rotation vector around the rotation group. In the case of the rotation matrix, the exponential map can be obtained from (\ref{eq:SO3_rotv_exponential3}) based on the fact that all skew-symmetric matrices verify that \nm{\rskew^2 = \vec r \, \vec r^T - \vec I_3} and \nm{\rskew^3 = - \rskew}, converting skew symmetric matrices into orthogonal ones:
\neweq{\vec R\lrp{\vec r} = exp\lrp{\rskew} = Exp\lrp{\vec r} = e^{\rskew} = \vec I_3 + \frac{\rskew}{\|\vec r\|} \sin \| \vec r\| + \frac{\rskew^2}{\|\vec r\|^2} \lrp{1 - \cos \|\vec r\|}}{eq:SO3_rotv_exponential2}
Geometrically, the skew symmetric matrix corresponds to an axis of rotation (via the mapping \nm{\vec n \rightarrow \nskew}) and the exponential map generates the rotation corresponding to rotating about that axis by an amount \nm{\phi} \cite{Murray1994}.
The angular velocity \nm{\vec \omega} however is in fact not required to be constant. Given a rotation matrix \nm{\vec R \in \mathbb{SO}\lrp{3}}, it can be proven that there exists a not necessarily unique vector \nm{\vec r \in \mathbb{R}^3} such that \nm{\vec R = e^{\rskew}}. The \emph{logarithmic map} \nm{\lrb{log\lrp{} : \mathbb{SO}(3) \rightarrow \mathfrak{so}(3) \ | \ \vec r^\wedge = log\lrp{\mathcal R}}} and its capitalized version \nm{\lrb{Log\lrp{} : \mathbb{SO}(3) \rightarrow \mathbb{R}^3 \ | \ \vec r = Log\lrp{\mathcal R}}} hence convert rigid body rotations into rotation vectors.
\neweq{\vec R = \RNB = \begin{bmatrix} \nm{R_{\sss11}} & \nm{R_{\sss12}} & \nm{R_{\sss13}} \\ \nm{R_{\sss21}} & \nm{R_{\sss22}} & \nm{R_{\sss23}} \\ \nm{R_{\sss31}} & \nm{R_{\sss32}} & \nm{R_{\sss33}} \end{bmatrix} \ \rightarrow \ \|\vec r \| = \arccos\lrp{\frac{trace\lrp{\vec R} - 1}{2}}, \ \vec r= \frac{\| \vec r\|}{2 \sin \| \vec r \|} \begin{bmatrix} \nm{R_{\sss32} - R_{\sss23}} \\ \nm{R_{\sss13} - R_{\sss31}} \\ \nm{R_{\sss21} - R_{\sss12}} \end{bmatrix}}{eq:SO3_rotv_logarithm}
Any rotation matrix can hence be realized by rotating a certain angle about a given axis, as indicated in (\ref{eq:SO3_rotv_definition}). The vector \nm{\vec n = \vec r / \|\vec r\|} indicates the rotation direction while \nm{\phi = \|\vec r\|} represents the turn angle. The exponential map described by (\ref{eq:SO3_rotv_exponential2}) is thus surjective (there is at least one rotation vector for every rotation matrix) but not injective, as a rotation of \nm{\lrp{\|\vec r\| + 2 \, k \, \pi} \forall \ k \in \mathbb{Z}} about \nm{\nm{\vec r / \|\vec r\|}} or a rotation of \nm{\lrp{- \|\vec r\| + 2 \, k \, \pi}} about \nm{-\vec r / \|\vec r\|} produce exactly the same rotation matrix.
Although inverting the rotation by means of the rotation vector is straightforward,
\neweq{\vec r_{\sss BN} = {\vec r_{\sss NB}}^{-1} = - \vec r_{\sss NB}}{eq:SO_rotv_inversion}
the different \nm{\mathbb{SO}(3)} actions (concatenation, point rotation, vector rotation), as well as the relationship between the rotation vector derivative with time and the angular velocities, are complex and rarely used.
\subsection{Unit Quaternion}\label{subsec:RigidBody_rotation_rodrigues}
The quaternions with unity norm, known as unit quaternions, comprise an additional representation of the rotation group \nm{\mathbb{SO}\lrp{3}}, as shown below. Quaternions in turn are generalizations of complex numbers in the same way that these are generalizations of real ones \cite{Soatto2001}. It is hence necessary to first describe the complex numbers in section \ref{subsubsec:RigidBody_rotation_rodrigues_complex} and the quaternions in section \ref{subsubsec:RigidBody_rotation_rodrigues_quat} before focusing on the unit quaternions in section \ref{subsubsec:RigidBody_rotation_rodrigues_unit_quat}.
\subsubsection{Complex Numbers}\label{subsubsec:RigidBody_rotation_rodrigues_complex}
The set of \emph{complex numbers} \nm{\mathbb{C}} is composed of two real numbers \nm{\lrb{\mathbb{C} = \mathbb{R} + \mathbb{R} \, i \ | \ i^2 = i \cdot i = - 1}}. Given two complex numbers \nm{c_1 = x_1 + y_1 \, i \in \mathbb{C}, c_2 = x_2 + y_2 \, i \in \mathbb{C}, \forall \ x_1, y_1, x_2, y_2 \in \mathbb{R}}, it is possible to define the operations of addition \nm{\lrb{+ : \mathbb{C} \times \mathbb{C} \rightarrow \mathbb{C}}} and multiplication \nm{\lrb{\cdot : \mathbb{C} \times \mathbb{C} \rightarrow \mathbb{C}}}.
\begin{eqnarray}
\nm{c_1 + c_2} & = & \nm{\lrp{x_1 + y_1 \, i} + \lrp{x_2 + y_2 \, i} = \lrp{x_1 + x_2} + \lrp{y_1 + y_2} \, i}\label{eq:SO3_complex_addition} \\
\nm{c_1 \cdot c_2} & = & \nm{c_1 \, c_2 = \lrp{x_1 + y_1 \, i} \cdot \lrp{x_2 + y_2 \, i} = \lrp{x_1 x_2 - y_1 y_2} + \lrp{x_1 y_2 + y_1 x_2} \, i}\label{eq:SO3_complex_multiplication}
\end{eqnarray}
The conjugate is defined as \nm{c^{\ast} = x - y \, i \in \mathbb{C}} and verifies that \nm{\lrp{c_1 \cdot c_2}^{\ast} = c_1^{\ast} \cdot c_2^{\ast}}, while the norm \nm{\|c\| = \sqrt{c \cdot c^{\ast}} = \sqrt{c^{\ast} \cdot c} = \sqrt{x^2 + y^2} \in \mathbb{R}} satisfies that \nm{\|c_1 \cdot c_2\| = \|c_1\| \cdot \|c_2\|}. The set of complex numbers \nm{\mathbb{C}} endowed with the operations of addition \nm{+} and multiplication \nm{\cdot} forms a field (not ordered), known as the field of complex numbers \nm{\langle \mathbb{C}, +, \cdot \rangle}, nearly always abbreviated to simply \nm{\mathbb{C}}. The additive identity is \nm{0 = 0 + 0 \, i} and the inverse \nm{- c = - x - y \, i}, while the multiplication identity is \nm{1 = 1 + 0 \, i} and the inverse \nm{c^{-1} = c^{\ast} / \| c \|^2}.
Complex numbers can always be written in polar form \nm{\lrp{c = r \lrp{\cos \phi + \sin \phi \, i} = r \, e^{\ds{i \, \phi}}}}, and as such are valid representations of the circle group or plane rotations group \nm{\mathbb{SO}\lrp{2}}, similarly to the case of rotation vectors in \nm{\mathbb{SO}\lrp{3}} described in section \ref{subsec:RigidBody_rotation_rotv}.
\subsubsection{Quaternions}\label{subsubsec:RigidBody_rotation_rodrigues_quat}
The set of \emph{quaternions} \nm{\mathbb{H}} is defined as \nm{\{\mathbb{H} = \mathbb{C} + \mathbb{C} \, j \ | \ j^2 = -1, \ i \cdot j = - j \cdot i\}}. A quaternion \nm{\vec q \in \mathbb{H}} has the form \nm{\vec q = q_0 + q_1 \, i + q_2 \, j + q_3 \, i \, j}, with \nm{q_i \in \mathbb{R}}. \emph{Pure quaternions} \nm{\vec q = q_1 \, i + q_2 \, j + q_3 \, i \, j \in \mathbb{H}_p} are those defined in the tridimensional imaginary subspace of \nm{\mathbb{H}}, and verify that \nm{\vec q = - \, \qast}.
There are many different conventions for the quaternion found in the literature \cite{Sola2017}. This article adopts the \emph{Hamilton convention}, characterized by locating the real part first (instead of last), being right handed (left), passive (rotates frames and not vectors as in active), and local to global rotations (global to local). Any variation to these choices would result in different expressions below, although the physical concepts do not vary.
The real plus imaginary notation \nm{\{1, i, j, i \, j\}} is not always the most convenient. A quaternion can also be expressed as the sum of a scalar plus a vector in the form \nm{\vec q = q_0 + \vec q_v}, where \nm{q_0} is the real or scalar part and \nm{\vec q_v = q_1 \, i + q_2 \, j + q_3 \, i \, j} is the imaginary or vector part. Quaternions are however mostly represented as 4-vectors \nm{\vec q = \lrsb{q_0, \vec q_v}^T = \lrsb{q_0, q_1, q_2, q_3}^T}, which enables the usage of matrix algebra for quaternion operations. It is also convenient to abuse the equal operator by combining general, real, and pure quaternions as in \nm{\vec q = q_0 + \vec q_v}, where \nm{q_0 = \lrsb{q_0, \vec 0_v }^T} and \nm{\vec q_v = \lrsb{0, \vec q_v }^T}.
The following expressions define the addition \nm{\lrb{+ : \mathbb{H} \times \mathbb{H} \rightarrow \mathbb{H}}} and inner product \nm{\lrb{\langle \cdot \, , \cdot \rangle: \mathbb{H} \times \mathbb{H} \rightarrow \mathbb{R}}} of two quaternions, which commute, as well as the scalar multiplication \nm{\lrb{\cdot : \mathbb{R} \times \mathbb{H} \rightarrow \mathbb{H}}}:
\begin{eqnarray}
\nm{\vec q + \vec p} & = & \nm{\lrsb{q_0, \vec q_v}^T + \lrsb{p_0, \vec p_v}^T = \lrsb{q_0 + p_0, q_1 + p_1, q_2 + p_2, q_3 + p_3}^T = \lrsb{q_0 + p_0, \vec q_v + \vec p_v}^T}\label{eq:SO3_quat_addition} \\
\nm{\langle \vec q , \vec p \rangle} & = & \nm{\vec q \cdot \vec p = {\vec q}^T \, \vec p = q_{\sss 0} \, p_{\sss 0} + q_{\sss 1} \, p_{\sss 1} + q_{\sss 2} \, p_{\sss 2} + q_{\sss 3} \, p_{\sss 3}} \label{eq:SO3_quat_inner_product} \\
\nm{a \cdot \vec q} & = & \nm{a \cdot \lrsb{q_0, \vec q_v}^T = \lrsb{a \, q_0, a \cdot \vec q_v}^T} \label{eq:SO3_quat_scalar_product}
\end{eqnarray}
The multiplication of quaternions \nm{\{\otimes : \mathbb{H} \times \mathbb{H} \rightarrow \mathbb{H}\}} is not commutative as it includes the cross product:
\neweq{\vec q \otimes \vec p = \begin{bmatrix}
\nm{q_0 \cdot p_0 - q_1 \cdot p_1 - q_2 \cdot p_2 - q_3 \cdot p_3} \\
\nm{q_1 \cdot p_0 + q_0 \cdot p_1 - q_3 \cdot p_2 + q_2 \cdot p_3} \\
\nm{q_2 \cdot p_0 + q_3 \cdot p_1 + q_0 \cdot p_2 - q_1 \cdot p_3} \\
\nm{q_3 \cdot p_0 - q_2 \cdot p_1 + q_1 \cdot p_2 + q_0 \cdot p_3} \end{bmatrix} =
\begin{bmatrix}
\nm{q_0 \, p_0 - {\vec q_v}^T \, \vec p_v} \\
\nm{q_0 \, \vec p_v + p_0 \, \vec q_v + \widehat{\vec q}_v \, \vec p_v} \end{bmatrix} } {eq:SO3_quat_product}
It is also bilinear \cite{Sola2017}:
\neweq{\vec q \otimes \vec p = [\vec q]_L \, \vec p = \begin{bmatrix}
\nm{+ q_0} & \nm{- q_1} & \nm{- q_2} & \nm{- q_3} \\
\nm{+ q_1} & \nm{+ q_0} & \nm{- q_3} & \nm{+ q_2} \\
\nm{+ q_2} & \nm{+ q_3} & \nm{+ q_0} & \nm{- q_1} \\
\nm{+ q_3} & \nm{- q_2} & \nm{+ q_1} & \nm{+ q_0} \end{bmatrix} \,
\begin{bmatrix} \nm{p_0} \\ \nm{p_1} \\ \nm{p_2} \\ \nm{p_3} \end{bmatrix} =
[\vec p]_R \, \vec q = \begin{bmatrix}
\nm{+ p_0} & \nm{- p_1} & \nm{- p_2} & \nm{- p_3} \\
\nm{+ p_1} & \nm{+ p_0} & \nm{+ p_3} & \nm{- p_2} \\
\nm{+ p_2} & \nm{- p_3} & \nm{+ p_0} & \nm{+ p_1} \\
\nm{+ p_3} & \nm{+ p_2} & \nm{- p_1} & \nm{+ p_0} \end{bmatrix} \,
\begin{bmatrix} \nm{q_0} \\ \nm{q_1} \\ \nm{q_2} \\ \nm{q_3} \end{bmatrix}} {eq:SO3_quat_product_matrices}
These expressions can be simplified for the common case of the multiplication of a quaternion \nm{\vec q} by a pure quaternion \nm{\vec p_v}, this is, \nm{\{\otimes : \mathbb{H} \times \mathbb{H}_p \rightarrow \mathbb{H}\}}:
\begin{eqnarray}
\nm{\vec q \otimes \vec p_v} & = & \nm{\begin{bmatrix}
\nm{- q_1 \cdot p_1 - q_2 \cdot p_2 - q_3 \cdot p_3} \\
\nm{+ q_0 \cdot p_1 - q_3 \cdot p_2 + q_2 \cdot p_3} \\
\nm{+ q_3 \cdot p_1 + q_0 \cdot p_2 - q_1 \cdot p_3} \\
\nm{- q_2 \cdot p_1 + q_1 \cdot p_2 + q_0 \cdot p_3} \end{bmatrix} =
\begin{bmatrix}
\nm{- {\vec q_v}^T \, \vec p_v} \\
\nm{q_0 \, \vec p_v + \widehat{\vec q}_v \, \vec p_v} \end{bmatrix} } \label{eq:SO3_quat_product_pure} \\
\nm{\vec q \otimes \vec p_v} & = & \nm{[\vec q]_{L3} \, \vec p_v = \begin{bmatrix}
\nm{- q_1} & \nm{- q_2} & \nm{- q_3} \\
\nm{+ q_0} & \nm{- q_3} & \nm{+ q_2} \\
\nm{+ q_3} & \nm{+ q_0} & \nm{- q_1} \\
\nm{- q_2} & \nm{+ q_1} & \nm{+ q_0} \end{bmatrix} \,
\begin{bmatrix} \nm{p_1} \\ \nm{p_2} \\ \nm{p_3} \end{bmatrix} =
[\vec p_v]_{R3} \, \vec q = \begin{bmatrix}
\nm{0} & \nm{- p_1} & \nm{- p_2} & \nm{- p_3} \\
\nm{+ p_1} & \nm{0} & \nm{+ p_3} & \nm{- p_2} \\
\nm{+ p_2} & \nm{- p_3} & \nm{0} & \nm{+ p_1} \\
\nm{+ p_3} & \nm{+ p_2} & \nm{- p_1} & \nm{0} \end{bmatrix} \,
\begin{bmatrix} \nm{q_0} \\ \nm{q_1} \\ \nm{q_2} \\ \nm{q_3} \end{bmatrix}} \label{eq:SO3_quat_product_pure_matrices}
\end{eqnarray}
The conjugate quaternion is defined as \nm{\qast = q_0 - \vec q_v \in \mathbb{H}} and verifies that \nm{\lrp{\vec q \otimes \vec p}^{\ast} = \vec p^{\ast} \otimes \qast}, while the quaternion norm \nm{\|\vec q\| = \sqrt{\langle \vec q \ , \ \qast\rangle} = \sqrt{\langle \qast \ , \ \vec q\rangle} = \sqrt{\vec q \otimes \qast} = \sqrt{\qast \otimes \vec q} \in \mathbb{R}} satisfies that \nm{\|\vec q \otimes \vec p\| = \|\vec p \otimes \vec q\| = \|\vec q\| \, \|\vec p\|}. Quaternions endowed with \nm{\otimes} form the non-commutative group \nm{\langle \mathbb{H}, \otimes \rangle}, where \nm{\vec{q_1} = 1 + \vec0_v} is the identity and \nm{\vec q^{-1} = \nicefrac{\qast}{\| \vec q \|^2}} the inverse \cite{Sola2017}. Additionally, quaternions endowed with addition \nm{+} and multiplication \nm{\otimes} form the ring \nm{\langle \mathbb{H}, +, \otimes \rangle} where \nm{\vec{q_0} = 0 + \vec0_v} is the addition identity and \nm{- \vec q} the addition inverse or negative.
The quaternion rotation operator or \emph{double product} is defined as \nm{\{\mathbb{H} \times \mathbb{R}^3 \rightarrow \mathbb{R}^3 \ | \ \vec q \in \mathbb{H},} \nm{\vec v \in \mathbb{R}^3 \rightarrow \vec q \otimes \vec v \otimes \qast \in \mathbb{R}^3\}}\footnote{It is easily proven that the double quaternion product results in \nm{\mathbb{R}^3} and not \nm{\mathbb{R}^4}.}, while the natural power of a quaternion \nm{\vec q^n, n \in \mathbb{N}} is obtained by multiplying the quaternion by itself \nm{n-1} times.
\subsubsection{Unit Quaternion}\label{subsubsec:RigidBody_rotation_rodrigues_unit_quat}
\emph{Unit quaternions} verify that \nm{\|\vec q\| = 1}, which implies that \nm{\vec q^{-1} = \qast}. They can always be written as
\neweq{\vec q = \cos \theta + \vec n \, \sin \theta}{eq:SO3_quat_unit}
where \nm{\vec n} is a unit vector and \nm{\theta} is a scalar. The exponential of a quaternion \nm{e^{\vec q}} is defined analogously to that of real numbers \cite{Sola2017}. For pure quaternions \nm{\vec q = \vec q_v}, if abusing notation with \nm{\vec q_v = \vec v = \vec n \; \theta} where \nm{\theta = \|\vec v\|} and \nm{\|\vec n \| = 1}, it can be proven that \nm{e^{\vec q_v} = \cos \theta + \vec n \, \sin \theta}, which can be considered an extension of the \nm{e^{\ds{i \theta}} = \cos \theta + i \, \sin \theta} expression for complex numbers introduced in section \ref{subsubsec:RigidBody_rotation_rodrigues_complex} \cite{Sola2017}. Notice that since \nm{\|e^{\vec q_v}\| = 1}, the exponential of a pure quaternion is a unit quaternion. If \nm{\|\vec q\| = 1}, it is easy to verify that \nm{log\lrp{\vec q} = log\lrp{e^{\vec n \; \theta}} = \vec q_v}, so the logarithm of a unit quaternion is a pure quaternion \cite{Sola2017}.
Unit quaternions endowed with \nm{\otimes} constitute a subgroup that represents a 3-sphere, this is, the three dimensional surface of the unit sphere of \nm{\mathbb{R}^4}, and is commonly noted as \nm{\mathbb{S}^3}. They comply with the orthogonality and handedness conditions required in section \ref{subsec:RigidBody_bases} for rigid body rotations, and hence their space \nm{\mathbb{SO}\lrp{3} = \{\vec q \in \mathbb{S}^3 \ | \ \qast \otimes \vec q = \vec q \otimes \qast = \vec{q_1}\}} possesses group structure under quaternion multiplication \nm{\{\otimes : \mathbb{S}^3 \times \mathbb{S}^3 \rightarrow \mathbb{S}^3 \ | \ \vec q_a \otimes \vec q_b \in \mathbb{S}^3, \forall \ \vec q_a, \ \vec q_b \in \mathbb{S}^3\}} \cite{Pinter1990}. While having dimension four, the special orthogonal group \nm{\mathbb{SO}\lrp{3}} defined by means of unit quaternions constitutes a three dimensional manifold to euclidean space \nm{\mathbb{E}^3}. Note that in this group \nm{\vec{q_1}} constitutes the identity and \nm{\qast} the inverse.
\begin{center}
\begin{tabular}{lcclcc}
\hline
\textbf{Concept} & \nm{\mathbb{SO}^3} & \nm{\mathbb{S}^3} & \textbf{Concept} & \nm{\mathbb{SO}^3} & \nm{\mathbb{S}^3} \\
\hline
Lie group element & \nm{\mathcal R} & \nm{\vec q} & Concatenation & \nm{\circ} & \nm{\otimes} \\
Identity & \nm{\mathcal {I_R}} & \nm{\vec{q_1}} & Inverse & \nm{\mathcal R^{-1}} & \nm{\qast} \\
Point rotation & \nm{\vec g_{\mathcal R}(\vec p)} & \nm{\vec q \otimes \vec p \otimes \qast} & Vector rotation & \nm{\vec g_{\mathcal R*}(\vec v)} & \nm{\vec q \otimes \vec v \otimes \qast} \\
\hline
\end{tabular}
\end{center}
\captionof{table}{Comparison between generic \nm{\mathbb{SO}(3)} and unit quaternion} \label{tab:Rotate_lie_unit_quat}
Coordinate transformation (point or vector rotation) and rotation concatenation are both linear:
\begin{eqnarray}
\nm{\vec g_{\mathcal R*}\lrp{\vec v}} & = & \nm{\vec q \otimes \vec v \otimes \qast} \label{eq::SO3_quat_transform} \\
\nm{\vec q_{\sss EB}} & = & \nm{\vec q_{\sss EN} \otimes \vec q_{\sss NB}} \label{eq:SO3_quat_concatenation}
\end{eqnarray}
\subsection{Half Rotation Vector as Tangent Space}\label{subsec:RigidBody_rotation_halfrotv}
It is interesting to point out that in the case of rotation matrices (section \ref{subsec:RigidBody_rotation_dcm}), the orthogonality (\nm{\vec R^T \, \vec R = \vec I_3}) and handedness \nm{\lrp{det\lrp{\vec R} = +1}} constraints constitute two different expressions, while in the case of quaternions both are contained within \nm{\lrp{\qast \otimes \vec q = \vec q \otimes \qast = \vec{q_1}}}. As in other Lie groups, the time derivation of this constraint results in the structure of the Lie algebra. Derivating leads to \nm{\qast \otimes \vec{\dot q} = - \lrp{\qast \otimes \vec{\dot q}}^{\ast}}, which indicates that \nm{\qast \otimes \vec{\dot q}} is in fact a pure quaternion, as is \nm{\vec{\dot q} \otimes \qast}. This results in the following particularizations of (\ref{eq:algebra_vE}) and (\ref{eq:algebra_vX}):
\begin{eqnarray}
\nm{\vec \Omega_{\sss {NB}}^{\sss N\wedge}} & = & \nm{\vec{\dot q}_{\sss {NB}} \otimes \qNB^{\ast} = - \qNB \otimes \vec{\dot q}_{\sss NB}^{\ast}} \label{eq:SO3_quat_Omega_space} \\
\nm{\vec \Omega_{\sss {NB}}^{\sss B\wedge}} & = & \nm{\qNB^{\ast} \otimes \qNBdot = - \vec{\dot q}_{\sss NB}^{\ast} \otimes \qNB} \label{eq:SO3_quat_Omega_body}
\end{eqnarray}
The Lie algebra velocity \nm{\vec v^\wedge} of \nm{\mathbb{S}^3} is known as the \emph{half angular velocity} \nm{\vec \Omega^\wedge} \cite{Sola2017}, and as shown in (\ref{eq:SO3_quat_Omega_space}) and (\ref{eq:SO3_quat_Omega_body}), has the structure of a pure quaternion because its negative coincides with its conjugate:
\neweq{\vec \Omega^\wedge\lrp{t} = \lrsb{0, \vec \Omega\lrp{t}}^T \in \mathbb{H}_p}{eq:SO3_quat_pure}
Inverting the previous equations results in the unit quaternion time derivative, which is linear:
\neweq{\qNBdot = \vec \Omega_{\sss {NB}}^{\sss N\wedge} \otimes \qNB = \qNB \otimes \vec \Omega_{\sss {NB}}^{\sss B\wedge}} {eq::SO3_quat_Omega_dot}
Notice that if \nm{\vec q\lrp{t_0} = \vec{q_1}}, then \nm{\vec{\dot q }\lrp{t_0} = \vec \Omega\lrp{t_0}}, and hence the pure quaternion \nm{\vec \Omega^\wedge\lrp{t_0}} provides a first order approximation of the unit quaternion around the identity \nm{\vec{q_1}}:
\neweq{\vec q\lrp{t_0 + \Deltat} \approx \vec q_1 +\vec \Omega^\wedge\lrp{t_0} \, \Deltat}{eq:SO3_quat_taylor}
The \emph{space of pure quaternions} \nm{\mathfrak{so}\lrp{3} = \{\vec \Omega^\wedge \in \mathbb{H}_p \ | \ \vec \Omega \in \mathbb{R}^3\}} is hence the \emph{tangent space} of the unit sphere \nm{\mathbb{S}^3} of quaternions at the identity \nm{\vec q_1}, denoted as \nm{T_{\vec q_1}{\mathcal R}}. The \emph{hat} \nm{\lrb{\cdot^\wedge: \mathbb{R}^3 \rightarrow \mathfrak{so}(3) \ | \ \vec \Omega \rightarrow \vec \Omega^\wedge}} and \emph{vee} \nm{\lrb{\cdot^\vee: \mathfrak{so}(3) \rightarrow \mathbb{R}^3 \ | \ \lrp{\vec \Omega^\wedge}^\vee \rightarrow \vec \Omega}} operators convert the half angular velocity vector into its pure quaternion form, and viceversa.
If \nm{\vec q\lrp{t_0} \neq \vec q_1}, the tangent space needs to be transported right multiplying by \nm{\qNB\lrp{t_0}} (in the case of space tangent space), or left multiplying for the local space:
\begin{eqnarray}
\nm{\qNB\lrp{t_0 + \Deltat}} & \nm{\approx} & \nm{\qNB\lrp{t_0} + \lrsb{\vec \Omega_{\sss NB}^{\sss N\wedge} \lrp{t_0} \, \Deltat} \otimes \qNB\lrp{t_0} = \lrsb{\vec q_1 + \vec \Omega_{\sss NB}^{\sss N\wedge} \lrp{t_0} \, \Deltat} \otimes \qNB\lrp{t_0} }\label{eq:SO3_quat_taylor_space} \\
\nm{\qNB\lrp{t_0 + \Deltat}} & \nm{\approx} & \nm{\qNB\lrp{t_0} + \qNB\lrp{t_0} \otimes \lrsb{\vec \Omega_{\sss NB}^{\sss B\wedge} \lrp{t_0} \, \Deltat} = \qNB\lrp{t_0} \otimes \lrsb{\vec q_1 + \vec \Omega_{\sss NB}^{\sss B\wedge} \lrp{t_0} \, \Deltat}}\label{eq:SO3_quat_taylor_body}
\end{eqnarray}
Note that the solution to the ordinary differential equation \nm{\vec{\dot x}\lrp{t} = \vec x\lrp{t} \otimes \vec \Omega^\wedge, \ \vec x\lrp{t} \in \mathbb{R}^4}, where \nm{\vec \Omega^\wedge} is constant, is \nm{\vec x\lrp{t} = \vec x\lrp{0} \, e^{\ds{\vec \Omega^\wedge t}}}. Based on it, assuming \nm{\vec q\lrp{0} = \vec{q_1}} as initial condition, and considering for the time being that \nm{\vec \Omega} is constant,
\neweq{\vec q\lrp{t} = e^{\ds{\vec \Omega^\wedge t}} = \vec{q_1} + \vec \Omega^\wedge t + \frac{\lrp{\vec \Omega^\wedge t}^2}{2!} + \dots + \frac{\lrp{\vec \Omega^\wedge t}^n}{n!} + \dots}{eq:SO3_quat_exponential3}
which is indeed a unit quaternion \cite{Sola2017}.
\begin{center}
\begin{tabular}{lcc}
\hline
\textbf{Concept} & \textbf{Lie Theory} & \nm{\mathbb{SO}^3} \\
\hline
Tangent space element & \nm{\vec \tau^\wedge} & \nm{\vec h^\wedge = \lrsb{0, \vec h}^T} \\
Velocity element & \nm{\vec v^\wedge} & \nm{\vec \Omega^\wedge = \lrsb{0, \vec \Omega}^T} \\
Structure & \nm{\wedge} & pure quaternion \\
\hline
\end{tabular}
\end{center}
\captionof{table}{Comparison between generic \nm{\mathbb{SO}(3)} and half rotation vector as tangent space} \label{tab:Rotate_lie_halfrotv}
Remembering that so far \nm{\vec \Omega^\wedge} is constant, (\ref{eq:SO3_quat_exponential3}) means that any rotation \nm{\vec q\lrp{t} = e^{\ds{\vec \Omega^\wedge t}}} can be realized by maintaining a constant half angular velocity \nm{\vec \Omega^\wedge} in \nm{\mathbb{H}_p} for a given time t. This is analogous to stating that any angular displacement \nm{\vec q\lrp{\theta} = e^{\ds{\vec n^\wedge \theta}}} can be achieved by rotating an angle \nm{\theta} about a fixed unitary rotation axis \nm{\vec n^\wedge \in \mathbb{H}_p}. It is customary to absorb \nm{t} into \nm{\vec \Omega} or \nm{\theta} into \nm{\vec n}, resulting in \emph{the half rotation vector} \nm{\vec h}:
\neweq{\vec h = \vec \Omega \, t = \vec n \, \theta \ \ \in \mathbb{R}^3}{eq:SO3_quat_half_velocity}
Expression (\ref{eq:SO3_quat_exponential3}) represents the \emph{exponential map} \nm{\{exp : \mathfrak{so}\lrp{3} \rightarrow \mathbb{SO}\lrp{3} | \ \vec h^\wedge \in \mathbb{H}_p \rightarrow exp\lrp{\vec h^\wedge} \in \mathbb{S}^3\}} \cite{Sola2017}, which transforms pure quaternions into unit quaternions. Before continuing, let's compare the similarities and differences between the exponential maps when applied to rotation matrices versus quaternions, as both represent maps between the tangent space \nm{\mathfrak{so}\lrp{3}} and the special orthogonal group \nm{\mathbb{SO}\lrp{3}}. In the case of rotation matrices, this translates to a map between skew-symmetric matrices and orthogonal ones, while for quaternions the exponential map converts pure quaternions into unitary ones. There is one additional difference, however. In the case of rotation matrices, the map encodes through the rotation vector \nm{\vec r = \vec \omega \, t = \vec n \, \phi}, this is, the axis of rotation \nm{n} and the rotated angle \nm{\phi}. In the case of quaternions, the encoding is through the half rotation vector \nm{\vec h = \vec \Omega \, t = \vec n \, \theta}. Since the rotation is accomplished by the double product \nm{\vec q \otimes \vec v \otimes \qast} as noted in (\ref{eq::SO3_quat_transform}), the vector \nm{\vec v} experiences a rotation that is twice that encoded in \nm{\vec q}, which means that \nm{\vec q} encodes half the intended rotation on \nm{\vec v}. This implies that the space of unit quaternions \nm{\mathbb{S}^3} is in fact a double covering of \nm{\mathbb{SO}\lrp{3}}, not \nm{\mathbb{SO}\lrp{3}} itself \cite{Sola2017}.
\begin{eqnarray}
\nm{\vec r} & = & \nm{2 \cdot \vec h} \label{eq:SO3_quat_equiv_r} \\
\nm{\omega} & = & \nm{2 \cdot \Omega} \label{eq:SO3_quat_equiv_omega} \\
\nm{\phi} & = & \nm{2 \cdot \theta} \label{eq:SO3_quat_equiv_phi}
\end{eqnarray}
Taking these relationships into consideration, it is possible to obtain a more practical expression for the exponential map \nm{\lrb{exp\lrp{} : \mathfrak{so}(3) \rightarrow \mathbb{SO}(3) \ | \ \mathcal R = exp\lrp{\vec h^\wedge} = exp\lrp{\vec r^\wedge / 2}}} and its capitalized form \nm{\lrb{Exp\lrp{} : \mathbb{R}^3 \rightarrow \mathbb{SO}(3) \ | \ \mathcal R = Exp\lrp{\vec h} = Exp\lrp{\vec r / 2}}} than (\ref{eq:SO3_quat_exponential3}) \cite{Sola2017}:
\neweq{\vec q\lrp{\vec h} = \vec q\lrp{\vec r / 2} = Exp\lrp{\vec h} = Exp\lrp{\vec r / 2} = e^{\ds{\vec \Omega^{\wedge} t}} = e^{\ds{\vec n^\wedge \theta}} = e^{\ds{\vec n^{\wedge} \phi / 2}} = \cos \dfrac{\phi}{2} + \vec n \, \sin \dfrac{\phi}{2}}{eq:SO3_quat_exponential2}
When regarded as a pure quaternion in \nm{\mathbb{H}_p}, the angle \nm{\theta} between a unit quaternion \nm{\vec q} and the identity \nm{\vec{q_1}} is \nm{\cos \theta = \vec{q_1}^T \, \vec q = q_0}. At the same time, the angle \nm{\phi} rotated by the unit quaternion \nm{\vec q} on objects in \nm{\mathbb{R}^3} satisfies (\ref{eq:SO3_quat_exponential2}), so the angle between a unit quaternion vector and the identify in \nm{\mathbb{H}_p} is half the angle rotated by the unit quaternion in \nm{\mathbb{R}^3} space, as depicted by (\ref{eq:SO3_quat_equiv_phi}).
So far the exponential map has been obtained based on the assumption of constant angular velocity, but this does not need to be the case. Given a unit quaternion \nm{\vec q \in \mathbb{S}^3}, there exists a not necessarily unique rotation vector \nm{\vec r = 2 \cdot \vec h \in \mathbb{R}^3} such that \nm{\vec q = Exp\lrp{\vec r / 2} = Exp\lrp{\vec n \, \phi / 2}}:
\begin{eqnarray}
\nm{\phi} & = & \nm{2 \, \arctan \lrp{\dfrac{\| \vec{q_v} \|}{q_0} }} \label{eq:SO3_theta_from_quat} \\
\nm{\vec n} & = & \nm{\dfrac{\vec{q_v}}{\| \vec{q_v} \|}} \label{eq:SO3_rotvec_from_quat}
\end{eqnarray}
This expression represents the capitalized \emph{logarithmic map} \nm{\{log : \mathbb{SO}\lrp{3} \rightarrow \mathfrak{so}\lrp{3} \ | \ \vec q \in \mathbb{S}^3 \rightarrow \vec h = \vec r / 2 \in \mathbb{R}^3\}} \cite{Sola2017}. As in the case of the rotation matrix described in section \ref{subsec:RigidBody_rotation_rotv}, the exponential map described by (\ref{eq:SO3_quat_exponential2}) is surjective but not injective, as a rotation of \nm{\lrp{\|\vec r\| + 2 \, k \, \pi} \forall \ k \in \mathbb{Z}} about \nm{\nm{\vec r / \|\vec r\|}} produces exactly the same unit quaternion. In contrast with the case of rotation matrices, a rotation of \nm{\lrp{- \|\vec r\| + 2 \, k \, \pi}} about \nm{-\vec r / \|\vec r\|} produces the opposite (negative) unit quaternion, although both represent the same rotation.
This shows that the map from rotation matrix to unit quaternion \nm{\{\mathbb{SO}\lrp{3} \rightarrow \mathbb{SO}\lrp{3} | \ \vec R \in \mathbb{R}^{3x3} \rightarrow \vec q \in \mathbb{S}^3\}} is also surjective but not injective, as there are two and only two quaternions corresponding to the same rotation matrix \nm{\lrp{\vec R\lrp{\vec q} = \vec R\lrp{- \vec q}}}. The reason is again the double covering of \nm{\mathbb{SO}\lrp{3}} by the unit quaternion \cite{Sola2017}. One quaternion induces a rotation in \nm{\mathbb{R}^3} that follows the shortest direction to the final angle \nm{\lrp{\phi < \pi / 2}}, while the opposite quaternion rotates the opposite way reaching the same final angle after rotating \nm{\lrp{\phi > \pi / 2}}.
As the vector \nm{\vec \Omega} represents half the angular velocity \nm{\vec \omega}, it is possible to adjust expressions (\ref{eq:SO3_quat_Omega_space}), (\ref{eq:SO3_quat_Omega_body}), and (\ref{eq::SO3_quat_Omega_dot}):
\begin{eqnarray}
\nm{\vec \omega_{\sss {NB}}^{\sss N\wedge}} & = & \nm{2 \ \vec{\dot q}_{\sss {NB}} \otimes \qNB^{\ast}} \label{eq:SO3_quat_omega_space} \\
\nm{\vec \omega_{\sss {NB}}^{\sss B\wedge}} & = & \nm{2 \ \qNB^{\ast} \otimes \qNBdot} \label{eq:SO3_quat_omega_body} \\
\nm{\qNBdot} & = & \nm{\dfrac{1}{2} \; \vec \omega_{\sss {NB}}^{\sss N\wedge} \otimes \qNB = \dfrac{1}{2} \; \qNB \otimes \vec \omega_{\sss {NB}}^{\sss B\wedge}} \label{eq::SO3_quat_omega_dot}
\end{eqnarray}
\subsection{Euler Angles}\label{subsec:RigidBody_rotation_euler}
All the previous representations have some type of redundancy as their dimension is higher than three. It is always possible, however, to pick three unitary vectors \nm{\lrp{\vec n_1, \vec n_2, \vec n_3}} forming a basis\footnote{The base vectors do not need to be orthogonal, just linearly independent.} and perform three consecutive rotations to define a map \nm{\{\mathbb{R}^3 \rightarrow \mathbb{SO}\lrp{3} \ | \ \lrp{\beta_1, \beta_2, \beta_3} \in \mathbb{R}^3 \rightarrow R = e^{\ds{\widehat {\vec n}_1 \beta_1}} \ e^{\ds{\widehat {\vec n}_2 \beta_2}} \ e^{\ds{\widehat {\vec n}_3 \beta_3}} \in \mathbb{SO}\lrp{3}\}} \cite{Soatto2001}.
In case the selected basis is orthonormal there are only twelve possible combinations or Euler angles, of which \nm{3-2-1} is the one employed in this document\footnote{This means to first rotate about the \third\ axis, then about the resulting \second\ axis, and finally about the ensuing \first\ axis.}. The rotation from the spatial or global frame \nm{F_{\sss N} = \{\vec O_{\sss {CR}}, \vec n_1, \vec n_2, \vec n_3\}} to the local or body frame \nm{F_{\sss B} = \{\vec O_{\sss {CR}}, \vec b_1, \vec b_2, \vec b_3\}} is performed by first rotating a \emph{yaw} angle (y) about \nm{\vec n_3}, followed by rotating a \emph{pitch} angle (p) about \nm{\vec n_2'}, and finally rotating a \emph{roll} angle (r) about \nm{\vec n_1''}, where {\nm{\vec n_2'} is the result of applying the first rotation to \nm{\vec n_2} and \nm{\vec n_1''}, which coincides with \nm{\vec b_1}, that of applying the first two rotations to \nm{\vec n_1}.
\neweq{ \vec R_1(r) = e^{\ds{\widehat {\vec n}_1 \ttt r}} = \begin{bmatrix} 1 & 0 & 0 \\ 0 & \nm{cr} & \nm{-sr}\\ 0 & \nm{sr} & \nm{cr} \end{bmatrix} \
\vec R_2(p) = e^{\ds{\widehat {\vec n}_2 \ttt p}} = \begin{bmatrix} \nm{cp} & 0 & \nm{sp} \\ 0 & 1 & 0 \\ \nm{-sp} & 0 & \nm{cp} \end{bmatrix} \
\vec R_3(y) = e^{\ds{\widehat {\vec n}_3 \ttt y}} = \begin{bmatrix} \nm{cy} & \nm{-sy} & 0 \\ \nm{sy} & \nm{cy} & 0 \\ 0 & 0 & 1 \end{bmatrix}} {eq:SO3_euler_individual}
where s and c stand for sine and cosine respectively. The complete map for the three rotations is:
\neweq{\RNB = \vec R_3(y) \; \vec R_2(p) \; \vec R_1(r) = \begin{bmatrix}
\nm{+ cp \cdot cy} &
\nm{- cr \cdot sy + sr \cdot sp \cdot cy} &
\nm{+ sr \cdot sy + cr \cdot sp \cdot cy} \\
\nm{+ cp \cdot sy} &
\nm{+ cr \cdot cy + sr \cdot sp \cdot sy} &
\nm{- sr \cdot cy + cr \cdot sp \cdot sy} \\
\nm{- sp} &
\nm{+ sr \cdot cp} &
\nm{+ cr \cdot cp} \end{bmatrix}} {eq:SO3euler_to_R}
In this document the Euler angles are denoted by \nm{\phiNB = \lrsb{y, \, p, \, r}^T}. They can also be obtained from the rotation matrix, but there are singular instances (\nm{p = \pm \pi/2}) where the angles can not be uniquely determined.
\neweq{y = \arctan \frac{R_{\sss{21}}}{R_{\sss{11}}} \ \ \ \ p = \arcsin\lrp{- R_{\sss{31}}} \ \ \ \ r = \arctan \frac{R_{\sss{32}}}{R_{\sss{33}}}} {eq:SO3_euler_from_R}
\subsection{Rotational Motion Algebraic Operations}\label{subsec:RigidBody_rotation_algebra}
The basic algebraic operations of addition, subtraction, multiplication, division, and exponentiation are not defined for objects of the special orthogonal group \nm{\mathbb{SO}\lrp{3}}, no matter if they are represented by a rotation matrix \nm{\vec R \in \mathbb{R}^{3x3}}, a rotation vector \nm{\vec r \in \mathbb{R}^3}, or a unit quaternion \nm{\vec q \in \mathbb{S}^3}. However, as members of the special orthogonal group \nm{\mathbb{SO}\lrp{3}}, all rotation representations are closed under a given operation that represents the concatenation of rotations, and define not only an identity rotation that represents the lack of rotation, but also an inverse operation representing the opposite rotation. The concatenation of rotations and the identity and inverse operations enable the definition of the power, exponential and logarithmic operators (section \ref{subsubsec:RigidBody_rotation_algebra_exp_log}), the spherical linear interpolation (section \ref{subsubsec:RigidBody_rotation_algebra_slerp}), and the perturbations together with the plus and minus operators (section \ref{subsubsec:RigidBody_rotation_algebra_plus_minus}).
\subsubsection{Powers, Exponentials and Logarithms}\label{subsubsec:RigidBody_rotation_algebra_exp_log}
Any rotation can be executed by rotating a given angle \nm{\phi} about a fixed rotation axis \nm{\vec n}, resulting in the rotation vector \nm{\vec r = \vec n \, \phi} (section \ref{subsec:RigidBody_rotation_rotv}) or its half \nm{\vec h = \vec n \, \phi / 2} (section \ref{subsec:RigidBody_rotation_halfrotv}). Taking a multiple or a fraction of a rotation vector is hence straightforward, as \nm{t \, \vec r = t \, \vec n \, \phi = \vec n \, \lrp{t \, \phi} \, \forall \, t \in \mathbb{R}, \vec r \in \mathfrak{so}\lrp{3}}. The exponential maps defined in (\ref{eq:SO3_rotv_exponential2}) and (\ref{eq:SO3_quat_exponential2}) are named that way because they comply with the behavior of the real exponential function \nm{exp^b\lrp{a} = exp\lrp{a \cdot b}\, \forall \, a, \, b \in \mathbb{R}}. As such, the exponential function \nm{\{exp(): \mathbb{R}^3 \times \mathbb{R} \rightarrow \mathbb{SO}\lrp{3} \ | \ \vec r \in \mathbb{R}^3, t \in \mathbb{R} \rightarrow {\mathcal R}^t = Exp\lrp{t \, \vec r} \in \mathbb{SO}\lrp{3}\}} is defined as:
\begin{eqnarray}
\nm{\vec R^{\ds t}\lrp{\vec r}} & = & \nm{\vec R\lrp{t \, \vec r} = Exp\lrp{t \, \vec r} = \vec I_3 + \frac{\rskew}{\|\vec r\|} \sin \|t \, \vec r\| + \frac{\rskew^2}{\|\vec r\|^2} \lrp{1 - \cos \|t \, \vec r\|}}\label{eq:SO3_rotv_exponential_fraction} \\
\nm{\vec q^{\ds t}\lrp{\vec h = \vec r / 2}} & = & \nm{\vec q\lrp{t \, \vec h = t \, \vec r / 2} = Exp\lrp{t \, \vec r / 2} = \cos \dfrac{t \, \phi}{2} + \vec n \, \sin \dfrac{t \, \phi}{2}}\label{eq:SO3_quat_exponential_fraction}
\end{eqnarray}
In a similar way, the logarithmic maps defined in (\ref{eq:SO3_rotv_logarithm}), (\ref{eq:SO3_theta_from_quat}), and (\ref{eq:SO3_rotvec_from_quat}) also comply with the behavior of the real logarithmic function \nm{b \cdot log\lrp{a} = log\lrp{a^b} \, \forall \, a, \, b \in \mathbb{R}}. As such, the logarithmic function \nm{\{log(): \mathbb{SO}\lrp{3} \times \mathbb{R} \rightarrow \mathbb{R}^3 \ | \ \mathcal R \in \mathbb{SO}\lrp{3}, t \in \mathbb{R} \rightarrow t \, \vec r = Log\lrp{{\mathcal R}^t} \in \mathbb{R}^3\}} is defined as:
\begin{eqnarray}
\nm{log\lrp{\vec R^{\ds t}\lrp{\vec r}}} & = & \nm{Log\lrp{Exp\lrp{t \, \vec r}} = t \, Log\lrp{\vec R\lrp{\vec r}} = t \, Log\lrp{Exp\lrp{\vec r}} = t \, \vec r}\label{eq:SO3_rotv_logarithmic_fraction} \\
\nm{log\lrp{\vec q^{\ds t}\lrp{\vec h = \vec r / 2}}} & = & \nm{Log\lrp{Exp\lrp{t \, \vec r / 2}} = t \, Log\lrp{\vec q\lrp{\vec r / 2}} = t \, Log\lrp{Exp\lrp{\vec r / 2}} = t \, \vec r / 2}\label{eq:SO3_quat_logarithmic_fraction}
\end{eqnarray}
\subsubsection{Spherical Linear Interpolation}\label{subsubsec:RigidBody_rotation_algebra_slerp}
Given two rotations \nm{\mathcal R_0, \, \mathcal R_1 \in \mathbb{SO}\lrp{3}}, \emph{spherical linear interpolation} (\hypertt{SLERP}) seeks to obtain a rotation function \nm{\mathcal R\lrp{t}, \, t \in \mathbb{R}} that linearly interpolates from \nm{\mathcal R\lrp{0} = \mathcal R_0} to \nm{\mathcal R\lrp{1} = \mathcal R_1} in such a way that the rotation occurs at constant angular velocity along a fixed axis \cite{Sola2017}.
If employing unit quaternions, \nm{\Delta \vec q} is according to (\ref{eq:SO3_quat_concatenation}) the full rotation required to go from \nm{\vec q_0} to \nm{\vec q_1}, such that \nm{\vec q_1 = \vec q_0 \otimes \Delta \vec q}, from where \nm{\Delta \vec q = \vec q_0^{\ast} \otimes \vec q_1}. The corresponding rotation vector is then \nm{\Delta \vec r = \vec n \, \Delta \phi = 2 \, Log\lrp{\Delta \vec q}}. Let's take a fraction of the full rotation \nm{\delta \phi = t \, \Delta \phi} and obtain the corresponding quaternion:
\begin{eqnarray}
\nm{\delta \vec q} & = & \nm{Exp\lrp{\dfrac{\vec n \, \delta \phi}{2}} = Exp\lrp{t \, \dfrac{\vec n \, \Delta \phi}{2}} = Exp\lrp{t \, \dfrac{\Delta \vec r}{2}}} \nonumber \\
& = & \nm{Exp\big(t \, Log\lrp{\Delta \vec q}\big) = Exp\big(t \, Log\lrp{\vec q_0^{\ast} \otimes \vec q_1}\big) = \lrp{\vec q_0^{\ast} \otimes \vec q_1}^{\ds t}} \label{eq:SO3_interp_quat_partial}
\end{eqnarray}
The interpolated unit quaternion is hence the following:
\neweq{\vec q\lrp{t} = \vec q_0 \otimes \lrp{\vec q_0^{\ast} \otimes \vec q_1}^{\ds t}}{eq:SO3_interp_quat}
Because of the double covering of \nm{\mathbb{SO}\lrp{3}} by the quaternion, only the interpolation between quaternions at acute angles \nm{\lrp{\Delta \theta = \Delta \phi / 2 \leq \pi / 2}}, is executed following the shortest path, which occurs if \nm{\cos \Delta \theta = {\vec q_0}^T \, \vec q_1 < 0}. If this is not the case, just replace \nm{\vec q_1} by \nm{- \vec q_1} and repeat the process.
A similar result is obtained when employing rotation matrices instead of unit quaternions:
\neweq{\vec R\lrp{t} = \vec R_0 \, \lrp{\vec R_0^T \, \vec R_1}^{\ds t}}{eq:SO3_interp_dcm}
\subsubsection{Plus and Minus Operators}\label{subsubsec:RigidBody_rotation_algebra_plus_minus}
A perturbed rigid body rotation \nm{\widetilde{\mathcal R} \in \mathbb{SO}\lrp{3}} can always be expressed as the composition of the unperturbed rotation \nm{\mathcal R} with a (usually) small perturbation \nm{\Delta \mathcal R}. Perturbations can be specified either at the local or body frame \nm{\FB}, this is, at the local vector space tangent to \nm{\mathbb{SO}\lrp{3}} at the actual orientation, in which case they are known as \emph{local perturbations}. They can also be specified at the global frame \nm{\FN}, which coincides with the vector space tangent to \nm{\mathbb{SO}\lrp{3}} at the origin; in this case they are known as \emph{global perturbations} \cite{Sola2017}. Local perturbations appear on the right hand side of the rotation composition, resulting in \nm{\widetilde{\mathcal R} = \mathcal R \circ \Delta\mathcal{R}^{\sss B}}, while global ones appear to the left, hence \nm{\widetilde{\mathcal R} = \Delta\mathcal{R}^{\sss N} \circ \mathcal R}.
The \emph{plus} and \emph{minus operators} are introduced in section \ref{subsubsec:algebra_lie_exp_plus} and enable operating with increments of the nonlinear \nm{\mathbb{SO}\lrp{3}} manifold expressed in the linear tangent vector space \nm{\mathfrak{so}\lrp{3}}. There exist right (\nm{\oplus, \, \ominus}) or left (\nm{\boxplus, \, \boxminus}) versions depending on whether the increments are viewed in the local frame (right) or the global one (left). It is important to remark that although perturbations and the plus and left operators are best suited to work with small rotation changes (perturbations), the expressions below are generic and work just the same no matter the size of the perturbation.
The right plus operator \nm{\{\oplus : \mathbb{SO}\lrp{3} \times \mathfrak{so}\lrp{3} \rightarrow \mathbb{SO}\lrp{3} \, | \, \widetilde{\mathcal R} = \mathcal R \oplus \ \Delta \vec r^{\sss B} = \mathcal R \circ Exp\lrp{\Delta \vec r^{\sss B}}\}} produces a rotation element \nm{\widetilde{\mathcal R}} resulting from the composition of a reference rotation \nm{\mathcal R} with an often small rotation \nm{\Delta \vec r^{\sss B} = \vec n^{\sss B} \, \Delta \phi}, contained in the tangent space to the reference rotation \nm{\mathcal R}, this is, in the local space \cite{Sola2017}. The left plus operator \nm{\{\boxplus : \mathfrak{so}\lrp{3} \times \mathbb{SO}\lrp{3} \rightarrow \mathbb{SO}\lrp{3} \, | \, \widetilde{\mathcal R} = \Delta \vec r^{\sss N} \boxplus \mathcal R = Exp\lrp{\Delta \vec r^{\sss N}} \circ \mathcal R\}} is similar but the often small rotation \nm{\Delta \vec r^{\sss N} = \vec n^{\sss N} \, \Delta \phi} is contained in the tangent space at the identify or global space. The expressions shown below are valid up to the first coverage of \nm{\mathbb{SO}\lrp{3}}, this is, \nm{\phi < \pi}. In the cases of rotation matrix and unit quaternion, the plus operator is defined as:
\begin{eqnarray}
\nm{\widetilde{\vec R}} & = & \nm{\vec R \oplus \Delta \vec r^{\sss B} = \vec R \ Exp\lrp{\Delta \vec r^{\sss B}} = \vec R \ \Delta \vec R^{\sss B}}\label{eq:SO3_dcm_plus} \\
\nm{\widetilde{\vec q}} & = & \nm{\vec q \oplus \Delta \vec r^{\sss B} = \vec q \otimes Exp\lrp{\Delta \vec r^{\sss B} / 2} = \vec q \otimes \Delta \vec q^{\sss B}}\label{eq:SO3_quat_plus} \\
\nm{\widetilde{\vec R}} & = & \nm{\Delta \vec r^{\sss N} \boxplus \vec R = Exp\lrp{\Delta \vec r^{\sss N}} \ \vec R = \Delta \vec R^{\sss N} \ \vec R}\label{eq:SO3_dcm_plus_left} \\
\nm{\widetilde{\vec q}} & = & \nm{\Delta \vec r^{\sss N} \boxplus \vec q = Exp\lrp{\Delta \vec r^{\sss N} / 2} \otimes \vec q = \Delta \vec q^{\sss N} \otimes \vec q}\label{eq:SO3_quat_plus_left}
\end{eqnarray}
The right minus operator \nm{\{\ominus : \mathbb{SO}\lrp{3} \times \mathbb{SO}\lrp{3} \rightarrow \mathfrak{so}\lrp{3} \, | \, \Delta \vec r^{\sss B} = \widetilde{\mathcal R} \ominus \mathcal R = Log\big(\mathcal R^{-1} \circ \widetilde{\mathcal R}\big)\}}, as well as the left \nm{\{\boxminus : \mathbb{SO}\lrp{3} \times \mathbb{SO}\lrp{3} \rightarrow \mathfrak{so}\lrp{3} \, | \, \Delta \vec r^{\sss N} = \widetilde{\mathcal R} \boxminus \mathcal R = Log\big(\widetilde{\mathcal R} \circ \mathcal R^{-1}\big)\}}, represent the inverse operations, returning the rotation vector difference \nm{\Delta \vec r} between two rotations \nm{\mathcal R} and \nm{\widetilde{\mathcal R}} expressed in either the local or global tangent spaces to \nm{\mathcal R}.
\begin{eqnarray}
\nm{\Delta \vec r^{\sss B} } & = & \nm{\widetilde{\vec R} \ominus \vec R = Log\lrp{\vec R^T \ \widetilde{\vec R}} = Log\lrp{\Delta \vec R^{\sss B}}}\label{eq:SO3_dcm_minus} \\
\nm{\Delta \vec r^{\sss B} } & = & \nm{\widetilde{\vec q} \ominus \vec q = 2 \ Log\lrp{\qast \otimes \widetilde{\vec q}} = 2 \ Log\lrp{\Delta \vec q^{\sss B}}}\label{eq:SO3_quat_minus} \\
\nm{\Delta \vec r^{\sss N} } & = & \nm{\widetilde{\vec R} \boxminus \vec R = Log\lrp{\widetilde{\vec R} \ \vec R^T} = Log\lrp{\Delta \vec R^{\sss N}}}\label{eq:SO3_dcm_minus_left} \\
\nm{\Delta \vec r^{\sss N} } & = & \nm{\widetilde{\vec q} \boxminus \vec q = 2 \ Log\lrp{\widetilde{\vec q} \otimes \qast} = 2 \ Log\lrp{\Delta \vec q^{\sss N}}}\label{eq:SO3_quat_minus_left}
\end{eqnarray}
If the \nm{\Delta \vec r} perturbation is small, the (\ref{eq:SO3_rotv_exponential3}) and (\ref{eq:SO3_quat_exponential3}) Taylor expansions can be truncated, resulting in the following expressions, valid for both the body frame (\nm{\Delta \vec r^{\sss B}}) or the global one (\nm{\Delta \vec r^{\sss N}}):
\begin{eqnarray}
\nm{\Delta \vec R = Exp\lrp{\Delta \vec r}} & \nm{\approx} & \nm{\vec I_3 + \Delta \vec r^\wedge = \vec I_3 + \Delta \phi \, \nskew}\label{eq:SO3_perturbation_dcm_truncated_local} \\
\nm{\Delta \vec q = exp\lrp{\Delta \vec r /2}} & \nm{\approx} & \nm{\vec {q_1} + \Delta \vec r^\wedge / 2 = \lrsb{1, \vec n \, \Delta \phi / 2}^T}\label{eq:SO3_perturbation_quat_truncated_local}
\end{eqnarray}
\subsection{Rotational Motion Time Derivative and Angular Velocity}\label{subsec:RigidBody_rotation_calculus_derivatives}
Let's consider a rotating rigid body \nm{\mathcal R\lrp{t} \in \mathbb{SO}\lrp{3}, t \in \mathbb{R}} and compute its derivative with time, which belongs to neither \nm{\mathbb{SO}\lrp{3}} nor \nm{\mathfrak{so}\lrp{3}} but to the Euclidean space of the chosen rotation representation, \nm{\mathbb{R}^{3x3}} for the rotation matrix and \nm{\mathbb{H}} for the unit quaternion:
\neweq{\dot{\mathcal{R}}\lrp{t} = \lim\limits_{\Delta t \to 0} \dfrac{\mathcal{R}\lrp{t + \Delta t} - \mathcal{R}\lrp{t}}{\Delta t}}{eq:SO3_time_derivative_def}
Considering the time modified rotation \nm{\mathcal{R}\lrp{t + \Delta t}} as the perturbed state (section \ref{subsubsec:RigidBody_rotation_algebra_plus_minus}), the resulting time derivatives for the rotation matrix and unit quaternion representations are the following:
\begin{eqnarray}
\nm{\vec {\dot R}\lrp{t}} & = & \nm{\lim\limits_{\Delta t \to 0} \dfrac{\vec R \, \Delta \vec R^{\sss B} - \vec R}{\Delta t} \ \nm{\approx} \lim\limits_{\Delta t \to 0}\dfrac{\vec R \, \lrsb{\lrp{\vec I_3 + \Delta \phi \ \nskew^{\sss B}} - \vec I_3}}{\Delta t} = \vec R \, \lim\limits_{\Delta t \to 0}\dfrac{\Delta \phi \, \nskew^{\sss B}}{\Delta t}}\label{eq:SO3_time_derivative_R1b} \\
\nm{\vec {\dot q}\lrp{t}} & = & \nm{\lim\limits_{\Delta t \to 0} \dfrac{\vec q \otimes \Delta \vec q^{\sss B} - \vec q}{\Delta t} \ \nm{\approx} \lim\limits_{\Delta t \to 0}\dfrac{\vec q \otimes \lrsb{\lrsb{1, \vec n^{\sss B} \, \Delta \phi / 2}^T - \vec{q_1}}}{\Delta t} = \vec q \otimes \lim\limits_{\Delta t \to 0}\dfrac{\lrsb{0, \vec n^{\sss B} \, \Delta \phi / 2}^T}{\Delta t}}\label{eq:SO3_time_derivative_q1b}
\end{eqnarray}
Similar expressions based on \nm{\vec r^{\sss N} = \Delta \phi \, \vec n^{\sss N}} can be obtained if left multiplying by the perturbation instead of right multiplying. The \nm{\vec {\dot R}\lrp{t}} and \nm{\vec {\dot q}\lrp{t}} expressions (\ref{eq:SO3_dcm_dot}) and (\ref{eq::SO3_quat_omega_dot}) are then directly obtained when defining the \emph{body angular velocity} \nm{\wNBB} as the time derivative of the rotation vector \nm{\vec r^{\sss B} = \vec n^{\sss B} \, \phi} when viewed in local or body frame \nm{\FB}, and the \emph{spatial angular velocity} \nm{\wNBN} as the time derivative of the rotation vector \nm{\vec r^{\sss N} = \vec n^{\sss N} \, \phi} when viewed in global or spatial frame \nm{\FN}:
\begin{eqnarray}
\nm{\wNBB\lrp{t}} & = & \nm{\Delta \vec{\dot r}^{\sss B}\lrp{t} = \lim\limits_{\Deltat \to 0} \frac{\Delta \vec r^{\sss B}}{\Deltat} = \lim\limits_{\Deltat \to 0} \frac{\vec n^{\sss B} \, \Delta \phi}{\Deltat}}\label{eq:SO3_time_derivative_wNBB} \\
\nm{\wNBN\lrp{t}} & = & \nm{\Delta \vec{\dot r}^{\sss N}\lrp{t} = \lim\limits_{\Deltat \to 0} \frac{\Delta \vec r^{\sss N}}{\Deltat} = \lim\limits_{\Deltat \to 0} \frac{\vec n^{\sss N} \, \Delta \phi}{\Deltat}}\label{eq:SO3_time_derivative_wNBN}
\end{eqnarray}
Note that the rotation of the angular velocity (relationship between \nm{\wNBN} and \nm{\wNBB}) is not given by the rotation action \nm{\vec g_{\mathcal R*}} (\ref{eq:Rotate_vector_action}) but by the adjoint map \nm{\vec{Ad}_{\mathcal R}} described in section \ref{subsec:RigidBody_rotation_adjoint}, although in the case of the \nm{\mathbb{SO}(3)} rotation group, both maps coincide.
\subsection{Rotational Motion Point Velocity}\label{subsec:RigidBody_rotation_velocity}
There exists a direct relationship between the velocity of a point belonging to a rigid body and the elements of its tangent space, this is, the angular velocity in \nm{\mathfrak{so}(3)} in the case of rotational motion. This relationship is independent of the \nm{\mathbb{SO}(3)} representation, although the rotation matrix is employed in the expressions below. As discussed in section \ref{subsec:RigidBody_bases}, the rotation actions have the same form for points as for vectors (\ref{eq:SO3_equivalence}). Hence, if \nm{\pB} are the fixed coordinates of a point belonging to the \nm{\FB} rigid body, the point spatial coordinates \nm{\pN} can be obtained by means of (\ref{eq:SO3_dcm_transform}):
\neweq{\pN\lrp{t} = \vec g_{\mathcal R_{NB}(t)}\lrp{\pB} = \RNB\lrp{t} \; \pB}{eq:SO3_velocity1}
The velocity of a point is the time derivative of its spatial or global coordinates. As \nm{\vec p} is fixed to \nm{F_{\sss B}}, its time derivative is zero \nm{\lrp{\vec {\dot p}^{\sss B} = \vec 0}}, so its velocity viewed in the spatial frame responds to:
\neweq{\vec v_{\sss p}^{\sss N}\lrp{t} = \vec {\dot p}^{\sss N}\lrp{t} = \vec {\dot R}_{\sss NB}\lrp{t} \; \pB}{eq:SO3_velocity2}
Although \nm{\RNBdot} maps the point body coordinates to its spatial velocity per (\ref{eq:SO3_velocity2}), its high dimension makes it inefficient \cite{Murray1994}. By making use of the spatial and body instantaneous angular velocities (\nm{\wNBNskew, \, \wNBBskew}) introduced in (\ref{eq:SO3_dcm_dot}), the velocity of a point \nm{\pB} viewed in \nm{\FN} can be obtained as follows:
\begin{eqnarray}
\nm{\vec v_{\sss p}^{\sss N}\lrp{t}} & = & \nm{\wNBNskew\lrp{t} \; \RNB\lrp{t} \; \pB = \wNBNskew\lrp{t} \; \pN\lrp{t}}\label{eq:SO3_velocity_n} \\
\nm{\vec v_{\sss p}^{\sss N}\lrp{t}} & = & \nm{\RNB\lrp{t} \; \wNBBskew\lrp{t} \; \pB}\label{eq:SO3_velocity_n_bis}
\end{eqnarray}
The velocity of \nm{\pB} viewed in \nm{\FB} can then be obtained by means of the vector action map:
\neweq{\vec v_{\sss p}^{\sss B}\lrp{t} = \vec g_{\mathcal R_{NB(t)}*}^{-1} \big(\vec v_{\sss p}^{\sss N}\lrp{t}\big) = \RNBtrans\lrp{t} \; \vec v_{\sss p}^{\sss N}\lrp{t} = \wNBBskew\lrp{t} \; \pB} {eq:SO3_velocity_b}
The point velocity is hence the result of the cross product between the angular velocity and the point coordinates (\ref{eq:SO3_velocity_n_bis}, \ref{eq:SO3_velocity_b}). Similar expressions are obtained if employing the unit quaternion (\nm{\vec v_{\sss p}^{\sss N}\lrp{t} = \vec \omega_{\sss NB}^{{\sss N}\wedge} \otimes \pN\lrp{t}}, \nm{\vec v_{\sss p}^{\sss B}\lrp{t} = \vec \omega_{\sss NB}^{{\sss B}\wedge} \otimes \pB}).
\subsection{Rotational Motion Adjoint}\label{subsec:RigidBody_rotation_adjoint}
The \emph{adjoint map} of a Lie group is defined in section \ref{subsubsec:algebra_lie_adjoint} as an action of the Lie group on its own Lie algebra that converts between the local tangent space and that at the identity. In the case of rotational motion, both the rotation vector and the angular velocity belong to the tangent space, so \nm{\lrb{\vec{Ad}\lrp{}: \mathbb{SO}(3) \times \mathfrak{so}(3) \rightarrow \mathfrak{so}(3) \ | \ \vec{Ad}_{\mathcal R}\lrp{\vec r^{\wedge}} = \mathcal R \circ \vec r^{\wedge} \circ \mathcal{R}^{-1}, \ \vec{Ad}_{\mathcal R}\lrp{\vec \omega^{\wedge}} = \mathcal R \circ \vec \omega^{\wedge} \circ \mathcal{R}^{-1}}}. This is equivalent to \nm{\vec q \otimes \vec \omega^{\wedge} \otimes \qast} for unit quaternions or \nm{\vec R \; \omegaskew \; \vec R^T} for rotation matrices, which represents the congruency transformation\footnote{Two square matrices \nm{\vec A} and \nm{\vec B} are called congruent if \nm{\vec B = {\vec P}^T \; \vec A \; \vec P} for some invertible square matrix \nm{\vec P}.} between the spatial and body angular velocities \nm{\wNBNskew} and \nm{\wNBBskew}:
\neweq{\wNBNskew = \RNB \; \wNBBskew \; \RNBtrans}{eq:SO3_dcm_velocity5}
The application of the vee operator results in the adjoint matrix coinciding with the rotation matrix itself \nm{\lrb{\vec{Ad}_{\mathcal R} \, \vec \omega = \vec R \, \vec \omega}}\footnote{The adjoint matrix is generic and applicable to all \nm{\mathbb{SO}(3)} representations.}, implying that elements of the \nm{\mathbb{SO}(3)} tangent space (both rotation vectors and angular velocities) can be transformed by means of the rotation action as any other free vector. Note that this result only applies to rotational motion, as for example the vector action and adjoint matrix of \nm{\mathbb{SE}(3)} discussed in section \ref{sec:Motion} do not coincide:
\neweq{\wNBN = \vec{Ad}_{\mathcal R_{NB}} \; \wNBB = \RNB \; \wNBB}{eq:SO3_dcm_velocity6}
A similar process leads to the inverse adjoint matrix (\nm{\vec{Ad}_{\mathcal R}^{-1} \, \vec \omega = \vec{Ad}_{\mathcal R^{-1}} \, \vec \omega = \vec R^T \, \vec \omega}):
\neweq{\wNBB = \vec{Ad}_{\mathcal R_{NB}}^{-1} \; \wNBN = \RNB^T \; \wNBN}{eq:SO3_dcm_velocity7}
\subsection{Rotational Motion Uncertainty and Covariance}\label{subsec:RigidBody_rotation_covariance}
Following the analysis of uncertainty on Lie groups presented in section \ref{subsubsec:algebra_lie_covariance}, the definitions of local and global autocovariances for \nm{\mathbb{SO}(3)} elements around a nominal or expected rotation \nm{E\lrsb{\mathcal R} = \vec \mu_{\mathcal R} \in \mathbb{SO}(3)} are the following:
\begin{eqnarray}
\nm{\vec C_{\mathcal R \mathcal R}^{\sss B}} & = & \nm{E\lrsb{\Delta \vec r^{\sss B} \, \Delta \vec r^{{\sss B}T}} = E\lrsb{\lrp{\mathcal R \ominus \vec \mu_{\mathcal R}}\lrp{\mathcal R \ominus \vec \mu_{\mathcal R}}^T} \ \ \in \mathbb{R}^{3x3}}\label{eq:SO3_covariance_right_def} \\
\nm{\vec C_{\mathcal R \mathcal R}^{\sss N}} & = & \nm{E\lrsb{\Delta \vec r^{\sss N} \, \Delta \vec r^{{\sss N}T}} = E\lrsb{\lrp{\mathcal R \boxminus \vec \mu_{\mathcal R}}\lrp{\mathcal R \boxminus \vec \mu_{\mathcal R}}^T} \ \ \in \mathbb{R}^{3x3}}\label{eq:SO3_covariance_left_def}
\end{eqnarray}
Note that although the notation refers to the covariance of the rotation manifold \nm{\mathcal R \in \mathbb{SO}(3)}, the definition in fact refers to the covariance of the rotation vectors \nm{\Delta \vec r^{\sss B}} or \nm{\Delta \vec r^{\sss N}} located in the tangent space, with its dimension (3) matching the number of degrees of freedom of the \nm{\mathbb{SO}(3)} manifold. The relationship between the local and global autocovariances responds to:
\neweq{\vec C_{\mathcal R \mathcal R}^{\sss N} = \vec{Ad}_{\mathcal R_{NB}} \ \vec C_{\mathcal R \mathcal R}^{\sss B} \ \vec{Ad}_{\mathcal R_{NB}}^T = \RNB \, \vec C_{\mathcal R \mathcal R}^{\sss B} \, \RNB^T} {eq:SO3_covariance_left_relationship}
Given a function \nm{\lrb{f: \mathcal{R} \rightarrow \mathcal {S} \ | \ \mathcal {S} = f\lrp{\mathcal {R}} \in \mathbb{SO}(3), \, \forall \mathcal {R} \in \mathbb{SO}(3)}} between two rotations, the covariances are propagated as follows:
\begin{eqnarray}
\nm{\vec C_{\mathcal S \mathcal S}^{\sss B}} & = & \nm{\vec J_{\ds{\oplus \; \mathcal R}}^{\ds{\oplus \; f\lrp{\mathcal R}}} \ \vec C_{\mathcal R \mathcal R}^{\sss B} \ \vec J_{\ds{\oplus \; \mathcal R}}^{{\ds{\oplus \; f\lrp{\mathcal R}}},T} \ \ \ \ \ \ \ \in \mathbb{R}^{3x3}} \label{eq:SO3_covariance_right_propagation} \\
\nm{\vec C_{\mathcal S \mathcal S}^{\sss N}} & = & \nm{\vec J_{\ds{\boxplus \; \mathcal R}}^{\ds{\boxplus \; f\lrp{\mathcal R}}} \ \vec C_{\mathcal R \mathcal R}^{\sss N} \ \vec J_{\ds{\boxplus \; \mathcal R}}^{{\ds{\boxplus \; f\lrp{\mathcal R}}},T} \ \ \ \ \ \ \ \in \mathbb{R}^{3x3}} \label{eq:SO3_covariance_left_propagation}
\end{eqnarray}
\subsection{Rotational Motion Jacobians}\label{subsec:RigidBody_rotation_calculus_jacobians}
Lie group jacobians are introduced in section \ref{subsec:algebra_lie_jacobians} based on the right and left Lie group derivatives of section \ref{subsubsec:algebra_lie_derivatives}, and in this section are customized for the \nm{\mathbb{SO}(3)} case, with table \ref{tab:RigidBody_rotation_jacobians} representing the particularization of table \ref{tab:algebra_lie_jacobians} to the case of rigid body rotations. The various jacobians listed in table \ref{tab:RigidBody_rotation_jacobians} have been obtained by means of the chain rule, the expressions already introduced in this article, and those of section \ref{subsec:algebra_lie}. Note that although in many cases the results include the rotation matrix, all jacobians are generic and do not depend on the specific \nm{\mathbb{SO}(3)} parameterization.
In addition to the adjoint matrix, two other jacobians are of particular importance as they appear repeatedly in table \ref{tab:RigidBody_rotation_jacobians}. These are the right and left jacobians of the capitalized exponential function, also known as simply the \emph{right jacobian} \nm{J_R\lrp{\vec r}} and the \emph{left jacobian} \nm{J_L\lrp{\vec r}}, and they evaluate the variation of the \nm{\mathfrak{so}(3)} tangent space provided by the output of the \nm{Exp\lrp{\vec r}} map (locally for \nm{J_R} and globally for \nm{J_L}) while moving along the \nm{\mathbb{SO}\lrp{3}} manifold with respect to the (Euclidean) variations within the original tangent space provided by \nm{\vec r}. Their closed forms as well as those of their inverses are included in table \ref{tab:RigidBody_rotation_jacobians}, and have been obtained from \cite{Chirikjian2012}; they verify that \nm{\vec J_L\lrp{\vec r} = \vec J_R^T\lrp{\vec r}}, and \nm{\vec J_L^{-1}\lrp{\vec r} = \vec J_R^{-T}\lrp{\vec r}}.
It is also worth noting the special importance of the \nm{\vec J_{\ds{+ \; \vec r}}^{\ds{+ \; g_{Exp\lrp{\vec r}*}(\vec v)}}} jacobian present at the bottom of table \ref{tab:RigidBody_rotation_jacobians}, which represents the derivative of a rotated vector with respect to perturbations in the Euclidean tangent space (not on the curved manifold) that generates the rotation, as it enables tangent space optimization by calculus methods designed exclusively for Euclidean spaces.
\renewcommand{\arraystretch}{1.5}
\begin{center}
\begin{tabular}{lcccll}
\hline
\textbf{Jacobian} & & \textbf{Table \ref{tab:algebra_lie_jacobians}} & & \multicolumn{1}{c}{\textbf{Expression}} & \textbf{Size} \\
\hline
\nm{\vec J_{\ds{\oplus \; \mathcal R}}^{\ds{\oplus \; \mathcal R}^{-1}}} & = & \nm{- \vec{Ad}_{\mathcal R}} & = & \nm{- \vec R} & \nm{\in \mathbb R^{3x3}} \\
\nm{\vec J_{\ds{\boxplus \; \mathcal R}}^{\ds{\boxplus \; \mathcal R}^{-1}}} & = & \nm{- \vec{Ad}_{\mathcal R}^{-1}} & = & \nm{- \vec R^T} & \nm{\in \mathbb R^{3x3}} \\
\nm{\vec J_{\ds{\oplus \; \mathcal R}}^{\ds{\oplus \; \mathcal R \circ \mathcal S}}} & = & \nm{\vec{Ad}_{\mathcal S}^{-1}} & = & \nm{\vec R_{\mathcal S}^T} & \nm{\in \mathbb R^{3x3}} \\
\nm{\vec J_{\ds{\boxplus \; \mathcal R}}^{\ds{\boxplus \; \mathcal R \circ \mathcal S}}} & = & \nm{\vec I} & = & \nm{\vec{I}_{3x3}} & \nm{\in \mathbb R^{3x3}} \\
\nm{\vec J_{\ds{\oplus \; \mathcal S}}^{\ds{\oplus \; \mathcal R \circ \mathcal S}}} & = & \nm{\vec I} & = & \nm{\vec{I}_{3x3}} & \nm{\in \mathbb R^{3x3}} \\
\nm{\vec J_{\ds{\boxplus \; \mathcal S}}^{\ds{\boxplus \; \mathcal R \circ \mathcal S}}} & = & \nm{\vec{Ad}_{\mathcal R}} & = & \nm{\vec R_{\mathcal R}} & \nm{\in \mathbb R^{3x3}} \\
\nm{\vec J_{\ds{+ \; \vec r}}^{\ds{\oplus \; Exp\lrp{\vec r}}}} & = & \nm{ J_R\lrp{\vec r}} & = & \nm{\vec I_3 - \frac{1 - \cos \|\vec r\|}{\|\vec r\|^2} \ \rskew + \frac{\|\vec r\| - \sin \|\vec r\|}{\|\vec r\|^3} \ \rskew^2} & \nm{\in \mathbb R^{3x3}} \\
\nm{J_R^{-1}\lrp{\vec r}} & & & = & \nm{\vec I_3 + \frac{\rskew}{2} + \lrp{\frac{1}{\|\vec r\|^2} - \frac{1 + \cos \|\vec r\|}{2 \, \|\vec r\| \, \sin \|\vec r\|}} \, \rskew^2} &\nm{\in \mathbb R^{3x3}} \\
\nm{\vec J_{\ds{+ \; \vec r}}^{\ds{\boxplus \; Exp\lrp{\vec r}}}} & = & \nm{J_L\lrp{\vec r}} & = & \nm{\vec I_3 + \frac{1 - \cos \|\vec r\|}{\|\vec r\|^2} \ \rskew + \frac{\|\vec r\| - \sin \|\vec r\|}{\|\vec r\|^3} \ \rskew^2} & \nm{\in \mathbb R^{3x3}} \\
\nm{J_L^{-1}\lrp{\vec r}} & & & = & \nm{\vec I_3 - \frac{\rskew}{2} + \lrp{\frac{1}{\|\vec r\|^2} - \frac{1 + \cos \|\vec r\|}{2 \, \|\vec r\| \, \sin \|\vec r\|}} \, \rskew^2} & \nm{\in \mathbb R^{3x3}} \\
\nm{\vec J_{\ds{\oplus \; \mathcal R}}^{\ds{+ \; Log\lrp{\mathcal R}}}} & = & \nm{J_R^{-1}\big(Log\lrp{\mathcal R}\big)} & & & \nm{\in \mathbb R^{3x3}} \\
\nm{\vec J_{\ds{\boxplus \; \mathcal R}}^{\ds{+ \; Log\lrp{\mathcal R}}}} & = & \nm{J_L^{-1}\big(Log\lrp{\mathcal R}\big)} & & & \nm{\in \mathbb R^{3x3}} \\
\nm{\vec J_{\ds{\oplus \; \mathcal R}}^{\ds{\oplus \; \mathcal R \oplus \vec r}}} & = & \nm{\vec{Ad}_{Exp\lrp{\vec r}}^{-1}} & = & \nm{\vec R^T\lrp{\vec r}} & \nm{\in \mathbb R^{3x3}} \\
\nm{\vec J_{\ds{\boxplus \; \mathcal R}}^{\ds{\boxplus \; \vec r \boxplus \mathcal R}}} & = & \nm{\vec{Ad}_{Exp\lrp{\vec r}}} & = & \nm{\vec R\lrp{\vec r}} & \nm{\in \mathbb R^{3x3}} \\
\nm{\vec J_{\ds{+ \; \vec r}}^{\ds{\oplus \; \mathcal R \oplus \vec r}}} & = & \nm{J_R\lrp{\vec r}} & & & \nm{\in \mathbb R^{3x3}} \\
\nm{\vec J_{\ds{+ \; \vec r}}^{\ds{\boxplus \; \vec r \boxplus \mathcal R}}} & = & \nm{J_L\lrp{\vec r}} & & & \nm{\in \mathbb R^{3x3}} \\
\nm{\vec J_{\ds{\oplus \; \mathcal R}}^{\ds{+ \; \mathcal S \ominus \mathcal R}}} & = & \nm{- J_L^{-1}\lrp{\mathcal S \ominus \mathcal R}} & & & \nm{\in \mathbb R^{3x3}} \\
\nm{\vec J_{\ds{\boxplus \; \mathcal R}}^{\ds{+ \; \mathcal S \boxminus \mathcal R}}} & = & \nm{- J_R^{-1}\lrp{\mathcal S \boxminus \mathcal R}} & & & \nm{\in \mathbb R^{3x3}} \\
\nm{\vec J_{\ds{\oplus \; \mathcal S}}^{\ds{+ \; \mathcal S \ominus \mathcal R}}} & = & \nm{J_R^{-1}\lrp{\mathcal S \ominus \mathcal R}} & & & \nm{\in \mathbb R^{3x3}} \\
\nm{\vec J_{\ds{\boxplus \; \mathcal S}}^{\ds{+ \; \mathcal S \boxminus \mathcal R}}} & = & \nm{J_L^{-1}\lrp{\mathcal S \boxminus \mathcal R}} & & & \nm{\in \mathbb R^{3x3}} \\
\nm{\vec J_{\ds{\oplus \; \mathcal R}}^{\ds{+ \; g_{\mathcal R*}(\vec v)}}} & & & = & \nm{- \vec R \, \widehat{\vec v}} & \nm{\in \mathbb R^{3x3}} \\
\nm{\vec J_{\ds{\boxplus \; \mathcal R}}^{\ds{+ \; g_{\mathcal R*}(\vec v)}}} & & & = & \nm{- \lrp{\vec R \, \vec v}^\wedge} & \nm{\in \mathbb R^{3x3}} \\
\nm{\vec J_{\ds{+ \; \vec v}}^{\ds{+ \; g_{\mathcal R*}(\vec v)}}} & & & = & \nm{\vec R} & \nm{\in \mathbb R^{3x3}} \\
\hline
\end{tabular}
\end{center}
\begin{center}
\begin{tabular}{lcccll}
\hline
\textbf{Jacobian} & & \textbf{Table \ref{tab:algebra_lie_jacobians}} & & \multicolumn{1}{c}{\textbf{Expression}} & \textbf{Size} \\
\hline
\nm{\vec J_{\ds{\oplus \; \mathcal R}}^{\ds{+ \; g_{\mathcal R*}^{-1}(\vec v)}}} & & & = & \nm{\lrp{\vec R^T \, \vec v}^\wedge} & \nm{\in \mathbb R^{3x3}} \\
\nm{\vec J_{\ds{\boxplus \; \mathcal R}}^{\ds{+ \; g_{\mathcal R*}^{-1}(\vec v)}}} & & & = & \nm{\vec R^T \, \vec v} & \nm{\in \mathbb R^{3x3}} \\
\nm{\vec J_{\ds{+ \; \vec v}}^{\ds{+ \; g_{\mathcal R*}^{-1}(\vec v)}}} & & & = & \nm{\vec R^T} & \nm{\in \mathbb R^{3x3}} \\
\nm{\vec J_{\ds{\oplus \; \mathcal R}}^{\ds{+ \; \vec{Ad}_{\mathcal R}(\vec \omega)}}} & & & = & \nm{- \vec R \, \widehat{\vec \omega}} & \nm{\in \mathbb R^{3x3}} \\
\nm{\vec J_{\ds{\boxplus \; \mathcal R}}^{\ds{+ \; \vec{Ad}_{\mathcal R}(\vec \omega)}}} & & & = & \nm{- \lrp{\vec R \, \vec \omega}^\wedge} & \nm{\in \mathbb R^{3x3}} \\
\nm{\vec J_{\ds{+ \; \vec \omega}}^{\ds{+ \; \vec{Ad}_{\mathcal R}(\vec \omega)}}} & = & \nm{\vec{Ad}_{\mathcal R}} & = & \nm{\vec R} & \nm{\in \mathbb R^{3x3}} \\
\nm{\vec J_{\ds{\oplus \; \mathcal R}}^{\ds{+ \; \vec{Ad}_{\mathcal R}^{-1}(\vec \omega)}}} & & & = & \nm{\lrp{\vec R^T \, \vec \omega}^\wedge} & \nm{\in \mathbb R^{3x3}} \\
\nm{\vec J_{\ds{\boxplus \; \mathcal R}}^{\ds{+ \; \vec{Ad}_{\mathcal R}^{-1}(\vec \omega)}}} & & & = & \nm{\vec R^T \, \vec \omega} & \nm{\in \mathbb R^{3x3}} \\
\nm{\vec J_{\ds{+ \; \vec \omega}}^{\ds{+ \; \vec{Ad}_{\mathcal R}^{-1}(\vec \omega)}}} & = & \nm{\vec{Ad}_{\mathcal R}^{-1}} & = & \nm{\vec R^T} & \nm{\in \mathbb R^{3x3}} \\
\nm{\vec J_{\ds{+ \; \vec r}}^{\ds{+ \; g_{Exp\lrp{\vec r}*}(\vec v)}}} & & & = & \nm{- \vec R\lrp{\vec r} \, \widehat{\vec v} \, J_R\lrp{\vec r}} = \nm{- \big(\vec R\lrp{\vec r} \, \vec v\big)^\wedge \, J_L\lrp{\vec r}} & \nm{\in \mathbb R^{3x3}} \\
\nm{\vec J_{\ds{+ \; \vec r}}^{\ds{+ \; g_{Exp\lrp{\vec r}*}^{-1}(\vec v)}}} & & & = & \nm{- \vec R^T\lrp{\vec r} \, \widehat{\vec v} \, J_L\lrp{\vec r}} = \nm{\big(\vec R^T\lrp{\vec r} \, \vec v\big)^\wedge \, J_R\lrp{\vec r}} & \nm{\in \mathbb R^{3x3}} \\
\hline
\end{tabular}
\end{center}
\captionof{table}{Rotational motion jacobians} \label{tab:RigidBody_rotation_jacobians}
\renewcommand{\arraystretch}{1.0}
\subsection{Rotational Motion Discrete Integration}\label{subsec:RigidBody_rotation_integration}
The discrete integration with time of an element of a Lie group based on its Lie algebra is discussed in detail in section \ref{subsec:algebra_integration}, which includes expressions for the Euler, Heun and Runge-Kutta methods. In the case of rotational motion, the state vector includes the rotation element \nm{\mathcal R \in \mathbb{SO}(3)} and its angular velocity \nm{\vec \omega \in \mathbb{R}^3} contained in the tangent space, viewed either in the local (\nm{\wNBB}) or global (\nm{\wNBN}) frames. The Euler method expressions equivalent to (\ref{eq:algebra_integration_comp_X_euler}) and (\ref{eq:algebra_integration_comp_X_euler_left}) are shown below. Expressions for other integration schemes can easily be derived from those in section \ref{subsec:algebra_integration}:
\begin{eqnarray}
\nm{\mathcal R_{k+1}} & \nm{\approx} & \nm{\mathcal R_k \oplus \lrsb{\Delta t \ \vec \omega_{{\sss NB}k}^{\sss B}} = \mathcal R_k \circ Exp\lrp{\Delta t \ \vec \omega_{{\sss NB}k}^{\sss B}}} \label{eq:SO3_integration_comp_X_euler} \\
\nm{\mathcal R_{k+1}} & \nm{\approx} & \nm{\lrsb{\Delta t \ \vec \omega_{{\sss NB}k}^{\sss N}} \boxplus \mathcal R_k = Exp\lrp{\Delta t \ \vec \omega_{{\sss NB}k}^{\sss N}} \circ \mathcal R_k} \label{eq:SO3_integration_comp_X_euler_left}
\end{eqnarray}
\subsection{Rotational Motion Gauss-Newton Optimization}\label{subsec:RigidBody_rotation_gauss_newton}
The minimization by means of the Gauss-Newton iterative method of the Euclidean norm of a non linear function whose input is a Lie group element is presented in section \ref{subsec:algebra_gradient_descent}. In the case of rotational motion, the resulting expressions for perturbations \nm{\Delta \vec r_{\sss NB}^{\sss N} \in \mathfrak{so}(3)} to an input rotation \nm{\mathcal R \in \mathbb{SO}(3)} viewed in the global frame \nm{\FN} are shown in (\ref{eq:SO3_gauss_newton_iterative_left}) and (\ref{eq:SO3_gauss_newton_solution_left}), which are equivalent to the generic (\ref{eq:algebra_gradient_descent_iterative_lie_left}) and (\ref{eq:algebra_gradient_descent_solution_lie_left}). Refer to section \ref{subsec:algebra_gradient_descent} for the meaning of the function jacobian \nm{\vec J} and to section \ref{subsec:RigidBody_rotation_calculus_jacobians} for that of the left jacobian \nm{\vec J_L}.
\begin{eqnarray}
\nm{\mathcal R_{k+1}} & \nm{\longleftarrow} & \nm{\Delta \vec r_{{\sss NB}k}^{\sss N} \boxplus \mathcal R_k = \Delta \vec r_{{\sss NB}k}^{\sss N} \circ Exp\lrp{\vec r_{{\sss NB}k}}} \label{eq:SO3_gauss_newton_iterative_left} \\
\nm{\Delta \vec r_{{\sss NB}k}^{\sss N}} & = & \nm{- \lrsb{\vec J_{Lk}^{-T} \, \vec J_k^T \, \vec J_k \, \vec J_{Lk}^{-1}}^{-1} \, \vec J_{Lk}^{-T} \, \vec J_k^T \, \vec{\mathcal E}_k} \label{eq:SO3_gauss_newton_solution_left}
\end{eqnarray}
If the perturbation is viewed in the local frame \nm{\FB}, (\ref{eq:algebra_gradient_descent_iterative_lie_right}) and (\ref{eq:algebra_gradient_descent_solution_lie_right}) are customized as follows, making use of the right jacobian \nm{\vec J_R} defined in section \ref{subsec:RigidBody_rotation_calculus_jacobians}:
\begin{eqnarray}
\nm{\mathcal R_{k+1}} & \nm{\longleftarrow} & \nm{\mathcal R_k \oplus \Delta \vec r_{{\sss NB}k}^{\sss B} = Exp\lrp{\vec r_{{\sss NB}k}} \circ \Delta \vec r_{{\sss NB}k}^{\sss B}} \label{eq:SO3_gauss_newton_iterative_right} \\
\nm{\Delta \vec r_{{\sss NB}k}^{\sss B}} & = & \nm{- \lrsb{\vec J_{Rk}^{-T} \, \vec J_k^T \, \vec J_k \, \vec J_{Rk}^{-1}}^{-1} \, \vec J_{Rk}^{-T} \, \vec J_k^T \, \vec{\mathcal E}_k} \label{eq:SO3_gauss_newton_solution_right}
\end{eqnarray}
\subsection{Rotational Motion State Estimation}\label{subsec:RigidBody_rotation_SS}
The adaptation of the \hypertt{EKF} state estimation introduced in section \ref{subsec:SS} to the case in which Lie group elements and their velocities are present is discussed in detail in section \ref{subsec:algebra_SS}. For rotational motion with local perturbations, it is necessary to replace \nm{\mathcal X \in \mathcal G} by \nm{\mathcal R \in \mathbb{SO}(3)}, \nm{\Delta \vec \tau^{\mathcal X} \in T_{\mathcal X}\mathcal G} by \nm{\Delta \vec r^{\sss B} \in \ \mathfrak{so}(3)}, \nm{\vec v^{\mathcal X} \in \mathbb{R}^m} by \nm{\vec \omega^{\sss B} \in \mathbb{R}^3}, \nm{\vec C_{{\mathcal {XX}}}^{\mathcal X} \in \mathbb{R}^{mxm}} by \nm{\vec C_{{\mathcal {RR}}}^{\sss B} \in \mathbb{R}^{3x3}}, and \nm{\vec J_{\ds{\oplus \; \mathcal X}}^{\ds{\oplus \; \mathcal X \oplus \vec \tau}}} by \nm{\vec J_{\ds{\oplus \; \mathcal R}}^{\ds{\oplus \; \mathcal R \oplus \vec r}}}. The particularizations for global perturbations are similar.
\subsection{Applications of the Various Rotation Representations}\label{subsec:RigidBody_rotation_applications}
This section discusses five different representations of the rotation or special orthogonal group \nm{\mathbb{SO}(3)}: the rotation matrix, the rotation vector, the unit quaternion, the half rotation vector, and the Euler angles. Although in theory all of them can be employed for each of the purposes described in this article, and the required expressions derived, each representation has its own advantages and disadvantages, being suited for certain purposes but not recommended for others.
\begin{itemize}
\item The rotation matrix \nm{\vec R} is the most natural representation, possesses an easy to obtain inverse, and linear expressions for composition and rotation. It provides a clear connection with the tangent space, together with the exponential and logarithmic maps, \hypertt{SLERP}, and plus and minus operators, which are not complex. Its main drawbacks are the storage costs associated with its high dimension (9) and the expense involved in maintaining orthogonality if allowed to deviate from the manifold \cite{Shuster1993, Baritzhack1969}. Its high cost precludes its use to track the rotation over its manifold, although most implementations continuously compute it if the adjoint matrix or the jacobian blocks are required.
\item The unit quaternion \nm{\vec q} is the preferred representation to track the rotation over its manifold, even if it is necessary to obtain the rotation matrix for the adjoint and jacobian blocks. Its advantages with respect to the rotation matrix are its small dimension (4) and ease to maintain unitary if allowed to deviate from the manifold. Unit quaternions are the least natural of the rotation representations, being necessary to convert to a different \nm{\mathbb{SO}\lrp{3}} representation for visualization \cite{Shuster1993}. While the inverse and concatenation are linear, the rotation action is bilinear, which presents a disadvantage with the rotation matrix. Unit quaternion expressions are slightly more complex than those of the rotation matrix, and present a slightly less obvious connection with the tangent space because in fact they represent a double covering of \nm{\mathbb{SO}\lrp{3}} instead of \nm{\mathbb{SO}\lrp{3}} itself.
\item The main advantage of the rotation vector \nm{\vec r} is that it belongs to the \nm{\mathfrak{so}(3)} tangent space while simultaneously being an \nm{\mathbb{SO}(3)} representation. It is hence indicated for those uses related with incremental rotation changes by means of the exponential map together with the plus and minus operators (periodically adding the perturbations to the unit quaternion tracking the rotation), such as discrete integration, optimization, and state estimation. The rotation vector norm is the most adequate metric for evaluating the rotated distance (or estimation error) between two rigid bodies. Although it benefits from its straightforward inverse, its geometric appeal, and its small dimension (3+1)\footnote{Strictly speaking the dimension is 3, although any usage requires computing the norm, which is often stored to accelerate the transformations.}, its usage for other applications is discouraged by its complex non linear kinematics, rotation action, and composition \cite{Shuster1993}, which are not shown in this article.
\item The half rotation vector \nm{\vec h} is so similar (half) to the rotation vector that its usage is not recommended in order to avoid confusion. Its only real application as the tangent space of the unit quaternion is in practice solved by dividing the rotation vector by two when necessary.
\item The Euler angles \nm{\vec \phi} have a long history and a clear physical meaning, which makes them the best choice for attitude visualization, and constitute the only representation in which its dimension (3) coincides with that of the manifold. However, they are not recommended for any other usage because of the presence of discontinuities, together with complex and non linear expressions for inversion, composition, and rotation action \cite{Shuster1993}.
\end{itemize}
\section{Motion of Rigid Bodies} \label{sec:Motion}
This section can be considered as a continuation of the analysis of the rotational motion of rigid bodies contained in section \ref{sec:Rotate}, in which its center of rotation \nm{\vec O_{\sss {CR}}} is not stationary but moves in the Euclidean space \nm{\mathbb{E}^3}. It follows a similar scheme, relying on Lie theory concepts discussed in sections \ref{subsec:algebra_lie} and \ref{subsec:algebra_lie_jacobians}. Table \ref{tab:Motion_lie_comparison} provides a comparison between the generic nomenclature employed in section \ref{sec:Algebra} and their rigid body motion equivalents. The different representations discussed in this section are summarized in Table \ref{tab:Motion_summary}.
\begin{center}
\begin{tabular}{lcclcc}
\hline
\textbf{Concept} & \textbf{Lie Theory} & \textbf{Motion} & \textbf{Concept} & \textbf{Lie Theory} & \textbf{Motion} \\
\hline
Lie group & \nm{\mathcal G} & \nm{\mathbb{SE}(3)} & Lie group element & \nm{\mathcal X, \, \mathcal Y} & \nm{\mathcal M, \, \mathcal N} \\
Concatenation & \nm{\circ} & \nm{\circ} & Lie algebra & \nm{\mathfrak{m}} & \nm{\mathfrak{se}(3)} \\
Identity & \nm{\mathcal E} & \nm{\mathcal {I_M}} & Inverse & \nm{\mathcal X^{-1}} & \nm{\mathcal M^{-1}} \\
Velocity & \nm{\vec v} & \nm{\vec \xi} & Tangent element & \nm{\vec \tau} & \nm{\vec \tau} \\
Local frame & \nm{\mathcal X} & B & Global frame & \nm{\mathcal E} & E \\
Point action & \nm{g_{\mathcal X}()} & \nm{\vec g_{\mathcal M}(\vec p)} & Vector action & \nm{g_{\mathcal X}()} & \nm{\vec g_{\mathcal M*}(\vec v)} \\
Adjoint & \nm{\vec{Ad}_{\mathcal X}\lrp{\vec \tau^{\wedge}}} & \nm{\vec{Ad}_{\mathcal M}\lrp{\vec \tau^{\wedge}}} & Adjoint matrix & \nm{\vec{Ad}_{\mathcal X} \, \vec \tau} & \nm{\vec{Ad}_{\mathcal M} \, \vec \tau} \\
\hline
\end{tabular}
\end{center}
\captionof{table}{Comparison between generic Lie elements and those of rigid body motions} \label{tab:Motion_lie_comparison}
This section begins with an introduction to rigid body motion in section \ref{subsec:RigidBody_motion}, followed by a description of the different rigid body motion Lie group representations: the affine representation (section \ref{subsec:RigidBody_motion_affine}), the homogeneous matrix (section \ref{subsec:RigidBody_motion_homogeneous}), the transform vector (section \ref{subsec:RigidBody_motion_transform_vector}), the unit dual quaternion (section \ref{subsec:RigidBody_motion_unit_dual_quaternion}), the half transform vector (section \ref{subsec:RigidBody_motion_halftransform_vector}), and the screw (section \ref{subsec:RigidBody_motion_screw}). Algebraic operations on rigid body motions are introduced in section \ref{subsec:RigidBody_motion_algebra}, such as powers, linear interpolation, and the plus and minus operators. Section \ref{subsec:RigidBody_motion_calculus_derivatives} presents the motion time derivative that leads to the definition of the twist or motion velocity in the tangent space. The velocity of the rigid body points is discussed in section \ref{subsec:RigidBody_motion_velocity}, followed by the adjoint map in section \ref{subsec:RigidBody_motion_adjoint}, which transforms elements of the tangent space while the motion progresses on its manifold, and by an analysis of uncertainty and covariances applied to rigid body motion (section \ref{subsec:RigidBody_motion_covariance}). An extensive analysis of the rigid body motion jacobians is presented in section \ref{subsec:RigidBody_motion_calculus_jacobians}. Sections \ref{subsec:RigidBody_motion_integration}, \ref{subsec:RigidBody_motion_gauss_newton}, and \ref{subsec:RigidBody_motion_SS} apply the discrete integration of Lie groups, the Gauss-Newton optimization of Lie group functions, and the state estimation of Lie groups contained in sections \ref{subsec:algebra_integration}, \ref{subsec:algebra_gradient_descent}, and \ref{subsec:algebra_SS} to the case of rigid body motions. Finally, the advantages and disadvantages of each motion representation are discussed in section \ref{subsec:RigidBody_motion_applications}.
\subsection{Special Euclidean (Lie) Group}\label{subsec:RigidBody_motion}
A rigid body can be represented with a cartesian frame attached to any of its points (the origin), with the basis vectors \nm{\vec e_1}, \nm{\vec e_2}, and \nm{\vec e_3} being simply unit vectors along the main axes. Rigid body motions can be combined and reversed, complying with the algebraic concept of group, but are not endowed with a metric, so they are not part of a metric or Euclidean space (section \ref{subsec:algebra_structures}). They do however comply with the axioms of a Lie group (section \ref{subsec:algebra_lie}), and hence the set of rigid body motions together with the operation of motion concatenation comprises \nm{\langle \mathbb{SE}(3), \circ \rangle}, known as the \emph{special Euclidean group} of \nm{\mathbb{R}^3} \cite{Soatto2001}, where its elements are denoted by \nm{\mathcal M}, the identify motion by \nm{\mathcal {I_M}}, and the inverse by \nm{\mathcal M^{-1}}. The group has two main actions, which are the motion of points \nm{\lrb{\vec g() : \mathbb{SE}^3 \times \mathbb{R}^3 \rightarrow \mathbb{R}^3 \ | \ \vec p \rightarrow \vec g_{\mathcal M}\lrp{\vec p}}} and that of vectors \nm{\lrb{\vec g_*() : \mathbb{SE}^3 \times \mathbb{R}^3 \rightarrow \mathbb{R}^3 \ | \ \vec v \rightarrow \vec g_{\mathcal M*}\lrp{\vec v}}}.
The movements of rigid bodies are introduced in section \ref{subsec:RigidBody_bases} as orthogonal transformations, this is, those that preserve orthogonality and handedness.
\begin{itemize}
\item Norm: \nm{\|\vec g_{\mathcal M*}\lrp{\vec v}\| = \|\vec v\|, \forall \, \vec v \in \mathbb{R}^3}
\item Cross product: \nm{\vec g_{\mathcal M*}\lrp{\vec u} \times \vec g_{\mathcal M*}\lrp{\vec v} = \vec g_{\mathcal M*}\lrp{\vec u \times \vec v}, \forall \, \vec u, \vec v \in \mathbb{R}^3}
\end{itemize}
It is also worth noting the relationship between the motions of vectors and points:
\neweq{\vec g_{\mathcal M*}\lrp{\vec v} = \vec g_{\mathcal M*}\lrp{\vec q - \vec p} = \vec g_{\mathcal M}\lrp{\vec q} - \vec g_{\mathcal M}\lrp{\vec p}}{eq:Motion_maps}
The \nm{\mathbb{SE}(3)} analysis below adopts the convention introduced in section \ref{subsec:algebra_lie}, in which all actions, including concatenation \nm{\lrb{\circ : \mathbb{SE}\lrp{3} \times \mathbb{SE}\lrp{3} \rightarrow \mathbb{SE}\lrp{3}}}, transform elements viewed in the local or body frame \nm{F_{\sss B} = \{\OB, \vec b_1, \vec b_2, \vec b_3\}} into elements viewed in the global or spatial frame \nm{\FE = \{\OECEF, \vec e_1, \vec e_2, \vec e_3\}} \nm{= \{\vec g_{\mathcal M}\lrp{\OB}, \vec g_{\mathcal M*}\lrp{\vec b_1}, \vec g_{\mathcal M*}\lrp{\vec b_2}, \vec g_{\mathcal M*}\lrp{\vec b_3}\}}\footnote{In contrast with the case of rotational motion described in section \ref{sec:Rotate}, the spatial frame is now named E as it usually corresponds to the \hypertt{ECEF} frame. The \hypertt{NED} case does not apply to this case as it shares origin with the body frame.}, which overlap each other before the motion takes place.
\begin{center}
\begin{tabular}{lccc}
\hline
\textbf{Representation} & \textbf{Symbol} & \textbf{Structure} & \textbf{Space} \\
\hline
Affine representation & \nm{(\mathcal R, \, \vec T)} & \nm{\mathbb{SO}(3)} \& free 3-vector & \nm{\mathbb{SE}(3)} \\
Homogeneous matrix & \nm{\vec M} & 4x4 matrix (\ref{eq:SE3_homogeneous_SE3}) & \nm{\mathbb{SE}(3)} \\
Twist & \nm{\vec \xi^\wedge} & 4x4 matrix (\ref{eq:SE3_twist_expression2}) & \nm{\mathfrak{se}(3)} \\
& \nm{\vec \xi = \lrsb{\vec \nu, \, \vec \omega}^T} & free 6-vector & \\
Transform vector & \nm{\vec \tau^\wedge} & 4x4 matrix (\ref{eq:SE3_twist_expression2}) & \nm{\mathbb{SE}(3)} \& \nm{\mathfrak{se}(3)} \\
& \nm{\vec \tau = \vec \xi \, t = \lrsb{\vec s, \, \vec r}^T = \lrsb{ \vec k \, \rho, \, \vec n \, \phi}^T} & free 6-vector & \\
Unit dual quaternion & \nm{\vec \zeta} & unit dual quaternion & \nm{\mathbb{SE}(3)} \\
Half twist & \nm{\vec \Upsilon^\wedge} & pure dual quaternion & \nm{\mathfrak{se}(3)} \\
& \nm{\vec \Upsilon = \vec \xi / 2} & free 6-vector & \\
Half transform vector & \nm{\vec \Psi^\wedge} & pure dual quaternion & \nm{\mathbb{SE}(3)} \& \nm{\mathfrak{se}(3)} \\
& \nm{\vec \Psi = \vec \Upsilon \, t = \vec \xi \, t / 2 = \vec \tau / 2} & free 6-vector & \\
Screw & \nm{\vec S^\wedge = (\phi^\diamond / 2, \, \vec{nm}^\diamond)} & dual number \& dual vector & \nm{\mathbb{SE}(3)} \& \nm{\mathfrak{se}(3)} \\
& \nm{\vec S = \lrsb{\vec n, \, \vec m, \, h, \, \phi}^T} & 8-vector & \\
\hline
\end{tabular}
\end{center}
\captionof{table}{Summary of rigid body motion representations} \label{tab:Motion_summary}
\subsection{Affine Representation}\label{subsec:RigidBody_motion_affine}
The motion of a rigid body can always be divided into a rotation plus a translation, in which the point motion action responds to:
\neweq{\vec g_{\mathcal M}\lrp{\vec p} = \vec g_{\mathcal R}\lrp{\vec p} + \vec T} {eq:SE3_affine_transform}
where \nm{\vec g_{\mathcal R}} is the point rotation action discussed in section \ref{sec:Rotate}, the point \nm{\vec p} is viewed in the local or body frame, and \nm{\vec T} represents the vector going from the origin of the global or spatial frame to that of the body frame, viewed in the global frame. Any \nm{\mathbb{SO}(3)} representation can be employed for the above expression, but the rotation matrix (section \ref{subsec:RigidBody_rotation_dcm}) and the unit quaternion (section \ref{subsec:RigidBody_rotation_rodrigues}) are the most common:
\begin{eqnarray}
\nm{\pE} & = & \nm{\REB \, \pB + \TEBE} \label{eq:SE3_affine_dcm_transform} \\
\nm{\pE} & = & \nm{\qEB \otimes \pB \otimes \qEBast + \TEBE} \label{eq:SE3_affine_quat_transform}
\end{eqnarray}
The set of all possible rigid body motions \nm{\mathbb{SE}\lrp{3} = \lrb{\mathcal M = \lrp{\mathcal R, \ \vec T} | \ \mathcal R \in \mathbb{SO}\lrp{3}, \ \vec T \in \mathbb{R}^3}}, coupled with the motion concatenation defined below, is a valid representation of the special Euclidean group. The inverse motion, as well as the concatenation operation, get slightly more complex because of the affine nature of the point action:
\begin{eqnarray}
\nm{\lrp{\mathcal R, \, \vec T}^{-1}} & = & \nm{\lrp{\mathcal R^{-1}, \, - g_{\mathcal R*}^{-1}\lrp{\vec T}}} \label{eq:SE3_affine_inverse} \\
\nm{\lrp{\mathcal R_{\sss EB}, \, \vec T_{\sss EB}^{\sss E}}} & = & \nm{\lrp{\mathcal R_{\sss EN}, \, \vec T_{\sss EN}^{\sss E}} \circ \lrp{\mathcal R_{\sss NB}, \, \vec T_{\sss NB}^{\sss N}} = \lrp{\mathcal R_{\sss EN} \circ \mathcal R_{\sss NB}, \, \vec g_{\mathcal{R}_{EN}*}\lrp{\vec T_{\sss NB}^{\sss N}} + \vec T_{\sss EN}^{\sss E}}} \label{eq:SE3_affine_concatenation}
\end{eqnarray}
\begin{center}
\begin{tabular}{lcclcc}
\hline
\textbf{Concept} & \nm{\mathbb{SE}^3} & \textbf{Affine} & \textbf{Concept} & \nm{\mathbb{SE}^3} & \textbf{Affine} \\
\hline
Lie group element & \nm{\mathcal M} & \nm{\lrp{\mathcal R, \, \vec T}} & Concatenation & \nm{\circ} & \nm{\circ} \\
Identity & \nm{\mathcal {I_M}} & \nm{\lrp{\mathcal {I_R}, \, \vec 0}} & Inverse & \nm{\mathcal M^{-1}} & \nm{\lrp{\mathcal R, \, \vec T}^{-1}} \\
Point motion & \nm{\vec g_{\mathcal M}(\vec p)} & \nm{\vec g_{\mathcal R}\lrp{\vec p} + \vec T} & Vector motion & \nm{\vec g_{\mathcal M*}(\vec v)} & \nm{\vec g_{\mathcal R*}\lrp{\vec v}} \\
\hline
\end{tabular}
\end{center}
\captionof{table}{Comparison between generic \nm{\mathbb{SE}(3)} and motion affine representation} \label{tab:RigidBody_motion_lie_affine}
As indicated in (\ref{eq:Motion_maps}), the effect of a rigid body motion on a vector requires a different map than that of points because the translational part has the same influence on both the vector initial and final points:
\neweq{\vec g_{\mathcal M*}\lrp{\vec v} = \vec g_{\mathcal M}\lrp{\vec q} - \vec g_{\mathcal M}\lrp{\vec p} = \lrp{\vec g_{\mathcal R}\lrp{\vec q} + \vec T} - \lrp{ \vec g_{\mathcal R}\lrp{\vec p} + \vec T} = \vec g_{\mathcal R}\lrp{\vec q - \vec p} = \vec g_{\mathcal R*}\lrp{\vec v} \neq \vec g_{\mathcal M}\lrp{\vec v}} {eq:SE3_affine_vector}
\subsection{Homogeneous Matrix}\label{subsec:RigidBody_motion_homogeneous}
\emph{Homogeneous coordinates} are introduced with the objective of replacing the affine transformation (\ref{eq:SE3_affine_dcm_transform}) representing the rigid body motion with a linear transformation. Given a point \nm{\vec p = \lrsb{p_1, p_2, p_3}^T \in \mathbb{R}^3}, its homogeneous representation is obtained adding a ``1'' as a fourth coordinate, so that \nm{\pbar = \lrsb{\vec p, 1 }^T = \lrsb{p_1, p_2, p_3, 1 }^T \in \mathbb{R}^4}. In the case of a vector \nm{\vec v = \vec q - \vec p \in \mathbb{R}^3}, its homogeneous coordinates are \nm{\vbar = \lrsb{\vec v, 0 }^T = \qbar - \pbar = \lrsb{v_1, v_2, v_3, 0 }^T \in \mathbb{R}^4}. The affine coordinate transformation (\ref{eq:SE3_affine_dcm_transform}) can then be converted into a linear transformation:
\neweq{\vec g_{\mathcal M}\lrp{\pbar} = \begin{bmatrix} \nm{\vec g_{\mathcal M}\lrp{\vec p}} \\ 1 \end{bmatrix} = \begin{bmatrix} \nm{\vec R} & \nm{\vec T} \\ 0 & 1 \end{bmatrix} \ \begin{bmatrix} \nm{\vec p} \\ 1 \end{bmatrix} = \vec M \; \pbar} {eq:SE3_homogeneous_transform}
where \nm{\vec M \in \mathbb{R}^{4x4}} is the homogeneous representation of \nm{\lrp{\vec R, \, \vec T} \in \mathbb{SE}\lrp{3}}. This enables a natural matrix representation of the special Euclidean group \cite{Soatto2001}:
\neweq{\mathbb{SE}\lrp{3} = \Bigg\{\vec M = \begin{bmatrix} \nm{\vec R} & \nm{\vec T} \\ 0 & 1 \end{bmatrix} \ \Bigg| \ \vec R \in \mathbb{R}^{3x3}, \ \vec T \in \mathbb{R}^3\Bigg\} \subset \mathbb{R}^{4x4}} {eq:SE3_homogeneous_SE3}
which has group structure under matrix multiplication \nm{\{\mathbb{R}^{4x4} \times \mathbb{R}^{4x4} \rightarrow \mathbb{R}^{4x4} \ | \ \Ma \, \Mb \in \mathbb{R}^{4x4}, \forall \ \Ma, \ \Mb \in \mathbb{R}^{4x4}\}} \cite{Pinter1990}. While having dimension sixteen, the special euclidean group \nm{\mathbb{SE}\lrp{3}} defined by means of homogeneous matrices constitutes a six dimensional manifold to euclidean space \nm{\mathbb{E}^6}. Note that in this group the identity element is given by the identity matrix \nm{\lrp{\vec I = \vec I_4}}.
\begin{center}
\begin{tabular}{lcclcc}
\hline
\textbf{Concept} & \nm{\mathbb{SE}^3} & \textbf{Homogeneous} & \textbf{Concept} & \nm{\mathbb{SE}^3} & \textbf{Homogeneous} \\
\hline
Lie group element & \nm{\mathcal M} & \nm{\vec M} & Concatenation & \nm{\circ} & Matrix product \\
Identity & \nm{\mathcal {I_M}} & \nm{\vec I_4} & Inverse & \nm{\mathcal M^{-1}} & \nm{\vec M^{-1}} \\
Point motion & \nm{\vec g_{\mathcal M}(\vec p)} & \nm{\vec M \, \pbar} & Vector motion & \nm{\vec g_{\mathcal M*}(\vec v)} & \nm{\vec M \, \vbar} \\
\hline
\end{tabular}
\end{center}
\captionof{table}{Comparison between generic \nm{\mathbb{SE}(3)} and homogeneous matrix} \label{tab:RigidBody_motion_lie_homogeneous}
The inversion and concatenation of transformations are linear when using homogeneous coordinates \cite{Soatto2001}:
\begin{eqnarray}
\nm{\vec M^{-1}} & \nm{\!\! =} & \nm{\!\! {\begin{bmatrix} \nm{\vec R} & \nm{\vec T} \\ 0 & 1 \end{bmatrix}}^{-1} = \begin{bmatrix} \nm{\vec R^T} & \nm{- \vec R^T \, \vec T} \\ 0 & 1 \end{bmatrix}} \label{eq:SE3_homogeneous_inverse} \\
\nm{\MEB} & \nm{\!\! =} & \nm{\!\! \begin{bmatrix} \nm{\vec R_{\sss EB}} & \nm{\vec T_{\sss EB}^{\sss E}} \\ 0 & 1 \end{bmatrix} = \MEN \, \MNB = \begin{bmatrix} \nm{\vec R_{\sss EN}} & \nm{\vec T_{\sss EN}^{\sss E}} \\ 0 & 1 \end{bmatrix} \!\! \begin{bmatrix} \nm{\vec R_{\sss NB}} & \nm{\vec T_{\sss NB}^{\sss N}} \\ 0 & 1 \end{bmatrix} = \begin{bmatrix} \nm{\vec R_{\sss EN} \vec R_{\sss NB}} & \nm{\vec R_{\sss EN} \vec T_{\sss NB}^{\sss N} + \vec T_{\sss EN}^{\sss E}} \\ 0 & 1 \end{bmatrix}} \label{eq:SE3_homogeneous_concatenation}
\end{eqnarray}
An advantage of the homogeneous representation is that the motion actions on points and vectors share the same expression:
\neweq{\vec g_{\mathcal M*}\lrp{\vbar} = \vec g_{\mathcal M}\lrp{\qbar} - \vec g_{\mathcal M}\lrp{\pbar} = \vec M \ \qbar - \vec M \ \pbar = \vec M \ \lrp{\qbar - \pbar} = \vec M \ \vbar = \vec g_{\mathcal M}\lrp{\vbar}} {eq:SE3_homogeneous_vector_linear}
\subsection{Transform Vector as Tangent Space}\label{subsec:RigidBody_motion_transform_vector}
As discussed in section \ref{subsubsec:algebra_lie_velocities}, the structure of the Lie algebra associated to \nm{\mathbb{SE}(3)} can be obtained by time derivating the Lie group inverse constraint, \nm{\vec M^{-1}\lrp{t} \, \vec M\lrp{t} = \vec M\lrp{t} \, \vec M^{-1}\lrp{t} = \vec I_4}, resulting in the following particularizations of (\ref{eq:algebra_vE}) and (\ref{eq:algebra_vX}):
\begin{eqnarray}
\nm{\xiEBEskew} & = & \nm{\MEBdot \; \MEBinv = - \MEB \; \MEBdotinv} \label{eq:SE3_homogeneous_twist_space} \\
\nm{\xiEBBskew} & = & \nm{\MEBinv \; \MEBdot = - \MEBdotinv \; \MEB} \label{eq:SE3_homogeneous_twist_body}
\end{eqnarray}
The Lie algebra velocity \nm{\vec v^\wedge} of \nm{\mathbb{SE}(3)} is known as the \emph{twist} \nm{\vec \xi^\wedge}, and has the following structure, derived from (\ref{eq:SE3_homogeneous_twist_space}) and (\ref{eq:SE3_homogeneous_twist_body}):
\neweq{\vec \xi^{\wedge}\lrp{t} = \begin{bmatrix} \nm{\omegaskew\lrp{t}} & \nm{\vec \nu\lrp{t}} \\ 0 & 0 \end{bmatrix} \subset \mathbb{R}^{4x4}} {eq:SE3_twist_expression2}
The twist \nm{\vec \xi} represents the motion velocity and is composed by the angular velocity \nm{\vec \omega} defined in section \ref{subsec:RigidBody_rotation_calculus_derivatives} and the \emph{linear velocity} \nm{\vec \nu}, defined in section \ref{subsec:RigidBody_motion_calculus_derivatives}. Inverting (\ref{eq:SE3_homogeneous_twist_space}) and (\ref{eq:SE3_homogeneous_twist_body}) results in the homogeneous matrix time derivative, which is linear:
\neweq{\MEBdot = \xiEBEskew \; \MEB = \MEB \; \xiEBBskew} {eq:SE3_homogeneous_dot}
Notice that if \nm{\vec M\lrp{t_0} = \vec I_4}, then \nm{\vec{\dot M}\lrp{t_0} = \vec \xi^{\wedge} \lrp{t_0}}, and hence the twist matrix \nm{\vec \xi^{\wedge}\lrp{t_0}} provides a first order approximation of the homogeneous matrix around the identity matrix \nm{\vec I_4}:
\neweq{\vec M\lrp{t_0 + \Deltat} \approx \vec I_4 + \vec \xi^{\wedge} \lrp{t_0} \, \Deltat}{eq:SE3_twist_taylor}
The space of matrices with the (\ref{eq:SE3_twist_expression2}) structure, \nm{\mathfrak{se}\lrp{3} = \{\vec \xi^{\wedge} = \begin{bmatrix} \omegaskew & \vec \nu; & \vec 0 & 0 \end{bmatrix} \subset \mathbb{R}^{4x4} \ | \ \omegaskew \in \mathfrak{so}(3), \ \vec \nu \in \mathbb{R}^3\}} is hence the \emph{tangent space} of \nm{\mathbb{SE}\lrp{3}} at the identity \nm{\vec I_4} \cite{Soatto2001}, denoted as \nm{T_{\vec I_4}{\mathcal M}}. With the twist cartesian coordinates defined as \nm{\vec \xi = \lrsb{\vec \nu, \, \vec \omega}^T \in \mathbb{R}^6}, the \emph{hat} \nm{\lrb{\cdot^\wedge: \mathbb{R}^6 \rightarrow \mathfrak{se}(3) \ | \ \vec \xi \rightarrow \vec \xi^\wedge}} and \emph{vee} \nm{\lrb{\cdot^\vee: \mathfrak{se}(3) \rightarrow \mathbb{R}^6 \ | \ \lrp{\vec \xi^\wedge}^\vee \rightarrow \vec \xi}} operators convert the cartesian vector form of the twist into its matrix form, and viceversa.
If \nm{\vec M\lrp{t_0} \neq \vec I_4}, the tangent space needs to be transported right multiplying by \nm{\MEB\lrp{t_0}} (in the case of space twist), or left multiplying for the local based twist:
\begin{eqnarray}
\nm{\MEB\lrp{t_0 + \Deltat}} & \nm{\approx} & \nm{\MEB\lrp{t_0} + \lrsb{\xiEBEskew \lrp{t_0} \, \Deltat} \, \MEB\lrp{t_0} = \lrsb{\vec I_4 + \xiEBEskew \lrp{t_0} \, \Deltat} \, \MEB\lrp{t_0} }\label{eq:SE3_twist_taylor_space} \\
\nm{\MEB\lrp{t_0 + \Deltat}} & \nm{\approx} & \nm{\MEB\lrp{t_0} + \MEB\lrp{t_0} \, \lrsb{\xiEBBskew \lrp{t_0} \, \Deltat} = \MEB\lrp{t_0} \, \lrsb{\vec I_4 + \xiEBBskew \lrp{t_0} \, \Deltat}}\label{eq:SE3_twist_taylor_body}
\end{eqnarray}
Note that the solution to the ordinary differential equation \nm{\vec{\dot x}\lrp{t} = \vec \xi^\wedge \, \vec x\lrp{t}, \ \vec x\lrp{t} \in \mathbb{R}^6}, where \nm{\vec \xi^\wedge} is constant, is \nm{\vec x\lrp{t} = e^{\vec \xi^\wedge t} \, \vec x\lrp{0}}. Based on it, assuming \nm{\vec M\lrp{0} = \vec I_4} as initial condition, and considering for the time being that \nm{\vec \xi^\wedge} is constant,
\neweq{\vec M\lrp{t} = e^{\vec \xi^\wedge t} = \vec I_4 + \vec \xi^\wedge t + \frac{\lrp{\vec \xi^\wedge t}^2}{2!} + \dots + \frac{\lrp{\vec \xi^\wedge t}^n}{n!} + \dots}{eq:SE3_twist_exponential3}
It can be proved by means of (\ref{eq:SO3_rotv_exponential2}) and the matrix exponential properties that (\ref{eq:SE3_twist_exponential3}) is indeed an homogeneous matrix that hence represents the special Euclidean \nm{\mathbb{SE}\lrp{3}} transformations.
\begin{center}
\begin{tabular}{lcc}
\hline
\textbf{Concept} & \textbf{Lie Theory} & \nm{\mathbb{SE}^3} \\
\hline
Tangent space element & \nm{\vec \tau^\wedge} & \nm{\vec \tau^\wedge = \big[\rskew \ \vec s; \ \vec 0 \ 0\big]^T} \\
Velocity element & \nm{\vec v^\wedge} & \nm{\vec \xi^\wedge = \big[\omegaskew \ \vec \nu; \ \vec 0 \ 0\big]^T} \\
Structure & \nm{\wedge} & (\ref{eq:SE3_twist_expression2}) \\
\hline
\end{tabular}
\end{center}
\captionof{table}{Comparison between generic \nm{\mathbb{SE}(3)} and transform vector as tangent space} \label{tab:Motion_lie_trfv}
Remembering that so far \nm{\vec \xi^\wedge} is constant, which is equivalent to both \nm{\omegaskew} and \nm{\vec \nu} within (\ref {eq:SE3_twist_expression2}) also being constant, (\ref{eq:SE3_twist_exponential3}) means that any rigid body motion \nm{\vec M\lrp{t} = e^{\ds{\vec \xi^\wedge t}}} can be realized by maintaining a constant twist \nm{\vec \xi^\wedge} for a given time \nm{t} \cite{Soatto2001}. The vectors \nm{\vec n = \vec \omega \, t / \| \vec \omega \, t \| = \vec r / \|\vec r\|} and \nm{\vec k = \vec \nu \, t / \| \vec \nu \, t \| = \vec s / \|\vec s\|} indicate the twist directions, while \nm{\phi = \|\vec r\|} and \nm{\rho = \|\vec s\|} represent the twist magnitudes, respectively. This enables the definition of the \emph{transform vector} \nm{\vec \tau}, also known as the \emph{exponential coordinates} of the \nm{\mathcal M} motion, as
\neweq{\vec \tau = \vec \xi \, t = \lrsb{\vec \nu \, t, \, \vec \omega \, t}^T = \lrsb{\vec s, \, \vec r}^T = \lrsb{ \vec k \, \rho, \, \vec n \, \phi}^T \in \mathbb{R}^6}{eq:SE3_transform_vector_definition}
Note that the transform vector \nm{\vec \tau} belongs to the tangent space as it is a multiple of the twist \nm{\vec \xi \in \mathfrak{se}\lrp{3}}, and hence tends to coincide with it as time tends to zero. The \emph{exponential map} \nm{\lrb{exp\lrp{} : \mathfrak{se}(3) \rightarrow \mathbb{SE}(3) \ | \ \mathcal M = exp\lrp{\vec \tau^\wedge}}} and its capitalized form \nm{\lrb{Exp\lrp{} : \mathbb{R}^6 \rightarrow \mathbb{SE}(3) \ | \ \mathcal M = Exp\lrp{\vec \tau}}} wrap the transform vector around the special Euclidean group. However, the twist \nm{\vec \xi^\wedge \lrp{t}} in fact is not required to be constant. Given a rigid body motion represented by its homogeneous matrix \nm{\vec M \in \mathbb{SE}\lrp{3}}, it can be proved that there exists a not necessarily unique transform vector \nm{\vec \tau = \lrsb{\vec s, \, \vec r}^T = \lrsb{\vec k \, \rho, \, \vec n \, \phi}^T} such that \nm{\vec M = e^{\vec \tau^\wedge}} \cite{Soatto2001,Murray1994}. The exponential map has the following form \cite{Soatto2001}:
\begin{eqnarray}
\nm{\vec M\lrp{\vec \tau}} & = & \nm{exp\lrp{\vec \tau^\wedge} = \begin{bmatrix} \nm{exp\lrp{\rskew}} & \nm{\dfrac{\lrsb{\vec I_3 - exp\lrp{\rskew}} \, \rskew \, \vec s + \vec r \, {\vec r}^T \, \vec s}{{\|\vec r\|}^2}} \\ 0 & 1 \end{bmatrix} \ \ \ \ \ \ \ \ \ \ \ \ \vec r \neq \vec 0}\label{eq:SE3_twist_exponential4_a} \\
\nm{\vec M\lrp{\vec \tau}} & = & \nm{exp\lrp{\vec \tau^\wedge} = \begin{bmatrix} \nm{\vec I_3} & \nm{\vec s} \\ 0 & 1 \end{bmatrix} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \vec r = \vec 0}\label{eq:SE3_twist_exponential4_b}
\end{eqnarray}
The exponential map described above is thus surjective but not injective, as in general there are infinitely many solutions to the map. The \emph{logarithmic map} \nm{\lrb{log\lrp{} : \mathbb{SE}(3) \rightarrow \mathfrak{se}(3) \ | \ \vec \tau^\wedge = log\lrp{\mathcal M}}} and its capitalized version \nm{\lrb{Log\lrp{} : \mathbb{SE}(3) \rightarrow \mathbb{R}^6 \ | \ \vec \tau = Log\lrp{\mathcal M}}} hence convert rigid body motions into transform vectors \cite{Soatto2001}. The rotation vector \nm{\vec r} is provided by (\ref{eq:SO3_rotv_logarithm}); if \nm{\vec r = \vec 0}, \nm{\vec s} coincides with \nm{\vec T}, while otherwise it is obtained by solving for \nm{\vec s} from the following linear system \cite{Soatto2001}, taken from (\ref{eq:SE3_twist_exponential4_a}):
\neweq{\lrsb{(\vec I_3 - e^{\rskew}) \, \rskew + \vec r \, {\vec r}^T} \, \vec s = \|\vec r\|^2 \, \vec T}{eq:SE3_twist_logarithm}
Unlike the case of the rotation vector (\ref{eq:SO_rotv_inversion}), the transform vector inverse coincides with its negative only if the motion is very small. The different \nm{\mathbb{SE}(3)} actions (concatenation, point motion, vector motion), the inverse, and the relationship between the transform vector derivative with time and the twist, are complex and rarely used.
\subsection{Unit Dual Quaternion}\label{subsec:RigidBody_motion_unit_dual_quaternion}
The dual quaternions with unity norm, known as unit dual quaternions, comprise an additional representation of the special euclidean group \nm{\mathbb{SE}\lrp{3}}, as shown below. Dual quaternions in turn are generalizations of quaternions in the same way as dual numbers are generalization or real ones \cite{Jia2013,Kenwright2012,Valverde2018}. For this reason, it is necessary to first describe the dual numbers and dual vectors in sections \ref{subsubsec:RigidBody_motion_dual_numbers} and \ref{subsubsec:RigidBody_motion_dual_vectors} before discussing the dual quaternions in section \ref{subsubsec:RigidBody_motion_dual_quat} and finally the unit dual quaternions in section \ref{subsubsec:RigidBody_motion_unit_dual_quat}.
\subsubsection{Dual Numbers}\label{subsubsec:RigidBody_motion_dual_numbers}
The set of \emph{dual numbers} \nm{\mathbb{D}} is defined as \nm{\lrb{\mathbb{D} = \mathbb{R} + \mathbb{R} \, \epsilon \ | \ \epsilon^2 = \epsilon \cdot \epsilon = 0}}. Given two dual numbers \nm{d_1^{\diamond} = x_1 + y_1 \, \epsilon \in \mathbb{D}, d_2^{\diamond} = x_2 + y_2 \, \epsilon \in \mathbb{D}, \forall \ x_1, y_1, x_2, y_2 \in \mathbb{R}}, it is possible to define the operations of addition \nm{\lrb{+ : \mathbb{D} \times \mathbb{D} \rightarrow \mathbb{D}}} and multiplication \nm{\lrb{\cdot : \mathbb{D} \times \mathbb{D} \rightarrow \mathbb{D}}} \cite{Jia2013}:
\begin{eqnarray}
\nm{d_1^{\diamond} + d_2^{\diamond}} & = & \nm{\lrp{x_1 + y_1 \, \epsilon} + \lrp{x_2 + y_2 \, \epsilon} = \lrp{x_1 + x_2} + \lrp{y_1 + y_2} \, \epsilon}\label{eq:SE3_dual_numbers_addition} \\
\nm{d_1^{\diamond} \cdot d_2^{\diamond}} & = & \nm{\lrp{x_1 + y_1 \, \epsilon} \cdot \lrp{x_2 + y_2 \, \epsilon} = \lrp{x_1 x_2} + \lrp{x_1 y_2 + y_1 x_2} \, \epsilon} \label{eq:SE3_dual_numbers_multiplication}
\end{eqnarray}
The set of dual numbers \nm{\mathbb{D}} endowed with the operations of addition \nm{+} and multiplication \nm{\cdot} forms a ring, known as the ring of dual numbers \nm{\langle \mathbb{D}, +, \cdot \rangle}, nearly always abbreviated to simply \nm{\mathbb{D}}. The additive identity is \nm{0^{\diamond} = 0 + 0 \, \epsilon} and the inverse \nm{- d^{\diamond} = - x - y \, \epsilon}, while the multiplication identity is \nm{1^{\diamond} = 1 + 0 \, \epsilon} and the inverse \nm{d^{\diamond-1} = 1 / x - y \, \epsilon / x^2}. Note that \nm{\mathbb{D}} is a ring instead of a field as the multiplicative inverse \nm{d^{\diamond-1}} is not defined when \nm{x = 0}. The conjugate of a dual number is obtained by switching the sign of its dual part (\nm{d^{\ast} = x - y \, \epsilon \in \mathbb D}).
The most useful property of dual numbers is the explicit relationship that exists between the value of any function evaluated at a dual number \nm{f\lrp{d^{\diamond}} = f\lrp{x + y \, \epsilon}} and its value when evaluated exclusively at its real part \nm{f\lrp{x}} \cite{Kenwright2012}. The Taylor expansion of \nm{f\lrp{x + y \, \epsilon}} around \nm{x} reads:
\neweq{f\lrp{d^{\diamond}} = f\lrp{x + y \, \epsilon} = f\lrp{x} + \pderpar{f}{d^{\diamond}}\lrp{x} \, \lrp{d^{\diamond} - \!x} + \frac{1}{2!} \, \dfrac{\partial^2{f}}{\partial{d^{\diamond 2}}}\lrp{x} \, \lrp{d^{\diamond} - \!x}^2 + \frac{1}{3!} \, \dfrac{\partial^3{f}}{\partial{d^{\diamond 3}}}\lrp{x} \, \lrp{d^{\diamond} - \!x}^3 + \cdots}{eq:SE3_dual_number_taylor1}
As \nm{\lrp{d^{\diamond} - x}^n = y^n \, \epsilon^n} is zero when \nm{n > 1}, this translates into:
\neweq{f\lrp{d^{\diamond}} = f\lrp{x + y \, \epsilon} = f\lrp{x} + \pderpar{f}{d^{\diamond}}\lrp{x} \, y \, \epsilon}{eq:SE3_dual_number_taylor}
\subsubsection{Dual Vectors}\label{subsubsec:RigidBody_motion_dual_vectors}
Dual vectors in three dimensions are formed by grouping three dual numbers \nm{\{\vec d^{\diamond} = \lrsb{d_1^{\diamond}, \, d_2^{\diamond}, \, d_3^{\diamond}}^T \in \mathbb{D}^3,} \nm{\forall \ d_1^{\diamond}, \, d_2^{\diamond}, \, d_3^{\diamond} \in \mathbb{D}\}}. It is then possible to define, \nm{\forall \ d^{\diamond} \in \mathbb{D}, \vec d^{\diamond}, \vec e^{\diamond} \in \mathbb{D}^3}, the scalar multiplication of a double number by a double vector \nm{\lrb{\cdot : \mathbb{D} \times \mathbb{D}^3 \rightarrow \mathbb{D}^3}}, the inner product between two double vectors \nm{\lrb{\langle \cdot \, , \cdot \rangle: \mathbb{D}^3 \times \mathbb{D}^3 \rightarrow \mathbb{D}}}, and the cross product between two double vectors \nm{\lrb{\times : \mathbb{D}^3 \times \mathbb{D}^3 \rightarrow \mathbb{D}^3}}. The results are similar to those of real numbers shown in section \ref{subsec:algebra_structures} \cite{Jia2013}:
\begin{eqnarray}
\nm{d^{\diamond} \cdot \vec d^{\diamond}} & = & \nm{\lrsb{d^{\diamond} \, d_1^{\diamond}, \, d^{\diamond} \, d_2^{\diamond}, \, d^{\diamond} \, d_3^{\diamond}}^T}\label{eq:SE3_dual_vector_multiplication} \\
\nm{\langle \vec d^{\diamond}, \vec e^{\diamond} \rangle} & = & \nm{\vec d^{\diamond} \cdot \vec e^{\diamond} = {\vec d^{\diamond}}^T \, \vec e^{\diamond} = \lrsb{d_1^{\diamond} \, e_1^{\diamond}, \, d_2^{\diamond} \, e_2^{\diamond}, \, d_3^{\diamond} \, e_3^{\diamond}}^T}\label{eq:SE3_dual_vector_scalar_product} \\
\nm{\vec d^{\diamond} \times \vec e^{\diamond}} & = & \nm{\widehat{\vec d^{\diamond}} \; \vec e^{\diamond} = \begin{bmatrix} \nm{0^{\diamond}} & \nm{- d_3^{\diamond}} & \nm{+ d_2^{\diamond}} \\ \nm{+ d_3^{\diamond}} & \nm{0^{\diamond}} & \nm{- d_{\sss 1}^{\diamond}} \\ \nm{- d_2^{\diamond}} & \nm{+ d_{\sss 1}^{\diamond}} & \nm{0^{\diamond}} \end{bmatrix} \begin{bmatrix} \nm{e_{\sss 1}^{\diamond}} \\ \nm{e_2^{\diamond}} \\ \nm{e_3^{\diamond}} \end{bmatrix} = \begin{bmatrix} \nm{d_2^{\diamond} \, e_3^{\diamond} - d_3^{\diamond} \, e_2^{\diamond}} \\ \nm{d_3^{\diamond} \, e_{\sss 1}^{\diamond} - d_{\sss 1}^{\diamond} \, e_3^{\diamond}} \\ \nm{d_{\sss 1}^{\diamond} \, e_2^{\diamond} - d_2^{\diamond} \, e_{\sss 1}^{\diamond}} \end{bmatrix} = - \vec e^{\diamond} \times \vec d^{\diamond} = - \widehat{\vec e^{\diamond}} \; \vec d^{\diamond}}\label{eq:SE3_dual_vector_cross_product}
\end{eqnarray}
\subsubsection{Dual Quaternions}\label{subsubsec:RigidBody_motion_dual_quat}
The set of dual quaternions \nm{\mathbb{H}_d} is defined as \nm{\{\mathbb{H_d} = \mathbb{H} + \mathbb{H} \, \epsilon \ | \ \epsilon^2 = -1\}}. A dual quaternion \nm{\vec \zeta \in \mathbb{H}_d} has the form \nm{\vec \zeta = \qr + \qd \, \epsilon}, with \nm{\qr, \qd \in \mathbb{H}}. The real plus dual notation \nm{\lrb{1, \epsilon}} is not always the most convenient. A dual quaternion can also be expressed as the sum of a dual number plus a dual vector in the form \nm{\vec \zeta = d_0^{\diamond} + \vec d_v^{\diamond}}, where \nm{d_0^{\diamond}} is the scalar part and \nm{\vec d_v^{\diamond} = d_1^{\diamond} \, i + d_2^{\diamond} \, j + d_3^{\diamond} \, i \, j} is the vector part. Dual quaternions are however mostly represented as 8-vectors \nm{\vec \zeta = \lrsb{\qr, \qd}^T = \lrsb{q_{0 r}, \vec q_{vr}, q_{0d}, \vec q_{vd}}^T = \lrsb{q_{0 r}, q_{1 r}, q_{2 r}, q_{3 r}, q_{0 d}, q_{1 d}, q_{2 d}, q_{3 d}}^T}, which enables the use of matrix algebra for quaternion operations \cite{Valverde2018}. It is also convenient to abuse the equal operator as required to combine general, real, and pure dual quaternions.
The dual quaternion addition \nm{\lrb{+ : \mathbb{H}_d \times \mathbb{H}_d \rightarrow \mathbb{H}_d}} and the scalar product \nm{\lrb{\cdot : \mathbb{D} \times \mathbb{H}_d \rightarrow \mathbb{H}_d}} are both straightforward and commutative:
\begin{eqnarray}
\nm{\vec \zeta_a + \vec \zeta_b} & = & \nm{\lrp{\vec q_{ra} + \vec q_{da} \, \epsilon} + \lrp{\vec q_{rb} + \vec q_{db} \, \epsilon} = \lrp{\vec q_{ra} + \vec q_{rb}} + \lrp{\vec q_{da} + \vec q_{db}} \, \epsilon = \begin{bmatrix} \nm{\vec q_{ra} + \vec q_{rb}} \\ \nm{\vec q_{da} + \vec q_{db}} \end{bmatrix}}\nonumber \\
& = & \nm{\lrp{d_{0a}^{\diamond} + \vec d_{va}^{\diamond}} + \lrp{d_{0b}^{\diamond} + \vec d_{vb}^{\diamond}} = \lrp{d_{0a}^{\diamond} + d_{0b}^{\diamond}} + \lrp{\vec d_{va}^{\diamond} + \vec d_{vb}^{\diamond}}}\label{eq:SE3_dual_quat_addition} \\
\nm{d^{\diamond} \cdot \vec \zeta} & = & \nm{\lrp{x + y \, \epsilon} \cdot \lrp{\qr + \qd \, \epsilon} = x \cdot \qr + \lrp{x \cdot \qd + y \cdot \qr} \, \epsilon}\label{eq:SE3_dual_quat_scalar}
\end{eqnarray}
The multiplication of dual quaternions \nm{\{\otimes : \mathbb{H}_d \times \mathbb{H}_d \rightarrow \mathbb{H}_d\}} is not commutative as it includes the dual vector cross product (\ref{eq:SE3_dual_vector_cross_product}). Depending on how it is expressed, the similarities with the multiplication of dual numbers (\ref{eq:SE3_dual_numbers_multiplication}) or that of quaternions (\ref{eq:SO3_quat_product}) are obvious \cite{Jia2013, Valverde2018, Daniilidis1998}:
\begin{eqnarray}
\nm{\vec \zeta_a \otimes \vec \zeta_b} & = & \nm{\lrp{\vec q_{ra} + \vec q_{da} \, \epsilon} \otimes \lrp{\vec q_{rb} + \vec q_{db} \, \epsilon} = \lrp{\vec q_{ra} \otimes \vec q_{rb}} + \lrp{\vec q_{ra} \otimes \vec q_{db} + \vec q_{da} \otimes \vec q_{rb}} \, \epsilon}\nonumber \\
& = & \nm{\begin{bmatrix} \nm{\vec q_{ra} \otimes \vec q_{rb}} \\ \nm{\vec q_{ra} \otimes \vec q_{db} + \vec q_{da} \otimes \vec q_{rb}} \end{bmatrix} = \lrp{d_{0a}^{\diamond} + \vec d_{va}^{\diamond}} \otimes \lrp{d_{0b}^{\diamond} + \vec d_{vb}^{\diamond}}} \nonumber \\
& = & \nm{\lrp{d_{0a}^{\diamond} \, d_{0b}^{\diamond} - {\vec d_{va}^{\diamond}}^T \, \vec d_{vb}^{\diamond}} + \lrp{d_{0a}^{\diamond} \, \vec d_{vb}^{\diamond} + d_{0b}^{\diamond} \, \vec d_{va}^{\diamond} + \widehat{\vec d}_{va}^{\diamond} \, \vec d_{vb}^{\diamond}}}\label{eq:SE3_dual_quat_product}
\end{eqnarray}
Dual quaternion multiplication is also bilinear \cite{Valverde2018}, based on the operators defined in (\ref{eq:SO3_quat_product_matrices}):
\neweq{\vec \zeta_a \otimes \vec \zeta_b = [\vec \zeta_a]_L \, \vec \zeta_b = \begin{bmatrix}
\nm{\lrsb{\vec a_r}_L} & \nm{\vec O_{4x4}} \\
\nm{\lrsb{\vec a_d}_L} & \nm{\lrsb{\vec a_r}_L} \end{bmatrix} \,
\begin{bmatrix} \nm{\vec b_r} \\ \nm{\vec b_d} \end{bmatrix} =
[\vec \zeta_b]_R \, \vec \zeta_a = \begin{bmatrix}
\nm{\lrsb{\vec b_r}_R} & \nm{\vec O_{4x4}} \\
\nm{\lrsb{\vec b_d}_R} & \nm{\lrsb{\vec b_r}_R} \end{bmatrix} \,
\begin{bmatrix} \nm{\vec a_r} \\ \nm{\vec a_d} \end{bmatrix}} {eq:SE3_dual_quat_product_matrices}
It is possible to define three different conjugates for a dual quaternion \cite{Jia2013}, based on whether it only switches the sign of the dual part as in the case of dual numbers (\ref{eq:SE3_dual_quat_conj1}), it employs the conjugates of the real and dual quaternion components (\ref{eq:SE3_dual_quat_conj2}), or a combination of both (\ref{eq:SE3_dual_quat_conj3}):
\begin{eqnarray}
\nm{\zetacirc = \qr - \qd \, \epsilon} & \nm{\rightarrow} & \nm{\lrp{\vec \zeta_a \otimes \vec \zeta_b}^{\circ} = \vec \zeta_a^{\circ} \otimes \vec \zeta_b^{\circ}}\label{eq:SE3_dual_quat_conj1} \\
\nm{\zetaast = \qrast + \qdast \, \epsilon} & \nm{\rightarrow} & \nm{\lrp{\vec \zeta_a \otimes \vec \zeta_b}^{\ast} = \vec \zeta_b^{\ast} \otimes \vec \zeta_a^{\ast}}\label{eq:SE3_dual_quat_conj2} \\
\nm{\zetabullet = \qrast - \qdast \, \epsilon} & \nm{\rightarrow} & \nm{\lrp{\vec \zeta_a \otimes \vec \zeta_b}^{\bullet} = \vec \zeta_b^{\bullet} \otimes \vec \zeta_a^{\bullet}}\label{eq:SE3_dual_quat_conj3}
\end{eqnarray}
\emph{Pure dual quaternions} \nm{\vec \zeta = 0^{\diamond} + \vec d_v^{\diamond} \in \mathbb{H}_{dp}} are those in which its dual number is zero (\nm{0^\diamond}), or in which both its real and dual parts are pure quaternions (\nm{\qr, \, \qd \in \mathbb{H}_p}), and verify that \nm{\vec \zeta = - \, \zetaast}. The dual quaternion norm is defined as \nm{\|\vec \zeta\| = \sqrt{\vec \zeta \otimes \zetaast} = \sqrt{\qr \otimes \qrast + \lrp{\qr \otimes \qdast + \qd \otimes \qrast} \, \epsilon} \in \mathbb{D}} \cite{Daniilidis1998}. Dual quaternions endowed with \nm{\otimes} do not form a group, because although \nm{\vec{\zeta_1} = \vec{q_1} + \vec0 \, \epsilon} is the identity, the inverse \nm{\vec \zeta^{-1} = {\qr}^{-1} - {\qr}^{-1} \otimes \qd \otimes {\qr}^{-1} \, \epsilon = {\qr}^{-1} \otimes \lrp{\vec q_1 - \qd \otimes {\qr}^{-1} \, \epsilon}} is not defined when \nm{\qr = \vec 0}. Dual quaternions endowed with addition \nm{+} and multiplication \nm{\otimes} however do form the non abelian ring \nm{\langle \mathbb{H}_d, +, \otimes \rangle}.
As in the case of quaternions described in section \ref{subsubsec:RigidBody_rotation_rodrigues_quat}, the natural power of a dual quaternion \nm{\vec \zeta^n, n \in \mathbb{N}} is obtained by multiplying the dual quaternion by itself \nm{n-1} times. The double product of a dual quaternion by a vector \nm{\{\mathbb{H}_d \times \mathbb{R}^3 \rightarrow \mathbb{R}^3\}} is defined as the product \nm{\otimes} of the dual quaternion by the vector by the dual quaternion conjugate, resulting in three different versions based on the conjugate definition (\ref{eq:SE3_dual_quat_conj1}, \ref{eq:SE3_dual_quat_conj2}, \ref{eq:SE3_dual_quat_conj3}).
\subsubsection{Unit Dual Quaternion}\label{subsubsec:RigidBody_motion_unit_dual_quat}
\emph{Unit dual quaternions} are those dual quaternions in which \nm{\vec \zeta \otimes \zetaast = \zetaast \otimes \vec \zeta = \vec{\zeta_1}}, which implies that the inverse and the conjugate coincide as \nm{\vec \zeta^{-1} = \qrast - \qrast \otimes \qd \otimes \qrast \epsilon = \qrast + \qdast \, \epsilon = \zetaast}. Note that the norm \nm{\|\vec \zeta\|} has a unity real part and a zero dual part \cite{Daniilidis1998}. Based on (\ref{eq:SE3_dual_quat_product}), this translates into the following two conditions:
\begin{eqnarray}
\nm{\qr \otimes \qrast} & = & \nm{1 \rightarrow \|\qr\| = 1}\label{eq:SE3_unit_dual_quat_cond1} \\
\nm{\qr \otimes \qdast + \qd \otimes \qrast} & = & \nm{0 \rightarrow \langle \qr, \, \qd \rangle = 0}\label{eq:SE3_unit_dual_quat_cond2}
\end{eqnarray}
In other words, unit dual quaternions are those in which the real part \nm{\qr} is a unit quaternion that is also orthogonal to the dual part \nm{\qd}.
The rigid body motion between a body frame \nm{\FB} and a spatial frame \nm{\FE} represented by the unit quaternion \nm{\qEB} and the translation \nm{\TEBE} (section \ref{subsec:RigidBody_motion_affine}) can always be represented by the following unit dual quaternion \nm{\zetaEB} \cite{Jia2013, Valverde2018}, where the notation is abused to consider the quaternion \nm{\TEBE = \lrsb{0, \, \TEBE}^T}:
\neweq{\zetaEB = \qEB + \dfrac{\epsilon}{2} \, \TEBE \otimes \qEB}{eq:SE3_unit_dual_quat_from_affine}
(\ref{eq:SE3_unit_dual_quat_from_affine}) is indeed a unit dual quaternion as \nm{\zetaEB \otimes \zetaEBast = \vec \zeta_1 = \vec q_1 = 1} based on \nm{\vec T_{\sss EB}^{{\sss E}*} = - \TEBE} as it is a pure quaternion. The opposite map providing the affine representation based on the unit dual quaternion is the following:
\begin{eqnarray}
\nm{\qEB} & = & \nm{\vec \zeta_{{\sss EB}r}}\label{eq:SE3_unit_dual_quat_to_affine_q} \\
\nm{\TEBE} & = & \nm{2 \, \vec \zeta_{{\sss EB}d} \otimes \vec \zeta_{{\sss EB}r}^{\ast}}\label{eq:SE3_unit_dual_quat_to_affine_T}
\end{eqnarray}
\begin{center}
\begin{tabular}{lcclcc}
\hline
\textbf{Concept} & \nm{\mathbb{SE}^3} & \nm{\mathbb{H}_d} & \textbf{Concept} & \nm{\mathbb{SE}^3} & \nm{\mathbb{H}_d} \\
\hline
Lie group element & \nm{\mathcal M} & \nm{\vec \zeta} & Concatenation & \nm{\circ} & \nm{\otimes} \\
Identity & \nm{\mathcal {I_M}} &\nm{\vec{\zeta_1}} & Inverse & \nm{\mathcal M^{-1}} & \nm{\zetaast} \\
Point motion & \nm{\vec g_{\mathcal M}(\vec p)} & \nm{\vec \zeta \otimes \vec \zeta_{\vec p} \otimes \zetabullet} & Vector motion & \nm{\vec g_{\mathcal M*}(\vec v)} & \nm{\vec \zeta \otimes \vec \zeta_{\vec v} \otimes \zetabullet} \\
\hline
\end{tabular}
\end{center}
\captionof{table}{Comparison between generic \nm{\mathbb{SE}(3)} and unit dual quaternion} \label{tab:RigidBody_motion_lie_unit_dual_quat}
The unit dual quaternion endowed with the double product can be employed to transform both points and vectors, verifying that it complies with the rigid body motion orthogonality and handedness conditions described in section \ref{subsec:RigidBody_bases}. Given a point \nm{\vec p = \lrsb{p_1, p_2, p_3}^T \in \mathbb{R}^3}, its dual quaternion representation \nm{\vec \zeta_{\vec p}} is obtained by combining the unit quaternion \nm{\vec q_1} as the real part and the point coordinates as the dual part, resulting in \nm{\vec \zeta_{\vec p} = \vec q_1 + \epsilon \, \vec p \in \mathbb{R}^8} \cite{Jia2013}. In the case of a vector \nm{\vec v = \vec q - \vec p \in \mathbb{R}^3}, its dual quaternion representation is \nm{\vec \zeta_{\vec v} = \vec \zeta_{\vec q} - \vec \zeta_{\vec p} = \lrp{\vec q_1 + \epsilon \, \vec q} - \lrp{\vec q_1 + \epsilon \, \vec p} = \epsilon \, \lrp{\vec q - \vec p} = \epsilon \, \vec v \in \mathbb{R}^8}. It is then possible, based on (\ref{eq:SE3_affine_quat_transform}) and (\ref{eq:SE3_affine_vector}), to employ the double product to transform both points and vectors between different frames:
\begin{eqnarray}
\nm{\vec \zeta_{\pE} = \zetaEB \otimes \vec \zeta_{\pB} \otimes \zetaEBbullet} & = & \nm{\lrp{\qEB + \dfrac{\epsilon}{2} \, \TEBE \otimes \qEB} \otimes \lrp{\vec q_1 + \epsilon \, \pB} \otimes \lrp{\qEBast + \dfrac{\epsilon}{2} \, \qEBast \otimes \TEBE}}\nonumber \\
& = & \nm{\vec q_1 + \epsilon \, \lrp{\qEB \otimes \pB \otimes \qEBast + \TEBE} = \vec q_1 + \epsilon \, \pE}\label{eq:SE3_unit_dual_quat_transform_point} \\
\nm{\vec \zeta_{\pE} = \zetaEB \otimes \vec \zeta_{\pB} \otimes \zetaEBbullet} & = & \nm{\lrp{\qr + \qd \, \epsilon} \otimes \lrp{\vec q_1 + \epsilon \, \pB} \otimes \lrp{\qrast - \qdast \, \epsilon}}\nonumber \\
& = & \nm{\vec q_1 + \epsilon \, \lrp{\qr \otimes \pB \otimes \qrast + \qd \otimes \qrast - \qr \otimes \qdast} = \vec q_1 + \epsilon \, \pE}\label{eq:SE3_unit_dual_quat_transform_point2} \\
\nm{\vec \zeta_{\vE} = \zetaEB \otimes \vec \zeta_{\vB} \otimes \zetaEBbullet} & = & \nm{\lrp{\qEB + \dfrac{\epsilon}{2} \, \TEBE \otimes \qEB} \otimes \epsilon \, \vB \otimes \lrp{\qEBast + \dfrac{\epsilon}{2} \, \qEBast \otimes \TEBE}}\nonumber \\
& = & \nm{\epsilon \, \lrp{\qEB \otimes \vB \otimes \qEBast} = \epsilon \, \vE}\label{eq:SE3_unit_dual_quat_transform_vector} \\
\nm{\vec \zeta_{\vE} = \zetaEB \otimes \vec \zeta_{\vB} \otimes \zetaEBbullet} & = & \nm{\lrp{\qr + \qd \, \epsilon} \otimes \epsilon \, \vB \otimes \lrp{\qrast - \qdast \, \epsilon} = \epsilon \, \lrp{\qr \otimes \vB \otimes \qrast} = \epsilon \, \vE}\label{eq:SE3_unit_dual_quat_transform_vector2}
\end{eqnarray}
A disadvantage of the unit dual quaternion as an \nm{\mathbb{SE}\lrp{3}} representation is that a different expression is required for the inverse transformation:
\begin{eqnarray}
\nm{\vec \zeta_{\pB} = \zetaEBast \otimes \vec \zeta_{\pE} \otimes \zetaEBcirc} & = & \nm{\lrp{\qEBast - \dfrac{\epsilon}{2} \, \qEBast \otimes \TEBE} \otimes \lrp{\vec q_1 + \epsilon \, \pE} \otimes \lrp{\qEB - \dfrac{\epsilon}{2} \, \TEBE \otimes \qEB}}\nonumber \\
& = & \nm{\vec q_1 + \epsilon \, \lrp{\qEBast \otimes \pE \otimes \qEB - \qEBast \otimes \TEBE \otimes \qEB} = \vec q_1 + \epsilon \, \pB}\label{eq:SE3_unit_dual_quat_transform_inv_point} \\
\nm{\vec \zeta_{\pB} = \zetaEBast \otimes \vec \zeta_{\pE} \otimes \zetaEBcirc} & = & \nm{\lrp{\qrast + \qdast \, \epsilon} \otimes \lrp{\vec q_1 + \epsilon \, \pE} \otimes \lrp{\qr - \qd \, \epsilon}}\nonumber \\
& = & \nm{\vec q_1 + \epsilon \, \lrp{\qrast \otimes \pE \otimes \qr + \qdast \otimes \qr - \qrast \otimes \qd} = \vec q_1 + \epsilon \, \pB}\label{eq:SE3_unit_dual_quat_transform_inv_point2} \\
\nm{\vec \zeta_{\vB} = \zetaEBast \otimes \vec \zeta_{\vE} \otimes \zetaEBcirc} & = & \nm{\lrp{\qEBast - \dfrac{\epsilon}{2} \, \qEBast \otimes \TEBE} \otimes \epsilon \, \vE \otimes \lrp{\qEB - \dfrac{\epsilon}{2} \, \TEBE \otimes \qEB}}\nonumber \\
& = & \nm{\epsilon \, \lrp{\qEBast \otimes \vE \otimes \qEB} = \epsilon \, \vB}\label{eq:SE3_unit_dual_quat_transform_inv_vector} \\
\nm{\vec \zeta_{\vB} = \zetaEBast \otimes \vec \zeta_{\vE} \otimes \zetaEBcirc} & = & \nm{\lrp{\qrast + \qdast \, \epsilon} \otimes \epsilon \, \vE \otimes \lrp{\qr - \qd \, \epsilon} = \epsilon \, \lrp{\qrast \otimes \vE \otimes \qr} = \epsilon \, \vB}\label{eq:SE3_unit_dual_quat_transform_inv_vector2}
\end{eqnarray}
The inverse transformation coincides with the dual quaternion conjugate provided by (\ref{eq:SE3_dual_quat_conj2}):
\begin {eqnarray}
\nm{\zetaBE = \vec \zeta_{\sss EB}^{-1} = \zetaEBast} & = & \nm{\qBE + \dfrac{\epsilon}{2} \, \TBEB \otimes \qBE = \qEBast - \dfrac{\epsilon}{2} \, \lrp{\qEBast \otimes \TEBE \otimes \qEB} \otimes \qEBast}\nonumber \\
& = & \nm{\qEBast - \dfrac{\epsilon}{2} \, \qEBast \otimes \TEBE = \qrast + \qdast \, \epsilon}\label{eq:SE3_unit_dual_quat_inverse}
\end{eqnarray}
The concatenation of transformations is straightforward based on (\ref{eq:SE3_affine_concatenation}):
\begin{eqnarray}
\nm{\vec \zeta_{\sss EB} = \vec \zeta_{\sss EN} \otimes \vec \zeta_{\sss NB}} & = & \nm{\lrp{\vec q_{\sss EN} + \dfrac{\epsilon}{2} \, \vec T_{\sss EN}^{\sss E} \otimes \vec q_{\sss EN}} \otimes \lrp{\vec q_{\sss NB} + \dfrac{\epsilon}{2} \, \vec T_{\sss NB}^{\sss N} \otimes \vec q_{\sss NB}}}\nonumber \\
& = & \nm{\vec q_{\sss EN} \otimes \vec q_{\sss NB} + \dfrac{\epsilon}{2} \, \lrp{\vec q_{\sss EN} \otimes \vec T_{\sss NB}^{\sss N} \otimes \vec q_{\sss NB} + \vec T_{\sss EN}^{\sss E} \otimes \vec q_{\sss EN} \otimes \vec q_{\sss NB}}}\nonumber \\
& = & \nm{\vec q_{\sss EB} + \dfrac{\epsilon}{2} \, \lrp{\vec q_{\sss EN} \otimes \vec T_{\sss NB}^{\sss N} \otimes \vec q_{\sss EN}^{\ast} + \vec T_{\sss EN}^{\sss E}} \otimes \vec q_{\sss EB} = \vec q_{\sss EB} + \dfrac{\epsilon}{2} \, \vec T_{\sss EB}^{\sss E} \otimes \vec q_{\sss EB}}\nonumber \\
& = & \nm{\lrp{\vec q_{{\sss EN}r} + \vec q_{{\sss EN}d} \, \epsilon} \otimes \lrp{\vec q_{{\sss NB}r} + \vec q_{{\sss NB}d} \, \epsilon}}\nonumber \\
& = & \nm{\lrp{\vec q_{{\sss EN}r} \otimes \vec q_{{\sss NB}r}} + \lrp{\vec q_{{\sss EN}r} \otimes \vec q_{{\sss NB}d} + \vec q_{{\sss EN}d} \otimes \vec q_{{\sss NB}r}} \, \epsilon}\label{eq:SE3_unit_dual_quat_concatenation}
\end{eqnarray}
Unit dual quaternions comply with the orthogonality and handedness conditions required in section \ref{subsec:RigidBody_bases} for rigid body motions, and hence their space \nm{\mathbb{SE}\lrp{3} = \{\vec \zeta = \qr + \qd \, \epsilon \in \mathbb{H}_d \ | \|\qr\| = 1, \langle \qr, \, \qd \rangle = 0\}} possesses group structure under dual quaternion multiplication \nm{\{\otimes : \mathbb{H}_d \times \mathbb{H}_d \rightarrow \mathbb{H}_d \ | \ \vec \zeta_a \otimes \vec \zeta_b \in \mathbb{H}_d, \forall \ \vec \zeta_a, \ \vec \zeta_b \in \mathbb{H}_d\}}. Because of the (\ref{eq:SE3_unit_dual_quat_cond1}) and (\ref{eq:SE3_unit_dual_quat_cond2}) constraints, although they have dimension eight, the special euclidean group \nm{\mathbb{SE}\lrp{3}} defined by means of unit dual quaternions constitutes a six dimensional manifold to euclidean space \nm{\mathbb{E}^6} called the \emph{image space of spatial displacements}, which can be visualized in \nm{\mathbb{R}^4} as follows. Expression (\ref{eq:SE3_unit_dual_quat_cond1}) defines a unit hypersphere of three dimensions, while (\ref{eq:SE3_unit_dual_quat_cond2}) defines the three dimensional hyperplane orthogonal to the normal at the point \nm{\qr} on the hypersphere. Thus, the image space consists of the hypersphere and all of its tangent spaces, which have been translated to contain the origin. Note that in this group \nm{\vec{\zeta_1}} constitutes the identity and \nm{\zetaast} the inverse.
The map from the affine representation (or homogeneous matrix) to the unit dual quaternion is surjective but not injective for the same reasons as that between the rotation matrix and the unit quaternion described in section \ref{subsubsec:RigidBody_rotation_rodrigues_unit_quat}, this is, the double covering of the \nm{\mathbb{SO}\lrp{3}} by the unit quaternion.
\subsection{Half Transform Vector as Tangent Space}\label{subsec:RigidBody_motion_halftransform_vector}
As indicated in section \ref{subsubsec:algebra_lie_velocities}, the structure of the Lie algebra \nm{\mathfrak{se}(3)} can be obtained by time derivating its \nm{\mathbb{SE}(3)} Lie group constraint, this is, \nm{\vec \zeta \otimes \zetaast = \zetaast \otimes \vec \zeta = \vec{\zeta_1}}, leading to \nm{\zetaast \otimes \vec{\dot \zeta} = - \big(\zetaast \otimes \vec{\dot \zeta}\big)^{\ast}}, which indicates that \nm{\zetaast \otimes \vec{\dot \zeta}} is in fact a pure dual quaternion, as is \nm{\vec{\dot \zeta} \otimes \zetaast}. This results in the following particularizations of (\ref{eq:algebra_vE}) and (\ref{eq:algebra_vX}):
\begin{eqnarray}
\nm{\vec \Upsilon_{\sss {EB}}^{\sss E\wedge}} & = & \nm{\vec{\dot \zeta}_{\sss {EB}} \otimes \zetaEB^{\ast} = - \zetaEB \otimes \vec{\dot \zeta}_{\sss EB}^{\ast}} \label{eq:SE3_Upsilon_space} \\
\nm{\vec \Upsilon_{\sss {EB}}^{\sss B\wedge}} & = & \nm{\zetaEB^{\ast} \otimes \vec{\dot \zeta}_{\sss {EB}} = - \vec{\dot \zeta}_{\sss EB}^{\ast} \otimes \zetaEB} \label{eq:SE3_Upsilon_body}
\end{eqnarray}
The Lie algebra velocity \nm{\vec v^\wedge} is known as the \emph{half twist} \nm{\vec \Upsilon^\wedge}, and as shown in (\ref{eq:SE3_Upsilon_space}) and (\ref{eq:SE3_Upsilon_body}), has the structure of a pure dual quaternion because its negative coincides with its conjugate:
\neweq{\vec \Upsilon^\wedge\lrp{t} = \lrsb{0^\diamond + \vec\Upsilon_v^\diamond\lrp{t}} \in \mathbb{H}_{dp}}{eq:SE3_Upsilon_pure}
Inverting the previous equations results in the unit dual quaternion time derivative, which is linear:
\neweq{\zetaEBdot = \vec \Upsilon_{\sss {EB}}^{\sss E\wedge} \otimes \zetaEB = \zetaEB \otimes \vec \Upsilon_{\sss {EB}}^{\sss B\wedge}} {eq::SE3_Upsilon_dot}
Notice that if \nm{\vec \zeta\lrp{t_0} = \vec{\zeta_1}}, then \nm{\vec{\dot \zeta}\lrp{t_0} = \vec \Upsilon\lrp{t_0}}, and hence the pure dual quaternion \nm{\vec \Upsilon^\wedge\lrp{t_0}} provides a first order approximation of the unit dual quaternion around the identity \nm{\vec{\zeta_1}}:
\neweq{\vec \zeta\lrp{t_0 + \Deltat} \approx \vec \zeta_1 +\vec \Upsilon^\wedge\lrp{t_0} \, \Deltat}{eq:SE3_Upsilon_taylor}
The \emph{space of pure dual quaternions} \nm{\mathfrak{se}\lrp{3} = \{\vec \Upsilon^\wedge \in \mathbb{H}_{dp} \ | \ \vec \Upsilon^\diamond \in \mathbb{D}^3\}} is hence the \emph{tangent space} of the unit dual quaternions at the identity \nm{\vec \zeta_1}, denoted as \nm{T_{\vec \zeta_1}{\mathcal M}}. The \emph{hat} \nm{\lrb{\cdot^\wedge: \mathbb{R}^6 \rightarrow \mathbb{D}^3 \rightarrow \mathfrak{se}(3) \ | \ \vec \Upsilon \rightarrow \vec \Upsilon^\wedge}} and \emph{vee} \nm{\lrb{\cdot^\vee: \mathfrak{se}(3) \rightarrow \mathbb{D}^3 \rightarrow \mathbb{R}^6 \ | \ \lrp{\vec \Upsilon^\wedge}^\vee \rightarrow \vec \Upsilon}} operators convert the half twist vector into its pure dual quaternion form, and viceversa.
If \nm{\vec \zeta\lrp{t_0} \neq \vec \zeta_1}, the tangent space needs to be transported right multiplying by \nm{\zetaEB\lrp{t_0}} (in the case of space tangent space), or left multiplying for the local space:
\begin{eqnarray}
\nm{\zetaEB\lrp{t_0 + \Deltat}} & \nm{\approx} & \nm{\zetaEB\lrp{t_0} + \lrsb{\vec \Upsilon_{\sss EB}^{\sss E\wedge} \lrp{t_0} \, \Deltat} \otimes \zetaEB\lrp{t_0} = \lrsb{\vec \zeta_1 + \vec \Upsilon_{\sss EB}^{\sss E\wedge} \lrp{t_0} \, \Deltat} \otimes \zetaEB\lrp{t_0} }\label{eq:SE3_Upsilon_taylor_space} \\
\nm{\zetaEB\lrp{t_0 + \Deltat}} & \nm{\approx} & \nm{\zetaEB\lrp{t_0} + \zetaEB\lrp{t_0} \otimes \lrsb{\vec \Upsilon_{\sss EB}^{\sss B\wedge} \lrp{t_0} \, \Deltat} = \zetaEB\lrp{t_0} \otimes \lrsb{\vec \zeta_1 + \vec \Upsilon_{\sss EB}^{\sss B\wedge} \lrp{t_0} \, \Deltat}}\label{eq:SE3_Upsilon_taylor_body}
\end{eqnarray}
Note that the solution to the ordinary differential equation \nm{\vec{\dot x}\lrp{t} = \vec x\lrp{t} \otimes \vec \Upsilon^\wedge, \ \vec x\lrp{t} \in \mathbb{R}^8}, where \nm{\vec \Upsilon^\wedge} is constant, is \nm{\vec x\lrp{t} = \vec x\lrp{0} \, e^{\ds{\vec \Upsilon^\wedge t}}}. Based on it, assuming \nm{\vec \zeta\lrp{0} = \vec{\zeta_1}} as initial condition, and considering for the time being that \nm{\vec \Upsilon} is constant,
\neweq{\vec \zeta\lrp{t} = e^{\ds{\vec \Upsilon^\wedge t}} = \vec{\zeta_1} + \vec \Upsilon^\wedge t + \frac{\lrp{\vec \Upsilon^\wedge t}^2}{2!} + \dots + \frac{\lrp{\vec \Upsilon^\wedge t}^n}{n!} + \dots}{eq:SE3_Upsilon_exponential3}
which is indeed a pure dual quaternion and coincides with half the twist defined in section \ref{subsec:RigidBody_motion_transform_vector}, as proven next based on (\ref{eq::SO3_quat_omega_dot}), (\ref{eq:SE3_unit_dual_quat_from_affine}), (\ref{eq:SE3_Upsilon_space}), (\ref{eq:SE3_time_derivative_twist_space}), and \nm{\TEBE} being a pure quaternion:
\begin{eqnarray}
\nm{\vec \Upsilon_{\sss {EB}}^{\sss E\wedge}} & = & \nm{\vec{\dot \zeta}_{\sss {EB}} \otimes \lrsb{\qEBast + \frac{\epsilon}{2} \, \lrp{\TEBE \otimes \qEB}^\ast} = \vec{\dot \zeta}_{\sss {EB}} \otimes \lrsb{\qEBast - \frac{\epsilon}{2} \, \qEBast \otimes \TEBE}} \nonumber \\
& = & \nm{\frac{\vec \omega_{\sss EB}^{\sss E\wedge}}{2} + \frac{\epsilon}{2} \, \lrsb{\vec {\dot T}_{\sss EB}^{\sss E} + \TEBE \otimes \frac{\vec \omega_{\sss EB}^{\sss E\wedge}}{2} - \frac{\vec \omega_{\sss EB}^{\sss E\wedge}}{2} \otimes \TEBE} = \frac{\vec \omega_{\sss EB}^{\sss E\wedge}}{2} + \frac{\epsilon}{2} \, \lrsb{\vec {\dot T}_{\sss EB}^{\sss E} - \frac{\vec \omega_{\sss EB}^{\sss E\wedge}}{2} \otimes \TEBE}} \nonumber \\
& = & \nm{\frac{1}{2} \, \lrp{\vec \omega_{\sss EB}^{\sss E\wedge} + \epsilon \ \vec \nu_{\sss EB}^{\sss E\wedge}} = \frac{\vec \xi_{\sss {EB}}^{\sss E\wedge}}{2}} \label{eq:SE3_Upsilon_space_proof}
\end{eqnarray}
A similar process employing (\ref{eq:SE3_Upsilon_body}) and (\ref{eq:SE3_time_derivative_twist_body}) leads to:
\neweq{\vec \Upsilon_{\sss {EB}}^{\sss B\wedge} = \frac{1}{2} \, \lrp{\vec \omega_{\sss EB}^{\sss B\wedge} + \epsilon \ \vec \nu_{\sss EB}^{\sss B\wedge}} = \frac{\vec \xi_{\sss {EB}}^{\sss B\wedge}}{2}}{eq:SE3_Upsilon_body_proof}
\begin{center}
\begin{tabular}{lcc}
\hline
\textbf{Concept} & \textbf{Lie Theory} & \nm{\mathbb{SE}^3} \\
\hline
Tangent space element & \nm{\vec \tau^\wedge} & \nm{\vec \Psi^\wedge = \lrsb{0^\diamond + \vec\Psi_v^\diamond}} \\
Velocity element & \nm{\vec v^\wedge} & \nm{\vec \Upsilon^\wedge = \lrsb{0^\diamond + \vec\Upsilon_v^\diamond}} \\
Structure & \nm{\wedge} & pure dual quaternion \\
\hline
\end{tabular}
\end{center}
\captionof{table}{Comparison between generic \nm{\mathbb{SE}(3)} and half transform vector as tangent space} \label{tab:Rotate_lie_halftransform_vector}
Remembering that so far \nm{\vec \Upsilon^\wedge} is constant, (\ref{eq:SE3_Upsilon_exponential3}) means that any rigid body motion \nm{\vec \zeta\lrp{t} = e^{\ds{\vec \Upsilon^\wedge t}}} can be realized by maintaining a constant half twist \nm{\vec \Upsilon^\wedge \in \mathbb{H}_{dp}} for a given time \nm{t}. The vectors \nm{\vec n = \vec \omega \, t / \| \vec \omega \, t \| = \vec r / \|\vec r\|} and \nm{\vec k = \vec \nu \, t / \| \vec \nu \, t \| = \vec s / \|\vec s\|} indicate the half twist directions, while \nm{\theta = \phi / 2 = \|\vec r\| / 2} and \nm{\rho / 2 = \|\vec s\| / 2} represent the half twist magnitudes, respectively. This enables the definition of the \emph{half transform vector} \nm{\vec \Psi}, also known as the \emph{exponential coordinates} of the \nm{\mathcal M} motion, as
\neweq{\vec \Psi = \vec \Upsilon \, t = \frac{1}{2} \, \lrsb{\vec \nu \, t, \, \vec \omega \, t}^T = \frac{1}{2} \, \lrsb{\vec s, \, \vec r}^T = \frac{1}{2} \, \lrsb{ \vec k \, \rho, \, \vec n \, \phi}^T = \frac{\vec \tau}{2} \in \mathbb{R}^6}{eq:SE3_halftransform_vector_definition}
Note that the half transform vector \nm{\vec \Psi} belongs to the tangent space as it is a multiple of the half twist \nm{\vec \Upsilon \in \mathfrak{se}\lrp{3}}, and hence tends to coincide with it as time tends to zero. The \emph{exponential map} \nm{\lrb{exp\lrp{} : \mathfrak{se}(3) \rightarrow \mathbb{SE}(3) \ | \ \mathcal M = exp\lrp{\vec \Psi^\wedge}}} and its capitalized form \nm{\lrb{Exp\lrp{} : \mathbb{R}^6 \rightarrow \mathbb{SE}(3) \ | \ \mathcal M = Exp\lrp{\vec \Psi}}} wrap the half transform vector around the special Euclidean group. However, the half twist \nm{\vec \Upsilon^\wedge \lrp{t}} in fact is not required to be constant. Given a rigid body motion represented by its unit dual quaternion \nm{\vec \zeta \in \mathbb{SE}\lrp{3}}, it can be proved that there exists a not necessarily unique half transform vector \nm{\vec \Psi = \vec \tau / 2 = \lrsb{\vec s, \, \vec r}^T / 2 = \lrsb{\vec k \, \rho, \, \vec n \, \phi}^T / 2} such that \nm{\vec \zeta = e^{\vec \Psi^\wedge}}. The exponential map is made up by a combination of (\ref{eq:SO3_quat_exponential2}), (\ref{eq:SE3_twist_exponential4_a}), (\ref{eq:SE3_twist_exponential4_b}), and (\ref{eq:SE3_unit_dual_quat_from_affine}).
The \emph{logarithmic map} \nm{\lrb{log\lrp{} : \mathbb{SE}(3) \rightarrow \mathfrak{se}(3) \ | \ \vec \Psi^\wedge = log\lrp{\mathcal M}}} and its capitalized version \nm{\lrb{Log\lrp{} : \mathbb{SE}(3) \rightarrow \mathbb{R}^6 \ | \ \vec \Psi = Log\lrp{\mathcal M}}} convert unit dual quaternions into half transform vectors. It is composed by (\ref{eq:SE3_unit_dual_quat_to_affine_q}), (\ref{eq:SO3_theta_from_quat}), and (\ref{eq:SO3_rotvec_from_quat}) for the rotation part, and (\ref{eq:SE3_unit_dual_quat_to_affine_T}) together with (\ref{eq:SE3_twist_logarithm}) for the translation part.
As the vector \nm{\vec \Upsilon} represents half the twist \nm{\vec \xi}, it is possible to adjust expressions (\ref{eq:SE3_Upsilon_space}), (\ref{eq:SE3_Upsilon_body}), and (\ref{eq::SE3_Upsilon_dot}):
\begin{eqnarray}
\nm{\vec \xi_{\sss {EB}}^{\sss E\wedge}} & = & \nm{2 \ \vec{\dot \zeta}_{\sss {EB}} \otimes \zetaEB^{\ast}} \label{eq:SE3_dual_quat_xi_space} \\
\nm{\vec \xi_{\sss {EB}}^{\sss B\wedge}} & = & \nm{2 \ \zetaEB^{\ast} \otimes \vec{\dot \zeta}_{\sss {EB}}} \label{eq:SE3_dual_quat_xi_body} \\
\nm{\zetaEBdot} & = & \nm{\dfrac{1}{2} \; \vec \xi_{\sss {EB}}^{\sss E\wedge} \otimes \zetaEB = \dfrac{1}{2} \; \zetaEB \otimes \vec \xi_{\sss {EB}}^{\sss B\wedge}} \label{eq::SE3_dual_quat_xi_dot}
\end{eqnarray}
\subsection{Screw as Tangent Space}\label{subsec:RigidBody_motion_screw}
In the section \ref{sec:Rotate} analysis of rotational motion there exists two different representations for the tangent space \nm{\mathfrak{so}\lrp{3}}. The first is the skew-symmetric angular velocity \nm{\vec \omega^\wedge = \vec \omega^\wedge}, which converts into the rotation vector \nm{\vec r^\wedge} when applied during a certain amount of time (section \ref{subsec:RigidBody_rotation_rotv}), and represents the origin of the exponential map \nm{exp\lrp{\vec r^\wedge}} (\ref{eq:SO3_rotv_exponential2}) that transforms it into the rotation matrix \nm{\vec R} (section \ref{subsec:RigidBody_rotation_dcm}). The second is the pure quaternion half angular velocity \nm{\vec \Omega^\wedge = \lrsb{0, \, \vec \omega / 2}^T}, which converts into the half rotation vector \nm{\vec h^\wedge = \lrsb{0, \, \vec r / 2}^T} (section \ref{subsec:RigidBody_rotation_halfrotv}), and represents the origin of the exponential map \nm{exp\lrp{\vec h^\wedge} = exp\lrp{\vec r^\wedge / 2}} (\ref{eq:SO3_quat_exponential2}) that transforms it into the unit quaternion \nm{\vec q} (section \ref{subsec:RigidBody_rotation_rodrigues}). Note that both representations of the tangent space \nm{\mathfrak{so}\lrp{3}} are so similar that for all practical purposes they are considered the same, resulting in two versions of the exponential map that convert the rotation vector \nm{\vec r} into either the rotation matrix \nm{\vec R} or the unit quaternion \nm{\vec q}.
So far the rigid body motion looks similar. There are two \nm{\mathfrak{se}\lrp{3}} velocities, the twist \nm{\vec \xi^\wedge} and the pure dual quaternion half twist \nm{\vec \Upsilon^\wedge}, which convert into the transform vector \nm{\vec \tau^\wedge} and the half transform vector \nm{\vec \Psi^\wedge} (sections \ref{subsec:RigidBody_motion_transform_vector} and \ref{subsec:RigidBody_motion_halftransform_vector}) when applied during an amount of time, and constitute the origins of the exponential maps that transform them into the homogeneous matrix \nm{\vec M} (section \ref{subsec:RigidBody_motion_homogeneous}), the affine representation (section \ref{subsec:RigidBody_motion_affine}), or the unit dual quaternion \nm{\vec \zeta} (section \ref{subsec:RigidBody_motion_unit_dual_quaternion}). As in the rotation case, both \nm{\mathfrak{se}\lrp{3}} representations are so similar that for all practical purposes they are considered the same and can be interchanged in the various exponential maps.
There exists however an additional \nm{\mathbb{SE}(3)} representation, which also belongs to its tangent space \nm{\mathfrak{se}\lrp{3}}, that enables the definition of a different exponential map into the unit dual quaternion that explicitly separates the influence of the motion direction from that of its magnitude, and is also indispensable for the rigid body motion powers, linear interpolation, and perturbations introduced in section \ref{subsec:RigidBody_motion_algebra}.
The origin of the screw \nm{\vec S} lies in the fact that every rigid body motion can be realized by a rotation about an axis combined with a translation parallel to that same axis \cite{Murray1994}. The rotation and translation can be executed simultaneously or one after another without modifying the result. It is however necessary to remark that in this case the axis, defined in section \ref{subsec:algebra_points_and_vectors}, does not necessarily pass though the origin of the frame characterizing the rigid body, in contrast to previous representations of rigid body motions in which the rotating axis, more appropriately called rotating direction, in all cases passed through the origin. Note that according to section \ref{subsec:algebra_points_and_vectors}, an axis is represented by \nm{\lrp{\vec n, \, \vec m}} and has four degrees of freedom, while if restricted to passing through the origin \nm{\vec m = \vec 0} and the degrees of freedom are two. The line point closest to the origin responds to \nm{\vec p_{\perp} = \widehat{\vec n} \, \vec m}.
A \emph{screw} \nm{\{\vec S = \lrsb{\vec n, \, \vec m, \, h, \, \phi}^T \in \mathbb{R}^8 \ | \ \vec n, \vec m \in \mathbb{R}^3, h, \phi \in \mathbb{R}\}} consists of an axis \nm{\lrp{\vec n, \, \vec m}}, a pitch \nm{h}, and a magnitude \nm{\phi} \cite{Murray1994}. As all other \nm{\mathbb{SE}\lrp{3}} representations, it contains six degrees of freedom as it shares the line redundancies described in section \ref{subsec:algebra_points_and_vectors}, this is, \nm{\|\vec n\| = 1} and \nm{\vec n^T \, \vec m = 0}. It represents a rotation by an amount \nm{\phi} about the axis \nm{\lrp{\vec n, \, \vec m}} combined by a translation by an amount \nm{d = h \, \phi} parallel to axis \nm{\lrp{\vec n, \, \vec m}}. If \nm{h = \infty}, the corresponding screw motion consists of a pure translation along the axis of the screw by a distance \nm{\phi}. Note that \nm{\vec n} and \nm{\phi} are indeed the direction and magnitude of the rotation vector \nm{\vec r = \vec n \, \phi} defined by (\ref{eq:SO3_rotv_definition}).
Given a rigid body motion represented by the combination of rotation and translation vectors \nm{\lrp{\vec r = \vec n \, \phi, \, \vec T}}, the map converting it into a screw has two versions:
\begin{itemize}
\item \nm{\vec r \neq \vec 0}. If the motion contains both rotation and translation components:
\begin{eqnarray}
\nm{\phi} & = & \nm{\|\vec r \| = \phi}\label{eq:SE_screw_magnitude} \\
\nm{\vec n} & = & \nm{\dfrac{\vec r}{\|\vec r\|}}\label{eq:SE3_screw_line} \\
\nm{\vec m} & = & \nm{\widehat{\vec p}_{\perp} \, \vec n = \frac{1}{2} \lrsb{\widehat{\vec T} \, \vec n + \cot \dfrac{\phi}{2} \, {\lrp{\widehat{\vec n} \, \vec T} \times \vec n}}}\label{eq:SE3_screw_moment} \\
\nm{d} & = & \nm{\vec T^T \, \vec n}\label{eq:SE3_screw_displacement} \\
\nm{h} & = & \nm{\dfrac{d}{\phi}}\label{eq:SE3_screw_pitch}
\end{eqnarray}
\item \nm{\vec r = \vec 0}. If the motion is only a translation, the screw definition changes so it contains an \nm{\infty} pitch, a magnitude equal to the translation amount, and an axis in the direction of {\nm{\vec T}} that passes through the origin. The displacement definition does not change though.
\begin{eqnarray}
\nm{h} & = & \nm{\infty}\label{eq:SE3_screw_pitch_norotation} \\
\nm{\phi} & = & \nm{\|\vec T \|}\label{eq:SE_screw_magnitude_norotation} \\
\nm{\vec n} & = & \nm{\vec T / \phi} \label{eq:SE3_screw_line_norotation} \\
\nm{\vec m} & = & \nm{\vec 0}\label{eq:SE3_screw_moment_norotation} \\
\nm{d} & = & \nm{\vec T^T \, \vec n} \label{eq:SE3_screw_displacement_norotation}
\end{eqnarray}
\end{itemize}
The opposite map, which provides the rotation and translation vectors from a screw, also has two versions:
\begin{itemize}
\item \nm{h \neq \infty}. The screw contains both translation and rotation components:
\begin{eqnarray}
\nm{\vec T} & = & \nm{\vec p_{\perp} - \sin \phi \ \widehat{\vec n} \, \vec p_{\perp} - \cos \phi \, \vec p_{\perp} + d \, \vec n}\label{eq:SE3_screw_translation} \\
\nm{\vec r} & = & \nm{\vec n \, \phi} \label{eq:SE3_screw_rotation}
\end{eqnarray}
\item \nm{h = \infty}. The screw does not rotate:
\begin{eqnarray}
\nm{\vec T} & = & \nm{\vec n \, \phi} \label{eq:SE3_screw_translation_norotation} \\
\nm{\vec r} & = & \nm{\vec 0} \label{eq:SE3_screw_rotation_norotation}
\end{eqnarray}
\end{itemize}
It is however the \emph{exponential map} between the screw and the unit dual quaternion the one that provides a different perspective to the motion of a rigid body. It is built based on the expressions for the axis moment (\ref{eq:SE3_screw_moment}) and the unit dual quaternion (\ref{eq:SE3_unit_dual_quat_from_affine}), first as the sum of two quaternions (\ref{eq:SE3_screw_dual_first}) and next as that of a dual number plus a dual vector (\ref{eq:SE3_screw_dual}):
\begin{eqnarray}
\nm{\vec \zeta} & = & \nm{\lrp{\cos \frac{\phi}{2} + \sin \frac{\phi}{2} \, \vec n} + \lrsb{- \frac{d}{2} \, \sin \frac{\phi}{2} + \sin \frac{\phi}{2} \, \vec m + \frac{d}{2} \, \cos \frac{\phi}{2} \, \vec n} \, \epsilon} \label{eq:SE3_screw_dual_first} \\
& = & \nm{\qr + \qd \, \epsilon = \lrsb{q_{0 r}, \vec q_{vr}}^T + \lrsb{q_{0d}, \vec q_{vd}}^T \epsilon} \nonumber \\
& = & \nm{\lrsb{\cos \frac{\phi}{2} - \frac{d}{2} \, \sin \frac{\phi}{2} \, \epsilon} + \lrsb{\sin \frac{\phi}{2} \, \vec n + \lrp{\sin \frac{\phi}{2} \, \vec m + \frac{d}{2} \, \cos \frac{\phi}{2} \, \vec n} \, \epsilon}}\label{eq:SE3_screw_dual}
\end{eqnarray}
This last expression can be modified based the application of the dual number Taylor expansion (\ref{eq:SE3_dual_number_taylor}) to the sine and cosine:
\begin{eqnarray}
\nm{\cos d^{\diamond}} & = & \nm{\cos \lrp{x + y \, \epsilon} = \cos x - y \, \epsilon \, \sin x}\label{eq:SE3_unit_dual_quat_cos} \\
\nm{\sin d^{\diamond}} & = & \nm{\sin \lrp{x + y \, \epsilon} = \sin x + y \, \epsilon \, \cos x}\label{eq:SE3_unit_dual_quat_sin}
\end{eqnarray}
This results in:
\neweq{\vec \zeta = exp\lrp{\vec S} = \cos \frac{\phi + d \, \epsilon}{2} + \lrp{\vec n + \vec m \, \epsilon} \cdot \sin \frac{\phi + d \, \epsilon}{2} = \cos \frac{\phi^{\diamond}}{2} + {\vec{nm}}^{\diamond} \cdot \sin \frac{\phi^{\diamond}}{2} = \cos \theta^{\diamond} + {\vec{nm}}^{\diamond} \cdot \sin \theta^{\diamond}}{eq:SE3_unit_dual_quat1}
where \nm{{\vec{nm}}^{\diamond} = \vec n + \vec m \, \epsilon} is a unit dual vector, this is, one in which its real part is a unit vector that its orthogonal to its dual part, and \nm{\theta^{\diamond} = \frac{\phi^\diamond}{2} = \frac{\phi + d \, \epsilon}{2}} is a dual number. In fact unit dual quaternions can always be written as (\ref{eq:SE3_unit_dual_quat1}), which is the equivalent to (\ref{eq:SO3_quat_unit}) for unit quaternions.
A process in some aspects similar to that described in section \ref{subsec:RigidBody_rotation_halfrotv} proves that (\ref{eq:SE3_unit_dual_quat1}) is indeed the exponential map \nm{\{exp() : \mathfrak{se}\lrp{3} \rightarrow \mathbb{SE}\lrp{3} | \ \vec S \in \mathbb{R}^8 \rightarrow exp\lrp{\vec S} \in \mathbb{H}_d\}}, which transforms screws into unit dual quaternions. Note that to do so, it is first necessary to represent the screw as the combination of a unit dual vector \nm{{\vec{nm}}^{\diamond} = \vec n + \vec m \, \epsilon} representing the screw axis and a dual number \nm{\theta^{\diamond} = \frac{\phi^\diamond}{2} = \frac{\phi + d \, \epsilon}{2}} containing the rotation angle and translation distance about the screw axis. This map algebraically separates the line information of the screw axis from the pitch and angle values, where the dual vector \nm{\vec{nm}^{\diamond}} represents the axis of the screw motion with its direction vector and the dual angle \nm{\theta^\diamond = \frac{\phi^\diamond}{2}} contains both the translation length and the angle of rotation \cite{Busam2017}. If there is no rotation (\nm{h = \infty}), the resulting unit dual quaternion responds to \nm{\vec \zeta = \qr + \qd \, \epsilon = \vec q_1 + \frac{\phi}{2} \, \vec n \, \epsilon}.
Obtainment of the logarithmic map \nm{\{log() : \mathbb{SE}\lrp{3} \rightarrow \mathfrak{se}\lrp{3} | \ \vec \zeta \in \mathbb{H}_d \rightarrow \vec S \in \mathbb{R}^8\}} is now straightforward, resulting in expressions (\ref{eq:SE3_unit_dual_quat_log_phi}) through (\ref{eq:SE3_unit_dual_quat_log_m}) for the case of rotation plus translation (\nm{\qr \neq \vec q_1}):
\begin{eqnarray}
\nm{\phi} & = & \nm{2 \, \arctan\frac{\|\vec q_{vr}\|}{q_{0 r}}}\label{eq:SE3_unit_dual_quat_log_phi} \\
\nm{\vec n} & = & \nm{\frac{\vec q_{vr}}{\|\vec q_{vr}\|}}\label{eq:SE3_unit_dual_quat_log_l} \\
\nm{d} & = & \nm{- 2 \, \frac{q_{0 d}}{\|\vec q_{vr}\|}}\label{eq:SE3_unit_dual_quat_log_d} \\
\nm{\vec m} & = & \nm{\lrp{\vec q_{vd} - \frac{d \, \, q_{0 r}}{2} \, \vec n} \, {\|\vec q_{vr}\|}^{-1}} \label{eq:SE3_unit_dual_quat_log_m}
\end{eqnarray}
In the no rotation case (\nm{\qr = \vec q_1}), the logarithmic map changes as described above. Note that both the exponential and logarithmic maps share the same surjective traits as those between the rotation vector and unit quaternion described in section \ref{subsubsec:RigidBody_rotation_rodrigues_unit_quat}.
Although inverting the motion by means of the screw is straightforward,
\neweq{\vec S^{-1} = \lrsb{\vec n, \, \vec m, \, h, \, - \phi}^T}{eq:SE3_screw_inversion}
the different \nm{\mathbb{SE}(3)} actions (concatenation, point rotation, vector rotation) are complex and rarely used.
\subsection{Rigid Body Motion Algebraic Operations}\label{subsec:RigidBody_motion_algebra}
As in the case of pure rotations described in section \ref{subsec:RigidBody_rotation_algebra}, the basic algebraic operations of addition, subtraction, multiplication, division, and exponentiation are not defined for objects of the special Euclidean group \nm{\mathbb{SE}\lrp{3}}. However, all rigid body motion representations are closed under a given operation that represents the concatenation of transformations, and define not only an identity transformation that represents the lack of motion, but also an inverse operation representing the opposite movement. The concatenation of transformations and the identity and inverse operations enable the definition of the power, exponential and logarithmic operators (section \ref{subsubsec:RigidBody_motion_algebra_exp_log}), the screw linear interpolation (section \ref{subsubsec:RigidBody_motion_algebra_sclerp}), and the perturbations together with the plus and minus operators (section \ref{subsubsec:RigidBody_motion_algebra_plus_minus}).
\subsubsection{Powers, Exponentials and Logarithms}\label{subsubsec:RigidBody_motion_algebra_exp_log}
Any rigid body motion can be executed by rotating an angle \nm{\phi} about a certain fixed axis \nm{\lrp{\vec n, \, \vec m}} combined with a translation of a distance \nm{d = h \cdot \phi} along that same axis, resulting in the screw \nm{\vec S = \lrsb{\vec n, \, \vec m, \, h, \, \phi}^T} (section \ref{subsec:RigidBody_motion_screw}). As the rotation and translation can be executed simultaneously or one after another, taking a fraction of a screw results in \nm{t \, \vec S = t \, \lrsb{\vec n, \, \vec m, \, h, \, \phi}^T = \lrsb{\vec n, \, \vec m, \, h, \, t \, \phi}^T \, \forall \, t \in \mathbb{R}, \vec S \in \mathfrak{se}\lrp{3}}.
The exponential map defined in (\ref{eq:SE3_unit_dual_quat1}) is named that way because it complies with the behavior of the real exponential function \nm{exp^b\lrp{a} = exp\lrp{a \cdot b} \, \forall \, a, \, b \in \mathbb{R}}. As such, the exponential function \nm{\{exp(): \mathfrak{se}\lrp{3} \times \mathbb{R} \rightarrow \mathbb{SE}\lrp{3} \ | \ \vec S \in \mathfrak{se}\lrp{3}, t \in \mathbb{R} \rightarrow {\mathcal M}^t = exp\lrp{t \, \vec S} \in \mathbb{SE}\lrp{3}\}} is defined as:
\neweq{\vec \zeta^t\lrp{\vec S} = \vec \zeta\lrp{t \, \vec S} = exp\lrp{t \, \vec S} = \cos \frac{t \, \phi^\diamond}{2} + {\vec{nm}}^{\diamond} \cdot \sin \frac{t \, \theta^\diamond}{2} = \cos \frac{t \, \phi + t \, d \, \epsilon}{2} + \lrp{\vec n + \vec m \, \epsilon} \cdot \sin \frac{t \, \phi + t \, d \, \epsilon}{2}}{eq:SE3_dual_exponential_fraction}
In a similar way, the logarithmic map defined in (\ref{eq:SE3_unit_dual_quat_log_phi}) through (\ref{eq:SE3_unit_dual_quat_log_m}) also complies with the behavior of the real logarithmic function \nm{b \cdot log\lrp{a} = log\lrp{a^b} \, \forall \, a, \, b \in \mathbb{R}}. As such, the logarithmic function \nm{\{log : \mathbb{SE}\lrp{3} \times \mathbb{R} \rightarrow \mathfrak{se}\lrp{3} \ | \ \mathcal M \in \mathbb{SE}\lrp{3}, t \in \mathbb{R} \rightarrow t \, \vec S = log\lrp{{\mathcal M}^t} \in \mathfrak{se}\lrp{3}\}} is defined as:
\neweq{log\lrp{\vec \zeta^t\lrp{\vec S}} = log\lrp{exp\lrp{t \, \vec S}} = t \, log\lrp{\vec \zeta\lrp{\vec S}} = t \, log\lrp{exp\lrp{\vec S}} = t \, \vec S}{eq:SE3_quat_logarithmic_fraction}
It is important to remark that although other exponential maps have been defined, with inputs either the transform vector (\ref{eq:SE3_twist_exponential4_a}) or the half transform vector, they can not be employed in the exponential function, as the multiple of a transform vector \nm{t \, \vec \tau = \lrp{t \, \vec s, \, t \, \vec r}} does not result in a uniform movement and hence its associated motion does not coincide with that of the same multiple of the screw \nm{t \, \vec S}.
\subsubsection{Screw Linear Interpolation}\label{subsubsec:RigidBody_motion_algebra_sclerp}
Given two rigid body motions \nm{\mathcal M_0, \, \mathcal M_1 \in \mathbb{SE}\lrp{3}}, \emph{screw linear interpolation} (\hypertt{ScLERP}) is an extension of \hypertt{SLERP} (section \ref{subsubsec:RigidBody_rotation_algebra_slerp}) that obtains a motion function \nm{\mathcal M\lrp{t}, \, t \in \mathbb{R}} that linearly interpolates from \nm{\mathcal M\lrp{0} = \mathcal M_0} to \nm{\mathcal M\lrp{1} = \mathcal M_1} in such a way that the motion occurs with constant rotation and translation velocities \cite{Jia2013}.
If employing unit dual quaternions, \nm{\Delta \vec \zeta} is according to (\ref{eq:SE3_unit_dual_quat_concatenation}) the full motion required to go from \nm{\vec \zeta_0} to \nm{\vec \zeta_1}, such that \nm{\vec \zeta_1 = \vec \zeta_0 \otimes \Delta \vec \zeta}, from where \nm{\Delta \vec \zeta = \vec \zeta_0^{\ast} \otimes \vec \zeta_1}. The corresponding screw is then \nm{\Delta \vec S = \lrsb{\vec n, \, \vec m, \, h, \, \Delta \phi}^T = log\lrp{\Delta \vec \zeta}}. Let's take a fraction of the full screw magnitude \nm{\delta \phi = t \, \Delta \phi} and obtain the corresponding unit dual quaternion:
\begin{eqnarray}
\nm{\delta \vec \zeta} & = & \nm{exp\big(\vec S \lrp{\vec n, \, \vec m, \, h, \, \delta \phi}\big) = exp\big(t \cdot \vec S \lrp{\vec n, \, \vec m, \, h, \, \Delta \phi}\big) = exp\lrp{t \, \Delta \vec S}}\nonumber \\
& = & \nm{exp\big(t \log\lrp{\Delta \vec \zeta}\big) = exp\big(t \, log\lrp{\vec \zeta_0^{\ast} \otimes \vec \zeta_1}\big) = \lrp{\vec \zeta_0^{\ast} \otimes \vec \zeta_1}^t}\label{eq:SE3_interp_partial}
\end{eqnarray}
The interpolated unit dual quaternion is hence the following, which relies on (\ref{eq:SE3_dual_exponential_fraction}) for its solution:
\neweq{\vec \zeta\lrp{t} = \vec \zeta_0 \otimes \lrp{\vec \zeta_0^{\ast} \otimes \vec \zeta_1}^t}{eq:SE3_interp}
The restrictions described in section \ref{subsubsec:RigidBody_rotation_algebra_slerp} intended to ensure that the rotation is executed following the shortest path are also applicable in this case.
\subsubsection{Plus and Minus Operators}\label{subsubsec:RigidBody_motion_algebra_plus_minus}
A perturbed rigid body motion \nm{\widetilde{\mathcal{M}} \in \mathbb{SE}\lrp{3}} can always be expressed as the composition of the unperturbed motion \nm{\mathcal M} with a (usually) small perturbation \nm{\Delta \mathcal{M}}. Perturbations can be specified either at the local or body frame \nm{\FB}, this is, at the local vector space tangent to \nm{\mathbb{SE}\lrp{3}} at the actual pose, in which case they are known as \emph{local perturbations}. They can also be specified at the global frame \nm{\FE}, which coincides with the vector space tangent to \nm{\mathbb{SE}\lrp{3}} at the origin; in this case they are known as \emph{global perturbations}. Local perturbations appear on the right hand side of the motion composition, resulting in \nm{\widetilde{\mathcal M} = \mathcal M \circ \Delta \mathcal{M}^{\sss B}}, while global ones appear to the left, hence \nm{\widetilde{\mathcal M} = \Delta\mathcal{M}^{\sss E} \circ \mathcal M}.
The \emph{plus} and \emph{minus operators} are introduced in section \ref{subsec:algebra_lie} and enable operating with increments of the nonlinear \nm{\mathbb{SE}\lrp{3}} manifold expressed in the linear tangent vector space \nm{\mathfrak{se}\lrp{3}}. There exist right (\nm{\oplus, \, \ominus}) or left (\nm{\boxplus, \, \boxminus}) versions depending on whether the increments are viewed in the local frame (right) or the global one (left). It is important to remark that although perturbations and the plus and left operators are best suited to work with small motion changes (perturbations), the expressions below are generic and work just the same no matter the size of the perturbation.
The right plus operator \nm{\{\oplus : \mathbb{SE}\lrp{3} \times \mathfrak{se}\lrp{3} \rightarrow \mathbb{SE}\lrp{3} \, | \, \widetilde{\mathcal M} = \mathcal M \oplus \ \Delta \vec \tau^{\sss B} = \mathcal M \circ Exp\lrp{\Delta \vec \tau^{\sss B}}\}} produces a motion element \nm{\widetilde{\mathcal M}} resulting from the composition of a reference motion \nm{\mathcal M} with an often small motion \nm{\Delta \vec \tau^{\sss B}}, contained in the tangent space to the reference motion \nm{\mathcal M}, this is, in the local space. The left plus operator \nm{\{\boxplus : \mathfrak{se}\lrp{3} \times \mathbb{SE}\lrp{3} \rightarrow \mathbb{SE}\lrp{3} \, | \, \widetilde{\mathcal M} = \Delta \vec \tau^{\sss E} \boxplus \mathcal M = Exp\lrp{\Delta \vec \tau^{\sss E}} \circ \mathcal M\}} is similar but the often small motion \nm{\Delta \vec \tau^{\sss E}} is contained in the tangent space at the identify or global space. The expressions shown below are valid up to the first coverage of \nm{\mathbb{SE}\lrp{3}}, this is, \nm{\phi < \pi}. In the cases of homogeneous matrix and unit dual quaternion, the plus operator is defined as:
\begin{eqnarray}
\nm{\widetilde{\vec M}} & = & \nm{\vec M \oplus \Delta \vec \tau^{\sss B} = \vec M \ Exp\lrp{\Delta \vec \tau^{\sss B}} = \vec M \ \Delta \vec M^{\sss B}}\label{eq:SE3_homogeneous_plus} \\
\nm{\widetilde{\vec \zeta}} & = & \nm{\vec \zeta \oplus \Delta \vec \tau^{\sss B} = \vec \zeta \otimes Exp\lrp{\Delta \vec \tau^{\sss B} / 2} = \vec \zeta \otimes \Delta \vec \zeta^{\sss B}}\label{eq:SE3_dual_quat_plus} \\
\nm{\widetilde{\vec M}} & = & \nm{\Delta \vec \tau^{\sss E} \boxplus \vec M = Exp\lrp{\Delta \vec \tau^{\sss E}} \ \vec M = \Delta \vec M^{\sss E} \ \vec M}\label{eq:SE3_homogeneous_plus_left} \\
\nm{\widetilde{\vec \zeta}} & = & \nm{\Delta \vec \tau^{\sss E} \boxplus \vec \zeta = Exp\lrp{\Delta \vec \tau^{\sss E} / 2} \otimes \vec \zeta = \Delta \vec \zeta^{\sss E} \otimes \vec \zeta}\label{eq:SE3_dual_quat_plus_left}
\end{eqnarray}
The right minus operator \nm{\{\ominus : \mathbb{SE}\lrp{3} \times \mathbb{SE}\lrp{3} \rightarrow \mathfrak{se}\lrp{3} \, | \, \Delta \vec \tau^{\sss B} = \widetilde{\mathcal M} \ominus \mathcal M = Log\big(\mathcal M^{-1} \circ \widetilde{\mathcal M}\big)\}}, as well as the left \nm{\{\boxminus : \mathbb{SE}\lrp{3} \times \mathbb{SE}\lrp{3} \rightarrow \mathfrak{se}\lrp{3} \, | \, \Delta \vec \tau^{\sss E} = \widetilde{\mathcal M} \boxminus \mathcal M = Log\big(\widetilde{\mathcal M} \circ \mathcal M^{-1}\big)\}}, represent the inverse operations, returning the transform vector difference \nm{\Delta \vec \tau} between two motions \nm{\mathcal M} and \nm{\widetilde{\mathcal M}} expressed in either the local or global tangent spaces to \nm{\mathcal M}.
\begin{eqnarray}
\nm{\Delta \vec \tau^{\sss B} } & = & \nm{\widetilde{\vec M} \ominus \vec M = Log\lrp{\vec M^{-1} \ \widetilde{\vec M}} = Log\lrp{\Delta \vec M^{\sss B}}}\label{eq:SE3_homogeneous_minus} \\
\nm{\Delta \vec \tau^{\sss B} } & = & \nm{\widetilde{\vec \zeta} \ominus \vec \zeta = 2 \ Log\lrp{\zetaast \otimes \widetilde{\vec \zeta}} = 2 \ Log\lrp{\Delta \vec \zeta^{\sss B}}}\label{eq:SE3_dual_quat_minus} \\
\nm{\Delta \vec \tau^{\sss E} } & = & \nm{\widetilde{\vec M} \boxminus \vec M = Log\lrp{\widetilde{\vec M} \ \vec M^{-1}} = Log\lrp{\Delta \vec M^{\sss E}}}\label{eq:SE3_homogeneous_minus_left} \\
\nm{\Delta \vec \tau^{\sss E} } & = & \nm{\widetilde{\vec \zeta} \boxminus \vec \zeta = 2 \ Log\lrp{\widetilde{\vec \zeta} \otimes \zetaast} = 2 \ Log\lrp{\Delta \vec \zeta^{\sss E}}}\label{eq:SE3_dual_quat_minus_left}
\end{eqnarray}
If the \nm{\Delta \vec \tau} perturbation is small, the (\ref{eq:SE3_twist_exponential3}) and (\ref{eq:SE3_Upsilon_exponential3}) Taylor expansions can be truncated, resulting in the following expressions, valid for both the body frame (\nm{\Delta \vec \tau^{\sss B}}) or the global one (\nm{\Delta \vec \tau^{\sss E}}):
\begin{eqnarray}
\nm{\Delta \vec M = Exp\lrp{\Delta \vec \tau}} & \nm{\approx} & \nm{\vec I_4 + \Delta \vec \tau^\wedge = \vec I_4 + \lrsb{\vec k \, \Delta \rho, \, \vec n \, \Delta \phi}^\wedge}\label{eq:SE3_perturbation_homogeneous_truncated_local} \\
\nm{\Delta \vec \zeta = exp\lrp{\Delta \vec \tau /2}} & \nm{\approx} & \nm{\vec {\zeta_1} + \Delta \vec \tau^\wedge / 2 = \big[\lrsb{1, \vec n \, \Delta \phi / 2}^T + \epsilon \, \vec k \, \Delta \rho /2\big]^\wedge} \label{eq:SE3_perturbation_dual_quat_truncated_local}
\end{eqnarray}
\subsection{Rigid Body Motion Time Derivative and Twist}\label{subsec:RigidBody_motion_calculus_derivatives}
Let's consider a moving rigid body \nm{\mathcal M\lrp{t} \in \mathbb{SE}\lrp{3}, t \in \mathbb{R}} and compute its derivative with time, which belongs to neither \nm{\mathbb{SE}\lrp{3}} nor \nm{\mathfrak{se}\lrp{3}} but to the Euclidean space of the chosen motion representation, \nm{\mathbb{R}^{4x4}} for the homogeneous matrix and \nm{\mathbb{H}_d} for the unit dual quaternion:
\neweq{\dot{\mathcal{M}}\lrp{t} = \lim\limits_{\Delta t \to 0} \dfrac{\mathcal{M}\lrp{t + \Delta t} - \mathcal{M}\lrp{t}}{\Delta t}}{eq:SE3_time_derivative_def}
Considering the time modified motion \nm{\mathcal{M}\lrp{t + \Delta t}} as the perturbed state (section \ref{subsubsec:RigidBody_motion_algebra_plus_minus}), the resulting time derivatives for the homogeneous matrix and unit dual quaternion representations are the following:
\begin{eqnarray}
\nm{\vec {\dot M}\lrp{t}} & = & \nm{\lim\limits_{\Delta t \to 0} \dfrac{\vec M \, \Delta \vec M^{\sss B} - \vec M}{\Delta t} \ \nm{\approx} \lim\limits_{\Delta t \to 0}\dfrac{\vec M \, \Big[\big(\vec I_4 + \lrsb{\vec k^{\sss B} \, \Delta \rho, \, \vec n^{\sss B} \, \Delta \phi}^\wedge\big) - \vec I_4\Big]}{\Delta t}} \nonumber \\
& = & \nm{\vec M \, \lim\limits_{\Delta t \to 0}\dfrac{\lrsb{\Delta \rho \, \vec k^{\sss B}, \, \Delta \phi \, \vec n^{\sss B}}^\wedge}{\Delta t}}\label{eq:SE3_time_derivative_M1b} \\
\nm{\vec {\dot \zeta}\lrp{t}} & = & \nm{\lim\limits_{\Delta t \to 0} \dfrac{\vec \zeta \otimes \Delta \vec \zeta^{\sss B} - \vec \zeta}{\Delta t} \ \nm{\approx} \lim\limits_{\Delta t \to 0}\dfrac{\vec \zeta \otimes \lrsb{\lrp{\lrsb{1, \vec n^{\sss B} \, \Delta \phi / 2}^T + \epsilon \, \vec k^{\sss B} \, \Delta \rho /2}^\wedge - \vec{\zeta_1}}}{\Delta t}} \nonumber \\
& = & \nm{\vec \zeta \otimes \lim\limits_{\Delta t \to 0}\dfrac{\lrsb{\vec n^{\sss B} \, \Delta \phi / 2 + \epsilon \, \vec k^{\sss B} \, \Delta \rho /2}^\wedge}{\Delta t}}\label{eq:SE3_time_derivative_zeta1b}
\end{eqnarray}
Similar expressions based on \nm{\vec \tau^{\sss E} = \lrsb{\Delta \rho \, \vec k^{\sss E}, \, \Delta \phi \, \vec n^{\sss E}}^T} can be found if left multiplying by the perturbation instead of right multiplying. The \nm{\vec {\dot M}\lrp{t}} and \nm{\vec {\dot \zeta}\lrp{t}} expressions (\ref{eq:SE3_homogeneous_dot}) and (\ref{eq::SE3_dual_quat_xi_dot}) are then directly obtained when defining the \emph{body twist} \nm{\vec \xi_{\sss EB}^{\sss B}} as the time derivative of the transform vector \nm{\vec \tau^{\sss B}} when viewed in local or body frame \nm{\FB}, and the \emph{spatial twist} \nm{\vec \xi_{\sss EB}^{\sss E}} as the time derivative of the transform vector \nm{\vec \tau^{\sss E}} when viewed in global or spatial frame \nm{\FE}:
\begin{eqnarray}
\nm{\vec \xi_{\sss EB}^{\sss B}\lrp{t}} & = & \nm{\Delta \vec{\dot \tau}^{\sss B}\lrp{t} = \lim\limits_{\Deltat \to 0} \frac{\Delta \vec \tau^{\sss B}}{\Deltat} = \lim\limits_{\Deltat \to 0} \frac{\lrsb{\vec k^{\sss B} \, \Delta \rho, \, \vec n^{\sss B} \, \Delta \phi}^T}{\Deltat}}\label{eq:SE3_time_derivative_xiEBB} \\
\nm{\vec \xi_{\sss EB}^{\sss E}\lrp{t}} & = & \nm{\Delta \vec{\dot \tau}^{\sss E}\lrp{t} = \lim\limits_{\Deltat \to 0} \frac{\Delta \vec \tau^{\sss E}}{\Deltat} = \lim\limits_{\Deltat \to 0} \frac{\lrsb{\vec k^{\sss E} \, \Delta \rho, \, \vec n^{\sss E} \, \Delta \phi}^T}{\Deltat}}\label{eq:SE3_time_derivative_xiEBE}
\end{eqnarray}
The twist \nm{\vec \xi = \lrsb{\vec \nu, \, \vec \omega}^T} represents the motion velocity and is composed by the angular velocity \nm{\vec \omega} defined in section \ref{subsec:RigidBody_rotation_calculus_derivatives} and the linear velocity \nm{\vec \nu}. The twist physical meaning is revealed by obtaining its expressions when viewed in both the local and spatial frames. The body twist \nm{\xiEBBskew \in \mathfrak{se}\lrp{3}} corresponding to the rigid body motion \nm{\MEB\lrp{t} \in \mathbb{SE}\lrp{3}} responds to (\ref{eq:SE3_homogeneous_twist_body}):
\neweq{\xiEBBskew = \begin{bmatrix} \nm{\wEBBskew} & \nm{\nuEBB} \\ 0 & 0 \end{bmatrix} = \MEBinv \; \MEBdot = \begin{bmatrix} \nm{\REBtrans} & \nm{- \REBtrans \, \TEBE} \\ 0 & 1 \end{bmatrix} \begin{bmatrix} \nm{\REBdot} & \nm{\vec {\dot T}_{\sss EB}^{\sss E}} \\ 0 & 0 \end{bmatrix} = \begin{bmatrix} \nm{\REBtrans \; \REBdot} & \nm{\REBtrans \; \vec {\dot T}_{\sss EB}^{\sss E}} \\ 0 & 0 \end{bmatrix}}{eq:SE3_time_derivative_twist_body}
Its physical interpretation is that the angular component \nm{\wEBB} is indeed the (\ref{eq:SO3_dcm_omega_body}) angular velocity \nm{\vec \omega_{\sss EB}} as viewed from the body frame, and the linear component \nm{\nuEBB} is the linear velocity of the body frame origin \nm{\vec {\dot T}_{\sss EB}} also viewed from the body frame (\ref{eq:SO3_dcm_transform}, \ref{eq:SO3_dcm_inverse}) \cite{Murray1994}. The global twist \nm{\xiEBEskew \in \mathfrak{se}\lrp{3}} is determined by means of (\ref{eq:SE3_homogeneous_twist_space}):
\begin{eqnarray}
\nm{\xiEBEskew} & = & \nm{\begin{bmatrix} \nm{\wEBEskew} & \nm{\nuEBE} \\ 0 & 0 \end{bmatrix} = \MEBdot \; \MEBinv = \begin{bmatrix} \nm{\REBdot} & \nm{\vec {\dot T}_{\sss EB}^{\sss E}} \\ 0 & 0 \end{bmatrix} \begin{bmatrix} \nm{\REBtrans} & \nm{- \REBtrans \, \TEBE} \\ 0 & 1 \end{bmatrix}} \nonumber \\
& = & \nm{\begin{bmatrix} \nm{\REBdot \; \REBtrans} & \nm{\vec {\dot T}_{\sss EB}^{\sss E} - \REBdot \; \REBtrans \; \vec T_{\sss EB}^{\sss E}} \\ 0 & 0 \end{bmatrix}} \label{eq:SE3_time_derivative_twist_space}
\end{eqnarray}
The \nm{\xiEBEskew} physical interpretation is not intuitive, however. While the angular component \nm{\wEBE} is the (\ref{eq:SO3_dcm_omega_space}) angular velocity \nm{\wEB} as viewed from the spatial frame, its linear component \nm{\nuEBE} is not the velocity of the body frame origin \nm{\vec {\dot T}_{\sss EB}} viewed in the \nm{\FE} frame (\nm{\vec {\dot T}_{\sss EB}^{\sss E}}), but the velocity, viewed in the spatial frame, of a possibly imaginary point of the rigid body which at time \nm{t} is traveling through the origin of the spatial frame \cite{Murray1994}.
Note that the transformation or motion of the twist (relationship between \nm{\xiEBE} and \nm{\xiEBB}), and hence that of the linear velocities \nm{\nuEBE} and \nm{\nuEBB}, is not given by the motion action \nm{\vec g_{\mathcal M*}} (\ref{eq:Motion_maps}) but by the adjoint map \nm{\vec{Ad}_{\mathcal M}} described in section \ref{subsec:RigidBody_motion_adjoint}. Unlike in the case of rotations, these two maps do not coincide.
\subsection{Rigid Body Motion Point Velocity}\label{subsec:RigidBody_motion_velocity}
There exists a direct relationship between the velocity of a point belonging to a rigid body and the elements of its tangent space, this is, the twist in \nm{\mathfrak{se}(3)}. This relationship is independent of the \nm{\mathbb{SE}(3)} representation, although the homogeneous matrix is employed in the expressions below. If \nm{\pBbar = \lrsb{\pB, \, 1}^T} are the fixed coordinates of a point belonging to the \nm{\FB} rigid body, the point spatial coordinates \nm{\pEbar = \lrsb{\pE, \, 1}^T} can be obtained by means of (\ref{eq:SE3_homogeneous_transform}):
\neweq{\pEbar\lrp{t} = \vec g_{\mathcal M_{EB}(t)}\lrp{\pBbar} = \MEB\lrp{t} \; \pBbar} {eq:SE3_velocity1}
The velocity of a point is the time derivative of its spatial or global coordinates. As \nm{\vec {\bar p}} is fixed to \nm{\FB}, its time derivative is zero \nm{\lrp{\pBbardot = \vec 0}}, so its velocity viewed in the spatial frame responds to:
\neweq{\vpEbar\lrp{t} = \pEbardot\lrp{t} = \MEBdot\lrp{t} \; \pBbar}{eq:SE3_velocity2}
Although \nm{\MEBdot} maps the point body coordinates to its spatial velocity per (\ref{eq:SE3_velocity2}), its high dimension makes it inefficient. By making use of the spatial and body twists (\nm{\vec \xi_{\sss EB}^{{\sss E}\wedge}, \, \vec \xi_{\sss EB}^{{\sss B}\wedge}}) introduced in (\ref{eq:SE3_homogeneous_dot}), the velocity of a point \nm{\pBbar} viewed in \nm{\FE} can be obtained as follows:
\begin{eqnarray}
\nm{\vpEbar\lrp{t}} & = & \nm{\xiEBEskew\lrp{t} \; \MEB\lrp{t} \; \pBbar = \xiEBEskew\lrp{t} \; \pEbar\lrp{t}} \label{eq:SE3_velocity_v_e} \\
\nm{\vpEbar\lrp{t}} & = & \nm{\MEB\lrp{t} \; \xiEBBskew\lrp{t} \; \pBbar} \label{eq:SE3_velocity_v_e_bis}
\end{eqnarray}
The velocity of \nm{\pBbar} viewed in \nm{\FB} can then be obtained by means of the vector action map:
\neweq{\vpBbar\lrp{t} = \vec g_{\mathcal M_{EB(t)}*}^{-1} \big(\vpEbar\lrp{t}\big) = \MEBinv\lrp{t} \; \vpEbar\lrp{t} = \xiEBBskew\lrp{t} \; \pBbar} {eq:SE3_velocity_v_b}
Returning to cartesian coordinates and introducing the angular and linear components of the twist results in:
\begin{eqnarray}
\nm{\vec v_{\sss p}^{\sss E}\lrp{t}} & = & \nm{\wEBEskew\lrp{t} \; \pE\lrp{t} + \nuEBE\lrp{t}}\label{eq:SE3_velocity_e} \\
\nm{\vec v_{\sss p}^{\sss B}\lrp{t}} & = & \nm{\wEBBskew\lrp{t} \; \pB + \nuEBB\lrp{t}}\label{eq:SE3_velocity_b}
\end{eqnarray}
The point velocity is hence the result of the sum of the linear velocity and the cross product between the angular velocity and the point coordinates.
\subsection{Rigid Body Motion Adjoint}\label{subsec:RigidBody_motion_adjoint}
The \emph{adjoint map} of a Lie group is defined in section \ref{subsubsec:algebra_lie_adjoint} as an action of the Lie group on its own Lie algebra that converts between the local tangent space and that at the identity. In the case of rigid body motion, both the transform vector and the twist belong to the tangent space, so \nm{\lrb{\vec{Ad}\lrp{}: \mathbb{SE}(3) \times \mathfrak{se}(3) \rightarrow \mathfrak{se}(3) \ | \ \vec{Ad}_{\mathcal M}\lrp{\vec \tau^{\wedge}} = \mathcal M \circ \vec \tau^{\wedge} \circ \mathcal{M}^{-1}, \ \vec{Ad}_{\mathcal M}\lrp{\vec \xi^{\wedge}} = \mathcal M \circ \vec \xi^{\wedge} \circ \mathcal{M}^{-1}}}. This is equivalent to \nm{\vec \zeta \otimes \vec \xi^{\wedge} \otimes \vec \zeta^{\ast}} for unit dual quaternions or \nm{\vec M \, \vec \xi^\wedge \, \vec M^{-1}} for homogeneous matrices, which represents the similarity transformation\footnote{Two square matrices \nm{\vec A} and \nm{\vec B} are called similar if \nm{\vec B = {\vec P}^{-1} \; \vec A \; \vec P} for some invertible square matrix \nm{\vec P}.} between the spatial and body twists \nm{\vec \xi_{\sss EB}^{{\sss E}\wedge}} and \nm{\vec \xi_{\sss EB}^{{\sss B}\wedge}}:
\neweq{\xiEBEskew = \MEB \; \xiEBBskew \; \MEBinv \rightarrow \left\{\begin{aligned} \nm{\wEBEskew} & = \nm{\REB \; \wEBBskew \; \REBtrans = \vec{Ad}_{\mathcal R_{EB}}\lrp{\wEBBskew}} \\ \nm{\nuEBE} & = \nm{\REB \; \nuEBB - \wEBEskew \; \TEBE = \REB \; \nuEBB + \TEBEskew \; \wEBE} \end{aligned} \right.}{eq:SE3_velocity_similarity}
The application of the vee operator results in the adjoint matrix:
\neweq{\xiEBE = \begin{bmatrix} \nm{\nuEBE} \\ \nm{\wEBE} \end{bmatrix} = \begin{bmatrix} \nm{\REB} & \nm{\TEBEskew \; \REB} \\ 0 & \nm{\REB} \end{bmatrix} \, \begin{bmatrix} \nm{\nuEBB} \\ \nm{\wEBB} \end{bmatrix} = \begin{bmatrix} \nm{\REB} & \nm{\TEBEskew \; \REB} \\ 0 & \nm{\REB} \end{bmatrix} \, \xiEBB = \vec{Ad}_{\mathcal M_{EB}} \, \xiEBB}{eq:SE3_velocity_similarity2}
As stated above, note that the adjoint map (\ref{eq:SE3_velocity_similarity2}) is different than the vector action \nm{\vec g_{\mathcal M*}} (\ref{eq:Motion_maps}), unlike the case of rotational motion described in section \ref{subsec:RigidBody_rotation_adjoint}, in which they coincide.
A similar process leads to the inverse adjoint matrix (\nm{\vec{Ad}_{\mathcal M}^{-1} \, \vec \xi = \vec{Ad}_{\mathcal M^{-1}} \, \vec \xi}):
\neweq{\xiEBB = \vec{Ad}_{\mathcal M_{EB}}^{-1} \; \xiEBE = \begin{bmatrix} \nm{\REB^T} & \nm{- \REB^T \, \TEBEskew} \\ 0 & \nm{\REB^T} \end{bmatrix} \; \xiEBE}{eq:SE3_dcm_velocity7}
\subsection{Rigid Body Motion Uncertainty and Covariance}\label{subsec:RigidBody_motion_covariance}
Following the analysis of uncertainty on Lie groups presented in section \ref{subsubsec:algebra_lie_covariance}, the definitions of local and global autocovariances for \nm{\mathbb{SE}(3)} elements around a nominal or expected rotation \nm{E\lrsb{\mathcal M} = \vec \mu_{\mathcal M} \in \mathbb{SE}(3)} are the following:
\begin{eqnarray}
\nm{\vec C_{\mathcal M \mathcal M}^{\sss B}} & = & \nm{E\lrsb{\Delta \vec \tau^{\sss B} \, \Delta \vec \tau^{{\sss B}T}} = E\lrsb{\lrp{\mathcal M \ominus \vec \mu_{\mathcal M}}\lrp{\mathcal M \ominus \vec \mu_{\mathcal M}}^T} \ \ \in \mathbb{R}^{6x6}}\label{eq:SE3_covariance_right_def} \\
\nm{\vec C_{\mathcal M \mathcal M}^{\sss E}} & = & \nm{E\lrsb{\Delta \vec \tau^{\sss E} \, \Delta \vec \tau^{{\sss E}T}} = E\lrsb{\lrp{\mathcal M \boxminus \vec \mu_{\mathcal M}}\lrp{\mathcal M \boxminus \vec \mu_{\mathcal M}}^T} \ \ \in \mathbb{R}^{6x6}}\label{eq:SE3_covariance_left_def}
\end{eqnarray}
Note that although the notation refers to the covariance of the rigid body motion manifold \nm{\mathcal M \in \mathbb{SE}(3)}, the definition in fact refers to the covariance of the transform vectors \nm{\Delta \vec \tau^{\sss B}} or \nm{\Delta \vec \tau^{\sss E}} located in the tangent space, with its dimension (6) matching the number of degrees of freedom of the \nm{\mathbb{SE}(3)} manifold. The relationship between the local and global autocovariances responds to:
\neweq{\vec C_{\mathcal M \mathcal M}^{\sss E} = \vec{Ad}_{\mathcal M_{EB}} \ \vec C_{\mathcal M \mathcal M}^{\sss B} \ \vec{Ad}_{\mathcal M_{EB}}^T} {eq:SE3_covariance_left_relationship}
Given a function \nm{\lrb{f: \mathcal{M} \rightarrow \mathcal {N} \ | \ \mathcal {N} = f\lrp{\mathcal {M}} \in \mathbb{SE}(3), \, \forall \mathcal {M} \in \mathbb{SE}(3)}} between two rigid body motions, the covariances are propagated as follows:
\begin{eqnarray}
\nm{\vec C_{\mathcal N \mathcal N}^{\sss B}} & = & \nm{\vec J_{\ds{\oplus \; \mathcal M}}^{\ds{\oplus \; f\lrp{\mathcal M}}} \ \vec C_{\mathcal M \mathcal M}^{\sss B} \ \vec J_{\ds{\oplus \; \mathcal M}}^{{\ds{\oplus \; f\lrp{\mathcal M}}},T} \ \ \ \ \ \ \ \in \mathbb{R}^{6x6}} \label{eq:SE3_covariance_right_propagation} \\
\nm{\vec C_{\mathcal N \mathcal N}^{\sss E}} & = & \nm{\vec J_{\ds{\boxplus \; \mathcal M}}^{\ds{\boxplus \; f\lrp{\mathcal M}}} \ \vec C_{\mathcal M \mathcal M}^{\sss E} \ \vec J_{\ds{\boxplus \; \mathcal M}}^{{\ds{\boxplus \; f\lrp{\mathcal M}}},T} \ \ \ \ \ \ \ \in \mathbb{R}^{6x6}} \label{eq:SE3_covariance_left_propagation}
\end{eqnarray}
\subsection{Rigid Body Motion Jacobians}\label{subsec:RigidBody_motion_calculus_jacobians}
Lie group jacobians are introduced in section \ref{subsec:algebra_lie_jacobians} based on the right and left Lie group derivatives of section \ref{subsubsec:algebra_lie_derivatives}, and in this section are customized for the \nm{\mathbb{SE}(3)} case, with table \ref{tab:RigidBody_motion_jacobians} representing the particularization of table \ref{tab:algebra_lie_jacobians} to the case of rigid body motions. The various jacobians listed in table \ref{tab:RigidBody_motion_jacobians} have been obtained by means of the chain rule, the expressions already introduced in this article, and those of section \ref{subsec:algebra_lie}. Note that although in many cases the results internally include the rotation matrix, all jacobians are generic and do not depend on the specific \nm{\mathbb{SE}(3)} parameterization.
In addition to the adjoint matrix, two other jacobians are of particular importance as they appear repeatedly in table \ref{tab:RigidBody_motion_jacobians}. These are the right and left jacobians of the capitalized exponential function, also known as simply the \emph{right jacobian} \nm{J_R\lrp{\vec \tau}} and the \emph{left jacobian} \nm{J_L\lrp{\vec \tau}}, and they evaluate the variation of the \nm{\mathfrak{se}(3)} tangent space provided by the output of the \nm{Exp\lrp{\vec \tau}} map (locally for \nm{J_R} and globally for \nm{J_L}) while moving along the \nm{\mathbb{SE}\lrp{3}} manifold with respect to the (Euclidean) variations within the original tangent space provided by \nm{\vec \tau}. Their closed forms as well as those of their inverses are included in table \ref{tab:RigidBody_motion_jacobians}, and have been obtained from \cite{Barfoot2014}; they are based on the \nm{\vec Q\lrp{\vec \tau}} matrix:
\begin{eqnarray}
\nm{\vec Q\lrp{\vec \tau}} & = & \nm{\vec Q\lrp{\vec s, \, \vec r} = \vec Q\lrp{\vec k \, \rho, \, \vec n \, \phi} = \dfrac{\widehat{\vec s}}{2} + \dfrac{\phi - \sin \phi}{\phi^3}\lrp{\widehat{\vec r} \, \widehat{\vec s} + \widehat{\vec s} \, \widehat{\vec r} + \widehat{\vec r} \, \widehat{\vec s} \, \widehat{\vec r}}} \nonumber \\
& \nm{-} & \nm{\dfrac{1 - \phi^2 /2 - \cos \phi}{\phi^4}\lrp{\widehat{\vec r}^2 \, \widehat{\vec s} + \widehat{\vec s} \, \widehat{\vec r}^2 - 3 \, \widehat{\vec r} \, \widehat{\vec s} \, \widehat{\vec r}}} \nonumber \\
& \nm{-} & \nm{\dfrac{1}{2} \lrsb{\dfrac{1 - \phi^2 /2 - \cos \phi}{\phi^4} - 3 \dfrac{\phi - \sin \phi - \phi^3 /6}{\phi^5}} \lrp{\widehat{\vec r} \, \widehat{\vec s} \, \widehat{\vec r}^2 + \widehat{\vec r}^2 \, \widehat{\vec s} \, \widehat{\vec r}} \ \ \ \in \mathbb{R}^{3x3}} \label{eq:SE3_jacobian_left_Q}
\end{eqnarray}
It is also worth noting the special importance of the \nm{\vec J_{\ds{+ \; \vec \tau}}^{\ds{+ \; g_{Exp\lrp{\vec \tau}}(\vec p)}}} jacobian present at the bottom of table \ref{tab:RigidBody_motion_jacobians}, which represents the derivative of a transformed point with respect to perturbations in the Euclidean tangent space (not on the curved manifold) that generates the motion, as it enables tangent space optimization by calculus methods designed exclusively for Euclidean spaces.
\renewcommand{\arraystretch}{1.5}
\begin{center}
\begin{tabular}{lcccll}
\hline
\textbf{Jacobian} & & \textbf{Table \ref{tab:algebra_lie_jacobians}} & & \multicolumn{1}{c}{\textbf{Expression}} & \textbf{Size} \\
\hline
\nm{\vec J_{\ds{\oplus \; \mathcal M}}^{\ds{\oplus \; \mathcal M}^{-1}}} & = & \nm{- \vec{Ad}_{\mathcal M}} & = & \nm{- \lrsb{\vec R \ \ \Tskew \; \vec R; \ \ \vec{0}_{3x3} \ \ \vec R}} & \nm{\in \mathbb R^{6x6}} \\
\nm{\vec J_{\ds{\boxplus \; \mathcal M}}^{\ds{\boxplus \; \mathcal M}^{-1}}} & = & \nm{- \vec{Ad}_{\mathcal M}^{-1}} & = & \nm{- \lrsb{\vec R^T \ \ - \vec R^T \, \Tskew; \ \ \vec{0}_{3x3} \ \ \vec R^T}} & \nm{\in \mathbb R^{6x6}} \\
\nm{\vec J_{\ds{\oplus \; \mathcal M}}^{\ds{\oplus \; \mathcal M \circ \mathcal N}}} & = & \nm{\vec{Ad}_{\mathcal N}^{-1}} & = & \nm{\lrsb{\vec R_{\mathcal N}^T \ \ - \vec R_{\mathcal N}^T \, \widehat{\vec T}_{\mathcal N}; \ \ \vec{0}_{3x3} \ \ \vec R_{\mathcal N}^T}} & \nm{\in \mathbb R^{6x6}} \\
\nm{\vec J_{\ds{\boxplus \; \mathcal M}}^{\ds{\boxplus \; \mathcal M \circ \mathcal N}}} & = & \nm{\vec I} & = & \nm{\vec{I}_{6x6}} & \nm{\in \mathbb R^{6x6}} \\
\nm{\vec J_{\ds{\oplus \; \mathcal N}}^{\ds{\oplus \; \mathcal M \circ \mathcal N}}} & = & \nm{\vec I} & = & \nm{\vec{I}_{6x6}} & \nm{\in \mathbb R^{6x6}} \\
\nm{\vec J_{\ds{\boxplus \; \mathcal N}}^{\ds{\boxplus \; \mathcal M \circ \mathcal N}}} & = & \nm{\vec{Ad}_{\mathcal M}} & = & \nm{\lrsb{\vec R_{\mathcal M} \ \ \widehat{\vec T}_{\mathcal M} \; \vec R_{\mathcal M}; \ \ \vec{0}_{3x3} \ \ \vec R_{\mathcal M}}} & \nm{\in \mathbb R^{6x6}} \\
\nm{\vec J_{\ds{+ \; \vec \tau}}^{\ds{\oplus \; Exp\lrp{\vec \tau}}}} & = & \nm{ J_R\lrp{\vec \tau}} & = & \nm{\vec J_L\lrp{- \vec \tau} = \vec J_L\lrp{- \vec s, \; - \vec r}} & \nm{\in \mathbb R^{6x6}} \\
\nm{J_R^{-1}\lrp{\vec \tau}} & & & = & \nm{\vec J_L^{-1}\lrp{- \vec \tau} = \vec J_L^{-1}\lrp{- \vec s, \; - \vec r}} & \nm{\in \mathbb R^{6x6}} \\
\nm{\vec J_{\ds{+ \; \vec \tau}}^{\ds{\boxplus \; Exp\lrp{\vec \tau}}}} & = & \nm{J_L\lrp{\vec \tau}} & = & \nm{\Big[\vec J_L\lrp{\vec r} \ \ \vec Q\lrp{\vec \tau}; \ \ \vec{0}_{3x3} \ \ \vec J_L\lrp{\vec r}\Big]} & \nm{\in \mathbb R^{6x6}} \\
\nm{J_L^{-1}\lrp{\vec \tau}} & & & = & \nm{\Big[\vec J_L^{-1}\lrp{\vec r} \ \ - \vec J_L^{-1}\lrp{\vec r} \; \vec Q\lrp{\vec \tau} \, \vec J_L^{-1}\lrp{\vec r}; \ \ \vec{0}_{3x3} \ \ \vec J_L^{-1}\lrp{\vec r} \Big]} & \nm{\in \mathbb R^{6x6}} \\
\nm{\vec J_{\ds{\oplus \; \mathcal M}}^{\ds{+ \; Log\lrp{\mathcal M}}}} & = & \nm{J_R^{-1}\big(Log\lrp{\mathcal M}\big)} & & & \nm{\in \mathbb R^{6x6}} \\
\nm{\vec J_{\ds{\boxplus \; \mathcal M}}^{\ds{+ \; Log\lrp{\mathcal M}}}} & = & \nm{J_L^{-1}\big(Log\lrp{\mathcal M}\big)} & & & \nm{\in \mathbb R^{6x6}} \\
\nm{\vec J_{\ds{\oplus \; \mathcal M}}^{\ds{\oplus \; \mathcal M \oplus \vec \tau}}} & = & \nm{\vec{Ad}_{Exp\lrp{\vec \tau}}^{-1}} & = & \nm{\lrsb{\vec R\lrp{\vec r}^T \ \ - \vec R\lrp{\vec r}^T \Tskew\lrp{\vec s, \vec r}; \ \ \vec{0}_{3x3} \ \ \vec R\lrp{\vec r}^T}} & \nm{\in \mathbb R^{6x6}} \\
\nm{\vec J_{\ds{\boxplus \; \mathcal M}}^{\ds{\boxplus \; \vec \tau \boxplus \mathcal M}}} & = & \nm{\vec{Ad}_{Exp\lrp{\vec \tau}}} & = & \nm{\lrsb{\vec R\lrp{\vec r} \ \ \Tskew\lrp{\vec s, \vec r} \vec R\lrp{\vec r}; \ \ \vec{0}_{3x3} \ \ \vec R\lrp{\vec r}}} & \nm{\in \mathbb R^{6x6}} \\
\nm{\vec J_{\ds{+ \; \vec \tau}}^{\ds{\oplus \; \mathcal M \oplus \vec \tau}}} & = & \nm{J_R\lrp{\vec \tau}} & & & \nm{\in \mathbb R^{6x6}} \\
\nm{\vec J_{\ds{+ \; \vec \tau}}^{\ds{\boxplus \; \vec \tau \boxplus \mathcal M}}} & = & \nm{J_L\lrp{\vec \tau}} & & & \nm{\in \mathbb R^{6x6}} \\
\nm{\vec J_{\ds{\oplus \; \mathcal M}}^{\ds{+ \; \mathcal N \ominus \mathcal M}}} & = & \nm{- J_L^{-1}\lrp{\mathcal N \ominus \mathcal M}} & & & \nm{\in \mathbb R^{6x6}} \\
\nm{\vec J_{\ds{\boxplus \; \mathcal M}}^{\ds{+ \; \mathcal N \boxminus \mathcal M}}} & = & \nm{- J_R^{-1}\lrp{\mathcal N \boxminus \mathcal M}} & & & \nm{\in \mathbb R^{6x6}} \\
\nm{\vec J_{\ds{\oplus \; \mathcal N}}^{\ds{+ \; \mathcal N \ominus \mathcal R}}} & = & \nm{J_R^{-1}\lrp{\mathcal N \ominus \mathcal M}} & & & \nm{\in \mathbb R^{6x6}} \\
\nm{\vec J_{\ds{\boxplus \; \mathcal N}}^{\ds{+ \; \mathcal N \boxminus \mathcal R}}} & = & \nm{J_L^{-1}\lrp{\mathcal N \boxminus \mathcal M}} & & & \nm{\in \mathbb R^{6x6}} \\
\nm{\vec J_{\ds{\oplus \; \mathcal M}}^{\ds{+ \; g_{\mathcal M}(\vec p)}}} & & & = & \nm{\lrsb{\vec R \ \ - \vec R \; \pskew}} & \nm{\in \mathbb R^{3x6}} \\
\nm{\vec J_{\ds{\boxplus \; \mathcal M}}^{\ds{+ \; g_{\mathcal M}(\vec p)}}} & & & = & \nm{\lrsb{\vec I_{3x3} \ \ - \lrp{\vec R \; \vec p}^\wedge - \widehat{\vec T}}} & \nm{\in \mathbb R^{3x6}} \\
\nm{\vec J_{\ds{+ \; \vec p}}^{\ds{+ \; g_{\mathcal M}(\vec p)}}} & & & = & \nm{\vec R} & \nm{\in \mathbb R^{3x3}} \\
\nm{\vec J_{\ds{\oplus \; \mathcal M}}^{\ds{+ \; g_{\mathcal M}^{-1}(\vec p)}}} & & & = & \nm{\lrsb{- \vec I_{3x3} \ \ \big(\vec R^T\lrp{\vec p - \vec T}\big)^\wedge}} & \nm{\in \mathbb R^{3x6}} \\
\nm{\vec J_{\ds{\boxplus \; \mathcal M}}^{\ds{+ \; g_{\mathcal M}^{-1}(\vec p)}}} & & & = & \nm{\lrsb{- \vec R^T \ \ \vec R^T \; \pskew}} & \nm{\in \mathbb R^{3x6}} \\
\nm{\vec J_{\ds{+ \; \vec p}}^{\ds{+ \; g_{\mathcal M}^{-1}(\vec p)}}} & & & = & \nm{\vec R^T} & \nm{\in \mathbb R^{3x3}} \\
\nm{\vec J_{\ds{\oplus \; \mathcal M}}^{\ds{+ \; \vec{Ad}_{\mathcal M}(\vec \xi)}}} & = & \multicolumn{3}{l}{\nm{\lrsb{- \lrp{\vec R \; \vec \omega}^\wedge \vec R \ \ - \vec R \; \widehat{\nu} - \Tskew \; \vec R \; \omegaskew; \ \ \vec{0}_{3x3} \ \ - \vec R \; \omegaskew}}} & \nm{\in \mathbb R^{6x6}} \\
\nm{\vec J_{\ds{\boxplus \; \mathcal M}}^{\ds{+ \; \vec{Ad}_{\mathcal M}(\vec \xi)}}} & = & \multicolumn{3}{l}{\nm{\lrsb{- \lrp{\vec R \; \vec \omega}^\wedge \ \ - \lrp{\vec R \; \vec \nu}^\wedge - \Tskew \lrp{\vec R \; \vec \omega}^\wedge + \lrp{\vec R \; \vec \omega}^\wedge \Tskew; \ \ \vec{0}_{3x3} \ \ - \lrp{\vec R \; \vec \omega}^\wedge}}} & \nm{\in \mathbb R^{6x6}} \\
\hline
\end{tabular}
\end{center}
\begin{center}
\begin{tabular}{lcccll}
\hline
\textbf{Jacobian} & & \textbf{Table \ref{tab:algebra_lie_jacobians}} & & \multicolumn{1}{c}{\textbf{Expression}} & \textbf{Size} \\
\hline
\nm{\vec J_{\ds{+ \; \vec \xi}}^{\ds{+ \; \vec{Ad}_{\mathcal M}(\vec \xi)}}} & = & \nm{\vec{Ad}_{\mathcal M}} & = & \nm{\lrsb{\vec R \ \ \Tskew \; \vec R; \ \ \vec{0}_{3x3} \ \ \vec R}} & \nm{\in \mathbb R^{6x6}} \\
\nm{\vec J_{\ds{\oplus \; \mathcal M}}^{\ds{+ \; \vec{Ad}_{\mathcal M}^{-1}(\vec \xi)}}} & = & \multicolumn{3}{l}{\nm{\lrsb{\vec R^T \omegaskew \; \vec R \ \ \lrp{\vec R^T \vec \nu}^\wedge - \lrp{\vec R^T \Tskew \; \vec \omega}^\wedge; \ \ \vec{0}_{3x3} \ \ \lrp{\vec R^T \vec \omega}^\wedge}}} & \nm{\in \mathbb R^{6x6}} \\
\nm{\vec J_{\ds{\boxplus \; \mathcal M}}^{\ds{+ \; \vec{Ad}_{\mathcal M}^{-1}(\vec \xi)}}} & = & \multicolumn{3}{l}{\nm{\lrsb{\vec R^T \omegaskew \ \ \vec R^T \widehat{\vec \nu} - \vec R^T \Tskew \; \omegaskew; \ \ \vec{0}_{3x3} \ \ \vec R^T \omegaskew}}} & \nm{\in \mathbb R^{6x6}} \\
\nm{\vec J_{\ds{+ \; \vec \xi}}^{\ds{+ \; \vec{Ad}_{\mathcal M}^{-1}(\vec \xi)}}} & = & \nm{\vec{Ad}_{\mathcal M}^{-1}} & = & \nm{\lrsb{\vec R^T \ \ - \vec R^T \, \Tskew; \ \ \vec{0}_{3x3} \ \ \vec R^T}} & \nm{\in \mathbb R^{6x6}} \\
\nm{\vec J_{\ds{+ \; \vec \tau}}^{\ds{+ \; g_{Exp\lrp{\vec \tau}}(\vec p)}}} & & & = & \nm{\Big[\vec J_L\lrp{\vec r} \ \ \vec Q\lrp{\vec \tau} - \lrsb{\vec R \, \vec p}^\wedge \, \vec J_L\lrp{\vec r} - \widehat{\vec T} \, \vec J_L\lrp{\vec r}\Big]} & \nm{\in \mathbb R^{3x6}} \\
\nm{\vec J_{\ds{+ \; \vec \tau}}^{\ds{+ \; g_{Exp\lrp{\vec \tau}}^{-1}(\vec p)}}} & & & = & \nm{\Big[- \vec J_L\lrp{- \vec r} \ \ - \vec Q\lrp{- \vec \tau} - \lrsb{\vec R^T \; \lrp{\vec T - \vec p}}^\wedge \, \vec J_L\lrp{- \vec r}\Big]} & \nm{\in \mathbb R^{3x6}} \\
\hline
\end{tabular}
\end{center}
\captionof{table}{Rigid body motion jacobians} \label{tab:RigidBody_motion_jacobians}
\renewcommand{\arraystretch}{1.0}
\subsection{Rigid Body Motion Discrete Integration}\label{subsec:RigidBody_motion_integration}
The discrete integration with time of an element of a Lie group based on its Lie algebra is discussed in detail in section \ref{subsec:algebra_integration}, which includes expressions for the Euler, Heun and Runge-Kutta methods. In the case of rigid body motion, the state vector includes the motion element \nm{\mathcal M \in \mathbb{SE}(3)} and its twist \nm{\vec \xi \in \mathbb{R}^6} contained in the tangent space, viewed either in the local (\nm{\xiEBB}) or global (\nm{\xiEBE}) frames. The Euler method expressions equivalent to (\ref{eq:algebra_integration_comp_X_euler}) and (\ref{eq:algebra_integration_comp_X_euler_left}) are shown below. Expressions for other integration schemes can easily be derived from those in section \ref{subsec:algebra_integration}:
\begin{eqnarray}
\nm{\mathcal M_{k+1}} & \nm{\approx} & \nm{\mathcal M_k \oplus \lrsb{\Delta t \ \vec \xi_{{\sss EB}k}^{\sss B}} = \mathcal M_k \circ Exp\lrp{\Delta t \ \vec \xi_{{\sss EB}k}^{\sss B}}} \label{eq:SE3_integration_comp_X_euler} \\
\nm{\mathcal M_{k+1}} & \nm{\approx} & \nm{\lrsb{\Delta t \ \vec \xi_{{\sss EB}k}^{\sss E}} \boxplus \mathcal M_k = Exp\lrp{\Delta t \ \vec \xi_{{\sss EB}k}^{\sss E}} \circ \mathcal M_k} \label{eq:SE3_integration_comp_X_euler_left}
\end{eqnarray}
\subsection{Rigid Body Motion Gauss-Newton Optimization}\label{subsec:RigidBody_motion_gauss_newton}
The minimization by means of the Gauss-Newton iterative method of the Euclidean norm of a non linear function whose input is a Lie group element is presented in section \ref{subsec:algebra_gradient_descent}. In the case of rigid body motion, the resulting expressions for perturbations \nm{\Delta \vec \tau_{\sss EB}^{\sss E} \in \mathfrak{se}(3)} to an input motion \nm{\mathcal M \in \mathbb{SE}(3)} viewed in the global frame \nm{\FE} are shown in (\ref{eq:SE3_gauss_newton_iterative_left}) and (\ref{eq:SE3_gauss_newton_solution_left}), which are equivalent to the generic (\ref{eq:algebra_gradient_descent_iterative_lie_left}) and (\ref{eq:algebra_gradient_descent_solution_lie_left}). Refer to section \ref{subsec:algebra_gradient_descent} for the meaning of the function jacobian \nm{\vec J} and to section \ref{subsec:RigidBody_motion_calculus_jacobians} for that of the left jacobian \nm{\vec J_L}.
\begin{eqnarray}
\nm{\mathcal M_{k+1}} & \nm{\longleftarrow} & \nm{\Delta \vec \tau_{{\sss EB}k}^{\sss E} \boxplus \mathcal M_k = \Delta \vec \tau_{{\sss EB}k}^{\sss E} \circ Exp\lrp{\vec \tau_{{\sss EB}k}}} \label{eq:SE3_gauss_newton_iterative_left} \\
\nm{\Delta \vec \tau_{{\sss EB}k}^{\sss E}} & = & \nm{- \lrsb{\vec J_{Lk}^{-T} \, \vec J_k^T \, \vec J_k \, \vec J_{Lk}^{-1}}^{-1} \, \vec J_{Lk}^{-T} \, \vec J_k^T \, \vec{\mathcal E}_k} \label{eq:SE3_gauss_newton_solution_left}
\end{eqnarray}
If the perturbation is viewed in the local frame \nm{\FB}, (\ref{eq:algebra_gradient_descent_iterative_lie_right}) and (\ref{eq:algebra_gradient_descent_solution_lie_right}) are customized as follows, making use of the right jacobian \nm{\vec J_R} defined in section \ref{subsec:RigidBody_motion_calculus_jacobians}:
\begin{eqnarray}
\nm{\mathcal M_{k+1}} & \nm{\longleftarrow} & \nm{\mathcal M_k \oplus \Delta \vec \tau_{{\sss EB}k}^{\sss B} = Exp\lrp{\vec \tau_{{\sss EB}k}} \circ \Delta \vec \tau_{{\sss EB}k}^{\sss B}} \label{eq:SE3_gauss_newton_iterative_right} \\
\nm{\Delta \vec \tau_{{\sss EB}k}^{\sss B}} & = & \nm{- \lrsb{\vec J_{Rk}^{-T} \, \vec J_k^T \, \vec J_k \, \vec J_{Rk}^{-1}}^{-1} \, \vec J_{Rk}^{-T} \, \vec J_k^T \, \vec{\mathcal E}_k} \label{eq:SE3_gauss_newton_solution_right}
\end{eqnarray}
\subsection{Rigid Body Motion State Estimation}\label{subsec:RigidBody_motion_SS}
The adaptation of the \hypertt{EKF} state estimation introduced in section \ref{subsec:SS} to the case in which Lie group elements and their velocities are present is discussed in detail in section \ref{subsec:algebra_SS}. For rigid body motion with local perturbations, it is necessary to replace \nm{\mathcal X \in \mathcal G} by \nm{\mathcal M \in \mathbb{SE}(3)}, \nm{\Delta \vec \tau^{\mathcal X} \in T_{\mathcal X}\mathcal G} by \nm{\Delta \vec \tau^{\sss B} \in \ \mathfrak{se}(3)}, \nm{\vec v^{\mathcal X} \in \mathbb{R}^m} by \nm{\vec \xi^{\sss B} \in \mathbb{R}^6}, \nm{\vec C_{{\mathcal {XX}}}^{\mathcal X} \in \mathbb{R}^{mxm}} by \nm{\vec C_{{\mathcal {MM}}}^{\sss B} \in \mathbb{R}^{6x6}}, and \nm{\vec J_{\ds{\oplus \; \mathcal X}}^{\ds{\oplus \; \mathcal X \oplus \vec \tau}}} by \nm{\vec J_{\ds{\oplus \; \mathcal M}}^{\ds{\oplus \; \mathcal M \oplus \vec \tau}}}. The particularizations for global perturbations are similar.
\subsection{Applications of the Various Motion Representations}\label{subsec:RigidBody_motion_applications}
This section discusses six different representations of the rigid body motion or special euclidean group \nm{\mathbb{SE}(3)}: the affine representation, the homogeneous matrix, the transform vector, the unit dual quaternion, the half transform vector, and the screw. Although in theory all of them can be employed for each of the purposes described in this article, and the required expressions derived, each representation has its own advantages and disadvantages, being suited for certain purposes but not recommended for others.
\begin{itemize}
\item The affine representation \nm{\lrp{\mathcal R, \, \vec T}} based on either the rotation matrix or the unit quaternion is the most natural representation for rigid body motion. It can be employed to track the motion over its manifold, although other options are preferred. Many difficulties arise from its complex nature as a composition between an \nm{\mathbb{SO}(3)} rotation and a \nm{\mathbb{R}^3} vector, such as the complex inverse and concatenation, the different nature of the point and vector actions, the lack of simple plus and minus operators to deal with perturbations, and the need to continuously keep track of both components when moving over the manifold.
\item The homogeneous matrix \nm{\vec M} is a generalization of the rotation matrix for the case of rigid body motion that not only linearizes the transformation of coordinates at the expense of bigger size, but also adopts matrix algebra for the inversion and concatenation of transformations. A second advantage is that the transformations of vectors and points share the same map when employing homogeneous coordinates. Additionally, it provides a clear connection with the tangent space, together with the exponential and logarithmic maps, and plus and minus operators, which are not complex. Its main inconvenients are the huge size (16), the expense of maintaining the internal rotation matrix orthogonal if allowed to deviate from the manifold, and the need to work with homogeneous coordinates for both points and vectors. Its high cost precludes its use to track the motion over its manifold, although most implementations continuously compute its components (the rotation matrix and the translation vector) if the adjoint matrix or the jacobian blocks are required.
\item The unit dual quaternion \nm{\vec \zeta} is the preferred representation to track the motion over its manifold, even if it is necessary to obtain the rotation matrix and the translation vector for the adjoint and jacobian blocks. It possesses a significant size advantage (8) with respect to the homogeneous matrix, although it is not cheap to recover its structure if allowed to deviate from the manifold. Unit dual quaternions are the least natural of the rigid body motion representations, being necessary to convert to a different \nm{\mathbb{SE}\lrp{3}} representation for visualization. While the inverse and concatenation are simple and linear, the motion actions for points and vectors are bilinear and require slightly different expressions when inverting them, which presents a disadvantage with the homogeneous matrix. A significant advantage is given by its direct relationship with the screw and associated \hypertt{ScLERP} capabilities. Unit dual quaternion expressions are significantly more complex than those of the homogeneous matrix, and present a slightly less obvious connection with the tangent space.
\item The main advantage of the transform vector \nm{\vec \tau} is that it belongs to the \nm{\mathfrak{se}(3)} tangent space while simultaneously being an \nm{\mathbb{SE}(3)} representation. It is hence indicated for those uses related with incremental motion changes by means of the exponential map together with the plus and minus operators (periodically adding the perturbations to the unit dual quaternion tracking the motion), such as discrete integration, optimization, and state estimation. The norms of its angular and linear components \nm{\lrp{\phi, \rho}} are the most adequate metrics for evaluating the distance (or estimation error) between two rigid bodies. Although it benefits from its straightforward inverse when used as a perturbation, its geometric appeal, and its small dimension (6), its usage for other applications is discouraged by its complex non linear kinematics, coordinate transformation, and composition, which are not shown in this article.
\item The half transform vector \nm{\vec \Psi} is so similar (half) to the transform vector that its usage is not recommended in order to avoid confusion. Its only real application as the tangent space of the unit dual quaternion is in practice solved by dividing the transform vector by two when necessary.
\item The screw \nm{\vec S}, in addition to simultaneously belonging to the \nm{\mathfrak{se}(3)} tangent space and the \nm{\mathbb{SE}(3)} manifold, has the advantage that it clearly separates the influence of the motion direction from that of its magnitude, and as such it enables the definition of powers and \hypertt{ScLERP}, which can not be obtained with any of the other representations. The dimension is not big (8) and the inversion is straightforward, but all other possible expressions, including motion and concatenation, are very complex and not shown in this article.
\end{itemize}
\bibliographystyle{ieeetr}
| {
"timestamp": "2022-05-26T02:12:02",
"yymm": "2205",
"arxiv_id": "2205.12572",
"language": "en",
"url": "https://arxiv.org/abs/2205.12572",
"abstract": "Classical mathematical techniques such as discrete integration, gradient descent optimization, and state estimation (exemplified by the Runge-Kutta method, Gauss-Newton minimization, and extended Kalman filter or EKF, respectively), rely on linear algebra and hence are only applicable to state vectors belonging to Euclidean spaces when implemented as described in the literature. This document discusses how to modify these methods so they can be applied to non-Euclidean state vectors, such as those containing rotations and full motions of rigid bodies. To do so, this document provides an in-depth review of the concept of manifolds or Lie groups, together with their tangent spaces or Lie algebras, their exponential and logarithmic maps, the analysis of perturbations, the treatment of uncertainty and covariance, and in particular the definitions of the Jacobians required to employ the previously mentioned calculus methods. These concepts are particularized to the specific cases of the SO(3) and SE(3) Lie groups, known as the special orthogonal and special Euclidean groups of R3, which represent the rigid body rotations and motions, describing their various possible parameterizations as well as their advantages and disadvantages.",
"subjects": "Robotics (cs.RO)",
"title": "The SO(3) and SE(3) Lie Algebras of Rigid Body Rotations and Motions and their Application to Discrete Integration, Gradient Descent Optimization, and State Estimation",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9770226260757067,
"lm_q2_score": 0.724870282120402,
"lm_q1q2_score": 0.7082146666015136
} |
https://arxiv.org/abs/2002.09433 | Embeddings using universal words in the free group of rank 2 | For a countable group G = <A | R> presented by its generators A and defining relations R we discuss a simple method to embed G into such a 2-generator group T that the images of generators from A are explicitly given in T, and the defining relations for T can automatically be deduced from the relations R. The obtained method is applied on particular examples of groups, and references of its application for embeddings of recursive groups into finitely presented groups are given. | \section{Introduction}
\noindent
The objective of this note is to suggest some simple rules for explicit embedding of any countable group $G$ given by its generators and defining relations into a $2$-generator group $T$ such that the defining relations of $T$ can easily be deduced from those of $G$, and they inherit certain features of the relations of $G$ needed for embeddings of recursive groups into finitely presented groups (see subsection~\ref{SU Preserving the structure}).
By the well-known theorem of
Higman, Neumann and
Neumann an arbitrary countable group $G$ is embeddable into a $2$-generated group $T$ \cite{HigmanNeumannNeumann}. This result
called in Robinson's textbook \cite{Robinson}
\textit{``probably the most famous of all embedding theorems''} was a starting step for
further research on embeddings into $2$-generator groups with related properties.
Typically, such research discusses cases when the embedding has an extra feature (is subnormal, is verbal, etc.), or when the group $T$ has a required property, including those inherited from $G$ (is soluble, is generalized soluble or generalized nilpotent, is linearly ordered, residualy has some property, is simple, etc.). For an outline of the topic see articles
\cite{Dark}--\cite{On abelian subgroups}
and the literature cited therein.
In fact, the original embedding method of \cite{HigmanNeumannNeumann} and some of other embedding constructions cited above already are explicit, and they do allow to compute the relations of $T$ based on the relations of $G$. However, we need a method that not only makes discovery of the relations of $T$ a simple, automated task, but also \textit{preserves certain features} in them required for study of embeddings of recursive groups into finitely presented groups (see references in \ref{SU Preserving the structure} below).
\medskip
We need the following notations to introduce the embedding.
In the free group
$F_2=\langle
x,y
\rangle$ of rank $2$ consider some \textit{universal words}:
\begin{equation}
\label{EQ definition of a_i(x,y)}
a_i(x,y) = y^{(x y^i)^{\,2}\, x^{\!-1}}
\!\! y^{-x}
\!\!,\quad\quad i=1,2,\ldots
\end{equation}
(conventional notations $x^y=y^{-1}xy$,\; $x^{-y}=(x^{-1})^y$ are used here).
Assume a generic countable group $G$ is given by its generators and defining relations as:
$$
G = \langle\, A \mathrel{|} R\, \rangle= \langle a_1, a_2,\ldots \mathrel{|} r_1, r_2,\ldots \,\rangle
$$
where the $s$'th relation
$r_s \in R$
is a word of length $k_s$ on letters, say,
$a_{i_{s,1}},\ldots,a_{i_{s,\,k_s}}\!\!\!\! \in A$.\,
If we replace in $r_s$
each $a_{i_{s,j}}$,\; $j=1,\ldots,k_s$,\; by the respective word
$a_{i_{s,j}}(x,y)$ defined above, we get a new word
\begin{equation}
\label{EQ definition of r'_s}
r'_s (x,y)=
r_s\big(a_{i_{s,1}}\!(x,y),\ldots,a_{i_{s,\,k_s}}\!(x,y)\big)
\end{equation}
on just two letters $x,y$ in the free group $F_2$.
In these terms:
\begin{Theorem}
\label{TH universal embedding}
For any countable group $G = \langle a_1, a_2,\ldots \mathrel{|} r_1, r_2,\ldots \,\rangle $ the map $\gamma: a_i \to a_i(x,y)$,\; $i=1,2,\ldots$\,, defines an injective embedding of $G$ into the $2$-generator group
$$
T_G=\big\langle x,y
\;\mathrel{|}\;
r'_1 (x,y),\; r'_2 (x,y),\ldots\,
\big\rangle
$$
given by its relations
$r'_s (x,y)$,\;
$s=1,2,\ldots$
\end{Theorem}
This is noting else but a new formulation of SQ-universality of $F_2$.
The proofs of Theorem~\ref{TH universal embedding}
and of its modification
Theorem~\ref{TH universal embedding torsion-free} for torsion-free groups
occupy subsections~\ref{SU The universal generators}--\ref{SU Some simplification for torsion free groups} below.
Examples of applications with this embedding can be found in
subsection~\ref{SU Examples of embeddings}.
\medskip
We would like to stress the following case related to the question of Bridson and
de la Harpe mentioned as Problem 14.10 (b) in the Kourovka Notebook~\cite{kourovka}:
\textit{``Find an explicit embedding of $\Q$ in a finitely generated group; such a group
exists by Theorem IV in \cite{HigmanNeumannNeumann}''}.
The required explicit embedding of $\Q$ into a $2$-generator group $T$ was given in \cite{On a Problem on Explicit Embeddings of Q} in two manners, using free constructions and wreath products.
In the current note we add one more feature:
$\Q$ can be explicitly embedded into such a group $T$ the defining relations of which
can also be explicitly listed. In Example~\ref{EX embedding of rational group}
we display an
explicit embedding of $\Q$
into the $2$-generator group with directly given defining relations:
$$
T_\Q =\big\langle x,y
\;\mathrel{|}\;
(y^s)^{(x y^s)^{\,2} x^{\!-1}}
\!y^{-(x y^{s-1})^{\,2} x^{\!-1}}
\!\!,\;\;\;\; s=2,3\ldots
\big\rangle.
$$
Among other recent research on embeddings of $\Q$
into finitely generated groups we would like to briefly stress the following:
\cite{Darbinyan Mikaelian}
continues the \textit{linear order} relation of rational numbers in $\Q$ onto the whole $2$-generator group $T$, and it shows that the embedding can be \textit{verbal}.
\cite{On abelian subgroups}
mentions that $\Q$ can never be embedded into a finitely generated \textit{metabelian} group $T$ (see Section 7 in \cite{On abelian subgroups} and also \cite{Finiteness conditions for soluble groups}).
\cite{Adian Atabekyan}
provides an explicit verbal embedding of $\Q$ into a $2$-generator group $T=A_\Q(m,n)$ such that the \textit{center} of $T$ coincides with the image of $\Q$, i.e., $Z(T)\cong \Q$.
One of the tasks in Problem 14.10 (a) \cite{kourovka} is to find an explicit embedding of $\Q$ into a ``natural'' \textit{finitely presented} group.
\cite{The Higman operations and embeddings}
describes how Higman's procedure could be modified for a family of groups that includes $\Q$ to explicitly embed each of such groups into a $2$-generator finitely presented group.
And the first direct solution to the above problem appeared recently in \cite{Belk Hyde Matucci}. Moreover, one of the finitely presented groups constructed in \cite{Belk Hyde Matucci} is the group $T\!\mathcal{A}$ which is $2$-generator and simple.
\medskip
In the final subsection~\ref{SU Preserving the structure}
we refer to the main motivation that brought us to the study of embeddings in
Theorem~\ref{TH universal embedding}
and
Theorem~\ref{TH universal embedding torsion-free}:
the constructive Higman embeddings \cite{Higman Subgroups in fP groups} of recursive groups into finitely presented groups.
\medskip
When the text of this note was uploaded to the arXiv.org I had an opportunity to discuss the topic with Prof. L.A. Bokut' who remarked interesting parallelism with
\cite{Shirshov 58} where A.I. Shirshov
constructed the elements
$$d_k = \Big[a \circ \big\{[\cdots (a \underbrace{\circ \, b)\circ b \cdots ]\circ b}_k\big \}\Big]\circ (a\circ b),$$
\vskip-2mm
\noindent
$k=1,2,\ldots$,
in the free associative algebra $A$ with two generators $a$ and $b$. Here $a\circ b$ denotes the Lie algebra product $ab-ba$, and for details see \S 4 in \cite{Shirshov 58}.
These elements freely generate a free Lie algebra $L(a,b)$ of countable rank.
They are used to define an embedding of any countably generated Lie algebra into a $2$-generator Lie algebra. The set
$\{d_k \mathrel{|} k=1,2,\ldots \}$ is ``distinguished'' in the sense of \cite{Shirshov 56}.
Meanwhile, the technique of our proof is very much different from
\cite{Shirshov 58, Shirshov 56}, which makes this parallelism even more interesting.
See also \cite{Bokut 72} and Remark~\ref{RE reference to HNN} below where we refer to a construction in the original work of Higman, Neumann and Neumann \cite{HigmanNeumannNeumann}.
The current work is supported by the joint grant 18RF-109 of RFBR and SCS MESCS RA, and by the 18T-1A306 grant of SCS MESCS RA.
\section{References and some auxiliary results}
\noindent
For general group-theoretical information we refer to \cite{Robinson, Kargapolov Merzljakov, Rotman}.
If $G = \langle\, A \mathrel{|} R \,\rangle$ is the presentation of the group $G$ by its generators $A$ and deifing relations $R$, then for an alphabet $B$ disjoint from $A$ and for any set $S\subseteq F_B$ of group words on $B$, we denote by
$\langle G, B \mathrel{|} S \,\rangle$ the group $\langle \,A \cup B \mathrel{|} R \cup S \,\rangle$.
If $\varphi:G\to H$ is a homomorphism defined on a group $G=\langle g_1,g_2,\ldots \rangle$ by the images $\varphi(g_1)=h_1$, $\varphi(g_2)=h_2,\ldots$\; of its generators, we may for briefness refer to $\varphi$ as the homomorphism \textit{sending}\, $g_1,g_2,\ldots$ \;to\; $h_1,h_2,\ldots$
\medskip
Our proofs in Section~\ref{SE Embeddings into 2-generator groups by ``universal'' generators} will be based on free constructions: the operations of free product of groups, of free product of a groups with amalgamated subgroup, and of HNN-extension of group by one or multiple stable letters (including the case with infinitely many stable letters).
Background information on these constructions can be found in \cite{Rotman, Lyndon Schupp, Bogopolski}. Also, we refer to our recent notes \cite{A modified proof, Subvariety structures, The Higman operations and embeddings} for specific notations that we also share here to write the
free product
$G*_{\varphi} H = \langle G, H \mathrel{|} a=a^{\varphi} \text{ for all $a\in A$}\, \rangle$ of groups $G$ and $H$ with subgroups $A$ and $B$ amalgamated under the isomorphism $\varphi : A \to B$;\;
and the HNN-extension
$G*_{\varphi} t=\langle G, t \mathrel{|} a^t=a^{\varphi} \text{ for all $a\in A$}\, \rangle$
of the base group $G$ by the stable letter $t$ with respect to the isomorphism
$\varphi : A \to B$ of the subgroups $A,B\le G$.\;
We also use HNN-extensions
$G *_{\varphi_1, \varphi_2, \ldots} (t_1, t_2, \ldots) = \langle G, t_1, t_2,\ldots \mathrel{|} a_1^{t_1}=a_1^{\varphi_1}\!\!,\; a_2^{t_2}=a_2^{\varphi_2}\!\!,\,\ldots\; \text{ for all $a_1\in A_1$, $a_2\in A_2,\ldots$}\, \rangle$
with multiple stable letters $t_1, t_2, \ldots$ \;
with respect to isomorphisms $\varphi_1: A_1 \to B_1,\; \varphi_2: A_2 \to B_2,\ldots$ for pairs of subgroups
$A_1,B_1;\; A_2,B_2; \ldots$ in $G$.
\medskip
We are going to use certain subgroups of free products with amalgamated subgroups.
Lemma~\ref{LE subgroups in amalgamated product} is a slight variation of Lemma 3.1 given on p.~465 of \cite{Higman Subgroups in fP groups} without a proof as \textit{``obvious from the normal form theorem for free products with an amalgamation''}. The proof can be found in subsection 2.5 of \cite{A modified proof}.
\begin{Lemma}
\label{LE subgroups in amalgamated product}
Let $\Gamma = G *_\varphi \!H$ be the free product of the groups $G$ and $H$ with amalgamated subgroups
$A \le G$ and $B \le H$
with respect to the isomorphism
$\varphi: A \to B$.
If $G', H'$ are subgroups of $G, H$ respectively, such that
for $A'=G'\cap A$ and $B'=H'\cap B$
we have
$\varphi (A') = B'$, then for the subgroup $\Gamma'=\langle G',H'\rangle$ of $\Gamma$ and for the restriction $\varphi'$ of $\varphi$ on $A'$ we have:
\begin{enumerate}
\item $\Gamma' = G'*_{\varphi'} H'$,
\item $\Gamma' \cap A = A'$ and $\Gamma' \cap B=B'$,
\item $\Gamma' \cap G = G'$ and $\Gamma' \cap H = H'$.
\end{enumerate}
\end{Lemma}
If the amalgamated subgroups are trivial in a given free product with amalgamation, then that product is nothing but the ordinary free product of the same groups. Applying this observation to the groups $G'$ and $H'$ with trivial intersections $A'$ and $B'$ we get:
\begin{Corollary}
\label{CO G*H free products}
In the notations of Lemma~\ref{LE subgroups in amalgamated product}:
\begin{enumerate}
\item if $A'=G'\cap A$ and $B'=H'\cap B$ both are trivial,
then $\Gamma'=\langle G',H'\rangle = G'*H'$,
\item if, moreover, $A'$ is a free group of rank $r_1$, and $B'$ is a free group of rank $r_2$, then $\Gamma'= F_r$ is a free group of rank $r=r_1+r_2$.
\end{enumerate}
\end{Corollary}
\begin{comment}
\begin{Corollary}
\label{CO G*H lemma corollary}
If $\Gamma = G*_{A} H$, then in $\Gamma$:
\begin{enumerate}
\item for any $G'\!\!\le\! G$, $H'\!\le\! H$, such that $G' \cap\, A = H' \cap\, A$\,, we have $\langle G'\!\!, H' \rangle= G' \!*_{G' \cap \;A} H'$\!,
\item for any $G'\!\le G$, $H'\le H$, both containing $A$\,, we have $\langle G', H' \rangle= G' *_{A} H'$.
\end{enumerate}
\end{Corollary}
\begin{Lemma}
\label{LE subgroups in HNN-extension}
Let $\Gamma=G *_{\varphi} t$ be the HNN-extension of the base group $G$ by the stable letter $t$
with respect to the isomorphism
$\varphi: A \to B$
of the subgroups
$A, B \le G$.
If $G'$ is a subgroup of $G$ such that
for $A'=G'\cap\, A$ and $B'=G'\cap\, B$
we have
$\varphi (A') = B'$, then for the subgroup $\Gamma'=\langle G',t\rangle$ of $\Gamma$ and for the restriction $\varphi'$ of $\varphi$ on $A'$ we have:
\begin{enumerate}
\item $\Gamma' =G' *_{\varphi'} t$,
\item $\Gamma' \cap G = G'$,
\item $\Gamma' \cap \,A = A'$ \;and\; $\Gamma' \cap B = B'$.
\end{enumerate}
\end{Lemma}
The proof can be conducted in analogy with the proof of Lemma~\ref{LE subgroups in amalgamated product}.
\begin{Corollary}
\label{CO G*t lemma corollary}
If $\Gamma = G*_{A} t$, then in $\Gamma$:
\begin{enumerate}
\item for any subgroup $G'$ of $G$ we have $\langle G', t \rangle= G' *_{G' \cap\; A} t$,
\item for any subgroup $G'$ of $G$ containing $A$ we have $\langle G', t \rangle= G' *_{A} t$.
\end{enumerate}
\end{Corollary}
\end{comment}
\section{Embeddings into $2$-generator groups by universal words}
\label{SE Embeddings into 2-generator groups by ``universal'' generators}
\subsection{The universal words in free group of rank $2$}
\label{SU The universal generators}
Let $F=F_A$ be a free group on a countable alphabet $A=\{a_1, a_2,\ldots\}$. Fix a new generator $a$, and in the free product $F'=F * \langle a \rangle$
set the isomorphisms of cyclic subgroups $\varphi_i:\langle a \rangle \to \langle a_i a \rangle$
sending $a$ to $a_i a$,\; $i=1,2,\ldots$
Define the respective HNN-extension:
$$
P\;=\;
F' *_{\varphi_1, \varphi_2,\ldots}
(t_1, t_2,\ldots)
\;=\;
\langle
F', t_1, t_2,\ldots \mathrel{|}\;
a^{t_i}= a_i a,\;\; i=1,2,\ldots\,
\rangle.
$$
Clearly, the stable letters $t_i$,\, $i=1,2,\ldots$\,,\, generate in $P$ a free subgroup $X$ of countable rank.
It is simple to pick an auxiliary 2-generator group with a subgroup isomorphic to $X$: in the free group $Y=\langle
y,z
\rangle$ the elements $t_i'=y^i z^i$\!,\, $i=1,2,\ldots$\,, freely generate the subgroup $X'=\langle
t_1', t_2',\ldots
\rangle$ of countable rank. Amalgamating $X$ and $X'$
according to the isomorphism
$\psi$ sending $t_1,t_2,\ldots$ to $t'_1,t'_2,\ldots$
we get the group
$$
Q\;=\;
P *_{\psi} Y \;=\;
\langle\,
P,\; y,z \mathrel{|}
t_i=t_i',\;\; i=1,2,\ldots\,
\rangle.
$$
This group can already be generated by three elements $a,y,z$ because its generators
$$
a_i=a_i a \cdot a^{-1} = a^{t_i}a^{-1}
= a^{t_i'}a^{-1}
\quad \text{and}\quad\;\;
t_i=t_i'=y^i z^i
\!\!,\;\;\;\;\;
i=1,2,\ldots
$$
all are in $\langle a,y,z \rangle$.
By construction of $P$ no non-trivial power of
$a$ is in $X$, and
by construction of $Y$
no non-trivial power of
$y$ is in $X'$, that is, the intersections
$\langle a \rangle \cap X$ and $\langle y \rangle\cap X'$ both are trivial. This by Corollary~\ref{CO G*H free products} implies that
$\langle
a,y \rangle$ is a free subgroup of rank $1+1=2$ in $Q$.
Introducing a new stable letter $x$ for the isomorphism $\pi:\langle y,z\rangle \to \langle a,y\rangle$ sending $y,z$ to $a,y$, we construct:
\begin{equation}
\label{EQ formula of F_2}
F_2 =Q *_{\pi} x =
\langle
Q,x \mathrel{|}
y^x\!=a,\; z^x\!=y
\rangle
=\Big(\big((F* \langle a\rangle) *_{\varphi_1, \varphi_2,\ldots}
\!(t_1, t_2,\ldots)\big)*_{\psi} S\Big)*_{\pi} x.
\end{equation}
This seems to cause confusion, as we earlier used $F_2$ to denote the free group of rank $2$ on the alphabet $\{x,y\}$. Let us verify that, in fact, \eqref{EQ formula of F_2} coincides with that free group.
Firstly, $F_2$ is generated by $\{x,y\}$ because $a$ and $z$ can be expressed as some words on $x,y$:
$$
a=y^{x}=a(x,y),\quad
z=y^{x^{-1}}\!\!=z(x,y).
$$
Using these we can also
express as words on $x,y$
\textit{all} the above discussed generators:
\vskip-4mm
$$
t_i = t_i' =\, y^i z^i
=\; y^i x y^i x^{-1}\!
=t'_i(x,y)=t_i(x,y),
$$
\vskip-5mm
\begin{equation}
\label{EQ formula for a_(x,y)}
a_i
= a^{t_i}a^{-1} \!
= y^{x y^i \! x y^i \! x^{\!-1}}
\!\! y^{-x}
= y^{(x y^i)^{\,2}\, x^{\!-1}}
\!\! y^{-x}
= \;
x (y^{-i} x^{-1})^2
y\,
(x\, y^i)^2 x^{-2} y^{-1} \! x =a_i(x,y).
\end{equation}
Secondly, $F_2$ is free on $\{x,y\}$ because all the relations demanded by our ``nested'' free construction (see the right-hand side in \eqref{EQ formula of F_2}) already hold on words
$
a(x,y),\,
a_i(x,y),\,
t_i(x,y),\,$
$
t'_i(x,y),\,
y,\,
z(x,y),\,
x
$
in $F_2$, as it is easy to verify:
\begin{equation}
\label{EQ deducng everything from x, y}
\begin{split}
& \quad \;\; a(x,y)^{t_i(x,y)}
= (y^x)^{\, y^i x y^i x^{-1}}\!\!\!
= y^{x y^i x y^i x^{-1}} \!\!\! y^{-x}\! y^{x}=
a_i(x,y)\, a(x,y),
\\
& t'_i(x,y)=t_i(x,y)
,\quad \quad
y^x = a(x,y),\quad\quad
z(x,y)^x =
\big(y^{x^{-1}}\big)^x = y.
\end{split}
\end{equation}
No relations actually needed ``to bind'' $x$ with $y$, and so \eqref{EQ formula of F_2} is free on $x,y$.
Clearly, $\bar F=\big\langle a_1(x,y), a_2(x,y), \ldots\big\rangle$ is an isomorphic copy of $F$ inside $F_2$,
and for any word $r(a_{i_1},\ldots,a_{i_k})\in F$ we have a word
$$r' (x,y)=
r\,\big(a_{i_1}(x,y),\ldots,a_{i_k}(x,y)\big)\in \bar F \le F_2$$ obtained by replacing each $a_{i_j}$ by $a_{i_j}(x,y)$,\; $j=1,\ldots,k$.
\subsection{The embedding construction}
\label{SU The embedding construction}
Assume a countable group $G$ is given as $G= \langle\, A \mathrel{|} R\, \rangle = \langle a_1, a_2,\ldots \mathrel{|} r_1, r_2,\ldots \,\rangle $ where the $s$'th relation
$r_s(a_{i_{s,1}},\ldots,a_{i_{s,\,k_s}}\!)$
is a word on $k_s$ letters, as mentioned in Introduction.
To such a relation we put into correspondence the word
$
r'_s (x,y)$
defined in $\bar F$ by
\eqref{EQ definition of r'_s}. The set of such words for all $s=1,2,\ldots$ form a subset $\bar R =\big\{r'_1 (x,y), r'_2 (x,y),\ldots\big\}$ with the normal closure $\bar N = \langle \bar R\rangle^{\bar F}$ in $\bar F$.
Since $\bar F \cong F$, we have $\bar N \cong N$ and $\bar F / \bar N \;\cong\; F/ N \;\cong\; G$.
Next define $\tilde N = \langle \bar R\rangle^{F_2}$ to be normal closure of $\bar R$ in the whole group $F_2$.
The natural homomorphism $\nu: F\to G \cong F/N$ is sending $a_i$ to $g_i=N a_i$, $i=1,2,\ldots$\,,\;
and we may consider the
``updated'' isomorphisms
$\varphi_i:\langle a \rangle \to \langle g_i a \rangle$ of cyclic subgroups in the free product
$G * \langle a \rangle$
(for simplicity we do not introduce new letters for these $\varphi_i$).
In analogy with \eqref{EQ formula of F_2} we
build another ``nested'' free construction:
\begin{equation}
\label{EQ formula of H}
H=
\Big(\big((G * \langle a \rangle) *_{\varphi_1, \varphi_2,\ldots}
\!(t_1, t_2,\ldots)\big)*_{\psi} Y\Big)*_{\pi} x
\end{equation}
by using the new isomorphisms
$\varphi_i$
and the same isomorphisms
$\psi$ and $\pi$
as used above to define $Q$ and $F_2$.
The natural homomorphism $\nu$ can be extended to a homomorphism $\bar \nu$ from the group $F_2$ onto $H$
by requiring $\bar\nu$ to agree with $\nu$ on $F$, and to fix each of the remaining generators
$a,t_1,t_2,\ldots,\,y,z,x$.
It turns out that the relations $\bar R$ already are enough to define the group $H$ because from \eqref{EQ formula of H} it is clear that all other equalities $a^{t_i} = g_i a$,\; $t_i=t'_i$, $y^x = a$,\; $z^x=y$ of $H$ do follow, like in \eqref{EQ deducng everything from x, y}, from representation of the generators via $x,y$.
Since $G$ trivially is embedded into $H$, the subgroup $ (\bar F \tilde N) / \tilde N $ of $F_2/\tilde N$ is isomorphic to $\bar F / \bar N\cong G$.
On the other hand we have $ (\bar F \tilde N) / \tilde N \cong \bar F /(\bar F \cap \tilde N )$.
But since $\bar N$ is the kernel of the natural homomorphism from $\bar F$ to $\bar F /\bar N \cong G$, we get that $\bar F \cap \tilde N \le \bar N$.
Since also $\bar N \le \bar F$ and $\bar N \le \tilde F$, we have $
\bar F \cap \tilde N = \bar N$.
This construction together with~\ref{SU The universal generators} prove the following technical lemma:
\begin{Lemma}
\label{LE universal elements in F_2}
Let $F_2=\langle x,y\rangle$ be a free group of rank $2$ with elements $a_i(x,y)$ defined in \eqref{EQ definition of a_i(x,y)}
generating the subgroup $\bar F=\big\langle a_i(x,y) \mathrel{|} i=1,2,\ldots \big\rangle$ in $F_2$.
For any subgroup $N$ in $\bar F$ let $\bar N$
and $\tilde N$ denote
the normal closures of $N$ in $\bar F$ and in $F_2$ respectively.
Then:
$$
\bar F \cap \tilde N = \bar N.
$$
\end{Lemma}
Now we can conclude the proof of Theorem~\ref{TH universal embedding} as follows.
Let the map
$\gamma$,
the group
$T_G=\big\langle x,y
\mathrel{|}
r'_1 (x,y),\; r'_2 (x,y),\ldots\,
\big\rangle$,
and the relations
$r'_s (x,y)=
r_s\big(a_{i_{s,1}}\!(x,y),\ldots,a_{i_{s,k_s}}\!(x,y)\big)$
be those mentioned in the theorem.
Since the elements $a_1(x,y), a_2(x,y), \ldots$ generate an isomorphic copy $\bar F$ of $F$ in $F_2$, we have $G \cong \bar F / \bar N$.
Since $\bar N= \bar F \cap \tilde N$ by Lemma~\ref{LE universal elements in F_2}, then:
$$
G\;\cong\; \bar F / \bar N \;=\; \bar F / (\bar F \cap \tilde N)
\;\cong\;
(\bar F \tilde N) /\tilde N.
$$
But $\bar F \tilde N$ is in the whole $F_2$, and so $(\bar F \tilde N) /\tilde N$ clearly is a subgroup in $F_2 /\tilde N \;\cong\; T$.
\medskip
Theorem~\ref{TH universal embedding} provides a very easy way to embed a countable group $G = \langle a_1, a_2,\ldots \mathrel{|} r_1, r_2,\ldots\, \rangle $ into a $2$-generator group $T_G$ the relations of which are trivially obtained by just replacing
in $r_1, r_2,\ldots$ all occurrences of the letters $a_1,a_2,\ldots$ by expressions $a_1(x,y),\; a_2(x,y),\ldots$
Notice that $T_G$ does depend on the particular presentation $G = \langle\, A \mathrel{|} R\, \rangle$, and for a different choice of $A$ and $R$ we may output another $2$-generator group. However, we do not want to note it as $T_{\langle\, A \,\mathrel{|} \,R\, \rangle}$ because this would bring to bulky notations in examples in subsection
\ref{SU Examples of embeddings}.
\subsection{Some simplification for torsion free groups}
\label{SU Some simplification for torsion free groups}
The isomorphisms $\varphi_i:\langle a \rangle \to \langle a_i a \rangle$
sending $a$ to $a_i a$ used in \ref{SU The universal generators} cannot, in general, be replaced by isomorphisms sending $a$ to $a_i$, $i=1,2,\ldots$, because when $g_i \!\in\! G$ from \ref{SU The embedding construction} is an element of \textit{finite} order, then $\langle a \rangle$ and $\langle g_i \rangle$ are \textit{not} isomorphic, and they can no longer be used as associated subgroups. This is the reason why we used $g_i a$ instead. But when
$G$
is \textit{torsion-free}, this obstacle is dropped, and we can replace
$a_i a$ by $a_i$.
This allows to replace $a_i(x,y)$ of \eqref{EQ formula for a_(x,y)} by a shorter word
\begin{equation}
\label{EQ definition of bar a_i(x,y)}
\bar a_i(x,y)
= a^{t_i}
=
y^{(x y^i)^{\,2} x^{\!-1}}
\!\!
,\quad\quad i=1,2,\ldots
\end{equation}
Replacing in $r_s$
each $a_{i_{s,j}}$ by
$\bar a_{i_{s,j}}(x,y)$, we get other, shorter than $r'_s (x,y)$ word
$$
r''_s (x,y)=
r_s\big(\bar a_{i_{s,1}}\!(x,y),\ldots,\bar a_{i_{s,\,k_s}}\!(x,y)\big)
$$
on letters $x,y$ in the free group $F_2$. We have the following analog of
Theorem~\ref{TH universal embedding}:
\begin{Theorem}
\label{TH universal embedding torsion-free}
For any torsion-free countable group $G\, = \,\langle a_1, a_2,\ldots \mathrel{|} r_1, r_2,\ldots \,\rangle $ the map $\gamma: a_i \to \bar a_i(x,y)$,\; $i=1,2,\ldots$\,, defines an injective embedding of $G$ into the $2$-generator group
$$
T_G=\big\langle x,y
\;\mathrel{|}\;
r''_1 (x,y),\; r''_2 (x,y),\ldots\,
\big\rangle
$$
given by its relations
$r''_s (x,y)$,\;
$s=1,2,\ldots$
\end{Theorem}
Adaptation of the proof in \ref{SU The universal generators} and in \ref{SU The embedding construction} for this case is trivial.
\begin{Remark}
\label{RE reference to HNN}
The reader may collate the above constructions with pages 252--254 in \cite{HigmanNeumannNeumann}. We used some ideas from there and from \cite{Higman Subgroups in fP groups}, but our proof is briefer, and we produced shorter words $a_i(x,y)$. Compare them with words
$
e_i=
a^{-1} b^{-1} a\, b^{-i} a\, b^{-1} a^{-1} b^{i}a^{-1} b\, a\, b^{-i} a\, b\, a^{-1} b^i
$
used in \cite{HigmanNeumannNeumann}.
And we have even shorter words $\bar a_i(x,y)$ for torsion-free groups.
\end{Remark}
\subsection{Examples of explicit embeddings}
\label{SU Examples of embeddings}
Here are some applications of the method with
Theorem~\ref{TH universal embedding}
and with
Theorem~\ref{TH universal embedding torsion-free}.
\begin{Example}
\label{EX embedding of free abellian into 2-generator group}
The free abelian group $G=\Z^\infty$ of contable rank can be given as
$$
G = \big\langle a_1, a_2,\ldots \mathrel{|} [a_k,a_l],\; k,l=1,2\ldots \big\rangle
$$
by its relations $r_s=r_{k,l}=[a_k,a_l]$.
Since $G$ is torsion-free, we can use the shorter formula \eqref{EQ definition of bar a_i(x,y)} to
map each $a_i$ to respective $\bar a_i (x,y)$. This defines an embedding of $\Z^\infty$ into the $2$-generator group:
$$
T_{\Z^\infty} =\big\langle x,y
\;\mathrel{|}\;
\big[
y^{(x y^k)^{\,2} x^{\!-1}}
\!\!\!\!\!,\,\,\,
y^{(x y^l)^{\,2} x^{\!-1}}
\big] ,\;\;\; k,l=1,2\ldots
\big\rangle.
$$
\end{Example}
\begin{Example}
\label{EX embedding of rational group}
The additive group of rational numbers $G=\Q$ can be presented \cite{Johnson} as:
$$
G = \big\langle a_1, a_2,\ldots \mathrel{|} a_s^s=a_{s-1},\; s=2,3\ldots \big\rangle
$$
where the generator $a_i$ corresponds to the fractional number $\displaystyle {1 \over i!}$ with $i=2,3\ldots$
Rewrite each $a_s^s=a_{s-1}$ as
$a_s^s\,a_{s-1}^{-1}$ and use the latter as the relation $r_s=r_s(a_{s-1},\, a_s)$ for each $s=2,3\ldots$
Since $G$ again is torsion-free, we can use the shorter formula \eqref{EQ definition of bar a_i(x,y)} to
map each $a_i=\displaystyle {1 \over i!}$ to a $\bar a_i (x,y)$.
After easy simplification
$\bar a_i (x,y)=\big(y^{(x y^i)^{\,2} x^{\!-1}} \big)^i
\big(y^{(x y^{i-1})^{\,2} x^{\!-1}}\big)^{-1}\!\!
=\,
(y^i)^{(x y^i)^{\,2} x^{\!-1}}
y^{-(x y^{i-1})^{\,2} x^{\!-1}}
$
we get an embedding of $\Q$ into the $2$-generator group:
$$
T_\Q =\big\langle x,y
\;\mathrel{|}\;
(y^s)^{(x y^s)^{\,2} x^{\!-1}}
y^{-(x y^{s-1})^{\,2} x^{\!-1}}
\!\!,\;\;\;\; s=2,3\ldots
\big\rangle.
$$
\end{Example}
\begin{Example}
\label{EX embedding of Pruefer group}
The quasicyclic Pr\"ufer $p$-group $G=\Co_{p^\infty}$ can be presented as:
$$
G = \big\langle a_1, a_2,\ldots \mathrel{|}
a_1^p,\;\;\, a_{s+1}^p\!=a_s
,\;\; s=1,2\ldots \big\rangle
$$
where the generator $a_i$ corresponds to the
primitive $(p^i)$'th root $\varepsilon_i$ of unity \cite{Kargapolov Merzljakov}.
As this group is \textit{not} torsion-free, we have to use the rather longer formula
$a_i(x,y)$ from \eqref{EQ formula for a_(x,y)} as the image of $a_i$.
For the first relation
$a_1^p$ of $G$ we get the new relation
$a_1(x,y)^p=\big(y^{(x y)^{\,2}\, x^{\!-1}}
\!\! y^{-x}\big)^p$.
Next, rewrite each $a_{s+1}^p=a_s$ as
$a_{s+1}^p a_s^{-1}$\!\!,\, and use this as the relation $r_s=r_s(a_s,\, a_{s+1})$, with $s=1,2\ldots$\;
The respective new relation will be
$$
\big(y^{(x y^{s+1})^{\,2}\, x^{\!-1}}
\!\! y^{-x}\big)^p
\big(y^{(x y^s)^{\,2}\, x^{\!-1}}
\!\! y^{-x}\big)^{-1}
=
\big(y^{(x y^{s+1})^{\,2}\, x^{\!-1}}
\!\! y^{-x}\big)^p
y^{x} y^{-(x y^s)^{\,2}\, x^{\!-1}}\!\!.
$$
And we have an embedding of $\Co_{p^\infty}$ into the $2$-generator group:
$$
T_{\Co_{p^\infty}} =\big\langle x,y
\;\mathrel{|}\;\;
\big(y^{(x y)^{\,2}\, x^{\!-1}}
\!\! y^{-x}\big)^p\!\!,\;\;\;\;
\big(y^{(x y^{s+1})^{\,2}\, x^{\!-1}}
\!\! y^{-x}\big)^p
y^{x} y^{-\,(x y^s)^{\,2}\, x^{\!-1}}
\!\!\!,\;\;\; s=1,2\ldots
\big\rangle.
$$
\end{Example}
\subsection{Usage in embeddings of recursive groups}
\label{SU Preserving the structure}
The main motivation why we needed the embeddings of
Theorem~\ref{TH universal embedding}
and
Theorem~\ref{TH universal embedding torsion-free}
concerns study of constructive embeddings of recursive groups into finitely presented groups, i.e., \textit{constructive} Higman embeddings \cite{Higman Subgroups in fP groups}
(see, in particular, the references to embeddings of $\Q$ into finitely presented groups related to Problem 14.10 (a) \cite{kourovka} mentioned in Introduction).
One of the steps of the embedding for a recursive group
$G = \langle a_1, a_2,\ldots \mathrel{|} r_1, r_2,\ldots \,\rangle$ (i.e., of a group with recursively enumerable relations $r_1, r_2,\ldots$) into a finitely presented group is the preliminary embedding to $G$ into a $2$-generator group $T=T_G$. And this group $T$ need also have
recursively enumerable set of relations.
The simple, automated embeddings that we built above do preserve that property.
Moreover, we need embeddings \textit{preserving special features} of relations.
For the details we refer to \cite{The Higman operations and embeddings}, and give just rough idea here.
Each relation of a $2$-generator group $T$ can be coded by means of a certain sequence of integers. This allows to study the recursively enumerable sets of relations by means of certain sets of sequences of integers
in \cite{Higman Subgroups in fP groups}. As we see in
\cite{The Higman operations and embeddings},
the embeddings of
Theorem~\ref{TH universal embedding}
and
Theorem~\ref{TH universal embedding torsion-free}
guarantee some close correlation between the
relations of $G$ and those sets of sequences, which allows us to build constructive embedding of $G$ into a finitely presented group.
In particular, compare Example~\ref{EX embedding of free abellian into 2-generator group} from this note to
Example~3.1,
Example~3.2 and
Example 4.11 with ``abacus machine'' in
\cite{The Higman operations and embeddings}.
| {
"timestamp": "2020-09-04T02:01:39",
"yymm": "2002",
"arxiv_id": "2002.09433",
"language": "en",
"url": "https://arxiv.org/abs/2002.09433",
"abstract": "For a countable group G = <A | R> presented by its generators A and defining relations R we discuss a simple method to embed G into such a 2-generator group T that the images of generators from A are explicitly given in T, and the defining relations for T can automatically be deduced from the relations R. The obtained method is applied on particular examples of groups, and references of its application for embeddings of recursive groups into finitely presented groups are given.",
"subjects": "Group Theory (math.GR)",
"title": "Embeddings using universal words in the free group of rank 2",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9770226260757067,
"lm_q2_score": 0.7248702821204019,
"lm_q1q2_score": 0.7082146666015134
} |
https://arxiv.org/abs/1604.02712 | On the number of mutually disjoint pairs of S-permutation matrices | This work examines the concept of S-permutation matrices, namely $n^2 \times n^2$ permutation matrices containing a single 1 in each canonical $n \times n$ subsquare (block). The article suggests a formula for counting mutually disjoint pairs of $n^2 \times n^2$ S-permutation matrices in the general case by restricting this task to the problem of finding some numerical characteristics of the elements of specially defined for this purpose factor-set of the set of $n \times n$ binary matrices. The paper describe an algorithm that solves the main problem. To do that, every $n\times n$ binary matrix is represented uniquely as a n-tuple of integers. | \section{Introduction and notation}
Let $n$ be a positive integer. By $[n]$ we denote the set $[n] =\left\{ 1,2,\ldots ,n\right\}$.
A \emph{binary} (or \emph{boolean}, or (0,1)-\emph{matrix}) is a matrix all of whose elements belong to the set $\mathfrak{B} =\{ 0,1 \}$. In this paper we will consider only square binary matrices. With $\mathfrak{B}_n$ we will denote the set of all $n \times n$ binary matrices. With $\mathfrak{B}_{n,k}$ we will denote the set of all $n\times n$ binary matrices containing exactly $k$ elements equal to 1.
Two $n\times n$ binary matrices $A=(a_{ij} )\in \mathfrak{B}_{n}$ and $B=( b_{ij} )\in \mathfrak{B}_{n}$ will be called \emph{disjoint} if there are not integers $i,j\in [n]$ such that $a_{ij} =b_{ij} =1$, i.e. if $a_{ij} =1$ then $b_{ij} =0$ and if $b_{ij} =1$ then $a_{ij} =0$.
Let $n$ be a positive integer and let $A\in \mathfrak{B}_{n^2}$ be a $n^2 \times n^2$ binary matrix. With the help of $n - 1$ horizontal lines and $n - 1$ vertical lines $A$ has been separated into $n^2$ of number non-intersecting $n\times n$ square sub-matrices $A_{kl}$, $1\le k,l\le n$, e.i.
\begin{equation}\label{matrA}
A =
\left[
\begin{array}{cccc}
A_{11} & A_{12} & \cdots & A_{1n} \\
A_{21} & A_{22} & \cdots & A_{2n} \\
\vdots & \vdots & \ddots & \vdots \\
A_{n1} & A_{n2} & \cdots & A_{nn}
\end{array}
\right] .
\end{equation}
The sub-matrices $A_{kl}$, $1\le k,l\le n$ will be called \emph{blocks}.
A matrix $A\in \mathfrak{B}_{n^2}$ is called an \emph{S-permutation} if in each row, in each column, and in each block of $A$ there is exactly one 1. Let the set of all $n^2 \times n^2$ S-permutation matrices be denoted by $\Sigma_{n^2}$.
The concept of S-permutation matrix was introduced by Geir Dahl \cite{dahl} in relation to the popular Sudoku puzzle. Sudoku is a very popular game. On the other hand, it is well known that Sudoku matrices are special cases of Latin squares in the class of gerechte designs \cite{Bailey}.
Obviously a square $n^2 \times n^2$ matrix $M$ with elements of $[n^2 ] =\{ 1,2,\ldots ,n^2 \}$ is a Sudoku matrix if and only if there are matrices $A_1 ,A_2 ,\ldots ,A_{n^2} \in\Sigma_{n^2}$, each two of them are disjoint and such that $P$ can be given in the following way:
\begin{equation}\label{disj}
M=1\cdot A_1 +2\cdot A_2 +\cdots +n^2 \cdot A_{n^2}
\end{equation}
Some algorithms for obtaining random Sudoku matrices and their valuation are described in detail in \cite{yordzhev_random} and \cite{Fontana}.
In \cite{Fontana} Roberto Fontana offers an algorithm which randomly gets a family of
$n^2 \times n^2$ mutually disjoint S-permutation matrices, where $n = 2, 3$. In $n = 3$ he
ran the algorithm 1000 times and found 105 different families of nine mutually
disjoint S-permutation matrices. Then using (\ref{disj}) he obtained $9! \cdot 105 = 38\; 102\; 400$ Sudoku matrices.
But it is known~\cite{Felgenhauer} that the total number of $9\times 9$ Sudoku matrices is
$$9! \cdot 72^2 \cdot 2^7 \cdot 27\; 704\; 267\; 971 = 6\; 670\; 903\; 752\; 021\; 072\; 936\; 960 $$
Thus, in relation with Fontana's algorithm, it looks useful to calculate the probability of two randomly generated S-permutation matrices to be disjoint.
So the question of enumerating all disjoint pairs of S-permutation matrices naturally arises. This work is devoted to this task.
As we have shown in \cite{Yordzhev20151}, with hand calculations of the so assigned task with small values of $n$ ($n=2,3$), it is convenient to use the apparatus of graph theory. Unfortunately, when $n\ge 4$ this approach is inefficient. In this article, we will use only the operations of matrix analysis, which are not difficult to process with computers.
\section{A representation of S-permutation matrices}\label{kikiki1}
Let $n$ be a positive integer. If $z_1 \; z_2 \; \ldots \; z_n$ is a permutation of the elements of the set $[n] =\left\{ 1,2,\ldots ,n\right\}$ and let us shortly denote $\sigma$ this permutation. Then in this case we will denote by $\sigma (i)$ the $i$-th element of this permutation, i.e. $\sigma (i) =z_i$, $i=1,2,\ldots ,n$.
\begin{definition}\label{defPin}
Let $\Pi_n$ denotes the set of all $n\times n$ matrices, constructed such that $\pi\in\Pi_n$ if and only if the following three conditions are true:
{\bf i)} the elements of $\pi$ are ordered pairs of numbers $\langle i,j\rangle$, where $1\le i,j\le n$;
{\bf ii)} if
$$\left[ \langle a_1 , b_1 \rangle \quad \langle a_2 ,b_2 \rangle \quad \cdots \quad \langle a_n ,b_n \rangle \right]$$
is the $i$-th row of $\pi$ for any $i\in [n] =\{ 1,2,\ldots ,n\}$, then $a_1 \; a_2 \; \ldots \; a_n$ in this order is a permutation of the elements of the set $[n]$;
{\bf iii)} if
$$\left[
\begin{array}{c}
\langle a_1 ,b_1 \rangle \\
\langle a_2 ,b_2 \rangle \\
\vdots \\
\langle a_n ,b_n \rangle \\
\end{array}
\right]
$$
is the $j$-th column of $\pi$ for any $j\in [n]$, then $b_1 ,b_2 ,\ldots , b_n$ in this order is a permutation of the elements of the set $[n]$.
\end{definition}
From Definition \ref{defPin}, it follows that we can represent each row and each column of a matrix $M\in\Pi_n$ with the help of a permutation of elements of the set $[n]$.
Conversely for every $(2n)$-tuple $$\langle \langle \rho_1 ,\rho_2 ,\ldots ,\rho_n \rangle ,\langle \sigma_1 ,\sigma_2 ,\ldots , \sigma_n \rangle \rangle,$$ where
$$\rho_i = \rho_i (1)\; \rho_i (2) \; \ldots \; \rho_i (n),\quad 1\le i\le n$$
$$\sigma_j = \sigma_j (1)\; \sigma_j (2)\; \ldots \; \sigma_j (n),\quad 1\le j\le n$$
are $2n$ permutations of elements of $[n]$ (not necessarily different), then the matrix
$$
\pi =
\left[
\begin{array}{cccc}
\langle \rho_1 (1),\sigma_1 (1)\rangle & \langle \rho_1 (2),\sigma_2 (1)\rangle & \cdots & \langle \rho_1 (n),\sigma_n (1)\rangle \\
\langle \rho_2 (1),\sigma_1 (2)\rangle & \langle \rho_2 (2),\sigma_2 (2)\rangle & \cdots & \langle \rho_2 (n),\sigma_n (2)\rangle \\
\vdots & \vdots & \ddots & \vdots \\
\langle \rho_n (1),\sigma_1 (n)\rangle & \langle \rho_n (2),\sigma_2 (n)\rangle & \cdots & \langle \rho_n (n),\sigma_n (n)\rangle
\end{array}
\right]
$$
is matrix of $\Pi_n$. Hence
\begin{equation}\label{|Pin|}
\left| \Pi_n \right| =\left( n! \right)^{2n}
\end{equation}
\begin{definition}
We say that matrices $\pi ' =\left[ {p'}_{ij} \right]_{n\times n} \in\Pi_n$ and $\pi '' =\left[ {p''}_{ij} \right]_{n\times n} \in\Pi_n$ are \emph{disjoint}, if ${p'}_{ij} \ne {p''}_{ij}$ for every $i,j\in[n]$.
\end{definition}
\begin{definition}
Let $\pi ' ,\pi '' \in\Pi_n$, $\pi ' =\left[ {p'}_{ij} \right]_{n\times n}$, $\pi '' =\left[ {p''}_{ij} \right]_{n\times n}$ and let the integers $i,j\in[n]$ are such that ${p'}_{ij} = {p''}_{ij}$. In this case we will say that ${p'}_{ij}$ and ${p''}_{ij}$ are \emph{component-wise equal elements}.
\end{definition}
Obviously two $\Pi_n$-matrices are disjoint if and only if they do not have component-wise equal elements.
\begin{example}\label{ex1}
\rm We consider the following $\Pi_3$-matrices:
\end{example}
$$
\pi' =\left[ p_{ij}' \right] =
\left[
\begin{array}{ccc}
\langle 3,1\rangle & \langle 2,1\rangle & \langle 1,2\rangle \\
\langle 2,3\rangle & \langle 3,2\rangle & \langle 1,1\rangle \\
\langle 3,2\rangle & \langle 1,3\rangle & \langle 2,3\rangle
\end{array}
\right]
$$
$$
\pi'' =\left[ p_{ij}'' \right] =
\left[
\begin{array}{ccc}
\langle 3,2\rangle & \langle 1,3\rangle & \langle 2,1\rangle \\
\langle 3,3\rangle & \langle 1,1\rangle & \langle 2,2\rangle \\
\langle 2,1\rangle & \langle 1,2\rangle & \langle 3,3\rangle
\end{array}
\right]
$$
$$
\pi''' =\left[ p_{ij}''' \right] =
\left[
\begin{array}{ccc}
\langle 3,1\rangle & \langle 1,3\rangle & \langle 2,2\rangle \\
\langle 2,2\rangle & \langle 3,1\rangle & \langle 1,1\rangle \\
\langle 2,3\rangle & \langle 1,2\rangle & \langle 3,3\rangle
\end{array}
\right]
$$
Matrices $\pi'$ and $\pi''$ are disjoint, because they do not have component-wise equal elements.
Matrices $\pi'$ and $\pi'''$ are not disjoint, because they have two component-wise equal elements: $p_{11}' =p_{11}''' =\langle 3,1\rangle$ and $p_{23}' =p_{23}''' =\langle 1,1\rangle$.
Matrices $\pi''$ and $\pi'''$ are not disjoint, because they have three component-wise equal elements: $p_{12}'' =p_{12}''' =\langle 1,3\rangle$, $p_{32}'' =p_{32}''' =\langle 1,2\rangle$, and $p_{33}' =p_{33}''' =\langle 3,3\rangle$.
The relationship between S-permutation matrices and the matrices from the set $\Pi_n$ are given by the following theorem:
\begin{thm}\label{l2fhgg}
Let $n$ be an integer, $n\ge 2$. Then there is one to one correspondence between the sets $\Sigma_{n^2}$ and $\Pi_n$.
\end{thm}
Proof. Let $A\in \Sigma_{n^2}$. Then $A$ is constructed with the help of formula (\ref{matrA}) and for every $i,j\in [n]$ in the block $A_{ij} $ there is only one 1 and let this 1 has coordinates $(a_i ,b_j )$. For every $i,j\in [n]$ we obtain ordered pairs of numbers $\langle a_i ,b_j \rangle$ corresponding to these coordinates. As in every row and every column of $A$ there is only one 1, then the matrix $\left[ \alpha_{ij} \right]_{n\times n}$, where $\alpha_{ij} =\langle a_i ,b_j \rangle $, $1\le i,j\le n$, which is obtained by the ordered pairs of numbers is matrix of $\Pi_n$, i.e. matrix for which the conditions i), ii) and iii) are true.
Conversely, let $\left[ \alpha_{ij} \right]_{n\times n} \in \Pi_n$, where $\alpha_{ij} =\langle a_i ,b_j \rangle $, $i,j \in [n]$, $a_i ,b_j \in [n]$. Then for every $i,j\in [n]$ we construct a binary $n\times n$ matrices $A_{ij}$ with only one 1 with coordinates $(a_i ,b_j )$. Then we obtain the matrix of type (\ref{matrA}). According to the properties i), ii) and iii), it is obvious that the obtained matrix is S-permutation matrix.
\hfill $\Box$
\begin{corollary}
The number of all pairs of disjoint matrices from $\Sigma_{n^2}$ is equal to the number of all pairs of disjoint matrices from $\Pi_n$.
\end{corollary}
Proof. It is easy to see that with respect of the described in Theorem \ref{l2fhgg} one to one correspondence, every pair of disjoint matrices of $\Sigma_{n^2}$ will correspond to a pair of disjoint matrices of $\Pi_n$ and conversely every pair of disjoint matrices of $\Pi_n$ will correspond to a pair of disjoint matrices of $\Sigma_{n^2}$.
\hfill $\Box$
\begin{corollary} {\rm \cite{dahl}}
The number of all $n^2 \times n^2 $ S-permutation matrices is equal to
\begin{equation}\label{fcrl2}
\left| \Sigma_{n^2} \right| = \left( n! \right)^{2n}
\end{equation}
\end{corollary}
Proof. It follows immediately from Theorem \ref{l2fhgg} and formula (\ref{|Pin|}).
\hfill $\Box$
\section{A formula for counting all disjoint pairs of $n^2 \times n^2$ S-permutation matrices}\label{kikiki2}
Let $A =[a_{ij} ]_{n\times n}\in \mathfrak{B}_n$. We define the following numerical characteristics of the binary matrix A:
\begin{description}
\item[$r_k (A)$] -- the number of rows in $A$ having exactly $k$ units, $k=0,1,2,\ldots ,n$;
\item[$c_k (A)$] -- the number of columns in $A$ having exactly $k$ units, $k=0,1,2,\ldots ,n$;
\item[$\psi_k (A)$] = $\displaystyle r_k (A)+c_k (A)$, $k=0,1,2,\ldots ,n$;
\item[$\varepsilon (A)$] -- the number of units in $A$.
\end{description}
Let $A,B\in \mathfrak{B}_n$. We will say that $A\sim B $, if $B$ is obtained from $A$ after dislocation of some of the rows of $A$. Obviously, the relation defined like that is an equivalence relation. The factor-set ${\mathfrak{B}_n}_{/_\sim}$, i.e. the set of equivalence classes on the above defined relation we denote with $\overline{\mathfrak{B}}_n$. If $A\in \mathfrak{B}_n$, then with $\overline{A}$ we will denote the set $\overline{A} =\{ B\in \mathfrak{B}_n \; |\; B\sim A\}$. Thus $|\overline{A}| =|\{ B\in \mathfrak{B}_n \; |\; B\sim A\} |$ is the cardinality of the set $\overline{A}$. By definition $\overline{\mathfrak{B}}_{n,k} ={\mathfrak{B}_{n,k}}_{/_\sim}$
Obviously if $A,B\in \mathfrak{B}_n$ and $A\sim B$, then $r_k (A)=r_k (B)$, $c_k (A)=c_k (B)$, $\psi_k (A)=\psi_k (B)$, $\varepsilon_k (A)=\varepsilon_k (B)$, $k=0,1,2,\ldots ,n$. So in a natural way we can define the functions $r_k$, $c_k$, $\psi_k$ and $\varepsilon$ in the factor-set $\overline{\mathfrak{B}}_n ={\mathfrak{B}_n}_{/_\sim}$ as $r_k (\overline{A})$, $c_k (\overline{A})$, $\psi_k (\overline{A})$ and $\varepsilon (\overline{A})$ will mean respectively $r_k (A)$, $c_k (A)$, $\psi_k (A)$ $\varepsilon (A)$, where $A$ is an arbitrary representative of the set $\overline{A} =\{ B\in \mathfrak{B}_n \; |\; B\sim A\}$.
\begin{lemma}\label{l3dskmat}
Let $\pi\in\Pi_n$. Then the number $q(n,k)$ of all matrices $\pi' \in\Pi_n$ (including $\pi$), having at least $k$, $k=0,1,\ldots ,n^2$ component-wise equal elements to the matrix $\pi$ is equal to
\begin{equation}\label{fl3gfgfg}
q(n,k)= \sum_{\overline{A}\in \overline{\mathfrak{B}}_{n,k} } |\overline{A} | \prod_{i=0}^{n-2} \left[ \left( n-i\right) ! \right]^{\psi_i (\overline{A})}
\end{equation}
\end{lemma}
Proof. Let $\pi =\left[ p_{ij} \right]_{n\times n} ,\pi' =\left[ p'_{ij} \right]_{n\times n} \in \Pi_n$ and let $\pi$ and $\pi'$ have exactly $k$ component-wise equal elements. Then we uniquely obtain the binary $n\times n$ matrix $A=\left[ a_{ij} \right]_{n\times n}$, such that $a_{ij} =1$ if and only if $p_{ij} =p'_{ij}$, $i,j\in [n]$.
Inversely, let $A=[a_{ij} ]_{n\times n} \in \mathfrak{B}_n$ and let $\pi =\left[ p_{ij} \right]_{n\times n}$ be an arbitrary matrix from $\Pi_n$.We search for the number $h(\pi ,A)$ of all matrices $\pi' =[p'_{ij} ]_{n\times n} \in \Pi_n$, such that $p'_{ij} =p_{ij}$, if $a_{ij}=1$. (It is assumed that there exist $s,t\in [n]$ such that $a_{st} =0$ and $p'_{st} =p_{st}$.)
Let us denote with $\gamma_s$ the number of 1 in $s$-th row of $A$ and let the $s$-th row of $\pi$ correspond to the permutation $\rho_s$ of the elements of $[n]$, $s=1,2,\ldots ,n$. Then there exist $(n-\gamma_s)!$ permutations $\rho'$ of the elements of $ [n]$, such that if $a_{st} =1$, then $\rho_s (t) =\rho' (t)$, $t\in [n]$. Likewise we also prove the respective statement for the columns of $\pi$. Therefore
$$\displaystyle h(\pi ,A) = \prod_{i=0}^n \left[ (n-i)! \right]^{r_i (A)} \prod_{i=0}^n \left[ (n-i)! \right]^{c_i (A)} = \prod_{i=0}^n \left[ (n-i)! \right]^{\psi_i (A)} .$$
From everything said so far it follows that for each $\pi\in\Pi_n$ there exist
$$q(n,k)=\sum_{A\in\mathfrak{B}_{n,k}} \prod_{i=0}^n \left[ (n-i)! \right]^{\psi_i (A)} =\sum_{\overline{A}\in\mathfrak{\overline{B}}_{n,k}} \left| \overline{A} \right| \prod_{i=0}^n \left[ (n-i)! \right]^{\psi_i (\overline{A})}$$
matrices from $\Pi_n$, which have at least $k$ elements that are component-wise equal to the respective elements of $\pi$.
And since $(n-n)!=0!=1$ and $[n-(n-1)]!=1!=1$, then we finally obtain formula (\ref{fl3gfgfg}).
\hfill $\Box$
\begin{lemma}\label{lmm2}
For every integer $n\ge 2$
$$q(n,0)=q(n,1)=(n!)^{2n} =\left| \Pi_n \right| =\left| \Sigma _{n^2} \right| .$$
\end{lemma}
Proof. Let $k=0$. Then ${\mathfrak{B}}_{n,0}$ contains only the matrix, all elements of which are equal to 0. So $|{\mathfrak{B}}_{n,0}|=1$ and if $ A \in {\mathfrak{B}}_{n,0}$ then $|\overline{A} |=1$, $\psi_0 (A) =2n$ and $\psi_i (A) =0$ when $i\ge 1$. Therefore $\displaystyle q(n,0)= \sum_{\overline{A}\in \overline{\mathfrak{B}}_{n,0} } |\overline{A} | \prod_{i=0}^{n-2} \left[ \left( n-i\right) ! \right]^{\psi_i (\overline{A})} = 1\cdot \left[(n-0)!\right]^{2n} \prod_{i=1}^{n-2} \left[ \left( n-i\right) ! \right]^0 = (n!)^{2n}$.
When $k=1$, there are $n^2$ matrices $A\in \mathfrak{B}_{n,1}$. It is easy to see that $|\mathfrak{\overline{B}} |=n$ and for every $\overline{A}\in\overline{\mathfrak{B}}_{n,1}$, $|\overline{A}|=n$, $\psi_0 (\overline{A})=2(n-1)$, $\psi_1 (\overline{A})=2$ and $\psi_i (\overline{A})=0$ for $i>1$. Therefore $q(n,1)=n^2 [(n-0)!]^{2n-2} [(n-1)!]^2 =(n!)^{2n-2} (n!)^2 =(n!)^{2n}$.
\hfill $\Box$
\begin{thm}\label{gl6_th1-bg}
Let $A\in \Sigma_{n^2}$. Then the number $\xi_n$ of all matrices $B\in \Sigma_{n^2}$ which are disjoint with $A$ does not depend on $A$ and is equal to
\begin{equation}\label{gl6_main}
\xi_{n} = \sum_{\overline{A}\in \overline{\mathfrak{B}}_n ,\; \varepsilon (\overline{A})\ge 2} \left( -1\right)^{\varepsilon (\overline{A} )} \left| \overline{A} \right| \prod_{i=0}^{n-2} \left[ \left( n-i\right) ! \right]^{\psi_i (\overline{A})}
\end{equation}
\end{thm}
Proof. Let $n\ge 2$ be an integer. Then applying Theorem \ref{l2fhgg}, Lemma \ref{l3dskmat}, Lemma \ref{lmm2} and the principle of inclusion and exclusion we obtain that the number $\xi_n$ of all matrices $B\in \Sigma_{n^2}$ which are disjoint with $A$ is equal to
$$
\begin{array}{ccc}
\xi_n & = & \displaystyle |\Pi_n | +\sum_{k=1}^{n^2} (-1)^k q(n,k) \\
& = & \displaystyle (n!)^{2n} -(n!)^{2n}+\sum_{k=2}^{n^2} (-1)^k q(n,k) \\
& = & \displaystyle \sum_{k=2}^{n^2} (-1)^k q(n,k),
\end{array}
$$
where the function $q(n,k)$ is calculated with the help of formula (\ref{fl3gfgfg}). Thus we obtain the proof to formula (\ref{gl6_main}).
\hfill $\Box$
\begin{corollary}\label{th2-gl6}
The cardinality $\eta_{n}$ of the set of all disjoint non-ordered pairs of $n^2 \times n^2$ S-permutation matrices is equal to
\begin{equation}\label{nonordereddisjointpair_gl6}
\eta_{n} =\frac{(n!)^{2n}}{2} \xi_n
\end{equation}
where $\xi_n$ is described using formula \ref{gl6_main}.
\end{corollary}
Proof. It follows directly from formula (\ref{fcrl2}) and having in mind that the ''disjoint'' relation is symmetric and antireflexive.
\hfill $\Box$
\begin{corollary}\label{th3_gl6}
The probability $p_n$ of two randomly generated $n^2 \times n^2$ S-permutation matrices to be disjoint is equal to
\begin{equation}\label{probbility_gl6}
p_n = \frac{\displaystyle \xi_n}{\displaystyle \left( n! \right)^{2n} -1} ,
\end{equation}
where $\xi_n$ is described using formula (\ref{gl6_main}).
\end{corollary}
Proof. Applying Corollary \ref{th2-gl6} and formula (\ref{fcrl2}), we obtain:
$$p_n= \frac{\displaystyle \eta_{n}}{\displaystyle {\left| \Sigma_{n^2} \right| \choose 2}} = \frac{\displaystyle \frac{(n!)^{2n}}{2} \xi_n}{\displaystyle \frac{\left( n! \right)^{2n} \left( \left( n! \right)^{2n} -1\right) }{2}} = \frac{\displaystyle \xi_n}{\displaystyle \left( n! \right)^{2n} -1} .$$
\hfill $\Box$
\section{An algorithm for counting}
There is one to one correspondence between the representation of the integers in decimal and in binary notations. So a square binary $n\times n$ matrix can be represented using ordered $n$-tuple of nonnegative integers, which belong to the closed interval $[0,\; 2^n -1]$. Let the integer $a\in [0,\; 2^n -1]$. Then $a$ is represented uniquely in the form:
$$a=\sum_{u=0}^{n-1} b_u (a) 2^u ,$$
where $b_u (a)\in \mathfrak{B} =\{ 0,1\}$, $u=0,1,\ldots ,n-1$. We assume that we have implemented an algorithm for calculating the functions $b_u (a)$ for every $u=0,1,\ldots ,n-1$ and for every $a\in [0,\; 2^n -1]$. For example, in the programming languages C ++ and Java, $b_u (a)$ can be calculated using the expression
\begin{center}
\verb"bu = (a & (1<<u))==0 ? 0 : 1"
\end{center}
Let $A\in {\mathfrak B}_n $. With $\rho (A)$ we will denote the ordered $n$-tuple
$$\rho (A)=\langle x_{1} ,x_{2} ,\ldots ,x_{n} \rangle ,$$
where $0\le x_{i} \le 2^n -1$, $i=1,2,\ldots n$ and $x_{i} $ is the integer written in binary notation with the help of the $i$-th row of $A$.
We consider the set:
$$\begin{array}{lll} {{\mathfrak R}_n } & {=} & {\left\{\langle x_{1} ,x_{2} ,\ldots ,x_{n} \rangle \; |\; 0\le x_{i} \le 2^n -1,\; i=1,2,\ldots n\right\}} \\ {} & {=} & {\left\{ \rho(A)\, |\; A\in {\mathfrak B}_n \right\}} \end{array}$$
Thus we define the mapping
$\rho: {\mathfrak B}_n \to {\mathfrak R}_n ,$
which is bijective and therefore
${\mathfrak B}_n \cong {\mathfrak R}_n .$
If $A\in \mathfrak{B}_n$ and $\rho (A)=\alpha \in \mathfrak{R}_n$, then by analogy we define the numerical characteristics of the element $\alpha\in\mathfrak{R}_n$: $r_k (\alpha )= r_k (A)$, $c_k (\alpha ) =c_k (A)$, $\psi_k (\alpha ) = r_k (\alpha )+c_k (\alpha) =\psi_k (A)$, $k=0,1,2,\ldots ,n$ and $\varepsilon (\alpha )=\varepsilon (A)$. We assume $|\alpha |=|\overline{A}|$, where $\overline{A} =\{ B\in \mathfrak{B}_n \; |\; B\sim A \}$.
Let $\alpha=\langle x_1 , x_2 ,\ldots , x_n \rangle \in \mathfrak{R}_n$ and let $s$ be the number of different elements in $\alpha =\langle x_1 , x_2 ,\ldots , x_n \rangle$. Then the set $X=\{ x_1 , x_2 ,\ldots , x_n \}$ can be divide into parts $$X=X_1 \cup X_2 \cup \cdots \cup X_s$$ such that for every $k\in [s]$ and every $i,j\in [n]$, $i\ne j$ the condition $x_i ,x_j \in X_k$ is satisfied if and only if $x_i =x_j$. We assume $$z_i =\left| X_i \right| ,\quad i=1,2, \ldots s.$$
It is easily seen that
$$\displaystyle |\alpha |=\frac{n!}{\displaystyle \prod_{i=1}^s z_i !} .$$
Let $$\mathfrak{\overline{R}}_n =\{ \langle x_1 , x_2 ,\ldots , x_n \rangle \; |\; 0\le x_1 \le x_2 \le \cdots \le x_n \le 2^n -1 \} \subset \mathfrak{R}_n . $$
It is easily seen that $\mathfrak{\overline{B}}_n \cong \mathfrak{\overline{R}}_n$, which gives the basis to construct the following algorithm for calculating $\xi_n$:
\begin{algorithm} \label{alg1}
Calculation of $\xi_n$.\\
begin\\
$\xi_n :=0$ ;
For every $\alpha =\langle x_1 x_2 ,\ldots , x_n \rangle \in \mathfrak{\overline{R}}_n$ do
\{
\hspace{0.5 cm} $s:=1$;
\hspace{0.5 cm} $\varepsilon (\alpha ):=0$;
\hspace{0.5 cm} For $i=1,2,\ldots ,n$ do
\hspace{0.5 cm} \{
\hspace{1 cm} $z_s := z_s +1$ ;
\hspace{1 cm} $t=0$;
\hspace{1 cm} For $u=0,1,\ldots , n-1$ do
\hspace{1 cm} \{
\hspace{1.5 cm} $t:=t+b_u (x_i )$;
\hspace{1 cm} \}
\hspace{1 cm} $r_t (\alpha ):=r_t (\alpha )+1$;
\hspace{1 cm} $\varepsilon (\alpha ) := \varepsilon (\alpha )+t$;
\hspace{1 cm} If $i<n$ and $x_i <x_{i+1}$ then $s:=s+1$;
\hspace{0.5 cm} \}
\hspace{0.5 cm} If $\varepsilon (\alpha ) = 0$ or $\varepsilon (\alpha )=1$ then go to next $\alpha$;
\hspace{0.5 cm} For $u= 0,1,\ldots ,n-1$ do
\hspace{0.5 cm} \{
\hspace{1 cm} $t:=0$;
\hspace{1 cm} For $i=1,2,\ldots ,n$ do
\hspace{1 cm} \{
\hspace{1.5 cm} $t=t+b_u (x_i )$;
\hspace{1 cm} \}
\hspace{1 cm} $c_t (\alpha ) :=c_t (\alpha )+1$;
\hspace{0.5 cm} \}
\hspace{0.5 cm} For $k=0,1,\ldots , n$ do
\hspace{0.5 cm} \{
\hspace{1 cm} $\psi_k (\alpha ):= r_k (\alpha ) +c_k (\alpha )$;
\hspace{0.5 cm} \}
\hspace{0.5 cm} $\displaystyle |\alpha |:=\frac{n!}{\displaystyle \prod_{i=1}^s z_i !}$;
\hspace{0.5 cm} $\displaystyle T(\alpha):= (-1)^{\varepsilon (\alpha )} |\alpha | \prod_{i=0}^{n-2} \left[ \left( n-i\right) ! \right]^{\psi_i (\alpha)}$;
$\xi_n :=\xi_n +T(\alpha)$;
\}\\
end.
\end{algorithm}
\section{Conclusion}
On the basis of algorithm \ref{alg1} with programming language Java, we made a computer program for calculating $\xi_n$, $\eta_n$ and $p_n$ and we received the following results:\\
$$\xi_2 = 7$$
$$\xi_3 = 17\; 972$$
$$\xi_4 = 41\; 685\; 061\; 617$$
$$\xi_5 =232\; 152\; 032\; 603\; 580\; 176\; 504$$
$$\xi_6 = 7\; 236\; 273\; 578\; 711\; 450\; 275\; 537\; 707\; 547\; 657\; 855$$
$$\eta_2 = 56$$
$$\eta_3 = 419\; 250\; 816$$
$$\eta_4 = 2\; 294\; 248\; 126\; 968\; 596\; 791\; 296$$
$$\eta_5=71\; 871\; 209\; 790\; 288\; 983\; 974\; 921\; 874\; 964\; 480\; 000\; 000\; 000$$
$$\eta_6 = 7\; 022\; 228\; 210\; 556\; 132\; 949\; 916\; 635\; 069\; 726\; 824\; 032\; 981\; 704\; 989\; 720\; 182\; 784 \; \cdot \; 10^{13} $$
$$p_2 = 0.4666666666666667$$
$$p_3 = 0.38521058836137606$$
$$p_4 = 0.3786958223051558$$
$$p_5 = 0.37493849344703684$$
$$p_6 = 0.3728421644517476$$
For n = 2 and n = 3, the results that we get here coincide with the calculations made by hand in \cite{Yordzhev20151}, where we used a graph theory approach.
| {
"timestamp": "2016-08-17T02:09:34",
"yymm": "1604",
"arxiv_id": "1604.02712",
"language": "en",
"url": "https://arxiv.org/abs/1604.02712",
"abstract": "This work examines the concept of S-permutation matrices, namely $n^2 \\times n^2$ permutation matrices containing a single 1 in each canonical $n \\times n$ subsquare (block). The article suggests a formula for counting mutually disjoint pairs of $n^2 \\times n^2$ S-permutation matrices in the general case by restricting this task to the problem of finding some numerical characteristics of the elements of specially defined for this purpose factor-set of the set of $n \\times n$ binary matrices. The paper describe an algorithm that solves the main problem. To do that, every $n\\times n$ binary matrix is represented uniquely as a n-tuple of integers.",
"subjects": "Combinatorics (math.CO); Discrete Mathematics (cs.DM)",
"title": "On the number of mutually disjoint pairs of S-permutation matrices",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9770226334351969,
"lm_q2_score": 0.7248702761768249,
"lm_q1q2_score": 0.7082146661291799
} |
https://arxiv.org/abs/1502.04982 | On the Attached prime ideals of local cohomology modules defined by a pair of ideals | Let $I$ and $J$ be two ideals of a commutative Noetherian ring $R$ and $M$ be an $R$-module of dimension $d$. If $R$ is a complete local ring and $M$ is finite, then attached prime ideals of $H^{d-1}_{I,J}(M)$ are computed by means of the concept of co-localization. Also, we illustrate the attached prime ideals of $H^{t}_{I,J}(M)$ on a non-local ring $R$, for $t= \dim M$ and $t= cd(I,J,M)$. | \section{\bf Introduction}
Throughout this paper, $R$ denotes a commutative
Noetherian ring, $M$ an $R$-module and $I$ and $J$ stand for two
ideals of $R$. For all $i\in \mathbb{N}_0$ the $i$-th
local cohomology functor with respect to $(I,J)$, denoted by
$H^{i}_{I,J}(-)$, defined by Takahashi et. all in \cite{TAK} as the
$i$-th right derived functor of the $(I,J)$- torsion functor $\Gamma
_{I,J}(-)$, where $$\Gamma _{I,J}(M):=\{x \in M : I^{n}x\subseteq Jx
\ \text {for} \ n\gg 1\}.$$ This notion coincides with the
ordinary local cohomology functor $H^{i}_{I }(-)$ when $J=0$, see
\cite{B-SH}.
The main motivation for this generalization comes from the study of
a dual of ordinary local cohomology modules $H^{i}_{I }(M)$
(\cite{sch}). Basic facts and more information about local
cohomology defined by a pair of ideals can be obtained from
\cite{TAK}, \cite{CH} and \cite{CH-W}.
The second section of this paper is devoted to study the attached prime ideals of local cohomology
modules with respect to a pair of ideals by means of
co-localization. The concept of co-localization introduced by
Richardson in \cite{RICH}.
Let $(R,\mathfrak{m})$ be local and $M$ be a
finite $R$-module of dimension $d$. If $c$ is a non-negative integer
such that $H^{i}_{I,J}(R) = 0$ for all $i> c$ and $H^{c}_{I,J} (R)$
is representable, then we illustrate the attached prime ideals of
$^{\mathfrak{p}}H^{c}_{I,J}(M)$ (see Theorem \ref{th mainI}). In addition if
$R$ is complete, then we have made use of Theorem \ref{th mainI} to
prove that in a special case
$$\mbox{Att}\,(H^{d-1}_{I,J}(M)) \subseteq T \cup \mbox{Assh}\,(M)\ \ \text{and} \ \ T\subseteq
\mbox{Att}\,(H^{d-1}_{I,J}(M)),$$ where $$T=\{\mathfrak{p} \in \mbox{Supp}\,(M) : \dim M/\mathfrak{p} M
= d -1, J\subseteq \mathfrak{p} \ and \ \sqrt{I + \mathfrak{p} }= \mathfrak{m}\},$$ (see Theorem
\ref{th mainII}).
In \cite[Theorem 2.1]{CH} the set of attached
prime ideals of $H^{dim M}_{I,J}(M)$ was computed on a local ring. We generalize
this theorem to the non-local case. Also, the authors in
\cite[2.4]{D-YII} specified a subset of attached prime ideals of
ordinary top local cohomology module $H^{cd(I, M)}_{I }(M)$. We improve it
for $H^{cd(I,J,M)}_{I,J}(M)$ over a not necessarily local ring,
where $\mbox{cd}\,(I,J,M)= \sup \{i\in \mathbb{N}_{0}: H^{i}_{I,J}(M)\neq 0
\}$ with the convention that $\mbox{cd}\,(I,M)=\mbox{cd}\,(I,0,M)$.
\section{\bf Attached prime ideals}
\vskip 0.4 true cm
In this section we study the set of attached prime ideals of
local cohomology modules with respect to a pair of ideals.
\begin{rem}\label{ric}
\emph{Following \cite{RICH}, for a multiplicatively
closed subset $S$ of the local ring $(R,\mathfrak{m})$, the co-localization
of $M$ relative to $S$ is defined to be the $S^{-1}R$-module
$S_{-1}(M):=D_{S^{-1}R}(S^{-1}D_{R}(M))$, where $D_R(-)$ is the Matlis
dual functor $\mbox{Hom}\,_R(- , E_R(R/\mathfrak{m}))$. If $S = R \setminus \mathfrak{p} $ for some} \emph{$\mathfrak{p} \in
\mbox{Spec}\,(R)$}\emph{, we write $^{\mathfrak{p}}M$ for $S_{-1}(M)$.}
\emph{Richardson in \cite[2.2]{RICH} proved that if $M$ is a
representable $R$- module, then so is $S_{-1}(M)$ and}
\emph{$\mbox{Att}\,(S_{-1}M) = \{S^{-1}\mathfrak{p} : \mathfrak{p} \in
\mbox{Att}\,(M)\}.$} \emph{Therefore, in order to get some results about
attached prime ideals of a module, it is convenient to study the
attached prime ideals of the co-localization of it.}\end{rem}
\begin{lem}\label{lem befor main}
Let $(R,\mathfrak{m})$ be a local ring, $\mathfrak{a}$ be an ideal of $R$ and $\mathfrak{p}\in
\mbox{Spec}\,(R)$ with $\mathfrak{a}\subseteq \mathfrak{p}$. Let $R'=R/\mathfrak{a}$ and
$\mathfrak{p}'=\mathfrak{p}/\mathfrak{a}$. Then for any $R'$-module $X$ and $R'_{\mathfrak{p}'}$-module $Y$,
the following isomorphisms hold:
$(i)$ \emph{$D_{R}(X)\cong D_{R'}(X) $} as
$R$-modules.
$(ii)$ \emph{$D_{R}(X)_{\mathfrak{p}}\cong D_{R'}(X)_{\mathfrak{p}'}$} as
$R_{\mathfrak{p}}$-modules.
$(iii)$ \emph{$D_{R_{\mathfrak{p}}}(Y) \cong D_{R'_{\mathfrak{p}'}}(Y)$} as
$R_{\mathfrak{p}}$-modules.
\end{lem}
In \cite[2.1 and 2.2]{EGH} the following theorems have been proved
for the attached prime ideals of $H^{d}_{I}(R)$ and $H^{d-1}_{I}(R)$ where $d=\dim R$.
Here, we
generalize these theorems for the local cohomology modules of $M$ with respect to a pair
of ideals when $M$ is a finite $R$-module with $\dim
M=d$.
\begin{thm}\label{th mainI}
Let $(R,\mathfrak{m})$ be a local ring, $M$ be a finite $R$-module, and
\emph{$\mathfrak{p} \in \mbox{Spec}\, (R)$}. Assume that $c=cd(I,J,R)$ and
$H^{c}_{I,J} (R)$ is representable. Then
\begin{enumerate}
\item $\mbox{Att}\,_{R_\mathfrak{p}}(^{\mathfrak{p}}H^{c}_{I,J}(M)) \subseteq \{\mathfrak{q} R_\mathfrak{p} : \dim M/\mathfrak{q} M
\geq c$, $\mathfrak{q} \subseteq \mathfrak{p}$, and $\mathfrak{q} \in \mbox{Spec}\, (R)\}$.
\item If $R$ is complete, then
\begin{eqnarray*}
\mbox{Att}\,_{R_\mathfrak{p}}(^{\mathfrak{p}}H^{dim M}_{I,J}(M)) =& \{&\mathfrak{q} R_\mathfrak{p} : \mathfrak{q}\in \mbox{Supp}\,(M),\dim M/\mathfrak{q} M = \dim M ,
J\subseteq \mathfrak{q}\subseteq \mathfrak{p},\\ & &and \sqrt{I + \mathfrak{q}} = \mathfrak{m}\}.
\end{eqnarray*}
\end{enumerate}
\end{thm}
\begin{proof}
$(1)$ Let $\mathfrak{q} R_\mathfrak{p}\in \mbox{Att}\,_{R_\mathfrak{p}} (^{\mathfrak{p}}H^{c}_{I,J}(M))$. By
\cite[3.1]{T-T-Y} and Remark $\ref{ric}$, we have
$H^{c}_{I,J}(M)$ is representable and $\mbox{Att}\,_{R_\mathfrak{p}} (^{\mathfrak{p}}H^{c}_{I,J}(M)) = \{\mathfrak{q} R_\mathfrak{p} : \mathfrak{q}\in
\mbox{Att}\,(H^{c}_{I,J}(M)) \ and \ \mathfrak{q}\subseteq \mathfrak{p}\}$. Also, using
\cite[6.1.8]{B-SH} and \cite[2.11]{AGH-MEL}
$$\begin{array}{ll}\mbox{Att}\,(H^{c}_{I,J}(M/\mathfrak{q} M))&= \mbox{Att}\,(H^{c}_{I,J}(M))\cap
\mbox{Supp}\,(R/\mathfrak{q}).\end{array}$$ This implies that $H^{c}_{I,J}(M/\mathfrak{q}
M)\neq 0$ and consequently $\dim M/\mathfrak{q} M\geq c$.
$(2)$ Let $\mathfrak{p}\in \mbox{Supp}\,(M)$. Put $d:=\dim M$, $\overline{R} = R/\mbox{Ann}\,_{R}M$, and $$T:= \{\mathfrak{q} R_\mathfrak{p} : \mathfrak{q}\in \mbox{Supp}\,(M),\dim M/\mathfrak{q} M=d, J\subseteq \mathfrak{q}\subseteq \mathfrak{p} \ and \ \sqrt{I + \mathfrak{q}} = \mathfrak{m}\}.$$
Since $\dim_{\overline{R}}M=\dim_{R}M$, \cite[2.7]{TAK} and Lemma
$\ref{lem befor main}$ imply that
$^{\overline{\mathfrak{p}}}H^{d}_{I\overline{R},J\overline{R}}(M)\cong$ $
^{\mathfrak{p}}H^{d}_{I,J}(M)$, as $R_{\mathfrak{p}}$-modules. Therefore, by
\cite[8.2.5]{B-SH}, $\mathfrak{q}\in
\mbox{Att}\,_{\overline{R}_{\overline{\mathfrak{p}}}}(^{\overline{\mathfrak{p}}}H^{d}_{I\overline{R},J\overline{R}}(M))$
if and only if $$\mathfrak{q}\cap R_{\mathfrak{p}}\in \mbox{Att}\,_{R_{\mathfrak{p}}}(^{\overline{\mathfrak{p}}}H^{d}_{I\overline{R},J\overline{R}}(M))
= \mbox{Att}\,_{R_\mathfrak{p}}(^{\mathfrak{p}}H^{d}_{I,J}(M)).$$
Now, without loss of generality, we may assume that $M$ is faithful
and $\dim R = d$.
If $H^{d}_{I,J}(M)=0$, then
$\mbox{Att}\,_{R_{\mathfrak{p}}}(^{\mathfrak{p}}H^{d}_{I,J}(M))=\emptyset$. Assume that $T\neq
\emptyset$ and $\mathfrak{q} R_{\mathfrak{p}}\in T$. Since $\dim M/\mathfrak{q} M = \dim R$, we
have $\dim R/\mathfrak{q}= d$. On the other hand, $\mathfrak{q}\in \mbox{Supp}\, (M/JM)$.
Thus, by \cite[Theorem 2.4]{CH}, $\dim R/(I + \mathfrak{q})> 0$ which
contradicts $\sqrt{I + \mathfrak{q}} = \mathfrak{m}$. So $T=\emptyset$.
Now, we assume that $H^{d}_{I,J}(M)\neq0$.
$\supseteq$: Let $\mathfrak{q} R_{\mathfrak{p}}\in T$. Since $H^{d}_{I,J}(M)$ is an Artinian $R$-module (cf. \cite[2.1]{CH-W}) so, by Remark $\ref{ric}$, it is enough to show that $\mathfrak{q}\in \mbox{Att}\,(H^{d}_{I,J}(M))$.
As $M/\mathfrak{q} M$ is $J$-torsion with dimension $d$ and $\sqrt{I + \mathfrak{q}}=
\mathfrak{m}$ , so by \cite[4.2.1 and 6.1.4]{B-SH}.
$$H^{d}_{I,J}(M/\mathfrak{q} M)\cong H^{d}_{I}(M/\mathfrak{q} M)\cong
H^{d}_{I(R/\mathfrak{q})}(M/\mathfrak{q} M)\cong H^{d}_{\mathfrak{m}/\mathfrak{q}}(M/\mathfrak{q} M)\neq 0.$$
Hence \cite[6.1.8]{B-SH} and \cite[2.11]{AGH-MEL} imply that
$\emptyset\neq \mbox{Att}\,(H^{d}_{I,J}(M/\mathfrak{q} M))= \mbox{Att}\,(H^{d}_{I,J}(M)) \cap
\mbox{Supp}\,(R/\mathfrak{q}).$
Let $\mathfrak{q}_{0}\in \mbox{Att}\,(H^{d}_{I,J}(M))$ be
such that $\mathfrak{q}\subset \mathfrak{q} _{0}$. So that $\dim M/\mathfrak{q}_{0}M < d$. On
the other hand, by Remark $\ref{ric}$, $\mathfrak{q}_{0}R_{\mathfrak{q}_0} \in
\mbox{Att}\,_{R_{\mathfrak{q}_0}} (^{\mathfrak{q}_0}H^{d}_{I,J}(M))$ and this implies that
$\dim M/\mathfrak{q}_{0}M \geq d$ which is a contradiction. So $\mathfrak{q}= \mathfrak{q}
_{0}$.
$\subseteq$: Let $\mathfrak{q} R_{\mathfrak{p}}\in \mbox{Att}\,_{R_\mathfrak{p}}
(^{\mathfrak{p}}H^{d}_{I,J}(M))$. As we have seen in the proof of part $(1)$,
$\dim M/\mathfrak{q} M = d$ and $\mathfrak{q}\subseteq \mathfrak{p}$. So by \cite[2.7]{TAK},
$$H^{d}_{IR/\mathfrak{q},JR/\mathfrak{q}}(M/\mathfrak{q} M)\cong H^{d}_{I,J}(M/\mathfrak{q} M)\neq 0.$$
Now, by \cite[Theorem 2.4]{CH}, there exists $\mathfrak{r}/\mathfrak{q} \in \mbox{Supp}\,(R/\mathfrak{q}\otimes_{R/\mathfrak{q}}
\frac{M/\mathfrak{q} M}{(JR/\mathfrak{q})(M/\mathfrak{q} M)})$ such that $\dim
\frac{R/\mathfrak{q}}{\mathfrak{r}/\mathfrak{q}}=d$ and $\dim \frac{R/\mathfrak{q}}{IR/\mathfrak{q}+\mathfrak{r}/\mathfrak{q}}=0 $.
Since $\mathfrak{q} R_{\mathfrak{p}} \in \mbox{Att}\,_{R_{\mathfrak{p}}} (^{\mathfrak{p}}H^{d}_{I,J}(M))$, we
have $\mathfrak{q} \in \mbox{Att}\,(H^{d}_{I,J}(M))$ and so $\mathfrak{q}\in \mbox{Supp}\,(M)\cap
V(J)$. Hence $\mathfrak{q}/\mathfrak{q}\in \mbox{Supp}\,_{R/\mathfrak{q}} (M/\mathfrak{q} M)$ and then
$$\dim R/\mathfrak{q}= \dim M/\mathfrak{q} M= d= \dim \frac{R/\mathfrak{q}}{\mathfrak{r}/\mathfrak{q}}=\dim R/\mathfrak{q}
.$$
Therefore, $\dim R/\mathfrak{q} = \dim R/\mathfrak{r} $ which shows that $\mathfrak{q}=\mathfrak{r}$.
Thus $\sqrt{I+\mathfrak{q}}=\mathfrak{m}$.
\end{proof}
\begin{rem}
\emph{The inclusion in Theorem $\ref{th mainI}$(1) is not an equality
in general. Let the assumption be as in Theorem $\ref{th mainI}$.
Assume that $H^{d}_{I,J}(M)=0$, $\mathfrak{p}\in \mbox{Min}\,(M)$ and $\dim M/\mathfrak{p}
M=d$. Then }\emph{$\mbox{Att}\,_{R_\mathfrak{p}}
(^{\mathfrak{p}}H^{d}_{I,J}(M))=\emptyset$}\emph{. But}
$$\emph{$\{\mathfrak{q} R_\mathfrak{p} : \dim M/\mathfrak{q} M = d,\mathfrak{q} \subseteq \mathfrak{p}$ and
$\mathfrak{q}\in \mbox{Supp}\, (M)\} = \{\mathfrak{p} R_{\mathfrak{p}}\}$.}$$
\end{rem}
\begin{thm}\label{th mainII}
Let $(R,\mathfrak{m})$ be a complete local ring and $M$ be a finite
$R$-module with dimension $d$. Assume that $H^{i}_{I,J}(R) = 0$ for
all $i
> d-1$ and $H^{d-1}_{I,J} (R)$ is representable. Then
\begin{enumerate}
\item
\begin{eqnarray*}
\mbox{Att}\,_R (H^{d-1}_{I,J}(M)) \subseteq &\{& \mathfrak{p} \in \mbox{Supp}\,(M) :
\dim M/\mathfrak{p} M = d -1, J\subseteq \mathfrak{p} \ and \ \sqrt{I+\mathfrak{p}} =
\mathfrak{m}\} \\ & & \cup \mbox{Assh}\,(M).
\end{eqnarray*}
\item
$$
\{\mathfrak{p} \in \mbox{Supp}\,(M) : \dim M/\mathfrak{p} M = d
-1, J\subseteq \mathfrak{p} \ and \ \sqrt{I + \mathfrak{p} }= \mathfrak{m}\}\subseteq \mbox{Att}\,
(H^{d-1}_{I,J}(M)).$$
\end{enumerate}
\end{thm}
\begin{proof}
$(1)$ First we note that, by \cite[4.8]{TAK} and \cite[3.1]{T-T-Y},
$H^{d-1}_{I,J}(M)$ is representable and $\mbox{Att}\,
(H^{d-1}_{I,J}(M))\subseteq \mbox{Supp}\, (M)$. Now, let $\mathfrak{p} \in
\mbox{Att}\,(H^{d-1}_{I,J}(M))$. Since $\mathfrak{p} R_\mathfrak{p} \in \mbox{Att}\,_{R_\mathfrak{p}}
(^{\mathfrak{p}}H^{d-1}_{I,J}(M))$, by Theorem $\ref{th mainI}$ $(1)$, $\dim
M/\mathfrak{p} M \geq d-1$.
If $\dim M/\mathfrak{p} M=d$, then $\dim R/\mathfrak{p}=d$ and so $\mathfrak{p} \in \mbox{Assh}\,(M)$.
Now, assume that $\dim M/\mathfrak{p} M = d-1$. Since $\mathfrak{p} \in
\mbox{Att}\,(H^{d-1}_{I,J}(M))$, $H^{d-1}_{IR/\mathfrak{p},JR/\mathfrak{p}}(M/\mathfrak{p} M)\cong
H^{d-1}_{I,J}(M/\mathfrak{p} M)\neq 0$. Thus, by \cite[Theorem 2.4]{CH}, there exists $\mathfrak{r}/\mathfrak{p}
\in \mbox{Supp}\,(\frac{M/\mathfrak{p} M}{(JR/\mathfrak{p})(M/\mathfrak{p} M)})$ such that $\dim
\frac{R}{\mathfrak{r}}=d$ and
$\dim \frac{R}{I+\mathfrak{r}}=0 $. Hence $\mathfrak{r}=\mathfrak{p} , J\subseteq \mathfrak{p},$ and $\sqrt{I+\mathfrak{p}}=\mathfrak{m}$.
$(2)$ Let $\mathfrak{p}\in \mbox{Supp}\,(M)$, $J\subseteq \mathfrak{p}$, $\dim M/\mathfrak{p} M = d-1$, and
$\sqrt{I + \mathfrak{p}}=\mathfrak{m}$. Then, by \cite[3.1]{T-T-Y} and Theorem
$\ref{th mainI}$ $(2)$, $H^{d-1}_{I,J}(M)$ is representable, $\mathfrak{p}
R_\mathfrak{p} \in \mbox{Att}\,_{R_\mathfrak{p}} (^{\mathfrak{p}}H^{d-1}_{I,J}(M/\mathfrak{p} M))$, and so
$\mathfrak{p}\in \mbox{Att}\, (H^{d-1}_{I,J}(M/\mathfrak{p} M))$. Now, the proof is complete by considering the epimorphism
\\$H^{d-1}_{I,J}(M)\rightarrow H^{d-1}_{I,J}(M/\mathfrak{p} M)$.
\end{proof}
In the rest of the paper, following \cite{TAK}, we use the notations
$$W(I,J):= \{\mathfrak{p}\in Spec(R): I^{n}\subseteq \mathfrak{p} + J\
for \ an \ integer \ n\geq1\}$$ and
$$\widetilde{W}(I,J):= \{\mathfrak{a}: \mathfrak{a} \ is \ an \ ideal \ of \ R; I^{n}\subseteq \mathfrak{a}+J \ for \ an \ integer \ n\geq
1\}.$$
The following lemma can be proved using \cite[3.2]{TAK}.
\begin{lem}\label{supp}
For any non-negative integer $i$ and $R$-module $M$,
$(i)$ \emph{$\mbox{Supp}\,(H^{i}_{I,J}(M))\subseteq \underset{\mathfrak{a} \in
\widetilde{W}(I,J)}{\bigcup}\mbox{Supp}\,(H^{i}_{\mathfrak{a}}(M))$}.
$(ii)$ \emph{$\mbox{Supp}\,(H^{i}_{I,J}(M))\subseteq \mbox{Supp}\,(M) \cap W(I,J)$}.
\end{lem}
\begin{cor}\label{att}
Let $M$ be an $R$-module and $c=cd(I,J,R)$. Assume that $M$ is
representable or $H^{c}_{I,J}(R)$ is finite. Then
\emph{$$\mbox{Att}\,(H^{c}_{I,J}(M))\subseteq \mbox{Att}\,(M)\cap W(I,J).$$}
\end{cor}
\begin{proof}
By \cite[4.8]{TAK}, \cite[2.11]{AGH-MEL},\cite[3.1]{T-T-Y} and Lemma
\ref{supp} $(ii)$, we have
$$\begin{array}{lll}\mbox{Att}\,(H^{c}_{I,J}(M))&= \mbox{Att}\,(M\otimes
H^{c}_{I,J}(R))&\subseteq \mbox{Att}\,(M)\cap
\mbox{Supp}\,(H^{c}_{I,J}(R))\\&&\subseteq \mbox{Att}\,(M)\cap W(I,J).\end{array}$$
\end{proof}
Applying the set of attached prime ideals of top local cohomology module in
\cite[Theorem 2.2]{CH}, we obtain another presentation for it.
\begin{prop}\label{hat}
Let $(R,\mathfrak{m})$ be a local ring and $\hat{R}$ denotes the $\mathfrak{m}-$adic
completion of $R$. Suppose that $M$ is a finite $R$-module of
dimension $d$. Then
\begin{eqnarray*}
\mbox{Att}\,_{R}(H^{d}_{I,J} (M))=&\{&\mathfrak{q}\cap R : \mathfrak{q} \in
\mbox{Supp}\,_{\hat{R}} (\hat{R}\otimes_{R}M/JM), \dim(\hat{R}/\mathfrak{q}) = d,\\
&&and \
\dim\hat{R}/(I\hat{R} + \mathfrak{q}) = 0\}.
\end{eqnarray*}
\end{prop}
\begin{proof}
Denote the set of right hand side of the assertion by $T$. It is
clear that by\cite[Theorem 2.4]{CH}, $H^{d}_{I,J} (M)= 0$ if and only if
$T=\emptyset$. Assume that $H^{d}_{I,J} (M)\neq 0$ and $\mathfrak{p}\in \mbox{Supp}\,
(M/JM)$ with the property that $\mbox{cd}\, (I,R/\mathfrak{p}) = d$. Let $\mathfrak{q}\in
\mbox{Ass}(M/JM)$ be such that $\mathfrak{q}\subseteq \mathfrak{p}$. Then $$d=\mbox{cd}\,(I,R/\mathfrak{p})
\leq \mbox{cd}\,(I,R/\mathfrak{q})\leq \dim R/\mathfrak{q}\leq \dim M/JM\leq \dim M= d $$
implies that $\mathfrak{p}=\mathfrak{q}\in \mbox{Ass}(M/JM)$ and $\dim M/JM=d$. Now the
claim follows from \cite[3.10]{T-T-Y} and \cite[Theorem 2.1]{CH}.
\end{proof}
The following lemma, which can be proved by using the similar
argument of \cite[4.3]{TAK}, will be applied in the rest of the
paper.
\begin{lem}\label{4.3tak}
Let $M$ be a finite $R$-module. Suppose that $J\subseteq J(R)$,
where $J(R)$ denotes the Jacobson radical of $R$, and \emph{$\dim
M/JM=d$} be an integer. Then $H^{i}_{I,J}(M)=0$ for all $i>d$.
\end{lem}
Using Lemma \ref{4.3tak}, we can compute $\mbox{Att}\,(H^{dim M}_{I,J}(M))$
in non-local case as a generalization of \cite[2.5]{DIV}.
\begin{prop}\label{j(r)}
Let $M$ be a finite $R$-module of dimension $d$ and $J\subseteq
J(R)$. Then
\emph{$$ \begin{array}{ll}\mbox{Att}\,(H^{d}_{I,J}(M))&=\mbox{Att}\,(H^{d}_{I}(M/JM))\\ &=\{\mathfrak{p}\in
\mbox{Ass}(M)\cap V(J): \mbox{cd}\,(I,R/\mathfrak{p}) = d\}.\end{array}$$}
\end{prop}
\begin{proof}
The assertion holds by applying Lemma $\ref{4.3tak}$ and using the
same method of the proof of \cite[Theorem 2.1 and Proposition
2.1]{CH}.
\end{proof}
\begin{cor}\label{quotient}
Suppose that $J\subseteq J(R)$ and $M$ is a finite $R$-module such
that $\dim M=d$. Then \emph{$$\mbox{Att}\,(\frac{H^{d}_{I,J}(M)}{J
H^{d}_{I,J}(M)}) =\{\mathfrak{p}\in \mbox{Supp}\, (M)\cap V(J):\mbox{cd}\, (I,R/\mathfrak{p}) =
d\}.$$}
\end{cor}
\begin{proof}
Let $\overline{R} = R/\mbox{Ann}\,_{R}M$. Using \cite[2.7]{TAK},
$H^{d}_{I,J}(M)\cong H^{d}_{I\overline{R},J\overline{R}}(M)$ and
also for a prime $\mathfrak{p}\in \mbox{Supp}\, (M)\cap V(J)$,
$\mbox{cd}\,(I\overline{R},\overline{R}/\mathfrak{p})=\mbox{cd}\,(I,R/\mathfrak{p})$. Thus we may
assume that $M$ is faithful and so $\dim R=d$. In virtue of
\cite[6.1.8]{B-SH}, $H^{d}_{I}(M/JM)\cong H^{d}_{I,J}(M/JM) \cong
\frac{H^{d}_{I,J}(M)}{J H^{d}_{I,J}(M)}$. Now, the assertion follows
by Proposition $\ref{j(r)}$.
\end{proof}
The final result of this section is a generalization of
\cite[2.4]{D-YII} in non-local case for local cohomology modules
with respect to a pair of ideals.
\begin{prop}\label{M-purity}
Let $J\subseteq J(R)$ and $M$ be a finite $R$-module. Then
$$\{\mathfrak{p}\in \mbox{Ass}(M)\cap V(J) : \mbox{cd}\,(I,R/\mathfrak{p})=\dim R/\mathfrak{p}=\mbox{cd}\,(I,J,M)\} \subseteq \mbox{Att}\,(H^{\mbox{cd}\,(I,J,M)}_{I,J}(M)).$$
Equality holds if
$\mbox{cd}\,(I,J,M) = \dim M$.
\end{prop}
\begin{proof}
The same proof of \cite[2.4]{D-YII} remains valid by using
Proposition $\ref{j(r)}$.
\end{proof}
| {
"timestamp": "2015-02-18T02:12:37",
"yymm": "1502",
"arxiv_id": "1502.04982",
"language": "en",
"url": "https://arxiv.org/abs/1502.04982",
"abstract": "Let $I$ and $J$ be two ideals of a commutative Noetherian ring $R$ and $M$ be an $R$-module of dimension $d$. If $R$ is a complete local ring and $M$ is finite, then attached prime ideals of $H^{d-1}_{I,J}(M)$ are computed by means of the concept of co-localization. Also, we illustrate the attached prime ideals of $H^{t}_{I,J}(M)$ on a non-local ring $R$, for $t= \\dim M$ and $t= cd(I,J,M)$.",
"subjects": "Commutative Algebra (math.AC)",
"title": "On the Attached prime ideals of local cohomology modules defined by a pair of ideals",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9770226334351969,
"lm_q2_score": 0.7248702761768248,
"lm_q1q2_score": 0.7082146661291798
} |
https://arxiv.org/abs/1911.07411 | Learning product graphs from multidomain signals | In this paper, we focus on learning the underlying product graph structure from multidomain training data. We assume that the product graph is formed from a Cartesian graph product of two smaller factor graphs. We then pose the product graph learning problem as the factor graph Laplacian matrix estimation problem. To estimate the factor graph Laplacian matrices, we assume that the data is smooth with respect to the underlying product graph. When the training data is noise free or complete, learning factor graphs can be formulated as a convex optimization problem, which has an explicit solution based on the water-filling algorithm. The developed framework is illustrated using numerical experiments on synthetic data as well as real data related to air quality monitoring in India. | \section{Introduction}
Leveraging the underlying structure in data is central to many machine learning and signal processing tasks~\cite{Bigdata,shuman2012emerging,smola2003kernels,cai2010graph}. In many cases, data resides on irregular (non-Euclidean) domains. Some examples include datasets from meteorological stations, traffic networks, social and biological networks, to name a few.
Graphs offer a natural way to describe the structure and explain complex relationships in such network datasets. More specifically, data (signal) is indexed by the nodes of a graph and the edges encode the relationship between the function values at the nodes. When the number of nodes in the graph is very large, signal processing or machine learning operations defined on graphs require more memory and computational resources, e.g., computing the graph Fourier transform~\cite{shuman2012emerging} using an eigendecomposition requires $O(N^3)$ operations for a graph with $N$ nodes~\cite{CooleyTukey}. Whenever the underlying graph can be factorized into two or more factor graphs with fewer nodes, the computational costs of these operations can be reduced significantly~\cite{Bigdata}. Product graphs can efficiently represent multidomain data. For instance, in brain imaging, the data is multidomain as each spatially distributed sensor gathers temporal data. Similarly, in movie recommender systems (such as Netflix) the rating data matrix has a user dimension as well as a movie dimension. The graph underlying such multidomain data can be often be factorized so that each graph factor corresponds to one of the domains.
Having a good quality graph is essential for most of the machine learning or signal processing problems over graphs. However, in some applications, the underlying graph may not readily available, and it has to be estimated from the available training data. Given the training data, the problem of estimating the graph Laplacian or the weighted adjacency matrix, assuming that the graph signals are smooth on the underlying topology has been considered in\cite{learnDong,kalofolias2016learn,chepuri2016learning,kalofolias2017learning,LearnGraphData,egilmez2017graph}. Even though the available training data might be multidomain, existing graph learning methods ignore the product structure in the graph. For example, when dealing with data collected from spatially distributed sensors over a period of time, existing methods learn a graph that best explains the spatial domain while ignoring the temporal structure. In \cite{kalofolias2017learning}, time-varying graphs are estimated from smooth signals, where the second domain is assumed to be regular.
In this paper, instead of ignoring the structure in any of the domains or treating one of the domains as regular, we propose to learn the underlying graphs related to each domain. This corresponds to learning the factors of the product graph. Concretely, the contributions of this paper are as follows.
We propose a framework for estimating the graph Laplacian matrices of the factors of the product graph from the training data. The product graph learning problem can be solved optimally using a {\it water-filling} approach. Numerical experiments based on synthetic and real data related to air quality monitoring are provided to demonstrate the developed theory. Since the real dataset has many missing entries, we present an alternating minimization algorithm for {\it joint matrix completion and product graph learning}.
Throughout this paper, we will use upper and lower case boldface letters to denote matrices and column vectors, respectively. We will denote sets using calligraphic letters. ${\bf 1}$ (${\bf 0}$) denotes the vector/matrix of all ones (zeros) of appropriate dimension. $\bbI_P $ denotes the identity matrix of dimension $P$. $\diag[\cdot]$ is a diagonal matrix with its argument along the main diagonal.
${\rm vec}(\cdot)$ denotes the matrix vectorization operation. $X_{ij}$ and
$x_i$ denote the $(i,j)$th element and $i$th element of $\bbX$ and $\bbx$, respectively. $\oplus$ represents the Kronecker sum and $ \otimes $ represents the Kronecker product.
\section{Product graph signals} \label{sec:productgraphs}
Consider a graph $\ccalG_{N} = (\ccalV_{N},\ccalE_{N})$ with $N$ vertices (or nodes), where $\ccalV_{N}$ denotes the set of vertices
and $\ccalE_{N}$ denotes the edge set. The structure of the graph with $N$ nodes is captured by the weighted adjacency matrix $\bbW \in \reals^{N \times N} $ whose $(i,j)$th entry denotes the weight of the edge between node $i$ and node $j$. When there is a no edge between node $i$ and node $j$, the $(i,j)$th entry of $\bbW$ is zero. We assume that the graph is undirected with positive edge weights. The corresponding graph Laplacian matrix is a symmetric matrix of size $ N $, given by $ \bbL_{N} = {\diag}[\bbd] -\bbW$, where $\bbd \in \reals^N$ is a degree vector given by $ \bbd = \bbW{\bf 1} $.
Consider two graphs $\ccalG_{P} = (\ccalV_{P},\ccalE_{P})$ and $\ccalG_{Q} = (\ccalV_{Q},\ccalE_{Q})$ with $P$ and $Q$ nodes, respectively. Let the corresponding graph Laplacian matrices be $ \bbL_{P} \in \reals^{P \times P}$ and $ \bbL_{Q} \in \reals^{Q \times Q}$. Let the Cartesian product of two graphs $ \ccalG_{P} $ and $ \ccalG_{Q}$ be denoted by $\ccalG_{N} $ with $ |\ccalV_{N}| = |\ccalV_{P}||\ccalV_{Q}|$ nodes, i.e., $PQ = N$. In other words, the graphs $\ccalG_{P}$ and $\ccalG_{Q}$ are the factors of $\ccalG_{N}$. Then the graph Laplacian matrix $\bbL_{N}$ can be expressed in terms of $\bbL_{P}$ and $\bbL_Q$ as
\begin{equation}\label{eq:cartesian_product}
\begin{aligned}
\bbL_{N} = \bbL_{P} \oplus \bbL_{Q} = \bbI_Q \otimes \bbL_{P} + \bbL_{Q} \otimes \bbI_P.
\end{aligned}
\end{equation}
Let us collect the set of graph signals $\{\bbx_{i} \}^T_{i=1}$ with $\bbx_i \in \reals^N$ defined on the product graph $\ccalG_N$ in an $N \times T$ matrix $\bbX = [\bbx_1, \bbx_2,...,\bbx_T]$. As each node in the product graph $\ccalG_N$ is related to a pair of vertices in its graph factors, we can reshape any product graph signal $\bbx_i$ as $\bbX_i \in \reals^{P \times Q}$, $i=1,2,\ldots,T$, such that $\bbx_i = {\rm vec}(\bbX_i)$.
This means that each product graph signal represents a multidomain graph signal where the columns and rows of $\bbX_i$ are graph signals associated to the graph factor $\ccalG_P$ and $\ccalG_Q$, respectively.
In this work, we will assume that $\bbX$ is smooth with respect to (w.r.t.) the graph $\ccalG_{N}$. The amount of smoothness is quantified by the Laplacian quadratic form ${\rm tr}(\bbX^T \bbL_{N} \bbX)$, where small values of ${\rm tr}(\bbX^T \bbL_{N} \bbX)$ imply that the data $\bbX$ is smooth on the graph $\ccalG_N$.
\section{Task-cognizant product graph learning}\label{ProbelmStatement}
Suppose we are given the training data $\bbX \in \reals^{N \times T}$ defined on the product graph $\ccalG_{N}$, where each column of $\bbX$, i.e., $\bbx_i \in \reals^{N}$, represents an multidomain graph data $\bbX_i \in \reals^{P \times Q}$.
Assuming that the given data is smooth on the graph $\ccalG_N$ and the product graph $\ccalG_N$ can be factorized as the Cartesian product of two graphs $\ccalG_P$ and $\ccalG_Q$ as in \eqref{eq:cartesian_product} we are interested in estimating the graph Laplacian matrices $\bbL_P$ and $\bbL_Q$. To do so, we assume that $P$ and $Q$ are known.
Typically, we might not have access to the original graph data $\bbX$, but we might observe data related to $\bbX$. Let us call this data $\bbY \in \reals^{N \times T}$. For example, $\bbY$ could be a noisy or an incomplete version of $\bbX$.
Given $\bbY$, the joint estimation of $\bbX$, $\bbL_P$ and $\bbL_Q$, may be mathematically formulated as the following optimization problem
\begin{equation}\label{eq:Problem_modeling}
\begin{aligned}
& \underset{{\bbL_{P} \in \ccalL_P,\bbL_{Q} \in \ccalL_Q, \bbX}}{{\rm minimize}}
& & f(\bbX,\bbY) + \alpha {\rm tr}(\bbX^{T}(\bbL_{P}\oplus \bbL_{Q})\bbX)\\
& && \> + \beta_{1} \|\bbL_P \|_{F}^{2} + \beta_2 \|\bbL_Q \|_{F}^{2},
\end{aligned}
\end{equation}
where $\ccalL_N := \{ \bbL \in \reals^{N \times N} | \bbL {\bf 1} = {\bf 0}, {\rm tr}(\bbL) = N, L_{ij} = L_{ji} \leq 0, i \neq j\}$ is the space of all the valid Laplacian matrices of size $N$. We use the trace equality constraint in this set to avoid a trivial solution. The loss function $f(\bbX,\bbY)$ is appropriately chosen depending on the nature of the observed data. More importantly, we learn the product graph suitable for the task of minimizing $f(\bbX,\bbY)$. For instance, if the observed graphs signals are noisy as $\bby_i = \bbx_i + \bbn_i$ for $1 \leq i \leq T$ with the noise vector $\bbn_i$, then $f(\bbX,\bbY)$ is chosen as $\|\bbX-\bbY\|_F^2$.
A smoothness promoting quadratic term is added to the objective function with a positive regularizer $\alpha$.
The squared Frobenius norm of the Laplacian matrices with tuning parameters $\beta_1$ and $\beta_2$ in the objective function controls the distribution of edge weights.
By varying $\alpha$, $\beta_1$ and $\beta_2$, we can control the sparsity (i.e., the number of zeros) of $\bbL_P$ and $\bbL_Q$, while setting $\beta_1 = \beta_2 = 0$ gives the sparsest graph.
\section{Solver by adapting existing works}\label{ExistingWorks}
Ignoring the structure of $\ccalG_{N}$ that it can be factorized as $\ccalG_P$ and $\ccalG_Q$, and given $\bbX$ (or its noisy version), one can learn the graph Laplacian matrix $\bbL_N$ using any one of the methods in~\cite{learnDong,kalofolias2016learn,chepuri2016learning,kalofolias2017learning}. In this section, we will develop a solver for the optimization problem \eqref{eq:Problem_modeling} based on the existing method in~\cite{learnDong}.
There exists several numerical approaches for factorizing product graphs, i.e., to find the graph factors $\bbL_P$ and $\bbL_Q$ from $\bbL_{N}$~\cite{Hammack_HandBookofGraphProducts,Imrich:2008:TGT:1608956}.
One of the simplest and efficient ways to obtain the graph factors is by solving the convex optimization problem
\begin{equation}\label{eq:Cartesian_Factorization}
\underset{{\bbL_P \in \ccalL_P,\bbL_{Q} \in \ccalL_Q}}{{\rm minimize}} \| \bbL_N - \bbL_P \oplus \bbL_Q\|^2_F
\end{equation}
The above problem can be solved using any one of the off-the-shelf convex optimization toolboxes. However, notice that this is a two-step approach, which requires computating a size-$N$ Laplacian matrix in the first step using~\cite{learnDong}, for instance. The first step is computationally much expensive as the number of nodes in the product graph $\ccalG_{N}$ increases exponentially with the increase in the number of nodes in either of the factor graphs $\ccalG_P$ and $\ccalG_Q$. In other words, finding the product graph Laplacian matrix and then factorizing it is computationally expensive and sometimes infeasible because of the huge memory requirement for storing the data. Therefore, in the next section, we propose a computationally efficient solution.
\section{Proposed solver}\label{ProposedSolver}
In this section, instead of a two-step approach, as discussed in Section~\ref{ExistingWorks}, we provide a one-step solution for estimating the Laplacian matrices of the factors of the product graph by leveraging the properties of the Cartesian graph product and the structure of the data.
Assuming that the training data is noise free and complete, that is, $\bbY = \bbX$, the quadratic smoothness promoting term ${\rm tr}(\bbX^T (\bbL_P \oplus \bbL_Q) \bbX)$ can be written as
\begin{equation}\label{eq:traceExpansion}
{\rm tr}(\bbX^T (\bbL_P \oplus \bbL_Q) \bbX)= \sum_{i=1}^{T} \bbx_i^T(\bbL_P \oplus \bbL_Q)\bbx_i.
\end{equation}
Using the property of the Cartesian product that for any given matrices $\bbA$, $\bbB$, $\bbC$ and $\bbX$ of appropriate dimensions, the equation $\bbA\bbX + \bbX\bbB = \bbC$ is equivalent to $(\bbA \oplus \bbB^T) \rm vec(\bbX) = \rm vec(\bbC)$. Using the property that ${\rm tr}(\bbA^T\bbX) = {\rm vec}(\bbA)^T {\rm vec}(\bbX)$, \eqref{eq:traceExpansion} simplifies to
\begin{equation}\label{eq:traceExpansion3}
\begin{aligned}
& \sum_{i=1}^{T} \bbx_i^T(\bbL_P \oplus \bbL_Q)\bbx_i &&=\sum_{i=1}^{T}{\rm vec}(\bbX_i)^T {\rm vec}(\bbL_P \bbX_i + \bbX_i \bbL_Q)\\ \nonumber
&&&=\sum_{i=1}^{T}{\rm tr}(\bbX_i^T \bbL_P \bbX_i) + {\rm tr}(\bbX_i \bbL_Q \bbX_i^T). \nonumber
\end{aligned}
\end{equation}
This means that the amount of smoothness of the multidomain signal $\bbx_i$ w.r.t. $\ccalG_N$ is equal to the sum of the amount of smoothness of the signals collected in the rows and columns of $\bbX_i$ w.r.t. $\ccalG_P$ and $\ccalG_Q$, respectively.
Therefore, with $\bbX$ available, solving \eqref{eq:Problem_modeling} is equivalent to solving
\begin{equation}\label{eq:EquivalentFormulation}
\begin{aligned}
& \underset{{\bbL_{P} \in \ccalL_P,\bbL_{Q} \in \ccalL_Q}}{{\rm minimize}}
&&\alpha \sum_{i=1}^{T}[ {\rm tr}(\bbX_i^T \bbL_P \bbX_i) + {\rm tr}(\bbX_i \bbL_Q \bbX_i^T)] \\
& && \> + \beta_{1} \|\bbL_P \|_{F}^{2} + \beta_2 \|\bbL_Q \|_{F}^{2}
\end{aligned}
\end{equation}
with variables $\bbL_{P}$ and $\bbL_Q$. The optimization problem \eqref{eq:EquivalentFormulation} has a unique minimizer as it is convex in $\bbL_P$ and $\bbL_Q$. In fact, it can be expressed as a quadratic program (QP) with fewer variables as compared to~\eqref{eq:EquivalentFormulation} by exploiting the fact that $\bbL_P$ and $\bbL_Q$ are symmetric matrices. In essence, we need to solve for only the lower or upper triangular elements of $\bbL_P$ and $\bbL_Q$.
Let the vectorized form of the lower triangular elements of $\bbL_P$ and $\bbL_Q$ be denoted by ${\rm vecl}(\bbL_P) \in \reals^{P(P+1)/2}$ and ${\rm vecl}(\bbL_Q) \in \reals^{Q(Q+1)/2}$, respectively. Furthermore, ${\rm vecl}(\bbL_P)$ and ${\rm vecl}(\bbL_Q)$, may be, respectively, related to ${\rm vec}(\bbL_P)$ and ${\rm vec}(\bbL_Q)$ as ${\rm vec}(\bbL_P) = \bbA {\rm vecl}(\bbL_P)$, ${\rm vec}(\bbL_Q) = \bbB {\rm vecl}(\bbL_Q)$ using matrices $\bbA$ and $\bbB$ of appropriate dimensions.
Now, using the fact that ${\rm tr}(\bbX^T \bbL_P \bbX) = {\rm vec}(\bbX\bbX^T)^T {\rm vec}(\bbL_P)$ and
${\rm tr}(\bbX \bbL_Q \bbX^T) = {\rm vec}(\bbX^T\bbX)^T {\rm vec}(\bbL_Q)$ and the properties of Frobenius norm, we may rewrite \eqref{eq:EquivalentFormulation} as
\begin{equation}\label{eq:QpFormulation}
\underset{{\bbz \in \reals^{K}}}{{\rm minimize}} \quad
\dfrac{1}{2} \bbz^T \bbP \bbz + \bbq^T\bbz
\quad \text{subject to } \quad \bbC\bbz = \bbd, \,\, \bbz \geq \bb0,
\end{equation}
where $\bbz = \left[{\rm vecl}^T(\bbL_P), {\rm vecl}^T(\bbL_Q)\right]^T$ is the optimization variable of length $K = 0.5(P^2+Q^2+ P+ Q)$. Here, $\bbP = {\rm diag}[2\beta_1 \bbA^T \bbA,$ $2\beta_2 \bbB^T \bbB]$ is the diagonal matrix of size $K$ and $\bbq =\alpha \sum_{i=1}^T \bbq_i$ with $\bbq_i = [{\rm vec}(\bbX_i\bbX^T_i)^T \bbA, {\rm vec}(\bbX^T_i\bbX_i)^T \bbB]^T \in \reals^{K}$. The matrix $\bbC$ and the vector $\bbd$ are defined appropriately to represent the trace equality constraints (in the constraint sets) in \eqref{eq:EquivalentFormulation}. The problem \eqref{eq:QpFormulation} is a QP in its standard form and can be solved optimally using any one of the off-the-shelf solvers such as CVX~\cite{cvx}.
To obtain the graph factors $\bbL_P$ and $\bbL_Q$ using the solver based on the existing methods as described in Section~\ref{ExistingWorks} requires solving a QP with $N(N+1)/2$ variables for $\bbL_N$~\cite{learnDong}, and subsequently, factorizing $\bbL_N$ as $\bbL_P$ and $\bbL_Q$ as in \eqref{eq:Cartesian_Factorization}, which requires solving one more QP with $K$ variables. In contrast, the proposed method requires solving only one QP with $K$ variables. Thus, the computation complexity of the proposed method is very less as compared to the solver based on the existing methods.
\begin{figure*}[!h]
\centering
\psfrag{celcius}{\footnotesize $^\circ$ C}
\psfrag{5}{\tiny 5} \psfrag{6}{\tiny 6} \psfrag{7}{\tiny 7} \psfrag{8}{\tiny 8} \psfrag{9}{\tiny 9} \psfrag{10}{\tiny 10} \psfrag{11}{\tiny 11}
\begin{subfigure}[]{0.3\textwidth}
\includegraphics[width=\columnwidth]{True_Lapalacian_P.eps}
\caption{Ground truth: $\bbL_P$}
\label{fig:Learned_Laplacian_P}
\end{subfigure
\begin{subfigure}[]{0.3\textwidth}
\includegraphics[width=\columnwidth]{Learned_Lapalacian_P_kala.eps}
\caption{Learned: $\bbL_P$ (Solver 2)}
\label{fig:Learned_Laplacian_Q}
\end{subfigure}
\begin{subfigure}[]{0.3\textwidth}
\includegraphics[width=\columnwidth]{Learned_Lapalacian_P.eps}
\caption{Learned: $\bbL_P$ (Solver 1)}
\label{fig:Learned_Laplacian_P}
\end{subfigure}%
\begin{subfigure}[]{0.3\textwidth}
\includegraphics[width=\columnwidth]{True_Lapalacian_Q.eps}
\caption{Ground truth: $\bbL_Q$}
\label{fig:Learned_Laplacian_P}
\end{subfigure
\begin{subfigure}[]{0.3\textwidth}
\includegraphics[width=\columnwidth]{Learned_Lapalacian_Q_kala.eps}
\caption{Learned: $\bbL_Q$ (Solver 2)}
\label{fig:Learned_Laplacian_Q}
\end{subfigure}
\begin{subfigure}[]{0.3\textwidth}
\includegraphics[width=\columnwidth]{Learned_Lapalacian_Q.eps}
\caption{Learned: $\bbL_Q$ (Solver 1)}
\label{fig:Learned_Laplacian_P}
\end{subfigure}%
\caption{\footnotesize{Product graph learning on synthetic data}}
\label{fig:Laplacians}
\vskip-5mm
\end{figure*}
\begin{figure*}[!h]
\centering
\psfrag{celcius}{\footnotesize $^\circ$ C}
\psfrag{5}{\tiny 5} \psfrag{6}{\tiny 6} \psfrag{7}{\tiny 7} \psfrag{8}{\tiny 8} \psfrag{9}{\tiny 9} \psfrag{10}{\tiny 10} \psfrag{11}{\tiny 11}
\begin{subfigure}[t]{0.4\textwidth}
\includegraphics[width=\columnwidth, height=2.5in]{Data_plot_space_graph_2.eps}
\caption{Graph factor $\ccalG_P$ }
\label{fig:Graph_Laplacian_1}
\end{subfigure}%
~
\begin{subfigure}[t]{0.4\textwidth}
\centering
\includegraphics[width=\columnwidth]{Data_plot_time_graph_2.eps}
\caption{Graph factor $\ccalG_Q$}
\label{fig:Graph_Laplacian_2}
\end{subfigure}
\caption{\footnotesize{\emph{Factor graph learning on air pollution data}: The colored dots indicates the ${\rm PM}_{2.5}$ values. (a) Graph factor $\ccalG_P$ represents the spatially distributed sensor graph. (b) Graph factor $\ccalG_Q$ is a graph capturing the temporal/seasonal variation of the $PM_{2.5}$ data across months.}}
\label{fig:factor_graphs}
\vskip-6mm
\end{figure*}
\section{Water-filling solution} \label{sec:waterfil}
In this section, we derive an explicit solution for the optimization problem \eqref{eq:QpFormulation} based on the well-known {\it water-filling} approach. By writing the KKT conditions and solving for $z_i$ leads to an explicit solution.
The Lagrangian function for \eqref{eq:QpFormulation} is given by
\begin{equation*}\label{eq:Lagrangian}
\ccalL(\bbz,\,\bblam,\,\bbmu) = \frac{1}{2} \bbz^T \bbP \bbz + \bbq^T \bbz + \bbmu^T(\bbd -\bbC\bbz) -\bblam^T \bbz,
\end{equation*}
where $\bblam$ and $\bbmu$ are the Lagrange multipliers corresponding to the inequality and equality constraints. Then the KKT conditions are given by
\begin{equation*}
\begin{aligned}
&P_{i,i}z_i^\star + q_i - \bbc_i^T \bbmu^\star - \lam_i^\star = 0, \\
\sum_{i=1}^{N} \bbc_i z_i^\star &= \bbd, \quad z_i^\star \geq 0, \quad \lam_i^\star \geq 0, \quad \lam_i^\star z_i^\star = 0,
\end{aligned}
\end{equation*}
where $\bbc_i$ is the $i$th column of $\bbC$.
We can now solve the KKT conditions to find $z_i^\star$, $\bblam^\star$ and $\bbmu^\star$. To do so, we eliminate the slack variable $\lambda_i^\star$ and solve for $z_i^\star$. This results in
\[
z_i^\star = {\rm max}\left\{0, P_{i,i}^{-1}(\bbc_i^T\bbmu^\star - q_i^\star)\right\}.
\]
To find $\bbmu^\star$, we may use $z_i^\star$ in the second KKT condition to obtain
$
\sum_{i=1}^{N} \bbc_i {\rm max}\{0, P_{i,i}^{-1}(\bbc_i^T\bbmu^\star - q_i^*)\}= \bbd.
$
Since this is a linear function in $\mu_i^\star$, we can compute $\mu_i^\star$ using a simple iterative method. This approach is similar to the water-filling algorithm with multiple water levels~\cite{SimpleWaterfill}. This method is computationally very cheap compared to solving a standard QP.
\section{Numerical results}
In this section, we will evaluate the performance of the proposed water-filling based solver (henceforth, referred to as Solver 1) and compare it with the solver based on the existing methods described in Section~\ref{ExistingWorks} (henceforth, referred to as Solver 2) on synthetic and real datasets.
\vspace*{-3mm}
\subsection{Results on synthetic data}
To evaluate the quantitative performance of the proposed method, we generate synthetic data on a known graph, which can be factorized. We will use those graph factors as a reference (i.e., ground truth) and compare the estimated graph factors with the ground truth in terms of F-measure, which is a measure of the percentage of correctly recovered edges~\cite{learnDong}.
Specifically, we generate a graph with $N = 150$ nodes, which is formed by the Cartesian product of two community graphs with $P= 10$ and $Q = 15$ nodes, respectively.
We generate $T = 50$ signals $\{\bbx_i\}^{50}_{i=1}$ that are smooth w.r.t. $\ccalG_{N}$ by using the factor analysis model described in \cite{learnDong}. Given $\bbX$, we learn the graph factors using Solver 1 (using the water-filling method) and Solver 2 (wherein we learn the complete graph Laplacian matrix of $N=150$ nodes as in \cite{learnDong} and then factorize it by solving \eqref{eq:Cartesian_Factorization}).
In Table~\ref{tab:my-table}, we show the best (in terms of the tuning parameters) F-measure scores obtained by Solver 1 and Solver 2. The F-measure of Solver 1 is close to $0.98$, which means that the proposed method learns the graph factors close to the ground truth.
In Fig. \ref{fig:Laplacians}, we plot the Laplacian matrices of the ground truth, learned Laplacian matrices from Solver 1 and Solver 2. We see that the graph factors estimated from Solver 1 are more consistent with the ground truth than the factors estimated using Solver 2.
Moreover, we stress the fact that the computational complexity of the proposed water-filling method is very less as compared to Solver 2.
\begin{table}[!t]
\centering
\begin{tabular}{|l|l|l|l|}
\hline
\textbf{Method} & $\bbL_{P}$ & $\bbL_{Q}$ & $\bbL_{N}$ \\ \hline
\textbf{Solver 1}& 0.9615 & 0.9841 & 0.9755 \\ \hline
\textbf{Solver 2}& 0.7556 & 0.7842 & 0.7612 \\ \hline
\end{tabular}
\caption{F-measure of the proposed solver (Solver 1) and the solver based on the existing methods (Solver 2).}
\label{tab:my-table}
\vspace*{-5mm}
\end{table}
\subsection{Joint matrix completion and learning graph factors}
We now test the performance on real data. For this purpose, we use the ${\rm PM}_{2.5}$ data collected over $40$ air quality monitoring stations in different locations in India for each day of the year $2018$~\cite{aqi}. However, there are many missing entries in this dataset. Given this multidomain data that has spatial and temporal dimensions, the aim is to learn the graph factors that best explain the data.
More precisely, we aim to learn the graph factors that capture the relationships between spatially distributed sensors and a graph that captures the seasonal variations of the ${\rm PM}_{2.5}$ data. Since the dataset has missing entries, we impute the missing entries using a graph Laplacian regularized nuclear norm minimization~\cite{kalofolias2014matrix}.
That is, we use $f(\bbX,\bbY) := \sum_{i=1}^{T} \|\ccalA (\bbX_i -\bbY_i)\|^2_F + \|\bbX_i\|_*$, where $\ccalA$ denotes the known observation mask that selects the available entries and $\|\cdot\|_*$ denotes the nuclear norm. As the problem \eqref{eq:Problem_modeling} is not convex in $\{\bbX, \bbL_P, \bbL_Q\}$ due to the coupling between the variables in the smoothness promoting terms, we use alternating minimization method. Specifically, in an alternating manner, we solve for $\{\bbL_P, \bbL_Q\}$ by fixing $\bbX$ using the solver described in Section~\ref{sec:waterfil}, and then solve for $\bbX$ by fixing $\{\bbL_P, \bbL_Q\}$ as
\begin{equation*}\label{eq:MatrixCompletion}
\underset{\{\bbX_i\}_{i=1}^T}{\text{minimize}} \,\,
f(\bbX,\bbY) \,\,
+ \,\, \alpha \sum_{i=1}^{T} [{\rm \tr}(\bbX_i^T \bbL_P \bbX_i) + {\rm \tr}(\bbX_i\bbL_Q\bbX_i^T)],
\end{equation*}
where recall that ${\rm vec}(\bbX_i)$ forms the $i$th column of $\bbX$.
The learned graph factors that represent the multidomain graph signals are shown in Fig.~\ref{fig:factor_graphs}. Fig.~\ref{fig:Graph_Laplacian_1} shows the graph factor that encodes the relationships between the spatially distributed air quality monitoring stations. Each colored dot indicates the concentration of ${\rm PM}_{2.5}$ on a random day across different locations. We can see from Fig.~\ref{fig:Graph_Laplacian_1} that the graph connections are not necessarily related to the geographical proximity of the cities, but based on the similarity of the ${\rm PM}_{2.5}$ values.
The concentration of ${\rm PM}_{2.5}$ is relatively lesser during the summer/monsoon months (i.e., June, July, and August) as compared to the other months. The graph capturing these temporal variations in the ${\rm PM}_{2.5}$ data is shown in Fig.~\ref{fig:Graph_Laplacian_2}.
Each color dot indicates the concentration of ${\rm PM}_{2.5}$ averaged over each month at a random location. We can see in Fig.~\ref{fig:Graph_Laplacian_2} that months with similar average concentration of ${\rm PM}_{2.5}$ are connected. Moreover, the months having low ${\rm PM}_{2.5}$ values (during the monsoon months) are clustered together and the remaining months are clustered into two groups based on their occurrence before and after the monsoon. This shows that the obtained time graph has clusters that capture the seasonal variations hence confirms the quality of the learned graph.
\section{Conclusions}
\vspace*{-5mm}
We developed a framework for learning graphs that can be factorized as the Cartesian product of two smaller graphs. The estimation of the Laplacian matrices of the factor graphs is posed as a convex optimization problem. The proposed solver is computationally efficient and has an explicit solution based on the water-filling method. The performance of the proposed method is significantly better than other intuitive ways to solve the problem. We present numerical experiments based on air pollution data collected across different locations in India. Since this dataset has missing entries, we developed a task-cognizant learning method to solve the joint matrix completion and product graph learning problem.
\pagebreak
\bibliographystyle{IEEEtran}
| {
"timestamp": "2019-11-20T02:09:12",
"yymm": "1911",
"arxiv_id": "1911.07411",
"language": "en",
"url": "https://arxiv.org/abs/1911.07411",
"abstract": "In this paper, we focus on learning the underlying product graph structure from multidomain training data. We assume that the product graph is formed from a Cartesian graph product of two smaller factor graphs. We then pose the product graph learning problem as the factor graph Laplacian matrix estimation problem. To estimate the factor graph Laplacian matrices, we assume that the data is smooth with respect to the underlying product graph. When the training data is noise free or complete, learning factor graphs can be formulated as a convex optimization problem, which has an explicit solution based on the water-filling algorithm. The developed framework is illustrated using numerical experiments on synthetic data as well as real data related to air quality monitoring in India.",
"subjects": "Signal Processing (eess.SP)",
"title": "Learning product graphs from multidomain signals",
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https://arxiv.org/abs/1612.03059 | A note on the Chermak-Delgado lattice of a finite group | In this note we describe the structure of finite groups G whose Chermak-Delgado lattice is the interval [G/Z(G)] = {H \in L(G) \mid Z(G)\leq H\leq G}. | \section{Introduction}
Let $G$ be a finite group and $L(G)$ be the subgroup lattice of $G$. The \textit{Chermak-Delgado measure} of a subgroup $H$ of $G$ is defined by
\begin{equation}
m_G(H)=|H||C_G(H)|.\nonumber
\end{equation}Let
\begin{equation}
m(G)={\rm max}\{m_G(H)\mid H\leq G\} \mbox{ and } {\cal CD}(G)=\{H\leq G\mid m_G(H)=m(G)\}.\nonumber
\end{equation}Then the set ${\cal CD}(G)$ forms a modular self-dual sublattice of $L(G)$,
which is called the \textit{Chermak-Delgado lattice} of $G$. It was first introduced by Chermak and Delgado \cite{5}, and revisited by Isaacs \cite{7}. In the last years
there has been a growing interest in understanding this lattice, especially for $p$-groups (see e.g. \cite{1,2,3,10}). The study can be naturally extended to nilpotent groups,
since by \cite{2} the Chermak-Delgado lattice of a direct product of finite groups decomposes as the direct product of the Chermak-Delgado lattices of the factors. Recall also that
if $H\in {\cal CD}(G)$, then $C_G(H)\in {\cal CD}(G)$ and $C_G(C_G(H))=H$. This implies that ${\cal CD}(G)$ is contained in the centralizer lattice $\mathfrak{C}(G)$ of $G$.
We remark that ${\cal CD}(G)=[G/Z(G)]$ for many finite groups $G$, such us $D_8$, $Q_8$, any abelian group, ... and so on. Thus, the study of finite groups satisfying this property is very natural. It is the goal of the current note. Our main result is stated as follows.
\begin{theorem}\label{th:C1}
Let $G$ be a finite group. Then ${\cal CD}(G)=[G/Z(G)]$ if and only if $G=G_1\times\cdots\times G_r\times A$, where
\begin{equation}
\gcd(|G_i|,|G_j|)=1=\gcd(|G_i|,|A|) \mbox{ for all } i\neq j,\nonumber
\end{equation}$A$ is an abelian group, and every $G_i$ is a $p$-group satisfying
\begin{equation}
[G_i/Z(G_i)] \mbox{ is modular and } G'_i \mbox{ is cyclic}.
\end{equation}
\end{theorem}
Note that the conditions (1) are equivalent with the conditions
\begin{equation}
G'_i=\langle a\rangle \mbox{ is cyclic and } [\langle a\rangle,G_i]\leq\langle a^4\rangle
\end{equation}by Theorem 9.3.19 of \cite{9} (see also \cite{4,8}).
\bigskip
The following corollary is an immediate consequence of Theorem 1.
\begin{corollary}
Every finite group $G$ satisfying ${\cal CD}(G)=[G/Z(G)]$ is nilpotent.
\end{corollary}
By Corollary 9.3.18 of \cite{9} (see also \cite{6}), we know that there are finite groups in which every subgroup is a centralizer. Theorem 1 shows that a similar result
does not hold for the Chermak-Delgado lattice.
\begin{corollary}
There is no finite non-trivial group $G$ such that ${\cal CD}(G)=L(G)$.
\end{corollary}
Also, Theorem 1 shows that the property ${\cal CD}(G)=[G/Z(G)]$ is inherited by subgroups.
\begin{corollary}
If\, $G$ is a finite group satisfying ${\cal CD}(G)=[G/Z(G)]$ and $H$ is a subgroup of $G$, then\, ${\cal CD}(H)=[H/Z(H)]$.
\end{corollary}
Observe that if for a finite group $G$ we have ${\cal CD}(G)=[G/Z(G)]$, then $\mathfrak{C}(G)=[G/Z(G)]$ and so $\mathfrak{C}(G)={\cal CD}(G)$ is a modular lattice.
Moreover, its length $l$ must be even by Lemma 9.3.10 of \cite{9}. Elementary examples of such groups are all abelian groups for $l=0$, and $D_8$, $Q_8$ for $l=2$.
A more general example is the following.
\bigskip\noindent{\bf Example.} Let $G$ be an extra-special group $G$ of order $p^{2n+1}$. Then
\begin{equation}
G/Z(G)\cong\mathbb{Z}_p^{2n} \mbox{ is modular and } G'\cong\mathbb{Z}_p \mbox{ is cyclic},\nonumber
\end{equation}and therefore ${\cal CD}(G)=[G/Z(G)]$ is a modular lattice of length $2n$.
\bigskip
Finally, we indicate a natural open problem concerning the above study.
\bigskip\noindent{\bf Open problem.} Describe the structure of finite groups $G$ such that ${\cal CD}(G)$ is an interval (not necessarily $[G/Z(G)]$) of $L(G)$.
\section{Proof of the main result}
We start by proving an auxiliary result.
\begin{lemma}
Let $G$ be a finite $p$-group. Then ${\cal CD}(G)=[G/Z(G)]$ if and only if $[G/Z(G)]$ is modular and $G'$ is cyclic.
\end{lemma}
\begin{proof}
If ${\cal CD}(G)=[G/Z(G)]$, then $\mathfrak{C}(G)=[G/Z(G)]$ and so $[G/Z(G)]$ is modular and $G'$ is cyclic by Theorem 9.3.19 of \cite{9}.
Conversely, if $[G/Z(G)]$ is modular and $G'$ is cyclic, then $\mathfrak{C}(G)=[G/Z(G)]$. By Lemma 4 of \cite{4} we infer that
\begin{equation}
m_G(H)=|H||C_G(H)|=|G||Z(G)|, \forall\, H\in [G/Z(G)],\nonumber
\end{equation}that is all subgroups in $[G/Z(G)]$ have the same Chermak-Delgado measure. This shows that ${\cal CD}(G)=[G/Z(G)]$, as desired.
\end{proof}
\noindent{\bf Remark.} Let $G$ be a non-abelian $p$-group of order $p^n$ satisfying ${\cal CD}(G)=[G/Z(G)]$. If $G$ contains an abelian subgroup
$M$ of order $p^{n-1}$ (as it happens for $D_8$ and $Q_8$), then $(G:Z(G))=p^2$.
Indeed, we have $M\subseteq C_G(M)$ because $M$ is abelian and thus
\begin{equation}
|G||Z(G)|=m(G)=|M||C_G(M)|\geq |M|^2.\nonumber
\end{equation}One obtains
\begin{equation}
p^2\leq(G:Z(G))\leq(G:M)^2=p^2,\nonumber
\end{equation}that is $(G:Z(G))=p^2$.
\bigskip
We are now able to prove our main theorem.
\begin{proof}[Proof of Theorem \ref{th:C1}]
Assume first that $G=G_1\times\cdots\times G_r\times A$, where $G_i$, $i=1,...,r$, and $A$ satisfy the conditions in Theorem 1. Then
\begin{equation}
{\cal CD}(G_i)=[G_i/Z(G_i)], \forall\, i=1,...,r,\nonumber
\end{equation}by Lemma 5. It follows that
\begin{align*}
{\cal CD}(G)
&={\cal CD}(G_1)\times\cdots\times{\cal CD}(G_r)\times\{A\}\\
&=[G_1/Z(G_1)]\times\cdots\times[G_r/Z(G_r)]\times\{A\}\\
&=[G/Z(G)].\nonumber
\end{align*}
Conversely, assume that ${\cal CD}(G)=[G/Z(G)]$. Since ${\cal CD}(G)\subseteq \mathfrak{C}(G)$, we infer that $\mathfrak{C}(G)=[G/Z(G)]$. Then Theorem 9.3.17 of \cite{9} implies that
$G=G_1\times\cdots\times G_r\times A$, where
\begin{equation}
\gcd(|G_i|,|G_j|)=1=\gcd(|G_i|,|A|) \mbox{ for all } i\neq j,\nonumber
\end{equation}$A$ is an abelian group, and every $G_i$ is either a $\{p,q\}$-group with $|G_i/Z(G_i)|=pq$ or a $p$-group satisfying
$\mathfrak{C}(G_i)=[G_i/Z(G_i)]$, $p$ and $q$ primes. Clearly, this leads to
\begin{equation}
{\cal CD}(G)={\cal CD}(G_1)\times\cdots\times{\cal CD}(G_r)\times\{A\}\nonumber
\end{equation}and
\begin{equation}
[G/Z(G)]=[G_1/Z(G_1)]\times\cdots\times[G_r/Z(G_r)],\nonumber
\end{equation}implying that
\begin{equation}
{\cal CD}(G_i)=[G_i/Z(G_i)], \forall\, i=1,...,r.\nonumber
\end{equation}If $G_i$ would be a $\{p,q\}$-group with $|G_i/Z(G_i)|=pq$ and $p<q$, then ${\cal CD}(G_i)$ would consists only of the unique subgroup of index $p$ contained in $[G_i/Z(G_i)]$, a contradiction. Consequently, $G_i$ is a $p$-group. On the other hand, it satisfies the conditions (1) by Theorem 9.3.19 of \cite{9}. This completes the proof.
\end{proof}
| {
"timestamp": "2016-12-12T02:06:42",
"yymm": "1612",
"arxiv_id": "1612.03059",
"language": "en",
"url": "https://arxiv.org/abs/1612.03059",
"abstract": "In this note we describe the structure of finite groups G whose Chermak-Delgado lattice is the interval [G/Z(G)] = {H \\in L(G) \\mid Z(G)\\leq H\\leq G}.",
"subjects": "Group Theory (math.GR)",
"title": "A note on the Chermak-Delgado lattice of a finite group",
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https://arxiv.org/abs/1311.0226 | Classifying matchbox manifolds | Matchbox manifolds are foliated spaces with totally disconnected transversals. Two matchbox manifolds which are homeomorphic have return equivalent dynamics, so that invariants of return equivalence can be applied to distinguish non-homeomorphic matchbox manifolds. In this work we study the problem of showing the converse implication: when does return equivalence imply homeomorphism? For the class of weak solenoidal matchbox manifolds, we show that if the base manifolds satisfy a strong form of the Borel Conjecture, then return equivalence for the dynamics of their foliations implies the total spaces are homeomorphic. In particular, we show that two equicontinuous $\mT^n$--like matchbox manifolds of the same dimension are homeomorphic if and only if their corresponding restricted pseudogroups are return equivalent. At the same time, we show that these results cannot be extended to include the "\emph{adic}-surfaces", which are a class of weak solenoids fibering over a closed surface of genus 2. | \section{Introduction} \label{sec-intro}
In this paper, we study the problem of when do the local dynamics and shape type of a matchbox manifold ${\mathfrak{M}}$ determine the homeomorphism type of ${\mathfrak{M}}$.
For example, it is folklore \cite{CHT2001,Keesling1973} that two connected compact abelian groups with the same shape (or even just isomorphic first \v{C}ech cohomology groups) are homeomorphic.
In another direction, for minimal, $1$--dimensional matchbox manifolds, Fokkink \cite[Theorems 3.7,4.1]{Fokkink1991}, and Aarts and Oversteegen \cite[Theorem 17]{AO1995} show that:
\begin{thm}\label{thm-one-dim}
Two orientable, minimal, $1$--dimensional matchbox manifolds are homeomorphic if and only if they are return equivalent.
\end{thm}
Since any non--orientable minimal, matchbox manifold admits an orientable double cover, this demonstrates that the local dynamics effectively determines the global topology in dimension one.
The local dynamics of a minimal matchbox manifold ${\mathfrak{M}}$ is defined using the pseudogroup ${\mathcal G}_W$ of local holonomy maps for a local transversal $W$ of ${\mathfrak{M}}$. In Section~\ref{sec-return} we show that this notion of return equivalence is well-defined for minimal matchbox manifolds, and show that if ${\mathfrak{M}}_1,{\mathfrak{M}}_2$ are any two homeomorphic minimal matchbox manifolds, then for any local transversals $W_i \subset {\mathfrak{M}}_i$ we have that ${\mathcal G}_{W_1}$ is return equivalent to ${\mathcal G}_{W_2}$.
It thus makes sense to ask to what extent two return equivalent minimal matchbox manifolds have the same topology for matchbox manifolds with leaves of dimension greater than one.
There are many difficulties in extending the topological classification for $1$-dimensional matchbox manifolds to the cases with higher dimensional leaves. In Section~\ref{sec-examples}, we show by way of examples, that such extensions are not always possible. Thus, one seeks sufficient conditions for which return equivalence implies topological conjugacy.
For example, the one-dimensional case uses implicitly the basic property of $1$-dimensional flows, that every cover of a circle is again a circle. This leads to the introduction of shape theoretic properties of matchbox manifolds, which imposes some broad constraints on the topology of the leaves that appear necessary. In this work, we impose the following notion, introduced by Alexandroff in \cite{Alexandroff1928}:
\begin{defn} \label{def-Ylike}
Let $Y$ be a compact metric space.
A metric space $X$ is said to be \emph{$Y$--like} if for every $\e>0$, there is a continuous surjection $f_\e \colon X \to Y$ such that the fiber $f_\e^{-1}(y)$ of each point $y\in Y$ has diameter less than $\e$.
\end{defn}
Recall that a $CW$-complex $Y$ is \emph{aspherical} if it is connected, and $\pi_n(Y)$ is trivial for all $n\geq 2$. Equivalently, $Y$ is aspherical if it is connected and its universal covering space is contractible. Let ${\mathcal A}$ denote the collection of $CW$-complexes which are aspherical.
Our first main result is an extension of a main result in \cite{ClarkHurder2013}.
\begin{thm}\label{thm-main1}
Suppose that ${\mathfrak{M}}$ is an equicontinuous $Y$-like matchbox manifold, where $Y \in {\mathcal A}$. Then ${\mathfrak{M}}$ admits a presentation as an inverse limit
\begin{equation}\label{eq-nilpresentation}
{\mathfrak{M}}\; \homeo \;\underleftarrow{\lim} \, \{\, q_{\ell+1} \colon B_{\ell +1} \to B_{\ell} \mid \ell \geq 0\}
\end{equation}
where each $B_{\ell +1}$ is a closed manifold with $B_{\ell +1} \in {\mathcal A}$, and each bonding map $q_{\ell}$ is a finite covering.
\end{thm}
We formulate our main results regarding the topological conjugacy of matchbox manifolds with leaves of arbitrary dimension $n \geq 1$. The first result is for the special case where $Y = {\mathbb T}^n$, which gives a direct generalization of the classification of $1$-dimensional matchbox manifolds.
\begin{thm}\label{thm-main2}
Suppose that ${\mathfrak{M}}_1$ and ${\mathfrak{M}}_2$ are equicontinuous ${\mathbb T}^n$-like matchbox manifolds.
Then ${\mathfrak{M}}_1$ and ${\mathfrak{M}}_2$ are return equivalent if and only if ${\mathfrak{M}}_1$ and ${\mathfrak{M}}_2$ are homeomorphic.
\end{thm}
As shown in Sections~\ref{sec-equicontinuous} and \ref{sec-prohomotopy}, an equicontinuous ${\mathbb T}^n$-like matchbox manifold ${\mathfrak{M}}$ is homeomorphic to an inverse limit of finite covering maps of the base ${\mathbb T}^n$. The possible homeomorphism types for such the inverse limit spaces known to be ``unclassifiable'', in the sense of descriptive set theory, as discussed in \cite{Kechris2000, Thomas2001, Thomas2003, HT2006}. Thus, it is not possible to give a classification for the family of matchbox manifolds obtained using the covering data in a presentation as the invariant. The notion of return equivalence provides an alternate approach to classification of these spaces.
In order to formulate a version of Theorem~\ref{thm-main2} for manifolds more general than ${\mathbb T}^n$, we use the \emph{Borel Conjecture} for higher dimensional aspherical closed manifolds, which characterizes their homeomorphism types in terms of their fundamental groups. As discussed in Section~\ref{sec-examples}, when combined with Definition~\ref{def-Ylike}, this yields a weak form of the self-covering property of the circle, for leaves of general matchbox manifolds.
Recall that the \emph{Borel Conjecture} is that if $Y_1$ and $Y_2$ are homotopy equivalent, aspherical closed manifolds, then a homotopy equivalence between $Y_1$ and $Y_2$ is homotopic to a homeomorphism between $Y_1$ and $Y_2$. The Borel Conjecture has been proven for many classes of aspherical manifolds, including:
\begin{itemize}
\item the torus ${\mathbb T}^n$ for all $n \geq 1$,
\item all \emph{infra-nilmanifolds} of dimension $n \geq 3$,
\item all closed Riemannian manifolds $Y$ with negative sectional curvatures,
\end{itemize}
where a compact connected manifold $Y$ is an \emph{infra-nilmanifold} if its universal cover $\widetilde{Y}$ is contractible, and the fundamental group of $M$ has a nilpotent subgroup with finite index.
The above list is not exhaustive. The history and current status of the Borel Conjecture is discussed in the surveys of Davis \cite{Davis2012} and L\"{u}ck \cite{Lueck2012}.
We introduce the notion of a \emph{strongly Borel} manifold.
\begin{defn} \label{def-borel}
A collection ${\mathcal A}_B$ of closed manifolds is called \emph{Borel} if it satisfies the conditions
\begin{itemize}
\item[1)] Each $Y \in {\mathcal A}_B$ is aspherical,
\item[2)] Any closed manifold $X$ homotopy equivalent to some $Y \in{\mathcal A}_B$ is homeomorphic to $Y$, and
\item[3)] If $Y \in {\mathcal A}_B,$ then any finite covering space of $Y $ is also in ${\mathcal A}_B.$
\end{itemize}
A closed manifold $Y$ is \emph{strongly Borel} if the collection ${\mathcal A}_Y \equiv \langle Y \rangle$ of all finite covers of $Y$ forms a Borel collection.
\end{defn}
Each class of manifolds in the above list is strongly Borel. Here is our second main result:
\begin{thm}\label{thm-main3}
Suppose that ${\mathfrak{M}}_1$ and ${\mathfrak{M}}_2$ are equicontinuous, $Y$--like matchbox manifolds, where $Y$ is a strongly Borel closed manifold.
Assume that each of ${\mathfrak{M}}_1$ and ${\mathfrak{M}}_2$ have a leaf which is simply connected.
Then ${\mathfrak{M}}_1$ and ${\mathfrak{M}}_2$ are return equivalent if and only if ${\mathfrak{M}}_1$ and ${\mathfrak{M}}_2$ are homeomorphic.
\end{thm}
The requirement that there exists a simply connected leaf implies that the global holonomy maps associated to each of these foliations are injective maps, as shown in Proposition~\ref{prop-conjugateactions}. This conclusion yields a connection between return equivalence for the foliations of ${\mathfrak{M}}_1$ and ${\mathfrak{M}}_2$ and the homotopy types of the approximating manifolds in a shape presentation. This requirement is not imposed for the case of $Y = {\mathbb T}^n$ in Theorem~\ref{thm-main2}, due to the algebraic properties of its fundamental group.
The equicontinuous hypotheses is defined in Section~\ref{sec-holonomy}, and is used to obtain towers of approximations in \eqref{eq-nilpresentation}.
Theorem~\ref{thm-one-dim} holds for more general matchbox manifolds. It remains an open question whether a more general form of Theorems~\ref{thm-main2} and \ref{thm-main3} can be shown for classes of matchbox manifolds which are not equicontinuous. The last Section~\ref{sec-conjectures} of this paper formulates other generalizations of these results which we conjecture may be true.
In Section~\ref{sec-examples} we give some basic examples of equicontinuous matchbox manifolds which are not $Y$-like, for any $CW$-complex $Y$, and which are return equivalent but not homeomorphic. These examples show the strong relation between the $Y$-like hypothesis,
and the property of a closed manifold $Y$ that it has the \emph{non-co-Hopfian Property}. This section also defines a class of examples, the \emph{adic}-surfaces, which are not $Y$-like yet it is possible to give a form of classification result as an application of the ideas of this paper. In general, the examples of this section show
that we cannot hope to generalize Theorem~\ref{thm-main2} to matchbox manifolds approximated by a sequence of arbitrary manifolds.
The rest of this paper is organized as follows.
Sections~\ref{sec-concepts} and \ref{sec-holonomy} below collect together some definitions and results concerning matchbox manifolds and their dynamical properties that we use in the paper. Then in Section~\ref{sec-return}, we introduce the basic notion of return equivalence of matchbox manifolds.
Section~\ref{sec-bundles} introduces the notion of \emph{foliated Cantor bundles}, which play a fundamental role in the study of equicontinuous matchbox manifolds. Various results related to showing that these spaces are homeomorphism are developed, and Proposition~\ref{prop-conjugateactions} gives the main technical result required.
Section~\ref{sec-equicontinuous} recalls the properties of equicontinuous matchbox manifolds, and especially the notion of a \emph{presentation} for such a space.
Section~\ref{sec-prohomotopy} contains technical results concerning the \emph{pro-homotopy groups} of equicontinuous matchbox manifolds.
The proofs of Theorems~\ref{thm-main2} and \ref{thm-main3} are given at the end of Section~\ref{sec-prohomotopy}.
Finally, in Section~\ref{sec-conjectures} we offer several conjectures based on the results of this paper. In particular, we formulate an analogue of the Borel Conjecture for weak solenoids.
We thank Jim Davis for helpful remarks on the Borel Conjecture and related topics, and Brayton Gray and Pete Bousfield for helpful discussions concerning pro-homotopy groups of spaces.
This work is part of a program to generalize the results of the thesis of Fokkink \cite{Fokkink1991}, started during a visit by the authors to the University of Delft in August 2009. The papers \cite{ClarkHurder2013,CHL2013a,CHL2013b} are the initial results of this study. The authors also thank Robbert Fokkink for the invitation to meet in Delft, and the University of Delft for its generous support for the visit. The authors' stay in Delft was also supported by a travel grant No.~040.11.132 from the \emph{Nederlandse Wetenschappelijke Organisatie}.
\section{Foliated spaces and matchbox manifolds} \label{sec-concepts}
In this section we present the necessary background needed for our analysis of matchbox manifolds. More details can be found in the works \cite{CandelConlon2000,ClarkHurder2013,CHL2013a,CHL2013b,MS2006}. Recall that a \emph{continuum} is a compact connected metrizable space.
\begin{defn} \label{def-fs}
A \emph{foliated space of dimension $n$} is a continuum ${\mathfrak{M}}$, such that there exists a compact separable metric space ${\mathfrak{X}}$, and
for each $x \in {\mathfrak{M}}$ there is a compact subset ${\mathfrak{T}}_x \subset {\mathfrak{X}}$, an open subset $U_x \subset {\mathfrak{M}}$, and a homeomorphism defined on the closure
${\varphi}_x \colon {\overline{U}}_x \to [-1,1]^n \times {\mathfrak{T}}_x$ such that ${\varphi}_x(x) = (0, w_x)$ where $w_x \in int({\mathfrak{T}}_x)$. Moreover, it is assumed that each ${\varphi}_x$ admits an extension to a foliated homeomorphism
${\widehat \varphi}_x \colon {\widehat U}_x \to (-2,2)^n \times {\mathfrak{T}}_x$ where ${\overline{U}}_x \subset {\widehat U}_x$.
\end{defn}
The subspace ${\mathfrak{T}}_x$ of ${\mathfrak{X}}$ is the \emph{local transverse model} at $x$.
Let $\pi_x \colon {\overline{U}}_x \to {\mathfrak{T}}_x$ denote the composition of ${\varphi}_x$ with projection onto the second factor.
For $w \in {\mathfrak{T}}_x$ the set ${\mathcal P}_x(w) = \pi_x^{-1}(w) \subset {\overline{U}}_x$ is called a \emph{plaque} for the coordinate chart ${\varphi}_x$. We adopt the notation, for $z \in {\overline{U}}_x$, that ${\mathcal P}_x(z) = {\mathcal P}_x(\pi_x(z))$, so that $z \in {\mathcal P}_x(z)$. Note that each plaque ${\mathcal P}_x(w)$ is given the topology so that the restriction ${\varphi}_x \colon {\mathcal P}_x(w) \to [-1,1]^n \times \{w\}$ is a homeomorphism. Then $int ({\mathcal P}_x(w)) = {\varphi}_x^{-1}((-1,1)^n \times \{w\})$.
Let $U_x = int ({\overline{U}}_x) = {\varphi}_x^{-1}((-1,1)^n \times int({\mathfrak{T}}_x))$.
Note that if $z \in U_x \cap U_y$, then $int({\mathcal P}_x(z)) \cap int( {\mathcal P}_y(z))$ is an open subset of both
${\mathcal P}_x(z) $ and ${\mathcal P}_y(z)$.
The collection of sets
$${\mathcal V} = \{ {\varphi}_x^{-1}(V \times \{w\}) \mid x \in {\mathfrak{M}} ~, ~ w \in {\mathfrak{T}}_x ~, ~ V \subset (-1,1)^n ~ {\rm open}\}$$
forms the basis for the \emph{fine topology} of ${\mathfrak{M}}$. The connected components of the fine topology are called leaves, and define the foliation $\F$ of ${\mathfrak{M}}$.
For $x \in {\mathfrak{M}}$, let $L_x \subset {\mathfrak{M}}$ denote the leaf of $\F$ containing $x$.
Note that in Definition~\ref{def-fs}, the collection of transverse models
$\{{\mathfrak{T}}_x \mid x \in {\mathfrak{M}}\}$ need not have union equal to ${\mathfrak{X}}$. This is similar to the situation for a smooth foliation of codimension $q$, where each foliation chart projects to an open subset of ${\mathbb R}^q$, but the collection of images need not cover ${\mathbb R}^q$.
\begin{defn} \label{def-sfs}
A \emph{smooth foliated space} is a foliated space ${\mathfrak{M}}$ as above, such that there exists a choice of local charts ${\varphi}_x \colon {\overline{U}}_x \to [-1,1]^n \times {\mathfrak{T}}_x$ such that for all $x,y \in {\mathfrak{M}}$ with $z \in U_x \cap U_y$, there exists an open set $z \in V_z \subset U_x \cap U_y$ such that ${\mathcal P}_x(z) \cap V_z$ and ${\mathcal P}_y(z) \cap V_z$ are connected open sets, and the composition
$$\psi_{x,y;z} \equiv {\varphi}_y \circ {\varphi}_x ^{-1}\colon {\varphi}_x({\mathcal P}_x (z) \cap V_z) \to {\varphi}_y({\mathcal P}_y (z) \cap V_z)$$
is a smooth map, where ${\varphi}_x({\mathcal P}_x (z) \cap V_z) \subset {\mathbb R}^n \times \{w\} \cong {\mathbb R}^n$ and ${\varphi}_y({\mathcal P}_y (z) \cap V_z) \subset {\mathbb R}^n \times \{w'\} \cong {\mathbb R}^n$. The leafwise transition maps $\psi_{x,y;z}$ are assumed to depend continuously on $z$ in the $C^{\infty}$-topology.
\end{defn}
A map $f \colon {\mathfrak{M}} \to {\mathbb R}$ is said to be \emph{smooth} if for each flow box
${\varphi}_x \colon {\overline{U}}_x \to [-1,1]^n \times {\mathfrak{T}}_x$ and $w \in {\mathfrak{T}}_x$ the composition
$y \mapsto f \circ {\varphi}_x^{-1}(y, w)$ is a smooth function of $y \in (-1,1)^n$, and depends continuously on $w$ in the $C^{\infty}$-topology on maps of the plaque coordinates $y$. As noted in \cite{MS2006} and \cite[Chapter 11]{CandelConlon2000}, this allows one to define smooth partitions of unity, vector bundles, and tensors for smooth foliated spaces. In particular, one can define leafwise Riemannian metrics. We recall a standard result, whose proof for foliated spaces can be found in \cite[Theorem~11.4.3]{CandelConlon2000}.
\begin{thm}\label{thm-riemannian}
Let ${\mathfrak{M}}$ be a smooth foliated space. Then there exists a leafwise Riemannian metric for $\F$, such that for each $x \in {\mathfrak{M}}$, $L_x$ inherits the structure of a complete Riemannian manifold with bounded geometry, and the Riemannian geometry depends continuously on $x$ . \hfill $\Box$
\end{thm}
Bounded geometry implies, for example, that for each $x \in {\mathfrak{M}}$, there is a leafwise exponential map
$\exp^{\F}_x \colon T_x\F \to L_x$ which is a surjection, and the composition $\exp^{\F}_x \colon T_x\F \to L_x \subset {\mathfrak{M}}$ depends continuously on $x$ in the compact-open topology on maps.
\begin{defn} \label{def-mm}
A \emph{matchbox manifold} is a continuum with the structure of a
smooth foliated space ${\mathfrak{M}}$, such that for each $x \in {\mathfrak{M}}$, the
transverse model space ${\mathfrak{T}}_x \subset {\mathfrak{X}}$ is totally disconnected, and for each $x \in {\mathfrak{M}}$, ${\mathfrak{T}}_x \subset {\mathfrak{X}}$ is a clopen (closed and open) subset.
\end{defn}
The maximal path-connected components of ${\mathfrak{M}}$ define the leaves of a foliation $\F$ of ${\mathfrak{M}}$.
All matchbox manifolds are assumed to be smooth, with a given leafwise Riemannian metric, and with a fixed choice of metric $\dM$ on ${\mathfrak{M}}$.
A matchbox manifold ${\mathfrak{M}}$ is \emph{minimal} if every leaf of $\F$ is dense.
We next formulate the definition of a \emph{regular covering} of ${\mathfrak{M}}$.
For $x \in {\mathfrak{M}}$ and $\e > 0$, let $D_{{\mathfrak{M}}}(x, \e) = \{ y \in {\mathfrak{M}} \mid \dM(x, y) \leq \e\}$ be the closed $\e$-ball about $x$ in ${\mathfrak{M}}$, and $B_{{\mathfrak{M}}}(x, \e) = \{ y \in {\mathfrak{M}} \mid \dM(x, y) < \e\}$ the open $\e$-ball about $x$.
Similarly, for $w \in {\mathfrak{X}}$ and $\e > 0$, let $D_{{\mathfrak{X}}}(w, \e) = \{ w' \in {\mathfrak{X}} \mid d_{{\mathfrak{X}}}(w, w') \leq \e\}$ be the closed $\e$-ball about $w$ in ${\mathfrak{X}}$, and $B_{{\mathfrak{X}}}(w, \e) = \{ w' \in {\mathfrak{X}} \mid d_{{\mathfrak{X}}}(w, w') < \e\}$ the open $\e$-ball about $w$.
Each leaf $L \subset {\mathfrak{M}}$ has a complete path-length metric, induced from the leafwise Riemannian metric:
$$\dF(x,y) = \inf \left\{\| \gamma\| \mid \gamma \colon [0,1] \to L ~{\rm is ~ piecewise ~~ C^1}~, ~ \gamma(0) = x ~, ~ \gamma(1) = y ~, ~ \gamma(t) \in L \quad \forall ~ 0 \leq t \leq 1\right\}$$
where $\| \gamma \|$ denotes the path-length of the piecewise $C^1$-curve $\gamma(t)$. If $x,y \in {\mathfrak{M}}$ are not on the same leaf, then set $\dF(x,y) = \infty$.
For each $x \in {\mathfrak{M}}$ and $r > 0$, let $D_{\F}(x, r) = \{y \in L_x \mid \dF(x,y) \leq r\}$.
The leafwise Riemannian metric $\dF$ is continuous with respect to the metric $\dM$ on ${\mathfrak{M}}$, but otherwise the two metrics have no relation.
The metric $\dM$ is used to define the metric topology on ${\mathfrak{M}}$, while the metric $\dF$ depends on an independent choice of the Riemannian metric on leaves.
For each $x \in {\mathfrak{M}}$, the {Gauss Lemma} implies that there exists $\lambda_x > 0$ such that $D_{\F}(x, \lambda_x)$ is a \emph{strongly convex} subset for the metric $\dF$. That is, for any pair of points $y,y' \in D_{\F}(x, \lambda_x)$ there is a unique shortest geodesic segment in $L_x$ joining $y$ and $y'$ and contained in $D_{\F}(x, \lambda_x)$. Then for all $0 < \lambda < \lambda_x$ the disk $D_{\F}(x, \lambda)$ is also strongly convex. As ${\mathfrak{M}}$ is compact and the leafwise metrics have uniformly bounded geometry, we obtain:
\begin{lemma}\label{lem-stronglyconvex}
There exists ${\lambda_{\mathcal F}} > 0$ such that for all $x \in {\mathfrak{M}}$, $D_{\F}(x, {\lambda_{\mathcal F}})$ is strongly convex.
\end{lemma}
It follows from standard considerations (see \cite{ClarkHurder2013,CHL2013a}) that a matchbox manifold admits a covering by foliation charts which satisfies additional regularity conditions.
\begin{prop}\label{prop-regular}\cite{ClarkHurder2013}
For a smooth foliated space ${\mathfrak{M}}$, given ${\epsilon_{{\mathfrak{M}}}} > 0$, there exist ${\lambda_{\mathcal F}}>0$ and a choice of local charts ${\varphi}_x \colon {\overline{U}}_x \to [-1,1]^n \times {\mathfrak{T}}_x$ with the following properties:
\begin{enumerate}
\item For each $x \in {\mathfrak{M}}$, $U_x \equiv int({\overline{U}}_x) = {\varphi}_x^{-1}\left( (-1,1)^n \times B_{{\mathfrak{X}}}(w_x, \e_x)\right)$, where $\e_x>0$.
\item Locality: for all $x \in {\mathfrak{M}}$, each ${\overline{U}}_x \subset B_{{\mathfrak{M}}}(x, {\epsilon_{{\mathfrak{M}}}})$.
\item Local convexity: for all $x \in {\mathfrak{M}}$ the plaques of ${\varphi}_x$ are leafwise strongly convex subsets with diameter less than ${\lambda_{\mathcal F}}/2$. That is, there is a unique shortest geodesic segment joining any two points in a plaque, and the entire geodesic segment is contained in the plaque.
\end{enumerate}
\end{prop}
By a standard argument, there exists a finite collection $\{x_1, \ldots , x_{\nu}\} \subset {\mathfrak{M}}$ where ${\varphi}_{x_i}(x_i) = (0 , w_{x_i})$ for $w_{x_i} \in {\mathfrak{X}}$, and regular foliation charts ${\varphi}_{x_i} \colon {\overline{U}}_{x_i} \to [-1,1]^n \times {\mathfrak{T}}_{x_i}$
satisfying the conditions of Proposition~\ref{prop-regular}, which form an open covering of ${\mathfrak{M}}$. Relabel the various maps and spaces accordingly, so that
${\overline{U}}_{i} = {\overline{U}}_{x_i}$ and ${\varphi}_{i} = {\varphi}_{x_i}$ for example, with transverse spaces ${\mathfrak{T}}_i = {\mathfrak{T}}_{x_i}$ and projection maps $\pi_i = \pi_{x_i} \colon {\overline{U}}_i \to {\mathfrak{X}}_i$.
Then the projection $\pi_i(U_i \cap U_j) = {\mathfrak{T}}_{i,j} \subset {\mathfrak{T}}_i$ is a clopen subset for all $1 \leq i, j \leq \nu$.
Moreover, without loss of generality, we can impose a uniform size restriction on the plaques of each chart. Without loss of generality, we can assume there exists $0 < \dFU < {\lambda_{\mathcal F}}/4$ so that for all $1 \leq i \leq \nu$
and $\omega \in {\mathfrak{T}}_i$ with plaque ``center point'' $x_{\omega} = \tau_{i}(\omega)\myeq {\varphi}_{i}^{-1}(0 , \omega)$, then
the plaque ${\mathcal P}_i(\omega)$ for ${\varphi}_{i}$ through $x_{\omega}$ satisfies the uniform estimate of diameters:
\begin{equation}\label{eq-Fdelta}
D_{\F}(x_{\omega} , \dFU/2) ~ \subset ~ {\mathcal P}_i(\omega) ~ \subset ~ D_{\F}(x_{\omega} , \dFU) .
\end{equation}
For each $1 \leq i \leq \nu$ the set ${\mathcal T}_{i} = {\varphi}_i^{-1}(0 , {\mathfrak{T}}_i)$ is a compact transversal to $\F$. Again, without loss of generality, we can assume that the transversals
$\displaystyle \{ {\mathcal T}_{1} , \ldots , {\mathcal T}_{\nu} \}$ are pairwise disjoint, so there exists a constant $0 < \e_1 < \dFU$ such that
\begin{equation}\label{eq-e1}
\dF(x,y) \geq \e_1 \quad {\rm for} ~ x \ne y ~, x \in {\mathcal T}_{i} ~ , ~ y \in {\mathcal T}_{j} ~ , ~ 1 \leq i, j \leq \nu .
\end{equation}
In particular, this implies that the centers of disjoint plaques on the same leaf are separated by distance at least $\e_1$.
We assume in the following that a regular foliated covering of ${\mathfrak{M}}$ as in Proposition~\ref{prop-regular} has been chosen.
Let ${\mathcal U} = \{U_{1}, \ldots , U_{\nu}\}$ denote the corresponding open covering of ${\mathfrak{M}}$.
We can assume that the spaces ${\mathfrak{T}}_i$ form a \emph{disjoint clopen covering} of ${\mathfrak{X}}$, so that
$\displaystyle {\mathfrak{X}} = {\mathfrak{T}}_1 \ \dot{\cup} \cdots \dot{\cup} \ {\mathfrak{T}}_{\nu}$.
A \emph{regular covering} of ${\mathfrak{M}}$ is a finite covering $\displaystyle \{{\varphi}_i \colon U_i \to (-1,1)^n \times {\mathfrak{T}}_i \mid 1 \leq i \leq \nu\}$ by foliation charts which satisfies these conditions.
A map $f \colon {\mathfrak{M}} \to {\mathfrak{M}}'$ between foliated spaces is said to be a \emph{foliated map} if the image of each leaf of $\F$ is contained in a leaf of $\F'$. If ${\mathfrak{M}}'$ is a matchbox manifold, then each leaf of $\F$ is path connected, so its image is path connected, hence must be contained in a leaf of $\F'$. Thus we have:
\begin{lemma} \label{lem-foliated1}
Let ${\mathfrak{M}}$ and ${\mathfrak{M}}'$ be matchbox manifolds, and $h \colon {\mathfrak{M}}' \to {\mathfrak{M}}$ a continuous map. Then $h$ maps the leaves of $\F'$ to leaves of $\F$. In particular, any homeomorphism $h \colon {\mathfrak{M}} \to {\mathfrak{M}}'$ of matchbox manifolds is a foliated map. \hfill $\Box$
\end{lemma}
A \emph{leafwise path} is a continuous map $\gamma \colon [0,1] \to {\mathfrak{M}}$ such that there is a leaf $L$ of $\F$ for which $\gamma(t) \in L$ for all $0 \leq t \leq 1$.
If ${\mathfrak{M}}$ is a matchbox manifold, and $\gamma \colon [0,1] \to {\mathfrak{M}}$ is continuous, then $\gamma$ is a leafwise path by Lemma~\ref{lem-foliated1}. In the following, we will assume that all paths are piecewise differentiable.
\section{Holonomy} \label{sec-holonomy}
The holonomy pseudogroup of a smooth foliated manifold $(M, \F)$ generalizes the induced dynamical systems associated to a section of a flow. The holonomy pseudogroup for a matchbox manifold $({\mathfrak{M}}, \F)$ is defined analogously to the smooth case.
A pair of indices $(i,j)$, $1 \leq i,j \leq \nu$, is said to be \emph{admissible} if the \emph{open} coordinate charts satisfy $U_i \cap U_j \ne \emptyset$.
For $(i,j)$ admissible, define clopen subsets ${\mathfrak{D}}_{i,j} = \pi_i(U_i \cap U_j) \subset {\mathfrak{T}}_i \subset {\mathfrak{X}}$. The convexity of foliation charts imply that plaques are either disjoint, or have connected intersection. This implies that there is a well-defined homeomorphism $h_{j,i} \colon {\mathfrak{D}}_{i,j} \to {\mathfrak{D}}_{j,i}$ with domain $D(h_{j,i}) = {\mathfrak{D}}_{i,j}$ and range $R(h_{j,i}) = {\mathfrak{D}}_{j,i}$.
The maps $\cGF^{(1)} = \{h_{j,i} \mid (i,j) ~{\rm admissible}\}$ are the transverse change of coordinates defined by the foliation charts. By definition they satisfy $h_{i,i} = Id$, $h_{i,j}^{-1} = h_{j,i}$, and if $U_i \cap U_j\cap U_k \ne \emptyset$ then $h_{k,j} \circ h_{j,i} = h_{k,i}$ on their common domain of definition. The \emph{holonomy pseudogroup} $\cGF$ of $\F$ is the topological pseudogroup modeled on ${\mathfrak{X}}$ generated by the elements of $\cGF^{(1)}$. The elements of $\cGF$ have a standard description in terms of the ``holonomy along paths'', which we next describe.
A sequence ${\mathcal I} = (i_0, i_1, \ldots , i_{\alpha})$ is \emph{admissible}, if each pair $(i_{\ell -1}, i_{\ell})$ is admissible for $1 \leq \ell \leq \alpha$, and the composition
\begin{equation}\label{eq-defholo}
h_{{\mathcal I}} = h_{i_{\alpha}, i_{\alpha-1}} \circ \cdots \circ h_{i_1, i_0}
\end{equation}
has non-empty domain.
The domain ${\mathfrak{D}}_{{\mathcal I}}$ of $h_{{\mathcal I}}$ is the \emph{maximal clopen subset} of ${\mathfrak{D}}_{i_0} \subset {\mathfrak{T}}_{i_0}$ for which the compositions are defined.
Given any open subset $U \subset {\mathfrak{D}}_{{\mathcal I}}$ we obtain a new element $h_{{\mathcal I}} | U \in \cGF$ by restriction. Introduce
\begin{equation}\label{eq-restrictedgroupoid}
\cGF^* = \left\{ h_{{\mathcal I}} | U \mid {\mathcal I} ~ {\rm admissible} ~ \& ~ U \subset {\mathfrak{D}}_{{\mathcal I}} \right\} \subset \cGF ~ .
\end{equation}
For $g \in \cGF^*$ denote its domain by ${\mathfrak{D}}(g)$ then its range is the clopen set ${\mathfrak{R}}(g) = g({\mathfrak{D}}(g)) \subset {\mathfrak{X}}$.
The orbit of a point $w \in {\mathfrak{X}}$ by the action of the pseudogroup $\cGF$ is denoted by
\begin{equation}\label{eq-orbits}
{\mathcal O}(w) = \{g(w) \mid g \in \cGF^* ~, ~ w \in {\mathfrak{D}}(g) \} \subset {\mathfrak{T}}_* ~ .
\end{equation}
Given an admissible sequence ${\mathcal I} = (i_0, i_1, \ldots , i_{\alpha})$ and any $0 \leq \ell \leq \alpha$, the truncated sequence ${\mathcal I}_{\ell} = (i_0, i_1, \ldots , i_{\ell})$ is again admissible, and we introduce the holonomy map defined by the composition of the first $\ell$ generators appearing in $h_{{\mathcal I}}$,
\begin{equation}\label{eq-pcmaps}
h_{{\mathcal I}_{\ell}} = h_{i_{\ell} , i_{\ell -1}} \circ \cdots \circ h_{i_{1} , i_{0}}~.
\end{equation}
Given $\xi \in D(h_{{\mathcal I}})$ we adopt the notation $\xi_{\ell} = h_{{\mathcal I}_{\ell}}(\xi) \in {\mathfrak{T}}_{i_{\ell}}$. So $\xi_0 = \xi$ and
$h_{{\mathcal I}}(\xi) = \xi_{\alpha}$.
\medskip
Given $\xi \in D(h_{{\mathcal I}})$, let $x = x_0 = \tau_{i_0}(\xi_0) \in L_x$. Introduce the \emph{plaque chain}
\begin{equation}\label{eq-plaquechain}
{\mathcal P}_{{\mathcal I}}(\xi) = \{{\mathcal P}_{i_0}(\xi_0), {\mathcal P}_{i_1}(\xi_1), \ldots , {\mathcal P}_{i_{\alpha}}(\xi_{\alpha}) \} ~ .
\end{equation}
Intuitively, a plaque chain ${\mathcal P}_{{\mathcal I}}(\xi)$ is a sequence of successively overlapping convex ``tiles'' in $L_0$ starting at $x_0 = \tau_{i_0}(\xi_0)$, ending at
$x_{\alpha} = \tau_{i_{\alpha}}(\xi_{\alpha})$, and with each ${\mathcal P}_{i_{\ell}}(\xi_{\ell})$ ``centered'' on the point $x_{\ell} = \tau_{i_{\ell}}(\xi_{\ell})$.
Recall that ${\mathcal P}_{i_{\ell}}(x_{\ell}) = {\mathcal P}_{i_{\ell}}(\xi_{\ell})$, so we also adopt the notation ${\mathcal P}_{{\mathcal I}}(x) \equiv {\mathcal P}_{{\mathcal I}}(\xi)$.
We next associate an admissible sequence ${\mathcal I}$ to a leafwise path $\gamma$, and thus obtain the holonomy map $h_{\gamma} = h_{{\mathcal I}}$ defined by $\gamma$.
Let $\gamma$ be a leafwise path, and
${\mathcal I}$ be an admissible sequence. For $w \in D(h_{{\mathcal I}})$, we say that $({\mathcal I} , w)$ \emph{covers} $\gamma$,
if the domain of $\gamma$ admits a partition $0 = s_0 < s_1 < \cdots < s_{\alpha} = 1$ such that the plaque chain ${\mathcal P}_{{\mathcal I}}(w_0) = \{{\mathcal P}_{i_0}(w_0), {\mathcal P}_{i_1}(w_1), \ldots , {\mathcal P}_{i_{\alpha}}(w_{\alpha}) \}$ satisfies
\begin{equation}\label{eq-cover}
\gamma([s_{\ell} , s_{\ell + 1}]) \subset int ({\mathcal P}_{i_{\ell}}(w_{\ell}) )~ , ~ 0 \leq \ell < \alpha, \quad \& \quad \gamma(1) \in int( {\mathcal P}_{i_{\alpha}}(w_{\alpha})).
\end{equation}
It follows that $h_{{\mathcal I}}$ is well-defined, with $w_0 = \pi_{i_0}(\gamma(0)) \in D(h_{{\mathcal I}})$. The map $h_{{\mathcal I}}$ is said to define the holonomy of $\F$ along the path $\gamma$, and satisfies $h_{{\mathcal I}}(w_0) = \pi_{i_{\alpha}}(\gamma(1)) \in {\mathfrak{T}}_{i_{\alpha}}$.
Given two admissible sequences, ${\mathcal I} = (i_0, i_1, \ldots, i_{\alpha})$ and
${\mathcal J} = (j_0, j_1, \ldots, j_{\beta})$, such that both $({\mathcal I}, w_0)$ and $({\mathcal J}, v_0)$ cover the leafwise path $\gamma \colon [0,1] \to {\mathfrak{M}}$, then
$$\gamma(0) \in int( {\mathcal P}_{i_0}(w_0)) \cap int( {\mathcal P}_{j_0}(v_0)) \quad , \quad \gamma(1) \in int({\mathcal P}_{i_{\alpha}}(w_{\alpha})) \cap int( {\mathcal P}_{j_{\beta}}(v_{\beta}) )$$ Thus both $(i_0 , j_0)$ and $(i_{\alpha} , j_{\beta})$ are admissible, and
$v_0 = h_{j_{0} , i_{0}}(w_0)$, $w_{\alpha} = h_{i_{\alpha} , j_{\beta}}(v_{\beta})$.
The proof of the following standard observation can be found in \cite{ClarkHurder2013}.
\begin{prop}\label{prop-copc}\cite{ClarkHurder2013}
The maps $h_{{\mathcal I}}$ and
$\displaystyle h_{i_{\alpha} , j_{\beta}} \circ h_{{\mathcal J}} \circ h_{j_{0} , i_{0}}$
agree on their common domains.
\end{prop}
Let $U, U', V, V' \subset {\mathfrak{X}}$ be open subsets with $w \in U \cap U'$. Given homeomorphisms $h \colon U \to V$ and $h' \colon U' \to V'$ with $h(w) = h'(w)$, then $h$ and $h'$ have the same \emph{germ at $w$}, and write $h \sim_w h'$, if there exists an open neighborhood $w \in W \subset U \cap U'$ such that $h | W= h' |W$. Note that $\sim_w$ defines an equivalence relation.
\begin{defn}\label{def-germ}
The \emph{germ of $h$ at $w$} is the equivalence class $[h]_w$ under the relation ~$\sim_w$. The map $h \colon U \to V$ is called a \emph{representative} of $[h]_w$.
The point $w$ is called the source of $[h]_w$ and denoted $s([h]_w)$, while $w' = h(w)$ is called the range of $[h]_w$ and denoted $r([h]_w)$.
\end{defn}
Given a leafwise path $\gamma$ and plaque chain ${\mathcal P}_{{\mathcal I}}(w_0)$ chosen as above,
we let $h_{\gamma} \in \cGF^*$ denote a representative of the germ $\displaystyle [h_{{\mathcal I}}]_{w_0}$. Then Proposition~\ref{prop-copc} yields:
\begin{cor}\label{cor-copc}
Let $\gamma$ be a leafwise path as above, and $({\mathcal I}, w_0)$ and $({\mathcal J}, v_0)$ be two admissible sequences which cover $\gamma$. Then $h_{{\mathcal I}}$
$\displaystyle h_{i_{\alpha} , j_{\beta}} \circ h_{{\mathcal J}} \circ h_{j_{0} , i_{0}}$ determine the same germinal holonomy maps, $\displaystyle [h_{{\mathcal I}}]_{w_0} = [h_{i_{\alpha} , j_{\beta}} \circ h_{{\mathcal J}} \circ h_{j_{0} , i_{0}}]_{w_0}$.
In particular, the germ of $h_{\gamma}$ is well-defined for the path $\gamma$.
\end{cor}
Two leafwise paths $\gamma , \gamma' \colon [0,1] \to {\mathfrak{M}}$ are homotopic if there exists a family of leafwise paths $\gamma_s \colon [0,1] \to {\mathfrak{M}}$ with $\gamma_0 = \gamma$ and $\gamma_1 = \gamma'$. We are most interested in the special case when $\gamma(0) = \gamma'(0) = x$ and $\gamma(1) = \gamma'(1) = y$. Then $\gamma$ and $\gamma'$ are \emph{endpoint-homotopic}
if they are homotopic with $\gamma_s(0) = x$ for all $0 \leq s \leq 1$, and similarly
$\gamma_s(1) = y$ for all $0 \leq s \leq 1$. Thus, the family of curves $\{ \gamma_s(t) \mid 0 \leq s \leq 1\}$ are all contained in a common leaf $L_{x}$ and we have:
\begin{lemma}\label{lem-homotopic}\cite{ClarkHurder2013}
Let $\gamma, \gamma' \colon [0,1] \to {\mathfrak{M}}$ be endpoint-homotopic leafwise paths. Then the holonomy maps $h_{\gamma}$ and $h_{\gamma'}$ admit representatives which agree on some clopen subset $U \subset {\mathfrak{T}}_*$. In particular, they determine the same germinal holonomy maps,
$\displaystyle [h_{{\mathcal I}}]_{w_0} = [h_{i_{\alpha} , j_{\beta}} \circ h_{{\mathcal J}} \circ h_{j_{0} , i_{0}}]_{w_0}$.
\end{lemma}
We next consider some properties of the pseudogroup $\cGF$. First, let
$W \subset {\mathfrak{T}}$ be an open subset, and define the restriction to $W$ of $\cGF^*$ by:
\begin{equation}
{\mathcal G}_W = \left\{ g \in \cGF^* \mid {\mathfrak{D}}(g) \subset W ~, ~ {\mathfrak{R}}(g) \subset W \right\} .
\end{equation}
Introduce the filtrations of $\cGF^*$ by word length.
For $\alpha \geq 1$, let $\cGF^{(\alpha)}$ be the collection of holonomy homeomorphisms $h_{{\mathcal I}} | U \in \cGF^*$ determined by admissible paths ${\mathcal I} = (i_0,\ldots,i_k)$ such that $k \leq \alpha$ and $U \subset {\mathfrak{D}}(h_{{\mathcal I}})$ is open.
Then for each $g \in \cGF^*$ there is some $\alpha$ such that $g \in \cGF^{(\alpha)}$. Let $\|g\|$ denote the least such $\alpha$, which is called the \emph{word length} of $g$. Note that $\cGF^{(1)}$ generates $\cGF^*$.
We note the following finiteness result, whose proof is given in \cite[Section~4]{CHL2013b}:
\begin{lemma}\label{lem-finitegen}
Let $W \subset {\mathfrak{X}}$ be an open subset. Then there exists an integer $\alpha_W$ such that ${\mathfrak{X}}$ is covered by the collection
$\{h_{{\mathcal I}} (W) \mid h_{{\mathcal I}} \in \cGF^{(\alpha_W)} \}$.
\end{lemma}
Finally, we recall the definition of an {equicontinuous} pseudogroup.
\begin{defn} \label{def-equicontinuous}
The action of the pseudogroup $\cGF$ on ${\mathfrak{X}}$ is \emph{equicontinuous} if for all $\epsilon > 0$, there exists $\delta > 0$ such that for all $g \in \cGF^*$, if $w, w' \in D(g)$ and $d_{{\mathfrak{X}}}(w,w') < \delta$, then $d_{{\mathfrak{X}}}(g(w), g(w')) < \epsilon$.
Thus, $\cGF^*$ is equicontinuous as a family of local group actions.
\end{defn}
Further dynamical properties of the pseudogroup $\cGF$ for a matchbox manifold are discussed in the papers \cite{ClarkHurder2013,CHL2013a,CHL2013b,Hurder2013a}.
\section{Return equivalence}\label{sec-return}
For an open subset $W \subset {\mathfrak{T}}_*$ the induced pseudogroup ${\mathcal G}_W$ is used to represent the local dynamics of a matchbox manifold ${\mathfrak{M}}$. We first introduce the key concept of \emph{return equivalence} between two such pseudogroups, and then study the properties of the equivalence relation obtained. Return equivalence is the analog for matchbox manifolds of the notion of \emph{Morita equivalence} for foliation groupoids, which is discussed by Haefliger in \cite{Haefliger1984}.
Let ${\mathfrak{M}}_1$ and ${\mathfrak{M}}_2$ be matchbox manifolds with transversals ${\mathfrak{T}}_*^1$ and ${\mathfrak{T}}_*^2$, respectively.
Given clopen subsets $U_1 \subset {\mathfrak{T}}_*^1$ and $U_2 \subset {\mathfrak{T}}_*^2$ we say that the restricted pseudogroups ${\mathcal G}_{U_1}$ and ${\mathcal G}_{U_2}$
are \emph{isomorphic} if there exists a homeomorphism $\phi \colon U_1 \to U_2$ such that the induced map $\Phi \colon {\mathcal G}_{U_1} \to {\mathcal G}_{U_2}$ is an isomorphism. That is, for all $g \in {\mathcal G}_{U_1}$ the map $\Phi(g) = \phi \circ g \circ \phi^{-1}$ defines an element of ${\mathcal G}_{U_2}$. Conversely, for all $h \in {\mathcal G}_{U_2}$ the map $\Phi^{-1}(h) = \phi^{-1} \circ h \circ \phi$ defines an element of ${\mathcal G}_{U_1}$.
\begin{defn}\label{def-return}
Let ${\mathfrak{M}}_1$ and ${\mathfrak{M}}_2$ be minimal matchbox manifolds, with transversals ${\mathfrak{T}}_*^1$ and ${\mathfrak{T}}_*^2$, respectively. Given clopen subsets $W_i \subset {\mathfrak{T}}_i$ for $i=1,2$, we say that the restricted pseudogroups ${\mathcal G}_{W_i}$ are \emph{return equivalent} if there are non-empty clopen sets $U_i \subset W_i$ and homeomorphism $\phi \colon U_1 \to U_2$ such that the induced map $\Phi \colon {\mathcal G}_{U_1} \to {\mathcal G}_{U_2}$ is an isomorphism.
\end{defn}
The properties of this definition are given in the following sequence of results, but we first make a general remark. Recall that if ${\mathfrak{M}}_i$ is minimal, then every leaf of $\F_i$ intersects the local section $\tau_i(W_i)$ for any open set $W_i \subset {\mathfrak{T}}_*^i$. As seen in the proof of Lemma~\ref{lem-return3} below, this property is used to show that return equivalence satisfies the transitive axiom of an equivalence relation. In contrast, recall that a matchbox manifold ${\mathfrak{M}}$ is \emph{transitive} if it contains a leaf $L$ with $\overline{L}={\mathfrak{M}}$. Definition~\ref{def-return} does not define a transitive equivalence relation for transitive spaces, as can be seen for particular transitive matchbox manifolds and suitably chosen clopen subsets.
\begin{lemma}\label{lem-return1}
Let ${\mathfrak{M}}$ be a minimal matchbox manifold with transversal ${\mathfrak{T}}_*$.
Let $W, W' \subset {\mathfrak{T}}_*$ be non--empty clopen subsets, then ${\mathcal G}_W$ and ${\mathcal G}_{W'}$ are return equivalent.
\end{lemma}
\proof
Let $w \in W$ with $x = \tau(w)$. Let $w' \in W'$ be a point such that $y = \tau(w') \cap L_x$ which exists as $L_x$ is dense in ${\mathfrak{M}}$.
Choose a path $\gamma$ with $\gamma(0) = x$ and $\gamma(1) = y$, and let ${\mathcal I} = (i_0, i_1, \ldots , i_{\alpha})$ define a plaque chain which covers $\gamma$, with $W \subset {\mathfrak{T}}_{i_0}$ and $W' \subset {\mathfrak{T}}_{i_{\alpha}}$. Observe that $\alpha \leq \alpha_W$. Let $g = h_{{\mathcal I}}$ denote the holonomy transformation defined by the admissible sequence ${\mathcal I}$, with domain a clopen set ${\mathfrak{D}}(g) \subset {\mathfrak{T}}_{i_0}$.
Chose a clopen set $U$ with $w \in U \subset W \cap {\mathfrak{D}}(g)$ and $V = h_g(U) \subset W' \cap {\mathfrak{T}}_{i_{\alpha}}$. Then the restriction $\phi = h_g | U \colon U \to V$ is
a homeomorphism which satisfies the conditions above, so induces an isomorphism of pseudogroups, $\Phi \colon {\mathcal G}_U \to {\mathcal G}_V$.
Thus, ${\mathcal G}_W$ and ${\mathcal G}_{W'}$ are return equivalent.
\endproof
\begin{cor}\label{cor-return1}
Let ${\mathfrak{M}}$ be a minimal matchbox manifold with transversal ${\mathfrak{T}}_*$.
Let $W \subset {\mathfrak{T}}_*$ be a non--empty clopen subset, then $\cGF$ and ${\mathcal G}_W$ are return equivalent.
\end{cor}
\begin{lemma}\label{lem-return2}
Let ${\mathfrak{M}}$ be a minimal matchbox manifold, and suppose we are given regular coverings
$\displaystyle \{{\varphi}_i \colon U_i \to (-1,1)^n \times {\mathfrak{T}}_i \mid 1 \leq i \leq \nu\}$ and
$\displaystyle \{{\varphi}_j' \colon U_j' \to (-1,1)^n \times {\mathfrak{T}}_j' \mid 1 \leq j \leq \nu'\}$ of ${\mathfrak{M}}$, with transversals ${\mathfrak{T}}_*$ and ${\mathfrak{T}}_*'$ respectively.
Let $W \subset {\mathfrak{T}}_*$ and $W' \subset {\mathfrak{T}}_*'$ be non--empty clopen subsets. Let ${\mathcal G}_W$ denote the restricted pseudogroup on $W$ for the first covering,
and ${\mathcal G}_{W'}$ the restricted pseudogroup for the second covering. Then ${\mathcal G}_W$ and ${\mathcal G}_{W'}$ are return equivalent.
\end{lemma}
\proof
Let $w \in W$ with $x = \tau(w)$. Let $w' \in W'$ be a point such that $y = \tau'(w') \cap L_x$ which exists as $L_x$ is dense in ${\mathfrak{M}}$.
Choose a path $\gamma$ with $\gamma(0) = x$ and $\gamma(1) = y$, and let ${\mathcal I} = (i_0, i_1, \ldots , i_{\alpha})$ define a plaque chain which covers $\gamma$, with $W \subset {\mathfrak{T}}_{i_0}$. Let $g = h_{{\mathcal I}}$ be the holonomy map defined by the admissible sequence ${\mathcal I}$, with domain a clopen set ${\mathfrak{D}}(g) \subset {\mathfrak{T}}_{i_0}$.
Then there exists $i_{\alpha'}$ such that $y \in {\mathcal T}'_{j_{\alpha'}} = \tau'({\mathfrak{T}}'_{j_{\alpha'}})$.
Thus, $y \in U_{i_{\alpha}} \cap U'_{j_{\alpha'}}$.
Also, we have that $W' \subset {\mathfrak{T}}'_{j_{\alpha'}}$, and define $W'' = \pi_{i_{\alpha}}(\pi_{j_{\alpha'}}^{-1}(W') \subset {\mathfrak{T}}_{i_{\alpha}}$.
Chose a clopen set $U$ with $w \in U \subset W \cap {\mathfrak{D}}(g)$ and $V' = h_g(U) \subset W'$.
The composition $\phi = \pi'_{j_{\alpha'}} \circ \tau_{i_{\alpha}} \circ h_g | U \colon U \to V \subset {\mathfrak{T}}'_{j_{\alpha'}}$ is
a homeomorphism which induces an isomorphism of restricted pseudogroups, $\Phi \colon {\mathcal G}_U \to {\mathcal G}_{V'}$.
Thus, ${\mathcal G}_W$ and ${\mathcal G}_{W'}$ are return equivalent.
\endproof
\begin{lemma}\label{lem-return3}
Let ${\mathfrak{M}}_1 , {\mathfrak{M}}_2 , {\mathfrak{M}}_3$ be minimal matchbox manifolds with regular coverings
defining transversals ${\mathfrak{T}}_*^1 , {\mathfrak{T}}_*^2 , {\mathfrak{T}}_*^3$ respectively.
Suppose there exists non-empty clopen subsets $W_1 \subset {\mathfrak{T}}_*^1$ and $W_2 \subset {\mathfrak{T}}_*^2$ such that the restricted pseudogroups ${\mathcal G}_{W_1}$ and ${\mathcal G}_{W_2}$ are return equivalent, and that there exists non-empty clopen subsets $W_2' \subset {\mathfrak{T}}_*^2$ and $W_3 \subset {\mathfrak{T}}_*^3$ such that the restricted pseudogroups ${\mathcal G}_{W_2'}$ and ${\mathcal G}_{W_3}$ are return equivalent. Then ${\mathcal G}_{W_1}$ and ${\mathcal G}_{W_3}$ are return equivalent.
\end{lemma}
\begin{proof}
By definition, there exists non--empty clopen sets $U_i \subset W_i$ ($i=1,2$) and a homeomorphism $\phi_{1} \colon U_1 \to U_2$, which induces a pseudogroup isomorphism $\displaystyle \Phi_{1} \colon {\mathcal G}_{U_1} \to {\mathcal G}_{U_2}$.
Similarly, there exists non--empty clopen subsets, $V_2 \subset W_2' \subset {\mathfrak{T}}_*^2$ and $V_3 \subset W_3 \subset {\mathfrak{T}}_*^3$ and a homeomorphism $\phi_{2} \colon V_2 \to V_3$, which induces a pseudogroup isomorphism $\displaystyle \Phi_{2} \colon {\mathcal G}_{V_2} \to {\mathcal G}_{V_3}$.
Choose $w_2 \in W_2$ and set $y = \tau^2(w_2) \in {\mathcal T}_*^2$, then by minimality of ${\mathfrak{M}}_2$ the leaf $L_y$ containing $y$ intersects the transverse set $\tau^2(W_2')$ in a point $y'$. Choose a path $\gamma$ with $\gamma(0) = y$ and $\gamma(1) = y'$. Then the holonomy for $\F_2$ along $\gamma$ defines a homeomorphism
$h_{\gamma} \colon X \to X'$ for clopen sets satisfying $y \in X \subset U_2 \subset W_2$ and $y' \in X' \subset W_2'$.
Set $Y = \phi_1^{-1}(X)$ and $Z = \phi_2(X')$. Then the composition $\phi_3 = \phi_2 \circ h_{\gamma} \circ \phi_2 \colon Y \to Z$ is a homeomorphism between the clopen subsets $Y \subset W_1$ and $Z \subset W_3$ which induces an isomorphism of pseudogroups, $\Phi_3 \colon {\mathcal G}_{Y} \to {\mathcal G}_{Z}$.
Thus, ${\mathcal G}_{W_1}$ and ${\mathcal G}_{W_3}$ are return equivalent, which was to be shown.
\end{proof}
\begin{prop}\label{prop-return}
\emph{Return equivalence} is an equivalence relation on the class of restricted pseudogroups obtained from minimal matchbox manifolds.
\end{prop}
\proof
It is immediate that return equivalence is reflexive and symmetric relation. That return equivalence is transitive follows from Lemmas~\ref{lem-return1}, \ref{lem-return2} and \ref{lem-return3}.
\endproof
\begin{defn}\label{def-return5}
Two minimal matchbox manifolds ${\mathfrak{M}}_i$ for $i=1,2$, are \emph{return equivalent} if there exists regular coverings of ${\mathfrak{M}}_i$ and non--empty, clopen transversals $W_i$ for each covering so that the restricted pseudogroups ${\mathcal G}_{W_i}$ for $i=1,2$ are return equivalent.
\end{defn}
We conclude this section by showing that homeomorphism implies return equivalence.
\begin{thm}\label{thm-topinv}
Let ${\mathfrak{M}}_1$ and ${\mathfrak{M}}_2$ be minimal matchbox manifolds. Suppose that there exists a homeomorphism $h \colon {\mathfrak{M}}_1 \to {\mathfrak{M}}_2$, then ${\mathfrak{M}}_1$ and ${\mathfrak{M}}_2$ are return equivalent.
\end{thm}
\begin{proof}
First note that the homeomorphism $h$ is a foliated map by Lemma~\ref{lem-foliated1}. This implies that $h$ is a homeomorphism between the leaves of ${\mathfrak{M}}_1$ and the leaves of ${\mathfrak{M}}_2$, and thus the leaves of $\F_1$ and $\F_2$ have the same dimensions. However, we do not assume that $h$ is smooth when restricted to leaves.
Choose a regular covering
$\displaystyle {\mathcal U} = \{{\varphi}_i \colon U_i \to (-1,1)^n \times {\mathfrak{T}}_i \mid 1 \leq i \leq \nu\}$ for ${\mathfrak{M}}_1$ with transversal ${\mathfrak{T}}_*$.
Also, choose a regular covering $\displaystyle \{{\varphi}_j' \colon U_j' \to (-1,1)^n \times {\mathfrak{T}}_j' \mid 1 \leq j \leq \nu'\}$ of ${\mathfrak{M}}_2$, with transversal ${\mathfrak{T}}_*'$.
Consider the open covering of ${\mathfrak{M}}_1$ by the inverse images $\displaystyle {\mathcal V} = \{ V_j = h^{-1}(U_j') \mid 1 \leq j \leq \nu'\}$. Let $\e_V > 0$ be a Lebesgue number for this covering.
Then choose a regular covering
$\displaystyle {\mathcal U}'' = \{{\varphi}_l'' \colon U_k'' \to (-1,1)^n \times {\mathfrak{T}}_k \mid 1 \leq k \leq \nu''\}$ for ${\mathfrak{M}}_1$ with transversal ${\mathfrak{T}}_*''$
as in Proposition~\ref{prop-regular}, with constant ${\epsilon_{{\mathfrak{M}}}} < \e_{{\mathcal V}}$ so that each chart each ${\overline{U}}_k'' \subset B_{{\mathfrak{M}}}(z_k, \e_{{\mathcal V}})$ where $z_k$ is the ``center point'' for $V_k$.
It follows that for each $1 \leq k \leq \nu''$ there exists $1 \leq \ell_k \leq \nu'$ with $\displaystyle {\overline{U}}_k'' \subset V_{\ell_k}$, and thus $h({\overline{U}}_k'') \subset U_{\ell_k}'$.
Choose a clopen set $X \subset {\mathfrak{T}}_1$ and a clopen set $Y \subset {\mathfrak{T}}_1''$.
Then by Lemma~\ref{lem-return2}, the restricted pseudogroups ${\mathcal G}_{X}$ and ${\mathcal G}_{Y}$ are return equivalent. That is,
there exists clopen subsets $X' \subset X$ and $Y' \subset Y$ and a homeomorphism $\phi_1 \colon X' \to Y'$ which induces an isomorphism
$\Phi_1 \colon {\mathcal G}_{X'} \to {\mathcal G}_{Y'}$.
Then the composition $\phi = \pi_{\ell_1}' \circ h \circ \tau_1 \circ \phi_1 \colon X' \to Z' \subset {\mathfrak{T}}_{\ell_1}'$ is well-defined,
and is a homeomorphism onto the clopen subset $Z'$, and induces an isomorphism $\Phi \colon {\mathcal G}_{X'} \to {\mathcal G}_{Z'}$. Set $Z = {\mathfrak{T}}_{\ell_1}'$, then it follows that ${\mathcal G}_{X}$ and ${\mathcal G}_{Z}$ are return equivalent, and so ${\mathfrak{M}}_1$ and ${\mathfrak{M}}_2$ are return equivalent.
\end{proof}
\section{Foliated bundles}\label{sec-bundles}
A matchbox manifold ${\mathfrak{M}}$ has the structure of a \emph{foliated bundle} if there is a closed connected manifold $B$ of dimension $n \geq 1$, and a fibration map $\pi \colon {\mathfrak{M}} \to B$ such that for each leaf $L \subset {\mathfrak{M}}$, the restriction $\pi \colon L \to B$ is a covering map. For each $b \in B$, the fiber $\mathfrak{F}_b = \pi^{-1}(b)$ is then a totally disconnected compact space. If $\mathfrak{F}_b$ is a Cantor space, then we say that ${\mathfrak{M}}$ is a \emph{foliated Cantor bundle}.
The standard texts on foliations, such as \cite{CandelConlon2000} and \cite{CN1985}, discuss the suspension construction for foliated manifolds, which adapts to the context of foliated spaces without difficulties.
Also, the seminal work by Kamber and Tondeur \cite{KamberTondeur1968} discusses general foliated bundles (there referred to as flat bundles).
In this section, we obtain conditions which are sufficient to imply that return equivalence implies homeomorphism, which yields a converse to Theorem~\ref{thm-topinv} for foliated Cantor bundles.
These results are used in the following sections to prove Theorem~\ref{thm-main2}.
We recall some of the basic properties of the construction of foliated bundles, as needed in the following.
Let $\mathfrak{F}$ be a compact topological space, and let $\Homeo(\mathfrak{F})$ denote its group of homeomorphisms. Given a closed manifold $B$,
choose a basepoint $b_0 \in B$ and let $\Lambda = \pi_1(B, b_0)$ be the fundamental group, whose elements are represented by the endpoint-homotopic classes of closed curves in $B$ with endpoints at $b_0$. Let $\widetilde{B}$ denote the universal covering of $B$, defined as the endpoint-homotopy classes of paths in $B$ starting at $b_0$. The group $\Lambda$ acts on the right on $\widetilde{B}$ by pre-composing paths representing elements of $\widetilde{B}$ with paths representing elements of $\Lambda$. This yields the action of $\Lambda$ on $\widetilde{B}$ by deck transformations.
Let ${\varphi} \colon \Lambda \to \Homeo(\mathfrak{F})$ be a representation, which defines a left-action of $\Lambda$ on $\mathfrak{F}$ by homeomorphisms. Define the quotient space
\begin{equation}\label{eq-suspension}
{\mathfrak{M}}_{{\varphi}} = (\widetilde{B} \times \mathfrak{F}) / \{(x \cdot \gamma, \omega) \sim (x , {\varphi}(\gamma) \cdot \omega \} \quad , \quad x \in \widetilde{B} ~ , ~ \omega \in \mathfrak{F} ~, ~ \gamma \in \Lambda
\end{equation}
The images of the slices $\widetilde{B} \times \{\omega\} \subset \widetilde{B} \times \mathfrak{F}$ in ${\mathfrak{M}}_{{\varphi}}$ form the leaves of the suspension foliation $\F_{{\varphi}}$ and gives ${\mathfrak{M}}_{{\varphi}}$ the structure of a foliated space. The projection ${\widetilde{\pi}} \colon \widetilde{B} \times \mathfrak{F} \to \widetilde{B}$ is equivariant with respect to the action of $\Lambda$, so descends to a fibration map
$\pi \colon {\mathfrak{M}}_{{\varphi}} \to B$. Thus, ${\mathfrak{M}}_{{\varphi}}$ is a foliated bundle. The next result implies that all foliated bundles are of this form.
\begin{prop}\label{prop-standardform}
Let $\pi \colon {\mathfrak{M}} \to B$ be a foliated bundle, $b_0 \in B$ a basepoint, and let $\mathfrak{F}_0 = \pi^{-1}(b_0)$ be the fiber. Then there is a well-defined \emph{global holonomy} map ${\varphi} \colon \Lambda \to \Homeo(\mathfrak{F}_0)$ and a natural homeomorphism of foliated bundles, $\Phi_{\F} \colon {\mathfrak{M}}_{{\varphi}} \to {\mathfrak{M}}$.
\end{prop}
\proof
We sketch the construction of the maps ${\varphi}$ and $\Phi_{\F}$, as the construction is standard. Given $\lambda \in \Lambda$, let $\gamma \colon [0,1] \to B$ denote a continuous path with $\gamma(0) = \gamma(1) = b_0$ representing $\lambda$. Given $\omega \in \mathfrak{F}_0$ let $L_{\omega}$ be the leaf of $\F$ containing $\omega$, then $\pi \colon L_{\omega} \to B$ is a covering, so there exists a lift ${\widetilde{\gamma}}_{\omega}$ of $\gamma$ with ${\widetilde{\gamma}}_{\omega}(0) = {\omega}$. Then set ${\varphi}(\lambda) \cdot \omega = {\widetilde{\gamma}}(1)$. By the properties of holonomy for foliations, this defines a homeomorphism of the fiber $\mathfrak{F}_0$. The properties of path lifting implies that ${\varphi} \colon \Lambda \to \Homeo(\mathfrak{F}_0)$ is a homomorphism.
Define $\Phi_{\F} \colon \widetilde{B} \times \mathfrak{F}_0 \to {\mathfrak{M}}$ as follows: for $\omega \in \mathfrak{F}_0$ and $\gamma_0 \in \widetilde{B}$ with $\gamma_0(0) = b_0$ and $\gamma_0(1) = b$, then let $\gamma_{\omega}$ be the lift of $\gamma_0$ to the leaf $L_{\omega} \subset {\mathfrak{M}}$ with $\gamma_{\omega}(0) = \omega$. Set ${\widetilde{\Phi}}_{\F}(b , \omega) = \gamma_{\omega}(1) \in {\mathfrak{M}}$.
Note that by the definition of ${\varphi}$ we have, for all $\gamma \in \pi_1(B, b_0)$, that
${\widetilde{\Phi}}_{\F}(x \cdot \gamma, \omega) = {\widetilde{\Phi}}_{\F}(x , {\varphi}(\gamma) \cdot \omega) \in {\mathfrak{M}}$
so ${\widetilde{\Phi}}_{\F}$ descends to a map $\Phi_{\F} \colon {\mathfrak{M}}_{{\varphi}} \to {\mathfrak{M}}$, which is checked to be a homeomorphism.
\endproof
Next we consider two types of maps between foliated bundles. The following results are proved using the path-lifting property of foliated bundles, in a manner similar to the proof of Proposition~\ref{prop-standardform}.
First, let $f \colon B' \to B$ be a diffeomorphism of closed manifolds $B$ and $B'$.
Let $b_0 \in B$ be a basepoint, and let $b_0' = f^{-1}(b_0) \in B_0'$ be the basepoint for $B_0'$. Set $\Lambda' = \pi_1(B_0' , b_0')$, and
let $f_{\#} \colon \Lambda' \to \Lambda$ be the induced isomorphism of fundamental groups.
Given a representation ${\varphi} \colon \Lambda \to \Homeo(\mathfrak{F})$, set ${\varphi}' = {\varphi} \circ f_{\#} \colon \Lambda' \to \Homeo(\mathfrak{F})$. Then we obtain an associated foliated bundle
$${\mathfrak{M}}_{{\varphi}'}' = (\widetilde{B}' \times \mathfrak{F}) / \{(x' \cdot \gamma', \omega) \sim (x' , {\varphi}'(\gamma')(\omega) \} \quad , \quad x' \in \widetilde{B}' ~ , ~ \omega \in {\mathfrak{C}} ~, ~ \gamma' \in \Lambda' .$$
\begin{prop}\label{prop-changeofbase}
There is a foliated bundle isomorphism $\displaystyle F \colon {\mathfrak{M}}_{{\varphi}'}' \to {\mathfrak{M}}_{{\varphi}}$. \hfill $\Box$
\end{prop}
Next, let $h \colon \mathfrak{F}' \to \mathfrak{F}$ be a homeomorphism, and let ${\varphi} \colon \Lambda \to \Homeo(\mathfrak{F})$ be a representation. Define the representation
${\varphi}^h \colon \Lambda \to \Homeo(\mathfrak{F}')$ by setting ${\varphi}^h = h^{-1} \circ {\varphi} \circ h$. Then have
\begin{prop}\label{prop-changeoffiber}
There is a foliated bundle isomorphism $\displaystyle F \colon {\mathfrak{M}}_{{\varphi}^h} \to {\mathfrak{M}}_{{\varphi}}$. \hfill $\Box$
\end{prop}
\medskip
In the case of foliated Cantor bundles, there is yet another method to induce homeomorphisms between their total spaces. This uses the following notion:
\begin{defn}\label{def-collapsible}
Let ${\mathfrak{M}}_{{\varphi}}$ be a foliated Cantor bundle, with projection map $\pi \colon {\mathfrak{M}}_{{\varphi}} \to B$, $b_0 \in B$ a basepoint, Cantor fiber $\mathfrak{F}_0 = \pi^{-1}(b_0)$, and
global holonomy map ${\varphi} \colon \Lambda \to \Homeo(\mathfrak{F}_0)$.
A clopen subset $W \subset \mathfrak{F}_0$ is \emph{collapsible} if $\tau(W)$ is a fiber of a bundle projection $\pi' \colon {\mathfrak{M}}_{{\varphi}} \to B'$ such that there is a finite covering map $\pi_W \colon B' \to B$ that makes the following diagram commute:
\begin{equation}\label{eq-collapse}
\xymatrix{
{\mathfrak{M}}_{{\varphi}} \ar[d]_{\pi}\ar[rd]^{\pi'}& \\
B&B'\ar[l]^{\pi_W}}
\end{equation}
We say that ${\mathfrak{M}}_{{\varphi}}$ is \emph{infinitely collapsible} if every clopen subset of $W \subset \mathfrak{F}_0$ contains a collapsible clopen subset.
\end{defn}
The following gives effective criteria for when a clopen set is collapsible.
\begin{prop}\label{prop-partitions}
Let ${\mathfrak{M}}_{{\varphi}}$ be a foliated Cantor bundle, with projection map $\pi \colon {\mathfrak{M}}_{{\varphi}} \to B$, $b_0 \in B$ a basepoint, Cantor fiber $\mathfrak{F}_0 = \pi^{-1}(b_0)$, and
global holonomy map ${\varphi} \colon \Lambda \to \Homeo(\mathfrak{F}_0)$. Then the clopen subset $W \subset \mathfrak{F}_0$ is collapsible if and only if the collection
$\displaystyle \{{\varphi}(\gamma) \cdot W \mid \gamma \in \Lambda \}$ is a finite partition of $\mathfrak{F}_0$ into clopen subsets.
\end{prop}
\proof
Suppose that the clopen subset $W \subset \mathfrak{F}_0$ is collapsible, and there is a diagram \eqref{eq-collapse}.
Label the points in the preimage of $b_0$ by $X_W = \pi_W^{-1}(b_0) = \{b_1, \ldots , b_k\}$, and the corresponding fibers of $\pi'$ by
$W_i = (\pi')^{-1}(b_i) \subset \mathfrak{F}_0$ for $1 \leq i \leq b_k$. We can assume without loss that that $W = W_1$. It follows from the commutativity of the diagram \eqref{eq-collapse} that these sets form a clopen partition of ${\mathfrak{X}}_0$,
$\displaystyle {\mathfrak{X}}_0 = W_1 \cup \cdots \cup W_k$.
Let $\Lambda_W \subset \Lambda$ be the covering group for $\pi_W$ which is the image of the map
$$(\pi_W)_{\#} \colon \pi_1(B', b_1) \to \pi_1(B, b_0) = \Lambda .$$
Then the monodromy action of $\Lambda_W$ on the fiber $\mathfrak{F}_0$ permutes the clopen sets $W_i$ for $1 \leq i \leq k$. It follows that there is a homeomorphism
\begin{equation}\label{eq-collapsedhomeo}
{\mathfrak{M}}_{{\varphi}} \cong (\widetilde{B} \times \mathfrak{F}_0) / \{(x \cdot \gamma, \omega) \sim (x , {\varphi}(\gamma) \cdot \omega \} \quad , \quad x \in \widetilde{B} ~ , ~ \omega \in W_1 ~, ~ \gamma \in \Lambda_W .
\end{equation}
Conversely, suppose that $W \subset \mathfrak{F}_0$ is a clopen set, such that the collection
$\displaystyle \{{\varphi}(\gamma) \cdot W \mid \gamma \in \Lambda \}$ is a finite partition of $\mathfrak{F}_0$ into clopen subsets.
Set $W_1 = W$, and choose $\gamma_i \in \Lambda$ for $1 < i \leq k$ so that for $W_i = {\varphi}(\gamma_i) \cdot W$, the collection
${\mathcal W} = \{W_1, W_2, \ldots , W_k\}$ is a clopen partition of ${\mathfrak{X}}_0$. Then define
\begin{equation}\label{eq-isotropysubgroup}
\Lambda_W = \{ \gamma \in \Lambda \mid {\varphi}(\gamma) \cdot W = W\} .
\end{equation}
Note that as the collection of clopen sets ${\mathcal W}$ is permuted by the action of $\Lambda$, the subgroup $\Lambda_W$ has finite index. Let $\pi_W \colon B' \to B$ be the finite covering of $B$ associated to $\Lambda_W$. Then projection along the fiber in the decomposition of ${\mathfrak{M}}$ in \eqref{eq-collapsedhomeo} yields a projection map
$\pi' \colon {\mathfrak{M}}_{{\varphi}} \to B'$ so that the diagram \eqref{eq-collapse} commutes, as was to be shown.
\endproof
Next consider the properties of return equivalence and collapsibility in the context of foliated Cantor bundles.
For $i =1,2$, let ${\mathfrak{M}}_{{\varphi}_i}$ be minimal foliated Cantor bundles over the common base $B$. Let $\pi_i \colon {\mathfrak{M}}_{{\varphi}_i} \to B$ be the corresponding projection maps, $b_0 \in B$ a basepoint, and define the Cantor fibers $\mathfrak{F}_i = \pi_i^{-1}(b_0)$, with global holonomy maps ${\varphi}_{i} \colon \Lambda \to \Homeo(\mathfrak{F}_i)$. Assume there clopen sets $W_i \subset \mathfrak{F}_i$ such that ${\mathcal G}_{W_1}$ and ${\mathcal G}_{W_2}$ are return equivalent. Let $U_i \subset W_i$ be clopen sets and $\phi \colon U_1 \to U_2$ be a homeomorphism which induces an isomorphism of the restricted pseudogroups, $\Phi \colon {\mathcal G}_{U_1} \to {\mathcal G}_{U_2}$.
\begin{lemma}\label{lem-collpasing}
Assume that the clopen set $U_1$ is collapsible, then the clopen set $U_2$ is collapsible.
\end{lemma}
\proof
By Proposition~\ref{prop-partitions}, it suffices to show that the collection $\{{\varphi}_2(\gamma) \cdot U_2 \mid \gamma \in \Lambda\}$
is a clopen partition of $\mathfrak{F}_0$.
First, let $\gamma \in \Lambda$ satisfy $U_2 \cap {\varphi}_2(\gamma) \cdot U_2 \ne \emptyset$.
By assumption, ${\varphi}_2(\gamma)$ is conjugate to some $g \in {\mathcal G}_{U_1}$ for which $U_1 \cap g \cdot U_1 \ne \emptyset$. As $U_1$ is collapsible, this implies $g \cdot U_1 = U_1$, and thus ${\varphi}_2(\gamma) \cdot U_2 = U_2$.
Next, suppose there exists $\gamma_1 , \gamma_2 \in \Lambda$ such that there exists $\displaystyle \{ {\varphi}_2(\gamma_1) \cdot U_2\} \cap \{{\varphi}_2(\gamma_1) \cdot U_2\} \ne \emptyset$. Then $U_2 \cap {\varphi}_2(\gamma_1^{-1} \gamma_2) \cdot U_2 \ne \emptyset$, so by the above we have
$\displaystyle {\varphi}_2(\gamma_1^{-1} \gamma_2) \cdot U_2 = U_2$ and thus ${\varphi}_2(\gamma_1) \cdot U_2 = {\varphi}_2(\gamma_1) \cdot U_2$.
The action ${\varphi}_2$ is assumed to be minimal, so the collection
$\{{\varphi}_2(\gamma) \cdot U_2 \mid \gamma \in \Lambda\}$ is an open covering of the compact space $\mathfrak{F}_2$, and thus admits a finite subcovering.
The covering is by disjoint closed sets, hence is a clopen covering, as was to be shown.
\endproof
The proof of Lemma~\ref{lem-collpasing} shows that
\begin{equation}\label{eq-isotropyequality}
\Lambda_{U_1} \equiv \{ \gamma \in \Lambda \mid {\varphi}_1(\gamma) \cdot U_1 = U_1\} =
\{ \gamma \in \Lambda \mid {\varphi}_2(\gamma) \cdot U_2 = U_2\} \equiv \Lambda_{U_2} .
\end{equation}
Proposition~\ref{prop-changeoffiber} for ${\varphi}_2 | U_2 = \phi \circ {\varphi}_1 | U_1 \circ \phi^{-1}$ and the decompositions \eqref{eq-collapsedhomeo} for ${\mathfrak{M}}_1$ and ${\mathfrak{M}}_2$ then yield:
\begin{prop}\label{prop-conjugates}
Let ${\mathfrak{M}}_1$ and ${\mathfrak{M}}_2$ be minimal foliated Cantor bundles over $B$ as above, which have conjugate restricted pseudogroups on the collapsible clopen set $U_1 \subset \mathfrak{F}_1$. Then $\phi \colon U_1 \to U_2$ induces a homeomorphism ${\widehat{\Phi}} \colon {\mathfrak{M}}_1 \to {\mathfrak{M}}_2$. \hfill $\Box$
\end{prop}
We next consider the applications of these ideas to proving that two minimal foliated Cantor bundles are homeomorphic.
Assume we are given, for $i = 1,2$, minimal foliated Cantor bundles ${\mathfrak{M}}_{{\varphi}_i}$. Let $B_i$ denote the associated base manifolds, with basepoint $b_i \in B_i$, $\Lambda_i = \pi_1(B_i,b_i)$, and representations ${\varphi}_i \colon \Lambda \to \Homeo(\mathfrak{F}_i)$.
Assume also that ${\mathfrak{M}}_{{\varphi}_1}$ and ${\mathfrak{M}}_{{\varphi}_2}$ are return equivalent, so there exists clopen subsets $U_1 \subset \mathfrak{F}_1$ and $U_2 \subset \mathfrak{F}_2$ and a homeomorphism $\phi \colon U_1 \to U_2$ which conjugates ${\mathcal G}_{U_1}$ to ${\mathcal G}_{U_2}$. Assume that $U_1$ is collapsible. Then observe that the proof of Lemma~\ref{lem-collpasing} does not require the base manifolds be the same, so we conclude that $U_2$ is also collapsible.
Thus, for $i=1,2$, we can define the isotropy subgroups and their restricted actions
\begin{equation}
\Lambda_{U_i} = \left\{ \gamma \in \Lambda_i \mid {\varphi}_i(\gamma) \cdot U_i = U_i \right\} \quad , \quad {\varphi}_i \colon \Lambda_{U_i} \to \Homeo(U_i)
\end{equation}
and the homeomorphism $\phi$ induces a conjugation on the \emph{images} of these maps.
Note that each subgroup $\Lambda_{U_i} \subset \Lambda_i$ has finite index, though it need not be normal. Let $B_i'$ be the finite covering associated to the subgroup $\Lambda_{U_i} \subset \Lambda_i$.
\begin{defn}\label{def-commonbase}
Let ${\mathfrak{M}}_1$ and ${\mathfrak{M}}_2$ be return equivalent, minimal matchbox manifolds. We say that they have a \emph{common base} if there is a homeomorphism $\phi \colon U_1 \to U_2$ between clopen subsets which induces an isomorphism $\Phi \colon {\mathcal G}_{U_1} \to {\mathcal G}_{U_2}$, and there is a homeomorphism
$h \colon B_1' \to B_2'$ such that for the induced map on fundamental groups,
$\displaystyle h_{\#} \colon \Lambda_{U_1} = \pi_1(B_1' , b_1') \to \pi_1(B_2' , b_2') = \Lambda_{U_2}$
we have
\begin{equation}\label{eq-conjuagteholonomy}
{\varphi}_2(h_{\#}(\gamma)) \cdot \omega = \phi ( {\varphi}_1(\gamma) \cdot \phi^{-1}(\omega)) ~ , ~ {\rm for ~ all}~ \gamma \in \Lambda_1 ~, ~ \omega \in U_1 .
\end{equation}
\end{defn}
The following technical result is used in Section~\ref{sec-prohomotopy} to establish the ``common base'' hypothesis.
\begin{prop}\label{prop-conjugateactions}
Let $\pi_1 \colon {\mathfrak{M}}_1 \to B_1$ and ${\mathfrak{M}}_2$ be minimal foliated Cantor bundles, and suppose that there exist a simply connected leaf $L_2 \subset {\mathfrak{M}}_2$.
If ${\mathfrak{M}}_1$ and ${\mathfrak{M}}_2$ are return equivalent, clopen sets $U_i \subset \mathfrak{F}_i$ with $U_1$ collapsible, and a homeomorphism $\phi \colon U_1 \to U_2$ which induces an isomorphism $\Phi \colon {\mathcal G}_{U_1} \to {\mathcal G}_{U_2}$, then there exists an isomorphism on fundamental groups,
\begin{equation}\label{eq-conjuagteholonomy2e}
{\mathcal H}_{\phi} \colon \Lambda_{U_1} = \pi_1(B_1' , b_1') \to \pi_1(B_2' , b_2') = \Lambda_{U_2}
\end{equation}
such that
\begin{equation}\label{eq-conjuagteholonomy3e}
{\varphi}_2({\mathcal H}_{\phi}(\gamma)) \cdot \omega = \phi ( {\varphi}_1(\gamma) \cdot \phi^{-1}(\omega)) ~ , ~ {\rm for ~ all}~ \gamma \in \Lambda_{U_1} ~, ~ \omega \in U_2 .
\end{equation}
\end{prop}
\proof
We are given a representation $\displaystyle {\varphi}_2 \colon \Lambda_{U_2} \to \Homeo(U_2)$ whose image is the restricted pseudogroup ${\mathcal G}_{U_2}$.
Suppose that $\gamma \in \Lambda_{U_2}$ is mapped by ${\varphi}_2$ to the identity, then $\gamma$ defines a closed loop in $B_i$ which lifts to a closed loop in each leaf that intersects $U_2$. In particular, as $\F_2$ has all leaves dense, it defines a closed loop ${\widetilde{\gamma}} \subset L_2$. As $L_2$ is simply connected, the lift ${\widetilde{\gamma}}$ must be homotopic to a constant map. The restricted projection $\pi_2 \colon L_2 \to B_2$ is a covering map, so $\gamma$ is also homotopic to a constant, hence is the trivial element of $\Lambda_{U_2}$. Thus the map ${\varphi}_2$ is injective on $\Lambda_{U_2}$.
Now, use the conjugation defined by $\phi$ between the images ${\varphi}_1(\Lambda_{U_1})$ and ${\varphi}_2(\Lambda_{U_2})$ to define ${\mathcal H}_{\phi}$. Then the property \eqref{eq-conjuagteholonomy3e} holds by definition.
\endproof
\section{Equicontinuous matchbox manifolds}\label{sec-equicontinuous}
The dynamics and topology of equicontinuous matchbox manifolds are studied in the work \cite{ClarkHurder2013} by the first two authors. We recall three main results from this paper, which will be used in the proof of Theorem~\ref{thm-main2}. Theorem~\ref{thm-equiconjugation} below may be of interest on its own.
Recall that a matchbox manifold ${\mathfrak{M}}$ is equicontinuous, as stated in Definition~\ref{def-equicontinuous}, if the action of $\cGF$ on the transversal space ${\mathfrak{X}}_*$ is equicontinuous for the metric $d_{{\mathfrak{X}}}$.
\begin{thm}[Theorem~4.2, \cite{ClarkHurder2013}] \label{thm-equiminimal}
An equicontinuous matchbox manifold ${\mathfrak{M}}$ is minimal.
\end{thm}
The next result is a direct consequence of Theorem~8.9 of \cite{ClarkHurder2013}:
\begin{thm}\label{thm-mCb}
Let ${\mathfrak{M}}$ be an equicontinuous matchbox manifold. Then ${\mathfrak{M}}$ is homeomorphic to a minimal foliated Cantor bundle. That is, there exists a Cantor space $\mathfrak{F}_0$, a compact triangulated topological manifold $B$ with basepoint $b_0 \in B$, fundamental group $\Lambda = \pi_1(B,b_0)$, and representation ${\varphi} \colon \Lambda \to \Homeo(\mathfrak{F}_0)$ such that ${\mathfrak{M}} \cong {\mathfrak{M}}_{{\varphi}}$. Moreover, there is a metric $d_{\mathfrak{F}_0}$ on $\mathfrak{F}_0$ such that the minimal action of ${\varphi}$ is equicontinuous with respect to $d_{\mathfrak{F}_0}$.
\end{thm}
The third main result follows from Theorem~8.9 of \cite{ClarkHurder2013} and its proof:
\begin{thm}\label{thm-invdomains}
Let ${\mathfrak{M}}_{{\varphi}}$ be a suspension matchbox manifold, whose global holonomy is a minimal action ${\varphi} \colon \Lambda \to \Homeo(\mathfrak{F}_0)$ which is equicontinuous with respect to the metric $d_{\mathfrak{F}_0}$ on $\mathfrak{F}_0$. Then for any open set $W \subset \mathfrak{F}_0$, there exists a sequence of clopen sets $U_i \subset W$ with $U_{i+1} \subset U_i$ for all $i \geq 1$, such that
the translates $\{ {\varphi}(\gamma) \cdot U_i \mid \gamma \in \Lambda \}$ form a finite covering of $\mathfrak{F}_0$ by disjoint clopen subsets. Moreover, for
$\displaystyle
\e_{i} = \max \left\{ \diam_{\mathfrak{F}_0}\, \{ {\varphi}(\gamma) \cdot U_i \} \mid \gamma \in \Lambda \right\}
$
we have $\displaystyle \lim_{i \to \infty} ~ \e_i = 0$.
\end{thm}
Proposition~\ref{prop-partitions} and Theorem~\ref{thm-invdomains} then imply:
\begin{cor}\label{cor-invdomains}
Let ${\mathfrak{M}}_{{\varphi}}$ be a suspension matchbox manifold, whose global holonomy is a minimal action ${\varphi} \colon \Lambda \to \Homeo(\mathfrak{F}_0)$ which is equicontinuous with respect to the metric $d_{\mathfrak{F}_0}$ on $\mathfrak{F}_0$. Then ${\mathfrak{M}}_{{\varphi}}$ is infinitely collapsible.
\end{cor}
The main result of \cite{ClarkHurder2013} follows from a combination of these results:
\begin{thm}[Theorem~1.4, \cite{ClarkHurder2013}] \label{thm-solenoid}
Let ${\mathfrak{M}}$ be an equicontinuous matchbox manifold. Then there exists a sequence of closed triangulated topological manifolds and triangulated covering maps,
$\displaystyle \{ q_{\ell +1} \colon B_{\ell +1} \to B_{\ell} \mid \ell \geq 0\}$ such that ${\mathfrak{M}}$ is homeomorphic to the inverse limit of this system of maps
\begin{equation}\label{eq-eqpresentation}
{\mathfrak{M}} ~ \homeo ~ \underleftarrow{\lim} \left\{ \, q_{\ell+1} \colon B_{\ell+1} \to B_\ell \mid \ell \geq 0 \right\} .
\end{equation}
\end{thm}
That is, an equicontinuous matchbox manifold ${\mathfrak{M}}$ is foliated homeomorphic to a \emph{weak solenoid} in the sense of McCord \cite{McCord1965} and Schori \cite{Schori1966}.
\medskip
\begin{thm}\label{thm-equiconjugation}
Let ${\mathfrak{M}}_1$ and ${\mathfrak{M}}_2$ be equicontinuous matchbox manifolds, which are return equivalent with a common base. Then there is a homeomorphism ${\widehat{\Phi}} \colon {\mathfrak{M}}_1 \to {\mathfrak{M}}_2$. \hfill $\Box$
\end{thm}
\proof
By Theorem~\ref{thm-equiminimal}, ${\mathfrak{M}}_1$ and ${\mathfrak{M}}_2$ are minimal. Theorem~\ref{thm-mCb} implies there are homeomorphisms ${\mathfrak{M}}_i \cong {\mathfrak{M}}_{{\varphi}_i}$ where for $i=1,2$, ${\mathfrak{M}}_{{\varphi}_i}$ is a foliated Cantor bundle with notation as in Theorem~\ref{thm-mCb}. Then by Corollary~\ref{cor-invdomains} each ${\mathfrak{M}}_{{\varphi}_i}$ is infinitely collapsible.
Finally, Propositions~\ref{prop-changeofbase} and \ref{prop-conjugates} imply that the local conjugacy $\phi$ over the common base induces a homeomorphism ${\widehat{\Phi}} \colon {\mathfrak{M}}_1 \to {\mathfrak{M}}_2$.
\endproof
\section{Shape and the common base}\label{sec-prohomotopy}
In this section, we obtain general conditions on equicontinuous matchbox manifolds ${\mathfrak{M}}_1$ and ${\mathfrak{M}}_2$ such that return equivalence implies that they have a common base. First, we introduce a generalization of the notion of $Y$-like, and introduce the notion of an \emph{aspherical} matchbox manifold.
\begin{defn} \label{def-Clike}
Let ${\mathcal C}$ denote a collection of compact metric spaces.
A metric space $X$ is said to \emph{${\mathcal C}$--like} if for every $\e>0$, there exists $Y \in {\mathcal C}$ and a continuous surjection $f_Y \colon X \to Y$ such that the fiber $f_Y^{-1}(y)$ of each point $y\in Y$ has diameter less than $\e$.
\end{defn}
If the collection ${\mathcal C} = \{Y\}$ is a single space, then Definition~\ref{def-Clike} reduces to Definition~\ref{def-Ylike}.
Marde\v{s}i\'{c} and Segal show in Theorem $1^\ast$ of \cite{MardesicSegal1963} the following key result:
\begin{thm}\label{thm-MardSeg}
Let ${\mathcal C}$ be a given class of finite polyhedra, and let $X$ be a continuum. Then $X$ is ${\mathcal C}$--like if and only if $X$ admits a presentation as an inverse limit
$X\; \homeo \;\underleftarrow{\lim}\,\{\, q_{\ell+1} \colon Y_{\ell+1} \to Y_\ell \mid \ell \geq 0\}$
in which the bonding maps $q_\ell$ are continuous surjections, and $Y_\ell \in {\mathcal C}$ for all $\ell$.
\end{thm}
Observe that in this result, the only conclusion about the bonding maps $q_\ell$ is that they are continuous surjections, and in general they satisfy no other conditions. In particular, they need not be coverings.
We apply Theorem~\ref{thm-MardSeg} to the collection ${\mathcal A}$ which consists of $CW$-complexes which are aspherical. Recall that a $CW$-complex $Y$ is \emph{aspherical} if it is connected, and $\pi_n(Y)$ is trivial for all $n\geq 2$. Equivalently, $Y$ is aspherical if it is connected and its universal covering space is contractible.
Note that if $Y$ is aspherical, then every finite covering $Y'$ of $Y$ is also aspherical.
Our first result concerns the presentations of weak solenoids which are ${\mathcal A}$--like.
\begin{prop}\label{prop-aspher}
Let ${\mathfrak{M}}$ is a matchbox manifold which is homeomorphic to a weak solenoid of dimension $n \geq 1$.
Assume that ${\mathfrak{M}}$ is ${\mathcal A}$--like, then ${\mathfrak{M}}$ admits a presentation
\begin{equation}\label{eq-aspherpresentation}
{\mathfrak{M}}\; \homeo \;\underleftarrow{\lim}\, \{\, q_{\ell+1} \colon B_{\ell+1} \to B_\ell \mid \ell \geq 0\}
\end{equation}
in which each bonding map $q_\ell$ is a covering map, and each $B_\ell$ is a closed aspherical $n$-manifold.
\end{prop}
\begin{proof}
We are given that ${\mathfrak{M}}$ admits a presentation as in \eqref{eq-aspherpresentation}, in which each bonding map $q_\ell$ is a covering map, and each $B_\ell$ is a closed $n$-manifold. We show that each $B_\ell$ is aspherical. It suffices to show that for some $\ell \geq 0$, the universal covering $\widetilde{Y_{\ell}}$ is contractible.
By Theorem~\ref{thm-MardSeg}, there is a presentation
\begin{equation}\label{eq-Apresentation}
{\mathfrak{M}}\; \homeo \;\underleftarrow{\lim}\,\{\, r_{\ell+1} \colon A_{\ell+1} \to A_{\ell} \mid \ell \geq 0\}
\end{equation}
in which each map $r_{\ell}$ is a continuous surjection, and $A_\ell \in {\mathcal A}$ for all $\ell$.
Thus, using the notation ${\mathfrak{M}}_1\myeq \underleftarrow{\lim}\,\{\, q_{\ell+1} \colon B_{\ell+1} \to B_\ell \mid \ell \geq 0\}$ and ${\mathfrak{M}}_2 \myeq \underleftarrow{\lim}\,\{\, r_{\ell+1} \colon A_{\ell+1} \to A_\ell \mid \ell \in {\mathbb N}_0\}$, we have two homeomorphisms $h_1\colon {\mathfrak{M}} \to {\mathfrak{M}}_1$ and $h_2 \colon {\mathfrak{M}} \to {\mathfrak{M}}_2$.
Fix a base point $x \in {\mathfrak{M}}$, and set $x_i \myeq h_i (x)$ for $i=1,2$.
Consider the pro-groups homotopy groups, for $k \geq 1$, denoted by {\it pro}-$\pi_k({\mathfrak{M}},x)$ and {\it pro}-$\pi_k({\mathfrak{M}}_i,x)$.
For each $k \geq 1$, these groups are shape (and thus topological) invariants of the pointed spaces $({\mathfrak{M}}_i,x_i)$, as shown in \cite[Chapter II, Theorem 6]{MardesicSegal1982}. In fact, a map with homotopically trivial fibers induces isomorphisms of the pro--homotopy groups \cite{Dydak1975}. Thus, the homeomorphism $h_1 \circ h_2^{-1}$ induces isomorphisms of the corresponding pro--homotopy groups.
One can find a general treatment of pro-homotopy groups in \cite{MardesicSegal1982} or \cite{BousfieldKan1972}. For our purposes, what is important is that these pro-groups can be obtained from any shape expansion of a space, such as is provided by the above inverse limit presentations in \eqref{eq-aspherpresentation} and \eqref{eq-Apresentation}, and that isomorphisms of these towers have the form as described below.
By Theorem~3, Chapter 1 of \cite{MardesicSegal1982}, we can represent the isomorphism of pro-groups induced by a homeomorphism by a level morphism of isomorphic inverse sequences, in which the terms and bonding maps are derived from the original sequences.
By Morita's lemma (see Chapter II, Theorem~5 in \cite{MardesicSegal1982} or $\S$2.1, Chapter III in \cite{BousfieldKan1972}), this means that for each $k \geq 1$, there are subsequences $\displaystyle \{ i_{(k,\ell)} \mid \ell \geq 1\}$ and $\displaystyle \{ j_{(k,\ell)} \mid \ell \geq 1\}$,
such that for each $\ell \geq 1$ we have a commutative diagram of homomorphisms:
\[D_{(k,\ell)}:
\xymatrix{
\pi_k(A_{j_{(k,\ell)}},x_{(2,j_{(k,\ell)})})\ar[d]^{h_{(k,\ell)}}
& \pi_k(A_{j_{(k,s)}},x_{(2,j_{(k,s)})})\ar[d]^{h_{(k,s)}}\ar[l]
\\
\pi_k(B_{i_{(k,\ell)}},x_{(1,i_{(k,\ell)})})
& \pi_k(B_{i_{(k,s)}},x_{(1,i_{(k,s)})})\ar@{-->}[ul]^{g_{(k,s)}}\ar[l]
}\]
Here $x_{(i,j)}$ denotes the projection of $x_i$ in the $j$--th factor space of the inverse sequence for ${\mathfrak{M}}_i$ and each $s$ is some index greater than $\ell$ that depends on both $k$ and $\ell.$
The horizontal maps are the homomorphisms induced from the composition of corresponding bonding maps, and the labeled maps are those resulting from the isomorphisms of $\mathrm{pro}$--groups.
The bottom horizontal maps are injections since they result from covering maps, and hence each $g_{(k,s)}$ is also injective.
Thus, for $k>1$, the groups $\pi_k(B_{i_{(k,s)}},x_{(1,i_{(k,s)})})$ as above are isomorphic to a subgroup of the group $\pi_k(A_{j_{(k,\ell)}},x_{(2,j_{(k,\ell)})})$. By the definition of the class of spaces ${\mathcal A}$, each of these latter groups if trivial for $k > 1$, and thus the groups $\pi_k(B_{i_{(k,s)}},x_{(1,i_{(k,s)})})$ are trivial as well.
\bigskip
We now show that all the spaces in the sequence $B_\ell$ are aspherical. Consider the universal covering $p:(\widetilde{B_0},\widetilde{x_0})\to (B_0,x_{(1,0)}).$ We first show that $\widetilde{B_0}$ is contractible. By the above, for each $k>1$ we know that for some $s,$ we have that $\pi_k(B_{i_{(k,s)}},x_{(1,i_{(k,s)})})$ is trivial. We then have for each $k$ a commutative diagram of covering maps
\[
\xymatrix{
(\widetilde{B_0},\widetilde{x_0})\ar[d]_p\ar[rd]
&
\\
(B_0,x_{(1,0)})
& (B_{i_{(k,s)}},x_{(1,i_{(k,s)})})\ar[l]
}\]
where the horizontal map is the composition of corresponding bonding maps and the diagonal covering map results from the universal property of $p.$ Since covering maps induce monomorphisms of the corresponding homotopy groups, this shows that for each $k>1$, the group $\pi_k(\widetilde{B_0},\widetilde{x_0})$ factors through the trivial group and is therefore trivial. Since $(\widetilde{B_0},\widetilde{x_0})$ is the universal covering space of a connected $CW$-complex, we then can conclude that it is contractible. Thus $B_0$ is aspherical, and hence so is each covering space of $B_0,$ including each $B_\ell.$
\end{proof}
\begin{defn} \label{def-amm}
A matchbox manifold ${\mathfrak{M}}$ is \emph{aspherical} if {\it pro}-$\pi_k({\mathfrak{M}},x) =0$ for all $k > 1$.
\end{defn}
The proof of Proposition~\ref{prop-aspher} also shows the following.
\begin{prop}\label{prop-amm}
Let ${\mathfrak{M}}$ be a matchbox manifold which is ${\mathcal A}$--like, then ${\mathfrak{M}}$ is aspherical. \hfill $\Box$
\end{prop}
One of the important features of aspherical manifolds is given by the following standard result:
\begin{prop}\label{prop-homeq}
If two closed aspherical manifolds $M_1$ and $M_2$ have isomorphic fundamental groups, then the isomorphism induces a homotopy equivalence between $M_1$ and $M_2$.
\end{prop}
\proof
The proof follows from standard obstruction theory for $CW$-complexes, as described in the proof of Theorem 2.1 in \cite{Lueck2012} for example.
\endproof
For the rest of this section, we consider the problem of showing that we have a homeomorphism between the bases of presentations for foliated Cantor bundles ${\mathfrak{M}}_1$ and ${\mathfrak{M}}_2$. Proposition~\ref{prop-homeq} is used to construct a homotopy equivalence between two bases, which must then be shown to yield a homeomorphism. While this conclusion is in general not true, it does hold for the special class of strongly Borel manifolds introduced in Definition~\ref{def-borel}. First, we show:
\begin{thm}\label{thm-borelpres}
Let ${\mathcal A}_B$ be a Borel collection of closed manifolds. If ${\mathfrak{M}}$ is an equicontinuous ${\mathcal A}_B$--like matchbox manifold, then ${\mathfrak{M}}$ admits a presentation ${\mathfrak{M}}\; \homeo \;\underleftarrow{\lim}\{\, q_{\ell+1}:B_{\ell+1} \to b_{\ell} \mid \ell \geq 0\}$ in which each bonding map $q_\ell$ is a covering map and $B_{\ell}\in {\mathcal A}_B$ for all $\ell.$
\end{thm}
\begin{proof} By Theorem~\ref{thm-solenoid}, the equicontinuous matchbox manifold ${\mathfrak{M}}$ admits a presentation \eqref{eq-eqpresentation} in which each bonding map $q_{\ell+1}$ is a covering map, and each factor space $B_\ell$ is a closed manifold. We shall show that each closed manifold $B_\ell$ is an element of ${\mathcal A}_B.$
Note that by Proposition~\ref{prop-aspher} and condition 1) in Definition~\ref{def-borel}, each $B_\ell$ in this presentation is aspherical.
Now consider the diagrams $D_{(1,\ell)}$ as in the proof of Proposition~\ref{prop-aspher}. The diagram implies that for some $s,$ $\pi_1(B_{i_s},x_{(1,i_s)})$ is isomorphic to a finite indexed subgroup of $\pi_1(A_{j_{(k,\ell)}},x_{(2,j_{(k,\ell)})})$ since the bottom horizontal map is an isomorphism onto a subgroup of $\pi_1(B_{i_{(k,\ell)}},x_{(1,i_{(k,\ell)})})$ of finite index. Therefore, by the classification of covering spaces, $\pi_1(B_{i_s},x_{(1,i_s)})$ is isomorphic to the fundamental group of a finite covering space of $A_{j_{(k,\ell)}}$. By conditions 2) and 3) in the definition of Borel collection, we can conclude that $B_{i_s}$ is homeomorphic to some element in ${\mathcal A}_B.$ By condition 2), we can conclude that for all $\ell\geq i_s,$ $M_\ell$ is homeomorphic to an element of ${\mathcal A}_B.$ Thus by truncating the terms before $i_s$ and replacing each $B_\ell$ for $\ell \geq i_s$ with a homeomorphic element of ${\mathcal A}_B$ and adjusting the bonding maps accordingly, we obtain the desired presentation.
\end{proof}
Using the observation that ${\mathcal A}_B=\langle{\mathbb T}^n\rangle$ is a Borel collection, we immediately obtain:
\begin{cor}\label{cor-toruspres}
If ${\mathfrak{M}}$ is an equicontinuous ${\mathbb T}^n$--like matchbox manifold, then ${\mathfrak{M}}$ admits a presentation
${\mathfrak{M}}\; \homeo \;\underleftarrow{\lim}\{\, q_{\ell+1}:{\mathbb T}^n \to {\mathbb T}^n \mid \ell \geq 0\}$ in which each bonding map $q_\ell$ is a covering map.
\end{cor}
Finally, we use the results shown previously to give the proofs of Theorems~\ref{thm-main2} and \ref{thm-main3}.
First, note that if ${\mathfrak{M}}_1$ and ${\mathfrak{M}}_2$ are homeomorphic, then they are return equivalent by Theorem~\ref{thm-topinv}, so it suffices to show the converse.
Assume that ${\mathfrak{M}}_1$ and ${\mathfrak{M}}_2$ are equicontinuous. Then Corollary~\ref{cor-invdomains} implies that both are infinitely collapsible, and Theorem~\ref{thm-solenoid}
implies there is a presentation for each as in \eqref{eq-eqpresentation}, which we label as:
\begin{eqnarray}
{\mathfrak{M}}_1 ~& \homeo & ~ \underleftarrow{\lim} \left\{ \, q_{\ell+1}^1 \colon B_{\ell+1}^1 \to B_{\ell}^1 \mid \ell \geq 0 \right\} \label{eq-presentation1}\\
{\mathfrak{M}}_2 ~ & \homeo ~ & \underleftarrow{\lim} \left\{ \, q_{\ell+1}^2 \colon B_{\ell+1}^2 \to B_{\ell}^2 \mid \ell \geq 0 \right\} \label{eq-presentation2}.
\end{eqnarray}
For $i=1,2$, let $b_i \in B_0^i$ be basepoints, let $\mathfrak{F}_i \subset {\mathfrak{M}}_i$ be the fiber over $b_i$ and let $\Lambda_i = \pi_1(B_i , b_i)$ denote their fundamental groups.
Let ${\varphi}_i \colon \Lambda_i \to \Homeo(\mathfrak{F}_i)$ be the global holonomy of each presentation.
The assumption that ${\mathfrak{M}}_1$ and ${\mathfrak{M}}_2$ are return equivalent implies there exists clopen sets
$U_i \subset \mathfrak{F}_i$ and a homeomorphism $\phi \colon U_1 \to U_2$ which induces an isomorphism $\Phi \colon {\mathcal G}_{U_1} \to {\mathcal G}_{U_2}$.
By Theorem~\ref{thm-invdomains}, Lemma~\ref{lem-collpasing} and Proposition~\ref{prop-partitions}, we can assume that $U_1$ and $U_2$ are collapsible, and so are invariant under the action of the subgroups $\Lambda_{U_1} \subset \Lambda_1$ and $\Lambda_{U_2} \subset \Lambda_2$ as defined by \eqref{eq-isotropysubgroup}.
Then by Theorem~\ref{thm-equiconjugation}, it suffices to show these restricted actions have a common base.
For $i=1,2$, let $B_i'$ denote the finite covering of $B_i$ associated to the subgroup $\Lambda_i$. That is, by Definition~\ref{def-commonbase}, we must show there exists a homeomorphism
$h \colon B_1' \to B_2'$ such that for the induced map on fundamental groups,
$\displaystyle h_{\#} \colon \Lambda_{U_1} = \pi_1(B_1' , b_1') \to \pi_1(B_2' , b_2') = \Lambda_{U_2}$
we have
\begin{equation}\label{eq-conjuagteholonomy2}
{\varphi}_2(h_{\#}(\gamma)) \cdot \omega = \phi ( {\varphi}_1(\gamma) \cdot \phi^{-1}(\omega)) ~ , ~ {\rm for ~ all}~ \gamma \in \Lambda_1 ~, ~ \omega \in U_1 .
\end{equation}
The idea is that we show the existence of a map
\begin{equation}\label{eq-conjuagteholonomy3}
{\mathcal H}_{\phi} \colon \Lambda_{U_1} = \pi_1(B_1' , b_1') \to \pi_1(B_2' , b_2') = \Lambda_{U_2}
\end{equation}
so that \eqref{eq-conjuagteholonomy2} holds for $h_{\#} = {\mathcal H}_{\phi}$, and then construct the homeomorphism $h$. To implement this, we require the assumption that ${\mathfrak{M}}_1$ is $Y$-like, for an appropriate choice of $Y$.
\subsection{Proof of Theorem~\ref{thm-main2}} We are given that ${\mathfrak{M}}_1$ and ${\mathfrak{M}}_2$ are ${\mathbb T}^n$-like, where $n \geq 1$ is the dimension of the leaves of $\F_i$. By Corollary~\ref{cor-toruspres}, each ${\mathfrak{M}}_i$ then admits a presentation as in \eqref{eq-presentation1} and \eqref{eq-presentation2}, where $B_{\ell}^i = {\mathbb T}^n$ for $\ell \geq 0$.
For $i =1,2$, introduce $\displaystyle {\mathcal K}_i = ker \{ {\varphi}_i \colon \Lambda_{U_i} \to \Homeo(U_i) \} \subset \Lambda_{U_i} \cong {\mathbb Z}^n$. For simplicity of notation, identify $\Lambda_{U_i} = {\mathbb Z}^n$. As ${\mathbb Z}^n$ is free abelian, ${\mathcal K}_i$ is a free abelian subgroup with rank $0 \leq r_i < n$.
The quotient ${\mathbb Z}^n/{\mathcal K}_i$ is abelian. Let ${\mathcal A}_i \subset {\mathbb Z}^n/{\mathcal K}_i$ denote the subgroup of torsion elements.
By the structure theory of abelian groups, ${\mathcal A}_i$ is an interior direct sum of cyclic subgroups, so there exists elements $\{a_1^i, \ldots , a_{d_i}^i\} \subset {\mathbb Z}^n$ whose images in ${\mathbb Z}^n/{\mathcal K}_i$ form a minimal basis for ${\mathcal A}_i$. Observe that $0 \leq d_i \leq r_i$.
The conjugacy $\phi$ maps torsion elements in the image ${\varphi}_1(\Lambda_{U_1}) \subset \Homeo(U_1)$ to torsion elements of ${\varphi}_2(\Lambda_{U_2}) \subset \Homeo(U_2)$, so ${\mathbb Z}^n/{\mathcal K}_1 \cong {\mathbb Z}^n/{\mathcal K}_2$ and thus $d_1 = d_2$ hence $r_1 = r_2$.
Define the group isomorphism $\displaystyle {\mathcal H}_{\phi} \colon {\mathbb Z}^n \to {\mathbb Z}^n$ by defining its value on bases of the domain and range as follows.
First, we can assume without loss of generality that ${\varphi}_1(a_{\ell}^1)$ and ${\varphi}_2(a_{\ell}^2)$ generate isomorphic cyclic subgroups under the conjugacy $\phi$, for $1 \leq \ell \leq d_i$. Then set ${\mathcal H}_{\phi}(a_{i}^1) = a_{i}^2$.
Next, for each $1=1,2$, choose elements $\{a_{d_i+1}^i, \ldots , a_{r_i}^i\} \subset {\mathcal K}_i \subset {\mathbb Z}^n$ which span the complement in ${\mathcal K}_i$ of the subgroup $\langle a_1^i, \ldots , a_{d_i}^i \rangle \cap {\mathcal K}_i$ generated by the torsion generators.
Then the span $\langle a_1^i, \ldots , a_{r_i}^i \rangle$ is a subgroup of ${\mathbb Z}^n$.
Note that the action ${\varphi}_i(a_{\ell}^i)$ is trivial for $d_i < \ell \leq r_i$, and we set
${\mathcal H}_{\phi}(a_{i}^1) = a_{i}^2$.
Finally, note that the quotient of ${\mathbb Z}^n$ by $\langle a_1^i, \ldots , a_{r_i}^i \rangle$ is free abelian with rank $n - r_i$.
As the quotient is free, each set $\{ a_1^i, \ldots , a_{r_i}^i \}$ admits an extension to a basis
of ${\mathbb Z}^n$. First, choose an extension for $i=1$, say $\{ a_1^1, \ldots , a_{n}^1 \}$. Then for $r_i < \ell \leq n$, choose $a_{\ell}^2 \in {\mathbb Z}^n$ so that
$\displaystyle {\varphi}_2(a_{\ell}^2) = \phi \circ {\varphi}_1(a_{\ell}^1)$. Then set
$\displaystyle {\mathcal H}_{\phi}(a_{\ell}^1) = a_{\ell}^2$. It follows from our choices that the map ${\mathcal H}_{\phi} \colon {\mathbb Z}^n \to {\mathbb Z}^n$ so defined satisfies the condition \eqref{eq-conjuagteholonomy2} holds for $h_{\#} = {\mathcal H}_{\phi}$.
Finally, the map ${\mathcal H}_{\phi}$ defines a linear map $\widehat{{\mathcal H}_{\phi}} \colon {\mathbb R}^n \to {\mathbb R}^n$, and so induces a diffeomorphism of the quotient spaces, $h \colon {\mathbb T}^n \to {\mathbb T}^n$ so that condition \eqref{eq-conjuagteholonomy2} holds. Thus, we have shown that the presentations \eqref{eq-presentation1} and \eqref{eq-presentation2} have a common base. Theorem~\ref{thm-main2} then follows from Theorem~\ref{thm-equiconjugation}.
\medskip
\subsection{Proof of Theorem~\ref{thm-main3}} We are given that $Y$ is strongly Borel, and ${\mathfrak{M}}$ is equicontinuous and $Y$-like.
In addition, it is assumed that each of ${\mathfrak{M}}_1$ and ${\mathfrak{M}}_2$ have a leaf which is simply connected. By Theorem~\ref{thm-equiminimal} each leaf is dense, so in particular, every transversal clopen set intersects a leaf with trivial fundamental group.
By the proof of Theorem~\ref{thm-borelpres}, each of ${\mathfrak{M}}_1$ and ${\mathfrak{M}}_2$ admits a presentation in which each bonding map is a finite covering map.
The assumption that ${\mathfrak{M}}_1$ and ${\mathfrak{M}}_2$ which are return equivalent, implies by Proposition~\ref{prop-conjugateactions} that there is an induced map ${\mathcal H}_{\phi} \colon \Lambda_{U_1} = \pi_1(B_1' , b_1') \to \pi_1(B_2' , b_2') = \Lambda_{U_2}$ such that \eqref{eq-conjuagteholonomy3}holds. It follows from Proposition~\ref{prop-homeq} that
for the covering $B_1' \to B_1$ associated to $\Lambda_{U_1} \subset \Lambda_1$ and the covering $B_2' \to B_2$ associated to $\Lambda_{U_2} \subset \Lambda_2$,
the isomorphism ${\mathcal H}_{\phi}$ on fundamental groups induces a homotopy equivalence $\widehat{h} \colon B_1' \to B_2'$ such that $\widehat{h}_{\#} = {\mathcal H}_{\phi}$. Each of the manifolds $B_1'$ and $B_2'$ can be assumed to be coverings of $Y$. As $Y$ is assumed to be strongly Borel, the homotopy equivalence induces a homeomorphism $h$ such that \eqref{eq-conjuagteholonomy2} is satisfied. That is, they have a common base, so it follows from Theorem~\ref{thm-solenoid} that ${\mathfrak{M}}_1$ and ${\mathfrak{M}}_2$ are homeomorphic. This proves the claim of Theorem~\ref{thm-main3}.
\medskip
\begin{remark}
{\rm
Given the choice of the clopen sets $U_1$ and $U_2$ in the above proofs, these sets are infinitely collapsible, so by refinement, we can assume that the conjugacy $\phi$ is induced on an arbitrary covering of ${\mathbb T}^n$ for Theorems~\ref{thm-main2}, or $Y$ for Theorems~\ref{thm-main3}. As remarked in \cite{Davis2012}, the homeomorphism $h$ that is obtained from the solutions of the Borel Conjecture can be assume to be smooth for a sufficiently large finite covering. Thus, we conclude that the homeomorphism $\Phi \colon {\mathfrak{M}}_1 \to {\mathfrak{M}}_2$ obtained above can be chosen to be smooth along leaves.
}
\end{remark}
\vfill
\eject
\section{Examples and counter-examples}\label{sec-examples}
In this section, we give applications of the results in the previous sections. First, we describe a general construction of examples of equicontinuous matchbox manifolds for which the hypotheses of Theorem~\ref{thm-main3} are satisfied.
These constructions are based on the notion of \emph{non-co-Hopfian} manifolds, which is closely related to the $Y$-like property of Definition~\ref{def-Ylike}. Using these ideas, it is then clear how to construct classes of examples of equicontinuous matchbox manifolds for which return equivalence does not imply homeomorphism, as the $Y$-like hypothesis Theorem~\ref{thm-main3} is not satisfied.
Recall that a group $G$ is \emph{co-Hopfian} if there does not exist an embedding of $G$ to a proper subgroup of itself, and \emph{non-co-Hopfian} otherwise. A closed manifold $Y$ is co-Hopfian if every covering map $\pi \colon Y \to Y$ is a diffeomorphism, and non-co-Hopfian if $Y$ admits proper self-coverings. Clearly, $Y$ is co-Hopfian if and only if its fundamental group is co-Hopfian.
The co-Hopfian concept for groups was first studied by Reinhold Baer in \cite{Baer1944}, where they are referred to as ``S-groups''.
More recently, the paper of Delgado and Timm \cite{DT2003} considers the co-Hopfian condition for the fundamental groups of connected finite complexes, and the paper by Endimioni and Robinson \cite{ER2005} gives some sufficient conditions for a group to be co-Hopfian or non-co-Hopfian. The paper by Belegradek \cite{Belegradek2003} considers which finitely-generated nilpotent groups are non-co-Hopfian.
A finitely generated infinite group $G$ is called \emph{scale-invariant} if there is a nested sequence of finite index subgroups $G_n$ that are all isomorphic to $G$ and whose intersection is a finite group.
The paper by Nekrashevych \cite{NP2011}
gives natural conditions for which the semi-direct product $G$ of a countable scale-invariant group $H$ with a countable automorphism group $A$ of $G$ is scale-invariant, providing classes of examples of non-co-Hopfian groups which do not have polynomial word growth.
The product $G = G_1 \times G_2$ of any group $G_1$ with a non-co-Hopfian group $G_2$ is again non-co-Hopfian, though it may happen that the product of two co-Hopfian groups is non-co-Hopfian \cite{Li2007}.
The paper by Ohshika and Potyagailo \cite{OP1998} gives examples of a freely indecomposable geometrically finite torsion-free non-elementary Kleinian group which are not co-Hopfian. The work of Delzant and Potyagailo \cite{DP2003} also studies which non-elementary geometrically finite Kleinian groups are co-Hopfian.
The question of which compact $3$-manifolds admit proper self-coverings has been studied in detail by Gonz{\'a}lez-Acu{\~n}a, Litherland and Whitten in the works \cite{GLiW1994} and \cite{GW1994}.
\begin{prop}\label{prop-ncH}
Let $G$ be a finitely generated, torsion-free group which admits a descending chain of groups $G_{\ell +1} \subset G_{\ell}$ each of finite index in $G$, whose intersection is the identity, and for some $\ell_0$ we have $G_{\ell}$ is isomorphic to $G_{\ell_0}$ for all $\ell > \ell_0$.
Let $B_0$ be a closed manifold whose fundamental group $G_0 = \pi(B_0, b_0)$ satisfies this condition.
Let $p_{\ell} \colon B_{\ell} \to B_0$ be the finite covering associated to the subgroup $G_{\ell}$, and set $Y = B_{\ell_0}$. Let $q_{\ell +1} \colon B_{\ell +1} \to B_{\ell}$ denote that covering induced by the inclusion $G_{\ell +1} \to G_{\ell}$.
Let ${\mathfrak{M}}$ denote the weak solenoid defined as the inverse limit of the sequence of maps $q_{\ell +1} \colon B_{\ell +1} \to B_{\ell}$ for $\ell \geq 0$, so
\begin{equation}\label{eq-nilpresentation2}
{\mathfrak{M}} ~ \equiv ~ \underleftarrow{\lim} \, \{\, q_{\ell+1} \colon B_{\ell +1} \to B_{\ell} \mid \ell \geq 0\} \quad \subset \quad \prod_{\ell \geq 0} ~ B_{\ell}.
\end{equation}
Then ${\mathfrak{M}}$ is an equicontinuous matchbox manifold which is $Y$-like, and each leaf of the foliation $\F$ on ${\mathfrak{M}}$ is simply-connected.
\end{prop}
\proof
Proposition~10.1 of \cite{ClarkHurder2013} shows that ${\mathfrak{M}}$ is an equicontinuous matchbox manifold.
For each $\ell \geq 0$, the definition of the inverse limit as a closed subset of the infinite product in \eqref{eq-nilpresentation2} yields projection maps onto the factors, $\pi_{\ell} \colon {\mathfrak{M}} \to B_{\ell}$. By the definition of the product metric topology, for all $b \in B_0$, the diameters of the fibers $\pi_{\ell}^{-1}(b)$ tend to zero as $\ell \to \infty$. Given that $Y \cong B_{\ell}$ for all $\ell \geq \ell_0$ it follows that ${\mathfrak{M}}$ is $Y$-like.
For a leaf $L \subset {\mathfrak{M}}$, its fundamental group is isomorphic to the intersection of the subgroups $G_{\ell} = \pi_1(B_{\ell}, b_{\ell})$ for $\ell \geq 0$, which is the trivial group by assumption.
\endproof
The proof of Proposition~\ref{prop-aspher} shows the close connection between the $Y$-like hypothesis and the non-co-Hopfian property for the fundamental groups in the presentation \eqref{eq-nilpresentation2}. In fact, the $Y$-like hypothesis on a solenoid is a type of homotopy version of the non-co-Hopfian property for manifolds.
\subsection{Examples for dimension $n =1$} The circle is the prototypical example of a non-co-Hopfian space, and Theorem~\ref{thm-main2} applies to the classical Vietoris solenoids with base ${\mathbb S}^1$. We examine this case in detail, recalling the classical classification of these spaces.
Let $\vec{m} = (m_1, m_2,\dots)$ denote a sequence of positive integers with each $m_i \geq 2$. Set $m_0=1$, then
there is then the corresponding profinite group
\begin{eqnarray}
\mathfrak{G}_{\vec{m}} &\myeq &\underleftarrow{\lim}~ \left\{ \, q_{\ell+1} \colon {\mathbb Z}/m_1\cdots m_{\ell+1}{\mathbb Z} \to {\mathbb Z}/m_0m_1\cdots m_{\ell}{\mathbb Z} \mid \ell \geq 1 \, \right\} \label{eq-Madicgroup} \\
&=&\underleftarrow{\lim}~ \left\{ {\mathbb Z}/{\mathbb Z} \xleftarrow{m_1} {\mathbb Z}/m_1{\mathbb Z} \xleftarrow{m_2} {\mathbb Z}/m_1m_2{\mathbb Z} \xleftarrow{m_3} {\mathbb Z}/m_1m_2m_3{\mathbb Z}\xleftarrow{m_4}\cdots \right\} \nonumber
\end{eqnarray}
where $q_{\ell+1}$ is the quotient map of degree $m_{\ell+1}$.
Each of the profinite groups $\mathfrak{G}_{\vec{m}}$ contains a copy of ${\mathbb Z}$ embedded as a dense subgroup by $z \to ([z]_0,[z]_1,...,[z]_k,...),$ where $[z]_k$ corresponds to the class of $z$ in the quotient group $\displaystyle {\mathbb Z}/m_0\cdots m_k{\mathbb Z}$. There is a homeomorphism $a_{\vec{m}} \colon \mathfrak{G}_{\vec{m}} \to \mathfrak{G}_{\vec{m}}$ given by ``addition of $1$'' in each finite factor group. The dynamics of $a_{\vec{m}}$ acting on $\mathfrak{G}_{\vec{m}}$ is referred to as an \emph{adding machine}, or equivalently as an \emph{odometer}.
For a given sequence $\vec{m}$ as above, there is a corresponding Vietoris solenoid
\begin{equation}
{\mathcal S}(\vec{m}) \myeq \underleftarrow{\lim} ~ \{\, p_{\ell+1}:{\mathbb S}^1 \to {\mathbb S}^1 \mid \ell \geq 0\}
\end{equation}
where $p_{\ell+1}$ is the covering map of ${\mathbb S}^1$ defined by multiplication of the covering space ${\mathbb R}$ by $m_{\ell+1}$.
It is well known that ${\mathcal S}(\vec{m})$ is homeomorphic to the suspension over ${\mathbb S}^1$ of the action by the map $a_{\vec{m}}$.
Let $\pi_{\vec{m}} \colon {\mathcal S}(\vec{m}) \to {\mathbb S}^1$ denote projection onto the first factor, then ${\mathcal S}(\vec{m})$ is isomorphic as a topological group to the subgroup $\ker(\pi_{\vec{m}})$ (for example, see \cite{AartsFokkink1991,McCord1965}.)
Accordingly, ${\mathcal S}(\vec{m}) $ is the total space of a principal $\mathfrak{G}_{\vec{m}}$--bundle $\xi_{\vec{m}} = ({\mathcal S}(\vec{m}) ,\pi_{\vec{m}},{\mathbb S}^1)$ over ${\mathbb S}^1$.
The solenoids ${\mathcal S}(\vec{m})$ are classified using the following function, as shown in \cite{BlockKeesling2004}.
\begin{defn}
Given a sequence of integers $\vec{m}$ as above, let $C_{\vec{m}}$ denote the function from the set of
prime numbers to the set of extended natural numbers $\{0, 1, 2, . . .,\infty\}$ given by
$$C_{\vec{m}}(p)=\sum_1^\infty p_i,$$
where $p_i$ is the power of the prime $p$ in the prime factorization of $m_i.$
\end{defn}
\begin{defn}\label{defn-MorEquvSequ}
Two sequences of integers $\vec{m}$ and $\vec{n}$ as above are \emph{return equivalent}, denoted $\vec{m} \, \mor \, \vec{n}$ if and only if the following two conditions hold:
\begin{enumerate}
\item For all but finitely many primes $p$, $C_{\vec{m}}(p)=C_{\vec{n}}(p)$ and
\item for all primes $p$, $C_{\vec{m}}(p)= \infty$ if and only if $C_{\vec{n}}(p)=\infty$.
\end{enumerate}
\end{defn}
The classical classification of the Vietoris solenoids up to homeomorphism then becomes:
\begin{thm}\cite{McCord1965,AartsFokkink1991,BlockKeesling2004}\label{thm-onedimSol}
The solenoids ${\mathcal S}(\vec{m}) $ and ${\mathcal S}(\vec{n}) $ are homeomorphic if and only if ~ $\vec{m} \, \mor \, \vec{n}$.
Thus, they are return equivalent if and only if ~ \, $\vec{m} \, \mor \, \vec{n}$.
\end{thm}
\subsection{Examples for dimension $n =2$}
The simplest examples of \emph{co-Hopfian} closed manifolds are the closed surfaces $\Sigma_g$ of genus $g \geq 2$. The surface $\Sigma_g$ has Euler characteristic $\chi(\Sigma_g) = 2-2g$, and the Euler characteristic is multiplicative for coverings. That is, if $\Sigma_g'$ is a $p$-fold covering of $\Sigma_g$ then $\chi(\Sigma_g') = p \cdot \chi(\Sigma_g)$. Thus, for $g > 1$, a proper covering $\Sigma_g'$ of $\Sigma_g$ is never homeomorphic to $\Sigma_g$. We use this remark to construct examples of weak solenoids with common base $\Sigma_2$ which are return equivalent but not homeomorphic.
Recall that the fundamental group of $\Sigma_g$ has the standard finite presentation, for basepoint $\mathbf{x_0} \in \Sigma_g$:
$$\pi_1(\Sigma_g,\mathbf{x_0})\simeq \left\langle\,
\alpha_1,\beta_1,\dots,\alpha_g,\beta_g \mid [\alpha_1\beta_1]\cdots[\alpha_g\beta_g]\,\right\rangle.$$
Define a homomorphism $h_0 \colon \pi_1(\Sigma_g,\mathbf{x_0}) \to {\mathbb Z}$ by setting $h_0(\alpha_1) = 1 \in {\mathbb Z}$, $h_0(\alpha_i) = 0$ for $1 < i \leq g$, and
$h_0(\beta_i) = 0$ for $1 \leq i \leq g$. Then $h_0$ is induced by a continuous map, again denoted $h_0 \colon \Sigma_g \to {\mathbb S}^1$, which maps $\mathbf{x_0}$ to the basepoint $\theta_0 \in {\mathbb S}^1$. We use the map $h_0$ to form induced minimal Cantor bundles over $\Sigma_g$ to obtain what we call \emph{$\vec{m}$--adic surfaces}, as defined in the following.
For a given sequence $\vec{m}$ as above, and orientable surface $\Sigma_g$ of genus $g\geq 1,$ define an action $A_{\vec{m}}$ of $\pi_1(\Sigma_g,\mathbf{x_0})$ on the Cantor set $\mathfrak{G}_{\vec{m}}$ by composing the homomorphism $h_0 \colon \pi_1(\Sigma_g,\mathbf{x_0}) \to {\mathbb Z}$ with the action $a_{\vec{m}}$ of ${\mathbb Z}$ on $\mathfrak{G}_{\vec{m}}$. Note that the induced representation $A_{\vec{m}} \colon \pi_1(\Sigma_g,\mathbf{x_0}) \to \Homeo(\mathfrak{G}_{\vec{m}})$ thus constructed is never injective.
\begin{defn}\label{def-madicsurface}
Given a closed, orientable surface $\Sigma_g$ of genus $g\geq 1$ and a sequence of integers $\vec{m}$ as above, the \emph{$\vec{m}$--\emph{adic} surface} ${\mathfrak{M}}(\Sigma_g , \vec{m})$ is the Cantor bundle defined by the suspension of the action $A_{\vec{m}}$ as in \eqref{eq-suspension}, with $B = \Sigma_g$ and $\mathfrak{F} = \mathfrak{G}_{\vec{m}}$.
As the action $a_{\vec{m}}$ is minimal, the matchbox manifold ${\mathfrak{M}}(\Sigma_g , \vec{m})$ is minimal.
\end{defn}
We next make some basic observations about the $\vec{m}$--\emph{adic} surfaces ${\mathfrak{M}}(\Sigma_g , \vec{m})$.
Recall that the homomorphism $h_0 \colon \pi_1(\Sigma_g,\mathbf{x_0}) \to {\mathbb Z}$ is induced by a topological map $h_0 \colon \Sigma_g \to {\mathbb S}^1$. Then by general bundle theory \cite{Husemoller1994,KamberTondeur1968}, the foliated Cantor bundle $\pi_* \colon {\mathfrak{M}}(\Sigma_g , \vec{m}) \to \Sigma_g$ is the pull-back of the Cantor bundle ${\mathcal S}(\vec{m}) \to {\mathbb S}^1$. The methods of Section~\ref{sec-bundles} then yield:
\begin{lemma}\label{lem-inducedMor}
Let ${\mathfrak{M}}$ be a minimal matchbox manifold ${\mathfrak{M}}$ that is the total space of foliated bundle $\eta=\{\pi_* \colon {\mathfrak{M}} \to B\}$.
Suppose that $f\colon B' \to B$ is a continuous map which induces a surjection of fundamental groups, where the dimensions of $B$ and $B'$ need not be the same.
Then the total space $f^*({\mathfrak{M}})$ of the induced bundle $f^*(\eta)$ over $B'$ is return equivalent to ${\mathfrak{M}}$.
\hfill $\Box$
\end{lemma}
\begin{cor}\label{cor-induced}
Given a closed, orientable surface $\Sigma_g$ of genus $g\geq 1$ and a sequence of integers $\vec{m}$ as above, then the minimal matchbox manifolds ${\mathcal S}(\vec{m})$ and ${\mathfrak{M}}(\Sigma_g , \vec{m})$ are return equivalent.
\end{cor}
The geometric meaning of Corollary~\ref{cor-induced} is that the restricted pseudogroup of the $\vec{m}$--\emph{adic} surface ${\mathfrak{M}}(\Sigma_g , \vec{m})$ does not ``see'' the trivial holonomy maps corresponding to loops in the base $\Sigma_g$ that represent the classes $\alpha_{i>1},\beta_j$.
Note that the dimensions of the leaves for ${\mathcal S}(\vec{m})$ and ${\mathfrak{M}}(\Sigma_g , \vec{m})$ differ, so they cannot possibly be homeomorphic.
We obtain examples with the same leaf dimensions by applying Lemma~\ref{lem-inducedMor}, Theorem~\ref{thm-onedimSol} and Proposition~\ref{prop-return} to obtain the following result.
\begin{cor}\label{cor-surfadicMor}
Given closed orientable surfaces $\Sigma_{g_1}$ and $\Sigma_{g_2}$ of genus $g_i \geq 1$ for $i=1,2$, and sequences $\vec{m}$ and $\vec{n}$, then
${\mathfrak{M}}(\Sigma_{g_1} , \vec{m})$ is return equivalent to ${\mathfrak{M}}(\Sigma_{g_2} , \vec{m})$ if and only if $\vec{m} \, \mor \, \vec{n}$.
\end{cor}
Corollary~\ref{cor-surfadicMor} poses the problem, given \emph{adic}-surfaces ${\mathfrak{M}}(\Sigma_{g_1} , \vec{m})$ and ${\mathfrak{M}}(\Sigma_{g_2} , \vec{n})$ with $\vec{m} \, \mor \, \vec{n}$, when are they homeomorphic as matchbox manifolds? First, consider the case of genus $g_1 = g_2=1$ so that $\displaystyle \Sigma_{g_1}= \Sigma_{g_2} = {\mathbb T}^2$. Then Theorem~\ref{thm-main2} and Corollary~\ref{cor-surfadicMor} yield:
\begin{thm}\label{thm-torusclass}
${\mathfrak{M}}({\mathbb T}^2, \vec{m})$ and ${\mathfrak{M}}({\mathbb T}^2, \vec{n})$ are homeomorphic if and only if $\vec{m} \, \mor\, \vec{n}$. \hfill $\Box$
\end{thm}
For the general case, where at least one base manifold has higher genus, we have:
\begin{thm}\label{thm-surfaces}
Let ${\mathfrak{M}}_1 = {\mathfrak{M}}(\Sigma_{g_1} , \vec{m})$ and ${\mathfrak{M}}_2 = {\mathfrak{M}}(\Sigma_{g_2} , \vec{n})$ be \emph{adic}-surfaces.
\begin{enumerate}
\item If $g_1 > 1$ and $g_2 = 1$, then ${\mathfrak{M}}_1$ and ${\mathfrak{M}}_2$ are never homeomorphic.
\item If $g_1 = g_2 > 1$, then ${\mathfrak{M}}_1$ and ${\mathfrak{M}}_2$ are homeomorphic if and only if $C_{\vec{m}}=C_{\vec{n}}$.
\item If $g_1 = g_2 > 1$, then there exists $\vec{m}$, $\vec{n}$ such that $\vec{m} \, \mor \, \vec{n}$, but ${\mathfrak{M}}_1 \not\approx {\mathfrak{M}}_2$.
\end{enumerate}
\end{thm}
\begin{proof}
First, consider the case where $g = g_1 = g_2 > 1$ and $C_{\vec{m}}=C_{\vec{n}}$. Then the Cantor bundles $\pi_{\vec{m}} \colon {\mathcal S}(\vec{m}) \to {\mathbb S}^1$ and $\pi_{\vec{n}} \colon {\mathcal S}(\vec{n}) \to {\mathbb S}^1$ are homeomorphic as bundles over ${\mathbb S}^1$ (see \cite[Corollary 2.8]{BlockKeesling2004}) and therefore their pull-back bundles under the map $h_0 \colon \Sigma_g \to {\mathbb S}^1$ are homeomorphic as bundles over $\Sigma_g$ which is a stronger conclusion than the statement that ${\mathfrak{M}}_1$ and ${\mathfrak{M}}_2$ are homeomorphic.
For the proofs of parts 1) and 3) and also to show the converse conclusion in 2), assume there is a homeomorphism ${\mathcal H} \colon {\mathfrak{M}}_1 \to {\mathfrak{M}}_2$.
By the results of Rogers and Tollefson in \cite{RT1971,RT1972}, the map ${\mathcal H}$ is homotopic to a homeomorphism ${\widetilde{\mathcal H}}$ which is induced by a map of the inverse limit representations of ${\mathfrak{M}}_1$ and ${\mathfrak{M}}_2$ as in \eqref{eq-nilpresentation2}. Let $X_j \equiv {\mathfrak{M}}(\Sigma_{g_1} , \vec{m}; j)$ denote the $j-th$ stage in \eqref{eq-nilpresentation2} of the inverse limit representation for ${\mathfrak{M}}(\Sigma_{g_1} , \vec{m})$, and similarly set $Y_j \equiv {\mathfrak{M}}(\Sigma_{g_2} , \vec{n}, j)$.
Then there exists an increasing integer-valued function $k \to \ell_k$ for $k \geq 0$, and covering maps ${\widetilde{\mathcal H}}_{k} \colon X_{\ell_k} \to Y_k$ where
the collection of maps $\{{\widetilde{\mathcal H}}_{k} \mid k \geq k_0\}$ form a commutative diagram:
\begin{equation}\label{eq-commutingdiagram}
\xymatrix{
X_{\ell_0} \ar[d]_{{\widetilde{\mathcal H}}_0}
& X_{\ell_1} \ar[l]_{f^{n_2}_{n_1}}\ar[d]^{\widetilde{h_1}}
& \cdots\ar[l]
& X_{\ell_k}\ar[d]^{{\widetilde{\mathcal H}}_k}\ar[l]
& X_{\ell_{k+1}} \ar[l]_{f^{n_{k+1}-1}_{n_k+1}}\ar[d]^{{\widetilde{\mathcal H}}{k+1}}
& \cdots\ar[l]\\
Y_0
& Y_1\ar[l]^{g_1}
&\cdots\ar[l]
& Y_k\ar[l]
& Y_{k+1} \ar[l]^{g_k}
& \cdots\ar[l]}
\end{equation}
where the $f_k$ and $g_k$ are the bonding maps in the inverse limit representation of ${\mathfrak{M}}_1$ and ${\mathfrak{M}}_2$ and $f^{n_{k+1}-1}_{n_k+1}$ denotes the corresponding composition of bonding maps $f_k$.
Note that all of the maps in the diagram \eqref{eq-commutingdiagram} are covering maps by construction. Thus, the Euler classes of all surfaces there are related by the covering degrees. For example, $\chi(X_{\ell_k}) = d_k \cdot \chi(Y_k)$ where $d_k$ is the covering degree of ${\widetilde{\mathcal H}}_k$.
To show 1) we assume that a homeomorphism ${\mathcal H}$ exists, and so we have diagram \eqref{eq-commutingdiagram} as above. Observe that $g_2 = 1$ implies that $\chi(\Sigma_2) = \chi({\mathbb T}^2) = 0$, hence the covering $\chi(Y_k) = 0$ for all $k \geq 0$. Then as $d_k \geq 1$ for all $k$,
we obtain $\chi(X_{\ell_k}) = 0$. But this contradicts the assumption that $g_1 > 1$ hence $\chi(X_{\ell_k}) < 0$ as $X_{\ell_k}$ is a covering of $\Sigma_1$ which has $\chi(\Sigma_1) < 0$. Thus ${\mathfrak{M}}_1 \not\approx {\mathfrak{M}}_2$.
To show the converse in 2) assume that a homeomorphism ${\mathcal H}$ exists, and suppose that for some prime $p$ we have $C_{\vec{m}}(p) \ne C_{\vec{n}}(p)$.
We assume without loss of generality that $C_{\vec{m}}(p) < C_{\vec{n}}(p)$. Then as $\chi(\Sigma_1) = \chi(\Sigma_2)$, for sufficiently large $k$ the prime factorization of the Euler characteristic $\chi(X_{\ell_k})$ contains a lower power of $p$ than the prime factorization of $\chi(Y_k)$.
But this contradicts that $\chi(X_{\ell_k}) =d_k \cdot \chi(Y_k)$ where $d_k$ is the covering degree of ${\widetilde{\mathcal H}}_k$.
Finally, to show 3) let $\Sigma = \Sigma_{g_1} = \Sigma_{g_2}$ where $g = g_1 = g_2 > 1$.
It suffices to define $\vec{m}$, $\vec{n}$ such that $\vec{m} \, \mor \, \vec{n}$, but $C_{\vec{m}} \ne C_{\vec{n}}$. It then follows from 2) that ${\mathfrak{M}}_1 \not\approx {\mathfrak{M}}_2$. Pick a prime $p_1 \geq 3$ and let $\vec{m}$ be any sequence such that $C_{\vec{m}}(p_1) = 0$. Then define $\vec{n}$ by setting
$n_1 = p_1$ and $n_{k+1} = m_k$ for all $k \geq 1$.
Note that $C_{\vec{m}}(p_1) = 0 \ne 1 = C_{\vec{n}}(p_1)$ so $C_{\vec{m}}(p) \ne C_{\vec{n}}(p)$ is satisfied. But clearly $\vec{m} \, \mor \, \vec{n}$, so the \emph{adic}-surfaces ${\mathfrak{M}}(\Sigma_{g} , \vec{m})$ and ${\mathfrak{M}}(\Sigma_{g} , \vec{n})$ are return equivalent by Corollary~\ref{cor-surfadicMor}, but are not homeomorphic by
part 2) above.
\end{proof}
\medskip
\begin{remark}
{\rm
The results of Theorem~\ref{thm-surfaces} are restricted to the case of the \emph{adic}-surfaces introduced in Definition~\ref{def-madicsurface}, which are inverse limits defined by a system of subgroups of finite index of $\Lambda = \pi_1(\Sigma_g,\mathbf{x_0})$ associated to the choice of the
homomorphism $h_0 \colon \pi_1(\Sigma_g,\mathbf{x_0}) \to {\mathbb Z}$ and the sequence of integers $\vec{m}$. The proof of Theorem~\ref{thm-surfaces}
uses the classification results of the $1$-dimensional case in an essential manner.
There is a more general construction of $2$-dimensional equicontinuous matchbox manifolds ${\mathfrak{M}}(\Sigma_g, {\mathcal L})$ obtained from a given infinite, partially-ordered collection of subgroups of finite index. Set
${\mathcal L} \equiv \{\Lambda_i \subset \Lambda \mid i \in {\mathcal I}\}$ where each $\Lambda_i$ is a subgroup of finite index in $\Lambda$. The partial order on ${\mathcal L}$ is defined by setting where $\Lambda_i \lesssim \Lambda_j$ if $\Lambda_i \subset \Lambda_j$. Then ${\mathfrak{M}}(\Sigma_g, {\mathcal L})$ is the inverse limit of the finite coverings $\Sigma_{g,i} \to \Sigma_g$ associated to the subgroups in ${\mathcal L}$. In particular, let ${\mathcal L}^*$ denote the \emph{universal} partially ordered lattice of subgroups, which includes all subgroups of $\Lambda$ of finite index. The space ${\mathfrak{M}}(\Sigma_g, {\mathcal L}^*)$ was introduced by Sullivan in \cite{Sullivan1988}, where it was called the \emph{universal Riemann surface lamination}, and used in the study of conformal geometries for Riemann surfaces. The techniques of this paper give no insights to the classification up to homeomorphism of these spaces, and suggest that a deeper understanding of their homeomorphism types will require fundamentally new techniques.
}
\end{remark}
\vfill
\eject
\subsection{Examples for dimension $n \geq 3$}
We briefly discuss the homeomorphism problem for the case of $n$-dimensional equicontinuous matchbox manifolds, for $n \geq 3$. The discussion above of the examples of non-co-Hopfian groups shows there are many classes of closed $n$-manifolds which are non-co-Hopfian and not covered by the torus ${\mathbb T}^n$, and are also strongly Borel.
The papers \cite{GLiW1994,GW1994} apply especially to the case of closed $3$-manifolds, where it seems that some of the above results for $n=2$ can be extended to this case. The Euler characteristic of a closed $3$-manifold is always zero, so the method above will not apply directly, as it used the Euler characteristic of the closed manifolds appearing in the inverse representation to show the matchbox manifolds defined by the inverse systems are not homeomorphic. On the other hand,
the paper by Wang and Wu \cite{WangWu1994} gives invariants of coverings of $3$-manifolds which give obstructions to a proper covering being diffeomorphic to its base, so it is likely this can be used to show the inverse limits are not homeomorphic in an analogous manner.
\medskip
\begin{remark}
{\rm We also note that it follows from the results of \cite{ClarkHurder2011a} that given $\epsilon>0$, each ${\mathfrak{M}}(\Sigma_{g} , \vec{m})$ for $g \geq 1$ occurs as the minimal set of a $C^\infty$ $\epsilon$--perturbation of the product foliation of $\Sigma_g \times {\mathbb D}^2$, where ${\mathbb D}^2$ is the unit $2$-dimensional disk. Thus, the examples we construct above are topologically wild, but not necessarily pathological, as they can occur naturally in the study of the dynamics of smooth foliations. See \cite{Hurder2013a} for a further discussion of this topic.
}
\end{remark}
\section{Concluding remarks and a solenoidal Borel Conjecture} \label{sec-conjectures}
One of the key results required for the proofs of Theorems~\ref{thm-main2} and \ref{thm-main3}, is a form of the Borel Conjecture for solenoids that are approximated by strongly Borel manifolds. Here we show how our considerations lead to a generalized Borel Conjecture for equicontinuous matchbox manifolds.
It is known that two equicontinuous ${\mathbb T}^n$--like matchbox manifolds ${\mathfrak{M}}_1$ and ${\mathfrak{M}}_2$ with equivalent shape (or even just isomorphic first \v{C}ech cohomology groups) are homeomorphic. Indeed, since these spaces admit an abelian topological group structure, the first \v{C}ech cohomology group of such a space is isomorphic to its character group, and Pontrjagin duality then shows that two such spaces are homeomorphic if and only if their first \v{C}ech cohomology groups are isomorphic. Considering this in a broader context leads to the following two related conjectures for the class ${\mathcal B}$ of closed aspherical manifolds to which the Borel conjecture applies. That is, any closed manifold $M$ homotopy equivalent to some $B\in{\mathcal B}$ is in fact homeomorphic to $B$. We can then formulate two conjectures that would naturally generalize the Borel conjecture for aspherical manifolds to the setting of equicontinuous matchbox manifolds.
Consider two equicontinuous matchbox manifolds ${\mathfrak{M}}_1$ and ${\mathfrak{M}}_2$ of the same leaf dimension $n \geq 2$ that are shape equivalent, which is the appropriate generalization to this setting of two closed manifolds being homotopy equivalent. The first problem we pose is a generalization of the classification of the compact abelian groups in terms of their shape, as mentioned in the introduction to this paper.
\begin{conj}\label{conj-1}
Let ${\mathcal A}_B$ be a Borel collection of compact manifolds of dimension $n \geq 1$.
If ${\mathfrak{M}}_1$ and ${\mathfrak{M}}_2$ are equicontinuous, ${\mathcal A}_B$--like matchbox manifolds that are shape equivalent, then ${\mathfrak{M}}_1$ and ${\mathfrak{M}}_2$ are homeomorphic.
\end{conj}
As indicated in Proposition~\ref{prop-homeq}, two aspherical manifolds with isomorphic fundamental groups are in fact homotopy equivalent. If one can show analogously that two equicontinuous ${\mathcal A}_B$--like matchbox manifolds ${\mathfrak{M}}$ and ${\mathfrak{M}}'$ that have isomorphic $\mathrm{pro}-\pi_1$ pro-groups are in fact shape equivalent, then a proof of the first conjecture would lead to a proof of the following stronger conjecture.
\begin{conj}\label{conj-2}
If ${\mathfrak{M}}_1$ and ${\mathfrak{M}}_2$ are equicontinuous, ${\mathcal A}_B$--like matchbox manifolds that have isomorphic $\mathrm{pro}-\pi_1$ pro-groups, then ${\mathfrak{M}}_1$ and ${\mathfrak{M}}_2$ are homeomorphic.
\end{conj}
The positive results we have obtained have been in the context of a class of matchbox manifolds that are the total space of a foliated bundle over the same base manifold. One of the shortcomings of using restricted pseudogroups for the classification problem, is that they do not distinguish paths that induce trivial maps in holonomy. This is seen in the hypothesis on Theorem~\ref{thm-main3} that there exists simply connected leaves, which eliminates this possibility. On the other hand, Theorem~\ref{thm-main2} does not impose this assumption, and uses the structure of free abelian groups to resolve the difficulties in the proof of homeomorphism which arise.
\begin{prob}\label{prob-holonomy}
Let ${\mathcal A}_B$ be a Borel collection of infra-nil-manifolds of dimension $n \geq 3$.
Show that if ${\mathfrak{M}}_1$ and ${\mathfrak{M}}_2$ are equicontinuous, ${\mathcal A}_B$--like matchbox manifolds which are return equivalent, then ${\mathfrak{M}}_1$ and ${\mathfrak{M}}_2$ are homeomorphic.
\end{prob}
The techniques of this paper are based on the reduction of the classification problem to that for minimal Cantor fibrations over a closed base manifold. This is a strong restriction, and does not generally hold for the minimal sets of \emph{Axiom A attractors} as discussed by Williams \cite{Williams1967,Williams1970,Williams1974}.
Even if one restricts oneself to the class of two--dimensional, orientable matchbox manifolds that occur as an expanding attractor of a diffeomorphism, Farrell and Jones in \cite{FarrellJones1981} show there are examples that do not fiber over any closed manifold.
In consideration of this more general situation, the authors established in \cite{CHL2013a,CHL2013b} the existence of decompositions of minimal matchbox manifolds ${\mathfrak{M}}$ that arises from the fibers of a projection onto a branched manifold $\pi \colon {\mathfrak{M}} \to B$.
\begin{prob}\label{prob-branched}
Use the projection of a matchbox manifold onto a branched manifold $\pi \colon {\mathfrak{M}} \to B$ to develop a classification of minimal matchbox manifolds without holonomy.
\end{prob}
Starting with Williams \cite[Section 7]{Williams1970}, there has been an attempt to use the fundamental group of the base $B$ in this setting to classify special classes of one--dimensional attractors that have the structure of a matchbox manifold. However, this technique, even in dimension one, is fraught with difficulties. As discovered in the erratum to \cite{BS2007} and explored more fully in the paper \cite{Swanson2012}, there are fundamental problems with these techniques. In general, the lack of a true bundle structure in this setting creates serious obstructions to applying the techniques we have developed in this work.
Finally, the return equivalence of $1$-dimensional equicontinuous solenoids is described by a sequence of integers as discussed in Section~\ref{sec-examples}. In the higher dimensional case, equicontinuous torus solenoids are described by sequences of integer matrices. By the results of \cite{Kechris2000} there is no reasonable way of describing the classification of the structures resulting from these sequences of matrices. However, it might nonetheless be possible to describe a condition in the spirit of Definition~\ref{defn-MorEquvSequ}.
\begin{quest}
Can one find a combinatorial description of return equivalence for sequences of integer matrices associated to equicontinuous toral solenoids of dimension $d>1$?
\end{quest}
| {
"timestamp": "2013-11-04T02:08:43",
"yymm": "1311",
"arxiv_id": "1311.0226",
"language": "en",
"url": "https://arxiv.org/abs/1311.0226",
"abstract": "Matchbox manifolds are foliated spaces with totally disconnected transversals. Two matchbox manifolds which are homeomorphic have return equivalent dynamics, so that invariants of return equivalence can be applied to distinguish non-homeomorphic matchbox manifolds. In this work we study the problem of showing the converse implication: when does return equivalence imply homeomorphism? For the class of weak solenoidal matchbox manifolds, we show that if the base manifolds satisfy a strong form of the Borel Conjecture, then return equivalence for the dynamics of their foliations implies the total spaces are homeomorphic. In particular, we show that two equicontinuous $\\mT^n$--like matchbox manifolds of the same dimension are homeomorphic if and only if their corresponding restricted pseudogroups are return equivalent. At the same time, we show that these results cannot be extended to include the \"\\emph{adic}-surfaces\", which are a class of weak solenoids fibering over a closed surface of genus 2.",
"subjects": "Dynamical Systems (math.DS); General Topology (math.GN)",
"title": "Classifying matchbox manifolds",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9770226307590189,
"lm_q2_score": 0.7248702761768249,
"lm_q1q2_score": 0.708214664189298
} |
https://arxiv.org/abs/1511.00151 | Group Isomorphism with Fixed Subnormal Chains | In recent work, Rosenbaum and Wagner showed that isomorphism of explicitly listed $p$-groups of order $n$ could be tested in $n^{\frac{1}{2}\log_p n + O(p)}$ time, roughly a square root of the classical bound. The $O(p)$ term is entirely due to an $n^{O(p)}$ cost of testing for isomorphisms that match fixed composition series in the two groups. We focus here on the fixed-composition-series subproblem and exhibit a polynomial-time algorithm that is valid for general groups. A subsequent paper will construct canonical forms within the same time bound. | \section{Introduction}
\label{intro}
\medskip
The complexity of testing isomorphism of groups of order $n$ (input via Cayley tables) has long been
cited as $\,n^{\log_p n + O(1)}$,
where $p$ is the smallest prime divisor of $n$; this follows
immediately from the fact that the group requires no more than $\,\log_p n\,$ generators
\cite{LSZ77}. Wagner \cite{Wa11} suggested that this might
be improved by a careful consideration of the isomorphisms that
match fixed composition series.
While composition series have long been a staple for such computational problems
(e.g., \cite{FN67}), Wagner's
insight was that this could lead to an analyzable advantage
in terms of a provable worst-case bound.
Indeed, using that approach,
Rosenbaum and Wagner \cite{RW15} were able to improve the
bound for $p$-groups to $n^{(1/2)\log_p n + O(p)}$. The $O(p)$
term is contributed by their
$n^{O(p)}$
complexity analysis of
fixed-composition-series isomorphism in $p$-groups.
Their main result then uses the fact that there are $n^{(1/2)\log_p n + O(1)}$
composition series to consider.
Rosenbaum \cite{Ros13} extended the result to
solvable groups, achieving an $n^{(1/2)\log_p n + O(\log n/\log\log n)}$ time
for isomorphism, the fixed-composition series subproblem contributing to the
$\,\log n/\log\log n\,$ term.
Making use of a canonical-form version of the
fixed-composition-series subproblem,
Rosenbaum \cite{RosarX13} subsequently showed that a ``collision'' method
yields a square-root improvement. That result is striking and there
is much to be appreciated in the methods. However,
the composition-series-isomorphism subproblem remains appealing in its own right
and it appears to be susceptible to established algebraic and computational
methods.
\medskip
Thus, we focus on the following problem.
\bigskip
\PROBLEM{
{\sc Comp\-Series\-Iso}\\[.5ex]
\hspace*{1.5 em} {\sc Given:} Groups $G_1,G_2$ given by Cayley tables;\\
\hspace*{5.25em} composition series\\
\hspace*{7.25em}$G_1=G_{1,0} \vartriangleright G_{1,1} \vartriangleright G_{1,2} \vartriangleright \cdots \vartriangleright G_{1,m} ={\bf 1}$, \\[.5ex]
\hspace*{7.25em}$G_2=G_{2,0} \vartriangleright G_{2,1} \vartriangleright G_{2,2} \vartriangleright \cdots \vartriangleright G_{2,m} ={\bf 1}$.\\[.5ex]
\hspace*{1em} {\sc Question:}
Is there an isomorphism $~f: G_1 \rightarrow G_2~$ such that \\[.5ex]
\hspace*{7.25em}
$f(G_{1,i})= G_{2,i}$, for $0\le i\le m$?
}
\bigskip
The treatments of {\sc Comp\-Series\-Iso}~for nilpotent and solvable groups
in \cite{Wa11}, \cite{RW15}, \cite{Ros13}, put great effort into reductions to instances of bounded-valence graph isomorphism so as to
plug in the main result of \cite{Luk82}. However, underlying the latter was a method for set-stabilizers in permutation groups
which we can apply directly in a natural approach to {\sc Comp\-Series\-Iso}. This results in both a better bound
and an extension to general groups.
Specifically,
\smallskip
\begin{theorem}
\label{mainprob}
{\sl {\sc Comp\-Series\-Iso}~is in polynomial time.}
\end{theorem}
If one separates the elements needed for just the solvable-group case, the proof
can be compressed to a few lines.
\medskip
It should be no surprise that the method actually returns {\it all\/} isomorphisms
in the form of a coset of the analogous automorphism group. In fact,
our discussion concentrates mostly on finding automorphism groups.
\bigskip
\PROBLEM{
{\sc Comp\-Series\-Auto}\\[.5ex]
\hspace*{1em} {\sc Given:} A group $G$ given by Cayley table;\\
\hspace*{4.75em} a composition series
$G=G_0 \vartriangleright G_1 \vartriangleright G_2 \vartriangleright \cdots \vartriangleright G_m ={\bf 1}.
$\\[.5ex]
\hspace*{1em} {\sc Find:}
(Generators for) the group $\{f\in {\rm Aut}(G) \mid f(G_{i})= G_{i} \mbox{ for } 0\le i\le m\}$.
}
\bigskip\noindent
We prove
\begin{theorem}
\label{mainprobauto}
{\sl {\sc Comp\-Series\-Auto}~is in polynomial time.}
\end{theorem}
\noindent
The application
to isomorphism then follows a quick and standard path.
\smallskip
\medskip
An immediate consequence of Theorem~\ref{mainprob} is the time bound
$\,n^{(1/2)\log_p n + O(1)}\,$ for testing isomorphism
of groups of order $n$,
where $p$ is the smallest prime divisor of $n$;
this is due to the upper bound of
$\, n^{(1+\log_p n)/2}\,$
on the number of composition series (credited to Babai in \cite[Lemma 3.1]{RW15}).
Rosenbaum \cite{RosarX13}
has already realized that timing for isomorphism
using ``bidirectional collision'', though his method comes
at a substantial cost in space.
Collision and other innovative methods in
\cite{RosarX13} also mesh well with our results, with
further implications for group isomorphism.
We will be pleased to borrow from those methods
in a follow-up paper which will first extend our {\sc Comp\-Series\-Iso}~algorithm
to the computation of canonical forms.
\medskip
We remark that Babai \cite{Ba12} has found a holomorph approach
to {\sc Comp\-Series\-Iso}~which also improves the earlier bounds and is in polynomial time for solvable groups.
\bigskip
Our method for {\sc Comp\-Series\-Auto}~involves repeated consideration
of a classic issue in automorphism-group computation.
Specifically, for $H\trianglelefteq G$, we are given ${\mathcal A} \le {\rm Aut}(G/H)$ and ${\mathcal B}\le {\rm Aut}(H)$,
and we want to determine the pairs $(\alpha,\beta)\in {\rm Aut}(G/H)\times {\rm Aut}(H)$
that ``lift'' to automorphisms of $G$, if any such exist. The obstruction to such lifting
is easy to formulate algebraically and, in general, it poses a difficult
computational problem. However, for the ${\mathcal A}$ and ${\mathcal B}$
that arise herein, we are able to view the obstruction in stages,
each of which is resolvable using methods that are in polynomial time
for solvable permutation groups.
We give two procedures for this. An elementary method, described in \S\ref{firstL},
is all one needs for the resolution of
{\sc Comp\-Series\-Auto}~for solvable groups, but it is not effective for
general groups. The method is then strengthened in \S\ref{secondL} to one that
is
generally applicable.
In \S\ref{nice}, we recall the divide-and-conquer method for set-stabilizer
in permutation groups and show that it is in polynomial time for
the groups we encounter since they are shown to have solvable subgroups
of ``small'' index.
There are two demonstrations of Theorem~\ref{mainprobauto}
in \S\ref{auto}.
They exhibit two ways of breaking the problem down into instances of the
``lift'' scenario that are amenable to a set-stabilizer approach
(assuming they are not already in polynomial time via brute-force enumeration).
Theorem~\ref{mainprob} is succinctly resolved in
\S\ref{Iso} by viewing it
as an extended application of the methods of \S\ref{auto}.
\bigskip\noindent
{\bf Notation.}
\medskip
Suppose $G$ is a group acting on the set $\Omega$.
For $g \in G$, we denote the image of $\alpha \in \Omega$ under the
action of $g$ by $\alpha^g$. For $\Delta\subseteq\Omega$, let
$
{G}_{\Delta} = \{g\in G \mid \Delta^g = \Delta\}
$.
The automorphism group of $G$ is denoted by ${\rm Aut}(G)$, which we view as
a subgroup of ${\rm Sym}(G)$. Thus, for $H\le G$, ${\rm Aut}(G)_H$ denotes the group
of automorphisms stabilizing (or normalizing) $H$.
Implicit in the statement that $G$ is given by a Cayley table is the assumption that
the elements of $G$ can be listed in polynomial time.
Permutation groups
are assumed to be input or output via generating sets. A coset of
a permutation group $J$ is input or output via generators for $J$ and
a single representative.
For other concepts and notation, we
refer the reader to \cite{DM96}. For background on
polynomial-time computability in
permutation groups, see \cite{KL90}, \cite{Luk82}, \cite{LM11}, \cite{Se03}.
\bigskip
\section{The key subproblem}
\label{key}
\medskip
Throughout this section, we assume $G$ is given by a Cayley table and $H\trianglelefteq G$.
Then ${\rm Aut}(G)_H$ can be viewed as a subgroup of the
wreath product ${\rm Sym}(H) \wr {\rm Sym}(G/H)$, and especially a subgroup of
${\rm Sym}(H) \wr {\rm Sym}(G/H)_{\{H/H\}}$.
(The subscript $\{H/H\}$ signifies that we fix this single ``point''
in the permutation domain $G/H$.)~
For the reader's convenience
we give a more explicit indication of
the latter group, namely,
$${\rm Sym}(G)_{H,G/H} := \{ \gamma\in{\rm Sym}(G)_H \mid
\gamma \mbox{ permutes the cosets of }H\}.$$
\noindent
There is a natural homomorphism
$$
\Theta_{G,H}: {\rm Sym}(G)_{H,G/H} \,\rightarrow\, {\rm Sym}(G/H)\times{\rm Sym}(H).
$$
\noindent
Inasmuch as $H\trianglelefteq G$ will always be clear in context, we let $\Theta:=\Theta_{G,H}$. For $x\in G$,
we let $\inn{x}$ denote the restriction to $H$ of the inner automorphism due on $x$, i.e., for $h\in H$
$h^{\inn{x}} = x^{-1} h x$; this is extended to $X\subseteq G$ by $\InnH{X} = \{ \InnH{x} \mid x\in X\}$.
Because of repeated usage, it is useful to
set ${\rm C} :=\{g\in G \mid \forall h\in H, gh=hg\}$, i.e., the centralizer of
$H$ in $G$.
\medskip
\subsection{Lifting automorphisms from $G/H$ and $H$}
\label{autlifting}
Noting that
$\Theta({\rm Aut}(G)_H)\,\le {\rm Aut}(G/H)\times{\rm Aut}(H)
$, we are concerned with the following problem.
\PROBLEM{
\medskip
{\sc AutLifting}\\[1ex]
\hspace*{1em} {\sc Given:} $H\trianglelefteq G$;~
${\mathcal A}\le {\rm Aut}(G/H)$;~ ${\mathcal B} \le {\rm Aut}(H)$.\\[1ex]
\hspace*{1em} {\sc Find:}
$\,{\rm Aut}(G)_H \cap \Theta^{-1}({\mathcal A}\times {\mathcal B}).\,$
\\[-.75ex]
}
\medskip
There are a couple of natural ways to reduce Theorem~\ref{mainprobauto} to
polynomial-time instances of {{\sc AutLifting}. We will describe these in \S\ref{bottomup},\ref{topdown}.
\bigskip
The reader may already recognize {\sc AutLifting}
as a frequent issue in studies of group automorphism/isomorphism, theoretical or computational
(see, e.g., \cite[\S8.9]{HEO}).
Special cases are attacked with varied machinery and success.
Our method is guided by properties of the relevant groups that enable polynomial-time steps.
\medskip
Our approaches to {\sc AutLifting} each involve defining a group ${\mathcal L}$ such that
\medskip
\hspace*{1em}\parbox{4in}{
\begin{itemize}
\item ${\rm Aut}(G)_H \le {\mathcal L} \le {\rm Sym}(G)_{H,G/H}$.\\[-.45em]
\item It is ``easy'' to find $\,\Theta({\mathcal L} )\cap ({\mathcal A}\times {\mathcal B}).$\\[-.45em]
\item It is ``easy'' then to lift to $\,\widehat{{\mathcal L}}={\mathcal L} \cap \Theta^{-1}({\mathcal A}\times {\mathcal B}).$\\[-.45em]
\item It is ``easy'' to find $\,\widehat{{\mathcal L}} \cap {\rm Aut}(G)$.
\end{itemize}
}
\medskip
\subsection{First choice of ${\mathcal L}$}
\label{firstL}
We consider the following supergroup of ${\rm Aut}(G)_H$.
$$
{\mathcal L}_1~=~
\{ \gamma\in {\rm Sym}(G)_{H,G/H} \,\mid\,
\forall g\in G, h\in H:~ (hg)^\gamma = h^\gamma g^\gamma\}.
$$
\medskip
\subsubsection{Theory}
\medskip
\begin{lemma}
\label{kerL1}
{\sl
{~}\\[-3ex]
\begin{enumerate}[\rm (i)]
\item
The kernel of $\,\restr{\Theta}{{\mathcal L}_1}\,$ is isomorphic to the direct product
of $~|G/H| -1~$ copies of $\,H$.
\item
(Generators for) $\,{\rm Ker}(\restr{\Theta}{{\mathcal L}_1})\,$ can be found in polynomial time.
\end{enumerate}
}
\end{lemma}
\goodbreak
\medskip\noindent
{\sc Proof:}
Let $\gamma\in{\rm Ker}(\restr{\Theta}{{\mathcal L}_1})$. Then $\forall h\in H, \, h^\gamma = h$ and
$\forall x\in G,\, (Hx)^\gamma = Hx$.
Consider the action induced by $\gamma$ on $Hg \ne H$.
We have $g^\gamma = gk$ for some $k\in H$ (since $gH=Hg$). Then for any $hg\in Hg$,
$(hg)^\gamma = h^\gamma g^\gamma = hgk$, that is, $\forall x\in gH$, $x^\gamma = xk$.
Thus, the action of ${\rm Ker}(\restr{\Theta}{{\mathcal L}_1})$ on each $Hg\ne H$
is the group of right multiplications by $H$, a group isomorphic to $H$. Since the actions are independent
across the cosets, (i) follows.
For (ii), we deal with each $Hg\ne H$ independently. Focussing on a coset $X$,
we form for each $h\in H$, the permutation $\phi_{X,h}$ of $G$ that fixes all $x\not\in X$
and maps $x\mapsto xh$ for $x\in X$. Then ${\rm Ker}(\restr{\Theta}{{\mathcal L}_1})$
is generated by $\{\phi_{X,h} \mid h\in H, X\in G/H \mbox{ with } X\ne H \}$.
Take the union of all these permutations for all $Hg\ne H$.
(A bit more economically, only use $h$ in a generating set for $H$.)
{\hfill $\Box$}
\bigskip\noindent
\begin{lemma}
\label{L1lift}
{\sl ${\rm Aut}(G/H)\times {\rm Aut}(H) \le \Theta({\mathcal L}_1)$.
Furthermore, given $(\alpha,\beta)\in{\mathcal A}\times{\mathcal B}$,
${\mathcal L}_1 \cap \Theta^{-1}(\alpha,\beta)$ can be constructed
in polynomial time.
}
\end{lemma}
\medskip\noindent
{\sc Proof:} It suffices to show,
for any $\alpha\in {\rm Aut}(G/H),\,\beta\in{\rm Aut}(H)$, we can construct a {\it single\/}
$\gamma\in{\mathcal L}_1$,
for which $\Theta(\gamma)=(\alpha,\beta)$ since, by Lemma~\ref{kerL1} the coset
$\,\gamma \,{\rm Ker}(\restr{\Theta}{{\mathcal L}_1})$ then comprises the set
of preimages of $(\alpha,\beta)$ in ${\mathcal L}_1$.
We construct $\gamma\in{\mathcal L}_1$ as follows. For $h\in H$,
define $h^\gamma := h^\beta$. For each coset of $X$ of $H$ in $G$ with $X\ne H$,
define $\gamma$ on $X$ as follows:
\hspace*{1em}\parbox[t]{4in}{
\begin{enumerate}[1.]
\item
Fix any $a\in X$ (so $X=Ha$).\\[-1.0em]
\item Fix {any} $~b\in X^\alpha$. \\[-1.0em]
\item For all $h\in H$, define $(ha)^\gamma := h^\beta b$.
\end{enumerate}
}
\vspace*{-1em}
{\hfill $\Box$}
\bigskip\bs
Using Lemmas \ref{kerL1} and \ref{L1lift}, we conclude
\medskip\noindent
\begin{proposition}
\label{L1find}
{\sl
{~}\\[-2ex]
\begin{enumerate}[\rm (i)]
\renewcommand{\theenumi}{\roman{enumi}}
\ite
Given $\,{\mathcal A}\le {\rm Aut}(G/H)$ and ${\mathcal B} \le {\rm Aut}(H)$,
$\widehat{{\mathcal L}}_1:={\mathcal L}_1 \cap \Theta^{-1}({\mathcal A}\times{\mathcal B})$ can be constructed in polynomial
time.
\smallskip
\ite
$\widehat{{\mathcal L}}_1$ is an extension of $\,{\mathcal A}\times{\mathcal B}\,$ by a direct product of
copies of $H$.
\end{enumerate}
}
\end{proposition}
\medskip\noindent
{\sc Proof:} For (i), $\widehat{{\mathcal L}}_1$ is generated by the lifts of the generators of $\,{\mathcal A} \times {\mathcal B}\,$
together with generators of $\,{\rm Ker}(\restr{\Theta}{{\mathcal L}_1})$. {\hfill $\Box$}
\medskip
\subsubsection{Algorithm}
\label{Algorithm1}
~\\
\noindent
{\bf Step 1.} $\widehat{{\mathcal L}}_1:={\mathcal L}_1 \cap \Theta^{-1}({\mathcal A}\times{\mathcal B})$
\medskip\noindent
{\sc Method:}
By Lemma~\ref{L1find}(i), this is in polynomial time for any ${\mathcal A}, {\mathcal B}$. {\hfill $\Box$}
\bigskip\noindent
{\bf Step 2.}
Find $~\{\gamma \in \widehat{{\mathcal L}}_1 \mid \gamma\in{\rm Aut}(G)\}$.
\medskip\noindent
{\sc Method:} This can be expressed as a set-stabilizer problem for the natural extension of
$\widehat{{\mathcal L}}_1 \le {\rm Sym}(G)$
to an action on $G\times G\times G$. The set to stabilize
is $\{(a,b,ab) \mid a,b\in G\}$. \hspace*{2em} {\hfill $\Box$}
\bigskip
\begin{remark}{\rm
In an inductive approach to {\sc Comp\-Series\-Auto}~for solvable groups the calls to {\sc AutLifting}~result
in a solvable $\widehat{{\mathcal L}}_1$, thus putting Step 2 in polynomial time.
(We provide this forecast in the expectation
that some readers would like to finish the solvable case
as an exercise.)
A more restrictive ${\mathcal L}$ will yield our main results for general groups,
thus making this section superfluous. Nevertheless,
we retain this discussion of ${\mathcal L}_1$. By switching back to ${\mathcal L}_1$ in \S\ref{topdown}
whenever $H$ is solvable, we limit the requisite machinery to the early paper \cite{Luk82}.}
\end{remark}
\medskip
\subsection{Second choice of ${\mathcal L}$}
\label{secondL}
Consider now a more restricted supergroup of ${\rm Aut}(G)_H$.
$$
{\mathcal L}_2~=~
\{ \gamma\in {\rm Sym}(G)_{H,G/H} \,\mid\,
\forall g\in G, h\in H,\, (hg)^\gamma = h^\gamma g^\gamma~\mbox{and}~ (gh)^\gamma = g^\gamma h^\gamma \}.
$$
\smallskip
\subsubsection{Theory}
\label{theory}
The advantage of cutting down from ${\mathcal L}_1$ to ${\mathcal L}_2$ is that
${\rm Ker}(\restr{\Theta}{{{\mathcal L}}_2})$ is abelian,
which plays a role in guaranteeing polynomial-time
set-stabilizers for
the instances of $\widehat{{\mathcal L}}_2$ that arise. (Actually, we only need
that kernel to be solvable.)~
However, since
${\mathcal A} \times {\mathcal B}$ may not be contained in
$\Theta({\mathcal L}_2)$, we will first have to determine the
liftable subgroup.
We are guided in this by some properties of $\Theta({\mathcal L}_2)$.
\medskip\noindent
\begin{lemma}
\label{gammaconj}
{\sl
Let $\gamma\in {\mathcal L}_2$. Then for $g\in G$, $h\in H$,
$(g^{-1}hg)^\gamma = (g^\gamma)^{-1}h^\gamma g^\gamma$.}
\end{lemma}
\medskip\noindent
{\sc Proof:}
For $g\in G$, $h\in H$, $\gamma\in{\mathcal L}_2$,
$$
g^\gamma(g^{-1}hg)^\gamma = (gg^{-1}hg)^\gamma
= (hg)^\gamma
= h^\gamma g^\gamma,
$$
the first and third equalities using properties of ${\mathcal L}_2$. {\hfill $\Box$}
\bigskip\noindent
\begin{lemma}
\label{kerL2}
{\sl
{~}\\[-3ex]
\begin{enumerate}[\rm (i)]
\item
The kernel of $\restr{\Theta}{{\mathcal L}_2}$ is isomorphic to the direct product
of $~|G/H| -1~$ copies of ${\rm Z}(H)$ (the center of $H$).
\smallskip
\item
(Generators for) ${\rm Ker}(\restr{\Theta}{{\mathcal L}_2})$ can be found in polynomial time.
\end{enumerate}
}
\end{lemma}
\medskip\noindent
{\sc Proof:}
Let $\gamma\in{\rm Ker}(\restr{\Theta}{{\mathcal L}_2})$.
In particular, $h^\gamma = h$ for $h\in H$.
Consider $Hg\ne H$.
As in the proof of Lemma~\ref{kerL1}, there is some $k\in H$
such that the action of $\gamma$ on $Hg=gH$ is right-multiplication by $k$.
It suffices then to show $k\in {\rm Z}(H)$. Using Lemma~\ref{gammaconj},
for any $h\in H$,
$$g^{-1}hg = (g^{-1}hg)^\gamma = (g^\gamma)^{-1}h^\gamma g^\gamma
= (gk)^{-1}h gk = k^{-1}(g^{-1}hg)k.$$
The normality of ${\rm Z}(H)$ in $G$ then implies $k\in {\rm Z}(H)$.
For (ii), we proceed as in Lemma~\ref{kerL1}(ii) but restrict the choice of maps $x\mapsto xh$ to
$h\in {\rm Z}(H)$ (or just to a generating set of ${\rm Z}(H)$).
{\hfill $\Box$}
\medskip
\goodbreak
\smallskip\noindent
\goodbreak
\bigskip
The next two lemmas give necessary
conditions on $(\alpha,\beta) \in {\rm Aut}(G/H) \times {\rm Aut}(H)$ for it
to be liftable to
${\mathcal L}_2$.
Recall that ${\rm C}$ denotes
the centralizer of
$H$ in $G$, and $\inn{x}$ denotes the restriction to $H$ of the inner automorphism
corresponding to $x$.
\bigskip\noindent
\begin{lemma}
\label{L2property}
{\sl
Let $\gamma\in {\mathcal L}_2$. Suppose that
$\Theta(\gamma) = (\alpha,\beta) \in {\rm Aut}(G/H) \times {\rm Aut}(H)$. Then
\begin{enumerate}[\rm (i)]
\renewcommand{\theenumi}{\roman{enumi}}
\item
$\alpha$ normalizes ${{H \C }}/{H}$, and therefore induces an automorphism of $G/{H \C }$.
\medskip
\item
$\forall g\in G:~\beta^{-1} \,\InnH{{H \C } g} \, \beta =\InnH{({H \C } g)^\alpha}.$
\end{enumerate}
}
\end{lemma}
\medskip\noindent
{\sc Proof:}
Lemma~\ref{gammaconj} implies that ${\rm C}^\gamma = {\rm C}$, so that
$(H{\rm C})^\gamma = H^\gamma {\rm C}^\gamma$, proving (i).\\
For (ii), first note that $\forall g\in G, h\in H$,
$$ h^{\beta^{-1} \InnH{g} \beta} = (g^{-1} h^{\beta^{-1}} g)^\beta
= (g^\gamma)^{-1} h g^\gamma = h^{\InnH{g^\gamma}},
$$
the second equality following from Lemma~\ref{gammaconj}.
In other words,
$$ \forall g\in G,~
\beta^{-1}\InnH{g} \beta = \InnH{g^\gamma}.$$
\vspace*{-3ex}
{\hfill $\Box$}
\bigskip\noindent
\begin{lemma}
\label{L2property2}
{\sl Suppose $(\alpha,\beta) \in {\rm Aut}(G/H) \times {\rm Aut}(H)$ satisfies {\rm (i),(ii)} in
Lemma~\ref{L2property}.
Then
\begin{equation}
\label{cosetconj}
\forall g\in G:~\beta^{-1} \left(\InnH{H g} \right) \beta =\InnH{(H g)^\alpha}.
\end{equation}
}
\end{lemma}
\medskip\noindent
{\sc Proof:}
For any $A\subseteq G$,
$\InnH{A {\rm C}}= \InnH{A}$. Also, by (i),
$\alpha$ permutes the cosets of $H{\rm C}/H$ in $G/H$ so that
$(H{\rm C} g)^\alpha = {\rm C}(Hg)^\alpha$.
{\hfill $\Box$}
\bigskip\noindent
\begin{remark}{\rm
This organization may seem convoluted seeing that
equation~(\ref{cosetconj})
could also be viewed as a direct consequence of
Lemma~\ref{gammaconj}. Our motive is that, in the process of cutting down to
``liftable''
$(\alpha,\beta)$,
our {\it algorithmic\/}
route runs through (\ref{cosetconj}) after first forcing (i),(ii) of Lemma~\ref{L2property}.}
\end{remark}
\goodbreak
\bigskip\noindent
\begin{lemma}
\label{L2find}
{\sl
Suppose $(\alpha,\beta) \in {\rm Aut}(G/H) \times {\rm Aut}(H)$ satisfies property~{\rm (\ref{cosetconj})}. \\
Then
$\,\Theta^{-1}(\alpha,\beta)\cap{\mathcal L}_2\,$ is nonempty and can be found in polynomial
time}.
\end{lemma}
\medskip\noindent
{\sc Method:}
Construct $\gamma\in\Theta^{-1}(\alpha,\beta)$ as follows.
For $h\in H$, $h^\gamma := h^\beta$.
To define $\gamma$ on a coset $X$ of $H$ with $X\ne H$. Fix $a \in X$.
Then, by (1), there is some $b\in X^\alpha$ such that
$~
\beta^{-1}\InnH{a}\beta = \InnH{b}.
~$
Fix any such $b$. For any $g\in X$, say $g=ka$ with $k\in H$,
set $g^\gamma := k^\beta b$.
It follows easily that
\begin{equation}
\forall h\in H, ~ (hg)^\gamma = h^\gamma g^\gamma
\end{equation}
Also
\begin{equation}
\beta^{-1}\InnH{g}\beta = \InnH{g^\gamma}
\end{equation}
\smallskip\noindent
since~
$\beta^{-1} \InnH{ka} \beta =
\beta^{-1} \InnH{k} \beta\, \beta^{-1} \InnH{a} \beta =
\InnH{k^\beta} \InnH{b} = \InnH{k^\beta b}.
$
\medskip
Having established (2) and (3) for $g$ in each coset $X$, they hold for
all $g\in G$.
\medskip
Then, for all $h\in H, g\in G$,
$$(gh)^\gamma = (h^{\inn{g}^{-1}}g)^\gamma = h^{{\inn{g}^{-1}}\beta}g^\gamma
= h^{\beta (\beta^{-1} \inn{g} \beta)^{-1}}g^\gamma =
h^{\beta \, \inn{g^{\gamma}}^{-1} }g^\gamma = g^\gamma h^\gamma.
$$
We conclude that $\gamma\in {\mathcal L}_2$.
\medskip
The complete set of preimages of
$(\alpha,\beta)$ is
$\gamma\, {\rm Ker}(\restr{\Theta}{{\mathcal L}_2})$, for which we apply
Lemma~\ref{kerL2}(ii). \mbox{\hspace*{3em}}
{\hfill $\Box$}
\bigskip
To summarize the main points of this subsection, we have the following
consequence of
Lemmas \ref{kerL2}, \ref{L2property}, \ref{L2find}.
\goodbreak
\bigskip\noindent
\begin{proposition}
\label{L2summary}
{\sl Suppose ${\mathcal M}\le {\rm Aut}(G/H) \times {\rm Aut}(H)$ such that
property~{\rm (\ref{cosetconj})} is satisfied
by all $(\alpha,\beta)\in {\mathcal M}$. Then
\begin{enumerate}[\rm (i)]
\item
${\mathcal M} \le \Theta({{\mathcal L}}_2)$
\smallskip
\item
${\mathcal L}_2\cap\Theta^{-1}({\mathcal M})$ is an extension of ${\mathcal M}$ by an abelian group.
\smallskip
\item
Given ${\mathcal M}$, ${\mathcal L}_2\cap\Theta^{-1}({\mathcal M})$ can be constructed in polynomial time.
\end{enumerate}
}
\end{proposition}
\medskip\noindent
{\sc Proof:} For (iii), ${\mathcal L}_2\cap\Theta^{-1}({\mathcal M})$ is generated by the lifts of the generators of $\,{\mathcal M}\,$
together with generators of $\,{\rm Ker}(\restr{\Theta}{{\mathcal L}_2})$. {\hfill $\Box$}
\medskip
\subsubsection{Algorithm}
\label{algorithm}
Steps 1-3 cut ${\mathcal A}\times{\mathcal B}$ to the subgroup ${\mathcal M}$ consisting of elements that
can be lifted to ${\mathcal L}_2$.
\bigskip\noindent
{\bf Step 1.}
${\mathcal A}\,:=\, \nrm{{\mathcal A}}{{{H \C }}/{H}}$.
\medskip\noindent
{\sc Method:}
Viewing ${\mathcal A}\le {\rm Sym}(G/H)$,
this can be viewed as stabilizing the subset ${H \C }/{H}$
of the polynomial-size domain $G/H$. {\hfill $\Box$}
\medskip
By Lemma~\ref{L2property}(i), Step 1 does not affect
$~\Theta({\mathcal L}_2) \cap ({\mathcal A}\times{\mathcal B})$.
\bigskip\noindent
{\bf Step 2.} ${\mathcal B} \,:=\, \nrm{{\mathcal B}}{\InnH{G}}$.
\medskip\noindent
{\sc Method:}
Here we consider ${\mathcal B} \subseteq {\rm Sym}(H)$ acting on
${\rm Aut}(H)\subseteq{\rm Sym}(H)$ via conjugation. So, at first glance, this
appears to be a normalizer problem for permutation groups.
However, the specific reductions of
our main problems to {\sc AutLifting}~will reveal that required
instances of Step 2 can be handled via set-stabilizers.
{\hfill $\Box$}
\medskip
By Lemma~\ref{L2property}(ii), Step 2 does not affect
$~\Theta({\mathcal L}_2) \cap ({\mathcal A}\times{\mathcal B})$.
\bigskip
Step 2 does not yet accomplish the
compatibility condition on $(\alpha,\beta)$ expressed in property (1)
and required in Lemma~\ref{L2find},
but it establishes a structure for getting there.
\bigskip
We now have that ${\mathcal A} \le {{\rm Aut}(G/H)}_{{H \C }/H}$, so there is an induced action
$${\mathfrak a}: {\mathcal A} \rightarrow {\rm Aut}\left(\frac{G}{{H \C }}\right).
$$
\noindent
We also now have ${\mathcal B}$ acting on $\InnH{G}$ and, since ${\mathcal B}\le{\rm Aut}(H)$
also normalizes $\InnH{H}$, there is an induced action
$${\mathfrak b}: {\mathcal B} \rightarrow {\rm Aut}\left( \frac{\InnH{G}}{\InnH{H}} \right)
$$
\noindent
But there is a natural identification
$$
\frac{G}{{H \C }} \,\,\,{\simeq}\,\,\,
\frac{\InnH{G}}{\InnH{H}}.$$
\bigskip\noindent
(For all $g\in G$, $g{H \C } \leftrightarrow\InnH{g}\InnH{H}$.)$~~$
Via this identification, property (1)
is expressible in the form
$$
{\mathfrak a}(\alpha)={\mathfrak b}(\beta).
$$
\medskip\noindent
This motivates
\goodbreak
\medskip\noindent
{\bf Step 3.} ${\mathcal M} \,:=\, \{(\alpha,\beta)\in{\mathcal A}\times {\mathcal B} \mid {\mathfrak a}(\alpha)={\mathfrak b}(\beta)\}$.
\medskip\noindent
{\sc Method:}
By the above, this is a permutation-group intersection problem, solvable for example as a
set-stabilizer problem: consider ${\mathfrak a}({\mathcal A})\times {\mathfrak b}({\mathcal B})$ acting on $\,\Omega\times \Omega\,$
where $\,\Omega = G/{H \C }$; the object then is
to stabilize the diagonal $\{(\omega,\omega) \mid \omega\in \Omega\}$. {\hfill $\Box$}
\bigskip
By Lemmas~\ref{L2property},\ref{L2property2} and Proposition~\ref{L2summary}(i),
$~\Theta({\mathcal L}_2) \cap ({\mathcal A}\times{\mathcal B})\,=\, {\mathcal M}$.
Hence, ${\mathcal L}_2\cap\Theta^{-1}{({\mathcal A}\times{\mathcal B})} \,=\, {\mathcal L}_2\cap\Theta^{-1}({\mathcal M})$.
So the next step is
\bigskip\noindent
{\bf Step 4.}
$\widehat{{\mathcal L}}_2 \,:=\ {\mathcal L}_2\cap\Theta^{-1}({\mathcal M})$.
\medskip\noindent
{\sc Method:}
This is in polynomial time by Proposition~\ref{L2summary}(iii). {\hfill $\Box$}
\bigskip
The final step is
\bigskip\noindent
{\bf Step 5.}
Find $~\{\gamma \in \widehat{{\mathcal L}}_2 \mid \gamma\in{\rm Aut}(G)\}$.
\medskip\noindent
{\sc Method:}
As in \S\ref{Algorithm1},
this can be expressed as a set-stabilizer problem for the action of
$\widehat{{\mathcal L}}_2$ on $G\times G\times G$. {\hfill $\Box$}
\bigskip\medskip
Thus the computational complexity of our use of
{\sc AutLifting}~rests on properties of ${\mathcal A}$ and ${\mathcal B}$ that will enable efficient
routines for Steps 1,2,3,5. The properties are
described in \S\ref{nice}.
\bigskip
\section{Nice groups}
\label{nice}
\subsection{A reminder on polynomial-time set stabilizers}
\label{setstab}
\medskip
In \cite{Luk82}, the author proposed
an algorithm for finding $G_\Delta$ where $G\le {\rm Sym}(\Omega)$
and $\Delta\subseteq \Omega$. For the reader's convenience in what follows, we offer a
brief description.\footnote{See also \cite{LM11} for an extended discussion of the
divide-and-conquer paradigm for this and other applications.}
\medskip
To accommodate a recursion,
the problem is generalized to cosets.
\bigskip
\medskip
\begin{minipage}{\textwidth}
\begin{quote}
For a $G$-stable subset
$\,\Pi \subseteq \Omega\,$ and a coset $\,X=Ga\,$ of $\,G$, define
$$X_{\Delta \mid \Pi} = \{ x\in X \mid (\Delta\cap\Pi)^x = \Delta\cap\Pi^a\}.
$$
So $\,X_{\Delta \mid \Pi}\,$ is either $\emptyset$ or a right coset of $\,G_{\Delta\cap\Pi}$.
If $\,\Pi = \Pi_1\dot\cup\Pi_2\,$ for $G$-stable $\,\Pi_i$
$$X_{\Delta \mid \Pi} ~:=~ (X_{\Delta \mid \Pi_1})_{\Delta\mid \Pi_2}
$$
If $G$ acts transitively on $\Pi$ and $|\Pi| >1$, find a minimal
decomposition $\,\Pi = \Pi_1 \dot\cup \cdots \dot\cup \Pi_m\,$ into blocks
of imprimitivity and decompose $\,G= \bigcup_{1\le i\le |G/H|} Ht_i\,$ where $H$
is the kernel of the action of $G$ on $\,\{\Pi_i\}_{1\le i\le m}$.
$$X_{\Delta \mid \Pi} ~:=~ \bigcup_{1\le i\le |G/H|} (Ht_ia)_{\Delta \mid \Pi}.
$$
If $\,\Pi =\{\pi\}\,$ then\\
\centerline{if $\,|\Delta \cap \{\pi,\pi^a\}|=1\,$ then $\,X_{\Delta \mid \Pi} := \emptyset\,$
else $\,X_{\Delta \mid \Pi} := X$.}
\end{quote}
\end{minipage}
\bigskip
\bigskip\noindent
The computation of $\,G_\Delta\,$ starts with $\,\Pi:=\Omega\,$ and $\,a:=1$.
The key to the complexity of the method lies in the sizes of the primitive
groups arising in the action on the $m$ blocks. The algorithm will run in polynomial time for an hereditary
class of groups if such induced subgroups of $S_m$
have order $O(m^{\rm constant})$.
\medskip
Notice that the ``set-stabilizer'' algorithm actually dealt with cosets. Thus, we can find set-stabilizers for a
group $A$ that has a small-index subgroup $B$ in a good class: we start by
breaking $A$ into cosets of $B$.
We will find ourselves in exactly
that situation in \S\ref{almostsol}.
\def{\mbox{\tiny -1}}{{\mbox{\tiny -1}}}
\bigskip\noindent
\begin{remark}
\label{cosetstab}
{\rm
If we were only given a black-box for set-stabilizers in groups with bounded non-cyclic composition factors, i.e.,
without knowing the algorithm, we could still use that for the coset problem. It is useful
for another purpose in \S\ref{bottomup} to express that as a ``set-transporter'' problem;
namely, given $\Delta,\Lambda \subseteq \Omega$, find
$$G_{\Delta \mapsto \Lambda } = \{g\in G \mid \Delta^g = \Lambda\}.$$
For this, consider
the natural action of $\widetilde{G}= G\wr C_2$
on $\Omega\,\dot\cup\, \Omega'$, where $\Omega'$ is a copy of $\Omega$.
The desired transporter can be deduced from
$\widetilde{G}_{\Delta \,\dot\cup \,\Lambda'}$ where $\Lambda'$ is the corresponding
copy of $\Lambda$. In particular, $(Ga)_\Delta =
G_{\Delta\mapsto \Delta^{a^{{\mbox{\tiny -1}}}} }\,a$.
}
\end{remark}
\medskip
\subsection{The groups that turn up}
\label{almostsol}
In effect, we will only rely on the polynomial boundedness
of primitive solvable groups. However, we deal with groups
that are not quite solvable.
\medskip\noindent
{\bf Notation.} For a group $X$, ${\rm Rad}(X)$ is the solvable radical of $X$, i.e., the maximum normal
solvable subgroup.
\medskip\noindent
{\bf Definition.} Let $X\le {\rm Sym}(\Omega)$. We call $X$ {\it almost-solvable\/}$\,$\footnote{
The author regrets using a term that has had other meanings. However, the usage
of ``almost-solvable'' is already inconsistent in the literature and he could not think of an
{\it unused\/} term with
similar connotation.}$\,$ if
$\,|X:{\rm Rad}(X)| \le |\Omega|^2$.
\bigskip
Almost-solvable~groups still enable the
polynomial-time divide-and-conquer paradigm. For example,
\medskip\noindent
\begin{lemma}
\label{nicestab}
{\sl Given an almost-solvable~$G\le{\rm Sym}(\Omega)$ and $\Delta\subseteq\Omega$,
$G_\Delta$ can be found in polynomial time.}
\end{lemma}
\medskip\noindent
{\sc Proof:}
By decomposing $\,G\,$ into cosets of $\,{\rm Rad}(G)\,$, we reduce to
$|\Omega|^2$ problems involving
solvable groups. {\hfill $\Box$}
\bigskip\noindent
\begin{remark}{\rm
Our breakdown to cosets of a nice group is reminiscent of the first use of the set-stabilizer method.
When \cite{Luk82} was written,
primitive groups in the relevant class, specifically the class of groups with bounded composition factors, were not known to
be polynomially bounded.
This difficulty was overcome by partitioning the primitive
group into cosets of a small-index $p$-subgroup. That additional
complication soon became unnecessary as polynomial bounds were
established
first for primitive solvable
groups (independently by P\'{a}lfy \cite{Pa82} and Wolf \cite{Wo82}), and ultimately for
primitive groups in a
class that even includes groups with bounded {\it non-cyclic\/} composition factors
(Babai {\sl et al.}~{\cite{BCP82}). We
point, however, to
a subtle difference between what happens in our current passage
to cosets of a nice group and the earlier use of that trick.
In \cite{Luk82}, after passing to cosets of a $p$-group acting on the blocks, the
divide-and-conquer narrows our window to a single, stabilized block. Since the group acting within the block is not necessarily
a $p$-group, we may again arrive at a primitive group that requires the cosets-of-$p$-group
decomposition.
By contrast, in the present
case there is a {\it single\/} such decomposition in the lifetime of the set-stabilizer process, i.e.,
we only visit solvable groups thereafter. }
}\end{remark}
\medskip\noindent
\begin{remark}{\rm
Lemma~\ref{nicestab} is a weak consequence of the
discussion in \S\ref{setstab}.
The good subgroup can be of any polynomially-bounded index,
need not be normal,
and need only be in the broader class
described in \cite{BCP82}.
However, ``almost-solvable'' is a good fit for our situation because we next see that the
property arises so conveniently.
Furthermore, in our present application, one can keep track of small-index normal
solvable subgroups
as we cut down the group or take preimages, so there
is no need even to implement a radical-finder. Nevertheless,
we do note that radicals of
permutation groups can be found in polynomial-time
(\cite[Lect.~6]{Luk93}, \cite[\S6.3.1]{Se03}).
}\end{remark}
\medskip
The relevance to our problem is seen in
\smallskip\noindent\goodbreak
\begin{lemma}
\label{L1L2sol}
{\sl
With reference to the groups in \S2,
suppose ${\mathcal A}\le {\rm Sym}(G/H)$ and ${\mathcal B}\le {\rm Sym}(H)$ are almost-solvable.
\begin{enumerate}[\rm (i)]
\item
If $H$ is solvable, then $\widehat{{\mathcal L}}_1\le{\rm Sym}(G)$ is almost-solvable.
\smallskip
\item
$\widehat{{\mathcal L}}_2\le{\rm Sym}(G)$ is almost-solvable.
\end{enumerate} }
\end{lemma}
\goodbreak
\medskip\noindent
{\sc Proof:}
We have $|{\mathcal A}\times{\mathcal B}:{\rm Rad}({\mathcal A}\times{\mathcal B})| =
| {\mathcal A}:{\rm Rad}({\mathcal A})| \cdot |{\mathcal B}:{\rm Rad}({\mathcal B}) | \le |G/H|^2|H|^2 = |G|^2$.
Thus, (i) follows from Proposition~\ref{L1find}(ii).
\smallskip
Using ${\mathcal M}$ as in \S\ref{algorithm} (Steps 3,4), $\,{\mathcal M}\le {\mathcal A}\times{\mathcal B}$ implies
$|{\mathcal M} : {\rm Rad}({\mathcal M})| \le |{\mathcal A}\times{\mathcal B}:{\rm Rad}({\mathcal A}\times{\mathcal B})| \le |G|^2$.
Thus (ii) follows from Proposition~\ref{L2summary}(ii). {\hfill $\Box$}
\bigskip
Note, as the methods for {\sc AutLifting}~are to be used repeatedly, the almost-solvability~of
$\,\widehat{{\mathcal L}}_i\,$ implies that of the subgroup $\,\widehat{{\mathcal L}}_i \cap {\rm Aut}(G)$.
For a base case, we need a consequence of the Classification of Finite Simple Groups.
Using the fact that any simple group $T$ can be generated by two elements (\cite{AB}),
we immediately have $|{\rm Aut}(T)| \le |T|^2$. Hence,
\medskip\noindent
\begin{lemma}
\label{simplenice}
{\sl If $T$ is simple then ${\rm Aut}(T)$, viewed as a subgroup of ${\rm Sym}(T)$, is
almost-solvable.}
\end{lemma}
\medskip\noindent
In fact, it is well known that the Classification
yields a stronger bound of the form $|{\rm Aut}(T)| = O( |T| \log |T|)$ (e.g., see \cite{ATLAS}).
However, the square bound in ``almost-solvable'' is easier to maintain
through our process.
\goodbreak
\section{Automorphisms stabilizing a composition series}
\label{auto}
\medskip
\begin{sloppy}
Applying the machinery of \S\S\ref{key}-\ref{nice}, we offer two
polynomial-time Turing reductions of {\sc Comp\-Series\-Auto}~to polynomial-time instances of {\sc AutLifting}.
Moreover, we show that the
deepest tool needed in either implementation is
set-stabilizer for solvable permutation groups.
Aside from reducing to the most basic tool, this extra effort will be useful
in a subsequent development of canonical forms. (See also \cite{BaLu83}
for an indication of how the divide-and-conquer method for set-stabilizer
translates to canonical set placement.)
\end{sloppy}
\begin{comment}
\PROBLEM{
{\sc Comp\-Series\-Auto}\\[.5ex]
\hspace*{1em} {\sc Given:} A group $G$ given by Cayley table;\\
\hspace*{4.75em} a composition series
$G=G_0 \vartriangleright G_1 \vartriangleright G_2 \vartriangleright \cdots \vartriangleright G_m ={\bf 1}.
$\\[.5ex]
\hspace*{1em} {\sc Find:}
${\rm Aut}(G)_{\{G_1,G_2,\ldots \}}\,$.
}
\end{comment}
\subsection{Bottom-up on given series}
\label{bottomup}
\medskip
We go from a solution for $G_i$ to a solution for $G_{i-1}$.
The group on top, $G_{i-1}/G_i$, is simple and so
one can enumerate ${\rm Aut}(G_{i-1}/G_i)$.
\medskip\noindent
\begin{remark}{\rm Even for a simple {\it permutation group\/} $T$ given
by generators, one can produce
{generators for} ${\rm Aut}(T)$ (which is all that is necessary for some of our subproblems).
This follows from Kantor's demonstration \cite{Kan85}
that one can obtain the ``natural'' representation
of $T$.
}\end{remark}
\medskip\noindent
\begin{remark}{\rm
Before proceeding further,
we can claim to have already established a poly\-no\-mial-time solution for this use of
{\sc AutLifting}. That is, by virtue of almost-solvability (\S\ref{nice}), there are citable polynomial-time methods for
Steps 1,2,3,5~in \S\ref{secondL}. Thus,
a message of this subsection is that we do not need the
full power of available polynomial-time tools. That leads to speculation
that there are more general problems to solve.
}\end{remark}
\medskip
Let us consider the steps of the algorithm in \S\ref{algorithm}.
Since there are more interesting things to say about Step 2, we will postpone that discussion and
dispense with the other steps.
\bigskip\noindent
{\bf Step 1:}
Since we always arrive at {\sc AutLifting}~with $G/H$ simple,
either $\,{H \C }=G\,$ or $\,{H \C }= H$. In either case, ${\mathcal A}$ already normalizes ${H \C }/H$ so there is nothing
more to do.
\bigskip\noindent
{\bf Step 3:}
This is a group intersection. So it would be in polynomial time if just one group is
almost-solvable~\cite[ \S4.2]{Luk82} and, as indicated,
just a polynomial-time {\it set-stabilizer\/} if both groups are almost-solvable, as is the case here.
However, in this situation,
none of the machinery is even needed because
${\mathcal A}$ is listable. Note also that this is trivial when $\,{H \C } = G\,$ (which means ${\mathcal M} = {\mathcal A}\times{\mathcal B}$).
\bigskip\noindent
{\bf Step 5:}
As indicated, this is a set-stabilizer for a group that we now know to be almost-solvable.
\bigskip Now for {\bf Step 2:}
\medskip\noindent
Three approaches will be indicated, winding up with just set-stabilizers.
We state these for a
broader class than almost-solvable~groups.
For an integer constant $d > 0$,
let $\,\Gamma_d\,$ denote the class of finite
groups all of whose nonabelian composition factors lie in
$\,S_d$. Also, bear in mind that the polynomial timing
for $\,\Gamma_d\,$ groups immediately extends to situations
where we are in possession of a $\,\Gamma_d\,$
subgroup of polynomial index (see \S\ref{nice}.)~
In particular, these methods apply to
almost-solvable~groups.
\PROBLEM{
\medskip
{\sc Problem (I)}\\[1ex]
\hspace*{1em} {\sc Given:} $X,Y\le {\rm Sym}(\Omega)$ with $X\in \Gamma_d$.\\[1ex]
\hspace*{1em} {\sc Find:}
$X_Y$. \hfill (Where $X$ is acting on ${\rm Sym}(\Omega)$ via conjugation.)
\\[-.75ex]
}
\medskip\noindent
{\it Method.}
This was shown to be in polynomial-time by Luks and Miyazaki \cite{LM11}.
\footnote{While $\Gamma_d$ is slightly smaller
than the class available just for set-stabilization \cite{BCP82}, the restriction
is still needed for this method.} {\hfill $\Box$}
\bigskip
In our situation, we are trying to normalize a {\it polynomial-size\/} $Y$ and there
is a more elementary approach to this special situation.
\PROBLEM{
\medskip
{\sc Problem (II)}\\[1ex]
\hspace*{1em} {\sc Given:} $X,Y\le {\rm Sym}(\Omega)$ with $X\in \Gamma_d$ and
$|Y|= {\rm O}(|\Omega^{\rm const}|)$.\\[1ex]
\hspace*{1em} {\sc Find:}
$X_Y$
\\[-.75ex]
}
\medskip\noindent
{\sc Method:}~
Let ${\rm Sym}(\Omega)$ act on
$\Omega\times\Omega$ diagonally. Also, for $s\in{\rm Sym}(\Omega)$, let
$$\Delta_s \,:=\, \{(\omega,\omega^s) \mid \omega\in\Omega\}.
$$
Then for $y,x\in{\rm Sym}(\Omega)$,
$$\Delta_y^x\,=\, \Delta_{x^{-1} y x}.
$$
Thus, $x$ normalizes $Y$ iff $x$ stabilizes the collection
$\{\Delta_y\}_{y\in Y}$. So, finding $X_Y$ is a matter of finding the
subgroup of $X$ inducing automorphisms of a hypergraph. Miller \cite{Mil83b}
has shown that to be in polynomial time for $X\in \Gamma_d$. {\hfill $\Box$}
\bigskip
\begin{comment}
That's fine but if we want to avoid hypergraphs when we get to {\it the canonization analogue},\footnote{
We do know that hypergraph canonization {\it can\/} be done. However, our version has not been
published and Miller gave no details of his own.}
it is worth showing that Step 2 requires no more than set-stabilizer.
There is just one place where this requires more scrutiny. This occurs in the
``bottom-up'' approach in Oct3Notes (\S3.2).\footnote{
The ``top-down'' approach has another alternative.} Consider then a still
more special situation.
\end{comment}
Our situation is even more special.
\PROBLEM{
\medskip
{\sc Problem (III)}\\[1ex]
\hspace*{1em} {\sc Given:} $X,Y\le {\rm Sym}(\Omega)$ with $X\in \Gamma_d$ and
$|Y|= {\rm O}(|\Omega|^{\rm const})$;\\
\hspace*{4.7em} $K< Y$ with $K\vartriangleleft \langle X,Y\rangle$ and
we are able to list ${\rm Aut}(Y/K)$.
\\[1ex]
\hspace*{1em} {\sc Find:}
$X_Y$.
\\[-.75ex]
}
\medskip\noindent
{\sc Method:}~
Note that $X_Y=X_{Y/K}$.
For each $\sigma\in{\rm Aut}(Y/K)$, we find those $x\in X$ such that conjugation
by $x$ induces $\sigma$. This will be case iff
\medskip
\centerline{$\forall y\in Y: x^{-1}y x \,\in \, (yK)^\sigma $}
\noindent
which is the case iff
\medskip
\centerline{$\forall y\in Y, \exists z \in(yK)^\sigma: \Delta_y^x = \Delta_{z}$}
\medskip\noindent
(See Remark~\ref{cosetstab} for viewing these
``set-transporters'' as set-stabilizers.)~ It is feasible to run through all $y$ and then all $z$. {\hfill $\Box$}
\goodbreak
\bigskip
To apply the {\sc Problem~(III)} method in
our situation, $Y=\InnH{G}$, $K=\InnH{H}$.
Furthermore, we are only concerned with the case ${H \C }=H$, else
$Y=\InnH{G}=\InnH{H}$
which is already
normalized by ${\mathcal B}$. Thus $Y/K$ is simple.
So we can not only list
${\rm Aut}(Y/K)$ but we can even shorten the process by checking that
$x$ conjugates correctly on just $y_1, y_2\in Y$, where $y_1K, y_2K$ generate
$Y/K$.
\medskip
A further savings on the number of set-stabilizer calls can be realized by using
a quotient-group method of Kantor and Luks \cite[problem {P7}(ii)]{KL90}.
Finding the $x\in X$ such that $(yK)^x = (yK)^\sigma$ can be accomplished
with a single set-stabilizer.
\subsection{Top-down on a refinement of a characteristic series}
\label{topdown}
\medskip\noindent
Recall that a subgroup $H$ of a group $G$ is called {\it characteristic\/} if it is invariant
under ${\rm Aut}(G)$. A group with no proper characteristic subgroups is called
{\it characteristically simple} and is necessarily a product of isomorphic simple groups.
A {\it characteristic series\/} in $G$ is a chain
$G=K_0 \vartriangleright K_1 \vartriangleright K_2 \vartriangleright \cdots \vartriangleright K_r ={\bf 1}$
for which each $K_i$ is characteristic in $G$.
A characteristic series can be constructed
with characteristically simple quotients\linebreak {${K_i/K_{i+1}}\,$}
even if $G$ is a permutation group given by generators; see, e.g., \cite{KL90}. For groups
given by a Cayley table, the construction is elementary: find the minimal normal subgroups
of $G$ by considering the normal closures of all elements; these will each be
characteristically simple, so select those whose simple factors are of designated type
and let them generate the subgroup $K$. Continue the process with $G/K$, etc.
\begin{lemma}
\label{CharSer}
{\sl
Without loss of generality, we may assume that the series $G_0,G_1,G_2,\ldots,G_m$
in {\sc Comp\-Series\-Auto}~has a characteristic subseries
$$G=K_0 \vartriangleright K_1 \vartriangleright K_2 \vartriangleright \cdots \vartriangleright K_r ={\bf 1}$$
where each $K_i/K_{i+1}$ is characteristically simple.\footnote{
The Wagner-Rosenbaum composition series are already of the special type.}}
\end{lemma}
\noindent
{\sc Proof:}
We are given a composition series
$$G=G_0 \vartriangleright G_1 \vartriangleright G_2 \vartriangleright \cdots \vartriangleright G_m ={\bf 1}.
$$
As indicated above, we construct a {\it characteristic\/} series
$G=K_0 , K_1 , K_2 , \ldots, K_r ={\bf 1}$
with characteristically simple quotients $K_i/K_{i+1}$.
Refine the series between each $K_i$ and $K_{i+1}$ by inserting
$$K_i =(K_i\cap G_0)K_{i+1} \trianglerighteq (K_i\cap G_1)K_{i+1}
\trianglerighteq (K_i\cap G_2)K_{i+1} \trianglerighteq \cdots \trianglerighteq (K_i\cap G_m)K_{i+1} =K_{i+1}$$
and eliminate duplicates.
We again have a composition series and
automorphisms stabilizing the original series will stabilize this new one. Having computed
the automorphisms stabilizing the new series, we have an almost-solvable~group, so
cutting down the result to
stabilize the original series is done with set stabilizers.
{\hfill $\Box$}
\bigskip\noindent
{\it Method for Theorem~\ref{mainprobauto}.}
Having reset the series as in Lemma~\ref{CharSer},
successively compute the appropriate subgroup of ${\rm Aut}(K_0/K_i)$, where $(K_i)_i$ is
the embedded normal series as in Lemma~\ref{CharSer}.
In the base case $K_0/K_0$ is trivial.
For the inductive step, the call to
{\sc AutLifting} involves $K_0/K_{i+1}\,$ and its normal subgroup $K_i/K_{i+1}$.
We arrive with the inductive input ${\mathcal A} \le {\rm Aut}(K_0/K_{i})$.
The group ${\mathcal B}$ should consist of
the automorphisms of the semisimple group $H := K_i/K_{i+1}$ that fix
the composition
series induced on this section.
If $H$ is nonabelian, ${\mathcal B}$ is the direct product of the
automorphism groups of the simple factors. If $H$ is a product
of cyclic groups of prime order $p$, ${\mathcal B}$
can be viewed as the upper triangular matrices over ${\rm GF}(p)$.
Using the ${\mathcal L}_2$ method at each stage,
the procedure is in polynomial time for {{\it all groups}, but that seemed
to require hypergraph stabilizer in Step 2.
So let us revisit that step.
Suppose $H$ is nonabelian. Then ${\mathcal B}$ (which now fixes
the simple factors) is of polynomial size, e.g. $|{\mathcal B}|\le |H|^2$ in which
case Step 2 can be carried out
by testing each element of ${\mathcal B}$. This is not the case if $H$ is abelian, but
then we simply revert to the
${\mathcal L}_1$ method of \S\ref{firstL} for this round. {\hfill $\Box$}
\medskip
\section{Isomorphism matching fixed composition series}
\label{Iso}
\medskip
We prove Theorem~\ref{mainprob} via
a familiar technique
in isomorphism studies, applying the automorphism-group result to finding isomorphisms.
For example, an algorithm for finding automorphism groups of graphs will find the isomorphisms
between two connected graphs $X_1,X_2$ by finding the automorphism group of the disjoint
union $X_1\dot\cup X_2$. An analogous construction works here.
\medskip
We are given composition series
$$G_1=G_{1,0} \vartriangleright G_{1,1} \vartriangleright G_{1,2} \vartriangleright \cdots \vartriangleright G_{1,m} ={\bf 1}$$
$$G_2=G_{2,0} \vartriangleright G_{2,1} \vartriangleright G_{2,2} \vartriangleright \cdots \vartriangleright G_{2,m}={\bf 1}$$
Form the single subnormal series
$$G_1\!\times\!G_2=G_{1,0}\!\times\! G_{2,0}\vartriangleright G_{1,1}\!\times\! G_{2,1} \vartriangleright
G_{1,2} \!\times\! G_{2,2} \vartriangleright \cdots \vartriangleright G_{1,m}\!\times\! G_{2,m} ={\bf 1}\times{\bf 1}.$$
We directly accommodate the setup of \S\ref{bottomup}~to this situation. (The
method of \S\ref{topdown} could be adapted as well.)~
We now want the automorphisms of $G_1\!\times\!G_2$ that not only fix the terms in this series but, in doing so,
fix or switch the factors. If, for any $i$, we find that there are no relevant automorphisms of
$G_{1,i}\!\times\! G_{2,i}$ that switch factors, we exit with a negative
response to {\sc Comp\-Series\-Iso}.
Calls to {\sc AutLifting}~will now
have $G/H$ as the product of two isomorphic simple groups. For ${\mathcal A}$ we take the automorphisms
that fix or switch the factors. (That would always be the case if the simple groups are nonabelian.)
The rest of the discussion, including the various methods for Step 2, proceed as before.
\begin{remark}
{\rm A trivial technicality.
Our weak version
of Lemma~\ref{simplenice} does not quite say
${\rm Aut}({T\times{}T})\allowbreak\le{\rm Sym} (T\times T)$ is almost-solvable~for
simple nonabelian $T$ since $|{\rm Aut}(T\times T)| = 2 |{\rm Aut}(T)|^2$.
As already noted, Lemma~\ref{simplenice} could be strengthened,
but it is clear that the theory is unaffected by an extra factor of 2.
}
\end{remark}
\bigskip\goodbreak
\centerline{\sc acknowledgements}
\bigskip
The author is delighted to acknowledge communications with Laci Babai and
James Wilson that stimulated the development of these ideas
and improved their exposition.
\bigskip\medskip
\def\underline{\hspace*{4em}}{\underline{\hspace*{4em}}}
\bibliographystyle{amsplain}
| {
"timestamp": "2015-11-03T02:09:28",
"yymm": "1511",
"arxiv_id": "1511.00151",
"language": "en",
"url": "https://arxiv.org/abs/1511.00151",
"abstract": "In recent work, Rosenbaum and Wagner showed that isomorphism of explicitly listed $p$-groups of order $n$ could be tested in $n^{\\frac{1}{2}\\log_p n + O(p)}$ time, roughly a square root of the classical bound. The $O(p)$ term is entirely due to an $n^{O(p)}$ cost of testing for isomorphisms that match fixed composition series in the two groups. We focus here on the fixed-composition-series subproblem and exhibit a polynomial-time algorithm that is valid for general groups. A subsequent paper will construct canonical forms within the same time bound.",
"subjects": "Computational Complexity (cs.CC); Group Theory (math.GR)",
"title": "Group Isomorphism with Fixed Subnormal Chains",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.977022630759019,
"lm_q2_score": 0.7248702761768249,
"lm_q1q2_score": 0.708214664189298
} |
https://arxiv.org/abs/1908.11631 | Approximation Algorithms for Partially Colorable Graphs | Graph coloring problems are a central topic of study in the theory of algorithms. We study the problem of partially coloring partially colorable graphs. For $\alpha \leq 1$ and $k \in \mathbb{Z}^+$, we say that a graph $G=(V,E)$ is $\alpha$-partially $k$-colorable, if there exists a subset $S\subset V$ of cardinality $ |S | \geq \alpha | V |$ such that the graph induced on $S$ is $k$-colorable. Partial $k$-colorability is a more robust structural property of a graph than $k$-colorability. For graphs that arise in practice, partial $k$-colorability might be a better notion to use than $k$-colorability, since data arising in practice often contains various forms of noise.We give a polynomial time algorithm that takes as input a $(1 - \epsilon)$-partially $3$-colorable graph $G$ and a constant $\gamma \in [\epsilon, 1/10]$, and colors a $(1 - \epsilon/\gamma)$ fraction of the vertices using $\tilde{O}\left(n^{0.25 + O(\gamma^{1/2})} \right)$ colors. We also study natural semi-random families of instances of partially $3$-colorable graphs and partially $2$-colorable graphs, and give stronger bi-criteria approximation guarantees for these family of instances. | \section{Partial $2$-Coloring in the Semi-random model}
In this section, we give an efficient approximation algorithm for partial $2$-coloring problem in the semi-random model with tighter guarantees. The following theorem formally states our guarantees for this setting.
\TwoColRandom*
The algorithm for the above theorem (described as Algorithm \ref{alg:alg-2col-random}) is quite similar to P$3$C-Random algorithm, but overall, the algorithm and its analysis are much simpler. We begin by describing the algorithm.
\begin{algorithm}
\SetAlgoLined
For every vertex $v \in V$, compute a greedy triangle count as follows: \\
\For{$v \in V$}
{
Let $G_v = G[N_G(v)]$ be the graph induced on the neighborhood of $v$\;
Construct a maximal matching $T(v)$ in $G_v$ using greedy algorithm\;
Set $t(v) \gets |T(v)|$\;
}
Let $S \gets \{v \in V: t(v) \ge 2\epsilon n\}$\;
Let $G_0 = G[V\setminus S]$\;
Let $G_1 \subseteq G$ be the independent set obtained using the $2$-factor approximation for Vertex Cover on $G$\;
\If{$|{\rm vert}(G_0)| \geq |{\rm vert}(G_1)|$ and $G_0$ is bipartite}
{
Output bipartite graph $G_0$\;
}
\Else{
Output independent set $G_1$\;
}
\caption{P$2$C-Random} \label{alg:alg-2col-random}
\end{algorithm}
The key difference here is that the algorithm uses triangles as forbidden subgraphs for identifying bad vertices instead of neighborhoods with short odd cycles. As before, the algorithm broadly addresses two cases depending on the size of the maximum matching in $G[V_{\rm good}]$. Suppose the subgraph $G[V_{\rm good}]$ contains a linear sized matching $M$. Then, for every bad vertex $v \in V_{\rm bad}$, with high probability, at least one of the matching edges from $M$ will appear in the neighborhood of $v$, which together will form a triangle, which can then be used to identify the bad vertices. On the other hand, if the size of maximum matching in $G[V_{\rm good}]$ is small, then the subgraph $G[V_{\rm good}]$ and consequently $G$ must admit a small sized vertex cover. Therefore, using the greedy approximation algorithm for vertex cover, we can find a small sized vertex cover, whose complement must be a large independent set (which is $1$-colorable).
\subsection{Proof of Theorem \ref{thm:2col-random}}
Let $M \subseteq G[V_{\rm good}]$ be a {\em fixed matching of maximum size} in $G[V_{\rm good}]$, and let $m^* := |M|$ denote the size of the maximum matching. {\em We point out that the matching $M^*$ is not affected by the realization of edges between $V_{\rm good}$ and $V_{\rm bad}$ (i.e, the $E_0$ and $E_1$ edges)}. As before, we break the analysis into two cases depending on whether $m^*$ is small or large.
{\bf Case (i): $m^* \ge (8\epsilon/p^2) n$}: This case is similar to case (i) of the proof of Theorem \ref{thm:3col-random}. We begin by stating and proving two lemmas which say that the greedy triangle count $t(v)$ is small for all the good vertices, and large for all the bad vertices.
\begin{lemma} \label{lem:2col-(a)}
For every good vertex $v \in V_{\rm good}$, we have $t(v) \leq \epsilon n$
\end{lemma}
\begin{proof}
Fix a good vertex $v \in V_{\rm good}$, and let $T(v)$ be a set of edges as constructed in the algorithm. Observe that every edge $(a,b) \in T(v)$ along with vertex $v$ induces a triangle in $G$. Furthermore, since $G[V_{\rm good}]$ is bipartite (and hence triangle free), any triangle $T \subseteq G$ must contain at least one bad vertex. Therefore, as the vertex $v$ is good, every edge $e \in T(v)$ must contain at least one bad vertex. Finally, we observe that the edges in $T(v)$ are vertex disjoint, and there are at most $\epsilon n$ bad vertices, which together implies that $t(v) = |T(v)| \leq \epsilon n$.
\end{proof}
\begin{lemma} \label{lem:2col-(b)}
With probability at least $1 - e^{-O(\epsilon n)}$, for every vertex $v \in V_{\rm bad}$, we have $t(v) \ge 2\epsilon n$.
\end{lemma}
\begin{proof}
Let $G'$ be the subgraph on $G$ consisting of edges from $E_0$ (i.e,. the randomly sampled set of edges). Recall that $M = \{(a_i,b_i)\}_{i \in [m^*]}\subseteq G[V_{\rm good}]$ is the fixed maximum matching in $G[V_{\rm good}]$ of size $m^*$. Let $Z_i := \mathbbm{1}\Big(\{a_i,b_i \in N_{G'}(v)\}\Big)$ be the indicator variable for the event that $a_i,b_i$ are neighbors of $v$ in the graph $G'$. Then,
\begin{equation}
\E_{G'}\left[\sum_{i \in [m^*]}Z_i\right] = \sum_{i \in [m^*]} \Pr_{G'}\left[\{a_i,b_i \in N_{G'}(v)\}\right] = m^*p^2 \ge 8\epsilon n
\end{equation}
Furthermore, since the edges in $M$ are vertex disjoint, the random variables $Z_1,\ldots,Z_{m^*}$ are independent and identical. Therefore using Chernoff bound we get
\begin{equation} \label{eq:match-size}
\Pr_{G'}\left[\sum_{ i \in [m^*]} Z_i \le 4\epsilon n\right] \le \Pr_{G'}\left[\sum_{ i \in [m^*]} Z_i \le \frac12 \E \sum_{i \in [m^*]} Z_i \right] \le e^{-O(\epsilon n)}
\end{equation}
Let $M_v = \{(a_i,b_i): i\in [m^*],Z_i = 1\}$ be the set of matching edges from $M^*$ appearing in the neighborhood of $v$ in the graph $G'$. Furthermore, let $\tilde{M}_v$ be a maximum matching in the subgraph $G_V: = G[N_G(v)]$ induced on the neighborhood of $v$ (which contains both $E_0$ and $E_1$ edges). Then, by definition we have $|\tilde{M}_v| \ge |M_v|$. On the other hand, by construction, the set $T(v)$ is a maximal matching in the induced subgraph $G_v$. Since a maximal matching is a $2$-approximation to the maximum matching, it follows that $|T(v)| \ge |\tilde{M}_v|/2 \ge |M_v|/2 \ge 2\epsilon n$.
Therefore, for a fixed bad vertex $v \in V_{\rm bad}$, with probability at least $1 - e^{-O(\epsilon n)}$, we have $t(v) \ge 2\epsilon n$. The claim now follows by taking a union bound over all vertices $v \in V_{\rm bad}$.
\end{proof}
Therefore, combining Lemmas \ref{lem:2col-(a)} and \ref{lem:2col-(b)}, we know that with probability at least $1 - e^{-O(\epsilon n)}$, we have $t(v) \le \epsilon n$ if and only if $v \in V_{\rm good}$. Conditioned on this event, the set $S$ must exactly be the set of bad vertices, in which case $G[V \setminus S] = G[V_{\rm good}]$ is bipartite.
{\bf Case (ii): $m^* \le (8\epsilon/p^2) n$}: Since the size of maximum matching in $G[V_{\rm good}]$ is at most $(8\epsilon/p^2) n$, and $G[V_{\rm good}]$ is bipartite, by K\"onig's theorem (Theorem 2.1.1~\cite{diestel2012graph}), it follows that the minimum vertex cover of $G[V_{\rm good}]$ has size at most $(8\epsilon/p^2) n$. Then $G$ has a vertex cover of size at most $(8\epsilon/p^2) n + \epsilon n \leq (10\epsilon/p^2) n$. Therefore, the greedy approximation algorithm for vertex cover returns a vertex cover $S'$ of size at most $(20\epsilon/p^2)n$, and consequently, $V\setminus S'$ will be an independent set of size at least $(1 - (20\epsilon/p^2))n$.
{\bf Putting things together}: In case (i), the algorithm throws away at most $\epsilon n$ vertices and returns a $2$-colorable graph, with probability at least $1 - e^{-O(\epsilon n)}$. In case (ii), the algorithm throws away at most $O(\epsilon/p^2)n$ vertices, and returns an indpendent set. Combining the two cases gives us the guarantees for Theorem \ref{thm:2col-random}.
\section{Approximation algorithm for General Setting}
In this section, we prove our approximation guarantees in the adversarial model, as formally stated in the following theorem:
\ColMain*
\begin{algorithm}[h!]
\SetAlgoLined
Set $\Delta = n^{3/4}$\;
Solve the Partial-$3$-Coloring SDP (SDP-P$3$C):
\begin{alignat*}{4}
\mbox{minimize } & \sum_{i\in V} w_i \\
\mbox{subject to } & \langle v_i, v_j \rangle \leq -\frac{1}{2}+\frac{3}{2}z_{ij} &\qquad & \forall \{i,j\}\in E \\
& z_{ij} \leq w_i+w_j &\qquad & \forall \{i,j\}\in E \\
& 0 \le z_{ij}\leq 1 &\qquad & \forall \{i,j\}\in E \\
& 0 \le w_{i} \leq 1 &\qquad & \forall i\in V \\
& \|v_i\|^2=1 &\qquad & \forall i \in V
\end{alignat*}\\
\nonl
{\it(i) Thresholding}: \\
Let $S \gets \left\{i \in V| w_i \ge \gamma/3 \right\}$\;
Let $G' \gets G[V \setminus S]$ be the graph obtained after deleting $S$\;
\nonl{\it(ii) Coloring Large Degree vertices}:\\
\While{$\exists i \in G'$ such that ${\rm deg}_{G'}(i) \ge \Delta$}
{
Color $G'[\{i\} \cup N_{G'}(i)]$ using $\tilde{O}(n^{C\sqrt{\gamma}})$ colors using the algorithm guaranteed by Corollary \ref{corr:coloring}\;
Remove $\{i\} \cup N_{G'}(i)$ from $G'$\;
}
\nonl{\it(iii) Coloring Low Degree vertices}: \\
Use {\em randomized rounding} from Theorem \ref{thm:appx-col} to color the remaining vertices in $G'$\;
\caption{Partial-$3$-Coloring} \label{alg:3col-main}
\end{algorithm}
The algorithm for the above theorem is described in Algorithm \ref{alg:3col-main}. In the following subsections, we prove the correctness of the above algorithm. The proof of Theorem \ref{thm:3col-main} can broken down into the analysis of steps (i),(ii) and (iii) of the Partial-$3$-Coloring algorithm. Broadly, we show the following: In step (i), we show that the optimal of the SDP-P$3$C is small (i.e., at most $\epsilon n$), therefore by averaging, the fraction of large $w$ vertices is small. Furthermore, the graph induced on the surviving vertices must satisfy the edge constraints from the SDP with small slack $\gamma$, and therefore must be approximately vector $3$-colorable. As is usual in coloring algorithms, we first iteratively color large degree (i.e., $\ge \Delta$) vertices and their neighborhoods using small number of colors until the graph has degree bounded by $\Delta$ (Claim \ref{cl:step(ii)}). Finally, the remaining graph is also approximately vector $3$-colorable, and has degree bounded by $\Delta$. Therefore, using a hyperplane based randomized rounding procedure to iteratively find large independent sets in $G'$, we can give a $\tilde{O}(\Delta^{1/3 + O(\sqrt{\gamma})})$ coloring of the remaining vertices (Theorem \ref{thm:appx-col}). In the following subsection, we formally prove the steps described above.
To begin with, we first show that the thresholding step throws away at most a small fraction of vertices.
\begin{claim}[Removing Large Slack Vertices] \label{cl:step(i)}
Let $S \subset V$ be as constructed in the thresholding step. Then $|S| \le 3\epsilon n/\gamma$.
\end{claim}
\begin{proof}
We begin by showing that the optimal of SDP-P$3$C is at most $\epsilon n$. Let $V = V_{\rm good} \cup V_{\rm bad}$ be any partition of the vertex sets into good and bad vertices such that (a) $G[V_{\rm good}]$ is $3$-colorable and (b) $|V_{\rm bad}| \le \epsilon n$. Using this partition we now construct a $2$-dimensional feasible solution $(\widehat{v},\widehat{w},\widehat{z})$ to SDP-P$3$C as follows. We set the $\widehat{w}_i$ and $\widehat{z}_{ij}$ variables as
\[
\widehat{w}_{i}=
\begin{cases}
0, & \text{if } i \in V_{\rm good}\\
1, & \text{otherwise}
\end{cases}
\qquad\mbox{ and }\qquad \widehat{z}_{ij}=
\begin{cases}
0, & \text{if } i,j \in V_{\rm good}\\
1, & \text{otherwise}
\end{cases}
\]
Furthermore, we set $\{\widehat{v}_i\}_{i \in V_{\rm good}}$ be a vector $3$-coloring of $G[V_{\rm good}]$, and for every $i \in V_{\rm bad}$ we set $\widehat{v}_i = [1 \quad 0]$. We quickly verify that the $\widehat{v},\widehat{w}$ and the $\widehat{z}$ variables constructed as above form a feasible solution to the SDP. By construction, for every $i \in V$ we have $\widehat{w}_i \in [0,1]$ and $\|\widehat{v}_i\|^2 = 1$, and for every edge $(i,j) \in E$ we have $z_{ij} \in [0,1]$. Furthermore, for any edge $(i,j)$ we also have
\begin{equation*}
\widehat{z}_{ij} = \mathbbm{1}\Big(\left\{i \in V_{\rm bad}\right\} \vee \left\{j \in V_{\rm bad}\right\} \Big) \leq \mathbbm{1}\big(\left\{i \in V_{\rm bad}\right\} \big) + \mathbbm{1}\big(\left\{j \in V_{\rm bad}\right\} \big) = \widehat{w}_i + \widehat{w}_j
\end{equation*}
All that remains to verify is that the variables also satisfy the approximate vector coloring constraints. We look at two cases: if $i,j \in V_{\rm good}$, then $\widehat{v}_i,\widehat{v}_j$ come from the vector $3$-coloring of $G[V_{\rm good}]$ and therefore they satisfy $\langle \widehat{v}_i, \widehat{v}_j \rangle \leq -\frac12 \leq -\frac12 + \widehat{z}_{ij}$. On the other hand if $i \in V_{\rm bad}$ or $j \in V_{\rm bad}$ then by construction we have $\widehat{z}_{ij} = 1$, and therefore $\langle \widehat{v_i},\widehat{v}_j \rangle \le \|\widehat{v}_i\|\|\widehat{v}_j\| = 1 = -\frac12 + \frac32\widehat{z}_{ij}$.
Therefore, we have established that $(\widehat{z},\widehat{w},\widehat{v})$ are a feasible solution for SDP-P$3$C. Since by construction $\widehat{w}_i = \mathbbm{1}\left\{i \in V_{\rm good} \right\}$, and the $|V_{\rm bad}| \le \epsilon n$, it follows that the SDP optimal $\sum_{i \in V} w_i$ is at most $ \sum_{i \in V} \widehat{w}_i \le \epsilon n$. Therefore, using Markov's inequality, we get
\begin{equation*}
|S| = n\cdot \Pr_{i \sim V}\Big[w_i \ge \gamma/3\Big] \le n\cdot\frac{3\sum_{i \in V} w_i}{n\gamma} = \frac{3\epsilon n}{\gamma}
\end{equation*}
\end{proof}
From the above claim, the graph $G' = G[V \setminus S]$ induced on the remaining vertices satisfies the following properties:
\begin{itemize}
\item[1.] The graph $G'$ contains at least $(1-3\epsilon/\gamma)n$ vertices.
\item[2.] The graph $G'$ is $(3,\gamma)$-vector colorable. In particular, the vectors $(v_i)_{i \in V \setminus S}$ themselves are a $(3,\gamma)$-vector coloring of $G'$.
\end{itemize}
The second point shall be used crucially in the analysis of the remaining two steps. The next claim bounds the number of colors used while coloring the large degree vertices in step (ii).
\begin{claim}[Degree Reduction] \label{cl:step(ii)}
In step (ii), over all the iterations of the while loop, the algorithm uses at most $(n/\Delta)\tilde{O}\left(n^{C\sqrt{\gamma}}\right)$ colors, where $C> 0$ is a constant.
\end{claim}
\begin{proof}
Fix any vertex $i \in G'$, and let $\tilde{G}_i = G'[N(i)]$ the graph induced on the neighborhood of vertex $i$. Since the graph $G'$ is $(3,\gamma)$-vector colorable, using Lemma \ref{lem:appx-2-col(a)} we know that $\tilde{G}_i$ is $(2, 4\gamma)$-vector colorable. Furthermore, from Lemma \ref{lem:appx-2-col(b)}, we know that $G'$ does not contain odd cycles of length at most $1/(8\sqrt{4\gamma})$. Therefore, we can use Corollary \ref{corr:coloring} to obtain a $\tilde{O}(n^{C\sqrt{\gamma}})$ coloring of $\tilde{G}_i \cup \{i\}$. Finally, note that each iteration of the for loop removes and colors at least $\Delta + 1$ vertices of the graph. Therefore, the total number of iterations of the for loop is bounded by $n/\Delta$. Since in each such iteration we can color the vertex and its neighborhood using $n^{C\sqrt{\gamma}}$ number of colors, the claim follows.
\end{proof}
After steps $(i)$ and $(ii)$, we are left with the graph $G'=(V',E')$ which is $(3,\gamma)$-vector colorable graph and has degree at most $\Delta$. In particular, for every edge $(i,j) \in E'$, the corresponding vectors satisfy $\langle v_i , v_j \rangle \le -\frac12 + \gamma$. Since the independent set based rounding technique~\cite{KMS98}~\cite{AC06} for coloring vector $3$-colorable graphs is \emph{robust}, we can still use it to round the vector coloring of approximately $3$-colorable graphs with similar guarantees, as formally stated in the following theorem.
\begin{theorem} \label{thm:appx-col}
Let $G = (V,E)$ be a graph with maximum degree $\Delta$ which is $(3,\alpha)$-vector colorable. Then there exists an efficient randomized algorithm that can color it using $O\left((\ln \Delta)^{1 / 2} \Delta^{\frac{\frac{3}{4}+\alpha-{{\alpha}^2}}{(\frac{3}{2}-\alpha)^2} }\ln{n}\right)$ colors.
In particular, if $\alpha \le 1/10$, then the algorithm uses at most $\tilde{O}\left((\ln \Delta)^{1 / 2} \Delta^{{\frac{1}{3}+10\alpha}} \right)$, where $\tilde{O}$ hides polylogarithmic factors in $n$.
\end{theorem}
The proof of the above theorem is an extension of the proofs from \cite{KMS98,AC06} to the setting of approximately vector $3$-colorable graphs. We defer the proof to Appendix \ref{sec:3col-indset}. Instantiating the above theorem with $G = G'$ and $\alpha = \gamma$, we get that $G'$ is colored using $\tilde{O}(\Delta^{1/3 + 10\gamma})$ colors. Overall, the algorithm throws away at most $3 \epsilon/\gamma$ fraction of vertices in step (i). Furthermore, it uses a total of $\tilde{O}\left((n/\Delta)n^{O(\sqrt{\gamma})} + \Delta^{1/3 + 10\gamma}\right)$ colors in steps (ii) and (iii) respectively. Setting $\Delta = n^{3/4}$ in the previous expression, we get that the algorithm uses at most $\tilde{O}(n^{1/4 + O(\sqrt{\gamma})})$ colors. This concludes the analysis of the Partial-$3$-Coloring algorithm and the proof of Theorem \ref{thm:3col-main}.
\section{Introduction}
Graph coloring problems are a central topic of study in the theory of algorithms
\cite{wigderson_1983,KMS98,AG11,kawarabayashi_thorup_2017}.
An undirected graph $G=(V,E)$ is said to be $k$-colorable if there exists an assignment of colors
$f:V \to [k]$ such that $f(u) \neq f(v)$ for each $\set{u,v} \in E$.
For a graph $G$, the minimum value of $k$ for which it is $k$-colorable is called its chromatic number.
Computing a $3$-coloring of a $3$-colorable graph is a fundamental NP-hard problem.
Efficiently computing a coloring of a $3$-colorable graph which only uses a few colors is a major
open problem in the study of algorithms.
The current best known algorithm colors a $3$-colorable graph on $n$ vertices using $O(n^{0.199})$ colors \cite{kawarabayashi_thorup_2017}.
We study the problem of coloring partially colorable graphs.
\begin{definition}
\label{def:pkc}
An undirected graph $G = (V,E)$ is defined to be $\alpha$-partially $k$-colorable, denoted by
$\alpha$-\pkc{k}, if there
exists a subset $V_{\rm good} \subset V$ such that $\Abs{V_{\rm good}} \geq \alpha \Abs{V}$ and the graph
induced on $V_{\rm good}$ is $k$-colorable. We will call such a set $V_{\rm good}$ the set of {\em good} vertices,
and $V_{\rm bad} \defeq V \setminus V_{\rm good}$ the set of {\em bad} vertices.
\end{definition}
We remark that for a given graph the partitioning of the vertex set $V$ into $V_{\rm good}$ and $V_{\rm bad}$ may not be unique. In such cases, the claims we make in this paper will hold for any such fixed partition.
It is well known that for a fixed $k$, the problem of determining whether a given graph is $k$-colorable
is an NP-hard problem \cite{Karp72}. Therefore, determining whether a graph belongs to $1$-\pkc{k}
is an NP-hard problem, and hence, computing the largest value of $\alpha$ for which a graph
belongs to $\alpha$-\pkc{k} is also an NP-hard problem.
Note that a graph that is $(1-\epsilon)$-partially $3$-colorable can have chromatic number
as large as $\Abs{V_{\rm bad}} = \epsilon n$. Therefore, the notion of the chromatic number of the graph does not
capture the structural property ($3$-colorability) satisfied by most of the graph.
Partial $k$-colorability is a more robust stuctural property than $k$-colorability.
Therefore, for graphs that arise in practice, partial $k$-colorability might be a better notion to
to use than $k$-colorability, since data arising in practice often contains various forms of noise;
the notion of bad vertices can be used to capture some types of noisy vertices in the graph.
\paragraph{Other notions of partial $k$-coloring.}Another related notion
of partial coloring is the following.
\begin{definition}
\label{def:pkc-edge}
An undirected graph $G = (V,E)$ is defined to be $\alpha$-partially $k$-colorable,
if there exists a coloring of the vertices $f : V \to [k]$ such that for at least
$\alpha \Abs{E}$ edges $\set{u,v}$, $f(u) \neq f(v)$.
\end{definition}
This definition, which asks that the coloring should ``satisfy'' at least $\alpha$
fraction of the edges, can be viewed as the {\em edge} version of partial $k$-colorability,
whereas Definition \ref{def:pkc} can be viewed as the {\em vertex} version of partial $k$-colorability.
For a fixed constant $k$, computing the maximum value of $\alpha$ for which the input graph satisfies
Definition \ref{def:pkc-edge} can be formulated as a Max-$2$-{\sf CSP} with alphabet size $k$; approximation algorithms
for Max-$2$-{CSP}s have been extensively studied in the literature \cite{R08,RS09a,BRS11} etc.
Therefore, we focus our attention on Definition \ref{def:pkc}.
\subsection{Our Results}
We give an efficient (bi-criteria) approximation algorithm for coloring partially $3$-colorable graphs.
\begin{restatable}{thm}{ColMain} \label{thm:3col-main}
There exists a polynomial time algorithm that takes as input
a $(1-\epsilon)$-\pkc{3} ~graph $G = (V,E)$ and any fixed choice of $\gamma \in [\epsilon, 1/100]$,
and produces a set $S \subset V$ such that $\Abs{S} \leq(3\epsilon/\gamma) \Abs{V}$
and a coloring of $V \setminus S$ using $\tilde{O}(n^{0.25 + O(\gamma^{1/2})})$ colors\footnote{
$\tilde{O}(\cdot)$ hides factors polylogarithmic in $n$.}.
\end{restatable}
We point out that the above theorem gives a bi-criteria approximation guarantee which exhibits the tradeoff between the size of the set $S$, and the number of colors used to color the remaining graph $G[V\setminus S]$. In particular, setting $\gamma = \sqrt{\epsilon}$ in the above theorem gives us the following guarantee. Given a $(1-\epsilon)$-\pkc{3} graph, one can color $(1 - \sqrt{\epsilon})$-fraction of its vertices using $\tilde{O}(n^{0.25 + \epsilon^{1/4}})$-colors. Using similar techniques we can give an efficient approximation algorithm for the partial $2$-coloring setting as well. For completeness, we formally state the result below{\footnote{We implicitly use the algorithm in the degree reduction step of the algorithm from Theorem \ref{thm:3col-main}. See Claim \ref{cl:step(ii)} for details.} :
\begin{restatable}{prop}{TwoColGen} \label{prop:2col-gen}
There exists a polynomial time algorithm that takes as input
a $(1-\epsilon)$-\pkc{2} graph $G = (V,E)$ and any fixed choice of $\gamma \in [\epsilon, 1/100]$,
and produces a set $S \subset V$ such that $\Abs{S} \leq(4\epsilon/\gamma) \Abs{V}$
and a coloring of $V \setminus S$ using $\tilde{O}(n^{2\gamma})$ colors.
\end{restatable}
We also study a semi-random family of partially colorable graphs $\alpha$-\pkcr{k}, which we define as follows.
\begin{definition}
\label{def:pkcr}
An instance of $\alpha$-\pkcr{k} is generated as follows.
\begin{enumerate}
\item Let $V$ be a set of $n$ vertices. Arbitrarily partition $V$ into sets $V_{\rm good}$ and $V_{\rm bad}$
such that $\Abs{V_{\rm good}} \geq \alpha n$.
\item Add edges between an arbitrary number of arbitrarily chosen pairs of vertices in
$V_{\rm good}$ such that the graph induced on $V_{\rm good}$ is $k$-colorable.
\item Add edges between an arbitrary number of arbitrarily chosen pairs of vertices in $V_{\rm bad}$.
\item Between each pair of vertices in $V_{\rm good} \times V_{\rm bad}$, independently add an edge with probability
$p$. We call this set of edges $E_0$.
\item Add arbitrary number of edges between pairs of vertices of $V_{\rm good} \times V_{\rm bad}$. We call this set of edges $E_1$.
\end{enumerate}
Output the resulting graph.
\end{definition}
In the study of approximation algorithms for NP-hard problems, there have been many works
studying algorithms random and semi-random instances of various problems \cite{BS95,FK01,KMM11,MMV12,MMV14}.
Random and semi-random instances are often good models
for instances arising in practice; designing algorithms specifically for such instances,
whose performance guarantee is significantly better than guarantees for general instances,
could have more applications in practice.
Moreover, from a theoretical perspective, designing algorithms for semi-random instances
helps us to better understand what aspects of a problem make it intractable.
We study our semi-random model $\alpha$-\pkcr{k} for the same reasons.
The following is our main result.
\begin{restatable}{thm}{ColRandom} \label{thm:3col-random}
Suppose there exists an efficient algorithm which colors a $3$-colorable graph using $n^{\theta}$ colors. Then the following holds for all choices of $\epsilon = \Omega(\log n/n)$ and $p \ge (\epsilon\theta^{-2})^{O(\theta)}$. There exists a polynomial time algorithm that takes as input a graph $G$ sampled from
$(1-\epsilon)$-\pkcr{3} and produces a set $S$ such that $\Abs{S} = O\paren{ \epsilon\theta^{-2} n p^{-(O(1/\theta))}}$
and a coloring of $V \setminus S$ using at most $n^{\theta}$ colors with high probability.
Moreover, the algorithm runs in time $n^{O(1/\theta)}{\rm poly}(n)$.
\end{restatable}
In particular, instantiating the above theorem with the algorithm from \cite{kawarabayashi_thorup_2017}, w.h.p., we can colors $(1- O(\epsilon))n$ fraction of vertices with $\tilde{O}(n^{0.199})$-colors. We also study the partial $2$-coloring problem in the semi-random setting. Our guarantees for this setting are as follows:
\begin{restatable}{thm}{TwoColRandom} \label{thm:2col-random}
Let $\epsilon = \Omega(\log n/n)$ and $p > \sqrt{\epsilon}$. Then, there exists a polynomial time algorithm that takes as input a graph $G$ sampled from
$(1-\epsilon)$-\pkcr{2}, and with high probability, produces a set $S \subseteq V$ such that $\Abs{S} = O\paren{ \epsilon n p^{-2}}$ and the induced subgraph on the remaining vertices $G[V\setminus S]$ is $2$-colorable.
\end{restatable}
In particular, in the above theorem the number of vertices removed is bounded by $O(\epsilon n)$ which is stronger than the best known bound of $O(\sqrt{\log n}.\epsilon n)$~\cite{ACMM05} in the adversarial setting.
\subsection{Related Work}
\label{sec:related}
\paragraph{$3$-colorable graphs.}
There is extensive literature on algorithms for coloring $3$-colorable graphs. Wigderson \cite{wigderson_1983} gave a simple combinatorial algorithm that used $O(n^{\frac{1}{2}})$ colors. Blum \cite{Blum94} improved the number of colors used to $\tilde O(n^\frac{3}{8})$. These algorithms used purely combinatorial techniques. Karger, Motwani and Sudan \cite{KMS98} used semidefinite programming to develop an algorithm, which when balanced with Wigderson's technique \cite{wigderson_1983} used $\tilde{O}({n}^\frac{1}{4})$ colors. Blum and Karger \cite{blum_karger_1997} improved the number of colors used to $\tilde{O}({n}^\frac{3}{14})$ by combining the techniques used in \cite{Blum94} and \cite{KMS98}. Arora, Chlamtac and Charikar \cite{AC06} got the bound down to $\tilde{O}({\Delta}^{0.21111})$ using techniques from the ARV algorithm \cite{arora2009expander}, which was further improved by Chlamtac \cite{chlamtac_2007} to $\tilde{O}({n}^{0.2072})$ using SDP hierarchies. Using new combinatorial techniques, Kawarabayashi and Thorup improved the approximation bound to $\tilde{O}({n}^{0.2049})$ in \cite{kawarabayashi_thorup_2012}. Subsequently, by combining their techniques with \cite{chlamtac_2007}, they were able to give a approximation of $\tilde{O}({n}^{0.19996})$ \cite{kawarabayashi_thorup_2017}, which is the current state of the art.
\paragraph{Partially $2$-colorable graphs.}
The partial $2$-coloring problem, better known as Odd Cycle Transversal (OCT) in the literature, has also been studied extensively. Formally, the setting here is as follows. We are given a $(1-\epsilon)$-partially $2$-colorable graph $G = (V,E)$ and the objective is to find a set $S$ of minimum size such that $G[V\setminus S]$ is $2$-colorable (i.e., odd cycle free). Yannakakis first showed that it is NP-Complete in \cite{Yann78}. Later, Khot and Bansal~\cite{KB09} showed that OCT is hard to approximate to any constant factor, assuming the Unique Games Conjecture. From the algorithmic side, via a reduction through the Min2CNF Deletion problem, \cite{GVY96} gave a $O(\log n)$ approximation for the problem. This was later improved to $O(\sqrt{\log n})$ by \cite{ACMM05} by using techniques from the Arora-Rao-Vazirani~\cite{arora2009expander} algorithm for sparsest cut. This problem has also been studied under the lens of parameterized complexity. In \cite{RVS04}, Reed et al. showed that OCT is fixed parameter tractable when parameterized by the number of bad vertices, following which a sequence of works \cite{KB09}\cite{NVS12}\cite{LNVS14} gave algorithms with improved running times.
\paragraph{Partially $3$-colorable graphs}
In contrast to the $3$-colorable setting, there has been very little work on coloring partially $3$-colorable graph. The paper which is closest to our setting is by Kumar, Louis and Tulsiani~\cite{kumar_louis_tulsiani}, which also addresses the partial $3$-coloring problem, albeit in a more restrictive setting. Assuming that the $(1-\epsilon)$-partially $3$-colorable graph has threshold rank $r$ and the $3$-coloring on the good vertices satisfies certain psuedorandomness properties, they give an algorithm which $3$-colors $1-O(\gamma + \epsilon)$ fraction of vertices in time $(r.n)^{O(r)}$.
\paragraph{Graph problems in Semi-random Models}
The semi-random model used in this paper is similar to semi-random models which have been considered for the Max-Independent Set problem~\cite{BS95}~\cite{FK01}~\cite{stein17}~\cite{MMT18}. Semi-random models offer a natural way of understanding the complexity of problems in settings which are less restrictive than worst case complexity, but are still far from being average case. While semi-random models were first introduced for studying graph coloring in ~\cite{BS95}, it has also subsequently been used to study several other fundamental problems such as Unique Games~\cite{KMM11}, Graph Partitioning~\cite{MMV12}, Clustering~\cite{MMV14}, to name a few.The problem of coloring $3$-colorable graphs has also been studied in average-case and planted models. Alon and Kahale \cite{AK97} gave an efficient algorithm that finds an exact $3$-Coloring of a random $3$-Colorable graph with high probability. David and Fiege~\cite{FR16} studied the complexity of finding a planted random/adversarial $3$-coloring for both adversarial and random host graphs.
\section{Algorithm for Semi-random instances}
In this section, we prove Theorem \ref{thm:3col-random}, which we again state here for convenience.
\ColRandom*
We begin by describing the algorithm for the semi-random setting:
\begin{algorithm}
\SetAlgoLined
Let $\mathcal{A}$ be the algorithm which can color $3$-colorable graphs using $n^\theta$ colors\;
Set $\delta = \theta/10$\;
\nonl\texttt{\\}
\nonl\{{\it Many short odd cycles}\}: \\
{
\For{every vertex $v \in V$}
{
Let $G_v := G[N_{G}(v)]$ the subgraph induced by the neighborhood of $G$\;
Greedily construct a maximal set $\mathcal{C}_v$ of vertex disjoint odd cycles of length at most $1/\delta$ in $G_v$\;
}
Construct set $S \gets \{v \in V : |\mathcal{C}_v| \ge 2\epsilon n \}$\;
Let $G_0 \gets G[V \setminus S]$ be the graph obtained after deleting $S$\;
Let $\sigma_1$ be the coloring of $V \setminus S$ obtained by running algorithm $\mathcal{A}$ on $G_0$. Let $L$ denote the number of colors used by the algorithm\;
}
\nonl\texttt{\\}
\nonl\{{\it Few short odd cycles}\}: \\
{
Compute a maximal set $\mathcal{C} = \left\{C_1,C_2,\ldots,C_m\right\}$ of vertex disjoint odd cycles in $G$ of length at most $1/\delta$ using greedy algorithm\;
Let $V' = V \setminus \left( \bigcup_{i \in [m]} {\rm vert}(C_i)\right)$\;
Use the algorithm guaranteed by Corollary \ref{corr:coloring} to give a $\tilde{O}\left({n^{2\delta}}\right)$ coloring $\sigma_2$ of $G[V']$\;
}
\nonl\texttt{\\}
\nonl\{{\it Output best coloring}\}: \\
\If{$|S| \le \epsilon n$ and $L \le n^\theta$}
{Output coloring $\sigma_1$ of $V \setminus S$}
\Else
{
Output coloring $\sigma_2$ of $V'$\;
}
\caption{P$3$C-Random}
\end{algorithm}
The algorithm proceeds case wise depending on whether there exists many vertex disjoint short odd cycles in $G$. If it does, then since $V_{\rm bad}$ is small, $G[V_{\rm good}]$ must also contain many vertex disjoint odd cycles. We show that these short cycles will show up in the neighborhood of the bad vertices with high probability, which can be used to identify them. On removing these vertices, we will be left with a $3$-colorable graph. On the other hand, if the number of short odd cycles is small, we can remove them. The remaining graph will still contain most of the vertices and will be short odd cycle free. We can then use Lemma \ref{lem:ramsey} to recover large independent sets. Finally, since the odd cycles we consider are of length at most $1/\delta$, we can work with a \emph{maximal} set of vertex disjoint odd cycles, instead of the largest cardinality set of vertex disjoint odd cycles, while only losing a factor of $1/\delta$ in our analysis.
\subsection{Correctness of the P$3$C-Random algorithm}
Let $\mathcal{C}^* = \{C^*_1,C^*_2,\ldots,C^*_{m^*}\}$ be a {\em fixed largest cardinality set of vertex disjoint odd cycles} of length at most $1/\delta$ in $G[V_{\rm good }]$. {\em In particular, $\mathcal{C}^*$ and consequently $m^*$, does not depend on the realization of the random and adversarial edges (i.e., the $E_0$ and $E_1$ edges) between $V_{\rm good}$ and $V_{\rm bad}$}. We break our analysis into two cases depending on whether $m^*$ is small or large.
{\bf Case (i)} $m^* > 4\epsilon n/(\delta p^{1/\delta})$ : For ease of exposition, we say that an odd cycle $C$ in graph $G$ is \emph{good} if it consists of only good vertices, otherwise we call it {\em bad}. The first claim shows the set $\mathcal{C}_v$ must be small for good vertices.
\begin{claim} \label{cl:3col-a}
For every good vertex $v \in V$, we have $|\mathcal{C}_v| \le \epsilon n$.
\end{claim}
\begin{proof}
Fix a good vertex $v \in V_{\rm good}$. We claim that a good cycle $C$ can never appear in the neighborhood of a good vertex. For contradiction, let $C$ be a good odd cycle appearing in the neighborhood of $v$. Let $\tilde{G} = G\Big[{\rm vert}(C) \cup \{v\}\Big]$ be the subgraph induced on the vertex $v$ and the vertices from cycle $C$. Since $\tilde{G} \subseteq G[V_{\rm good}]$, the subgraph $\tilde{G}$ is also $3$-colorable. Hence, the neighborhood of $v$ in the induced subgraph $\tilde{G}$ must be $2$-colorable, and therefore it cannot contain odd cycles, and in particular $C$. This gives us the contradiction.
Hence, any odd cycle which appears in the neighborhood $N_G(v)$ must be bad. Since the number of bad vertices is bounded by $\epsilon n$, and the cycles in $\mathcal{C}_v$ are vertex disjoint, the claim follows.
\end{proof}
On the other hand, with high probability, we show that $|\mathcal{C}_v|$ is large for all the bad vertices.
\begin{claim} \label{cl:3col-b}
With probability at least $1 - e^{-O(\epsilon n)}$, every vertex $v \in V_{\rm bad}$ satisfies $|\mathcal{C}_v| \ge 2\epsilon n$.
\end{claim}
\begin{proof}
Consider the subgraph $G'(V,E_0)$ consisting of edges from $E_0$ (i.e., the randomly distributed set of edges). Fix a bad vertex $v \in V_{\rm bad}$, and let $G_v = G[N_{G}(v)]$ denote the subgraph induced by the neighborhood of $v$. We shall first give a high probability lower bound on the number of odd cycles from $\mathcal{C}^*$ which can appear in $N_{G}(v)$. Recall that $|\mathcal{C}^*| = m^*$. We also point out again that the choice of $\mathcal{C}^*$ is not affected by the choice of $E_0$ and $E_1$ edges, and can be fixed ahead.
For every $i \in [m^*]$, we define $Z_i := \mathbbm{1}\Big({\rm vert}(C^*_i) \subseteq N_{G'}(v)\Big)$ to be the indicator random variable that the $i^{th}$ cycle appears in the neighborhood of vertex $v$ in the graph $G'$. Note that these random variables depend only on the realization of the $E_0$ edges. Then we have
\begin{eqnarray*}
\E_{G}[Z_i] \ge \Pr_{E_0}\Big[{\rm vert}(C^*_i) \subseteq N_{G}(v) \Big] &\geq& \Pr_{E_0}\Big[{\rm vert}(C^*_i) \subseteq N_{G'}(v) \Big] \\
&=& \Pr_{E_0}\Big[\forall j \in {\rm vert}(C^*_i), j \in N_{G'}(v)\Big] \\
&\ge& p^{|C^*_i|} \ge p^{1/\delta}
\end{eqnarray*}
Here the last step uses the fact that any cycle $C^*_i \in \mathcal{C^*}$ has length at most $1/\delta$. It follows that
\begin{equation}
\E_G\left[\sum_{i \in [m^*]} Z_i\right] = \sum_{i \in [m^*]}\E_G[Z_i] \ge m^*p^{1/\delta} \ge (4\epsilon/\delta)n
\end{equation}
Furthermore, since the cycles $C^*_1,C^*_2,\ldots,C^*_{m^*}$ are vertex disjoint, the corresponding random variables $Z_1,Z_2,\ldots,Z_{m^*}$ are also independent. Therefore using Chernoff bound we get that
\begin{equation} \label{eq:cycle-bound}
\Pr_G\left[\sum_{i \in [m^*]} Z_i < (2\epsilon/\delta)n \right] \le \Pr_G\left[\sum_{i \in [m^*]} Z_i < \frac12 \E\Big[\sum_{i \in [m^*]} Z_i \Big]\right] \le e^{-\epsilon n/4\delta}
\end{equation}
Now let $\mathcal{C}^*_v = \{C^*_i : i \in [m^*], Z_i = 1\}$ be the set of cycles from $\mathcal{C}^*$ which appear in the neighborhood of $v$ in graph $G$ due to the $E_0$ edges. Furthermore, let $\widetilde{\mathcal{C}}_v$ be a \emph{largest cardinality} set of vertex disjoint odd cycles of length at most $1/\delta$ in $G_v$ (which contains edges from both $E_0$ and $E_1$). Then by definition we have $|\widetilde{\mathcal{C}}_v| \ge |\mathcal{C}^*_v|$. On the other hand, by construction, the set $\mathcal{C}_v$ is a \emph{maximal set} of such vertex disjoint odd cycles in $G_v$, and therefore, it must be a $\delta$-approximation to the largest cardinality set $\widetilde{\mathcal{C}}_v$ i.e., $|\mathcal{C}_v| \ge \delta|\widetilde{\mathcal{C}}_v|$ (see Proposition \ref{prop:max-bound}). Therefore using Equation \ref{eq:cycle-bound}, with probability at least $1 - e^{-\epsilon n/4\delta}$ we have
\begin{equation*}
|\mathcal{C}_v| \ge \delta|\widetilde{\mathcal{C}}_v| \ge \delta|\mathcal{C}^*_v| \ge 2\epsilon n
\end{equation*}
Hence, for any fixed vertex $v \in V_{\rm bad}$, w.h.p. we have $|\mathcal{C}_v| \ge 2 \epsilon n$. Therefore, by a union bound and using the lower bound on $\epsilon$, we get that $\Pr_{G}\Big[\exists v \in V_{\rm bad} : |\mathcal{C}_v| < 2\epsilon n\Big] \leq \epsilon n e^{-\epsilon n/4\delta} \leq e^{-\epsilon n/8\delta}$.
\end{proof}
Combining the two claims above, it follows that w.h.p. the set $(V\setminus S)$ must exactly be the set of good vertices, and therefore $G[V\setminus S]$ must be $3$-colorable. Hence algorithm $\mathcal{A}$ will give a $n^{\theta}$ coloring of $G[V \setminus S]$.
{\bf Case (ii)} $m^* \le 4\epsilon n/(\delta p^{1/\delta})$. Let $\mathcal{C} = \mathcal{C}_{\rm good} \uplus \mathcal{C}_{\rm bad}$ be the partition of $\mathcal{C}$ into the set of good and bad cycles respectively. Then, since $\mathcal{C}_{\rm good}$ is a set of vertex disjoint odd cycles of length at most $1/\delta$ in $G[V_{\rm good}]$, it follows that $|\mathcal{C}_{\rm good}| \le |\mathcal{C^*}| \le 4\epsilon n/(\delta p^{1/\delta})$. Furthermore, by arguments similar to the proof of Claim \ref{cl:3col-a}, we have $|\mathcal{C}_{\rm bad}| \le \epsilon n$. Therefore, combining the two bounds, we have $|\mathcal{C}| \leq 5\epsilon n/(\delta p^{1/\delta})$. Since every cycle $C \in \mathcal{C}$ contains at most $1/\delta$ vertices, the total number of vertices thrown away at this step is at most $5\epsilon n/(\delta^2 p^{1/\delta})$. Furthermore, using the {\em maximality} of $\mathcal{C}$, we know that the induced subgraph $G' = G[V']$ must be free of odd cycles of length at most $1/\delta$. Therefore, using Corollary \ref{corr:coloring}, we can color $G'$ using $\tilde{O}(n^{2\delta})$ colors. This concludes the analysis of case (ii).
{\bf Putting Things Together}: If case (i) holds, then w.h.p., in the {\em Many short odd cycles} block of the algorithm, the set $S$ constructed is identical to $V_{\rm bad}$, in which case the algorithm $\mathcal{A}$ will find a $n^\theta$-coloring of $G[V\setminus S] = G[V_{\rm good}]$. In particular, this implies that the conditions of the "if" block will be satisfied and the algorithm will return a $n^\theta$-coloring of $(1-\epsilon)n$ vertices.
On the other hand, if case (ii) holds, we know that $m \leq 5\epsilon n/(p^{1/\delta}\delta)$, and the {\em Few short odd cycles} block deletes at most $5\epsilon n/(p^{1/\delta}\delta^2)$ vertices, and colors the remaining vertices using $\tilde{O}(n^{2\delta})$ colors. Then the {\em else} block of the algorithm will return a $\tilde{O}(n^{2\delta})$ coloring of $\Big(1 - 5\epsilon n/(p^{1/\delta}\delta^2)\Big)n$ vertices. Since the else block is evaluated only when the conditions of the {\em if} block are not satisfied, it follows that in this case, the algorithm will throw away at most $\max\left(\epsilon n, 5\epsilon n/(\delta^2p^{1/\delta})\right) = O(\epsilon n /\delta^2p^{1/\delta})$ vertices, and color the remaining graph with at most $\max\left(n^{\theta},\tilde{O}(n^{2\delta})\right) = n^\theta$ colors.
Combining the two above cases gives us Theorem \ref{thm:3col-random}.
\section{Proof of Theorem \ref{thm:appx-col}} \label{sec:3col-indset}
The proof of the theorem is adapted from \cite{KMS98,AC06} and the rounding algorithm used is identical to theirs. The proof of the theorem goes through the following lemma which says that there exists a randomized algorithm which can find large sized independent sets in approximately vector $3$-colorable graphs with bounded degree.
The proof presented here is adapted from the proof for the exact $3$-colorable case in Section 13.2 \cite{williamson2011design}.
\begin{lemma} \label{lem:3col-indset}
Let $G = (V,E)$ be a graph on $n$ vertices with maximum degree $\Delta$ which is $(3,\alpha)$-vector colorable, where $\alpha \leq 1/2$. Then there exists an efficient randomized algorithm which finds an independent set of size $ \Omega\left(n(\ln \Delta)^{-1 / 2} \Delta^{-\frac{\frac{3}{4}+\alpha-{{\alpha}^2}}{(\frac{3}{2}-\alpha)^2} }\right)$ with high probability.
\end{lemma}
\begin{proof}
Let $v_1,v_2,\ldots,v_n \in \mathbbm{R}^d$ be a $(3,\alpha)$-vector coloring of $G$. Consider the following randomized hyperplane rounding procedure for finding a independent set:
\begin{itemize}
\item[1.] Draw a random vector $r \sim N(0,1)^d$ by picking each coordinate independently from a standard Gaussian.
\item[2.] Compute the sets ${S(\beta)=\left\{i \in V : \langle v_{i} , r \rangle \geq \beta\right\}}$ and ${S^{\prime}(\beta)=\left\{i \in S(\beta) : \forall(i, j) \in E, j \notin S(\beta)\right\}}$.
\item[3.] Return the set $S'(\beta)$.
\end{itemize}
Here $\beta = \beta(\alpha,\Delta) > 0 $ is a quantity which depends on $\alpha$ and $\Delta$ which will be fixed later. Clearly, by construction, the above procedure returns an independent set. The rest of the proof will just involve lower bounding the expected size of the set $S'(\beta)$.
Let $\Phi(\cdot)$ denote the gaussian CDF function, and let $\overline{\Phi}(\cdot) \overset{\rm def}{=} 1 - \Phi(\cdot)$.
Firstly, we observe that for any $i \in V,$ the probability that $i \in S(\beta)$ is $\overline{\Phi}(\beta)$, where $\Phi(\cdot)$ is the gaussian CDF function. This implies $E[|S(\beta)|]=n \overline{\Phi}(\beta)$. Now consider the probability that a vertex $i$ is in $S(\beta)$ but not in $S^{\prime}(\beta)$. Observe that
$\operatorname{Pr}\left[i \notin S^{\prime}(\beta) | i \in S(\beta)\right]=\operatorname{Pr}\left[\exists(i, j) \in E : \langle v_{j} , r \rangle \geq \beta | \langle v_{i} , r \rangle \geq \beta\right]$. Now fix a pair of neighbouring vertices $(i,j)$. We know that $v_j$ can we written as
\begin{equation*}
v_j=\left(-\frac{1}{2}+\alpha_{ij}\right)v_i+\left(\sqrt{\frac{3}{4}+\alpha_{ij}-{{\alpha}^2_{ij}}}\right)u
\end{equation*} where $u$ is a unit vector orthogonal to $v_i$ and $0 \le \alpha_{ij} \le \alpha$. Going forward, we shall assume that $\alpha_{ij} = \alpha$; it can be easily verified that the same bound holds when $\alpha_{ij} < \alpha$. Rearranging, $u$ can be written as
\begin{equation*}
u=\frac{(\frac{1}{2}-\alpha)}{\sqrt{\frac{3}{4}+\alpha-{{\alpha}^2}}}v_i+\frac{1}{\sqrt{\frac{3}{4}+\alpha-{{\alpha}^2}}}v_j
\end{equation*}
Now, consider the inner product of $u$ and $r$, assuming $\alpha\leq\frac12$
\begin{equation} \label{eqn:bound1}
\langle u, r \rangle \geq \frac{\left(\frac{1}{2}-\alpha\right)}{\sqrt{\frac{3}{4}+\alpha-{{\alpha}^2}}}\beta+\frac{1}{\sqrt{\frac{3}{4}+\alpha-{{\alpha}^2}}}\beta= \frac{\left(\frac{3}{2}-\alpha\right)}{\sqrt{\frac{3}{4}+\alpha-{{\alpha}^2}}}\beta
\end{equation}
where we use the fact that conditioned on the event $\{i\in S(\beta)\} \wedge \{j \in S(\beta)\}$, we must have $\langle v_i, r \rangle \ge \beta$ and $\langle v_j, r \rangle \ge \beta$ by definition of the set $S(\beta)$. Also, by our choice of parameters we have $\beta > 0$. The above sequence of observations implies the following: conditioned on the event $i \in \beta$, the probability of the event $j \in \beta$ is upper bounded by the probability of event that Eq. \ref{eqn:bound1} holds. Finally, recall that the maximum degree of the graph is at most $\Delta$. Now we proceed to bound the desired probability as follows:
\begin{eqnarray*}
\operatorname{Pr}\Big[\exists(i, j) \in E : \langle v_{j} , r \rangle \geq \beta | \langle v_{i} , r \rangle \geq \beta \Big]
&\leq& \sum_{j :(i, j) \in E} \operatorname{Pr}\Big[ \langle v_{j} , r \rangle \geq \beta | \langle v_{i} , r \rangle \geq \beta\Big] \\
&\leq& \sum_{j :(i, j) \in E} \operatorname{Pr}\Big[ \left\{\mbox{Eq. \ref{eqn:bound1} holds} \right\}\Big] \\
&\leq& \Delta \overline{\Phi}\left(\frac{\frac{3}{2}-\alpha}{\sqrt{\frac{3}{4}+\alpha-{{\alpha}^2}}}\beta\right)
\end{eqnarray*}
where the last step follows from the fact that for any unit vector $u$, and a gaussian vector $r \sim N(0,1)^d$, the random variable $\langle u,r \rangle$ is distributed as a gaussian, and the definition of $\overline{\Phi}(\cdot)$.
Recall that $\Delta$ is the maximum degree of the graph. Observe that if we choose $\beta$ such that $\overline{\Phi}\left(\frac{\frac{3}{2}-\alpha}{\sqrt{\frac{3}{4}+\alpha-{{\alpha}^2}}}\beta\right)\leq\frac{1}{2 \Delta},$ then the probability that $i \notin S^{\prime}(\beta)$ given that $i \in S(\beta)$ is at most $\frac{1}{2}$. This would imply that the expected size of $S^{\prime}(\beta)$ is least half
the expected size of $S(\beta),$ which is $\frac{n}{2} \overline{\Phi}(\beta).$\newline
Now, set $\beta=\sqrt{2\ln{\Delta}}\left(\frac{\sqrt{\frac{3}{4}+\alpha-{{\alpha}^2}}}{\frac{3}{2}-\alpha}\right)$. Next we use the following fact about standard normal distributions.
\begin{fact}[Lemma 13.8 \cite{williamson2011design}]
For $x>0$, $\frac{x}{1+x^{2}} \phi(x) \leq \overline{\Phi}(x) \leq \frac{1}{x} \phi(x)$, where $\phi(x) := \frac{1}{\sqrt{2\pi}}e^{-x^2/2}$ is the pdf of the standard gaussian distribution.
\end{fact}
From the above fact, for our choice of $\beta$, we get the following upper bound.
\begin{equation*}
\overline{\Phi}\left(\frac{\frac{3}{2}-\alpha}{\sqrt{\frac{3}{4}+\alpha-{{\alpha}^2}}}\beta\right)\leq\frac{{\sqrt{\frac{3}{4}+\alpha-{{\alpha}^2}}}}{\beta\Big(\frac{3}{2}-\alpha\Big)} e^{\frac{-\frac12\left(\frac{3}{2}-\alpha\right)^2\beta^2}{\frac{3}{4}+\alpha-{{\alpha}^2}}} \leq \frac{1}{2 \Delta}\quad (1)
\end{equation*}
On the other hand, since $\beta \geq 1$, we have $\frac{\beta}{1 + \beta^2} \geq \frac{\beta}{2\beta^2} = \frac{1}{2\beta}$, and hence we can lower bound $\overline{\Phi}(\beta)$ as follows
\begin{equation*}
\overline{\Phi}(\beta)\geq \frac{1}{2 \beta} \frac{1}{\sqrt{2 \pi}} e^{-(\ln \Delta)\frac{\frac{3}{4}+\alpha-{{\alpha}^2}}{(\frac{3}{2}-\alpha)^2} }=\Omega\left((\ln \Delta)^{-1 / 2} \Delta^{-\frac{\frac{3}{4}+\alpha-{{\alpha}^2}}{(\frac{3}{2}-\alpha)^2} }\right)\quad(2)
\end{equation*}
Thus,
\begin{eqnarray*}
E\left[ | S^{\prime}(\beta) |\right]=\sum_{i \in V} \operatorname{Pr}\Big[i \in S^{\prime}(\beta) \Big| i \in S(\beta)\Big] \operatorname{Pr}\Big[i \in S(\beta)\Big] \geq \frac{n}{2} \Omega\left((\ln \Delta)^{-1 / 2} \Delta^{-\frac{\frac{3}{4}+\alpha-{{\alpha}^2}}{(\frac{3}{2}-\alpha)^2} }\right).
\end{eqnarray*}
which gives us the desired lower bound on the expected size of the independent set output by the randomized rounding procedure.
\end{proof}
Now using the above lemma, we complete the proof of theorem. Consider algorithm ApproxHyperplaneColoring for rounding approximate $3$-vector colorings of graphs, which is analyzed in Claim \ref{cl:color}.
\begin{algorithm}
\label{alg:}
\SetAlgoLined
\KwIn{$(3,\alpha)$-vector coloring $\{v_i\}_{i \in V}$ of graph $G = (V,E)$}
Initialize $t \gets 1$,$G_1 \gets G$,$N =\frac{10}{\rho}\ln n$\;
\While{$G_t \neq \phi$}{
Let $I^1,I^2,\ldots,I^N$ be independent sets returned by $N$ i.i.d invocations of Lemma \ref{lem:3col-indset}\;
Let $I_t \gets \arg\max_{{I^i : i \in [N]}} |I^i|$, and set $G_{t+1} \gets G_t \setminus I_t$\;
Update $t \gets t + 1$\;
}
Output coloring $I_1 \uplus I_2 \uplus \cdots \uplus I_t$\;
\caption{ApproxHyperplaneColoring}
\end{algorithm}
\begin{claim} \label{cl:color}
For any iteration $t$, we have $|I_t| \geq \rho n|{\rm vert}(G_t)|/2$ with probability at least $1 - 1/n^3$.
\end{claim}
\begin{proof}
Consider one invocation of Lemma \ref{lem:3col-indset}. For any iteration $t$, let $n_t = \left|{\rm vert}(G_t)\right|$ denote the number of surviving vertices in the graph $G_t$. Then we have have $\E\big[|I|\big] \geq \rho n_t$ or $\E\big[|\overline{I}|\big] \leq (1 - \rho) n_t$. Therefore, we have
\begin{equation*}
\Pr\Big[\forall i \in [N], |\overline{I}^i| \leq (1 - \rho/2)n_t \Big] = \left(\Pr\Big[|\overline{I}| \leq (1 - \rho/2)n_t \Big]\right)^N
\overset{1}{\leq} \left(\frac{1 - \rho}{1 - \rho/2}\right)^N
\overset{2}{\leq} (1 - \rho/2)^N \overset{3}{\leq} n^{-3}
\end{equation*}
where step $1$ uses Markov's inequality, and step $2$ uses the fact $(1 - \rho/2)^2 \geq 1 - \rho$ and step $3$ follows from our choice of $N$.
\end{proof}
Therefore, at iteration $t$, with probability at least $1 - n^{-3}$, the size of the independent set returned is at least $\rho |{\rm Vert}(G_t)|/2$, and therefore, the number of surviving vertices drops by a factor of $(1 - \rho/2)$. Hence, with probability at least $1 - 1/n^2$, in $t^* = {O}\left(\frac1\rho\ln n\right)$ iterations, every vertex will be accounted for i.e., they will be part of some independent set. Since each independent set forms a color class, the total number colors used is $O\left(\frac1\rho\ln n\right) = \tilde{O}\left((\ln \Delta)^{1 / 2} \Delta^{\frac{\frac{3}{4}+\alpha-{{\alpha}^2}}{(\frac{3}{2}-\alpha)^2} }\right)$ colors.
\input{twocol-gen}
\section{Maximal and Maximum Short Odd Cycle sets}
\begin{proposition} \label{prop:max-bound}
For any graph $G:=(V,E)$, and parameter $\delta \in (0,1)$ the following holds. Let $\mathcal{C}$ be a maximal set of vertex disjoint odd cycles of length at most $1/\delta$, and let $\tilde{\mathcal{C}}$ be a set of largest cardinality of vertex disjoint odd cycles of length at most $1/\delta$. Then $|\mathcal{C}| \geq \delta|\tilde{\mathcal{C}}|$.
\end{proposition}
\begin{proof}
Since $\mathcal{C}$ is a maximal set of vertex disjoint odd cycles of length at most $1/\delta$, for every odd cycle $\tilde{C} \in \tilde{\mathcal{C}}$, there exists an odd cycle $C \in \mathcal{C}$ such that $C \cap \tilde{C} \neq \emptyset$ i.e,. $\tilde{C}$ is hit by $C$. Now we observe that (i) the cycles in $\mathcal{C}$ are vertex disjoint and (ii) each cycle $C \in \mathcal{C}$ has size at most $1/\delta$, it follows that any cycle $C \in \mathcal{C}$ hits at most $1/\delta$ cycles in $\tilde{\mathcal{C}}$. Since every cycle in $\tilde{\mathcal{C}}$ is hit by some cycle in $\mathcal{C}$, we must have $|\mathcal{C}| \geq \frac{|\tilde{\mathcal{C}}|}{1/\delta} = \delta|\tilde{\mathcal{C}}|$.
\end{proof}
\section{Identifying the set of Good Vertices is NP-hard}
\begin{fact} \label{fact1}
For all $k \in \mathbbm{N}$, given a graph $\alpha$-partially $k$-colorable graph $G = (V,E)$ it is NP-Hard to identify a set $V_{\rm good} \subset V$ of size at least $\alpha n$ such that $G[V_{\rm good}]$ is $k$-colorable
\end{fact}
\begin{proof}
For $\alpha = 1 - 1/2n$, this is exactly the $k$-Coloring problem which is NP-Hard \cite{Karp72}.
\end{proof}
\subsection{Discussion and Proof Overview}
\label{sec:overview}
{\bf Adversarial Model}: The key component in most approximation algorithms for $3$-coloring involves solving a SDP relaxation of the $3$-coloring problem, and followed by a randomized rounding procedure for coloring the graph. The standard SDP relaxation for $3$-coloring is the following which was introduced in \cite{KMS98}:
\begin{SDP}[Exact $3$-Coloring SDP]
\label{SDP:kms}
\begin{eqnarray*}
\text{minimize} & 0 & \\
\text{subject\space to}& v_i\cdot v_j \le -\frac{1}{2} & \quad \forall \{i,j\}\in E \\
& \|v_i\|^2=1 &\forall i \in V
\end{eqnarray*}\\
\end{SDP}
SDP \ref{SDP:kms} doesn't optimize any objective function,
it finds a feasible solution which satisfies all the constraints of SDP. The intended solution to the above SDP is as follows. Let $\sigma:V \to \set{1,2,3}$ be any legal coloring of $G$. Furthermore, let $u_1,u_2,u_3 \in \mathbbm{R}^2$ be any three unit vectors satisfying $\langle u_i , u_j \rangle = -1/2$ for every $i,j \in \{1,2,3\}, i \neq j$. We identify the vector $u_i$ with the color $i$, and assign $v_j = u_{\sigma(j)}$ for every $j \in V$. It can be easily verified that this is a feasible solution to the above SDP. As is usual, while the SDP in general may not return the above vector coloring, one can round a feasible vector coloring to color the graph using not too many colors \cite{KMS98}.
The approximation guarantee is usually of the form $\Delta^c$ (for some $c \in (0,1)$), where $\Delta$ is the maximum degree of the graph.
Since in general, one cannot hope to have a degree bound on the graph, the above step is usually preceded by a {\em degree reduction} sub-routine. Note that if a graph is $3$-colorable (more generally $k$-colorable), then the graph induced on the neighbours of any vertex $v$ is $2$-colorable (more generally $k-1$ colorable). Since a $2$-colorable graph can be colored with $2$ colors efficiently, the graph induced on any vertex and its neighbours can be colored efficiently with $3$ colors.
Therefore, fixing a threshold $\Delta$, this procedure iteratively removes vertices (and their neighbours) having degree larger than $\Delta$ from the graph while coloring them with few colors, and terminates when maximum degree of the remaining graph is at most $\Delta$.
In particular, if the degree reduction step uses $f(n,\Delta)$ colors, then the total number of colors used by the algorithm is at most $f(n,\Delta) + \Delta^c$. Then one can optimize the choice of $\Delta$ for giving the best possible approximation guarantee. This degree reduction approach and its variants, first studied by Wigderson \cite{wigderson_1983}, has been subsequently used in almost all known approximation algorithms for graph coloring
In translating the above template to the setting of partially $3$-colorable graphs, we face several
immediate challenges. SDP \ref{SDP:kms} is guaranteed to return a feasible solution only for $3$-colorable graphs,
it might be infeasible if the graph is not $3$-colorable.
If we could compute the set of good vertices then we could use
SDP \ref{SDP:kms} only on the set of good vertices.
However, in general, the problem of identifying the set of good vertices is NP-hard (Fact \ref{fact1}).
Finally, the preprocessing steps for degree reduction rely heavily on the combinatorial
structural properties of the neighborhood of vertices in {\em exactly} $3$-colorable graphs,
which, in general, may not be satisfied by {\em partially} $3$-colorable graphs.
Our approach is to begin with an SDP relaxation that tries to solve both problems together: identifying
the set of bad vertices, and coloring the set of good vertices.
We introduce variables $w_1,w_2,\ldots,w_n$ where the $i^{th}$ variable $w_i$ is
meant to indicate if vertex $i$ is bad. Additionally, for every edge $(i,j) \in E$, we introduce
slack variables $z_{ij}$ which are meant to indicate if at least one of the vertices $i,j$ is bad.
Using the slack variables we relax the edge constraints as $\langle v_i, v_j \rangle \le -1/2 + (3/2)z_{ij}$.
Finally, we connect the edge indicator variables with vertex indicator variables using constraints
of the form $z_{ij} \le w_i + w_j$.
Since we want the
set of bad vertices to be small, our objective function will be to minimize $\sum_{i \in V} w_i$.
Our SDP relaxation is the following.
\begin{SDP}[Partial $3$-Coloring SDP]
\label{SDP:p3c}
\begin{eqnarray*}
\text{minimize} & \sum_{i\in V} {w_i} & \\
\text{subject\space to}& \langle v_i, v_j \rangle \le -\frac{1}{2}+\frac{3}{2}z_{ij} & \quad \forall \{i,j\}\in E \\
& z_{ij} \leq w_i+ w_j & \forall \{i,j\}\in E \\
& 0 \leq z_{ij} \leq 1 & \forall \{i,j\}\in E \\
& 0 \leq w_{i} \leq 1 &\forall i\in V \\
& \|v_i\|^2=1 &\forall i \in V
\end{eqnarray*}\\
\end{SDP}
Since the optimal ``integer solution'' forms a feasible solution to the SDP relaxation,
it is easy to show that for a $(1-\epsilon)$-partially $3$-colorable graph, the optimal of
the above SDP is at most $\epsilon n$. Therefore by Markov's inequality, we get that for a
large fraction of $i \in [n]$, the $w_i$ variables are small. Let $V' \subset V$ be the set
of vertices with small $w_i$. Since $\Abs{V\setminus V'} = O(\epsilon n)$, we can focus on
coloring the induced subgraph $G' = G[V']$. $G'$ has the following nice property:
{\em for every edge $(i,j)$ in G', the corresponding edge constraint is approximately satisfied
i.e., $\langle v_i , v_j \rangle \le -1/2 + o_\epsilon(1)$, where the second term goes to $0$ as
$\epsilon$ goes to $0$}. We call such graphs as being approximately vector $3$-colorable (See Definition \ref{defn:appx-vect-col} for a formal description).
We use this property crucially in designing our preprocessing step.
We observe that the neighborhood of any vertex in an approximately vector $3$-colorable graph is
approximately vector $2$-colorable. Furthermore, we show that approximately vector $2$-colorable graphs
are {\em short odd cycle} free. Graphs having this property are known to have large independent sets
which can be found efficiently \cite{RS85ramsey}. Thus one can find such large independent sets recursively to
color the neighborhood of large degree vertices using a small number of colors.
For the randomized rounding step, we observe that hyperplane rounding based procedures are naturally robust to small perturbations, and the arguments for analyzing the guarantees of such procedures hold even when the edge constraints are approximately satisfied. In particular, we can use known randomized rounding algorithm as is, while adapting the analysis to account for the edge constraints being satisfied approximately.
{\bf Semi-random model}: While the guarantees of our algorithm from the adversarial setting also apply to the semi-random instances, here we seek to achieve the best known approximation bounds for exactly $3$-colorable graphs. We begin by describing two distinct classes of instances which illustrate the technical challenges in designing such an algorithm.
In this setting, the adversary is free to choose $G[V_{\rm bad}]$ in a way such that it is noisy and has large chromatic number (e.g, graphs sampled from Erdos Renyi random model). For such instances, it is easy to see that the only way an algorithm can have good approximation guarantees is when it can eliminate a significant fraction of from $V_{\rm bad}$. Then, for a start, one can hope to address this setting by first using a preprocessing step that deletes $V_{\rm bad}$ and then running the best possible approximation algorithm on the graph induced on the remaining vertices.
On the other hand, the adversary can also choose $G[V_{\rm bad}]$ in a way so that it is {\em structurally indistinguishable} from the good subgraph $G[V_{\rm good}]$. For instance, suppose the good subgraph $G[V_{\rm good}]$ is a randomly sampled unbalanced bipartite graph, where the smaller side (which we call $V_S$) has size at most $\epsilon n$. Then the adversary can choose $V_{\rm bad}$ to be an independent set, in which case the entire graph is $3$-colorable. In particular, it is information theoretically impossible to distinguish the set $V_{S}$ from $V_{\rm bad}$, since they are both independent sets and the edges incident on them are identically distributed. While the instances constructed here make it difficult to identify $V_{\rm good}$, they are also naturally easy instances for us. In particular, these instances are also $(1-\epsilon)$-partially $2$-colorable, and one can use tools for coloring partially $2$-colorable graphs to color these instances with small number of colors.
However, the two cases above clearly do not cover the full range of instances that we can encounter in our model. Therefore, we need a way to relax the above two characterizations which allows for a seamless transition from one class of instances to other. It turns out that we can robustly characterize both classes of instances by the number of {\em vertex disjoint short odd cycles} present in the graph. Informally, if the number of short odd cycles is large, then with high probability, they will show up in the neighborhood of the bad vertices, and therefore this can be used to identify and eliminate $V_{\rm bad}$. We can then simply run the best known approximation algorithm on the remaining induced graph $G[V_{\rm good}]$. On the other hand, if the number of short odd cycles is small, by eliminating a small fraction of vertices, we can make the graph short odd cycle free. Finally, as discussed in the adversarial model setting, such graphs can be colored efficiently using a small number of colors by recursively finding large independent sets~\cite{RS85ramsey}.
\section{Conclusion}
In this work we consider the problem of coloring partial $3$-colorable graphs in adversarial and semi-random settings. In the adversarial setting, we give an efficient approximation algorithm which can color $(1 - O(\epsilon^c))$-fraction of vertices using $\tilde{O}(n^{0.25 + \epsilon^{c'}})$ colors. On the other hand, the best known approximation guarantees for $3$-colorable graphs is $n^{0.199}$~\cite{kawarabayashi_thorup_2017}. An obvious open question here is to achieve analogous approximation bounds for partially $3$-colorable graphs as well.
One direct way to improve on our approximation bounds in the adversarial setting is through the use of more efficient degree reduction mechanisms as typically done in the exact $3$-coloring setting \cite{blum_karger_1997},\cite{kawarabayashi_thorup_2012,kawarabayashi_thorup_2017} using combinatorial techniques like Blum's coloring tools~\cite{Blum}. However, these tools rely on fragile combinatorial properties present in $3$-colorable graphs (e.g. two vertices whose common neighborhood is not an independent set must have the same color in any legal coloring), and as such, it is not obvious how to extend these techniques to the setting of partially $3$-colorable graphs.
In the semi-random model, we show how any efficient algorithm for exact $3$-coloring that uses $n^{\theta}$ colors can be leveraged to obtain an efficient algorithm in this setting which uses the same number of colors with high probability and also does not remove too many vertices. An obvious next step would be to see if similar results can also be obtained for partially $k$-colorable graphs with $k>3$. Another interesting question would be to see if one can design efficient approximation algorithms with similar guarantees, where the adversary can also delete the randomly sampled edges.
\section*{Acknowledgements}
AL was supported in part by SERB Award ECR/2017/003296. SG would like to thank Pasin Manurangsi for pointing him to the Odd Cycle Transversal problem.
\bibliographystyle{alpha}
\section{Preliminaries}
We introduce some notation used frequently in this paper. Throughout the paper, for a $(1-\epsilon)$-partially $3$-colorable graph $G = (V,E)$, we will write $V = V_{\rm good} \uplus V_{\rm bad}$ where $V_{\rm good}$ and $V_{\rm bad}$ are the set of good vertices and bad vertices as defined in Definition \ref{def:pkc}. For a subset $V' \subseteq V$, we use $G[V']$ to denote the subgraph induced on the set of vertices $V'$. For a subgraph $G' \subseteq G$, we shall use ${\rm vert}(G')$ to denote the vertex set of $G'$. Additionally, for any vertex $i \in {\rm vert}(G')$, we use $N_{G'}(i)$ denote the set of neighbors of $i$ in the graph $G'$. We use $\mathbbm{1}(\cdot)$ to denote the indicator function, and $\tilde{O}(\cdot)$ to hide terms which are polylogarithmic in the number of vertices.
\paragraph{Approximate Vector Coloring.}
We begin by recalling the notion of vector coloring of a graph which was introduced in \cite{KMS98}.
\begin{definition}[Vector Coloring]
\label{defn:vect-col}
Given a positive integer $k\in \mathbbm{N}$, we say that a graph $G = (V,E)$ is $k$-vector colorable
if there exists unit vectors $v_1,v_2,\ldots,v_n \in \mathbbm{R}^d$ for some $d \in \mathbbm{N}$ which satisfy
\begin{equation*}
\langle v_i , v_j \rangle \leq -\frac{1}{k-1} \qquad \qquad \forall \set{i,j} \in E.
\end{equation*}
\end{definition}
We will use the notion of {\em approximate vector colorings} of a graph, which
we define as follows.
\begin{definition}[Approximate Vector Coloring]
\label{defn:appx-vect-col}
Given a positive integer $k\in \mathbbm{N}$ and a $\gamma > 0$, we say that a graph $G = (V,E)$ is
$(k,\gamma)$-vector colorable if there exists unit vectors $v_1,v_2,\ldots,v_n \in \mathbbm{R}^d$
for some $d \in \mathbbm{N}$ which satisfy
\begin{equation*}
\langle v_i , v_j \rangle \le -\frac{1}{k-1} + \gamma \qquad \qquad \forall \set{i,j} \in E.
\end{equation*}
\end{definition}
Observe that a graph that $(k,0)$ vector colorable is vector-$k$-colorable. We now state a couple of lemmas which illustrate some useful properties of approximate vector colorings. In \cite{KMS98}, it was observed that the vector chromatic number of sub-graph induced on the neighborhood of a vertex is strictly less than the vector chromatic number of the actual graph. In the following lemma, we observe that this property can be extended to approximate vector colorings as well.
\begin{lemma} \label{lem:appx-2-col(a)}
Let $G = (V,E)$ be $(3,\gamma)$-vector colorable, for some $0 < \gamma <1/10$. Then for any vertex $i \in V$, the graph induced on $N(i)$ is $(2,4\gamma)$-vector colorable.
\end{lemma}
\begin{proof}
The proof of this lemma follows along the lines of Lemma 4.3 from \cite{KMS98}, which says that subgraphs induced by neighborhoods of vertices in vector $3$-colorable graphs are vector $2$-colorable. Without loss of generality, let $N_G(i) = \{1,2,\ldots,r\}$ and let $\{v_1,v_2,\ldots,v_r\}$ be the set of vectors which are a $(3,\gamma)$-vector coloring of $N_G(i)$. For every $j \in [r]$, we can write $v_j = v^{\|}_j + v^\perp_j$ where $v^{\|}_j$ and $v^\perp_j$ are the projections of $v_j$ along $v_i$ and $({\rm span}(v_i))^\perp$ respectively. Finally, for every $j \in [r]$ we define $\tilde{v}_j := v^\perp_j/\|v^\perp_j\|$ to be unit vector given by the projection of $v_j$ on the subspace $({\rm span}(v_i))^\perp$. It can be easily verified that $\tilde{v}_1,\tilde{v}_2,\ldots,\tilde{v}_r$ is a $(2,4\gamma)$-vector coloring of the graph induced on $N(v)$. To see this, fix any $j \in V$. By construction, we have $\|v^{\|}_j\| = |\langle v_i , v_j \rangle | \geq \frac12 - \gamma$, and therefore $\|v^\perp_j\| = \sqrt{1 - \|v^{\|}_j\|^2} \leq \sqrt{\frac34 + \gamma - \gamma^2}$. Therefore for any $j,j' \in [r]$ such that $(j,{j'}) \in E$, using the orthonormal decomposition of $v_j$ and $v_{j'}$ we have
\begin{eqnarray*}
\langle \tilde{v}_j , \tilde{v}_{j'} \rangle = \left\langle \frac{v^\perp_j}{\|v^\perp_j\|}, \frac{v^\perp_{j'}}{\|v^\perp_{j'}\|} \right\rangle
&=& \frac{1}{\|v^\perp_j \| \|v^\perp_{j'}\|}\Big(\langle v_j, v_{j'} \rangle - \langle v^{\|}_{j} , v^{\|}_{j'} \rangle\Big) \\
&=& \frac{1}{\|v^\perp_j \| \|v^\perp_{j'}\|}\Big(\langle v_j, v_{j'} \rangle - \langle v_i, v_{j} \rangle\langle v_i, v_{j'} \rangle \Big) \\
&\le& \frac{1}{\Big(\frac34 + \gamma - \gamma^2\Big)}\Big(-1/2 + \gamma - \Big(\frac12 - \gamma\Big)^2\Big) \\
&\le& - 1 + 4 \gamma
\end{eqnarray*}
Since the above holds for any pair of vertices $j,j' \in [r]$ which forms an edge, the claim follows.
\end{proof}
The next lemma says that approximately vector $2$-colorable graphs cannot contain short odd cycles.
\begin{lemma} \label{lem:appx-2-col(b)}
Let $G = (V,E)$ be a $(2,\gamma)$-vector colorable, where $\gamma \le 1/16$. Then $G$ does not contain odd cycles of length at most $1/8\sqrt{\gamma}$.
\end{lemma}
\begin{proof}
Let $v_1,v_2,\ldots,v_n$ be the $(2,\gamma)$-vector coloring of $G$. For contradiction, let $C$ be an odd cycle in $G$ of length $r \le 1/(8\sqrt{\gamma})$. Without loss of generality, let $C = \{1,2,\ldots,r\}$, such that for every $i \in [r]$, the pair $\{i,({i\mod r}) + 1\}$ forms an edge. Let $r = 2k + 1$. Now for any $i \in [r]$, we have $-1 \le \langle v_i,v_{i+1} \rangle \le - 1 + \gamma$. Since $v_i, v_{i+1}$ are unit vectors, we have
\begin{equation}
\|v_i + v_{i+1}\|^2 = \|v_i\|^2 + \|v_{i+1}\|^2 + 2 \langle v_i, v_{i+1} \rangle \le 2\gamma
\end{equation}
which implies that $\|v_i + v_{i+1}\| \le 2\sqrt{\gamma}$ i.e, any consecutive pair of vectors are {\em almost anti-podal}. Then, for any $i \in [r]$ we also get that
\begin{equation} \label{eq:vec-bound}
\|v_i - v_{i+2}\| \le \|v_i + v_{i+1}\| + \|v_{i+1} + v_{i+2}\| \le 4\sqrt{\gamma}
\end{equation}
We shall now use the above observations to arrive at a contradiction. From the upper bound on $r$, we have $k \leq (r-1)/2 \leq 1/(16\sqrt{\gamma})$, and hence using Eq. \ref{eq:vec-bound} we get that
\begin{eqnarray}
\|v_1 - v_r\| \le \sum_{j = 0}^{k-1} \|v_{1 + 2j} - v_{1 + 2(j+1) } \| \le 4k\sqrt{\gamma} < 1/4
\end{eqnarray}
But on the other hand, since $v_1,v_r$ are consecutive vertices in the cycles $C$, we also have $\langle v_1 ,v_r \rangle \le -1 + \gamma$ which implies that $\|v_1 - v_r\| \ge \sqrt{4 - 4 \gamma} > 1$, which give us the contradiction.
\end{proof}
\paragraph{Coloring graphs without short odd cycles}
A key combinatorial tool used in our paper is the following Ramsey theoretic result which says that graphs without short odd cycles contain large independent sets which can be found efficiently.
\begin{lemma}\cite{RS85ramsey} \label{lem:ramsey}
There exists a constant $\epsilon_0 \in (0,1)$ such that for every choice of $0 < \epsilon < \epsilon_0$ the following holds. Let $G = (V,E)$ be a graph without odd cycles of length at most $1/\epsilon$. Then, $G$ contains an independent set of size at least $|V|^{1 - 2\epsilon}$. Furthermore, there exists a polynomial time algorithm which finds such an independent set.
\end{lemma}
Consequently, given a graph without short odd cycles, one can color it efficiently using a small number of colors, as stated in the following corollary.
\begin{corollary} \label{corr:coloring}
There exists a constant $\epsilon_0 \in (0,1)$ for which the following holds. Given a graph $G = (V,E)$ which does not contain odd cycles of length at most $1/\epsilon$ where $\epsilon < \epsilon_0$, there exists a polynomial time algorithm which can compute a coloring of $G$ using $\tilde{O}(n^{2\epsilon})$ colors.
\end{corollary}
Establishing the above corollary using Lemma \ref{lem:ramsey} is straightforward, and just uses the fact that one can keep removing large independent sets in the graph using Lemma \ref{lem:ramsey}, and recurse on the remaining vertices. For the sake of completeness, we include the proof here.
\begin{proof}
\begin{algorithm}
\label{alg:ind-set}
\SetAlgoLined
\KwIn{Graph $G = (V,E)$}
Initialize $t \gets 1$ and $G_1 \gets G$\;
\While{$G_t \neq \phi$}{
Let $I_t$ be the independent set from Lemma \ref{lem:ramsey} instantiated with $G_t$\;
Set $G_{t+1} \gets G_t \setminus I_t$\;
Update $t \gets t + 1$\;
}
Output coloring $I_1 \uplus I_2 \uplus \cdots \uplus I_t$\;
\caption{IndSetColoring}
\end{algorithm}
Consider Algorithm IndSetColoring for coloring by iteratively finding large independent sets. Here, we use Lemma \ref{lem:ramsey} to iteratively remove independent sets $I_1,I_2,\ldots,I_t$, where each independent set forms a color class. Let $G_t = G[V \setminus (I_1 \cup I_2 \cup \cdots I_t)]$ denote the graph on the surviving vertices after $t$ iterations. We claim that in every $T = n^{2\epsilon}$ applications of Lemma \ref{lem:ramsey} at least a constant fraction of vertices are removed, i.e., for any iteration $t$, we have $|{\rm Vert}(G_{t + T})| \leq ( 1- 1/2^{1-2\epsilon})|{\rm Vert}(G_t)|$.
This can be shown as follows. Let $n_t = |{\rm Vert}(G_t)|$ denote the number of vertices in graph $G_t$. Then, we can assume that $|{\rm vert}(G_{t + T})| > n_t/2$ (otherwise we are done). Then, in $T$ iterations the number of vertices removed can be lower bounded by
\begin{equation}
\sum_{j = 1}^{T}|I_{j+T}|\geq \sum_{j = 1}^{T}|{\rm Vert}(G_{t + j})|^{1 - 2\epsilon} \geq n^{2\epsilon}(n_t/2)^{1 - 2\epsilon} \ge n_t/2^{(1 - 2\epsilon)}
\end{equation}
where the first inequality follows from the guarantee of Lemma \ref{lem:ramsey}. Therefore, in $\tilde{O}(n^{2\epsilon}) $ iterations, all the vertices will be accounted for.
\end{proof}
\section{Partial $2$-coloring in the Adversarial Model}
In this section we prove Proposition \ref{prop:2col-gen}, which we recall here for convenience.
\TwoColGen*
The algorithm for the above proposition (described in Algorithm \ref{alg:2col-main})is basically the same as Algorithm 2, with the following key differences: (i) we solve the SDP for approximate vector $2$-coloring with slack constraints (instead of approximate vector $3$-coloring) and (ii) we can directly use Corollary \ref{corr:coloring} to round the vector solution without going through the degree reduction step.
\begin{algorithm}[h!]
\SetAlgoLined
Solve the Partial-$2$-Coloring SDP (SDP-P$2$C):
\begin{alignat*}{4}
\mbox{minimize } & \sum_{i\in V} w_i \\
\mbox{subject to } & \langle v_i, v_j \rangle \leq -{1} + 2z_{ij} &\qquad & \forall \{i,j\}\in E \\
& z_{ij} \leq w_i+w_j &\qquad & \forall \{i,j\}\in E \\
& 0 \le z_{ij}\leq 1 &\qquad & \forall \{i,j\}\in E \\
& 0 \le w_{i} \leq 1 &\qquad & \forall i\in V \\
& \|v_i\|^2=1 &\qquad & \forall i \in V
\end{alignat*}
{\it(i) Thresholding}: \;
Let $S \gets \left\{i \in V| w_i \ge \gamma/4 \right\}$\;
Let $G' \gets G[V \setminus S]$ be the graph obtained after deleting $S$\;
\nonl{\it(ii) Round the approximate vector $2$-coloring}: \\
Use the algorithm from Corollary \ref{corr:coloring} to color the remaining vertices in $G'$\;
\caption{Partial-$2$-Coloring} \label{alg:2col-main}
\end{algorithm}
We give a proof sketch of the correctness of the above algorithm.
\subsection{Proof of Proposition \ref{prop:2col-gen}}
To begin with, the following claim shows that in step (i) we do not throw away too many vertices.
\begin{claim} \label{cl:2colstep(i)}
Let $S \subset V$ be the set of vertices constructed in step (i) of the algorithm. Then $|S| \leq 4\epsilon n/\gamma$.
\end{claim}
The proof of the above claim is almost identical to that of Claim \ref{cl:step(i)}, hence we omit it here. Therefore after step $(i)$, the subgraph $G' = G[V\setminus S]$ satisfies the following properties:
\begin{enumerate}
\item The graph $G'$ contains at least $(1 - 4\epsilon/\gamma)n$ vertices.
\item The graph $G'$ is again $(2,\gamma)$-vector colorable since the vectors $(v_i)_{i \in V \setminus S}$ themselves give a $(2,\gamma)$-vector coloring of the graph.
\end{enumerate}
Now from Lemma \ref{lem:appx-2-col(b)} we know that $G'$ cannot contain odd cycles of length at most $1/8\sqrt{\gamma}$. Hence we can use Corollary \ref{corr:coloring} to color $G'$ using $\tilde{O}(n^{2\gamma})$ colors. This concludes the proof of Proposition \ref{prop:2col-gen}.
| {
"timestamp": "2019-09-02T02:08:31",
"yymm": "1908",
"arxiv_id": "1908.11631",
"language": "en",
"url": "https://arxiv.org/abs/1908.11631",
"abstract": "Graph coloring problems are a central topic of study in the theory of algorithms. We study the problem of partially coloring partially colorable graphs. For $\\alpha \\leq 1$ and $k \\in \\mathbb{Z}^+$, we say that a graph $G=(V,E)$ is $\\alpha$-partially $k$-colorable, if there exists a subset $S\\subset V$ of cardinality $ |S | \\geq \\alpha | V |$ such that the graph induced on $S$ is $k$-colorable. Partial $k$-colorability is a more robust structural property of a graph than $k$-colorability. For graphs that arise in practice, partial $k$-colorability might be a better notion to use than $k$-colorability, since data arising in practice often contains various forms of noise.We give a polynomial time algorithm that takes as input a $(1 - \\epsilon)$-partially $3$-colorable graph $G$ and a constant $\\gamma \\in [\\epsilon, 1/10]$, and colors a $(1 - \\epsilon/\\gamma)$ fraction of the vertices using $\\tilde{O}\\left(n^{0.25 + O(\\gamma^{1/2})} \\right)$ colors. We also study natural semi-random families of instances of partially $3$-colorable graphs and partially $2$-colorable graphs, and give stronger bi-criteria approximation guarantees for these family of instances.",
"subjects": "Data Structures and Algorithms (cs.DS)",
"title": "Approximation Algorithms for Partially Colorable Graphs",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9770226300899744,
"lm_q2_score": 0.7248702761768249,
"lm_q1q2_score": 0.7082146637043275
} |
https://arxiv.org/abs/1506.02991 | The full exceptional collections of categorical resolutions of curves | This paper gives a complete answer of the following question: which (singular, projective) curves have a categorical resolution of singularities which admits a full exceptional collection? We prove that such full exceptional collection exists if and only if the geometric genus of the curve equals to 0. Moreover we can also prove that a curve with geometric genus equal or greater than 1 cannot have a categorical resolution of singularities which has a tilting object. The proofs of both results are given by a careful study of the Grothendieck group and the Picard group of that curve. | \section{Introduction}\label{Introduction}
For a triangulated category $\mathcal{C}$, having a full exceptional collection is a very good property.
Recall that the definition of full exceptional collection is as follows.
\begin{defi}\label{defi:full excep coll}
A full exceptional collection of a
triangulated category $\mathcal{C}$ is a collection $\{A_1\ldots A_n\}$ of objects such that
\begin{enumerate}
\item for all $i$ one has $\Hom_{\mathcal{C}}(A_i,A_i)=k$ and $\Hom_{\mathcal{C}}(A_i,A_i[l])=0$ for all $l\neq 0$;
\item for all $1\leq i<j\leq n$ one has $\Hom_{\mathcal{C}}(A_j,A_i[l])=0$ for all $l\in \mathbb{Z}$;
\item the smallest triangulated subcategory of $\mathcal{C}$ containing
$A_1,\ldots, A_n$ coincides with $\mathcal{C}$.
\end{enumerate}
\end{defi}
However it is not very common that a triangulated category $\mathcal{C}$ has a full exceptional collection. In algebraic geometry, it is well-known that for a smooth projective curve $X$ over an algebraically closed field $k$, its bounded derived category of coherent sheaves $\rD^b(\coh(X))$ has a full exceptional collection if and only if the genus of $X$ equals to $0$.
Moreover for a singular projective curve $X$ and a (geometric) resolution of singularities $\widetilde{X}\to X$, the geometric genus of $\widetilde{X}$ and $X$ are equal, hence it is clear that $\rD^b(\coh(\widetilde{X}))$ has a full exceptional collection if and only if the geometric genus of $X$ equals to $0$.
In this paper we would like to consider the categorical resolution of $X$, which is introduced in \cite{kuznetsov2008lefschetz}.
\begin{defi}\label{defi: categorical resolution}[\cite{kuznetsov2008lefschetz} Definition 3.2 or \cite{kuznetsov2014categorical} Definition 1.3]
A categorical resolution of a scheme $X$ is a smooth, cocomplete,
compactly generated, triangulated category $\mathscr{T}$ with an adjoint pair of triangulated
functors
$$
\pi^*: \rD(X)\to \mathscr{T} \text{ and } \pi_*: \mathscr{T}\to \rD(X)
$$
such that
\begin{enumerate}
\item $\pi_*\circ \pi^*=id$;
\item both $\pi_*$ and $\pi^*$ commute with arbitrary direct sums;
\item $\pi_*(\mathscr{T}^c)\subset \rD^b(\coh(X))$ where $\mathscr{T}^c$ denotes the full subcategory of $\mathscr{T}$ which consists of compact objects.
\end{enumerate}
\end{defi}
\begin{rmk}
The first property implies that $\pi^*$ is fully faithful and the second property implies that $\pi^*(\rD^{\perf}(X))\subset \mathscr{T}^c$.
\end{rmk}
\begin{rmk}
The categorical resolution of $X$ is not necessarily unique.
\end{rmk}
\begin{rmk}
In this paper we will not discuss further on the smoothness of a triangulated category and the interested readers may refer to \cite{kuznetsov2014categorical} Section 1. Moreover, the main result in this paper does not depend on the smoothness, see Corollary \ref{coro: non-exist of full excep collection for factor through reduced case} and Corollary \ref{coro: non-exist of full excep collection for factor through general case} below.
\end{rmk}
We are interested in the question that when does $\mathscr{T}^c$ have a full exceptional collection. If $X$ is an projective curve of geometric genus $g=0$, it can be deduced from the construction in \cite{kuznetsov2014categorical} that there exists a categorical resolution $(\mathscr{T},\pi^*,\pi_*)$ of $X$ such that $\mathscr{T}^c$ has a full exceptional collection. See Proposition \ref{prop: g=0 has a full exc coll} below.
The main result of this paper is the following theorem, which rules out the possibility for any categorical resolution of a curve with geometric genus $g \geq 1$ has a full exceptional collection.
\begin{thm}\label{thm: non-exist of full excep collection in the introduction}[See Theorem \ref{thm: non-exist of full excep collection general case} below]
Let $X$ be a projective curve over an algebraically closed field $k$. Let $(\mathscr{T},\pi^*,\pi_*)$ be a categorical resolution of $X$. If the geometric genus of $X$ is $\geq 1$, then $\mathscr{T}^c$ cannot have a full exceptional collection.
In other words, $X$ has a categorical resolution which admits a full exceptional collection if and only if the geometric genus of $X$ equals to $0$.
\end{thm}
\begin{rmk}\label{rmk: extist of f.d. algebra for g=0}
In a recent paper \cite{burban2015singular} a result which is related to the above claim has been proved. Actually it has been proved that if $X$ is a reduced rational curve, then there exists a categorical resolution $(\mathscr{T},\pi^*,\pi_*)$ of $X$ such that $\mathscr{T}^c$ has a tilting object, which in general does not come from an exceptional collection. See \cite{burban2015singular} Theorem 7.4.
\end{rmk}
Recall that the definition of tilting object is given as follows.
\begin{defi}\label{defi: tilting object}
Let $\mathcal{C}$ be a triangulated category. A tilting object is an object $L$ of $\mathcal{C}$ which satisfies the following properties.
\begin{enumerate}
\item $L$ is a compact object of $\mathcal{C}$;
\item $\Hom_{\mathcal{C}}(L,L[i])=0$ for any non-zero integer $i$;
\item the smallest thick triangulated subcategory of $\mathcal{C}$ which contains $L$ is $\mathcal{C}$ itself.
\end{enumerate}
For a tilting object let $\Lambda=\text{End}_{\mathcal{C}}(L)$. Then it can be shown that we have equivalence of triangulated categories
$$\mathcal{C}\cong \rD^b(\Lambda-\text{mod})$$
where $\rD^b(\Lambda-\text{mod})$ is the derived category of bounded complexes of finitely generated $\Lambda$-modules.
\end{defi}
Actually we can also prove a related result in the $g\geq 1$ case. (thanks to Igor Burban for pointing it out)
\begin{thm}\label{thm: non-exist of f.d algebra as a cat resolution in the introduction} [See Theorem \ref{thm: non-exist of f.d algebra as a cat resolution} below]
Let $X$ be a projective curve over an algebraically closed field $k$ of geometric genus $\geq 1$. Let $(\mathscr{T},\pi^*,\pi_*)$ be a categorical resolution of $X$. Then $\mathscr{T}^c$ cannot have a tilting object, moreover there cannot be a finite dimensional $k$-algebra $\Lambda$ of finite global dimension such that
$$
\mathscr{T}^c\cong \rD^b(\Lambda-\text{mod})
$$
\end{thm}
The proofs of both theorems depend on a careful study of various Grothendieck groups of $X$. In particular we will investigate the natural map $K_0(\rD^{\perf}(X))\to K_0(\rD^b(\coh(X)))$ and show that if $g\geq 1$ then the image is not finitely generated, of which Theorem \ref{thm: non-exist of full excep collection in the introduction} and \ref{thm: non-exist of f.d algebra as a cat resolution in the introduction} will be a direct consequence.
\section{Some generalities on K-theory and the Picard group}
In this section we quickly review the K-theory and the Picard group of schemes. For reference see \cite{weibel2013k} Chapter II.
Let $\mathcal{A}$ be an abelian category (or more generally an exact category). The Grothendieck group $K_0(\mathcal{A})$ is defined as an abelian group with generators $[A]$ for each isomorphism class of objects $A$ in $\mathcal{A}$ and subjects to the relation that
$$
[A_2]=[A_1]+[A_3]
$$
for any short exact sequence $0\to A_1\to A_2\to A_3\to 0$ in $\mathcal{A}$.
Similarly let $\mathcal{C}$ be a triangulated category. The Grothendieck group $K_0(\mathcal{C})$ is defined as an abelian group with generators $[C]$ for each isomorphism class of objects $C$ in $\mathcal{C}$ and subjects to the relation that
$$
[C_2]=[C_1]+[C_3]
$$
for any exact triangle $C_1\to C_2\to C_3\to C_1[1]$ in $\mathcal{C}$.
\begin{prop}\label{prop: Grothen group of full excep coll}
If a triangulated category $\mathcal{C}$ has a full exceptional collection $\{A_1\ldots A_n\}$, then the Grothendieck group of $\mathcal{C}$, $\mathrm{K}_0(\mathcal{C})$, is isomorphic to $\mathbb{Z}^n$.
\end{prop}
\begin{proof}
It is an immediate consequence of Definition \ref{defi:full excep coll}.
\end{proof}
\begin{defi}\label{defi: groth gps of perf and coh}
Let $X$ be an Noetherian scheme, follow the standard notation (see for example \cite{srinivas1996algebraic} Section 5.6 or \cite{weibel2013k} Chapter II) we denote the Grothendieck group of $\rD^{\perf}(X)$ by $K_0(X)$ and the Grothendieck group of $\rD^b(\coh(X))$ by $G_0(X)$.
Notice that in some literatures, say \cite{berthelot1966seminaire} Expos\'{e} IV or \cite{manin1969lectures}, $K_0(X)$ is denoted by $K^0(X)$ and $G_0(X)$ is denoted by $K_0(X)$. Nevertheless in this paper we will use the previous notation.
\end{defi}
\begin{rmk}\label{rmk: K naive and K}
In the literature people also define $K^{\text{na\"{i}ve}}_0(X)$ to be the Grothendieck group of the exact category $VB(X)$ and $G^{\text{na\"{i}ve}}_0(X)$ to be the Grothendieck group of the abelian category $\coh(X)$.
Nevertheless $G^{\text{na\"{i}ve}}_0(X)$ is isomorphic to $G_0(X)$ for any Noetherian scheme $X$ (\cite{berthelot1966seminaire}, Expos\'{e} IV, 2.4) and $K^{\text{na\"{i}ve}}_0(X)$ is isomorphic to $K_0(X)$ for any quasi-projective scheme $X$ (\cite{berthelot1966seminaire}, Expos\'{e} IV, 2.9). Since we always work with quasi-projective schemes in this paper, we can identify $G^{\text{na\"{i}ve}}_0(X)$ and $G_0(X)$ as well as $K^{\text{na\"{i}ve}}_0(X)$ and $K_0(X)$.
\end{rmk}
\begin{defi}\label{defi: cartan homomorphism}
Let $X$ be a Noetherian scheme. The inclusion $\rD^{\perf}(X)\hookrightarrow \rD^b(\coh(X))$ gives a group homomorphism
$$
c: K_0(X)\to G_0(X)
$$
which is called the Cartan homomorphism.
\end{defi}
\begin{prop}\label{prop: Cartan map is a module map}
For a Noetherian scheme $X$, the tensor product gives $K_0(X)$ a ring structure and $G_0(X)$ a $K_0(X)$-module structure. Moreover, the Cartan homomorphism $c: K_0(X)\to G_0(X)$ is a morphism of $K_0(X)$-modules.
\end{prop}
\begin{proof}
See \cite{manin1969lectures} 1.5 and 1.6.
\end{proof}
\begin{prop}\label{prop: K=G for regular schemes}
If $X$ is a regular Noetherian scheme, then the Cartan homomorphism is an isomorphism, i.e. we have
$$
c: K_0(X)\overset{\cong}{\to} G_0(X)
$$
\end{prop}
\begin{proof}
See \cite{weibel2013k} Chapter II Theorem 8.2.
\end{proof}
Smooth schemes are regular hence the Cartan homomorphism is an isomorphism for any smooth scheme.
\begin{rmk}\label{rmk: Cartan is not an isom in general}
For general $X$ the Cartan homomorphism is not an isomorphism, actually it is not even injective in general.
\end{rmk}
Next we talk about the functorial properties of $K_0$ and $G_0$, which are more involved. First we have the following definition.
\begin{defi}\label{defi: pull back of K_0 and G_0}
Let $f: X\to Y$ be a morphism of schemes, then the derived pull-back $Lf^*$ functor induces the map
$$
f^*: K_0(Y)\to K_0(X).
$$
See \cite{berthelot1966seminaire} Expos\'{e} IV, 2.7.
If $f: X\to Y$ is a flat morphism between Noetherian schemes, or more generally $f$ is of finite Tor-dimension. Then $Lf^*$ is a functor $D^b(\coh(Y))\to D^b(\coh(X))$ and induces the map
$$
f^*: G_0(Y)\to G_0(X).
$$
See \cite{berthelot1966seminaire} Expos\'{e} IV, 2.12.
\end{defi}
We can also define the push-forward map for $G_0(-)$ for proper morphisms.
\begin{defi}\label{defi: push forward for G_0}
Let $f: X\to Y$ be a proper morphism of Noetherian schemes, then the derived push-forward functor $Rf_*$ induces the map
$$
f_*: G_0(X)\to G_0(Y).
$$
\end{defi}
We will also need some results on the relationship between the Grothendieck group and the Picard group. Let $\Pic(X)$ denote the Picard group of $X$ and we have the following proposition.
\begin{prop}\label{prop: K_0 to Pic det}
There is a determinant map
$$
\det: K_0(X)\to \Pic(X)
$$
which is a surjective group homomorphism. Moreover, the determinant map commutes with the restriction map, i.e. we have the following commutative diagram
$$
\begin{CD}
K_0(X) @>\det>> \Pic(X)\\
@VVrV @VVrV\\
K_0(U) @>\det>> \Pic(U)
\end{CD}
$$
\end{prop}
\begin{proof}
For an $n$-dimensional vector bundle $\mathcal{E}$ we can take its determinant line bundle, i.e. the top exterior power $\wedge^n \mathcal{E}$ and we call it $\det(\mathcal{E})$. Moreover, for a short exact sequence of vector bundles $0\to \mathcal{E}\to \mathcal{F}\to \mathcal{G}\to 0$ we have $\det(\mathcal{F})\cong \det(\mathcal{E})\otimes \det(\mathcal{G})$ hence we get a well-defined group homomorphism $\det: K_0(X)\to \Pic(X)$.
The above diagram commutes because the construction of the determinant map is natural. The surjectivity of det also comes from the construction since we could pick $\mathcal{E}$ to be any line bundle and hence $\det(\mathcal{E})=\mathcal{E}$.
\end{proof}
\section{The irreducible and reduced case of the main theorem}\label{section: the reduced case}
To illustrate the idea, we focus on the case that $X$ is an irreducible, reduced, projective curve over $k$ in this section.
In this case let $p: \widetilde{X}\to X$ be a (geometric) resolution of singularity and we can obtain more information on $\Pic(\widetilde{X})$. First we have
\begin{thm}[\cite{liu2002algebraic} Corollary 7.4.41]\label{thm: picard group of curves genus >1}
Let $\widetilde{X}$ be a smooth, connected, projective curve over an algebraically closed field $k$, of genus $g$. Let $\Pic^0(\widetilde{X})$ denote the subgroup of $\Pic(\widetilde{X})$ consisting of divisors of degree $0$. Let $n\in \mathbb{Z}$ be non-zero and $\Pic^0(\widetilde{X})[n]$ denote the kernel of the multiplication by $n$ map.
\begin{enumerate}
\item If $(n,\text{char} (k))=1$, then $\Pic^0(\widetilde{X})[n]\cong (\mathbb{Z}/n\mathbb{Z})^{2g}$;
\item If $p=\text{char} (k)>0$, then there exists an $0\leq h\leq g$ such that for any $n=p^m$, we have $\Pic^0(\widetilde{X})[n]=(\mathbb{Z}/n\mathbb{Z})^h$.
\end{enumerate}
\end{thm}
\begin{coro}\label{coro: Picard is infinitely generated}
Let $\widetilde{X}$ be a smooth, connected, projective curve over an algebraically closed field $k$ of genus $g\geq 1$, then $\Pic^0(\widetilde{X})$ and hence $\Pic(\widetilde{X})$ are not finitely generated as an abelian group. Moreover, for any non-zero integer $n$, $n\Pic(\widetilde{X})$ is not finitely generated.
\end{coro}
\begin{proof}
It is an immediate consequence of Theorem \ref{thm: picard group of curves genus >1}.
\end{proof}
\begin{rmk}\label{rmk: picard group non-algebraic closed case}
If the base field $k$ is not algebraically closed, then $\Pic^0(\widetilde{X})$ may be finitely generated. For example if $k=\mathbb{Q}$ and $\widetilde{X}$ is a smooth elliptic curve, then by Mordell theorem, $\Pic^0(\widetilde{X})$ is a finitely generated abelian group.
\end{rmk}
Let $Z$ be the closed subset consisting of singular points of $X$ and $U=X-Z$. Since $p: \widetilde{X}\to X$ is a resolution of singularity, the restriction of $p$
$$
p|_{p^{-1}(U)}: p^{-1}(U)\overset{\cong}{\to} U
$$
is an isomorphism.
We want to understand the picard group of $U$. In fact we have the following result
\begin{lemma}\label{lemma: Picard gp of U is infi generated}
Let $\widetilde{X}$ be a smooth and connected projective curve with genus $g\geq 1$ over an algebraically closed field $k$. Let $U$ be a non-empty open subset of $\widetilde{X}$. Then $\Pic(U)$ is not finitely generated. Moreover, for any non-zero integer $n$, $n\Pic(U)$ is not finitely generated.
\end{lemma}
\begin{proof}
This is actually part of \cite{liu2002algebraic} Exercise 7.4.9. Thanks to Georges Elencwajg for helping with the proof. Actually we can write $U=X\backslash \{p_1,\ldots ,p_l\}$. It follows that the kernel of the natural homomorphism $\Pic^0(X)\to \Pic(U)$ is the subgroup of $\Pic^0(X)$ generated by $[p_i]-[p_j]$, hence is finitely generated. Then this lemma is a consequence of Corollary \ref{coro: Picard is infinitely generated}.
\end{proof}
It is also necessary to know the relation between the Picard group of a non-smooth curve $X$ and its non-empty subscheme $U$, which is given in the following lemma.
\begin{lemma}\label{lemma: ext of line bundles on singular curves}
Let $X$ be a (not necessarily smooth) curve over an algebraically closed field $k$. Let $U$ be an open subscheme of $X$.
Let $\mathcal{L}$ be a line bundle on $U$. Then we can always extend $\mathcal{L}$ to a line bundle on $X$. As a result, the restriction map of the Picard groups
$$
r: \Pic(X)\to \Pic(U)
$$
is surjective
\end{lemma}
\begin{proof}
One way to proof this result (thanks to K\c{e}stutis \v{C}esnavi\v{c}ius for pointing it out) is to first find a Cartier divisor $D$ on $U$ whose associated line bundle is $\mathcal{L}$. The existence of such $D$ is guaranteed by \cite{grothendieck1967elements} Proposition 21.3.4 (a). Then apply \cite{grothendieck1967elements} Proposition 21.9.4 we can extend $D$ to a Cartier divisor $D^{\prime}$ on $X$, whose associated line bundle $\mathcal{L}^{\prime}$ gives an extension of $\mathcal{L}$.
\end{proof}
The next Proposition is the key step of our proof.
\begin{prop}\label{prop: Cartan homo has infinite image, reduced case}
Let $X$ be a reduced, irreducible, projective curve of geometric genus $g\geq 1$ over an algebraically closed field $k$, then the image of the Cartan homomorphism
$$
c: K_0(X)\to G_0(X)
$$
is not finitely generated.
\end{prop}
\begin{proof}
First let $Z$ be the closed subset consisting of singular points of $X$ and $U=X-Z$ be the smooth open subscheme. We have the restriction maps $r: K_0(X)\to K_0(U)$ and $r: G_0(X)\to G_0(U)$ and they give the commutative diagram
$$
\begin{CD}
K_0(X) @>c>> G_0(X)\\
@VVrV @VVrV\\
K_0(U) @>c>> G_0(U)
\end{CD}
$$
Since $U$ is smooth, by Proposition \ref{prop: K=G for regular schemes} the bottom map is an isomorphism.
Now assume the image of the top map is finitely generated, then the image of the composition $r\circ c: K_0(X)\to G_0(U)$ is also finitely generated. Since we have the isomorphism $c: K_0(U)\overset{\cong}{\to}G_0(U)$, the left vertical map $r: K_0(X)\to K_0(U)$ must also have finitely generated image. Therefore the image of the composition
$$
K_0(X)\overset{r}{\to} K_0(U)\overset{\det}{\to}\Pic(U)
$$
is finitely generated.
On the other hand we consider the commutative diagram
$$
\begin{CD}
K_0(X) @>\det>> \Pic(X)\\
@VVrV @VVrV\\
K_0(U) @>\det>> \Pic(U)
\end{CD}
$$
By Proposition \ref{prop: K_0 to Pic det} and Lemma \ref{lemma: ext of line bundles on singular curves}, the top and the right vertical map of the above diagram are surjective and so does their composition. As a result $\Pic(U)=\Pic(p^{-1}(U))$ is finitely generated, which is contradictory to Lemma \ref{lemma: Picard gp of U is infi generated}.
\end{proof}
\begin{coro}\label{coro: non-exist of full excep collection for factor through reduced case}
Let $X$ be a reduced, irreducible, projective curves of geometric genus $g\geq 1$ over an algebraically closed field $k$. If the inclusion $\rD^{\perf}(X)\to \rD^b(\coh(X))$ factors through a triangulated category $\mathcal{S}$, then $\mathcal{S}$ cannot have a full exceptional collection.
\end{coro}
\begin{proof}
The composition
$$
K_0(X)\to K_0(\mathcal{S})\to G_0(X)
$$
coincides with the Cartan homomorphism $c: K_0(X)\to G_0(X)$. By Proposition \ref{prop: Cartan homo has infinite image, reduced case}, the image of the Cartan homomorphism is not finitely generated, hence $ K_0(\mathcal{S})$ is not finitely generated. Then by Proposition \ref{prop: Grothen group of full excep coll}, $\mathcal{S}$ cannot have a full exceptional collection.
\end{proof}
\begin{coro}\label{coro: non-exist of full excep collection reduced case}
Let $X$ be a reduced, irreducible, projective curves of geometric genus $g\geq 1$ over an algebraically closed field $k$. Let $(\mathscr{T},\pi_*,\pi^*)$ be a categorial resolution of $X$. Then $\mathscr{T}^c$ cannot have a full exceptional collection.
\end{coro}
\begin{proof}
By the definition of categorical resolution, the composition
$$
\rD^{\perf}(X)\overset{\pi^*}{\to}\mathscr{T}^c \overset{\pi_*}{\to}\rD^b(\coh(X))
$$
is the same as the inclusion $\rD^{\perf}(X)\hookrightarrow \rD^b(\coh(X))$. Therefore the composition
$$
K_0(X)\to K_0(\mathscr{T}^c)\to G_0(X)
$$
coincides with the Cartan homomorphism $c: K_0(X)\to G_0(X)$. Then it is a direct consequence of Corollary \ref{coro: non-exist of full excep collection for factor through reduced case}.
\end{proof}
\section{The general case of the main theorem}\label{section: the general case}
In this section we consider the case that $X$ is not irreducible nor reduced. In this case we still want to show that the image of the Cartan homomorphism $c: K_0(X)\to G_0(X)$ is not finitely generated but the proof is more involved.
Let $X_{\red}$ denote the associated reduced scheme of $X$ and $i:X_{\red}\to X$ the natural closed immersion. Then $X_{\red}$ is a reduced, projective curve with the same geometric genus as $X$.
First we investigate the $g=0$ case, which is the following Proposition.
\begin{prop}\label{prop: g=0 has a full exc coll}
Let $X$ be a projective curve over an algebraically closed field $k$ of geometric genus $g=0$, then $X$ has a categorical resolution $(\mathscr{T},\pi^*,\pi_*)$ such that $\mathscr{T}^c$ has a full exceptional collection.
\end{prop}
\begin{proof}
As we mentioned in the Introduction, the result in this Proposition is a direct consequence of the construction of categorical resolution in \cite{kuznetsov2014categorical}, although it is not explicitly stated in \cite{kuznetsov2014categorical}.
First \cite{kuznetsov2014categorical} Equation (59) in page 69 gives a chain
\begin{equation}
\xymatrix{X_m \ar[r] &X_{m-1}\ar[r]&\ldots \ar[r]&X_1\ar[r]&X_0\ar@{=}[r]&X\\
& Z_{m-1}\ar@{^{(}->}[u] & &Z_1\ar@{^{(}->}[u]&Z_0\ar@{^{(}->}[u]}
\end{equation}
where each $X_{i+1}$ is the blowup of $X_i$ at the center $Z_i$ and $(X_m)_{\red}$ is smooth.
Moreover \cite{kuznetsov2014categorical} Equation (61) in page 71 tells us that there exists a categorical resolution $\mathscr{T}$ of $X$ such that its subcategory $\mathscr{T}^c$ has the following semiorthogonal decomposition
\begin{equation}
\begin{split}
\mathscr{T}^c=&\langle\underbrace{\rD^b(\coh(Z_0))\ldots \rD^b(\coh(Z_0))}_{n_0 \text{ times}},\ldots,\\
&\underbrace{\rD^b(\coh(Z_{m-1}))\ldots \rD^b(\coh(Z_{m-1}))}_{n_{m-1} \text{ times}},\\
&\underbrace{\rD^b(\coh((X_m)_{\red}))\ldots \rD^b(\coh((X_m)_{\red}))}_{n_m \text{ times}}\rangle
\end{split}
\end{equation}
where the $n_i$'s are certain multiples given in \cite{kuznetsov2014categorical} after Equation (61) and we do not need their precise definition.
Since $X$ is of dimension $1$, each of the $Z_i$ is $0$-dimensional hence $\rD^b(\coh(Z_i))$ has a full exceptional collection. Moreover since $X$ is of genus $0$, we have $(X_m)_{\red}$ is a finite product of $\mathbb{P}^1$'s hence $\rD^b(\coh((X_m)_{\red}))$ also has a full exceptional collection. As a result $\mathscr{T}^c$ has a full exceptional collection.
\end{proof}
Then we consider the $g\geq 1$ case. By Definition \ref{defi: pull back of K_0 and G_0} and \ref{defi: push forward for G_0} we have the natural map
$$
i^*:K_0(X)\to K_0(X_{\red})
$$
and
$$
i_*:G_0(X_{\red})\to G_0(X).
$$
For $i_*$ we have the following "devissage" theorem.
\begin{thm}\label{thm: devissage}[\cite{weibel2013k} Chapter II Corollary 6.3.2]
Let $X$ be a Noetherian scheme, and $X_{\red}$ the associated reduced
scheme. Then $
i_*:G_0(X_{\red})\to G_0(X)
$ is an isomorphism.
\end{thm}
\begin{proof}
See \cite{weibel2013k} Chapter II Corollary 6.3.2.
\end{proof}
However, the following diagram
$$
\begin{CD}
K_0(X) @>c>> G_0(X)\\
@VVi^*V @A\cong Ai_*A\\
K_0(X_{\red}) @>c>> G_0(X_{\red})
\end{CD}
$$
does not commute. Hence we cannot directly apply the result in Section \ref{section: the reduced case} and need to find another way.
Let $X=\cup_{i=1}^m X_i$ be the decomposition into irreducible components, hence $X_{\red}=\cup_{i=1}^m (X_i)_{\red}$ (Do not confused with the $X_i$'s in the proof of Proposition \ref{prop: g=0 has a full exc coll}). Since $X$ has geometric genus $\geq 1$, at least one of the irreducible components $X_i$'s also has geometric genus $\geq 1$, say $X_1$.
For an non-empty, open, irreducible subscheme $U$ of $X_1$ we also consider $U_{\red}$. We can make $U$ small enough so that both $U$ and $U_{\red}$ are affine and $U_{\red}$ is smooth. Let $U=\Spec(A)$ and $U_{\red}=\Spec(A/I)$ where $I$ is the nilpotent radical of $A$ with $I^{l+1}=0$. Since $U$ is irreducible, $I$ is also the minimal prime ideal of $A$. Let $\mathcal{I}$ denote the associated sheaf on $U$.
Let us consider the diagram
$$
\begin{CD}
K_0(U) @>c>> G_0(U)\\
@VVi^*V @AAi_*A\\
K_0(U_{\red}) @>c>> G_0(U_{\red})
\end{CD}
$$
Again it does not commute. Nevertheless we will prove that it is not too far from commutative.
First let us fix the notations. Let $e_U$ denote the element $[\mathcal{O}_U]$ in $G_0(U)$ and $e_{U_{\red}}$ denote the element $[\mathcal{O}_{U_{\red}}]$ in $G_0(U_{\red})$.
\begin{lemma}\label{lemma: n times reduced in G_0}
We can choose $U$ small enough such that there is a non-zero integer $n$ such that
$$
e_U=n\,i_*(e_{U_{\red}}).
$$
\end{lemma}
\begin{proof}
By Theorem \ref{thm: devissage}, $i_*$ is an isomorphism so it is sufficient to prove
$$
i_*^{-1}(e_U)=n \,e_{U_{\red}}
$$
in $G_0(U_{\red})$.
It is clear that in $G_0(U_{\red})$ we have
\begin{equation}\label{equa: decompose of e_U}
i_*^{-1}(e_U)=e_{U_{\red}}+[\mathcal{I}/\mathcal{I}^2]+\ldots+[\mathcal{I}^{l-1}/\mathcal{I}^l]+[\mathcal{I}^l].
\end{equation}
Each of the $\mathcal{I}^{k-1}/\mathcal{I}^k$ is a coherent sheaf on the smooth scheme $U_{\red}$ hence we have a resolution of finite length
$$
0\to \mathcal{P}^{m_k}_k\to \mathcal{P}^{m_k-1}_k\to \ldots \to \mathcal{P}^{0}_k\to \mathcal{I}^{k-1}/\mathcal{I}^k ~\text{ for }1\leq k\leq l+1.
$$
where the $\mathcal{P}^{m_k-j}_k$'s are locally free sheaves on $U_{\red}$. We can shrink $U$ further to make all the $\mathcal{P}^{m_k-j}_k$'s are free sheaves on $U_{\red}$. Hence for each $k$ there is an integer $n_k$ such that
$$
[\mathcal{I}^{k-1}/\mathcal{I}^k]=n_k e_{U_{\red}}
$$
and as a result there is an integer $n$ such that
$$
i_*^{-1}(e_U)=n \,e_{U_{\red}}
$$
in $G_0(U_{\red})$.
We still need to show that $n \neq 0$. This can be achieved by localizing to the generic point of $U$. Recall that $I$ is the minimal prime ideal of $A$ hence $I$ corresponds to the generic point of $U$.
Let us denote $A_I$, the localization of $A$ at $I$ by $B$ and denote the ideal $IB$ by $J$. Moreover we denote $\Spec(B)$ by $V$ and similarly denote $\Spec(B/J)$ by $V_{\red}$. Let $f: V\to U$, $f_{\red}: V_{\red}\to U_{\red}$, and $j: V_{\red}\to V$ be the natural maps.
\begin{comment}
\begin{lemma}\label{lemma: base chage for G_0}
Let
$$
\begin{CD}
X^{\prime} @>g^{\prime}>> X\\
@Vf^{\prime}VV @VVfV\\
Y^{\prime} @>g>>Y
\end{CD}
$$
be a fiber product diagram of quasi-projective schemes. Assume that $f$ is proper and $g$ has finite flat dimension, and that that $X$ and $Y^{\prime}$ are Tor-independent over $Y$, i.e. for $q>0$ and all $x\in X$, $y^{\prime}\in Y^{\prime}$ and $y\in Y$ with $y=f(x)=g(y^{\prime})$ we have
$$
\text{Tor}_q^{\mathcal{O}_{Y,y}}(\mathcal{O}_{X,x},\mathcal{O}_{Y^{\prime},y^{\prime}})=0.
$$
Then
$$
g^*f_*=f^{\prime}_*g^{\prime *}
$$
as maps $G_0(X)\to G_0(Y^{\prime})$.
\end{lemma}
\begin{proof}[Proof of Lemma \ref{lemma: base chage for G_0}] See \cite{srinivas1996algebraic} Proposition 5.13.
\end{proof}
Now we consider the diagram
$$
\begin{CD}
V_{\red}@>f_{\red}>> U_{\red}\\
@VjVV @VViV\\
V @>f>> U
\end{CD}
$$
Since $i$ is a closed immersion and $f$ is flat, it satisfies the requirement of Lemma \ref{lemma: base chage for G_0}. Hence we have the following commutative diagram
$$
\begin{CD}
G_0(U_{\red}) @>i_*>> G_0(U)\\
@Vf_{\red}^*VV @VVf^*V\\
G_0(V_{\red}) @>j_*>> G_0(V)
\end{CD}
$$
\end{comment}
Since $f: V\to U$ is flat, we can define the pull-back map $f^*: G_0(U)\to G_0(V)$.
Let us denote the class $[\mathcal{O}_V]$ in $G_0(V)$ by $e_V$. By definition $f^*(e_U)=e_V$. If $e_U=0$ then we have $e_V=0$ and $j_*^{-1}(e_V)=0$.
On the other hand $B/J=A_I/I_I\cong \Frac(A/I)$ is a field hence $G_0(V_{\red})=G_0(B/J)\cong \mathbb{Z}$. Similar to Equation (\ref{equa: decompose of e_U}) we have
$$
j_*^{-1}(e_V)=[B/J]+[J/J^2]+\ldots +[J^{l-1}/J^l]+[J^l].
$$
Each of the $J^{m-1}/J^m$ is a vector space over the field $B/J$ hence the right hand side cannot be zero in $G_0(V_{\red})$.
\end{proof}
\begin{prop}\label{prop: n times reduced for general elements in G_0}
Let $U$ and $n$ be as in Lemma \ref{lemma: n times reduced in G_0}. Then for any element $a\in K_0(U)$ we have
$$
c(a)=n\,i_*c\,i^*(a),
$$
i.e. the diagram
$$
\begin{CD}
K_0(U) @>c>> G_0(U)\\
@VVn\,i^*V @A\cong Ai_*A\\
K_0(U_{\red}) @>c>> G_0(U_{\red})
\end{CD}
$$
commutes.
\end{prop}
\begin{proof}
We need the following lemma.
\begin{lemma}\label{lemma: i_* is a module map}
For any Noetherian scheme $U$, $G_0(U_{\red})$ has a $K_0(U)$-module structure. Moreover, the map $i_*: G_0(U_{\red})\to G_0(U)$ is a morphism of $K_0(U)$-modules.
\end{lemma}
\begin{proof}[Proof of Lemma \ref{lemma: i_* is a module map}]
First the $K_0(U)$-module structure on $G_0(U_{\red})$ is given by composing with $i^*$. More explicitly, for $a\in K_0(U)$ and $m\in G_0(U_{\red})$ we define
$$
a\cdot m= i^*(a)\cdot m
$$
where the right hand side uses the $K_0(U_{\red})$-module structure on $G_0(U_{\red})$.
Then we need to show that $i_*$ is a $K_0(U)$-module map, i.e.
$$
i_*(i^*(a)\cdot m)=a\cdot i_*(m).
$$
But this is exactly the projection formula.
\end{proof}
Now we can prove Proposition \ref{prop: n times reduced for general elements in G_0}. Let us denote $[\mathcal{O}_U]\in K_0(U)$ by $1_U$ and $[\mathcal{O}_{U_{\red}}]\in K_0(U_{\red})$ by $1_{U_{\red}}$. Then it is clear that
$$
c(1_U)= e_U \text{ and } c(1_{U_{\red}})=e_{U_{\red}}.
$$
Then for any $a\in K_0(U)$ we have
\begin{equation*}
\begin{split}
c(a)=&c(a\cdot 1_U)\\
=&a\cdot e_U\\
=&a\cdot (n i_*(e_{U_{\red}})) (\text{ Lemma } \ref{lemma: n times reduced in G_0})\\
=&n(a\cdot i_*(e_{U_{\red}}))\\
=&ni_*(i^*(a)\cdot e_{U_{\red}}) (\text{ Lemma } \ref{lemma: i_* is a module map})\\
=&n\,i_*c\,i^*(a).
\end{split}
\end{equation*}
\end{proof}
Now we are ready to prove the following Proposition, which is the general version of Proposition \ref{prop: Cartan homo has infinite image, reduced case}.
\begin{prop}\label{prop: Cartan homo has infinite image, general case}
Let $X$ be a projective curves of geometric genus $g\geq 1$ over an algebraically closed field $k$, then the image of the Cartan homomorphism
$$
c: K_0(X)\to G_0(X)
$$
is not finitely generated.
\end{prop}
\begin{proof}
First let $U$ be as in Lemma \ref{lemma: n times reduced in G_0} and Proposition \ref{prop: n times reduced for general elements in G_0}. By Proposition \ref{prop: n times reduced for general elements in G_0} and Theorem \ref{thm: devissage}there is a non-zero integer $n$ such that the following diagram commutes
$$
\begin{CD}
K_0(U) @>c>> G_0(U)\\
@VVn\,i^*V @VV(i_*)^{-1}V\\
K_0(U_{\red}) @>c>> G_0(U_{\red})
\end{CD}
$$
hence the diagram
$$
\begin{CD}
K_0(X) @>c>> G_0(X)\\
@VrVV @VVrV\\
K_0(U) @>c>> G_0(U)\\
@VVn\,i^*V @VV(i_*)^{-1}V\\
K_0(U_{\red}) @>c>> G_0(U_{\red})
\end{CD}
$$
commutes. For short we have
$$
\begin{CD}
K_0(X) @>c>> G_0(X)\\
@VVn\,i^* rV @VV(i_*)^{-1} rV\\
K_0(U_{\red}) @>c>> G_0(U_{\red})
\end{CD}
$$
Now assume the image of $c: K_0(X) \to G_0(X)$ is finitely generated. Since $U_{\red}$ is smooth, the $c: K_0(U_{\red}) \to G_0(U_{\red})$ in the above diagram is an isomorphism, hence the image of $n\,i^* r$ is also finitely generated.
Next we observe that we have the commutative diagrams
$$
\begin{CD}
K_0(X) @>n\,i^*>> K_0(X_{\red})\\
@VVrV @VV rV\\
K_0(U) @>n\,i^*>> G_0(U_{\red})
\end{CD}
$$
and
$$
\begin{CD}
K_0(X) @>\det>> \Pic(X)\\
@Vn\,i^*VV @VVn\,i^*V\\
K_0(X_{\red}) @>\det>> \Pic(X_{\red})\\
@VVrV @VVrV\\
K_0(U_{\red}) @>\det>> \Pic(U_{\red})
\end{CD}
$$
From the left-bottom composition of the above diagram we know that the image of $\det\circ r\circ (n\,i^*)$ is finitely generated.
On the other hand we will study the top-right composition of the above diagram. By Proposition \ref{prop: K_0 to Pic det} the map det is surjective and by Lemma \ref{lemma: ext of line bundles on singular curves} the map $r$ is also surjective. As for the map $i^*$ we need the following lemma.
\begin{lemma}\label{lemma: Pic is surj when pull back to reduced}[\cite{liu2002algebraic} Lemma 7.5.11]
Let $X$ be a connected projective curve over an algebraically closed field $k$, Then $i^*: \Pic(X)\to \Pic(X_{\red})$ is surjective.
\end{lemma}
\begin{proof}[Proof of Lemma \ref{lemma: Pic is surj when pull back to reduced}] See \cite{liu2002algebraic} Lemma 7.5.11.
\end{proof}
Then it is clear that the image of $r\circ(ni^*)\circ \det$ is $n\Pic(U_{\red})$. Compare with the left-bottom composition we get the conclusion that $n\Pic(U_{\red})$ is finitely generated, which is contradictory to Lemma \ref{lemma: Picard gp of U is infi generated}.
\end{proof}
\begin{coro}\label{coro: non-exist of full excep collection for factor through general case}
Let $X$ be a projective curves of geometric genus $g\geq 1$ over an algebraically closed field $k$. If the inclusion $\rD^{\perf}(X)\to \rD^b(\coh(X))$ factors through a triangulated category $\mathcal{S}$, then $\mathcal{S}$ cannot have a full exceptional collection.
\end{coro}
\begin{proof}
The proof is almost the same as that of Corollary \ref{coro: non-exist of full excep collection for factor through reduced case} except that we use Proposition \ref{prop: Cartan homo has infinite image, general case} instead of Proposition \ref{prop: Cartan homo has infinite image, reduced case}.
\end{proof}
\begin{thm}\label{thm: non-exist of full excep collection general case}[See Theorem \ref{thm: non-exist of full excep collection in the introduction}]
Let $X$ be a projective curve over an algebraically closed field $k$. Let $(\mathscr{T},\pi^*,\pi_*)$ be a categorical resolution of $X$. If the geometric genus of $X$ is $\geq 1$, then $\mathscr{T}^c$ cannot have a full exceptional collection.
In other words, $X$ has a categorical resolution which admits a full exceptional collection if and only if the geometric genus of $X$ equals to $0$.
\end{thm}
\begin{proof}
Since we have Proposition \ref{prop: g=0 has a full exc coll}, it is sufficient to prove the first claim of the theorem, which is a direct consequence of Corollary \ref{coro: non-exist of full excep collection for factor through general case}.
\end{proof}
\begin{rmk}\label{rmk: haven't use T is smooth}
In the proof we did not use the fact the $\mathscr{T}$ is a smooth triangulated category.
\end{rmk}
\begin{rmk}\label{rmk: the proof does not work for non-algebraically closed field}
The proof of Theorem \ref{thm: non-exist of full excep collection general case} fails if the base field $k$ is not algebraically closed. The main reason is when $k$ is not algebraically closed, the picard group may be finitely generated. See Remark \ref{rmk: picard group non-algebraic closed case} after Corollary \ref{coro: Picard is infinitely generated}.
Nevertheless, we expect that the result of Theorem \ref{thm: non-exist of full excep collection general case} is still true in the non-algebraically closed case. We believe that a proof could be achieved through a systematic study of the behavior of categorical resolution under scalar extension and we will leave this topic for a future paper.
\end{rmk}
It is worthwhile to mention that we have another application of Proposition \ref{prop: Cartan homo has infinite image, general case} (thanks to Igor Burban for pointing it out).
\begin{thm}\label{thm: non-exist of f.d algebra as a cat resolution}
Let $X$ be a projective curve over an algebraically closed field $k$ of geometric genus $\geq 1$. Let $(\mathscr{T},\pi^*,\pi_*)$ be a categorical resolution of $X$. Then $\mathscr{T}^c$ cannot have a tilting object, moreover there cannot be a finite dimensional $k$-algebra $\Lambda$ of finite global dimension such that
$$
\mathscr{T}^c\cong \rD^b(\Lambda-\text{mod})
$$
where $\rD^b(\Lambda-\text{mod})$ is the derived category of bounded complexes of finitely generated $\Lambda$-modules.
\end{thm}
\begin{proof}
With Proposition \ref{prop: Cartan homo has infinite image, general case} it is sufficient to prove that the Grothendieck group $K_0(\rD^b(\Lambda-\text{mod}))$ is finitely generated. The proof is as follows: Since $\Lambda$ is finite dimensional, it is a finitely generated Artinian $k$-algebra, hence every finitely generated $\Lambda$-module has a composition series. Moreover the set of isomorphic classes of simple $\Lambda$-module is finite. We get the desired result.
\end{proof}
\begin{rmk}
Again in the proof we did not use that fact that $\Lambda$ is of finite global dimension, which corresponds to the smoothness of $\mathscr{T}$ .
\end{rmk}
\section*{Acknowledgement}
The author first wants to thank Valery Lunts for introducing him to this topic and for very helpful comments and suggestions during this work. Igor Burban suggests the author to investigate the non-irreducible case and shares the ideas on tilting object and the author is grateful to him too. Moreover the author would like to thank Dave Anderson,
K\c{e}stutis \v{C}esnavi\v{c}ius , Georges Elencwajg, Volodimir Gavran, Adeel Khan, S\'{a}ndor Kov\'{a}cs, and Jason Starr for their help on algebraic K-theory and the theory of algebraic curves.
| {
"timestamp": "2016-03-14T01:03:53",
"yymm": "1506",
"arxiv_id": "1506.02991",
"language": "en",
"url": "https://arxiv.org/abs/1506.02991",
"abstract": "This paper gives a complete answer of the following question: which (singular, projective) curves have a categorical resolution of singularities which admits a full exceptional collection? We prove that such full exceptional collection exists if and only if the geometric genus of the curve equals to 0. Moreover we can also prove that a curve with geometric genus equal or greater than 1 cannot have a categorical resolution of singularities which has a tilting object. The proofs of both results are given by a careful study of the Grothendieck group and the Picard group of that curve.",
"subjects": "Algebraic Geometry (math.AG); Category Theory (math.CT); K-Theory and Homology (math.KT)",
"title": "The full exceptional collections of categorical resolutions of curves",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.977022634773286,
"lm_q2_score": 0.7248702702332475,
"lm_q1q2_score": 0.7082146612921113
} |
https://arxiv.org/abs/1401.1793 | No finite $5$-regular matchstick graph exists | A graph $G=(V,E)$ is called a unit-distance graph in the plane if there is an injective embedding of $V$ in the plane such that every pair of adjacent vertices are at unit distance apart. If additionally the corresponding edges are non-crossing and all vertices have the same degree $r$ we talk of a regular matchstick graph. Due to Euler's polyhedron formula we have $r\le 5$. The smallest known $4$-regular matchstick graph is the so called Harborth graph consisting of $52$ vertices. In this article we prove that no finite $5$-regular matchstick graph exists. | \section{Prologue}
\noindent
One of the possibly best known problems in combinatorial geometry asks how often the same distance can occur among
$n$ points in the plane. Via scaling we can assume that the most frequent distance has length $1$. Given any set $P$ of points in the plane, we can define the so called unit-distance graph in the plane, connecting two elements of $P$ by an edge if their distance is one. The known bounds for the maximum number $u(n)$ of edges of a unit-distance graph in the plane, see e.~g.{} \cite{1086.52001}, are given by
\[
\Omega\!\left(ne^{\frac{c\log n}{\log\log n}}\right)\le u(n)\le O\!\left(n^{\frac{4}{3}}\right).
\]
For $n\le 14$ the exact numbers of $u(n)$ were determined in \cite{schade}, see also \cite{1086.52001}.
If we additionally require that the edges are non-crossing, then we obtain another class of geometrical and combinatorial objects:
\begin{definition}
A \textbf{matchstick graph} $\mathcal{M}$ consists of a graph $G=(V,E)$
and an injective embedding $g:V\rightarrow\mathbb{R}^2$ in the plane which fulfill the following conditions:
\begin{itemize}
\item[(1)] $G$ is a planar (simple) graph,
\item[(2)] for all edges $\{i,j\}\in E$ we have $\left\Vert g(i),g(j)\right\Vert_2=1$, where $\Vert x,y\Vert_2$
denotes the Euclidean distance between the points $x$ and $y$,
\item[(3)] $g(i)\neq g(j)$ for $i\neq j$, and
\item[(4)] if $\left\{i_1,j_1\right\},\left\{i_2,j_2\right\}\in E$ for pairwise different vertices
$i_1,i_2,j_1,j_2\in V$ then the line segments $\overline{g\left(i_1\right)g\left(j_1\right)}$
and $\overline{g\left(i_2\right)
g\left(j_2\right)}$ do not have a common point.
\end{itemize}
\end{definition}
\noindent
For matchstick graphs the known bounds for the maximum number $\tilde{u}(n)$ of edges, see e.~g.{} \cite{1086.52001}, are given by
\[
\left\lfloor 3n-\sqrt{12n-3}\right\rfloor\le \tilde{u}(n)\le 3n-O\!\left(\sqrt{n}\right),
\]
where the lower bound is conjectured to be exact.
We call a matchstick graph $r$-regular if every vertex has degree $r$. In \cite{matchsticks_in_the_plane} the authors consider $r$-regular matchstick graphs with the minimum number $m(r)$ of vertices. Obviously we have $m(0)=1$, $m(1)=2$, and $m(2)=3$, corresponding to a single vertex, a single edge, and a triangle, respectively.
The determination of $m(3)$ is an entertaining amusement. At first we observe that $m(3)$ must be even since every graph contains an even number of vertices with odd degree. For $n\le 8$ vertices the set of $3$-regular planar graphs is fairly assessable, consisting of one graph of order, i.~e.{} the number of vertices, $4$, one graph of order $6$, and three graphs of order $8$. Utilizing area arguments rules out all but one graph of order $8$, so that we have $m(3)=8$. For degree $r=4$ the exact determination of $m(4)$ is unsettled so far. The smallest known example is the so called Harborth graph, see e.~g.{} \cite{gerbracht}, yielding $m(4)\le 52$.
Due to the Eulerian polyhedron formula every finite planar graph contains a vertex of degree at most five so that we have $m(r)=\infty$ for $r\ge 6$. We would like to remark that the regular triangular lattice is the unique example of an infinite matchstick graph with degree at least $6$.
For degree $5$ it is announced at several places that no finite $5$-regular matchstick graph does exist, see e.~g.{} Ivars Peterson's MathTrek ``Match Sticks in the Summer'' \cite{peterson} most likely referring to personal communication with Heiko Harborth, the discoverer of the $4$-regular matchstick graph. Indeed Heiko Harborth posed the question whether there exists a $5$-regular matchstick graph, or not at an Oberwolfach meeting and was aware of a preprint claiming the non-existence proof (personal communication). After a while he found out that there were some mistakes in the proof, so that to his knowledge there is no correct proof of the non-existence. Asking the author of this preprint about the state of this problem he replied too, that the problem is still open (at this point in time the concerning was untraceable).
Wolfram Research's MathWorld, see http://mathworld.wolfram.com/MatchstickGraph.html, refers to Erich Friedman, who himself maintains a webpage \cite{friedman} stating the non-existence of a $5$-regular matchstick graph. Asking Erich Friedman for a proof of this claim he responded
that unfortunately he lost his reference but also thinks that such a graph cannot exist.
In \cite{1063.05036} the authors list five publications where they have mentioned an unpublished proof for the non-existence of a $5$-regular matchstick graph and state that up to their knowledge the problem is open.
After the submission of this paper one of the referees came up with a very short proof for the non-existence \cite{short}. In this paper we would like to give a different, admittedly more complicated, proof using a topological equation that needs some explanation.
So in some parts this (unpublished) paper coincides with \cite{short}.
In the meantime Heiko Harborth could recover the lost preprint and send it to the original author and myself. Curiously enough, after
rectifying some annoying minor mistakes, the preprint turns out to be basically correct. So the author of this article took the opportunity to retype and slightly modify the original manuscript, see \cite{manuscript}. It even turned out that the underlying method can be easily adopted to prove the following stronger result: No finite matchstick graph with minimum degree $5$ does exist. So up to now there are at least three proofs of Theorem~\ref{thm_main} whose pairwise intersections are non-empty.
We would like to mention the recent proof of the Higuchi conjecture \cite{1117.05026} -- a result of similar flavor where related techniques are applied.
\section{Main Theorem}
\begin{theorem}
\label{thm_main}
No finite $5$-regular matchstick graph does exist.
\end{theorem}
\noindent
In order to prove Theorem \ref{thm_main} we denote the number of $i$-gons, i.~e.{} a face consisting of $i$ vertices, of a given matchstick graph $\mathcal{M}$ by $F_i$, where we also count the outer face. In the example in Figure \ref{fig_matchstick_graph} we have $F_3=3$, $F_4=2$, $F_7=1$, and $F_i=0$ for all other $i$.
\begin{figure}[htp]
\begin{center}
\includegraphics{regular_matchstick_graphs_1_0.pdf}
\caption{A matchstick graph.}
\label{fig_matchstick_graph}
\end{center}
\end{figure}
\begin{lemma}
\label{lemma_euler_sum}
For a finite $5$-regular matchstick graph we have
\[
\sum_{i=3}^{\infty}(10-3i)F_i=F_3-2F_4-5F_5-8F_6-11F_7-\dots=20.
\]
\end{lemma}
\begin{proof}
Due to the Eulerian polyhedral formula we have $V+F-E=2$, where $V$ denotes the number of vertices, $F$ the number
of faces, and $E$ the number of edges. The number of faces is given by
\[
F=\sum_{i=3}^\infty F_i.
\]
Since every edge is part of two faces and every vertex is part of $5$ faces we have
\[
2E=\sum_{i=3}^{\infty} iF_i\quad\text{and}\quad
5V=\sum_{i=3}^{\infty} iF_i.
\]
Inserting yields the proposed formula.
\end{proof}
\begin{definition}
The \textbf{face set} $f(v)$ of a vertex $v$ contained in a planar graph is a multiset containing the number of corners
of the adjacent faces.
\end{definition}
In the example in Figure \ref{fig_matchstick_graph} we have $f(v)=\{3,3,3,4,4\}$. For a $5$-regular matchstick graph the face sets $f(v)$ of the vertices have cardinality $5$, i.~e. there exist integers $a_{v,1},\dots,a_{v,5}\ge 3$ with $f(v)=\left\{a_{v,1},\dots,a_{v,5}\right\}$. Due to the angle sum of $2\pi$ at a vertex we have $a_{v,i}\ge 4$ for at least one index $i$ for every vertex $v\in V$.
\begin{definition}
For an integer $a\ge 3$ we set $c(a)=\frac{10-3a}{a}$ and use this to define the \textbf{contribution} $c(v)$ of
a vertex with face set $f(v)=\left\{a_{v,1},\dots,a_{v,5}\right\}$ as
\[
c(v)=c\left(\left\{a_{v,1},\dots,a_{v,5}\right\}\right)=\sum_{i=1}^5c\!\left(a_{v,i}\right)=
\sum_{i=1}^5\frac{10-3a_{v,i}}{a_{v,i}}.
\]
For $U\subseteq V$ we set $c(U)=\sum\limits_{u\in U}c(u)$.
\end{definition}
Now we can relate this definition to Lemma \ref{lemma_euler_sum} and state two easy lemmas.
\begin{lemma}
\label{lemma_20}
If $\mathcal{M}$ is a (finite) $5$-regular matchstick graph we have $c(V)=20$.
\end{lemma}
\begin{proof}
For $j\ge 3$ we have $\left|\left\{a_{v,i}=j\mid v\in V,\, 1\le i\le 5\right\}\right|=jF_j$. Applying this
to the definition of $c(V)$ yields the left hand side of the formula of Lemma \ref{lemma_euler_sum}.
\end{proof}
We would like to remark that for an infinite $5$-regular matchstick graph we would have $c(V)=0$.
\begin{lemma}
\label{lemma_non_negative}
The face sets with non-negative contribution are given by
\begin{eqnarray*}
c\left(\left\{3,3,3,3,4\right\}\right)=\frac{5}{6}, && c\left(\left\{3,3,3,4,4\right\}\right)=0,\\
c\left(\left\{3,3,3,3,5\right\}\right)=\frac{1}{3}, && c\left(\left\{3,3,3,3,6\right\}\right)=0.
\end{eqnarray*}
\end{lemma}
\noindent
Our strategy for proving Theorem \ref{thm_main} will be to partition the vertex set $V$ into subsets each having non-positive contribution, which yields a contradiction to Lemma \ref{lemma_20}.
To determine a suitable partition of $V$ into subsets for a given matchstick graph $\mathcal{M}$ we consider the set $\mathcal{TQ}$ of triangles and quadrangles with internal angles in $\left\{\frac{1}{3}\pi,\frac{2}{3}\pi\right\}$ and define an equivalence relation $\sim$ on $\mathcal{TQ}$. Whenever there exist $x,y\in\mathcal{TQ}$ sharing an edge we require $x\sim y$. We complete this relation to an equivalence relation by taking the transitive closure.
Since we prove that no finite $5$-regular matchstick graphs exists it is impossible to give examples to illustrate our definitions. Therefore we define them (mostly) for matchstick graphs $\mathcal{M}$ where the degrees of its vertices
are at most $5$.
\begin{definition}
For a given (possible infinite) matchstick graph $\mathcal{M}$ with vertex degrees at most $5$ we call an
equivalence class $x\sim =\left\{y\in\mathcal{TQ}\mid y\sim x\right\}$ of the above defined equivalence
relation a $\mathcal{TQ}$-class.
\end{definition}
So a $\mathcal{TQ}$-class is an edge-connected union of triangles and quadrangles whose vertices are situated on a suitable common regular triangular lattice. In Figure \ref{fig_honeycomb_components} we have depicted an example, where we have marked the $\mathcal{TQ}$-classes by different face colors.
\begin{figure}[htp]
\begin{center}
\includegraphics{regular_matchstick_graphs_2_0.pdf}
\caption{$\mathcal{TQ}$-classes of a matchstick graph.}
\label{fig_honeycomb_components}
\end{center}
\end{figure}
\noindent
In the next lemma we summarize some easy facts on $\mathcal{TQ}$-classes.
\begin{lemma}
Let $\mathcal{B}$ be a $\mathcal{TQ}$-class, then the following holds:
\begin{itemize}
\item[(1)] The faces of $\mathcal{B}$ are edge-to-edge connected, meaning that the dual graph is connected.
\item[(2)] The vertices of $\mathcal{B}$ are situated on a suitable regular triangular lattice.
\end{itemize}
\end{lemma}
\noindent
For brevity we associate with a $\mathcal{TQ}$-class $\mathcal{B}$ the set of its vertices $V(\mathcal{B})$ and the set of its edges $E(\mathcal{B})$ so that we can utilize the notation $c(\mathcal{B})$ for the contribution of the vertex set of $\mathcal{B}$. Now our aim is to show that the contribution $c\!\left(\mathcal{B}\right)$ for every $\mathcal{TQ}$-class $\mathcal{B}$ of a $5$-regular matchstick graph is at most zero. Since some vertices may be contained in more than one $\mathcal{TQ}$-class the corresponding vertex sets can not be used directly to partition the vertex set $V$. Instead we choose the union $\mathcal{C}=\cup_i\mathcal{B}_i$ of all $\mathcal{TQ}$-classes. Here every vertex $v\in V$ occurs at most once in $\mathcal{C}$, nevertheless it may be contained in several $\mathcal{TQ}$-classes $\mathcal{B}_i$. In order to prove $c(\mathcal{C})\le 0$ we have to perform some bookkeeping during the proof of $c(\mathcal{B})\le 0$, which will be the topic of the next section. Once we haven proven this we can state:
\bigskip
\noindent
\textbf{Proof of the main theorem.} Let $\mathcal{M}$ be a finite $5$-regular matchstick graph and $\mathcal{C}$ be the union of all $\mathcal{TQ}$-classes of $\mathcal{M}$. Using Corollary \ref{cor_non_negativ_all} and the fact that all vertices with positive contribution are contained in a $\mathcal{TQ}$-class, see Lemma \ref{lemma_non_negative}, we conclude
\[
c(V)\,\,\,\,=\,\,\,\,c(\mathcal{C})+\sum_{v\in V\backslash\mathcal{C}} c(v)\,\,\,\,\le\,\,\,\, 0+
\sum_{v\in V\backslash\mathcal{C}}0\,\,\,\,=\,\,\,\,0.
\]
This contradicts Lemma \ref{lemma_20}.
\hfill{$\square$}
\section{Contribution of $\mathcal{TQ}$-classes of $5$-regular matchstick graphs}
\noindent
In this section we want to study the contribution $c(\mathcal{B})$ of a finite $\mathcal{TQ}$-class $\mathcal{B}$ in a $5$-regular matchstick graph $\mathcal{M}$. Since we show that no finite $5$-regular matchstick graph exists $\mathcal{M}$ has to be infinite in this context. Nevertheless there may be some finite $\mathcal{TQ}$-classes in $\mathcal{M}$.
But since we are only interested in finite matchstick graphs we introduce another concept and consider incomplete finite parts of $5$-regular matchstick graphs.
\begin{definition}
Let $\mathcal{M}$ be a finite matchstick graph with maximum vertex degree at most $5$ and $\mathcal{B}$ a
$\mathcal{TQ}$-class of $\mathcal{M}$ which induces a planar graph $\mathcal{G}$ containing the vertices, edges,
and faces of $\mathcal{B}$. We call a vertex $v$ in $\mathcal{G}$ (or $\mathcal{B}$) an \textbf{inner vertex} if
all faces being adjacent to $v$ in $\mathcal{G}$ are contained in $\mathcal{B}$. All other vertices $v$ in
$\mathcal{G}$ (or $\mathcal{B}$) are called \textbf{outer vertices}. Now we can call the $\mathcal{TQ}$-class
$\mathcal{B}$ \textbf{prospective $\mathbf{5}$-regular} exactly if all inner vertices of $\mathcal{G}$ have
degree $5$ and all outer vertices of $\mathcal{G}$ have degree at most $5$.
\end{definition}
\begin{figure}[htp]
\begin{center}
\includegraphics{regular_matchstick_graphs_3_0.pdf}
\caption{A prospective $5$-regular $\mathcal{TQ}$-class $\mathcal{B}$
of a matchstick graph $\mathcal{M}$
and its induced planar graph $\mathcal{G}$
.}
\label{fig_prospective}
\end{center}
\end{figure}
\noindent
In Figure \ref{fig_prospective} we have depicted a matchstick graph with maximum vertex degree at most $5$ on the left hand side. The faces of the, in this case uniquely, contained $\mathcal{TQ}$-class $\mathcal{B}$ are filled with green color. If we build up a planar graph out of the vertices, edges, and faces of $\mathcal{B}$ we obtain the right hand side of Figure \ref{fig_prospective}. In $\mathcal{G}$ vertex $v$ is the only vertex which is not adjacent to the outer
face. Thus $v$ is an inner vertex in $\mathcal{B}$ and the remaining $7$ vertices of $\mathcal{B}$ are outer vertices.
\medskip
Since every $\mathcal{TQ}$-class $\mathcal{B}$ of a given finite $5$-regular matchstick graph $\mathcal{M}$ is prospective $5$-regular, we now study prospective $5$-regular $\mathcal{TQ}$-classes. Therefore we want to specify them by some parameters $\sigma$, $k$, $\tau$, $b_1$ and $b_2$. (For brevity we forego to use notations as $\sigma(\mathcal{B})$, \dots whenever the corresponding $\mathcal{TQ}$-class is evident from the context.)
\medskip
The parameter $\sigma$ stands for the contribution of the faces of $\mathcal{B}$. To become more precisely we have to introduce a new technical notation. A given vertex $v$ is adjacent to five faces denoted as $(v,1),\dots,(v,5)$, where face $(v,i)$ is an $a_{v,i}$-gon. Since there exist pairs of vertices $u\neq v$ and indices $1\le i,j\le 5$ with $(v,i)=(u,j)$ we introduce the notation $[v,i]$ addressing the arc of face $(v,i)$ at vertex $v$. The contribution of such a face arc $[v,i]$ is defined to be $c\!\left(a_{v,i}\right)$. If $\mathcal{A}$ is the set of all face arcs $[v,i]$ where the face $(v,i)$ is contained in $\mathcal{B}$, then $\sigma$ is given by $\sum\limits_{\alpha\in\mathcal{A}}c(\alpha)$.
\medskip
Let us look at the four $\mathcal{TQ}$-classes of Figure \ref{fig_honeycomb_components} (it is easy to check that they are all prospective $5$-regular). The blue $\mathcal{TQ}$-class consists of a single triangle. So it has three \textit{triangle}-arcs and a contribution of $\sigma_{\text{blue}}=\frac{1}{3}+\frac{1}{3}+\frac{1}{3}=1$. It is easy to figure out that in general a triangle contributes $1$ and a quadrangle contributes $-2$ to $\sigma$ so that we have $\sigma_{\text{red}}=3$, $\sigma_{\text{green}}=0$, and $\sigma_{\text{brown}}=16$.
We have already remarked that every $e\in E$ edge of a planar graph is contained in two faces. Whenever both faces are contained in a given $\mathcal{TQ}$-class $\mathcal{B}$ we say that $e$ is an inner edge. If only one face is contained in $\mathcal{B}$ we call $e$ an outer edge. By $k$ we count the outer edges of a $\mathcal{TQ}$-class $\mathcal{B}$. In our example we have $k_{\text{blue}}=3$, $k_{\text{red}}=5$, $k_{\text{green}}=10$, and $k_{\text{brown}}=20$.
If we consider the vertices and edges of a prospective $5$-regular $\mathcal{TQ}$-class as a subgraph, then every vertex $v$ has a degree $\delta(v)$ at most $5$. By $\tau(v)$ we denote the gap $5-\delta(v)$ (free valencies) and by $\tau(\mathcal{B})$ the sum over all $\tau(v)$ where $v$ is in $\mathcal{B}$. So in our example we have $\tau_{\text{blue}}=9$, $\tau_{\text{red}}=11$ , $\tau_{\text{green}}=20$ , and $\tau_{\text{brown}}=22$.
In a $5$-regular planar graph every vertex $v$ is contained in exactly five different faces $(v,i)$. Whenever all five faces are contained in a given $\mathcal{TQ}$-class $\mathcal{B}$ we say that $v$ in $\mathcal{B}$ is an inner vertex. If the number of faces $(v,i)$ which are contained in $\mathcal{B}$ is between one and four we call $v$ an outer vertex. This coincides with our previous definition of inner and outer vertices of a prospective $5$-regular $\mathcal{TQ}$-class. Clearly we have $\tau(v)=0$ for all inner vertices $v$. If $e=\{v,u\}$ is an outer edge in $\mathcal{B}$ then $v$ and $u$ are outer vertices of $\mathcal{B}$. For the other direction we have that for an outer vertex $v$ in $\mathcal{B}$ there exist at least two outer edges $e_1$, $e_2$ in $\mathcal{B}$ being adjacent to $v$. More precisely the number of outer edges being adjacent to $v$ is either $2$ or $4$.
\begin{figure}[htp]
\begin{center}
\includegraphics{regular_matchstick_graphs_4_0.pdf}
\caption{Vertices of special type.}
\label{fig_special_situation}
\end{center}
\end{figure}
For a given vertex $v$ in $\mathcal{B}$ we consider the faces $(v,i)$ which are contained in a given $\mathcal{TQ}$-class $\mathcal{B}$ as vertices of a graph $H$. Two vertices of $H$ are connected via an edge whenever the corresponding faces have a common edge in $\mathcal{M}$. If the graph $H$ is connected we call $v$ of normal type. Otherwise we say that vertex $v$ is of special type. In Figure \ref{fig_special_situation} we depict all cases of vertices of special type up to symmetry. Here vertex $v$ is marked blue and the faces of $\mathcal{B}$ are marked green. Alternatively we could also define a vertex of special type as an outer vertex of a prospective $5$-regular $\mathcal{TQ}$-class which is adjacent to exactly $4$ outer edges of $\mathcal{B}$. We would like to remark that from a local point of view at a vertex of special type it seems that the faces of the $\mathcal{TQ}$-class are not edge connected.
\medskip
By $b_1$ we count the number of vertices $v$ in $\mathcal{B}$ of special type. In our example of Figure \ref{fig_honeycomb_components} no vertex of special type exists and we have $b_1=0$. In Figure \ref{fig_honeycomb_components_2} we have depicted another example of a $\mathcal{TQ}$-class, where we have $b_1=1$ -- the blue vertex.
\begin{figure}[htp]
\begin{center}
\includegraphics{regular_matchstick_graphs_5_0.pdf}
\caption{Another $\mathcal{TQ}$-class of a matchstick graph.}
\label{fig_honeycomb_components_2}
\end{center}
\end{figure}
If a face $f\in\mathcal{B}$ of a $\mathcal{TQ}$-class is a quadrangle then there are two inner angles of $\frac{2\pi}{3}$ and two inner angles of $\frac{\pi}{3}$. By $b_2$ we count the number of inner angles of $\frac{2\pi}{3}$ where the corresponding vertex $v$ is an outer vertex. It may happen that an outer vertex
$v$ is part of two quadrangles in $\mathcal{B}$ each having an inner angle of $\frac{2}{3}\pi$ at $v$. Thus
for a prospective $5$-regular $\mathcal{TQ}$-class $\mathcal{B}$ we may write $b_2(v)\in\{0,1,2\}$ for the
number of inner angles of $\frac{2\pi}{3}$ at an outer vertex $v$. For inner vertices we set $b_2(v)=0$ so
that we can set $b_2=\sum_{v\in V(\mathcal{B})}b_2(v)$. In the example
of Figure \ref{fig_honeycomb_components} we have $b_2=4$ in the green class, $b_2=2$ in the brown class,
and $b_2=0$ in the red and blue class. We remark that due to an angle sum of $2\pi$ there is exactly one
face with an inner angle of $\frac{2\pi}{3}$ at every inner vertex of $\mathcal{B}$.
\medskip
Now we want to prove the equation
\begin{equation}
\label{eqn_paramater}
\sigma-k+\frac{\tau-k}{3}+\frac{5}{3}b_1+\frac{5}{3}b_2=0.
\end{equation}
relating the parameters $\sigma$, $k$, $\tau$, $b_1$ and $b_2$. This equation even holds for more general objects
than prospective $5$-regular $\mathcal{TQ}$-classes. A $\mathcal{TR}$-class $\mathcal{B}$ is a union of
triangles and quadrangles with inner angles in $\left\{\frac{\pi}{3},\frac{2\pi}{3}\right\}$ on a regular triangular grid
such that each inner vertex has degree $5$ and each outer vertex has degree at most $5$. Clearly we can transfer the
definitions of the parameters $\sigma$, $k$, $\tau$, $b_1$ and $b_2$ from (prospective $5$-regular) $\mathcal{TQ}$-classes to $\mathcal{TR}$-classes. The crucial difference between $\mathcal{TQ}$-classes and $\mathcal{TR}$-classes is that the dual graph of a $\mathcal{TR}$-class (considered as a planar graph) is not connected in general.
\begin{figure}[htp]
\begin{center}
\includegraphics{regular_matchstick_graphs_6_0.pdf}
\caption{Examples of $\mathcal{TR}$-classes.}
\label{fig_tr_classes}
\end{center}
\end{figure}
In Figure \ref{fig_tr_classes} we depict some extraordinary examples of $\mathcal{TR}$-classes to demonstrate the whole variety being covered by the definition of a $\mathcal{TR}$-class. Here the faces of a $\mathcal{TR}$-class $\mathcal{B}$ are filled in green. Those faces which are not contained in $\mathcal{B}$ are left in white. In the left example of Figure \ref{fig_tr_classes} we have $\sigma=3$, $k=9$, $\tau=12$, $b_1=3$, and $b_2=0$. In the second example we have $\sigma=4$, $k=12$, $\tau=16$, $b_1=4$, and $b_2=0$. In the third example of Figure \ref{fig_tr_classes}
we have $\sigma=5$, $k=11$, $\tau=14$, $b_1=3$, and $b_2=0$. For the central vertex $v$ of the right example of Figure \ref{fig_tr_classes} we have $b_2(v)=2$. Additionally $v$ is of special type. The parameters are given by $\sigma=-4$, $k=8$, $\tau=19$, $b_1=1$, and $b_2=4$. Thus in all four cases Equation (\ref{eqn_paramater}) is valid. If we consider those four examples as a single example on the same triangular grid again Equation (\ref{eqn_paramater}) is valid.
\begin{lemma}
\label{lemma_parameter}
For a $\mathcal{TR}$-class $\mathcal{B}$ Equation~(\ref{eqn_paramater}) is valid.
\end{lemma}
\begin{proof}
We prove by induction on the number of faces of $\mathcal{B}$. For a single triangle we have $\sigma=1$, $k=3$,
$\tau=9$, and $b_1=b_2=0$. Thus the left hand side of (\ref{eqn_paramater}) sums to zero. For a quadrangle we
have $\sigma=-2$, $k=4$, $\tau=12$, $b_1=0$, and $b_2=2$, again summing up to zero in (\ref{eqn_paramater}). We
remark that Equation~(\ref{eqn_paramater}) is also valid for an empty $\mathcal{TR}$-class.
For the induction step we now
assume that Equality (\ref{eqn_paramater}) is valid and we show that it remains valid after adding another
single triangle or quadrangle to $\mathcal{B}$. Here we only consider those additions which do not alter
faces of $\mathcal{B}$ (one might think of inserting an edge into a quadrangle of $\mathcal{B}$ resulting in
two triangles and destroying the initial quadrangle). In general it might happen that adding the edges of one
new face produces a second new face in one step. An example arises by adding an edge into a quadrangle which
is not contained in $\mathcal{B}$, see the second and the third example of Figure \ref{fig_tr_classes}.
Here, in one step we only add one of the resulting triangles to $\mathcal{B}$. If the other triangle should also
be added to $\mathcal{B}$ we do this in a second step where we only have a new face and no new vertices or edges.
\begin{figure}[htp]
\begin{center}
\includegraphics{regular_matchstick_graphs_7_0.pdf}
\caption{Adding a triangle.}
\label{fig_adding_a_triangle}
\end{center}
\end{figure}
Adding a triangle results in eight (main) cases, see Figure \ref{fig_adding_a_triangle}, and adding a quadrangle
results in nineteen (main) cases, see Figure \ref{fig_adding_a_quadrangle}. In the different (main) cases we
depict the new edges by dotted lines, the new vertices by empty circles, and the old edges and vertices in black.
We have a closer look on the change of the
parameters. If $p$ is the parameter of $\mathcal{B}$ and $p'$ the corresponding parameter after adding a triangle
(or a quadrangle), then we denote the change by $\Delta(p):=p'-p$. In order to prove the induction step it
suffices to verify
\begin{equation}
\label{eqn_paramater_delta}
\Delta(\sigma)-\Delta(k)+\frac{\Delta(\tau)-\Delta(k)}{3}+\frac{5}{3}\Delta\!\left(b_1\right)+
\frac{5}{3}\Delta\!\left(b_2\right)= 0.
\end{equation}
We remark that all black edges in the (main) cases of Figure \ref{fig_adding_a_triangle} and Figure
\ref{fig_adding_a_quadrangle} are outer edges before adding the new triangle or quadrangle. After adding the
new face the black edges become inner edges and the dotted edges become outer edges.
\begin{table}[!ht]
\begin{center}
\begin{tabular}{r|r|r|r|r}
(main) case & $\Delta(\sigma)$ & $\Delta(k)$ & $\Delta(\tau)$ & $\Delta\!\left(b_1\right)+\Delta\!\left(b_2\right)$\\
\hline
(1) & 1 & 3 & 9 & 0\\
(2) & 1 & 3 & 4 & 1\\
(3) & 1 & 3 & -1 & 2\\
(4) & 1 & 3 & -6 & 3\\
(5) & 1 & 1 & 1 & 0\\
(6) & 1 & 1 & -4 & 1\\
(7) & 1 & -1 & -2 & -1\\
(8) & 1 & -3 & 0 & -3\\
\end{tabular}
\caption{$\Delta(\cdot)$-values for the eight (main) cases of Figure \ref{fig_adding_a_triangle}.}
\label{table_delta_triangles}
\end{center}
\end{table}
In order to shorten our calculation and case distinction we have a look at the situation from a vertex point
of view. For a fixed vertex $a$ (in Figure \ref{fig_adding_a_triangle}) up to symmetry there are four different
(vertex) cases: If $a$ is a new (empty) vertex then also the two adjacent edges of the new triangle have to be
dotted. Otherwise if $a$ is an old (black) vertex the number of dotted adjacent edges of the new triangle can be two,
one, or zero. These four possibilities show up as (main) cases (1), (2), (5), and (7) in Figure
\ref{fig_adding_a_triangle} and we denote them as (vertex) cases (i), (ii), (iii), and (iv). Clearly, also for
vertex $b$ and vertex $v$ we have the same four possibilities. So we restrict our considerations on vertex $a$.
\begin{figure}[htp]
\begin{center}
\includegraphics{regular_matchstick_graphs_8_0.pdf}
\caption{Adding a quadrangle.}
\label{fig_adding_a_quadrangle}
\end{center}
\end{figure}
By $a_1$ we denote the vertex $a$ before the addition of the new face and by $a_2$ we denote the vertex $a$ after
the addition of the new face.
In (vertex) case (i), occurring in (main) case (1), vertex $a_1$ does not exist and $a_2$ is an outer vertex.
Obviously $a_2$ is of normal type and $b_2\!\left(a_2\right)=0$. In (vertex) case (ii), occurring in (main) case
(2) both $a_1$ and $a_2$ are outer vertices. We have $b_2\!\left(a_1\right)
=b_2\!\left(a_2\right)$. Since $a$ is adjacent to two red edges of the new face vertex $a_2$ is of special type
and $a_1$ is of normal type, so that the $b_1$-count increases by one. In (vertex) case (iii)
both $a_1$ and $a_2$ are outer vertices. So $b_2\!\left(a_1\right)=b_2\!\left(a_2\right)$ and
$a_1$ is of special type if and only if $a_2$ is of special type. Thus the $b_1$- and the $b_2$-counts
do not change. In (vertex) case (iv)
two different things could happen at vertex $a$. At first we mention that $a_1$ is an outer
vertex. If also $a_2$ is an outer vertex then $a_1$ is of special type and $a_2$ is of normal type. In this
situation we have $b_2\!\left(a_1\right)=b_2\!\left(a_2\right)$. The other possibility is that $a_2$ is an inner
vertex. Since the degree of $a_2$ is $5$ this is only possible if $b_2\!\left(a_1\right)=1$ and if $a_1$ is of
normal type. And since $a_2$ is an inner vertex it is of normal type and we have $b_2\!\left(a_2\right)=0$. Thus
the sum of the $b_1$-count and the $b_2$-count decreases by one.
With the above we are able to give the $\Delta(\cdot)$-values for the (main) cases of Figure
\ref{fig_adding_a_triangle} in Table \ref{table_delta_triangles}. It is easy to check that in all eight (main) cases
Equation~(\ref{eqn_paramater_delta}) is valid.
\medskip
\begin{table}[ht]
\begin{center}
\begin{tabular}{r|r|r|r|r}
(main) case & $\Delta(\sigma)$ & $\Delta(k)$ & $\Delta(\tau)$ & $\Delta\!\left(b_1\right)+\Delta\!\left(b_2\right)$\\
\hline
(9) & -2 & 4 & 12 & 2 \\
(10) & -2 & 4 & 7 & 3 \\
(11) & -2 & 4 & 7 & 3 \\
(12) & -2 & 4 & 2 & 4 \\
(13) & -2 & 4 & 2 & 4 \\
(14) & -2 & 4 & 2 & 4 \\
(15) & -2 & 4 & -3 & 5 \\
(16) & -2 & 4 & -8 & 6 \\
(17) & -2 & 2 & 4 & 2 \\
(18) & -2 & 2 & -1 & 3 \\
(19) & -2 & 2 & -1 & 3 \\
(20) & -2 & 2 & -6 & 4 \\
(21) & -2 & 0 & -4 & 2 \\
(22) & -2 & 0 & 1 & 1 \\
(23) & -2 & 0 & 1 & 1 \\
(24) & -2 & 0 & -4 & 2 \\
(25) & -2 & 0 & -4 & 2 \\
(26) & -2 & -2 & -2 & 0 \\
(27) & -2 & -4 & 0 & -2 \\
\end{tabular}
\caption{$\Delta(\cdot)$-values for the nineteen (main) cases of Figure \ref{fig_adding_a_quadrangle}.}
\label{table_delta_quadrangles}
\end{center}
\end{table}
Next we can deal with the nineteen (main) cases of Figure \ref{fig_adding_a_quadrangle}. In every (main) case
the situation of vertex $a$ and vertex $w$ has an equivalent in Figure \ref{fig_adding_a_triangle}. So up to
symmetry we have to consider the situation at vertex $v$. Again there are four possibilities to consider showing
up in (main) cases (9), (14), (21), and (27) of Figure \ref{fig_adding_a_quadrangle}. We denote the corresponding
(vertex) cases by (v), (vi), (vii), and (viii), respectively. Similarly as before we denote the vertex $v$ before
the addition of the quadrangle by $v_1$ and afterwards by $v_2$.
\medskip
In (vertex) case (v), occurring in (main) case (9), $v_1$ does not exist and $v_2$ is of normal type and we
have $b_2\!\left(v_2\right)=1$ so that the sum of the $b_1$- and the $b_2$-count increases by one. In (vertex)
case (vi)
both $v_1$ and $v_2$ are outer vertices.
Vertex $v_2$ is of special type and $v_1$ has to be of normal type. For the $b_2$-value we have
$b_2\!\left(v_2\right)=b_2\!\left(v_1\right)+1$ so that the sum of the $b_1$- and the $b_2$-count increases by
two. In (vertex) case (vii)
both $v_1$ and $v_2$ are outer vertices. Vertex $v_1$ is of special type if and only if $v_2$
is of special type. Since $b_2\!\left(v_2\right)=b_2\!\left(v_1\right)+1$ the sum
of the $b_1$- and the $b_2$-count increases by one. In (vertex) case (viii)
If $v_2$ is an inner vertex than $v_1$, $v_2$ are of normal type, $b_2\!\left(v_1\right)=0$, and
$b_2\!\left(v_2\right)=0$. If $v_2$
is an outer vertex then $v_1$ is of special type, $v_2$ is of normal type, and
$b_2\!\left(v_2\right)=b_2\!\left(v_1\right)+1$. Thus in both cases the sum of the $b_1$- and
the $b_2$-count does not change.
\medskip
With the above we are able to give the $\Delta(\cdot)$-values for the (main) cases of Figure
\ref{fig_adding_a_quadrangle} in Table \ref{table_delta_quadrangles}. It is easy to check that in all
nineteen (main) cases Equation~(\ref{eqn_paramater_delta}) is valid.
\end{proof}
\begin{corollary}
\label{cor_parameter}
For a (finite) prospective $5$-regular $\mathcal{TQ}$-class of a matchstick graph we have
\[
\sigma-k+\frac{\tau-k}{3}+\frac{5}{3}b_1+\frac{5}{3}b_2=0.
\]
\end{corollary}
\noindent
Now we are ready to prove $c(\mathcal{B})\le 0$ for every (finite) prospective $5$-regular $\mathcal{TQ}$-class of a matchstick graph with maximum vertex degree at most $5$. To be more precisely we need to consider finite matchstick graphs $\mathcal{M}$ with maximum vertex degree at most $5$ and a prospective $5$-regular $\mathcal{TQ}$-class $\mathcal{B}$ of $\mathcal{M}$. In $\mathcal{M}$ all vertices of $\mathcal{B}$ must have vertex degree exactly $5$.
In order to be able to speak of a contribution $c(\mathcal{B})$ for every vertex $v$ in $\mathcal{B}$ there must be five faces $(v,i)$, unequal to the outer face, completely being contained in $\mathcal{M}$. If a finite $5$-regular matchstick graph would exist, it would certainly be such a graph. Otherwise we can only look at local parts of such a (possible infinite) graph.
As mentioned in the previous section we have to perform some bookkeeping in order to prove $c\!\left(\cup_i\mathcal{B}_i\right)\le 0$ for a set of prospective $5$-regular $\mathcal{TQ}$-classes. To every face arc $[v,i]$ where $v$ is contained in a given prospective $5$-regular $\mathcal{TQ}$-class $\mathcal{B}$ we will assign a real weight $\omega([v,i])\ge c([v,i])$ which fulfills
\[
\sum_{v\in\mathcal{B}}\sum_{i=1}^5 \omega([v,i])\le 0.
\]
We start with the face arcs $[v,i]$ where the face $(v,i)$ is contained in $\mathcal{B}$. We call those arcs inner arcs and set $\omega([v,i])=c([v,i])$. With this the sum $\sum\limits_{[v,i]\text{ inner arc of }\mathcal{B}}\omega([v,i])$ over the inner arcs equals the parameter $\sigma$ from Corollary \ref{cor_parameter}.
\begin{figure}[htp]
\begin{center}
\includegraphics{regular_matchstick_graphs_9_0.pdf}
\caption{A part of a $5$-regular matchstick graph.}
\label{fig_honeycomb_components_3}
\end{center}
\end{figure}
Now we consider the remaining face arcs $[v,i]$ which are given by a vertex $v$ and two edges $e$, $e'$ being adjacent to $v$. We say that the edges $e$ and $e'$ are associated to $[v,i]$.
Let $v$ be an outer vertex of $\mathcal{B}$ and $e$ be an edge (in $\mathcal{M}$) being adjacent to $v$.
If $e$ is not an outer edge of $\mathcal{B}$ we call $e$ a \textit{leaving edge}. This is only possible if $e$ is contained in $\mathcal{M}$ but not in $\mathcal{B}$.
To emphasize that we consider edge $e$ being rooted at vertex $v$ we also speak of \textit{leaving half edges}.
It may happen that a leaving edge $e=\{u,v\}$ corresponds to two leaving half edges being rooted at vertex $u$ and $v$, respectively.
We remark that the number of outer edges equals $k$ and that the number of leaving half edges equals $\tau$ in Corollary \ref{cor_parameter}.
In Figure \ref{fig_honeycomb_components_3} we have depicted the outer edges in blue and red. The leaving half edges are depicted by dashed lines.
If $[v,i]$ is a face arc where both associated edges $e$ and $e'$ correspond to leaving half edges then we set $\omega([v,i])=\frac{1}{3}$ being greater or equal to $c([v,i])$.
For the remaining face arcs we consider the outer edges of $\mathcal{B}$. They form a union of simple cycles, i.~e.{} cycles without repeated vertices. In the example of Figure \ref{fig_honeycomb_components_2} we have a cycle of length $6$ and a cycle of length $15$. We will treat each cycle $C$ separately. Such a cycle $C$ divides the plane into two parts. We call the part containing the faces of $\mathcal{B}$ the interior of $C$ and the other part the exterior of $C$. When we speak of leaving half edges then we only want to address those which go into the exterior of $C$. So the blue cycle of Figure \ref{fig_honeycomb_components_3} does not contain a leaving half edge whereas the red cycle contains $19$ leaving half edges.
It may happen that such a cycle $C$ does not contain any leaving half edges at all, like the blue cycle of Figure \ref{fig_honeycomb_components_3}. In this case $C$ consists of the single face $(v,i)$, where $[v,i]$ is an arbitrary face arc being associated to an edge of $C$. Since $(v,i)$ is not contained in $\mathcal{B}$ it is neither a triangle nor a quadrangle. If $(v,i)$ is a pentagon we set $\omega([v,i])=c([v,i])=-1$ for all associated face arcs $[v,i]$ otherwise we set $\omega([v,i])=-\frac{4}{3}\ge c([v,i])$.
If $C$ does contain leaving half edges the face which is adjacent to a given outer edge $e$ and not contained in $\mathcal{B}$ is bordered by two leaving half edges. In Figure \ref{fig_weights} we have depicted the possible cases.
In principle it would be possible that $C$ contains only one leaving half edge $e$. In such a situation we consider the two half edges the cases of Figure \ref{fig_weights} as being identified to $e$.
\begin{figure}[htp]
\begin{center}
\includegraphics{regular_matchstick_graphs_10_0.pdf}
\caption{Assigning weights.}
\label{fig_weights}
\end{center}
\end{figure}
In case (1) a single outer edge is bordered by two leaving half edges. Since the outer face is not part of $\mathcal{B}$ it cannot be a triangle. So it is at least a quadrangle and we can set $\omega([v,i])=-\frac{1}{2}\ge c([v,i])$. In cases (2a) and (2b) we consider a path $P$ of length two which is bordered by two leaving half edges. Since the outer face is not part of $\mathcal{B}$ it can not be a triangle or a quadrangle. If it is a pentagon we set $\omega([v,i])=-\frac{2}{3}\ge c([v,i])$ for the ends of $P$ and $\omega([v,i])=-1\ge c([v,i])$ for the central vertex. If the outer face has at least $6$ edges we set $\omega([v,i])=-\frac{1}{2}\ge c([v,i])$ for the ends of $P$ and $\omega([v,i])=-\frac{4}{3}\ge c([v,i])$ for the central vertex. In the remaining cases we consider paths of length at least $3$ in case (3a), (3b), paths of length $4$ in case (4a), (4b), and paths of length at least $5$ in case (5). Again the outer face cannot be a triangle or a quadrangle. In case (5) it even cannot be a pentagon. Here we do the assignments of the weights as depicted in Figure \ref{fig_weights}. We can easily check that we have $\omega([v,i])\ge c([v,i])$.
\begin{lemma}
Let $\mathcal{B}$ be a given prospective $5$-regular $\mathcal{TQ}$-class of matchstick graph $\mathcal{M}$ with
vertex degree at most $5$. If $c(\mathcal{B})$ can be evaluated within $\mathcal{M}$ then the weight function
$\omega$ constructed above fulfills
\begin{equation}\label{eqn_non_negativ}\sum_{v\in\mathcal{B}}\sum_{i=1}^5 \omega([v,i])\le 0.\end{equation}
and $\omega([v,i])\ge c([v,i])$ for all vertices $v\in\mathcal{B}$, $1\le i\le 5$.
\end{lemma}
\begin{proof}
By construction we have $\omega([v,i])\ge c([v,i])$ for all $v\in\mathcal{B}$, $1\le i\le 5$. To prove Equation
(\ref{eqn_non_negativ}) we utilize a booking technique. We want to book face arcs $[v,i]$, outer edges, leaving
half edges and vertices of special type.
By $\Omega$ we denote the sum of weights $\omega([v,i])$ over the arcs $[v,i]$
which are booked. If we book an arc $[v,i]$ we also want to book half of its both associated edges and half edges
each. Therefore we define $\eta(e)=1$ if edge or half edge $e$ was never booked, $\eta(e)=\frac{1}{2}$ if $e$ was
booked only one time, and $\eta(e)=0$ if $e$ was booked two times. By $\mathcal{K}$ we denote the sum over $\eta(e)$,
where the edges $e$ are outer edges of $\mathcal{B}$ and by $T$ we denote the sum over all leaving half edges. By
$B_1$ we count the number of vertices of special type which are not booked so far.
As done in the construction of $\omega$ we start with those face arcs $[v,i]$, where the corresponding face $(v,i)$
is contained in $\mathcal{B}$. Here we set $\omega([v,i])=c([v,i])$. After this initialization we have $\Omega=\sigma$,
$\mathcal{K}=k$, $T=\tau$, and $B_1=b_1$ using the notation of Lemma \ref{lemma_parameter}. With
this we have
\begin{equation}
\label{eqn_bookkeeping}
\Omega-\mathcal{K}+\frac{T-\mathcal{K}}{3}+\frac{5}{3}B_1\le 0
\end{equation}
since $b_2\ge 0$. Now we book the remaining face arcs and keep track that Inequality (\ref{eqn_bookkeeping}) endures.
If $[v,i]$ is a face arc between two leaving half edges $e$ and $e'$, then we book these edges and assign
$\omega([v,i])=\frac{1}{3}$. By this booking step $\Omega$ increases by $\frac{1}{3}$ and $T$ decreases by $1$.
Thus Inequality (\ref{eqn_bookkeeping}) remains valid.
Next we consider the simple cycles $C$ of the outer edges. We start with the cases where $C$ does not contain any
leaving half edges. In this case $C$ consists of the single face $(v,i)$, where $[v,i]$ is an arbitrary face arc
being associated to an edge of $C$. If $(v,i)$ is a pentagon both $\Omega$ and $\mathcal{K}$ decrease by $5$.
We can easily check that on a regular triangular grid there does not exist an equilateral pentagon where all
inner angles are at most $\frac{2\pi}{3}$. Thus face $(v,i)$ does contain an inner angle of at least $\pi$ at a vertex
$u$. This vertex $u$ must be of special type due to its angle sum of $2\pi$. Thus we can book the special type
and decrease $B_1$ by one so that Inequality (\ref{eqn_bookkeeping}) remains valid. If $(v,i)$ contains $l\ge 6$
edges we set $\omega([v,i])=-\frac{4}{3}$ at the corresponding face arcs. Thus $\mathcal{K}$ decreases by $l$ and
$\Omega$ decreases by $\frac{4}{3}l$ so that Inequality (\ref{eqn_bookkeeping}) remains valid.
Next we consider the cases of Figure \ref{fig_weights}. In all cases $T$ decreases by one. The decreases of
$\mathcal{K}$ and $\Omega$ vary in the different cases, but we can easily check that Inequality
(\ref{eqn_bookkeeping}) remains valid.
A the end of this procedure we have $\mathcal{K}=T=0$ since all edges are booked properly. Since all vertices
of special type are booked at most once we have $B_1\ge 0$. Thus we can conclude
\[
\sum_{v\in\mathcal{B}}\sum_{i=1}^5 \omega([v,i])=\Omega\le 0
\]
from Inequality (\ref{eqn_bookkeeping}) in the end.
\end{proof}
\begin{corollary}
For a prospective $5$-regular $\mathcal{TQ}$-class $\mathcal{B}$ of a matchstick graph with maximum
vertex degree $5$ we have
\[
c(\mathcal{B})\le 0.
\]
\end{corollary}
As depicted in Figure \ref{fig_honeycomb_components} it may happen that a vertex $v$ is contained in more than one prospective $5$-regular $\mathcal{TQ}$-class $\mathcal{B}$. In this cases it belongs to exactly two prospective $5$-regular $\mathcal{TQ}$-classes due to an vertex degree of five. In Figure \ref{fig_two_tq_classes} we have depicted the possible cases. Here one prospective $5$-regular $\mathcal{TQ}$-class is depicted in green and the other one is depicted in blue. In case (1) the angles of face arcs $[v,3]$ and $[v,5]$ need not to be multiples of $\frac{\pi}{3}$. Similarly also in case (2) the angles of the face arcs $[v,2]$, $[v,3]$, and $[v,5]$ need not to be multiples of $\frac{\pi}{3}$. In the next lemma we show that the assignment of our weight functions do not underestimate the contribution of vertex $v$.
\begin{figure}[htp]
\begin{center}
\includegraphics{regular_matchstick_graphs_11_0.pdf}
\caption{Two prospective $5$-regular $\mathcal{TQ}$-classes with a common vertex.}
\label{fig_two_tq_classes}
\end{center}
\end{figure}
\begin{lemma}
Let $\mathcal{B}_1$ and $\mathcal{B}_2$ be two different prospective $5$-regular $\mathcal{TQ}$-classes of a
matchstick graph with maximum vertex degree at most $5$ and weight functions $\omega_1$ and $\omega_2$,
respectively. If a vertex $v$ is contained in both $\mathcal{B}_1$ and $\mathcal{B}_2$ then we have
$$
\sum_{i=1}^5 \omega_1([v,i])+\omega_2([v,i])\ge \sum_{i=1}^5 c([v,i]).
$$
\end{lemma}
\begin{proof}
W.l.o.g.\ the prospective $5$-regular $\mathcal{TQ}$-class $\mathcal{B}_1$ is depicted green in Figure
\ref{fig_two_tq_classes} and $\mathcal{B}_2$ is depicted blue.
At first we consider case (i). Here we have $\omega_1([v,1])\ge c[(v,1])$, $\omega_1([v,2])\ge c[(v,2])$, and
$\omega_2([v,4])\ge c([v,4])$. Next we look at the face arcs which are bordered by two leaving half edges. Here we
have $\omega_2([v,1])=\frac{1}{3}$, $\omega_2([v,2])=\frac{1}{3}$, and $\omega_1([v,4])=\frac{1}{3}$. Due to symmetry
it suffices to show
\begin{equation}
\label{eq_overcount}
\omega_1([v,3])+\omega_2([v,3])\ge c([v,3])-\frac{1}{2}.
\end{equation}
Since at vertex $v$ there are leaving half edges both in $\mathcal{B}_1$ and $\mathcal{B}_2$ we have to be in one
of the cases of Figure \ref{fig_weights} for the determination of the weight of face arc $[v,3]$. If for
$\mathcal{B}_1$ face arc $[v,3]$ is in one of the cases (1), (2b), (3b), (4b), or (5) then we have
$\omega_1([v,3])=-\frac{1}{2}$ and Inequality~(\ref{eq_overcount}) is valid. Due to symmetry the same holds if
for $\mathcal{B}_2$ face arc $[v,3]$ is in one of the cases (1), (2b), (3b), (4b), or (5). In all other cases
$(v,i)$ is a pentagon so that $c([v,i])= -1$.
We remark that $[v,3]$ can not be in case (3a) both in $\mathcal{B}_1$ and $\mathcal{B}_2$. Thus we have
$\omega_1([v,3])+\omega_2([v,3])\ge -\frac{5}{6}-\frac{2}{3}=-1-\frac{1}{2}\ge c([v,3])-\frac{1}{2}$.
To finish the proof we consider case (ii). Here we have $\omega_1([v,1])\ge c[(v,1])$, $\omega_1([v,2])\ge c[(v,2])$,
$\omega_2([v,4])\ge c([v,4])$, $\omega_2([v,3])\ge c([v,3])$, $\omega_1([v,3])=\frac{1}{3}$,
$\omega_1([v,4])=\frac{1}{3}$,
$\omega_2([v,1])=\frac{1}{3}$, and $\omega_2([v,2])=\frac{1}{3}$. Thus it suffices to show
$$
\omega_1([v,5])+\omega_2([v,5])\ge c([v,5])-\frac{4}{3}.
$$
Since $\omega_1([v,5]),\omega_2([v,5])\ge-\frac{5}{6}$ and $c([v,5])\le -\frac{1}{2}$ this inequality is true.
\end{proof}
\begin{corollary}
\label{cor_non_negativ_all}
If $\mathcal{B}_1$, $\mathcal{B}_2$, $\dots$ are all prospective $5$-regular$\mathcal{TQ}$-classes of a
(finite) matchstick graph with maximum vertex degree at most $5$ we have
$$
c\!\left(\underset{i}{\cup}\mathcal{B}_i\right)\le 0.
$$
\end{corollary}
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\end{picture}
\end{center}
\caption{Infinite $5$-regular match stick graph.}
\label{fig_infinite}
\end{figure}
\section{Conclusion}
\noindent
In this paper we have proven that no finite $5$-regular matchstick graph exists. In Figure \ref{fig_infinite} we have depicted a fraction of an infinite $5$-regular matchstick graph. Since we can shift single rows of such a construction there exists an uncountable number of these graphs (even in the combinatorial sense). As mentioned by Bojan Mohar there are several further examples.
Starting with the infinite $6$-regular triangulation we can delete several vertices at different rows and obtain an example by suitably removing horizontal edges to enforce degree $5$ for all vertices. Taking an arbitrary $\mathcal{TQ}$-class and continueing the boundary vertices with trees gives another set of examples.
In the context of regular matchstick graphs the only remaining question is the determination of $m(4)$, i~e. the smallest $4$-regular matchstick graph.
We strongly believe that there exists a more general topological interpretation of Equation~(\ref{eqn_paramater}). We have simply discovered it going along the concept of a potential function in order to prove Theorem~\ref{thm_main}.
\providecommand{\href}[2]{#2}
| {
"timestamp": "2014-01-09T02:11:39",
"yymm": "1401",
"arxiv_id": "1401.1793",
"language": "en",
"url": "https://arxiv.org/abs/1401.1793",
"abstract": "A graph $G=(V,E)$ is called a unit-distance graph in the plane if there is an injective embedding of $V$ in the plane such that every pair of adjacent vertices are at unit distance apart. If additionally the corresponding edges are non-crossing and all vertices have the same degree $r$ we talk of a regular matchstick graph. Due to Euler's polyhedron formula we have $r\\le 5$. The smallest known $4$-regular matchstick graph is the so called Harborth graph consisting of $52$ vertices. In this article we prove that no finite $5$-regular matchstick graph exists.",
"subjects": "Combinatorics (math.CO)",
"title": "No finite $5$-regular matchstick graph exists",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9770226347732858,
"lm_q2_score": 0.7248702702332475,
"lm_q1q2_score": 0.7082146612921111
} |
https://arxiv.org/abs/2209.03543 | Resolutions of local face modules, functoriality, and vanishing of local $h$-vectors | We study the local face modules of triangulations of simplices, i.e., the modules over face rings whose Hilbert functions are local $h$-vectors. In particular, we give resolutions of these modules by subcomplexes of Koszul complexes as well as functorial maps between modules induced by inclusions of faces. As applications, we prove a new monotonicity result for local $h$-vectors and new results on the structure of faces in triangulations with vanishing local $h$-vectors. | \section{Introduction}
In this paper, we study the modules over face rings, introduced by Athanasiadis and Stanley, whose Hilbert functions are the relative local $h$-vectors of quasi-geometric homology triangulations of simplices, a broad class of formal subdivisions that includes all geometric triangulations and is natural from the point of view of combinatorial commutative algebra. See Section 2.1 for the precise definition and further references.
Fix an infinite field $k$. Let $\sigma \colon \Gamma \to 2^V$ be a quasi-geometric homology triangulation of a simplex, and let $E$ be a face of $\Gamma$. Say that a face $G \in \Gamma$ is \emph{interior} if $\sigma(G) = V$, and let $I$ be the ideal in the face ring $k[\lk_\Gamma(E)]$ generated by the faces that are interior relative to $E$, i.e.,
\[
I = (x^F : F \sqcup E \mbox{ is interior} ).
\]
Let $d = |V|-|E|$, which is the Krull dimension of $k[\lk_\Gamma(E)]$, and let $\theta_1, \ldots, \theta_{d}$ be a special l.s.o.p., as in \cite{Stanley92, Athanasiadis12b}. See also \mathcal{S}\ref{ss:sr}, where we recall the definition and construction of special l.s.o.p.s.
\begin{definition}
The \emph{local face module} $L(\Gamma,E)$ is the image of $I$ in $k[\lk_\Gamma(E)]/(\theta_1, \ldots, \theta_d)$.
\end{definition}
\noindent Note that $L(\Gamma,E)$ is a finite dimensional graded $k$-vector space. The \emph{local $h$-vector} is its Hilbert function:
\[
\ell(\Gamma, E) := (\ell_0, \ldots, \ell_{d}), \quad \quad \mbox{ where } \ \ell_i := \dim L(\Gamma,E)_i.
\]
The local face module $L(\Gamma,E)$ depends on the choice of a special l.s.o.p., but $\ell(\Gamma,E)$ is an invariant of the triangulation with the symmetry $\ell_i = \ell_{d- i}$. See \mathcal{S}\ref{sec:triangulations} for details and references. In the past few years, there has been significant research activity on the combinatorics of local $h$-vectors and relations to intersection homology \cite{Athanasiadis16, KatzStapledon16, Stapledon17, deCataldoMiglioriniMustata18}. Recent advances include a proof that every non-negative integer vector satisfying $\ell_0 = 0$ and $\ell_i = \ell_{d-i}$ is the local $h$-vector of a quasi-geometric triangulation for $E = \emptyset$ \cite{JKMS}, and a relative hard Lefschetz theorem that yields unimodality of local $h$-vectors for regular subdivisions in a more general setting (for regular nonsimplicial polyhedral subdivisions that are not necessarily rational) \cite{Karu19}.
\medskip
Here, we investigate the local face modules $L(\Gamma, E)$ using methods of combinatorial commutative algebra. In particular, we describe natural combinatorial resolutions of these modules as well as natural maps of $k[\lk_\Gamma(E)]$-modules, $L(\Gamma,E) \to L(\Gamma, E')$, for $E \subset E'$. Our first theorem gives explicit generators for the kernel of the natural map $I \to k[\lk_\Gamma(E)]/(\theta_1, \ldots, \theta_d)$. Moreover, we extend this to an exact sequence of graded $k[\lk_\Gamma(E)]$-modules in which each term is a direct sum of degree-shifted monomial ideals.
Label the vertices of the simplex $V = \{ v_1, \ldots, v_n \}$. For a subset $U \subset V$, let $U^c := V \smallsetminus U$. After relabeling, we may assume that $\sigma(E)^c = \{v_1, \ldots, v_b\}$. Given $S \subset \{v_1, \ldots, v_d\}$,
we define the ideal $I_S \subset k[\lk_\Gamma(E)]$ by
\begin{equation*}
I_S := ( x^F : \, \sigma(F \sqcup E)^c \subset S).
\end{equation*}
Note that $I_{S'} \subset I_{S}$ for $S' \subset S$, and $I_S$ depends only on $S \cap \{v_1, \ldots, v_b\}$. For instance, $I_{\emptyset} = I$ and $I_{S} = k[\lk_{\Gamma}(E)]$ if $\{v_1, \ldots, v_b \} \subset S$. By the definition of a special l.s.o.p. (Definition~\ref{def:special}), after reordering, we may assume
\[
\supp(\theta_i) \subset \{ w \in \lk_\Gamma(E) : v_i \in \sigma(w) \},
\]
for $1 \leq i \leq b$. As a consequence, for any $v_i \in S$, multiplication by $\theta_i$ induces a degree 1 map $\lambda_i \colon I_S \to I_{S \smallsetminus \{v_i\}}$.
\begin{theorem}\label{thm:resolution}
There is an exact sequence of graded $k[\lk_{\Gamma}(E)]$-modules
$$0 \to k[\lk_{\Gamma}(E)][-d] \to \bigoplus_{ |S| = d - 1 } I_{S}[-(d-1)] \to \dotsb \to \bigoplus_{|S| = 1} I_{S}[-1] \to I \to L(\Gamma,E) \to 0,$$
where,
for $S= \{v_{i_0}, \ldots, v_{i_k}\}$, with $i_0 < \cdots < i_k$, the differential restricted to $I_S$ is $\oplus_{j = 0}^k (-1)^j \lambda_{i_j}$.
\end{theorem}
\begin{corollary}\label{cor:presentation}
The kernel of the surjection $I \to L(\Gamma, E)$ is the ideal $J$ generated by
$$ \left \{ \theta_i \cdot x^{F} : F \sqcup E \mbox{ is interior } \right \} \cup \left \{ \theta_{j} \cdot x^G : \sigma(G \sqcup E) = \{v_j\}^c, \mbox{ for } 1 \leq j \leq b \right \}.$$
\end{corollary}
We also construct maps between local face modules, as follows. Given an inclusion of simplicial complexes $\Delta' \subset \Delta$, there is a natural inclusion of face rings $k[\Delta'] \to k[\Delta]$, given by $x^F \mapsto x^F$.
In particular, for faces $E \subset E'$ in $\Gamma$, let $\Star(E' \smallsetminus E)$ denote the closed star of $E' \smallsetminus E$ in $\lk_\Gamma(E)$. Then we identify $k[\lk_\Gamma(E')]$ with the subalgebra of $k[\Star(E' \smallsetminus E)]$ supported on $\lk_\Gamma(E')$.
\begin{theorem}\label{thm:maps}
Let $E \subset E'$ be faces of $\Gamma$, with $$d = n - \vert E \vert, \quad
d' = n - \vert E' \vert, \quad
\mbox{ and } \quad b' = n - |\sigma(E')|.$$ Let $\{\theta_1, \dotsc, \theta_{d } \}$ be a special l.s.o.p. for $k[\lk_{\Gamma}(E)]$,
and let
$\theta_i' := \theta_i|_{\Star(E' \smallsetminus E)}$. Then there is a
unique homomorphism of graded $k[\lk_{\Gamma}(E')]$-algebras
\[
\phi\colon k[\lk_{\Gamma}(E)]/(\theta_1, \dotsc, \theta_d) \to k[\lk_{\Gamma}(E')]/(k[\lk_\Gamma(E')] \cap (\theta_1', \dotsc, \theta_{d}'))
\]
whose kernel contains $\{[x^F] : F \not \in \Star(E' \smallsetminus E)\}$. Moreover, there is a special l.s.o.p. $\zeta_1, \ldots, \zeta_{d'}$ for $k[\lk_\Gamma(E')]$ such that $(\zeta_1, \ldots, \zeta_{d'}) = k[\lk_\Gamma(E')] \cap (\theta'_1, \ldots, \theta'_{d})$ and, up to reordering, we have $\theta_i|_{\lk_\Gamma(E')} = \zeta_i$, for $1 \leq i \leq b'$. With this choice of special l.s.o.p., $\phi(L(\Gamma,E)) \subset L(\Gamma,E')$.
\end{theorem}
\begin{remark}
Theorem~\ref{thm:maps} may be viewed as a functoriality statement for local face modules. Start by fixing the special l.s.o.p. $\theta_1, \ldots, \theta_d$. Then $L(\Gamma, E)$ is well-defined. For $E' \supset E$ the special l.s.o.p. $\zeta_1, \ldots, \zeta_{d'}$ depends on some choices, but the ideal that it generates does not, nor does the map $\phi \colon L(\Gamma,E) \to L(\Gamma,E')$. Moreover, for $E'' \supset E'$, one readily checks that the maps $\phi' \colon L(\Gamma, E') \to L(\Gamma, E'')$ and $\phi '' \colon L(\Gamma,E) \to L(\Gamma, E'')$ are independent of all choices and satisfy $\phi'' = \phi' \circ \phi$. Thus one obtains a functor from the poset of faces of $\Gamma$ that contain $E$ to graded vector spaces, given by $E' \mapsto L(\Gamma, E')$.
\end{remark}
We now give two applications of the above theorems. One motivation for investigating local face modules is a decades old problem posed by Stanley, who introduced and studied local $h$-vectors in the special case where $E = \emptyset$ and asked for a characterization of triangulations for which they vanish \cite[Problem~4.13]{Stanley92}. This problem remains open, and is of enduring interest \cite[Problem~2.12]{Athanasiadis16}. The extension to the case where $E$ is not empty is particularly relevant for applications to the monodromy conjecture \cite{Igusa78, DenefLoeser98, Stapledon17}. In \cite{LarsonPayneStapledon}, we prove a theorem on the structure of geometric triangulations with vanishing local $h$-vectors that is tailored to this purpose, and we use it to prove the monodromy conjectures for all singularities that are nondegenerate with respect to a simplicial Newton polyhedron. See Theorems~1.1.1, 1.4.3, and 4.1.3 in loc. cit.
Here, we apply Theorem~\ref{thm:resolution} to prove another theorem on the structure of faces in triangulations with vanishing local $h$-vectors. Let $F \in \lk_\Gamma(E)$ be a face such that $F\sqcup E$ is interior. Following terminology from the monodromy conjecture literature (see, e.g., \cite{LemahieuVanProeyen11}), we say that $F$ is a \emph{pyramid with apex $w \in F$} if $(F \sqcup E ) \smallsetminus w$ is not interior. Let $$\mathcal{A}_F := \{ w \in F : F \mbox{ is a pyramid with apex } w \}, \mbox{ \ and \ } V_w := \sigma( (F \sqcup E) \smallsetminus w)^c.$$
The elements of $V_w$ correspond to the \emph{base directions} of $F$, i.e., the facets of $2^V$ that contain the base of $F$, when viewed as a pyramid with apex $w$. We say $F$ is a $U$-pyramid if there is an apex $w \in \mathcal{A}_F$ such that $|V_w| = 1$. In other words, a $U$-pyramid is a pyramid with a unique base direction, for some choice of apex.
\begin{definition}
Let $F \in \lk_{\Gamma}(E)$ be a face. An \emph{interior partition} of $F$ is a decomposition
\[
F = F_1 \sqcup F_2 \sqcup \mathcal{A}_F
\]
such that $F_1 \sqcup \mathcal{A}_F \sqcup E$ and $F_2 \sqcup \mathcal{A}_F \sqcup E$ are both interior.
\end{definition}
\begin{theorem}\label{thm:interiornonvanish}
Suppose $\ell(\Gamma,E) = 0$ and $F \in \lk_\Gamma(E)$ has an interior partition $F = F_1 \sqcup F_2 \sqcup \mathcal{A}_F$ such that $|F_1| \leq 2$. Then $F$ is a $U$-pyramid.
\end{theorem}
\noindent See Remark~\ref{r:specialcase} for a short proof in a special case that illustrates the naturality of the
$U$-pyramid condition.
The method of proof breaks down when $|F_i| \geq 3$. See Example~\ref{example:nonrestriction}.
\begin{remark}
The analogous theorem in \cite{LarsonPayneStapledon} requires that the triangulation be geometric and that the interior partition satisfies the additional condition $\sigma(F_2 \sqcup E)^c = \bigcup_{w \in \mathcal{A}_F} V_w$. But then the hypothesis that $|F_1| \leq 2$ is dropped entirely. So, even for geometric triangulations, there are cases of Theorem~\ref{thm:interiornonvanish} that are not necessarily covered by \cite[Theorem~4.1.3]{LarsonPayneStapledon}. It should be interesting to look for a common generalization of these vanishing results, and to pursue further progress on Stanley's problem of characterizing triangulations with vanishing local $h$-vector more generally.
\end{remark}
We also apply Theorem~\ref{thm:maps} to prove the following monotonicity property for local $h$-vectors.
\begin{theorem}\label{thm:increase}
Let $E \subset E'$ be faces of $\Gamma$ such that $\sigma(E) = \sigma(E')$. Then $\ell (\Gamma, E) \geq \ell(\Gamma, E')$.
\end{theorem}
\noindent The inequality in Theorem~\ref{thm:increase} is term by term, i.e., $\dim L(\Gamma, E)_i \geq \dim L(\Gamma,E')_i$ for all $i$. The proof is by showing that the map $\phi \colon L(\Gamma, E) \to L(\Gamma,E')$ given by Theorem~\ref{thm:maps} is surjective.
\begin{remark}
To the best of our knowledge, all of the theorems stated in the introduction are new even for regular triangulations. The reader who prefers to do so may safely restrict attention to geometric or even regular triangulations. However, while the structure results for triangulations with vanishing local $h$-vectors in \cite{dMGPSS20} and \cite{LarsonPayneStapledon} rely on special properties of geometric triangulations, the proofs presented here work equally well for quasi-geometric homology triangulations, and we find it natural to work in this level of generality.
\end{remark}
We conclude the introduction with an example illustrating the above theorems.
\begin{example} \label{ex:triforce}
Let $\Gamma$ be the \emph{triforce} triangulation, which figures prominently in \cite{dMGPSS20} and in the adventures of hero protagonist Link in the video game series The Legend of Zelda.
\medskip
\begin{center}
\begin{tikzpicture}[scale=2]
\draw (0.5,0.8) node[above] { $u$ } --
(1,0) node[right] { $v$ } --
(0,0) node[left] { $w$ } -- (0.5,0.8);
\draw (0.75,0.4) node[right] { $c$ } --
(0.5,0) node[below] { $a$ } --
(0.25,0.4) node[left] { $b$ } -- (0.75,0.4) ;
\draw (-.75,0.4) node {$\Gamma$ };
\end{tikzpicture}
\end{center}
Consider first $E = \emptyset$. The face ring is
\[
k[\lk_\Gamma(E)] = k[x^{a},x^{b},x^{c},x^{u},x^{v},x^{w}]/(x^{a}x^{u},x^{b}x^{v},x^{c}x^{w},x^{u}x^{v},x^{u}x^{w},x^{v}x^{w}),
\]
and its ideal of interior faces is
\[
I = (x^ax^b, x^ax^c, x^bx^c).
\]
A special l.s.o.p. is of the form $\theta_1, \theta_2, \theta_3$, with
\[
\supp(\theta_1) = \{ b,c,u\}, \quad \quad \supp(\theta_2) = \{ a,c,v\}, \quad \quad \supp(\theta_3) = \{ a,b,w\},
\]
subject to the condition that the restrictions (of the corresponding affine linear functions) to the face $\{a, b, c\}$ are linearly independent. Our resolution of the local face module $L(\Gamma, E)$ also involves the monomial ideals
\[
\begin{array}{ccc}
I_a = (x^a, x^bx^c), & I_b = (x^b, x^ax^c), & I_c = (x^c, x^ax^b), \\ I_{ab} = (x^a, x^b, x^w), & I_{ac} = (x^a, x^c, x^v), & I_{bc} = (x^b, x^c, x^u).
\end{array}
\]
The resolution given by Theorem~\ref{thm:resolution} is then
\[
0 \to k[\lk_\Gamma(E)] \xrightarrow{\begin{bsmallmatrix}
\theta_1 \\
-\theta_2 \\
\theta_3
\end{bsmallmatrix}} I_{bc} \oplus I_{ac} \oplus I_{ab} \xrightarrow{\begin{bsmallmatrix}
0 & -\theta_3 & -\theta_2\\
-\theta_3 & 0 & \theta_1\\
\theta_2 & \theta_1 & 0\\
\end{bsmallmatrix} } I_a \oplus I_b \oplus I_c \xrightarrow{\begin{bsmallmatrix}
\theta_1 &
\theta_2 &
\theta_3
\end{bsmallmatrix}} I \to L(\Gamma, E) \to 0
\]
In particular, we have $L(\Gamma, E) \cong I/J$, where
\[
J = ( \theta_1 \cdot x^a, \theta_2 \cdot x^b, \theta_3 \cdot x^c ).
\]
Since $\theta_1$, $\theta_2$, and $\theta_3$ restrict to linearly independent functions on $\{a, b,c\}$, the generators of $J$ span the 3-dimensional subspace $\langle x^a x^b, x^a x^c, x^bx^c \rangle$ of $k[\lk_\Gamma(E)]$. Hence $I = J$ and $L(\Gamma, E) = 0$.
Next, consider $E' = \{c\}$. Then
\[
k[\lk_\Gamma(E')] = k[x^a, x^b, x^u, x^v] / (x^ax^u, x^bx^v, x^ux^v).
\]
A special l.s.o.p. is any l.s.o.p. of the form $\zeta_1, \zeta_2$, where $\supp(\zeta_1) \subset \{a, b\}$. The ideal of interior faces in this case is $I' = (x^a, x^b)$, and the resolution given by Theorem~\ref{thm:resolution} is
\[
0 \to k[\lk_\Gamma(E')] \xrightarrow{\begin{bsmallmatrix}
-\zeta_2 \\
\zeta_1 \\
\end{bsmallmatrix}} k[\lk_\Gamma(E')] \oplus I' \xrightarrow{\begin{bsmallmatrix}
\zeta_1 &
\zeta_2
\end{bsmallmatrix}} I' \to L(\Gamma,E') \to 0.
\]
Note, in particular, that $L(\Gamma,E') \cong I'/J'$, where $J' = (\zeta_1, \zeta_2 x^a, \zeta_2 x^b)$. Thus one sees that $L(\Gamma,E')$ has dimension 1 in degree 1, i.e., $\ell(\Gamma, E') = (0,1,0)$.
Let us now consider Theorem~\ref{thm:maps} in this example. Let $\theta'_i$ denote the restriction of $\theta_i$ to $k[\Star(E' \smallsetminus E)]$. Note that $\zeta_1 := \theta'_3$ is supported on $\lk_\Gamma(E')$. Extend $\{ \zeta_1 \}$ to a basis for $k[\lk_\Gamma(E)] \cap (\theta'_1, \theta'_2, \theta'_3)$, e.g., by choosing $\zeta_2$ to be a linear combination of $\theta'_1$ and $\theta'_2$ in which the coefficient of $x^c$ vanishes. Then $\zeta_1, \zeta_2$ is a special l.s.o.p. for $k[\lk_\Gamma(E')]$, and the map $\phi$ in Theorem 1.4 is given as follows. First, we set
\[
\phi(x^a) = x^a, \quad \phi(x^b) = x^b, \quad \phi(x^u) = x^u, \quad \phi(x^v) = x^v, \quad \phi(x^w) = 0.
\]
Then, writing $\theta_2 = \lambda_c x^c + \lambda_a x^a + \lambda_v x^v$, with all three coefficients nonzero, we set
\[
\phi(x^c) = \frac{-1}{\lambda_c} (\lambda_ax^a + \lambda_v x^v).
\]
Note that there is no subset of $\{ \theta_1, \theta_2, \theta_3 \}$ whose restrictions to $k[\lk_\Gamma(E')]$ form an l.s.o.p. This explains and motivates our two-step process for constructing the map: first restricting to $\Star(E' \smallsetminus E)$ and then intersecting with $k[\lk_\Gamma(E')]$ to produce the special l.s.o.p. that yields the functorial map $\phi \colon L(\Gamma,E) \to L(\Gamma,E')$.
Let also describe how Theorems~\ref{thm:interiornonvanish} and \ref{thm:increase} manifest in this example. For Theorem~\ref{thm:interiornonvanish}, observe that the face $F = \{a, b\}$ in $\lk_\Gamma(E')$ has an interior partition $F = \{a\} \sqcup \{b\}$. The proof in this case shows that the classes of both $x^a$ and $x^b$ are nonzero in $L(\Gamma,E')$, for any choice of special l.s.o.p.
Finally, note that $L(\Gamma,E) = 0$ and $L(\Gamma, E') \neq 0$, so there is no surjective map of graded vector space $L(\Gamma,E) \to L(\Gamma,E')$. In this case, $\sigma(E) \neq \sigma(E')$. Thus, we see that the hypothesis $\sigma(E) = \sigma(E')$ cannot be dropped in Theorem~\ref{thm:increase}.
\end{example}
\noindent \textbf{Acknowledgments.} The work of ML is supported by an NDSEG fellowship and the work of SP is supported in part by NSF DMS--2001502 and DMS--2053261.
\section{Preliminaries} \label{sec:sr}
We begin by recalling definitions and background results that will be used throughout, following \cite[Chapter~III]{Stanley96} and \cite{Athanasiadis16}. We work over a field $k$. In particular, all rings are commutative $k$-algebras and singular homology is computed with coefficients in $k$.
\subsection{Triangulations of simplices} \label{sec:triangulations}
In this section only, for the purposes of providing context, we allow that the field $k$ may be finite, and the triangulation $\sigma \colon \Gamma \to 2^V$ is not necessarily quasi-geometric.
\medskip
We recall the notion of a homology triangulation, following \cite{Athanasiadis12}. A $d$-dimensional simplicial complex $\Gamma$ with trivial reduced homology is a \emph{homology ball}
of dimension $d$ if there is a subcomplex $\partial \Gamma \subset \Gamma$ such that
\begin{itemize}
\item $\partial \Gamma$ is a homology sphere of dimension $d -1$,
\item $\lk_\Gamma(F)$ is a homology sphere of dimension $d - |F|$ for $F \not \in \partial \Gamma$.
\item $\lk_\Gamma(F)$ is a homology ball of dimension $d - |F|$ for all nonempty $F \in \partial \Gamma$.
\end{itemize}
The \emph{interior faces} of a homology ball $\Gamma$ are the faces not contained in $\partial \Gamma$. A \emph{homology triangulation} of the simplex $2^V$ is a finite simplicial complex $\Gamma$ and a map $\sigma\colon \Gamma \to 2^V$ such that for every non-empty $U \subset V$,
\begin{itemize}
\item the simplicial complex $\Gamma_U := \sigma^{-1}(2^U)$ is a homology ball of dimension $\vert U \vert - 1$.
\item $\sigma^{-1}(U)$ is the set of interior faces of the homology ball $\sigma^{-1}(2^U)$.
\end{itemize}
\noindent Note that the Betti numbers of a simplicial complex, and hence the property of being a homology ball, depend only on the characteristic of the field $k$. Homology triangulations are a special case of the (strong) formal subdivisions of Eulerian posets considered in \cite[\mathcal{S} 7]{Stanley92} and \cite[\mathcal{S} 3]{KatzStapledon16}.
The \emph{carrier} of a face $F \in \Gamma$ is $\sigma(F)$. A homology triangulation $\sigma\colon \Gamma \to 2^V$ is \emph{quasi-geometric} if there is no face $F \in \Gamma$ and $U \subset V$ such that the dimension of $\Gamma_U$ is strictly smaller than the dimension of $F$ and the carrier of every vertex in $F$ is contained in $U$. A homology triangulation is \emph{geometric} if it can be realized in $\mathbb{R}^n$ as the subdivision of a geometric simplex into geometric simplices. Every geometric homology triangulation is quasi-geometric.
The local $h$-vector, which we have defined in the introduction as the Hilbert function of the local face module, can be expressed in terms of $h$-vectors of subcomplexes of links of faces in the homology balls $\Gamma_U$:
\begin{equation} \label{eq:localh}
\ell(\Gamma, E) = \sum_{U \supset \sigma(E)} (-1)^{|V| - \vert U \vert} h(\lk_{\Gamma_U}(E)).
\end{equation}
Note that \eqref{eq:localh} makes sense even when $k$ is finite or $\sigma \colon \Gamma \to 2^V$ is not quasi-geometric, and should be taken as the definition of the local $h$-vector in this broader context.
\begin{theorem}[\cite{Stanley92, Athanasiadis12, KatzStapledon16}]\label{t:localproperties}
Let $\sigma \colon \Gamma \to 2^V$ be a homology triangulation, let $E$ be a face of $\Gamma$ and let $d = |V| - |E|$. Then the local $h$-vector $(\ell_0, \ldots, \ell_d)$ satisfies: \\
\begin{tabular}{lll}
\quad \quad $\bullet$ & \emph{(symmetry)} &
$\ell_i = \ell_{d-i};$ \\
\quad \quad $\bullet$ & \emph{(non-negativity)} & if $\Gamma$ is quasi-geometric, then $\ell_i \geq 0;$\\
\quad \quad $\bullet$ & \emph{(unimodality)} &
if $\Gamma$ is regular, then $\ell_0 \leq \ell_1 \leq \cdots \leq \ell_{\lfloor d/2 \rfloor}$.
\end{tabular}
\end{theorem}
\noindent Note that the proof of non-negativity for quasi-geometric triangulations, due to Stanley and Athanasiadis, is via the identification with the Hilbert function of the local face module. It suffices to consider the case where $k$ is infinite, since \eqref{eq:localh} is invariant under field extensions.
\subsection{Face rings and special l.s.o.p.s}\label{ss:sr}
Here, and for the remainder of the paper, the field $k$ is fixed and infinite, and all triangulations are quasi-geometric homology triangulations.
Given a finite simplicial complex $\Gamma$ with vertex set $V = \{v_1, \ldots, v_n\}$, let $k[\Gamma]$ denote the \emph{face ring}. In other words, for each subset $F \subset V$, let $x^F$ be the corresponding squarefree monomial in the polynomial ring $k[x_1, \ldots, x_n]$, i.e.,
$
x^F:= \prod_{v_i \in F} x_i.
$
Then the face ring is
\[
k[\Gamma] := k[x_1, \ldots, x_n] / (x^F : F \mbox{ is not a face in } \Gamma).
\]
Given a subcomplex $\Gamma'$ of $\Gamma$, we have a natural restriction map $k[\Gamma] \rightarrow k[\Gamma']$, taking $x^F$ to $x^F$ if $F \in \Gamma'$ and to 0 otherwise. Given $\theta \in k[\Gamma]$, let $\theta|_{\Gamma'}$ denote the image of $\theta$ in $k[\Gamma']$. In particular, each $F$ in $\Gamma$ may be viewed as a subcomplex, and we write $\theta|_F$ for the restriction of $\theta$ to this subcomplex.
Note that $k[\Gamma]$ is graded by degree. By definition, a linear system of parameters (l.s.o.p.) for a finitely generated graded $k$-algebra $R$ of Krull dimension $d$ is a sequence of elements $\theta_1, \ldots, \theta_d$ in $R_1$ such that $R/(\theta_1, \ldots, \theta_d)$ is a finite-dimensional $k$-vector space. If $\Gamma$ is a Cohen-Macaulay complex (i.e., if $k[\Gamma]$ is a Cohen-Macaulay ring) and $\theta_1, \ldots, \theta_d$ is an l.s.o.p.\ for $k[\Gamma]$, then $(\theta_1, \ldots, \theta_d)$ is a regular sequence and the $h$-polynomial of $\Gamma$ is the Hilbert series of $k[\Gamma]/(\theta_1, \ldots, \theta_d)$. Links of faces in triangulations of simplices are Cohen-Macaulay \cite{Reisner76}.
Suppose $\Gamma$ has dimension $d-1$, so $k[\Gamma]$ has Krull dimension $d$. Then a sequence of elements $\theta_1, \ldots, \theta_d$ in $k[\Gamma]_1$ is an l.s.o.p. for $k[\Gamma]$ if and only if the following condition is satisfied \cite[Lemma~2.4(a)]{Stanley96}:
\renewcommand{\labelitemi}{$(*)$}
\begin{itemize}
\item For every face $F \in \Gamma$ (or equivalently, for every facet $F \in \Gamma$), the restrictions $\theta_1|_F, \ldots, \theta_d|_F$ span a vector space of dimension $|F|$.
\end{itemize}
\noindent This characterization provides flexibility in constructing l.s.o.p.s in which the linear functions have specified support, where the \emph{support} of $\theta = \sum a_i x_i$ is $\supp(\theta) := \{ v_i : a_i \neq 0 \}$.
\begin{lemma} \label{lemma:lsopexistence}
Let $S_1, \ldots, S_d$ be subsets of the vertices of $\Gamma$. Then there is an l.s.o.p. $\theta_1, \ldots, \theta_d$ for $k[\Gamma]$ such that $\supp (\theta_i) = S_i$ for $1 \leq i \leq d$ if and only if, for every face $F \in \Gamma$,
\begin{equation} \label{eq:marriageinequality}
| \{ S_i : S_i \cap F \neq \emptyset \}| \geq |F|.
\end{equation}
\end{lemma}
\begin{proof}
The argument is similar to that given by Stanley in \cite[Corollary~4.4]{Stanley92}. The necessity of \eqref{eq:marriageinequality} follows immediately from (*). We now prove its sufficiency. Suppose $S_1, \ldots, S_d$ are chosen such that \eqref{eq:marriageinequality} holds for every
$F \in \Gamma$.
Let $N = |S_1| + \cdots + |S_d|$, and consider the space $k^N$
parametrizing tuples $(\theta_1, \ldots, \theta_d)$
with $\supp (\theta_i) \subset S_i$.
Fix $F = \{v_1, \dotsc, v_k\} \in \Gamma$.
Let $X_F \subset k^N$ parametrize the tuples
whose restrictions to $F$
span a vector space of dimension $|F|$.
Note that $X_F$ is Zariski open.
By Hall's Marriage Theorem, there is a permutation $\sigma \in \mathfrak{S}_d$ such that $v_i \in S_{\sigma(i)}$. If we set $\theta_{\sigma(i)} = x_i$ for $1 \le i \le k$, and $\theta_{\sigma(i)} = 0$ for $i > k$, then $\theta \in X_F$, and hence $X_F$ is nonempty. Also, the subset of $k^N$ where all coordinates are nonzero is Zariski open and nonempty. Since $k$ is infinite, the intersection of these nonempty Zariski open subsets of $k^N$ is nonempty, and hence there is an l.s.o.p. $\theta_1, \ldots, \theta_d$ with $\supp(\theta_i)= S_i$.
\end{proof}
Let $\sigma \colon \Gamma \to 2^V$ be a quasi-geometric homology triangulation, and let $E \in \Gamma$ be a face.
\begin{definition}[\cite{Stanley92, Athanasiadis12b}] \label{def:special} A linear system of parameters $\theta_1, \dotsc, \theta_{d}$ for $k[\lk_{\Gamma}(E)]$ is \textit{special} if, for each vertex $v \in V$ with $v \not \in \sigma(E)$, there is an element $\theta_v$ of the l.s.o.p. such that $\supp(\theta_v)$ consists of vertices in $\lk_{\Gamma}(E)$ whose carrier contains $v$, and such that $\theta_v \not= \theta_{v'}$ for $v \not= v'$.
\end{definition}
In other words, after reordering so that $\sigma(E)^c = \{v_1, \ldots, v_b\}$, an l.s.o.p. for $k[\lk_\Gamma(E)]$ is special if we can order it $\theta_1, \ldots, \theta_d$ such that
\[
\supp(\theta_i) \subset \{ w \in \lk_\Gamma(E) : v_i \in \sigma(w)\},
\]
for $1 \leq i \leq b$. The existence of special l.s.o.p.s is well-known to experts and the proof is similar to Stanley's argument in the case $E = \emptyset$. For completeness, we provide a short proof.
\begin{proposition} \label{prop:speciallsopexistence}
Suppose $k$ is infinite. Let $\sigma\colon \Gamma \to 2^V$ be a quasi-geometric homology triangulation of a simplex, and let $E$ be a face of $\Gamma$. Then there is a special l.s.o.p. for $k[\lk_{\Gamma}(E)]$.
\end{proposition}
\begin{proof}
Let $V = \{v_1, \ldots, v_n\}$. After renumbering, we may assume that $\sigma(E)^c = \{v_1, \dotsc, v_b\}$. Fix $d = n - \vert E \vert$. Note that $b \leq d$. We define subsets $S_1, S_2, \dotsc, S_{d}$ of the vertices in $\lk_{\Gamma}(E)$, as follows. For $i \le b$, let $S_i$ be the set of vertices $w$ such that $v_i \in \sigma(w)$.
For $i > b$, let $S_i$ be the set of all vertices of $\lk_{\Gamma}(E)$. Because $\sigma$ is quasi-geometric, for each face $F$ of $\lk_{\Gamma}(E)$, the union of the sets $\sigma(w) \subset V$, as $w$ ranges over vertices of $E \sqcup F$, has size at least $|E| + |F|$. It follows that $|\{i \leq b : S_i \cap F \neq \emptyset \}| \geq |F| - (d-b)$. Since $S_j \cap F \neq \emptyset$ for $j > b$, we conclude that $|\{i : S_i \cap F \neq \emptyset \}| \geq |F|$. Hence, by Lemma~\ref{lemma:lsopexistence}, there is an l.s.o.p. $\theta_1, \ldots, \theta_d$ for $k[\lk_\Gamma(E)]$ with $\supp(\theta_i) = S_i$.
\end{proof}
\section{A resolution of the local face module}
In this section, we prove Theorem~\ref{thm:resolution}, giving an explicit resolution of the local face module $L(\Gamma,E)$ by a subcomplex of the Koszul resolution of $k[\lk_{\Gamma}(E)]/(\theta_1, \dotsc, \theta_{d})$. We continue to use the notation established above. In particular, $\sigma\colon \Gamma \to 2^V$ is a quasi-geometric homology triangulation of the simplex with vertex set $V = \{v_1, \ldots, v_n\}$. We consider a face $E \in \Gamma$ with $d = n - |E|$ and $b = n - |\sigma(E)|$. After reordering, we assume $\sigma(E)^c = \{v_{1}, \dotsc, v_{b}\}$. For $S \subset \{v_1, \ldots, v_d\}$,
we consider the ideal $I_S \subset k[\lk_\Gamma(E)]$ given by
\begin{equation*
I_S := ( x^F : \, \sigma(F \sqcup E)^c \subset S).
\end{equation*}
Let $\theta_1, \ldots \theta_d$ be a special l.s.o.p. for $k[\lk_\Gamma(E)]$. We may assume that
\[
\supp(\theta_i) \subset \{ w \in \lk_\Gamma(E) : v_i \in \sigma(w) \},
\]
for $1 \leq i \leq b$. For any $v_i \in S$, multiplication by $\theta_i$ gives a map $\lambda_i \colon I_{S} \to I_{S \smallsetminus \{v_i\}}$, and we consider the complex of graded $k[\lk_{\Gamma}(E)]$-modules
\begin{equation} \label{eq:resolution}
0 \to k[\lk_{\Gamma}(E)][-d] \to \bigoplus_{ |S| = d - 1 } I_{S}[-(d-1)] \to \dotsb \to \bigoplus_{|S| = 1} I_{S}[-1] \to I \to L(\Gamma,E) \to 0,
\end{equation}
in which
the differential restricted to $I_S$, for $S= \{v_{i_0}, \ldots, v_{i_k}\}$, with $i_0 < \cdots < i_k$, is $\oplus_{j = 0}^k (-1)^j \lambda_{i_j}$.
\begin{example}
If $E$ is an interior face of $\Gamma$ then every l.s.o.p. is special, $I_S = k[\lk_{\Gamma}(E)]$ for all $S$, and \eqref{eq:resolution} is the Koszul resolution of $L(\Gamma,E) = k[\lk_{\Gamma}(E)]/(\theta_1,\ldots,\theta_{d})$.
\end{example}
\begin{proof}[Proof of Theorem \ref{thm:resolution}]
We must show \eqref{eq:resolution} is exact.
We begin by considering two complexes of $k[\lk_{\Gamma}(E)]$-modules
studied by Stanley and Athanasiadis. Recall that, for $U \subset V$, we write $\Gamma_U := \sigma^{-1}(2^U)$.
Say $U \supset \sigma(E)$ and $U \smallsetminus \sigma(E) = \{v_{i_0}, \ldots, v_{i_k}\}$, with $i_0 < \cdots < i_k$. For $0 \leq j \leq k$, let $\rho_j \colon k[\lk_{\Gamma_U}(E)] \to k[\lk_{\Gamma_{U \smallsetminus \{v_{i_j}\}}}(E)]$ be the restriction map. The first complex we consider is
\small
\begin{equation}\label{eq:modcomp}
\begin{tikzcd}
k[\lk_{\Gamma}(E)] \arrow[r] & \bigoplus\limits_{\substack{U \supset \sigma(E) \\ \vert U \vert = n - 1}} k[\lk_{\Gamma_U}(E)] \arrow[r] &\bigoplus\limits_{\substack{U \supset \sigma(E) \\ \vert U \vert = n - 2}} k[\lk_{\Gamma_U}(E)] \arrow[r] &\cdots \arrow[r] & k[\lk_{\Gamma_{\sigma(E)}}(E)] \arrow[r] & 0,
\end{tikzcd}
\end{equation}
\normalsize
in which the differential restricted to $k[\lk_{\Gamma_U}(E)]$ is $\bigoplus_j (-1)^j \rho_j$. Next, we consider its quotient by $(\theta_1,\ldots,\theta_{d})$:
\begin{equation}\label{eq:quotientedcomplex}
\begin{tikzcd}
\frac{k[\lk_{\Gamma}(E)]}{(\theta_1, \dotsc, \theta_{d})} \arrow[r] & \bigoplus\limits_{\mathclap{\substack{U \supset \sigma(E) \\ \vert U \vert = n-1}}} \frac{k[\lk_{\Gamma_{U}}(E)]}{(\theta_1, \dotsc, \theta_{d})} \arrow[r] & \bigoplus\limits_{\mathcal{\substack{U \supset \sigma(E) \\ \vert U \vert = n-2}}} \frac{k[\lk_{\Gamma_{U}}(E)]}{(\theta_1, \dotsc, \theta_{d})} \arrow[r] &\cdots \arrow[r] & \frac{k[\lk_{\Gamma_{\sigma(E)}}(E)]}{(\theta_1, \dotsc, \theta_{d})} \arrow[r] &0.
\end{tikzcd}
\end{equation}
For any $U \subset V$, with $U \supset \sigma(E)$, let $S_U$ be defined as
\[
S_U := (U \cap \{v_1, \ldots, v_b\}) \cup \{ v_{b+1}, \ldots, v_{d}\}.
\]
Then $\dim k[\lk_{\Gamma_U}(E)] = |S_U|$ and it follows that the restriction of $\theta_i$ to $\lk_{\Gamma_U}(E)$ is nonzero if and only if $v_i \in S_U$. Furthermore, $\{ \theta_i|_{\lk_{\Gamma_U}(E)} : v_i \in S_U \}$ is a special l.s.o.p. for $k[\lk_{\Gamma_{U}}(E)]$. Stanley and Athanasiadis proved that both \eqref{eq:modcomp} and \eqref{eq:quotientedcomplex} are exact, and the kernel of the first arrow in \eqref{eq:quotientedcomplex} is $L(\Gamma,E)$. (We will recall the proofs below.) Using the additivity of Hilbert functions in exact sequences, they deduced that the Hilbert function of $L(\Gamma, E)$ satisfies \eqref{eq:localh} \cite{Stanley92, Athanasiadis12}.
With the goal of proving that \eqref{eq:resolution} is exact, we take Koszul resolutions of each term in \eqref{eq:quotientedcomplex} to build a double complex of $k[\lk_{\Gamma}(E)]$-modules. Since $k[\lk_{\Gamma_{U}}(E)]$ is Cohen-Macauley, the special l.s.o.p. $\{ \theta_i|_{\lk_{\Gamma_U}(E)} : v_i \in S_U \}$ is a regular sequence. Hence the corresponding Koszul complex $K^{\bullet}_U$
\begin{center}
\begin{tikzcd}[column sep = small]
0 \arrow[r] &k[\lk_{\Gamma_U}(E)]_{S_U} \arrow[r] &\bigoplus\limits_{\mathclap{\substack{S \subset S_U \\ \vert S \vert = \vert S_U \vert - 1}}} k[\lk_{\Gamma_U}(E)]_S \arrow[r]& \cdots \arrow[r] &\bigoplus\limits_{\mathclap{\substack{S\subset S_U\\ \vert S \vert = 1}}} k[\lk_{\Gamma_U}(E)]_S \arrow[r] &k[\lk_{\Gamma_U}(E)] \arrow[r]& \frac{k[\lk_{\Gamma_U}(E)]}{(\theta_1, \dotsc, \theta_{d})} \arrow[r] &0,
\end{tikzcd}
\end{center}
is exact. Here, for a graded module $M$ and a finite set $S$, we write $M_{S} := M[-|S|]$. Replacing each term in \eqref{eq:quotientedcomplex} with its corresponding Koszul resolution, gives a complex of complexes
\begin{equation}\label{eq:koszul}
\begin{tikzcd}
K_{V}^{\bullet} \arrow[r] & \bigoplus\limits_{\mathclap{\substack{U \supset \sigma(E) \\ \vert U \vert = n - 1}}} K_U^{\bullet} \arrow[r] & \bigoplus\limits_{\mathclap{\substack{U \supset \sigma(E) \\ \vert U \vert = n - 2}}} K_U^{\bullet} \arrow[r] & \cdots \arrow[r] & K_{\sigma(E)}^{\bullet} \arrow[r] & 0,
\end{tikzcd}
\end{equation}
which may be expanded as the commuting double complex shown in Figure~\ref{fig:doublecx}.
\begin{figure}[h!]
\begin{center}
\begin{tikzcd}[column sep = tiny]
0 &0 &0 & \cdots & 0 \\
\frac{k[\lk_{\Gamma}(E)]}{(\theta_1, \dotsc, \theta_{d})} \arrow[r] \arrow[u] & \bigoplus\limits_{\mathclap{\substack{U \supset \sigma(E) \\ \vert U \vert = n-1}}} \frac{k[\lk_{\Gamma_{U}}(E)]}{(\theta_1, \dotsc, \theta_{d})} \arrow[r] \arrow[u] & \bigoplus\limits_{\mathcal{\substack{U \supset \sigma(E) \\ \vert U \vert = n-2}}} \frac{k[\lk_{\Gamma_{U}}(E)]}{(\theta_1, \dotsc, \theta_{d})} \arrow[r] \arrow[u] &\cdots \arrow[r] & \frac{k[\lk_{\Gamma_{\sigma(E)}}(E)]}{(\theta_1, \dotsc, \theta_{d})} \arrow[r] \arrow[u] &0 \\
k[\lk_{\Gamma}(E)] \arrow[r] \arrow[u] & \bigoplus\limits_{\substack{U \supset \sigma(E) \\ \vert U \vert = n - 1}} k[\lk_{\Gamma_U}(E)] \arrow[r] \arrow[u] &\bigoplus\limits_{\substack{U \supset \sigma(E) \\ \vert U \vert = n - 2}} k[\lk_{\Gamma_U}(E)] \arrow[r] \arrow[u] &\cdots \arrow[r] & k[\lk_{\Gamma_{\sigma(E)}}(E)] \arrow[r] \arrow[u] & 0 \\
\bigoplus\limits_{\mathclap{\substack{\vert S \vert = 1}}}k[\lk_{\Gamma}(E)]_S \arrow[r] \arrow[u] & \bigoplus\limits_{\substack{U \supset \sigma(E) \\ \vert U \vert = n - 1}} \quad \bigoplus\limits_{\mathclap{\substack{S \subset S_U \\ \vert S \vert = 1}}} k[\lk_{\Gamma_U}(E)]_S \arrow[r] \arrow[u] &\bigoplus\limits_{\substack{U \supset \sigma(E) \\ \vert U \vert = n - 2}} \quad \bigoplus\limits_{\mathclap{\substack{S \subset S_U \\ \vert S \vert = 1}}} k[\lk_{\Gamma_U}(E)]_S \arrow[r] \arrow[u] &\cdots \arrow[r] & \bigoplus\limits_{\mathclap{\substack{S \subset S_{\sigma(E)} \\ \vert S \vert = 1}}} k[\lk_{\Gamma_{\sigma(E)}}(E)]_S \arrow[r] \arrow[u] & 0 \\
\bigoplus\limits_{\mathclap{\substack{\vert S \vert = 2}}}k[\lk_{\Gamma}(E)]_S \arrow[r] \arrow[u] & \bigoplus\limits_{\substack{U \supset \sigma(E) \\ \vert U \vert = n - 1}} \quad \bigoplus\limits_{\mathclap{\substack{S \subset S_U \\ \vert S \vert = 2}}} k[\lk_{\Gamma_U}(E)]_S \arrow[r] \arrow[u] & \bigoplus\limits_{\substack{U \supset \sigma(E) \\ \vert U \vert = n - 2}} \quad \bigoplus\limits_{\mathclap{\substack{S \subset S_U \\ \vert S \vert = 2}}} k[\lk_{\Gamma_U}(E)]_S \arrow[r] \arrow[u] &\cdots \arrow[r] & \bigoplus\limits_{\mathclap{\substack{S \subset S_{\sigma(E)} \\ \vert S \vert = 2}}} k[\lk_{\Gamma_{\sigma(E)}}(E)]_S \arrow[r] \arrow[u] & 0. \\
\vdots \arrow[u] & \vdots \arrow[u] &\vdots \arrow[u] &\cdots & \vdots \arrow[u] \\
\bigoplus\limits_{\mathclap{\substack{\vert S \vert = d -1}}}k[\lk_{\Gamma}(E)]_S \arrow[r] \arrow[u] & \bigoplus\limits_{\mathclap{\substack{U \supset \sigma(E) \\ \vert U \vert = n - 1}}}k[\lk_{\Gamma}(E)]_{S_U} \arrow[u]\arrow[r] &0 \arrow[u] \\
k[\lk_{\Gamma}(E)]_{\{v_1, \ldots, v_d\}} \arrow[u] \arrow[r] &0 \arrow[u] \\
0 \arrow[u]
\end{tikzcd}
\end{center}
\vspace{-15 pt}
\caption{The double complex obtained by taking the Koszul resolution of \eqref{eq:quotientedcomplex}.} \label{fig:doublecx}
\end{figure}
The columns of this complex are exact by construction. We claim that the rows are also exact, and prove this using ideas from \cite[Theorem~4.6]{Stanley92}. First, we show that all rows except for the top row are exact. Choose a subset $S$ of $\{v_1, \ldots, v_d\}$, and consider the piece of the complex indexed by $S$:
\begin{equation}\label{eq:generalS}
\begin{tikzcd}[column sep = small]
k[\lk_{\Gamma}(E)]_S \arrow[r] & \bigoplus\limits_{\substack{ S \subset S_U \\ \vert U \vert = n - 1}} k[\lk_{\Gamma_U}(E)]_S \arrow[r] &\bigoplus\limits_{\substack{S \subset S_U \\ \vert U \vert = n - 2}} k[\lk_{\Gamma_U}(E)]_S \arrow[r] &\cdots
\arrow[r] & 0.
\end{tikzcd}
\end{equation}
When $S = \emptyset$, we obtain (\ref{eq:modcomp}). Observe that the complex (\ref{eq:generalS}) is multigraded by $\mathbb{N}^m$, where $m$ is the number of vertices of $\lk_{\Gamma}(E)$. Explicitly, $\deg x_1^{\alpha_1} \dotsb x_m^{\alpha_m} = (\alpha_1, \dotsc, \alpha_m)$. Therefore it suffices to show exactness on graded pieces. Fix $\alpha = (\alpha_1, \dotsc, \alpha_m)$. By the definition of the face ring, every term of (\ref{eq:generalS}) will have $0$ in the graded piece corresponding to $\alpha$ unless the set of vertices with $\alpha_i \not= 0$ forms a face $F$, in which case the $\alpha$-graded part can be identified with the augmented cochain complex of a simplex, indexed by all $U$ that contain $\sigma(E) \cup \sigma(F) \cup S$, and hence is exact.
We now recall the proof that the top row of the double complex, \eqref{eq:quotientedcomplex}, is exact.
\begin{center}
\begin{tikzcd}
\frac{k[\lk_{\Gamma}(E)]}{(\theta_1, \dotsc, \theta_{d})} \arrow[r] & \bigoplus\limits_{\mathclap{\substack{U \supset \sigma(E) \\ \vert U \vert = n-1}}} \frac{k[\lk_{\Gamma_{U}}(E)]}{(\theta_1, \dotsc, \theta_{d})} \arrow[r] & \bigoplus\limits_{\mathcal{\substack{U \supset \sigma(E) \\ \vert U \vert = n-2}}} \frac{k[\lk_{\Gamma_{U}}(E)]}{(\theta_1, \dotsc, \theta_{d})} \arrow[r] &\cdots \arrow[r] & \frac{k[\lk_{\Gamma_{\sigma(E)}}(E)]}{(\theta_1, \dotsc, \theta_{d})} \arrow[r] &0
\end{tikzcd}
\end{center}
The proof involves showing that the quotients of (\ref{eq:modcomp}) by $(\theta_{d}, \dotsc, \theta_{d - (r-1)})$ is exact by induction on $r$. The case of $r = 0$ is the exactness of the second row.
Now assume that (\ref{eq:modcomp}) remains exact after quotienting by $(\theta_{d}, \dotsc, \theta_{d - (r-1)})$. Let $C^i$ denote the $i$th term of (\ref{eq:modcomp}) tensored with $k[\lk_{\Gamma}(E)]/(\theta_{d}, \dotsc, \theta_{d - (r - 1)})$. By the induction hypothesis, we have an exact sequence
\begin{equation*}
C^{\bullet}\colon \enskip C^0 \to C^1 \to \dotsb \to C^{b} \to 0.
\end{equation*}
Set $m = d - r$. Recall that
$\theta_i = 0 \in k[\lk_{\Gamma_{U}}(E)]$ if $v_i \notin S_U$, and that $\{ \theta_i|_{\lk_{\Gamma_U}(E)} : v_i \in S_U \}$ is a special l.s.o.p. for $k[\lk_{\Gamma_{U}}(E)]$.
Also, for $\sigma(E) \subset U$, $v_m \notin S_U$ if and only if $v_m \notin U$. Hence, we have an exact sequence
\begin{equation} \label{eq:multbym}
0 \to B^{\bullet} \to C^{\bullet} \xrightarrow{\theta_{m}} C^{\bullet} \to C^{\bullet}/(\theta_{m}) \to 0,
\end{equation}
where
\[
B^{i} = \bigoplus_{\substack{U \supset \sigma(E), \enskip \vert U \vert = n-i \\ v_m \not \in U}}k[\lk_{\Gamma_U}(E)]/(\theta_{d}, \dotsc, \theta_{m + 1}).
\]
For example, when $m > b$, $v_m \in \sigma(E)$ and $B^{\bullet} = 0$.
Up to signs and a degree shift, we can then identify $ B^{\bullet}$ with the complex (\ref{eq:modcomp}) for $\Gamma|_{\{v_m\}^c}$ quotiented by $(\theta_{d}, \dotsc, \theta_{m + 1})$. Then $ B^{\bullet}$ is exact by the induction hypothesis applied to $\Gamma|_{\{v_m\}^c}$. By breaking (\ref{eq:multbym}) up into two short exact sequences we see that $H^i(C^{\bullet}/(\theta_{m})) \cong H^{i + 2}(B^{\bullet}) = 0$ as desired.
Now that we know the exactness of (\ref{eq:koszul}), let
\begin{equation*} \begin{split}
A^{\bullet} &= \ker \Bigg( K_{V}^{\bullet} \to \bigoplus\limits_{\mathclap{\substack{U \supset \sigma(E) \\ \vert U \vert = n - 1}}} K_U^{\bullet} \Bigg).
\end{split}\end{equation*}
Then, by construction, we have an exact sequence of complexes
\begin{equation*}
\begin{tikzcd}
0 \arrow[r] & A^{\bullet} \arrow[r] &K_{V}^{\bullet} \arrow[r] & \bigoplus\limits_{\mathclap{\substack{U \supset \sigma(E) \\ \vert U \vert = n - 1}}} K_U^{\bullet} \arrow[r] & \bigoplus\limits_{\mathclap{\substack{U \supset \sigma(E) \\ \vert U \vert = n - 2}}} K_U^{\bullet} \arrow[r] & \cdots \arrow[r] & K_{\sigma(E)}^{\bullet} \arrow[r] & 0.
\end{tikzcd}
\end{equation*}
As above, we repeatedly apply the long exact sequence on cohomology to see that $A^{\bullet}$ is exact.
We may then identify $A^{\bullet}$ with the exact sequence
$$0 \to k[\lk_{\Gamma}(E)][-n] \to \oplus_{ |S| = d - 1 } I_{S}[-(n-1)] \to \dotsb \to \oplus_{|S| = 1} I_{S}[-1] \to I \to A^0 \to 0.$$ Since $I$ surjects onto $A^0$ and $A^0 \subset k[\lk_{\Gamma}(E)]/(\theta_1, \dotsc, \theta_{d})$, we conclude that $A^0 = L(\Gamma,E)$, as required.
\end{proof}
\begin{remark}\label{r:specialcase}
Let $\sigma\colon \Gamma \to 2^V$ be a quasi-geometric homology triangulation of a simplex, and let $E$ be a face of $\Gamma$. Let $F \in \lk_\Gamma(E)$ such that $F \sqcup E$ is interior, and suppose that $F = \mathcal{A}_F$ is an interior partition of $F$, i.e., with $F_1 = F_2 = \emptyset$. Suppose that $F$ is not a $U$-pyramid.
By Corollary \ref{cor:presentation}, $x^{F} \in I$ is not in the image of $J$, and hence $x^F$ is nonzero in $L(\Gamma,E)$. This proves Theorem~\ref{thm:interiornonvanish} in the special case when $F_1 = F_2 = \emptyset$.
\end{remark}
\section{Functorial properties of local face modules}
In this section, we prove Theorem~\ref{thm:maps}, giving natural maps between local face modules.
Consider a quasi-geometric homology triangulation $\sigma\colon \Gamma \to 2^V$, and let $E \subset E'$ be faces of $\Gamma$.
\begin{lemma}\label{lemma:lsop}
Let $R$ be a graded $k$-algebra with $R_0 = k$. Let $\{ \theta_1, \dotsc, \theta_n \}$ be an l.s.o.p. for $R[x_1, \dotsc, x_m]$, where each $x_j$ has degree $1$. Then there is a unique graded $R$-algebra isomorphism
\[
\phi \colon R[x_1, \dotsc, x_m]/(\theta_1, \dotsc, \theta_n) \rightarrow R/R \cap (\theta_1, \dotsc, \theta_n).
\]
Moreover, any $k$-basis for $R_1 \cap (\theta_1, \dotsc, \theta_n)$ is an l.s.o.p. for $R$ and generates $R \cap (\theta_1, \dotsc, \theta_n)$.
\end{lemma}
\begin{proof}
Consider the exact sequence of $k$-linear maps
\[
0 \rightarrow R_1 \rightarrow R[x_1, \dotsc, x_m]_1 \rightarrow (x_1,\dotsc, x_m)_1 \rightarrow 0,
\]
where the right hand map takes $r + \sum_i \alpha_i x_i$ to $\sum_i \alpha_i x_i$, for any $r \in R_1$ and $\alpha_i \in k$. This restricts to an exact sequence of $k$-linear maps
\[
0 \rightarrow R_1 \cap (\theta_1, \dotsc, \theta_n)_1 \rightarrow (\theta_1, \dotsc, \theta_n)_1 \rightarrow (x_1,\dotsc, x_m)_1 \rightarrow 0,
\]
where the surjectivity of the right-hand map follows from the fact that $\theta_1, \dotsc, \theta_n$ is an l.s.o.p. Hence, for $1 \le i \le m$, we can write $x_i = r_i + s_i$, for some $r_i \in R_1$ and $s_i \in (\theta_1, \dotsc, \theta_n)_1$. For any $R$-algebra map $\phi \colon R[x_1, \dotsc, x_m]/(\theta_1, \dotsc, \theta_n) \rightarrow R/R \cap (\theta_1, \dotsc, \theta_n),$ we must have that $\phi(x_i) = r_i$, so there is a unique such map. On the other hand, the $R$-algebra homomorphism defined by $\phi(x_i) = r_i$ is well-defined, since if $x_i = r_i' + s_i'$, for some $r_i' \in R_1$ and $s_i' \in (\theta_1, \dotsc, \theta_n)_1$, then $r_i - r_i' \in R_1 \cap (\theta_1, \dotsc, \theta_n)_1$. Note that the unique $R$-algebra homomorphism from $R/R \cap (\theta_1, \dotsc, \theta_n)$ to $R[x_1, \dotsc, x_m]/(\theta_1, \dotsc, \theta_n)$ is the inverse of $\phi$.
Since $\phi$ is an isomorphism and factors through $R/(R_1 \cap (\theta_1, \dotsc, \theta_n)_1)$, we conclude that the $R$-ideal $R \cap (\theta_1, \dotsc, \theta_n)$ is generated in degree $1$ and hence any $k$-basis for $R_1 \cap (\theta_1, \dotsc, \theta_n)$ is an l.s.o.p. for $R$.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{thm:maps}]
Note that $\Star(E' \smallsetminus E)$ is the join of $E' \smallsetminus E$ with $\lk_{\Gamma}(E')$. The face ring $k[\Star(E' \smallsetminus E)]$ is therefore a polynomial ring over $k[\lk_\Gamma(E')]$. Its Krull dimension is equal to $d = \dim k[\lk_\Gamma(E)]$, and hence the restrictions $\theta'_1, \ldots, \theta'_d$ form an l.s.o.p., where $\theta'_i := \theta_i|_{\Star(E' \smallsetminus E)}$. By Lemma~\ref{lemma:lsop}, there is a unique graded $k[\lk_\Gamma(E')]$-algebra homomorphism $k[\Star(E' \smallsetminus E)]/(\theta'_1, \ldots, \theta'_d) \to k[\lk_\Gamma(E')]/ k[\lk_\Gamma(E')] \cap (\theta'_1, \ldots, \theta'_d)$, which lifts to the unique homomorphism $\phi$ in the statement of the theorem. It remains to construct a special l.s.o.p. for $k[\lk_\Gamma(E')]$ with the specified properties.
After reordering, we may assume that
\[
\sigma(E)^c = \{ v_1, \ldots, v_b\}, \quad \supp(\theta_i) \subset \{ w : v_i \in \sigma(w) \}, \ \mbox{ for } 1 \leq i \leq b, \quad \mbox{ and } \sigma(E')^c = \{v_1, \ldots, v_{b'}\}.
\]
Note, in particular, that $\theta'_i$ is supported on vertices in the link of $E'$, for $1 \leq i \leq b'$. By Lemma~\ref{lemma:lsop}, any $k$-basis for $k[\lk_\Gamma(E')] \cap (\theta'_1, \ldots, \theta'_d)$ is an l.s.o.p. for $k[\lk_\Gamma(E')]$. Set $\zeta_i = \theta_i|_{\lk_\Gamma(E')}$, for $1 \leq i \leq b'$, and note that $\{\zeta_1, \ldots, \zeta_{b'}\}$ is linearly independent. Extending this independent set to a basis produces a special l.s.o.p. for $k[\lk_\Gamma(E')]$. It remains to verify that $\phi(L(\Gamma, E)) \subset L(\Gamma, E')$. Let $F \in \lk_{\Gamma}(E)$ be a face with $F \sqcup E$ interior. If $F$ is not in $\Star(E' \smallsetminus E)$, then $\phi(x^F) = 0$. Otherwise, $F$ can be written uniquely as the join of possibly empty faces $F_1 \subset E' \smallsetminus E$ and $F_2 \in \lk_{\Gamma}(E')$. Then $F_2 \sqcup E'$ is interior, and $\phi(x^F) = \phi(x^{F_1})x^{F_2} \in (x^{F_2})$. Hence $\phi(x^F) \in L(\Gamma,E')$, as required.
\end{proof}
\begin{proof}[Proof of Theorem \ref{thm:increase}]
Let $E \subset E'$ be faces of a
quasi-geometric homology triangulation $\Gamma$ of a simplex, and assume that $\sigma(E) = \sigma(E')$. It is enough to show that the induced map $\phi \colon L(\Gamma, E) \to L(\Gamma,E')$ given by Theorem~\ref{thm:maps} is surjective. Note that $L(\Gamma,E')$ is generated by the monomials $x^F$ such that $F \in \lk_\Gamma(E')$ and $F \sqcup E'$ is interior. If $F$ is such a face, then it is also in the link of $E$ and, since $\sigma(E) = \sigma(E')$, the face $(F \sqcup E) < (F \sqcup E')$ is also interior. Then $\phi(x^F) = x^F$, and the theorem follows. \end{proof}
\section{Restrictions of local face modules}
In this section, we use the resolution found in Theorem~\ref{thm:resolution} to show that the vanishing of a local face module $L(\Gamma, E)$ implies the vanishing of a \emph{restricted local face module} $L(\Gamma, \mathcal{A}_F \sqcup E)|_{F_1 \sqcup F_2},$ for certain interior partitions $F_1 \sqcup F_2 \sqcup \mathcal{A}_F$. We then develop algebraic arguments, inspired by ideas from \cite{dMGPSS20}, to show that $F$ being a $U$-pyramid is necessary for the vanishing of the restricted local face module when $\vert F_1 \vert \le 2$ and thus prove Theorem~\ref{thm:interiornonvanish}.
We use the notation introduced in the introduction. Let $\Delta$ be a subcomplex of $\lk_{\Gamma}(E)$. For any $k[\lk_{\Gamma}(E)]$-module $M$, the \emph{restriction} of $M$ to $\Delta$ is $M|_{\Delta} := M \otimes_{k[\lk_{\Gamma}(E)]} k[\Delta]$, where $k[\Delta]$ is a $k[\lk_{\Gamma}(E)]$-module via the restriction map. By the resolution of $L(\Gamma,E)$ in Theorem~\ref{thm:resolution} and the right exactness of tensoring with $k[\Delta]$, we have an exact sequence
\begin{equation}\label{e:restrict}
\bigoplus_{|S| = 1} I_{S}|_\Delta[-1] \to I|_\Delta \to L(\Gamma,E)|_\Delta \to 0.
\end{equation}
Recall from Corollary~\ref{cor:presentation} that $L(\Gamma, E) \cong I/J$, where $J$ is the ideal generated by $\{\theta_i x^{F} : F \sqcup E \mbox{ is interior}\}$ and $\{\theta_{j} x^G : \sigma(G \sqcup E) = \{v_j\}^c \}$. Hence, $L(\Gamma,E)|_\Delta \cong I|_\Delta/J|_\Delta$, where $I|_\Delta,J|_\Delta$ are the $k[\Delta]$-ideals
\begin{equation}\label{eq:Ires}
I|_\Delta = (x^H : H \subset \Delta, \sigma(H \sqcup E) = V),
\end{equation}
\begin{equation}\label{eq:Jres}
J|_\Delta = (\theta_1|_\Delta,\ldots,\theta_{d}|_\Delta) \cdot I|_\Delta + (\theta_{j}|_{\Delta} x^G: \enskip G \subset \Delta, \enskip \sigma(G \sqcup E) = \{v_j\}^c).
\end{equation}
For example, if $F$ is a face of $\lk_{\Gamma}(E)$, then $k[F]$ is a polynomial ring with variables indexed by the vertices of $F$, and
$L(\Gamma, E)|_{F}$ is identified with a quotient of ideals in this polynomial ring.
\begin{lemma}\label{lem:restriction}
Let $\sigma\colon \Gamma \to 2^V$ be a quasi-geometric homology triangulation of a simplex, and let $E$ be a face of $\Gamma$. Let $F \in \lk_{\Gamma}(E)$ be a face with $F \sqcup E$ interior. Assume that $F$ is not a $U$-pyramid.
Then there is a surjective graded $k[F]$-module homomorphism
$$L(\Gamma, E)|_{F} \to L(\Gamma, \mathcal{A}_F \sqcup E)|_{F \smallsetminus \mathcal{A}_F}[-\vert \mathcal{A}_F \vert],$$
where the second term is a $k[F]$-module via the restriction map $k[F] \mapsto k[F \smallsetminus \mathcal{A}_F]$.
\end{lemma}
\begin{proof}
If $\Delta$ is a subcomplex of $\lk_{\Gamma}(E)$ contained in the closed star of $\mathcal{A}_F$, then $x^{\mathcal{A}_F}$ is a non-zero divisor in $k[\Delta]$. In particular, $x^{\mathcal{A}_F}$ is a non-zero divisor in $k[F]$ (this is also clear since $k[F]$ is a polynomial ring). Note that every face of $F$ with carrier codimension at most $1$ contains $\mathcal{A}_F$. Thus $I|_F = x^{\mathcal{A}_F} \cdot M$ and $J|_F = x^{\mathcal{A}_F} \cdot N$, where
$M$ and $N$ are the ideals in $k[F]$
\[
M = (x^H: \enskip H \subset F \smallsetminus \mathcal{A}_F, \enskip \sigma(H \sqcup \mathcal{A}_F \sqcup E) = V),
\]
\[
N = (\theta_1|_F,\ldots,\theta_{d}|_F) \cdot M + (\theta_{j}|_{F} x^G: \enskip G \subset F \smallsetminus \mathcal{A}_F, \enskip \sigma(G \sqcup \mathcal{A}_F \sqcup E) = \{v_j\}^c).
\]
Then we have surjective graded $k[F]$-module homomorphisms
\[
I|_F/J|_F \rightarrow M /N [-|\mathcal{A}_F|] \rightarrow M|_{F \smallsetminus \mathcal{A}_F}/N|_{F \smallsetminus \mathcal{A}_F}[-|\mathcal{A}_F|],
\]
where the first map is the isomorphism taking $x^{\mathcal{A}_F} x^H \mapsto x^H$ and the second map is restriction. Finally the right hand term can be identified with $L(\Gamma, \mathcal{A}_F \sqcup E)|_{F \smallsetminus \mathcal{A}_F}[-\vert \mathcal{A}_F \vert]$.
\end{proof}
We will derive Theorem~\ref{thm:interiornonvanish} from the following more technical statement.
\begin{theorem}\label{thm:localizedcase}
Let $\sigma\colon \Gamma \to 2^V$ be a quasi-geometric homology triangulation, and let $E$ be a face. Let $F \in \lk_{\Gamma}(E)$ be a face with $F \sqcup E$ interior. Suppose $\mathcal{A}_F = \emptyset$ and $F$ admits an interior partition $F = F_1 \sqcup F_2$. Assume that $F$ has no faces $G$ with $G \sqcup E$ interior and $ \vert G \vert < \vert F_1 \vert$. If $\vert F_1 \vert \le 2$, then $L(\Gamma, E)|_{F}$ is non-zero in degree $\vert F_1 \vert$.
\end{theorem}
\begin{example}\label{example:nonrestriction}
The conclusion of Theorem~\ref{thm:localizedcase} can fail when $\vert F_1 \vert \ge 3$, even for $\mathcal{A}_F = E = \emptyset$. Consider a geometric triangulation $\sigma \colon \Gamma \to 2^V$, where $V = \{v_1, \ldots, v_6 \}$ with a face $F = \{w_1, \ldots, w_6\}$ such that
\[
\begin{array}{lll}
\sigma(w_1) = \{ v_1, v_3, v_6 \} & \sigma(w_2) = \{ v_1, v_4, v_5 \} & \sigma(w_3) = \{ v_2, v_3, v_5 \} \\
\sigma(w_4) = \{ v_2, v_4, v_6 \} & \sigma(w_5) = \{ v_3, v_4, v_5 \} & \sigma(w_6) = \{ v_3, v_5, v_6 \}
\end{array}
\]
Then $\mathcal{A}_F = \emptyset$, and $F$ admits an interior partition given by $F_1 = \{w_1, w_4, w_5\}$, $F_2 = \{w_2, w_3, w_6\}$. Then \eqref{e:restrict} gives generators and relations for $L(\Gamma, \emptyset)|_{F}$, and a linear algebra computation shows that $L(\Gamma, \emptyset)|_{F} = 0$.
\end{example}
Before proceeding with the proof of Theorem~\ref{thm:localizedcase}, we show how Theorem~\ref{thm:interiornonvanish} follows from it.
\begin{proof}[Proof of Theorem~\ref{thm:interiornonvanish}]
We may assume that $F = F_1' \sqcup F_2' \sqcup \mathcal{A}_F$ is an interior partition of $F$ with $\vert F_1' \vert$ minimal among all possible interior partitions of $F$. In particular, if $\vert F_1' \vert = 2$, then there is no vertex $v \in F \smallsetminus \mathcal{A}_F$ such that $\{v\} \sqcup \mathcal{A}_F \sqcup E$ is interior, as then $\{v\} \sqcup (F_1' \sqcup F_2' \smallsetminus \{v\}) \sqcup \mathcal{A}_F$ would be an interior partition. Hence there are no faces $G$ of $F \smallsetminus \mathcal{A}_F$ with $G \sqcup \mathcal{A}_F \sqcup E$ interior and with cardinality smaller than $\vert F_1' \vert$.
By Theorem \ref{thm:localizedcase}, $L(\Gamma, \mathcal{A}_F \sqcup E)|_{F_1' \sqcup F_2'}$ is non-zero in degree $\vert F_1' \vert$. Then, by Lemma~\ref{lem:restriction}, $L(\Gamma, E)$ is nonzero in degree $\vert F_1' \vert + \vert \mathcal{A}_F \vert$.
\end{proof}
We now proceed with the proof of Theorem~\ref{thm:localizedcase}. We begin with a series of three lemmas. Inspired by the results of \cite{dMGPSS20} in the case $E = \emptyset$, we consider the \emph{internal edge graph} of a subcomplex $\Delta \subset \lk_{\Gamma}(E)$. This is the graph contained in the $1$-skeleton of $\lk_{\Gamma}(E)$ consisting of edges $e \subset \Delta$ with $e \sqcup E$ interior.
\begin{lemma}\label{lem:intedge}
Assume $\sigma(E)$ has codimension at least $2$. Let $\Delta$ be a subcomplex of $\lk_{\Gamma}(E)$, and assume $\Delta$ has no vertices $v$ with $\{v\} \sqcup E$ interior. If $L(\Gamma, E)|_{\Delta}$ is zero in degree $2$, then each connected component of the internal edge graph of $\Delta$ satisfies one of the following.
\begin{enumerate}
\item The component is a tree, and it has at most one vertex $v$ with $\{v\} \sqcup E$ having carrier codimension more than $1$.
\item The component has a unique cycle, and the carrier codimension of $\{w\} \sqcup E$ is equal to $1$ for every vertex $w$ in the component.
\end{enumerate}
\end{lemma}
\begin{proof}
From \eqref{e:restrict}, we have the following exact sequence for the degree $2$ part of $L(\Gamma, E)|_{\Delta}$.
$$ \bigoplus_{\vert S \vert = 1} (I_S)_1\otimes_{k[\lk_{\Gamma}(E)]} k[\Delta] \to I_2\otimes_{k[\lk_{\Gamma}(E)]} k[\Delta] \to (L(\Gamma, E)|_{\Delta})_2 \to 0.$$
Because $(L(\Gamma, E)|_{\Delta})_2 = 0$, the first map in the above complex is surjective. As $\Delta$ has no vertices $v$ with $\{v\} \sqcup E$ interior, we see that
\begin{equation}\label{eq:intedge}
(x^{e}: e \subset \Delta, \enskip e \sqcup E \text{ is interior })_2 = (x^{v} \theta_{i}: v \subset \Delta, \enskip \sigma(\{v\} \sqcup E) = \{v_i\}^c)_2.
\end{equation}
Thus the number of edges $e$ with $e \sqcup E$ interior is less than or equal to the number of vertices $w$ with the carrier codimension of $\{w\} \sqcup E$ equal to $1$. If $\sigma(\{v\} \sqcup E) = \{v_i\}^c$ and $\theta_{i} = \sum_{w_j} a_{i,j}x^{w_j}$, then
$$x^{v} \theta_{i} = \sum_{(v, w_j) \sqcup E \text{ interior }} a_{i,j}x^{v}x^{w_j}.$$ In particular, both vector spaces in (\ref{eq:intedge}) naturally decompose into a direct sum of vector spaces indexed by the connected components of the internal edge graph. Therefore, in each connected component of the internal edge graph, the number of edges $e$ with $e \sqcup E$ interior is less than or equal to the number of vertices $v$ with $\{v\} \sqcup E$ of carrier codimension $1$. As the only connected graphs $(V, E)$ where $\vert E \vert \le \vert V \vert$ are either trees or contain a unique cycle, the result follows.
\end{proof}
\begin{lemma}\label{lem:nofourcycle}
Assume $\sigma(E)$ has codimension at least $2$. Let $F \subset \lk_{\Gamma}(E)$ be a face. Assume $F$ has no vertices $v$ with $\{v\} \sqcup E$ interior. If $L(\Gamma, E)|_{F}$ is zero in degree $2$, then no component of the internal edge graph of $F$ contains a cycle of length $4$.
\end{lemma}
\begin{proof}
Suppose a component of the internal edge graph contains a $4$-cycle of vertices $F = \{t_1, t_2, u_1, u_2\}$. By Lemma~\ref{lem:intedge}, this is the unique cycle in this component and every vertex $w \in F$ has $\{w\} \sqcup E$ of carrier codimension $1$. Because $F$ is a face and there are no $3$-cycles in this component of the internal edge graph, we may assume that $\sigma(\{t_i\} \sqcup E) = \{v_1\}^c$ and $\sigma(\{u_i\} \sqcup E) = \{v_2\}^c$. Restricting to $F$ and using that $(L(\Gamma, E)|_{F})_2 = 0$, we have that
$$(x^{t_1}x^{u_1}, x^{u_1}x^{t_2}, x^{t_2}x^{u_2}, x^{u_2}x^{t_1}) = (x^{t_1} \theta_{2}, x^{t_2} \theta_{2}, x^{u_1} \theta_{1}, x^{u_2} \theta_{1}).$$
The relation $\theta_{1} \theta_{2} - \theta_{2} \theta_{1} = 0$ expands into a relation between the generators of the right-hand side. But the left-hand side is $4$-dimensional, a contradiction.
\end{proof}
\begin{lemma}\label{lem:cd1case}
Assume $\sigma(E)$ has codimension $1$. Let $\Delta \subset \lk_{\Gamma}(E)$ be a subcomplex. Then
$$\dim (L(\Gamma, E)|_{\Delta})_1 \ge |\{v \in \Delta: \{v\} \sqcup E \text{ interior}\}| - 1.$$
\end{lemma}
\begin{proof}
By considering the degree $1$ part of \eqref{e:restrict}, as the codimension of $\sigma(E)$ is $1$,
we get the following exact sequence.
\begin{center}
\begin{tikzcd}[column sep = large]
k \arrow[r] & \bigoplus\limits_{\mathclap{\substack{w \in \Delta \\ \{w\} \sqcup E \text{ interior }}}} k \cdot x^w \arrow[r] & (L(\Gamma, E)|_{\Delta})_1 \arrow[r] & 0,
\end{tikzcd}
\end{center}
and the result follows.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{thm:localizedcase}]
We must show that $L(\Gamma, E)|_{F}$ is non-zero in degree $|F_1|$.
Recall that $L(\Gamma, E)|_{F}$ is isomorphic to $I|_{F}/J|_{F}$, where $I|_{F}$ and $J|_{F}$ are described in \eqref{eq:Ires} and \eqref{eq:Jres} respectively. First we handle the cases when $\vert F_1 \vert \le 1$. If $F_1 = \emptyset$, then $E$ is interior and $x^{\emptyset} = 1$, but $J|_{F}$ is a proper ideal as it is generated by elements of positive degree, so $x^{F_1} \not \in J|_{F}$. If $F_1 = \{v\}$, then we assume that $E$ is not an interior face. Then $J|_{F}$ is generated by elements of degree at least $2$, so $x^{F_1} \not \in J|_{F}$.
Suppose $\vert F_1 \vert = 2$. We assume that there are no vertices $v$ with $\{v\} \sqcup E$ interior and $E$ is not interior. If $\sigma(E)$ has codimension $1$, then both $F_1$ and $F_2$ must have a vertex $v$ with $\{v\} \sqcup E$ interior. Then by Lemma~\ref{lem:cd1case}, we see that $\dim L(\Gamma, E)|_{F} \ge 1$. Hence we may assume that $\sigma(E)$ has codimension at least $2$.
Let $F_1 = \{u, t\}$ and assume that $L(\Gamma, E)|_{F}$ has no non-zero elements in degree $2$. Consider the connected component of the internal edge graph containing $F_1$. By Lemma~\ref{lem:intedge}, we may assume that $\sigma(\{u\} \sqcup E) = \{v_1\}^c$. Note that $v_1 \in \sigma(t)$. There is a vertex $t' \in F_2$ such that $v_1 \in \sigma(t')$, so $\{u, t'\} \sqcup E$ is interior. Therefore either $\{t\} \sqcup E$ or $\{t'\} \sqcup E$ has carrier codimension $1$.
If $\sigma(\{t\} \sqcup E) = \{v_2\}^c$, then there is a vertex $u' \in F_2$ such that $v_2 \in \sigma(u')$. First assume $u'$ and $t'$ are distinct. Since at least one of $\{u'\} \sqcup E$ and $\{t'\} \sqcup E$ has carrier codimension $1$, it follows that $\{ u', t'\} \sqcup E$ is interior. Then $\{u, t, u', t'\}$ forms a $4$-cycle, contradicting Lemma~\ref{lem:nofourcycle}.
If $u' = t'$, then the internal edge graph contains a cycle and hence every vertex $w$ in it (including $t$) has $\{w\} \sqcup E$ of carrier codimension $1$. As $F_2$ is interior and $\{u'\} \sqcup E$ has carrier codimension $1$, there is a vertex $w \in F_2$ such that $\{u', w\} \sqcup E$ is interior. But then either $\{u, w\} \sqcup E$ or $\{t, w\} \sqcup E$ is interior, contradicting the uniqueness of the cycle in Lemma~\ref{lem:intedge}.
If $\{t\} \sqcup E$ does not have carrier codimension $1$, then we may assume that $\sigma(\{t'\} \sqcup E) = \{v_2\}^c$. Choose a vertex $u' \in F_2$ with $v_2 \in \sigma(u')$. Then $\{t', u'\} \sqcup E$ is interior, so $\{u'\} \sqcup E$ has carrier codimension $1$. If $v_1 \in \sigma(u')$, then $\{u, u'\} \sqcup E$ is interior. If $v_1 \not \in \sigma(u')$, then $\{t, u'\} \sqcup E$ is interior. In either case, there is a cycle and a vertex $v$ with $\{v\} \sqcup E$ of carrier codimension more than $1$ in the internal edge graph, contradicting Lemma~\ref{lem:intedge}.
\end{proof}
\begin{remark}\label{rem:newnonvanish}
One can use the same overall strategy more generally to show that other combinatorial types of faces cannot appear in triangulations with vanishing local $h$-vectors. For instance, suppose $V = \{v_1, \ldots, v_6\}$ and $\sigma \colon \Gamma \to 2^V$ is a geometric triangulation with a facet $F = \{w_1, \ldots, w_6\}$ such that
\[
\begin{array}{lll}
\sigma(w_1) = \{ v_1\} & \sigma(w_2) = \{ v_2 \} & \sigma(w_3) = \{ v_3 \} \\
\sigma(w_4) = \{ v_1, v_4, v_5 \} & \sigma(w_5) = \{ v_2, v_4, v_6 \} & \sigma(w_6) = \{ v_3, v_5, v_6 \}
\end{array}
\]
Then the interior $2$-faces of $F$ are $\{w_1, w_5, w_6\}$, $\{w_2, w_4, w_6\}$, $\{w_3, w_4, w_5\}$, and $\{w_4, w_5, w_6\}$. But $F$ has no interior vertices or edges, and it has only three edges with carrier codimension one, namely $\{w_4, w_5\}, \{w_4, w_6\},$ and $\{w_5, w_6\}$. Thus $L(\Gamma, \emptyset)|_{F}$ is non-zero in degree three. Note that $F$ is not a pyramid and does not admit an interior partition.
\end{remark}
| {
"timestamp": "2022-09-09T02:05:49",
"yymm": "2209",
"arxiv_id": "2209.03543",
"language": "en",
"url": "https://arxiv.org/abs/2209.03543",
"abstract": "We study the local face modules of triangulations of simplices, i.e., the modules over face rings whose Hilbert functions are local $h$-vectors. In particular, we give resolutions of these modules by subcomplexes of Koszul complexes as well as functorial maps between modules induced by inclusions of faces. As applications, we prove a new monotonicity result for local $h$-vectors and new results on the structure of faces in triangulations with vanishing local $h$-vectors.",
"subjects": "Combinatorics (math.CO)",
"title": "Resolutions of local face modules, functoriality, and vanishing of local $h$-vectors",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9770226267447513,
"lm_q2_score": 0.7248702761768248,
"lm_q1q2_score": 0.7082146612794746
} |
https://arxiv.org/abs/1507.05775 | Compression of Fully-Connected Layer in Neural Network by Kronecker Product | In this paper we propose and study a technique to reduce the number of parameters and computation time in fully-connected layers of neural networks using Kronecker product, at a mild cost of the prediction quality. The technique proceeds by replacing Fully-Connected layers with so-called Kronecker Fully-Connected layers, where the weight matrices of the FC layers are approximated by linear combinations of multiple Kronecker products of smaller matrices. In particular, given a model trained on SVHN dataset, we are able to construct a new KFC model with 73\% reduction in total number of parameters, while the error only rises mildly. In contrast, using low-rank method can only achieve 35\% reduction in total number of parameters given similar quality degradation allowance. If we only compare the KFC layer with its counterpart fully-connected layer, the reduction in the number of parameters exceeds 99\%. The amount of computation is also reduced as we replace matrix product of the large matrices in FC layers with matrix products of a few smaller matrices in KFC layers. Further experiments on MNIST, SVHN and some Chinese Character recognition models also demonstrate effectiveness of our technique. | \section{Abstract}
In this paper we propose and study a technique to reduce the number of
parameters and computation time in fully-connected layers of neural networks
using Kronecker product, at a mild cost of the prediction quality. The technique
proceeds by replacing Fully-Connected layers with so-called Kronecker
Fully-Connected layers, where the weight matrices of the FC layers are
approximated by linear combinations of multiple Kronecker products of smaller
matrices. In particular, given a model trained on SVHN dataset, we are able
to construct a new KFC model with 73\% reduction in total number of parameters,
while the error only rises mildly. In contrast, using low-rank method can only achieve
35\% reduction in total number of parameters given similar quality degradation
allowance.
If we only compare the KFC layer with its counterpart fully-connected layer, the
reduction in the number of parameters exceeds 99\%. The amount of
computation is also reduced as we replace matrix product of the large matrices
in FC layers with matrix products of a few smaller matrices in KFC layers.
Further experiments on MNIST, SVHN and some Chinese Character recognition models also demonstrate effectiveness of our technique.
\section{Introduction}
Model approximation aims at reducing the number of parameters and amount of
computation of neural network models, while keeping the quality of prediction
results mostly the same.\footnote{In some circumstances, as less number
of model parameters reduce the effect of overfitting, model approximation sometimes
leads to more accurate predictions.} Model approximation is important for real
world application of neural network to satisfy the time and storage constraints
of the applications.
In general, given a neural network $\tilde{f}(\cdot;\tilde{\theta})$, we want to
construct another neural network $f(\cdot;\theta)$ within some pre-specified
resource constraint, and minimize the differences between the outputs of two
functions on the possible inputs.
An example setup is to directly minimize the differences between the
output of the two functions:
\begin{align}
\label{align:model_approximation}
\inf_{\theta} \sum_i d(f(x_i;\theta), \tilde{f}(x_i;\tilde{\theta}))
\text{,}
\end{align}
where $d$ is some distance function and $x_i$ runs over all input data.
The formulation \ref{align:model_approximation} does not give any constraints
between the structure of $f$ and $\tilde{f}$, meaning that any model can be used
to approximate another model. In practice, a structural similar model is often
used to approximate another model. In this case, model approximation may be
approached in a modular fashion {w.r.t.\ } to each layer.
\subsection{Low Rank Model Approximation}
Low rank approximation in linear regression dates back to
\cite{anderson1951estimating}. In \cite{sainath2013low, liao2013large,
xue2013restructuring, zhang2014extracting, denton2014exploiting}, low rank
approximation of fully-connected layer is used; and
\cite{jaderberg2014speeding, rigamonti2013learning, DBLP:journals/corr/LebedevGROL14} considered low rank
approximation of convolution layer. \cite{zhang2014efficient} considered
approximation of multiple layers with nonlinear activations.
We first outline the low rank approximation method below. The fully-connected
layer widely used in neural network construction may be formulated as:
\begin{align}
{\bf L}_{a} = h({\bf L}_{a-1} {\bf M}_a + {\bf b}_a) \text{,}
\end{align}
where ${\bf L}_i$ is the output of the $i$-th layer of the neural network,
${\bf M}_a$ is often referred to as ``weight term'' and ${\bf b}_a$ as ``bias term'' of
the $a$-th layer.
As the coefficients of the weight term in the fully-connected layers are
organized into matrices, it is possible to perform low-rank approximation of
these matrices to achieve an approximation of the layer, and consequently the
whole model.
Given Singular Value Decomposition of a matrix ${\bf M} = {\bf U} {\bf D} {\bf V}^*$, where ${\bf U},
{\bf V}$ are unitary matrices and ${\bf D}$ is a diagonal matrix with the diagonal made
up of singular values of ${\bf M}$, a rank-$k$ approximation of ${\bf M}\in{\mathbb F}^{m\times
n}$ is:
\begin{align}
{\bf M} \approx {\bf M}_k = \tilde{{\bf U}} \tilde{{\bf D}} \tilde{{\bf V}}^*\text{ , where }
\tilde{{\bf U}}\in{\mathbb F}^{m\times k}\text{, }\tilde{{\bf D}}\in{\mathbb F}^{k\times
k}\text{, }\tilde{{\bf V}}\in{\mathbb F}^{n\times k}
\text{,}
\end{align}
where $\tilde{{\bf U}}$ and $\tilde{{\bf V}}$ are the first $k$-columns of the ${\bf U}$ and
${\bf V}$ respectively, and $\tilde{{\bf D}}$ is a diagonal matrix made up of the largest
$k$ entries of ${\bf D}$.
In this case approximation by SVD is optimal in the sense that the following
holds \cite{horn1991topics}:
\begin{align}
{\bf M}_r = \inf_{{\bf X}} \|{\bf X} - {\bf M}\|_F \text{ s.t. } \operatorname{rank}({\bf X}) \le r \text{,}
\end{align}
and
\begin{align}
{\bf M}_r = \inf_{{\bf X}} \|{\bf X} - {\bf M}\|_2 \text{ s.t. } \operatorname{rank}({\bf X}) \le r \text{.}
\end{align}
The approximate fully connected layer induced by SVD is:
\begin{align}
{\bf Z} & = {\bf L}_{a-1} \tilde{{\bf U}} \\
{\bf L}_{a} & = h({\bf Z} {\bf D} {\bf V}^* + {\bf b}_a) \text{,}
\end{align}
In the modular representation of neural network, this means that the original
fully connected layer is now replaced by two consequent fully-connected layers.
However, the above post-processing approach only ensures getting an optimal
approximation of ${\bf M}$ under the rank constraint, while there is still no
guarantee that such an approximation is optimal {w.r.t.\ } the input data.
I.e., the optimum of the following may well be different from the rank-$r$
approximation ${\bf M}_r$ {w.r.t.\ } some given input ${\bf X}$:
\begin{align}
\inf_{{\bf Z}} \|{\bf Z}{\bf X} - {\bf M}{\bf X}\|_F \text{ s.t. } \operatorname{rank}({\bf Z}) \le r \text{.}
\end{align}
Hence it is often necessary for the resulting low-rank model
$\tilde{\mathcal{M}}$ to be trained for a few more epochs on the input,
which is also known as the ``fine-tuning'' process.
Alternatively, we note that the rank constraint can be enforced by the
following structural requirement for ${\bf X}\in{\mathbb F}^{m\times n}$:
\begin{align}
\operatorname{rank}({\bf X}) \le r \Leftrightarrow \exists {\bf A}\in{\mathbb F}^{m\times r},\,
{\bf B}\in{\mathbb F}^{r\times n}\text{ s.t. } {\bf X} = {\bf A} {\bf B}\text{.}
\end{align}
In light of this, if we want to impose a rank constraint on a fully-connected
layer $L({\bf M}, g)$ in a neural network where ${\bf M}\in{\mathbb F}^{m\times n}$, we can
replace that layer with two consecutive layers $L_1({\bf B}, g_1)$ and $L_2({\bf A},
g_2)$, where $g_1(x) = x$, $g_2 = g$, and ${\bf M} = {\bf A} {\bf B}$ where ${\bf A}\in{\mathbb F}^{m\times
r},\,{\bf B}\in{\mathbb F}^{r\times n}$, and then train the structurally constrained neural
network on the training data.
As a third method, a regularization term inducing low rank matrices may be
imposed on the weight matrices. In this case, the training of a $k$-layer model is modified
to be:
\begin{align}
\inf_{\theta} \sum_i f(x_i;\theta) + \sum_{j=1}^k r_j(\theta)
\text{,}
\end{align}
where $r$ is the regularization term. For the weight term of the FC layers,
conceptually we may use the matrix rank function as the regularization term. However, as the rank function is only well-defined for infinite-precision
numbers, nuclear norm may be used as its convex proxy
\cite{jaderberg2014speeding, Recht:2010:GMS:1958515.1958520}.
\section{Model Approximation by Kronecker Product}
Next we propose to use Kronecker product of matrices of particular shapes for
model approximation in Section~\ref{subsec:kronecker-product}. We also outline
the relationship between the Kronecker product approximation and low-rank
approximation in Section~\ref{subsec:rel-kp-lowrank}.
Below we measure the reduction in amount of computation by number of floating
point operations. In particular, we will assume the computation complexity of
two matrices of dimensions $M\times K$ and $K\times N$ to be $O(M K N)$, as many
neural network implementations \cite{Bastien-Theano-2012,
bergstra+al:2010-scipy, jia2014caffe, collobert2011torch7} have not used
algorithms of lower computation complexity for the typical inputs of the neural
networks. Our analysis is mostly immune to the ``hidden constant'' problem in
computation complexity analysis as the underlying computations of the
transformed model may also be carried out by matrix products.
\subsection{Weight Matrix Approximation by Kronecker Product}
\label{subsec:kronecker-product}
We next discuss how to use Kronecker product to approximate weight matrices of
FC layers, leading to construction of a new kind of layer which we call
Kronecker Fully-Connected layer. The idea originates from the observation that
for a matrix ${\bf M} \in {\mathbb F}^{m\times n}$ where the dimensions are not prime
\footnote{In case any of $m$ and $n$ is prime, it is possible to add some extra dummy
feature or output class to make the dimensions dividable.}, we have
approximations like:
\begin{align}
{\bf M} = {\bf M}_1 \otimes {\bf M}_2 \text{,}
\end{align}
where $m = m_1 m_2$, $n = n_1 n_2$, ${\bf M}_1\in {\mathbb F}^{m_1\times n_1}$, ${\bf M}_2\in
{\mathbb F}^{m_2\times n_2}$.
Any factors of $m$ and $n$ may be selected as $m_1$ and $n_1$ in the above
formulation. However, in a Convolutional Neural Network, the input to a FC layer
may be a tensor of order 4, which has some natural shape constraints that we will try
to leverage in \ref{subsubsec:kp-approx-fc-4d-tensor}. Otherwise, when the input is
a matrix, we do not have natural choices of $m_1$ and $n_1$. We will explore
heuristics to pick $m_1$ and $n_1$ in \ref{subsubsec:kp-approx-fc-matrix}.
\subsubsection{Kronecker product approximation for fully-connected
layer with 4D tensor input}
\label{subsubsec:kp-approx-fc-4d-tensor}
In a convolutional layer processing images, the input data ${\bf L}_{a-1}$ may be
a tensor of order 4 as ${\mathcal T}_{nchw}$ where $n=1,2,\cdots,N$ runs over $N$
different instances of data, $c=1,2,\cdots,C$ runs over $C$ channels of the
given images, $h=1,2,\cdots,H$ runs over $H$ rows of the images, and
$w=1,2,\cdots,W$ runs over $W$ columns of the images. ${\mathcal T}$ is often reshaped
into a matrix before being fed into a fully connected layer as ${\bf D}_{nj}$, where
$n=1,2,\cdots,N$ runs over the $N$ different instances of data and
$j=1,2,\cdots,CHW$ runs over the combined dimension of channel, height, and
width of images. The weights of the fully-connected layer would then be a matrix
${\bf M}_{jk}$ where $j=1,2,\cdots,CHW$ and $k=1,2,\cdots,K$ runs over output number
of channels. I.e., the layer may be written as:
\begin{align}
{\bf D} & = \text{Reshape}({\bf L}_{a-1}) \\
{\bf L}_a & = h({\bf D} {\bf M} + {\bf b})
\text{.}
\end{align}
Though the reshaping transformation from ${\mathcal T}$ to ${\bf D}$ does not incur any loss in
pixel values of data, we note that the dimension information of the tensor of
order 4 is lost in the matrix representation. As a consequence, ${\bf M}$ has $CHWK$ number
of parameters.
Due to the shape of ${\bf M}$, we may propose a few kinds of structural constraint on
${\bf M}$ by requiring ${\bf M}$ to be Kronecker product of matrices of particular shapes.
\subsubsection{Formulation I}
In this formulation, we require ${\bf M} = {\bf M}_1 \otimes {\bf M}_2 \otimes {\bf M}_3$, where
${\bf M}_1\in{\mathbb F}^{C\times K_1}, {\bf M}_2 \in {\mathbb F}^{H\times K_2}, {\bf M}_3 \in {\mathbb F}^{W\times
K_3}$, and $K=K_1 K_2 K_3$.
The number of parameters is reduced to $CK_1 + HK_2 + WK_3$. The underlying
assumption for this model is that the transformation is invariant across rows and
columns of the images.
\subsubsection{Formulation II}
In this formulation, we require ${\bf M} = {\bf M}_1 \otimes {\bf M}_2$, where
${\bf M}_1\in{\mathbb F}^{C\times K_1}, {\bf M}_2 \in {\mathbb F}^{HW\times K_2}$, and $K = K_1 K_2$.
The number of parameters is reduced to $CK_1 + HWK_2$. The underlying assumption
for this model is that the channel transformation should be decoupled from the
spatial transformation.
\subsubsection{Formulation III}
In this formulation, we require ${\bf M} = {\bf M}_1 \otimes {\bf M}_2$, where
${\bf M}_1\in{\mathbb F}^{CH\times K_1}, {\bf M}_2 \in {\mathbb F}^{W\times K_2}$, and $K = K_1 K_2$.
The number of parameters is reduced to $CHK_1 + WK_2$. The underlying
assumption for this model is that the transformation {w.r.t.\ } columns may be
decoupled.
\subsubsection{Formulation IV}
In this formulation, we require ${\bf M} = {\bf M}_1 \otimes {\bf M}_2$, where
${\bf M}_1\in{\mathbb F}^{CW\times K_1}, {\bf M}_2 \in {\mathbb F}^{H\times K_2}$, and $K = K_1 K_2$.
The number of parameters is reduced to $CWK_1 + HK_2$. The underlying
assumption for this model is that the transformation {w.r.t.\ } rows may be
decoupled.
\subsubsection{Combined Formulations}
Note that the above four formulations may be linearly combined to produce more
possible kinds of formulations. It would be a design choice with respect to
trade off between the number of parameters, amount of computation and the
particular formulation to select.
\subsubsection{Kronecker product approximation for matrix input}
\label{subsubsec:kp-approx-fc-matrix}
For fully-connected layer whose input are matrices, there does not exist natural
dimensions to adopt for the shape of smaller weight matrices in KFC. Through
experiments, we find it possible to arbitrarily pick a decomposition of input
matrix dimensions to enforce the Kronecker product structural constraint. We
will refer to this formulation as KFCM.
Concretely, when input to a fully-connected layer is ${\bf X} \in {\mathbb F}^{N\times C}$
and the weight matrix of the layer is ${\bf W}\in {\mathbb F}^{C\times K}$, we can construct
approximation of ${\bf W}$ as:
\begin{align}
\tilde{{\bf W}} = {\bf W}_1 \otimes {\bf W}_2 \approx {\bf W} \text{,}
\end{align}
where $C = C_1 C_2$, $K=K_1 K_2$, ${\bf W}_1 \in {\mathbb F}^{C_1\times K_1}$ and ${\bf W}_2 \in
{\mathbb F}^{C_2\times K_2}$.
The computation complexity will be reduced from $O(N C K)$ to $O(N C K
(\frac{1}{K_2} + \frac{1}{C_1})) = O(N C_2 C_1 K_1 + N C_2 K_1 K_2)$,
while the number of parameters will be reduced from $C K$ to $C_1 K_1 + C_2
K_2$.
Through experiments, we have found it sensible to pick $C_1 \approx \sqrt{C}$
and $K_1 \approx \sqrt{K}$.
As the choice of $C_1$ and $K_1$ above is arbitrary, we may use linear
combination of Kronecker products if matrices of different shapes for
approximation.
\begin{align}
\label{align:kp-multiple-shape}
\tilde{{\bf W}} = \sum_{j=1}^{J} {\bf W}_{1j} \otimes {\bf W}_{2j} \approx {\bf W} \text{,}
\end{align}
where ${\bf W}_{1j} \in {\mathbb F}^{C_{1j}\times K_{1j}}$ and ${\bf W}_{2j} \in
{\mathbb F}^{C_{2j}\times K_{2j}}$.
\subsection{Relationship between Kronecker Product Constraint and Low Rank
Constraint}
\label{subsec:rel-kp-lowrank}
It turns out that factorization by Kronecker product is closely related to the
low rank approximation method. In fact, approximating a matrix ${\bf M}$ with
Kronecker product ${\bf M}_1 \otimes {\bf M}_2$ of two matrices may be casted into a Nearest Kronecker product Problem:
\begin{align}
\inf_{{\bf M}_1, {\bf M}_2} \|{\bf M} - {\bf M}_1 \otimes {\bf M}_2\|_F \text{.}
\end{align}
An equivalence relation in the above problem is given in
\cite{van1993approximation, van2000ubiquitous} as:
\begin{align}
\label{align:nkp-rank1}
\arg\inf_{{\bf M}_1, {\bf M}_2} \|{\bf M} - {\bf M}_1 \otimes {\bf M}_2\|_F =
\arg \inf_{{\bf M}_1, {\bf M}_2} \|\mathcal{R}({\bf M}) - \operatorname{vec}{{\bf M}_1} (\operatorname{vec}{{\bf M}_2})^{\top}\|_F
\text{,}
\end{align}
where $\mathcal{R}({\bf M})$ is a matrix formed by a fixed reordering of entries
${\bf M}$.
Note the right-hand side of formula~\ref{align:nkp-rank1} is a rank-1
approximation of matrix $\mathcal{R}({\bf M})$, hence has a closed form solution.
However, the above approximation is only optimal {w.r.t.\ } the parameters of the
weight matrices, but not {w.r.t.\ } the prediction quality over input data.
Similarly, though there are iterative algorithms for rank-1 approximation of
tensor \cite{friedland2013best, Lathauwer:2000:BRR:354353.354405}, the
optimality of the approximation is lost once input data distribution is taken
into consideration.
Hence in practice, we only use the Kronecker Product constraint to construct KFC
layers and optimize the values of the weights through the training process on
the input data.
\subsection{Extension to Sum of Kronecker Product}
Just as low-rank approximation may be extended beyond rank-1 to arbitrary number
of ranks, one could extend the Kronecker Product approximation to Sum of
Kronecker Product approximation. Concretely, one not the following
decomposition of ${\bf M}$:
\begin{align}
{\bf M} = \sum_{i=1}^{\operatorname{rank}(\mathcal{R}({\bf M}))} {\bf A}_i \otimes {\bf B}_i
\text{.}
\end{align}
Hence it is possible to find $k$-approximations:
\begin{align}
{\bf M} \approx \sum_{i=1}^{k} {\bf A}_i \otimes {\bf B}_i
\text{.}
\end{align}
We can then generalize Formulation~I-IV in \ref{subsec:kronecker-product} to the
case of sum of Kronecker Product.
We may further combine the multiple shape formulation of
\ref{align:kp-multiple-shape} to get the general form of KFC layer:
\begin{align}
\label{align:kp-multiple-shape-multiple-sum}
{\bf M} \approx \sum_{j=1}^{J} \sum_{i=1}^{k} {\bf A}_{ij} \otimes {\bf B}_{ij}
\text{.}
\end{align}
where ${\bf A}_{ij} \in {\mathbb F}^{C_{1j}\times K_{1j}}$ and ${\bf B}_{ij} \in
{\mathbb F}^{C_{2j}\times K_{2j}}$.
\section{Empirical Evaluation of Kronecker product method}
We next empirically study the properties and efficacy of the Kronecker product method
and compare it with some other common low rank model approximation methods.
To make a fair comparison, for each dataset, we train a covolutional neural network with a fully-connected layer
as a baseline. Then we replace the fully-connected layer with different layers according to different methods and train
the new network until quality metrics stabilizes. We then compare KFC method
with low-rank method and the baseline model in terms of number of parameters and
prediction quality. We do the experiments based on implementation of KFC layers
in Theano\cite{bergstra+al:2010-scipy, Bastien-Theano-2012} framework.
As the running time may depend on particular implementation details of the KFC
and the Theano work, we do not report running time below. However, there is no
noticeable slow down in our experiments and the complexity analysis suggests
that there should be significant reduction in amount of computation.
\subsection{MNIST}
The MNIST dataset\cite{lecun1998gradient} consists of $28\times28$ grey scale images
of handwritten digits. There are 60000 training images and 10000 test images. We
select the last 10000 training images as validation set.
Our baseline model has $8$ layers and the first $6$ layers consist of four
convolutional layers and two pooling layers.
The 7th layer is the fully-connected layer and the 8th is the softmax output.
The input of the fully-connected layer is of size $32\times3\times3$, where $32$ is the number of channel and $3$ is the side length of
image patches(the mini-batch size is omitted). The output of the fully-connected layer is of size $256$, so the weight matrix
is of size $288\times256$.
CNN training is done with Adam\cite{kingma2014adam} with weight decay of 0.0001. Dropout\cite{hinton2012improving} of 0.5 is used
on the fully-connected layer and KFC layer. $y=|\tanh{x}|$ is used as activation
function. Initial learning rate is $1e-4$ for Adam.
Test results are listed in Table~\ref{tab:mnist_approximation}. The number of layer parameters means the number of parameters of the fully-connected
layer or its counterpart layer(s). The number of model parameters is the number of the parameters of the whole model. The test error is the min-validation
model's test error.
In Cut-96 method, we use 96 output neurons instead of 256 in fully-connected layer. In the LowRank-96 method, we replace the fully-connected layer with two fully-connected layer where the first FC layer output size is 96 and the second FC layer output size is 256. In the KFC-II method, we replace the fully-connected layer with KFC layer using
formulation II with $K_1=64$ and $K_2=4$. In the KFC-Combined method, we replace the fully-connected layer with KFC layer and linear combined the formulation II, III and IV($K_1=64, K_2=4$ in formulation II, $K_1=128, K_2=2$ in formulation III and IV).
\begin{table}[!ht] \centering \small
\caption{Comparison of using Low-Rank method and using KFC layers on MNIST
dataset}
\begin{center}
\begin{tabular}{p{2cm} p{5cm} p{5cm} p{1.5cm}}
\toprule Methods & \# of Layer Params(\%Reduction) & \# of Model Params(\%Reduction) & Test Error \\
\midrule Baseline & 74.0K & 99.5K & 0.51\% \\
\midrule Cut-96 & 27.8K(62.5\%) & 51.7K(48.1\%) & 0.58\% \\
\midrule LowRank-96 & 52.6K(39.0\%) & 78.1K(21.6\%) & 0.54\% \\
\midrule KFC-II & 2.1K(97.2\%) & 27.7K(72.2\%) & 0.76\%\\
\midrule KFC-Combined & 27.0K(63.51\%) & 52.5K(47.2\%) & 0.57\%\\
\hline
\end{tabular}
\end{center} \label{tab:mnist_approximation}
\end{table}
\subsection{Street View House Numbers}
The SVHN dataset\cite{netzer2011reading} is a real-world digit recognition dataset
consisting of photos of house numbers in Google Street View images. The
dataset comes in two formats and we consider the second format: 32-by-32 colored images centered
around a single character. There are 73257 digits for training, 26032 digits
for testing, and 531131 less difficult samples which can be used as extra training
data. To build a validation set, we randomly select 400 images per class from
training set and 200 images per class from extra training set
as \cite{sermanet2012convolutional, goodfellow2013maxout} did.
Here we use a similar but larger neural network as used in MNIST to be the baseline. The input of the fully-connected layer
is of size $256\times5\times5$. The fully-connected layer has $256$ output neurons. Other implementation details are not changed.
Test results are listed in Table~\ref{tab:svhn_approximation}. In the Cut-$N$ method, we use $N$ output neurons instead of 256 in fully-connected layer. In the LowRank-$N$ method, we replace the fully-connected layer with two fully-connected layer where the first FC layer output size is $N$ and the second FC layer output size is 256. In the KFC-II method, we replace the fully-connected layer with KFC layer using formulation II with $K_1=64$ and $K_2=4$. In the KFC-Combined method, we replace the fully-connected layer with KFC layer and linear combined the formulation II, III and IV($K_1=64, K_2=4$ in formulation II, $K_1=128, K_2=2$ in formulation III and IV). In the KFC-Rank$N$ method, we use KFC formulation II with $K_1=64,K_2=2$ and extend it to rank $N$ with as described above.
\begin{table}[!ht] \centering \small
\caption{Comparison of using Low-Rank method and using KFC layers on SVHN dataset}
\begin{center}
\begin{tabular}{p{2cm} p{5cm} p{5cm} p{1.5cm}}
\toprule Methods & \# of Layer Params(\%Reduction) & \# of Model Params(\%Reduction) & Test Error \\
\midrule Baseline & 1.64M & 2.20M & 2.57\% \\
\midrule Cut-128 & 0.82M(50.0\%) & 1.38M(37.3\%) & 2.79\% \\
\midrule Cut-64 & 0.41(25.0\%) & 0.97(55.9\%) & 3.19\% \\
\midrule LowRank-128 & 0.85M(48.2\%) & 1.42M(35.7\%) & 3.02\% \\
\midrule LowRank-64 & 0.43M(73.7\%) & 0.99M(55.1\%) & 3.67\% \\
\midrule KFC-II & 0.016M(99.0\%) & 0.58M(73.7\%) & 3.33\% \\
\midrule KFC-Combined & 0.34M(79.3\%) & 0.91M(58.6\%) & 2.60\% \\
\midrule KFC-Rank10 & 0.17M(89.6\%) & 0.73M(66.8\%) & 3.19\% \\
\hline
\end{tabular}
\end{center} \label{tab:svhn_approximation}
\end{table}
\subsection{Chinese Character Recognition}
We also evaluate application of KFC to a Chinese character recognition model.
Our experiments are done on a private dataset for the moment and may extend to
other established Chinese character recognition datasets like
HCL2000(\cite{zhang2009hcl2000}) and CASIA-HWDB(\cite{liu2013online}).
For this task we also use a convolutional neural network. The distinguishing
feature of the neural network is that following the convolution and pooling
layers, it has two FC layers, one with 1536 hidden size, and the other with more
than 6000 hidden size.
The two FC layers happen to be different type. The 1st FC layer accepts tensor
as input and the 2nd FC layer accepts matrix as input. We apply KFC-I
formulation to 1st FC and KFCM to 2nd FC.
\begin{table}[!ht] \centering \small
\caption{Effect of using KFC layers on a Chinese
recognition dataset}
\begin{center}
\begin{tabular}{p{2cm} p{2cm} p{2cm} p{2cm} p{1.5cm}}
\toprule Methods & \%Reduction of 1st FC Layer Params & \%Reduction of 2nd
FC Layer Params & \%Reduction of Total Params & Test Error \\
\midrule Baseline & 0\% & 0\% & 0\% & 10.6\% \\
\midrule KFC-II & 99.3\% & 0\% & 36.0\% & 11.6\% \\
\midrule KFC-KFCM-rank1 & 98.7\% & 99.9\% & 94.5\% & 21.8\% \\
\midrule KFC-KFCM-rank-10 & 93.3\% & 99.1\% & 91.8\% & 13.0\% \\
\hline
\end{tabular}
\end{center} \label{tab:svhn_approximation}
\end{table}
It can be seen KFC can significantly reduce the number of parameters. However,
in case of ``KFC and KFCM (rank=1)'', this also leads to serious degradation of
prediction quality. However, by increasing the rank from 1 to 10, we are able to
recover most of the lost prediction quality. Nevertheless, the rank-10 model is
still very small compared to the baseline model.
\section{Conclusion and Future Work}
In this paper, we propose and study methods for approximating the weight
matrices of fully-connected layers with sums of Kronecker product of smaller
matrices, resulting in a new type of layer which we call Kronecker
Fully-Connected layer.
We consider both the cases when input to the fully-connected layer is a tensor
of order 4 and when the input is a matrix. We have found that using the KFC
layer can significantly reduce the number of parameters and amount of
computation in experiments on MNIST, SVHN and Chinese character recognition.
As future work, we note that when weight parameters of a convolutional layer is
a tensor of order 4 as $\mathcal{T}\in {\mathbb F}^{K\times C\times H\times W}$, it can
be represented as a collection of $H\times W$ matrices $\mathbf{T}_{hw}$. We
can then approximate each matrix by Kronecker products as $\mathbf{T}_{hw} =
{\bf A}_{hw}\otimes{\bf B}_{hw}$ following KFCM formulation, and apply the other
techniques outlined in this paper. It is also noted that the Kronecker product
technique may also be applied to other neural network architectures like
Recurrent Neural Network, for example approximating transition matrices with
linear combination of Kronecker products.
\bibliographystyle{spmpsci}
| {
"timestamp": "2015-07-23T02:08:34",
"yymm": "1507",
"arxiv_id": "1507.05775",
"language": "en",
"url": "https://arxiv.org/abs/1507.05775",
"abstract": "In this paper we propose and study a technique to reduce the number of parameters and computation time in fully-connected layers of neural networks using Kronecker product, at a mild cost of the prediction quality. The technique proceeds by replacing Fully-Connected layers with so-called Kronecker Fully-Connected layers, where the weight matrices of the FC layers are approximated by linear combinations of multiple Kronecker products of smaller matrices. In particular, given a model trained on SVHN dataset, we are able to construct a new KFC model with 73\\% reduction in total number of parameters, while the error only rises mildly. In contrast, using low-rank method can only achieve 35\\% reduction in total number of parameters given similar quality degradation allowance. If we only compare the KFC layer with its counterpart fully-connected layer, the reduction in the number of parameters exceeds 99\\%. The amount of computation is also reduced as we replace matrix product of the large matrices in FC layers with matrix products of a few smaller matrices in KFC layers. Further experiments on MNIST, SVHN and some Chinese Character recognition models also demonstrate effectiveness of our technique.",
"subjects": "Neural and Evolutionary Computing (cs.NE); Computer Vision and Pattern Recognition (cs.CV); Machine Learning (cs.LG)",
"title": "Compression of Fully-Connected Layer in Neural Network by Kronecker Product",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9770226260757066,
"lm_q2_score": 0.7248702761768248,
"lm_q1q2_score": 0.708214660794504
} |
https://arxiv.org/abs/1708.07761 | Cubulated moves for 2-knots | In this paper, we prove that given two cubical links of dimension two in ${\mathbb R}^4$, they are isotopic if and only if one can pass from one to the other by a finite sequence of cubulated moves. These moves are analogous to the Reidemeister and Roseman moves for classical tame knots of dimension one and two, respectively. | \section{Introduction}
In \cite{BHV} it was shown that any smooth knot ${K}^n:{\mathbb S}^n\hookrightarrow{\mathbb R}^{n+2}$ can be deformed isotopically into the $n$-skeleton
of the canonical cubulation of ${\mathbb R}^{n+2}$ and this isotopic copy is called \emph{cubical $n$-knot}. In particular, every smooth 1-knot
$\mathbb S^1\subset{\mathbb R}^3$ is isotopic to a cubical knot.
\noindent There are two types of elementary ``cubulated moves''. The first one (M1) is obtained by dividing each cube of the original cubulation of $\mathbb{R}^3$ into $m^3$ cubes,
which means that each edge of the knot is subdivided into $m$ equal segments. The second one (M2) consists in exchanging a connected set of edges in a face of the
cubulation
with the complementary edges in that face. If two cubical knots $K_1$ and $K_2$ are such that we can convert $K_1$ into $K_2$ using a finite sequence of cubulated moves then we say that they are \emph{equivalent via cubulated moves} and is denoted by $K_1\overset{c}\sim K_2$.
\noindent In \cite{HVV} it was proved the following:
\begin{case}\label{case}
Given two cubical knots $K_1$ and $K_2$ in $\mathbb{R}^{3}$, $K_1\cong K_2 \cong \mathbb{S}^{1}$,
they are isotopic if and only if $K_1$ is equivalent to $K_2$ by a finite sequence of cubulated moves; {\emph{i.e.}}, $K_1\sim K_2 \iff K_1\overset{c}\sim K_2$.
\end{case}
\noindent Theorem 0 is analogous to the Reidemeister moves of classical tame knots for cubical knots.
\begin{figure}[h]
\begin{center}
\includegraphics[height=4cm]{cubtre.jpg}
\end{center}
\caption{\sl Two isotopic knots are equivalent via cubulated moves.}
\label{M1}
\end{figure}
\noindent Notice that cubulated moves can be extended to cubical 2-knots in a natural way:
The first one (M1) is obtained by dividing each hypercube of the original cubulation of
$\mathbb{R}^4$ into $m^4$ hypercubes, and the second one (M2) consists in exchanging a connected set of squared faces homeomorphic to a
disk $\mathbb{D}^2$ in a cube of the cubulation (or a subdivision of the cubulation) with the complementary faces in that cube.
\noindent The study of 2-knots in $\mathbb{R}^4$ has been considered by various authors, for instance in
\cite{CRS}, \cite{CS}, \cite{K} and \cite{Roseman}.
\noindent Our goal is to extend the Theorem 0 for cubical knots of dimension two:
\begin{main}\label{main}
Given two cubical 2-knots $K^2_1$ and $K^2_2$ in $\mathbb{R}^{4}$,
then they are isotopic if and only if $K^2_1$ is equivalent to $K^2_2$ by cubulated moves; {\emph{i.e.}}, $$K^2_1\sim K^2_2 \iff K^2_1\overset{c}\sim K^2_2.$$
\end{main}
\section{Preliminaries}
\subsection{Cubulations of $\mathbb{R}^{4}$}
\noi The regular hypercubic honeycomb whose Schl\"afli symbol is $\{4,3,3,4\}$, is called a {\it cubulation} of $\mathbb{R}^{4}$. In other words, a cubulation of $\mathbb{R}^{4}$ is a decomposition into a collection of right-angled $4$-dimensional
hypercubes $\{4,3,3\}$ called $\textit{cells}$ such that any two are
either disjoint or meet in one common $k-$face of some dimension $k$. This provides $\mathbb{R}^{4}$ with the structure of a cubical
complex whose category is similar to the simplicial category PL.
\noi The combinatorial structure of the regular Euclidean honeycomb $\{4,3,3,4\}$ is the following: around each vertex there are 8 edges, 24 squares, 32 cubes and 16 hypercubes. Around each edge there are 6 squares, 12 cubes and 8 hypercubes. Around each square there are 4 cubes and 4 hypercubes. Finally around each cube there are 2 hypercubes.
\begin{figure}
\centering
\includegraphics[scale=0.3]{Cubichoneycomb.pdf}
\begin{center}
{{\bf Figure 1.} The 3-dimensional cubical kaleidoscopic honeycomb $\{4,3,4\}$.
This figure is courtesy of Roice Nelson \cite{RN}.}
\end{center}
\end{figure}
\noindent The canonical hypercubic honeycomb $\cal C$ of $\mathbb{R}^{4}$ is its decomposition into hypercubes which are the images of the unit hypercube:
$$\{4,3,3\}=I^4=[0,1]^4=\{(x_{1},x_{2},x_{3},x_{4})\in \mathbb{R}^4 \,|\,0\leq x_{i}\leq 1\}$$ by translations by vectors with integer coefficients. Then all vertices of $\cal C$ have integer coordinates.
\noindent Any regular hypercubic honeycomb $\{4,3,3,4\}$ or cubulation of $\mathbb{R}^{4}$ is obtained from the canonical cubulation by applying a conformal transformation to the canonical cubulation. Remember that a conformal transformation is of the form
$x\mapsto{\lambda{A(x)}+a},$ where $\lambda\neq0,\,\,a\in \mathbb{R}^{4},\,\,\,A\in{SO(4)}$.
\begin{definition} The $k${\it-skeleton} of $\cal C$, denoted by $\mathcal{S}^k$, consists of the union of the $k$-skeletons of the hypercubes in $\cal C$,
{\it i.e.,} the union of all cubes of dimension $k$ contained in the faces of the $4$-cubes in $\cal C$.
We will call the 2-skeleton $\mathcal{S}^2$ of $\cal C$ the {\it canonical scaffolding} of $\mathbb{R}^{4}$.
\end{definition}
\noindent Notice that all the previous definitions can be extended in a natural way to $\mathbb{R}^{n+2}$.
\subsection{Cubical and smooth 2-knots}
\noindent In classical knot theory, a subset $K$ of a space $X$ is a {\it knot} if $K$ is homeomorphic to a $p$-dimensional sphere
$\mathbb{S}^{p}$ embedded in either the Euclidean $n$-space $\mathbb{R}^{n}$ or the $n$-sphere $\mathbb{S}^{n}=\mathbb{R}^{n}\cup\{\infty\}$, where $p<n$. Two knots $K_1$, $K_2$ are {\it equivalent} or {\it isotopic} if there is a homeomorphism $h:X\hookrightarrow X$ such that $h(K_1)=K_2$;
in other words $(X,K_1)\cong (X,K_2)$. However, a knot $K$ is sometimes defined to be an embedding
$K:\mathbb{S}^{p}\hookrightarrow\mathbb{S}^{n}\cong\mathbb{R}^{n} \cup \{\infty \}$ (see \cite{mazur}, \cite{rolfsen}).
We shall also find this convenient at times and will use the same symbol to
denote either the map $K$ or its image $K(\mathbb{S}^{p})$ in $\mathbb{S}^{n}$.
\begin{definition}
Let $K^2$ be a $2$-dimensional knot in $\mathbb{R}^{4}$. If $K^2$ is contained in $\mathcal{S}^2$, we say that $K^{2}$ is a \emph{cubical knot}.
\end{definition}
\begin{definition} If $K^2$ is a smooth knot and $\widehat{K^2}$ denotes a cubical knot which is isotopic to $K^2$, {\it i.e.,} $K^2 \sim \widehat{K^2}$, we say that $\widehat{K^2}$ is a \emph{cubical diagram} of the knot $K^2$.
\end{definition}
\noi Given two parametrized $2$-dimensional smooth knots $K^2_1$, $K^2_2:\mathbb{S}^2\hookrightarrow\mathbb{R}^{4}$ we say they are \textit{smoothly isotopic} if there exists a
smooth isotopy $H:\mathbb{S}^{2}\times\mathbb{R}\rightarrow\mathbb{R}^{4}$ such that
$$
H(x,t)= \left\{ \begin{array}{ll}
K^2_1(x) & \mbox{if $t\leq1$},\\
K^2_2(x) & \mbox{if $t\geq 2$},
\end{array} \right.
$$
and $H(\ \cdot \ ,t)$ is an embedding of $\mathbb{S}^2$ for all $t\in\mathbb{R}$.
\begin{definition}\label{cylinder}
We will say
$J^3=\{(H(x,t),t)\in\mathbb{R}^{5}\,|\,x\in\mathbb{S}^{2},\,\,t\in\mathbb{R}\}$ is the {\emph {isotopic cylinder}} of $K^2_1$ and $K^2_2$.
\end{definition}
\noi Note that $J^3$ is a smooth noncompact submanifold of codimension two in $\mathbb{R}^{5}$.
\begin{definition}
Let $p:\mathbb{R}^{5}\hookrightarrow\mathbb{R}$ be the projection onto the last coordinate. Let $M$ be a connected subset of $\mathbb{R}^{5}$ such that $p^{-1}(c)\cap M$ (or $p|_M^{-1}(c)$) is connected for all $c\in\mathbb{R}$,
we say that $M$ is {\emph{sliced by connected level sets of $p$}}.
\end{definition}
\noindent Observe that there is no restriction on the dimension of $M$.
\noindent In \cite{HVV} it was proved the following result.
\begin{theo}\label{trace}
The isotopic cylinder $J^3$ can be cubulated. In other words, there exists an isotopic copy $\widehat{J^3}$ of $J^3$ contained
in the $3$-skeleton of the canonical cubulation of $\mathbb{R}^{5}$.
Moreover $\widehat{J^3}$ can be chosen to be sliced by connected level sets of $p$.
\end{theo}
\noi Consider $\widehat{J^3}$ as above; so $\widehat{J^3}$ is a cubical $3$-manifold and there exist integer numbers
$m_1$ and $m_2$ and cubical knots $\widehat{K^2_1}$ and $\widehat{K^2_2}$ isotopic to $K^2_{1}$ and $K^2_{2}$, respectively;
such that $p|_{\widehat{J^3}}^{-1}(t)= \widehat{K^2_1}$ for all $t\leq m_1$ and
$p|_{\widehat{J^3}}^{-1}(t)= \widehat{K^2_2}$ for all $t\geq m_2$.
\subsection{Cubulated moves}
\begin{definition} The following are the allowed {\emph{cubulated moves}}:
\begin{description}
\item[{\bf{M1}}] {\emph{Subdivision:}} Given an integer $m>1$, consider the subcubulation ${\cal{C}}_m$ of $\cal{C}$ by subdividing each $k$-dimensional cell of $\cal{C}$ in
$m^k$ congruent $k$-cells in ${\cal{C}}_m$, in particular each hypercube in $\cal{C}$ is subdivided in $m^4$ congruent hypercubes in ${\cal{C}}_m$. Moreover as a cubical complex, each $k$-dimensional face of the 2-knot $K^2$ is subdivided into $m^k$ congruent
$k$-faces. Since ${\cal{C}}\subset {\cal{C}}_{m}$, then
$K^2$ is contained in the scaffolding ${\cal{S}}^2_m$ (the $2$-skeleton) of ${\cal{C}}_m$.
\item[{\bf{M2}}] {\emph{Face Boundary Moves:}} Suppose that $K^2$ is contained in some subcubulation ${\cal{C}}_m$ of
the canonical cubulation ${\cal{C}}$ of $\mathbb{R}^{4}$.
Let $Q^4\in {\cal{C}}_m$ be a $4$-cube such that
$A^2=K^2\cap Q^4$ contains a $2$-face. We can assume, up to applying the elementary $(M1)$-move if necessary, that $A^2$ consists of either one,
two, or three
squares such that it is a connected surface and it is contained in the boundary of a 3-cube
$F^3\subset Q^4$; in other words $A^2$ is a cubical disk contained in the boundary of
$F^3$. The boundary $\partial F^3$ is divided by $\partial A^2$ into two cubulated surfaces, one of them is $A^2$ and we denote the other by $B^2$. Observe
that both cubulated surfaces share a common circle boundary. The face boundary move consists in replacing $A^2$ by $B^2$ (see Figure \ref{M1}). There are three types of face boundary moves depending of the number of 2-faces in each $A^2$ and $B^2$. If $A^2$ has $p$ squares then $B^2$ has $6-p$ squares.
\begin{figure}[h]
\begin{center}
\includegraphics[height=5cm]{2moves.jpg}
\end{center}
\caption{\sl The three types of face boundary moves.}
\label{M1}
\end{figure}
\end{description}
\end{definition}
\begin{rem}\label{equivalencia}
It can be easily shown that the $(M2)$-move can be extended to an ambient isotopy of $\mathbb{R}^4$ .
\end{rem}
\begin{definition}
Given two cubical $2$-knots $K^2_1$ and $K^2_2$ in $\mathbb{R}^{4}$. We say that
$K^2_1$ is {\emph{equivalent}} to $K^2_2$ {\emph{by cubulated moves}}, denoted by
$K^2_1\overset{c}\sim K^2_2$, if we can transform $K^2_1$ into $K^2_2$ by a finite number
of cubulated moves.
\end{definition}
\section{Main theorem}
\noi We are now ready to prove our main theorem. Notice that this proof is a generalization of the one given in \cite{HVV} for the one dimensional case.
\begin{main}\label{main}
Given two cubical 2-knots $K^2_1$ and $K^2_2$ in $\mathbb{R}^{4}$,
then they are isotopic if and only if $K^2_1$ is equivalent to $K^2_2$ by cubulated moves; {\emph{i.e.}}, $$K^2_1\sim K^2_2 \iff K^2_1\overset{c}\sim K^2_2.$$
\end{main}
\begin{proof} \noi First, note that if $K^2_1$ and $K^2_2$ are equivalent by cubulated moves, then
these 2-knots are isotopic by Remark \ref{equivalencia}. Hence, what remains to be proved is
that two cubical 2-knots that are isotopic must also be equivalent by cubulated
moves.
\noi Our strategy is like the one used for cubic knots of dimension one (see \cite{HVV}). First, for $i \in \{1,2\}$, we will smooth each $K^2_i$ to obtain $\widetilde{K^2_i}$, and then cubulate these two 2-knots to obtain $\widehat{K^2_i}$ in such a way that
A) $K^2_i \overset{c}\sim \widehat{K^2_i} \quad$ and
B) $\widehat{K^2_1}\overset{c}\sim \widehat{K^2_2}$.
\noindent In \cite{DHVV} was proved the following.
\noindent{\bf Theorem.} \emph{Any compact, closed, oriented, cubical surface $M^2$ in $\mathbb{R}^{4}$ is smoothable. More precisely, $M^2$
admits a global continuous transverse field of 2-planes and therefore by a theorem of J. H. C. Whitehead there is an arbitrarily small topological
isotopy that moves $M^2$ onto a smooth surface $\widetilde{M^2}$ in $\mathbb{R}^4$ (see \cite{pugh}, \cite{whitehead})}.
\noindent As a consequence, we have that given a cubical knot $K^2$, there exists a smooth knot $\widetilde{K^2}$ isotopic to
$K^2$ such that $\widetilde{K^2}$ is ${\cal{C}}^0$-arbitrarily close to $K^2$.
\noi Let $J^3$ be the isotopic cylinder (see Definition \ref{cylinder}) of $\widetilde{K^2_1}$ and $\widetilde{K^2_2}$.
Then $J^3$ is a smooth submanifold of codimension two in $\mathbb{R}^{5}$.
By Theorem \ref{trace}, there exists an isotopic copy of $J^3$, say $\widehat{J^3}$, contained in the $3$-skeleton of the canonical
cubulation $\cal{C}$ of $\mathbb{R}^{5}$. Recall that $\widehat{J^3}$ is sliced by connected level sets of $p$ and there exist integer numbers
$m_1$ and $m_2$ such that $p^{-1}(t)\cap \widehat{J^3} =\widehat{K^2_1}$ for all $t\leq m_1$ and
$p^{-1}(t)\cap \widehat{J^3} = \widehat{K^2_2}$ for all $t\geq m_2$, where $\widehat{K^2_1}$ and $\widehat{K^2_2}$ are cubical knots which are isotopic to $\widetilde{K^2_{1}}$ and $\widetilde{K^2_{2}}$, respectively.
\noindent Now, we will use the following two results which will be proved in Sections \ref{smooth} and \ref{cubic}, respectively.
\begin{LA}\label{smooth1}
Given a cubical 2-knot $K^2$, we can choose a small cubulation ${\cal{C}}_{m}$ fine enough so that
$N^4_{K}=\{Q^4\in {\cal{C}}_{m}\,|\,Q^4\cap K^2\neq\emptyset\}$ is a closed tubular neighborhood of $K^2$ and
$Q^4\cap K^2$ is equal to either a vertex, one square, two squares sharing an edge (neighboring 2-faces) or three neighboring squares (two by two neighboring 2-faces).
We can also choose $\widetilde{K^2}$ isotopic to $K^2$ such that $\widetilde{K^2}$ is
${\cal{C}}^0$-arbitrarily close to $K^2$ and $\widetilde{K^2}\subset \mbox{Int}(N^4_{K})$. Let
$\widehat{K^2}$ be an isotopic copy of $\widetilde{K^2}$ contained in $\partial N^4_{K}$, then $K^2\overset{c}\sim \widehat{K^2}$; {\emph{i.e.}}, we can
go from $K^2$ to $\widehat{K^2}$ by a finite sequence of cubulated moves.
\end{LA}
\begin{rem}\label{conditions}
\begin{enumerate}
\item The existence of $\widehat{K^2}$ is proved on Theorem 3.1 in \cite{BHV} (see also the proof of Theorem \ref{trace}).
\item We may assume, using a subdivision move if necessary, that the intersection $Q^4\cap K^2$ is equal to either a vertex, one square, two squares sharing an edge (neighboring 2-faces) or three neighboring squares (two by two neighboring 2-faces).
\end{enumerate}
\end{rem}
\begin{LB}\label{cubemoves}
Given two cubical knots $K^{2}_1$ and $K^{2}_2$, we obtain $\widehat{K^{2}_{1}}$ and $\widehat{K^{2}_{2}}$ as in Lemma A. Then
there exists a finite sequence
of cubulated moves that carries $\widehat{K^{2}_{1}}$ into $\widehat{K^{2}_{2}}$. In other words, $\widehat{K^{2}_{1}}$ is equivalent to $\widehat{K^{2}_{2}}$ by cubulated moves: $\widehat{K^{2}_{1}}\overset{c}\sim \widehat{K^{2}_{2}}$
\end{LB}
\noindent If we go back to the proof of the Theorem 1, we have by Lemma A that there exist a finite sequence of cubulated moves that carries $K^{2}_{1}$ into $\widehat{K^{2}_{1}}$ and also a finite
sequence of cubulated moves that transforms $K^{2}_{2}$ into $\widehat{K^{2}_{2}}$. By Lemma B,
there exists a finite sequence of cubulated moves that converts $\widehat{K^{2}_1}$ into $\widehat{K^{2}_2}$. As a consequence there exists a finite sequence
of cubulated moves that carries $K^{2}_{1}$ onto ${K}^{2}_{2}$.
\end{proof}
\subsection{$K^{2}_i$ is equivalent to $\widehat{K^{2}_i}$ by cubulated moves}\label{smooth}
\noindent Our goal is to prove Lemma A.
\begin{LA}
Given a cubical 2-knot $K^2$, we can choose a small cubulation ${\cal{C}}_{m}$ fine enough so that
$N^4_{K}=\{Q^4\in {\cal{C}}_{m}\,|\,Q^4\cap K^2\neq\emptyset\}$ is a closed tubular neighborhood of $K^2$ and
$Q^4\cap K^2$ is equal to either a vertex, one square, two squares sharing an edge (neighboring 2-faces) or three neighboring squares (two by two neighboring 2-faces).
We can also choose $\widetilde{K^2}$ isotopic to $K^2$ such that $\widetilde{K^2}$ is
${\cal{C}}^0$-arbitrarily close to $K^2$ and $\widetilde{K^2}\subset \mbox{Int}(N^4_{K})$. Let
$\widehat{K^2}$ be an isotopic copy of $\widetilde{K^2}$ contained in $\partial N^4_{K}$, then $K^2\overset{c}\sim \widehat{K^2}$; {\emph{i.e.}}, we can
go from $K^2$ to $\widehat{K^2}$ by a finite sequence of cubulated moves.
\end{LA}
\noindent \emph{Proof.}
The knots $K^2$ and $\widehat{K^2}$ are isotopic and both are contained in the 2-skeleton of $N^4_{K}$; however $K^2\subset \mbox{Int}(N^4_{K})$ and $\widehat{K^2}\subset\partial N^4_{K}$. Our goal is to construct a connected 3-manifold ${\cal {M}}^3$ such that it will be contained in the 3-skeleton of $N^4_{K}$ and its boundary
will consist of two connected components, namely $K^2$ and $\widehat{K^2}$.
\noindent Let $B^4=\{Q^4\subset N^4_{K} \,|\,\,\,Q^4\cap \widehat{K^2}\neq\emptyset\}$.
\noindent Since all hypercubes in $N^4_{K}$ intersect $K^2$, $B^4$ consists
of a finite number of hypercubes, say $m$, such that all of them also intersect $\widehat{K^2}$.
By orienting $\widehat{K^2}$ we can enumerate the cubes in $B^4$ in such a way that
consecutive numbers belong to neighboring hypercubes (hypercubes sharing a common 3-face).
To construct the 3-manifold ${\cal {M}}^3$, we will look at all possible cases of $Q^4_j\in B^4$
and find pieces $M_j^3$ which will be consist of the union of some 3-faces of $Q^4_j$ such that they are two by two neighboring faces. The union of all $M_j^3$ will be ${\cal {M}}^3$.
Notice that the boundary of these $M_j^3$'s will
intersect both $K^2$ and $\widehat{K^2}$, hence 3-faces corresponding to neighboring 4-cubes will share a common face.
\noindent In order to describe the cubes $M_j^3$'s in this proof, we will use the following notation for the 3-faces of a given hypercube $Q^4_j$.
\begin{figure}[h]
\begin{center}
\includegraphics[height=4.6cm]{notationM.jpg}
\caption{\sl Notation for the cubes in a hypercube $Q^4_j$.}
\end{center}
\label{not}
\end{figure}
\noindent Next, we will construct ${\cal {M}}^3$ considering
all possible cases of both $K^2\cap Q^4_j$ and $\widehat{K^2}\cap Q^4_j$.
\noindent {\bf Claim 1:} Let $Q^4_j\in B^4$ and suppose that $K^2\cap Q^4_j$ consists of three squares (two by two neighboring 2-faces).
Then, using face boundary moves if necessary, we can assume that
$\widehat{K^2}\cap Q^4_j$ is a connected 2-disk consisting of the union of at most three squares.
\noindent {\it Proof of Claim 1.} Suppose that $K^2\cap Q^4_j$ consists of three neighboring 2-faces $F_1^2$, $F_2^2$ and $F_3^2$. Observe that
$F_1^2\cup F_2^2\cup F_3^2$
intersects fifteen distinct 2-faces of $Q^4_j$, hence $\widehat{K^2}\cap Q^4_j$ must be contained in the remaining six 2-faces. Notice that
these six 2-faces are contained in a 3-face $C^3$ of $Q^4_j$ and applying (M2)-moves if necessary, we can assume that at most three 2-faces of
$\widehat{K^2}\cap Q^4_j$ lie on $C^3$.
Therefore $K^2\cap Q^4_j$ consists of at most three 2-faces. See Figure \ref{3s}.1.
\begin{figure}[h]
\begin{center}
\includegraphics[height=2.5cm]{3s.jpg}
\end{center}
\caption{\sl $K^2\cap Q^4_j$ consists of three 2-faces.}
\label{3s}
\end{figure}
\noindent This proves claim 1. $\square$
\noindent {\bf Case 1.} Suppose that $K^2\cap Q^4_j$ consists of three neighboring 2-faces $F_1^2$, $F_2^2$ and $F_3^2$. By the above claim, we have that
$\widehat{K^2}\cap Q^4_j$ is a connected 2-disk consisting of the union of at most three 2-faces; therefore $\widehat{K^2}\cap Q^4_j$
consists of at most three neighboring faces $E_1^2$, $E_2^2$ and $E_3^2$ such that $E_i^2$ is parallel
to $F_i^2$ ($i=1,2,3$). Then the only possibility up to a face boundary moves $(M2)$ is shown in Figure \ref{3s}.2. Each $K^2\cap Q^4_j$ and $\widehat{K^2}\cap Q^4_j$ consist of three neighboring 2-faces.
Thus $M_j^3=M_3^3\cup M_{-1}^3\cup M_{-4}^3$.
\noindent {\bf Claim 2:} \emph{Let $Q^4_j\in B^4$ and suppose that $K^2\cap Q^4_j$ consists of two squares sharing an edge (neighboring 2-faces). Then,
using face boundary moves if necessary, we can assume that $\widehat{K^2}\cap Q^4_j$ is a connected 2-disk consisting of the union of at most four faces.}
\noindent {\it Proof of Claim 2.} Suppose that $K^2\cap Q^4_j$ consists of two neighboring 2-faces $F_1$ and $F_2$. Observe that the union
$F_1\cup F_2$ intersects
fifteen distinct 2-faces of $Q^4_j$, then $\widehat{K^2}\cap Q^4_j$ must be contained in the remaining seven 2-faces.
Notice that six of these seven 2-faces are contained in a 3-face $C^3$ of $Q^4_j$ and
applying (M2)-moves if necessary, we can assume that three 2-faces of $\widehat{K^2}\cap Q^4_j$ lie on $C^3$. See Figure \ref{2s}.1.
Therefore $K^2\cap Q^4_j$ consists of at most four 2-faces.
\begin{figure}[h]
\begin{center}
\includegraphics[height=5cm]{2s.jpg}
\end{center}
\caption{\sl $K^2\cap Q^4_j$ consists of two 2-faces.}
\label{2s}
\end{figure}
\noindent This proves claim 2. $\square$
\noindent {\bf Case 2.} Suppose that $K^2\cap Q^4_j=F_1^2\cup F_2^2$ where $F_1^2$ and $F_2^2$ are two neighboring 2-faces. Since $\widehat{K^2}\cap Q^4_j$ is a connected 2-disk consisting of the union of at most four 2-faces, and $K^2\cap \widehat{K^2}=\emptyset$; we have that $\widehat{K^2}\cap Q^4_j$ is contained in the union of seven 2-faces
of $Q^4_j$, such that $F_1^2$, $F_2^2,\,\ldots,\,F_6^2$ belong to a 3-face $C^3$ and $F_7^2$ belongs to the opposite 3-face $\bar{C^3}$. See Figure \ref{2s}.1. Since $\widehat{K^2}$ intersects some neighbor cubes of $Q^4_j$, we have three possibilities:
\begin{description}
\item [$(a)$] Suppose that $\widehat{K^2}\cap Q^4_j$ consists of the union of four squares. Hence $F_7^2$ belongs to $\widehat{K^2}\cap Q^4_j$. Thus, there exist three 3-faces $M_1^3$, $M_{-1}^3$ and $M_{-4}^3$ that
contained two 2-faces of $(K ^2\cup \widehat{K^2})\cap Q^4_j$ (see Figure \ref{2s}.2).
Then $M_j^3=M_1^3\cup M_{-1}^3\cup M_{-4}^3$.
\item [$(b)$] Suppose that $\widehat{K^2}\cap Q^4_j$ consists of the union of three neighboring faces. Then considering all the possibilities
satisfying that $K^2\cap Q^4_j$ can be extended, we have that $\widehat{K^2}\cap Q^4_j$ consists of $F_1^2$, $F_2^2$ and $F_3^2$ such that
$F_1^2$ and $F_3^2$ are disjoint, $F_1^2$ and $F_2^2$ share an edge and so does
$F_2^2$ and $F_3^2$ (see Figure \ref{2s}.3). So $M_j^3=M_1^3\cup M_{-1}^3\cup M_{-4}^3$.
\item [$(c)$] Suppose that $\widehat{K^2}\cap Q^4_j$ consists of the union of two neighboring faces. So considering all the options which $K^2\cap Q^4_j$ can be extended, we have two possibilities for $\widehat{K^2}\cap Q^4_j$:
\begin{description}
\item [$(i)$] $\widehat{K^2}\cap Q^4_j$ is the union of the two 2-faces $F_7^2$ and $F_4^2$ (see Figure \ref{2s}.4). Then there exist two 3-faces $M_4^3$ and $M_{-2}^3$ containing two 2-faces of
$(K ^2\cup \widehat{K^2})\cap Q^4_j$. So $M_j^3=M_4^3\cup M_{-2}^3$.
\item [$(ii)$] $\widehat{K^2}\cap Q^4_j$ is the union of the two 2-faces $F_2^2$ and $F_3^2$ (see Figure \ref{2s}.5). Then
there exist two 3-faces $M_{-1}^3$ and $M_{-4}^3$ containing two 2-faces: one from $K ^2\cap Q^4_j$ and the other from $\widehat{K^2}\cap Q^4_j$.
So $M_j^3=M_{-1}^3\cup M_{-4}^3$.
\end{description}
\end{description}
\noindent {\bf Claim 3:} \emph{Let $Q^4_j\in B^4$ and suppose that $K^2\cap Q^4_j$ consists of one 2-face $F^2$. Then, using face boundary moves if necessary, we can assume that
$\widehat{K^2}\cap Q^4_j$ is a connected 2-disk consisting of the union of at most five 2-faces.}
\noindent {\it Proof of Claim 3.} Observe that $F^2$ intersects twelve distinct 2-faces of $Q^4_j$, then $\widehat{K^2}\cap Q^4_j$ must be contained
in the remaining eleven 2-faces. These eleven 2-faces are distributed in the following way: six of them are contained in a 3-face $C^3$ and the
remaining faces lie on a neighboring 3-face $D^3$ such that $C^3\cap D^3$ consists of one 2-face (see Figure \ref{1s}.1). Since $\widehat{K^2}\cap Q^4_j$ is a connected disk,
then using (M2)-moves if necessary, we can assume that three 2-faces of $\widehat{K^2}\cap Q^4_j$ lie on $C^3$ and two 2-faces lie on $D^3$.
\begin{figure}[h]
\begin{center}
\includegraphics[height=11cm]{1s.jpg}
\end{center}
\caption{\sl $K^2\cap Q^4_j$ consists of one 2-face.}
\label{1s}
\end{figure}
\noindent This proves claim 3. $\square$
\noindent {\bf Case 3.} Suppose that $K^2\cap Q^4_j$ consists of one 2-face $F^2$. By claim 3, we can assume that at most three 2-faces
$F_1^2$, $F_2^2$, $F_3^2$ of $\widehat{K^2}\cap Q^4_j$ lie on $C^3$ and two 2-faces $F_4^2$, $F_5^2$ lie on $D^3$. We have five possibilities:
\begin{description}
\item [$(a)$] Suppose that $\widehat{K^2}\cap Q^4_j$ consists of the union of five 2-faces. Then we have only one possibility for $\widehat{K^2}\cap Q^4_j$
(see Figure \ref{1s}.2).
There exist three 3-faces $M_1^3$, $M_{-2}^3$ and $M_{-1}^3$ which
contain two 2-faces of $(K ^2\cup \widehat{K^2})\cap Q^4_j$. Take $M_j^3=M_1^3\cup M_{-2}^3\cup M_{-1}^3$.
\item [$(b)$] Suppose that $\widehat{K^2}\cap Q^4_j$ consists of the union of four 2-faces. In this case, we have three possibilities:
\begin{description}
\item [$(i)$] $\widehat{K^2}\cap Q^4_j$ is the union of the four 2-faces $F_1^2$, $F_2^2$, $F_3^2$ and $F_4^2$. Then we have only one possibility (see Figure \ref{1s}.3).
Let $M_{-2}^3$ be the 3-face of $Q^4_j$ such that $(K^2\cap Q^4_j)\cup F_2^2\subset M_{-2}^3$ and let $M_1^3$ and $M_{-1}^3$ be the 3-faces of $Q^4_j$ such that each of them is a
neighboring 3-face of $M_{-2}^3$ and $F_1^2\subset M_1^3$ and $F_4^2\subset M_{-1}^3$. Then $M_j^3=M_1^3\cup M_{-1}^3\cup M_{-2}^3$.
\item [$(ii)$] $\widehat{K^2}\cap Q^4_j$ is the union of the 2-faces $F_1^2$, $F_2^2$ and $F_4^2$, $F_5^2$. Thus the only possible configuration is shown in
Figure \ref{1s}.4. Let $M_1^3$, $M_{-2}^3$ and $M_{-1}^3$ be as above and let $M_{-4}^3$ be the neighboring 3-face such that $(K^2\cap Q^4_j)\cup F_5^2\subset M_{-4}^3$
so $M_j^3=M_1^3\cup M_{-2}^3\cup M_{-1}^3\cup M_{-4}^3$.
\item [$(iii)$] This configuration is similar to the one described in case $3b (i)$, but we exchange the cubes $C^3$ and $D^3$. Then, we get only one possibility
(see Figure \ref{1s}.5 ). Let $M_j^3=M_1^3\cup M_{-1}^3\cup M_{-4}^3$.
\end{description}
\item [$(c)$] Suppose that $\widehat{K^2}\cap Q^4_j$ consists of the union of three 2-faces. Let $F_1^2$, $F_2^2$, $F_3^2$ be 2-faces contained in a
cube $C^3$ and let $F_4^2$, $F_5^2$, $F_6^2$ be 2-faces contained in the 3-cube $D^3$. Then we have three possibilities:
\begin{description}
\item [$(i)$] $\widehat{K^2}\cap Q^4_j$ is the union of the 2-faces $F_1^2$, $F_2^2$ and $F_3^2$.
Let $M_1^3$, $M_{-1}^3$ and $M_{-2}^3$ be as above, so $M_j^3=M_1^3\cup M_{-1}^3\cup M_{-2}^3$ (see Figure \ref{1s}.6).
\item [$(ii)$] $\widehat{K^2}\cap Q^4_j$ is the union of the 2-faces $F_2^2$, $F_3^2$ and $F_6^2$. Thus $M_j^3=M_1^3\cup M_{-2}^3$
(see Figure \ref{1s}.7).
\item [$(iii)$] $\widehat{K^2}\cap Q^4_j$ is the union of the 2-faces $F_4^2$, $F_5^2$ and $F_6^2$. Let $M_{2}^3$ be the 3-face which contains the three squares $\widehat{K^2}\cap Q^4_j$. Then $M_j^3= M_2^3\cup M_{-4}^3 $
(see Figure \ref{1s}.8).
\end{description}
\item [$(d)$] Suppose that $\widehat{K^2}\cap Q^4_j$ consists of the union of two neighboring faces. Let $F_1^2$, $F_2^2$, $F_3^2$ be 2-faces contained in the
cube $C^3$ and let $F_4^2$, $F_5^2$, $F_6^2$ be 2-faces contained in the cube $D^3$. We have two possibilities:
\begin{description}
\item [$(i)$] $\widehat{K^2}\cap Q^4_j$ is the union of the 2-faces $F_1^2$ and $F_2^2$. Thus
$M_j^3=M_1^3\cup M_{-2}^3$
(see Figure \ref{1s}.9).
\item [$(ii)$] $\widehat{K^2}\cap Q^4_j$ is the union of the 2-faces $F_3^2$ and $F_4^2$. Thus
$M_j^3=M_2^3\cup M_{-4}^3$ (see Figure \ref{1s}.10).
\end{description}
\item [$(e)$] Suppose that $\widehat{K^2}\cap Q^4_j$ consists of one square. Then we have two possibilities:
\begin{description}
\item [$(i)$] $\widehat{K^2}\cap Q^4_j$ is the square $F_4^2$. Thus
$M_j^3=M_{-2}^3\cup M_4^3$
(see Figure \ref{1s}.11).
\item [$(ii)$] $\widehat{K^2}\cap Q^4_j$ is the 2-face $F_2^2$. Thus
$M_j^3=M_{-2}^3$
(see Figure \ref{1s}.12).
\end{description}
\end{description}
\noindent {\bf Claim 4:} \emph{Let $Q^4_j\in B^4$ and suppose that $K^2\cap Q^4_j$ consists of an edge $e$. Then, using face boundary moves if necessary, we can assume that
$\widehat{K^2}\cap Q^4_j$ is a connected 2-disk consisting of the union of at most six faces.}
\noindent {\it Proof of Claim 4.} Suppose that $K^2\cap Q^4_j$ consists of an edge $e$. See Figure \ref{1e}. Then $\widehat{K^2}$ must turn on $Q^4_j$; in other words,
$\widehat{K^2}\cap Q^4_j$ must contain two
faces $F_1^2$ and $F_2^2$ such that $F_1^2=v+\{ae_{i_1}+be_j\,:\,0\leq a,\,b\leq 1\}$ and $F_2^2=v+\{ce_{i_2}+de_j\,:\,0\leq c,\,d\leq 1\}$, where $e_{i_1}$, $e_{i_2}$ and
$e_{j}$ are distinct canonical vectors and $v$ is a vector with integer coordinates. Notice that $e_j$ is parallel to the edge $e$ and only
nine 2-faces of $Q^4_j$ satisfy both they do not intersect $K^2$ and they have an edge parallel to $e_j$;
hence $\widehat{K^2}\cap Q^4_j$ must be contained in the union of these nine 2-faces. Since
$\widehat{K^2}\cap Q^4_j$ is a connected disk it follows, up to applying face boundary moves (M2) if necessary, that it consists of the union of at most six faces.
\noindent This proves claim 4. $\square$
\begin{figure}[h]
\begin{center}
\includegraphics[height=11cm]{1e.jpg}
\end{center}
\caption{\sl $K^2\cap Q^4_j$ is an edge.}
\label{1e}
\end{figure}
\noindent {\bf Case 4.} Suppose that $K^2\cap Q^4_j$ consists of an edge $e$. By claim 4,
$\widehat{K^2}\cap Q^4_j$ is contained in three 3-faces $M_3^3$, $M_{-2}^3$ and $M_{-4}^3$ such that any two of them share a 2-face.
\begin{description}
\item [$(a)$] $\widehat{K^2}\cap Q^4_j$ consists of the union of six 2-faces. Then, applying (M2)-moves if necessary, we have that $\widehat{K^2}\cap Q^4_j$ consists of the
union of two 2-faces contained in $M_3^3$, three 2-faces in $M_{-3}^3$ and one 2-face in $M_{-4}^3$ (see Figure \ref{1e}.2). Thus $M_j^3=M_3^3\cup M_{-3}^3\cup M_{-4}^3$.
\item [$(b)$] $\widehat{K^2}\cap Q^4_j$ consists of the union of five 2-faces. Then, applying (M2)-moves if necessary, as in the previous case
we have that $M_j^3=M_3^3\cup M_{-3}^3\cup M_{-4}^3$. (see Figure \ref{1e}.3).
\item [$(c)$] $\widehat{K^2}\cap Q^4_j$ consists of the union of four 2-faces. We have five possibilities:
\begin{description}
\item [$(i)$] $\widehat{K^2}\cap Q^4_j$ consists of two 2-faces contained in $M_4^3$ and three 2-faces in $M_{-3}^3$.
Thus $M_j^3=M_{-3}^3\cup M_2^3\cup M_{-4}^3$ (see Figure \ref{1e}.4).
\item [$(ii)$] $\widehat{K^2}\cap Q^4_j$ consists of one 2-face contained in $M_{-4}^3$ and three 2-faces in $M_{-3}^3$.
Then $M_j^3=M_{-3}^3\cup M_{-4}^3$ (see Figure \ref{1e}.5).
\item [$(iii)$] $\widehat{K^2}\cap Q^4_j$ consists of one square contained in $M_{-4}^3$ and three 2-faces in $M_{-3}^3$ (see Figure \ref{1e}.6). Then $M_j^3=M_{-3}^3\cup M_{-4}^3$.
\item [$(iv)$] $\widehat{K^2}\cap Q^4_j$ consists of two 2-faces contained in $M_{-3}^3$ and two 2-faces in $M_2^3$ (see Figure \ref{1e}.7).
Then $M_j^3=M_{-3}^3\cup M_{2}^3\cup M_{-4}^3$.
\item [$(v)$] $\widehat{K^2}\cap Q^4_j$ consists of three 2-faces contained in $M_2^3$ and one square in $M_{-4}^3$ (see Figure \ref{1e}.8).
So $M_j^3= M_2^3\cup M_{-4}^3$.
\end{description}
\item [$(d)$] $\widehat{K^2}\cap Q^4_j$ consists of the union of three 2-faces. We have two possibilities:
\begin{description}
\item [$(i)$] $\widehat{K^2}\cap Q^4_j$ consists of one square contained in $M_2^3$ and two 2-faces in $M_{-3}^3$ (see Figure \ref{1e}.9).
Then $M_j^3=M_2^3\cup M_{-3}^3\cup M_{-4}^3$.
\item [$(ii)$] $\widehat{K^2}\cap Q^4_j$ consists of two 2-faces contained in $M_{-3}^3$ and one 2-face in $M_{-4}^3$ (see Figure \ref{1e}.10).
Then $M_j^3=M_{-3}^3\cup M_{-4}^3$.
\end{description}
\item [$(e)$] $\widehat{K^2}\cap Q^4_j$ consists of the union of two neighboring 2-faces. We have two possibilities:
\begin{description}
\item [$(i)$] $\widehat{K^2}\cap Q^4_j$ consists of two 2-faces contained in $M_4^3$ (see Figure \ref{1e}.11).
Then $M_j^3=M_4^3\cup M_3^3$.
\item [$(ii)$] $\widehat{K^2}\cap Q^4_j$ consists of two 2-faces contained in $M_{-4}^3$ (see Figure \ref{1e}.12).
Thus $M_j^3=M_{-4}^3$.
\end{description}
\end{description}
\noindent {\bf Claim 5:} \emph{Let $Q^4_j\in B^4$ and suppose that $K^2\cap Q^4_j$ consists of a vertex $v$. Then, using face boundary moves if
necessary, we can assume that
$\widehat{K^2}\cap Q^4_j$ is a connected 2-disk consisting of the union of at most six 2-faces.}
\noindent {\it Proof of Claim 5.}
Since $K^2\cap Q^4_j$ consists of a vertex $v$, then $v$ must be a \emph{corner} of $K^2$ \emph{i.e.} $v$ is a common vertex of three
neighboring faces of
$K^2$. Hence $\widehat{K^2}$ must have a corner at some vertex of $Q^4_j$. Since $K^2\cap\widehat{K^2} =\emptyset$,
it follows that $\widehat{K^2}\cap Q^4_j$ can be contained in either one, two or three 3-faces of $Q^4_j$ such that
these 3-faces do not contained $v$. Suppose that $\widehat{K^2}\cap Q^4_j$ is a connected disk consisting of at least six 2-faces,
then by a combinatorial analysis considering all the possible
descriptions of $\widehat{K^2}$, we should have that
four of these 2-faces are contained in some 3-face of $Q^4_j$; hence applying a face boundary move (M2), we get that
$\widehat{K^2}\cap Q^4_j$ can be reduced to at most six 2-faces. (see Figure \ref{1v}.1.)
\begin{figure}[h]
\begin{center}
\includegraphics[height=5cm]{1v.jpg}
\end{center}
\caption{\sl $\widehat{K^2}\cap Q^4_j$ consists of a vertex.}
\label{1v}
\end{figure} This proves claim 5. $\square$
\noindent {\bf Case 5.} Suppose that $K^2\cap Q^4_j$ consists of a vertex $v$. By the above claim,
$\widehat{K^2}\cap Q^4_j$ is the union of at least two 2-faces of $Q^4_j$ and its boundary must be
contained in the union of the three neighboring 3-faces containing $v$.
\begin{description}
\item [$(a)$] $\widehat{K^2}\cap Q^4_j$ is the union of six faces. Then $\widehat{K^2}\cap Q^4_j$
is contained in three 3-faces $M_{-2}^3$, $M_{3}^3$ and $M_{4}^3$ of $Q^4_j$ (see Figure \ref{1v}.2).
Since $\widehat{K^2}\cap Q^4_j$ is homeomorphic to a disk, we have that
each 3-face $M_{i}^3$ must contain two 2-faces of it and two of these three 3-faces contain $v$.
Then $M_j^3=M_{-2}^3 \cup M_{3}^3\cup M_{4}^3$.
\item [$(b)$] $\widehat{K^2}\cap Q^4_j$ is the union of five 2-faces. Then $\widehat{K^2}\cap Q^4_j$
is contained in the union of two 3-faces $M_{-4}^3$ and $M_{3}^3$ of $Q^4_j$ (see Figure \ref{1v}.3). So $M_j^3= M_3^3\cup M_{-4}^3$.
\item [$(c)$] $\widehat{K^2}\cap Q^4_j$ is the union of four 2-faces. Again, $\widehat{K^2}\cap Q^4_j$
is contained in a 3-face $M_{4}^3$ of $Q^4_j$ and its boundary must be
contained in the union of the three neighboring 3-faces containing $v$. Hence, one 3-face $M_{3}^3$, $M_{-1}^3$, $M_{-2}$ must contain one 2-face. The only possibility is that four 2-faces lie in some 3-face $M_{4}^3$ of $Q^4_j$ (see Figure \ref{1v}.4).
In this case $M_j^3=M_{4}^3$.
\item [$(d)$] $\widehat{K^2}\cap Q^4_j$ is the union of three neighboring 2-faces. These three 2-faces must be contained in some 3-face $M_{-4}^3$.
Then $M_j^3=M_{-4}^3$
(see Figure \ref{1v}.5).
\item [$(e)$] $\widehat{K^2}\cap Q^4_j$ is the union of two neighboring 2-faces. These two 2-faces must be contained in some 3-face $M_{-4}^3$.
Then $M_j^3=M_{-4}^3$.
\end{description}
\noindent Next, we will construct our 3-manifold ${\cal {M}}^3$.
\noindent Let ${\cal {M}}^3=\cup{M_j^3}$. By construction, each $M_j^3$ consists of the union of 3-faces contained in the 3-skeleton of ${N}_K^4$,
hence ${\cal{M}}^3$ is contained in the 3-skeleton of ${{N}}_K^4$.
\noindent As we mention before, if we apply an (M2)-move on a hypercube $Q^4_j$, then this movement does not affect the corresponding choice of $M_{l}^3$ on its neighbor hypercube $Q^4_{l}$; in other words,
the choice of $M_j^3$ depends only of the configuration of $\widehat{K^2}\cap Q^4_j$ and $K^2\cap Q^4_j$ in $Q^4_j$. Observe that the boundary of each $M_j^3$ is composed by 2-faces belonging to $K^2$ and $\widehat{K^2}$ and
some disjoint faces $F^j_i$ ($i=1,2,\ldots ,s_j$) that do not belong to
neither $K^2$ nor $\widehat{K^2}$. Thus, if we take a neighbor hypercube of $Q^4_j$, say $Q^4_{j_{l}}\in B^4$, and we construct $M_j^3$ and $M_{j_{l}}^3$, then
the intersection $M_j^3\cap M_{j_l}^3$ consists of the union of some $F^{j}_{r_i}$; notice that each of these 2-faces $F^j_i$ belongs to a neighbor hypercube of $Q^4_j$,
$Q^4_{j_{l_s}}\in B$. Hence ${\cal{M}}^3$ is a 3-manifold whose boundary
consists of two connected components, namely $K^2$ and $\widehat{K^2}$.
\noindent Now, we will carry the knot $\widehat{K^2}$ onto the knot $K^2$ via a finite number of cubulated moves. Notice that ${\cal{M}}^3$ is
the union
of $m$ components $M_1^3,\,\ldots M_m^3$ which are enumerated in such a way that $M_{n+1}^3$ is a
neighbor of $M_n^3$ for all $n$, \emph{i.e.}, $M_{n+1}^3\cap M_n^3$ consists of a 2-face $F_n^2$.
\noindent We will use induction on $m$. Consider $M_1^3$. We apply (M2)-moves on $M_1^3$ in such a way that the faces (of any dimension)
belonging to $K^2$ are replaced by
those belonging to $\widehat{K^2}$ (by construction $M_1^3\cap K^2\neq\emptyset$ and $M_1^3\cap \widehat{K^2}\neq \emptyset$).
Next, we consider $M_2^3$. By the previous step, $M_1^3$ and $M_2^3$ share 2-faces belonging to $\widehat{K^2}$. Then we apply again (M2)-moves,
the faces (of any dimension) belonging to $K^2$ are replaced by those belonging to $\widehat{K^2}$. We continue this finite process in this way.
Notice that if $l\subset K^2$
then $l\subset \partial {\cal{M}}^3$, thus $l$ is not a common 2-face of some pair $M_i^3$, $M_j^3$ belonging to ${\cal{M}}^3$, hence if
$l$ is replaced by a 2-face belonging to $\widehat{K^2}$, then this replacement will be kept in the following steps. Therefore, the result follows. $\square$
\subsection{$\widehat{K^2_1}$ is equivalent to $\widehat{K^2_2}$ via cubulated moves}\label{cubic}
In this section, we shall prove the Lemma B which says that $\widehat{K^2_1}$ is equivalent to $\widehat{K^2_2}$ by cubulated moves.
\noi Consider again the projection $p:\mathbb{R}^{5}\hookrightarrow\mathbb{R}$ on the last coordinate. We call \emph{horizontal hyperplane} to an affine hyperplane
parallel to the space $\mathbb{R}^{4}\times\{0\}$.
Thus $p^{-1}(t)=\mathbb{R}^{4}_t$ is a horizontal hyperplane. Observe that each hyperplane
$\mathbb{R}^{4}_t$ has a canonical cubulation
given by the restriction of the canonical cubulation of $\mathbb{R}^{5}$ to it.
\begin{definition}
A $2$-cell ($2$-face) $F^2$ of the canonical cubulation $\cal{C}$ of $\mathbb{R}^{5}$ is called \textit{horizontal} if $p(F^2)$ is
a constant number in $\mathbb{N}$.
A $2$-cell $F^2$ is \textit{vertical} if $p(F^2)=[m,m+1]$ for some $m\in\mathbb{N}$.
\end{definition}
\begin{definition}
Let $\Sigma^3$ be a cubulated $3$-manifold and $P^4$ be a horizontal hyperplane in $\mathbb{R}^{5}$.
We say that $P^4$ \emph{intersects transversally to} $\Sigma^3$, denoted by $P^4\pitchfork\Sigma^3$, if $P^4$ intersects transversally each $k$-cube of $\Sigma^3$, $k\geq 1$.
\end{definition}
\begin{lem}
Let $\mathbb{R}^{4}_t \subset \mathbb{R}^5$, $t\not\in\mathbb{Z}$ be an affine hyperplane. Then $\mathbb{R}^{4}_t$ intersects $\widehat{J^3}$ transversally.
\end{lem}
\noindent{\it Proof.} By Theorem \ref{trace},
$\mathbb{R}^{4}_t\cap \widehat{J^3}$ is connected. Let $x\in \mathbb\mathbb{R}^{4}_t\cap \widehat{J^3}$. Then
$x\in Q^3_i$, where $Q^3_i$ is a 3-face of the cubulation of $\mathbb{R}^{5}$. Notice that $Q^3_i$ is a vertical $3$-face, since $t\not\in\mathbb{Z}$.
So, we have two possibilities:
either $x\in\mbox{Int}(Q^3_i)$ or $x\in F^2\setminus\partial F^2$ where $F^2$ is a 2-face of $Q^3_i$, or $x$ belongs to an edge.
\begin{enumerate}
\item If $x\in\mbox{Int}(Q^3_i)$ then $\mathbb{R}^{4}_t\cap Q^3_i=E^2$, where $E^2$ is a square parallel to a 2-face and $x\in E^2$.
\item If $x$ belongs to $F^2\setminus\partial F^2$, then there exists another vertical $3$-face $Q^3_j$ such that $x\in Q^3_i\cap Q^3_j$. Thus
$\mathbb{R}^{4}_t\cap Q^3_i$ must be a square $E_i^2$ such that it is parallel to a 2-face, and analogously $\mathbb{R}^{4}_t\cap Q^3_j$ is also a square $E_j^2$,
thus $x\in E_i^2\cap E_j^2$.
\item If $x$ belongs to an edge $l$, then there exist vertical $3$-faces $Q^3_{i_r}$ $r=1,2,3$ such that $x\in Q^3_j\cap_{r=1}^3 Q^3_{i_r}$. As in the previous case,
$\mathbb{R}^{4}_t\cap Q^3_{i_r}$ must be a square $E_{i_{r}}^2$ parallel to a 2-face, and analogously $\mathbb{R}^{4}_t\cap Q^3_j$ is also a square $E_j^2$, hence
$x\in E_j^2\cap_{r=1}^4 E_{i_r}^2$.
\end{enumerate}
\noindent Therefore, the result follows. $\square$
\begin{coro}
For $x\not\in\mathbb{Z}$, the set $p^{-1}(x)\cap \widehat{J^3}$ is a $2$-knot.
\end{coro}
\noindent{\it Proof.} By the above, $p^{-1}(x)\cap \widehat{J^3}$ is a cubulated compact connected surface. Since
$\widehat{J^3}$ is homeomorphic to $\mathbb{S}^2\times\mathbb{R}$, it follows that $p^{-1}(x)\cap \widehat{J^3}$ is homeomorphic to $\mathbb{S}^2$. $\square$
\noindent Now, for each $n\in\mathbb{N}$ we define
$$
K^2_{n-}:= p^{-1}\left(n-\frac{1}{2}\right)\cap \widehat{J^3}
$$
and
$$
K^2_{n+}:= p^{-1}\left(n+\frac{1}{2}\right)\cap \widehat{J^3}.
$$
Observe that $K^2_{n-}$ and $K^2_{n+}$ are cubical $2$-knots.
\noi Let ${\cal{C}}^3$ be the set of 3-cubes ($3$-cells) belonging to the canonical cubulation of $\mathbb{R}^{5}$. Consider the three spaces:
$$
M^3_{n-}:=\{Q^3\in \mathcal{C} ^3 \,|\,Q^3 \cap K^2_{n-}\neq\emptyset\},
$$
$$
M^3_{n+}:=\{Q^3\in \mathcal{C} ^3 \,|\,Q^3 \cap K^2_{n+}\neq\emptyset\}
$$
and $M^3_n:= p^{-1}(n)\cap \widehat{J^3}$.
\noi By construction $M^3_{n-}= K^2_{n-}\times [0,1]$
and $M^3_{n+}= K^2_{n+}\times [0,1]$.
\noindent Let $M^3:=M^3_{n-}\cup M^3_{n}\cup M^3_{n+}$. Hence
$M^3=\mbox{Cl}(p^{-1}(n-1,n+1)\cap \widehat{J^3})$, where Cl denotes closure.
\begin{lem}\label{circle}
The space $M^3$ is homeomorphic to $\mathbb{S}^{2}\times [0,1]$.
\end{lem}
\noindent{\it Proof.} Since $M^3$ is a compact submanifold of $\widehat{J^3}$, then by Lemma \ref{trace}, it is also connected. Now $\widehat{J^3}$ is
homeomorphic to $\mathbb{S}^{2}\times\mathbb{R}$, and $\widehat{J^3}-M^3$ has two connected components, hence the result follows. $\square$
\begin{lem}
$M^3_n$ has the homotopy type of $\mathbb{S}^{2}$.
\end{lem}
\noindent{\it Proof.} Consider the set $M^3$. Then by the previous Lemma, we have that $M^3\cong \mathbb{S}^{2}\times [0,1]$. Notice that
$K^2_{n-}\times\{0\}\cong \mathbb{S}^2\times\{0\}$ and $K^2_{n+}\times\{0\}\cong \mathbb{S}^2\times\{1\}$.
Hence $\widetilde{M^3}=M^3/(K^2_{n-}\times\{0\})\cup (K^2_{n+}\times\{1\})$ is homeomorphic to $\mathbb{S}^3$.
By Alexander duality,
we have that $\widetilde{H}_0(\widetilde{M^3}-M^3_n,\mathbb{Z})=\widetilde{H}^{2} (M_n^3,\mathbb{Z})$, but $\widetilde{M^3}-M_n^3$ has two simply connected components, so
$\widetilde{H}_{0} (\widetilde{M^3},\mathbb{Z})\cong \mathbb{Z}$. Since $M_n^3\subset M^3\cong \mathbb{S}^{2}\times [0,1]$, we have that either
$\pi_2 (M_n^3)\cong \{0\}$ or $\pi_2 (M_n^3)\cong \mathbb{Z}$, but $\tilde{H}^{2}(M_n^3,\mathbb{Z})\cong\mathbb{Z}$, hence $\pi_2 (M_n^3)\cong \mathbb{Z}$.
\noi Again by Alexander duality, we have that $\widetilde{H}_1(\widetilde{M^3}-M_n^3,\mathbb{Z})=\widetilde{H}^{1} (M_n^3,\mathbb{Z})$, but
$\widetilde{H}_1(\widetilde{M^3}-M_n^3,\mathbb{Z})\cong \{0\}$ hence $\widetilde{H}^{1} (M_n^3,\mathbb{Z})\cong \{0\}$. This implies that
$\pi_1 (M_n^3)\cong \{0\}$.
Therefore, $M_n^3$ has the homotopy type of $\mathbb{S}^2$. $\square$
\begin{lem}
The space $M^3$ retracts strongly to $M_n^3$.
\end{lem}
\noindent{\it Proof.} Since $M^3=M^3_{n-}\cup M^3_{n}\cup M^3_{n+}$ is homeomorphic to $\mathbb{S}^2\times [0,1]$,
and $M^3_{n-}= K^2_{n-}\times [0,1]$ and $M^3_{n+}= K^2_{n+}\times [0,1]$,
we have that
$M^3_{n-}= K^2_{n-}\times [0,1]$ retracts strongly to $K^2_{n-}\times \{1\}$ and
$M^3_{n+}= K^2_{n+}\times [0,1]$ retracts strongly to $K ^2_{n+}\times \{0\}$. Now
$\partial M_n^3=(K^2_{n-}\times\{1\})\cup (K^2_{n+}\times\{0\})$. Therefore, the result follows. $\square$
\noindent Next, we are going to describe the subset $M_n^3$. Notice that the squares of $M_n^3$ are of four types, which we will denote
by $T_{-}$, $T_{+}$, $T_{\pm}$ and $T$.
\begin{itemize}
\item A square $F^2\subset M_n^3$ belongs to $T_{-}$ if $F^2\subset M^3_{n-}$ but $F^2\not\subset M^3_{n+}$.
\item A square $F^2\subset M_n^3$ belongs to $T_{+}$ if $F^2\subset M^3_{n+}$ but $F^2\not\subset M^3_{n-}$.
\item A square $F^2\subset M_n^3$ belongs to $T_{\pm}$ if $F^2\subset M^3_{n-}\cap M^3_{n+}$.
\item A square $F^2\subset M_n^3$ belongs to $T$ if $F^2\not\subset M^3_{n+}\cup M^3_{n-}$.
\end{itemize}
\noi By the above Lemma, there are copies of $K^2_{n-}$ and $K^2_{n+}$ contained in $\partial M_n^3$. By abuse of notation we will
denote them in the same way. Notice that $K^2 _{n-}$ is the union of squares of types $T_{-}$ and $T_{\pm}$, and
$K^2_{n+}$ is the union of squares of types $T_{+}$ and $T_{\pm}$.
\begin{lem}\label{fns}
$K^2_{n-}\overset{c}\sim K^2_{n+}:$ There exists a finite sequence of cubulated moves that carries the 2-knot $K^2_{n-}$ into the 2-knot $K^2_{n+}$.
\end{lem}
\noi {\it Proof.} We will show it by cases.
\noi Case 1. Suppose that $K^2_{n-}=K^2_{n+}$. Clearly, the result is true.
\noi Case 2. Suppose that $K^2_{n-}\cap K^2_{n+}=\emptyset$. In other words, $K^2_{n-}$ and
$K^2_{n+}$
do not have squares of type $T_{\pm}$.
Remember that $M_n^3$ is a cubical compact 3-manifold whose fundamental group is isomorphic to $\mathbb{Z}$. Thus $\partial M_n^3$
has two connected components; namely $K^2_{n-}$ and $K^2_{n+}$, such that their intersection is empty.
Hence $M_n^3$ is the union of a finite number of cubes (3-faces) belonging to the 3-skeleton of the cubulation $\cal C$, whose 2-faces are of any of the types $T_{-}$, $T_{+}$ and $T$.
\noi Next, we will carry the $2$-knot $K^2_{n-}$ onto the $2$-knot $K^2_{n+}$ via a finite number of cubulated moves; {\it i.e.} we will
carry the squares of type $T_{-}$ onto the squares of type $T_{+}$. Let $Q^3$ be a cube contained in $M_n^3$. We can assume, up to $(M1)$-move, that if a square
$F^2\subset Q^3$ belongs to $T_{-}$, then $Q^3\cap K^2_{n+}=\emptyset$ and
$Q^3\cap K^2_{n-}$ consists of either a square, two neighboring 2-faces or three neighboring 2-faces. Analogously, if
$F^2\subset Q^3$ belongs to $T_{+}$, then $Q^3\cap K^2_{n-}=\emptyset$ and
$Q^3\cap K^2_{n+}$ consists of a square, two neighboring 2-faces or three neighboring 2-faces.
\noi The compact space $M_n^3$ is the union of a finite number of cubes, say $m$. We will enumerate them by levels in the following way. Remember that $M_n^3$ is a cubical compact 3-manifold, so we may assume that $M_n^3\subset\mathbb{R}_+^3$.
Let $q:\mathbb{R}^3\hookrightarrow \mathbb{R}$ be the projection
on the last coordinate. We define the level $k$, $M_k^3:=M_n^3\cap q^{-1}([k,k+1])$, for $k\in\mathbb{N}$. Notice that there exists $k_1$ and $k_2$ positive integers such that $M_k^3=\emptyset$ for $k_1\leq k\leq k_2$. Then we start enumerating the cubes $Q^3\in M^3$ by levels. We start at the level $k_1$. The first cube
$Q_1^3$ contains a 2-face of type $T_{-}$, and given the cube $Q_n^3$, the cube $Q_{n+1}^3$ shares a 2-face $F_n^2$ with $Q_n^3$ and whenever
it is possible, we choose $Q_{n+1}^3$ in such a way that $F_n^2$ is parallel to $F_{n-1}^2$; otherwise we choose $Q_{n+2}^3$ such that $F^2_{n+1}$ is parallel
to $F^2_{n-1}$, at the end we continue on the level $k_1+1$ and so on.
\noi We will use induction on $m$. Consider the cube $Q_1^3$. We apply the $(M2)$-move to $Q_1^3$ replacing the 2-faces of type $T$ by 2-faces of type $T_{-}$.
We consider $Q_2^3$. Observe that $Q_1^3$ and $Q_2^3$ share a 2-face of type $T_{-}$. Then we apply again the $(M2)$-move replacing the 2-faces of type
$T$ by 2-faces of type $T_{-}$. We continue inductively.
\noi Notice that if $F^2\subset M_n^3$ is a 2-face of type $T_{+}$ then $F^2\subset \partial M_n^3$;
so $F^2$ is not a common 2-face of two cubes $Q_i^3$ and $Q^3_j$ in $M_n^3$; hence if $F^2$ is replaced by a 2-face of type $T_{-}$ then this replacement is not modified in
any other next step. Therefore, the result follows.
\noi Case 3. Suppose that the intersection $K^2_{n-}\cap K^2_{n+}$ contains a finite number of 2-faces.
\noi The 3-manifold $M_n^3$ consists of connected components $C_i^3$, $i=1,\ldots,r$ such that each $C_i^3$ is the union of cubes
$Q_{i_1}^3, \ldots, Q_{i_{m_i}}^3\in{\cal{C}}$ and the intersection $C_i^3\cap C_j^3$ is either empty or a square belonging to
$K^2_{n-}\cap K^2_{n+}$. Therefore, we apply the previous argument to each $C_i^3$.
\noi Case 4. Suppose that the intersection $K^2_{n-}\cap K^2_{n+}$ contains a square of type $T_{\pm}$. The 3-manifold $M_n^3$ consists of 3-dimensional connected components $C_i^3$, $i=1,\ldots,r$
and cubical 2-disks $\gamma_{ij}$. As before, each $C_i^3$ is a union of cubes $Q_{i_1}^3, \ldots, Q_{i_{m_i}}^3\in{\cal{C}}$, and $\gamma_{ij}$ is a
cubical disk (or edge) joining the component $C_i^3$ with the component $C_j^3$.
Observe that if $\gamma_{ij}$ is the union of 2-faces of type $T_{\pm}$, hence $\gamma_{ij}\subset K^2_{n-}\cap K^2_{n+}$.
Moreover $K^2_{n-}\cap K^2_{n+}= \gamma_{ij}$ and $\partial M_n^3=K^2_{n-}\cup K^2_{n+}$.
\noi Since $\pi_2 (M_n^3)\cong\mathbb{Z}$, then $C_i^3$ the homotopy type either the 2-sphere or the 3-ball.
Suppose that $C_i^3$ has the homotopy type of the 2-sphere,
then by hypothesis $\partial C_i^3=\partial M_n^3=K^2_{n-}\cup K^2_{n+}$ contains a face $F^2$ of type $T_{\pm}$, but $F^2$ does not belong to
any cube $Q^3$ of $M_n^3$; so $F^2$ does not belong to $C_i^3$. This is a contradiction, hence $C_i^3$ has the homotopy type of the 3-ball.
\noi By the above, $\partial C_i^3$ is homeomorphic to $\mathbb{S}^2$ and consists of 2-faces of type $T_{-}$ and $T_{+}$. Moreover, $\partial C_i^3$ consists of two disks
$D_-^2$ and $D_+^2$, such that $D_-^2$ is the union of faces of type $T_{-}$ and $D_+^2$ is the union of faces of type $T_{+}$. Now we apply the
argument of the case 2, so $D_+^2$ is replaced by $D_-^2$. Since we have a finite number of components $C_i^3$,
the result follows. $\square$
\begin{LB}
$\widehat{K^2_1}\overset{c}\sim \widehat{K^2_2}$: There exists a finite sequence
of cubulated moves that carries $\widehat{K^2_1}$ into $\widehat{K^2_2}$. In other words, $\widehat{K^2_1}$ is equivalent to $\widehat{K^2_2}$ by cubulated moves. \end{LB}
\noindent{\it Proof of Lemma B.} Recall that there exist integer numbers
$m_1$ and $m_2$ such that $p^{-1}(t)\cap \widehat{J^3} = \widehat{K^2_1}$ for all $t\leq m_1$ and
$p^{-1}(t)\cap \widehat{J^3} = \widehat{K^2_2}$ for all $t\geq m_2$. Consider the integer $m_1+1$. By
Lemma \ref{fns} there exists a finite number of cubulated moves that
carries the 2-knot $\widehat{K^2_1}$ into the 2-knot $K^2_{(m_1+1)-}$. We continue inductively, and again by
Lemma \ref{fns}
there exists a finite number of cubulated moves that
carries the knot $K^2_{(m_2-1)+}$ into the knot $\widehat{K^2_2}$. Since, we have a finite number of integers contained in the interval $[m_1,m_2]$, then
there exists a finite sequence of cubulated moves that carries $\widehat{K^2_1}$ into $\widehat{K^2_2}$. $\square$
| {
"timestamp": "2017-08-28T02:05:51",
"yymm": "1708",
"arxiv_id": "1708.07761",
"language": "en",
"url": "https://arxiv.org/abs/1708.07761",
"abstract": "In this paper, we prove that given two cubical links of dimension two in ${\\mathbb R}^4$, they are isotopic if and only if one can pass from one to the other by a finite sequence of cubulated moves. These moves are analogous to the Reidemeister and Roseman moves for classical tame knots of dimension one and two, respectively.",
"subjects": "Geometric Topology (math.GT)",
"title": "Cubulated moves for 2-knots",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9770226320971079,
"lm_q2_score": 0.7248702702332475,
"lm_q1q2_score": 0.7082146593522294
} |
https://arxiv.org/abs/2110.03677 | Large Learning Rate Tames Homogeneity: Convergence and Balancing Effect | Recent empirical advances show that training deep models with large learning rate often improves generalization performance. However, theoretical justifications on the benefits of large learning rate are highly limited, due to challenges in analysis. In this paper, we consider using Gradient Descent (GD) with a large learning rate on a homogeneous matrix factorization problem, i.e., $\min_{X, Y} \|A - XY^\top\|_{\sf F}^2$. We prove a convergence theory for constant large learning rates well beyond $2/L$, where $L$ is the largest eigenvalue of Hessian at the initialization. Moreover, we rigorously establish an implicit bias of GD induced by such a large learning rate, termed 'balancing', meaning that magnitudes of $X$ and $Y$ at the limit of GD iterations will be close even if their initialization is significantly unbalanced. Numerical experiments are provided to support our theory. | \section{Background and Related Work}\label{sec:background}
\noindent {\bf Notations}. $\|v \|$ is $\ell_2$ norm of a column or row vector $v$. $\fnorm{M}$ is the Frobenius norm of a matrix $M$.
\paragraph{Sharp and flat minima in \eqref{problem:intro_matrix_factorization}}
We discuss the curvatures at global minima of \eqref{problem:intro_matrix_factorization}. To ease the presentation, consider simplified versions of \eqref{problem:intro_matrix_factorization} with either $n = 1$ or $d = 1$. In this case, $X$ and $Y$ become vectors and we denote them as $x, y$, respectively. We show the following proposition characterizing the spectrum of Hessian at a global minimum.
\begin{proposition}
\label{pro:eigenvalue_hessian}
When $n=1, d\in\mathbb{N}^+$ or $d = 1, n\in\mathbb{N}^+$ in \eqref{problem:intro_matrix_factorization}, the largest eigenvalue of Hessian at a global minimum $(x,y)$ is $\norm{x}^2+\norm{y}^2$ and the smallest eigenvalue is $0$.
\end{proposition}
Homogenity implies global minimizers of \eqref{problem:intro_matrix_factorization} are not isolated, which is consistent with the 0 eigenvalue.
On the other hand, if the largest eigenvalue $\norm{x}^2+\norm{y}^2$ is large (or small), then the curvature at such a global minimum is sharp (or flat), in the direction of the leading eigenvector. Meanwhile, note this sharpness/flatness is an indication of the balancedness between magnitudes of $x, y$ at a global minimum. To see this, singular value decomposition (SVD) yields that at a global minimum, $(x, y)$ satisfies $\fnorm{xy^\top}^2=\sigma_{\max}^2(A)$. Therefore, large $\norm{x}^2+\norm{y}^2$ is obtained when $|\|x\|-\|y\||$ is large, i.e., $x$ and $y$ magnitudes are unbalanced, and small $\norm{x}^2+\norm{y}^2$ is obtained when balanced.
\paragraph{Large learning rate}
We study smoothness properties of \eqref{problem:intro_matrix_factorization} and demonstrate that our learning rate is well beyond conventional optimization theory. We first define the smoothness of a function.
\begin{definition}[$L$-smooth]
\label{def:l_smooth}
A function $f\in\mathcal{C}^1$ defined on $\mathbb{R}^N$ is {\it $L$-smooth} if for all $u_1,u_2\in \mathbb{R}^N$,
\begin{align}
\label{eqn:l-smooth}
\|\nabla f(u_1)-\nabla f(u_2)\|\le L\|u_1-u_2\|.
\end{align}
If further $f\in\mathcal{C}^2$, then $\nabla^2 f\preceq L I$. Moreover, if we have~\eqref{eqn:l-smooth} for $u_1,u_2\in \mathcal{X}\subseteq \mathbb{R}^N$, we call it locally $L$-smooth.
\end{definition}
In traditional optimization \citep{nesterov2003introductory,polyak1987introduction,nesterov1983method,polyak1964some,beck2009fast}, most analyzed objective functions often satisfy (i) (some relaxed form of) convexity or strong convexity, and (ii) $L$-smoothness. Choosing a step size $h < 2/L$ guarantees the convergence of GD to a minimum by the existing theory (reviewed in Appendix~\ref{app:gd_converge_2/L}). Our choice of learning rate $h \approx 4/L$ (more precisely, $4/L_0$; see below) goes beyond the classical analyses.
Besides, in our problem \eqref{problem:intro_matrix_factorization}, the regularity is very different. Even simplified versions of \eqref{problem:intro_matrix_factorization}, i.e., with either $n = 1$ or $d = 1$, suffer from (i) non-convexity and (ii) unbounded eigenvalues of Hessian, i.e., no global smoothness (see Appendix~\ref{app:scalar_non_convex} for more details). As shown in \citet{NEURIPS_Du2018algorithmic,ye2021global,ma2021beyond}, decaying or infinitesimal learning rate ensures that the GD trajectory stays in a locally smooth region. However, the gap between the magnitudes of $X, Y$ can only be maintained in that case. We show, however, that larger learning rate can shrink this gap. More precisely, if initial condition is well balanced, there is no need to use large learning rate; otherwise, we can use learning rate as large as approximately $4/L$, and within this range, larger $h$ provides smaller gap between $X$ and $Y$ at the limit (i.e. infinitely many GD iterations).
\paragraph{Related work}
Matrix factorization problems in various forms have been extensively studied in the literature. The version of \eqref{problem:intro_matrix_factorization} is commonly known as the low-rank factorization, although here $d$ can arbitrary and we consider both $d\leq \text{rank}(A)$ and $d>\text{rank}(A)$ cases.
\citet{baldi1989neural,li2019symmetry, valavi2020landscape} provide landscape analysis of \eqref{problem:intro_matrix_factorization}. \citet{ge2017no,tu2016low} propose to penalize the Frobenius norm of $\fnorm{X^\top X - Y^\top Y}^2$ to mitigate the homogeneity and establish global convergence guarantees of GD for solving \eqref{problem:intro_matrix_factorization}. \citet{cabral2013unifying,li2019non} instead penalize individual Frobenius norms of $\fnorm{X}^2 + \fnorm{Y}^2$. We remark that penalizing $\fnorm{X}^2 + \fnorm{Y}^2$ is closely related to nuclear norm regularization, since the variational formula for nuclear norm $\nucnorm{Z}^2 = \min_{Z = XY^\top} \fnorm{X}^2 + \fnorm{Y}^2$. More recently, \citet{liu2021noisy} consider using injected noise as regularization to GD and establish a global convergence (see also \cite{zhou2019toward,liu2022noise}. More specifically, by perturbing the GD iterate $(X_k, Y_k)$ with Gaussian noise, GD will converge to a flat global optimum. On the other hand, \citet{NEURIPS_Du2018algorithmic,ye2021global,ma2021beyond} show that even without explicit regularization, when the learning rate of GD is infinitesimal, i.e., GD approximating gradient flow, $X$ and $Y$ maintain the gap in their magnitudes. Such an effect is more broadly recognized as implicit bias of learning algorithms \citep{neyshabur2014search, gunasekar2018implicit, soudry2018implicit,Li2020Implicit,li2021implicit}. Built upon this implicit bias, \citet{NEURIPS_Du2018algorithmic} further prove that GD with diminishing learning rates converges to a bounded global minimum of \eqref{problem:intro_matrix_factorization}, and this conclusion
is recently extended to the case of a constant small learning rate \citep{ye2021global}. Our work goes beyond the scopes of these milestones and considers matrix factorization with much larger learning rates.
Additional results exist that demonstrate large learning rate can improve performance in various learning problems. Most of them involve non-constant learn rates. Specifically,
\citet{li2019towards} consider a two-layer neural network setting, where using learning rate annealing (initially large, followed by small ones) can improve classification accuracy compared to training with small learning rates. \citet{nakkiran2020learning} shows that the observation in \citet{li2019towards} even exists in convex problems. \citet{lewkowycz2020large} study constant large learning rates, and demonstrate distinct algorithmic behaviors of large and small learning rates, as well as empirically illustrate large learning rate yields better testing performance on neural networks. Their analysis is built upon the neural tangent kernel perspective \citep{jacot2018neural}, with a focus on the kernel spectrum evolution under large learning rate.
Worth noting is,
\citet{NEURIPS2020_1b9a8060} also study constant large learning rates, and show that large learning rate provides a mechanism for GD to escape local minima, alternative to noisy escapes due to stochastic gradients.
\section{General matrix factorization}
\label{sec:general_matrix_factorization}
In this section, we consider problem \eqref{problem:intro_matrix_factorization} with an arbitrary matrix $A \in \mathbb{R}^{n \times n}$. We replicate the problem formulation here for convenience,
\begin{align}
\label{eqn:general}
\min_{X,Y\in\mathbb{R}^{n\times d}} \frac{1}{2}\fnorm{A-X Y^\top }^2.
\end{align}
Note this is the most general case with $n,d\in\mathbb{N}^+$ and any square matrix $A$. Due to this generalization, we no longer utilize the convergence analysis and instead, establish the balancing theory via stability analysis of GD as a discrete time dynamical system.
Let $\mu_1\ge\mu_2\ge\cdots\ge\mu_n\ge 0$ be the singular values of $A$. Assume for technical convenience $\fnorm{A}^2 = \sum_{i=1}^n \mu_i^2$ being independent of $d$ and $n$. We denote the singular value decomposition of $A$ as $A = U D V^\top$, where $U, D, V \in \mathbb{R}^{n \times n}$, $U,V$ are orthogonal matrices and $D$ is diagonal. Then we establish the following balancing effect.
\begin{theorem}
\label{thm:general_case}
Given almost all the initial conditions, for any learning rate $h$ such that GD for~\eqref{eqn:general} converges to a point $(X,Y)$, there exists $c=c(n, d) > c_0$ with constant $c_0 > 0$ independent of $h$, $n$, and $d$, such that $(X, Y)$ satisfies
\begin{align*}
c( \fnorm{X}^2+\fnorm{Y}^2)< \frac{2}{h},
\end{align*}
and the extent of balancing is quantified by $
\fnorm{X-(UV^\top) Y}^2< \frac{2}{ch}-2\sum_{i=1}^{\min\{d,n\}}\mu_i$, which means
\[
\big|\fnorm{X}-\fnorm{Y}\big|^2 < \frac{2}{ch}-2\sqrt{\sum_{i=1}^{\min\{d,n\}}\mu_i^2} .
\]
In particular, when $d=1$, i.e., rank-$1$ factorization of an arbitrary $A$, the constant $c$ equals $1$.
\end{theorem}
We observe that the extent of balancing can be quantified under some rotation of $Y$. This is necessary, since for factorizing a general matrix $A$ (which can be asymmetric), at a global minimum, $X, Y$ may only align after a rotation (which is however fixed by $A$, independent of initial or final conditions). Figure~\ref{fig:over_parameterized_balancing} illustrates an example of the balancing effect under different learning rates. Evidently, larger learning rate leads to
a more balanced global minimizer. Additional experiments with various dimensions, learning rates, and initializations can be found in Appendix~\ref{app:experiments}; a similar balancing effect is also shown for additional problems including matrix sensing and matrix completion there.
\begin{figure}[ht]
\centering
\includegraphics[trim=5cm 1.3cm 5cm 1.3cm,width=0.85\textwidth]{general_matrix_factorization_balancing-eps-converted-to.pdf}
\caption{Balancing effect of general matrix factorization. We independently generate elements in $A\in\mathbb{R}^{6\times 6}$ from a Gaussian distribution. We choose $X, Y \in \mathbb{R}^{6 \times 100}$ and randomly pick a pair of initial point $(X_0, Y_0)$ with $\fnorm{X_0}=1$ and $\fnorm{Y_0}=9$.}
\label{fig:over_parameterized_balancing}
\end{figure}
Different from previous sections, Theorem \ref{thm:general_case} builds on stability analysis by viewing GD as a dynamical system in discrete time. More precisely, the proof of Theorem \ref{thm:general_case} (see Appendix \ref{app:general_matrix_factorization_balancing} for details) consists of two parts: (i) the establishment of an easier but equivalent problem via the rotation of $X$ and $Y$, (ii) stability analysis of the equivalent problem.
For (i), by singular value decomposition (SVD), $A=UDV^\top $, where $U,D,V\in\mathbb{R}^{n\times n}$, $U$ and $V$ are orthogonal matrices, and $D$ is a non-negative diagonal matrix. Let $X_k=UR_k,\ Y_k=VS_k.$ Then
\[
\begin{cases}
&X_{k+1}=X_k+h(A-X_kY_k^\top )Y_k\\
&Y_{k+1}=Y_k+h(A-X_kY_k^\top )^\top X_k
\end{cases}
\]
\[
\Leftrightarrow
\begin{cases}
&U R_{k+1}=U R_k+h(UDV^\top -U R_kS_k^\top V^\top )VS_k\\
&VS_{k+1}=VS_k+h(UDV^\top -U R_kS_k^\top V^\top )^\top U R_k
\end{cases}
\]
\[
\Leftrightarrow
\begin{cases}
&R_{k+1}= R_k+h(D-R_kS_k^\top )S_k\\
&S_{k+1}=S_k+h(D- R_kS_k^\top )^\top R_k
\end{cases}.
\]
Therefore, GD for problem \eqref{eqn:general} is equivalent to GD for the following problem
\begin{align*}
\min_{R,S\in\mathbb{R}^{n\times d}} \frac{1}{2} \fnorm{D-R S^\top }^2
\end{align*}
and it thus suffices to work with diagonal non-negative $A$.
For (ii), here is a brief description of the idea of stability analysis:
consider each iteration of GD as a mapping $\psi$ from $u_k \sim (X_k, Y_K)$ to $u_{k+1} \sim (X_{k+1}, Y_{k+1})$, where matrices $X$ and $Y$ are flattened and concatenated into a vector so that $\psi$ is a closed map on vector space $\mathbb{R}^{2dn}$. GD iteration is thus a discrete time dynamical system on state space $\mathbb{R}^{2dn}$ given by
\begin{align*}
u_{k+1} = \psi(u_k) = u_k - h \nabla f(u_k),
\end{align*}
where $f$ is the objective function $f(u_k) = \frac{1}{2} \fnorm{A - X_kY_k^\top}^2$, and gradient returns a vector that collects all component-wise partial derivatives.
It's easy to see that any stationary point of $f$, denoted by $u^*$, is a fixed point of $\psi$, i.e., $u^*=\psi(u^*)$. What fixed point will the iterations of $\psi$ converge to? For this, the following notions are helpful:
\begin{proposition}\label{def:stable}
Consider a fixed point $u^*$ of $\psi$. If all the eigenvalues of Jacobian matrix $\nabla \psi(u^*)$ are of complex modulus less than $1$, it is a {\it stable} fixed point.
\end{proposition}
\begin{proposition}
Consider a fixed point $u^*$ of $\psi$. If at least one eigenvalue of Jacobian matrix $\nabla \psi(u^*)$ is of complex modulus greater than $1$, it is an {\it unstable} fixed point.
\end{proposition}
Roughly put, the stable set of an unstable fixed point is of negligible size when compared to that of a stable fixed point, and thus what GD converges to is a stable fixed point for almost all initial conditions \citep{alligood1996chaos}. Thus, we investigate the stability of each global minimum of $f$ (each saddle of $f$ is an unstable fixed point of $\psi$ and thus is irrelevant). By a detailed evaluation of $\nabla \psi$'s eigenvalues (Appendix~\ref{app:general_matrix_factorization_balancing}), we see that a global minimum $(X,Y)$ of $f$ corresponds to a stable fixed point of GD iteration if $\left|1 - c h \left(\fnorm{X}^2 + \fnorm{Y}^2\right)\right| < 1$, i.e., it is balanced as in Thm. \ref{thm:general_case}.
\section{Introduction}
Training machine learning models such as deep neural networks involves optimizing highly nonconvex functions. Empirical results indicate an intimate connection between training algorithms and the performance of trained models \citep{le2011optimization, bottou2018optimization, zhang2021understanding, soydaner2020comparison, zhou2020towards}. Especially for widely used first-order training algorithms (e.g., GD and SGD), the learning rate is of essential importance and has received extensive focus from researchers \citep{smith2017cyclical, jastrzkebski2017three, smith2018disciplined, gotmare2018closer, liu2019variance, li2019exponential}. A recent perspective is that large learning rates often lead to improved testing performance compared to the counterpart trained with small learning rates \citep{smith2019super, yue2020salr}.
Towards explaining the better performance, a common belief is that large learning rates encourage the algorithm to search for flat minima, which often generalize better and are more robust than sharp ones \citep{seong2018towards, lewkowycz2020large}.
Despite abundant empirical observations, theoretical understandings of the benefits of large learning rate are still limited for non-convex functions, partly due to challenges in analysis. For example, the convergence (of GD or SGD) under large learning rate is not guaranteed. Even for globally smooth functions, very few general results exist if the learning rate exceeds certain threshold \citep{NEURIPS2020_1b9a8060}. Besides, popular regularity assumptions such as global smoothness for simplified analyses are often absent in homogeneous models, including commonly used ReLU neural networks.
This paper theoretically studies the benefits of large learning rate in a matrix factorization problem
\begin{align}
\label{problem:intro_matrix_factorization}
\min_{X,Y}~ \frac{1}{2} \fnorm{A-XY^\top}^2,\quad \text{where}\ A\in\mathbb{R}^{n\times n},\ X,Y\in\mathbb{R}^{n\times d}.
\end{align}
We consider Gradient Descent (GD) for solving \eqref{problem:intro_matrix_factorization}: at the $k$-th iteration, we have
\begin{align*}
X_{k+1} = X_k+h(A-X_kY_k^\top )Y_k \quad \text{and} \quad
Y_{k+1} = Y_k+h(A^\top -Y_kX_k^\top )X_k,
\end{align*}
where $h$ is the learning rate. Despite its simple formula, problem~\eqref{problem:intro_matrix_factorization} serves as an important foundation of a variety of problems, including matrix sensing \citep{chen2015fast, bhojanapalli2016dropping, tu2016low}, matrix completion \citep{keshavan2010matrix, hardt2014understanding}, and linear neural networks \citep{ji2018gradient, gunasekar2018implicit}.
Problem \eqref{problem:intro_matrix_factorization} possesses several intriguing properties. Firstly, the objective function is non-convex, and critical points are either global minima or saddles (see e.g., \citet{baldi1989neural,li2019symmetry,pmlr-v108-valavi20a,chen2018landscape}). Secondly, problem \eqref{problem:intro_matrix_factorization} is homogeneous in $X$ and $Y$, meaning that rescaling $X, Y$ to $aX, a^{-1}Y$ for any $a \neq 0$ will not change the objective's value. This property is shared by commonly used ReLU neural networks. A direct consequence of homogeneity is that global minima of \eqref{problem:intro_matrix_factorization} are non-isolated and can be unbounded. The curvatures at these global minima are highly dependent on the magnitudes of $X, Y$. When $X, Y$ have comparable magnitudes, the largest eigenvalue of Hessian is small, and this corresponds to a flat minimum; on the contrary, unbalanced $X$ and $Y$ give a sharp minimum. Last but not the least, the homogeneity impairs smoothness conditions of \eqref{problem:intro_matrix_factorization}, rendering the gradient being not Lipschitz continuous unless $X, Y$ are bounded. See a formal discussion in Section \ref{sec:background}.
Existing approaches for solving \eqref{problem:intro_matrix_factorization} often uses explicit regularization \citep{ge2017no,tu2016low,cabral2013unifying,li2019non}, or infinitesimal (or diminishing) learning rates for controlling the magnitudes of $X, Y$ \citep{NEURIPS_Du2018algorithmic,ye2021global}. In this paper, we go beyond the scope of aforementioned works, and analyze GD with a large learning rate for solving \eqref{problem:intro_matrix_factorization}. In particular, we allow the learning rate $h$ to be as large as approximately $4/L$ (see more explanation in Section~\ref{sec:background}), where $L$ denotes the largest eigenvalue of Hessian at GD initialization. In connection to empirical observations, we provide positive answers to the following two questions:
\begin{center}
\it Does GD with large learning rate converge at least for some cases of \eqref{problem:intro_matrix_factorization}? \\
Does larger learning rate biases toward flatter minima (i.e., $X, Y$ with comparable magnitudes)?
\end{center}
We theoretically show the convergence of GD with large learning rate for the two situations $n=1,d\in\mathbb{N}^+$ or $d=1,n\in\mathbb{N}^+$ with isotropic $A$. We also observe a, perhaps surprising, ``balancing effect'' for general matrix factorization (i.e., any $d$, $n$, and $A$), meaning that when $h$ is sufficiently large, the difference between $X$ and $Y$ shrinks significantly at the convergence of GD compared to its initial, even if the initial point is close to an unbalanced global minimum. In fact, with a proper large learning rate $h$, $\fnorm{X_k-Y_k}^2$ may decrease by an arbitrary factor at its limit. The following is a simple example of our theory for $n=1$ (i.e. scalar factorization), and more general results will be presented later with a precise bound for $h$ depending on the initial condition and $A$.
\begin{theorem}[Informal version of Thm.\ref{thm:scalar_convergence} \& \ref{thm:scalar_balancing}]
Given scalar $A$ and initial condition $X_0,Y_0\in \mathbb{R}^{1\times d}$ chosen almost everywhere, with learning rate $h \lesssim 4/L$, GD converges to a global minimum $(X,Y)$ satisfying $
\|X\|_{\sf F}^2+\|Y\|_{\sf F}^2\le \frac{2}{h}$.
Consequently, its extent of balancing is quantified by $
\|X-Y\|_{\sf F}^2\le \frac{2}{h}-2A$.
\label{cor:scalarFactorizationBalancing}
\end{theorem}
We remark that having a learning rate $h \approx 4/L$ is far beyond the commonly analyzed regime in optimization.
Even for globally $L$-smooth objective, traditional theory requires $h < 2/L$ for GD convergence and $h = 1/L$ is optimal for convex functions \citep{boyd2004convex}, not to mention that our problem \eqref{problem:intro_matrix_factorization} is never globally $L$-smooth due to homogeneity. Modified equation provides a tool for probing intermediate learning rates (see \citet[Chapter 9]{hairer2006geometric} for a general review, and \citet[Appendix A]{NEURIPS2020_1b9a8060} for the specific setup of GD), but the learning rate here is too large for modified equation to work (see Appendix \ref{sec:modified_eqn}). In fact, besides blowing up, GD with large learning rate may have a zoology of limiting behaviors (see e.g., Appendix~\ref{sec:periodic_orbits} for convergence to periodic orbits under our setup, and \citet{NEURIPS2020_1b9a8060} for convergence to chaotic attractors).
Our analyses (of convergence and balancing) leverage various mathematical tools, including a proper partition of state space and its dynamical transition (specifically invented for this problem), stability theory of discrete time dynamical systems \citep{alligood1996chaos}, and geometric measure theory~\citep{federer2014geometric}.
The rest of the paper is organized as: Section \ref{sec:background} provides the background of studying \eqref{problem:intro_matrix_factorization} and discusses related works; Section \ref{sec:scalar_factorization} presents convergence and balancing results for scalar factorization problems; Section \ref{sec:rank_one_approx_isotropic} generalizes the theory to rank-$1$ matrix approximation; Section \ref{sec:general_matrix_factorization} studies problem \eqref{problem:intro_matrix_factorization} with arbitrary $A$ and its arbitrary-rank approximation; Section \ref{sec:discussion} summarizes the paper and discusses broadly related topics and future directions.
\input{background.tex}
\input{scalar_factorization.tex}
\input{rank1_factorization.tex}
\input{general_factorization.tex}
\section{Conclusion and Discussion}\label{sec:discussion}
In this paper, we demonstrate an implicit regularization effect of large learning rate on the homogeneous matrix factorization problem solved by GD. More precisely, a phenomenon termed as ``balancing'' is theoretically illustrated, which says the difference between the two factors $X$ and $Y$ may decrease significantly at the limit of GD, and the extent of balancing can increase as learning rate increases. In addition, we provide theoretical analysis of the convergence of GD to the global minimum, and this is with large learning rate that can exceed the typical limit of $2/L$, where $L$ is the largest eigenvalue of Hessian at GD initialization.
For the matrix factorization problem analyzed here, large learning rate avoids bad regularities induced by the homogeneity between $X$ and $Y$. We feel it is possible that such balancing behavior can also be seen in problems with similar homogeneous properties, for example, in tensor decomposition \citep{kolda2009tensor}, matrix completion \citep{keshavan2010matrix, hardt2014understanding}, generalized phase retrieval \citep{candes2015phase, sun2018geometric}, and neural networks with homogeneous activation functions (e.g., ReLU). Besides the balancing effect, the convergence analysis under large learning rate may be transplanted to other non-convex problems and help discover more implicit regularization effects.
In addition, factorization problems studied here are closely related to two-layer linear neural networks. For example, one-dimensional regression via a two-layer linear neural network can be formulated as the scalar factorization problem \eqref{eqn:scalarFactorizationObjective}: Suppose we have a collection of data points $(x_i, y_i) \in \mathbb{R} \times \mathbb{R}$ for $i = 1, \dots, n$. We aim to train a linear neural network $y = (u^\top v) x$ with $u, v \in \mathbb{R}^d$ for fitting the data. We optimize $u, v$ by minimizing the quadratic loss,
\begin{align}\label{eq:twolayer_NN}
(u^*, v^*) \in \arg\min_{u, v}~ \frac{1}{n} \sum_{i=1}^n \left(y_i - (u^\top v) x_i \right)^2 = \arg\min_{u, v}\frac{1}{n} \left(\frac{\sum_{i=1}^n x_i y_i}{\sum_{i=1}^n x_i^2} - u^\top v\right)^2.
\end{align}
As can be seen, taking $\mu = \frac{\sum_{i=1}^n x_i y_i}{\sum_{i=1}^n x_i^2}$ recovers \eqref{eqn:scalarFactorizationObjective}. In this regard, our theory indicates that training of $u, v$ by GD with large learning rate automatically balances $u, v$, and the obtained minimum is flat. This may provide some initial understanding of the improved performance brought by large learning rates in practice. Note that \eqref{eq:twolayer_NN} generalizes to arbitrary data distribution of training a two-layer linear network with atomic data (i.e., $x = 1$ and $y = 0$) in \citet{lewkowycz2020large}.
It is important to clarify, however, that there is a substantial gap between this demonstration and extensions to general neural networks, including deep linear and nonlinear networks. Although we suspect that large learning rate leads to similar balancing effect of weight matrices in the network, rigorous theoretical analysis is left as a future direction.
\newpage
\section*{Acknowledgments}
We thank anonymous reviewers and area chair for suggestions that improved the quality of this paper. The authors are grateful for partial supports from NSF DMS-1847802 (YW and MT) and ECCS-1936776 (MT).
\section{Rank-1 approx. of isotropic $A$ (an under-parameterized case)}\label{sec:rank_one_approx_isotropic}
Given insights from scalar factorization, we consider rank-$1$ factorization of an isotropic matrix $A$, i.e., $A=\mu I_{n\times n}$ with $\mu>0$, $d = 1$, and $n \in \mathbb{N}^+$. The corresponding optimization problem is
\begin{align}
\label{problem:rank_one_isotropic}
\min_{x,y\in\mathbb{R}^{n\times 1}} \frac{1}{2} \fnorm{\mu I_{n\times n}-xy^\top}^2.
\end{align}
Although similar at an uncareful glance, Problems \eqref{problem:rank_one_isotropic} and \eqref{eqn:scalarFactorizationObjective} are rather different unless $n = d = 1$. First of all, Problem \eqref{problem:rank_one_isotropic} is under-parameterized for $n > 1$, while \eqref{eqn:scalarFactorizationObjective} is overparameterized. More importantly, we'll show that, when $(x, y)$ is a global minimum of \eqref{problem:rank_one_isotropic}, $x, y$ must be aligned, i.e., $x = \ell y$ for some $\ell > 0$. In the scalar factorization problem, however, no such alignment is required. As a result, the set of global minima of \eqref{problem:rank_one_isotropic} is an $n$-dimensional submanifold embedded in a $2n$-dimensional space, while in the scalar factorization problem the set of global minimum is a $(2d-1)$-dimensional submanifold --- one rank deficient --- in a $2d$-dimensional space. We expect the convergence in \eqref{problem:rank_one_isotropic} is more complicated than that in \eqref{eqn:scalarFactorizationObjective}, since searching in \eqref{problem:rank_one_isotropic} is demanding.
To prove the convergence of large learning rate GD for \eqref{sec:rank_one_approx_isotropic}, our theory consists of two steps: (i) show the convergence of the alignment between $x$ and $y$ (this is new); (ii) use that to prove the convergence of the full iterates (i.e., $x$ \& $y$). Step (i) first:
\begin{theorem}[Alignment]
\label{thm:alignment}
Given $(x_0,y_0)\in\ (\mathbb{R}^n \times \mathbb{R}^n) \backslash\mathcal{B}$, where $\mathcal{B}$ is some Lebesgue measure-0 set, when learning rate $h\le \min\left\{\frac{4}{\|x_0\|^2+\|y_0\|^2+4\sqrt{7}\mu},\ \frac{1}{2\sqrt{7}\mu}\right\} $, the iterator $(x_k, y_k)$ of GD at the $k$-th iteration satisfies $|\cos(\angle(x_k,y_k))| \to 1$ as $k\to\infty$.
\end{theorem}
\begin{proof}[Proof sketch]
A sufficient condition for the convergence of $|\cos(\angle(x_k,y_k))|$ is $\norm{x_k}^2\norm{y_k}^2 - (x_k^\top y_k)^2 \to 0$. To ease the presentation, let $U_k=x_k^\top y_k,\ V_k=x_k^\top x_k$, and $W_k=y_k^\top y_k$. By the GD update and some algebraic manipulation, we derive
\begin{align*}
V_{k+1}W_{k+1}-U_{k+1}^2=r_k \cdot (V_{k}W_{k}-U_{k}^2) ,
\end{align*}
where $r_k = (1 - h (V_k + W_k) + h^2 (V_k W_k - \mu^2))^2$. When $k$ is sufficiently large, we can show a uniform upper bound on $r_k < 1 - c$ for some constant $c > 0$. In this way, we deduce that $V_{k}W_{k}-U_{k}^2$ will exponentially decay and converge to $0$. More details are provided in Appendix \ref{app:rank1_convergence_balancing}.
\end{proof}
Theorem \ref{thm:alignment} indicates that GD iterations will converge to the neighbourhood of $\{(x,y):x=\ell y, \text{for some} ~\ell\in\mathbb{R}\backslash\{0\}\}$. This helps establish the global convergence as stated in Step (ii).
\begin{theorem}[Convergence]
\label{thm:rank1_convergence}
Under the same initial conditions and learning rate $h$ as Theorem~\ref{thm:alignment}, GD for \eqref{problem:rank_one_isotropic} converges to a global minimum.
\end{theorem}
Similar to the over-parametrized scalar case, this convergence can also be split into two phases where phase 1 motivates the balancing behavior with the decrease of $\norm{x_k}^2+\norm{y_k}^2$, and phase 2 ensures the convergence. The following balancing theorem is thus obtained.
\begin{theorem}[Balancing]
\label{thm:rank_one_isotropic_norm_balancing}
Under the same initial conditions and learning rate $h$ as Theorem~\ref{thm:alignment}, GD for \eqref{problem:rank_one_isotropic} converges to a global minimizer that obeys
\begin{align*}
\|x\|^2+\|y\|^2\le \frac{2}{h},
\end{align*}
and its extent of balancing is quantified by
\begin{align*}
\|x-y\|^2\le \frac{2}{h}-2\mu.
\end{align*}
\end{theorem}
This conclusion is the same as the one in Section~\ref{sec:scalar_factorization}. A quantitatively similar corollary like Corollary~\ref{cor:unbalanced_to_balanced} can also be obtained from the above theorem, namely, if $x_0$ and $y_0$ start from an unbalanced point near a minimum, the limit will be a more balanced one.
\section{Overparameterized scalar factorization}
\label{sec:scalar_factorization}
In order to provide intuition before directly studying the most general problem, we begin with a simple special case, namely factorizing a scalar by two vectors. It corresponds to \eqref{problem:intro_matrix_factorization} with $n = 1$ and $d\in\mathbb{N}^{+}$, and this overparameterized problem is written as
\begin{equation}
\min_{x,y \in \mathbb{R}^{1\times d}}~ \frac{1}{2} (\mu-x y^\top)^2,
\label{eqn:scalarFactorizationObjective}
\end{equation}
where $\mu$ is assumed without loss of generality to be a positive scalar. Problem \eqref{eqn:scalarFactorizationObjective} can be viewed as univariate regression using a linear two-layer neural network with the quadratic loss, which is studied in \citet{lewkowycz2020large} with atomic data distribution. Yet our analysis in the sequel can be used to study arbitrary univariate data distributions; see details in Section \ref{sec:discussion}.
Although simplified, problem \eqref{eqn:scalarFactorizationObjective} is still nonconvex and exhibits the same homogeneity as \eqref{problem:intro_matrix_factorization}. The convergence of its large learning rate GD optimization was previously not understood, let alone balancing. Many results that we will obtain for \eqref{eqn:scalarFactorizationObjective} will remain true for more general problems.
We first prove that GD converges despite of $>2/L$ learning rate and for almost all initial conditions:
\begin{theorem}[Convergence]
\label{thm:scalar_convergence}
Given $(x_0^\top,y_0^\top)\in\ (\mathbb{R}^d \times \mathbb{R}^d) \backslash\mathcal{B}$ where $\mathcal{B}$ is some Lebesgue measure-0 set, when the learning rate $h$ satisfies $$h\le \min\left\{\frac{4}{\|x_0\|^2+\|y_0\|^2+4\mu},\ \frac{1}{3\mu}\right\},$$ GD converges to a global minimum.
\label{thm:scalarFactorizationConvergence}
\end{theorem}
Theorem \ref{thm:scalarFactorizationConvergence} says that choosing a constant learning rate depending on GD initialization guarantees the convergence for almost every starting point in the whole space. This result is even stronger than the already nontrivial convergence under small learning rate with high probability over random initialization \citep{ye2021global}. Furthermore, the upper bound on $h$ is sufficiently large: on the one hand, suppose GD initialization $(x_0, y_0)$ is close to an unbalanced global minimum. By Proposition~\ref{pro:eigenvalue_hessian}, we can check that the largest eigenvalue $L(x_0,y_0)$ of Hessian $\nabla^2 f(x_0, y_0)$ is approximately $ \norm{x_0}^2 + \norm{y_0}^2$. Consequently, our upper bound of $h$ is almost $4/L$, which is beyond $2/L$ (see Section~\ref{sec:background} for more details). On the other hand, we observe numerically that the $4/L$ upper bound is actually very close to the stability limit of GD when initialized away from the origin (see more details in Appendix~\ref{app:scalar_stability_limit_h}).
\begin{figure}[!htb]
\centering
\includegraphics[trim=5cm 1.3cm 5cm 5cm,width=0.6\textwidth]{two_phase_figure.pdf}
\caption{The dynamics of GD under different learning rate $h$}
\label{fig:two_phase}
\end{figure}
The convergence in Theorem \ref{thm:scalarFactorizationConvergence} has an interesting searching-to-converging transition as depicted in Figure \ref{fig:two_phase}, where we observe two phases. In Phase $1$, large learning rate drives GD to search for flat regions, escaping from the attraction of sharp minima. After some iterations, the algorithm enters the vicinity of a global minimum with more balanced magnitudes in $x, y$. Then in Phase $2$, GD converges to the found balanced global minimum. We remark that the searching-to-converging transition also appears in \citet{lewkowycz2020large}. However, the algorithmic behaviors are not the same. In fact, in our searching phase (phase 1), the objective function does not exhibit the blow-up phenomenon. In our convergence phase (phase 2), the analysis relies on a detailed state space partition (see Line 192) due to nonconvex nature of \eqref{eqn:scalarFactorizationObjective}, while the analysis in \citet{lewkowycz2020large} is akin to monotone convergence in a convex problem.
In comparison with the dynamics of small learning rate, we note that the searching phase (Phase $1$) is vital to the convergence analysis. Meanwhile, the searching phase induces a balancing effect of large learning rate. The following theorem explicitly quantifies the extent of balancing.
\begin{theorem}[Balancing]
\label{thm:scalar_balancing}
Under the same initial condition and learning rate $h$ as Theorem~\ref{thm:scalarFactorizationConvergence}, GD for \eqref{eqn:scalarFactorizationObjective} converges to a global minimizer $(x,y)$ satisfying
\[
\|x\|^2+\|y\|^2\le \frac{2}{h}.
\]
Consequently, its extent of balancing is quantified by
\begin{align*}
\|x-y\|^2\le \frac{2}{h}-2\mu.
\end{align*}
\end{theorem}
One special case of Theorem~\ref{thm:scalar_balancing} is the following theorem, which states that no matter how close to a global minimum does GD start, if this minimum does not correspond to well-balanced norms, a large learning rate will take the iteration to a more balanced limit. We also demonstrate a sharp shrinkage in the distance between $x$ and $y$.
\begin{corollary}[From `unbalanced' to `balanced']
\label{cor:unbalanced_to_balanced}
For any $\delta \in (0, \mu)$, let the GD initialization satisfy $$(x_0,y_0)\in\big\{(u,v): |uv^\top-\mu|<\delta,~ \|u\|^2+\|v\|^2>8\mu\big\}\backslash\mathcal{B},$$ where $\mathcal{B}$ is some Lebesgue measure-0 set. When the learning rate $h$ satisfies $h=\frac{4}{\|x_0\|^2+\|y_0\|^2+4\mu},$
the extent of balancing at the limiting point $(x,y)$ of GD obeys
\[
\|x-y\|^2 < \frac{1}{2}\|x_0-y_0\|^2 + 2\mu.
\]
\end{corollary}
Both Theorem~\ref{thm:scalar_balancing} and Corollary~\ref{cor:unbalanced_to_balanced} suggest that larger learning rate yields better balancing effect, as $\norm{x - y}^2$ at the limit of GD may decrease a lot and is controlled by the learning rate. We remark that the balancing effect is a consequence of large learning rate, as small learning rate can only maintain the difference in magnitudes of $x, y$ \citep{NEURIPS_Du2018algorithmic}.
In addition, the actual balancing effect can be quite strong with $\norm{x_k-y_k}^2$ decreasing to be almost 0 at its limit under a proper choice of large learning rate. Figure~\ref{fig:perfect_balance_example} illustrates an almost perfect balancing case when $h=\frac{4}{\norm{x_0}^2+\norm{y_0}^2+4}\approx 0.0122$ is chosen as the upper bound. The difference $\norm{x_k-y_k}$ decreases from approximately $17.9986$ to $0.0154$ at its limit. Additional experiments with various learning rates and initializations can be found in Appendix~\ref{app:experiments}.
\begin{figure}[ht]
\centering
\includegraphics[trim=5cm 1.3cm 5cm 4cm,width=0.3\textwidth]{perfect_balancing_example-eps-converted-to.pdf}
\caption{The objective function is $(1-xy^\top)^2/2$, where $x^\top,y^\top\in\mathbb{R}^{10}$. Highly unbalanced initial condition is uniformly randomized, with the norms to be $\norm{x_0}=18,\norm{y_0}=0.09$.}
\label{fig:perfect_balance_example}
\end{figure}
\paragraph{Technical Overview}
We sketch the main ideas behind Theorem \ref{thm:scalarFactorizationConvergence}, which lead to the balancing effect in Theorem \ref{thm:scalar_balancing}. Full proof is deferred to Appendix \ref{app:scalar_convergence_balancing}.
The convergence is proved by handling Phase $1$ and $2$ separately (see Fig.\ref{fig:two_phase}). In Phase $1$, we prove that $\norm{x_k}^2 + \norm{y_k}^2$ has a decreasing trend as GD searches for flat minimum. We show that $\norm{x_k}^2 + \norm{y_k}^2$ may not be monotone, i.e., it either decreases every iteration or decreases every other iteration.
In Phase $2$, we carefully partition the state space and show GD at each partition will eventually enter a monotone convergence region. Note that the partition is based on detailed understanding of the dynamics and is highly nontrivial. Attentive readers may refer to Appendix \ref{app:scalar_convergence_balancing} for more details. The combination of Phase 1 \& 2 is briefly summarized as a proof flow chart in Figure \ref{fig:flow_chart_scalar_decomposition_proof}.
\begin{figure}[ht]
\centering
\includegraphics[width=0.75\textwidth]{pure_scalar_decomposition_proof_flow_chart.pdf}
\caption{Proof overview of Theorem~\ref{thm:scalarFactorizationConvergence}. At the $k$-th iteration, we denote $(x_k, y_k)$ as the iterate and $s_k$ is defined as $x_{k+1} y_{k+1}^\top-\mu=s_k(x_k y_k^\top-\mu)$.}
\label{fig:flow_chart_scalar_decomposition_proof}
\end{figure} | {
"timestamp": "2022-03-01T02:18:22",
"yymm": "2110",
"arxiv_id": "2110.03677",
"language": "en",
"url": "https://arxiv.org/abs/2110.03677",
"abstract": "Recent empirical advances show that training deep models with large learning rate often improves generalization performance. However, theoretical justifications on the benefits of large learning rate are highly limited, due to challenges in analysis. In this paper, we consider using Gradient Descent (GD) with a large learning rate on a homogeneous matrix factorization problem, i.e., $\\min_{X, Y} \\|A - XY^\\top\\|_{\\sf F}^2$. We prove a convergence theory for constant large learning rates well beyond $2/L$, where $L$ is the largest eigenvalue of Hessian at the initialization. Moreover, we rigorously establish an implicit bias of GD induced by such a large learning rate, termed 'balancing', meaning that magnitudes of $X$ and $Y$ at the limit of GD iterations will be close even if their initialization is significantly unbalanced. Numerical experiments are provided to support our theory.",
"subjects": "Machine Learning (cs.LG); Dynamical Systems (math.DS); Optimization and Control (math.OC)",
"title": "Large Learning Rate Tames Homogeneity: Convergence and Balancing Effect",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9770226314280634,
"lm_q2_score": 0.7248702702332475,
"lm_q1q2_score": 0.7082146588672589
} |
https://arxiv.org/abs/2103.04501 | Large Deviations for High Minima of Gaussian Processes with Nonnegatively Correlated Increments | In this article we prove large deviations principles for high minima of Gaussian processes with nonnegatively correlated increments on arbitrary intervals. Furthermore, we prove large deviations principles for the increments of such processes on intervals $[a,b]$ where $b-a$ is either less than the increment or twice the increment, assuming stationarity of the increments. As a chief example, we consider fractional Brownian motion and fractional Gaussian noise for $H\geq 1/2$. | \section{Introduction}
Let $X(t)$ be a centered continuous Gaussian process with covariance function $R(s,t)=E[X(t)X(s)]$. There has been an abundance of literature on large deviations principles for the maxima of $X(t)$ on an interval $[a,b]$. See e.g. \cite{Berman-1,Berman-2,Berman-3,Berman-4,Berman-5,Berman-6,Berman-7,Kobelkov}. There has however been less work done on large deviations for the minima of $X(t)$ on an interval $[a,b]$. The problem was introduced in \cite{Gennady-High-Level-Sets}, with follow up papers considering the smooth and nonsmooth cases in \cite{Gennady-Asymptotic} and \cite{Gennady-Non-Smooth} respectively. More precisely, setting some high threshold $u>0$, we are interested in the asymptotic behavior of
\begin{equation*}
P\left(\min_{t\in [a,b]} X(t)>u\right),
\end{equation*}
as $u\to\infty$.
In \cite{Gennady-High-Level-Sets} it is shown that
\begin{equation}\label{eq:LDP}
\lim_{u\to \infty} \frac{1}{u^2}\log \left(P\left(\min_{t\in [a,b]} X(t)>u\right)\right)=-\frac{1}{2\sigma_\ast^2(a,b)},
\end{equation}
where
\begin{equation}\label{eq:optimization}
\sigma_\ast^2(a,b):=\inf_{\mu\in \mathcal M_1([a,b])}\int_a^b\int_a^b R(s,t)\mu(ds)\mu(dt),
\end{equation}
where $\mathcal M_1$ denotes the space of Borel probability measures on $[a,b]$. This problem has applications in the structure of the high level excursion sets of Gaussian random processes and fields (see \cite{Gennady-Asymptotic,Gennady-High-Level-Sets}). It is also of interest to compute small ball probabilities.
In addition, the optimization problem $\sigma^2_\ast(a,b)$ is of independent interest and has been generalized in \cite{Minimum-energy-measures}, e.g., by replacing $R(s,t)$ by a general symmetric kernel on some compact set $\mathcal K\subset \mathbb R^d$, and considering different classes of measures instead of $\mathcal M_1$. In \cite{Minimum-energy-measures}, several numerics are given.
In this article, we specialize to a class of processes including fractional Brownian motion and fractional Gaussian noise, which have found wide applications in many fields. For an introduction to fBm, see e.g. \cite{Nourdin-fBm}. FBm finds applications in Queueing theory - see e.g. \cite{Assaf-Queue,Delgado-Queueing,Yamnenko-Queueing}. Fbm also has applications in finance, see e.g. \cite{Oksendal-fBm,Hu-fBm,Gatheral-volatility,Gatheral-volatility-2}. Fractional Gaussian noise has applications in signal processing, see e.g. \cite{Barton-signal}.
Our contributions are as follows. First, we compute $\sigma_\ast^2(a,b)$ in the case where $X$ is a centered continuous Gaussian process with nonnegatively correlated increments on aribitrary interval $[a,b]$. Second, we compute $\sigma_\ast^2(a,b)$ in the case $X$ is the increment of a centered continuous Gaussian process with nonnegatively correlated increments and satisfying some technical conditions on interval $[a,b]$ where $b-a$ is less than the increment. Third, we compute $\sigma_\ast^2(a,b)$ in the case $X$ is the increment of a centered continuous Gaussian process with nonnegatively correlated increments and satisfying some technical conditions on the interval $[a,b]$ where $b-a$ is twice the increment. As a chief example, we compute $\sigma_\ast^2(a,b)$ for fractional Brownian motion and fractional Gaussian noise for $H\geq 1/2$.
\section*{Acknowledgements}
ZS would like to acknowledge helpful conversations with Gennady Samorodnitsky and Harsha Honnappa.
\section{Results}
In order to compute the value $\sigma_\ast^2(a,b)$, we need a technical theorem that allows us to check whether a probability measure is a minimizer in the problem \eqref{eq:optimization}. In this section $X(t)$ will be a centered continuous Gaussian process with covariance function $R(s,t)=E[X(t)X(s)]$. The following result, Theorem 4.3 in \cite{Gennady-High-Level-Sets}, provides a check for the optimal measure in the minimization problem $\sigma_\ast^2(a,b)$.
\begin{theorem}(Theorem 4.3 \cite{Gennady-High-Level-Sets})\label{theorem:check}
Let $X(t)$ be a continuous centered Gaussian process on $[a,b]$ with covariance function $R(s,t)$. Then $\mu^\ast$ is an optimal measure in problem $\sigma_\ast^2(a,b)$ defined in equation \eqref{eq:optimization} if and only if
\begin{align*}
\min_{u\in [a,b]}\phi(u)&:= \min_{u\in [a,b]} \int_a^b R(s,u)\mu^\ast(ds)\\
&=\int_a^b \int_a^b R(s,t) \mu^\ast(ds)\mu^\ast(dt)\\
&=\phi(t),
\end{align*}
for $\mu^\ast$-a.e. $t\in [a,b]$, where $$\phi(t)=\int_a^b R(s,t)\mu^\ast(ds).$$
\end{theorem}
\begin{remark}
The measure $\mu^\ast$ in Theorem \ref{theorem:check} should be thought of as the distribution of the location of the minima under the limit $u\to \infty$. For an example of a precise statement, see \cite{Gennady-Non-Smooth} Theorem 4.1.
\end{remark}
This section will be broken up into two subsections. The first subsection will handle the case when $X$ is a centered continuous Gaussian process with nonnegatively correlated increments. The second subsection will handle the case where $X$ is the increment of a centered continuous Gaussian process with nonnegatively correlated stationary increments.
\subsection{Processes with Nonnegatively Correlated Increments}
In this subsection, we consider continuous centered Gaussian processes with nonnegatively correlated increments.
\begin{proposition}\label{proposition:non-increment}
Let $X(t)$ be a continuous centered Gaussian process on the interval $[a,b]$ with covariance function $R(s,t)$ and $X(0)=0$ a.s. Assume that $X$ has nonnegatively correlated increments. That is, for all quadruples $(s_1,t_1,s_2,t_2)$ with $a\leq s_1\leq t_1\leq s_2\leq t_2\leq b$ we have that
$$E\left[(X(t_1)-X(s_1))(X(t_2)-X(s_2))\right]\geq 0.$$
Then, the optimal measure $\mu^\ast=\delta_a$.
\end{proposition}
\begin{proof}
By Theorem \ref{theorem:check}, we must show the following two things
\begin{equation*}
\min_{u\in [a,b]}\phi(u):= \min_{u\in [a,b]} \int_a^b R(s,u)\mu^\ast(ds)=\int_a^b \int_a^b R(s,t) \mu^\ast(ds)\mu^\ast(dt)
\end{equation*}
and
\begin{equation*}
\min_{u\in [a,b]}\phi(u)=\phi(t),
\end{equation*}
for $t=\{a\}$. To this aim, we compute that
\begin{equation*}
\int_a^b R(s,t) \delta_a(ds)=R(a,t).
\end{equation*}
Taking a minimum achieves that
\begin{align*}
\min_{t\in [a,b]}R(a,t)&=\min_{t\in [a,b]}E[X(t)X(a)]\\
&=\min_{t\in [a,b]}E[(X(t)-X(a)+X(a))X(a)]\\
&=\min_{t\in [a,b]}E[(X(t)-X(a))(X(a)-X(0))]+E[X^2(a)].
\end{align*}
By the assumption that the increments are nonnegatively correlated, we conclude that this minima is achieved at $t=a$.
\end{proof}
There is a partial converse to Proposition \ref{proposition:non-increment}.
\begin{proposition}\label{proposition:converse-1}
Let $X(t)$ be a continuous centered Gaussian process with covariance function $R(s,t)$ and $X(0)=0$ a.s. Suppose that on any interval $[a,b]$ the optimal measure $\mu^\ast=\delta_a$. Then $X$ satisfies
\begin{equation*}
E[(X(t)-X(a))(X(a)-X(0))]\geq 0,
\end{equation*}
for all $0\leq a\leq t$.
\end{proposition}
\begin{proof}
By Theorem \ref{theorem:check}, we have that for any interval $[a,b]$
\begin{align*}
\min_{t\in [a,b]}R(a,t)&=\min_{t\in [a,b]}E[X(t)X(a)]\\
&=\min_{t\in [a,b]} E[(X(t)-X(a)+X(a))X(a)]\\
&=\min_{t\in [a,b]}E[(X(t)-X(a))(X(a)-X(0))]+E[X^2(a)]\\
&=E[X^2(a)]+\min_{t\in [a,b]}E[(X(t)-X(a))(X(a)-X(0))]\\
&=E[X^2(a)].
\end{align*}
Therefore, we have that for $0\leq a\leq t$ that
\begin{equation*}
E[(X(t)-X(a))(X(a)-X(0))]\geq 0.
\end{equation*}
\end{proof}
\subsection{Increment of Processes with Nonnegatively Correlated Increments}
In this subsection we find the optimal measure for the increment process, $X_Y(t)=Y(t+h)-Y(t)$ of a continuous centered Gaussian process $Y(t)$ with stationary increments whose increment function $f_Y^h(t)=E[Y^2(t+h)]-E[Y^2(t)]$ satisfies some growth conditions which are easily numerically verified for examples chiefly including fractional Gaussian noise for $H\geq 1/2$.
First, we need a technical lemma which will let us handle the case where $[a,b]$ is an interval such that $b-a\leq h$.
\begin{lemma}\label{lemma:function}
Let $\rho \in C^1((0,h),\mathbb R)$ be a continuously differentiable function such that $\rho'>0$ and $\rho'$ is decreasing on the interval $(0,h)$ for some $h>0$. Then for $b\in (0,h)$ the function $\phi(t):=\rho(t)+\rho(b-t)$ has a unique extrema on $(0,b)$ at $t=b/2$, which is a maxima.
\end{lemma}
\begin{proof}
As $\rho$ and thus $\phi$ is differentiable, we have that
$$\phi'(t)=\rho'(t)-\rho'(b-t).$$
Therefore, $\phi$ has a local extrema if and only if
$$\rho'(t)=\rho'(b-t).$$
If $t\neq b/2$ satisfies the above, this contradicts the monotonicity of $\rho'$. Therefore $t=b/2$ is the only solution. By concavity, it is a maxima.
\end{proof}
The following lemma decomposes the covariance function of an increment process into terms involving the increment function $f_Y^h$. This will ease analysis and make useful Lemma \ref{lemma:function} in the computations to come.
\begin{lemma}\label{lemma:covariance}
Let $Y(t):[0,\infty)\to \mathbb R$ be a continuous centered Gaussian process which has stationary increments with covariance function $R_Y(s,t)$ and variance function $V_Y(t)=R_Y(t,t)$. We also extend the variance process to all of $\mathbb R$ by imposing that $V_Y(t)=V_Y(-t)$ for all $t\in \mathbb R$. Consider the increment process $X_Y(t)=Y(t+h)-Y(t)$ for fixed $h>0$. Then $X_Y$ is a continuous centered Gaussian process with covariance function
\begin{equation}
\Gamma(t-s)=R(s,t)=\frac12(f_Y^h(t-s)+f_Y^h(s-t)),
\end{equation}
where we have denoted $f_Y^h(t):=V_Y(t+h)-V_Y(t).$
\end{lemma}
\begin{proof}
$X_Y$ is clearly a continuous centered Gaussian process, so we only need to compute the covariance.
\begin{align*}
R(s,t)&=E[X_Y(s)X_Y(t)]\\
&=E[(Y(s+h)-Y(s))(Y(t+h)-Y(t))]\\
&=R_Y(s+h,t+h)-R_Y(s,t+h)-R_Y(t,s+h)+R_Y(s,t).
\end{align*}
By stationarity of increments, we have that
$$R_Y(u,v)=\frac{1}{2}\left(V_Y(u)+V_Y(v)-V_Y(u-v)\right).$$
Using this expression for $R_Y$ implies that
\begin{align*}
2R(s,t)&=V_Y(s+h)+V_Y(t+h)-V_Y(s-t)-V_Y(s)-V_Y(t+h)+V_Y(s-t-h)\\
&~-V_Y(t)-V_Y(s+h)+V_Y(t-s-h)+V_Y(s)+V_Y(t)-V_Y(t-s)\\
&=-V_Y(s-t)+V_Y(s-t-h)+V_Y(t-s-h)-V_Y(t-s).
\end{align*}
Recalling that we imposed $V_Y(u)=V_Y(-u)$ for $u\in \mathbb R$ concludes that
\begin{align*}
2R(s,t)&=-V_Y(t-s)+V_Y(h+t-s)+V_Y(t-s-h)-V_Y(t-s)\\
&=-V_Y(t-s)+V_Y(h+t-s)+V_Y(-t+s+h)-V_Y(t-s)\\
&=f_Y^h(t-s)+f_Y^h(s-t).
\end{align*}
\end{proof}
In light of the above decomposition of $\Gamma$ into increment functions $f_Y^h$, the following lemma describes the relevant behavior of the increment function of processes in interest, and motivates our key Assumption \ref{assumption:for-first case}.
\begin{lemma}
Let $Y(t):[0,\infty)\to \mathbb R$ be a centered continuous Gaussian process with stationary nonnegatively correlated increments and covariance function $R_Y(s,t)$ and variance function $V_Y(t)=R_Y(t,t)$. Assume that $V_Y$ is differentiable on $(0,\infty)$. Also assume that $Y(0)=0$ a.s. Then the increment function $f_Y^h(t):\mathbb R\to \mathbb R$ defined by $f_Y^h(t)=V_Y(t+h)-V_Y(t)$, where we have extended $V_Y(t)=V_Y(-t)$ for $t\in (-\infty,0)$ is increasing on $\mathbb R$.
\end{lemma}
\begin{proof}
Let $t_2>t_1>0$. Using stationary increments of $Y$ and thus the relation $V_Y(t+h)-V_Y(t)=2R_Y(t+h,h)-V_Y(h)$, we arrive at
\begin{align*}
f_Y^h(t_2)-f_Y^h(t_1)&=V_Y(t_2+h)-V_Y(t_2)-V_Y(t_1+h)+V_Y(t_1)\\
&=2R_Y(t_2+h,h)-V_Y(h)-2R_Y(t_1+h,h)+V_Y(h)\\
&=2 E[(Y(t_2+h)-Y(t_1+h))Y(h)]\\
&=2 E[(Y(t_2+h)-Y(t_1+h))(Y(h)-Y(0)]\\
&\geq 0,
\end{align*}
where the last inequality is because $Y$ has nonnegatively correlated increments.\\
Now, let $-h<t<0$. Then
$$f_Y^h(t)=V_Y(h+t)-V_Y(-t),$$
where $h+t>0$ and $-t>0$. Therefore by assumption, we may differentiate $f_Y^h$ to get that
$$(f_Y^h)'(t)=V_Y'(h+t)+V_Y'(-t)>0+0=0.$$
By Proposition \ref{proposition:non-increment}, we know that $V_Y'(u)>0$ for $u>0$. \\
Finally, let $t<-h$. Then
$$f_Y^h(t)=V_Y(-t-h)-V_Y(-t)=-f_Y^h(-t).$$
Again, differentiation shows that
$$(f_Y^h)'(t)>0.$$
\end{proof}
In light of our decomposition of $\Gamma$ Lemma \ref{lemma:covariance} and the above lemma, the relevant assumptions are on the increment function $f_Y^h$. Therefore we state our assumptions on the process $Y$, its increment $Y(t+h)-Y(t)$ and its increment function $f_Y^h$.
\begin{assumption}\label{assumption:for-first case}
Let $Y(t):[0,\infty)\to \mathbb R$ be a centered continuous Gaussian process with stationary increments and covariance function $R_Y(s,t)$ and variance function $V_Y(t)$, where we extend $V_Y(t)=V_Y(-t)$ for $t\in (-\infty,0)$. Let $X_Y(t)=Y(t+h)-Y(t)$ denote the increment process for fixed $h>0$ with covariance function $\Gamma(t-s)=R(s,t)$. We assume that the increment function $f_Y^h(t)=V_Y(t+h)-V_Y(t)$ is $C^2((0,\infty),\mathbb R)$. We assume that $(f_Y^h)'(t)>0$ for all $t\in \mathbb R$. We also assume that for all $b\in [0,h]$ $(f_Y^h)''(t)+(f_Y^h)''(t-b)<0$, for all $t\in [0,\infty)/\{b\}$.
\end{assumption}
\begin{remark}
The last line in Assumption \ref{assumption:for-first case} is easily verified numerically. It is difficult in general to prove from the nonnegatively correlated increments of $Y$. It is true for fractional Gaussian noise.
\end{remark}
With the above assumption and lemmas, we are able to state and prove first our result for processes satisfying Assumption \ref{assumption:for-first case} on the interval $[a,b]$ with $b-a\leq h$. Later, we will state our result on the interval $[a,b]$ with $b-a=2h$.
\begin{theorem}
Let $Y$ be a function satisfying the assumption \ref{assumption:for-first case} with increment process $X_Y(t)$. Consider the interval $[a,b]$ with $b-a\leq h$. Then the optimal measure associated to the optimization problem $\sigma_\ast^2(a,b)$ defined in \eqref{eq:optimization} is $$\mu^\ast=\frac{1}{2}\left(\delta_a+\delta_b\right).$$
\end{theorem}
\begin{proof}
By Assumption \ref{assumption:for-first case}, the process $X_Y$ is stationary. Therefore without loss of generality we may assume that $a=0$ and $b\leq h$. Recall that by Theorem \ref{theorem:check} we only have to verify that
\begin{equation*}
\min_{u\in [0,b]} \int_0^b R(s,u) \mu^\ast(ds)=\int_0^b\int_0^b R(s,t) \mu^\ast(ds) \mu^\ast(dt),
\end{equation*}
and
\begin{equation*}
\min_{u\in [0,b]} \int_0^b R(s,u) \mu^\ast(ds)=\int_0^b R(s,t) \mu^\ast(ds)
\end{equation*}
for $t\in \{0,b\}$. To verify both properties we note that
\begin{equation*}
\phi(u):=\int_0^b R(s,u) \mu^\ast(ds)=\frac{1}{2}\left(\Gamma(u)+\Gamma(b-u)\right).
\end{equation*}
Recalling from Lemma \ref{lemma:covariance} that $\Gamma(u)=f_Y^h(u)+f_Y^h(-u)$, we may rewrite $\phi$ as
\begin{equation*}
\phi(u)=\frac{1}{4}\left(f_Y^h(u)+f_Y^h(-u)+f_Y^h(b-u)+f_Y^h(u-b)\right).
\end{equation*}
Using the notation of Lemma \ref{lemma:function}, we write
\begin{equation*}
\rho(t)=\frac{1}{4}(f_Y^h(u)+f_Y^h(u-b)).
\end{equation*}
By assumption, we have that $\rho'(t)>0$ and $\rho''(t)<0$ for $t\in [0,b]$. Thus Lemma \ref{lemma:function} says there is a unique extreme point of $\phi(t)$ on $[0,b]$ at $t=b/2$. Therefore
\begin{equation*}
\min_{u\in [0,b]} \phi(u)=\min \{\phi(0),\phi(b)\}=\phi(0)=\phi(b),
\end{equation*}
which verifies both properties.
\end{proof}
In order to handle the case where $b-a=2h$, we need one final assumption.
\begin{assumption}\label{assumption:for-second-case}
Let $Y$ be a process satisfying Assumption \ref{assumption:for-first case} with increment process $X_Y$. We also assume that
\begin{equation}
C^\ast:=1+\frac{\Gamma(h)-\Gamma(2h)}{\Gamma(h)-\Gamma(0)}>0,
\end{equation}
and
\begin{equation}
\gamma(t)=\Gamma(t)+C^\ast\Gamma(h-t)+\Gamma(2h-t)
\end{equation}
has at most one critical point, a maxima on $(0,h)$.
\end{assumption}
Now we can state and prove our result for processes satisfying Assumption \ref{assumption:for-first case} on the interval $[a,b]$ where $b-a=2h$.
\begin{theorem}\label{theorem:second-case}
Let $Y$ be a process satisfying Assumptions \ref{assumption:for-first case} and \ref{assumption:for-second-case} with increment process $X_Y$. Consider the interval $[a,b]$ with $b-a= 2h$. Then the optimal measure associated to the optimization problem $\sigma_\ast^2(a,b)$ defined in \eqref{eq:optimization} for $X_Y$ is $$\mu^\ast =\frac{1}{2+C^\ast}(\delta_a+C^\ast \delta_{a+h}+\delta_b).$$ \end{theorem}
\begin{proof}
By Assumption \ref{assumption:for-first case}, $X_Y$ is a stationary process. Therefore without loss of generality we may work on the interval $[0,2h]$. Recall that by Theorem \ref{theorem:check} we only have to verify that
\begin{equation*}
\min_{u\in [0,2h]} \int_0^b R(s,u) \mu^\ast(ds)=\int_0^b\int_0^b R(s,t) \mu^\ast(ds) \mu^\ast(dt),
\end{equation*}
and
\begin{equation*}
\min_{u\in [0,2h]} \int_0^b R(s,u) \mu^\ast(ds)=\int_0^b R(s,t) \mu^\ast(ds)
\end{equation*}
for $t\in \{0,h,2h\}$. We use the definition of $\mu^\ast$ to get that
\begin{equation*}
\phi(t):=\int_0^b R(s,u) \mu^\ast(ds)=\frac{1}{2+C^\ast}\left(\Gamma(t)+C^\ast \Gamma(h-t)+\Gamma(2h-t)\right).
\end{equation*}
By assumption, we can verify that $\phi(t)$ has no minima on $(0,h)$. Thus by symmetry $\phi$ can have at most one extrema on the interval $(h,2h)$ which would also be a maxima, as well. Therefore there are no minima in the interval $(h,2h)$ either. Then we just need to check that
\begin{equation*}
\phi(0)=\phi(h)=\phi(2h).
\end{equation*}
Again, as $\phi(t)=\phi(2h-t)$ we only need to compute $\phi(0)$ and $\phi(h)$. They are
\begin{equation*}
\phi(0)=\frac{1}{2+C^\ast}\left(\Gamma(0)+C^\ast \Gamma(h)+\Gamma(2h)\right)
\end{equation*}
and
\begin{equation*}
\phi(h)=\frac{1}{2+C^\ast}\left(\Gamma(h)+C^\ast \Gamma(0)+\Gamma(h)\right).
\end{equation*}
By definition of $C^\ast$, we have that
\begin{equation*}
\phi(0)-\phi(h)=\frac{1}{2+C^\ast}\left(\Gamma(0)+\Gamma(2h)-2\Gamma(h)+C^\ast(\Gamma(h)-\Gamma(0)\right)=0
\end{equation*}
\end{proof}
\begin{remark}
The assumption that $\gamma(t)$ has no minima on $(0,h)$ in Theorem \ref{theorem:second-case} can be verified numerically easily. It is however difficult to verify in generality. See the figures at the end of the article for numerical verification.
\end{remark}
\section{Fractional Brownian Motion and Fractional Gaussian Noise}\label{section:Examples}
In this section we give examples of Gaussian processes that satisfy Assumptions \ref{assumption:for-first case} and \ref{assumption:for-second-case}.
\begin{example}
An example of a continuous centered Gaussian process with nonnegatively correlated increments is fractional Brownian motion, $B_H$ with Hurst index $H\geq 1/2$. Fractional Brownian motion is a process whose covariance function is
\begin{equation*}
R(s,t)=\frac{1}{2}\left(t^{2H}+s^{2H}-|t-s|^{2H}\right) ,
\end{equation*}
and variance function is $V(t)=|t|^{2H}$. When $H=1/2$, one recovers standard Brownian motion. It is well known that fractional Brownian motion has stationary nonnegatively correlated increments for $H\geq 1/2$ - see e.g. \cite{Nourdin-fBm}. Therefore Proposition \ref{proposition:non-increment} applies and the optimal measure for $B_H$ on the interval $[a,b]$ is $\mu^\ast=\delta_a$. Therefore the rate function for the optimization problem $\sigma_\ast^2(a,b)$ is
\begin{equation*}
\lim_{u\to \infty} \frac{1}{u^2}\log \left(P\left(\min_{t\in [a,b]} B_H(t)>u\right)\right)= -\frac{1}{2a^{2H}}.
\end{equation*}
\end{example}
\begin{example}
The increment of fractional Gaussian Brownian motion is called fractional Gaussian noise. The covariance function for fractional Gaussian noise is
\begin{equation*}
\Gamma(t-s)=\frac{1}{2}\left( |t-s-h|^{2H}-2|t-s|^{2H}+|t-s+h|^{2H} \right),
\end{equation*}
and increment function
\begin{equation*}
f_Y^h(t)=|t+h|^{2H}-|t|^{2H},
\end{equation*}
whose first derivative is
\begin{equation*}
(f_Y^h)'(t)=2H\left(\frac{t+h}{|t+h|^{2-2H}}-\frac{t}{|t|^{2-2H}}\right)
\end{equation*}
and second derivative is
\begin{equation*}
(f_Y^h)''(t)=2H(2H-1)\left(\frac{(t+h)^{2}}{|t+h|^{4-2H}}-\frac{t^2}{|t|^{4-2H}}\right)
\end{equation*}
Assumptions \ref{assumption:for-first case} and \ref{assumption:for-second-case} can be verified numerically in the case of fractional Gaussian noise, as the following figures will illustrate. In this case, we have that
\begin{equation*}
\sigma_\ast^2(a,b)=\frac{1}{C^\ast+2}\left(\Gamma(0)+C^\ast \Gamma(h)+\Gamma(2h)\right),
\end{equation*}
and
\begin{equation*}
\lim_{u\to \infty} \frac{1}{u^2}\log \left(P\left(\min_{t\in [a,b]} B_H(t+h)-B_H(t)>u\right)\right)= -\frac{C^\ast+2}{2(\Gamma(0)+C^\ast \Gamma(h)+\Gamma(2h))}.
\end{equation*}
\end{example}
\section{Further Questions}
We conclude the paper by asking some questions for possible future directions of research.
\begin{question}
Can Proposition \ref{proposition:converse-1} be strengthened by adding additional constraints on $X$, for example if $X$ has stationary increments?
\end{question}
\begin{question}
Can the assumption that $\gamma(t)$ has no minima on $(0,h)$ in Theorem \ref{theorem:second-case} be verified in generality under Assumption \ref{assumption:for-first case}?
\end{question}
\begin{question}
Does the last line of Assumption \ref{assumption:for-first case} hold for general Gaussian processes with stationary nonnegatively correlated increments?
\end{question}
\begin{question}
What is the optimal measure for the increment of processes $Y$ satisfying Assumption \ref{assumption:for-first case} on intervals $(0,nh)$ for $n\in \{3,4,...\}$?
\end{question}
\begin{question}
What is the optimal measure for the increment of processes $Y$ satisfying Assumption \ref{assumption:for-first case} on intervals $(0,b)$ for $b\neq nh$ and $b>h$?
\end{question}
\FloatBarrier
\begin{figure}[p]
\centering
\includegraphics[width=90mm]{F-half.png}
\caption{The increment process $f_Y^h$ for fBm with $2H=1.5$, $h=1$}
\end{figure}
\begin{figure}[p]
\centering
\includegraphics[width=90mm]{F-half-fir-der.png}
\caption{The first derivative of the increment process $f_Y^h$ for fBm with $2H=1.5$, $h=1$}
\end{figure}
\begin{figure}[p]
\centering
\includegraphics[width=90mm]{F-half-sec-der.png}
\caption{The second derivative of the increment process $f_Y^h$ for fBm with $2H=1.5$, $h=1$}
\end{figure}
\begin{figure}[p]
\centering
\includegraphics[width=90mm]{Gamma-half.png}
\caption{The covariance function $\Gamma$ for fBm with $2H=1.5$, $h=1$}
\end{figure}
\begin{figure}[p]
\centering
\includegraphics[width=90mm]{little-gamma-half.png}
\caption{The function $\gamma$ for fBm with $2H=1.5$, $h=1$}
\end{figure}
\begin{figure}[p]
\centering
\includegraphics[width=90mm]{little-gamma-der-half.png}
\caption{The derivative of the function $\gamma$ for fBm with $2H=1.5$, $h=1$. Notice how there is one critical point on $(0,h)=(0,1)$.}
\end{figure}
\FloatBarrier
\newpage
\bibliographystyle{plain}
| {
"timestamp": "2021-03-09T02:29:53",
"yymm": "2103",
"arxiv_id": "2103.04501",
"language": "en",
"url": "https://arxiv.org/abs/2103.04501",
"abstract": "In this article we prove large deviations principles for high minima of Gaussian processes with nonnegatively correlated increments on arbitrary intervals. Furthermore, we prove large deviations principles for the increments of such processes on intervals $[a,b]$ where $b-a$ is either less than the increment or twice the increment, assuming stationarity of the increments. As a chief example, we consider fractional Brownian motion and fractional Gaussian noise for $H\\geq 1/2$.",
"subjects": "Probability (math.PR)",
"title": "Large Deviations for High Minima of Gaussian Processes with Nonnegatively Correlated Increments",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9770226294209298,
"lm_q2_score": 0.7248702702332475,
"lm_q1q2_score": 0.7082146574123473
} |
https://arxiv.org/abs/1801.02923 | Wirtinger Numbers for Virtual Links | The Wirtinger number of a virtual link is the minimum number of generators of the link group over all meridional presentations in which every relation is an iterated Wirtinger relation arising in a diagram. We prove that the Wirtinger number of a virtual link equals its virtual bridge number. Since the Wirtinger number is algorithmically computable, it gives a more effective way to calculate an upper bound for the virtual bridge number from a virtual link diagram. As an application, we compute upper bounds for the virtual bridge numbers and the quandle counting invariants of virtual knots with 6 or fewer crossings. In particular, we found new examples of nontrivial virtual bridge number one knots, and by applying Satoh's Tube map to these knots we can obtain nontrivial weakly superslice links. | \section{Introduction}
Virtual knots were introduced by Kauffman \cite{MR1721925} as a generalization of classical knot theory, and since then many invariants have been developed to help distinguish virtual knots. One can represent virtual knots geometrically as knots in thickened surfaces up to stable equivalence. Therefore, an oriented virtual knot invariant is also an invariant of a knot in a thickened surface. Among these invariants, the virtual bridge number has been studied in \cite{MR3334661,MR2468375,MR3024023,MR2812268,MR3105303,MR3431020}. A naive way to determine an upper bound for the virtual bridge number is to consider a virtual knot diagram from a knot table, and count the number of overbridges in the diagram. However, since diagrams from knot tables are crossing number minimizing and not necessarily bridge number minimizing, one can get upper bounds that are much larger than the actual virtual bridge numbers. Previously, more accurate upper bounds were obtained by performing a sequence of extended Reidemeister moves on virtual knot diagrams to reduce the number of overbridges. Finding such a sequence of moves can be time-consuming and difficult. This motivates the search for an alternative way to obtain stronger upper bounds from the diagrams without having to perform any extended Reidemeister moves.
In \cite{blair2017wirtinger}, the authors defined the Wirtinger number of a classical link in 3-space to be the minimum number of meridional generators of the link group where all the relations in the group presentation are iterated Wirtinger relations in the link diagram and showed that it equals the bridge number of the link. This result has some beneficial consequences. First, the Wirtinger number of a link is bounded below by the meridional rank of the link group. Therefore, the main theorem of \cite{blair2017wirtinger} gave rise to an alternative approach to Cappell and Shaneson's Meridional Rank Conjecture \cite{(Ed.)95problemsin}, which asks if the bridge number of a knot equals the meridional rank of the knot group. In particular, the conjecture is true if every link admits a minimal meridional presentation in which all relations arise as iterated Wirtinger relations in a diagram. Furthermore, the Wirtinger number is algorithmically computable and gives rise to a useful combinatorial tool to obtain strong upper bounds on classical bridge numbers from knot diagrams without having to perform Reidemeister moves. This allowed the authors in \cite{blair2017wirtinger} to determine the bridge numbers of nearly half a million classical knots.
In this paper, we extend the notion of the Wirtinger number to virtual links. We show that the Wirtinger number equals the virtual bridge number. As an application, we compute upper bounds for the virtual bridge numbers of virtual knots with 6 or fewer crossings. From these upper bounds, we can obtain further information about other virtual and classical knot invariants. For instance, if $X$ is a finite quandle, then an upper bound for the virtual bridge number of a knot is also an upper bound for the number of coloring of the knot by $X$. In particular, for virtual bridge number one knots, there are only $|X|$ colorings of the knot by $X$, where $|X|$ denotes the order of $X$. Moreover, by applying Satoh's Tube map to these knots we can obtain interesting embeddings of unknotted ribbon tori in the 4-sphere, and considering cross-sections of these tori leads to diagrams of nontrivial weakly superslice link. The proofs presented here are inspired by \cite{blair2017wirtinger}.
\section{Preliminaries}
\subsection{Virtual Links}
In this section, we recall several equivalent definitions of virtual links. The first definition is in terms of a virtual link diagram. A \textit{virtual link diagram} is an immersion of $n$ circles into the 2-sphere such that each double point is marked as either a classical crossing or a virtual crossing (see Figure 1). A \textit{virtual link} is an equivalence class of virtual link diagrams under planar isotopies and the \textit{extended Reidemeister moves} shown in Figure 2. \\
\begin{figure}[!ht]
\centering
\includegraphics[width=0.4\textwidth]{virtualcrossing}
\caption{(Left) A classical crossing. (Right) A virtual crossing.}
\end{figure}
\begin{figure}[!ht]
\centering
\includegraphics[width=0.7\textwidth]{extendedrei}
\caption{Extended Reidemeister moves.}
\end{figure}
A virtual link diagram can be represented as a link diagram in an oriented surface $\Sigma$ by adding handles to the sphere where the diagram is drawn to \textit{desingularize} the virtual crossings (see Figure 3). We may assume that $\Sigma$ is connected because we can take the connected sum of the components if $\Sigma$ is not connected after desingularization.
It is shown in \cite{MR1905687} that one can regard a virtual link as a link diagram in $\Sigma$ up to Reidemeister moves on the diagram, orientation-preserving homeomorphisms of the surface, stabilizations, and destabilizations. The stabilization operation consists of removing two open disks in $\Sigma$ disjoint from the link diagram, and then joining the resulting boundary components by an annulus. The destabilization operation consists of cutting $\Sigma$ along a simple closed curve disjoint from the link diagram, and then capping off the resulting boundary components with a pair of disks. It is well-known that one can also regard a virtual link as an embedded link in thickened surfaces up to ambient isotopies, stabilizations, and destabilizations. Furthermore, Kuperberg \cite{MR1997331} showed that there exists a unique link in a thickened surface of minimum genus corresponding to each virtual link.
\begin{figure}[!ht]
\centering
\includegraphics[width=0.5\textwidth]{tube}
\caption{Desingularizing virtual crossings.}
\end{figure}
The final definition is in terms of Gauss diagrams. Given an oriented virtual link diagram $p:S^1\sqcup S^1 \sqcup \cdots \sqcup S^1\rightarrow \mathbb{R}^2$, its \textit{Gauss diagram} $D$ is a decoration of the oriented circles in the domain of $p$ such that the pre-images of the classical crossings are connected by \textit{chords}, which are signed arrows starting from the over crossing to the under crossing. The sign of the arrow indicates the sign of a crossing using the right hand rule. The classical Reidemeister moves can be translated to moves on the Gauss diagrams. Virtual links are then in one-to-one correspondence with Gauss diagrams modulo the Reidemeister moves \cite{goussarov2000finite}. See Figure 4 for an example of a virtual link diagram, and its corresponding Gauss diagram. It is well-known that a Gauss diagram does not always represent a classical link, but every Gauss diagram corresponds to some virtual link. In a sense, Gauss diagrams give simpler representations of virtual links than virtual link diagrams since virtual crossings are not present. Therefore, we state our results mostly in terms of Gauss diagrams.
\begin{figure}[!ht]
\centering
\includegraphics[width=0.8\textwidth]{vwhitehead}
\caption{A virtual link diagram and its Gauss diagram.}
\end{figure}
\subsection{Virtual Bridge Number}
Let $D$ be a Gauss diagram for a virtual link. A \textit{strand} is a subarc of a circle component from one arrowhead to the next. Observe that a strand contains a finite number (possibly zero) of arrowtails, but does not contain any arrowheads. Two strands are said to be \textit{adjacent} if they are separated by an arrowhead. An \textit{overbridge} is a strand with at least one arrowtail on it. The \textit{bridge number} of $D$ is the number of overbridges of $D$, denoted vb$(D).$ If $L$ is a virtual link, then the \textit{virtual bridge number} of $L$, denoted $\text{vb}(L)$ is the minimum bridge number taken over all Gauss diagrams $D$ of $L$. For example, the Gauss diagram in Figure 4 has two overbridges. It is a well-established fact that there is only one classical link $L$ with $\text{vb}(L) =1$, but there are infinitely many virtual knots whose virtual bridge numbers are equal to one \cite{MR2468375}.
\begin{rem}
In this paper, we only consider Gauss diagrams where each circle component contains at least one arrowtail. If there is a circle component with no arrowtails, we can always add a trivial overbridge by performing the first Reidemeister move on the circle component. In particular, if $L$ is the $n$-component unlink, then $\text{vb}(L) = n$.
\end{rem}
\subsection{Link Group}
Given a Gauss diagram $D$ for a link $L$ with $n$ strands, a presentation of the link group $G_L$ is given by the following construction. The generators of $G_L$ consist of the strands of $D$. Each chord gives rise to a relation. Suppose that a circle component contains $m$ arrowheads coming from chords $c_1,c_2,...,c_m$. These $m$ chords divide the circle component into $m$ strands $a_1,...,a_m$. We order the chords $c_i$ and strands $a_i$ consistently so that the arrowhead of $c_i$ separates $a_i$ from $a_{i+1}$, modulo $n$. If the arrowtail of $c_i$ lies on the strand $b$, we impose the relation $a_{i+1}=b^{\epsilon_i}a_ib^{-\epsilon_i}$, where $\epsilon_i$ is the sign of $c_i$.
\subsection{Wirtinger Number}
We say that $D$ is \textit{$k$-partially colored} if $k$ distinct colors have been assigned to a subset of the strands of $D$. Suppose that $D_1$ is $k$-partially colored. Let $c_p$ be a chord in $D_1$ whose arrowtail lies on a colored strand $a_r$. Suppose further that a strand $a_p$ on one side of the arrowhead of $c_p$ is colored, and the strand $a_q$ on other side of the arrowhead of $c_p$ is not colored. Then, we may extend the color on $a_p$ to $a_q$ to obtain a new $k$-partially colored diagram $D_2$. The process of extending a color in this fashion is called a \textit{coloring move}, which we denote by $D_1 \rightarrow D_2$. Figure 5 demonstrates this process where each chord can take any signs.
For a Gauss diagram $D$ with $n$ strands, we say that $D$ is \textit{$k$-meridionally colorable} if there exists a $k$-partially colored diagram $D_0$ with $k$ colored strands $a_{i_1},...,a_{i_k}$, and a sequence of coloring moves $D_0 \rightarrow D_1 \rightarrow \cdots \rightarrow D_{n-k}.$ We call the strands $a_{i_1},...,a_{i_k}$ of $D_0$ the \textit{seed strands}. The \textit{Wirtinger number} of $D$, denoted $\omega(D)$, is the minimum value of $k$ such that $D$ is $k$-meridionally colorable. Now, let $L$ be a virtual link. The \textit{Wirtinger number} of $L$, denoted $\omega(L)$, is the minimal value of $\omega(D)$ over all Gauss diagrams $D$ representing $L$.\\
\begin{figure}[!ht]
\centering
\includegraphics[width=0.6\textwidth]{colormove}
\caption{A coloring move.}
\end{figure}
It is useful to record the order in which strands are colored. Suppose that $D$ is $k$-meridionally colorable with seed strands $\lbrace a_{i_1},a_{i_2},...,a_{i_k}\rbrace$, and a sequence of coloring moves $D_0 \rightarrow D_1 \rightarrow \cdots \rightarrow D_{n-k}.$ We associate to these coloring moves the \textit{coloring sequence} $\lbrace \alpha_j \rbrace_{j=1}^{n}$ given by $\alpha_j = a_{i_j}$ for $1 \leq j \leq k$. For, $k+1 \leq j \leq n$, we define $\alpha_j$ to be the strand that is colored in $D_{j-k}$, but not colored in $D_{j-(k+1)}$. Furthermore, given a coloring sequence $\{ \alpha_j\}$ we associate to it a \textit{height function} $h : \{ \text{{strands} of $D$} \} \rightarrow \mathbb{R}$ by $h(\alpha_j) = \frac{1}{j+1}$. Given a set of strands $\{ a_i \}_{i=1}^n$ ordered by adjacency, we say that $h$ has a \textit{local maximum} at a strand $a_j$ if the function $h':\lbrace 1,2,...,n \rbrace \rightarrow \mathbb{R}$ defined by $h'(i) = h(a_i)$ has a local maximum at $j$. Observe that the \textit{seed strands} $ a_{i_1},...,a_{i_k}$ generate the link group via iterated application of Wirtinger relations in $D$.
\section{Wirtinger Number and Bridge Number}
In this section, we prove that for a virtual link $L$, its Wirtinger number equals its virtual bridge number. Since a Gauss diagram represents some virtual link, we extend the results proved in \cite{blair2017wirtinger} and rephrase them in terms of Gauss diagrams. We begin by studying the case of knots.
\begin{lem}
Let $D$ be a Gauss diagram of a nontrivial virtual knot. Suppose that the arrowhead of a chord $c_p$ separates $a_p$ from $a_{q}$, and the arrowtail of $c_p$ lies on the strand $a_r$. Further, suppose that both $a_p$ and $a_q$ get assigned the same color $j$ at the end of the coloring process. If $h(a_r) \leq \min\{h(a_p),h(a_{q})\}$, then $\omega(D) = 1$, and $c_p$ is the unique chord with the property that $h(a_r) \leq \min\{h(a_p),h(a_q)\}$.
\end{lem}
\begin{proof}
Since $D$ represents a nontrivial virtual knot, $a_p \neq a_{q}$. Suppose that $h(a_p) > h(a_q) \geq h(a_r)$. Let $\delta_{q-1}$ denote the stage right before $a_q$ receives a color. Since $a_r$ is not colored at the stage $\delta_{q-1}$, this implies that $a_{q}$ must receive its color from some other strand $a_l$. Note that $a_q$ is adjacent to both $a_l$ and $a_p$. The condition that $a_{r}$ is not colored at stage $\delta_{q-1}$, and the fact that the strands of a Gauss diagram of a virtual knot lie on a circle force $a_l$ to be distinct from $a_p$. It is an easy exercise to check that at any stage of the coloring process, the set of strands of $D$ that receive the same color must be connected. Therefore, at stage $\delta_{q-1}$, the strands $a_l$ and $a_p$ will both be colored $j$. In particular, all strands of $D$ except $a_q$ are assigned the color $j$. Now, if $a_r\neq a_q$, we arrive at a contradiction because $h(a_r) \leq h(a_q)$ by assumption. Thus, $a_r$ and $a_q$ are the same strand. This implies that $D$ is 1-meridionally colorable. Furthermore, $a_q$ is the last strand in the Gauss diagram that gets colored. Thus, $c_p$ is the only chord with the property that $h(a_r) \leq \min\{h(a_p),h(a_q)\}$.
\end{proof}
We would like to have a result similar to Lemma 3.1 for virtual links as well. A Gauss diagram $D$ is called \textit{cut-split} if there exist two strands $a_p$ and $a_q$ that are adjacent at some arrowhead of $D$ such that $a_p = a_q$ or if $D$ contains a circle with no chords. For example, the Gauss diagram in Figure 4 is cut-split. The following lemma is the analog of Lemma 3.1 for Gauss diagrams that are not cut-split.
\begin{lem}
Let $D$ be a Gauss diagram of a virtual link that is not cut-split. Suppose that the arrowhead of a chord $c_p$ separates $a_p$ from $a_{q}$, and the arrowtail of $c_p$ lies on the strand $a_r$. If both $a_p$ and $a_q$ get assigned the same color $j$ at the end of the coloring process, then one of the following holds:\\
\begin{enumerate}
\item $h(a_r) > \min\{h(a_p),h(a_{q})\}$
\item The strands that get assigned the color $j$ at the end of the coloring process form one circle component $U$ of the Gauss diagram, and $c_p$ is the unique chord having an arrowhead on $U$ with the property that $h(a_r) \leq \min\{h(a_p),h(a_{q})\}$
\end{enumerate}
\end{lem}
\begin{proof}
Since $D$ is not cut-split, $a_p \neq a_{q}$. Suppose that $h(a_p) > h(a_q) \geq h(a_r)$. Let $\delta_{q-1}$ denote the stage right before $a_q$ receives a color. Since $a_r$ is not colored at the stage $\delta_{q-1}$, this implies that $a_{q}$ must receive its color from some strand $a_l$. Note that $a_q$ is adjacent to both $a_l$ and $a_p$.
If $a_p=a_l$, then the set of strands that are assigned the color $j$ form one circle component $U$ of $D$. Furthermore, there are exactly two arrowheads $c_p$ and $c_p'$ that touch $U$. By the definition of the coloring move, the strand $a_s$ that the arrowtail of $c_p'$ touches is already colored at the stage $\delta_{q-1}$. Therefore, we get the situation \textit{(2)} described in the statement of the lemma.
Suppose now that $a_p\neq a_l$. Then, there are more than two arrowheads touching $U$. As in the proof of Lemma 3.1, at the stage $\delta_{q-1}$, all strands of $U$ except $a_q$ are colored $j$ because otherwise, the strands that get assigned the color $j$ form a disconnected subset of $U$, which cannot happen. Thus, the stage $\delta_q$ is the first stage where every strand in $U$ gets assigned the color $j$. Now we show that $c_p$ is the unique chord incident to $U$ with the property that $h(a_r) \leq \min\{h(a_p),h(a_{q})\}$.
Suppose that there exists a chord $c_p'$, whose arrowhead separates $a_p'$ from $a_{q}'$, and the arrowtail of $c_p'$ lies on the strand $a_r'$. Suppose also that $a_p'$ and $a_q'$ get assigned the same color $j$ at the end of the coloring process, and $h(a_r') \leq \min\{h(a_p'),h(a_{q}')\}$. We can apply the argument in the previous paragraph and see that $a_{q}'$ must receive its color from some other strand $a_l'$, and that $\delta_{q'}$ is the first stage where every strand in $U$ gets assigned the color $j$. This implies that $\delta_q = \delta_{q'}$ and $a_q=a_{q}'$. Suppose $c_p \neq c_p'$. Now, $a_q$ is the arc that connects the arrowhead of $c_p$ and the arrowhead of $c_p'$. Then, at the stage $\delta_q$, $a_r'$ is already colored by the definition of coloring move and $a_r$ is uncolored by assumption. But on the other hand, since $h(a_r') \leq \min\{h(a_p'),h(a_{q}')\}$, $a_r'$ is not yet colored at stage $\delta_q$. This is a contradiction. Thus, $c_p = c_p'$.
\end{proof}
Observe that if $D$ is a $k$-meridionally colorable Gauss diagram of a virtual link $L$ that is not cut-split, then $h$ attains a unique local maximum along each color at the seed strand. This fact was proved rigorously in \cite{blair2017wirtinger} for classical links, and it is not difficult to see that this fact generalizes to virtual links.
\begin{thm}
Let $L$ be a virtual link. The Wirtinger number and the virtual bridge number of $L$ are equal.
\end{thm}
\begin{proof}
Suppose that $L$ is an $N$-component virtual link with $\text{vb}(L)=k$. Let $D$ be a Gauss diagram such that $\text{vb}(D) = \text{vb}(L)$. We assign $k$ distinct colors to the overbridges. We pick a point on one of the overbridges, say $b_1$, and travel along the circle until we encounter an arrowhead of the chord $c_1$. If the strand $b_2$ adjacent to $b_1$ is already colored, we do nothing. If $b_2$ is not yet colored, we can use the coloring move to extend the color from $b_1$ to $b_2$ since the arrowtail of $c_1$ is on some overbridge which has already received a color. Then, we start at a point on $b_2$, and follow the same procedure to make sure that the strand $b_3$ adjacent to $b_2$ receives a color. Continuing in this manner, we can color the whole circle component containing $b_1$. We can apply this procedure to every component of $D$ so that every strand of $D$ receives a color. This shows that the overbridges are the seed strands. Therefore, we have that $\omega(L) \leq \text{vb}(L)$ as desired.
We establish the other inequality by induction on $N$.
First, we consider the case where $N=1.$ Suppose that a virtual knot $L$ admits a Gauss diagram $D$ with $c(D)$ chords, which is $k$-meridionally colorable. We can obtain a knot diagram $\widehat{D}$ on an oriented surface $\Sigma$ of genus $g$ from $D$. Let $I = [-1,1] \subset \mathbb{R}$. Let $f:\Sigma\times I \rightarrow \mathbb{R}$ be the standard Morse function. We will construct a smooth embedding of $L$ in $\Sigma \times I$ with exactly $k$ maxima. To that end, for $t\in \mathbb{R}$, let $\Sigma_t = \Sigma \times \lbrace t \rbrace$ and $\Sigma_{[s,t]} = \Sigma \times [s,t]$. First, we embed the diagram $\widehat{D}$ in the level surface $\Sigma_0$. Next, we embed a copy $\widetilde{a_i}$ of each strand $a_i$ of $\widehat{D}$ in the level $\Sigma_{h(a_i)}$ in such a way that the orthogonal projection to $\Sigma_0$ maps $\widetilde{a_i}$ to $a_i$. At the moment, we have an embedding of a collection of disconnected line segments in $\Sigma \times I$. To obtain the knot $K$, we will construct arcs $a_{ij}$ connecting $\widetilde{a_i}$ to $\widetilde{a_j}$ for adjacent strands $a_i$ and $a_j$ in $\widehat{D}$.
Since $a_i$ is adjacent to $a_j$, they are the under-strands of some crossing $c_{ij}$ in $\widehat{D}$ with the over-strand $a_k$. For a small enough $\epsilon > 0$, let $B_{\epsilon}(c_{ij})$ be an open disk centered at $c_{ij}$ with $B_{\epsilon}(c_{ij}) \cap \widehat{D} = \lbrace a_i,a_j, a_k \rbrace.$ Consider the cylinder $B_{\epsilon}(c_{ij}) \times [0,1]$ transverse to the level surfaces of $\Sigma \times I$. It follows that $(B_{\epsilon}(c_{ij}) \times [0,1]) \cap \{\widetilde{a_1},...,\widetilde{a_{c(D)}} \} = \{\widetilde{a_i},\widetilde{a_j},\widetilde{a_k}\}$ for a small enough $\epsilon > 0$. To construct the arc $a_{ij}$, we need to consider two cases:\\
\textbf{Case I:} $L$ is a virtual knot with $\omega(L) > 1$.\\
\textbf{Case II:} $L$ is a virtual knot with $\omega(L) = 1$.\\
\textbf{Case I:} There are two subcases:\\
Subcase I: Suppose that $a_i$ and $a_j$ get assigned the same color $\mu$. Since $\omega(D) \neq 1$, Lemma 3.1 implies that $h(a_k) > \min\lbrace h(a_i),h(a_j)\rbrace$. Then $a_{ij}$ can be chosen so that the orthogonal projection of $\widetilde{a_k} \cup (\widetilde{a_i} \cup a_{ij} \cup \widetilde{a_j})$ to the level $\Sigma_0$ is the subset of $D$. (see Figure 6).
Subcase II: Suppose that $a_i$ and $a_j$ get assigned distinct colors. Let $x_{ij}$ be a point in $(B_{\epsilon}(c_{ij}) \times [0,1]) \cap \Sigma_{1/(c(D)+2)}$ so that when we orthogonally project $x_{ij}$ to the plane $\Sigma_0$, $x_{ij}$ gets mapped to the crossing $c_{ij}$. Then, we construct $a_{ij} \subset B_{\epsilon}(c_{ij}) \times [0,1]$ as the union of a smooth arc connecting $\widetilde{a_i}$ to $x_{ij}$ and another smooth arc connecting $x_{ij}$ to $\widetilde{a_j}$. As $h(a_k) \geq 1/(c(D)+1) > 1/(c(D)+2)$, $a_{ij}$ can be chosen so that the orthogonal projection of $\widetilde{a_k} \cup (\widetilde{a_i} \cup a_{ij} \cup \widetilde{a_j})$ to the level $\Sigma_0$ is the subset of $\widehat{D}$ (see Figure 6).\\
\begin{figure}
\centering
\begin{subfigure}[b]{0.475\textwidth}
\includegraphics[width=\textwidth]{embed1}
\end{subfigure}
\begin{subfigure}[b]{0.5\textwidth}
\includegraphics[width=\textwidth]{embed2}
\end{subfigure}
\caption{At left, the construction of $a_{ij}$ in Subcase I. At right, The construction of $a_{ij}$ in Subcase II.}
\end{figure}
Once we construct $a_{ij}$ for each crossing, we obtain an embedding $\widetilde{L}$ of $L$ in $\Sigma \times I$. To see that $\widetilde{L}$ has exactly $k$ local maxima, we perturb the knot slightly. Suppose that $a_i$ is adjacent to $a_{i-1}$ and $a_{i+1}$. Let $\widetilde{c_{ij}}$ be the point in $a_{ij}$ that orthogonally projects to $c_{ij}$ in $\Sigma_0$. If $a_i$ is a seed strand, we perturb the subarc $a_{(i-1)i} \cup \widetilde{a_i} \cup a_{i(i+1)}$ from $\widetilde{c_{(i-1)i}}$ to $\widetilde{c_{i(i+1)}}$ to obtain a smooth arc that monotonically increases to the midpoint of $\widetilde{a_i}$ and monotonically decreases from there.
On the other hand, if $a_i$ is not a seed strand, then either $a_i$ has the same color as $a_{i-1}$ and $h(a_i) < h(a_{i-1})$, or it has the same color as $a_{i+1}$ and $h(a_i) < h(a_{i+1})$. As $\omega(D) > 1$, it follows from the connectedness of the set of strands having the same color that if $a_{i-1}$ and $a_{i+1}$ have the same color, then $h(a_{i+1}) < h(a_i) < h(a_{i-1})$ or vice versa. This allows us to isotope the knot in the following way regardless of the ways $a_{i+1}$ and $a_{i-1}$ are colored: we perturb the subarc $a_{(i-1)i} \cup \widetilde{a_i} \cup a_{i(i+1)}$ from $\widetilde{c_{(i-1)i}}$ to $\widetilde{c_{i(i+1)}}$ to obtain a smooth arc that is strictly increasing if $h(a_{i-1}) < h(a_{i+1})$ or strictly decreasing if $h(a_{i-1}) > h(a_{i+1})$. \\
\textbf{Case II:} Since $\omega(D) = 1$, we may not have the property that $h(a_k) > \min\lbrace h(a_i),h(a_j)\rbrace$ for all crossings on the knot diagram. But by Lemma 3.1, there is only one crossing $c_{ij}$ on the knot diagram with the property that $h(a_k) \leq \min\lbrace h(a_i),h(a_j)\rbrace$. This means that at every crossing except $c_{ij}$, we can construct $a_{ij}$ in the same way as in Subcase I of Case I. Now, let $x_{ij}$ be a point in $(B_{\epsilon}(c_{ij}) \times [0,1]) \cap \Sigma_{1/(c(D)+2)}$ so that when we orthogonally project $x_{ij}$ to the plane $\Sigma_0$, $x_{ij}$ gets mapped to the crossing $c_{ij}$. We construct $a_{ij}$ as the union of two smooth, monotonic arcs, connecting $x_{ij}$ to endpoints of $\widetilde{a_i}$ and $\widetilde{a_j}$. These two monotonic arcs can be chosen so that the orthogonal projection of $\widetilde{a_k} \cup (\widetilde{a_i} \cup a_{ij} \cup \widetilde{a_j})$ to the level $\Sigma_0$ is a subset of $\widehat{D}$. Then, the arc $a_{ij}$ contains the unique local minimum of the constructed embedding, and we obtain an embedding $\widetilde{L}$ with one maxima.\\
It follows that $\widetilde{L}$ has a projection onto $\Sigma_0$ with $k$ overbridges. Therefore, the Gauss diagram corresponding to the projection has $k$ overbridges, and $\text{vb}(D) = k$.\\
Suppose now that $N > 1$. Suppose that $\omega(L') = \text{vb}(L')$ for all links $L'$ of fewer than $N$ components. Let $D$ be a Wirtinger number minimizing Gauss diagram for $L$. We consider two cases:\\
\textbf{Case A:} $D$ is not cut-split.\\
\textbf{Case B:} $D$ is cut-split.\\
\textbf{Case A:} We will construct a smooth embedding of $L$ in $\Sigma \times I$ with exactly $k$ maxima. There are three subcases:\\
Subcase I: If $a_i$ and $a_j$ get assigned the same color, and $h(a_k) > \min \lbrace h(a_i),h(a_j) \rbrace$, then we follow the construction of $a_{ij}$ in Subcase I of Case I in the virtual knot case.\\
Subcase II: If $a_i$ and $a_j$ get assigned distinct colors, then we follow the construction of $a_{ij}$ in Subcase II of Case I in the virtual knot case.\\
Subcase III: If $a_i$ and $a_j$ get assigned the same color, say $\mu$, and $h(a_k) \leq \min \lbrace h(a_i),h(a_j) \rbrace$, then by Lemma 3.2, the strands that get assigned the color $\mu$ form a circle component of $D$, and $c_{ij}$ is the unique crossing with the property $h(a_k) \leq \min \lbrace h(a_i),h(a_j) \rbrace$. We construct $a_{ij}$ as in Subcase II of Case I of the knot case. Namely, we let $x_{ij}$ be a point in $(B_{\epsilon}(c_{ij}) \times [0,1]) \cap \Sigma_{1/(c(D)+2)}$ so that when we orthogonally project $x_{ij}$ to the plane $\Sigma_0$, $x_{ij}$ gets mapped to the crossing $c_{ij}$. We construct $a_{ij}$ as the union of two smooth, monotonic arcs, connecting $x_{ij}$ to endpoints of $\widetilde{a_i}$ and $\widetilde{a_j}$. These two monotonic arcs can be chosen so that the orthogonal projection of $\widetilde{a_k} \cup (\widetilde{a_i} \cup a_{ij} \cup \widetilde{a_j})$ to the level $\Sigma_0$ is a subset of $\widehat{D}$. Then, the arc $a_{ij}$ contains the unique local minimum in the color $\mu$ of the constructed embedding. \\
After we perform the construction above at every crossing, we obtain a smooth embedding of $L$ in $\Sigma \times I$. Furthermore, the standard height function $f:\Sigma \times I \rightarrow \mathbb{R}$ restricts to a Morse function on $L$ with exactly $k$ minima and $k$ minima. This implies that $\widetilde{L}$ has a projection onto $\Sigma_0$ with $k$ overbridges. Therefore, the Gauss diagram corresponding to the projection has $k$ overbridges, and $\text{vb}(D) = k$.\\
\textbf{Case B:} Let $D$ be the Wirtinger number minimizing Gauss digram for $L$ that is cut-split. That is, there exist two strands $a_p$, and $a_q$ that are adjacent at some arrowhead of $D$ such that $a_p = a_q$ or if $D$ contains a circle with no chords. Let $U$ be a component of $D$ that contains $a_p=a_q$. Observe that $\omega(D\backslash U) = \omega(D) - 1$ because $U$ cannot arise as a result of a coloring move. Also, $\text{vb}(D\backslash U) = \text{vb}(D) - 1$ because $U$ has one overbridge. Now, by the induction hypothesis, $\omega(D) - 1 =\omega(D \backslash U) = \text{vb}(D \backslash U) = \text{vb}(D)-1$ Thus, $\omega(D) = \text{vb}(D)$. Since $D$ is Wirtinger minimizing, $\omega(L) = \text{vb}(L).$ This completes the proof of Theorem 3.3.\\
\end{proof}
\section{Applications}
In this section, we present some applications of Theorem 3.3
\subsection{Computations of the Virtual Bridge Numbers}
\begin{exmp}
We will demonstrate the procedure in the proof of Theorem 3.3 with a specific example. For integers $a$ and $b$ with $a \geq 1$, $b\geq 2$, and $b$ even, Satoh and Tomiyama gave an example of a family of virtual knots $K_2(a,b)$ (see Figure 7) whose real crossing numbers equal to $a+b$ \cite{MR2833547}.
\begin{figure}[!ht]
\includegraphics[width=1\textwidth]{ex1}
\caption{Satoh and Tomiyama's example}
\end{figure}
\begin{figure}
\centering
\begin{subfigure}[b]{0.49\textwidth}
\includegraphics[width=\textwidth]{tor}
\end{subfigure}
\begin{subfigure}[b]{0.49\textwidth}
\includegraphics[width=\textwidth]{shadow}
\end{subfigure}
\caption{}
\end{figure}
Observe that the diagram in Figure 7 has $b$ overbridges. But since the blue strand is a seed strand, we can obtain another diagram of $K_2(a,b)$ with a unique overbridge as follows. We start by $K_2(a,b)$ as a knot diagram $D_2(a,b)$ on a torus $T$. This is demonstrated on the left of Figure 8. Thinking of $K_2(a,b)$ as a knot in $T \times [-1,1]$, we can start by embedding a copy of the blue seed strand in $T \times \{1/2\}$. We then embed copies of the remaining strands of $D_2(a,b)$ on different level surfaces dictated by the coloring move. At the end of the embedding process, we have copies of disconnected arcs, whose orthogonal projections to $T \times \{ 0 \}$ is $D_2(a,b)$. During the coloring process, there is no crossing where locally the overstrand gets assigned a color last. Therefore, we can connect the disjoint copies of strands lying above $T \times \{ 0 \}$ to form a knot in such a way that no addition critical points with respect to the standard Morse function $f:T \times [-1,1] \rightarrow \mathbb{R}$ restricted to $K_2(a,b)$ are created. After a slight perturbation, we see that $f_{K_2(a,b)}$ has a unique maximum. A shadow of the bridge disk corresponding to such a maximum is drawn on the right of Figure 8 in blue.
\end{exmp}
In \cite{MR3334661}, Boden and Gaudreau showed how to use other virtual invariants to compute lower bounds for the bridge numbers. We summarize some of their results here. Suppose that the knot group $G_K$ of a virtual knot $K$ has a presentation with $n$ generators. Then, one can form the Alexander matrix $A$ associated to $G_K$, whose $(i,j)$ entry is the Laurent polynomial obtained from taking the Fox derivative of the $i$-th relation arising in the presentation of $G_K$ with respect to the $j$-th generator and substituting $t$ for each generator. The $k$-th elementary ideal $E_k$ of $G_k$ is the ideal generated by all the $(n-k) \times (n-k)$ minors of $A$. It follows that if $K$ has meridional rank $k$, then, $E_k = (1)$. Using this fact, we can bound the meridional rank and hence, the bridge number of $K$ from below.
\begin{prop}[\cite{MR3334661}]
If $K$ is a virtual knot whose $k$-th elementary ideal is proper and nontrivial, then the knot group $G_K$ has meridional rank at least $k+1$ and $K$ has bridge number at least $k+1$.
\end{prop}
Another lower bound for vb$(K)$ considered by Boden and Gaudreau comes from the Gaussian parity, which is defined in terms of Gauss diagrams as follows. Let $\mathcal{C}(D)$ denote the set of chords in a Gauss diagram $D$ and take $c \in \mathcal{C}(D)$. Then, the \textit{Gaussian parity} is a function $f:\mathcal{C}(D) \rightarrow \mathbb{Z}_2$ where $f(c)$ is the number of elements in $\mathcal{C}(D)$ that intersects $c$ mod 2. Now, given a Gauss diagram $D$ we define its \textit{projection} $P_f(D)$ to be a Gauss diagram obtained from $D$ be erasing all chords $c$ in $\mathcal{C}(D)$ such that $f(c) = 1$. By considering the behavior of $f$ under Reidemeister moves, one can check that if $D_1$ and $D_2$ are equivalent Gauss diagrams, then $P_f(D_1)$ will be equivalent to $P_f(D_2).$ Since $P_f(D)$ is obtain from $D$ by erasing some chords, vb($K$) is bounded below by vb($P_f(K))$. We can now combine these techniques with Theorem 3.3 to determine bridge numbers of more virtual knots.
\begin{exmp}
Figure 9 shows a virtual knot $K$ together with its projection. In \cite{MR3334661}, the authors used $K$ as an example of a knot whose upper and lower quandles are trivial, but has vb$(K)> 1$. Combining with Theorem 3.3, we can conclude that vb$(K)$ = 2. More specifically, one can verify that the first elementary ideal of the knot group of $P_f(K)$ is $E_1 = (t+1,3)$. Hence, vb($K$) $\geq$ vb($P_f(K)) = 2$. On the other hand, the green strand and the blue strand in the Gauss diagram in Figure 9 on the left are seed strands. Therefore, we have that vb($K) \leq 2$.
\end{exmp}
\begin{figure}
\centering
\begin{subfigure}[b]{0.35\textwidth}
\includegraphics[width=\textwidth]{2pcolor}
\end{subfigure}
\begin{subfigure}[b]{0.34\textwidth}
\includegraphics[width=\textwidth]{projection}
\end{subfigure}
\caption{}
\end{figure}
\newpage
\begin{exmp}
The authors in \cite{blair2017wirtinger} wrote a program to calculate $\omega(D)$ for a Gauss diagram $D$ representing a classical knot. The original code is available at \cite{pv}. Starting with $k=2$, the program runs across all subsets of size $k$ of the set of strands $s(D)$ and determines whether $D$ is $k$-meridionally colorable. If not, the program repeats the process with all subsets of size $k+1$ of $s(D)$. The algorithm terminates once the first valid coloring is found. The program can be used to calculate $\omega(D)$ for a Gauss diagram $D$ representing a virtual knot if we modify the program to start at $k=1$. This allows us to compute $\omega(D)$ for all Gauss diagrams of virtual knots up to 6 crossings from Jeremy Green's virtual knot table. The spreadsheets containing the results are available at \cite{pp}.
From this table of data, we can also get some information about the quandle counting invariants of the knots. More precisely, for a finite quandle $X$, a \textit{quandle coloring} of a knot diagram $D$ is an assignment of elements of $X$ to the strands of $D$ such that the quandle relation is satisfied at each crossing. The quandle counting invariant of a virtual knot $K$, denoted $Col_X(K)$, is the number of quandle colorings of $K$. If $D$ is $k$-meridionally colorable, then we know that $k$ strands generate the coloring of the whole diagram. So if $|X|$ denotes the order of the quandle, then there are $|X|$ possible choices of elements to assign to each seed strand. Thus, there are $|X|^k$ ways of coloring the whole diagram if we start with the seeds and generate the coloring by a sequence of coloring moves. But some of these colorings may not be quandle colorings. Since there are always $|X|$ trivial quandle colorings of a virtual knot, it follows that $|X| \leq Col_X(K) \leq |X|^k$. This implies that, for instance, virtual bridge number one knots only admit trivial quandle colorings.
\end{exmp}
\subsection{Weakly Superslice Links}
Let $D$ and $D'$ be virtual knot diagrams. We say that $D$ is \textit{welded equivalent} to $D'$ if one can be obtained from the other by a sequence of extended Reidemeister moves, planar isotopies, and welded moves (See Figure 10).
\begin{figure}[!ht]
\centering
\includegraphics[width=0.5\textwidth]{wequiv}
\caption{Welded move.}
\end{figure}
In \cite{MR1758871}, Satoh proved that any ribbon torus in $\mathbb{R}^4$ can be represented by a virtual knot diagram through the correspondence in Figure 11. We denote the ribbon torus presented by $K$ as $Tube(K)$. Furthermore, Satoh showed that if $K$ is welded equivalent to $K'$, then the corresponding ribbon tori are ambient isotopic. Using Satoh's correspondence, virtual bridge number one knots correspond to a particularly simple ribbon torus.
\begin{figure}
\centering
\begin{subfigure}[b]{0.35\textwidth}
\includegraphics[width=\textwidth]{tubereal}
\end{subfigure}
\begin{subfigure}[b]{0.34\textwidth}
\includegraphics[width=\textwidth]{tubevirt}
\end{subfigure}
\caption{}
\end{figure}
\begin{prop}
Let $K$ be a virtual bridge number one knot. Then, Tube(K) is unknotted.
\end{prop}
\begin{proof}
Since vb$(K) = 1,$ there exists a Gauss diagram for $K$ with one overbridge, which is a sequence of arrowtails without any arrowheads. We can then repeatedly apply the welded move to unhook every pair of crossed arrowtails one by one to obtain a Gauss diagram with only parallel chords. By a repeated applications of the Reidemeister I move, we obtain the empty Gauss diagram for the unknot. This means that $K$ is welded equivalent to the unknot, and by Satoh's result, $Tube(K)$ is ambient isotopic to the unknotted torus.
\end{proof}
A $\mu$-component link $L$ in $S^3$ is said to be \textit{weakly superslice} if it is bounds a smooth planar surface properly embedded $F$ in $B^4$ such that the double of $F$ along $L$ produces an unknotted surface of genus $\mu-1$ in $S^4$. Now, from our table of upper bounds for virtual bridge numbers, we can select a knot $K$ with vb$(K) = 1$. By Proposition 4.5, $Tube(K)$ is unknotted, and interesting nontrivial links can arise as its cross-sections.
\begin{exmp}
Consider the virtual bridge number one knot $K$ with Gauss code O1-O2-O3-U1-O4-U3-O5-U6-U2-U5-U4-O6-. The equatorial cross-section $L$ of $Tube(K)$ is depicted in Figure 12. SnapPy identified $L$ as L13n2916 from the link table. It follows by Proposition 4.5, that L13n2916 is weakly superslice.
\begin{figure}[!ht]
\centering
\includegraphics[width=0.6\textwidth]{path929}
\caption{L13n2916.}
\end{figure}
\end{exmp}
\subsection*{Acknowledgments} I thank all reviewers for their thorough critiques. I am grateful to Maggy Tomova, and Ryan Blair for helpful discussions; and to Katie Burke for helping me create the table of upper bounds for the Wirtinger numbers of virtual knots.
\bibliographystyle{plain}
| {
"timestamp": "2019-11-12T02:31:58",
"yymm": "1801",
"arxiv_id": "1801.02923",
"language": "en",
"url": "https://arxiv.org/abs/1801.02923",
"abstract": "The Wirtinger number of a virtual link is the minimum number of generators of the link group over all meridional presentations in which every relation is an iterated Wirtinger relation arising in a diagram. We prove that the Wirtinger number of a virtual link equals its virtual bridge number. Since the Wirtinger number is algorithmically computable, it gives a more effective way to calculate an upper bound for the virtual bridge number from a virtual link diagram. As an application, we compute upper bounds for the virtual bridge numbers and the quandle counting invariants of virtual knots with 6 or fewer crossings. In particular, we found new examples of nontrivial virtual bridge number one knots, and by applying Satoh's Tube map to these knots we can obtain nontrivial weakly superslice links.",
"subjects": "Geometric Topology (math.GT)",
"title": "Wirtinger Numbers for Virtual Links",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9770226294209298,
"lm_q2_score": 0.7248702702332475,
"lm_q1q2_score": 0.7082146574123473
} |
https://arxiv.org/abs/2110.14072 | Efficient solvers for Armijo's backtracking problem | Backtracking is an inexact line search procedure that selects the first value in a sequence $x_0, x_0\beta, x_0\beta^2...$ that satisfies $g(x)\leq 0$ on $\mathbb{R}_+$ with $g(x)\leq 0$ iff $x\leq x^*$. This procedure is widely used in descent direction optimization algorithms with Armijo-type conditions. It both returns an estimate in $(\beta x^*,x^*]$ and enjoys an upper-bound $\lceil \log_{\beta} \epsilon/x_0 \rceil$ on the number of function evaluations to terminate, with $\epsilon$ a lower bound on $x^*$. The basic bracketing mechanism employed in several root-searching methods is adapted here for the purpose of performing inexact line searches, leading to a new class of inexact line search procedures. The traditional bisection algorithm for root-searching is transposed into a very simple method that completes the same inexact line search in at most $\lceil \log_2 \log_{\beta} \epsilon/x_0 \rceil$ function evaluations. A recent bracketing algorithm for root-searching which presents both minmax function evaluation cost (as the bisection algorithm) and superlinear convergence is also transposed, asymptotically requiring $\sim \log \log \log \epsilon/x_0 $ function evaluations for sufficiently smooth functions. Other bracketing algorithms for root-searching can be adapted in the same way. Numerical experiments suggest time savings of 50\% to 80\% in each call to the inexact search procedure. | \section{Introduction}
Backtracking is an inexact line search technique typically used in the context of descent direction algorithms for solving non-linear optimization problems \citep{luenberger, boyd, gill}. After a descent direction is computed, a step size must be chosen by solving an inexact line searching problem that can be written as
\begin{equation}\label{eq:problem}
\text{Find } \hat{x} \in \mathbb{R}_+ \text{ such that } g(\hat{x}) \leq 0;
\end{equation}
for some $g:\mathbb{R}_+ \to \mathbb{R}$ such that $g(x)\leq 0$ for all $x$ less than or equal to an unknown turning point $x^*\in \mathbb{R}_+$ and $g(x)>0$ otherwise. The condition $g(x)\leq 0$ expresses some acceptable criteria for a descent method to attain desired convergence properties, such as the well known Armijo's condition \citep{armijo}, Wolfe's condition \citep{wolfe}, amongst others \citep{burachik,shi,boukis,calatroni,truong,vaswani}. The backtracking procedure, initiated with some pre-specified values of $x_0\geq x^*$ and $\beta \in (0,1)$, sequentially verifies and returns $\hat{x}$ as the first value of the sequence $x_0,\ x_0\beta,\ x_0\beta^2, ...$ that satisfies the inequality in (\ref{eq:problem}), i.e. it usually takes no more than three lines (within a larger routine) as described in Algorithm \ref{alg:back_track}.\\ \\
\begin{algorithm}[H]
$\tilde{x} \leftarrow x_0$\;
\While{$g(\tilde{x})>0$}
{
$\tilde{x} \leftarrow \beta\tilde{x}$\;
}
\caption{Backtracking\label{alg:back_track}}
\end{algorithm} \emph{ }\\[-3mm]
Notwithstanding the relevance of Algorithm \ref{alg:back_track} as a component of a large variety of nonlinear optimization algorithms, the literature has not focused on its study yet. The working principles of the traditional backtracking algorithm are examined here, and a new class of methods for inexact line search with enhanced performance is proposed.
It is shown that the traditional backtracking delivers a $\lceil \log_{\beta} \epsilon/x_0 \rceil $ upper-bound on the number of function evaluations to terminate, where $\epsilon$ is a lower bound on $x^*$. The simplest method belonging to the class proposed here, which is based on the traditional bisection algorithm for root-searching, completes the same task with at most $\lceil \log_2 \log_{\beta} \epsilon/x_0 \rceil$ function evaluations. The same upper bound is provided by another method based on a recent bracketing algorithm for root-searching \cite{oliveira1}, which requires asymptotically only $\sim \log \log \log \epsilon/x_0 $ function evaluations in the case of sufficiently smooth functions. Other root-searching bracketing algorithms can be adapted similarly for performing inexact line searches efficiently.
Numerical experiments are provided, suggesting 50\% to 80\% of function evaluation savings in each call to the inexact search procedure.
\section{Analysis of Traditional Backtracking}
The procedure described in Algorithm \ref{alg:back_track} enjoys the following guarantees:
\begin{theorem}\label{the:back_track}
Assume that $x^*>\epsilon>0$, $\beta \in (0,1)$, $x_0 > x^*$, and $g(x) > 0$ iff $x > x^*$. Then, the backtracking algorithm finds a solution $\hat{x}$ such that $g(\hat{x})\leq 0$ in at most $\lceil\log_\beta \epsilon/x_0\rceil$ iterations, and the solution $\hat{x}$ satisfies $\beta x^* < \hat{x} \leq x^*$.
\end{theorem}
Theorem \ref{the:back_track} is often an unstated and implicit motivation to employ backtracking, since it both guarantees a finite termination in $\lceil\log_\beta \epsilon/x_0\rceil$ iterations\footnote{The exact number of iterations can be more precisely expressed as a function of $x^*$ with the relation $n =\lceil\log_\beta x^*/x_0\rceil$. The solution-independent bound requires $x^*$ to be bounded away from zero, since otherwise, backtracking may require arbitrarily many iterations the closer $x^*$ is to zero.} and gives a guarantee on the location of $\hat{x}$. The more the value of $\beta$ approximates $1.0$ the closer $\hat{x}$ is guaranteed to be to $x^*$, which is the maximum possible step-size within the guarantees associated with $g(x)\leq 0$. The property that $\beta x^*< \hat{x}\leq x^*$ is often key in ensuring that the parent algorithm ``makes the most out of'' each descent direction expensively computed throughout its iterations. Of course, arbitrarily fast procedures could easily be devised that find $g(x)\leq 0$ by taking faster converging sequences to $0$ if this requirement were to be dropped. Hence implicit to applications that make use of backtracking is the requirement that the solution to problem (\ref{eq:problem}) must be ``not too far from $x^*$''.
Of a similar nature to the requirement that $\hat{x}$ is ``not too far from $x^*$'' is the requirement that $x^*$ is ``not too close to zero''. Without this, the algorithm could take arbitrarily long to find $\hat{x}$ the closer $x^*$ is to zero. This second requirement is, again, implicit in the formulation of backtracking procedures and, at times, it is even entailed by the construction of the parent algorithm. For example, assume the stopping criteria of the parent algorithm verifies stagnation in the domain of the objective function $f(\cdot)$. Then, by construction, when the parent algorithm finds one instance of (\ref{eq:problem}) such that ``$x^*$ is too close to zero'', it terminates. Thus, with the exception of the very last iteration, every other iteration will satisfy $x^*\geq \epsilon$.
In practice, any backtracking procedure should include a stopping condition that is activated when the iteration count $i$ becomes greater than an allowed maximum $i_{max}$, in order to guarantee its termination. This is equivalent to the condition $x_0 \beta^i < \epsilon$ for $\epsilon = x_0 \beta^{i_{max}}$. Hence, the assumption that $x^*\geq \epsilon$ for some pre-specified $\epsilon$ seems to be a hypothesis on (\ref{eq:problem}) that applications that make use of backtracking must assume, either explicitly or implicitly.
Both of these requirements, extracted from Theorem \ref{the:back_track} and found implicitly or explicitly in the literature, are now stated formally for the sake of clarity. We require that: \\[-3mm]
\\
\textit{\textbf{Condition 1.} For some pre-specified $\beta$ in $(0,1)$, the solution $\hat{x}$ to problem (\ref{eq:problem}) must satisfy $\beta x^*<\hat{x}$.}\\
\textit{\textbf{Condition 2.} For some pre-specified $\epsilon >0$, the turning point $x^*$ of problem (\ref{eq:problem}) satisfies $\epsilon<x^*$.}\\
\section{Bracketing-based inexact line search}
The following general algorithm is proposed here:\\[-2mm]
\begin{algorithm}[H]
$a \leftarrow \epsilon$\;
$b \leftarrow x_0$\;
\While{$a\leq \beta b$}
{
chose $\tilde{x}$ in $(a,b)$ and evaluate $g(\tilde{x})$\;
update $(a,b)$ according to (\ref{eq:update})\;
}
return $\hat{x} = a$;
\caption{Fast-tracking\label{alg:fast_back_track}}
\end{algorithm}
\emph{ }\\
The update rule in line 5 is defined by:
\begin{equation}
\left\{
\begin{array}{l}
a \leftarrow \tilde{x} \mbox{ if } g(\tilde{x})<0 \\[1mm]
b \leftarrow \tilde{x} \mbox{ if } g(\tilde{x})>0 \\[1mm]
a \leftarrow \tilde{x} \mbox{ and } b \leftarrow \tilde{x} \mbox{ if } g(\tilde{x}) = 0
\end{array}
\right.
\label{eq:update}
\end{equation}
Algorithm \ref{alg:fast_back_track} defines a class of bracketing-based methods for inexact line search because both the turning point $x^*$ and the final solution $\hat{x}$ are kept inside the interval $[a,b]$ throughout the iterations. Different instances of this algorithm are defined by different choices of $\tilde{x}$ in line 4.
\subsection{Geometric bisection fast tracking}
\noindent
Consider the instance of Algorithm \ref{alg:fast_back_track} with the choice of $\tilde{x}$ in line 4 performed according to the choice rule (\ref{eq:choice}):
\begin{equation}
\tilde{x} \equiv \sqrt{ab}
\label{eq:choice}
\end{equation}
This procedure enjoys the following guarantees:
\begin{theorem}\label{the:fast_back_tracking_worst}
Fast-tracking with $\tilde{x}$ given by (\ref{eq:choice}) finds a solution $\hat{x}$ such that $g(\hat{x})\leq 0$ in at most $ \lceil \log_2 \log_{\beta} \epsilon/x_0 \rceil$ iterations and the solution $\hat{x}$ satisfies $\beta x^* < \hat{x}\leq x^*$.
\end{theorem}
\begin{proof}
The proof follows from the fact that the inequalities $\beta x^* < \hat{x}\leq x^*$ are equivalent to $\log_2 \beta < \log_2 \hat{x} - \log_2x^* \leq 0$, which in turn implies that $|\log_2 \beta |> |\log_2 \hat{x} - \log_2x^*|$. Therefore, to produce an estimate $\hat{x}$ to $x^*$ with \emph{relative} precision of at least $\beta$, is equivalent to searching for an estimate $\hat{X} = \log_2 \hat{x}$ of $X^* = \log_2 x^*$ with an \emph{absolute} error of at most $-\log_2 \beta$. Under this logarithmic scale, the bisection method is guaranteed to perform the search task with minmax optimality guarantees. What remains is, quite simply, to translate the bisection method from the logarithmic to the standard scale. This is done as follows: Define $A = \log_2 a$ and $B = \log_2b$; thus, if the bisection method takes the midpoint $X_{1/2}=(A+B)/2$ in each iteration on the logarithmic scale, then, in the standard scale this translates to $X_{1/2} = (\log_2 a+\log_2 b)/2 = (\log_2 ab)/2 = \log_2 (ab)^{1/2}$. Thus, we have that $\tilde{x}$ must be taken to be equal to $\sqrt{ab}$ in the standard scale.
We now verify that when $B-A\leq -\log_2 \beta$, the lower estimate produced by $A = \log_2 a$ satisfies Condition 1, i.e. that for any value of $x^*$ in $(a,b)$ we must have that $\beta x^*<a$. For this, notice that $B-A\leq -\log_2 \beta \implies \log_2b/a \leq \log_2 \beta^{-1} \implies b/a\leq \beta^{-1}$ which in turn implies that $\beta b\leq a$. And, since $x^*$ is less than $b$ the inequality in Condition 1 holds. In fact, we express the condition $B-A\leq -\log_2 \beta$ as $a\leq \beta b$ in the standard scale. What is left now is to verify the number of iterations required by the bisection method over the logarithmic scale.
The bisection method requires at most $n_{1/2}\equiv \lceil\log_2 (B_0-A_0)/\delta \rceil$ iterations to reduce the interval $(A,B)$ to one of length $B-A\leq \delta$. Thus, given that $A_0 = \log_2\epsilon$ and that $B_0 = \log_2x_0$ and $\delta = -\log_2 \beta$ we find that $n_{1/2}$ is equal to $\lceil \left(\log_2 (B_0-A_0)/\log_2 \beta\right) \rceil = \lceil\log_2\left(( \log_2 x_0-\log_2\epsilon)/\log_2 \beta\right) \rceil = \lceil\log_2 \log_{\beta} x_0/\epsilon \rceil$.
\end{proof}
Thus, an immediate consequence of Theorem \ref{the:fast_back_tracking_worst} is that naive backtracking procedures unnecessarily fall short in terms of worst case performance. They require exponentially more iterations on the worst case when compared to simple binary searching applied to the logarithmic scale. Of course, the $\lceil \log_\beta \epsilon/x_0 \rceil$ upper-bound of standard backtracking can, and often is, carefully minimized by choosing $x_0$ as ``near as possible'' to $x^*$ by means of interpolation bounds. However, the same procedures that minimize $\lceil \log_\beta \epsilon/x_0 \rceil$ can also be used to minimize the tighter $\lceil \log \log_\beta \epsilon/x_0 \rceil$ upper-bound of {\em geometric bisection fast-tracking}. Notice that backtracking for an estimate with relative precision $\beta$ is equivalent to grid searching with a fixed step size on the logarithmic scale: the relative inefficiencies of grid searching when compared to binary searching are well documented in the literature \citep{press}. Thus, this improvement is attained with no appeal to additional assumptions on the conditions of Problem (\ref{eq:problem}), nor on the use of additional function or derivative evaluations. It is attained solely at the cost of computing the method itself, which for choice rule (\ref{eq:choice}) is the additional computation of one square-root per iteration.
The application of the bisection method on the logarithmic scale seems to be an often forgotten technique within the different communities that make use of numerical solvers, and it is certainly under-represented in the literature. We surveyed popular numerical analysis and optimization textbooks, including \citet{press,chapra,boyd,luenberger,gill}, and found no reference to this technique, despite the existence of scattered references in computational forums \footnote{Some early external references to ``geometric bisection'' can be found in \url{codeforces.com/blog/entry/49189}, \url{math.stackexchange.com/questions/3877202/bisection-method-with-geometric-mean} and \url{github.com/SimpleArt/solver/wiki/Binary-Search}} and other isolated references to ``geometric bisection'' in the context of eigenvalue computation \cite{ralha1,ralha2}. In fact, it is easy to find textbook examples that recommend the use of relative error stopping criteria in conjunction with bisection method on a linear scale (see pseudo-code in Figure 5.11 of \citep{chapra} and chapter 9.1 of \citep{press}). This gives rise to the same inefficiency as the one caused by the use of naive backtracking. Similar remarks can be made concerning the use of golden-section searching / Fibonacci-searching for a minimum using relative error stopping criteria. Of course, the underlying metric behind floating point arithmetic most certainly prioritizes relative over absolute error in numerical representations \citep{press,chapra}, hence it is natural to recommend upper-bounding relative errors and, for the same reasons, the proper logarithmic scaling should be recommended before the use of bisection type methods, specifically when the initial interval $(a,b)$ can span several orders of magnitude.
A noticeable exception to the ``inefficiency gap'' between the use of arithmetic and geometric averages in the bisection method, is when the search is already initiated with a small interval $(a,b)$ with\footnote{This way, if the standard bisection method runs till $\beta b \leq a$ for some $\beta$ near one, it will take a number of iterations $n_{1/2} $ of the order of $\sim \log_2 (b_0-a_0)/[a_0(1-\beta)] = \log_2 (b_0-a_0)/a_0 - \log_2 (1-\beta)$, and since $\Delta / x \approx \log_2 (x+\Delta ) - \log x$, we find that $n_{1/2}$ is of the order of $\approx \log_2 (\log_2 b_0 -\log_2 a_0) + \log_2(1-\beta)/\beta = \log_2 \log_2 b_0/ a_0 + \log_2\log_2 \beta$
which simplifies to $\approx \log_2 \log_\beta b_0/ a_0$, the complexity of the bisection method applied to the logarithmic scale.} $a,b>0$ and with $b-a\ll a$, and thus a value of $\beta$ close to $1$. However, standard conditions under which backtracking is used can hardly be expected to satisfy this condition since the further into the run of a descent direction algorithm, the closer $x^*$ is expected to be to zero, and hence, quite the opposite is expected. That is, we find that throughout the run of a standard descent direction algorithm $b-a = x_0-\epsilon$ tends to be much greater than $a = \epsilon$. Furthermore, the choice of $\beta$ near one defeats the purpose of employing inexact searching, since it is often intended as a reduction to the computational cost of exact searching. Instead of choosing $\beta$ near one, in this case one might as well employ exact one dimensional minimization techniques to dictate the step-size.
\subsection{Fast tracking with multi-logarithmic speed-up}
Asymptotic bounds are also improved when the proper scale is adopted. This is shown in the following by making explicit the estimated number of iterations when a hybrid technique for the construction of $\tilde{x}$ is used. The exact construction of $\tilde{x}$ is a straightforward application of the ITP root-searching method, described in \citep{oliveira1}, on the logarithmic scale, and is omitted for brevity.
\begin{corollary}\label{cor:ITP}
If $\tilde{x}$ in line 4 of Algorithm \ref{alg:fast_back_track} is taken as the ITP estimate on the logarithmic scale (instead of the bisection method), then, the same guarantees as Theorem \ref{the:fast_back_tracking_worst} hold; and, if furthermore $g(x)$ is $C^1$ with $x^*$ a simple root, then asymptotically the number of iterations is of the order of $\sim \log \log \log_{\beta} \epsilon/x_0$.
\end{corollary}
\begin{proof}
Follows immediately from the properties of the ITP method \citep{oliveira1}.
\end{proof}
Corollary \ref{cor:ITP} makes use of standard assumptions on the smoothness of $g$, under which even faster convergence can be guaranteed. The ITP method mentioned therein is an efficient first order root-searching method that in the likes of Ridders' rule, Brent's method or Dekker's method, attains a superlinear order of convergence when employed to solve one dimensional root searching problems. However, unlike the aforementioned methods it is the only one known to retain the minmax optimal performance of the bisection method. The exact inner-workings of the ITP method are beyond the scope of this paper. A reader more familiar with other hybrid methods (such as Ridders', Brent's or Dekker's method) may substitute the ITP method for the solver of preference, albeit with weaker worst case guarantees. The point being that \emph{multi-logarithmic speed-ups can be attained} with interpolation based strategies while retaining the logarithmic speed-up on the worst case performance.
\section{Experiments}
Quick numerical comparisons between standard backtracking and fast-tracking are performed here under the optimization set-up in which inexact searching is typically employed. For this we implement a standard gradient descent algorithm with Armijo's condition, from which the corresponding function $g(x)$ is derived, to minimize ten different loss functions $f: \mathbb{R}^{10} \mapsto \mathbb{R}$ described in Table \ref{tab:examples}. Both methods were initiated at $\boldsymbol{x} = [1, 1, ... 1]^T$ with $\beta = 0.8$, $\epsilon = 10^{-10},x_0 = 1$ and were compared after twenty gradient descent iterations. All functions chosen contain at least one local minimum not too far from the initial guess, and thus both implementations produced approximately the same path, hence ensuring the comparison is made on as-similar-as-possible conditions. We report here the results using a fixed upper-bound step value for $x_0$ that does not depend on the size of the gradient, i.e. our standard backtracking sequentially searches for the first term in the sequence $\{ \boldsymbol{x}+\beta^k \nabla f(\boldsymbol{x})/ \|\nabla f(\boldsymbol{x})\| \ \text{ for } k=1,2,...\}$, and fast-tracking calls an external root-searching solver on the logarithmic scale. We use the ITP method\footnote{The ITP parameters used were of $\kappa_1 = 0.1; \kappa_2 = 2$; and, a slack parameter of $N_0 = 0.99$ applied on the rescaled root-searching problem made to satisfy $b-a\leq 1$ in order to benefit of the guarantees of \citep{oliveira1}.}, however other non-linear solvers could have been used with slightly weaker guarantees. Under the conditions here considered the simple ``geometric average'' bisection method would require exactly 7 function evaluations in each iteration if exact arithmetic were used, hence we use this number as a reference point to which standard backtracking and fast-tracking are compared.
Figure \ref{fig:evolution} focuses on the first function considered, and displays the evolution of the number of iterations required by each inexact searching procedure as a function of the gradient-descent iteration. And, as can be seen, fast-tracking tends to reduce the number of iterations the further into the run while backtracking increases the number of iterations the further into the run. This is because interpolation guarantees of the ITP method are improved with the progression of the gradient run (since it is initialized closer to the final solution), while standard backtracking will require more iterations as the ratio of $x^*/x_0$ is reduced the further into the run. In fact, we observe this pattern of progression of both backtracking and of fast-tracking in most runs.
\begin{figure}
\centering
\includegraphics[width=12cm]{untitled.jpg}
\caption{Evolution of number of iterations after each gradient calclation
\label{fig:evolution}}
\end{figure}
\begin{table}[ht]
\caption{\label{tab:examples} Average number of function evaluations required to solve the inexact line-search problem in each iteration of a vanilla gradient descent for different loss functions. The numbers reported are the average obtained after 20 gradient steps under conditions where the minmax ``geometric-average'' binary-searching procedure would require exactly 7 iterations. Bellow, the symbol $V$ stands for an identity matrix plus the Vandermonde matrix obtained in interpolation problems on $n$ Chebyshev points; the vector $\boldsymbol{n}$ is defined as $[1, 2, ..., n]^T$, and every operation on $\boldsymbol{n}$ is done element-wise. }
\begin{tabular}{lc|cc|} & & \rotatebox[origin=c]{-90}{Backtracking \hspace{7pt}} & \rotatebox[origin=c]{-90}{Fast-tracking \hspace{8pt}} \\
\multicolumn{2}{l}{\textbf{Functions -}} \textbf{ } &\hspace{-22.3pt}\raisebox{4pt}{|} \hspace{-4.55pt}\raisebox{-4pt}{|} \textbf{ } & \textbf{ } \\
Simple Quadratic & $\sum_i x_i^2$ & 12.2 & 4.0 \\
High Degree Polynomial & $\sum_i x_i^{2i}$ & 10.8 & 4.8 \\
Vandermonde Interpolation & $\boldsymbol{x}^TV\boldsymbol{x}$ & 16.2 & 3.9 \\
Trigonometric 1 & $\sum_i i\cos(x_i)$ & 7.0 & 4.8 \\
Trigonometric 2 & $\sum_i i\cos(\cos(x_i))$ & 12.0 & 4.0 \\
Log-Poly & $2\log ||\boldsymbol{x}-\boldsymbol{n}^{1/\boldsymbol{n}}||_2$ & 27.2 & 2.7 \\
Quartic & $\tfrac{1}{n}(\sum_i x_i)^4+|\sqrt{\boldsymbol{n}^Tx}|$ & 24.7 & 4.2 \\
Interpolation w/ Regularizer & $\boldsymbol{x}^TV\boldsymbol{x}+||\boldsymbol{x}-\sqrt{\boldsymbol{n}}||_1$ & 49.5 & 3.5 \\
Noisy Quadratic Hard & $||\boldsymbol{x}||^2_2+10^{-3}\sum_i \sin(i/x_i)$ & 26.8 & 3.0 \\
Noisy Quadratic Easy & $||\boldsymbol{x}||^2_2+10^{-3}\sum_i \sin(10^3ix_i)$ & 35.6 & 2.3 \\
& & \textbf{ } & \textbf{ } \\
\textbf{Global Average} & & \textbf{22.2} & \textbf{3.7} \\
\textbf{Global Worst Case} & & \textbf{147} & \textbf{8} \\
\end{tabular}
\end{table}
As can be seen in Table \ref{tab:examples}, fast-tracking vastly improves on standard backtracking under both average and worst-case performance. The global average of fast-tracking is roughly 50\% that of the minmax guarantee of 7 iterations, and, since the ITP solver called made use of the $0.99$ slack variable, the worst case performance over the test set is at most $\lceil 0.99\rceil = 1$ iteration more than the minmax guarantee, i.e. at most $7+1 = 8$ iterations. Standard backtracking only attained a number of iterations equal to the minmax on one instance, and was outperformed by vanilla ``geometric average'' binary searching on every other instance.
Furthermore, by varying the values of $\beta$ and the initial estimate $\boldsymbol{x}$, we verify that the differences in performance are affected too. Our preliminary estimates suggest that for values of $\beta$ near $0.5$ backtracking performs much worse than fast-tracking than what is reported in Table \ref{tab:examples}, multiplying by a factor of $10$ the difference in average iteration count at it's peak value. For $\beta$ near $0$ or $1$ the differences are kept roughly in the range of the ones reported in Table \ref{tab:examples}. Concerning the effect of the initial estimate for $\boldsymbol{x}$, our experiments suggest that the the closer the initial estimate is to the stationary point $\boldsymbol{x}^*$ to which the gradient method converges, the greater the benefit of fast-tracking over backtracking, and when initiated far from $\boldsymbol{x}^*$ the difference in performance is reduced, but not reversed. Finally, analogous experiments were also performed providing the solvers with additional interpolation information and different values of $x_0$ and found no significant difference in the comparative performance reported above. Thus, these results have been kept out for brevity.
\section{Discussion}
The emphasis of ``backtracking papers'' does not typically lie on the three lines that construct and verify which point in the sequence $x_0, x_0\beta ...$ first satisfies $g(x)\leq 0$. In fact, the construction of $g(\cdot)$, and the guarantees associated with the criteria $g(\cdot)\leq 0$, is typically where the contributions of those papers are found. Thus, perhaps justifiably so, it seems that not much research effort has been devoted to those three lines since they, informally speaking, ``get the job done'' and ``some other paper can deal with it''. This is that paper.
Here, we show a simple and proper construction of a procedure that finds $g(x)\leq 0$, and does so with optimal guarantees. A logarithmic speed-up is attained with respect to worst case, and a multi-logarithmic speed-up is attained with respect to asymptotic performance if hybrid interpolation based techniques are employed. These speed-ups are well reflected in experiments achieving roughly 50\% to 80\% time savings in each call to the inexact line-searching procedure.
\begin{acks}
This paper was written when the first author was a graduate student at the Federal University of Minas Gerais.
\end{acks}
\bibliographystyle{ACM-Reference-Format}
| {
"timestamp": "2021-10-28T02:07:59",
"yymm": "2110",
"arxiv_id": "2110.14072",
"language": "en",
"url": "https://arxiv.org/abs/2110.14072",
"abstract": "Backtracking is an inexact line search procedure that selects the first value in a sequence $x_0, x_0\\beta, x_0\\beta^2...$ that satisfies $g(x)\\leq 0$ on $\\mathbb{R}_+$ with $g(x)\\leq 0$ iff $x\\leq x^*$. This procedure is widely used in descent direction optimization algorithms with Armijo-type conditions. It both returns an estimate in $(\\beta x^*,x^*]$ and enjoys an upper-bound $\\lceil \\log_{\\beta} \\epsilon/x_0 \\rceil$ on the number of function evaluations to terminate, with $\\epsilon$ a lower bound on $x^*$. The basic bracketing mechanism employed in several root-searching methods is adapted here for the purpose of performing inexact line searches, leading to a new class of inexact line search procedures. The traditional bisection algorithm for root-searching is transposed into a very simple method that completes the same inexact line search in at most $\\lceil \\log_2 \\log_{\\beta} \\epsilon/x_0 \\rceil$ function evaluations. A recent bracketing algorithm for root-searching which presents both minmax function evaluation cost (as the bisection algorithm) and superlinear convergence is also transposed, asymptotically requiring $\\sim \\log \\log \\log \\epsilon/x_0 $ function evaluations for sufficiently smooth functions. Other bracketing algorithms for root-searching can be adapted in the same way. Numerical experiments suggest time savings of 50\\% to 80\\% in each call to the inexact search procedure.",
"subjects": "Optimization and Control (math.OC); Numerical Analysis (math.NA)",
"title": "Efficient solvers for Armijo's backtracking problem",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9770226267447513,
"lm_q2_score": 0.7248702702332476,
"lm_q1q2_score": 0.7082146554724652
} |
https://arxiv.org/abs/1912.09743 | Groups whose prime graph on class sizes has a cut vertex | Let $G$ be a finite group, and let $\Delta(G)$ be the prime graph built on the set of conjugacy class sizes of $G$: this is the simple undirected graph whose vertices are the prime numbers dividing some conjugacy class size of $G$, two vertices $p$ and $q$ being adjacent if and only if $pq$ divides some conjugacy class size of $G$. In the present paper, we classify the finite groups $G$ for which $\Delta(G)$ has a cut vertex. | \section{Introduction}
Let \(\Delta\) be a graph with \(n\) connected components; denoting by \(V\) the vertex set of \(\Delta\), an element \(v\in V\) is called a \emph{cut vertex} of \(\Delta\) if the number of connected components of the subgraph induced by \(V\setminus\{v\}\) in \(\Delta\) (i.e., the graph obtained by removing the vertex $v$ and all edges incident to $v$ from $\Delta$) is larger than \(n\).
If $\Delta$ is connected and it has a cut-vertex, then $\Delta$ is said to be \emph{$1$-connected}.
Now, given a finite group \(G\), we consider the graph $\Delta(G)$ defined as follows:
its vertex set \(\V G\) consists of the prime numbers dividing the size of some conjugacy class of $G$, and two vertices $p$ and $q$ are adjacent in $\Delta(G)$ if and only if there exists a conjugacy class of $G$ having size divisible by the product $p q$. A well-established research field in the theory of finite groups investigates the interplay between graph-theoretical properties of \(\Delta(G)\) and the structure of \(G\) itself (see, for instance, the items in the References), and the present paper is a contribution in this framework; more specifically, our aim here is to describe the finite groups \(G\) such that the graph \(\Delta(G)\) has a cut vertex.
Note that, under this assumption, \(\Delta(G)\) is in fact $1$-connected. This follows from Theorem~4 of \cite{D0}: if \(\Delta(G)\) is disconnected then (it has two connected components and) the connected components are complete subgraphs, so \(\Delta(G)\) cannot have any cut vertex in this case.
We will show that $\Delta(G)$ has at most two cut vertices, and we will provide a
complete characterization of the structure of the group $G$, as well as of the graph \(\Delta(G)\), in both the cases when \(\Delta(G)\) has either one or two cut vertices. In the following statements, given a graph $\Delta$ with vertex set $V$, for $r\in V$ we denote by $\Delta -r$ the subgraph induced by \(V\setminus\{r\}\) in \(\Delta\); recall also that the vertex \(r\) of \(\Delta\) is called \emph{complete} if it is adjacent to all the other vertices of~\(\Delta\).
\begin{thmA}
Let $G$ be a finite group such that \(\Delta(G)\) has a cut vertex \(r\). Then the following conclusions hold.
\begin{enumeratei}
\item \(G\) is a solvable group whose Fitting height is at most \(3\).
\item $\Delta(G) - r$ is a graph with two connected components, that are both complete graphs.
\item If \(r\) is a complete vertex of \(\Delta(G)\), then it is the unique complete vertex and the unique cut vertex of \(\Delta(G)\). If \(r\) is non-complete, then \(\Delta(G)\) is a graph of diameter \(3\), and it can have at most two cut vertices.
\end{enumeratei}
\end{thmA}
In order to state Theorem~B, which provides a much deeper description of the finite groups \(G\) such that \(\Delta(G)\) is \(1\)-connected, we need to introduce some terminology. We say that a finite group \(G\) is \emph{reduced} if it does not have any non-trivial normal (equivalently, central) subgroup \(Z\) with \(G'\cap Z=1\). As one quickly realizes (see Proposition~\ref{rid}), the set of conjugacy class sizes of $G$ is the same as the set of conjugacy class sizes of the factor group $G/Z$ by any such subgroup \(Z\); moreover, it is not difficult to see that if \(Z\) is maximal with respect to the above property, then \(G/Z\) is reduced. In view of these remarks, it is sensible and not restrictive to focus on reduced groups.
\begin{ThmB}
Let $G$ be a finite reduced group. Then the graph $\Delta(G)$ has a cut vertex $r$ if and only if, denoting by \(\alpha\) and \(\beta\) the vertex sets of the two complete connected components of \(\Delta(G)-r\), we have
$G = ABR$ where $A \in \hall{\alpha}G$, $B \in \hall{\beta}G$, $R \in \syl rG$ are all non-trivial, $AB$ is an $r$-complement of $G$, $A$ and $B$ are abelian, and (up to interchanging \(\alpha\) and \(\beta\)) one of the following holds.
\medskip
\begin{enumeratei}
\item[{\bf{(I)}}] The Fitting subgroup $\fit G$ is $R$, and the set $\beta$ consists of a single prime $q$. Also, \(\fit{AB}=A\) is cyclic, and $|B| = q$, so \(G\) is nilpotent by metacyclic, of Fitting height \(3\). Furthermore,
for all $x \in R$, either
\begin{description}
\item[(i)] $A^y \leq \cent Gx$ for some $y \in R$ or
\item[(ii)] $B^g \leq \cent Gx$ for some $g \in G$ and $\cent Ax \leq \zent {AB}$.
\end{description}
\bigskip
\item[{\bf{(II)}}] $\fit G = A \times R$ (so \(G\) is nilpotent by abelian, in fact metabelian if \(R\) is abelian), $Z = \zent {AB} < B$, $AB/Z$ is a Frobenius group with kernel \(AZ/Z\), and
either
\begin{description}
\item[(IIa)] $R$ is abelian, $\cent BR = 1$, $Z \neq 1$ and $\cent Bx \leq Z$ for every non-trivial $x \in R$; or
\item[(IIb)] $R$ is non-abelian and either
\begin{description}
\item[(IIb(i))] $G = R \times AB$; or
\item[(IIb(ii))] $\cent Bx \leq Z$ for all $x \in R$ such that $\cent Gx R < G$.
\end{description}
\end{description}
\bigskip
\item[{\bf{(III)}}]
Up to replacing \(R\) by a \(G\)-conjugate of it, we have that \(BR\) is a nilpotent subgroup of \(G\); furthermore, $\fit G = A \times R_0$ with $R_0 < R$, $\cent AR = 1$,
and $[A,B]BR/R_0$ is a Frobenius group with kernel \([A,B]R_0/R_0\). In particular, \(G\) is metanilpotent, in fact metabelian if \(R\) is abelian; in this case, we also have $R_0 = 1$ and $\cent AB \neq~1$.
\end{enumeratei}
\end{ThmB}
In Section~5 we will discuss the various types of groups that appear in Theorem~B, and we will describe the structure of the relevant graphs. We will also see, in Example~\ref{ex}, that the graphs having a cut vertex, which can be realized as $\Delta(G)$ for some finite group $G$, are precisely the $1$-connected graphs whose vertices are covered by two complete subgraphs.
It turns out that the finite reduced groups $G$ for which $\Delta(G)$ has two cut vertices constitute a subclass of the groups described in {\bf(IIa)} of Theorem~B (with respect to one of the cut vertices, whereas they are a subclass of {\bf(III)} with respect to the other; see Remark~\ref{thmC}). In the following statement, we refer to the notation introduced in Theorem~B.
\begin{ThmC}
Let $G$ be a finite reduced group. Then the graph $\Delta(G)$ has two distinct cut vertices \(r\) and \(t\) if and only if, with respect to \(r\) (say), the following holds.
\begin{enumeratei}
\item \(G\) is as in {\bf(IIa)} of Theorem~{\rm B}, with \(t\) lying in \(\beta\).
\item Denoting by \(B^*\) the Hall \(t'\)-subgroup of \(B\), we have \(B^*\leq Z\), and \(B\) acts fixed-point freely (by conjugation) on \([R,B^*]\).
\end{enumeratei}
\end{ThmC}
We briefly digress with the following remark. One problem that may be of interest, concerning the graph \(\Delta(G)\), is to \emph{understand the situation when \(\Delta(G)\) does not contain any cycle}. This property clearly holds if the graph has at most two vertices; moreover, as we have a complete control of the case when \(\Delta(G)\) is disconnected (via Theorem~4 of \cite{D0}), the relevant question in this context is to \emph{classify the finite reduced groups \(G\) such that \(\Delta(G)\) is acyclic, connected, with at least three vertices}. It turns out that this problem is strongly related to our present discussion.
In fact, since the vertices of \(\Delta(G)\) can be partitioned in two subsets each inducing a complete subgraph (\cite[Corollary B]{DPSS}), it is easily seen that \(\Delta(G)\) has at most four vertices if it is acyclic. Therefore, the purpose is to describe the groups \(G\) for which \(\Delta(G)\) is a path of length two or three. In both cases \(\Delta(G)\) has a cut vertex (actually two of them in the latter case), therefore Theorem~B and Theorem~C enable us to complete this classification.
To close with, we mention that the study of cut vertices for the \emph{character degree graph} (i.e. the graph obtained by considering the degrees of irreducible characters, instead of the sizes of the conjugacy classes) has been carried out by M.L. Lewis and Q. Meng in~\cite{LM}.
All the groups considered in the following discussion are tacitly assumed to be finite groups.
\section{Preliminary results}
For a positive integer \(n\), we define \(\pi(n)\) to be the set of prime divisors of \(n\); if \(G\) is a group, \(\pi(G)\) will stand for \(\pi(|G|)\).
Next, we gather some well-known facts concerning conjugacy class sizes of a group.
Given an element $x$ of the group $G$, denote by $x^G$ the conjugacy class of $x$ in $G$, and by \(\pi_G(x)\) the set of prime divisors of \(|x^G|\): if $N$ is a normal subgroup of $G$ then, for any $x \in G$, we have
$\pi_{G/N}(xN) \subseteq \pi_G(x)$ and, for $y \in N$, we have $\pi_N(y) \subseteq \pi_G(y)$. Another elementary remark is that a prime number $p$ does not belong to \(\V G\) if and only if $G$ has a central Sylow $p$-subgroup. Furthermore, the following holds.
\begin{proposition}
\label{rid}
Let \(Z\) be a normal subgroup of \(G\) such that \(G'\cap Z=1\). Then $Z \leq \zent G$ and the set of conjugacy class sizes of \(G/Z\) is the same as the set of conjugacy class sizes of \(G\).
\end{proposition}
\begin{proof} As $[G, Z ] \leq G'\cap Z$, it is clear that $Z$ is contained in the center of $G$. It will be enough to show that, for every \(x\in G\), we have \(\cent{G/Z}{xZ}=\cent G x/Z\). In fact, if \(yZ\) lies in \(\cent{G/Z}{xZ}\), we get \([x,y]\leq G'\cap Z=1\), and therefore \(y\) lies in \(\cent G x\); this proves that \(\cent{G/Z}{xZ}\subseteq \cent G x/Z\), and equality clearly holds.
\end{proof}
As mentioned in the Introduction, the group \(G\) is said to be \emph{reduced} if it does not have any non-trivial subgroup \(Z\) as in the hypothesis of the above proposition, and it not restrictive to focus on reduced groups for the purposes of this paper. Note that, for a reduced group \(G\), we have \(\V G=\pi(G)\).
In the following proposition, we recall the description of the groups $G$ such that $\Delta(G)$ is disconnected.
\begin{proposition}[\mbox{\cite[Theorem~4]{D0}}]
\label{disconnected}
Let $G$ be a group. Then
the graph $\Delta(G)$ is disconnected if and only if $G=AB$, where
$A\trianglelefteq G$ and $B$ are abelian Hall subgroups of $G$ of coprime order, and
$G/Z$, where $Z = \zent G$, is a Frobenius group with Frobenius kernel \(AZ/Z\).
In this case $\Delta(G)$ has two connected components, with vertex sets \(\pi(AZ/Z)\) and \(\pi(BZ/Z)\) respectively, that are both complete.
\end{proposition}
The next lemma is well known and easy to prove. After that, we recall some statements that will come into play, dealing with non-complete vertices of \(\Delta(G)\).
\begin{lemma} \label{product}
Let $G$ be a group and let $x,y \in G$ be such that one of the following holds.
\begin{enumeratei}
\item $x$ and $y$ have coprime orders and they commute.
\item $x \in X $ and $y \in Y$, where $X$ and $Y$ are normal subgroups of $G$ such that $X \cap Y = 1$.
\end{enumeratei}
Then \(\pi_G(x) \cup \pi_G(y) \subseteq \pi_G(xy). \)
\end{lemma}
Given a prime \(p\), as customary, we say that a group is \(p\)-nilpotent if it has a normal Hall \(p'\)-subgroup.
\begin{proposition}
\label{nilpotency}
Let \(G\) be a group; then the following holds.
\begin{enumeratei}
\item Let \(p\), \(q\) be non-adjacent vertices of \(\Delta(G)\). Then \(G\) is either \(p\)-nilpotent or \(q\)-nilpotent, with both abelian Sylow \(p\)-subgroups and Sylow \(q\)-subgroups.
\item If $\pi$ is a set of vertices which are all non-adjacent to a vertex $p$ in $\Delta(G)$,
then $G$ is $\pi$-solvable with abelian Hall $\pi$-subgroups, and the vertices in \(\pi\) are pairwise adjacent.
\end{enumeratei}
\end{proposition}
\begin{proof}
Part (a) comes from~ \cite[Lemma~5 and Theorem~B]{CD2} and part (b) from~\cite[Theorem~C]{CDPS13}.
\end{proof}
We remark that the last conclusion in part (b) of Proposition~\ref{nilpotency} follows from a much more general fact, that will be crucial in our discussion, and that was already mentioned in the Introduction. This is Corollary~B in \cite{DPSS}:
\begin{theorem}
\label{ultimo}
Let \(G\) be a group. Then the vertex set of \(\Delta(G)\) can be partitioned into two subsets, each inducing a complete subgraph of \(\Delta(G)\).
\end{theorem}
\begin{lemma}\label{three}
Let $p,r,q$ be three distinct primes and let $G = PRQ$, where $P \in \syl pG$, $R \in \syl rG$,
$Q \in \syl qG$, $RQ \leq G$, and both $P$ and $PR$ are normal subgroups of $G$. If $\{ p, q\}$ is not an edge of
$\Delta(G)$, then $R$ centralizes either $P$ or $Q$.
\end{lemma}
\begin{proof}
Note that, as \(PR\trianglelefteq\, G\), we have \(R=PR\cap RQ\trianglelefteq\, RQ\). Also, we can assume that both $p$ and $q$ are vertices of $\Delta(G)$, as otherwise either $P$ or $Q$ are
central in $G$. Now, Theorem 24 of~\cite{BDIP} yields that either $R \trianglelefteq\, G$, and hence $[R, P] = 1$,
or $PQ \trianglelefteq\, G$. In the latter case, as above, we have \(Q=PQ\cap RQ\trianglelefteq\, RQ\); therefore both $R$ and $Q$ are normal subgroups of $RQ$, and $[R,Q] = 1$.
\end{proof}
The following lemma introduces an important characteristic subgroup of \(G\), that we denote by \(K_p(G)\), associated to a non-complete vertex \(p\) of $\Delta(G)$. Before stating it, we introduce some more notation.
\begin{defn}
For a group \(G\), we denote by $\nu(G)$ the set of the primes $t\in\pi(G)$ such that
$G$ has a \emph{normal} Sylow $t$-subgroup.
\end{defn}
\begin{lemma}{\cite[Lemma~2.3]{DPSS}.} \label{vertex}
Let $G$ be a group, let $p$ be a non-complete vertex of $\Delta(G)$ and
$P$ a Sylow $p$-subgroup of $G$.
Then $G$ is $p$-solvable, $P$ is abelian, and $[G, P]$ has a normal
$p$-complement $K_p(G)$.
Furthermore, $[K_p(G), P] = K_p(G)$ and, if $ p \not\in \nu(G)$, then there are elements $x$ in $K_p(G)$ such that
$p \in \pi_G(x)$.
\end{lemma}
We note that, using the bar convention in a factor group $\overline {G} = G/N$ (for $N\trianglelefteq\, G$),
we have $\overline{[G,P]} = [\overline{G}, \overline{P}]$, so the image of $K_p(G)$ along the canonical projection is the normal $p$-complement of $[\overline{G}, \overline{P}]$. In particular, if \(p\) is a non-complete vertex also for \(\Delta(\o G)\), then $\overline{K_p(G)} = K_p(\overline{G})$ holds.
We also observe that $p \in \nu(G)$ if and only if $K_p(G) = 1$.
Further, we need a basic result related to the existence of regular orbits in coprime actions
of abelian groups.
\begin{lemma}{\cite[Lemma~2.4]{DPSS}.}\label{action}
Let $G$ be a group such that $G/\fit G$ is abelian. Then there exists an element $g \in G$ such that the set of all prime divisors of $|G/\fit G|$ is contained in $\pi_G(g)$.
\end{lemma}
Finally, we are ready to state a key preliminary result. We refer to the notation introduced in Lemma~\ref{vertex}.
\begin{proposition}\label{gamma}
Let $G$ be a group. Assume that $p$ and $q$ are non-adjacent vertices of $\Delta(G)$, and denote by $P$ and \(Q\) a Sylow $p$-subgroup and a Sylow $q$-subgroup of $G$, respectively. Assume further that $M = K_p(G)$ is a minimal normal subgroup of \(G\), and that $Q$ is not normal in $G$.
Then $M$ is abelian, it has a complement in \(G\), and the following conclusions hold.
\begin{enumeratei}
\item \(\oh q G=Q\cap\cent G M\).
\item
$\o G = G/\cent GM$ is a \(q\)-nilpotent group, \(\fit{\o G}\) is a cyclic group acting fixed-point freely and irreducibly on $M$, and \(\o G/\fit{\o G}\) is cyclic as well. Also, $1\neq \o{P} \leq \fit{\o G}$ and $\o{Q} \cap \fit{\o G} = 1$.
\item Setting \(|M|=r^m\), we have that $|\o Q|$ divides $m$; also, \(q\) does not divide \(r^m-1\), and
$(r^m -1)/(r^{m/|\o Q|} -1)$ divides $|\fit{\o{G}}|$.
\item If \(N\) is a normal subgroup of G such that \(N\cap M=1\), then
\(Q\leq\cent G N\).
\item \(\oh p G=P\cap\zent G\).
\end{enumeratei}
\end{proposition}
\begin{proof} This is a reformulation of Proposition~3.1 in \cite{CDPS13} and Proposition~2.5 in \cite{DPSS}; the proof of \cite[Proposition~2.5]{DPSS} includes an explanation of the fact that the hypotheses of \cite[Proposition~3.1]{CDPS13} are fulfilled under our assumptions.
\end{proof}
We conclude this preliminary section with an application of the tools introduced so far.
\begin{proposition}
\label{pieces}
Let $G$ be a group, and let $\alpha$, $\beta$ be non-empty and disjoint vertex subsets of $\Delta(G)$ such that there are no edges of \(\Delta(G)\) having one extreme in \(\alpha\) and the other in \(\beta\).
Assume also that $\nu(G) \cap \alpha= \emptyset = \nu(G) \cap \beta$.
Then, up to interchanging $\alpha$ and $\beta$, there exists a normal subgroup $K$ of $G$ such that
$K=K_p(G)$ for all $p \in \alpha$ and $K < K_q(G)$ for all $q \in \beta$.
\end{proposition}
\begin{proof}
For $p \in \alpha$ and $q \in \beta$, consider the subgroups \(K_p=K_p(G)\) and \(K_q=K_q(G)\):
we will first show that, say, $K_p < K_q$.
Set \(N=K_p\cap K_q\) and assume, working by contradiction, that $N$ is a proper subgroup of both
$K_p$ and $K_q$. In particular, \(p\) and \(q\) are both (non-complete) vertices of \(\Delta(G/N)\) as well, therefore, as remarked in the paragraph following Lemma~\ref{vertex}, we have $K_p/N = K_p(G/N)$ and $K_q/N = K_q(G/N)$. Now, an application of Lemma~\ref{vertex} to the factor group \(G/N\) yields that there exist two elements \(x\in K_{p}\), \(y\in K_{q}\) such that \(p\in\pi_{G/N}(xN)\) and \(q\in\pi_{G/N}(yN)\); by Lemma~\ref{product}(b), we see that \(pq\) divides \(|(xyN)^{G/N}|\), thus it divides \(|(xy)^G|\) contradicting the fact that \(p\) and \(q\) are non-adjacent in \(\Delta(G)\). We conclude that (say) \(K_{p}=N\), whence \(K_{p}\leq K_{q}\). Also, if \(L\) is a normal subgroup of \(G\) such that \(K_{p}/L\) is a chief factor of \(G\) (so, as above, $K_p/L = K_p(G/L)$), then we can apply Proposition~\ref{gamma}(b) to the group \(G/L\), obtaining that $\o G = G/\cent G{K_{p}/L}$ has a normal Sylow \(p\)-subgroup, and a (non-trivial) Sylow \(q\)-subgroup intersecting \(\fit{\o G}\) trivially. In particular, the roles of \(p\) and \(q\) are not symmetric, and therefore the inclusion of \(K_{p}\) in \(K_{q}\) must be proper. Up to interchanging $p$ and $q$, we thus have $K_p < K_q$.
Next, we claim that \(K_{p_0} < K_{q}\) for every choice of $p_0 \in \alpha$.
In fact, assuming this does not hold, the paragraph above yields \(K_{q} < K_{p_0}\); working in the factor group $\o G = G/\cent G{K_{p}/L}$ as above,
by Lemma~\ref{action} we have that $p_0$ does not divide $|\o G /\fit{\o G}|$, so
$\o{K_{p_0}} = 1$, a contradiction as $\o{K_{p_0}} \geq \o{K_q} > 1$.
Note that, by essentially the same argument, we can see that \(K_{p} < K_{q_0}\) holds as well for every choice of \(q_0 \in \beta \).
We work now to show that, for every choice of $p , p_0 \in \alpha$, we have \(K_{p} = K_{p_0}\).
First, let us see that one of these two subgroups is contained in the other. For a proof by contradiction, assume that \(N=K_{p}\cap K_{p_0}\) is properly contained in both \(K_{p}\) and \(K_{p_0}\). So, we can take normal subgroups \(L\) and \(L_0\) of \(G\), containing \(N\), such that \(K_{p}/L\) and \(K_{p_0}/L_0\) are chief factors of \(G\). Let \(Q\) be a Sylow \(q\)-subgroup of \(G\), where \(q\) lies in \(\beta\); by Proposition~\ref{gamma}(d) applied to the factor group \(G/L\), the normal subgroup \(K_{p_0}L/L\) (which intersects \(K_{p}/L\) trivially) is centralized by \(QL/L\), therefore \([K_{p_0}, Q]\leq L\). But clearly \([K_{p_0}, Q]\) also lies in \(K_{p_0}\), hence it lies in \(N\). In particular, \(QL_0/L_0\) centralizes \(K_{p_0}/L_0\), and thus Proposition~\ref{gamma}(a) (applied to $G/L_0$) yields $QL_0/L_0 \trianglelefteq\, G/L_0$, so $K_q \leq L_0 \leq K_p$, a contradiction by the previous paragraph. Our conclusion so far is that (say) \(K_{p}\leq K_{p_0}\), and it remains to show that equality holds. To this end, setting \(\o G=G/\cent{G}{K_{p}/L}\), observe first that \(\o{K_{p_0}}=1\). Otherwise, setting \(P_0\) to be a Sylow \(p_0\)-subgroup of \(G\), \(\o {K_{p_0}}\) would be a non-trivial (normal) \(p_0'\)-subgroup of \([\o G, \o{P_0}]\), thus \(\o G\) would not have a normal Sylow \(p_0\)-subgroup, yielding \(p_0\mid |\o G/\fit {\o G}|\); but Proposition~\ref{gamma}(b) ensures that also \(q\) divides \(|\o G/\fit {\o G}|\), so that (by Lemma~\ref{action}) \(p_0q\) divides the size of some conjugacy class of \(\o G\), a contradiction. Finally, we know by Proposition~\ref{gamma} that \(K_{p}/L\) has a complement \(H/L\) in \(G/L\), so, in particular, \(K_{p_0}=K_{p}(K_{p_0}\cap H)\); as \((K_{p_0}\cap H)/L\) is normal in \(H/L\) and it centralizes \(K_{p}/L\), we get that \((K_{p_0}\cap H)/L\) is a normal subgroup of \(G/L\) intersecting \(K_{p}/L\) trivially. An application of Proposition~\ref{gamma}(d) to the factor group \(G/L\) gives \([K_{p_0}\cap H, Q]\leq L\), whence \([K_{p_0},Q]=[K_{p}(K_{p_0}\cap H), Q]\leq K_{p}\). But now, if \(K_{p_0}\) is strictly larger than \(K_{p}\), we can take a subgroup \(L_0\) of \(G\), containing \(K_{p}\), such that \(K_{p_0}/L_0\) is a chief factor of \(G\). Proposition~\ref{gamma}(a) applied to \(G/L_0\) yields \([K_{p_0},Q]\not\leq L_0\), a contradiction. We conclude that, in fact, \(K_{p_0}=K_{p}\) holds.
Therefore, we have proved that $K = K_ p < K_q$ for all $p \in \alpha$ and $q \in \beta$.
\end{proof}
\section{Proof of Theorem A}
In this section we prove Theorem~A, whose statement is recalled next. Our proof relies essentially on Theorem~\ref{ultimo}, and on some easy graph-theoretical considerations.
\begin{thmA}
Let $G$ be a group such that \(\Delta(G)\) has a cut vertex \(r\). Then the following conclusions hold.
\begin{enumeratei}
\item \(G\) is a solvable group whose Fitting height is at most \(3\).
\item $\Delta(G) - r$ is a graph with two connected components, that are both complete subgraphs.
\item If \(r\) is a complete vertex of \(\Delta(G)\), then it is the unique complete vertex and the unique cut vertex of \(\Delta(G)\). If \(r\) is non-complete, then \(\Delta(G)\) is a graph of diameter \(3\), and it can have at most two cut vertices.
\end{enumeratei}
\end{thmA}
\begin{proof}
By Theorem~\ref{ultimo}, the vertex set \(\V G\) of \(\Delta(G)\) can be partitioned in two subsets, each inducing a complete subgraph of \(\Delta(G)\): we write the part containing \(r\) as \(\{r\}\cup\alpha\), and we denote by \(\beta\) the other one (note that both \(\alpha\) and \(\beta\) are non-empty in this situation).
Since the graph \(\Delta(G)-r\) is not connected, there are no edges of \(\Delta(G)\) having one extreme in \(\alpha\) and the other extreme in \(\beta\). We conclude that \(\Delta(G)-r\) is a graph whose connected components are the two cliques \(\alpha\) and \(\beta\), so (b) is proved.
On the other hand, since the existence of a cut vertex \(r\) for \(\Delta(G)\) implies that \(\Delta(G)\) is connected by Proposition~\ref{disconnected}, \(r\) must be adjacent to some vertex of \(\beta\), and we have the following dichotomy that proves (c).
\begin{enumeratei}
\item[\(\bullet\)]{\sl The cut vertex \(r\) is a complete vertex}. Then, \(r\) is obviously the unique complete vertex and the unique cut vertex of \(\Delta(G)\).
\item[\(\bullet\)]{\sl The graph \(\Delta(G)\) has no complete vertices at all}. In this situation, it follows at once that a minimal path connecting a vertex in \(\alpha\) to a vertex (in \(\beta\)) not adjacent to \(r\) has length \(3\). Recalling that, whenever \(\Delta(G)\) is connected, its diameter is at most \(3\) (and a characterization of groups for which the bound is attained can be found in \cite{CD}), the claim of (c) concerning the diameter is proved. Also, if \(t\) is another cut vertex of \(\Delta(G)\), then it is easily seen that \(t\) lies in \(\beta\), and \(\{r,t\}\) is the unique edge of \(\Delta(G)\) involving a vertex in \(\{r\}\cup\alpha\) and a vertex in \(\beta\). As a consequence, \(\Delta(G)\) has at most two cut vertices.
\end{enumeratei}
Finally note that, in both the situations described above, the graph \(\Delta(G)\) has at most one complete vertex. Therefore we can apply Theorem~A of \cite{CDPS12}, which yields conclusion (a) and completes the proof.
\end{proof}
\section{Proof of Theorem B}
We will now tackle the substantial part of our analysis. Before proving Theorem~B in full, we will treat separately one of the cases that may occur (namely, the situation that leads to conclusion \({\bf (I)}\) in the statement of Theorem~B). Recall that, for a group \(G\), we defined $\nu(G)$ as the set of the primes $t\in\pi(G)$ such that $G$ has a normal Sylow $t$-subgroup; also, in the following statement, \(\frat G\) denotes the Frattini subgroup of the group \(G\).
\begin{theorem}\label{completecut1} Let \(G\) be a reduced group such that \(\Delta(G)\) has a cut vertex \(r\), and let \(R\) be a Sylow \(r\)-subgroup of \(G\). Denoting by \(\alpha\) and \(\beta\) the vertex sets of the two complete connected components of \(\Delta(G)-r\), assume that
$\nu(G) \cap \alpha = \emptyset = \nu(G) \cap \beta$. Then, up to interchanging \(\alpha\) and \(\beta\), the following conclusions hold.
\begin{enumeratei}
\item \(\fit G = R\).
\item Set \(\Phi=\frat R\) and \(K=K_p(G)\), for some \(p\in\alpha\). Then we have $R/\Phi = K\Phi/\Phi \times \zent{G/\Phi}$, and \(K\Phi/\Phi\) is a chief factor of \(G\) whose centralizer in \(G\) is \(R\).
Furthermore, setting \(\o G=G/R\), we have that \(\fit{\o G}\) is cyclic, it is the \(\alpha\)-Hall subgroup of \(\o G\), and it acts fixed-point freely and irreducibly on $K\Phi/\Phi$. Finally, \(\beta\) consists of a single prime \(q\), \(G\) is \(q\)-nilpotent and \(|\o G/\fit{\o G}|=q\).
\end{enumeratei}
\end{theorem}
\begin{proof}
\medskip
An application of Proposition~\ref{pieces} to the sets $\alpha$ and $\beta$ yields (up to interchanging \(\alpha\) and \(\beta\)) that $K=K_p(G) \trianglelefteq\, G$ for all $p \in \alpha$, and $K < K_q(G)$ for all $q \in \beta$.
As $\pi(G) = \{ r \}\cup\alpha \cup \beta$, and $K_t(G)$ is a $t'$-subgroup for all
$t \in \alpha \cup \beta$, we see that $K$ is an $r$-group, so $K \leq R $.
Let now \(L\trianglelefteq G\) be such that \(K/L\) is a chief factor of \(G\). An application of Proposition~\ref{gamma}(b) to the factor group \(G/L\) (together with Theorem~2.1 of \cite{MW}) yields that $\o G = G/\cent G{K/L}$ is a subgroup of the group of semilinear maps $\Gamma(K/L)$ on \(K/L\), with the cyclic group \(\fit{\o G}\) lying in the subgroup $\Gamma_0(K/L)$ of multiplication maps, and acting (fixed-point freely and) irreducibly on \(K/L\). Also, we get that \(\fit{\o G}\) is the \(\alpha\)-Hall subgroup of \(\o G\) and, taking into account Lemma~\ref{action}, \(\beta\subseteq \pi(\o G/\fit{\o G})\subseteq \beta\cup\{r\}\). As we will see, it turns out that \(\cent G{K/L}\) is in fact \(R\). We proceed through a number of steps.
\smallskip
{\bf{Step 1.}} {\sl The order of \(\o G/\fit{\o G}\) is a power of a prime \(q\) in~\(\beta\)} (hence \(\beta\) consists of a single prime).
For a proof by contradiction, assume that \(|\o G/\fit{\o G}|\) is divisible by two distinct primes \(q\) and \(t\) (where \(q\in\beta\) and possibly \(t=r\)), let \(\o Q\) be a Sylow \(q\)-subgroup and \(\o T\) a Sylow \(t\)-subgroup of \(\o G\); setting \(|K/L|=r^m\),
we observe that \(\o T\) is cyclic, hence the order of \(\cent{\Gamma_0(K/L)}{\o T}\) is \(r^{m/|\o T|}-1\) (see \cite[Lemma~3(i)]{D}). Observe also that there exists a primitive prime divisor \(s\) of \(r^{m/|\o T|}-1\): in fact, this is not the case only if \(m/|\o T|=2\) or \(r^{m/{|\o T|}}=2^6\). But in the former situation, by Proposition~\ref{gamma}(c), we have \(q=2\) against the fact that \(q\) does not divide \(r^m-1\); on the other hand, if \(r^{m/{|\o T|}}=2^6\), then \(q=3\) divides \(2^6-1\), again a contradiction. Now, \(s\) is certainly a divisor of \(r^m-1\), but in fact it also divides \((r^m-1)/(r^{m/|\o Q|}-1)\); otherwise, \(s\) is a common divisor of \(r^{m/|\o T|}-1\) and \(r^{m/|\o Q|}-1\), thus it divides \(r^{d}-1\) where $d={\rm{g.c.d.}}(m/|\o Q|,m/|\o T|)$ and (since \(s\) is a primitive prime divisor of \(r^{m/|\o T|}-1\)) we get that \({m/|\o T|}\) divides \({m/|\o Q|}\), a clear contradiction. Again by Proposition~\ref{gamma}(c), it follows that \(s\) divides \(|\fit{\o G}|\), i.e., there exists an element \(\o x\) of \(\fit{\o G}\) whose order is \(s\); recalling that \(\Gamma_0(K/L)\) is cyclic and it has a unique subgroup of order \(s\), we deduce that \(\o x\) is centralized by \(\o T\). Since, as already observed, \(s\) does not divide \(r^{m/|\o Q|}-1=|\cent{\Gamma_0(K/L)}{\o Q}|\), we deduce that \(\o x\) is not centralized by any Sylow \(q\)-subgroup of \(\o G\), whence \(q\) lies in \(\pi_{\o G}(\o x)\). Also, if \(\o y\) is a generator of \(\o T\), certainly \(\o y\) does not centralize \(\fit{\o G}\); as a consequence, \(\pi_{\o G}(\o y)\) contains a prime \(p\) in \(\alpha\). We conclude that \(|(\o {xy})^{\o G}|\) is divisible by \(pq\), which is not the case. This contradiction shows that \(|\o G/\fit{\o G}|\) is a power of \(q\in\beta\), as claimed.
\smallskip
{\bf{Step 2.}} {\sl \(G\) is \(q\)-nilpotent.}
In fact, if we assume the contrary, then \(G\), and hence \(\o G\), is \(p\)-nilpotent for every \(p\) in \(\alpha\) (see Proposition~\ref{nilpotency}), but this implies that \(\fit{\o G}\) is central in \(\o G\), which is definitely not the case.
\smallskip
{\bf{Step 3.}} {\sl The order of \(\o G/\fit{\o G}\) is \(q\).}
For a proof by contradiction, assume \(|\o G/\fit{\o G}|=q^a\) with \(a>1\). Let \(\o Q\) be a Sylow $q$-subgroup of \(\o G\) and consider a subgroup \(\o{Q_0}\) of \(\o Q\) such that \(|\o{Q_0}|=q^{a-1}\). Writing \(m=q^ab\), we have \(|\cent{\Gamma_0(K/L)}{\o {Q_0}}|=r^{bq}-1\), whereas \(|\cent{\Gamma_0(K/L)}{\o {Q}}|=r^{b}-1\). If \(|\cent{\fit{\o G}}{\o {Q}}|\) is strictly smaller than \(|\cent{\fit{\o G}}{\o {Q_0}}|\), then we can choose \(\o x\in \cent{\fit{\o G}}{\o {Q_0}}\) whose conjugacy class size in \(\o G\) is divisible by~\(q\); on the other hand, a generator \(\o y\) of \(\o {Q_0}\) does not centralize \(\fit{\o G}\), hence its conjugacy class in \(\o G\) has a size divisible by a prime \(p\in\alpha\). But now we get the contradiction that \(pq\) divides \(|(\o{xy})|^{\o G}\). In view of this, it will be enough to show that \(|\cent{\fit{\o G}}{\o {Q}}|<|\cent{\fit{\o G}}{\o {Q_0}}|\) holds.
Recalling that \(\Gamma_0(K/L)\) is a cyclic group, what we need to prove is \[{\rm{g.c.d.}}(r^b-1,\;|\fit{\o G}|)\neq{\rm{g.c.d.}} \left((r^b-1)\left(\dfrac{r^{bq}-1}{r^b-1}\right),\;|\fit{\o G}|\right).\] Assuming the contrary, and considering that \((r^{bq}-1)/(r^b-1)\) is a divisor of \(|\fit{\o G}|\) by Proposition~\ref{gamma}(c), we would get that \((r^{bq}-1)/(r^b-1)\) divides \(r^b-1\), hence \(r^{bq}-1\) divides \((r^b-1)^2\). Since it is not difficult to see, as we did above, that \(r^{bq}-1\) has a primitive prime divisor, we reached a contradiction, and our claim is proved.
\smallskip
{\bf{Step 4.}} {\sl \(R\) is a normal subgroup of \(G\).}
Recalling that every prime of \(\alpha\) is not adjacent to \(q\) in \(\Delta(G)\), Proposition~\ref{nilpotency}(b) yields that there exists an \(\alpha\)-Hall subgroup \(A\) of \(G\), and \(A\) is abelian. Observe also that \(AK\) is a normal subgroup of \(G\), as $KP = [G, P] P \trianglelefteq\, G$ for every $P \in \syl pG$ and $p \in \alpha$. Choosing (again) \(L\trianglelefteq G\) such that \(K/L\) is a chief factor of \(G\), for our purposes (and for this step only) we can clearly assume that the \(r\)-subgroup \(L\) is trivial. By Proposition~\ref{gamma}, we know that \(K\) has a complement \(H\) in \(G\), and this \(H\) can be chosen to contain \(A\), so that \(A=AK\cap H\) is a normal subgroup of \(H\). Setting \(A_0=A\cap\cent H K\) we observe that, for every \(p\in\alpha\), we have \(\oh p {A_0}\trianglelefteq H\) because \(A_0\trianglelefteq H\); but \(\oh p {A_0}\) is clearly normalized by \(K\) as well, so we have \(\oh p {A_0}\leq\oh p G\). Now, Proposition~\ref{gamma}(e) yields that \(\oh p{A_0}\) lies in \(\zent G\) and, as this holds for every choice of \(p\in\alpha\), we deduce that \(A_0\leq\zent G\); in particular, \(A_0\) centralizes a Sylow \(r\)-subgroup \(R_0\) of \(\cent H K\). Recalling that \(H/\cent H K\) is an \(r'\)-group (because \(\o G/\fit{\o G}\) is a \(q\)-group by step 1), we have that \(R_0K\) is a Sylow \(r\)-subgroup of \(G\), and it is enough to show that \(R_0\) is normal in \(\cent H K\) (thus in \(H\)) in order to get \(R_0K\trianglelefteq G\). But the normality of \(R_0\) in \(\cent H K\) follows at once from the fact that \(\cent H K\) is \(q\)-nilpotent, with normal \(q\)-complement \(R_0\times A_0\).
\smallskip
{\bf{Step 5.}} {\sl We have \(\fit G=R\).}
Let \(U\) be a complement for \(R\) in \(G\): we have to show that \(\fit G\cap U=1\). Setting \(S=\fit G\cap U\), our first remark is that \(S\) lies in \(\zent G\). In fact, \(S\) is certainly normal in \(G\); thus, writing \(S=S_q\times S_{\alpha}\) as a direct product of its Sylow \(q\)-subgroup and its Hall \(\alpha\)-subgroup, we have that both \(S_q\) and \(S_{\alpha}\) are normal in \(G\). But \(G\) is \(q\)-nilpotent with abelian Sylow \(q\)-subgroups, therefore \(S_q\) is central in \(G\). On the other hand, considering the usual normal subgroup \(L\) of \(G\) such that \(K/L\) is a chief factor of \(G\), we have that \(S_{\alpha}L/L\) is a normal subgroup of \(G/L\) intersecting \(K/L\) trivially, so Proposition~\ref{gamma}(d) yields \([S_{\alpha},Q]\leq L\) where \(Q\) is a Sylow \(q\)-subgroup of \(G\). Now, \([S_{\alpha},Q]\leq L\cap S_{\alpha}=1\), thus \(S_{\alpha}\) is centralized by a Sylow \(q\)-subgroup of \(G\). Since \(G\) has abelian Hall \(\alpha\)-subgroups and \(S_{\alpha}\) centralizes \(R\), we conclude that \(S_{\alpha}\) lies in \(\zent G\) as well, so \(S\leq \zent G\). This step can be concluded by observing that no prime divisor of \(|S|\) can divide \(|G'\cap\zent G|\), because \(G\) has abelian Sylow subgroups for each of these primes (see \cite[Theorem~5.3]{I}); as a consequence, \(G'\cap S=1\), and our assumption that \(G\) is a reduced group forces \(S=1\). Thus, we proved claim~(a) of our statement.
\smallskip
{\bf{Step 6.}}
The last step is devoted to the proof of claim (b). We start by observing that, for every prime \(p\) in \(\alpha\) and \(P\in\syl p G\), we have \(K=[R,P]\). In fact, we know that \(K=[K,P]\leq[R,P]\); on the other hand, \([R,P]\) is a \(p'\)-subgroup of \([G,P]\), and it is therefore contained in the normal \(p\)-complement \(K\) of \([G,P]\). Taking into account that, as remarked in step 4, an \(\alpha\)-Hall subgroup \(A\) of \(G\) is abelian, we thus get \(K=[R,A]\). Also, an application of \cite[Proposition~3.1]{CDPS13} to the factor group \(G/\Phi\) (recall that here \(\Phi\) is defined as \(\frat R\)) yields that \(K\Phi/\Phi\) is a minimal normal subgroup of \(G/\Phi\), so \(L\) can be chosen to be \(\Phi\cap K\), and \(K\Phi/\Phi\) is isomorphic to \(K/L\) as a \(G\)-module. Now, by Fitting's decomposition we have \(R/\Phi=K\Phi/\Phi\times Z/\Phi\), where \(Z/\Phi\) is set to be \(\cent{R/\Phi}A\) (note that \(Z\) is a normal subgroup of \(G\), as \(AR\trianglelefteq\, G\)); but since \(Z/L\) is a normal subgroup of \(G/L\) intersecting \(K/L\) trivially, Proposition~\ref{gamma}(d) yields that \([Z,Q]\leq L\) (where \(Q\) is a Sylow \(q\)-subgroup of \(G\)) and, in particular, \(Q\) centralizes \(Z/\Phi\). We conclude that \(Z/\Phi\) lies in \(\zent{G/\Phi}\) (in fact, equality clearly holds), and \(\cent G{K/L}=\cent G {K\Phi/\Phi}=\cent G{R/\Phi}=R\). Now all the remaining claims in (b) follow by the description of \(\o G=G/R\) that we made in the previous parts of this proof.
\end{proof}
\begin{rem} \label{rem} Assume that \(G\) is a reduced group satisfying the hypothesis of the previous result, so, \(\Delta(G)\) has a cut vertex \(r\) and \(G\) does not have any normal Sylow subgroup except (eventually) for the prime \(r\). We can summarize the conclusions of Theorem~\ref{completecut1}, taking into account the notation introduced therein, as follows.
Writing \(Z/\frat R\) for the center of \(G/\frat R\), the factor group \(G/Z\) is isomorphic to a subgroup of the affine semilinear group \({\rm A}\Gamma(R/Z)\). Also, the Fitting subgroup of \(G/Z\) is \(R/Z\) and, if \(F/Z\) is the second Fitting subgroup of \(G/Z\), then \(F/R\) is the cyclic Hall \(\alpha\)-subgroup of \(G/R\); as for the top section \(G/F\), it is a group of order~\(q\). Furthermore, we observe that \(\Delta(G/Z)\) is the same as \(\Delta(G)\).
So, the groups appearing as an output in conclusion {\bf(I)} of Theorem~B are well understood (at least as concerns the section over the Frattini subgroup of their normal Sylow \(r\)-subgroup), and they are essentially certain groups of affine semilinear maps.
\end{rem}
\bigskip
We are now ready to prove Theorem~B, that was stated in the Introduction.
\begin{proof}[Proof of Theorem~B]
We start by assuming that $G$ is a reduced group whose graph $\Delta(G)$ has a cut vertex $r$ and, as usual, we denote by \(\alpha\) and \(\beta\) the vertex sets of the two complete connected components of \(\Delta(G)-r\). By Theorem~A, we know that $G$ is solvable.
Let $A \in \hall{\alpha}G$, $B \in \hall{\beta}G$ and $R \in \syl rG$ be such that $ AB$ and $AR$ are subgroups of $G$.
Since no vertex of $\alpha$ is adjacent in $\Delta(G)$ to any vertex of $\beta$, Proposition~\ref{nilpotency}
yields that both $A$ and $B$ are abelian.
\smallskip
Recalling that $\nu(G)$ is the set of the prime divisors $t$ of $|G|$ such that $G$ has a normal Sylow $t$-subgroup, let us first assume that $\nu(G) \cap (\alpha \cup \beta) = \emptyset$. Our aim is to show that conclusion \({\bf (I)}\) holds in this case. By Theorem~\ref{completecut1}, we have that \(R=\fit G\); moreover, \(H = AB\) has a Sylow \(q\)-subgroup $B$ of order \(q\), where $\{ q \} = \beta$, and a cyclic normal \(q\)-complement \(A=\fit H\).
Now, assume that \(x\in R\) does not centralize any conjugate \(A^y\) with \( y \in R\) (hence, any \(G\)-conjugate of \(A\) at all). As a consequence, there exists a prime \(p\in\pi(A)\) which divides the size of \(x^G\). Since, as remarked above, \(p\) is not adjacent to \(q\), certainly \(x\) is centralized by a Sylow \(q\)-subgroup of \(G\); moreover, if there exists an element \(w\) in \(\cent A x\setminus\zent H\), then some prime in \(\pi(H)\) has to divide \(|w^H|\), and this prime is certainly \(q\) because \(A\) is abelian. But now \(q\) divides \(|w^G|\) as well (because \(H\) is isomorphic to \(G/R)\), so \(pq\) divides \(|(xw)^G|\), a contradiction. We deduce that \(\cent A x\) lies in \(\zent H\), and we get case {\bf(I)}.
\smallskip
Assume now, by the symmetry of $\alpha$ and $\beta$, that there exists a prime $ t \in \nu(G) \cap \alpha$.
Note that this implies, by Proposition~\ref{nilpotency}(a), that $G$ is $q$-nilpotent for all $q \in \beta$. Hence, the $\beta$-complement $AR$ is a normal subgroup of $G$, and $G' \leq AR$.
Observe first that $\nu(G) \cap \beta = \emptyset$. In fact, if $q \in \nu(G) \cap \beta$, then
$G = \cent GT \cup \cent GQ$, where $T \in \syl tG$ and $Q \in \syl qG$, which is not possible.
Next, we claim that \(\alpha\subseteq\nu(G)\). In fact assume, working by contradiction, that \(\pi=\alpha\setminus\nu(G)\) is non-empty; then, as shown in step 5 of the proof of \cite[Theorem~A]{DPSS}, we have \(K_q(G)<K_p(G)\) for all \(q\in\beta\) and \(p\in\pi\).
Also, Proposition~\ref{pieces} yields that there exists \(K\trianglelefteq\, G\) such that $K_q(G) = K$ for all $q \in \beta$.
In particular, this implies that $\pi(K) \subseteq \nu(G) \cup \{ r \}$.
Let now $L \leq K$ be a normal subgroup of $G$ such that $K/L$ is a chief factor of $G$ and let
$\o G = G/ \cent G{K/L}$. Observe that, as the Fitting subgroup of $G$ centralizes every chief
factor of $G$, the group $\o G$ is a $\nu(G)'$-group.
By Proposition~\ref{gamma}(b), for all $p \in \pi$ the Sylow $p$-subgroup $\o P$ of $\o G$ intersects
$\fit {\o G}$ trivially, and $\o{B}$ acts fixed-point freely on $K/L$. As $\o B$ is central in $\o G$ (because \(G\) is \(q\)-nilpotent for every \(q\) in \(\beta\)) and $\o A$ is
abelian, it follows that $K_p(\o{G}) = [\o G, \o P]$ is an $r$-group. Hence, $r$ does not divide $|K/L|$, so
$K/L$ is a $t$-group for some $t \in \nu(G)$. For any non-trivial $ xL \in K/L$, we have
$\pi(\o B ) \subseteq \pi_G(x)$, so $x$ is centralized by a Sylow $p$-subgroup $P_0$
of $G$. Since $P_0$ is not contained in $\cent G{K/L}$, there exists $ y \in P_0$ such that $t \in \pi_G(y)$, and hence
$\pi_{G}(xy)$ contains both $\pi(\o B)$ and $t$, a contradiction.
Hence, $\alpha \subseteq \nu(G)$ and $A$ is a normal subgroup of $G$.
We will show, next, that either $R$ or $AB$ is a normal subgroup of $G$.
We first observe that, for every $q \in \beta$, there exists $Q \in \syl qG$ such that $RQ$ is a subgroup of $G$. As $G_0 = PRQ$ is isomorphic to
a normal section of $G$, the graph $\Delta(G_0)$ is a subgraph of $\Delta(G)$ so, in particular, \(\{p,q\}\) is not an edge of \(\Delta(G_0)\).
As both $P$ and $PR$ are normal subgroups of $G_0$, by Lemma~\ref{three}, $R$ commutes with either $P$ or $Q$; in the first case $R$ is normal in $G_0$ and in the second case $PQ$ is normal in $G_0$.
Thus the subgroup $AB$ is non-abelian; otherwise, either $P$ or $Q$ would be central in $G$, a contradiction.
So, by a suitable choice of $p \in \alpha$ and $q \in \beta$, we can assume that $[P, Q] \neq 1$. Since $PQ$ is either a normal subgroup of $G_0$ or isomorphic to a quotient of $G_0$, there are elements $x \in P$ and $y \in Q$ such that $q \in \pi_{G_0}(x) \subseteq \pi_G(x)$ and
$p \in \pi_{G_0}(y) \subseteq \pi_G(y)$.
Assume first that $[R,P] = 1$ (so $R \trianglelefteq\, G_0$) and let $t \in \alpha$ and $T \in \syl tG$.
If $[R, T] \neq 1$, we consider $w \in R$, $w \not\in \cent RT$ and get
$\{t, q\} \subseteq \pi_G(xw)$, a contradiction. So, in this case, $R$ commutes with $A$ and
hence $R$ is a normal subgroup of $G$.
Assume, on the other hand, $[R, Q] = 1$.
Let $\o G = G/A$. If $[\o R, \o B] \neq 1$, then there is $w \in R$ and $t \in \beta$ such that $t \in \pi_{\o G}(\o w)$.
Thus, $\{t, p\} \subseteq \pi_G(yw)$, a contradiction. Therefore, in this case,
$G/A \simeq R \times B$ and $AB$ is the normal $r$-complement of $G$.
\smallskip
We now suppose that $R$ is normal in $G$. Hence, $AR = A \times R = \fit G$, because
$B \cap \fit G \leq \zent G$ has trivial intersection with $G'$ and $G$ is reduced.
Let $Z = \zent {AB}$ and note that $Z \cap A=\cent A B$ is (by Fitting's decomposition) a central direct factor of $G$, so $Z = \oh{\beta}{AB} \leq B$ as $G$ is reduced. Note that $Z < B$, as otherwise $A$ would be central in $G$.
Let $b \in B \setminus Z$ and $a \in \cent Ab$.
If $a \neq 1$, then there exists $q \in \beta$ such that $q \in \pi_G(a)$.
Also, there is $p \in \alpha$ such that $p \in \pi_G(b)$, so we get the contradiction
$\{p, q \} \subseteq \pi_G(ab)$.
Hence $AB/Z$ is a Frobenius group, with kernel $AZ/Z$.
If $[R, B] = 1$, then $G = R \times AB$ and $R$ is non-abelian; so we are in case {\bf(IIb(i))}.
If $R$ is abelian, then $\cent RB$ is a central direct factor of $G$ and hence $\cent RB = 1$ as
$G$ is reduced. So, for every non-trivial $x\in R$ we have $\pi_G(x) \cap \beta \neq \emptyset$ and
hence $\cent Bx \leq \cent BA = Z$ by Lemma~\ref{product}. Note also that in this case $Z \neq 1$,
as otherwise the graph $\Delta(G)$ would be disconnected by Proposition~\ref{disconnected}.
Finally, as $\cent BR \leq Z$ we see that $\cent BR \leq \zent G$. Since $B \cap G' = 1$ and $G$
is reduced, we see that $\cent BR = 1$.
Thus, we have case {\bf(IIa)}.
Assume now that $R$ is non-abelian and that $[R, B] \neq 1$.
Consider an element $x \in R$ such that $\cent Gx R < G$, i.e. such that $\cent Gx$ does
not contain any conjugate of $B$ in $G$. Then there exists a prime $q \in \beta$ such that
$q \in \pi_G(x)$ and again Lemma~\ref{product} implies that $\cent Bx \leq Z$. So, we have case {\bf(IIb(ii))}.
\smallskip
For the last case, assume that $AB$ is the normal $r$-complement of $G$.
By the Frattini argument we can choose $R \leq \norm GB$; therefore, $BR$ is a subgroup of $G$ and, since \(AR\trianglelefteq\, G\), we have \(R=AR\cap RB\trianglelefteq\, BR\). As a consequence, \(B\) and \(R\) are direct factors of \(BR\) (i.e., \(BR\) is nilpotent).
Let $R_0 = \oh rG$, and observe
that we can assume that $R_0 < R$, as otherwise $G = R \times AB$ and we are again in case {\bf(IIb(i))}.
As above we observe that, as $G$ is reduced, we have $\fit G = A \times R_0$. So, $R_0 = \cent {BR}A$.
Write $A = A_0 \times C$, where $A_0 = [A,B]$ and $C = \cent AB$.
We show that $A_0BR/R_0$ is a Frobenius group. In fact, if $x \in A_0\setminus\{1\}$,
then $q \in \pi_G(x)$ for some $q \in \beta$, and hence Lemma~\ref{product} implies that
$\cent {BR}x \leq \cent{BR}A = R_0$.
Moreover, if $R$ is abelian, then
$R_0 = 1$ as $G$ is reduced. Hence, if $C = 1$, then $\Delta(G)$ would be disconnected by Proposition~\ref{disconnected}, against our assumptions, and we reached conclusion {\bf(III)}.
\bigskip
We now start proving the ``if part" of Theorem~B.
We recall that, if $\pi_1, \pi_2, \ldots, \pi_n$ are disjoint sets of primes and $g$ is an element of $G$,
one can uniquely write $g = g_{\pi_1}g_{\pi_2}\cdots g_{\pi_n}$, where each $g_{\pi_i}$ is a
$\pi_i$-element and a power of $g$; we call this the \emph{standard decomposition} of $g$ (with respect to $\pi_1, \pi_2, \ldots, \pi_n$). Note that then $\cent Gg = \bigcap_{i = 1}^n \cent G{g_{\pi_i}}$.
\medskip
Let us assume {\bf(I)}: in this case $B = Q$ is a Sylow \(q\)-subgroup of $G$.
We first show that \(q\) is not adjacent in \(\Delta(G)\) to any prime in $\alpha$.
What we have to prove is that, for a fixed \(p\in\pi(A)\) and \(g\in G\), the size of \(g^G\) is not divisible by \(pq\).
We have the standard decomposition \(g=g_rg_{\alpha}g_q\), where we can assume, up to conjugation in \(G\), that \(g_r\in R\), \(g_{\alpha}\in A\) and \(g_q\in Q_0\), for some $Q_0 \in \syl qG$. If \(g_q\neq 1\), then \(\langle g_q\rangle=Q_0\) (recall that \(|Q_0|=q\)) centralizes \(g\), therefore \(q\nmid |g^G|\). To the end of showing that \(|g^G|\) is not divisible by \(pq\) we will therefore assume \(g_q=1\).
Let us consider the case when \(g_r\) is centralized by a conjugate \(A^v\) of \(A\), with \(v\in R\). Since \(g_{\alpha}\) is a \(\pi(A)\)-element of \(\cent G {g_r}\) and \(A^v\) is a Hall \(\pi(A)\)-subgroup of \(\cent G {g_r}\), there exists \(c\in \cent G {g_r}\) such that \(g_{\alpha}\) lies in \(A^{vc}\). But \(A^{vc}\) is abelian, so \(g_{\alpha}\) is centralized by \(A^{vc}\), as well as \(g_r\). The conclusion is that \(g=g_rg_{\alpha}\) is centralized by the Hall \(\pi(A)\)-subgroup \(A^{vc}\) of \(G\), whence \(p\nmid |g^G|\) and we are done in this case.
The last situation that has to be considered is when \(g_r\) is not centralized by \(A^v\) for any \(v\in R\). Set $H = AB$. Then, by our assumptions, a \(G\)-conjugate \(Q^u\) of \(Q\) lies in \(\cent G {g_r}\), and \(\cent A {g_r}\leq\zent H\); in particular, we get \(g_{\alpha}\in\zent H\), thus \(o(g_{\alpha})\mid|\zent H|\). Choose now an \(r\)-complement \(H_1\) of \(\cent G {g_r}\) which contains \(Q^u\), and let \(A_1\) be the (cyclic) \(\pi(A)\)-Hall subgroup of \(H_1\). Since \(g_{\alpha}\) is a \(\pi(A)\)-element of \(\cent G {g_r}\), there exists \(c\in\cent G {g_r}\) such that \(g_{\alpha}\) lies \(A_1^c\). Observe that \(o(g_{\alpha})\) divides the order of \(\zent {H_1^c}\leq A_1^c\) and, \(A_1^c\) being cyclic, its unique subgroup of order \(o(g_{\alpha})\) (i.e., \(\langle g_{\alpha}\rangle\)) is forced to lie in \(\zent{H_1^c}\). We conclude that \(g_{\alpha}\) lies in \(\zent{H_1^c}\), and therefore \(g_{\alpha}\) is centralized by \(Q^{uc}\). But \(Q^u\) lies in \(\cent G {g_r}\), so the same holds for \(Q^{uc}\) (recall that \(c\in \cent G {g_r}\)) and \(Q^{uc}\) centralizes \(g_r\) as well. As a consequence, in this situation the size of the conjugacy class of \(g=g_rg_{\alpha}\) in \(G\) is not divisible by \(q\).
So we finished the proof that \(q\) is not adjacent in \(\Delta(G)\) to any prime in \(\alpha\), which also implies (by Proposition~\ref{nilpotency}(b)) that the vertices in \(\alpha\) are pairwise adjacent in \(\Delta(G)\).
Finally, we observe that $r$ is a complete vertex of $\Delta(G)$. In fact, assuming the contrary, our graph would have no complete vertices, and therefore \(G\) would be metabelian by Theorem~C of \cite{CDPS12}. But this is not the case, as \(G\) has Fitting height~\(3\). We conclude that \(r\) is a cut vertex of \(\Delta(G)\) and we are done.
\smallskip
Let us assume now case {\bf(II)}: $\fit G = A \times R$, $Z = \zent{AB} <B$, and $AB/Z$ is a Frobenius group with kernel \(AZ/Z\) (note that \(Z= \cent BA\) and \(AZ/Z\simeq A\)).
\smallskip
{\bf(IIa)} ($R$ is abelian, $\cent BR = 1$ and $\cent Bx \leq Z \neq 1$ for every non-trivial $x \in R$).
Note that, as $G$ is reduced, the vertex set of $\Delta(G)$ is $\alpha \cup \beta \cup \{ r \}$.
We first show that, for $p \in \alpha$ and $q \in \beta$, $p$ and $q$ are non-adjacent in $\Delta(G)$. In fact,
let $g \in G$ and consider the standard decomposition $g = g_{\alpha}g_r g_{\beta}$, with $g_{\alpha} \in A$, $g_r \in R$ and, up to
conjugation, $g_{\beta} \in B$. Assuming that \(pq\) divides \(|g^G|\), we clearly have \(p\in\pi_G(g_{\beta})\), which implies \(g_{\beta}\not\in Z\). Since \(AB/Z\) is a Frobenius group with kernel \(AZ/Z\), and \(g_{\beta}\) commutes with \(g_{\alpha}\), we deduce that \(g_{\alpha}\) must be trivial and so \(g_r\neq 1\) (otherwise \(g=g_{\beta}\) would not lie in a conjugacy class having size divisible by \(q\)). But now we get \(g_{\beta}\in\cent B{g_r}\leq Z\), a contradiction.
As in case {\bf(I)}, this also implies that both \(\alpha\) and \(\beta\) induce complete subgraphs of \(\Delta(G)\).
Finally, we observe that $\Delta(G)$ is connected by Proposition~\ref{disconnected}, so $r$ is a cut vertex of $G$, as wanted.
Note also that, as easily seen, every element in \(B\setminus Z\) has a $G$-conjugacy class size divisible by \(r\) and by all the primes in \(\alpha\), therefore \(\alpha\cup\{r\}\) induces a complete subgraph of \(\Delta(G)\).
\smallskip
{\bf(IIb(i))} ($G = R \times AB$).
In this case, it is clear that $\Delta(G)$ is the join of a graph with one vertex $r$ and a
disconnected graph with connected components of vertex sets $\alpha$ and $\beta$.
\smallskip
{\bf(IIb(ii))} ($R$ is non-abelian, and $\cent Bx \leq Z$ for all $x \in R$ with $\cent Gx R <G$).
Let $g \in G$, and write \(g\) in its standard decomposition as $g_{\alpha}g_r g_{\beta}$, with $g_{\alpha} \in A$, $g_r \in R$ and, up to
conjugation, $g_{\beta} \in B$. Assume, working by contradiction, that $\{p, q\} \subseteq \pi_G(g)$
for some $p \in \alpha$ and $q \in \beta$;
then $p \in \pi_G(g_{\beta})$. Thus we have $g_{\beta} \not\in Z$, and hence $g_{\alpha} = 1$, because \(g_{\alpha}\) commutes with \(g_{\beta}\) and $AB/Z$ is a Frobenius group with kernel \(AZ/Z\). But also \(g_r\) commutes with \(g_{\beta}\), therefore, by our assumptions, we have \(\cent G{g_r}R=G\); in particular, there exists a Hall \(\beta\)-subgroup \(B_0\) of \(G\) lying in \(\cent G{g_r}\). Now, \(g_{\beta}\) is a \(\beta\)-element of \(G\) contained in \(\cent G{g_r}\), and so there exists \(c\in\cent G{g_r}\) such that \(g_{\beta}\) lies in \(B_0^c\) (which is abelian). As a consequence, \(B_0^c\) centralizes \(g=g_rg_{\beta}\), and in particular \(q\not\in\pi_G(g)\), contradicting our assumptions. As in case {\bf(I)}, \(G\) being not metabelian, \(r\) is a complete vertex of \(\Delta(G)\) and it is therefore a cut vertex of \(\Delta(G)\), as wanted.
\smallskip
Let us assume the last case {\bf(III)}: \(BR\) is a nilpotent subgroup of \(G\); also, $\fit G = A \times R_0$, with $R_0 < R$, $\cent AR = 1$,
and $[A,B]BR/R_0$ is a Frobenius group with kernel \([A,B]R_0/R_0\). In the case when $R$ is abelian, in addition we have $R_0 = 1$ and $C = \cent AB \neq 1$.
As before, let $g \in G$ and consider the standard decomposition $g = g_{\alpha} g_{\{ r \} \cup \beta}$, with $g_{\alpha} \in A$, and, up to
conjugation, $g_{\{ r \} \cup \beta} \in BR$. Assume, working by contradiction, that $\{p, q\} \subseteq \pi_G(g)$,
for some $p \in \alpha$ and $q \in \beta$.
Then $p \in \pi_G(g_{\{ r \} \cup \beta})$.
As $A = [A,B] \times C$, write also $g_{\alpha} = g_0 g_1$ with $g_0 \in [A, B]$ and $g_1 \in C$, and note that \(g_{\{ r \} \cup \beta}\) centralizes both \(g_0\) and \(g_1\), because \([A,B]\) and \(C\) are normal subgroups of \(G\).
Since $g_{\{ r \} \cup \beta} \not\in R_0 = \cent{BR}A$, our assumptions imply that $g_0 = 1$, so
$g_{\alpha} \in C$ and hence \(q\not\in\pi_G(g)\), a contradiction. As usual, what we proved implies also that both \(\alpha\) and \(\beta\) induce complete subgraphs of \(\Delta(G)\). Finally we observe that, by Proposition~\ref{disconnected}, $\Delta(G)$ is connected both when
$R$ is non-abelian (in which case \(r\), as in {\bf (I)}, is a complete vertex of \(\Delta(G)\)) and when $R$ is abelian; in fact, in the latter case, we get that $\zent G = 1$ and $G$ is not a Frobenius group. Thus $r$ is a cut vertex of $\Delta(G)$, and the proof is complete.
We also note that every non-trivial element in \([A,B]\) has a $G$-conjugacy class size divisible by \(r\) and by all the primes in \(\beta\), therefore \(\beta\cup\{r\}\) induces a complete subgraph of \(\Delta(G)\).
\end{proof}
\section{Discussion of the cases of Theorem~B, and proof of Theorem~C}
Next, we take time for a closer look at the groups that appear in Theorem~B, also deriving some more detailed information about the associated graphs. As a consequence of this discussion, we will get Theorem~C (see Remark~\ref{thmC}). We will also determine, in Example~\ref{ex}, which \(1\)-connected graphs can occur as \(\Delta(G)\) for a finite group \(G\).
So, let \(G\) be a reduced group such that \(\Delta(G)\) has a cut vertex \(r\). As in Theorem~B, we denote by \(\alpha\) and \(\beta\) the vertex sets of the two connected components of the graph \(\Delta(G)-r\) (the description being given up to interchanging \(\alpha\) and \(\beta\)).
First of all we stress that, in this setting, the groups as in {\bf(I)} are characterized by the fact that they have a normal Sylow subgroup only for the prime \(r\). The structure of these groups has been already discussed in Remark~\ref{rem}, and we do not comment further on that.
As regards the groups in classes {\bf (II)} and {\bf(III)}, they share the property of having normal Sylow subgroups for all the primes in \(\alpha\), whereas the Sylow subgroups for the primes in \(\beta\) are all non-normal. If, in this situation, the group has an abelian normal Sylow \(r\)-subgroup, then it lies in {\bf(IIa)}; if it has a non-abelian normal Sylow \(r\)-subgroup, then we are in case {\bf (IIb)}. On the other hand, if the group does not have a normal Sylow \(r\)-subgroup, then it belongs to class {\bf(III)}.
Some more remarks:
\medskip
${\bullet}$
For a group \(G\) as in {\bf(IIa)}, the cut vertex \(r\) need not be a complete vertex of \(\Delta(G)\). If it is not, as observed in Theorem~A, the graph \(\Delta(G)\) has diameter \(3\).
More specifically, \(r\) is adjacent to all the primes in \(\alpha\), but it can be non-adjacent to some prime in \(\beta\): in order to have a better understanding of \(\Delta(G)\) in this case, we characterize next the set \(\beta^*\subseteq\beta\) of the vertices of our graph that are non-adjacent to \(r\).
Let \(R\) be the Sylow \(r\)-subgroup of \(G\) and, for $q \in \beta$, let $Q$ be in $\syl qB$. We claim that $q$ lies in $\beta^*$ if and only if
$Q \leq Z= \cent BA$ and $B$ acts fixed point-freely on $[R,Q]$.
In fact, if $Q \not\leq Z$, then $q \in \pi_G(x)$ for some element $x\in A$.
Consider a non-trivial element $y \in Z$ (recall that $Z \neq 1$); then $r \in \pi_G(y) $ and hence $\{r, q\} \subseteq \pi_G(xy)$.
If, on the other hand, there exist non-trivial and commuting elements $x \in [R, Q]$ and $y \in B$, then $\pi_G(xy) \supseteq \pi_G(x) \cup \pi_G(y) \supseteq \{r,q\}$ (recall that $\cent BR = 1$).
Conversely, let $g = g_{\alpha}g_rg_{\beta}$ be the standard decomposition of \(g\), where we can assume, up to conjugation, $g_{\alpha} \in A$, $g_r \in R$ and $g_{\beta} \in B$. Assume that
$\{r, q\} \subseteq \pi_G(g)$ and that $B$ acts fixed point-freely on $[R,Q]$. As $r \in \pi_G(g)$, then $g_{\beta} \neq 1$; so, using the Fitting decomposition of the abelian group $R$ with respect to the action of $Q$, we get $g_r \in \cent RQ$. Thus $q \in \pi_G(g)$ implies $Q \not\leq \cent B{g_{\alpha}}$, and hence
$Q \not\leq Z$.
\medskip
${\bullet}$ For the groups in {\bf (IIb)} (as well as for those as in {\bf (I)}), the cut vertex $r$ is a complete vertex of $\Delta(G)$.
\medskip
$\bullet$ Finally, let \(G\) be as in {\bf(III)}. Then the cut vertex \(r\) is always adjacent in \(\Delta(G)\) to all the vertices in \(\beta\), and it is a complete vertex if a Sylow \(r\)-subgroup \(R\) of \(G\) is non-abelian. On the other hand, if $R$ is abelian, \(r\) can be non-adjacent to some prime in \(\alpha\) (and, if this happens, then \(\Delta(G)\) has diameter \(3\)): as we did for class {\bf(IIa)}, we characterize next the set \(\alpha^*\subseteq\alpha\) of the vertices of \(\Delta(G)\) that are non-adjacent to \(r\) in this case.
For $p \in \alpha$ and $P \in \syl pA$, we show that $p \in \alpha^*$ if and only if $P \leq C=\cent AB$ and $\cent Rx \leq \cent RP$ for all non-trivial $x \in A$.
In fact, if $P \not\leq C$, then there exists $y \in B$ such that $p \in \pi_G(y)$. Considering a non-trivial $x \in C$, we have $r \in \pi_G(x)$ (as $\cent AR = 1$) and
hence $\{p, r \} \subseteq \pi_G(xy)$. If, on the other hand, there exist non-trivial elements $y \in R \setminus \cent RP$ and $x \in \cent Ay$, then again $\{p, r \} \subseteq \pi_G(xy)$.
Conversely, let $g = g_{\alpha}g_rg_{\beta}$ be the standard decomposition of \(g\), where we can assume, up to conjugation, $g_{\alpha} \in A$, $g_r \in R$ and $g_{\beta} \in B$. Assume that
$\{p, r \} \subseteq \pi_G(g)$ and that $\cent Rx \leq \cent RP$ for all non-trivial $x \in A$. As $R$ does not centralize $g$, then $g_{\alpha} \neq 1$ and hence $g_r \in \cent RP$.
So $g_{\beta} \not\in \cent GP$ and hence $P \not\leq C$.
\begin{rem}\label{thmC}
Observe that Theorem~C is an immediate consequence of (Theorem~B and) the analysis carried out above. In fact, the reduced groups whose related graph has two cut vertices are easily seen to be those lying in class {\bf(IIa)} such that the second cut vertex \(t\) is the unique element in \(\beta\setminus\beta^*\) (or, equivalently, the groups lying in class {\bf (III)} such that the second cut vertex \(t\) is the unique element in \(\alpha\setminus\alpha^*\)).
\end{rem}
We close this section by showing that \emph{every $1$-connected graph which is covered by two complete subgraphs does in fact occur as the graph \(\Delta(G)\) for a suitable group~$G$.} (Conversely, every graph of the kind \(\Delta(G)\) which has a cut vertex is \(1\)-connected, as observed in the Introduction, and it is covered by two complete subgraphs by Theorem~\ref{ultimo}.)
\begin{example}\label{ex}
Let $n, m_1$ be positive integers and $m_0$ a non-negative integer.
Let $b_0 = q_1q_2\cdots q_{m_0}$ and
$b_1 = t_1t_2\cdots t_{m_1}$ where the $q_i$ and the $t_j$ are distinct primes
(meaning also $q_i \neq t_j$, for all $i, j$).
Let $r$, $p_1, p_2, \ldots, p_n$ be distinct primes such that
$r \equiv 1 \pmod{b_0b_1}$ and $p_i \equiv 1 \pmod{b_1}$
for all $1 \leq i \leq n$; note that they exist by Dirichlet's Theorem on primes in an arithmetic progression.
Let $B_0$ and $B_1$ be cyclic groups of order $b_0$ and $b_1$,
and $R$ and $A$ cyclic groups of order $r$ and $p_1p_2\cdots p_n$, respectively.
Consider the semidirect product $G = (A \times R) \rtimes (B_0 \times B_1)$
with respect to a Frobenius action of $B_0 \times B_1$ on $R$ and
of $B_1$ on $A$, while $B_0$ acts trivially on $A$.
Then it is easily seen that the graph $\Delta(G)$ is covered by two complete subgraphs (on the sets $\{r, p_1, \ldots, p_n\}$ and $\{q_1, \ldots, q_{m_0}, t_1,\ldots , t_{m_1}\}$), and that $r$ is a cut vertex of $\Delta(G)$ which is
adjacent exactly to the primes $\{ t_1,\ldots , t_{m_1}\}$ (see Figure~1).
\begin{figure}[h]
\label{example1}
\centering
{\includegraphics[width=.4\textwidth]{example1.png}}
\caption{Example 5.2}
\end{figure}
Observe that $r$ is complete if and only if $G = (A \times R) \rtimes B_1$, and that there
are two cut vertices if and only if $m_1 = 1$. \end{example}
\section*{Acknowledgements}
This research has been carried out during a visit of the second and fourth authors at the Dipartimento di Matematica e Informatica ``Ulisse Dini" (DIMAI) of Universit\`a degli Studi di Firenze. They wish to thank the DIMAI for the hospitality.
| {
"timestamp": "2019-12-23T02:08:40",
"yymm": "1912",
"arxiv_id": "1912.09743",
"language": "en",
"url": "https://arxiv.org/abs/1912.09743",
"abstract": "Let $G$ be a finite group, and let $\\Delta(G)$ be the prime graph built on the set of conjugacy class sizes of $G$: this is the simple undirected graph whose vertices are the prime numbers dividing some conjugacy class size of $G$, two vertices $p$ and $q$ being adjacent if and only if $pq$ divides some conjugacy class size of $G$. In the present paper, we classify the finite groups $G$ for which $\\Delta(G)$ has a cut vertex.",
"subjects": "Group Theory (math.GR)",
"title": "Groups whose prime graph on class sizes has a cut vertex",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9770226341042415,
"lm_q2_score": 0.72487026428967,
"lm_q1q2_score": 0.7082146550001311
} |
https://arxiv.org/abs/2105.11946 | Quantum Approximate Optimization Algorithm with Adaptive Bias Fields | The quantum approximate optimization algorithm (QAOA) transforms a simple many-qubit wavefunction into one which encodes a solution to a difficult classical optimization problem. It does this by optimizing the schedule according to which two unitary operators are alternately applied to the qubits. In this paper, the QAOA is modified by updating the operators themselves to include local fields, using information from the measured wavefunction at the end of one iteration step to improve the operators at later steps. It is shown by numerical simulation on MaxCut problems that, for a fixed accuracy, this procedure decreases the runtime of QAOA very substantially. This improvement appears to increase with the problem size. Our method requires essentially the same number of quantum gates per optimization step as the standard QAOA, and no additional measurements. This modified algorithm enhances the prospects for quantum advantage for certain optimization problems. | \section{MaxCut Problem} \label{model}
MaxCut is a standard problem in optimal graph theory~\cite{mezard2009information}. Let an undirected graph be denoted by $G(V,E)$, where $V$ is the $n$-vertex set and $E$ is the edge set. The edges may or may not be assigned weights. If they are, then the weights are chosen uniformly at random from the interval $[0,1]$. In the unweighted version we wish to partition $V$ into two subsets $V_1$, $V_2$ in such a way as to make the number of edges connecting $V_1$ and $V_2$ as large as possible. In the weighted version, the total weight of the partition is maximized.
We convert MaxCut to an $n$-vertex Ising model as follows. Define a Pauli matrix $Z_j$ to act on the $j$th vertex and use the eigenstates $|0\rangle$, $|1\rangle$ of the $Z_j$ to represent $V_1$ and $V_2$. Thus, in operator language, the MaxCut problem Hamiltonian for $n$ qubits is $H = E_{0} - H_{C}$, where
\begin{equation}
H_{C} = \sum_{\langle v_1,v_2\rangle \in E} \frac{\omega_{v_1,v_2}}{2} Z_{v_1}Z_{v_2} \; .
\label{eq:H}
\end{equation}
The constant
$E_0 = \sum \omega_{v_1,v_2}/2$
plays no role in the partition of the graph, but enters the calculations of the accuracy $r$ as defined in the main text. The ground state has an obvious $\mathbb{Z}_2$ symmetry. The ground state of $H_C$ in Eq.~\eqref{eq:H} encodes the solutions to the original MaxCut problems.
%
We consider weighted 3-regular graphs with $\omega_{v_1,v_2}$ chosen uniformly at random in $[0,1]$ (w3r graphs), and unweighted 3-regular graphs with $\omega_{v_1,v_2}=1$ (u3r graphs).
%
An example of an unweighted graph is shown in Fig.~\ref{fig:maxcut}.
\begin{figure}[ht]
\centering
\includegraphics[scale=0.48]{supp_maxcut.pdf}
\caption{MaxCut problem on an unweighted $3$-regular $6$-vertex graph. Different colors give the different states $|0\rangle$ and $|1\rangle$, and represent the two different subsets $V_1$ and $V_2$ of the vertex set. The problem is to find the division of the vertices that maximizes the number of edges connecting the two subsets. The dashed edges in the figure represent the cut in this case.} \label{fig:maxcut}
\end{figure}
\section{Computational Details of the ab-QAOA}\label{abQAOA Details}
In this section we give further details of how the algorithm for the ab-QAOA differs from that of the QAOA. For a $p$-level QAOA, as stated in the main text, the mixing Hamiltonian is $H_M^\mathrm{s}=\sum_j X_j$. The quantum processor is initialized in $|\psi_{0}^{\mathrm{s}}\rangle$, the ground state of $H_M^{\mathrm{s}}$. Then alternately apply problem Hamiltonian $H_C$ and mixing Hamiltonian $H_M^\mathrm{s}$ to generate the final state,
\begin{equation}
\label{eq:s}
|\psi_{f}^{\mathrm{s}} \rangle=
\prod_{k=1}^{p}
\mathrm{e}^{-i\beta_{k}H_{M}^\mathrm{s}} \mathrm{e}^{-i\gamma_{k}H_{C}}|\psi_{0}^{\mathrm{s}}\rangle,
\end{equation}
where the level $p$ is the number of times the unitary operators corresponding to $H_M^\mathrm{s}$ and $H_C$ are applied to the initial state to move it to the final state. The scheduling parameters $\{\gamma_k\}$, $\{\beta_k\}$ in the operators are determined by optimizing
\begin{align}
\langle H_C \rangle(\{\gamma_k\},\{\beta_k\})=\langle\psi_{f}^{\mathrm{s}} |H_C|\psi_{f}^{\mathrm{s}}\rangle.
\end{align}
For clarity, we drop the range of $k$ and other indices in the following.
Note that for the original QAOA~\cite{farhi,lukin}, $|\psi_0^{\mathrm{s}}\rangle$ is $|+\rangle^{\otimes n}$, but in our description, $|\psi_0^{\mathrm{s}}\rangle$ is $|-\rangle^{\otimes n}$, the ground state of $H_M^{\mathrm{s}}$. If we denote the QAOA final state from $|+\rangle^{\otimes n}$ by $|\psi_f^\mathrm{s+}\rangle$, the final state from $|-\rangle^{\otimes n}$ by $|\psi_f^\mathrm{s-}\rangle$ and define $\Tilde{Z}=\prod_j Z_j$, it is easy to prove that
\begin{align}
\begin{split}
|\psi_f^\mathrm{s-}\left(\{\gamma_k\},\{\beta_k\}\right)\rangle &=
\prod_{k=1}^{p}
\mathrm{e}^{-i\beta_{k}H_{M}^\mathrm{s}} \mathrm{e}^{-i\gamma_{k}H_{C}}\Tilde{Z}|+\rangle^{\otimes n}\\
&=\Tilde{Z}\prod_{k=1}^{p}
\mathrm{e}^{i\beta_{k}H_{M}^\mathrm{s}} \mathrm{e}^{-i\gamma_{k}H_{C}}|+\rangle^{\otimes n}\\
&=\Tilde{Z}|\psi_f^\mathrm{s+}\left(\{\gamma_k\},\{-\beta_k\}\right)\rangle.
\end{split}
\end{align}
There is no difference in the classical optimization for both $|\psi_f^\mathrm{s-}\rangle$ and $|\psi_f^\mathrm{s+}\rangle$.
There are two differences in the quantum part of the two algorithms. 1. In the $p$-level ab-QAOA, $H_M^\mathrm{ab}$ contains local longitudinal fields as well as the usual global transverse field, $H_{M}^\mathrm{ab}(\{h_j\}) = \sum_{j=1}^n (X_j-h_{j}Z_{j})$. 2. The wavefunction at the initial stage of each learning step is chosen to be the ground state of the updated $H_M^\mathrm{ab}$. Hence both the longitudinal fields in the mixing Hamiltonian and the "re-initialized" wavefunction change during the course of the ab-QAOA algorithm. Thus the final state of ab-QAOA is
\begin{equation}
|\psi_{f}^{\mathrm{ab}} \rangle=
\prod_{k=1}^{p}
\mathrm{e}^{-i\beta_{k}H_{M}^\mathrm{ab}(\{h_j\})}
\mathrm{e}^{-i\gamma_{k}H_{C}}|\psi_{0}^{\mathrm{ab}}(\{h_j\})\rangle.\label{eq:psiab}
\end{equation}
To avoid being trapped in the local optimum as far as possible, we start the optimization from $R$ initial points and find the point with the best energy, as was done in Ref. \cite{lukin}. To reach level $p$, we start from level $1$ and find the point with the best energy from $R$ initial points after the optimization. In level-2, $R$ initial points are generated by adding some random numbers to the best point in level $1$. Then repeat the optimization and initial point generation procedure with increasing level $p^\prime$ until $p^\prime=p$. We use $p$ to represent the target level and $p^\prime$ to represent the inner levels. The update of $\{\gamma_k\},\{\beta_k\},\{h_j\}$ until convergence for a fixed level $p^\prime$ is the inner loop of the algorithm while the whole process from level $1$ to level $p$ including the initial points generation is the outer loop of the algorithm. The detailed procedure follows and is also illustrated in Fig.~\ref{fig:update}. The same loops are used for the QAOA except that all the $h_j$ are set to $0$.
This more elaborate classical optimization is not strictly necessary to demonstrate the advantages of the ab-QAOA over the QAOA, but it does mean that the results can be compared more directly with those of Ref. \cite{lukin}. The sampling parameter $R$ was set to $10$ in our calculations. We note once more that formally the QAOA can be considered as the limit of the ab-QAOA when $h_j \rightarrow 0$ . This means that from a formal standpoint the ab-QAOA is guaranteed to be at least as good as the QAOA.
For the current problem, it turns out that the change in the $\{\gamma_k\},\{\beta_k\}$ parameters are relatively smooth as $k$ increases. In Ref. \cite{lukin}, it was found that optimization of the Fourier components of these parameters is more efficient than optimization of $\{\gamma_k\},\{\beta_k\}$ themselves. We follow this procedure, which again helps to compare our results to previous work. So we actually minimize the energy
$ \langle H_C \rangle $ with respect to their Fourier transforms $\{ u_l\},\{v_l\}$, then determine the new $\{\gamma_k\},\{\beta_k\}$ in level $p^\prime$ by
\begin{equation}
\begin{split}
\gamma_{k} &=\sum_{l=1}^{p^\prime} u_{l} \sin \left[\left(l-\frac{1}{2}\right)\left(k-\frac{1}{2}\right) \frac{\pi}{p^\prime}\right],\\
\beta_{k} &=\sum_{l=1}^{p^\prime} v_{l} \cos \left[\left(l-\frac{1}{2}\right)\left(k-\frac{1}{2}\right) \frac{\pi}{p^\prime}\right].
\end{split}\label{eq:uv}
\end{equation}
The inner loop of the algorithm (fixed $p^\prime$) is as follows.
\begin{table}[ht]
\centering
\begin{tabular}{p{\linewidth}}
\toprule
\textbf{Inner Loop Update Procedure for level $p^\prime$} \\
\hline
\end{tabular}
\end{table}
\begin{itemize}
\item Initialization
\begin{enumerate}
\item Initialize 2 $p^\prime$-element sets $\{u_l\}$ and $\{v_l\}$ that are used to update $\{\gamma_k\}$ and $\{\beta_k\}$.
\item Initialize the $n$-element local field set $\{h_j\}$.
\item Set a learning rate $\ell$, a global parameter defined in Step 6 in optimization procedure.
\end{enumerate}
\end{itemize}
\begin{itemize}
\item Optimization
\begin{enumerate}
\item Set $\{\gamma_k\}$ and $\{\beta_k\}$ according to the discrete Fourier transforms of $\{u_l\}$ and $\{v_l\}$.
\label{item_start}
\item Construct the mixing Hamiltonian with bias fields:
\begin{align}
H_{M}^\mathrm{ab}(\{h_j\}) = \sum_{j=1}^n (X_j-h_{j}Z_{j}).\nonumber
\end{align}
\item Prepare $|\psi_{0}^{\mathrm{ab}}\rangle$, the product ground state of $H_{M}^\mathrm{ab}(\{h_j\})$.
\item Compute the final state for this step using Eq.~\eqref{eq:psiab}:
\begin{equation}
|\psi_{f}^{\mathrm{ab}} \rangle=
\prod_{k=1}^{p^\prime}
\mathrm{e}^{-i\beta_{k}H_{M}^\mathrm{ab}(\{h_j\})}
\mathrm{e}^{-i\gamma_{k}H_{C}}|\psi_{0}^{\mathrm{ab}}\rangle\nonumber.
\end{equation}
\item Using projective measurements, obtain the gradients of the energy:
\begin{equation}
\quad
\quad
\frac{\partial \langle\psi_{f}^{\mathrm{ab}}| H_{C}|\psi_{f}^{\mathrm{ab}} \rangle}{\partial \vec{u}} \quad \textrm{and} \quad
\frac{\partial \langle\psi_{f}^{\mathrm{ab}}| H_{C}|\psi_{f}^{\mathrm{ab}} \rangle}{\partial \vec{v}}\nonumber
\end{equation}
and the quantity
\begin{equation}
\delta h_{j} = h_{j} - \langle\psi_{f}^{\mathrm{ab}}| Z_{j}|\psi_{f}^{\mathrm{ab}} \rangle.\nonumber
\end{equation}
\item Update $\{v_l\}$, $\{u_l\}$ using the Adam gradient-based stochastic optimization algorithm~\cite{adam}. Update $\{h_j\}$ with learning rate $\ell$\label{learning_rate} according to
$ h_j \rightarrow h_j - \ell\delta h_j$. The update of $\{h_j\}$ feeds back into both the mixing Hamiltonian in Step 2 and the wavefunction in Step 3.
\item Measure the expectation value of the energy/cost function
$E(\{u_l\}, \{v_l\}, \{h_j\}) = \langle\psi_{f}^{\mathrm{ab}}| H_{C} |\psi_{f}^{\mathrm{ab}} \rangle $.
\label{item_end}
\item Repeat steps 1-7 until convergence with a fixed tolerance.
Output the final energy $E_{f}(\{u_l\}, \{v_l\}, \{h_j\})$, and a measurement of $|\psi_{f}^{\mathrm{ab}} \rangle$ in the computational basis. Allowing for the constant term, the optimized energy is $E^{\mathrm{opt}}_{p^\prime}=E_0-E_f$.
\end{enumerate}
\end{itemize}
\begin{table}[ht]
\centering
\begin{tabular}{p{\linewidth}}
\hline
\quad
\end{tabular}
\end{table}
The outer loop of the algorithm ($1 \rightarrow p$) is as follows.
\begin{table}[ht]
\centering
\begin{tabular}{p{\linewidth}}
\toprule
\textbf{Outer Loop from level $1$ to $p$} \\
\hline
\end{tabular}
\end{table}
\begin{enumerate}
\item In level $1$, we generate $R$ initial "$0$" points $\left(\{u_l\}_1^{0,s},\{v_l\}_1^{0,s},\{h_j\}_1^{0,s}\right)$, where the elements of $\{u_l\}_1^{0,s}$ and $\{v_l\}_1^{0,s}$ are random numbers drawn from a uniform distribution and all elements of $\{h_j\}_1^{0,s}$ are initialized to be $1$. The subscripts refer to the ab-QAOA level in the outer loop, and the $s$ superscript ranges from $1$ to $R$ representing the different initial points. Using the update procedure above we get the optimal "$\mathrm{B}$" point $\left(\{u_l\}_1^{\mathrm{B}},\{v_l\}_1^{\mathrm{B}},\{h_j\}_1^{\mathrm{B}}\right)$ with the best optimal energy $E_1^\mathrm{B}$ from $R$ points for this level.
\item \label{alg}In level $p^\prime$ greater than 1, we use the best point $\left(\{u_l\}_{p^\prime-1}^{\mathrm{B}},\{v_l\}_{p^\prime-1}^{\mathrm{B}},\{h_j\}_{p^\prime-1}^{\mathrm{B}}\right)$ in level $p^\prime-1$ to construct $R$ initial points $\left(\{u_l\}_{p^\prime}^{0,s},\{v_l\}_{p^\prime}^{0,s},\{h_j\}_{p^\prime}^{0,s}\right)$. The $s$ superscript refers to the elements of the following random selection procedure, representing the different points.
\begin{equation}\label{update}
\begin{split}
\{u_l\}_{p^\prime}^{0,s}&=\left\{
\begin{array}{ll}
\{u_l\}_{p^\prime-1}^{\mathrm{B}}\cup\{0\},&s=1\\
\{u_l+\alpha \mathrm{Ran}^s[u_l]\}_{p^\prime-1}^\mathrm{B}\cup\{0\},&2\leq s\leq R
\end{array}
\right.\\%\nonumber\\
\{v_l\}_{p^\prime}^{0,s}&=\left\{
\begin{array}{ll}
\{v_l\}_{p^\prime-1}^{\mathrm{B}}\cup\{0\},&s=1\\
\{v_l+\alpha \mathrm{Ran}^s[v_l]\}_{p^\prime-1}^\mathrm{B}\cup\{0\},&2\leq s\leq R
\end{array}
\right.\\%\nonumber\\
\{h_j\}_{p^\prime}^{0,s}&=\left\{
\begin{array}{ll}
\{h_j\}_{p^\prime-1}^{\mathrm{B}},&s=1\\
\{h_j+\alpha \mathrm{Ran}^s[h_j]\}_{p^\prime-1}^\mathrm{B},&2\leq s\leq R
\end{array}
\right.\\%\nonumber\\
\end{split}
\end{equation}
$\{u_l\}_{p^\prime}^{0,s}$ or $\{v_l\}_{p^\prime}^{0,s}$ is a $p^\prime$-element set whose $p^\prime$th element is zero. The random number $\mathrm{Ran}^s[a]$ is the $s$-th selection from a normal distribution with average $0$ and variance $a^2$, \textit{i.e.}, $\mathrm{Ran}^s[a]=\mathrm{Norm}(0,a^2)$. We optimize these $R$ initial points to find the best point $\left(\{u_l\}_{p^\prime}^{\mathrm{B}},\{v_l\}_{p^\prime}^{\mathrm{B}},\{h_j\}_{p^\prime}^{\mathrm{B}}\right)$ with the best energy $E_{p^\prime}^\mathrm{B}$. The update parameter $\alpha$ was set to $ \alpha = 0.6$.
\item Repeat step 2 until $p^\prime$ reaches the target level $p$.
\item Output all energies $E_{p^\prime}^\mathrm{B}$ from level $1$ to $p$.
\end{enumerate}
\begin{table}[ht]
\centering
\begin{tabular}{p{\linewidth}}
\hline
\quad
\end{tabular}
\end{table}
\begin{figure}[ht]
\centering
\includegraphics[scale=0.4]{supp_update.pdf}
\caption{Schematics of the outer loop of ab-QAOA. Using Eq.~\eqref{update}, we generate $R$ initial points in level $p^\prime$ from the best point in level $p^\prime-1$. After the optimization of these $R$ points, we get the point with the best energy. We do this procedure iteratively until the target level $p$.}\label{fig:update}
\end{figure}
\section{Computation time}
Here we give the analysis that leads to the conclusion in the main text that the total computation time is $O(p^2)$. We assume that the quantum part of the algorithm dominates the time. This will be true for the foreseeable future. The MaxCut cost Hamiltonian $H_{C}$ is defined on an $n$-vertex $\mathcal{R}$-regular graph, and a $p$-level QAOA and ab-QAOA are implemented with optimization to find a target state. In our calculations in the main text $\mathcal{R} = 3.$
We denote the iterations needed for convergence by $N_{\mathrm{ite}}$. In each iteration of the optimization in our calculation, we need to calculate the expectation of the problem Hamiltonian $\langle H_C\rangle$ $2p+1$ times to get gradients of the input parameters. In both of these two QAOA the gradient of $E_p$, the energy in one iteration for the $p$ level QAOA or ab-QAOA, with respect to the $u_{l^\prime}$ is
\begin{align}
\begin{split}
\frac{\partial E_p(\{u_l\},\{v_l\},\{h_j\})}{\partial u_{l^\prime}}&=\frac{E_p(\{u_l\}^\prime,\{v_l\},\{h_j\})-E_p(\{u_l\},\{v_l\},\{h_j\})}{\epsilon_g},\\
\{u_l\}^\prime&=\{u_1,u_2,\cdots,u_{l^\prime}+\epsilon_g,\cdots\},
\end{split}
\end{align}
where $\epsilon_g$ is a small quantity. There are $p$ $u_l$, so $p$ $E_p(\{u_l\}^\prime,\{v_l\},\{h_j\})$ and one $E_p(\{u_l\},\{v_l\},\{h_j\})$ are needed. As a result, $2p+1$ calculations of $\langle H_C\rangle$ are needed.
In a single calculation of $\langle H_C\rangle$, one needs to measure $n\mathcal{R} /2$ different $ZZ$ terms of $H_C$. $|\psi_f\rangle$ (either $|\psi_f^{\mathrm{s}}\rangle$ or $|\psi_f^{\mathrm{ab}}\rangle$ in the main text)
is prepared $M_{ZZ}$ times to get an accurate expectation value for the $ZZ$ term.
In the ab-QAOA, unlike the QAOA, knowledge of the $Z$ term is also needed to guide $\{h_j\}$ in the flowing iteration. However, this does not require an additional measurement, since if we have the value of $\langle ZZ\rangle$ measured in the computational basis, we automatically also know $\langle Z\rangle$, as we now show. Consider a single $ZZ$ term, $Z_{v_1}Z_{v_2}$. It has a spectral decomposition
\begin{equation}
\begin{split}
Z_{v_1}Z_{v_2}&=|0_{v_1}\rangle\langle 0_{v_1}|\otimes|0_{v_2}\rangle\langle 0_{v_2}|-|0_{v_1}\rangle\langle 0_{v_1}|\otimes|1_{v_2}\rangle\langle 1_{v_2}|\\
&-|1_{v_1}\rangle\langle 1_{v_1}|\otimes|0_{v_2}\rangle\langle 0_{v_2}|+|1_{v_1}\rangle\langle 1_{v_1}|\otimes|1_{v_2}\rangle\langle 1_{v_2}|,
\end{split}
\end{equation}
where $|1_{v_1}\rangle\langle 1_{v_1}|\otimes|1_{v_2}\rangle\langle 1_{v_2}|$ is short for
$\mathbb{I}\otimes\cdots\otimes\underbrace{|1\rangle\langle 1|}_{v_1}\otimes\cdots\otimes\underbrace{|1\rangle\langle 1|}_{v_2}\otimes\cdots\otimes\mathbb{I}$, which is denoted as $T_{11}^{v_1v_2}$, so as for $T_{10}^{v_1v_2}$, $T_{01}^{v_1v_2}$ and $T_{00}^{v_1v_2}$. Once these four $T$ operators are measured then $\langle Z\rangle$ can be obtained:
\begin{align}
\begin{split}
\langle Z_{v_1}\rangle&=\langle T_{00}^{v_1v_2}\rangle+\langle T_{01}^{v_1v_2}\rangle-\langle T_{10}^{v_1v_2}\rangle-\langle T_{11}^{v_1v_2}\rangle,\\
\langle Z_{v_2}\rangle&=\langle T_{00}^{v_1v_2}\rangle+\langle T_{10}^{v_1v_2}\rangle-\langle T_{01}^{v_1v_2}\rangle-\langle T_{11}^{v_1v_2}\rangle.
\end{split}
\end{align}
As a result, there are no additional measurements needed in the ab-QAOA compared to the QAOA.
In one preparation of $|\psi_f\rangle$, the operator $\exp(-i\gamma_k H_C)$ is applied $p$ times and $\exp(-i\beta_k H_M^\mathrm{s})$ or $\exp(-i\beta_k H_M^\mathrm{ab})$ is applied $p$ times. The operator $\exp(-i\gamma_k H_C)$ can be decomposed into $3$ quantum gates while $\exp(-i\beta_k H_M^\mathrm{ab})$ can be represented by one, as shown in Fig.~\ref{fig:circuit}, so $p(3n\mathcal{R}/2+n)$ quantum gates are needed. In the meanwhile, for ab-QAOA $n$ $R_y$ rotation gates around $\hat{y}$ axis are needed for the starting state preparation from $|0\rangle^{\otimes n}$, and for the QAOA, $n$ Hadamard gates are needed.
\begin{figure}[!ht]
\centerline{
\Qcircuit @C=0.5em @R=0.5em {
\lstick{\ket{0}}&\gate{R_{y}(\theta_1)}&\ctrl{1}&\qw&\qw&\ctrl{1}&\qw&\qw&\qw&\qw&\qw&\qw&\qw&\ctrl{3}&\qw&\ctrl{3}&\gate{R_{xz}(\beta_1,h_1)}&\qw\\
\lstick{\ket{0}}&\gate{R_{y}(\theta_2)}&\targ&\qw&\gate{R_z(\xi_{12})}&\targ&\ctrl{1}&\qw&\qw&\ctrl{1}&\qw&\qw&\qw&\qw&\qw&\qw&\gate{R_{xz}(\beta_1,h_2)}&\qw\\
\lstick{\ket{0}}&\gate{R_{y}(\theta_3)}&\qw&\qw&\qw&\qw&\targ&\qw&\gate{R_z(\xi_{23})}&\targ&\ctrl{1}&\qw&\ctrl{1}&\qw&\qw&\qw&\gate{R_{xz}(\beta_1,h_3)}&\qw\\
\lstick{\ket{0}}&\gate{R_{y}(\theta_4)}&\qw&\qw&\qw&\qw&\qw&\qw&\qw&\qw&\targ&\gate{R_z(\xi_{34})}&\targ&\targ&\gate{R_{z}(\xi_{14})}&\targ&\gate{R_{xz}(\beta_1,h_4)}&\qw \gategroup{1}{2}{4}{2}{.7em}{--}
}
}
\caption{Quantum circuit for $1$-level ab-QAOA on 2-regular graphs with 4 vertices. $\xi_{v_{1}v_{2}}$ is the real coefficient of $Z_{v_{1}}Z_{v_{2}}$ appearing in $\exp(-i\gamma_1 H_C)$. $R_y$ and $R_z$ are the rotation operators around the $\hat{y}$ and $\hat{z}$ axis respectively while $R_{xz}(\beta_1,h_j)=\exp[-i\beta_1 (X_j-h_j Z_j)]$. When $h_j=0$, $R_{xz}$ is the rotation operator around the $\hat{x}$ axis. The gates in the dashed box prepare the starting state for ab-QAOA. There are $4+3\times 4+4=20$ gates in the circuit. }\label{fig:circuit}
\end{figure}
In conclusion, there are
\begin{equation}
N_{\mathrm{gate}}=N_{\mathrm{ite}}(2p+1)M_{ZZ}\frac{n\mathcal{R}}{2}[p(\frac{3n\mathcal{R}}{2}+n)+n]
\end{equation}
quantum gates for a $p$-level QAOA or ab-QAOA with full optimization, $N_{\mathrm{gate}}\sim O(N_{\mathrm{ite}} p^2n^2\mathcal{R}^2)$.
In our simulation, there are two kinds of initial points. One kind is the randomly generated points in level $1$, and the other one is the points generated with the above outer loop in the other levels. Since $p^{*}$ is always larger than $1$ for $r^{*}=0.99$, we focus on $N_{\mathrm{ite}}$ when the level $p\geq 2$. In this case, the iterations are similar among different graphs and between the QAOA and ab-QAOA as shown in Fig.~\ref{fig:ite_level2}. So we conclude $N_{\mathrm{ite}}$ is the same constant for different levels and for both algorithms, so $N_{\mathrm{gate}}\sim O(p^2n^2\mathcal{R}^2)$. The additional classical cost for the ab-QAOA is only a small constant. This is because essentially the entire classical cost is in the optimization routine, which does not depend on whether bias fields are included, since these fields are not optimized over.
\begin{figure}[ht]
\centering
\subfigure[Iterations for w3r graphs in standard QAOA]
{
\begin{minipage}[h]{0.45\linewidth}
\centering
\includegraphics[scale=0.48]{supp_ite_o_w3r.pdf}
\end{minipage}
}
\subfigure[Iterations for w3r graphs in ab-QAOA]
{
\begin{minipage}[h]{0.45\linewidth}
\centering
\includegraphics[scale=0.48]{supp_ite_b_w3r.pdf}
\end{minipage}
}\\
\subfigure[Iterations for u3r graphs in standard QAOA]
{
\begin{minipage}[h]{0.45\linewidth}
\centering
\includegraphics[scale=0.48]{supp_ite_o_u3r.pdf}
\end{minipage}
}
\subfigure[Iterations for u3r graphs in ab-QAOA]
{
\begin{minipage}[h]{0.45\linewidth}
\centering
\includegraphics[scale=0.48]{supp_ite_b_u3r.pdf}
\end{minipage}
}
\caption{Iterations needed for convergence $N_{\mathrm{ite}}$ in u3r graphs and w3r graphs for points generated by the above outer loop. $N_{\mathrm{ite}}$ is the average over $R$ samples. The classical optimizer is the Adam gradient-based stochastic optimization algorithm mentioned above. $N_{\mathrm{ite}}$ is very similar for different graphs and for the two different algorithms.}\label{fig:ite_level2}
\end{figure}
Of course this analysis assumes that there is no error correction. It also assumes that 2-qubit gates can be applied to any pair of qubits, thus avoiding the necessity of SWAP gates. These considerations apply equally to QAOA and ab-QAOA, so they should not affect the speedup that is defined in the main text since it is a \textit{relative} speedup. Similarly, $\mathcal{R}$ and $n$ are the same for the two algorithms and the same reasoning may be applied. For a given accuracy and problem size, only $p$ is different.
\section{Effect of bias fields for level $p=1$.}
In this section, we illustrate the effect of bias fields for the smallest non-trivial graphs and only at level $p=1$. This isolates the effect of having these fields in the mixing Hamiltonian. We simply repeat the evolution, measuring $\langle H_C\rangle$ at the end of each step and then update the bias fields in the mixing Hamiltonian and the starting wavefunction using the prescription given above. Of course this leads to lower fidelities than for the full algorithm presented in the main text.
In the ab-QAOA, the mixing Hamiltonian with bias field is:
\begin{equation}
H_{M}^{\mathrm{ab}}(\{h_j\}) = \sum_{j=1}^{n} (X_{j}-h_{j}Z_{j}).
\end{equation}
Once we know one product ground state $|\psi_{\max}^{\alpha}\rangle$ of the MaxCut problem Hamiltonian [Eq.\eqref{eq:H}] (whose ground states are always degenerate and $\alpha$ is used to eliminate this degeneracy), then we have expectation value of each $Z_j$. If $h_j$ is fixed to $\langle \psi_{\max}^{\alpha}|Z_j|\psi_{\max}^{\alpha}\rangle$ in our ab-QAOA, then $|\psi_{0}^{\mathrm{ab}}\rangle$ is closer to $|\psi_{\max}^{\alpha}\rangle$ than $|-\rangle^{\otimes n}$ (the starting state of the standard QAOA), leading to a higher accuracy for the ab-QAOA.
The bias field parameter $h_j$ is updated according to
\begin{align}
h_j \rightarrow h_j-\ell(h_j-\langle Z_j \rangle).
\end{align}
This update strategy will bring $h_{j}$ closer to $\langle \psi_{\max}^{\alpha}|Z_{j}|\psi_{\max}^{\alpha}\rangle$ and the starting state $|\psi_{0}^{\mathrm{ab}}\rangle$ closer to $|\psi_{\max}^{\alpha}\rangle$. In realistic calculations, prior knowledge of $|\psi_{\max}^{\alpha}\rangle$ may not be available. It turns out that we can still find $|\psi_{\max}^{\alpha}\rangle$ faster than the QAOA even without prior knowledge of $|\psi_{\max}^{\alpha}\rangle$, as we now show.
To illustrate this, we calculate the fidelity $\sum_\alpha|\langle \psi_{0}^{\mathrm{ab}}|\psi_{\max}^{\alpha}\rangle|^2$, where $|\psi_{0}^{\mathrm{ab}}\rangle$ is the ground state of $H_{M}^\mathrm{ab}(\{h_j\})$ in the best optimization iterations from $R$ initial points in the level-$1$ ab-QAOA for the u3r graphs with $8$ vertices [Fig.~\ref{fig:u3r8}]. The sum is over the ground states that ab-QAOA steers the starting state to. For comparison, we also calculate $\sum_\alpha|\langle \psi_{0}^{\mathrm{s}}|\psi_{\max}^{\alpha}\rangle|^2$, $\sum_\alpha|\langle \psi_{f}^{\mathrm{s}}|\psi_{\max}^{\alpha}\rangle|^2$ and $\sum_\alpha|\langle \psi_{f}^{\mathrm{ab}}|\psi_{\max}^{\alpha}\rangle|^2$ in the best optimization iterations in Fig.~\ref{fig:bias}, where $|\psi_{0}^{\mathrm{s}}\rangle$ is the starting state of the standard QAOA, $|\psi_{f}^{\mathrm{s}}\rangle$ is the output state produced by the QAOA and $|\psi_{f}^{\mathrm{ab}}\rangle$ is the output state produced by the ab-QAOA.
\begin{figure}[h]
\centering
\includegraphics[scale=0.48]{supp_3_8.pdf}
\caption{All the $5$ different u3r graphs with 8 vertices. These graphs are labeled as $1,2,\cdots$ in sequence.}\label{fig:u3r8}
\end{figure}
It is clear that the bias field will bring the starting state closer to one or more ground states of $H_{C}$. There are some iterations for which both the starting state and output state curves of ab-QAOA grow rapidly and that is when the bias field brings the starting state $|\psi_0^\mathrm{ab}\rangle$ close to one or more ground states. The operations $\exp(-i\beta_k H_M^\mathrm{ab})$ and $\exp(-i\gamma_k H_c)$ bring $|\psi_f^\mathrm{ab}\rangle$ closer to the target states than $|\psi_0^\mathrm{ab}\rangle$. Note that the fidelity approaches $0.5$ for the output state of the ab-QAOA. This is bounded above by the ab-QAOA driven by fixed bias fields $h_j=\langle\psi_{\max}^\alpha|Z_j|\psi_{\max}^\alpha\rangle$ with $\ell=0$.
\begin{figure}
\centering
\subfigure[ Graph $1$]
{
\begin{minipage}[h]{0.45\linewidth}
\centering
\includegraphics[scale=0.48]{supp_F0.pdf}
\end{minipage}\label{fig:bias_a}
}
\subfigure[ Graph $2$]
{
\begin{minipage}[h]{0.45\linewidth}
\centering
\includegraphics[scale=0.48]{supp_F1.pdf}
\end{minipage}\label{fig:bias_b}
}\\
\subfigure[ Graph $3$]
{
\begin{minipage}[h]{0.45\linewidth}
\centering
\includegraphics[scale=0.48]{supp_F2.pdf}
\end{minipage}\label{fig:bias_c}
}
\subfigure[ Graph $4$]
{
\begin{minipage}[h]{0.45\linewidth}
\centering
\includegraphics[scale=0.48]{supp_F3.pdf}
\end{minipage}\label{fig:bias_d}
}
\end{figure}
\begin{figure}
\subfigure[ Graph $5$]
{
\begin{minipage}[h]{0.45\linewidth}
\centering
\includegraphics[scale=0.48]{supp_F4.pdf}
\end{minipage}\label{fig:bias_e}
} \\
\caption{The fidelity between the target ground state and the starting or output states from different QAOA. The calculations are on the $5$ u3r graphs in Fig.~\ref{fig:u3r8} with level-$1$ QAOA or ab-QAOA. $|\psi_{\max}^{\alpha}\rangle$ for these graphs are $|00101101\rangle$, $|10101010\rangle$, $|00101011\rangle$ ($|01101010\rangle$ or $|10101010\rangle$),$|01101001\rangle$, and $|00101101\rangle$ respectively. The bias field will steer the starting state to these states in ab-QAOA. In the last subfigure, Fig.~\ref{fig:bias_e}, the yellow curve, starting state QAOA coincides with some part of red one, starting state QAOA, which is not clear in the figure.}\label{fig:bias}
\end{figure}
To better investigate how the bias fields work, we also plot $\{h_j\}$ of graph $1$ from level $1$ ab-QAOA in the best optimization iterations (graph $1$ in Fig.~\ref{fig:bias_a}) as shown in Fig.~\ref{fig:h1}. There are four regions in Fig.~\ref{fig:h1}. In Region A, all $\{h_j\}$ decrease to $0$ quickly in the first 5 iterations. In Region B, from the $5$th iteration to the $80$th iteration, all $\{h_j\}$ are near $0$. In Region C, from about the $80$th iteration to the $100$th iteration, the $\{h_j\}$ diverge and each $h_j$ tries to find its true value, $\langle\psi_{\max}^\alpha|Z_j|\psi_{\max}^\alpha\rangle$. In Region D, in the last half of the optimization, the values of $\{h_j\}$ don't change. The behavior of the fidelity in Fig.~\ref{fig:h1} is related to $\{h_j\}$ in Fig.~\ref{fig:h1}. The divergence of $\{h_j\}$ from $0$ implies a sharp rise in the fidelity. The two platforms in Figs.~\ref{fig:bias_b} and \ref{fig:bias_d} mean that not all $\{h_j\}$ begin to diverge in the same iteration.
\begin{figure}
\includegraphics[scale=0.48]{supp_h1.pdf}
\caption{The bias fields $\{h_j\}$ of graph $1$ in the best optimization iterations in ab-QAOA. The results are obtained from level-$1$ QAOA or ab-QAOA. $h_1$, $h_2$, $h_4$ and $h_7$ are greater than $0$ at the end of optimization. }\label{fig:h1}
\end{figure}
For each of these four regions, we choose four specific points and plot the energy landscape using ab-QAOA as shown in Fig.~\ref{fig:landscape}. Note that for region B, the landscape is in close agreement with that of the QAOA since all $\{h_j\}$ are small. In region A, due to the "wrong" bias fields, it is harder to find the target state using ab-QAOA than the QAOA, so the QAOA output state can be regarded as the optimal state of ab-QAOA's. As a result, each $h_j$ moves towards $0$ fast in region A. In region B, although all $\{h_j\}$ are small, their effects accumulate until the bias fields can have a significant effect on the cost function. In region C, all $\{h_j\}$ changes quickly because of the accumulation in region B. When updating $\{h_j\}$, all $\{h_j\}$ are getting closer to their true values, so it is easier to find the target state in this region, which can be explained by the smaller lowest energy in the landscape. In region D, the output energy nearly meets the convergence criterion and each $h_j$ finds its true value, $\langle\psi_{\max}^\alpha|Z_j|\psi_{\max}^\alpha\rangle$, so the lowest energy is smaller than in the other regions.
\begin{figure}[h]
\subfigure[Energy landscape in region A ($0$th iteration)]
{
\begin{minipage}[h]{0.45\linewidth}
\centering
\includegraphics[scale=0.48]{supp_landscape1_0.pdf}
\end{minipage}
}
\subfigure[Energy landscape in region B ($50$th iteration)]
{
\begin{minipage}[h]{0.45\linewidth}
\centering
\includegraphics[scale=0.48]{supp_landscape1_50.pdf}
\end{minipage}
}
\end{figure}
\begin{figure}
\subfigure[Energy landscape in region C ($95$th iteration)]
{
\begin{minipage}[h]{0.45\linewidth}
\centering
\includegraphics[scale=0.48]{supp_landscape1_95.pdf}
\end{minipage}
}
\subfigure[Energy landscape in region D (final iteration)]
{
\begin{minipage}[h]{0.45\linewidth}
\centering
\includegraphics[scale=0.48]{supp_landscape1_f.pdf}
\end{minipage}
}
\caption{The energy landscape of graph $1$. The $0$th, $50$th, $95$th, and final iteration belong to the region A, B, C and D. As analysed in Ref.~\cite{lukin}, $\gamma_1$, $\beta_1$ can be restricted to $[-\pi/2,\pi/2]$ and $[-\pi/4,\pi/4]$ respectively, so $u_1$ and $v_1$ are restricted to $[-\sqrt{2}\pi/2,\sqrt{2}\pi/2]$ and $[-\sqrt{2}\pi/4,\sqrt{2}\pi/4]$ according to Eq.\eqref{eq:uv}. }\label{fig:landscape}
\end{figure}
\newpage
\section{Fitting parameters}
Here we list the fitting parameters defined in the main text.
\begin{table}[h!]
\centering
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline
{ }&$n=8$&$n=10$&$n=12$&$n=14$&$n=16$&$n=18$\\
\hline
$p_0$&$0.42800977$&$0.62230354$&$0.73323484$&$0.80231316$&$0.90685782$&$0.9443149$\\
\hline
$c$&$0.10739954$&$-0.12764388$&$-0.23251733$&$-0.26348911$&$-0.39672828$&$-0.35855954$\\
\hline
\end{tabular}
\caption{Fitting parameters of standard QAOA for the accuracy in w3r graphs }
\end{table}
\begin{table}[h!]
\centering
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline
{ }&$n=8$&$n=10$&$n=12$&$n=14$&$n=16$&$n=18$\\
\hline
$p_0$&$0.17327266$&$0.17580396$&$0.17704141$&$0.17957186$&$0.18424291$&$0.23922264$\\
\hline
$c$&$-0.24508518$&$-0.27957327$&$-0.26115427$&$-0.27881266$&$-0.24826492$&$-0.67913148$\\
\hline
\end{tabular}
\caption{Fitting parameters of ab-QAOA for the accuracy in w3r graphs }
\end{table}
\begin{table}[h!]
\centering
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline
{ }&$n=8$&$n=10$&$n=12$&$n=14$&$n=16$&$n=18$\\
\hline
$p_0$&$8.43984294$&$10.17259833$&$12.81448118$&$18.54647523$&$29.51964065$&$35.31026806$\\
\hline
$c$&$0.0340003$&$0.08045262$&$0.08814327$&$0.06948011$&$0.04850331$&$0.04527017$\\
\hline
\end{tabular}
\caption{Fitting parameters of standard QAOA for the infidelity in w3r graphs }
\end{table}
\begin{table}[h!]
\centering
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline
{ }&$n=8$&$n=10$&$n=12$&$n=14$&$n=16$&$n=18$\\
\hline
$p_0$&$0.50210827$&$0.61290557$&$0.75486606$&$1.00539136$&$1.57615813$&$2.53419367$\\
\hline
$c$&$1.01553842$&$0.79839955$&$0.6930013$&$0.58993449$&$0.51400306$&$0.37161959$\\
\hline
\end{tabular}
\caption{Fitting parameters of ab-QAOA for the infidelity in w3r graphs }
\end{table}
\begin{table}[h!]
\centering
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline
{ }&$n=8$&$n=10$&$n=12$&$n=14$&$n=16$&$n=18$\\
\hline
$p_0$&$1.30228564$&$2.07538215$&$2.46757535$&$2.7788049$&$3.03328213$&$3.15617106$\\
\hline
$c$&$-0.55076766$&$-1.01623816$&$-1.16513946$&$-1.24119178$&$-1.26713441$&$-1.26322867$\\
\hline
\end{tabular}
\caption{Fitting parameters of standard QAOA for the accuracy in u3r graphs }
\end{table}
\begin{table}[h!]
\centering
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline
{ }&$n=8$&$n=10$&$n=12$&$n=14$&$n=16$&$n=18$\\
\hline
$p_0$&$0.0481013$&$0.04482199$&$0.05410145$&$ 0.05651275$&$0.04743622$&$0.0476715 $\\
\hline
$c$&$2.41506594$&$2.5535474$&$1.93841831$&$1.89568572$&$2.39503391$&$2.39867199$\\
\hline
\end{tabular}
\caption{Fitting parameters of ab-QAOA for the accuracy in u3r graphs }
\end{table}
\begin{table}[h!]
\centering
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline
{ }&$n=8$&$n=10$&$n=12$&$n=14$&$n=16$&$n=18$\\
\hline
$p_0$&$1.2630754$&$ 2.47189574$&$3.33176173$&$4.93148061$&$5.8239181$&$6.96389395$\\
\hline
$c$&$1.18876273$&$0.55371133$&$0.42748629$&$0.28974544$&$0.27341578$&$0.24360132$\\
\hline
\end{tabular}
\caption{Fitting parameters of standard QAOA for the infidelity in u3r graphs }
\end{table}
\begin{table}[h!]
\centering
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline
{ }&$n=8$&$n=10$&$n=12$&$n=14$&$n=16$&$n=18$\\
\hline
$p_0$&$0.04960161$&$0.04422032$&$0.0519072$&$0.05735311$&$0.04898178$&$0.0498607$\\
\hline
$c$&$4.14004776$&$4.58598679$&$4.17168606$&$4.08843585$&$4.6724513$&$4.70423705$\\
\hline
\end{tabular}
\caption{Fitting parameters of ab-QAOA for the infidelity in u3r graphs }
\end{table}
\newpage
Using these fitting parameters and redefining the vertical axes of Figs. 1 and 2 of the main text, we can collapse the graphs for the accuracy and infidelity onto straight lines, as shown in Fig.~\ref{fig:fito} and Fig.~\ref{fig:fitb}.
\begin{figure}[ht]
\centering
\subfigure[Accuracy for w3r graphs]
{
\begin{minipage}[h]{0.45\linewidth}
\centering
\includegraphics[scale=0.4]{supp_r_w3r_fit_o.pdf}
\end{minipage}
}
\subfigure[Infidelity for w3r graphs]
{
\begin{minipage}[h]{0.45\linewidth}
\centering
\includegraphics[scale=0.4]{supp_F_w3r_fit_o.pdf}
\end{minipage}
}\\
\subfigure[Accuracy for u3r graphs]
{
\begin{minipage}[h]{0.45\linewidth}
\centering
\includegraphics[scale=0.4]{supp_r_u3r_fit_o.pdf}
\end{minipage}
}
\subfigure[Infidelity for u3r graphs]
{
\begin{minipage}[h]{0.45\linewidth}
\centering
\includegraphics[scale=0.4]{supp_F_u3r_fit_o.pdf}
\end{minipage}
}
\caption{Fitting for standard QAOA. The dashed lines in the the four subplots represent $p=p_0[c-\ln(1-r)]^2$, $p=p_0[c-\ln(1-F)]$, $p=p_0[c-\ln(1-r)]$ or $p=p_0[c-\ln(1-F)]$ which is equivalent to the fitting functions in the main text.}\label{fig:fito}
\end{figure}
\begin{figure}[h!]
\centering
\subfigure[Accuracy for w3r graphs]
{
\begin{minipage}[h]{0.45\linewidth}
\centering
\includegraphics[scale=0.4]{supp_r_w3r_fit_b.pdf}
\end{minipage}
}
\subfigure[Infidelity for w3r graphs]
{
\begin{minipage}[h]{0.45\linewidth}
\centering
\includegraphics[scale=0.4]{supp_F_w3r_fit_b.pdf}
\end{minipage}
}\\
\subfigure[Accuracy for u3r graphs]
{
\begin{minipage}[h]{0.45\linewidth}
\centering
\includegraphics[scale=0.4]{supp_r_u3r_fit_b.pdf}
\end{minipage}
}
\subfigure[Infidelity for u3r graphs]
{
\begin{minipage}[h]{0.45\linewidth}
\centering
\includegraphics[scale=0.4]{supp_F_u3r_fit_b.pdf}
\end{minipage}
}
\caption{Fitting for ab-QAOA. The dashed lines in the the four subplots represent $p=p_0[c-\ln(1-r)]^2$ or $p=p_0[c-\ln(1-F)]^2$, which is equivalent to the fitting functions in the main text.}\label{fig:fitb}
\end{figure}
\section{$p^*$ in Speedup}
Here we list $p^*$ for the calculation of the speedup $S(n)$ shown in Fig. 4 in the main text. For the QAOA, $p^*$ is obtained from the fitting function. For the ab-QAOA, $p^*$ is obtained directly from the numerical simulation. For clarity of the speedup, all $p^{*}$ are rounded to integers.
\begin{table}[h!]
\centering
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline
{ }&$n=8$&$n=10$&$n=12$&$n=14$&$n=16$&$n=18$\\
\hline
standard QAOA&$10$&$12$&$14$&$15$&$16$&$17$\\
\hline
ab-QAOA&$3$&$3$&$3$&$3$&$3$&$3$\\
\hline
\end{tabular}
\caption{$p^*$ for w3r graphs }
\end{table}
\begin{table}[h!]
\centering
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline
{ }&$n=8$&$n=10$&$n=12$&$n=14$&$n=16$&$n=18$\\
\hline
standard QAOA&$5$&$7$&$8$&$9$&$10$&$11$\\
\hline
ab-QAOA&$3$&$3$&$3$&$3$&$3$&$3$\\
\hline
\end{tabular}
\caption{$p^*$ for u3r graphs }
\end{table}
\end{document}
| {
"timestamp": "2021-12-23T02:07:23",
"yymm": "2105",
"arxiv_id": "2105.11946",
"language": "en",
"url": "https://arxiv.org/abs/2105.11946",
"abstract": "The quantum approximate optimization algorithm (QAOA) transforms a simple many-qubit wavefunction into one which encodes a solution to a difficult classical optimization problem. It does this by optimizing the schedule according to which two unitary operators are alternately applied to the qubits. In this paper, the QAOA is modified by updating the operators themselves to include local fields, using information from the measured wavefunction at the end of one iteration step to improve the operators at later steps. It is shown by numerical simulation on MaxCut problems that, for a fixed accuracy, this procedure decreases the runtime of QAOA very substantially. This improvement appears to increase with the problem size. Our method requires essentially the same number of quantum gates per optimization step as the standard QAOA, and no additional measurements. This modified algorithm enhances the prospects for quantum advantage for certain optimization problems.",
"subjects": "Quantum Physics (quant-ph)",
"title": "Quantum Approximate Optimization Algorithm with Adaptive Bias Fields",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9770226260757067,
"lm_q2_score": 0.7248702702332475,
"lm_q1q2_score": 0.7082146549874946
} |
https://arxiv.org/abs/2006.15379 | Cops and an Insightful Robber | The 'Cheating Robot' version of Cops and Robbers is played on a finite, simple, connected graph. The players move in the same time period. However, before moving, the robot observes to which vertices the cops are moving and it is fast enough to complete its move in the time period. The cops also know that the robot will use this information. More cops are required to capture a robot than to capture a robber. Indeed, the minimum degree is a lower bound on the number of cops required to capture a robot. Only on a tree is one cop guaranteed to capture a robot, although two cops are sufficient to capture both a robber and a robot on outerplanar graphs. In graphs where retracts are involved, we show how cop strategies against a robber can be modified to capture a robot. This approach gives exact numbers for hypercubes, and $k$-dimensional grids in general. | \section{Introduction}
In Pursuit-Evasion games such as Cops and Robbers, the players move alternately. This is not true in `real life'. Also, the Bad Guys often have access to information (via bribery or bugging offices and equipment) about the Good Guys strategies. How this changes the Good Guys strategies is the subject of this paper.
For the \emph{Cheating Robot} (CR) variant of Cops and Robbers, the board is a finite, simple, connected graph. One player controls $k$ tokens, called \textit{cops}, and the other player controls one token, called the \textit{robot}. In accordance with usual practice, the cops are female and the robot is male. Initially, the cops place their tokens on vertices, then the robot places their token. A vertex can have more than one token. Thereafter, they move each of their tokens to another vertex in its closed neighbourhood. The players move in the same time period, however, the robot moves immediately after divining the cops' moves.
Play can be regarded as alternating, and we will adopt this point of view. However, the capturing rules must be changed to reflect the timing of the moves.
A cop \textit{captures} the robot either (i) if, at the end of the robot's move, a cop and the robot are on the same vertex, or (ii) the robot traverses
an edge traversed by a cop on their last move. We assume that the cops know that the robot has the information about their moves when they are determining their strategy.
This model was first introduced within the general context of combinatorial games in~\cite{Huggan,HugganRJN}. For context, the name,
Cheating Robot, comes from the Japanese robot that wins Rock-Paper-Scissors 100\% of the time against humans.
It cheats by having processors fast enough to both identify the human's move and respond appropriately \cite{ItoSYI2016,ShortHVS2010}. Even though the players are supposed to move in the same time period, the robot is moving second every time.
The original game of Cops and Robbers was first considered in \cite{NowaW,Quilliot} and extended in \cite{AignerF}. The players moved alternately, there is perfect information and the robber is caught if, at any point in the game, a cop and the robber are on the same vertex. The game has been extensively studied with many outstanding questions. See \cite{BonatoN} for the foundations of this area of research. Cops and Robbers belong to a wider class called pursuit-evasion games.
For a graph theory reference, see \cite{West}. The following definitions are standard.
\begin{definition}
For a graph $G$, let $c(G)$ and $c_{cr}(G)$ be the least number of cops required to capture the robber and robot, respectively, on $G$.
\end{definition}
When referring to results, to help make the distinction clear, results about a `robber' will always be referring to the Cops and Robbers game and results about a `robot' will refer to the Cheating Robot variant. Knowing the behaviour of parameters on subgraphs is often helpful. We show that subgraphs with minimum degree constraints give a lower bound for the number of cops needed to capture a robot, Theorem \ref{thm: k-core}. This also leads to showing that $c_{cr}(G)=1$ if and only if $G$ is a tree, Theorem \ref{thm: trees}.
It is not known if $c_{cr}(H) \leq c_{cr}(G)$ for $H$ an induced subgraph of $G$ but Theorem \ref{thm:retract} shows that $c_{cr}(H) \leq c_{cr}(G)$ if $H$ is a retract. Retracts are useful. Apart from providing a lower bound, they play an important part when considering products of graphs. Theorem \ref{thm:cartesianproduct} proves that the Cartesian product of two graphs requires no more cops than the total needed for two graphs individually. This immediately gives the exact number for hypercubes and $k$-dimensional grids, Corollary \ref{cor:cartesiangrid}. In Cops and Robbers,
the strong product of graphs was relatively easy to analyse. The same is not true in the Cheating Robot model. Even for paths,
although the two-dimensional case is relatively straightforward, Theorem \ref{thm: strong grid base case}, in general, only bounds are known,
Theorem \ref{thm: k-dim strong grids}.
For outerplanar graphs, again, retracts help to show that only 2 cops are required to capture the robot, Theorem \ref{thm:outerplanar}. We close with some open questions.
Closely related game variants to the Cheating Robot are: (1) simultaneously moving cops and robbers~\cite{Konstantinidis2016} and (2) surrounding cops and robbers (\cite{BradshawH}, \cite{BurgessCCDFJP}). The former investigates a variant where players are all moving simultaneously. Naturally, this is a probabilistic approach. The latter examines a similar ruleset to ours with the caveat that the robber and cop are allowed to cross an edge and the game can end after a cop move. In comparing the model from \cite{BurgessCCDFJP} and ours, a natural question arises: How closely related are the models? More specifically, do the cop numbers differ by at most $1$ for all classes of graphs? This currently remains an open question.
\section{Degree Constraints}
For any graph $G$, comparing strategies available to the robot and those available to the robber, we obtain a lower bound on the
required number of cops.
\begin{thm} For any graph $G$, $c_{cr}(G)\geq c(G)$.
\end{thm}
\begin{proof} If there are $c(G)-1$ cops, the robot can use the robber's strategy in $G$.
\end{proof}
\begin{cor}\label{cor: robot win}
Let $G$ be a graph and the cop player is controlling $k$ cops. If the robber can win on $G$ against $k$ cops then so can the robot.
\end{cor}
\begin{proof}
The robot adopts the robber's strategy.
\end{proof}
The converse of Corollary~\ref{cor: robot win} is false. For example, one cop on $C_3$ will capture the robber but not the robot because the robot will always have an escape move. The minimum degree of a graph provides a lower bound for the number of cops required to capture. This idea is extended in the next result.
\begin{definition} Let $G$ be a graph, $k$ a positive integer, and $\delta_{G}$ be the minimum degree of $G$. An induced subgraph $H$ of $G$ is a $k$-core if
$\delta_H\geq k$.
\end{definition}
It is known that to find a $k$-core it is sufficient to iteratively choose a vertex $x$ with degree less than $k$, delete $x$,
and continue in the reduced graph.
\begin{thm}\label{thm: k-core} Let $G$ be a graph with a $k$-core for some $k$. Then $c_{cr}(G)\geq k$.
\end{thm}
\begin{proof} Let $H$ be a $k$-core of $G$. The robot restricts himself to only moving in $H$.
Suppose there are $k-1$ cops and suppose at some move the robot is on vertex $x\in V(H)$. By assumption,
the degree of $x$ in $H$ is at least $k$. If none of the cops move to $x$ the robot does not move. If $i$ of the cops move
from, say $y_1,y_2,\ldots,y_i$ to $x$ then the robot will be captured if he moves to any $y_j$. However, there are only $k-i-1$ cops to cover the other, at least $k-i$, neighbours of $x$. Consequently, one neighbour will be unoccupied and the robot moves to it. \end{proof}
\begin{cor}\label{cor: cycles}
For any cycle $C_n$, $c_{cr}(C_n)=2$.
\end{cor}
\begin{proof}
A cycle is a 2-core and hence at least two cops are required. Two cops can capture by starting on the same vertex and moving around the cycle in different directions.
\end{proof}
To characterize when one cop will suffice is now straightforward.
\begin{thm}\label{thm: trees} Let $G$ be a connected graph. Then $c_{cr}(G)=1$ if and only if $G$ is a tree.
\end{thm}
\begin{proof}
If $G$ is a tree, then the cop places herself somewhere close to the centre of the tree, and the robot places himself anywhere. Since $G$ is a tree, there exists a unique shortest path between the cop and the robot. The cop moves along the first edge of this path. The robot cannot pass the cop and is eventually forced onto a leaf and is caught on the next move.
If $G$ is not a tree then there exists a cycle in $G$. This cycle is a 2-core and, by Theorem \ref{thm: k-core}, $c_{cr}(G) >1$.
\end{proof}
\section{Retracts}
A typical technique to capture the robber is to capture his image on a subgraph first, see the papers \cite{AignerF,BI,NowaW} and the book \cite{BonatoN} for others. The following definitions are standard and can be found in \cite{BonatoN}.
\begin{definition}\normalfont{\cite{BonatoN}} Given a graph $G$, an induced subgraph $R$ is a \textit{retract} if there is a \textit{retraction}
map $f: V(G)\rightarrow V(R)$
where (i) if $x\sim y$ then $f(x)\sim f(y)$ or $f(x)=f(y)$, and (ii) if $x\in V(R)$ then $f(x)=x$.
\end{definition}
For example, let $P_m=\{a_1,a_2,\ldots, a_m\}$ and $P_n=\{b_1,b_2,\ldots,b_n\}$. In both cases, if $G=P_m\square P_n$, the Cartesian grid, or $G=P_m\boxtimes P_n$, the strong grid, then for each $i$, $\{a_i\}\times P_n$ is a retract of $G$ with $f((a_r,b_s))=(a_i,b_s)$. Similarly, $P_m\times \{b_j\}$ is also a retract of $G$ with $g((a_r,b_s)) = (a_r,b_j)$. Unless otherwise specified, these will be the retraction maps used when dealing with paths.
\begin{definition}\normalfont{\cite{BonatoN}}
Given a graph $G$, a retract $R$, and a retraction map $f:G\rightarrow R$, then $f(x)$ is the \textit{shadow} of vertex $x$.
\end{definition}
A retraction map is \textit{edge preserving} in that edges are mapped to edges or the endpoints are mapped to the same vertex.
This means that the shadow of the robot in a retract traces out a walk on the retract. Retracts are well known in the Cops and Robbers literature (see \cite{NowaW,BI} for example). Retracts are also important substructures in the Cheating Robot model.
\begin{thm}\label{thm:retract} If $R$ is a retract of $G$ then $c_{cr}(R)\leq c_{cr}(G)$.
\end{thm}
\begin{proof} Let $f:G\rightarrow R$ be a retraction map. We restrict the robot to playing on $R$ but allow $c_{cr}(G)$ cops to play on $G$. Now, consider the situation with $c_{cr}(G)$ cops on
$R$ where each cop in $R$ is the image of the corresponding cop in $G$. Note that, because $f$ is edge-preserving, any move by a cop in $G$ is a legal move for the image of the cop. The cops on $G$ have a winning strategy. On the last move of this winning strategy,
the cops move to occupy all, except for possibly one, neighbouring vertices to the robot and the one cop moves to the vertex containing the robot,
moving from this unoccupied vertex if it exists. The robot has no escape, in $G$, and so has no escape in $R$. These vertices include all the vertices in $R$ and a cop has moved on to the vertex occupied by the robot. The robot has been caught by the $c_{cr}(G)$ cops in $R$.
\end{proof}
The follow-up result in \cite{BI} has $c(G)\leq \max\{c(R),c(G\setminus R)+1\}$. However, this is not true in the Cheating Robot
model. In Figure~\ref{fig: retraction example}, $R=C_6=G\setminus R$, $c_{cr}(R) = c_{cr}(G\setminus R)=2$, and $R$ is a retract of $G$.
However, every vertex of $G$ has degree 5, i.e., $G$ is a 5-core, thus $c_{cr}(G)\geq 5>3=\max\{c(R),c(G\setminus R)+1\}$.
\begin{figure}[htb]
\begin{center}
\scalebox{0.65}{
\begin{tikzpicture}
\draw [line width=1.pt] (-1.,5.)-- (1.,5.);
\draw [line width=1.pt] (-1.,5.)-- (-2.,3.);
\draw [line width=1.pt] (-2.,3.)-- (-1,1.);
\draw [line width=1.pt] (-1.,1.)-- (1,1.);
\draw [line width=1.pt] (2.,3)-- (1,1.);
\draw [line width=1.pt] (2.,3.)-- (1,5.);
\draw [line width=1.pt] (-2.,7.)-- (2.,7.);
\draw [line width=1.pt] (-2.,7.)-- (-4.,3.);
\draw [line width=1.pt] (-4.,3.)-- (-2,-1.);
\draw [line width=1.pt] (-2.,-1.)-- (2,-1.);
\draw [line width=1.pt] (4.,3)-- (2,-1.);
\draw [line width=1.pt] (4.,3.)-- (2,7.);
\draw [ line width=1.pt] (-1.,5.)-- (-2.,7.);
\draw [line width=1.pt] (1.,5.)-- (2.,7.);
\draw [line width=1.pt] (-4.,3.)-- (-2,3.);
\draw [line width=1.pt] (4.,3.)-- (2,3.);
\draw [line width=1.pt] (1.,1)-- (2,-1.);
\draw [line width=1.pt] (-1.,1.)-- (-2,-1.);
\draw [line width=1.pt] (-1.,5.)-- (2.,7.);
\draw [line width=1.pt] (1.,5.)-- (-2.,7.);
\draw [line width=1.pt] (1,5.)-- (4,3.);
\draw [line width=1.pt] (2.,7.)-- (2,3.);
\draw [line width=1.pt] (1.,1)-- (-2,-1.);
\draw [line width=1.pt] (-1.,1.)-- (2,-1.);
\draw [line width=1.pt] (-1,5.)-- (-4,3.);
\draw [line width=1.pt] (-2.,7.)-- (-2,3.);
\draw [line width=1.pt] (1.,1)-- (4,3.);
\draw [line width=1.pt] (2.,3.)-- (2,-1.);
\draw [line width=1.pt] (-1.,1)-- (-4,3.);
\draw [line width=1.pt] (-2.,3.)-- (-2,-1.);
\begin{scriptsize}
\draw [fill=black] (-1.,5.) circle (4.0pt);
\draw [fill=black] (1,5.) circle (4.0pt);
\draw [fill=black] (-1.,1.) circle (4.0pt);
\draw [fill=black] (1,1.) circle (4.0pt);
\draw [fill=black] (-2.,3.) circle (4.0pt);
\draw [fill=black] (2,3.) circle (4.0pt);
\draw [fill=black] (-2.,7.) circle (2.0pt);
\draw [fill=black] (2,7.) circle (2.0pt);
\draw [fill=black] (-2.,-1.) circle (2.0pt);
\draw [fill=black] (2,-1.) circle (2.0pt);
\draw [fill=black] (-4.,3.) circle (2.0pt);
\draw [fill=black] (4,3.) circle (2.0pt);
\end{scriptsize}
\end{tikzpicture}}
\caption{A graph $G$ with retract, $R$, which is the graph induced by the bold vertices.}\label{fig: retraction example}
\end{center}
\end{figure}
Despite this, for this paper, the method by which the cops capture the robot is important when considering products and we highlight this as a result in its own right.
\begin{lemm}\label{lem:retract} Given a graph $G$, a retract $R$, a retraction map $f:G\rightarrow R$ and $c_{cr}(R)$ cops. If the cops, but not the robot, are restricted to $R$ then the cops can capture the robot's shadow. At the completion of the capture move and every move thereafter, only one cop is needed so as to move onto the shadow as part of the cops' turn. \end{lemm}
\begin{proof}
Since $f$ is edge-preserving then the shadow of the robot, on a turn, can only move to an adjacent vertex. By assumption then,
$c_{cr}(R)$ cops suffice to capture the shadow and at capture one cop, say $c_1$, is on the same vertex as the shadow. However, the robot is not the shadow and may move so the shadow has moved to an adjacent vertex. The cop $c_1$, as part of the cops turn, can move on to the shadow.
\end{proof}
If each vertex in $R$ is adjacent to at most one vertex in $G\setminus R$ then the cop that follows the robot's shadow blocks the robot's entry from
$G\setminus R$ back into $R$. In other cases, more cops are needed to follow the shadow in order to prevent the robot entering $R$. The first case is typical of Cartesian products which are considered next.
\section{Cartesian Product of Graphs}
When $G$ is a product then $G$ inherits some properties of the multiplicands. These properties have been useful when determining
the cop-number of graphs.
In this section we consider the Cartesian product of graphs and the strong product in the next. See \cite{IK} for more on graph products.
\begin{definition} The Cartesian product of graphs $G$ and $H$, written $G\square H$, has $V(G\square H)=\{(g,h): g\in V(G), h\in V(H)\}$
and $(g,h)$ is adjacent to $(g',h')$ if either $g=g'$ and $h$ is adjacent to $h'$ or $g$ is adjacent to $g'$ and $h=h'$.
\end{definition}
This definition is easily extended to the product of more than two graphs. In general, a vertex of the product is $(a_1,a_2,\ldots, a_n)$ where
$a_i$ is in the $i^\text{th}$-coordinate.
Given a graph $F$ with $c_{cr}(F)$ cops, consider the last move by the cops in their winning strategy. All vertices adjacent to the robot
will be occupied except for possibly those from which the cops will move to the robot's vertex. The robot has nowhere safe to move. However, if
these cops do not move, but the other cops do, then the robot is not captured but neither can he move. We refer to this as the \textit{surrounding} strategy.
It is an integral part of capturing the robot on the Cartesian product.
For connected graphs $G$ and $H$, To\v{s}i\'{c} \cite{tos} showed that $c(G\square H)\leq c(G)+c(H)$. The same is true for the Cheating Robot. For an example, see Figure~\ref{fig: cartesian product example}.
\begin{figure}
\begin{subfigure}{.5\textwidth}
\begin{center}
\scalebox{0.6}{
\begin{tikzpicture}
\draw [->,line width=2.pt] (0.75,2.25)-- (2.75,2.25);
\draw [->,line width=2.pt] (7.25,-1.75)--(6.25,0);
\draw [line width=1.pt] (0.,2.)-- (0.,0.);
\draw [line width=1.pt] (0.,2.)-- (0.,0.);
\draw [line width=1.pt] (3.,2.)-- (3.,0.);
\draw [line width=1.pt] (6.,2.)-- (6.,0.);
\draw [line width=1.pt] (-1.,-2.)-- (0.,0.);
\draw [line width=1.pt] (1.,-2.)-- (0.,0.);
\draw [line width=1.pt] (2.,-2.)-- (3.,0.);
\draw [line width=1.pt] (4.,-2.)-- (3.,0.);
\draw [line width=1.pt] (5.,-2.)-- (6.,0.);
\draw [line width=1.pt] (7.,-2.)-- (6.,0.);
\draw [line width=1.pt] (0.,0.)-- (3.,0.);
\draw [line width=1.pt] (3.,0.)-- (6.,0.);
\draw [line width=1.pt] (0.,2.)-- (3,2.);
\draw [line width=1.pt] (3.,2.)-- (6,2.);
\draw (-1,-2) to[out=280,in=260,loop, distance =1cm] (2,-2);
\draw (2,-2) to[out=280,in=260,loop, distance =1cm] (5,-2);
\draw (1,-2) to[out=45,in=135,loop, distance =1cm] (4,-2);
\draw (4,-2) to[out=45,in=135,loop, distance =1cm] (7,-2);
\begin{scriptsize}
\draw [fill=black] (0.,2.) circle (2.0pt);
\draw [fill=black] (3.,2.) circle (2.0pt);
\draw [fill=black] (6.,2.) circle (2.0pt);
\draw [fill=black] (0.,0.) circle (2.0pt);
\draw [fill=black] (3.,0.) circle (2.0pt);
\draw [fill=black] (6.,0.) circle (2.0pt);
\draw [fill=black] (-1.,-2.) circle (2.0pt);
\draw [fill=black] (1.,-2.) circle (2.0pt);
\draw [fill=black] (2.,-2.) circle (2.0pt);
\draw [fill=black] (4.,-2.) circle (2.0pt);
\draw [fill=black] (5.,-2.) circle (2.0pt);
\draw [fill=black] (7.,-2.) circle (2.0pt);
\draw[color=black] (3.25,0.25) node {\Large $R$};
\draw[color=black] (0.25,2.25) node {\Large $c_{1}$};
\draw[color=black] (7.35,-2.25) node {\Large $c_{2}$};
\end{scriptsize}
\end{tikzpicture}}
\caption{Initial Placement.}
\end{center}
\end{subfigure}
\begin{subfigure}{.5\textwidth}
\begin{center}
\scalebox{0.6}{
\begin{tikzpicture}
\draw [->,line width=2.pt] (2.75,2.25)-- (0.75,2.25);
\draw [->,line width=2.pt] (5.75,0.25)--(3.25,0.25);
\draw [line width=1.pt] (0.,2.)-- (0.,0.);
\draw [line width=1.pt] (3.,2.)-- (3.,0.);
\draw [line width=1.pt] (6.,2.)-- (6.,0.);
\draw [line width=1.pt] (-1.,-2.)-- (0.,0.);
\draw [line width=1.pt] (1.,-2.)-- (0.,0.);
\draw [line width=1.pt] (2.,-2.)-- (3.,0.);
\draw [line width=1.pt] (4.,-2.)-- (3.,0.);
\draw [line width=1.pt] (5.,-2.)-- (6.,0.);
\draw [line width=1.pt] (7.,-2.)-- (6.,0.);
\draw [line width=1.pt] (0.,0.)-- (3.,0.);
\draw [line width=1.pt] (3.,0.)-- (6.,0.);
\draw [line width=1.pt] (0.,2.)-- (3,2.);
\draw [line width=1.pt] (3.,2.)-- (6,2.);
\draw (-1,-2) to[out=280,in=260,loop, distance =1cm] (2,-2);
\draw (2,-2) to[out=280,in=260,loop, distance =1cm] (5,-2);
\draw (1,-2) to[out=45,in=135,loop, distance =1cm] (4,-2);
\draw (4,-2) to[out=45,in=135,loop, distance =1cm] (7,-2);
\begin{scriptsize}
\draw [fill=black] (0.,2.) circle (2.0pt);
\draw [fill=black] (3.,2.) circle (2.0pt);
\draw [fill=black] (6.,2.) circle (2.0pt);
\draw [fill=black] (0.,0.) circle (2.0pt);
\draw [fill=black] (3.,0.) circle (2.0pt);
\draw [fill=black] (6.,0.) circle (2.0pt);
\draw [fill=black] (-1.,-2.) circle (2.0pt);
\draw [fill=black] (1.,-2.) circle (2.0pt);
\draw [fill=black] (2.,-2.) circle (2.0pt);
\draw [fill=black] (4.,-2.) circle (2.0pt);
\draw [fill=black] (5.,-2.) circle (2.0pt);
\draw [fill=black] (7.,-2.) circle (2.0pt);
\draw[color=black] (0.25,0.25) node {\Large $R$};
\draw[color=black] (3.25,2.25) node {\Large $c_{1}$};
\draw[color=black] (6.35,0.25) node {\Large $c_{2}$};
\end{scriptsize}
\end{tikzpicture}}
\caption{Round 1.}
\end{center}
\end{subfigure}
\begin{subfigure}{0.5\textwidth}
\begin{center}
\scalebox{0.6}{
\begin{tikzpicture}
\draw [->,line width=2.pt] (-0.25,1.75)-- (-0.25,0.25);
\draw [->,line width=2.pt] (3.25,-0.25)--(4.0,-1.75);
\draw [line width=1.pt] (0.,2.)-- (0.,0.);
\draw [line width=1.pt] (3.,2.)-- (3.,0.);
\draw [line width=1.pt] (6.,2.)-- (6.,0.);
\draw [line width=1.pt] (-1.,-2.)-- (0.,0.);
\draw [line width=1.pt] (1.,-2.)-- (0.,0.);
\draw [line width=1.pt] (2.,-2.)-- (3.,0.);
\draw [line width=1.pt] (4.,-2.)-- (3.,0.);
\draw [line width=1.pt] (5.,-2.)-- (6.,0.);
\draw [line width=1.pt] (7.,-2.)-- (6.,0.);
\draw [line width=1.pt] (0.,0.)-- (3.,0.);
\draw [line width=1.pt] (3.,0.)-- (6.,0.);
\draw [line width=1.pt] (0.,2.)-- (3,2.);
\draw [line width=1.pt] (3.,2.)-- (6,2.);
\draw (-1,-2) to[out=280,in=260,loop, distance =1cm] (2,-2);
\draw (2,-2) to[out=280,in=260,loop, distance =1cm] (5,-2);
\draw (1,-2) to[out=45,in=135,loop, distance =1cm] (4,-2);
\draw (4,-2) to[out=45,in=135,loop, distance =1cm] (7,-2);
\begin{scriptsize}
\draw [fill=black] (0.,2.) circle (2.0pt);
\draw [fill=black] (3.,2.) circle (2.0pt);
\draw [fill=black] (6.,2.) circle (2.0pt);
\draw [fill=black] (0.,0.) circle (2.0pt);
\draw [fill=black] (3.,0.) circle (2.0pt);
\draw [fill=black] (6.,0.) circle (2.0pt);
\draw [fill=black] (-1.,-2.) circle (2.0pt);
\draw [fill=black] (1.,-2.) circle (2.0pt);
\draw [fill=black] (2.,-2.) circle (2.0pt);
\draw [fill=black] (4.,-2.) circle (2.0pt);
\draw [fill=black] (5.,-2.) circle (2.0pt);
\draw [fill=black] (7.,-2.) circle (2.0pt);
\draw[color=black] (1.25,-2.25) node {\Large $R$};
\draw[color=black] (0.25,2.25) node {\Large $c_{1}$};
\draw[color=black] (3.35,0.25) node {\Large $c_{2}$};
\end{scriptsize}
\end{tikzpicture}}
\caption{Round 2.}
\end{center}
\end{subfigure}
\begin{subfigure}{0.5\textwidth}
\begin{center}
\scalebox{0.6}{
\begin{tikzpicture}
\draw [->,line width=2.pt] (0.25,-0.25)-- (1.,-1.75);
\draw [line width=1.pt] (0.,2.)-- (0.,0.);
\draw [line width=1.pt] (3.,2.)-- (3.,0.);
\draw [line width=1.pt] (6.,2.)-- (6.,0.);
\draw [line width=1.pt] (-1.,-2.)-- (0.,0.);
\draw [line width=1.pt] (1.,-2.)-- (0.,0.);
\draw [line width=1.pt] (2.,-2.)-- (3.,0.);
\draw [line width=1.pt] (4.,-2.)-- (3.,0.);
\draw [line width=1.pt] (5.,-2.)-- (6.,0.);
\draw [line width=1.pt] (7.,-2.)-- (6.,0.);
\draw [line width=1.pt] (0.,0.)-- (3.,0.);
\draw [line width=1.pt] (3.,0.)-- (6.,0.);
\draw [line width=1.pt] (0.,2.)-- (3,2.);
\draw [line width=1.pt] (3.,2.)-- (6,2.);
\draw (-1,-2) to[out=280,in=260,loop, distance =1cm] (2,-2);
\draw (2,-2) to[out=280,in=260,loop, distance =1cm] (5,-2);
\draw (1,-2) to[out=45,in=135,loop, distance =1cm] (4,-2);
\draw (4,-2) to[out=45,in=135,loop, distance =1cm] (7,-2);
\begin{scriptsize}
\draw [fill=black] (0.,2.) circle (2.0pt);
\draw [fill=black] (3.,2.) circle (2.0pt);
\draw [fill=black] (6.,2.) circle (2.0pt);
\draw [fill=black] (0.,0.) circle (2.0pt);
\draw [fill=black] (3.,0.) circle (2.0pt);
\draw [fill=black] (6.,0.) circle (2.0pt);
\draw [fill=black] (-1.,-2.) circle (2.0pt);
\draw [fill=black] (1.,-2.) circle (2.0pt);
\draw [fill=black] (2.,-2.) circle (2.0pt);
\draw [fill=black] (4.,-2.) circle (2.0pt);
\draw [fill=black] (5.,-2.) circle (2.0pt);
\draw [fill=black] (7.,-2.) circle (2.0pt);
\draw[color=black] (1.25,-2.25) node {\Large $R$};
\draw[color=black] (0.35,0.35) node {\Large $c_{1}$};
\draw[color=black] (4.35,-2.25) node {\Large $c_{2}$};
\end{scriptsize}
\end{tikzpicture}}
\caption{Round 3; Cops win on the next round.}
\end{center}
\end{subfigure}
\caption{Example of game play on $P_{3}\square S_{3}$, using the strategy from Theorem~\ref{thm:cartesianproduct}. Arrows indicate where the cops ($c_{1}$, $c_{2}$) will move on the next round; the robot ($R$) is aware of this information before he is required to move.}\label{fig: cartesian product example}
\end{figure}
\begin{thm}\label{thm:cartesianproduct} If G and H are each connected then $c_{cr}(G\square H)\leq c_{cr}(G)+c_{cr}(H)$.
\end{thm}
\begin{proof}
Let $r=c_{cr}(G)$, $s=c_{cr}(H)$ and let $G'$ and $H'$ be subgraphs of $G$ and $H$ respectively with $|V(G')|=r$ and $|V(H')|=s$. In $G\square H$, place cops on each vertex of $G'\square\{h\}$ and $\{g\}\square H'$ for some $g\in V(G)$ and $h\in V(H)$. Call these the $G$-cops and the $H$-cops respectively.
We will show that this number of cops suffices to capture the robot. There are two parts to the strategy.
First, the $G$-cops ($H$-cops) move to
eventually have their second (first) coordinate the same as that of the robot.
Next, they follow the surrounding strategy until both the $G$- and $H$-cops have the robot surrounded.
\textit{Phase 1:} Assume that the robot is at vertex $(a,x)$. If $h\ne x$ then the $G$-cops move to $G'\square\{h'\}$ where $h'$ is closer to $x$ than $h$. If $g\ne a$ then the $H$-cops move to $\{g'\}\square H'$ where $g'$ is closer to $a$ than $g$. In response, the robot moves to
$(a',x')$ where at most one of the coordinates has changed, say to $(a',x)$. The $G$-cops have moved closer in the second coordinate
and the $H$-cops have maintained the distance in the first. Similarly, the $H$-cops will have moved closer if the robot moves to $(a,x')$.
\textit{Phase 2:} Thus, on every move one group of cops gets closer (or both if the robot remains on the same vertex)
and eventually, without loss of generality, the $G$-cops and the robot will be on $G\square \{x\}$. The $G$-cops play their surrounding strategy
and the $H$-cops move as in Phase 1. If the robot moves to change the first coordinate, we are still in Phase 2 and the $H$-cops have moved closer in the second coordinate. If the robot moves to change the second coordinate we are back in Phase 1 but the $G$-cops have taken one move toward surrounding the
robot and the $H$-cops are closer in the first coordinate.
Eventually, without loss of generality, we can assume that the $G$-cops have surrounded the robot on $G\square \{x\}$, say at $(a,x)$. We also assume that the $H$-cops are not on $\{a\}\square H$.
\textit{Phase 3:} The $G$-cops do not move. The $H$-cops move as in Phase 1. The robot can only not move or change the second coordinate thus $H$-cops have moved closer in the first coordinate.
On the next move, the $G$-cops surround the robber on some $G\square \{y\}$ and the $H$-cops are closer in the first coordinate. Eventually, the robot will be on some $(a',x')$ and the $G$-cops will have surrounded the robot on $G\square \{x'\}$ and the $H$-cops are on $\{a'\}\square H$. Again, the robot can only not move or change the second coordinate, thus we are playing in $\{a'\}\square H$. Eventually, the robot will be at some $(a',z)$ surrounded by the $G$-cops on $G\square \{z\}$ and surrounded by the $H$-cops on $\{a'\}\square H$. The robot has nowhere safe to move and will be captured on the next turn.
\end{proof}
This result give exact bounds for some Cartesian products.
\begin{cor}\label{cor:cartesiangrid}
Let $G$ be a graph, $\{T_{n_1}, T_{n_2},\ldots,T_{n_k}\}$ and $\{C_{m_1}, C_{m_2},\ldots,C_{m_\ell}\}$ be sets of trees and cycles respectively.
\begin{enumerate}
\item If $G=\square_{i=1}^kT_{n_i}$ then $c_{cr}(G) =k$.
\item If $G=\square_{j=1}^\ell C_{m_j}$ then $c_{cr}(G) =2\ell$.
\end{enumerate}
\end{cor}
\begin{proof} First note that $G=\square_{i=1}^kT_{n_i}$ has minimum degree $k$ so at least $k$ cops are required. Now, on a
tree one cop suffices to capture the robot therefore, by Theorem \ref{thm:cartesianproduct}, $k$ cops suffice on $\square_{i=1}^kT_{n_i}$.
Similarly, at least $2\ell$ cops are required since this is the minimum degree. Also, $c_{cr}(C_{m_{i}})=2$ for any cycle $C$, thus $2\ell$ cops are suffice
by Theorem \ref{thm:cartesianproduct}.
\end{proof}
Note that the first result of Corollary \ref{cor:cartesiangrid} covers hypercubes when the trees are each $P_{2}$ and $k$-dimensional grids when the trees are paths in general.
\section{Strong Products of Paths}
\begin{definition} The strong product of graphs $G$ and $H$, written $G\boxtimes H$, has $V(G\boxtimes H)=\{(g,h): g\in V(G), h\in V(H)\}$
and $(g,h)$ is adjacent to $(g',h')$ if $g$ is equal or adjacent to $g'$ and $h$ is equal or adjacent to $h'$.
\end{definition}
As with the Cartesian product, this definition is easily extended to the product of more than two graphs. In general, a vertex of the product is $(a_1,a_2,\ldots, a_n)$ where
$a_i$ is in the $i^\text{th}$-coordinate.
If $G$ is the strong product of paths, a \emph{corner} is a vertex where for each $i$, the $i^\text{th}$-coordinate is a leaf in $P_{i}$.
For example, there are $4$ corners in a grid, $P_m\boxtimes P_n$, where $P_{i}$ is a path with $i$ vertices.
\begin{thm}\label{thm: strong grid base case}
Let $G$ be a strong grid, $G = P_{m} \boxtimes P_{n}$. Then
\[c_{cr}(G) = \begin{cases}
4, \text{ if $m,n>3$;}\\
3, \text{ if $\min\{m,n\} = 2$ or $3$.}\\
\end{cases}
\]
\end{thm}
\begin{proof}
Suppose $G=P_2\boxtimes P_n$. The minimum degree of $G$ is 3, so at least 3 cops are required. The following algorithm shows that 3 cops suffice.
Place two cops on $(1,1)$ and one on $(2,1)$, the robot is on some vertex $(i,j)$ with $j>1$. Move the first two cops to $(1,2)$ and $(2,2)$, the third stays on $(2,1)$. The robot cannot move to $(1,1)$. The cop on $(2,1)$ now moves to $(2,2)$ and after the robot moves he is on some vertex $(i',j')$ with $j'>2$. The cops repeat until the robot is caught on $(1,n)$ or $(2,n)$.
Suppose $G=P_3\boxtimes P_n$, $n\geq 3$. The minimum degree of $G$ is 3, so at least 3 cops are required. The following algorithm shows that 3 cops suffice.
In general, we assume that cops are on $(1,i)$, $(2,i)$ and $(3,i)$, starting with $i=1$. The robot is on some $(k,j)$ with $j>1$.
On the cops move, if the robot is on $(k,j)$, $j>i+1$ then the cops increase their second coordinate by 1.
If the robot is on $(1,i+1)$ then the cop on $(3,i)$ moves to $(2,i+1)$. There are two cases. One, if the robot passes then the
cop on $(2,i)$ moves to $(1,i+1)$ and now he has to move or be captured. He has to move to $(k,i+2)$, for $k=1$ or $2$, and the other two cops now move to
$(2,i+1)$ and $(3,i+1)$. Two, if the robot moves then
he must move to $(k,i+2)$, for some $k$ or else be captured. The other two cops move to $(1,i+1)$ and $(3,i+1)$. In both cases, the cops have increased their second coordinate and the robot is on smaller subgraph.
If the robot is on $(2,i+1)$ then the cop on $(3,i)$ moves to $(2,i+1)$. There are three cases. One, if the robot moves to $(k,i+2)$ then cops move to occupy $(1,i+1)$, $(2,i+1)$ and $(3,i+1)$. Two, the robot moves to $(1,i+1)$ and this situation is covered in the previous paragraph. Three, the robot moves to $(3,i+1)$ whence the other two cops move to $(2,i)$ and $(3,i)$ giving the symmetric case to case 2. In all cases, the end result is that the cops have increased their second coordinate and the robot is on smaller subgraph.
Eventually, the robot is captured in the next sequence after the cops occupy $(1,n-1)$, $(2,n-1)$ and $(3,n-1)$.
Now suppose that $m,n>3$. There exists an induced $4$-core in $G$ (from deleting the four corners), hence four cops are necessary.
We must show that $4$ cops is also sufficient. To do so, we present an algorithm for the cops to capture the robot. Again, we will take the alternate move interpretation. In this algorithm, `capture of the shadow' by the cops means that after the cops' move, the robot and some cop are in the same column. The robot still has a move to try and evade the cops.
Let the strong grid be of size $m \times n$. The vertices, indexed by the rows and columns, are $(i,j)$, $1\leq i\leq m$, $1\leq j \leq n$. Increasing the first coordinate corresponds to \textit{moving up} a row. Similarly, increasing the second coordinate corresponds to \textit{moving right}.
A \textit{4-line} of cops, $(i, \langle j\rangle)$, has the cops on $(i, j-1)$, $(i, j)$, $(i, j+1)$, and $(i,j+2)$. With this notation, if $s>n$ then $s$ is taken as $n$
and if $s<1$ then $s$ is taken as 1.
We refer to $(i, j)$ and $(i, j+1)$ as the \textit{interior} vertices of the $4$-line.
Placement: Place the cops on the vertices of $(1,\langle 2\rangle)$. The robot places himself on $(i,j)$.
\begin{enumerate}
\item[Step 1:] The cops move the $4$-line along the first row as a group until the shadow of the robot is either (i) captured on an interior vertex of the $4$-line or (ii) the robot is either on $(i, n)$ or $(i, 1)$ and the $4$-line is at $(1, \langle n-2\rangle)$ or $(1,\langle 2\rangle)$, respectively. After this shadow capture, move to Step 2.
\item[Step 2:] The robot moves but, regardless of the move, its shadow is still caught. If the robot is at least two rows above the cops, move the $4$-line up one row to keep the shadow on an interior vertex.
If the $4$-line is on a row end, then he moves up.
Repeat Step 2 until the cops are one row away from the robot.
\item[Step 3:] It is now the robot's move, where the $4$-line is at $(i,\langle j\rangle)$ and the robot is at $(i+1, j)$ or $(i+1, j+1)$.
There are a two cases to consider: (i) If the robot moves up one row then the $4$-line follows keeping the shadow on an interior vertex. (ii) If the robot does not move up then he moves to (without loss of generality) to either $(i+1,j)$ or $(i+1,j-1)$. In the former, only the cop on $(i, j+2)$ moves and moves to $(i+1, j+1)$. In the latter, the cop on $(i, j+2)$ moves $(i+1, j+1)$ and the others move one column to the left. In both cases the situation is now, the robot is on $(i+1,k)$ and
the cops are on $(i+1,k+1)$ and $(i,k-1)$, $(i,k)$, and $(i,k+1)$. If the robot does not move up a row he can only pass or move to $(i,k-1)$. If he
passes, the cop on $(i+1,k+1)$ moves to $(i+1,k)$, the others pass. To avoid capture, now the robot must move up or left on the same row. If the robot moves up, in all cases the cops can move the four cops to form a $4$-line on the $i+1$ row with the robot's shadow captured by an interior vertex. If the robot moves left then all the cops also move left and the situation is repeated. Eventually, either the robot is on $(i,1)$ and must move up or is on $(m,1)$ and is captured when the cop on $(m,2)$ moves to $(m,1)$.
\end{enumerate}
Thus four cops is sufficient for capture on a strong grid.
\end{proof}
Next, we generalize Theorem~\ref{thm: strong grid base case} to higher dimensions. In a $k$-dimensional strong grid, $G=\boxtimes_{i=1}^kP_{n_i}$, for brevity, a vertex can represented by $(i_1, i_2, \dots, i_k)$, where $1\leq i_j\leq n_j$. Implicitly, we are setting $V(P_{n_i})=\{1,2,\ldots,n_i\}$. The paths are disjoint and the context will indicate to which path a vertex belongs.
\begin{thm}\label{thm: k-dim strong grids}
Let $G$ be a $k$-dimensional strong grid, $G = \boxtimes_{i=1}^{k} P_{n_{i}}$. Then
\begin{equation*}
3\cdot2^{k-1} -2 \leq c_{cr}(\boxtimes_{i=1}^{k} P_{n_{i}}) \leq 3^{k}.
\end{equation*}
\end{thm}
\begin{proof}
The vertices of minimum degree in $G$ are the `corners', i.e. the vertices where all the coordinates are all leaves in the original paths. Their degrees are $2^k-1$. Let $L$ be the vertices of $G$ where all $k$ coordinates are leaves in the original paths and let $H$ be the induced graph on $V(G)\setminus L$.
The vertices of minimum degree in $H$ are those with $k-1$ coordinates being leaves in the original paths and the other coordinate being adjacent to leaves. Their degrees in $G$ are $3\cdot2^{k-1} - 1$ and, in addition, the adjacent corner has been removed leaving the degree as $3\cdot2^{k-1} -2$ in $H$.
To capture the robot we give an algorithm based on the cop-win strategy.
To avoid special cases when the cops are close to the edges or corners of this strong hypergrid, let
$H=\boxtimes_{i=1}^kP'_{n_i}$ where $V(P'_{n_i})= \{0,1,2,\ldots,n_i, n_{i}+1\}$, that is $P_{n_i}$ extended by a new leaf at each end. Note that $P_{n_i}$ is a retract of $P'_{n_i}$ via $f_i(j) = j$ if $1\leq j\leq n_i$, $f_i(0)=1$ and $f_i(n_i+1)=n_i$. In turn, $G$ is a retract of $H$ via $f(v_1,v_2,\ldots,v_k) = (f_1(v_1),f_2(v_2),\ldots,f_k(v_k))$. We now use Theorem \ref{thm:retract} and restrict the robot to $G$ but let the cops play on $H$.
Again, we consider the alternating model with the appropriate restrictions and winning condition. Place the cops on the vertices $(x_1,x_2,\ldots,x_k)$, $1\leq x_i\leq 3$ for each $i$. The cop on $(2,2,\ldots,2)$ is the \textit{central} cop and the others occupy every vertex of its neighbourhood.
The central cop plays the cop-win strategy on $H$. When the central cop moves from $(y_1,y_2,\ldots,y_k)$ to $(y'_1,y'_2,\ldots,y'_k)$ each coordinate changes by at most 1. The non-central cops now move to change their coordinates by
$(y'_1-y_1,y'_2-y_2,\ldots,y'_k-y_k)$ and they still occupy every vertex of the central cop's neighbourhood. Thus, when the central cop is on the same vertex as the robot, the robot has nowhere safe to move.
\end{proof}
In the proof of the upper bound in Theorem \ref{thm: k-dim strong grids}, note that the robot will have been captured before the central cop captures via the cop-win strategy. This indicates that the upper bound is probably not tight. Indeed, in the 2-dimensional case, the lower bound is correct.
\section{Outerplanar}
An \textit{outerplanar} graph can be drawn so that the vertices can be arranged in a circle and all edges can be placed on or outside of the circle with no crossing edges.
We will call edges not lying directly on the circle, \textit{chords}. Let $V(G)=\{v_0,v_1,\ldots,v_n\}$ where the vertices are placed in that order around the circle. The \textit{length} of a chord $v_iv_j$, $i<j$, is $\min\{|j-i|,|n+1+i-j|\}$, i.e., the shortest arc along the circle.
Clarke \cite{Clarke} showed that two cops were sufficient to capture a robber. We show that two are sufficient to capture a robot.
\begin{thm}\label{thm:outerplanar}
Let $G$ be a connected outerplanar graph. If $G$ is a tree then $c_{cr}(G) = 1$, otherwise $c_{cr}(G) = 2$.
\end{thm}
\begin{proof} Theorem \ref{thm: trees} covers the case when $G$ is a tree. Thus, we may assume that $G$ is not a tree and therefore,
again by Theorem \ref{thm: trees}, we know $c_{cr}(G)\geq 2$. It remains to show that two cops suffice.
We proceed by induction on $|V(G)|$. If $G$ has one vertex then one cop suffices. We now assume that for some $n\geq 1$, 2 cops suffice to
capture the robot on any connected outerplanar graph with $k\leq n$ vertices. Let $G$ be a connected outerplanar graph on $n+1$ vertices. Suppose $G$ contains a cut-vertex $x$ and $G\setminus\{x\}$ has a connected component $X'$ such that $X'\cup\{x\}$ is a tree. Let $f:G\rightarrow G\setminus X'$ by $f(y) = y$ if $y\in G\setminus X$ and $f(y)=x$ otherwise. Now $f$ is a retraction map and
thus $G\setminus X$ is a retract of $G$. Moreover, $G\setminus X$ is connected and outerplanar. By induction, 2 cops suffice to capture the shadow on $G\setminus X$. If the shadow is not on $x$ then the robot has been captured. If the shadow is on $x$ the robot is not captured then he is on $X'$. Now one cop remains on $x$, which prevents the robot moving off $X'$, and by Theorem \ref{thm: trees}, the other cop can capture the robot.
We may suppose that all vertices are of degree 2 or greater. If $G$ is a cycle then two cops suffice. If $G$ is not a cycle then
there is a chord and let $v_iv_j$, $i<j$, be a chord of shortest length. Note $j\geq i+2$. Without loss of generality, we may assume that there is no chord $v_rv_s$ with $i\leq r<s\leq j$ unless $i=r$ and $s=j$. Let $R=G\setminus \{v_{i+1},v_{i+2},\ldots, v_{j-1}\}$. Set $f:G\rightarrow R$ by
$f(x) = x$, if $x\in R$, and $f(x) = v_i$ otherwise. Now $f$ is a retraction map and $R$ is a retract. The graph $R$ is a connected, outerplanar graph and by induction, 2 cops can capture the robot's shadow on $R$.
Once captured, either the robot is captured or
the shadow is on $v_i$ and the robot is on one of $v_{i+1},v_{i+2},\ldots, v_{j-1}$. The game is over in the former situation so we suppose the latter. Note that if the robot had been on $v_i$ then he would have moved to $v_{i+1}$. Since the shadow has been captured in $R$ on $v_i$ then $v_i$ only has two neighbours in $R$, those being $v_{i-1}$ and $v_j$. Hence on the cops move which captured the shadow, one cop was on either $v_{j+1}$ or $v_j$ and the other on $v_{i-1}$ or $v_i$. One capturing move is to move the cop on $v_{i-1}$ to $v_i$, or remain on $v_i$, and the other to move to or remain on $v_j$. The robot is now trapped on the path $v_{i+1},v_{i+2},\ldots, v_{j-1}$. The cops can capture by having the one on $v_i$ move successively along this path. \end{proof}
\section{Open Problems}
Given the similarities between the surrounding model \cite{BurgessCCDFJP} and the Cheating Robot, we observe that for any graph $G$ we have $c_{cr}(G) \leq s(G)$, where $s(G)$ is the surrounding cop number. For paths with $4$ or more vertices, outerplanar graphs and complete graphs, we know that $0\leq s(G)-c_{cr}(G) \leq 1$. \emph{Is it true that $s(G)$ and $c_{cr}(G)$ differ by at most 1?}
An important Cops and Robbers result is that three cops suffice to capture a robber on a planar graph \cite{AignerF}. For the surrounding model the upper bound is 7 \cite{BradshawH}. The icosahedron is regular of degree 5, hence there is at least one planar graph that requires at least $5$ cops for capture. We ask: \textit{Is it true that for all planar graphs $G$, $c_{cr}(G)\leq 5$?}
Let $H$ be a connected subgraph of $G$. For Cops and Robbers, there are examples where $c(H)>c(G)$. For the Cheating Robot, \textit{is it always true that
$c_{cr}(H)\leq c_{cr}(G)$?} Theorem \ref{thm:retract} shows the inequality is true when $H$ is a retract.
\section*{Acknowledgements}
\noindent
Melissa A. Huggan was supported by the Natural Sciences and Engineering Research Council of Canada (funding reference number PDF-$532564-2019$).
Richard J. Nowakowski was supported by the Natural Sciences and Engineering Research Council of Canada (funding reference number 2019-04914).
| {
"timestamp": "2021-03-12T02:06:28",
"yymm": "2006",
"arxiv_id": "2006.15379",
"language": "en",
"url": "https://arxiv.org/abs/2006.15379",
"abstract": "The 'Cheating Robot' version of Cops and Robbers is played on a finite, simple, connected graph. The players move in the same time period. However, before moving, the robot observes to which vertices the cops are moving and it is fast enough to complete its move in the time period. The cops also know that the robot will use this information. More cops are required to capture a robot than to capture a robber. Indeed, the minimum degree is a lower bound on the number of cops required to capture a robot. Only on a tree is one cop guaranteed to capture a robot, although two cops are sufficient to capture both a robber and a robot on outerplanar graphs. In graphs where retracts are involved, we show how cop strategies against a robber can be modified to capture a robot. This approach gives exact numbers for hypercubes, and $k$-dimensional grids in general.",
"subjects": "Combinatorics (math.CO); Discrete Mathematics (cs.DM)",
"title": "Cops and an Insightful Robber",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
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https://arxiv.org/abs/1506.08679 | Analysis of a slow-fast system near a cusp singularity | This paper studies a slow-fast system whose principal characteristic is that the slow manifold is given by the critical set of the cusp catastrophe. Our analysis consists of two main parts: first, we recall a formal normal form suitable for systems as the one studied here; afterwards, taking advantage of this normal form, we investigate the transition near the cusp singularity by means of the blow up technique. Our contribution relies heavily in the usage of normal form theory, allowing us to refine previous results. | \section{Introduction}\label{sec:intro}
A \emph{slow-fast system} (SFS) is a singularly perturbed ordinary differential equation of the form
\eq{\label{intro:sf1}
\dot x &= f(x,z,\varepsilon)\\
\varepsilon\dot z &= g(x,z,\varepsilon),
}
where $x\in\mathbb{R}^m$, $z\in\mathbb{R}^n$ are local coordinates and where $\varepsilon>0$ is a small parameter. The over-dot denotes the derivative with respect to the time parameter $t$. Throughout this text, we assume that the functions $f$ and $g$ are of class $\mathcal{C}^{\infty}$. In applications (e.g \cite{Zeeman1}), $z(t)$ represents states or measurable quantities of a process while $x(t)$ stands for control parameters. The parameter $\varepsilon$ models the difference of the rates of change between the variables $z$ and $x$. That is why systems like \cref{intro:sf1} are often used to model phenomena with two time scales. Observe that the smaller $\varepsilon$ is, the faster $z$ evolves with respect to $x$. Therefore we refer to $x$ (resp. $z$) as the \emph{slow} (resp. \emph{fast}) variable. The time parameter $t$ is known as the \emph{slow time}. For $\varepsilon\neq 0$, we can define a new time parameter $\tau$ by the relation $t=\varepsilon\tau$. With this time reparametrization \cref{intro:sf1} can be written as
\eq{\label{intro:sf2}
x' &= \varepsilon f(x, z,\varepsilon)\\
z' &= g(x,z,\varepsilon),
}
where now the prime denotes the derivative with respect to the rescaled time parameter $\tau$, which we call \emph{the fast time}. Since we consider only autonomous systems, we often omit to indicate the time dependence of the variables. In the rest of this document, we prefer to work with slow-fast systems presented as \cref{intro:sf2}.\smallskip
Observe that as long as $\varepsilon\neq 0$ and $f$ is not identically zero, systems \cref{intro:sf1} and \cref{intro:sf2} are equivalent. A first approach to understand the qualitative behavior of slow-fast systems is to study the limit $\varepsilon\to0$. The slow equation \cref{intro:sf1} restricted to $\varepsilon=0$ reads as
\eq{\label{intro:cde1}
\dot x &= f(x,z,0)\\
0 &= g(x,z,0).
}
A system of the form \cref{intro:cde1} is called \emph{constrained differential equation} (CDE) \cite{Jardon1,Takens1}. On the other hand, in the limit $\varepsilon\to0$, a system given by \cref{intro:sf2} becomes
\eq{\label{intro:layer}
x' &= 0\\
z' &= g(x,z,0),
}
which is called \emph{the layer equation}. Associated to both systems, \cref{intro:cde1} and \cref{intro:layer}, the slow manifold $S$ is defined by
\eq{
S=\left\{ (x,z)\in\mathbb{R}^m\times\mathbb{R}^n\, | \, g(x,z,0)=0 \right\},
}
which serves as the phase space of the CDE \cref{intro:cde1} and as the set of equilibrium points of the layer equation \cref{intro:layer}. In the latter context, it is useful to recall the concept of Normally Hyperbolic Invariant Manifold (NHIM).
\begin{definition}[Normally Hyperbolic Invariant Manifold] Consider a slow-fast system given by a vector field of the form
\eq{X_\varepsilon=\varepsilon f(x,z,\varepsilon)\parc{x}+g(x,z,\varepsilon)\parc{z}.
}
The associated slow (invariant) manifold $S=\left\{ g(x,z,0)=0\right\}$ is said to be normally hyperbolic if each point of $S$ is a hyperbolic equilibrium point of $X_0$.
\end{definition}
NHIMs are relevant in the context of the geometric study of slow-fast systems, see for example \cite{Fenichel}. It is known that compact NHIMs persist under $\mathcal C^1$ small perturbation of the vector field \cite{Jones,Kaper}. In the particular context presented above, a normally hyperbolic compact subset of the slow manifold $S$ persists as an invariant manifold of the slow-fast system $X_\varepsilon$. We show in \cref{fig:intro1} a schematic of the previous description.
\begin{figure}[htbp]\centering
\includegraphics{qual1.pdf}
\caption{A schematic representation of the persistence of a NHIM under the perturbation of the corresponding vector field. $S$ denotes the slow manifold. Left-above: $S$ is a set of hyperbolic equilibrium points of the layer equation. Left-below: $S$ is the phase space of the constrained equation. Right: since $S$ is a NHIM, it persists as an invariant manifold $S_\varepsilon$ under small perturbations of the vector field. }
\label{fig:intro1}
\end{figure}
After this intruduction, we turn into the subject of this paper. Our goal is to understand the dynamics of a particular slow-fast system which has one fast and two slow variables given as
\eq{\label{intro:eqcusp}
X_\varepsilon=\varepsilon(1+\varepsilon f_1)\parc{x_1}+\varepsilon^2f_2\parc{x_2}-\left( z^3+x_2z+x_1+\varepsilon f_3 \right)\parc{z},
}
where the functions $f_i=f_i(x_1,x_2,z)$, for $i=1,2,3$, are smooth and vanish at the origin. The corresponding slow manifold is defined by
\eq{
S=\left\{ (x_1,x_2,z)\in\mathbb{R}^3 \,| \, z^3+x_2z+x_1=0 \right\}.
}
\begin{remark}
The slow manifold $S$ can be regarded as the critical set of the cusp (or $A_3$) catastrophe, which is given as \rm\cite{Arnold_singularities,Brocker}\eq{\label{eqV}
V(x_1,x_2,z)=\frac{1}{4}z^4+\frac{1}{2}x_2z^2+x_1z.
}
\end{remark}
We denote by $\Delta$ the set of points in $S$ at which $S$ is tangent to the fast direction, that is
\eq{\label{eqD}
\Delta=\left\{ (x_2,z)\in S \, | \, 3z^2+x_2=0\right\}.
}
In other words, $\Delta$ is the set of degenerate critical points of \cref{eqV}. See figure \cref{fig:qual1} for a description of the slow manifold and the set $\Delta$.
\begin{figure}[htbp]\centering
\begin{tikzpicture}
\pgftext{\includegraphics{cusp1.pdf}}
\node at (1,1.4) {$S$};
\node at (-2,0.25) {$\Delta$};
\node at (1.6,.75) {$C$};
\node at (2.2,.25) {$z$};
\node at (1.2,-1) {$x_2$};
\node at (-1.2,-1) {$x_1$};
\end{tikzpicture}
\caption{The manifold $S$ is two dimensional and can be defined as the critical set of the potential function $V(x_1,x_2,z)=\tfrac{1}{4}z^4 + \tfrac{1}{2}x_2z^2+x_1z$. The curve $\Delta$ is defined by the set of degenerate critical points of $V$. Geometrically, $B$ is the set of point of $S$ where $S$ is tangent to the fast direction, and $C$ denotes the cusp point.}
\label{fig:qual1}
\end{figure}
Our interest in studying \cref{intro:eqcusp} is due to the fact that the origin $(x_1,x_2,z)=(0,0,0)$ is a \emph{non-hyperbolic equilibrium point} of $X_0$. This implies that a compact subset, around the origin, of the slow manifold $S$ is not a NHIM of $X_0$, and therefore, the Geometric Singular Perturbation Theory \cite{Fenichel,Jones,Kaper} is not enough.
\subsection{Motivation}
There have been several studies, e.g. \cite{Krupa3,Krupa20102841}, dealing with a SFS of the form
\eq{
X_\varepsilon=\varepsilon(1+f_1)\parc{x_1}-\left( z^2+x_1+ \varepsilon h \right)\parc{z},
}
whose slow manifold is the critical set of the fold catastrophe. The next natural step is to consider the following case in the Thom list \cite{Stewart1}, i.e., a slow-fast system induced by the cusp catastrophe. That is
\eq{\label{mot:eqcusp}
X_\varepsilon=\varepsilon(1+f_1)\parc{x_1}+\varepsilon f_2\parc{x_2}-\left( z^3+x_2z+x_1+\varepsilon f_3 \right)\parc{z}.
}
In \cite{BKK}, the system \cref{mot:eqcusp} is studied in a qualitative way. Here, however, we aim to refine the results by heavily using techniques from normal form theory. Moreover, we remark that the methods presented here are applicable to a larger class of slow-fast system given by
\eq{
X_\varepsilon=\varepsilon(1+f_1)\parc{x_1}+\sum_{i=2}^{k-1}\varepsilon f_i\parc{x_i}-\left( z^k+\sum_{j=1}^{k-1}x_jz^{j-1}-\varepsilon f_k \right)\parc{z},
}
which is called (regular) $A_k$-SFS, see \cite{JardonThesis}.
\subsection{Statement}
We shall study the SFS
\eq{\label{s1}
X_\varepsilon=\varepsilon (1+f_1)\parc{x_1}+\varepsilon f_2\parc{x_2}-\left( z^3+x_2z+x_1 + \varepsilon f_3 \right)\parc{z},
}
where the functions $f_i=f_i(x_1,x_2,z,\varepsilon)$ are smooth. To avoid working with an $\varepsilon$-parameter family of vector fields as \cref{s1}, it is customary to extend \cref{s1} by adding the trivial equation $\varepsilon'=0$, and thus consider a smooth vector field in $\mathbb{R}^4$ which reads as
\eq{\label{s2}
X=\varepsilon (1+ f_1)\parc{x_1}+\varepsilon f_2\parc{x_2}-\left( z^3+x_2z+x_1 + \varepsilon f_3 \right)\parc{z} + 0\parc{\varepsilon}.
}
We regard \cref{s2} as a perturbation of ``the principal part'' $F$ which is given as
\eq{\label{s3}
F=\varepsilon \parc{x_1}+0\parc{x_2}-\left( z^3+x_2z+x_1 \right)\parc{z} + 0\parc{\varepsilon}.
}
Note that in a qualitative sense, $F$ contains the essential elements of $X$. To state our main result, we first define the sections
\eq{
\Sigma^-=\left\{ (x_1,x_2,z,\varepsilon)\in\mathbb{R}^4\, |\, x_1=-x_1^{i} \right\}\\
\Sigma^-=\left\{ (x_1,x_2,z,\varepsilon)\in\mathbb{R}^4\, |\, x_1=x_1^{f} \right\},
}
where $x_1^{i}>0$ and $x_1^{f}>0$ are arbitrarily large constants. For $\varepsilon>0$ but sufficiently small, the sections $\Sigma^{-}$ and $\Sigma^+$ are transversal to the flow of $X_\varepsilon$. Next, let $\Pi:\Sigma^{-}\to\Sigma^{+}$ be the Poincaré map induced by the flow of $X_\varepsilon$. We shall prove the following.
\paragraph{{\bfseries{Transition along the cusp}} ({\rm see \cref{teo:main}})} {
Consider a slow-fast system given by \cref{s2}. Let $\Sigma^-$, $\Sigma^+$ and $\Pi:\Sigma^-\to\Sigma^+$ be defined as above. Then, we can choose coordinates in $\Sigma^-$ and in $\Sigma^+$ such that the map $\Pi$ reads as
\eq{
\Pi(X_2,Z,\varepsilon)=(\tilde X_2,\tilde Z,\tilde \varepsilon),
}
where $\tilde X_2=X_2+H(X_2,\varepsilon)$ (with $H$ flat at $(X_2,\varepsilon)=(0,0)$), $\tilde\varepsilon=\varepsilon$ and where
\eq{\label{s4}
\tilde Z=\Phi(X_2,\varepsilon)+Z\exp\left( -\frac{1}{\varepsilon}(A(X_2,\varepsilon)+\varepsilon\Psi(X_2,Z,\varepsilon)) \right),
}
where $A(X_2,0)>0$. Details of the functions $\Phi$, $A$, and $\Psi$ are given in \cref{teo:main}. In an heuristic way, this result is described in \cref{figcusp}.
\begin{figure}\centering
\begin{tikzpicture}
\pgftext{\includegraphics[scale=1.5]{cusp_qual2.pdf}}
\node at (1,2) {$S_\varepsilon$};
\node at (-3.3,-0.1) {$S$};
\node at (-2,2.5) {$\Sigma^-$};
\node at (3,-0.1) {$\Sigma^+$};
\end{tikzpicture}
\caption{Description of our main result. We may choose appropriate coordinates at the sections $\Sigma^-$ and $\Sigma^+$ under which the invariant manifold $S_\varepsilon$ is given by $Z=0$. Moreover form \cref{s3} we have that all other trajectories starting at $\Sigma^-$ are exponentially attracted to the invariant manifold $S_\varepsilon$. In this paper we provide quantitative information regarding this exponential contraction.}
\label{figcusp}
\end{figure}
}
\subsection{Idea of the proof}
Our proof consists of two main steps.
\begin{enumerate}
\item From \cite{Jardon2}, it is known that there exists a formal transformation bringing \cref{s2} into
\eq{\label{s5}
F=\varepsilon \parc{x_1}+0\parc{x_2}-\left( z^3+x_2z+x_1 \right)\parc{z} + 0\parc{\varepsilon}.
}
Then, by Borel's lemma \cite{Brocker}, the vector field $F$ can be realized as a smooth normal form $X^N=F+R$ of \cref{s2} and where $R$ is flat at $(x_1,x_2,z,\varepsilon)=(0,0,0,0)$. See more details in \cref{sec:formal_nf}.
\item Based the previous normalization, next we use the geometric desingularization or blow up method (as introduced in \cite{DumRou2}) to study the flow of the normal form $X^N=F+R$. This is detailed in \cref{sec:GeomDes}.
\end{enumerate}
\begin{remark} With this document we aim at two goals:
\begin{enumerate}
\item To refine the results of \cite{BKK}. This is, we do not only provide a qualitative description of the transition $\Pi$, but details on the differentiability of such a map is also presented.
\item To prepare a framework for the geometric desingularization of $A_k$ slow-fast systems. These are a generalization of \cref{s2} given as
\eq{
X=\varepsilon(1+ f_1)\parc{x_1} + \sum_{i=1}^{k-1}\varepsilon f_i\parc{x_i} - \left( z^k+\sum_{j=1}^{k-1}x_jz^{j-1}+\varepsilon f_k \right)\parc{z}+0\parc{\varepsilon}.
}
\end{enumerate}
\end{remark}
The rest of this document is arranged as follows: in \cref{sec:preliminaries} we provide a brief recollection of preliminary results that will simplify our later studies. Next, in \cref{sec:GeomDes} we pose our result and prove it by means of the geometric desingularization method and the results of \cref{sec:preliminaries}. For readability purposes, many technicalities have been put in the appendix.
\section{Preliminaries of slow-fast systems}\label{sec:preliminaries}
In this section, we provide a number preliminary results that will be used later in \cref{sec:GeomDes}. First of all, we consider slow-fast systems along normally hyperbolic regions of the slow manifold. Afterwards, we recall a result from \cite{Jardon2} dealing with the normal form of \cref{s2}. We remark that we only consider SFS with one fast variable. Let us be more precise with the type of SFS that we shall study first.
\begin{definition} A slow-fast system is said to be (locally) regular around a point $p_0$, if its corresponding slow manifold is normally hyperbolic in a some neighborhood of $p_0$.
\end{definition}
\subsection{The slow vector field}
Let us consider a slow-fast system given by
\eq{\label{sdi1}
X_\varepsilon = \sum_{i=1}^m \varepsilon f_i(x,z,\varepsilon)\parc{x_i}+H(x,z,\varepsilon)\parc{z},
}
where $x\in\mathbb{R}^m$, $z\in\mathbb{R}$, and as usual $0<\varepsilon\ll 1$. Furthermore, assume that $f(0,0,0)\neq 0$, $H(0,0,0)=0$ and $\tparcs{H}{z}(0,0,0)<0$. Thus $X_\varepsilon$ is regular around $0\in\mathbb{R}^{m+2}$. The slow manifold associated to \cref{sdi1} is defined by
\eq{
S=\left\{ (x,z)\in\mathbb{R}^{m+1}\, | \, H(x,z,0)=0 \right\}.
}
From the defining assumptions of \cref{sdi1}, we have that $S$ is a NHIM in a neighborhood of the origin. By looking at the Jacobian of $X_\varepsilon$ at $0$, it follows that there exists an $m+1$ dimensional a center manifold. Since $X$ is smooth, we can choose a $\mathcal{C}^{\ell}$ center manifold $\mathcal{W}^{^C}$ for any $\ell<\infty$. The manifold $\mathcal{W}^{^C}$ is given as a graph $z=\phi(x,\varepsilon)$ where $\phi$ is a $\mathcal{C}^{\ell}$ function.
\begin{remark} Along the rest of the document we frequently make use of a finite class of differentiability. As it is customary in the present context, when we say that a manifold (or a map) is $\mathcal{C}^{\ell}$, we mean that such a manifold (or map) is $\ell$-differentiable for $\ell$ as large as necessary.
\end{remark}
The slow manifold $S$ is naturally given by the restriction $\mathcal{W}^{^C}|_{\varepsilon=0}=S$. Next, let us consider the vector field $\frac{1}{\varepsilon}X_\varepsilon(x,\phi,\varepsilon)$. Since $\mathcal{W}^{^C}$ is locally invariant, it follows that $\frac{1}{\varepsilon}X_\varepsilon$ is tangent to $\mathcal{W}^{^C}$. Therefore the vector field
\eq{
X^{slow}=\lim_{\varepsilon\to 0}\frac{1}{\varepsilon}X_\varepsilon(x,\phi,\varepsilon),
}
is tangent to $S$ at each point of $S$, and we call it \emph{the slow vector field}. We remark that the slow vector field $X^{slow}$ is only well defined whenever $\phi$ is invertible.
\subsubsection{The slow divergence integral}\label{sec:sdi}
Associated to a regular slow-fast system and the corresponding slow vector field, the \emph{slow divergence integral} is defined here. For this, let $\Sigma^-$ and $\Sigma^+$ be two sections which are transversal to the flow of $X_\varepsilon$ given by \cref{sdi1}. For $\varepsilon\neq 0$ but sufficiently small, these sections are also transversal to the slow manifold $S$. Let $\gamma_\varepsilon$ be a solution curve of $X_\varepsilon$ chosen along a center manifold $\mathcal{W}^{^C}$, thus $\gamma_\varepsilon$ is transversal to the sections $\Sigma^-$ and $\Sigma^+$. In the limit $\varepsilon=0$, the curve $\gamma_0$ is a curve along the slow manifold $S$. The idea now is to borrow the well-known divergence theorem \cite{Spivak} to get some sense on how the trajectories of $X_\varepsilon$ are attracted to $S$ (recall that we made the assumption $\tparcs{H}{z}<0$). The divergence of $X_\varepsilon$ (given by \cref{sdi1}) reads as
\eq{
\Div X_\varepsilon &= \parcs{H(x,z,\varepsilon)}{z}+O(\varepsilon).
}
We can now take the integral of $\Div X_\varepsilon$ along the orbit $\gamma_\varepsilon$ of $X_\varepsilon$ parametrized by the fast time $\tau$, we have
\eq{\label{sdi2}
\int_{\gamma_\varepsilon}\Div X_\varepsilon \, d\tau=\int_{\gamma_\varepsilon} \left( \parcs{H(x,z,\varepsilon)}{z}+O(\varepsilon) \right)d\tau.
}
The \emph{slow divergence integral} is defined by
\eq{
I(t)=\int_{\gamma_0} \Div X_0 \, dt,
}
where $t$ is the slow time defined by the slow vector field $X^{slow}$. Our goal then is to relate the divergence integral \cref{sdi2} with $I$.
\begin{proposition} Under the assumptions made in this section, we have that
\eq{
\int_{\gamma_\varepsilon}\Div X_\varepsilon \, d \tau=\frac{1}{\varepsilon}\left( I(t)+o(1)\right),
}
where $I(t)$ is the slow divergence integral.
\end{proposition}
\begin{proof}
Recall that the slow vector field reads as $X^{slow}=\lim_{\varepsilon\to 0}\frac{1}{\varepsilon}X_\varepsilon(x,\phi,\varepsilon)$, where $\phi=\phi(x,\varepsilon)$ is a $\mathcal{C}^{\ell}$ function. By our assumptions, the curve $\gamma_\varepsilon$ is transversal to the sections $\Sigma^-$ and $\Sigma^+$ for $\varepsilon$ small enough. Without loss of generality we can assume that $\gamma_\varepsilon$ is parametrized by $x_1$. Then let $x_1^-$ and $x_1^+$ be defined by $\gamma_\varepsilon(x_1^{-})=\gamma_\varepsilon\cap\Sigma^-$ and $\gamma_\varepsilon(x_1^{+})=\gamma_\varepsilon\cap\Sigma^+$. Next, the integral of the divergence of $X_\varepsilon$ along $\gamma_\varepsilon$ from $\Sigma^-$ to $\Sigma^+$ reads as
\eq{
\int_{\gamma_\varepsilon}\Div X_\varepsilon \, d\tau &=\frac{1}{\varepsilon}\int_{x_1^-}^{x_1^+} \left(\parcs{H(x,z,0)}{z}+O(\varepsilon)\right) \frac{dx_1}{f_1(x,z,0)+o(1)}\\
&=\frac{1}{\varepsilon} \left(\int_{x_1^-}^{x_1^+} \parcs{H(x,z,0)}{z}\frac{dx_1}{f_1(x,z,0)} + o(1)\right)\\
&=\frac{1}{\varepsilon} \left(\int_{\gamma_0}\Div X_0\, dt+o(1)\right),
}
where $t$ is the slow time induced by $X^{slow}$, which in coordinates means that $\frac{dx_1}{dt}=f_1$.
\end{proof}
Observe that the slow divergence integral is a first order approximation of the divergence along orbits of $X_\varepsilon$. This will be useful when presenting our main result in \cref{sec:GeomDes}.
\subsubsection{Normal form and transition of a regular slow-fast system}\label{sec:nf_reg}
Now we consider the problem of finding a suitable normal form of a regular SFS. The following is a well-known result but we recall it here for completeness.
\begin{proposition}\label{prop:nf_reg}
Consider a regular slow-fast system on $\mathbb{R}^{m+3}$ given by
\eq{\label{eqr}
X_\varepsilon=\varepsilon(1+ f_1)\parc{u}+\sum_{j=1}^m\varepsilon g_j\parc{v_j}+H\parc{z},
}
where $(u,v_1,\ldots,v_m,z,\varepsilon)\in\mathbb{R}^{m+3}$; where the functions $f_1=f_1(u,v,z,\varepsilon)$ and $g_j=g_j(u,v,z,\varepsilon)$, for $2\geq j\geq k-1$, are smooth and where the function $H=H(u,v,z,\varepsilon)$ is smooth with $H(0,0,0,0)=0$ and $\tparcs{H}{z}(0,0,0,0)<0$. Then, the vector field $X$ is $\mathcal{C}^{\ell}$-equivalent to a normal form given by
\eq{
X_{\varepsilon}^N=\varepsilon\parc{U} + \sum_{j=1}^m 0\parc{V_j}-Z\parc{Z},
}
where $\left\{ Z=0\right\}$ corresponds to a choice of the center manifold $\mathcal{W}^{^C}$ of $X_\varepsilon$.
\end{proposition}
\begin{proofof}{\cref{prop:nf_reg}}
The first step is to divide the vector field $X$ by $1+f_1$. In a sufficiently small neighborhood of the origin this is a smooth equivalence relation. That is $Y=\tfrac{1}{1+f_1}X$ reads as
\eq{
Y= \varepsilon\parc{u}+\sum_{j=1}^m\varepsilon \tilde g_j\parc{v_j}+\tilde H\parc{z},
}
where $\tilde g_j$, for $2\geq j\geq k-1$, and $\tilde H$ are smooth with $\tilde H(0)=0$ and $\tparcs{\tilde H}{z}(0)<0$. Now we note that the origin of $\mathbb{R}^{m+3}$ is a semyhyperbolic equilibrium point with $(u,v,\varepsilon)$ being center coordinates and $z$ being the hyperbolic coordinate. We can now use Takens-Bonckaert results on normal forms of partially hyperbolic vector fields \cite{Bonckaert1,Bonckaert2,Takens_partially}. Thus, there exists a $\mathcal{C}^{\ell}$ change of coordinates (maybe respecting some constraints if required) under which $Y$ is conjugated to
\eq{
\bar Y=\varepsilon\parc{U}+\sum_{j=1}^m\varepsilon \bar G_j\parc{V_j}+\bar H Z\parc{Z},
}
where $\bar G_j=\bar G_j(U,V,\varepsilon)$, for $2\geq j\geq k-1$, and $\bar H=\bar H(U,V,\varepsilon)$ are $\mathcal{C}^{\ell}$ functions, and where $\left\{ Z=0\right\}$ corresponds to a choice center manifold which we denote by $\mathcal{W}^{^C}$. We remark that in the vector field $\bar Y$, the functions $\bar G_j$ and $\bar H$ are independent of $Z$. Furthermore we have
\eq{
\bar H(0,0,0)=\parcs{\tilde H}{z}(0,0,0,0)<0.
}
This means that in a small neighborhood of the origin $\bar Y$ can be divided by $|\bar H|$. In other words, $\bar Y$ is $\mathcal{C}^{\ell}$-equivalent to
\eq{
\mathcal Y=\varepsilon\mathcal G \parc{U}+\sum_{j=1}^m\varepsilon \bar K_j\parc{V_j}-Z\parc{Z},
}
where $\mathcal G(0,0,0)\neq 0$ and $\bar K_j=\bar K_j(U,V,\varepsilon)$, for $2\geq j\geq k-1$, are $\mathcal{C}^{\ell}$. Next, since $\mathcal{W}^{^C}=\left\{ Z=0 \right\}$ is invariant under the flow of $\mathcal Y$, we can study the restriction $\mathcal Y|_{Z=0}$. This is
\eq{
\mathcal Y|_{Z=0}=\varepsilon\mathcal G \parc{U}+\sum_{j=1}^m\varepsilon \bar K_j\parc{V_j}.
}
For $\varepsilon\neq 0$, the vector field $\mathcal Y|_{Z=0}$ is regular because $\mathcal G(0,0,0)\neq 0$. Thus, by the flow-box theorem, there exists a change of coordinates, depending in a $\mathcal{C}^{\ell}$ way on $\varepsilon$, under which $\mathcal Y|_{Z=0}$ can be written as
\eq{
\varepsilon\parc{U}+\sum_{j=1}^{m}0\parc{V_j}.
}
This implies that $\mathcal Y$ is $\mathcal{C}^{\ell}$-equivalent to
\eq{
X_{reg}^N=\varepsilon \parc{U}+\sum_{j=1}^m0\parc{V_j}-Z\parc{Z},
}
as stated in the proposition.
\end{proofof}
Motivated by \cref{prop:nf_reg} let us now discuss the dynamics of the vector field
\eq{\label{r1} X_{reg}^N=\varepsilon \parc{U}+\sum_{j=1}^m0\parc{V_j}-Z\parc{Z}.
}
The slow manifold $S$, corresponding to the normal form \cref{r1}, is given by
\eq{
S=\left\{ \varepsilon=0, \, Z =0 \right\}.
}
Furthermore, we can parametrize the solution of \cref{r1} by $U$. Let us define the sections
\eq{\label{sec_reg}
\Sigma^{-} &= \left\{ (U,V,Z,\varepsilon)\in\mathbb{R}\times\mathbb{R}^m\times\mathbb{R}\times\mathbb{R}\, | \, U=U^- \right\}\\
\Sigma^{+} &= \left\{ (U,V,Z,\varepsilon)\in\mathbb{R}\times\mathbb{R}^m\times\mathbb{R}\times\mathbb{R}\, | \, U=U^+ \right\},
}
where $U^-<U^+$. The sections $\Sigma^{-}$ and $\Sigma^{+}$ are transversal to the manifold $S$ and therefore, for $\varepsilon\neq 0$, are also transversal to the flow of \cref{r1}. Associated to these sections, we define the transition
\eq{\label{regtr}
\Pi &:\Sigma^-\to\Sigma^+\\
&(V,Z,\varepsilon)\mapsto (\tilde V,\tilde Z,\tilde\varepsilon).
}
To compute the component $\tilde Z$ we only need to integrate $\tfrac{dZ}{dU}=-\tfrac{1}{\varepsilon}Z$. Then it follows that $\tilde Z=Z(T)$, where $T$ is the time to go from $\Sigma^-$ to $\Sigma^+$, which is $T=U_f-U_i$. Then it follows that
\eq{
\tilde V&= V\\
\tilde Z&=Z\exp\left(-\frac{1}{\varepsilon}(U_f-U_i)\right)\\
\tilde\varepsilon&=\varepsilon.
}
Observe the particular format of the transition $\Pi$. The $Z$ component is an \emph{exponential} contraction towards the center manifold $\left\{ Z=0 \right\}$. Maps with this characteristic appear frequently in our text and also in several other cases where slow-fast systems are studied. Therefore, in \cref{sec:Exp_trans} we discuss in a rather general way, the properties of such maps.
\subsection{Formal normal form of $A_k$ slow-fast systems}\label{sec:formal_nf}
In this section we recall a normal form of the so-called $A_k$ slow-fast systems. A proof can be found in \cite{Jardon2}. This normalization is important since it eliminates many unwanted terms from the system being studied here.
\begin{definition}\label{def:AkSFS} Let $k\in\mathbb{N}$ with $k\geq 2$. An $A_k$ slow-fast system ($A_k$-SFS) is an ODE of the form
\eq{\label{formal1}
x_1' &= \varepsilon(1+ f_1)\\
x_j' &= \varepsilon f_j\\
z' &= -\left( z^k+\sum_{i=1}^{k-1} x_iz^{i-1} \right) + \varepsilon f_k\\
\varepsilon' &= 0,
}
where $j=2,\ldots,k-1$, and where the functions $f_i=f_i(x_1,\ldots,x_{k-1},z,\varepsilon)$, for $1\leq i\leq k$, are smooth.
\end{definition}
\begin{remark}\leavevmode
\begin{itemize}
\item The system investigated in this work is an $A_3$-SFS.
\item The slow manifold associated to an $A_k$-SFS is defined by
\eq{
S=\left\{ (x,z)\in\mathbb{R}^k\, | \, z^k+\sum_{i=1}^{k-1} x_iz^{i-1}=0 \right\}.
}
The manifold $S$ can equivalently be defined as the critical set of an $A_k$ catastrophe \cite{Arnold_singularities}. Hence the name $A_k$-SFS.
\end{itemize}
\end{remark}
Locally, we can regard \cref{formal1} as $X=F+P$ where $F$ and $P$ are smooth vector fields of the form
\eq{\label{formalF}
F &= \varepsilon\parc{x_1}+\sum_{j=2}^{k-1}0\parc{x_j}+g\parc{z}+0\parc{\varepsilon}
}
and
\eq{\label{formalP}
P &=\sum_{i=1}^{k-1}\varepsilon f_i\parc{x_i}+\varepsilon f_k\parc{z}+0\parc{\varepsilon},
}
respectively and where $g=-\left( z^k+\sum_{i=1}^{k-1} x_iz^{i-1} \right)$. We refer to $F$ as the ``principal part'' and to $P$ as the ``perturbation''. Briefly speaking we want to eliminate, via a change of coordinates, the perturbation. The procedure of normalizing the vector field $X$ is motivated by \cite{Sto10}, where normal forms of analytic perturbations of quasihomogeneous vector fields are investigated. The relevant result is the following
\begin{theorem}[Formal normal form \cite{Jardon2}]\label{prop:formal_nf} Let $k\geq 2$ and let $X=F+P$ be a smooth vector field where
\eq{
F=\varepsilon\parc{x_1}+\sum_{i=2}^{k-1}0\parc{x_i}-\left( z^k+\sum_{j=1}^{k-1}x_jz^{j-1} \right)\parc{z}+0\parc{\varepsilon}.
}
and where
\eq{
P=\sum_{i=1}^{k-1}P_i\parc{x_i}+P_k\parc{z}+0\parc{\varepsilon},
}
where each $P_i=P_i(x_1,\ldots,x_{k-1},z,\varepsilon)$ is a smooth function. Assume that the following conditions are satisfied
\begin{enumerate}
\item $P_i(x_1,\ldots,x_{k-1},z,0)=0$,
\item $\rho(\hat P_i)\geq 2k-i+1$,
\end{enumerate}
where $\hat P_i$ denotes the Taylor expansion of $P_i$ and $\rho(\hat P_i)$ is the quasihomogeneous order of the polynomial $\hat P_i$.
Then, there exists a formal diffeomorphism $\hat\Phi$ such that $\hat\Phi_*\hat X=F$.
\end{theorem}
In words, \cref{prop:formal_nf} shows that $\hat X$ and $F$ are conjugated via $\hat\Phi$. It follows that, by Borel's lemma \cite{Brocker}, the formal vector field $\hat X^N=F$ can be realized as a smooth vector field $X^N=F+\tilde P$ where $\tilde P$ is \emph{flat} at $(x,z,\varepsilon)=(0,0,0)$. This has important consequences in the geometric desingularization of an $A_3$-SFS, presented in the following section.
\section{Geometric desingularization of a slow-fast system near a cusp singularity}\label{sec:GeomDes}
\renewcommand{\a}{a}
\renewcommand{\b}{b}
\newcommand{z}{z}
In this section we study an $A_3$ slow-fast system based on: a) the techniques introduced in \cref{sec:preliminaries} and in \cref{sec:Exp_trans}, and b) the blow up method. To simplify the notation, let us now write the $A_3$-SFS as
\eq{\label{eq:cusp1}
X &=\varepsilon(1+f_1)\parc{\a}+\varepsilon f_2\parc{\b}-(z^3+\b z+\a+\varepsilon f_3)\parc{z} + 0\parc{\varepsilon},
}
where thanks to \cref{prop:formal_nf}, the smooth functions $f_i=f_i(a,b,z,\varepsilon)$ are flat at the origin of $\mathbb{R}^4$. We invetigate the transition associated to \cref{eq:cusp1} between the sections
\eq{\label{def:cuspSectionsfar}
\Sigma^{-} &= \left\{ (\a,\b,z,\varepsilon)\in\mathbb{R}^4\, | \, \a=-a^-, \, z>0 \right\}\\
\Sigma^{+} &= \left\{ (\a,\b,z,\varepsilon)\in\mathbb{R}^4\, | \, \a=a^+, \, z<0 \right\},
}
where $a^->0$ and $a^+>0$ are arbitrarily large constants. However, since the trajectories of $X$ spend a long time along regular parts of $S$, it will be useful to define the ``entry'' and ``exit'' sections
\eq{\label{def:cuspSections}
\Sigma^{\en} &= \left\{ (\a,\b,z,\varepsilon)\in\mathbb{R}^4\, | \, \a=-\a_0, \, z>0 \right\}\\
\Sigma^{\ex} &= \left\{ (\a,\b,z,\varepsilon)\in\mathbb{R}^4\, | \, \a=\a_0, \, z<0 \right\},
}
where $\a_0$ is a positive but sufficiently small constant, for reference see \cref{fig:rr}.
\begin{figure}[htbp]\centering
\begin{tikzpicture}
\pgftext{\includegraphics[scale=1.5]{cusp_qual.pdf}}
\node at (-3.3,-.25) {$S$};
\node at (-1.3,2.5) {$S_{\varepsilon}^{-}$};
\node at (3,0.4) {$S_{\varepsilon}^{+}$};
\node at (1.8,1.4) {$\mathcal{M}_{\varepsilon}$};
\node at (2.1,0.75) {$\Sigma^{\ex}$};
\node at (.5,2.5) {$\Sigma^{\en}$};
\end{tikzpicture}
\caption{ Qualitative representation of the investigation performed in this section. The sections $\Sigma^{\en}$ and $\Sigma^{\ex}$ are arbitrarily close to the cusp point. On the other hand the sections $\Sigma^{-}$ and $\Sigma^{+}$ (not shown) are parallel to $\Sigma^{\en}$ and $\Sigma^{\ex}$ but far away from the cusp point. In a qualitative sense, we will construct an invariant manifold $\mathcal{M}_\varepsilon$ and then extend it all the the way up to the sections $\Sigma^{-}$ and $\Sigma^{+}$. Our analysis aims for simplicity and thus depends extensively on the usage of normal forms. This, of course, makes our results coordinate-dependant.}
\label{fig:rr}
\end{figure}
It will be clear from our analysis in the blow up space \cref{Ken} that the section $\Sigma^-$ needs to be partitioned as follows.
\begin{definition}[The inner layer and the lateral regions]\label{def:layers} Let $0<L<M<\infty$ be constants. The inner layer $\Sigma^{\inn}\subset\Sigma^-$ is defined as
\eq{
\Sigma^-\supset\Sigma^{\inn}=\left\{ (b,z,\varepsilon)\in\Sigma^-\, | \, |b|<M\varepsilon^{2/5} \right\}.
}
On the other hand, the lateral regions are defined as
\eq{
\Sigma^-\supset\Sigma^{+b}&=\left\{ (b,z,\varepsilon)\in\Sigma^-\, | \, b>L\varepsilon^{2/5} \right\}\\
\Sigma^-\supset\Sigma^{-b}&=\left\{ (b,z,\varepsilon)\in\Sigma^-\, | \, -b>L\varepsilon^{2/5} \right\}.
}
Note that the set $\left\{\Sigma^{\inn},\Sigma^{+b}, \Sigma^{-b} \right\}$ is an open cover of $\Sigma^-$, see \cref{fig:part0}
\begin{figure}[htbp]\centering
\begin{tikzpicture}
\pgftext{\includegraphics[scale=1.5]{region_partition0.pdf}}
\node at (1.25,2.9) {$\Sigma^{\inn}$};
\node at (4.1,-2.1) {$\Sigma^{+b}$};
\node at (-4.1,-2.1) {$\Sigma^{-b}$};
\node at (0,-3) {$b$};
\node at (-3,0) {$\varepsilon$};
\end{tikzpicture}\hfill
\caption{ The section $\Sigma^{-}$ needs to be partitioned into three subsections: the inner layer $\Sigma^{\inn}$ and the lateral regions $\Sigma^{+b}$, $\Sigma^{-b}$. From a qualitative point of view, these three layers correspond to three different types of trajectories: 1. Trajectories starting at $\Sigma^{\inn}$ pass close to the cusp point. Observe that $\lim_{\varepsilon\to 0}(\Sigma^{\inn})=\left\{ b=0 \right\}$ and then corresponds to a solution of the associated $CDE$ passing exactly through the cusp point. 2. Trajectories starting at $\Sigma^{+b}$ pass sufficiently away from the cusp point along the regular side of the manifold $S$. 3. Trajectories starting at $\Sigma^{-b}$ pass sufficiently away from the cusp point along the folded side of the manifold $S$. }
\label{fig:part0}
\end{figure}
\end{definition}
We are now in position to present our main result. In the following theorem, we characterize the transition $\Pi:\Sigma^-\to\Sigma^+$ under a suitable choice of coordinates at the section $\Sigma^-$ and $\Sigma^+$. Furthermore, we give details on the differentiability of this map according to the cover of $\Sigma^-$, see \cref{def:layers}.
\begin{theorem}[Transition map of an $A_3$-SFS]\label{teo:main} Let $X$ be an $A_3$ slow-fast system. This is, $X$ is a vector field defined by
\eq{\label{eq:mainteo}
X=\varepsilon(1+ f_1)\parc{\a}+\varepsilon f_2\parc{\b}-\left(z^3+\b z+\a+\varepsilon f_3\right)\parc{z} + 0\parc{\varepsilon},
}
where each $f_i=f_i(\a,\b,z,\varepsilon)$, $i=1,2,3$, is smooth. Let the sections $\Sigma^{-}$, $\Sigma^{+}$ be defined as above. Then we can choose suitable $\mathcal{C}^{\ell}$-coordinates $(B,Z,\varepsilon)$ in $\Sigma^{-}$ and $\mathcal{C}^{\ell}$-coordinates $(\tilde B,\tilde Z,\tilde\varepsilon)$ in $\Sigma^{+}$ such that the transition $\Pi:(B,Z,\varepsilon)\mapsto (\tilde B,\tilde Z,\tilde\varepsilon)$ is an exponential type map of the form
\eq{\label{eq:expmainteo}
\Pi(B,Z,\varepsilon)=\left( B+h, \, \phi(B,\varepsilon)+Z\exp\left( -\frac{A(B,\varepsilon)+\Psi(B,Z,\varepsilon)}{\varepsilon} \right),\varepsilon \right),
}
where $h$ is flat at the origin, $A>0$ is $\mathcal{C}^{\ell}$, $\phi$ is $\mathcal{C}^{\ell}$-admissible with $\phi(B,0)=0$, and $\Psi$ is $\mathcal{C}^{\ell}$-admissible with $\Psi(B,Z,0)=0$, see \cref{sec:Exp_trans} for the definition of $\mathcal{C}^{\ell}$-admissible. Moreover, we have the following properties of the function $A$, $\phi$ and $\Psi$.
\begin{enumerate}
\item $-A(B,0)=I(B)$ where $I$ is the slow divergence integral associated to \cref{eq:mainteo}.
\item Restricted to $(B,Z,\varepsilon)\in\Sigma^{\inn}$, there are functions $\tilde \phi$ and $\tilde \Psi$ such that
\eq{
\phi(B,\varepsilon) &=\tilde\phi\left(\mu,\varepsilon^{1/5}\right)\\
\Psi(B,Z,\varepsilon) &= \tilde\Psi\left(|B|^{1/2},\varepsilon^{1/5}, \varepsilon\ln\varepsilon, \mu,Z\right),
}
where $\tilde \phi$ and $\tilde \Psi$ are $\mathcal{C}^{\ell}$-functions with respect to monomials (see \cref{def:mon}) with $\mu=B\varepsilon^{-2/5}$. Note that in this domain, $\mu$ is well defined in the sense that $\mu$ is bounded by a constant as $\varepsilon\to 0$.
\item Restricted to $(B,Z,\varepsilon)\in\Sigma^{+b}$, there is a function $\tilde\Psi$ such that
\eq{
\phi(B,\varepsilon) &= 0\\
\Psi(B,Z,\varepsilon) &= \tilde\Psi\left(|B|^{1/2},\varepsilon^{1/5}, \varepsilon\ln(|B|), \sigma,Z\right),
}
where $\tilde\Psi$ is a $\mathcal{C}^{\ell}$-function with respect to monomials (see \cref{def:mon}) with $\sigma=\varepsilon|B|^{-5/2}$. Note that in this domain, $\sigma$ is well defined since $|B|>0$.
\item Restricted to $(B,Z,\varepsilon)\in\Sigma^{-b}$, there are functions $\tilde \phi$ and $\tilde \Psi$ such that
\eq{
\phi(B,\varepsilon) &=\tilde\phi\left(|B|^{1/2},\sigma\right)\\
\Psi(B,Z,\varepsilon) &= \tilde\Psi\left(|B|^{1/2},\varepsilon^{1/5}, \varepsilon\ln(|B|), \sigma\right),
}
where $\tilde \phi$ and $\tilde \Psi$ are $\mathcal{C}^{\ell}$-functions with respect to monomials (see \cref{def:mon}) with $\sigma=\varepsilon|B|^{-5/2}$. Note that in this domain, $\sigma$ is well defined since $|B|>0$.
\end{enumerate}
\end{theorem}
\paragraph{Sketch of the proof.} { The first step is to recall \cref{prop:formal_nf}, which shows that $X$ is formally conjugate to
\eq{
F=\varepsilon\parc{\a}+0\parc{\b}-\left(z^3+\b z+\a\right)\parc{z} + 0\parc{\varepsilon}.
}
Next, by means of the Borel's lemma \cite{Brocker}, the vector field $F$ can be realized as a smooth vector field $X^N=F+\varepsilon H$ where $H$ is flat at $(a,b,z,\varepsilon)=(0,0,0,0)$. Thus, from now on, we only treat an $A_3$-SFS given as
\eq{
X=\varepsilon(1+\varepsilon \tilde f_1)\parc{\a}+\varepsilon^2 \tilde f_2\parc{\b}-\left(z^3+\b z+\a+\varepsilon \tilde f_3\right)\parc{z} + 0\parc{\varepsilon},
}
where each $\tilde f_i=\tilde f_i(a,b,z,\varepsilon)$ is flat at $(a,b,z,\varepsilon)=(0,0,0,0)$.
Another important ingredient of the proof is the blow up technique, which is described in \cref{sec:blowup}. This method provides several local vector fields whose corresponding transitions are of exponential type, refer to \cref{sec:Exp_trans}. Later all these local transitions are composed to produce an exponential type transition between the sections $\Sigma^-$ and $\Sigma^+$. Along the analysis of the local vector fields (in the blow up space) we will take advantage of the flatness of the higher order terms of $X$. The complete proof follows \cref{sec:blowup,Ken,Ke,Kex,Kpm} and is given in \cref{sec:proofofmain}.
Now, assuming that the transition $\Pi$ is of the form \cref{eq:expmainteo}, we can show that $A(B,0)$ is given by the slow divergence integral of $X$. For this, let us recall the Poincaré-Leontovich-Sotomayor formula \cite{DeMaesschalck2005}, which in general is given as follows.
\begin{proposition} Let $X$ be a vector field on a manifold $M^n$ with a volume form $\Omega$. Let $\Sigma^-$ and $\Sigma^+$ be two open sections of $M$ and transverse to the flow of $X$. Let $\gamma_\varepsilon$ be an orbit of $X$ along a center manifold $\mathcal{W}^{^C}$ of $X$, starting at $p=\gamma_\varepsilon\cap\Sigma^-$ and reaching $q=\gamma_\varepsilon\cap\Sigma^+$ in finite time. Let $\Pi:\Sigma^-\to\Sigma^+$ be the transition map defined in a neighborhood of $p$. If $\psi^-:U\to\Sigma^-$ and $\psi^+:V\to\Sigma^+$, with $U\subset\mathbb{R}^{n-1}$ and $V\subset\mathbb{R}^{n-1}$, are coordinates in $\Sigma^-$ and in $\Sigma^+$ respectively, then
\eq{\label{aux:sdi}
\det\left( D\left( (\psi^{+})^{-1}\circ\Pi\circ\psi^- \right) \right)(s^-)=\frac{\langle \Omega(p), D\psi^-(s^-)\times X(p) \rangle}{\langle \Omega(q), D\psi^+(s^+)\times X(p) \rangle}\exp\left( \int_{\gamma_\varepsilon}\Div_{\Omega}X \, d\tau \right),
}%
where $s^-=(\psi^-)^{-1}(p)$ and $s^+=(\psi^+)^{-1}(q)$. The integral is taken along the orbit $\gamma_\varepsilon$ from $p$ to $q$ parametrized by the fast time $\tau$.
\end{proposition}
So we have the following.
\begin{proposition} \label{prop:slowdiv} Consider an $A_3$-SFS and assume that the transition $\Pi:\Sigma^-\to\Sigma^+$ is given by \cref{eq:expmainteo}. Then $-A(B,0)=I(B)$, where $I(B)$ is the slow divergence integral associated to the $A_3$-SFS.
\end{proposition}
\begin{proof} The only relevant component is $Z$, so denote by $\Pi_Z$ the $Z$-component of $\Pi$. The factor multiplying the exponential in \cref{aux:sdi} can be taken as a constant $C>0$. Then we have that \cref{aux:sdi} for the vector field of \cref{teo:main} reads as
\eq{\frac{\partial\Pi_Z}{\partial Z}= C\exp\left( \int_{\gamma_\varepsilon}\Div_{\Omega}X \, d\tau \right). }
Using the properties of the slow divergence integral described in \cref{sec:sdi}, and since $C\neq 0$, we have
\eq{\label{mainau0}
\frac{\partial\Pi_Z}{\partial Z} &= C\exp\left( \int_{\gamma_\varepsilon}\Div_{\Omega}X \, d\tau \right)\\
&=\exp\left( \frac{1}{\varepsilon}\left(\int_{\gamma_0}\Div X_0 \, dt+\varepsilon\ln C+o(1) \right) \right)\\
&=\exp\left( \frac{1}{\varepsilon}\left(I +O(\varepsilon) \right) \right),
}
where $I$ is the slow divergence integral of $X$ along a curve in the slow manifold $S$ from $\Sigma^-$ to $\Sigma^+$. In principle, the limit $\varepsilon\to 0$ of \cref{mainau0} is not well defined. However, according to our \cref{teo:main}, we have by differentiating \cref{eq:expmainteo} w.r.t. $Z$
\eq{\label{mainau1}
\frac{\partial\Pi_Z}{\partial Z}=\exp\left( -\frac{A(B,\varepsilon)+\varepsilon \Psi(B,Z,\varepsilon)}{\varepsilon} \right).
}
Identifying \cref{mainau0} with \cref{mainau1} and taking the limit $\varepsilon\to 0$ we have indeed that
\eq{
\lim_{\varepsilon\to 0} (I+O(\varepsilon))=\lim_{\varepsilon\to 0} (-A(B,\varepsilon)+\varepsilon\Psi(B,Z,\varepsilon)),
}
which shows the claim. Note that the slow divergence integral in the coordinates $(a,b,z)$ reads as
\eq{
I(b)=\tilde I(b,\zeta^+)-\tilde I(b,\zeta^-),
}
where straightforward computations show that
\eq{\tilde I(b,\zeta)=\frac{9}{5}\zeta^5+2\zeta^3b+b^2\zeta,
}
and where $\zeta^\pm$ is a constant defined by $(a^{\pm},b,\zeta^{\pm})\in\Sigma^{\pm}\cap S$.
On the other hand, in normal coordinates and along regular parts of the slow manifold, the $A_3$-SFS can be written as (see \cref{sec:nf_reg})
\eq{
X(A,B,Z,\varepsilon)=\varepsilon\parc{A}+0\parc{B}-Z\parc{Z}+0\parc{\varepsilon}.
}%
In these coordinates the slow divergence integral reads as
\eq{
I=A^+-A^-,
}%
where $A^+$ and $A^-$ are the corresponding parametrizations of $\Sigma^+$ and $\Sigma^-$ (respectively) in the coordinates $(A,B,Z,\varepsilon)$.
\end{proof}
}
\subsection{Blow-up and charts}\label{sec:blowup}
Let us briefly recall the blow up technique, for more details see e.g. \cite{DumRou1,DumRou2,Krupa1}. The vector field $X$ \cref{eq:cusp1} is quasihomogeneous \cite{Arnold_singularities,Jardon2}. Therefore, it is convenient to use the \emph{quasihomogeneous blow up}. This technique consists on performing a coordinate transformation defined by
\eq{\label{blowup}
\a =r^{3}\bar \a , \, \b = r^{2}\bar \b, \, z=r\bar z, \, \varepsilon=r^{5}\bar\varepsilon,
}
which is called the blow up map, and where $\bar \a^2+ \bar \b^2+\bar z^2+\bar\varepsilon^2=1$ and $r\in[0,+\infty)$. That is $(\bar a, \bar b, \bar z,\bar\varepsilon,r)\in S^{3}\times\mathbb{R}^+$. Since $\varepsilon\geq 0$, we can restrict the coordinates to $\bar\varepsilon\geq 0$. Note that $S^{3}\times\left\{ 0 \right\}$ is mapped, via the blow up map \cref{blowup}, to the origin of $\mathbb{R}^{4}$. The powers or weights of the blow up map \cref{blowup} are obtained from the type of quasihomogeneity of $X$.
Let us denote by $\Phi(\bar \a,\bar\b, \bar z,\bar \varepsilon)$ the blow up map \cref{blowup}. This map induces a smooth vector field $\tilde X$ on $S^{3}\times\mathbb{R}^+$ defined by $\Phi_*\tilde X=X$. It is often the case in which the vector field $\tilde X$ is degenerate along $S^{3}\times\left\{ 0 \right\}$. Then one defines another vector field $\bar X$ by $\bar X=\frac{1}{r^m}\tilde X$ for a well chosen positive integer $m$ so that $\bar X$ is non-degenerate along $S^{3}\times\left\{ 0 \right\}$. Since $r\in\mathbb{R}^+$, the phase portraits of $\tilde X$ and $\bar X$ are equivalent outside $S^{3}\times\left\{ 0 \right\}$, and therefore it is equally useful to study $\bar X$ instead of $\tilde X$. One obtains a complete description of the local flow of $X$ near the the cusp point by studying the flow of $\bar X$ for $(\bar \a,\bar\b, \bar z,\bar\varepsilon,r)\in S^{3}\times [0,r_0)$ with $r_0>0$ sufficiently small.
For problems of dimension greater than $2$, performing computations in spherical coordinates becomes tedious. Therefore, it is more convenient to consider charts which parametrize hemispheres of the ball $S^{3}\times[0,r_0)$. In the present context, the useful charts are
\eq{\label{charts}
K_{\en}=\left\{ \bar \a=-1 \right\}, \; K_{\ex}=\left\{ \bar\a=1 \right\}, \; K_{\bar\varepsilon}=\left\{ \bar\varepsilon=1 \right\},\; K_{\pm}=\left\{ \bar\b=\pm 1 \right\}
}
and we always keep $r\in[0,r_0)$. The previous setting is also known as directional blow up. A qualitative picture of the charts is given in \cref{fig:charts}.
Briefly speaking, our analysis goes as follows: first, we perform a local analysis on each chart given in \cref{charts}. Next, we compose (``glue'') the local results to provide a full description of the flow of $X$ \cref{eq:cusp1} in a small neighborhood of the cusp point. In this way, we construct an invariant manifold from $\Sigma^{\en}$ to $\Sigma^{\ex}$. Later we ``push away'' this invariant manifold all the way up to the sections $\Sigma^-$ and $\Sigma^+$ along regular parts of the slow manifold $S$.
\begin{figure}[htbp]\centering
\begin{tikzpicture}
\pgftext{\includegraphics[]{blowup.pdf}}
\node at (2.5,1.8) {$S^3\times [0,r_0)$};
\node at (1.5,2.5) {$S^3$};
\node at (-.2,2.5) {$\bar z$};
\node at (-2,-0.8) {$\bar \varepsilon$};
\node at (1.8,-0.5) {$\bar a$};
\node at (-2,-2.6) {$K_{\bar \varepsilon}$};
\node at (1.8,-2.3) {$K_{\bar a}$};
\end{tikzpicture}
\caption{The blow up space and the charts. Each chart $K_\ell$ parametrizes a region of the ball $S^3\times [0,r_0)$. A local analysis in the charts provides a full picture of the dynamics of the vector field $\bar X$.}
\label{fig:charts}
\end{figure}
To avoid confusion of the coordinates we adopt the following notation. Any object $O$ defined in the chart $K_{\en}$ is denoted by $O_1$. Similarly any object defined in the chart $K_{\ex}$ is denoted by $O_3$. Finally, an object $O$ defined in either of the charts $K_{\bar\varepsilon}$ or $K_{\pm}$ is denoted by $O_{2}$.
\subsection{Analysis in the chart $K_{\en}$}\label{Ken}
Taking into account our notation convention, the blow-up map in this chart is given by
\eq{\label{blowupKen}
a=-r_1^3, \; b=r_1^2b_1, \; z=r_1^3z_1, \; \varepsilon=r_1^5\varepsilon_1.
}
\renewcommand{\r}{r_1}
\newcommand{a_1}{a_1}
\newcommand{b_1}{b_1}
\renewcommand{z}{z_1}
\newcommand{\ve_1}{\varepsilon_1}
\renewcommand{\mathcal{W}^{^C}}{\mathcal{W}_1^{^C}}
The corresponding vector field in this chart (after multiplication by $3$) has the form
\eq{X_{\en}: \begin{cases}
\r' &= - \ve_1\r\left( 1+ \tilde f_1\right)\\
b_1' &= 2 \eb_1\left( 1+ \tilde f_1\right)+\r^6\ve_1^2\tilde f_2\\
z' &= -3\left( z^3+\yz-1-\frac{1}{3}\ez \right)+\r^2\ve_1\tilde f_3\\
\ve_1' &= 5\ve_1^2\left( 1+ \tilde f_1\right)
\end{cases}
}
where the functions $\tilde f_i=f_i(\r,b_1,z,\ve_1)$ are flat along $\r=0$, recall that $S^3\times\left\{ r=0 \right\}\mapsto 0\in\mathbb{R}^4$ via the blow up map. We study a transition $\Pi_1:\Delta_{1}^{\en}\to\Delta_1^{\ex}$ where
\eq{
\Delta_{1}^{\en} &= \left\{ (\r,b_1,z,\ve_1)\in\mathbb{R}^4 \, | \, \r=r_0, \ve_1<\delta, \, z>0 \right\}\\
\Delta_1^{\ex} &= \left\{ (\r,b_1,z,\ve_1)\in\mathbb{R}^4 \, | \, \ve_1=\delta, \r<r_0 \right\},
}
where $r_0$ and $\delta$ are sufficiently small positive constants.
\begin{remark}
The section $\Delta_1^{\en}$ corresponds to $\Sigma^{\en}$ in the blow-up space, that is $\Sigma^{\en}=\Phi(\Delta_1^{\en})$, where $\Phi$ is the blow-up map \cref{blowupKen}. This implies that trajectories of $X$ crossing $\Sigma^{\en}$ correspond to trajectories of $X_{\en}$ crossing $\Delta_1^{\en}$.
\end{remark}
Before going any further, let us provide a qualitative description of $X_{\en}$ as in \cite{BKK}. This process can be repeated, following similar arguments, in all the local charts; however, for brevity we only detail it for the current one.
\paragraph{Qualitative description of the flow of $X_{\en}$}{
The subspaces $\left\{ \r=0 \right\}$, $\left\{ \ve_1=0 \right\}$ and $\left\{ \r=0 \right\}\cap\left\{ \ve_1=0 \right\}$ are invariant. Therefore, it is useful to study the flow of $X_{\en}$ restricted to the aforementioned subspaces.
\begin{description}[leftmargin=0cm]
\item[Restriction to $\left\{ \r=0 \right\}\cap\left\{ \ve_1=0 \right\}$.] In this space $X_{\en}$ is reduced to
\eq{\label{eq:K_en1}
b_1' & = 0\\
z' & = -3\left( z^3+\yz-1 \right).\\
}
The set
\eq{\label{gamma1}
\gamma_1 = \left\{ (b_1,z)\, | \, z^3+\yz-1=0 \right\}
}
is a curve of equilibrium points. The phase portrait of \eqref{eq:K_en1} is shown in figure \ref{fig:Ken1}.
\begin{figure}[htbp]\centering
\begin{tikzpicture}
\pgftext{\includegraphics[scale=1]{cusp2_2.pdf}}
\node at (2.2,0.2) {$b_1$};
\node at (0.1,2.3) {$z_1$};
\end{tikzpicture}
\caption{The phase portrait of $X_{\en}$ restricted to the invariant space $\left\{ \r=0 \right\}\cap\left\{ \ve_1=0 \right\}$. The shown curve is $\gamma_1$ and it comprises a set of equilibrium points. Note that locally, all trajectories with initial condition $z_1(0)>0$ are attracted to $\gamma_1|_{\left\{z>0\right\}}$.}
\label{fig:Ken1}
\end{figure}
\begin{remark} All the trajectories of \cref{eq:K_en1} restricted to an initial condition $z_0>0$ are attracted to the curve $\gamma_1|_{z_1>0}$. Furthermore, due to our definition of $\Delta_1^{\en}$, we are interested \emph{only} in trajectories satisfying this initial condition. Thus, from now on, we restrict our analysis to the subspace $\left\{ z>0 \right\}$.
\end{remark}
\item[Restriction to $\left\{ \ve_1=0 \right\}$.] In this space $X_{\en}$ is reduced to
\eq{\label{eq:K_en2}
\r' & = 0\\
b_1' & = 0\\
z' & = -3\left( z^3+\yz-1\right).\\
}
The set $\Gamma_1 = \left\{ (\r,b_1,z)\, | \, z^3+\yz-1=0 \right\}$ is a surface of equilibrium points given by $\Gamma_1=(r_1,\gamma_1)$. Since $\r'=0$, the phase space of \cref{eq:K_en2} is foliated by two dimensional leaves in which the flow looks like \cref{fig:Ken1}.
\item[Restriction to $\left\{ \r=0 \right\}$.] In this space $X_{\en}$ is reduced to
\eq{\label{eq:K_en3}
b_1' & = 2\eb_1 \\
z' & = -3\left( z^3+\yz-1-\frac{1}{3}\ez \right) \\
\ve_1' & = 5\ve_1^2,
}
Once again, the set $\gamma_1 = \left\{ (b_1,z,\ve_1)\, | \, \ve_1=0, \, z>0,\, z^3+\yz-1=0 \right\}$ is a curve of equilibrium points. The Jacobian of \cref{eq:K_en3} evaluated along $\gamma_1$ shows that, for small enough $\ve_1$, there exists an invariant center manifold that passes through $\gamma_1$. Furthermore, the non-zero eigenvalue corresponding to the $z$-direction is negative along $\gamma_1$. The phase portrait of \eqref{eq:K_en3} is shown in figure \ref{fig:Ken3}.\\
\begin{figure}[htbp]\centering
\begin{tikzpicture}
\pgftext{\includegraphics[scale=1]{cusp3_2.pdf}}
\node at (2.5,0) {$b_1$};
\node at (0.2,2.1) {$z_1$};
\node at (-1.5,-1.6) {$\varepsilon_1$};
\end{tikzpicture}
\caption{Phase portrait of \eqref{eq:K_en3} restricted to $z>0$. The shown surface is an invariant center manifold, which is attracting in the $z_1$-direction. }
\label{fig:Ken3}
\end{figure}
Observe that the $b_1$ and the $\ve_1$ directions are expanding. It is important to know the relation between such two expanding variables. We have
\eq{
\frac{db_1}{d\ve_1}=\frac{2}{5}\frac{b_1}{\ve_1},
}
which has the solution
\eq{
b_1=b_1^*\left( \frac{\ve_1}{\ve_1^*} \right)^{2/5},
}
where $b_1^*\leqb_1$ and $\ve_1^*\leq\ve_1$ are the initial conditions, that is $(b_1^*,\ve_1^*)=(b_1,\ve_1)|_{\Delta_1^{\en}}$. It is important to look at the ratio of initial conditions $\tfrac{b_1^*}{\left(\ve_1^*\right)^{2/5}}$. This ratio tells us that $b_1$ is bounded as $\ve_1\to 0$ (and therefore as $\ve_1^*\to 0$) if and only if $b_1^*\in O\left( \left(\ve_1^*\right)^{2/5} \right)$. In other words, if the initial condition $b_1^*$ is not of order $O((\ve_1^*)^{2/5})$ then the value of $b_1$ at $\Delta_1^{\ex}$ blows up as $\ve_1^*\to 0$. This leads us to partition the section $\Delta_1^{\en}$ into three open regions as follows.
\eq{
\Delta_1^{\en,\inn} &= \Delta_1^{\en}|_{|b_1|<M\ve_1^{2/5}}\\
\Delta_1^{\en,b_1} &= \Delta_1^{\en}|_{b_1>K\ve_1^{2/5}}\\
\Delta_1^{\en,-b_1} &= \Delta_1^{\en}|_{-b_1>K\ve_1^{2/5}},
}
where $0<K<M<\infty$. Observe that the open sets $\Delta_1^{\en,\inn}$, $\Delta_1^{\en,b_1}$ and $\Delta_1^{\en,-b_1}$ form an open cover of $\Delta_1^{\en}$. Accordingly, these sets induce an open cover of the entry section $\Sigma^{\en}$ via the blow up map \cref{blowupKen}. See \cref{fig:region_partition} for a representation of the aforementioned partition.\bigskip
\begin{figure}[htbp]\centering
\begin{tikzpicture}\node at (-4,0){
\pgftext{\includegraphics[scale=1]{region_partition.pdf}}};
\node at (-4,1) {$\Delta_1^{\en,\inn}$};
\node at (-5,-.5) {$\Delta_1^{\en,-b_1}$};
\node at (-3,-.5) {$\Delta_1^{\en,b_1}$};
\node at (0,-1.5) {$ $};
\end{tikzpicture}
\caption{Partition of $\Delta_1^{\en}$. Trajectories crossing through $\Delta_1^{\en,\ve_1}$ corresponding to the inner wedge area, have a continuation on the chart $K_{\bar\varepsilon}$. On the other hand, outside $\Delta_1^{\en,\ve_1}$ we must consider the lateral regions $\Delta_1^{\en,b_1}$ and $\Delta_1^{\en,-b_1}$. }
\label{fig:region_partition}
\end{figure}
Based on the partition of the entry section $\Delta_1^{\en}$, we define three transitions as follows
\eq{\label{trKen}
\Pi_1^{\inn}&:\Delta_1^{\en,\inn} \to \Delta_1^{\ex}\\
\Pi_1^{+b_1}&:\Delta_1^{\en,+b_1} \to \Delta_1^{\ex,+b_1}\\
\Pi_1^{-b_1}&:\Delta_1^{\en,-b_1} \to \Delta_1^{\ex,-b_1},
}
where
\eq{
\Delta_1^{\ex} &= \left\{ (\r,b_1,z,\ve_1)\in\mathbb{R}^4 \, | \, \ve_1=\delta, \r<r_0 \right\},\\
\Delta_1^{\ex,\pmb_1} &=\left\{ (\r,b_1,z,\ve_1)\in\mathbb{R}^4 \, | \, b_1=\pm\eta, \r<r_0 \right\}.
}
To finish with the qualitative description, note that there exists a (non-unique) $3$-dimensional center manifold $\mathcal{W}^{^C}$, which is shown to exist by evaluating the Jacobian of $X_{\en}$ all along the surface
\eq{
\Gamma_1=\left\{ (\r,b_1,z,\ve_1)\, | \, \ve_1=0, \, z>0 \,z^3+\yz-1=0 \right\}.
}
Moreover, by the analysis provided above, the center manifold $\mathcal{W}^{^C}|_{z>0}$ is attracting for $\ve_1$ small enough. Note that $\mathcal{W}^{^C}|_{\ve_1=0}=\Gamma_1$. This means that $\mathcal{W}^{^C}$ can be interpreted as a perturbation of the slow manifold $S$, written in the coordinates of the current chart. See \cref{fig:Ken2} for a representation of the previous exposition.
\begin{figure}[htbp]\centering
\begin{tikzpicture}
\pgftext{\includegraphics{foldKen2.pdf}}
\node at (3,-.5) {$b_1$};
\node at (-2,-2.8) {$\r$};
\node at (0.25,2.8) {$\ve_1$};
\node at (-1,-2.4) {$\Delta_1^{\en}$};
\node at (3.25,1) {$\Delta_1^{\ex,+b_1}$};
\node at (-3.,1) {$\Delta_1^{\ex,-b_1}$};
\node at (1,2.15) {$\Delta_1^{\ex,\ve_1}$};
\end{tikzpicture}
\caption{Phase portrait of the trajectories of $X_{\en}$ depending on their initial condition. If the trajectories satisfy the estimate $y\in O(\varepsilon^{2/5})$, then they arrive to $\Delta_1^{\ex,\varepsilon_1}$ in finite time. If the estimate $y\in O(\varepsilon^{2/5})$ is not satisfied, then we must choose one of the outgoing sections $\Delta_1^{\ex,\pm b}$ in order to have a well defined transition map. }
\label{fig:Ken2}
\end{figure}
\end{description}
Let us recall that the vector field $X_{\en}$ is of the form
\eq{\label{eq:Ken_original} X_{\en}: \begin{cases}
\r' &= - \ve_1\r\left( 1+ \tilde f_1\right)\\
b_1' &= 2 \eb_1\left( 1+ \tilde f_1\right)+\r^6\ve_1^2\tilde f_2\\
z' &= -3\left( z^3+\yz-1-\frac{1}{3}\ez \right)+\r^2\ve_1\tilde f_3\\
\ve_1' &= 5\ve_1^2\left( 1+ \tilde f_1\right)
\end{cases}
}
We now proceed to describe the transitions $\Pi_1$ given by \cref{trKen}. For this, first we write \cref{eq:Ken_original} in a suitable normal form. Next, based on this normal form, we compute the corresponding transition. \bigskip
First of all, let us move the origin to the point $(\r,\b,z,\ve_1)=(0,0,1,0)$. This is done by defining a new variable $\zeta_1$ by $\zeta_1=z_1-1$. With this variable we have a new local vector field $Y_{\en}$ which is defined by
\eq{\label{eq:Ken_normal0}
Y_{\en}:\begin{cases}
\r' &= - \ve_1\r\left( 1+ \tilde f_1\right)\\
b_1' &= 2 \eb_1\left( 1+ \tilde f_1\right)+\r^6\ve_1^2\tilde f_2\\
\ve_1' &= 5\ve_1^2\left( 1+ \tilde f_1\right)\\
\zeta_1' &= -3 G(b_1,\ve_1,\zeta_1)+\ve_1\tilde h,
\end{cases}
}
where $G(0,0,0)=0$ and $\tparcs{G}{\zeta_1}(0,0,0)=3$. Now, we want to write $Y_{\en}$ in a suitable normal form. From \cref{prop:nf_semihyp}, we know that $Y_{\en}$ is $\mathcal{C}^{\ell}$ equivalent to
\renewcommand{b_1}{B_1}
\eq{\label{eq:Ken_normal1}X_{\en}^N: \begin{cases}
\r' &= - \ve_1\r\\
b_1' &= 2 \eb_1 \\
\ve_1' &= 5\ve_1^2\\
Z_1' &= -9(1+H_1(\r,b_1,\ve_1))Z_1,
\end{cases}
}
where $H_1$ is a $C^\ell$-function vanishing at the origin. This normal form $X_{\en}^{N}$ is convenient since the chosen center manifold $\mathcal{W}^{^C}$ is now simply given by $\mathcal{W}^{^C}=\left\{ Z_1=0 \right\}$. Furthermore, from the format of $X_{\en}^{N}$, it is evident the ``hyperbolic nature'' of the flow restricted to the center manifold: the restriction of $X_{\en}^N$ to the center manifold $\mathcal{W}^{^C}$ has a simple structure, namely
\eq{
X_{\en}^N|_{\mathcal{W}^{^C}}: \begin{cases}
\r' &= - \ve_1\r\\
b_1' &= 2 \eb_1 \\
\ve_1' &= 5\ve_1^2.
\end{cases}
}
Note that for $\ve_1\neq 0$, the vector field $\tfrac{1}{\ve_1}X_{\en}^N|_{\mathcal{W}^{^C}}$ is hyperbolic.\bigskip
\newcommand{\tilde{r}_1}{\tilde{r}_1}
\newcommand{\tilde{B}_1}{\tilde{B}_1}
\newcommand{\tilde{\ve}_1}{\tilde{\varepsilon}_1}
\newcommand{\tilde{Z}_1}{\tilde{Z}_1}
The vector field $X_{\en}^N$ is of the form studied in \cref{prop:tr_semihyp}, therefore we have that the transition
\eq{
\Pi_1^{\inn}:(b_1,\ve_1,z)\mapsto(\tilde{r}_1,\tilde{B}_1,\tilde{Z}_1)
}
is of the form
\eq{
\tilde{r}_1 &=r_0\left( \frac{\ve_1}{\delta} \right)^{1/5}\\
\tilde{B}_1 &=b_1\left( \frac{\delta}{\ve_1} \right)^{2/5}\\
\tilde{Z}_1&=Z_1\exp\left( -\frac{9}{5\ve_1}(1+\alpha_1\ve_1\ln\ve_1+\ve_1 G_1) \right),
}
where $\alpha_1=\alpha_1(r_0|b_1|^{1/2},r_0\ve_1^{1/5})$ and $ G_1= G_1(r_0|b_1|^{1/2}, r_0\ve_1^{1/5},\mu)$ where $\mu=b_1\ve_1^{-2/5}$. Recall that for this transition we have the condition $b_1\in O(\ve_1^{2/5})$ so $\mu$ is well defined.\bigskip
On the other hand, the transition
\eq{
\Pi_1^{\pmb_1}:(b_1,\ve_1,Z_1)\mapsto(\tilde{r}_1,\tilde{\ve}_1,\tilde Z_1)
}
is (see \cref{prop:tr_semihyp}) of the form
\eq{
\tilde{r}_1 &=r_0\left( \frac{b_1}{\eta} \right)^{1/2}\\
\tilde{\ve}_1 &=\ve_1\left( \frac{\eta}{b_1} \right)^{5/2}\\
\tilde{Z}_1&=Z_1\exp\left( -\frac{9}{5\ve_1}(1+\beta_1\ve_1\ln(|b_1|)+\ve_1 H_1) \right),
}
where $\beta_1=\beta_1(r_0|b_1|^{1/2},r_0\ve_1^{1/5})$ and $H_1=H_1(r_0|b_1|^{1/2},r_0\ve_1^{1/5},\sigma)$, where $\sigma=\ve_1|b_1|^{-5/2}$. Note that since $b_1\notin O(\ve_1^{2/5})$, $\sigma$ is well defined. We observe that the transitions $\Pi_1^{\ve_1}$ and $\Pi_1^{\pmb_1}$ are exponential type maps.
\subsection{Analysis in the chart $K_{\bar\varepsilon}$}\label{Ke}
Taking into account our notation convention, the blow-up map in this chart is given by
\eq{
a=r_2^3a_2, \; b=r_2^2b_2, \; z=r_2^3z_2, \; \varepsilon=r_2^5.
}
\newcommand{\bar\ve}{\bar\varepsilon}
\renewcommand{\r}{r_2}
\renewcommand{a_1}{a_2}
\renewcommand{b_1}{b_2}
\renewcommand{z}{z_2}
\renewcommand{\ve_1}{\varepsilon_2}
Then, the blown up vector field reads as
\eq{ \label{eq:X2}X_{\bar\ve}:\begin{cases}
\r' &=0\\
a_1' &=1+\tilde g_1\\
b_1' &=r^6\tilde g_2\\
z' &=-\left(z^3+\yz+a_1\right)+\tilde g_3,
\end{cases}
}
where the function $\tilde g_i=\tilde g_i(\r,a_1,b_1,z)$ are flat along $\r=0$. Note that in this chart $\r$ acts as a parameter and that the flow is regular. Furthermore, note that $X_{\bar\ve}$ is not a slow-fast system, but a regular vector field. From the equation $a_1'=1+\tilde g_1$, we define the following ``entry'' and ``exit'' sections.
\eq{
\Delta_2^{\en,\bar\ve} &=\left\{ (\r,a_1,b_1,z)\, | \, a_1=-A_0, \, z\geq 0 \right\},\\
\Delta_2^{\ex,\bar\ve} &=\left\{ (\r,a_1,b_1,z)\, | \, a_1=A_0, \, z\leq 0 \right\}.\\
}
\renewcommand{\tilde{r}_1}{\tilde{r}_2}
\newcommand{\tilde{a}_2}{\tilde{a}_2}
\renewcommand{\tilde{B}_1}{\tilde{b}_2}
\renewcommand{\tilde{Z}_1}{\tilde{z}_2}
\renewcommand{\tilde{\ve}_1}{\tilde{\varepsilon}_2}
Therefore, we define a transition $\Pi_2^{\bar\ve}$ as
\eq{
\Pi_2^{\bar\ve} : &\Delta_2^{\en,\bar\ve}\to\Delta_2^{\en,\bar\ve}\\
&(\r,b_1,z)\mapsto (\tilde{r}_1,\tilde{B}_1,\tilde{Z}_1).
}
Since \cref{eq:X2} is regular, by the flow box theorem all trajectories starting at $\Delta_2^{\en,\bar\ve}$ arrive at $\Delta_2^{\ex,\bar\ve}$ in finite time. Moreover, the transition $\Pi_2^{\bar\ve}$ is a diffeomorphism and then, from \cref{eq:X2} we have that $\Pi_2^{\bar\ve}$ reads as
\eq{
\Pi_2{\bar\ve}(\r,b_1,z) &= (\tilde{r}_1,\tilde{B}_1,\tilde{Z}_1) \\
&=(\r,b_1+h_{b_1},\phi_1(\r,b_1)+ \phi_2(\r,b_1)(1+\phi_3(\r,b_1,z))z),
}
where the $\phi_i$'s are smooth functions. Observe that in this chart, the transition is not an exponential type map.
\subsection{Analysis in the chart $K_{\ex}$}\label{Kex}
Taking into account our notation convention, the blow-up map in this chart is given by
\eq{
a=r_3^3, \; b=r_3^2b_3, \; z=r_3^3z_3, \; \varepsilon=r_3^5\varepsilon_3.
}
\renewcommand{\r}{r_3}
\renewcommand{a_1}{a_3}
\renewcommand{b_1}{b_3}
\renewcommand{z}{z_3}
\renewcommand{\ve_1}{\varepsilon_3}
\renewcommand{\mathcal{W}^{^C}}{\mathcal{W}_3^{^C}}
\renewcommand{\tilde{r}_1}{\tilde{r}_3}
\renewcommand{\tilde{a}_2}{\tilde{a}_3}
\renewcommand{\tilde{B}_1}{\tilde{b}_3}
\renewcommand{\tilde{Z}_1}{\tilde{z}_3}
\renewcommand{\tilde{\ve}_1}{\tilde{\varepsilon}_3}
\renewcommand{\bar\ve}{\bar\varepsilon}
Then, the blown up vector field reads as
\eq{X_{\ex}: \begin{cases}
\r' &= \ve_1\r\left( 1+ \tilde f_1\right)\\
b_1' &= -2 \eb_1\left( 1+ \tilde f_1\right)+\r^6\ve_1^2\tilde f_2\\
z' &= -3\left( z^3+\yz+1+\frac{1}{3}\ez \right)+\r^2\ve_1\tilde f_3\\
\ve_1' &= -5\ve_1^2\left( 1+ \tilde f_1\right)
\end{cases}
}
where the function $\tilde f_i=\tilde f_i(\r,b_1,\ve_1,z)$ are flat along $\r=0$. Observe that the vector field $X_{\ex}$ resembles the vector field $X_{\en}$. Therefore, we have a similar behavior of the trajectories, the main difference is that in the case of $X_{\ex}$, there is one expanding ($\r$) and three contracting ($b_1$, $\ve_1$ and $z$) directions. The flow of $X_{\ex}$ is obtained following similar arguments as for the flow of $X_{\en}$.\bigskip
From the fact that $X_{\ex}$ has three contracting and one expanding direction, we define the entry sections
\eq{
\Delta_{3}^{\en,\bar\ve} &=\left\{ (\r,b_1,\ve_1,z)\, : \, \ve_1=\delta,\, z<0, \, \r<r_0\right\}\\
\Delta_{3}^{\en,+b_1} &=\left\{(\r,b_1,\ve_1,z)\, : \,b_1=\eta ,\, z<0, \, \r<r_0\right\}\\
\Delta_{3}^{\en,-b_1} &=\left\{(\r,b_1,\ve_1,z)\, : \,b_1=-\eta,\, z<0, \, \r<r_0 \right\},
}
where all the constants are positive and sufficiently small, and the exit section
\eq{
\Delta_{3}^{\ex} &=\left\{ (\r,b_1,\ve_1,z)\, : \, \r=r_0, \, z<0,\, \ve_1<\delta, \, |b_1|<\eta \right\}.
}
Then, accordingly, we define three transition maps as follows
\eq{\label{Kextrs}
\Pi_3^{\ve_1} &:\Delta_{3}^{\en,\bar\ve} \to \Delta_{3}^{\ex}\\
&:(\r,b_1,z)\mapsto(\tilde{B}_1,\tilde{\ve}_1,\tilde{Z}_1)\\[2ex]
\Pi_3^{+b_1} &:\Delta_{3}^{\en,+b_1}\to \Delta_{3}^{\ex}\\
&:(\r,\ve_1,z)\mapsto(\tilde{B}_1,\tilde{\ve}_1,\tilde{Z}_1)\\[2ex]
\Pi_3^{-b_1} &:\Delta_{3}^{\en,-b_1}\to \Delta_{3}^{\ex}\\
&:(\r,\ve_1,z)\mapsto(\tilde{B}_1,\tilde{\ve}_1,\tilde{Z}_1).
}
\renewcommand{b_1}{B_3}
\renewcommand{\tilde{B}_1}{\tilde{B}_3}
Now we proceed to write $X_{\ex}$ in a normal form just as we did with $X_{\en}$ in \cref{Ken}. Following \cref{prop:nf_semihyp} we have that $X_{\ex}$ is $\mathcal{C}^{\ell}$ equivalent to
\eq{X_{\ex}^N:\begin{cases}
\r' &= \ve_1\r\\
b_1' &= -2 \eb_1\\
\ve_1' &= -5\ve_1^2\\
Z_3' &= -9(1+H_3)Z_3,
\end{cases}
}
where $H_3=H_3(\r,b_1,\ve_1)$ is a $C^\ell$ function vanishing at the origin. Just as in the chart $K_{\en}$, there exists a three dimensional center manifold $\mathcal{W}^{^C}$ associated to $X_{\ex}^N$ and which has been chosen such that $\mathcal{W}^{^C}=\left\{ Z_3=0 \right\}$. Since $\r$ is the only expanding direction, we take as transition time $T_3=\ln\left( \tfrac{r_0}{\r} \right)$. This transition time is computed from the dynamics restricted to $\mathcal{W}^{^C}$, that is, from the equation $\r'=\r$. In contrast to what happened in the chart $K_{\en}$, the time $T_3$ is well defined for all the three transitions $\Pi_3^{\ve_1}$, $\Pi_3^{+b_1}$ and $\Pi_3^{-b_1}$. Following \cref{prop:tr_semihyp} we have
\eq{
\tilde{B}_1 &= b_1 \left( \frac{\r}{r_0} \right)^{2}\\
\tilde{\ve}_1 &=\ve_1 \left( \frac{\r}{r_0} \right)^{5}\\
\tilde Z_3 &= Z_3\exp\left( -\frac{9}{5\ve_1}\left( \left( \frac{r_0}{\r} \right)^{5}-1+\alpha_3\ve_1\ln\r+\ve_1 H_3 \right) \right),
}
where $\alpha_3=\alpha_3(\r|b_1|^{1/2}, \r\ve_1^{1/5})$ and $H_3=H_3(\r|b_1|^{1/2}, \r\ve_1^{1/5},\r)$. Therefore, by taking the definitions of the entry sections we have
\eq{
\Pi_3^{\varepsilon_3}(r_3,B_3,Z_3) &= \left( b_1 \left( \frac{\r}{r_0} \right)^{2}, \, \delta \left( \frac{\r}{r_0} \right)^{5},\, Z_3\exp\left( -\frac{9}{5\delta}\left( \left( \frac{r_0}{\r} \right)^{5}-1+\alpha_3\delta\ln\r+\delta H_3 \right) \right) \right)\\
\Pi_3^{\pm b_3}(r_3,\varepsilon_3,Z_3) &= \left( \pm\eta \left( \frac{\r}{r_0} \right)^{2}, \, \varepsilon_3 \left( \frac{\r}{r_0} \right)^{5},\, Z_3\exp\left( -\frac{9}{5\varepsilon_3}\left( \left( \frac{r_0}{\r} \right)^{5}-1+\alpha_3\varepsilon_3\ln\r+\varepsilon_3 H_3 \right) \right) \right).
}
Observe that these transitions are of exponential type.
\subsection{Analysis in the charts $K_{\pm\bar b}$}\label{Kpm}
In this section we study the local flow at the charts $K_{+\bar b}$ and $K_{-\bar b}$. In a qualitative sense, these charts come into play when the initial condition $b_0=b|_{\Sigma^{\en}}$ does not satisfy the estimate $b_0\in O(\varepsilon^{2/5})$. This implies that the corresponding trajectory passes away from the cusp point. The chart $K_{+\bar b}$ ``sees'' trajectories with initial condition $b|_{\Sigma^{\en}}>0$ while $K_{-\bar b}$ ``sees'' trajectories with initial condition $b|_{\Sigma^{\en}}<0$.
\subsection*{Analysis in the chart $K_{+\bar{b}}$}
In this chart the blow-up maps reads
\eq{
a=r_2^3a_2, \, b=r_2^2, \, z=r_2z_2, \, \varepsilon = r_2^5\varepsilon_2.
}
\renewcommand{\r}{r_2}
\renewcommand{a_1}{a_2}
\renewcommand{b_1}{b_2}
\renewcommand{z}{z_2}
\renewcommand{\ve_1}{\varepsilon_2}
\renewcommand{\mathcal{W}^{^C}}{\mathcal{W}_2^{^C}}
\renewcommand{\tilde{r}_1}{\tilde{r}_2}
\renewcommand{\tilde{a}_2}{\tilde{a}_2}
\renewcommand{\tilde{B}_1}{\tilde{b}_2}
\renewcommand{\tilde{Z}_1}{\tilde{z}_2}
\renewcommand{\tilde{\ve}_1}{\tilde{\varepsilon}_2}
Then we have that the blow-up vector field is given by
\eq{
X_{+\bar{b}}:\begin{cases}
\r' &= \ve_1\bar f_r\\
a_1' &= \ve_1(1+\bar f_{a_1})+\ve_1\bar g_{a_1}\\
\ve_1' &= -\ve_1\bar f_{\ve_1}\\
z' &= -(z^3+z+a_1)+\ve_1\bar f_{z}
\end{cases}
}
where all the functions $\bar f_\ell$ are flat along $\left\{ \r=0 \right\}$. Observe that the set
\eq{\label{gamma2}
\Gamma_2=\left\{ (\r,a_1,\ve_1,z) \, |\, \ve_1=0,\, z^3+z+a_1=0 \right\}
}
is a NHIM of $X_{+\bar{b}}$. However, $X_{+\bar{b}}$ is not exactly a slow-fast system since $\ve_1'\neq 0$, but the restriction of $X_{+\bar{b}}$ to $\left\{\r=0\right\}$ is indeed a slow-fast system. This restriction reads as
\eq{\label{Kb1}
X_{+\bar{b}}|_{\left\{ \r=0 \right\}}:\begin{cases}
a_1' &= \ve_1\\
\ve_1' &= 0\\
z' &= -(z^3+z+a_1).
\end{cases}
}
\begin{remark} The subspace $\left\{\r=0\right\}$ is invariant. Moreover, since $X_{+\bar{b}}$ is a flat perturbation of $X_{+\bar{b}}|_{\left\{ \r=0 \right\}}$, it is equally useful to study the restriction $X_{+\bar{b}}|_{\left\{ \r=0 \right\}}$. After all, by regular perturbation theory, their flows are equivalent.
\end{remark}
The slow manifold of $X_{+\bar{b}}|_{\left\{ \r=0 \right\}}$ is defined by $\Gamma_2|_{\r=0}$ and is normally hyperbolic. Let us define the sections
\eq{\label{Kbsections}
\Delta_2^{\en,+b_1} &= \left\{ (\r,a_1,\ve_1,z)\in\mathbb{R}^4\, | \, a_1=-A_0 \right\}\\
\Delta_2^{\ex,+b_1} &= \left\{ (\r,a_1,\ve_1,z)\in\mathbb{R}^4\, | \, a_1=A_0 \right\}.
}
Accordingly, we study the transition
\eq{
\Pi_2^{+b_1}: & \Delta_2^{\en,+b_1}\to\Delta_2^{\ex,+b_1}\\
&(\r,\ve_1,z)\mapsto (\tilde{r}_1,\tilde{\ve}_1,\tilde{Z}_1).
}
For a qualitative description of $X_{+\bar{b}}|_{\left\{ \r=0 \right\}}$ and the objects defined above see \cref{fig:Ky1}.
\begin{figure}[htbp]\centering
\begin{tikzpicture}
\pgftext{\includegraphics[scale=1]{ky1.pdf}}
\node at (1.75,0) {$a_1$};
\node at (0,1.6) {$z$};
\node at (1.75+4,0) {$a_1$};
\node at (0+4,1.6) {$z$};
\node at (1.75-3.9,0) {$a_1$};
\node at (0-3.9,1.6) {$z$};
\node at (3,1.5) {$\Delta_2^{\en,+b_1}$};
\node at (5.5,-1.5) {$\Delta_2^{\ex,+b_1}$};
\end{tikzpicture}
\caption{Left: phase portrait of the corresponding layer equation of $X_{+\bar{b}}|_{\left\{ \r=0 \right\}}$. Center: phase portrait of the corresponding CDE of $X_{+\bar{b}}|_{\left\{ \r=0 \right\}}$. Right: Since the critical manifold is regular, by Fenichel theory we know that the manifold $\Gamma_2$ is perturbed to an invariant manifold $\Gamma_{2,\varepsilon_2}$ which is at distance of order $O(\varepsilon_2)$ from $\Gamma_2$. }
\label{fig:Ky1}
\end{figure}
We know from \cref{sec:nf_reg} that for sufficiently small $\ve_1$, there exists a $C^\ell$ change of coordinates that transforms $X_{+\bar{b}}|_{\left\{ \r=0 \right\}}$ into the vector field
\eq{Y^N:\begin{cases}
a_1' &= \ve_1\\
\ve_1' &= 0\\
Z_2' &= -Z_2,
\end{cases}
}
From the definition of the entry and exit sections \cref{Kbsections}, the time of integration is $T=2A_0$. To obtain the component $Z_2$ of the transition $\Pi_2^{+b_1}|_{\left\{ \r=0 \right\}}$ we need to integrate
\eq{
Z_2' = -\frac{1}{\ve_1}Z_2,
}
and then $\tilde Z_2=Z_2(T)$. Therefore we have that after choosing a center manifold $\mathcal{W}^{^C}$, the transition $\Pi_2^{+b_1}$ reads as
\eq{
\Pi_2^{+b_1}(0,\ve_1,Z_2) &=\left( 0,\ve_1, Z_2\exp\left( -\frac{2A_0}{\ve_1}\right)\right).
}
Note that $\Pi_2^{+b_1}$ is an exponential type map.
\subsection*{Analysis in the chart $K_{-\bar b}$}
In this chart the blow-up maps reads
\eq{
a=r_2^3a_2, \, b=-r_2^2, \, z=r_2z_2, \, \varepsilon = r^5\varepsilon_2.
}
\renewcommand{\r}{r_2}
\renewcommand{a_1}{a_2}
\renewcommand{b_1}{b_2}
\renewcommand{z}{z_2}
\renewcommand{\ve_1}{\varepsilon_2}
\renewcommand{\mathcal{W}^{^C}}{\mathcal{W}_2^{^C}}
\renewcommand{\tilde{r}_1}{\tilde{r}_2}
\renewcommand{\tilde{a}_2}{\tilde{a}_2}
\renewcommand{\tilde{B}_1}{\tilde{b}_2}
\renewcommand{\tilde{Z}_1}{\tilde{z}_2}
\renewcommand{\tilde{\ve}_1}{\tilde{\varepsilon}_2}
Then we have that the blow-up vector field is given by
\eq{
X_{-\bar{b}}:\begin{cases}
\r' &= -\ve_1\bar f_r\\
a_1' &= \ve_1(1+\bar f_{a_1})+\ve_1\bar g_{a_1}\\
\ve_1' &= \ve_1\bar f_{\ve_1}\\
z' &= -(z^3-z+a_1)+\ve_1\bar f_{z}
\end{cases}
}
where all the functions $\bar f_\ell$ and $\bar g_{a_1}$ are flat along $\left\{ \r=0 \right\}$. Observe that, as in the previous section, the subspace $\left\{ \r=0 \right\}$ is invariant. The restriction of $X_{-\bar{b}}$ to this subspace reads as
\eq{
X_{-\bar{b}}|_{\left\{ \r=0 \right\}}:\begin{cases}
a_1' &= \ve_1\\
\ve_1' &= 0\\
z' &= -(z^3-z+a_1).
\end{cases}
}
The flow of $X_{-\bar{b}}$ is a flat perturbation of the flow of $X_{-\bar{b}}|_{\left\{ \r=0 \right\}}$. Therefore, let us continue our analysis restricted to the invariant space $\left\{ \r=0 \right\}$.\bigskip
The manifold $\Gamma_2$, which is defined by
\eq{
\Gamma_2=\left\{ (\r,a_1,\ve_1,z) \, |\, \r=0,\, \ve_1=0, \, z^3-z+a_1=0 \right\}
}
is normally hyperbolic except at the two points $p_{\pm}=\pm\left( \frac{2}{3\sqrt{3}},\frac{1}{\sqrt{3}} \right)$. Let us define the sections
\eq{
\Delta_2^{\en,-b_1} &= \left\{ (\r,a_1,\ve_1,z)\in\mathbb{R}^4\, | \, a_1=-A_0 \right\}\\
\Delta_2^{\ex,-b_1} &= \left\{ (\r,a_1,\ve_1,z)\in\mathbb{R}^4\, | \, a_1=A_0 \right\},
}
where $A_0>0$ is a sufficiently large constant. We are interested in the transition
\eq{
\Pi_2^{-b_1}: & \Delta_2^{\en,-b_1}\to\Delta_2^{\ex,-b_1}\\
&(\r,\ve_1,z)\mapsto (\tilde{r}_1,\tilde{\ve}_1,\tilde{Z}_1).
}
For a qualitative description of $X_{-\bar{b}}|_{\left\{ \r=0 \right\}}$ and the objects defined above see \cref{fig:Ky2}.
\begin{figure}[htbp]\centering
\begin{tikzpicture}
\pgftext{\includegraphics[scale=1]{ky2.pdf}}
\node at (1.75,0) {$a_1$};
\node at (0,1.6) {$z$};
\node at (1.75+4,0) {$a_1$};
\node at (0+4,1.6) {$z$};
\node at (1.75-3.9,0) {$a_1$};
\node at (0-3.9,1.6) {$z$};
\node at (3,1.5) {$\Delta_2^{\en,+b_1}$};
\node at (5.5,-1.5) {$\Delta_2^{\ex,+b_1}$};
\end{tikzpicture}
\caption{Left: phase portrait of the corresponding layer equation of $X_{-\bar{b}}|_{\left\{ \r=0 \right\}}$. Center: phase portrait of the corresponding CDE of $X_{-\bar{b}}|_{\left\{ \r=0 \right\}}$. Right: The expected perturbed invariant manifold obtained from the flow of the corresponding CDE and layer equation.}
\label{fig:Ky2}
\end{figure}
Away from the fold points $p_\pm$, the manifold $\Gamma_2$ is regular and thus, Fenichel's theory applies. However, we need to take care of the transition near the fold point $p_+$. The local transition of a slow-fast system near a fold point is investigated in e.g. \cite{Krupa3}. However, in our current problem this transition is not essential. By this we mean that the passage through the fold point is seen as a flat perturbation of the trajectory along the stable branch of $\Gamma_2$. In a qualitative sense, this is due to the fact that the transition $\Pi_2^{-b_1}$ goes along a large NHIM, which fails to be normally hyperbolic only at one point.
\begin{proposition} We can choose appropriate coordinates $(Z_2,\ve_1)$ in $\Delta_2^{\en,-b_1}$ such that the transition $\Pi_2^{-b_1}: \Delta_2^{\en,-b_1}\to\Delta_2^{\ex,-b_1}$, restricted to $\r=0$, is an exponential type map of the form
\eq{
\Pi_2^{-b_1}(0,\varepsilon_2,Z_2)=\left( 0,\ve_1,\phi_2(\ve_1)+Z_2\exp\left( -\frac{1}{\ve_1}(A_0+\ve_1\psi_2(Z_2,\ve_1)) \right)\right),
}
where $\phi_2$ are flat at $\ve_1=0$, $\psi_2$ is $\mathcal{C}^{\ell}$-admissible, and where $A_0$ is given by the slow divergence integral of $X_{-\bar{b}}|_{\left\{ \r=0 \right\}}$.
\end{proposition}
\begin{proof}
To prove that $A_0$ is given by the slow divergence integral we proceed along the same reasoning as in \cref{prop:slowdiv}, so we do not repeat it here. In figure \cref{fig:ky3} we see the three transitions that we must consider.
\begin{figure}[htbp]\centering
\begin{tikzpicture}
\pgftext{\includegraphics[scale=1.5]{ky3.pdf}}
\node at (0,2.5) {$z_2$};
\node at (2.5,0) {$a_2$};
\node at (-2.25,2.1) {$\Delta_2^{\en,-b_2}$};
\node at (2.25,-2.1) {$\Delta_2^{\ex,-b_2}$};
\node at (-.45,.35) {$\Omega^{\en}$};
\node at (1.5,0.3) {$\Omega^{\ex}$};
\end{tikzpicture}
\caption{The three different transitions in which $\Pi_2^{-b_1}$ is decomposed. The central transitions is locally an $A_2$ problem. The other two transitions at the sides are regular.}
\label{fig:ky3}
\end{figure}
The three transitions are defined as
\eq{
\Pi_2^{reg_1} &:\Delta_2^{\en,-b_2}\to \Omega^{\en}\\
\Pi_2^{fold} &: \Omega^{\en}\to \Omega^{\ex}\\
\Pi_2^{reg_2} &:\Omega^{\ex}\to \Delta_2^{\ex,-b_2},
}
where we define $\Omega^{\en}$ and $\Omega^{\en}$ as
\eq{
\Omega^{\en} &=\left\{ (a_1,\ve_1,Z_2)\in\mathbb{R}^3 \, | \, a_1=-a_{2,\en}\right\}\\
\Omega^{\ex} &=\left\{ (a_1,\ve_1,Z_2)\in\mathbb{R}^3 \, | \, Z_2=-Z_{2,\ex}\right\},
}
where $a_{2,\en}$ and $Z_{2,\ex}$ are sufficiently small positive constants. The total transition $\Pi_2^{+b_1}$ is given by $\Pi_2^{b_1}=\Pi_2^{reg_2}\circ\Pi_2^{fold}\circ\Pi_2^{reg_1}$. Recall from \cref{sec:Exp_trans} that if we want to write the transition $\Pi_2^{+b_1}$ as an exponential type map, we require that $\Pi_2^{reg_1}$ is expressed as an exponential type map with no shift. The transition $\Pi_2^{fold}$ is studied in e.g. \cite{JardonThesis,Krupa3}. In \cite{JardonThesis} is proved that there are local coordinates $(\bar Z_2,\varepsilon)$ in $\Omega^{\en}$, and $(\tilde a_2,\tilde\varepsilon)$ in $\Omega^{\ex}$, such that the transition $\Pi_2^{fold}$ is given by
\eq{
\Pi_2^{fold}(\bar Z_2,\ve_1)&=(\tilde a_2,\tilde\ve_1)\\
&=\left(\ve_1^{2/3}+O(\ve_1), \ve_1\right).
}
Assume now that we have characterized an invariant manifold $\mathcal{M}_{\ve_1}^{fold}$ from $\Omega^{\en}$ to $\Omega^{\ex}$ via the map $\Pi_2^{fold}$. Now we want to ``extend'' $\mathcal{M}_{\ve_1}^{fold}$ all the way up to the sections $\Delta_2^{\en,-b_2}$ and $\Delta_2^{\ex,-b_2}$ via transitions along normally hyperbolic regions of $\Gamma_2$. For this, it is more convenient to regard $\mathcal{M}_{\ve_1}^{fold}$ as a graph $\zeta_2=\phi_{\ve_1}(A_2)$ where $(\zeta_2,A_2)$ are local coordinates around the fold point $p_+$ and where $\phi_{\ve_1}$ is a diffeomorphism for $\ve_1>0$. In this way we can equivalently express the map $\Pi_2^{fold}$ as
\eq{
\Pi_2^{fold}(\zeta,\ve_1) &=(\tilde\zeta_2,\tilde\ve_1)\\
&=(\psi_{\ve_1}(\zeta),\ve_1)
}
where $\psi_{\ve_1}$ is a diffeomorphism for $\ve_1>0$ and only a homeomorphism for $\ve_1=0$. Next, following \cref{sec:nf_reg} we can find coordinates $(Z_2,\ve_1)$ in $\Delta_2^{\en,-b_2}$, and coordinates $(\tilde Z_2,\ve_1)$ in $\Delta_2^{\ex,-b_2}$ in such a way that the transitions $\Pi_2^{reg_1}$ and $\Pi_2^{reg_2}$ are given as
\eq{
\Pi_2^{reg_1}(Z_2,\ve_1) &=\left( Z_2\exp\left( -\frac{1}{\ve_1}(A_0-a_{2,\en}) \right) \right) = (\bar Z_2,\ve_1)\\
\Pi_2^{reg_2}(-Z_{2,\ex},\ve_1) &=\left( -Z_{2,\ex}\exp\left( -\frac{1}{\ve_1}(A_0-\tilde a_2) \right) \right) = (\tilde Z_2,\ve_1).
}
\begin{remark}
Recall that along normally hyperbolic slow manifolds, it is possible to make a normal form transformation in such a way that this transformation respects certain constraint or structure of the vector field, \cite{Bonckaert1,Bonckaert2}. In this particular case, we respect the choice of the invariant manifold $\mathcal{M}_{\ve_1}^{fold}$.
\end{remark}
Next, we can compute the composition $\Pi_2^{-b_1}=\Pi_2^{reg_2}\circ\Pi_2^{fold}\circ\Pi_2^{reg_1}$ by following \cref{sec:Exp_trans} and it thus follows that
\eq{
\Pi_2^{-b_1}(0,Z_2,\ve_1)=\left(0,\bar\psi_{\ve_1}+Z_2\exp\left( -\frac{1}{\ve_1}(A_1+A_3+\ve_1\psi_2) \right),\ve_1 \right),
}%
where $\bar\psi_{\ve_1}=\psi_{\ve_1}(0)\exp\left( -\frac{A_3}{\ve_1} \right)$ and where $\psi_2=\psi_2(Z,\ve_1)$ is a $\mathcal{C}^{\ell}$-admissible function. Note that $\bar\psi_{\ve_1}$ is flat at $\ve_1=0$.
\end{proof}
\subsection{Proof of \cref{teo:main}}\label{sec:proofofmain}
\renewcommand{\bar\ve}{\bar{\varepsilon}}
\newcommand{\bar{b}}{\bar{b}}
\newcommand{\mathcal{A}}{\mathcal{A}}
Let us first recall that, within the blow up space, we have three types of transitions according to the initial condition $b_1|_{\Delta_1^{\en}}$, namely
\begin{itemize}
\item If $b_1|_{\Delta_1^{\en}}\in O(\varepsilon_1^{2/5})$ then we construct a transition passing through the charts $K_{\en}\to K_{\bar\ve}\to K_{\ex}$.
\item If $b_1|_{\Delta_1^{\en}}\notin O(\varepsilon_1^{2/5})$ and $b_1|_{\Delta_1^{\en}}>0$ then we construct a transition passing through the charts $K_{\en}\to K_{+\bar{b}}\to K_{\ex}$.
\item If $b_1|_{\Delta_1^{\en}}\notin O(\varepsilon_1^{2/5})$ and $b_1|_{\Delta_1^{\en}}<0$ then we construct a transition passing through the charts $K_{\en}\to K_{-\bar{b}}\to K_{\ex}$.
\end{itemize}
In \cref{fig:cusp_comp} we give a qualitative diagram of the local transitions obtained and their relationship.
\tikzstyle{stuff_fill}=[rectangle,draw,preaction={fill=white}]
\begin{figure}[htbp]\centering
\begin{tikzpicture}
\pgftext{\includegraphics[scale=0.75]{cusp_comp}}
\node at (-5,-2.3) {$K_{\en}$};
\node at (0,-2.3) {$K_{\bar\ve}$};
\node at (5,-2.3) {$K_{\ex}$};
\node at (0,2.7) {$K_{+\bar{b}}$};
\node at (0,-7.6) {$K_{-\bar{b}}$};
\node at (-6.05,-.8) {\scriptsize{$r_1$}};
\node at (-4.3,1.75) {\scriptsize{$b_1$}};
\node at (-2.4,0) {\scriptsize{$\varepsilon_1$}};
\node at (-6.05+8.475,.05) {\scriptsize{$\varepsilon_3$}};
\node at (-4.3+8.65,1.75) {\scriptsize{$b_3$}};
\node at (-2.4+8.45,-.9) {\scriptsize{$r_3$}};
\node at (-5.6,-1.45) {\scriptsize{$\Delta_1^{\en}$}};
\node at (-3.7,-1.65) {\scriptsize{$\Delta_1^{\ex,-b_1}$}};
\node at (-3.65, 0.95) {\scriptsize{$\Delta_1^{\ex,+b_1}$}};
\node at (-2.75, -.8) {\scriptsize{$\Delta_1^{\ex,\varepsilon_1}$}};
\node at (3,-.85) {\scriptsize{$\Delta_3^{\en,\varepsilon_3}$}};
\node at (5.6,-1.65) {\scriptsize{$\Delta_3^{\en,-b_3}$}};
\node at (5.6, 0.95) {\scriptsize{$\Delta_3^{\en,+b_3}$}};
\node at (5.8, -.1) {\scriptsize{$\Delta_3^{\ex}$}};
\node at (-2.25,.85) [stuff_fill] {\scriptsize{$M_{\en}^{\bar\ve}$}};
\node at (-3.65,3.85) [stuff_fill] {\scriptsize{$M_{\en}^{+\bar{b}}$}};
\node at (3.45,2.95) [stuff_fill] {\scriptsize{$M_{+\bar{b}}^{\ex}$}};
\node at (1.5,0.25) [stuff_fill] {\scriptsize{$M_{\bar\varepsilon}^{\ex}$}};
\node at (2.5,-3.95) [stuff_fill] {\scriptsize{$M_{-\bar{b}}^{\ex}$}};
\node at (-3.5,-2.95) [stuff_fill] {\scriptsize{$M_{\en}^{-\bar{b}}$}};
\node at (-.6,1.3) {\scriptsize{$\Delta_2^{\en,\varepsilon_2}$}};
\node at (1,-1.3) {\scriptsize{$\Delta_2^{\ex,\varepsilon_2}$}};
\end{tikzpicture}
\caption{All the transitions obtained in the charts. We have to compose all such transitions through the matching maps $M_i^j$. A matching map $M_i^j$ relates the coordinates between the charts $K_i$ and $K_j$. }
\label{fig:cusp_comp}
\end{figure}
Let us only detail the transition through the inner layer $\Delta^{\inn}$ corresponding to $b_1|_{\Delta_1^{\en}}\in O(\varepsilon_1^{2/5})$, the other cases follow the same lines.
The transition $\Pi^{\inn}:\Delta_1^{\inn}\to\Delta_2^{\ex}$ is given as
\renewcommand{a_1}{a}
\renewcommand{b_1}{b}
\eq{
\Pi^{\inn} &= \Pi_{3}^{\varepsilon_3}\circ M_{\bar\varepsilon}^{\ex}\circ\Pi_{2}^{\varepsilon_2}\circ M_{\en}^{\bar\varepsilon}\circ\Pi_1^{\inn}
}
where the matching maps are obtained from the blow-up map. For example, to obtain the matching map from the chart $K_{\en}$ to the chart $K_{\bar\ve}$ we relate the two directional blow-up maps
\eq{
a_1=-r_1^3, \; b_1 = r_1^2 b_1, \; z= r_1z_1, \; \varepsilon=r_1^5\varepsilon_1
}
and
\eq{
a_1=r_2^3 a_2, \; b_1 = r_2^2 b_2, \; z= r_2z_2, \; \varepsilon=r_2^5.
}
Let us work out only with the $z$-component of the transitions as it is the only relevant one. Recall from \cref{Ken} that $\Pi_1^{\inn}$ is an exponential type map with no shift. Next, the composition $\Pi^{central}=M_{\bar\varepsilon}^{\ex}\circ\Pi_{2}^{\varepsilon_2}\circ M_{\en}^{\bar\varepsilon}$ yields a diffeomorphism as $\Pi_2^{\varepsilon_2}$ is a diffeomorphism, and the matching maps are also diffeomorphisms on their domain of definition. Next, the last transition $\Pi_3^{\varepsilon_3}$ is an exponential type map with no shift, see \cref{Kex}. Therefore, following \cref{sec:Exp_trans} we have that $\Pi_3^{\varepsilon_3}\circ\Pi^{central}\circ\Pi_1^{\inn}$ is an exponential type map of the form
\eq{
\Pi^{\inn}_{Z_1}=\bar\phi(B_1,\varepsilon_1)+Z_1\exp\left( -\frac{1}{\varepsilon_1}\left( \bar \mathcal{A}(B_1,\varepsilon_1)+\varepsilon_1\bar\Psi(B_1,\varepsilon_1,Z_1) \right) \right),
}
where $\bar \mathcal{A}>0$ and $\phi$ and $\Psi$ are $\mathcal{C}^{\ell}$ admissible functions. The differentiability of $\phi$ and $\Psi$ with respect to monomials is evident from the results of \cref{Ken}. By blowing down we obtain that the transition $\Pi^{\inn}:\Sigma^{\en}\to\Sigma^{\ex}$ (in a small neighborhood of the cusp point and within the inner layer as domain) reads as
\eq{
\Pi^{inner}_{Z}=\phi(B,\varepsilon)+Z\exp\left( -\frac{1}{\varepsilon}\left( \mathcal{A}(B,\varepsilon)+\varepsilon\bar\Psi(B,\varepsilon,Z) \right) \right).
}
To obtain the transition $\Pi:\Sigma^-\to\Sigma^+$ we now need to compose $\Pi^{\inn}_{Z}$ with exponential type maps on the left and on the right corresponding to
\eq{
\Pi^-:\Sigma^-\to\Sigma^{\en}\\
\Pi^+:\Sigma^{\ex}\to\Sigma^+.
}
However, we must proceed with care. In order to express the transition $\Pi$ as an exponential type map, we need to choose appropriate coordinates on $\Sigma^{-}$ and on $\Sigma^+$ that respect the already chosen coordinates in $\Sigma^{\en}$ and in $\Sigma^{\ex}$. Fortunately, this is possible with the extensions of Bonckaert \cite{Bonckaert1,Bonckaert2} to the normalization results of Takens \cite{Takens_partially}.
For sake of clarity, let $(B_{\en},Z_{\en})$ be coordinates in $\Sigma^{\en}$ and $(B_{\ex}, Z_{\ex})$ be coordinates in $\Sigma^{\ex}$. We have shown that these coordinates can be chosen in such a way that the ``vertical'' component of the transition map $\Pi^{\inn}:\Sigma^{\en}\to\Sigma^{\ex}$ reads as
\eq{
\Pi_{Z_{\en}}(B_{\en},Z_{\en},\varepsilon)&=Z_{\ex}\\
&=\phi(B_{\en},\varepsilon)+Z_{\en}\exp\left( -\frac{1}{\varepsilon}\left( \mathcal{A}(B_{\en},\varepsilon)+\varepsilon\bar\Psi(B_{\en},\varepsilon,Z_{\en}) \right) \right).
}
In this case the invariant manifold, say $\mathcal{M}_{\varepsilon}$, is given by $Z_{\en}=0$. Using \cite{Bonckaert1,Bonckaert2} we can find suitable coordinates $(B_-,Z_-)$ in $\Sigma^-$ in such a way that
\eq{
\Pi^-_{Z_-}(B_-,Z_-,\varepsilon)=Z_-\exp\left( -\frac{1}{\varepsilon}(A_0) \right)=Z_{\en}.
}
In other words, there is a change of coordinates respecting the invariant manifold $\mathcal{M}_{\varepsilon}$ under which the transition $\Pi^-$ is an exponential type map with no shift and linear. Similar arguments hold for the choice of coordinates in $\Sigma^+$. Finally, following \cref{sec:Exp_trans}, the composition $\Pi^+_{Z_+}\circ\Pi_{Z_{\en}}\circ\Pi^-_{Z_-}$ leads to the result.
| {
"timestamp": "2015-06-30T02:18:20",
"yymm": "1506",
"arxiv_id": "1506.08679",
"language": "en",
"url": "https://arxiv.org/abs/1506.08679",
"abstract": "This paper studies a slow-fast system whose principal characteristic is that the slow manifold is given by the critical set of the cusp catastrophe. Our analysis consists of two main parts: first, we recall a formal normal form suitable for systems as the one studied here; afterwards, taking advantage of this normal form, we investigate the transition near the cusp singularity by means of the blow up technique. Our contribution relies heavily in the usage of normal form theory, allowing us to refine previous results.",
"subjects": "Dynamical Systems (math.DS)",
"title": "Analysis of a slow-fast system near a cusp singularity",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.977022630759019,
"lm_q2_score": 0.7248702642896702,
"lm_q1q2_score": 0.7082146525752789
} |
https://arxiv.org/abs/1904.06141 | Low-rank binary matrix approximation in column-sum norm | We consider $\ell_1$-Rank-$r$ Approximation over GF(2), where for a binary $m\times n$ matrix ${\bf A}$ and a positive integer $r$, one seeks a binary matrix ${\bf B}$ of rank at most $r$, minimizing the column-sum norm $||{\bf A} -{\bf B}||_1$. We show that for every $\varepsilon\in (0, 1)$, there is a randomized $(1+\varepsilon)$-approximation algorithm for $\ell_1$-Rank-$r$ Approximation over GF(2) of running time $m^{O(1)}n^{O(2^{4r}\cdot \varepsilon^{-4})}$. This is the first polynomial time approximation scheme (PTAS) for this problem. |
\section{Applications}
\label{sec:applications}
In this section we explain the impact of Theorem~\ref{thm:kclustering} about {\sc Binary Constrained $k$-Center}\xspace to other problems around low-rank matrix approximation.
We would like to mention
that {\sc Binary Constrained $k$-Center}\xspace is very similar to the {\sc Binary $\cR$-Clustering}\xspace problem from \cite{DBLP:journals/corr/abs-1807-07156}. In {\sc Binary Constrained $k$-Center}\xspace we want to minimize the maximum distance of a vector from the input set of vectors to the closest center, whereas in {\sc Binary $\cR$-Clustering}\xspace the sum of distances is minimized. While these problems are different, the reduction we explain here, except a few details, are identical to the ones described in \cite{DBLP:journals/corr/abs-1807-07156}.
For reader's convenience, we give one reduction (Lemma~\ref{lem:matrixFas}) in full details and skip all other reductions, which are similar.
In the following lemma we show that
\textsc{$\ell_1$-Rank-$r$ Approximation over \GF}\xspace\ is a special case of {\sc Binary Constrained $k$-Center}\xspace.
\begin{lemma}\label{lem:matrixFas}
There is an algorithm that given an instance $(\bfA,r)$ of \textsc{$\ell_1$-Rank-$r$ Approximation over \GF}\xspace,
where $\bfA$ is an $m\times n$-matrix and $r$ is an integer,
runs in time ${\cal O}(m+n+2^{2r})$, and outputs an instance $J=(X,k=2^r,\cR)$ of {\sc Binary Constrained $k$-Center}\xspace\
with the following property.
Given any $\alpha$-approximate solution $C$ to $J$, an $\alpha$-approximate solution $\bfB$ to $(\bfA,r)$ can be constructed in time ${\cal O}(rmn)$ and vice versa.
\end{lemma}
\begin{proof}
Notice that if ${\rm{GF}}(2)\text{{\rm -rank}}\xspace(\bfB)\leq r$, then $\bfB$ has at most $2^r$ distinct columns, because each column is a linear combination of at most $r$ vectors of a basis of the column space of $\bfB$. Moreover, \textsc{$\ell_1$-Rank-$r$ Approximation over \GF}\xspace{} can also be stated as follows: find vectors $\bfs_1,\ldots,\bfs_r\in\{0,1\}^m$ such that
$\max_{i\in[n]} d_H(\bfa_i,S)$
is minimum, where
$\bfa_1,\ldots,\bfa_n$ are the columns of $\bfA$ and $S=\{\bfs \in \{0,1\}^m \colon \bfs~\text{is a linear combination of}~\bfs_1,\ldots,\bfs_r~\text{over}~{GF}(2)\xspace{}\}$.
To encode an instance of \textsc{$\ell_1$-Rank-$r$ Approximation over \GF}\xspace\ as an instance of {\sc Binary Constrained $k$-Center}\xspace,
we construct the following relation $R$. Set $k=2^r$.
Let $\Lambda=(\mathbf{\lambda}_1,\ldots,\mathbf{\lambda}_k)$ be the $k$-tuple composed of all
distinct vectors in $\{0,1\}^r$. Thus, each element $\lambda_i\in \Lambda$ is a binary $r$-vector. We define
$R=\{(x^\intercal \mathbf{\lambda}_1,\ldots,x^\intercal\mathbf{\lambda}_k)\mid x\in\{0,1\}^r\}.$ Thus, $R$ consists of $k=2^r$ $k$-tuples and every $k$-tuple in $R$ is a row of the matrix $\Lambda^\intercal \cdot \Lambda$.
Now we define $X$ to be the set of columns of $\bfA$ and for each $i\in [m]$, $R_i=R$. Our algorithm outputs the instance $J=(X,k,{\cal R}=(R_1,\ldots,R_m))$.
To show that the instance $(\bfA,r)$ of \textsc{$\ell_1$-Rank-$r$ Approximation over \GF}\xspace{} is equivalent to the constructed instance $J$, assume first that the vectors $\bfs_1,\ldots,\bfs_r\in \{0,1\}^m$
compose an (approximate) solution of \textsc{$\ell_1$-Rank-$r$ Approximation over \GF}\xspace.
For every $i\in[k]$ define the vector
$$\bfc_i=\lambda_i[1] \bfs_1\oplus\cdots\oplus\lambda_i[r] \bfs_r,$$
where $\mathbf{\lambda}_i^\intercal=(\lambda_i[1],\ldots,\lambda_i[r])$, $\oplus$ denotes the sum over {GF}(2)\xspace, and define the tuple $C=(\bfc_1,\ldots,\bfc_k)$. That is, $C$ contains all linear combinations of $\bfs_1,\ldots,\bfs_r$.
For every $i\in[k]$ and $j\in[m]$, we have that $\bfc_i[j]=(\bfs_1[j],\ldots,\bfs_r[j])\mathbf{\lambda}_i$. Therefore, $(\bfc_1[j],\ldots,\bfc_k[j])\in R$ for all $j\in[m]$. Thus, $C$ is a solution to $J$ of cost $\max_{\mathbf{x}\in X}d_H(\mathbf{x},C)$.
For the opposite direction, assume that $C=(\bfc_1,\ldots,\bfc_k)$ is an (approximate) solution to $J$. We construct the vectors $\bfs_1,\ldots,\bfs_r$ as follows.
Let $j\in[m]$. We have that $(\bfc_1[j],\ldots,\bfc_k[j])\in R$. Therefore, there is $\mathbf{x}\in\{0,1\}^r$ such that
$(\bfc_1[j],\ldots,\bfc_k[j])=(\mathbf{x}^\intercal\mathbf{\lambda}_1,\ldots,\mathbf{x}^\intercal\mathbf{\lambda}_k)$.
We set $\bfs_i[j]=\mathbf{x}[i]$ for $i\in[r]$.
Observe that vectors in $C$ are linear combinations of the vectors $\bfs_1,\ldots,\bfs_r$.
This immediately implies that for any $\alpha$-approximate solution $C$ of $J$ an $\alpha$-approximate solution $\bfB$ of $(\bfA,r)$ can be constructed in time ${\cal O}(rmn)$.
\end{proof}
Thus, Theorem~\ref{thm:norm1PTAS} follows from Theorem~\ref{thm:kclustering} and Lemma~\ref{lem:matrixFas}.
\paragraph{Low Boolean-Rank Approximation.}
Let $\bfA$ be a binary $m\times n$ matrix. Now we consider the elements of $\bfA$ to be \emph{Boolean} variables.
The \emph{Boolean rank} of $\bfA$ is the minimum $r$ such that $\bfA=\mathbf{U}\wedge \bfV$ for a Boolean $m\times r$ matrix $\mathbf{U}$ and a Boolean $r\times n$ matrix $\bfV$, where the product is Boolean, that is, the logical $\wedge$ plays the role of multiplication and $\vee$ the role of sum. Here $0\wedge 0=0$, $0 \wedge 1=0$, $1\wedge 1=1$ , $0\vee0=0$, $0\vee1=1$, and $1\vee 1=1$.
Thus the matrix product is over the Boolean semi-ring $({0, 1}, \wedge, \vee)$. This can be equivalently expressed
as the normal matrix product with addition defined as $1 + 1 =1$. Binary matrices equipped with such algebra are called \emph{Boolean
matrices}.
In \textsc{Boolean $\ell_1$-Rank-$r$ Approximation}\xspace, we are given an $m\times n$ binary data matrix $\bfA$ and a positive integer $r$, and we seek a binary matrix $\bfB$ optimizing
\begin{eqnarray}\label{eq_PCA_1111}
\text{ minimize } \|\bfA-\bfB\|_1 \nonumber \\
\text{ subject to } {\rm rank}\xspace(\bfB) \leq r. \nonumber
\end{eqnarray}
Here, by the rank of binary matrix $\bfB$ we mean its Boolean rank, and norm $\| \cdot\|_1$ is
the \emph{column sum norm}. Similar to Lemma~\ref{lem:matrixFas}, one can prove that \textsc{Boolean $\ell_1$-Rank-$r$ Approximation}\xspace is a special case of {\sc Binary Constrained $k$-Center}\xspace, where $k = 2^r$. Thus, we get the following corollary from Theorem~\ref{thm:kclustering}.
\begin{corollary}
\label{cor:booleancase}
There is an algorithm for \textsc{Boolean $\ell_1$-Rank-$r$ Approximation}\xspace that given an instance $I=(\bfA,r)$ and $0<\varepsilon<1$, runs in time
$m^{{\cal O}(1)}n^{{\cal O}(2^{4r}/\varepsilon^4)}$,
and outputs a $(1+\varepsilon)$-approximate solution with probability at least $1-2n^{-2}$.
\end{corollary}
\paragraph*{Projective $k$-center.}
The {\sc Binary Projective $k$-Center} problem is a variation of the {\sc Binary $k$-Center} problem, where
the centers of clusters are linear subspaces of bounded dimension $r$. (For $r=1$ this is {\sc Binary $k$-Center} and for $k=1$ this is \textsc{$\ell_1$-Rank-$r$ Approximation over \GF}\xspace.)
Formally, in
{\sc Binary Projective $k$-Center} we are given a set $X\subseteq \{0,1\}^m$ of $n$ vectors and positive integers $k$ and $r$. The objective is to find a family of $r$-dimensional linear subspaces $C=\{C_1,\ldots,C_k\} $ over {GF}(2)\xspace minimizing \(\max_{\mathbf{x}\in X} d_H(\mathbf{x},\bigcup_{i=1}^kC).\)
To see that {\sc Binary Projective $k$-Center} is a special case of {\sc Binary Constrained $k$-Center}\xspace, we observe that the condition that $C_i$ is an $r$-dimensional subspace over {GF}(2)\xspace can be encoded (as in Lemma~\ref{lem:matrixFas}) by $2^r$ constraints. This observation leads to the following lemma.
\begin{lemma}\label{lem:proj}
There is an algorithm that given an instance $(X,r,k)$ of {\sc Binary Projective $k$-Center},
runs in time ${\cal O}(m+n+2^{{\cal O}(rk)})$, and outputs an instance $J=(X,k'=2^{kr},\cR)$ of {\sc Binary Constrained $k$-Center}\xspace\
with the following property.
Given any $\alpha$-approximate solution $C$ to $J$, an $\alpha$-approximate solution $C'$ to $(X,r,k)$ can be constructed in time ${\cal O}(rkmn)$ and vice versa.
\end{lemma}
Combining Theorem~\ref{thm:kclustering} and Lemma~\ref{lem:proj} together, we get the following corollary.
\begin{corollary}
\label{cor:booleancase}
There is an algorithm for {\sc Binary Projective $k$-Center} that given an instance $I=(X,r,k)$ and $0<\varepsilon<1$, where $X\subseteq \{0,1\}^m$ is a set of $n$ vectors and $r,k\in {\mathbb N}$, runs in time
$m^{{\cal O}(1)}n^{{\cal O}(2^{4kr}/\varepsilon^4)}$,
and outputs a $(1+\varepsilon)$-approximate solution with probability at least $1-2n^{-2}$.
\end{corollary}
\section{Linear programming}\label{section:lp}
\section{Proof of Lemma~\ref{lem:partitioncreation}}
\label{sec:dimred}
In this section we prove Lemma~\ref{lem:partitioncreation}. The main idea is to map the given instance to a low-dimensional space while approximately preserving distances, then try all possible tuples of centers in the low-dimensional space, and construct an instance of {\sc Binary Constrained Partition Center}\xspace by taking the optimal partition of the images with respect to a fixed tuple of centers back to the original vectors.
To implement the mapping, we employ the notion of $(\delta,\ell,h)$-distorted maps, introduced by Ostrovsky and Rabani \cite{OstrovskyR02}. Intuitively, a $(\delta, \ell, h)$-distorted map approximately preserves distances between $\ell$ and $h$, does not shrink distances larger than $h$ too much, and does not expand distances smaller than $\ell$ too much. In what follows we make the definitions formal.
A metric space is a pair $(P, d)$ where $P$ is a set (whose elements are called
points), and $d$ is a distance function $d : P \times P \rightarrow {\mathbb R}$ (called a metric), such that
for every $p_1,p_2,p_3 \in P$ the following conditions hold:
$(i)$ $d(p_1,p_2 )\geq 0$,
$(ii)$ $d(p_1,p_2)=d(p_2,p_1)$,
$(iii)$ $d(p_1,p_2) =0$ if and only if $p_1 = p_2$, and
$(iv)$ $d(p_1,p_2 )+d(p_2, p_3) \geq d(p_1,p_3)$.
Condition $(iv)$ is called the triangle inequality.
The pair $(\{0,1\}^m,d_H)$, binary vectors of lentgh $m$ and the Hamming distance, is a metric space.
\begin{definition}[\cite{OstrovskyR02}]
\label{def:distortion}
Let $(P,d)$ and $(P',d')$ be two metric spaces. Let $X,Y \subseteq P$.
Let $\delta,\ell,h$ be such that $\delta > 0$ and $h>\ell\geq 0$. A mapping $\psi : P \rightarrow P'$
is $(\delta,\ell,h)$-distorted on $(X,Y)$ if and only if there exists $\alpha > 0$ such that for every
$x \in X$ and $y \in Y$, the following conditions hold.
\begin{enumerate}
\item If $d(x,y) <\ell$, then $d(\psi(x),\psi(y)) < (1 + \delta)\alpha \ell$.
\item If $d(x,y) >h$, then $d(\psi(x),\psi(y)) > (1-\delta)\alpha h$.
\item If $\ell \leq d(x, y) \leq h$, then $(1-\delta)\alpha d(x,y) \leq d(\psi(x), \psi(y)) \leq (1 + \delta)\alpha d(x, y)$.
\end{enumerate}
If $X=Y$, then we say that $\psi$ is $(\delta,\ell,h)$-distorted on $X$.
\end{definition}
For any $r,r'\in {\mathbb N}$ and $\varepsilon>0$, ${\cal A}_{r,r'}(\varepsilon)$ denotes a distribution over $r'\times r$ binary matrices $M\in \{0,1\}^{r'\times r}$, where entries are independent, identically distributed, random $0/1$ variables with $\Pr[1]=\varepsilon$.
\begin{proposition}[\cite{OstrovskyR02}]
\label{prop:distortion0}
Let $m,\ell\in {\mathbb N}$, and let $X\subseteq \{0,1\}^m$ be a set of $n$ vectors.
For every $0 < \epsilon \leq 1/2$, there exists a mapping $\phi : X \rightarrow \{0,1\}^{m'}$, where $m' = {\cal O}(\log n/\epsilon^4)$, which is $(\epsilon, \ell/4, \ell/2\epsilon)$-distorted on $X$ (with respect to the Hamming distance in both spaces).
More precisely, for every $\gamma > 0$ there exists $\lambda > 0$, such that, setting $m' = \lambda \log n/\epsilon^4$, the linear map $\mathbf{x} \mapsto A\mathbf{x}$, where $A$ is a random matrix drawn from ${\cal A}_{m,m'}(\epsilon^2/\ell)$, is $(\epsilon, \ell/4, \ell/2\epsilon)$-distorted on $X$ with probability at least $1-n^{-\gamma}$.
\end{proposition}
Now we are ready to prove Lemma~\ref{lem:partitioncreation}. We restate it for convenience.
\lempartcreation*
\begin{proof}
Without loss of generality, we may assume ${\sf OPT}(J) > 0$. If ${\sf OPT}(J) = 0$, there are at most $k$ distinct vectors in $X$, and we trivially construct a single instance of {\sc Binary Constrained Partition Center}\xspace by grouping equal vectors together.
Let $n=\vert X\vert$ and $n'=n+k$.
Let $\lambda=\lambda(\gamma)$ be the constant mentioned in Proposition~\ref{prop:distortion0}, and $m' = \lambda \log n'/\epsilon^4$. Then, for each $\ell\in [m]$\footnote{For an integer $n\in {\mathbb N}$, we use $[n]$ as a shorthand for $\{1,\ldots,n\}$.}, we construct
the collection ${\cal I}_{\ell}$ of $n^{{\cal O}(k/\epsilon^4)}$ {\sc Binary Constrained Partition Center}\xspace instances
as follows.
\begin{itemize}
\item Start with ${\cal I}_{\ell}:=\emptyset$.
\item Randomly choose a matrix $A^{\ell}$ from the distribution ${\cal A}_{m,m'}(\epsilon^2/\ell)$.
\item For each choice of $k$ vectors $\bfc_1',\ldots,\bfc_k'\in \{0,1\}^{m'}$, construct a partition $X_1\uplus\ldots\uplus X_k$ of $X$
such that for each $\mathbf{x}\in X_i$, $\bfc_i'$ is one of the closest vectors to $A^{\ell}\mathbf{x}$ among $C'=\{\bfc_1',\ldots,\bfc_k'\}$. Then, add $(k,X=X^i_1\uplus\ldots X^i_k,{\cal R})$ to ${\cal I}_{\ell}$.
\end{itemize}
Finally, our algorithm outputs ${\cal I}=\bigcup_{\ell\in [m]} {\cal I}_{\ell}$ as the required collection of {\sc Binary Constrained Partition Center}\xspace instances.
Notice that for any $\ell\in [m]$, $\vert {\cal I}_{\ell}\vert=2^{m'k}=n^{{\cal O}(k/\epsilon^4)}$. This implies that the cardinality of $\cal I$ is upper bounded by $m\cdot n^{{\cal O}(k/\epsilon^4)}$, and the construction of ${\cal I}_{\ell}$ takes time $m\cdot n^{{\cal O}(k/\epsilon^4)}$. Thus, the total running time of the algorithm is $m^2\cdot n^{{\cal O}(k/\epsilon^4)}$.
Next, we prove the correctness of the algorithm. Let $\ell={\sf OPT}(J)$ and $C=(\bfc_1,\ldots,\bfc_k)$ be an optimum solution of $J$.
Let $Y_1,\ldots,Y_k$ be the clusters corresponding to $C$. Consider the step in the algorithm where we constructed ${\cal I}_{\ell}$.
By Proposition~\ref{prop:distortion0}, the map $\psi \colon \mathbf{x} \mapsto A^{\ell}\mathbf{x}$ is $(\epsilon, \ell/4, \ell/2\epsilon)$-distorted on $X\cup C$ with probability at least $1-n^{-\gamma}$.
In the rest of the proof, we assume that this event happened.
Let $\bfc_1'=A^{\ell}\bfc_1,\ldots,\bfc_k'=A^{\ell}\bfc_k$.
Consider the {\sc Binary Constrained Partition Center}\xspace instance constructed for the choice of vectors $\bfc_1',\ldots,\bfc'_k$. That is, let $X_1,\ldots,X_k$ be the partition of $X$ such that for each $\mathbf{x}\in X_i$, $\bfc_i'$ is one of the closest vector to $A^{\ell}\mathbf{x}$ from $C'=\{\bfc_1',\ldots,\bfc_k'\}$.
Let $J'$ be the instance $(k,X=X^i_1\uplus\ldots X^i_k,{\cal R})$ of {\sc Binary Constrained Partition Center}\xspace.
Now, we claim that $C$ is a solution to $J'$ with cost at most $(1+4\epsilon)\ell=(1+4\epsilon){\sf OPT}(J)$. Since $C$ satisfies ${\cal R}$, $C$ is a solution of $J'$.
To prove ${\sf OPT}(J')\leq (1+4\epsilon)\ell$, it is enough to prove that for each $i\in [k]$ and $\mathbf{x}\in X_i$, $d_H(\mathbf{x},\bfc_i)\leq (1+4\epsilon)\ell$. Fix an index $i\in [k]$ and $\mathbf{x}\in X_i$. Suppose $\mathbf{x}\in Y_i$.
Since $C$ is an optimum solution of $J$ with corresponding clusters $Y_1,\ldots Y_k$, we have that $d_H(\bfy,\bfc_i)\leq \ell$ for all $\bfy\in Y_i\cap X_i$. Thus, $d_H(\mathbf{x},\bfc_i)\leq \ell$. So, now consider the case $\mathbf{x}\in Y_j$ for some $j\neq i$. Notice that if $d_H(\mathbf{x},\bfc_i)\leq \ell$, then we are done. We have the following two subcases.
\paragraph*{Case 1: $d_H(\mathbf{x},\bfc_i)\leq \frac{\ell}{2\epsilon}$.}
We know that the map $\psi \colon \mathbf{x} \mapsto A^{\ell}\mathbf{x}$ is $(\epsilon, \ell/4, \ell/2\epsilon)$-distorted on $X\cup C$, and let $\alpha>0$ be the number such that conditions of Definition~\ref{def:distortion} hold.
Since $\mathbf{x}\in X_i$, we have that $(a)$ $d_H(\psi(\mathbf{x}),\psi(\bfc_i))\leq d_H(\psi(\mathbf{x}),\psi(\bfc_j))$. Since $d_H(\mathbf{x},\bfc_j)\leq \ell$ (because $\mathbf{x}\in Y_j$) and $\psi$ is $(\epsilon, \ell/4, \ell/2\epsilon)$-distorted on $X\cup C$, we have that
$(b)$ $d_H(\psi(\mathbf{x}),\psi(\bfc_j))\leq (1+\epsilon)\alpha \ell$. Since $\ell<d_H(\mathbf{x},\bfc_i)\leq \frac{\ell}{2\epsilon}$, and $\psi$ is $(\epsilon, \ell/4, \ell/2\epsilon)$-distorted on $X\cup C$, we have that $(c)$ $(1-\epsilon)\alpha d_H(\mathbf{x},\bfc_i)\leq d_H(\psi(\mathbf{x}),\psi(\bfc_i))$. The statements $(a)$, $(b)$, and $(c)$ imply that
\[
d_H(\mathbf{x},\bfc_i) \leq \frac{1+\epsilon}{1-\epsilon}\ell \leq (1+4\epsilon)\ell,
\]
where the last inequality holds since $\epsilon \leq 1/4$.
\paragraph*{Case 2: $d_H(\mathbf{x},\bfc_i)> \frac{\ell}{2\epsilon}$.} We prove that this case is impossible by showing a contradiction. Since $\epsilon \leq 1/4$, in this case, we have that $d_H(\mathbf{x},\bfc_i)>2\ell$. Since $\psi$ is $(\epsilon, \ell/4, \ell/2\epsilon)$-distorted on $X\cup C$, $d_H(\mathbf{x},\bfc_i)>2\ell$, and
$d_H(\mathbf{x},\bfc_j)\leq \ell$, we have that
\begin{eqnarray*}
(1-\epsilon) \alpha \cdot 2\ell \leq d_H(\psi(\mathbf{x}),\psi(\bfc_i))\leq d_H(\psi(\mathbf{x}),\psi(\bfc_j))\leq (1+\epsilon) \alpha \cdot \ell.
\end{eqnarray*}
Then $2(1-\epsilon)\leq (1+\epsilon)$ and thus $\epsilon\geq 1/3$, which contradicts the assumption that $\epsilon \le 1/4$.
This completes the proof of the lemma.
\end{proof}
\section{Introduction}\label{sec:intro}
Low-rank matrix approximation is the method of compressing a matrix by reducing its dimension. It is the basic component of various methods in data analysis including Principal Component Analysis (PCA),
one of the most popular and successful techniques used for dimension reduction in data analysis and machine learning \cite{pearson1901liii,hotelling1933analysis,eckart1936approximation}. In low-rank matrix approximation
one seeks the best low-rank approximation of data matrix $\bfA$ with matrix $\bfB$ solving
\begin{eqnarray}\label{eq_PCA}
\text{ minimize } \|\bfA-\bfB\|_\nu \\
\text{ subject to } {\rm rank}\xspace(\bfB) \leq r. \nonumber
\end{eqnarray}
Here $\| \cdot \|_\nu$ is some matrix norm. The most popular matrix norms studied in the literature are the
\emph{Frobenius}
$||\bfA||_F^2 = \sum_{i, j} a_{ij}^2$
and the \emph{spectral} $\|\bfA\| _2=\sup_{x\neq 0}\frac{\|\bfA\mathbf{x}\| _{2}}{\|\mathbf{x}\| _2}$ norms.
By the Eckart-Young-Mirsky theorem \cite{eckart1936approximation,MR0114821}, \eqref{eq_PCA} is efficiently solvable via
Singular Value Decomposition (SVD) for these two norms.
The spectral norm is an “extremal” norm---it measures the worst-case stretch of the matrix. On the other hand, the Frobenius norm is “averaging”.
Spectral norm is usually applied in the situation when one is interested in actual columns for the subspaces they define and is of greater interest in scientific computing and numerical linear algebra. The Frobenius norm is widely used in statistics and machine learning, see the survey of Mahony~\cite{MahonyM11} for further discussions.
Recently there has been considerable interest in developing algorithms for low-rank matrix approximation problems for binary (categorical) data.
Such variants of dimension reduction for high-dimensional data sets with binary attributes
arise naturally in applications involving binary data sets, like latent semantic analysis \cite{berry1995using}, pattern discovery for gene expression\cite{Shen2009}, or web search models \cite{DBLP:journals/jacm/Kleinberg99}, see \cite{DanHJWZ15,Jiang2014,GutchGYT12,Koyuturk2003,PainskyRF16,Yeredor11} for other applications.
In many such applications it is much more desirable to approximate a binary matrix $\bfA$ with a binary matrix $\bfB$ of small ({GF}(2)\xspace or Boolean) rank because it
could provide a deeper insight into
the semantics associated with the original matrix. There is a big body of work done on binary and Boolean low-rank matrix approximation, see \cite{Bartl2010,BelohlavekV10,DanHJWZ15,LuVAH12,MiettinenMGDM08,DBLP:conf/kdd/MiettinenV11,Mitra:2016,VaidyaAG07,DBLP:conf/icde/Vaidya12} for further discussions.
Unfortunately, SVD is not applicable for the binary case which makes such problems computationally much more challenging. For binary matrix, its Frobenius norm is equal to the number of its $1$-entries, that is
$\|\bfA\| _F= \sum_{j=1}^n \sum_{i=1}^m | a_{ij} |$. Thus, the value $\|\bfA\ -\bfB\| _F$ measures the total Hamming distance from points (columns) of $\bfA$ to the subspace spanned by the columns of $\bfB$. For this variant of the low-rank binary matrix approximation, a number of approximation algorithms were developed, resulting in efficient polynomial time approximation schemes (EPTASes) obtained in
\cite{BanBBKLW19, DBLP:journals/corr/abs-1807-07156}.
However, the algorithmic complexity of the problem for any vector-induced norm, including the spectral norm, remained open.
For binary matrices, the natural ``extremal'' norm to consider is the
$\| \cdot\|_1$ norm, also known as
\emph{column-sum norm},
operator $1$-norm, or H\"older matrix 1-norm. That is,
for a matrix $\bfA$,
\[\|\bfA\| _1=\sup_{x\neq 0}\frac{\|\bfA\mathbf{x}\| _{1}}{\|\mathbf{x}\| _1} = \max_{1 \leq j \leq n} \sum_{i=1}^m | a_{ij} |.\]
In other words, the column-sum norm is the maximum number of $1$-entries in a column in $\bfA$, whereas the Frobenius norm is the total number of $1$-entries in $\bfA$.
The column-sum norm is analogous to the spectral norm, only it is induced by the $\ell_1$ vector norm, not the $\ell_2$ vector norm.
We consider the problem, where for an $m\times n$
binary data matrix $\bfA$ and a positive integer $r$, one seeks a binary matrix $\bfB$ optimizing
\begin{eqnarray}\label{eq_PCA_1}
\text{ minimize } \|\bfA-\bfB\|_1 \\
\text{ subject to } {\rm rank}\xspace(\bfB) \leq r. \nonumber
\end{eqnarray}
Here, by the rank of the binary matrix $\bfB$ we mean its {GF}(2)\xspace-rank.
We refer to the problem defined by \eqref{eq_PCA_1} as to \textsc{$\ell_1$-Rank-$r$ Approximation over \GF}\xspace.
The value $\|\bfA\ -\bfB\| _1$ is the maximum Hamming distance from each of the columns of $\bfA$ to the subspace spanned by columns of $\bfB$ and thus, compared to approximation with the Frobenius norm, it could provide a more accurate dimension reduction.
It is easy to see by the reduction from the \textsc{Closest String} problem, that already for $r=1$, \textsc{$\ell_1$-Rank-$r$ Approximation over \GF}\xspace is NP-hard.
The main result of this paper is that \eqref{eq_PCA_1} admits a polynomial time approximation scheme (PTAS).
More precisely, we prove the following theorem.
\begin{theorem}
\label{thm:norm1PTAS} For every $\varepsilon\in (0, 1)$, there is a {randomized} $(1+\varepsilon)$-approximation algorithm for \textsc{$\ell_1$-Rank-$r$ Approximation over \GF}\xspace of running time $m^{{\cal O}(1)}n^{{\cal O}(2^{4r}\cdot \varepsilon^{-4})}$.
\end{theorem}
In order to prove Theorem~\ref{thm:norm1PTAS} we obtain a PTAS for a more general problem, namely {\sc Binary Constrained $k$-Center}\xspace. This problem has a strong expressive power and can be used to obtain PTASes for a number of problems related to \textsc{$\ell_1$-Rank-$r$ Approximation over \GF}\xspace. For example, for the variant, when the rank of the matrix $\bfB$ is not over {GF}(2)\xspace but is Boolean. Or a variant of clustering, where we want to partition binary vectors into groups, minimizing the maximum distance in each of the group to some subspace of small dimension. We provide discussions of other applications of our work in Section~\ref{sec:applications}.
\paragraph*{Related work.}
The variant of \eqref{eq_PCA} with both matrices $\bfA$ and $\bfB$ binary, and $\|\cdot\|_{\nu}$ being the Frobenius norm, is known as \ProblemName{Low GF(2)-Rank Approximation}.
Due to numerous applications,
various heuristic algorithms for \ProblemName{Low GF(2)-Rank Approximation} could be found in the literature
\cite{DBLP:conf/icdm/JiangH13a,Jiang2014,fu2010binary,Koyuturk2003,Shen2009}.
When it concerns rigorous algorithmic analysis of \ProblemName{Low GF(2)-Rank Approximation},
Gillis and Vavasis ~\cite{GillisV15} and Dan et al. \cite{DanHJWZ15} have shown that \ProblemName{Low GF(2)-Rank Approximation} is NP-complete for every $r\geq1$. A subset of the authors studied parameterized algorithms for \ProblemName{Low GF(2)-Rank Approximation} in \cite{FominGP18}. The first approximation algorithm for \ProblemName{Low GF(2)-Rank Approximation} is due to
Shen et al. \cite{Shen2009}, who gave a $2$-approximation algorithm for the special case of $r=1$. For rank $r>1$, Dan et al. \cite{DanHJWZ15} have shown that a $(r/2 +1 +\frac{r}{2(2^r-1)})$-approximate solution can be formed from $r$ columns of the input matrix $\bfA$. Recently, these algorithms were significantly improved in \cite{BanBBKLW19, DBLP:journals/corr/abs-1807-07156}, where efficient polynomial time approximation schemes (EPTASes) were obtained.
Also note that for general (non-binary) matrices a significant amount of work is devoted to $L1$-PCA, where one seeks a low-rank matrix $\bfB$ approximating given matrix $\bfA$ in \emph{entrywise} $\ell_1$ norm, see e.g. \cite{SongWZ17}.
\medskip
While our main motivation stems from low-rank matrix approximation problems, \textsc{$\ell_1$-Rank-$r$ Approximation over \GF}\xspace extends \textsc{Closest String}, very well-studied problem about strings.
Given a set of binary strings $S = \{s_1 , s_2 , \dots , s_n \}$, each of length $m$, the \textsc{Closest String} problem is to find the smallest $d$ and a string $s$ of length $m$ which is within Hamming distance $d$ to each $s_i \in S$.
A long history of algorithmic improvements for \textsc{Closest String} was concluded by
the PTAS of running time $n^{{\cal O}(\epsilon^{-5})}$ by
Li, Ma, and Wang
\cite{LiMW02}, which running time was later improved to $n^{{\cal O}(\epsilon^{-2})}$~\cite{MaS09}. Let us note that \textsc{Closest String} can be seen as a special case of \textsc{$\ell_1$-Rank-$r$ Approximation over \GF}\xspace for $r=1$. Indeed,
\textsc{Closest String} is exactly the variant of \textsc{$\ell_1$-Rank-$r$ Approximation over \GF}\xspace, where columns of $\bfA$ are strings of $S$ and approximating matrix $\bfB$ is required to have all columns equal. Note that in a binary matrix $\bfB$ of rank $1$ all non-zero columns are equal. However, it is easy to construct an equivalent instance of \textsc{Closest String} by attaching to each string of $S$ a string $1^{m+1}$, such that the solution to \textsc{$\ell_1$-Rank-$r$ Approximation over \GF}\xspace for $r=1$ does not have zero columns.
Cygan et al. \cite{cygan_et_al:LIPIcs:2016:6023} proved that the existence of an EPTAS for \textsc{Closest String},
that is $(1+\varepsilon)$-approximation in time $n^{{\cal O}(1)} \cdot f(\varepsilon)$, for any computable function $f$,
is unlikely, as it would imply that FPT$=$W[1], a highly unexpected collapse in the hierarchy of parameterized complexity classes. They also showed that the existence of a PTAS for \textsc{Closest String} with running time $f(\varepsilon) n^{o(1/\varepsilon)}$, for any computable function $f$, would contradict the Exponential Time Hypothesis. The result of Cygan et al. implies that \textsc{$\ell_1$-Rank-$r$ Approximation over \GF}\xspace also does not admit EPTAS (unless FPT$=$W[1]) already for $r=1$.
A generalization of \textsc{Closest String},
\textsc{$k$-closest strings} is also known to admit a PTAS
\cite{JiaoXL04,GasieniecJL04}.
This problem corresponds to the variant of \textsc{$\ell_1$-Rank-$r$ Approximation over \GF}\xspace, where approximating matrix $\bfB$ is required to have at most $k$ different columns. However, it is not clear how solution to this special case can be adopted to solve
\textsc{$\ell_1$-Rank-$r$ Approximation over \GF}\xspace.
\paragraph*{Our approach.}
The usual toolbox of techniques to handle NP-hard variants of low-rank matrix approximation problems like sketching~\cite{Woodr14}, sampling, and dimension reduction~\cite{BlumHK17} is based on randomized linear algebra. It is very unclear whether any
of these techniques can be used to solve even the simplest case of \textsc{$\ell_1$-Rank-$r$ Approximation over \GF}\xspace with $r=1$. For example for sampling, the presence of just one outlier outside of a sample, makes all information
we can deduce from the sample about the column sum norm of the matrix,
completely useless. This is exactly the reason why approximation algorithms for \textsc{Closest String}
do not rely on such techniques.
On the other hand, randomized dimension reduction appears to be very helpful as a ``preprocessing'' procedure whose application allows us to solve \textsc{$\ell_1$-Rank-$r$ Approximation over \GF}\xspace by applying linear programming techniques similar to the ones developed for the \textsc{Closest String}.
From a very general perspective, our algorithm consists of three steps. While each of these steps is based on the previous works, the way to combine these steps, as well as the correctness proof, is a non-trivial task. We start with a high-level description of the steps and then provide more technical explanations.
\medskip\noindent\textbf{Step~1.} In order to solve \textsc{$\ell_1$-Rank-$r$ Approximation over \GF}\xspace, we encode it as the {\sc Binary Constrained $k$-Center}\xspace problem.
This initial step is almost identical to the encoding used in \cite{DBLP:journals/corr/abs-1807-07156} for \ProblemName{Low GF(2)-Rank Approximation}.
Informally, {\sc Binary Constrained $k$-Center}\xspace is defined as follows. For a given set of binary vectors $X$, a positive integer $k$, and a set of constraints, we want to find $k$ binary vectors $C=(\bfc_1,\dots, \bfc_k)$ satisfying the constraints and minimizing $\max_{\mathbf{x} \in X} d_H(\mathbf{x},C)$, where $d_H(\mathbf{x},C)$ is the Hamming distance between
$\mathbf{x}$ and the closest vector from $C$.
For example, when $k=1$ and there are no constraints, then this is just
the \textsc{Closest String} problem over binary alphabet.
In the technical description below we give a formal definition of this encoding and in Section~\ref{sec:applications} we prove that \textsc{$\ell_1$-Rank-$r$ Approximation over \GF}\xspace is a special case of {\sc Binary Constrained $k$-Center}\xspace. Now on, we are working with {\sc Binary Constrained $k$-Center}\xspace.
\medskip\noindent\textbf{Step~2.} We give an approximate Turing reduction which allows to find a partition of vector set $X$ into clusters $X_1, \dots, X_k$ such that if we find a tuple of vectors
$C=(\bfc_1,\dots, \bfc_k)$ satisfying the constraints and minimizing
$\max_{1\leq i\leq k,\mathbf{x}\in X_i} d_H(\mathbf{x},\{\bfc_i\})$, then the same tuple $C$ will be a good approximation to {\sc Binary Constrained $k$-Center}\xspace. In order to obtain such a partition, we use the dimension reduction technique of Ostrovsky and Rabani~\cite{OstrovskyR02}. While this provides us with important structural information, we are not done yet. Even with a given partition, the task of finding the corresponding tuple of ``closest strings'' $C$ satisfying the constraints, is non-trivial.
\medskip\noindent\textbf{Step~3.} In order to find the centers, we implement the approach used by Li, Ma, and Wang
in \cite{LiMW02} to solve \textsc{Closest String}. By brute-forcing, it is possible to reduce the solution of the problem to special instances, which loosely speaking, have a large optimum.
Moreover, {\sc Binary Constrained $k$-Center}\xspace\ has an Integer Programming (IP) formulation.
Similar to \cite{LiMW02}, for the reduced instance of {\sc Binary Constrained $k$-Center}\xspace (which has a ``large optimum'')
it is possible to prove that the randomized rounding of the corresponding Linear Program (LP) relaxation of this IP, provides a good approximation.
\section{Proof of Lemma~\ref{lem:partkclustering}}
\label{sec:lp}
For a set of positions $P \subset [m]$, let us define the Hamming distance restricted to $P$ by
\[d_H^P(\mathbf{x}, \bfy) = \sum_{i \in P} |x_i - y_i|.\]
We use the following lemma in our proof.
\begin{lemma}
\label{lemma:ragree}
Let $Y = \{\bfy_1, \cdots, \bfy_l\} \subset \{0, 1\}^m$ be a set of vectors and $\bfc^*\in \{0,1\}^m$ be a vector.
Let $d^*={\operatorname{cost}}(Y,\{\bfc^*\})=\max_{\bfy\in Y} d_H(\bfy,\bfc^*)$. For any $r \in \mathbb{N}$, $r > 2$, there exist indices $i_1$, \ldots, $i_r$ such that for any $\mathbf{x} \in Y$
\[d_H^P(\mathbf{x}, \bfy_{i_1}) - d_H^P(\mathbf{x}, \bfc^*) \le \frac{1}{r - 1}d^*,\]
where $P$ is any subset of $Q_{i_1, \ldots, i_r}$ and $Q_{i_1, \ldots, i_r}$ is the set of positions where all of $\bfy_{i_1}, \ldots, \bfy_{i_r}$ coincide (i.e., $Q_{i_1, \ldots, i_r}=\{j\in[m] \colon \bfy_{i_1}[j]=\bfy_{i_2}[j]=\ldots=\bfy_{i_r}[j]\}$).
\end{lemma}
\begin{proof}
For a vector $\mathbf{x}=\bfy_{\ell'}\in Y$ and $P\subseteq Q_{i_1, \ldots, i_r}$, let
\begin{eqnarray*}
J_P(\ell') &=& \left\{j \in P \colon \bfy_{i_1}[j] \ne \mathbf{x}[j] \text{ and } \bfy_{i_1}[j] \ne \bfc^*[j]\right\}, \mbox{ and } \\
J(\ell') &=& \left\{j \in Q_{i_1, \ldots, i_r} \colon \bfy_{i_1}[j] \ne \mathbf{x}[j] \text{ and } \bfy_{i_1}[j] \ne \bfc^*[j]\right\}.
\end{eqnarray*}
To prove the lemma it is enough to prove that $\vert J_P(\ell')\vert \leq \frac{1}{r - 1}d^*$. Also, since $J_P(\ell')\subseteq J(\ell')$,
to prove the lemma, it is enough to prove that $\vert J(\ell')\vert \leq \frac{1}{r - 1}d^*$.
Recall that for any $s\in [\ell]$ and $1\leq i_1, \ldots, i_s\leq \ell$, $Q_{i_1, \ldots, i_s}$ is the set of positions where all of $\bfy_{i_1}, \ldots, \bfy_{i_s}$ coincide. For any $2\leq s \le r + 1$ and $1\leq i_1, \ldots, i_s\leq \ell$, let $p_{i_1, \ldots, i_s}$ be the number of mismatches between
$\bfy_{i_1}$ and $\bfc^*$ at the positions in $Q_{i_1, \ldots, i_s}$. Let
\[
\rho_s=\min_{1\leq i_1, \ldots, i_s\leq n} \frac{p_{i_1, \ldots, i_s}}{d^*}.
\]
Notice that for any $2\leq s \le r + 1$, $\rho_s\leq 1$.
\begin{claim}[Claim~2.2~\cite{LiMW02}]\footnote{We remark that Claim~2.2 in \cite{LiMW02} is stated for a vector $\bfc$ such that $d^*={\operatorname{cost}}(Y,\{\bfc\})=\min_{\bfc'}{\operatorname{cost}}(Y,\{\bfc'\})$. But the steps of the same proof work in our case as well.}
For any $s$ such that $2\leq s \leq r$, there are indices $1\leq i_1,i_2,\ldots,i_r\leq \ell$ such that for any $\mathbf{x}=\bfy_{\ell'}\in Y$, $\vert J(\ell')\vert \leq (\rho_s-\rho_{s+1})d^*$.
\end{claim}
\begin{proof}
Consider indices $1\leq i_1,\ldots,i_s\leq \ell$ such that $p_{i_1, \ldots, i_s}=\rho_s \cdot d^*$. Next arbitrarily pick $r-s$ indices $i_{s+1},i_{s+2},\ldots,i_r$ from $[\ell]\setminus \{i_1,\ldots,i_s\}$. Next we prove that $i_1,i_2,\ldots,i_r$ are the required set of indices.
Towards that, fix $\mathbf{x}=\bfy_{\ell'}\in Y$,
\begin{eqnarray}
J(\ell') &=& \vert \left\{j \in Q_{i_1, \ldots, i_r} \colon \bfy_{i_1}[j] \ne \mathbf{x}[j] \text{ and } \bfy_{i_1}[j] \ne \bfc^*[j]\right\}\vert \nonumber\\
&\leq& \vert \left\{j \in Q_{i_1, \ldots, i_s} \colon \bfy_{i_1}[j] \ne \mathbf{x}[j] \text{ and } \bfy_{i_1}[j] \ne \bfc^*[j]\right\}\vert \qquad\quad
(\mbox{Because } Q_{i_1, \ldots, i_r}\subseteq Q_{i_1, \ldots, i_s}) \nonumber\\
&=& \vert \left\{j \in Q_{i_1, \ldots, i_s} \colon \bfy_{i_1}[j] \ne \bfc^*[j]\right\}\setminus \left\{j \in Q_{i_1, \ldots, i_s} \colon \bfy_{i_1}[j] = \mathbf{x}[j] \text{ and } \bfy_{i_1}[j] \ne \bfc^*[j]\right\}\vert \nonumber\\
&=& \vert \left\{j \in Q_{i_1, \ldots, i_s} \colon \bfy_{i_1}[j] \ne \bfc^*[j]\right\}\setminus \left\{j \in Q_{i_1, \ldots, i_s,{\ell'}} \colon \bfy_{i_1}[j] \ne \bfc^*[j]\right\}\vert \quad (\mbox{Because }\mathbf{x}=\bfy_{\ell'}) \nonumber\\
&=& \vert \left\{j \in Q_{i_1, \ldots, i_s} \colon \bfy_{i_1}[j] \ne \bfc^*[j]\right\}\vert - \vert \left\{j \in Q_{i_1, \ldots, i_s,{\ell'}} \colon \bfy_{i_1}[j] \ne \bfc^*[j]\right\}\vert \label{eqoneof}\\
&=& p_{i_1, \ldots, i_s}-p_{i_1, \ldots, i_s,\ell'} \qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad (\mbox{By definition})\nonumber\\
&\le&(\rho_s-\rho_{s+1})d^* \label{ineqlityother}
\end{eqnarray}
The equality \eqref{eqoneof} holds since $Q_{i_1, \ldots, i_s}\supseteq Q_{i_1, \ldots, i_s,\ell'}$. The inequality
\eqref{ineqlityother} holds because $p_{i_1, \ldots, i_s}=\rho_s \cdot d^*$ by the choice of $i_1$, \ldots, $i_s$, and $\rho_{s + 1} d^* \le p_{i_1, \ldots, i_s, \ell'}$ by definition.
This completes the proof of the claim.
\end{proof}
Notice that
\(
(\rho_2-\rho_{3})+(\rho_3-\rho_{4})+\ldots+(\rho_r-\rho_{r+1})=(\rho_2-\rho_{r+1})\leq \rho_2\leq 1.
\)
Thus, one of $(\rho_2-\rho_{3}),(\rho_3-\rho_{4}),\ldots,(\rho_r-\rho_{r+1})$ is at most $1/(r-1)$. This completes the proof of the lemma.
\end{proof}
Consider the instance $J=(k,X=X^i_1\uplus\ldots X^i_k,{\cal R})$ of {\sc Binary Constrained Partition Center}\xspace.
Let $C^* = (\bfc^*_1, \cdots, \bfc^*_k)\subset \{0,1\}^m$ be an optimum solution to $J$.
Let $d_{opt}={\sf OPT}(J)=\max_{i \in [k], \mathbf{x} \in X_i} d_H(\mathbf{x}, \bfc^*_i)$.
For each $i \in [k]$ and $r\geq 2$, by Lemma~\ref{lemma:ragree}, there exist $r$ elements $\mathbf{x}^{(1)}_i$, \ldots, $\mathbf{x}^{(r)}_i$ of $X_i$ such that for any $\mathbf{x}\in X_i$,
\begin{equation}
\label{eqn:partialxis}
d_H^P(\mathbf{x}, \mathbf{x}^{(1)}_{i}) - d_H^P(\mathbf{x}, \bfc_i^*) \le \frac{1}{r - 1}d_{opt},
\end{equation}
where $P$ is any subset of $Q_i$, and $Q_i$ is the set of coordinates on which $\mathbf{x}^{(1)}_i$, \ldots, $\mathbf{x}^{(r)}_i$ agree.
Let us denote as $Q$ the intersection of all $Q_i$ from which the positions not satisfying $\mathcal{R}$ are removed. That is,
\[Q = \left\{j \in \bigcap_{i \in [k]} Q_i \colon (\mathbf{x}^{(1)}_1[j],\mathbf{x}^{(1)}_2[j],\ldots,\mathbf{x}^{(1)}_k[j]) \in R_j\right\}.\]
Because of \eqref{eqn:partialxis}, there is an approximate solution where the coordinates $j\in Q$ are identified using $\mathbf{x}^{(1)}_1,\ldots,\mathbf{x}^{(1)}_k$.
Let $\overline{Q}=[m] \setminus Q$. Now the idea is to solve the problem restricted to $\overline{Q}$ separately, and then complement the solution on $Q$ by the values of $\mathbf{x}^{(1)}_i$. We prove that for the `subproblem' restricted on $\overline{Q}$, the optimum value is {\em large}. Towards that we first prove the following lemma.
\begin{lemma}
\label{lem:firstphase}
Let $J=(k,X=X^i_1\uplus\ldots X^i_k,{\cal R})$ be an instance of {\sc Binary Constrained Partition Center}\xspace. Let $({\bfc}^*_1,\ldots,{\bfc}^*_k)$ be an optimal solution for $J$, and $r\geq 2$ be an integer. Then, there exist $\{\mathbf{x}^{(1)}_1$, \ldots, $\mathbf{x}^{(r)}_1\} \subset X_1, \ldots, \{\mathbf{x}^{(1)}_k$, \ldots, $\mathbf{x}^{(r)}_k\} \subset X_k$ with the following properties. For each $i\in [k]$, let $Q_i$ be the set of coordinates on which $\mathbf{x}^{(1)}_i$, \ldots, $\mathbf{x}^{(r)}_i$ agree, \(Q = \left\{j \in \bigcap_{i \in [k]} Q_i \colon (\mathbf{x}^{(1)}_1[j],\mathbf{x}^{(1)}_2[j],\ldots,\mathbf{x}^{(1)}_k[j]) \in R_j\right\}\), and $\overline{Q}=[m] \setminus Q$.
\begin{itemize}
\item For any $i\in [k]$ and $\mathbf{x}\in X_i$, $d_H^Q(\mathbf{x}, \mathbf{x}^{(1)}_{i}) - d_H^Q(\mathbf{x}, \bfc_i^*) \le \frac{1}{r - 1}{\sf OPT}(J)$, and
\item $|\overline{Q}| \le rk \cdot {\sf OPT}(J)$.
\end{itemize}
\end{lemma}
\begin{proof}
Fix an integer $i\in [k]$. By substituting $Y=X_i$ and $\bfc^*=\bfc_i^*$ in Lemma~\ref{lemma:ragree}, we get
$\{\mathbf{x}^{(1)}_i$, \ldots, $\mathbf{x}^{(r)}_i\} \subset X_i$ such that for any $\mathbf{x}\in X_i$, $d_H^Q(\mathbf{x}, \mathbf{x}^{(1)}_{i}) - d_H^Q(\mathbf{x}, \bfc_i^*) \le \frac{1}{r - 1}{\sf OPT}(J)$. That is, we have proved
the first condition in the lemma. The second condition is proved in the following claim.
\begin{claim}
$|\overline{Q}| \le rk \cdot {\sf OPT}(J)$.
\end{claim}
\begin{proof}
We claim that for each position $j \in \overline{Q}$ there exist $i \in [k]$ and $s \in [r]$ such that $\mathbf{x}^{(s)}_i[j] \ne {\bfc}^*_i[j]$. There are two kinds of positions in $\overline{Q}$. First, positions, where for some $i \in [k]$ vectors $\mathbf{x}^{(1)}_i$, \ldots, $\mathbf{x}^{(r)}_i$ do not agree, in this case certainly one of them does not agree with the corresponding position in $\bfc^*_i$. Second, positions $j$ where for any $i \in [k]$, $\mathbf{x}^{(1)}_i[j]=\mathbf{x}^{(2)}_i[j] = \cdots = \mathbf{x}^{(r)}_i[j]$, but $(\mathbf{x}^{(1)}_1[j],\ldots,\mathbf{x}^{(1)}_k[j]) \notin R_j$. Then, there exists $i \in [k]$ such that $\mathbf{x}^{(1)}_i[j] \ne {\bfc}^*_i[j]$ because $({\bfc}^*_i[j],\ldots,{\bfc}^*_k[j]) \in R_j$.
Now, for any $i \in [k]$ and $s \in [r]$, $\mathbf{x}^{(s)}_i$ contributes at most ${\sf OPT}(J)$ positions to $\overline{Q}$, since $d_H(\mathbf{x}^{(s)}_i, {\bfc}^*_i) \le {\sf OPT}(J)$. Thus, in total there are at most $rk \cdot {\sf OPT}(J)$ positions in $\overline{Q}$.
\end{proof}
This completes the proof of the lemma.
\end{proof}
As mentioned earlier, we fix the entries of our solution in positions $j$ of $Q$ with values in $\mathbf{x}^{(1)}_1[j],\ldots,\mathbf{x}^{(1)}_k[j]$.
Towards finding the entries of our solution in positions of $\overline{Q}$, we define the following problem and solve it.
\defproblem{{\sc Binary Constrained Partition Center$^{\star}$}\xspace}{A positive integer $k$, a set $X\subseteq \{0,1\}^m$ of $n$ vectors partitioned into $X_1\uplus\ldots\uplus X_k$, a tuple of $k$-ary relations
$\cR=(R_1, \dots, R_m)$, and for all $\mathbf{x}\in X$, $d_{\mathbf{x}}\in {\mathbb N}$ }{Among all tuples $C=(\bfc_1,\ldots,\bfc_k)$ of vectors from $\{0,1\}^m$ satisfying $\cR$, find a tuple $C$ that minimizes the integer $d$ such that
for all $i\in [k]$ and $\mathbf{x}\in X_i$, $d_H(\mathbf{x},\bfc_i)\leq d-d_{\mathbf{x}}$.}
\begin{restatable}{lemma}{lemlargeopt}
\label{lemma:largeopt}
Let $J'=(k,X=X_1\uplus\ldots X_k,{\cal R}, (d_\mathbf{x})_{\mathbf{x} \in X})$ be an instance of {\sc Binary Constrained Partition Center$^{\star}$}\xspace, ${\sf OPT}(J')\geq \frac{m}{c}$ for some integer $c$, and $0<\delta<1/c$. Then, there is an algorithm which runs in time $m^{{\cal O}(1)}n^{{\cal O}(c^2k/\delta^2)}$, and outputs a solution $C$ of $J'$, of cost at most $(1 + \delta) {\sf OPT}(J')$ with probability at least $1-n^{-2}$.
\end{restatable}
Before proving Lemma~\ref{lemma:largeopt}, we explain how all these results puts together to form a proof of Lemma~\ref{lem:partkclustering}. We restate Lemma~\ref{lem:partkclustering} for the convenience of the reader.
\lempartitionprob*
\begin{proof}
Let $J=(k,X=X^i_1\uplus\ldots X^i_k,{\cal R})$ be the input instance of {\sc Binary Constrained Partition Center}\xspace, and $0<\epsilon<\frac{1}{2}$ be the error parameter.
Let $({\bfc}^*_1,\ldots,{\bfc}^*_k)$ be an optimal solution for $J$. Let $r\geq 2$ be an integer which we fix later.
First, for each $i \in [k]$ we obtain $r$ vectors $\mathbf{x}^{(1)}_i$, \ldots, $\mathbf{x}^{(r)}_i \in X_i$ which satisfy the conditions of Lemma~\ref{lem:firstphase}. Their existence is guaranteed by Lemma~\ref{lem:firstphase}, and we guess them in time $n^{{\cal O}(rk)}$ over all $i \in [k]$.
For each $i\in [k]$, let $Q_i$ be the set of coordinates on which $\mathbf{x}^{(1)}_i$, \ldots, $\mathbf{x}^{(r)}_i$ agree, \(Q = \left\{j \in \bigcap_{i \in [k]} Q_i \colon (\mathbf{x}^{(1)}_1[j],\mathbf{x}^{(1)}_2[j],\ldots,\mathbf{x}^{(1)}_k[j]) \in R_j\right\}\), and $\overline{Q}=[m] \setminus Q$. Next, we construct a solution $C=(\bfc_1,\ldots,\bfc_k)$ as follows.
For each $i\in [k]$ and $j\in Q$, we set $\bfc_i[j]=\mathbf{x}_i^{(1)}[j]$.
Towards finding the entries of vectors $\bfc_1,\ldots,\bfc_k$ at the coordinates in $\overline{Q}$, we use Lemma~\ref{lemma:largeopt}. Let $J'$ be the instance of {\sc Binary Constrained Partition Center$^{\star}$}\xspace, which is a natural restriction of $J$ to $\overline{Q}$.
That is, $J'=(k,X'=X'_1\uplus\ldots X'_k,{\cal R}|_{\overline{Q}}, (d_{\mathbf{x}|_{\overline{Q}}})_{\mathbf{x} \in X'})$, where
for each $i\in [k]$, $X_i'=\{\mathbf{x}|_{\overline{Q}} \colon \mathbf{x}\in X_i\}$ and for each $\mathbf{x} \in X_i$, $d_{\mathbf{x}|_{\overline{Q}}} = d_H^Q(\mathbf{x}, \mathbf{x}^{(1)}_i)$. By the second condition in Lemma~\ref{lem:firstphase},
we have that $|\overline{Q}| \le rk \cdot {\sf OPT}(J)$.
\begin{claim}
\label{cliamoptcom}
${\sf OPT}(J)\leq {\sf OPT}(J')\leq \left(1+\frac{1}{r-1}\right) {\sf OPT}(J).$
\end{claim}
\begin{proof}
First, we prove that ${\sf OPT}(J)\leq {\sf OPT}(J')$.
Towards that we show that we can transform a solution $C' = (\bfc'_1, \cdots, \bfc'_k)$ of $J'$ with the objective value $d$ to a solution $C$ of $J$ with the same objective value. For each $i \in [k]$, consider $\widehat{\bfc}_i$ which is equal to $\mathbf{x}^{(1)}_i$ restricted to $Q$, and to $\bfc'_i$ restricted to $\overline{Q}$, and the solution $\widehat{C} = (\widehat{\bfc}_1, \cdots, \widehat{\bfc}_k)$. Clearly, $\widehat{C}$ satisfies ${\cal R}$ since on $\overline{Q}$ it is guaranteed by $C'$ being a solution to $J'$, and on $Q$ by construction of $Q$.
The objective value of $C$ is
\begin{eqnarray*}
\max_{i \in [k],\mathbf{x} \in X_i} d_H(\mathbf{x}, \bfc_i) &=& \max_{i \in [k],\mathbf{x} \in X_i} \left(d_H^{\overline{Q}}(\mathbf{x}, \bfc_i) + d_H^Q (\mathbf{x}, \bfc_i)\right) \\
&=& \max_{i \in [k],\mathbf{x} \in X_i} \left(d_H(\mathbf{x}|_{\overline{Q}}, \bfc'_i) + d_H^Q (\mathbf{x}, \mathbf{x}^{(1)}_i)\right)\\
&=& \max_{i \in [k],\mathbf{x} \in X_i} \left(d_H(\mathbf{x}|_{\overline{Q}}, \bfc'_i) + d_{\mathbf{x}|_{\overline{Q}}}\right) = d.
\end{eqnarray*}
Thus, ${\sf OPT}(J)\leq {\sf OPT}(J')$.
Next, we prove that ${\sf OPT}(J')\leq\left(1+\frac{1}{r-1}\right) {\sf OPT}(J)$. Recall that $({\bfc}^*_1,\ldots,{\bfc}^*_k)$ is an optimal solution for $J$. Then, $({\mathbf{e}}^*_1,\ldots,{\mathbf{e}}^*_k)$, where each $\mathbf{e}^*_i$ is the restriction of $\bfc^*_i$ on $\overline{Q}$, is a solution for $J'$. For each $i\in [k]$ and $\mathbf{x}\in X_i$,
\begin{eqnarray*}
d_H(\mathbf{x}|_{\overline{Q}}, \mathbf{e}^*_i) + d_{\mathbf{x}|_{\overline{Q}}}&=&d_H^{\overline{Q}}(\mathbf{x}, \bfc^*_i) + d_H^Q (\mathbf{x}, \mathbf{x}^{(1)}_i)\\
&\leq& d_H^{\overline{Q}}(\mathbf{x}, \bfc^*_i)+d_H^Q(\mathbf{x}, \bfc^*_i) + \frac{1}{r - 1} {\sf OPT}(J) \qquad\qquad(\mbox{By Lemma~\ref{lem:firstphase}})\\
&\leq& d_H(\mathbf{x}, \bfc^*_i) + \frac{1}{r - 1} {\sf OPT}(J) \\
&\leq& \left(1+ \frac{1}{r - 1}\right) {\sf OPT}(J)
\end{eqnarray*}
This completes the proof of the claim.
\end{proof}
Since $|\overline{Q}| \le rk \cdot {\sf OPT}(J)$ and by Claim~\ref{cliamoptcom}, we have that ${\sf OPT}(J')\geq \frac{|\overline{Q}|}{rk}=\frac{|\overline{Q}|}{c}$, where $c=rk$. Let $0<\delta<\frac{1}{c}$ be a number which we fix later.
Now we apply Lemma~\ref{lemma:largeopt} on the input $J'$ and $\delta$, and let $C'=(\bfc'_1$, \ldots, $\bfc'_k)$ be the solution for $J'$ obtained. We know that the cost $d'$ of $\bfc'$ is at most $(1+\delta){\sf OPT}(J')$ with probability at least $1-n^{-2}$.
For the rest of the proof we assume that the cost $d'\leq (1+\delta){\sf OPT}(J')$.
Recall that we have partially computed the entries of the solution $\bfc=(\bfc_1,\ldots,\bfc_k)$ for the instance $J$. That is, for each $j\in Q$ and $i\in [k]$, we have already set the value of $\bfc_i[j]$. Notice that $C'\subseteq \{0,1\}^{\vert \overline{Q}\vert}$. Since $J'$ is obtained from $J$ by restricting to $\overline{Q}$, there is a natural bijection $f$ from $\overline{Q}$ to $[\vert \overline{Q}\vert]$ such that for
each $\mathbf{x}\in X$ and $j\in {\overline{Q}}$, $\mathbf{x}[j]=\bfy[f(j)]$, where $\bfy=\mathbf{x}|_{\overline{Q}}$. Now for each $i\in [k]$ and $j\in \overline{Q}$, we set $\bfc_i[j]=\bfc'_i[f(j)]$.
In Claim~\ref{cliamoptcom}, we have proven that the solution $C$ of $J$ obtained in this way has cost at most $d'$.
By Lemma~\ref{lemma:largeopt}, we know that $d'\leq (1+\delta){\sf OPT}(J')$. By Claim~\ref{cliamoptcom}, ${\sf OPT}(J')\leq (1+\frac{1}{r-1}){\sf OPT}(J)$. Thus, we have that the cost of the solution $C$ of $J$ is at most $(1+\delta)(1+\frac{1}{r-1}){\sf OPT}(J)$. Now we fix $r=(1+\frac{4}{\epsilon})$ and $\delta=\frac{\epsilon}{(2\epsilon+8)k}$. Then the cost of $C$ is at most $(1+\epsilon){\sf OPT}(J)$.
\paragraph*{Running time analysis.}
The number of choices for $\{\mathbf{x}^{(1)}_1$, \ldots, $\mathbf{x}^{(r)}_1\} \subset X_1, \ldots, \{\mathbf{x}^{(1)}_k$, \ldots, $\mathbf{x}^{(r)}_k\} \subset X_k$ is at most $n^{{\cal O}(rk)}=n^{{\cal O}(k/\epsilon)}$. For each such choice, we run the algorithm of Lemma~\ref{lemma:largeopt} which takes time at most $m^{{\cal O}(1)}n^{{\cal O}(c^2k/\delta^2)}=m^{{\cal O}(1)}n^{{\cal O}((k/\epsilon)^4)}$. Thus, the total running time is $m^{{\cal O}(1)}n^{{\cal O}((k/\epsilon)^4)}$.
\end{proof}
Now the only piece left is the proof of Lemma~\ref{lemma:largeopt}. We use the following tail inequality (a variation of Chernoff bound) in the proof of Lemma~\ref{lemma:largeopt}.
\begin{proposition}[Lemma 1.2~\cite{LiMW02}]
\label{prop:chernoff}
Let $X_1,\ldots,X_n$ be $n$ independent $0$-$1$ random variables, $X=\sum_{i=1}^n X_i$, and $0<\epsilon\leq 1$.
Then, \(\Pr[X>E[X]+\epsilon n]\leq e^{-\frac{1}{3}n\epsilon^2}.\)
\end{proposition}
Finally, we prove Lemma~\ref{lemma:largeopt}.
\begin{proof}[Proof of Lemma~\ref{lemma:largeopt}]
First, assume that $m < 9 c^2 \log n/\delta^2$.
If this is the case, we enumerate all possible solutions for $J'$ and output the best solution. The number of solutions is
at most $2^{k\cdot m}=n^{{\cal O}(c^2 k/\delta^2)}$. Thus, in this case the algorithm is exact and deterministic, and the running time bound holds.
For the rest of the proof we assume that $m \ge 9 c^2 \log n/\delta^2$.
{\sc Binary Constrained Partition Center$^{\star}$}\xspace\ can be formulated as a $0$-$1$ optimization problem as explained below. For each $j\in [m]$ and tuple $t\in R_j$, we use a $0$-$1$ variable $y_{j,t}$ to indicate whether the $j^{th}$ entries of a solution form a tuple $t\in R_j$ or not. For any $i\in [k]$, $\mathbf{x}\in X_i$, $j\in [m]$ and $t\in R_j$, denote $\chi_i(\mathbf{x}[j],t)=0$ if $\mathbf{x}[j]=t[i]$ and
$\chi_i(\mathbf{x}[j],t)=1$ if $\mathbf{x}[j]\neq t[i]$. Now {\sc Binary Constrained Partition Center$^{\star}$}\xspace\ can be defined as the following $0$-$1$ optimization problem.
\begin{eqnarray}
&&\min d \nonumber\\
&&\mbox{subject to} \nonumber \\
&&\sum_{t\in R_j} y_{j,t}=1, \qquad\qquad\qquad\qquad\qquad\quad\; \mbox{for all } j\in [m] ; \label{eqn:obtprob}\\
&&\sum_{j\in [m]} \sum_{t\in R_j} \chi_i(\mathbf{x}[j],t)\cdot y_{j,t}\leq d-d_{\mathbf{x}}, \qquad \mbox{for all } i\in [k] \mbox{ and } \mathbf{x} \in X_i \nonumber\\
&&y_{j,t}\in \{0,1\}, \qquad\qquad\qquad\qquad\qquad\quad\; \mbox{for all } j\in [m] \mbox{ and } t\in R_j \nonumber.
\end{eqnarray}
Any solution $y_{j,t}$ ($j\in [m]$ and $t\in R_j$) to \eqref{eqn:obtprob} corresponds to the solution $C=(\bfc_1,\ldots,\bfc_k)$ where for all $j\in [m]$ and $t\in R_j$ such that $y_{j,t}=1$, we have $(\bfc_1[j],\ldots,\bfc_k[j])=t$.
Now, we solve the above optimization problem using linear programming relaxation and obtain a fractional solution
$y^{\star}_{j,t}$ ($j\in [m]$ and $t\in R_j$) with cost $d'$. Clearly, $d'\leq d_{opt}={\sf OPT}(J')$.
Now, for each $j\in [m]$, independently with probability $y^{\star}_{j,t}$, we set $y'_{j,t}=1$ and $y'_{j,t'}=0$, for any $t'\in R_j\setminus \{t\}$. Then $y'_{j,t}$ ($j\in [m]$ and $t\in R_j$) form a solution to \eqref{eqn:obtprob}. Next we construct the solution $C=(\bfc_1,\ldots,\bfc_k)$ to {\sc Binary Constrained Partition Center$^{\star}$}\xspace, corresponding to $y'_{j,t}$ ($j\in [m]$ and $t\in R_j$). That is, for all $j\in [m]$ and $t\in R_j$ such that $y_{j,t}=1$, we have $(\bfc_1[j],\ldots,\bfc_k[j])=t$.
For the running time analysis, notice that solving the linear program and performing the random rounding takes polynomial time in the size of the problem \eqref{eqn:obtprob}. And the size of \eqref{eqn:obtprob} is polynomial in the size of $J'$, so the running time bound is satisfied. It remains to show that the constructed solution has cost at most $(1 + \delta) {\sf OPT}(J')$ with probability at least $1 - n^{-2}$.
For any $j\in [m]$, the above random rounding procedure ensures that there is exactly one tuple $t\in R_j$
such that $y'_{j,t}=1$. This implies that for any $j\in [m]$, $i\in [k]$ and $\mathbf{x}\in X_i$, $\sum_{t\in R_j} \chi_i(\mathbf{x}[j],t)\cdot y'_{j,t}$ is a $0$-$1$ random variable. Since for each $j\in [m]$ the rounding procedure is independent, we have that for any $i\in [k]$ and $\mathbf{x}\in X_i$ the random variables $(\sum_{t\in R_1} \chi_i(\mathbf{x}[1],t)\cdot y'_{1,t}), \ldots, (\sum_{t\in R_m} \chi_i(\mathbf{x}[m],t)\cdot y'_{j,t})$ are independent. Hence, for any $i\in [k]$ and $\mathbf{x}\in X_i$, the Hamming distance between $\mathbf{x}$ and $\bfc_i$, $d_H(\mathbf{x},\bfc_i)=\sum_{j\in [m]}\sum_{t\in R_j} \chi_i(\mathbf{x}[j],t)\cdot y'_{j,t}$, is the sum of $m$ independent $0$-$1$ random variables. For each $i\in [k]$ and $\mathbf{x}\in X_i$, we upper bound the expected value of $d_H(\mathbf{x},\bfc_i)$ as follows.
\begin{eqnarray*}
E[d_H(\mathbf{x},\bfc_i)]&=&E\left[\sum_{j\in [m]}\sum_{t\in R_j} \chi_i(\mathbf{x}[j],t)\cdot y'_{j,t}\right]\\
&=&\sum_{j\in [m]}\sum_{t\in R_j} \chi_i(\mathbf{x}[j],t)\cdot E[y'_{j,t}]\\
&=&\sum_{j\in [m]}\sum_{t\in R_j} \chi_i(\mathbf{x}[j],t)\cdot y^{\star}_{j, t}\\
&\leq& d'-d_{\mathbf{x}} \qquad\qquad\qquad (\mbox{By the constraints of \eqref{eqn:obtprob}} )
\end{eqnarray*}
Fix $\epsilon=\frac{\delta}{c}$.
Then, by Proposition~\ref{prop:chernoff}, for all $i\in [k]$, and $\mathbf{x}\in X_i$,
\[
\Pr[d_H(\mathbf{x},\bfc_i)>d'-d_{\mathbf{x}}+\epsilon m] \leq e^{-\frac{1}{3}m\epsilon^2}.
\]
Therefore, by the union bound,
\begin{equation}
\label{eqn:unionboundstar}
\Pr[\mbox{There exist }i\in [k] \mbox{ and } \mathbf{x}\in X_i \mbox{ such that }d_H(\mathbf{x},\bfc_i)>d'-d_{\mathbf{x}}+\epsilon m] \leq n\cdot e^{-\frac{1}{3}m\epsilon^2}
\end{equation}
We remind that $m\geq 9c^2\log n/\delta^2 = 9\log n/\epsilon^2$ and so $n\cdot e^{-\frac{1}{3}m\epsilon^2} \le n^{-2}$. Thus, by \eqref{eqn:unionboundstar},
\begin{equation}
\label{eqn:case1}
\Pr[\mbox{There exist }i\in [k] \mbox{ and } \mathbf{x}\in X_i \mbox{ such that }d_H(\mathbf{x},\bfc_i)>d'-d_{\mathbf{x}}+\epsilon m] \leq n^{-2}.
\end{equation}
Since $d'\leq {\sf OPT}(J')$ and ${\sf OPT}(J')\geq m/c$, $d' + \epsilon m \le (1 + c \epsilon) {\sf OPT}(J')$. Then, the probability that
there exist $i\in [k]$ and $\mathbf{x}\in X_i$ such that $d_H(\mathbf{x},\bfc_i)>(1+c\epsilon){\sf OPT}(J')-d_{\mathbf{x}}$ is at most $n^{-2}$ by \eqref{eqn:case1}.
Since $c\epsilon=\delta$, the proof is complete.
\end{proof}
\section{Preliminaries}
| {
"timestamp": "2019-04-15T02:14:19",
"yymm": "1904",
"arxiv_id": "1904.06141",
"language": "en",
"url": "https://arxiv.org/abs/1904.06141",
"abstract": "We consider $\\ell_1$-Rank-$r$ Approximation over GF(2), where for a binary $m\\times n$ matrix ${\\bf A}$ and a positive integer $r$, one seeks a binary matrix ${\\bf B}$ of rank at most $r$, minimizing the column-sum norm $||{\\bf A} -{\\bf B}||_1$. We show that for every $\\varepsilon\\in (0, 1)$, there is a randomized $(1+\\varepsilon)$-approximation algorithm for $\\ell_1$-Rank-$r$ Approximation over GF(2) of running time $m^{O(1)}n^{O(2^{4r}\\cdot \\varepsilon^{-4})}$. This is the first polynomial time approximation scheme (PTAS) for this problem.",
"subjects": "Data Structures and Algorithms (cs.DS)",
"title": "Low-rank binary matrix approximation in column-sum norm",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9770226274137961,
"lm_q2_score": 0.72487026428967,
"lm_q1q2_score": 0.7082146501504262
} |
https://arxiv.org/abs/cs/0605007 | Systematic Topology Analysis and Generation Using Degree Correlations | We present a new, systematic approach for analyzing network topologies. We first introduce the dK-series of probability distributions specifying all degree correlations within d-sized subgraphs of a given graph G. Increasing values of d capture progressively more properties of G at the cost of more complex representation of the probability distribution. Using this series, we can quantitatively measure the distance between two graphs and construct random graphs that accurately reproduce virtually all metrics proposed in the literature. The nature of the dK-series implies that it will also capture any future metrics that may be proposed. Using our approach, we construct graphs for d=0,1,2,3 and demonstrate that these graphs reproduce, with increasing accuracy, important properties of measured and modeled Internet topologies. We find that the d=2 case is sufficient for most practical purposes, while d=3 essentially reconstructs the Internet AS- and router-level topologies exactly. We hope that a systematic method to analyze and synthesize topologies offers a significant improvement to the set of tools available to network topology and protocol researchers. |
\subsection{$dK$-graph-constructing algorithms}
\label{sec:dk-constructions}
We classify existing approaches to constructing $0K$- and $1K$-graphs
into the following categories: {\em stochastic}, {\em pseudograph},
{\em matching}, and two types of {\em rewiring}: {\em randomizing\/}
and {\em targeting}. We attempted to extend each of these techniques
to general $dK$-graph construction. In this section, we qualitatively
discuss the relative merits of each of these approaches before
presenting a more quantitative comparison in Section~\ref{sec:results}.
\subsubsection{Stochastic}
\label{sec:dk-stochastic}
The simplest and most convenient for theoretical analysis is the
stochastic approach. For $0K$, reproducing an $n$-sized graph with a
given expected average degree~$\bar{k}$ involves connecting every pair
of $n$~nodes with probability~\mbox{$p_{0K}=\bar{k}/n$}. This
construction forms the classical (Erd\H{o}s-R\'{e}nyi) random
graphs~$\mathcal{G}_{n,p}$~\cite{ErRe59}. Recent efforts have
extended this stochastic approach to $1K$~\cite{ChLu02} and
$2K$~\cite{BoPa03,dorogovtsev03}. In these cases, one
first labels all nodes~$i$ with their expected degrees~$q_i$ drawn
from the distribution~$P(k)$ and then connects pairs of nodes~$(i,j)$
with probabilities $p_{1K}(q_i,q_j) = q_iq_j/(n\bar{q})$ or
$p_{2K}(q_i,q_j) = (\bar{q}/n) P(q_i,q_j)/(P(q_i)P(q_j))$ reproducing
the expected values of~$1K$- or~$2K$-distributions,
respectively.
In theory, we could generalize this
approach for any~$d$ in two stages: 1)~{\em extraction}: given a
graph~$G$, calculate the frequencies of all (including disconnected)
$d$-sized subgraphs in $G$, and 2)~{\em construction}: prepare an
$n$-sized set of $q_i$-labeled nodes and connect their $d$-sized
subsets into different subgraphs with (conditional) probabilities based on the
calculated frequencies. In practice, we find the stochastic
approach performs poorly even for $1K$ because of high statistical
variance. For example, many nodes with expected degree~$1$ wind
up with degree~$0$ after the construction phase, resulting in
many tiny connected components.
\subsubsection{Pseudograph}
\label{sec:dk-pseudograph}
The pseudograph (also known as {\em configuration}) approach is
probably the most popular and widely used class of
graph-generating algorithms.
In its original form~\cite{AiChLu00,MolRee95}, it applies only to the
$1K$ case. Relative to the stochastic approach, it reproduces a given
degree distribution exactly, but does not necessarily construct
simple graphs. That is, it may construct graphs with both ends of
an edge connected to the same node (self-loops) and with multiple
edges between the same pair of nodes (loops).
It operates as follows. Given the number of nodes, $n(k)$, of degree $k$,
$n=\sum_{k=1}^{k_{\max}}n(k)$, first prepare $n(k)$~nodes with
$k$~stubs attached to each node, $k=1,\ldots,k_{\max}$, and then
randomly choose pairs of stubs and connect them to form edges. To
obtain a simple connected graph, remove all loops and extract the
largest connected component.
We extended this algorithm to $2K$ as follows. Given the
number~$m(k_1,k_2)$ of edges between $k_1$- and $k_2$-degree
nodes, $m=\sum_{k_1,k_2=1}^{k_{\max}}m(k_1,k_2)$, we first prepare
lists of~$m(k_1,k_2)$ disconnected edges and label the both ends of
each edge by~$k_1$ and~$k_2$, $k_1,k_2=1,\ldots,k_{\max}$.
Next, for each~$k$, $k=1,\ldots,k_{\max}$, we create a list of all
edge-ends labeled with~$k$. From this list, we randomly select groups
of $k$~edge-ends to form the $k$-degree nodes in the final graph.
The pseudograph algorithm works well for~\mbox{$d=2$}.
Unfortunately, we could not easily generalize it for~\mbox{$d>2$}
because starting at~\mbox{$d=3$},
$d$-sized subgraphs overlap over edges. Such overlapping introduces a series
of topological constraints and non-local dependencies among different
subgraphs, and we could not find a simple technique to
preserve these combinatorial constraints during the construction phase.
\subsubsection{Matching}
The matching approach differs from the pseudograph approach in
avoiding loops during the construction phase. In the $1K$ case,
the algorithm works exactly as its pseudograph counterpart but skips
pairs of stubs that form loops if connected.
We extend the matching approach to~$2K$ in a manner similar to our
$2K$~pseudograph approach.
Unfortunately, loop avoidance suffers from various forms of
deadlock for both $1K$ and $2K$. In both cases, the algorithms can end up in
incomplete configurations when not all edges are formed, and the
graph cannot be completed because there are no suitable stub pairs remaining
that can be connected without forming loops. We devised several
techniques to deal with these problems.
With these additional techniques, we obtained good results for $2K$ graphs.
As in the pseudograph case however, we could not generalize matching
for~\mbox{$d>2$} for essentially the same reasons
related to subgraphs' overlapping and non-locality.
\subsubsection{Rewiring}
\label{subsec:dk-rewiring}
The rewiring approaches are generalizable to any~$d$ and
work well in practice. They involve $dK$-preserving rewiring
as illustrated in Figure~\ref{fig:dk-rewiring}.
\begin{figure}
\centerline{\includegraphics[width=2.5in]{graphs/dk-rewiring.eps}}
\caption{\footnotesize \bf $dK$-preserving rewiring for~\mbox{$d=0,1,2$}.}
\label{fig:dk-rewiring}
\end{figure}
The main idea is to rewire {\em random\/} (pairs of) edges
preserving an existing form of the $dK$-distribution. For
\mbox{$d=0$}, we rewire a random edge to a random pair of nodes,
thus preserving~$\bar{k}$. For~\mbox{$d=1$}, we rewire two random
edges that do not alter~$P(k)$, as shown in Figure~\ref{fig:dk-rewiring}. If, in
addition, there are at least two nodes of equal degrees adjacent to the
different edges in the edge pair, then the same rewiring
leaves~$P(k,k')$ intact. Due to the inclusion property of the
$dK$-series, \mbox{$(d+1)K$}-rewirings form a subset
of~\mbox{$dK$}-rewirings for~\mbox{$d>0$}. For example, to
preserve~$3K$, we permit a $2K$-rewiring only if it also preserves the
wedge and triangle distributions.
The {\em $dK$-randomizing rewiring algorithm\/} amounts to performing
$dK$-preserving rewirings a sufficient number of times for some
$dK$-graph.
A ``sufficient number'' means enough rewirings for this process to lead to
graphs that do not change their properties even if we subject them to
additional rewirings.
In other words, this rewiring process {\em converges}
after some number of steps, producing
random graphs having property~$\mathcal{P}_d$.
Even for~\mbox{$d=1$}, there are no known rigorous results
regarding how quickly this process converges, but~\cite{GkaMiZe03a}
shows that this process is an irreducible, symmetric and aperiodic
Markov chain and demonstrates experimentally that it takes $O(m)$~steps
to converge.
In our
experiments in Section~\ref{sec:results}, we employ the following
strategy applicable for any~$d$. We first calculate the number of
possible initial $dK$-preserving rewirings. By ``initial rewirings'' we
mean rewirings we can perform on a given graph~$G$, to differentiate
them from rewirings we can apply to graphs obtained from~$G$ after its
first (and subsequent) rewirings. We then subtract the number of
rewirings that leave the graph isomorphic. For example, rewiring of
any two \mbox{$(1,k)$}- and \mbox{$(1,k')$}-edges is a $dK$-preserving
rewiring, for any~$d$, and more strongly, the graph before rewiring is
isomorphic to the graph after rewiring. We multiply this difference by~$10$,
and perform that number of random rewirings. At
the end of our rewiring procedure, we explicitly verify that
randomization is indeed complete and the process has converged by further increasing the number of rewirings and checking that
all graph characteristics remain unchanged.
One obvious problem with $dK$-randomization
is that it requires an original graph~$G$ as input to construct its $dK$-random versions.
It cannot start with a description of the $dK$-distribution
to generate random $dK$-graphs as is possible with the
other construction approaches discussed above.
To address this limitation, we consider the inverse process of {\em
$dK$-targeting $d'K$-preserving rewiring}, also known as {\em
Metropolis dynamics\/}~\cite{Metro53}. It incorporates the
following modification to $d'K$-preserving rewiring: every rewiring
step is accepted only if it moves the graph ``closer''
to~$\mathcal{P}_{d}$.
In practice, we can then employ targeting rewiring to construct
$dK$-graphs with high values of~$d$ by beginning with any $d'K$-graph
where~\mbox{$d'<d$}. Recall that we can always compute~$\mathcal{P}_{d'}$
from~$\mathcal{P}_{d}$ due to the inclusion
property of the $dK$-series. For instance, we can start with a graph
having a given degree distribution
(\mbox{$d'=1$})~\cite{ViLa05-cocoon}, and then move it toward
a~$dK$-graph via $dK$-targeting $1K$-preserving rewiring.
The definition of ``closer'' above requires further explanation.
We require a set of distance metrics that
quantitatively differentiate two graphs based on the values
of their $dK$-distributions.
In our experiments, we use the sum of squares of differences
between the existing and target numbers of subgraphs of a given
type. For example, in the \mbox{$d=2$}~case, we measure the distance
between the target graph's JDD and the JDD of the current graph being rewired by $\mathcal{D}_2=\sum_{k_1,k_2}
\left[m_{\mathrm{current}}(k_1,k_2)-m_{\mathrm{target}}(k_1,k_2)\right]^2$,
and at each rewiring step, we accept the rewiring only if it decreases
this distance. Note that $\mathcal{D}_2$ is non-negative and equals
zero only when reaching the target JDD. For \mbox{$d=3$}, this distance~$\mathcal{D}_3$
is a sum of squares of differences between the current and target numbers of
wedges and triangles, and we can generalize it to~$\mathcal{D}_d$ for any~$d$.
A potential problem with $dK$-targeting
$d'K$-preserving rewiring is that it can be nonergodic, meaning that
there might be no chain of $d'K$-preserving $\mathcal{D}_d$-decreasing rewirings leading from the
initial $d'K$-graph to the target $dK$-graph. In other
words, we cannot be sure beforehand that any two $d'K$-graphs are connected by
a sequence of $d'K$-preserving and $\mathcal{D}_d$-decreasing rewirings.
To address this problem we
note that the $d'K$-randomizing and $dK$-targeting $d'K$-preserving
rewirings are actually two extremes of an entire family of rewiring
processes. Indeed, let \mbox{$\Delta \mathcal{D}_d =
\mathcal{D}_{d,\mathrm{after}}-\mathcal{D}_{d,\mathrm{before}}$} be
the difference of distance to the target $dK$-distribution computed
before and after a $d'K$-preserving rewiring step. As with the usual
$dK$-targeting rewiring, we accept a rewiring step if~\mbox{$\Delta
\mathcal{D}_d < 0$}, but even if~\mbox{$\Delta \mathcal{D}_d \geq 0$},
we also accept this step with probability $e^{-\Delta\mathcal{D}_d/T}$,
where~$T>0$ is some parameter that we call {\em
temperature\/} because of the similarity of the process to
simulated annealing.
In the \mbox{$T \to 0$}~limit, this probability goes to~$0$, and we
have the standard $dK$-targeting $d'K$-preserving rewiring
process. When~\mbox{$T \to \infty$}, the probability approaches~$1$,
yielding the standard $d'K$-randomizing rewiring process. To verify
ergodicity, we can start with a high temperature and then gradually
cool the system while monitoring any metric known to have different
values in $dK$- and $d'K$-graphs. If this metric's value forms a
continuous function of the temperature, then our rewiring process is
ergodic. Maslov {\it et al.}~performed these experiments
in~\cite{MaSneZa04} and demonstrated ergodicity in the case with
\mbox{$d'=1$} and \mbox{$d=2$}. In our experiments in
Section~\ref{sec:results} where~$d$ and~$d'$ are below~$4$,
we always obtain a good match for all target graph metrics.
Thus, we perform rewiring at zero temperature without
further considering ergodicity. If however in some future experiments
one detects the lack of a smooth convergence of rewiring procedures,
then one should first verify ergodicity using the methodology described above.\\
For all the algorithms discussed in this section, we do not check for graph
connectedness at
each step of the algorithm
since: 1)~it is an expensive operation
and 2)~all resulting graphs always have giant connected components~(GCCs)
with characteristics similar to the whole disconnected graphs.
\subsection{$dK$-random graphs}
\label{sec:dk-random-graphs}
\begin{table*}[tb]
\begin{centering}
\caption{The summary of $dK$-series.
\label{tab:dk}
}
\begin{tabular}{|p{.15in}|p{.45in}|p{.73in}|p{1.02in}|p{1.65in}|p{2.05in}|}\hline
Tag $dK$ & Property symbol & $dK$-distribution & $\mathcal{P}_d$ defines $\mathcal{P}_{d-1}$ &
Edge existence probability in stochastic constructions
& Maximum entropy value of $(d+1)K$-distribution in $dK$-random graphs\\
\hline $0K$ & $\mathcal{P}_0$ & $\bar{k}$ & & $p_{0K}=\bar{k}/n$
& $P_{0K}(k) = e^{-\bar{k}}\bar{k}^k/k!$\\
\hline $1K$ & $\mathcal{P}_1$ & $P(k)$ & $\bar{k}=\sum kP(k)$ & $p_{1K}(q_1,q_2)
= q_1q_2/(n\bar{q})$
& $P_{1K}(k_1,k_2)=k_1P(k_1)k_2P(k_2)/\bar{k}^2$\\
\hline $2K$ & $\mathcal{P}_2$ & $P(k_1,k_2)$ & $P(k) =
(\bar{k}/k)\sum_{k'}P(k,k')$ & $p_{2K}(q_1,q_2) = (\bar{q}/n)
P(q_1,q_2)/(P(q_1)P(q_2))$ & See~\cite{dorogovtsev04} for clustering in $2K$-random graphs \\
\hline $3K$ & $\mathcal{P}_3$ & $P_\wedge(k_1,k_2,k_3)$ $P_\triangle(k_1,k_2,k_3)$ &
\multicolumn{3}{l|}{\parbox[t]{4.72in}{
By counting edges, we get
$P(k_1,k_2) \sim \sum_k \left\{ P_\wedge(k,k_1,k_2) + P_\triangle(k,k_1,k_2) \right\} / (k_1-1)
\sim \sum_k \left\{ P_\wedge(k_1,k_2,k) + P_\triangle(k_1,k_2,k) \right\} / (k_2-1)$,
where we omit normalization coefficients.}}\\
\hline
\ldots&\ldots&\ldots&\ldots&\ldots&\ldots\\
\hline
$nK$ & $\mathcal{P}_n$ & $G$ & & &\\
\hline
\end{tabular}
\end{centering}
\end{table*}
No $dK$-graph-generating algorithm can quickly construct the set of
{\em all\/} $dK$-graphs because: 1)~such sets are too large,
especially for small~$d$; and, less obviously, 2)~all algorithms try
to produce graphs having property~$\mathcal{P}_d$ while remaining {\em
unbiased\/} (random) with respect to all other properties. One can
check directly that the last characteristic applies to all the
algorithms we have discussed above.
As a consequence, the $dK$-graph construction algorithms
result in non-uniform sampling of graphs with different values of properties
that are not fully defined by~$\mathcal{P}_d$. More specifically, two generated $dK$-graphs
having different forms of a $d'K$-distribution with~\mbox{$d'>d$}
can appear as the output of these
algorithms with drastically different probabilities. Some $dK$-graphs
have such a small probability of being constructed
that we can safely assume they never arise.
For example, consider the simplest $0K$ stochastic
construction, i.e., the classical random
graphs~$\mathcal{G}_{n,p}$. Using a probabilistic argument, one can
show that the naturally-occurring $1K$-distribution~(degree distribution)
in these gra\-phs has a specific form: binomial, which is
closely approximated by the Poisson distribution: \mbox{$P_{0K}(k) =
e^{-\bar{k}}\bar{k}^k/k!$}~\cite{DorMen-book03}.
The $0K$ stochastic algorithm may produce a graph with a different
$1K$-distribution, e.g., the
power-law~\mbox{$P(k) \sim k^{-\gamma}$}, but the probability of such
an outcome is extremely low.
Indeed, suppose \mbox{$n \sim 10^4$},
\mbox{$\bar{k} \sim 5$}, and \mbox{$\gamma \sim 2.1$}, so
that the characteristic maximum degree is \mbox{$k_{\max} \sim 2000$}
(we chose these values to reflect measured values for Internet AS
topologies).
In this case, the probability that a
$\mathcal{G}_{n,p}$-graph contains at least one node with degree equal
to~$k_{\max}$ is dominated by \mbox{$1/2000! \sim 10^{-6600}$}, and
the probability that the remaining degrees simultaneously match those
required for a power law is much lower.
It is thus natural to introduce a set of graphs that correspond
to the graphs most likely to be generated by $dK$-graph constructing
algorithms.
We call such graphs the {\em $dK$-random graphs}.
These graphs have property~$\mathcal{P}_d$ but
are unbiased with respect to any other more constraining property.
In this sense, the
$dK$-random graphs are the {\em maximally random\/} or {\em
maximum-entropy\/} $dK$-graphs. Our term {\em maximum entropy\/}
here has the following justification. As we have just seen,
$0K$-random graphs have the maximum-entropy value of
the $1K$-distribution since their node degree distribution is the
distribution with the maximum entropy among all the distributions with
a fixed average.\footnote{The entropy of a discrete
distribution~$P(x)$ is $H[P(x)]=-\sum_x P(x) \log P(x)$. If the sample
space is also finite, then among all the distributions with a fixed average,
the binomial distribution maximizes entropy~\cite{harrem01}.}
The $1K$-random graphs have the
maximum-entropy value of the $2K$-distribution since their joint
degree distribution,
\mbox{$P_{1K}(k_1,k_2)=\tilde{P}(k_1)\tilde{P}(k_2)$}, where
\mbox{$\tilde{P}(k)=kP(k)/\bar{k}$}~\cite{DorMen-book03}, is the distribution with the
maximum joint entropy (minimum mutual information)\footnote{The mutual
information of a joint distribution~$P(x,y)$ is
$I[P(x,y)]=H[P(x)]+H[P(y)]-H[P(x,y)]$, where $P(x)$ and $P(y)$ are
the marginal distributions.} among all the joint distributions with
fixed marginal distributions.\footnote{In reality, the last statement
generally applies only to the class of all (not necessarily connected)
pseudographs. Narrowing the class of graphs to simple connected
graphs introduces topological constraints affecting the
maximum-entropy form of the $2K$-distribution.}
The main point we extract from these observations is that in trying to
construct $dK$-graphs, we generally obtain graphs from subsets
of the $dK$-space.
We call these subsets $dK$-random graphs and
schematically depict them as centers of the $dK$-circles in
Figure~\ref{fig:dk}. These graphs do have property~$\mathcal{P}_d$ and,
consequently, properties~$\mathcal{P}_i$ with \mbox{$i<d$},
but they might not ever display property~$\mathcal{P}_j$ with \mbox{$j>d$}
since their $jK$-distributions have specific, maximum-entropy values
that may be different from the $jK$-distribution values in the original graph.
\subsection{$dK$-space explorations}
\label{sec:dk-exploraitons}
Often we wish to analyze the topological constraints
a given graph~$G$ appears to obey.
In other cases, we are interested in
exploring the structural diversity among $dK$-graphs.
If we are attempting to determine the minimum~$d$ such that
all $dK$-graphs are similar to~$G$,
we can start with a small value of~$d$, generate $dK$-graphs, and
measure their ``distance'' from~$G$.
If the distance is too great, we can increase~$d$ and repeat
the process.
On the other hand, to explore structural diversity
among all $dK$-graphs, we must generate $dK$-graphs that are not random.
These non-random $dK$-graphs are still constrained by~$\mathcal{P}_d$ but
have extremely low probabilities of being generated by unperturbed
$dK$-graph constructing algorithms.
We cannot construct all $dK$-graphs, so we need to use
heuristics to generate some $dK$-graphs and adjust them
according to a distance metric that draws us closer to
the types of $dK$-graphs we seek. One such
heuristic is based on the inclusion feature of the
$dK$-series. Because all $dK$-graphs have the same values
of $dK$- but not of $(d+1)K$-distributions, we look for simple metrics
fully defined by~$\mathcal{P}_{d+1}$ but not by~$\mathcal{P}_d$.
While identifying such metrics can be
challenging for high~$d$'s, we can always retreat to the following two
simple extreme metrics:
\begin{itemize}
\item the correlation of degrees of nodes located at distance~\mbox{$d$};
\item the concentration of $d$-simplices (cliques of size~\mbox{$d+1$}).
\end{itemize}
These metrics are ``extreme'' in the sense that they correspond
to the \mbox{$(d+1)$}-sized subgraphs with, respectively,
the maximum~($d$) and minimum~($1$) possible diameter.
We can then construct $dK$-graphs with extreme values,
e.g., the smallest or largest possible, for these (extreme) metrics. The
$dK$-random graphs have the values of these metrics lying somewhere in between the extremes.
If the goal is to find the smallest~$d$ that results in sufficiently
constraining graphs, we can compute the difference between the extreme values of
these metrics, as well as of other metrics we might consider. If this
difference is too large, then the selected value of~$d$ is not
constraining enough and we need to increase it.
If the goal is to visit exotic locations in a large space of $dK$-graphs, then
such $dK$-space exploration may be used to move beyond the relatively
small circle of $dK$-random graphs.
To illustrate this approach in practice, we consider $1K$-
and $2K$-space explorations. For $1K$, the simplest metric defined
by~$\mathcal{P}_2$ is any scalar summary statistics of the $2K$-distribution,
such as likelihood~$S$ (cf.~Section~\ref{sec:metrics}). To construct graphs
with the maximum value of~$S$, we can run a form of targeting
$1K$-preserving rewiring that accepts each rewiring step only if it
increases~$S$. We can perform the opposite to minimize~$S$. This type
of experiment was at the core of recent work that led the authors
of~\cite{LiAlWiDo04} to conclude that \mbox{$d=1$} was not
constraining enough for the topology they considered.
To perform $2K$-space explorations, we need to find simple scalar
metrics defined by~$\mathcal{P}_3$. Since the $3K$-distribution is
actually two distributions, $P_\wedge(k_1,k_2,k_3)$ and
$P_\triangle(k_1,k_2,k_3)$, we should have two independent scalar
metrics. The {\em second-order likelihood}~$S_2$ is one such metric
for~$P_\wedge(k_1,k_2,k_3)$. We define it as the sum of the products
of degrees of nodes located at the ends of wedges,
$S_2 \sim \sum_{k_1,k_2,k_3} k_1k_3P_\wedge(k_1,k_2,k_3)$.
Any graphs with the same~$P_\wedge(k_1,k_2,k_3)$ have the same~$S_2$.
For the $P_\triangle(k_1,k_2,k_3)$ component, average clustering
$\bar{C} \sim \sum_{k_1,k_2,k_3}k_1P_\triangle(k_1,k_2,k_3)$ is an
appropriate candidate. We note that these two metrics are also
the two extreme metrics in the sense defined above: $S_2$~measures the
properly normalized correlation of degrees of nodes located at distance~$2$,
while $\bar{C}$~describes the concentration of $2$-simplices (triangles).
The $2K$-explorations amount then to performing
the following two types of targeting~$2K$-preserv\-ing rewiring: accept a $2K$-rewiring
step only if it maximizes or minimizes: 1)~$S_2$, or 2)~$\bar{C}$.
\section{Introduction}
\label{sec:intro}
\input{intro}
\section{Important Topology Metrics}
\label{sec:metrics}
\input{metrics}
\section{{\it\rm\large\lowercase{d}K}-series and {\it\rm\large\lowercase{d}K}-graphs}
\label{sec:degcor}
\input{degcor}
\section{Constructing {\it\rm\large\lowercase{d}K}-graphs}
\label{sec:methodology}
\input{methodology}
\section{Evaluation}
\label{sec:results}
\input{results}
\section{Discussion and Future Work}
\label{sec:discussion}
\input{discussion}
\section{Conclusions}
\label{sec:conclusion}
\input{conclusion}
\section*{Acknowledgements}
\input{ack}
\bibliographystyle{abbrv}
\vspace*{0.05in}
\scriptsize
\subsection{Algorithmic Comparison}
\label{subsec:algo-comp}
\begin{figure*}[tbh]
\centerline{
\subfigure[Clustering in skitter for different $2K$~algorithms]
{\includegraphics[width=2.1in]{graphs/skitter_allalgos_clus.eps}
\label{fig:sk-all-clus} }
\hfill
\hfill
\subfigure[Distance distribution in HOT for different $2K$~algorithms]
{\includegraphics[width=2.1in]{graphs/hot_allalgos_dist.eps}
\label{fig:hot-all-dist} }
\hfill
\subfigure[Distance distribution in HOT for different $3K$~algorithms]
{\includegraphics[width=2.1in]{graphs/hot_3k_dist.eps}
\label{fig:3k-dist-hot} }
}
\caption{\footnotesize {\bf Comparison of $2K$- and $3K$-graph-constructing algorithms.} }
\label{fig:comp-all-algos}
\end{figure*}
\begin{table}
\begin{center}
\caption{\footnotesize{\bf Scalar metrics for $2K$-random HOT graphs generated using different techniques.}}
\begin{tabular}{|l|l|l|l|l|l|l|}
\hline
Met- & Stoch- & Pseu- & Match- & $2K$- & $2K$- & Orig. \\
ric & astic & dogr. & ing & rand. & targ. & HOT \\
\hline
$\bar{k}$ & 2.87 & 2.19 & 2.22 & 2.18 & 2.18 & 2.10 \\
$r$ & -0.22 & -0.24 & -0.21 & -0.23 & -0.24 & -0.22 \\
$\bar{d}$ & 4.99 & 6.25 & 6.22 & 6.32 & 6.35 & 6.81 \\
$\sigma_{d}$ & 0.85 & 0.75 & 0.74 & 0.70 & 0.70 & 0.57 \\
\hline
\end{tabular}
\label{table:all-algos-hot}
\end{center}
\end{table}
\begin{table}
\caption{\footnotesize{\bf Scalar metrics for $3K$-random HOT graphs generated using different techniques.} }
\begin{tabular}{|l|l|l|l|}
\hline
Metric & $3K$-randomizing & $3K$-targeting & Original \\
& rewiring & rewiring & HOT \\
\hline
$\bar{k}$ & 2.10 & 2.13 & 2.10 \\
$r$ & -0.22 & -0.23 & -0.22 \\
$\bar{d}$ & 6.55 & 6.79 & 6.81 \\
$\sigma_{d}$ & 0.84 & 0.72 & 0.57 \\
\hline
\end{tabular}
\label{table:3k-algos-hot}
\end{table}
We first compare the different graph generation algorithms discussed in
Section~\ref{sec:dk-constructions}.
All the algorithms give consistent results,
except the stochastic approach, which suffers from the problems related to high statistical variance
discussed in Section~\ref{sec:dk-stochastic}. This conclusion immediately follows from
Figure~\ref{fig:comp-all-algos} and Tables~\ref{table:all-algos-hot} and \ref{table:3k-algos-hot}
showing graph metric values for the different $2K$ and $3K$~algorithms
described in Section~\ref{sec:dk-constructions}.
In our experience, we find that $dK$-randomizing rewiring is easiest to use.
However, it requires the original graph as input. If
only the target $dK$-distribution is
available and if~\mbox{$d \leq 2$}, we find the pseudograph algorithm most appropriate
in practice. We note that our $2K$ version results in fewer pseudograph ``badnesses'',
i.e., (self-)loops and small connected components~(CCs),
than PLRG~\cite{AiChLu00}, its commonly-known $1K$ counterpart.
This improvement is due to the additional constraints introduced by the $2K$ case.
For example, if there is only one node of high degree~$x$ and one node of
another high degree~$y$ in the original graph, then there can be only one
link of type~\mbox{$(x,y)$}. Our $2K$~modification of the pseudograph algorithm
must consequently produce exactly one link between these two $x$- and $y$-degree nodes,
whereas in the $1K$~case, the algorithm tends to create many such links.
Similarly, a $1K$ generator tends to produce many pairs of isolated
$1$-degree nodes connected to each another.
Since the original graph does not have such pairs, i.e., \mbox{(1,1)}-links,
our $2K$~generator, as opposed to~$1K$, does not form these small $2$-node CCs either.
While the pseudograph algorithm is a good $2K$-random graph
generator, we could not generalize it for~\mbox{$d \geq 3$} (see Section~\ref{sec:dk-pseudograph}). Therefore, to generate
$dK$-random graphs with $d \geq 3$ when an original graph is
unavailable, we use $dK$-targeting rewiring. We first bootstrap
the process by constructing $1K$-random graphs using the
pseudograph algorithm and then apply $2K$-targeting
$1K$-preserving rewiring to obtain $2K$-random graphs. To produce
$3K$-random graphs, we apply $3K$-targeting $2K$-preserving
rewiring to the $2K$-random graphs obtained at the previous step.
\subsection{Topology Comparisons}
We next test the convergence of our $dK$-series for the skitter and HOT graphs.
Since all $dK$-graph constructing algorithms yield consistent results, we selected
the simplest one, the $dK$-randomizing rewiring from Section~\ref{subsec:dk-rewiring},
to obtain $dK$-random graphs in this section.
The number of possible initial $dK$-randomizing rewirings is a good preliminary
indicator of the size of the $dK$-graph space.
We show these numbers for the HOT graph in Table~\ref{table:hot-initial-rewirings}.
If we discard rewirings leading to obvious isomorphic graphs,
cf.~Section~\ref{subsec:dk-rewiring},
then the number of possible initial rewirings is even smaller.
\begin{table}
\caption{\footnotesize{\bf Numbers of possible initial $dK$-randomizing rewirings for the HOT graph.} }
\begin{tabular}{|p{0.25in}|l|l|}
\hline
$d$ & Possible initial & Possible initial rewirings,\\
& rewirings & ignoring obvious isomorphisms\\
\hline
0 & 435,546,699 & - \\
1 & 477,905 & 440,355 \\
2 & 326,409 & 268,871 \\
3 & 146 & 44 \\
\hline
\end{tabular}
\label{table:hot-initial-rewirings}
\end{table}
\begin{figure*}[tbh]
\centerline{
\subfigure[Distance distribution]
{\includegraphics[width=2.1in]{graphs/skitter_dist.eps}
\label{fig:sk-dist} }
\hfill
\subfigure[Betweenness]
{\includegraphics[width=2.1in]{graphs/skitter_bet.eps}
\label{fig:sk-bet} }
\hfill
\subfigure[Clustering]
{\includegraphics[width=2.1in]{graphs/skitter_clus.eps}
\label{fig:sk-clus} }
}
\caption{\footnotesize \bf Comparison of $dK$-random and skitter graphs.}
\label{fig:sk-met}
\end{figure*}
\begin{table}
\caption{\footnotesize{\bf Comparing scalar metrics for $dK$-random and skitter graphs.}}
\begin{tabular}{|p{0.4in}|p{0.39in}|p{0.39in}|p{0.39in}|p{0.4in}|p{0.4in}|}
\hline
Metric & $0K$ & $1K$ & $2K$ & $3K$ & skitter \\
\hline
$\bar{k}$ & 6.31 & 6.34 & 6.29 & 6.29 & 6.29 \\
$r$ & 0 & -0.24 & -0.24 & -0.24 & -0.24\\
$\bar{C}$ & 0.001 & 0.25 & 0.29 & 0.46 & 0.46\\
$\bar{d}$ & 5.17 & 3.11 & 3.08 & 3.09 & 3.12 \\
$\sigma_{d}$ & 0.27 & 0.4 & 0.35 & 0.35 & 0.37 \\
$\lambda_{1}$ & 0.2 & 0.03 & 0.15 & 0.1 & 0.1\\
$\lambda_{n-1}$ & 1.8 & 1.97 & 1.85 & 1.9 & 1.9 \\
\hline
\end{tabular}
\label{table:skitter}
\end{table}
We compare the skitter topology with its $dK$-random counterparts,
\mbox{$d=0,\ldots,3$},
in Table~\ref{table:skitter} and Figure~\ref{fig:sk-met}.
We report all the metrics calculated for the giant connected component~(GCCs).
Minor discrepancies between values of average degree~$\bar{k}$
and~$r$ result from GCC extractions. If we do not extract the
GCC, then~$\bar{k}$ is the
same as that of the original graph for all~\mbox{$d=0,\ldots,3$}, and~$r$ is exactly the same for~\mbox{$d>1$}.
We do not show degree distributions for brevity. However, degree distributions match
when considering the entire graph and are very similar for the GCCs for all~\mbox{$d>0$}.
When $d=0$, the degree distribution is binomial, as expected.
We see that all other metrics gradually converge to those in the original graph as~$d$ increases. A value of
$d=1$ provides a reasonably good description of the skitter topology, while $d=2$
matches all properties except clustering. The $3K$-random graphs are identical to the original
for all metrics we consider, including clustering.
\begin{figure*}[tbh]
\begin{minipage}[t]{2in}
\centerline{
\includegraphics[width=2in]{graphs/skitter_varyclus.eps}
}
\caption{\footnotesize \bf Varying clustering in $2K$-graphs for skitter.
}
\label{fig:sk-vary-clus}
\end{minipage}
\hfill
\begin{minipage}[t]{2in}
\centerline{
\includegraphics[width=2in]{graphs/hot_dist.eps}
}
\caption{\footnotesize \bf Distance distribution for $dK$-random and HOT graphs
}
\label{fig:hot-dist}
\end{minipage}
\hfill
\begin{minipage}[t]{2in}
\centerline{
\includegraphics[width=2in]{graphs/hot_bet.eps}
}
\caption{\footnotesize \bf Betweenness for $dK$-random and HOT graphs
}
\label{fig:hot-bet}
\end{minipage}
\end{figure*}
We perform the $2K$-space explorations described in Section~\ref{sec:dk-exploraitons}
to check if $d=2$ is indeed sufficiently constraining for the skitter topology. We observe small variations of
clustering~$\bar{C}$, second-order likelihood~$S_2$, and spectrum, shown
in Table~\ref{table:sk-exploration}
and Figure~\ref{fig:sk-vary-clus}. All other metrics do not change,
so we do not
show plots for them. We conclude that $d=2$, i.e., the joint degree distribution
provides a reasonably accurate description of observed AS-level topologies.
\begin{table}
\begin{center}
\caption{ \footnotesize \bf{Scalar metrics for $2K$-space explorations for skitter.}}
\begin{tabular}{|l|l|l|l|l|l|l|}
\hline
Metric & Min & Max & Min & Max & $2K$- & Skit-\\
& $\bar{C}$ & $\bar{C}$ & $S_2$ & $S_2$ & rand. & ter \\
\hline
$\bar{k}$ &6.29 & 6.29 & 6.29 & 6.29 & 6.29 & 6.29 \\
$r$ & -0.24 & -0.24 & -0.24 & -0.24 & -0.24 & -0.24\\
$\bar{C}$ & 0.21 & 0.47 & 0.4 & 0.4 & 0.29 & 0.46 \\
$\bar{d}$ & 3.06 & 3.12 & 3.12 & 3.10 & 3.08 & 3.12 \\
$\sigma_{d}$ & 0.33 & 0.38 & 0.37 & 0.36 & 0.35 & 0.37 \\
$\lambda_{1}$ & 0.25 & 0.11 & 0.11 &0.1 & 0.15 & 0.1\\
$\lambda_{n-1}$ & 1.75 & 1.89 & 1.89 &1.89 & 1.85 & 1.9 \\
$S_{2}/S_{2}^{\max}$ & 0.988 & 0.961 & 0.955 & 1.000 & 0.986 & 0.958\\
\hline
\end{tabular}
\label{table:sk-exploration}
\end{center}
\end{table}
The HOT topology is
more complex than AS-level topologies.
Earlier work argues that
this topology cannot be accurately modeled using degree
distributions alone~\cite{LiAlWiDo04}.
We therefore selected the HOT topology graph
as a difficult case for our approach.
A preliminary inspection of visualizations in Figure~\ref{fig:hot-cool-pic} indicates
that the $dK$-series converges at a reasonable rate even for the HOT graph.
The $0K$-random graph is a classical random graph and lacks high-degree nodes,
as expected.
The $1K$-random graph has all the high-degree nodes we desire,
but they are crowded toward the core,
a property absent in the HOT graph. The $2K$~constraints start pushing
the high-degree nodes away to the periphery, while the lower-degree nodes
migrate to the core, and the $2K$-random graph
begins to resemble the HOT graph.
The $3K$-random topology looks
remarkably similar to the HOT topology.
Of course, visual inspection of a small number of randomly generated graphs
is insufficient to demonstrate our ability to capture important metrics
of the HOT graph. Thus, we compute the different metric values for each of the $dK$-random
graph and compare them with the corresponding value
for the original HOT graph.
In Table~\ref{table:hot}
and Figures~\ref{fig:hot-dist} and~\ref{fig:hot-bet} we see that the $dK$-series converges more slowly for
HOT than for skitter. Note that we do not show clustering plots because
clustering is almost zero everywhere:
the HOT topology has very few cycles; it is almost a tree.
The $1K$-random graphs yield a poor approximation
of the original topology, in agreement with the main argument in~\cite{LiAlWiDo04}.
Both Figures~\ref{fig:hot-cool-pic}
and~\ref{fig:hot-bet} indicate that starting with \mbox{$d=2$}, low- but not high-degree
nodes form the core: betweenness is approximately as high for nodes of
degree $\sim 10$ as for high-degree nodes.
Consequently, the $2K$-random graphs provide a better approximation,
but not nearly as good as it was for skitter.\footnote{The speed of $dK$-series
convergence depends both on the structure and size of an original graph.
It must converge faster for smaller input graphs of similar structure. However, here we see that
the graph structure plays a more crucial role than its size. The $dK$-series
converges slower for HOT than for skitter, even though the former graph is an order
of magnitude smaller than the latter.}
However, the $3K$-random graphs match the original HOT topology
{\em almost exactly}.
We thus conclude that the $dK$-series captures the
essential characteristics of
even particularly difficult topologies, such as HOT,
by sufficiently increasing~$d$,
in this case to~3.
\begin{table}
\caption{\footnotesize {\bf Comparing scalar metrics for $dK$-random and HOT graphs.}}
\begin{tabular}{|p{0.4in}|p{0.4in}|p{0.4in}|p{0.4in}|p{0.4in}|p{0.4in}|}
\hline
Metric & $0K$ & $1K$ & $2K$ & $3K$ & HOT \\
\hline
$\bar{k}$ & 2.47& 2.59 & 2.18 & 2.10 & 2.10\\
$r$ & -0.05 & -0.14 & -0.23 & -0.22 & -0.22 \\
$\bar{C}$ & 0.002& 0.009& 0.001& 0 & 0 \\
$\bar{d}$ & 8.48 & 4.41 & 6.32 & 6.55 & 6.81\\
$\sigma_{d}$ & 1.23 & 0.72 & 0.71 & 0.84 & 0.57\\
$\lambda_{1}$ & 0.01 & 0.034 & 0.005 & 0.004 & 0.004\\
$\lambda_{n-1}$ & 1.989 & 1.967 & 1.996 & 1.997 & 1.997\\
\hline
\end{tabular}
\label{table:hot}
\end{table}
| {
"timestamp": "2006-07-29T07:26:59",
"yymm": "0605",
"arxiv_id": "cs/0605007",
"language": "en",
"url": "https://arxiv.org/abs/cs/0605007",
"abstract": "We present a new, systematic approach for analyzing network topologies. We first introduce the dK-series of probability distributions specifying all degree correlations within d-sized subgraphs of a given graph G. Increasing values of d capture progressively more properties of G at the cost of more complex representation of the probability distribution. Using this series, we can quantitatively measure the distance between two graphs and construct random graphs that accurately reproduce virtually all metrics proposed in the literature. The nature of the dK-series implies that it will also capture any future metrics that may be proposed. Using our approach, we construct graphs for d=0,1,2,3 and demonstrate that these graphs reproduce, with increasing accuracy, important properties of measured and modeled Internet topologies. We find that the d=2 case is sufficient for most practical purposes, while d=3 essentially reconstructs the Internet AS- and router-level topologies exactly. We hope that a systematic method to analyze and synthesize topologies offers a significant improvement to the set of tools available to network topology and protocol researchers.",
"subjects": "Networking and Internet Architecture (cs.NI); Statistical Mechanics (cond-mat.stat-mech); Physics and Society (physics.soc-ph)",
"title": "Systematic Topology Analysis and Generation Using Degree Correlations",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9381240160063031,
"lm_q2_score": 0.7549149923816048,
"lm_q1q2_score": 0.7082038843963988
} |
https://arxiv.org/abs/1701.08833 | Pre-kites: Simplices having a regular facet | The investigation of the relation among the distances of an arbitrary point in the Euclidean space $\mathbb{R}^n$ to the vertices of a regular $n$-simplex in that space has led us to the study of simplices having a regular facet. Calling an $n$-simplex with a regular facet an $n$-pre-kite, we investigate, in the spirit of [4], [10], [9], and [15], and using tools from linear algebra, the degree of regularity implied by the coincidence of any two of the classical centers of such simplices. We also prove that if $n \ge 3$, then the intersection of the family of $n$-pre-kites with any of the four known special families is the family of $n$-kites, thus extending the result in [18]. A basic tool is a closed form of a determinant that arises in the context of a certain Cayley-Menger determinant, and that generalizes several determinants that appear in [9], [15], and [16]. Thus the paper is a further testimony to the special role that linear algebra plays in higher dimensional geometry. | \section{Introduction} \label{Int}
The distances $t_1,\cdots,t_{n+1}$ between the vertices of a regular $n$-simplex $S$ of edge length $t_0$ and an arbitrary point $P$ in its affine hull are related by the elegant relation
\begin{eqnarray} \label{REL}
(n+1) \sum_{j=0}^{n+1} t_j^4 &=& \left( \sum_{j=0}^{n+1} t_j^2 \right)^2;
\end{eqnarray}
see \cite{Bentin-1} for a very short proof, and see \cite{HHSh} for a proof that (\ref{REL}) is essentially the only relation that exists among the quantities $t_0,\cdots, t_{n+1}$, for fixed $t_0$. Figure 1 below illustrates the case when $d=2$, i.e., when the regular simplex is an equilateral triangle.
\begin{center}
~~\psset{unit=.3cm}
\begin{pspicture}(-11,-5)(11,20)
\psline[linestyle=dashed](-15,8)(-10,0)
\psline[linestyle=dashed](-15,8)(10,0)
\psline[linestyle=dashed](-15,8)(0,17.32)
\rput(-13.5,4){\small $t_2$} \rput(-2.5,3){\small $t_3$} \rput(-8.5,13.1){\small $t_1$}
\psline(10,0)(0,17.32)
\psline(-10,0)(0,17.32)
\psline(-10,0)(10,0)
\rput(0,-1){\small $t_0$} \rput(-6,8.7){\small $t_0$} \rput(6,8.7){\small $t_0$}
\rput(0,18){\small $\mathbf{v}}\newcommand{\yy}{\mathbf{y}_1$} \rput(-10.5,-1){\small $\mathbf{v}}\newcommand{\yy}{\mathbf{y}_2$} \rput(10.5,-1){\small $\mathbf{v}}\newcommand{\yy}{\mathbf{y}_3$}
\rput(-16,8){\small $P$}
\rput(0,-2.5){\small \bf Figure 1}
\rput(0,-4.5){\small \bf Illustrating (\ref{REL}) in the case $d=2$}
\end{pspicture}
\end{center}
The relation (\ref{REL}) has been a source of fascination, and its special case $n=2$ has been a source of inspiration to problem lovers. Problems that refer to Figure 1 and that give numerical values of three of the variables $t_0,t_1,t_2,t_3$ and ask about the fourth variable can, in the absence of the relation (\ref{REL}), be thought of as challenging. Such problems have appeared in \cite{500}, \cite{Graham}, \cite{Wagon}, \cite{CMJ}, and possibly others. One of the features of Figure 1 is the fact that the lengths $t_1, t_2, t_3$ can serve as the side lengths of a triangle, i.e., they satisfy the triangle inequality. This non-obvious and interesting fact is attributed to the Romanian mathematician Dimitrie Pompeiu (1873--1954), and now carries his name. It, too, appeared, with different proofs quite frequently; see, for example, \cite{Titu}, \cite{Titu-2}, \cite{Fomin}, \cite{Alsina}, \cite{mini}, \cite{Delights}, \cite{Roy}, and \cite{Putnam}.
Moving the point $P$ outside the affine hull of the regular $n$-simplex $S$ results in an $(n+1)$-simplex $(S,P)$ having $S$ as a facet. This led us to consider such simplices. Thus we call an $(n+1)$-simplex having a regular facet an $(n+1)$-{\it pre-kite}, and we investigate their properties. When the edges emanating from the point $P$ to the vertices of $S$ are equal, then the pre-kite $(S,P)$ is what was called a {\it kite} in \cite{RM-ortho} and in \cite{kites}.
In this paper, we find a closed form of the Cayley-Menger determinant of a pre-kite. A main determinant that comes up generalizes several determinants that have appeared in \cite{RM-ortho}, \cite{RM-balloon}, and \cite{impurity},
and possibly other places. By setting the Cayley-Menger determinant $M$ equal to 0, we obtain the relation (\ref{REL}) mentioned above, and we use this relation to give a new proof of Pompeiu's theorem. We also investigate the degree of regularity implied by the coincidence of two of the classical centers of a pre-kite, and we find the intersection of the family of pre-kites with any of the known special families of orthocentric, circumscriptible, isodynamic, and tetra-isogonic simplices.
\bigskip
The paper is organized as follows. In Section 2, we define
the Cayley-Menger and the inner Cayley-Menger determinants $\mathcal{C}} \newcommand{\DDD}{\mathcal{D}}\newcommand{\EEE}{\mathcal{E}$ and $\DDD$ of a simplex $S$, and we recall the formulas that give the volume and circumradius of $S$ in terms of $\mathcal{C}} \newcommand{\DDD}{\mathcal{D}}\newcommand{\EEE}{\mathcal{E}$ and $\DDD$. In Section 3, we find a closed form of a certain determinant that will be used to evaluate $\mathcal{C}} \newcommand{\DDD}{\mathcal{D}}\newcommand{\EEE}{\mathcal{E}$ and $\DDD$. This determinant has, as special cases, several determinants that have appeared in the earlier literature. Section 4 applies the formulas in Section 3 to pre-kites to give a new derivation of (\ref{REL}) and to give a new proof of Pompeiu's theorem. In Section 5, we prove that a non-regular pre-kite cannot have more than two different regular facets, and we also characterize the positive numbers that can serve as edge lengths of a two-apexed pre-kite.
In Section 6, we prove that if the circumcenter and centroid of a pre-kite coincide, then it is regular. The same holds if the circumcenter and incenter coincide. We also prove that if $n \le 5$, and if the centroid and the incenter of an $n$-pre-kite coincide, then it is regular, and we provide examples of non-regular $n$-pre-kites, $n \ge 6$, in which the centroid and incenter coincide. We also prove that if the Fermat-Torricelli point of a pre-kite coincides with either the centroid or the circumcenter, then it is regular. In Section 7, we prove that if a pre-kite of dimension $n \ge 3$ belongs to any of the four known special families of orthocentric, circumscriptible, isodynamic, and tetra-isogonic $n$-simplices, then it is a kite.
\bigskip It is not unusual to use tools from linear algebra, such as properties
of certain matrices and determinants, in investigations pertaining to higher-dimensional geometry. The importance of such tools is manifested, for example, in the proofs of the higher dimensional analogues of the theorems of Pythagoras, as in \cite{Lin} and \cite{Bhatia}, Napoleon, as in \cite{Weiss} and \cite{Spirova}, the law of sines, as in \cite{Leng}, coincidences of centers, as in \cite{ET}, \cite{BAG-1}, \cite{RM-ortho}, and \cite{RM-balloon}, and the open mouth theorem, as in \cite{OMT}.
\section{The Cayley-Menger determinant, and formulas for the volume and circumradius of an $n$-simplex} This section puts together known formulas for
the volume and circumradius of an $n$-simplex in terms of the Cayley-Menger and inner Cayley-Menger determinants of $S$.
We recall that an {\it $n$-simplex}, $n \ge 2$, is defined to be the convex hull $S = [A_0,\cdots, A_n]$ of $n+1$ affinely independent points $A_0, \cdots, A_n$ in a Euclidean space $\mathbb{R}^m$, $m \ge n$. For $0 \le j\le n$, $A_j$ is called the $j$-th {\it vertex} of $S$, and the $(n-1)$-simplex obtained from $S$ by removing the vertex $A_j$ is called the $j$-th {\it facet} of $S$ and is denoted by $S_j$. The simplex obtained from $S$ by removing any number of vertices is called a {\it face} of $S$.
Let $S = [A_0,\cdots,A_n]$ be an $n$-simplex, and let
\begin{eqnarray} \label{aij} \| A_i - A_j \|^2 = a_{i,j}, ~~0 \le i, j \le n.\end{eqnarray}
The {\it Cayley-Menger determinant} $\mathcal{C}} \newcommand{\DDD}{\mathcal{D}}\newcommand{\EEE}{\mathcal{E} = \mathcal{C}} \newcommand{\DDD}{\mathcal{D}}\newcommand{\EEE}{\mathcal{E} (S)$ of $S$ is the $(n+2) \times (n+2)$ determinant whose entries $c_{i,j}$, $-1 \le i, j \le n$, are defined by
\begin{eqnarray} \label{cij} c_{i,j} &=& \left\{
\begin{array}{ll}
0 & \mbox{if $i=j$},\\
1 & \mbox{if $i=-1$ and $j \ne-1$},\\
1 & \mbox{if $j=-1$ and $i \ne-1$},\\
a_{i,j} & \mbox{otherwise};
\end{array}
\right.
\end{eqnarray}
see, for example, \cite[\S 9.7.3.1, pp.~237--238]{Berger}. Thus
\begin{eqnarray}
\mathcal{C}} \newcommand{\DDD}{\mathcal{D}}\newcommand{\EEE}{\mathcal{E}=\mathcal{C}} \newcommand{\DDD}{\mathcal{D}}\newcommand{\EEE}{\mathcal{E} (S) &=& \left|
\begin{array}{cccccccc}
0&1&1&1&\cdots&\cdots&1&1\\
1&0&a_{0,1}&a_{0,2}&\cdots&\cdots&a_{0,n-1}&a_{0,n}\\
1&a_{1,0}&0&a_{1,2}&\cdots&\cdots&a_{1,n-1}&a_{1,n}\\
\cdots &\cdots &\cdots &\cdots &\cdots&\cdots &\cdots &\cdots\\
\cdots &\cdots &\cdots &\cdots &\cdots&\cdots &\cdots &\cdots\\
1&a_{n-1,0}&a_{n-1,1}&a_{n-1,2}&\cdots&\cdots&0&a_{n-1,n}\\
1&a_{n,0}&a_{n,1}&a_{n,2}&\cdots&\cdots&a_{n,n-1}&0\\
\end{array}
\right|. \label{CCC}
\end{eqnarray}
If $\VVV = \VVV (S)$ is the volume (i.e., the $n$-dimensional Lebesgue measure or content) of $S$, then it is well known that
\begin{eqnarray}\label{V}
(-1)^{n+1} 2^n (n!)^2 \VVV^2 &=& \mathcal{C}} \newcommand{\DDD}{\mathcal{D}}\newcommand{\EEE}{\mathcal{E};
\end{eqnarray}
see, for example, \cite[(5.1), \S 5, Chapter VIII, p.~125]{Sommerville} and \cite{Ivanoff}.
The determinant obtained from $\mathcal{C}} \newcommand{\DDD}{\mathcal{D}}\newcommand{\EEE}{\mathcal{E}$ by deleting the uppermost row and the leftmost column will be denoted by $\DDD = \DDD (S)$, and will be referred to as the {\it inner Cayley-Menger determinant} of $S$.
Thus
\begin{eqnarray}
\DDD = \DDD (S) &=& \left|
\begin{array}{ccccccc}
0&a_{0,1}&a_{0,2}&\cdots&\cdots&a_{0,n-1}&a_{0,n}\\
a_{1,0}&0&a_{1,2}&\cdots&\cdots&a_{1,n-1}&a_{1,n}\\
\cdots &\cdots &\cdots &\cdots&\cdots &\cdots &\cdots\\
\cdots &\cdots &\cdots &\cdots&\cdots &\cdots &\cdots\\
a_{n-1,0}&a_{n-1,1}&a_{n-1,2}&\cdots&\cdots&0&a_{n-1,n}\\
a_{n,0}&a_{n,1}&a_{n,2}&\cdots&\cdots&a_{n,n-1}&0\\
\end{array}
\right|. \label{DDD}
\end{eqnarray}
The determinants $\mathcal{C}} \newcommand{\DDD}{\mathcal{D}}\newcommand{\EEE}{\mathcal{E} = \mathcal{C}} \newcommand{\DDD}{\mathcal{D}}\newcommand{\EEE}{\mathcal{E} (S)$ and $\DDD = \DDD (S)$ are used in \cite{Ivanoff} to express the circumradius $\RRR = \RRR (S)$ of $S$ as
\begin{eqnarray}\label{R}
\RRR^2 &=& \frac{- \DDD}{2 \mathcal{C}} \newcommand{\DDD}{\mathcal{D}}\newcommand{\EEE}{\mathcal{E}}.
\end{eqnarray}
\section{A special determinant} \label{special-det}
In this section, we consider the determinant $\mathbb{K}(n;z;x_1,\cdots,x_n;y_1,\cdots,y_n;a;b)$ defined by (\ref{K}) below, and we evaluate it in closed form in Theorem \ref{KK}. This will then be used in Theorems \ref{CjDj} and \ref{CCCDDD} to find formulas for the volumes and circumradii of pre-kites and their facets. These formulas will in turn be used to determine the degree of regularity implied by the coincidence of any two of the classical centers of a pre-kite. Note that the special cases of $\mathbb{K}(n;z;x_1,\cdots,x_n;y_1,\cdots,y_n;a;b)$ when
$[z=1,~x_i=y_i]$, $[z=0,~x_i=y_i,~a=1,~b=-1]$, and $[z=0,~x_i=y_i]$
have appeared in \cite{RM-ortho}, \cite{RM-balloon}, and \cite{impurity}, respectively, where they were instrumental in establishing the results there.
\bigskip
We start with defining the special determinants $\mathbb{J}$ and $\mathbb{K}$. In all that follows, $a, b, z, x_j, y_j$ stand for real numbers for all non-negative integers $j$, but the treatment may still hold over other rings.
\begin{define} \label{DefJK}
{\em The determinant $\mathbb{J} (n;z;a;b)$, $n \in \mathbb{N}$, is the $n \times n$ determinant that has $b$ on every entry on the main diagonal and $a$ everywhere else. The determinant $\mathbb{K} (n;z;x_1,\cdots,x_n;y_1,\cdots,y_n;a;b)$ is the $(n+1) \times (n+1)$ determinant $\left(d_{i,j}\right)_{0 \le i, j \le n}$ whose 0-th row is $[z,y_1,\cdots,y_n]$, whose 0-th column is $[z,x_1,\cdots,x_n]^t$, and whose subdeterminant $\left(d_{i,j}\right)_{1 \le i, j \le n}$ is $\mathbb{J} (n;a;b)$. More formally, the entries $d_{i,j}, 0 \le i, j \le n,$ are given by
\begin{eqnarray}
d_{i,j} &=& \left\{
\begin{array}{cl}
z & \mbox{if $i=j=0$},\\
x_i & \mbox{if $j=0$ and $1 \le i \le n$},\\
y_j & \mbox{if $i=0$ and $1 \le j \le n$},\\
b & \mbox{if $1 \le i = j \le n$},\\
a & \mbox{otherwise}.
\end{array} \right. \label{K-formal}
\end{eqnarray}
Thus
\begin{eqnarray}
\mathbb{J} (n;a;b) &=& \left|
\begin{array}{ccccc}
b&a&a&\cdots&a\\a&b&a&\cdots&a\\a&a&b&\cdots&a\\
\cdots &\cdots &\cdots&\cdots &\cdots \\
a&a&a&\cdots&b
\end{array}
\right|,~~~\mbox{(of size $n\times n$)} \nonumber\\
&=& ((n-1)a+b) (b-a)^{n-1}, \mbox{~by Lemma 3.1 of \cite{impurity}}. \label{J}
\end{eqnarray}
The formula above has also appeared as Lemma 7.1 in \cite{RM-balloon} and as Lemma 3.6 in \cite{RM-ortho}. Also,
\begin{eqnarray}
\mathbb{K} (n;z;\xx;\yy;a;b) &=&\mathbb{K} (n;z;x_1,\cdots,x_n;y_1,\cdots,y_n;a;b) \nonumber\\ && \nonumber\\
&=& \left|
\begin{array}{ccccccc}
z&y_1&y_2&\cdots&y_j&\cdots&y_n\\
x_1&b&a&\cdots&a&\cdots&a\\
x_2&a&b&\cdots&a&\cdots&a\\
\cdots &\cdots &\cdots &\cdots &\cdots&\cdots &\cdots \\
x_j&a&a&\cdots&b&\cdots&a\\
\cdots &\cdots &\cdots &\cdots &\cdots&\cdots &\cdots \\
x_n&a&a&\cdots&a&\cdots&b
\end{array}
\right|. \label{K} \end{eqnarray}
}\end{define}
\begin{thm} \label{KK}
Let $n \ge 1$, and let
\begin{eqnarray} \label{xxyy}
\xx = (x_1,\cdots,x_n),~~\yy = (y_1,\cdots, y_n),
\end{eqnarray}
and
\begin{eqnarray} \label{MxMyMxy}
\MMM_{\xx} = \sum_{j=1}^n x_j,~~\MMM_{\yy} = \sum_{j=1}^n y_j,~~
\MMM_{\xx\yy} = \sum_{j=1}^n x_jy_j. \end{eqnarray}
Let $\mathbb{K} = \mathbb{K} (n;z;\xx;\yy;a;b)$ be the determinant defined by (\ref{K}).
Then
\begin{eqnarray}
&& \mathbb{K} (n;z;\xx;\yy;a;b) \nonumber\\
&=& (b-a)^{n-2} \left[
((n-1)a+b)
\left(z(b-a) - \MMM_{\xx\yy}\right) + a \MMM_{\xx} \MMM_{\yy}\right].
\label{K=} \end{eqnarray}
\end{thm}
\vspace{.0cm} \noindent {\it Proof.}
We proceed by induction. For $n=1$, the statement is trivial, being nothing but
\begin{eqnarray*}\mathbb{K} (1;z;x_1;y_1;a;b) &=& \left| \begin{array}{cc} z&y_1\\x_1&b \end{array} \right|= zb-x_1y_1.
\end{eqnarray*}
Suppose now that (\ref{K=}) holds for $n= r$ for some $r \ge 1$. We are to show that it holds for $n = r+1$. Thus we let
\begin{eqnarray}
\xx' &=& \left(x_1,\cdots,x_{r+1}\right),~~
\yy' ~=~ \left(y_1,\cdots,y_{r+1}\right),\label{x'}\\
\MMM_{\xx}' &=& \sum_{j=1}^{r+1} x_j,~~\MMM_{\yy}' ~=~ \sum_{j=1}^{r+1} y_j,~~
\MMM_{\xx\yy}' ~=~ \sum_{j=1}^{r+1} x_jy_j,\label{M'} \end{eqnarray}
and we show that
\begin{eqnarray}
&&\mathbb{K}(r+1;z;\xx';\yy';a;b) \nonumber \\
&=& (b-a)^{r-1}\left[
(ra+b) \left(z(b-a) - \MMM_{\xx\yy}'\right) + a \MMM_{\xx}'
\MMM_{\yy}' \right]. \label{M-r+1}
\end{eqnarray}
For simplicity, we denote $\mathbb{K}(r+1;z;\xx',\yy';a;b)$ by $K$, and we refer to its rows and columns as the 0-th, the first, etc. Thus the 0-th row of $K$ is $[z,y_1,\cdots,y_{r+1}]$, and the 0-th column is $[z,x_1,\cdots,x_{r+1}]^t$. Expanding $K$ along the 0-th row, we obtain
\begin{eqnarray}
K &=& z C_0 +\sum_{j=1}^{r+1} (-1)^{j} y_j C_j, \label{C0Cj}
\end{eqnarray}
where $C_j$ is the $(0,j)$-th minor of $K$.
Since $C_0$ is the case $n = r+1$ of the determinant given in (\ref{J}), it is clear that
\begin{eqnarray} \label{C0-1}
C_0 &=& \mathbb{J} (r+1;a;b) ~=~
(ra+b)(b-a)^r, \mbox{~by (\ref{J})}.
\end{eqnarray}
To calculate $C_j$, $1 \le j \le r+1$, we recall that $C_j$ is obtained from $K$ by deleting the 0-th row and $j$-th column, and we let $R_1, \cdots, R_{r+1}$ be the rows of $C_j$. Notice that $R_1=[x_1,b,a,\cdots,a]$ and $R_j = [x_j,a,a,\cdots,a]$, for $j > 1$. Let $E_j$ be the determinant obtained from $C_j$ by moving $R_j$ to the very top (with $E_1=C_1$). Thus the rows of $E_j$ are $R_j,R_1,\cdots,R_{j-1},R_{j+1},\cdots,R_{r+1}$, i.e., $R_{\sigma (1)},\cdots,R_{\sigma(r+1)}$, where $\sigma$ is the cyclic permutation $(1~~j~~j-1~~j-2~~\cdots~~2)$. Thus
$C_j = (-1)^{j-1} E_j$. Also, the uppermost row of $E_j$ is $[x_j,a,\cdots,a]$, the leftmost column is $[x_j, x_1,\cdots,x_{j-1},x_{j+1},\cdots,x_{r+1}]^t$, and the remaining entries form $\mathbb{J}(r;a;b)$.
Therefore
\begin{eqnarray}
C_j &=& (-1)^{j-1} E_j \nonumber\\
&=& (-1)^{j-1} \mathbb{K}(r;x_j;x_1,\cdots,x_{j-1},x_{j+1},\cdots,x_{r+1};a,\cdots,a;a;b)\nonumber\\
&=& (-1)^{j-1} (b-a)^{r-2} \left[((r-1)a+b)(x_j (b-a)-a(\MMM_{\xx}' - x_j)) + a (\MMM_{\xx}' - x_j)(ra)\right]\nonumber \\
&=& (-1)^{j-1} (b-a)^{r-2} \left[((r-1)a+b)x_j (b-a)-a(\MMM_{\xx}' - x_j) ((r-1)a+b-ra)\right] \nonumber\\
&=& (-1)^{j-1} (b-a)^{r-2+1} \left[((r-1)a+b)x_j-a(\MMM_{\xx}' - x_j)\right]
\nonumber\\
&=& (-1)^{j-1} (b-a)^{r-1} \left[(ra+b)x_j-a\MMM_{\xx}'\right].\label{Cj-1}
\end{eqnarray}
Using (\ref{C0Cj}), (\ref{C0-1}), and (\ref{Cj-1}), we obtain
\begin{eqnarray*}
K&=&z(ra+b)(b-a)^r+\sum_{j=1}^{r+1}(-1)^{j}y_j(-1)^{j-1}(b-a)^{r-1}\left[ (ra+b)x_j-a\MMM_{\xx}'\right] \\
&=&(b-a)^{r-1}\left[ z(ra+b)(b-a)-\sum_{j=1}^{r+1}y_j\left( (ra+b)x_j-a \MMM_{\xx}'\right) \right] \\
&=&(b-a)^{r-1}\left[ (ra+b)\left( z(b-a)-\sum_{j=1}^{r+1}x_jy_j\right) + a\MMM_{\xx}' \sum_{j=1}^{r+1}y_j \right] \\
&=&(b-a)^{r-1}\left[ (ra+b)\left( z(b-a)-\MMM_{\xx\yy}'\right) + a\MMM_{\xx}'\MMM_{\yy}' \right],
\end{eqnarray*}
as desired. This completes the proof. \hfill $\Box$
\section{Pre-kites and formulas for their volumes and circumradii} \label{prekites}
In this section, we introduce the new family of pre-kites, and we use Theorem \ref{KK} to derive formulas for the volumes and circumradii of these simplices and of their facets. These formulas will be used in Section \ref{coincidence} to investigate the degree of regularity implied by the coincidence of two of the classical centers of an $n$-pre-kite $S$, $n \ge 2$.
We shall call the $n$-simplex $S=[A_0,\cdots,A_n]$ an {\it $n$-pre-kite} if one of the facets $S_j$ is a regular $(n-1)$-simplex. In this case, we call $A_j$ an {\it apex}, and $S_j$ a {\it base} of $S$. Actually there will be no harm in referring to these as ``the" apex and ``the" base, although an $n$-pre-kite can have more than one apex (and hence more than one base), as we shall see later in Section \ref{Jaws}.
Notice that if the $n$-simplex $S=[A_0,\cdots,A_n]$ is an $n$-pre-kite with apex $A_0$, and if the lengths of the edges that emanate from $A_0$ are all equal, then $S$ is what was called an $n$-kite in \cite{kites} and in other papers. Notice also that all the facets (and hence all the faces) of a pre-kite are also pre-kites.
If the $n$-simplex $S=[A_0,\cdots,A_n]$ is an $n$-pre-kite with apex $A_0$, and if
\begin{eqnarray} \label{v}
\|A_i - A_j\|^2 = \left\{
\begin{array}{cl}
v_i & \mbox{~~if $j=0$ and $1 \le i \le n$},
\\
u & \mbox{~~$i \ne j$ and $1 \le i, ~ j \le n$},
\end{array}
\right.
\end{eqnarray}
then we shall denote $S$ by $PK[n;u;v_1,\cdots,v_n]$.
Note that this $n$-pre-kite
is an $n$-kite precisely when $v_1 = \cdots = v_n$.
\begin{thm} \label{CCCDDD} Let
\begin{eqnarray} \label{vv} \mathbf{v}}\newcommand{\yy}{\mathbf{y} &=& (v_1,\cdots,v_n), \end{eqnarray} and let $S=[A_0,\cdots,A_n]$ be the $n$-pre-kite with apex $A_0$ defined by
\begin{eqnarray} \label{pre-k-S} S&=& PK[n;u;\mathbf{v}}\newcommand{\yy}{\mathbf{y}] ~=~ PK[n;u;v_1,\cdots,v_n],
\end{eqnarray}
where $n \ge 3$ and $u > 0$. Then the Cayley-Menger determinant of $S$ is given by
\begin{eqnarray}
\mathcal{C}} \newcommand{\DDD}{\mathcal{D}}\newcommand{\EEE}{\mathcal{E} (PK[n;u;\mathbf{v}}\newcommand{\yy}{\mathbf{y}]) &=&
(-u)^{n-2} \left[n (u^2 + v_1^2 + \cdots + v_n^2) - (u + v_1 + \cdots + v_n)^2 \right], \label{CCC-PK-2}
\end{eqnarray}
and the inner Cayley-Menger determinant of $S$ is given by
\begin{eqnarray}
\DDD (PK[n;u;\mathbf{v}}\newcommand{\yy}{\mathbf{y}]) &=&
(-u)^{n-1} \left[
(n-1) (v_1^2+\cdots + v_n^2) - (v_1+\cdots+v_n)^2 \right].
\label{DDD-PK-2}
\end{eqnarray}
\end{thm}
\vspace{.2cm} \noindent {\it Proof.} The Cayley-Menger determinant of the pre-kite $PK[n;u;\mathbf{v}}\newcommand{\yy}{\mathbf{y}]$ is given by
\begin{eqnarray}
\mathcal{C}} \newcommand{\DDD}{\mathcal{D}}\newcommand{\EEE}{\mathcal{E} (PK[n;u;\mathbf{v}}\newcommand{\yy}{\mathbf{y}]) &=& \left|
\begin{array}{cccccccc}
0&1&1&1&\cdots&\cdots&1&1\\
1&0&v_1&v_{2}&\cdots&\cdots&v_{n-1}&v_{n}\\
1&v_{1}&0&u&\cdots&\cdots&u&u\\
\cdots &\cdots &\cdots &\cdots &\cdots&\cdots &\cdots &\cdots\\
\cdots &\cdots &\cdots &\cdots &\cdots&\cdots &\cdots &\cdots\\
1&v_{n-1}&u&u&\cdots&\cdots&0&u\\
1&v_{n}&u&u&\cdots&\cdots&u&0\\
\end{array}
\right|. \label{CCC-PK}
\end{eqnarray}
Multiplying the uppermost row by $u$ and interchanging it with the next row, and then multiplying the leftmost column by $u$ and interchanging it with the next column, we obtain
\begin{eqnarray}
u^2 \mathcal{C}} \newcommand{\DDD}{\mathcal{D}}\newcommand{\EEE}{\mathcal{E} (PK[n;u;\mathbf{v}}\newcommand{\yy}{\mathbf{y}]) &=& \left|
\begin{array}{cccccccc}
0&u&v_1&v_{2}&\cdots&\cdots&v_{n-1}&v_{n}\\
u&0&u&u&\cdots&\cdots&u&u\\
v_1&u&0&u&\cdots&\cdots&u&u\\
\cdots &\cdots &\cdots &\cdots &\cdots&\cdots &\cdots &\cdots\\
\cdots &\cdots &\cdots &\cdots &\cdots&\cdots &\cdots &\cdots\\
v_{n-1}&u&u&u&\cdots&\cdots&0&u\\
v_{n} &u&u&u&\cdots&\cdots&u&0\\
\end{array}
\right| \nonumber
\\ \vspace{.08cm} \nonumber\\
&=& \mathbb{K} (n+1;0;u,v_1,\cdots,v_n;u,v_1,\cdots,v_n;u,0)
\nonumber \\ \vspace{.03cm} \nonumber\\
&=& (-u)^{n-1} \left[(-nu)(u^2 + v_1^2 + \cdots + v_n^2) + u
(u + v_1 + \cdots + v_n)^2\right] \nonumber\\ \vspace{.03cm} \nonumber\\
&=& (-u)^{n} \left[n (u^2 + v_1^2 + \cdots + v_n^2) - (u + v_1 + \cdots + v_n)^2 \right]. \nonumber
\end{eqnarray}
Therefore
\begin{eqnarray*}
\mathcal{C}} \newcommand{\DDD}{\mathcal{D}}\newcommand{\EEE}{\mathcal{E} (PK[n;u;\mathbf{v}}\newcommand{\yy}{\mathbf{y}]) &=&
(-u)^{n-2} \left[n (u^2 + v_1^2 + \cdots + v_n^2) - (u + v_1 + \cdots + v_n)^2 \right],
\end{eqnarray*}
as desired.
We now calculate the inner Cayley-Menger determinant $\DDD (PK[n;u;\mathbf{v}}\newcommand{\yy}{\mathbf{y}])$ of the $n$-pre-kite $PK[n;u;\mathbf{v}}\newcommand{\yy}{\mathbf{y}]$.
\begin{eqnarray*}
\DDD (PK[n;u;\mathbf{v}}\newcommand{\yy}{\mathbf{y}]) &=& \left|
\begin{array}{ccccccc}
0&v_{1}&v_{2}&\cdots&\cdots&v_{n-1}&v_{n}\\
v_{1}&0&u&\cdots&\cdots&u&u\\
\cdots &\cdots &\cdots &\cdots&\cdots &\cdots &\cdots\\
\cdots &\cdots &\cdots &\cdots&\cdots &\cdots &\cdots\\
v_{n-1}&u&u&\cdots&\cdots&0&u\\
v_{n}&u&u&\cdots&\cdots&u&0\\
\end{array}
\right| \nonumber \\ \vspace{.12cm} \nonumber \\
&=& \mathbb{K} (n;0;v_1,\cdots,v_n;v_1,\cdots,v_n;u,0) \nonumber \\ \vspace{.05cm}
&=&
(-u)^{n-2} \left[
(n-1) (-u) (v_1^2+\cdots + v_n^2) +u (v_1+\cdots+v_n)^2 \right] \nonumber
\\ &=&
(-u)^{n-1} \left[
(n-1) (v_1^2+\cdots + v_n^2) - (v_1+\cdots+v_n)^2 \right].
\end{eqnarray*}
This completes the proof. \hfill $\Box$
\bigskip
The next theorem is immediate, but we record it for ease of reference.
\begin{thm} \label{CjDj} Let $n \ge 3$, and let $S$ be the $n$-pre-kite defined by
\begin{eqnarray*} S&=& PK[n;u;\mathbf{v}}\newcommand{\yy}{\mathbf{y}] ~=~ PK[n;u;v_1,\cdots,v_n].
\end{eqnarray*}
For $0 \le j \le n$, let $S_j$ be the $j$-th facet of $S$, and let $\mathcal{C}} \newcommand{\DDD}{\mathcal{D}}\newcommand{\EEE}{\mathcal{E}_j$ and $\DDD_j$ be the Cayley-Menger and the inner Cayley-Menger determinants of $S_j$. Let
\begin{eqnarray} \label{alphabeta}
\alpha = u + v_1 + \cdots + v_n,~~\beta = u^2+v_1^2+\cdots+v_n^2.
\end{eqnarray}
Then for $1 \le j \le n$, we have
\begin{eqnarray}
\mathcal{C}} \newcommand{\DDD}{\mathcal{D}}\newcommand{\EEE}{\mathcal{E}_0 &=& (-1)^n n u^{n-1}. \label{C0}\\
\mathcal{C}} \newcommand{\DDD}{\mathcal{D}}\newcommand{\EEE}{\mathcal{E}_j &=& (-u)^{n-3} \left[ - \alpha^2 + (n-1) \beta - n v_j^2 + 2 \alpha v_j \right]. \label{Cj} \\
\DDD_0 &=& (-1)^{n+1} u^n (n-1).\label{D0} \\
\DDD_j &=& (-u)^{n-2} [ (n-2)\beta -\alpha^2 + 2 \alpha u - (n-1) u^2 - (n-1) v_j^2 \nonumber \\&& + 2 \alpha v_j - 2u v_j ].\label{Dj}
\end{eqnarray}
\end{thm}
\vspace{.2cm} \noindent {\it Proof.} Observing that
\begin{eqnarray}
S_0&=& PK[n-1;u;u,\cdots,u]\\
S_j&=& PK[n-1;u;v_1,\cdots,v_{j-1},v_{j+1},\cdots,v_n] \mbox{~~if $1\le j \le n$},
\end{eqnarray}
and using Theorem \ref{CCCDDD}, we obtain
\begin{eqnarray*}
\mathcal{C}} \newcommand{\DDD}{\mathcal{D}}\newcommand{\EEE}{\mathcal{E}_0 &=& (-u)^{n-3}
\left[(n-1) nu^2 - n^2u^2 \right] \nonumber\\
&=& (-1)^n n u^{n-1},\\
\mathcal{C}} \newcommand{\DDD}{\mathcal{D}}\newcommand{\EEE}{\mathcal{E}_j
&=& (-u)^{n-3} \left[(n-1) ( \beta - v_j^2) - (\alpha - v_j)^2 \right] \\
&=& (-u)^{n-3} \left[(n-1) \beta - (n-1) v_j^2
- \alpha^2 - v_j^2 + 2 \alpha v_j\right] \\
&=& (-u)^{n-3} \left[- \alpha^2 + (n-1) \beta - n v_j^2 + 2 \alpha v_j\right], \\
\DDD_0
&=& (-u)^{n-2} \left[(n-2)u^2 (n-1) - (n-1)^2 u^2 \right] \\
&=& (-1)^{n+1} u^{n} (n-1), \\
\DDD_j
&=& (-u)^{n-2} \left[(n-2)(\beta -v_j^2 - u^2)-(\alpha - v_j-u)^2\right] \\
&=& (-u)^{n-2} \left[ (n-2)\beta - (n-2) v_j^2 - (n-2) u^2 -\alpha^2 - v_j^2 - u^2 + 2 \alpha v_j + 2 \alpha u - 2u v_j
\right] \\
&=& (-u)^{n-2} \left[
(n-2)\beta -\alpha^2 + 2 \alpha u - (n-1) u^2
- (n-1) v_j^2 + 2 \alpha v_j - 2u v_j
\right].
\end{eqnarray*}
This completes the proof. \hfill $\Box$
\bigskip
The next theorem uses Theorem \ref{CCCDDD} to provide another derivation of the relation (\ref{REL}) mentioned earlier.
\begin{thm} \label{Bentin} Let $S = [A_1,\cdots,A_{n+1}]$ be a regular $n$-simplex of edge length $t_0$, and let $P$ be a point in its affine hull. Let $t_j$, $1 \le j \le n+1$, denote the distance from $P$ to the vertex $A_j$. Then
\begin{eqnarray} \label{RELREL}
(n+1) \sum_{j=0}^{n+1} t_j^4 &=& \left( \sum_{j=0}^{n+1} t_j^2 \right)^2.
\end{eqnarray}
\end{thm}
\vspace{.1cm} \noindent {\it Proof.}
Since the point $P$ lies in the affine hull of the regular $n$-simplex
$S = [A_1,\cdots,A_{n+1}]$, then the $(n+1)$-pre-kite (having $P$ as an apex) is degenerate, and hence has volume 0. Equivalently, its Cayley-Menger determinant is 0. Since $t_0$ is not 0, the relation (\ref{RELREL}) follows immediately from Theorem \ref{CCCDDD} and (\ref{V}).\hfill $\Box$
\bigskip We end this section by giving a new proof of Pompeiu's theorem.
The proof uses the last part of the next lemma; the other parts of the lemma will be used later.
\begin{lem} \label{uvh}
Let $S = [A_0, \cdots, A_n]$ be a regular $n$-simplex with center $I$, and let $u$ and $R$ be its edge length and circumradius, respectively. Let $G$ be the center of the (regular) $(n-1)$-simplex $S_0 = [A_1, \cdots, A_n]$. Then
\begin{eqnarray}
\frac{R^2}{u^2} &=& \frac{n}{2(n+1)}, \label{(i)}\\
\|A_0 - G\| &=& \frac{(n+1)R}{n} = \sqrt{\frac{n+1}{2n}}u. \label{(ii)}
\end{eqnarray}
If $P$ is an arbitrary point in the affine hull of $S$ with $\|P-I\| = \rho$, then
\begin{eqnarray}
\|P-A_0\|^2 + \cdots + \|P-A_n\|^2 &=& (n+1) (\rho^2 + R^2), \label{(iii)}
\end{eqnarray}
and therefore $P$ lies on the circumsphere of $S$ if and only if
\begin{eqnarray}
\|P-A_0\|^2 + \cdots + \|P-A_n\|^2 &=& 2(n+1) R^2 ~=~ n u^2. \label{(iv)}
\end{eqnarray}
\end{lem}
\vspace{.3cm} \noindent {\it Proof.} For (\ref{(i)}) and (\ref{(ii)}), see Proposition 4.6 (p. 281) of \cite{RM-ortho}. For (\ref{(iii)}),
assume, without loss of generality, that $I$ is the origin $\mathcal{O}$. Then
$\|A_j\| = R$ for $0 \le j \le n$, and $\|P \| = \rho$. Also $A_0 + \cdots + A_n = \mathcal{O}$, and hence $P \cdot A_0 + \cdots + P\cdot A_n = 0$. Therefore
\begin{eqnarray*}
\|P-A_0\|^2 + \cdots + \|P-A_n\|^2 &=& (n+1)\rho^2 + (n+1) R^2,
\end{eqnarray*}
as desired. For (\ref{(iv)}), we use (\ref{(iii)}) and (\ref{(i)}).
\hfill $\Box$
\begin{thm} \label{Pompeiu} Let $T$ be an equilateral triangle with circumcircle $\Gamma$, and let $P$ be an arbitrary point in its plane. Then the distances from $P$ to the vertices of $T$ can serve as the side lengths of a triangle $T_P$. Also, $T_P$ is degenerate if and only if $P$ lies on $\Gamma$.
\end{thm}
\vspace{.15cm} \noindent {\it Proof.}
Let $a$ be the side lengths of $T$, and let $x$, $y$, and $z$ be the distances from $P$ to the vertices of $T$. By the case $n=2$ of (\ref{REL}), we have
\begin{eqnarray} \label{g}
g : = 3(a^4+x^4+y^4+z^4) - \left(a^2+x^2+y^2+z^2\right)^2 &=& 0.
\end{eqnarray}
This polynomial $g$ simplifies into
\begin{eqnarray*}
g &=& 2[a^4 -a^2 (x^2+y^2+z^2)]+3(x^4+y^4+z^4)- (x^2+y^2+z^2)^2\nonumber\\
&=& 2\left( a^2 - \frac{x^2+y^2+z^2}{2}\right)^2
- \frac{3}{2} (x^2+y^2+z^2)^2
+3(x^4+y^4+z^4) \nonumber\\
&=& 2\left( a^2 - \frac{x^2+y^2+z^2}{2}\right)^2
- \frac{3}{2}
\left[2(x^2y^2+y^2z^2+z^2x^2) - (x^4+y^4+z^4)\right]. \nonumber\\
\end{eqnarray*}
Since $g=0$, it follows that
\begin{eqnarray}
h:= 2(x^2y^2+y^2z^2+z^2x^2) - (x^4+y^4+z^4) &\ge& 0, \label{ti}
\end{eqnarray}
with equality if and only if
\begin{eqnarray} \label{equality}
a^2 &=& \frac{x^2+y^2+z^2}{2}.
\end{eqnarray}
By the last part of Lemma \ref{uvh}, this is equivalent to saying that $P$ is on $\Gamma$. Thus we assume that $h > 0$, i.e.,
\begin{eqnarray} \label{ti-2}
(x+y+z)(-x+y+z)(x-y+z)(x+y-z) &>& 0.
\end{eqnarray}
Since $x+y+z > 0$, $h >0$ is equivalent to
\begin{eqnarray} \label{ti-3}
(-x+y+z)(x-y+z)(x+y-z) &>& 0.
\end{eqnarray}
Since the sum of any two of the terms $-x+y+z$, $x-y+z$, and $x+y-z$ is non-negative, it follows that at most one of these terms
is negative. Thus if one of them is negative, then the other two are non-negative, contradicting (\ref{ti-3}).
Therefore the three terms are non-negative, i.e., positive. Therefore
\begin{eqnarray} \label{333}
y+z > x,~~z+x > y, ~~ x+y > z,
\end{eqnarray}
proving that $x$, $y$, and $z$ can serve as the side lengths of a (non-degenerate) triangle.
Thus if $h=0$, $P$ lies on $\Gamma$, and $x$, $y$, and $z$ form the side lengths of a degenerate triangle; if $h > 0$, $P$ does not lie on $\Gamma$, and $x$, $y$, and $z$ form the side lengths of a non-degenerate triangle. This is what we were to prove.
\hfill $\Box$
\section{Two-apexed pre-kites and the limitations on their edge lengths} \label{Jaws}
We define a {\it two-apexed $n$-pre-kite} to be an $n$-simplex $S = [A_0,\cdots,A_n]$, $n \ge 3$, in which two of its facets are regular $(n-1)$-simplices. Notice that a two-apexed $n$-pre-kite is nothing but an $n$-pre-kite with two apexes. It will also be proved in Lemma \ref{1-2}
that a non-regular $n$-pre-kite cannot have more than 2 apexes.
The main theorem in this section, namely Corollary \ref{cor}, gives necessary and sufficient conditions on given positive numbers so that they can serve as the edge lengths of a two-apexed $n$-pre-kite.
\bigskip
\begin{lem} \label{1-2}
A non-regular $n$-pre-kite, $n \ge 3$, can have at most two apexes.
\end{lem}
\vspace{.15cm} \noindent {\it Proof.} Let $S = [A_0, \cdots, A_n]$ be an $n$-pre-kite, and suppose that $A_0$, $A_1$, and $A_2$ are three apexes. We are to prove that $S$ is regular. Since $A_2A_3$
is an edge in both facets $S_0$ and $S_1$, and since these facets are regular, it follows that the facets $S_0$ and $S_1$ have the same edge length. Similary, we show that
the facets $S_0$ and $S_2$ have the same edge length.
But every edge in $S$ is an edge in one of the facets $S_0$, $S_1$, and $S_2$. Thus all edges of $S$ have the same length. \hfill $\Box$
\begin{thm} \label{limitations}
Let $S = [A_0, \cdots, A_n]$ be a regular $n$-simplex that lies in $\mathbb{R}^m$ for some $m \ge n+1$, and let its side length be $u$ and its circumradius be $R$. Let $Q_0$ be the reflection of $A_0$ about the affine hull $H_0$ of the facet $S_0 = [A_1,\cdots,A_n]$, and let $\Omega$ be the set of all points $Q$ in $\mathbb{R}^m$ for which the $n$-simplex $[Q, A_1, \cdots, A_n]$ is regular.
Then
\begin{eqnarray}
\{\|Q - A_0\| : Q \in \Omega\} &=& \left[ 0, \sqrt{\frac{2(n+1)}{n}} u\right] \label{lim}\\
&=& \left[ 0, \frac{2(n+1)}{n} R \right], \label{limm}
\end{eqnarray}
with the extreme values taken at $Q=A_0$ and at $Q=Q_0$.
\end{thm}
\vspace{.15cm} \noindent {\it Proof.} Without loss of generality, we assume that the center of $S$ lies at the origin $\mathcal{O}$ of $\mathbb{R}^m$.
Let $\mathcal{C}} \newcommand{\DDD}{\mathcal{D}}\newcommand{\EEE}{\mathcal{E}$ be the collection of all $(n+1)$-dimensional subspaces of $\mathbb{R}^m$ that contain $S$, and for any $V \in \mathcal{C}} \newcommand{\DDD}{\mathcal{D}}\newcommand{\EEE}{\mathcal{E}$, let $L_V = \{\|Q - A_0\| : Q \in \Omega \cap V\}$. Let $LHS$ and $RHS$ stand for the left and right hand sides of (\ref{lim}), respectively. If we could prove that $L_V = RHS$, then we will be done. This is because every point in $\mathbb{R}^m$ belongs to some $V \in \mathcal{C}} \newcommand{\DDD}{\mathcal{D}}\newcommand{\EEE}{\mathcal{E}$, and hence $LHS$ is the union of all $L_V$, where $V$ ranges in $\mathcal{C}} \newcommand{\DDD}{\mathcal{D}}\newcommand{\EEE}{\mathcal{E}$, and since $RHS$ depends on $S$ only. Thus we take any $V \in \mathcal{C}} \newcommand{\DDD}{\mathcal{D}}\newcommand{\EEE}{\mathcal{E}$, and we are to prove that $L_V = RHS$. In other words, $$\mbox{we assume that $m = n+1$, and we set $\mathbb{R}^m = V$}.$$
Let $G$ be the center of $S_0$, and let $H_0$ be the affine hull of $S_0$. Let $W = \{ P \in V : (P-G) \perp H_0\}$. Thus $W$ is the shifted orthogonal complement of $H_0$, namely $W - G = (H_0 - G)^{\perp}$. Thus $\mbox{dim} W = \mbox{dim} V - \mbox{dim} H_0 = (n+1)-(n-1) = 2,$ i.e., $W$ is a plane. We have used the facts that
if $A$ is a closed subspace of a Hilbert space $B$, then $B = A \oplus A^{\perp}$ (\cite[Theorem 3.3-4, p.~146]{Kry}), and that every finite dimensional subspace of a normed space is closed (\cite[Theorem 2.4-5, p.~74]{Kry}).
Since $G$ is the center of the facet $S_0$ of the regular $n$-simplex $S$, it follows that $A_0 - G$ is an altitude of $S$, i.e., $(A_0-G) \cdot H_0 = 0$, and hence $A_0 \in W$. Similarly, $Q_0 \in W$.
Let $\Gamma$ be the circle in $W$ centered at $G$ and passing through $A_0$ (and $Q_0$). We claim that $\Omega = \Gamma$. To see this, let $P \in \Gamma$, and let $1 \le i \le n$. Then $P \in W$ and hence $(G-P) \perp H_0$. Therefore
\begin{eqnarray*}
\|P-A_i\|^2 &=& \|P- G\|^2 + \| G-A_i\|^2, \mbox{~by Pythagoras' theorem}\\
&=& \|A_0 - G\|^2 + \| G- A_i\|^2, \mbox{~because $P \in \Gamma$}\\
&=& \|A_0- A_i\|^2, \mbox{~by Pythagoras' theorem}.
\end{eqnarray*}
This shows that $[P,A_1,\cdots,A_n]$ is regular, and therefore $P \in \Omega$. Conversely, let $P \in \Omega$. Thus $T=[P, A_1,\cdots,A_n]$ is regular. Since $G$ is the center of the facet $S_0$ of $T$, and since $T$ is regular,
it follows that
$(P-G)\perp H_0$, and hence $P \in W$. In particular, $A_0, Q_0 \in W$.
Therefore
\begin{eqnarray*}
\|P-G\|^2 &=& \|P- A_1\|^2 - \| A_1 -G\|^2, \mbox{~by Pythagoras' theorem}\\
&=& \|A_2- A_1\|^2 - \| A_1 -G\|^2, \mbox{~because $[P,A_1,\cdots,A_n]$ is regular}\\
&=& \|A_0- A_1\|^2 - \| A_1 -G\|^2, \mbox{~because $[A_0,A_1,\cdots,A_n]$ is regular}\\
&=& \|A_0- G\|^2, \mbox{~by Pythagoras' theorem}.
\end{eqnarray*}
Therefore $P \in \Gamma$. Thus we have shown that $\Gamma = \Omega$.
Since $S=[A_0,\cdots,A_n]$ is regular and since $G$ is the center of its facet
$S_0 = [A_1,\cdots,A_n]$, it follows that $A_0G$ is perpendicular to $H_0$. Also $A_0Q_0$ is perpendicular to $H_0$. Therefore $A_0$, $G$, and $Q_0$ are collinear. Since $A_0$ and $Q_0$ lie on $\Gamma$, and since $G$ is the center of $\Gamma$, it follows that $A_0Q_0$ is a diameter of $\Gamma$ (with midpoint $G$). Therefore, as $Q$ moves on $\Gamma$, $\|Q-A_0\|$ takes all values between $0$ and $\|Q_0-A_0\|$. Thus our proof will be complete if we prove that $$\|Q_0-A_0\| = \frac{2(n+1)}{n} R ~=~ \sqrt{\frac{2(n+1)}{n}} u.$$
But this follows immediately from the formula for $\| A_0 - G\|$ given in (\ref{(ii)}) and the fact that
\begin{eqnarray*}
\|A_0 - Q_0\| &=& 2 \|A_0 - G\|.
\end{eqnarray*}
This completes the proof.
\hfill $\Box$
\bigskip The following corollary will be used in a later section.
\begin{cor} \label{cor} There exists an $n$-simplex having one edge of length $v > 0$ and having all the remaining edges of lengths $u > 0$ if and only if
\begin{eqnarray}\label{uv}
0 < \frac{v}{u} < \sqrt{\frac{2n}{n-1}}.
\end{eqnarray}
In other words, there exists a two-apexed $n$-pre-kite $PK[n;u;v_1,\cdots, v_n]$, with $v_1=\cdots,v_{n-1}=u$ and $v_n=v$ if and only if $u$ and $v$ satisfy
(\ref{uv}).
\end{cor}
\vspace{.2cm} \noindent {\it Proof.} The previous theorem shows that
there exists an $(n+1)$-simplex having one edge of length $v > 0$ and having all the remaining edges of lengths $u > 0$ if and only if
\begin{eqnarray}
0 < \frac{v}{u} < \sqrt{\frac{2(n+1)}{n}}.
\end{eqnarray}
The desired result is obtained by replacing $n+1$ by $n$. \hfill $\Box$
\section{Coincidence of two of the classical centers of a pre-kite} \label{coincidence}
The classical centers of an $n$-simplex $S=[A_0,\cdots,A_n]$ refer to the circumcenter, the incenter, and the centroid
of $S$. The {\it circumcenter} $\QQQ = \QQQ (S)$ of $S$ is the center of the $(n-1)$-sphere that passes through the vertices of $S$. The {\it incenter} $\III = \III (S)$ of $S$ is the center of the $(n-1)$-sphere that touches the facets internally, i.e., at points that lie in the convex hulls of the facets. The {\it centroid} $\GGG = \GGG (S)$ of $S$ is defined inductively to be the intersection of the medians of $S$, where a {\it median} of $S$ is the line segment joining a vertex of $S$ to the centroid of the opposite facet. It is also defined by the simple formula
$$\GGG (S) = \frac{A_0 + \cdots + A_n}{n+1}.$$
Theorem \ref{O=G} proves that if the circumcenter and the centroid of an $n$-pre-kite $S$, $n \ge 2$, coincide, then it is regular.
Theorem \ref{O=I} proves that if the circumcenter and the incenter of an $n$-pre-kite $S$, $n \ge 2$, coincide, then it is regular.
Theorem \ref{I=G} proves that if the incenter and the centroid of an $n$-pre-kite $S$ coincide, and if $n \le 5$, then it is regular, and exhibits examples of non-regular $n$-pre-kite $S$, $n \ge 6$, in which the incenter and the centroid coincide.
The main tools in proving Theorems \ref{O=G}, \ref{O=I}, and \ref{I=G}
are the formulas established in Theorem \ref{CjDj}, together with
the following theorem, proved in \cite[Theorem 3.2, p.~496]{BAG-1}. We recall that an $n$-simplex is said to be {\it equiareal} if its facets have equal volumes, i.e., equal $(n-1)$-dimensional Lebesgue measures.
It is said to have {\it well-distributed} edge lengths if its facets have equal variance, i.e., if the sum of squares of the edge lengths of a facet is the same for all facets. It is said to be {\it equiradial} if its facets have equal circumradii.
\begin{thm} \label{BAG-1}
Let $S = [A_0,\cdots,A_n]$ be an $n$-simplex.
Then
\begin{enumerate}
\item[(i)] The circumcenter $\QQQ$ and the centroid $\GGG$ of $S$ coincide if and only if $S$ has {\it well-distributed} edge lengths.
\item[(ii)] The circumcenter $\QQQ$ and the incenter $\III$ of $S$ coincide if and only if $\QQQ$ is interior and $S$ is equiradial.
\item[(iii)] The centroid $\GGG$ and the incenter $\III$ of $S$ coincide if and only if $S$ is equiareal.
\item[(iv)] The centroid $\GGG$, the circumcenter $\QQQ$, and the incenter $\III$ of $S$ coincide if and only if two of the conditions
\begin{itemize}
\item[(a)] $S$ has well-distributed edge lengths,
\item[(b)] $S$ is equiradial,
\item[(c)] $S$ is equiareal
\end{itemize}
hold. When this happens, the third condition also holds.
\end{enumerate}
\end{thm}
\begin{thm} \label{O=G}
Let $S = [A_0,\cdots,A_n]$ be an $n$-pre-kite, and suppose that $n \ge 2$. If the circumcenter $\QQQ$ and the centroid $\GGG$ of $S$ coincide,
then $S$ is regular. \end{thm}
\vspace{.1cm}\noindent {\it Proof.}
Suppose that the circumcenter $\QQQ$ and the centroid $\GGG$ of $S$ coincide. By Theorem \ref{BAG-1}, $S$ has well-distributed edge lengths. Then the sum $L_j$, $0 \le j \le n$, of squares of the lengths of the edges of the facet $S_j$ does not depend on $j$. Let $L$ be
the sum of squares of the lengths of the edges of $S$, and let
$M_j$, $0 \le j \le n$, be the sum of squares of the lengths of the edges that emanate from $A_j$. Since $M_j = L - L_j$, it follows that $M_j$ does not depend on $j$.
Assuming that $A_0$ is the apex of the pre-kite $S$, it follows from (\ref{v}) that
\begin{eqnarray}
M_0 &=& v_1^2 + \cdots + v_n^2, \label{00}\\
M_j &=& v_j^2 + (n-1) u^2 \mbox{~~for $0 \le j \le n$.} \label{jj}
\end{eqnarray}
It follows from (\ref{jj}) that
$v_1 = \cdots = v_n$. If $v$ is the common value of
$v_1, \cdots, v_n$, then it follows by subtracting (\ref{00}) from (\ref{jj}) and using that their left hand sides are equal that $(n-1)u^2 = (n-1) v^2$, and hence $v=u$. This shows that $S$ is regular, and ends the proof. \hfill $\Box$
\begin{thm} \label{O=I}
Let $S = [A_0,\cdots,A_n]$ be an $n$-pre-kite, and suppose that $n \ge 2$. If the circumcenter $\QQQ$ and the incenter $\III$ of $S$ coincide,
then $S$ is regular. \end{thm}
\vspace{.1cm}\noindent {\it Proof.}
Suppose that the circumcenter $\QQQ$ and the incenter $\III$ of $S$ coincide. By Theorem \ref{BAG-1}, $S$ is equiradial. Let $R_j$, $0 \le j \le n$, be the circumradius of the $j$-th facet. Let $1 \le j \le n$. Using (\ref{R}), we see that $R_0 = R_j$ if and only if
\begin{eqnarray} \label{CD01}
\mathcal{C}} \newcommand{\DDD}{\mathcal{D}}\newcommand{\EEE}{\mathcal{E}_0 \DDD_j &=& \mathcal{C}} \newcommand{\DDD}{\mathcal{D}}\newcommand{\EEE}{\mathcal{E}_j \DDD_0.
\end{eqnarray}
By Theorem \ref{CjDj}, (\ref{CD01}) simplifies into
\begin{eqnarray} \label{j-ind}
2(\alpha - nu)v_j&=&\alpha^2 + \beta - 2n\alpha u+n(n-1)u^2.
\end{eqnarray}
If $\alpha = nu$, then it follows from (\ref{j-ind}) that $\beta = nu^2$. By (\ref{CCC-PK-2}), the Cayley-Menger determinant $\mathcal{C}} \newcommand{\DDD}{\mathcal{D}}\newcommand{\EEE}{\mathcal{E} (PK[n;u;\mathbf{v}}\newcommand{\yy}{\mathbf{y}])$ of $S$ is 0. Thus $S$ is degenerate, which we discard. Therefore $\alpha \neq nu$. It now follows from (\ref{j-ind}) that $v_j$ does not depend on $j$. Therefore $v_1=\cdots=v_n$. This means that $S$ is a $n$-kite. By Lemma 4.5 of \cite{RM-ortho},
$S$ is regular.
\hfill $\Box$
\begin{thm}\label{I=G}
Let $S = [A_0,\cdots,A_n]$, $n \ge 2$, be an $n$-pre-kite.
Suppose that the incenter $\III$ and the centroid $\GGG$ of $S$ coincide, i.e., $S$ is equiareal.
If $n\le 5$, then $S$ is regular.
If $n \ge 6$, then $S$ is not necessarily regular; i.e. there exist non-regular $n$-pre-kites, in fact two-apexed $n$-pre-kites, in which the incenter and centroid coincide.
\end{thm}
\vspace{.15cm}\noindent {\it Proof.} Suppose that $S=[A_0,\cdots,A_n]$ is a non-regular equiareal $n$-pre-kite with apex $A_0$, say
$$S = PK [n; u; v_1,\cdots,v_n].$$
Let $\VVV_j$ be the volume of the $j$-th facet $S_j$ of $S$,
and let $\mathcal{C}} \newcommand{\DDD}{\mathcal{D}}\newcommand{\EEE}{\mathcal{E}_j$ be the Cayley-Menger determinant of $S_j$, as defined in (\ref{CCC}). By Theorem \ref{CjDj},
$$
\mathcal{C}} \newcommand{\DDD}{\mathcal{D}}\newcommand{\EEE}{\mathcal{E}_j = \left\{ \begin{array}{ll}
(-u)^{n-3} \left[ - \alpha^2 + (n-1) \beta - n v_j^2 + 2 \alpha v_j \right]
& \mbox{if $1\le j\le n$},\\
\vspace{.15cm}
(-1)^n n u^{n-1} & \mbox{if $j = 0$}.
\end{array} \right.
$$
By (\ref{V}), $\VVV_i = \VVV_j \ifff \mathcal{C}} \newcommand{\DDD}{\mathcal{D}}\newcommand{\EEE}{\mathcal{E}_i = \mathcal{C}} \newcommand{\DDD}{\mathcal{D}}\newcommand{\EEE}{\mathcal{E}_j$.
We prove first that $v_1, \cdots, v_n$ cannot be all equal. In fact, if $v_1=\cdots=v_n (= x, \mbox{~say})$, then the condition $\VVV_1 = \VVV_0$
yields $u=x$ as follows:
\begin{eqnarray*}
\VVV_1 = \VVV_0 &\ifff& nu^2 - \alpha^2 + (n-1)\beta - nx^2 + 2 \alpha x = 0\\
&\ifff& 2(n-1)u^2 + 2ux (1-n) = 0\\
&\ifff& 2(n-1)u (u-x) = 0\\
&\ifff& u = x.
\end{eqnarray*}
Then $S$ is regular, contradicting the assumptions.
Thus we assume that $v_1,\cdots,v_n$ are not all equal.
Next, we prove that there do not exist three distinct indices $i, j, k$ in $\{1, 2, \cdots, n\}$ such that $v_i$, $v_j$, and $v_k$ are pairwise different. This is because the existence of such indices contradicts the assumption $\VVV_i = \VVV_j = \VVV_k$. In fact, \begin{eqnarray*}
&&\VVV_i = \VVV_j = \VVV_k ~\ifff~ \mathcal{C}} \newcommand{\DDD}{\mathcal{D}}\newcommand{\EEE}{\mathcal{E}_i = \mathcal{C}} \newcommand{\DDD}{\mathcal{D}}\newcommand{\EEE}{\mathcal{E}_j = \mathcal{C}} \newcommand{\DDD}{\mathcal{D}}\newcommand{\EEE}{\mathcal{E}_k\\
&\ifff& - n v_i^2 + 2 \alpha v_i = - n v_j^2 + 2 \alpha v_j = - n v_k^2 + 2 \alpha v_k \\
&\ifff& (v_i - v_j) \left[- n (v_i + v_j) + 2 \alpha\right] =
(v_j - v_k) \left[- n (v_j + v_k) + 2 \alpha \right] = 0
\\
&\ifff& - n (v_i + v_j) + 2 \alpha=- n (v_j + v_k) + 2 \alpha = 0\\
&\Longrightarrow& - n (v_i + v_j) = -n (v_j + v_k)
\\ &\Longrightarrow& v_i = v_k,
\end{eqnarray*}
a contradiction. Therefore no three of the numbers $v_1,\cdots,v_n$ are pairwise distinct.
Thus there are two different numbers $x$ and $y$ and an index $t \in \{1, \cdots, n\}$ such that
\begin{eqnarray}
\mbox{$v_j = x$ if $1 \le j \le t$, and $v_j = y$ if $t < j \le n$}. \end{eqnarray}
Let $s = n-t$. We may clearly assume that $t \ge s$. Thus $S$ is of the form
\begin{eqnarray}
S &=& PK[n;u;\overbrace{x,\cdots,x}^t,\overbrace{y,\cdots,y}^s], \end{eqnarray}
where $x$ is repeated $t$ times and $y$ is repeated $s = n-t$ times.
Let $i$ and $j$ be such that $v_i = x$ and $v_j = y$.
Then
\begin{eqnarray*}
\VVV_i = \VVV_j &\ifff& - n x^2 + 2 \alpha x = - n y^2 + 2 \alpha y\\
&\ifff& (x-y) \left[- n (x+y) + 2 \alpha \right]=0\\
&\ifff& - n (x+y) + 2 \alpha =0\\
&\ifff& - n (x+y) + 2 (tx + sy) + 2 u =0\\
&\ifff& (-n + 2t) x+ (-n+2s) y +2u =0\\
&\ifff& (t-s) x+ (s-t) y +2u =0\\
&\ifff& (t-s)(y-x) = 2u.
\end{eqnarray*}
Also,
\begin{eqnarray*}
\VVV_i = \VVV_0 &\ifff& - \alpha^2 + (n-1)\beta - nx^2 + 2 \alpha x = -nu^2\\
&\ifff& nu^2 - \alpha^2 + (n-1)\beta - nx^2 + 2 \alpha x = 0.
\end{eqnarray*}
Therefore
\begin{eqnarray}
\mbox{$S$ is equiareal} &\ifff& \VVV_i = \VVV_0 \mbox{~and~} \VVV_i= \VVV_j\nonumber \\
&\ifff& (i)~ nu^2 - \alpha^2 + (n-1)\beta - nx^2 + 2 \alpha x = 0, \mbox{~and~}\nonumber \\
&&(ii) ~(t-s)(y-x) = 2u. \label{i-ii}
\end{eqnarray}
Thus equiareality of $S$ is equivalent to fulfilment of (i) and (ii) of (\ref{i-ii}).
Since $u \ne 0$, these imply that
\begin{eqnarray}
t &\ne& s. \label{ts}
\end{eqnarray}
Let us first treat the case $$s=1,~~t=n-1.$$ In this case,
\begin{eqnarray*}
&&\mbox{$S$ is equiareal}\\
&\ifff& (i) ~2nu^2 - \alpha^2 + (n-1)\beta - nx^2 + 2 \alpha x = 0, \mbox{~and~}\\
&&(ii)~ (n-2)(y-x) = 2u.
\end{eqnarray*}
Plugging
$2u = (n-2)(y-x)$ in (i), we obtain
$(n-1) (n-2) (x-y) (x n - y n + 2 y)=0$, i.e., $x n - y n + 2 y=0$. Solving this with $2u = (n-2)(y-x)$, we obtain $x=u$. Thus one of the edge lengths of $S$ is
$$\frac{un}{n-2},$$ and each other edge length is $u$.
In view of Corollary \ref{cor},
\begin{eqnarray*}
\mbox{such an $n$-simplex exists} &\ifff& \frac{n}{n-2} < \sqrt{\frac{2n}{n-1}}\\
&\ifff& n^2 (n-1) < 2n (n-2)^2 \\
&\ifff& n^2 - 7n + 8 > 0\\
&\ifff& n > \frac{7 + \sqrt{17}}{2} \approx 5.6\\
&\ifff& n \ge 6.
\end{eqnarray*}
Thus if $n \ge 6$, there are non-regular equiareal $n$-pre-kites. These can even be chosen to be of the form $PK[n;u;v_1,\cdots,v_n]$ with $v_1=\cdots=v_{n-1}=u$, i.e., a two-apexed $n$-pre-kite.
If $n \le 5$, then a non-regular equiareal $n$-pre-kite must have
\begin{eqnarray}
t, s &\ge& 2. \label{ge2}
\end{eqnarray}
In view of (\ref{ts}), this is possible only if $(n,t,s) = (5,3,2)$. We show now that this cannot happen either. In fact, this assumption would imply that
$$\alpha = u + 3x + 2y,~~\beta = u^2 + 3x^2 + 2y^2,~~2u = y- x.$$
Substituing these values of $\alpha$, $\beta$, and $u$ in (ii) and factorizing, we obtain $$(y-x)(y-2x) = 0.$$
Since $y \ne x$, it follows that $y=2x$ and $u = x/2$. Thus our $5$-pre-kite is of the form $PK[5;u;2u,2u,2u,4u,4u]$. Using (\ref{CCC-PK-2}), we see that the Cayley-Menger determinant of such a pre-kite is
$$(-u)^{3} [5(u^2+3(4u^2)+2(16u^2)) - (u + 3(2u)+2(4u))^2] = (-u)^{3} [(5)(45u^2) - (15u)^2)] = 0.$$
Thus this pre-kite, if it exists, is degenerate, which we reject.
This completes the proof. \hfill $\Box$
\begin{que}
{\rm The {\it Fermat-Torricelli point} $\mathcal{F}} \newcommand{\GGG}{\mathcal{G}} \newcommand{\HHH}{\mathcal{H}}\newcommand{\III}{\mathcal{I} = \mathcal{F}} \newcommand{\GGG}{\mathcal{G}} \newcommand{\HHH}{\mathcal{H}}\newcommand{\III}{\mathcal{I} (S)$ of an $n$-simplex $S$ is the point whose distances from the vertices of $S$ have a minimal sum. It is often thought of as a semi-classical (or even a classical) center. Thus it is natural to investigate the degrees of regularity implied by the coincidences
$\mathcal{F}} \newcommand{\GGG}{\mathcal{G}} \newcommand{\HHH}{\mathcal{H}}\newcommand{\III}{\mathcal{I}=\GGG$, $\mathcal{F}} \newcommand{\GGG}{\mathcal{G}} \newcommand{\HHH}{\mathcal{H}}\newcommand{\III}{\mathcal{I}=\III$, and $\mathcal{F}} \newcommand{\GGG}{\mathcal{G}} \newcommand{\HHH}{\mathcal{H}}\newcommand{\III}{\mathcal{I}=\QQQ$. In this regard, we recall Theorem 3.1, p. 496, of \cite{BAG-1}. This states that if any two of the three centers $\mathcal{F}} \newcommand{\GGG}{\mathcal{G}} \newcommand{\HHH}{\mathcal{H}}\newcommand{\III}{\mathcal{I}$, $\QQQ$, and $\GGG$ coincide, then all the three coincide. Thus each of the coincidences $\mathcal{F}} \newcommand{\GGG}{\mathcal{G}} \newcommand{\HHH}{\mathcal{H}}\newcommand{\III}{\mathcal{I}=\GGG$ and $\mathcal{F}} \newcommand{\GGG}{\mathcal{G}} \newcommand{\HHH}{\mathcal{H}}\newcommand{\III}{\mathcal{I}=\QQQ$ implies that $\QQQ=\GGG$, and hence regularity (by Theorem \ref{O=G} above). This leaves us with the question about the degree of regularity implied by the coincidence $\mathcal{F}} \newcommand{\GGG}{\mathcal{G}} \newcommand{\HHH}{\mathcal{H}}\newcommand{\III}{\mathcal{I}=\III$. We leave this open.}
\end{que}
\section{Pre-kites in the four special families of simplices} In this section, we shall see how the new family of $n$-pre-kites is related to the four known special families of orthocentric, circumscriptible, isodynamic, and tetra-isogonic $n$-simplices.
We recall that an $n$-simplex $S = [A_1,\cdots,A_{n+1}]$, $n \ge 2$, is
said to be {\it orthocentric} if the altitudes of $S$ are concurrent.
It is said to be {\it circumscriptible} (or {\it edge-incentric} or {\it balloon}) if there is an $(n-1)$-sphere that touches all its edges internally.
It is said to be {\it isodynamic} if the incentral cevians are concurrent. Here, an {\it incentral cevian} is the cevian that joins a vertex and the incenter of the opposite facet. It is said to be {\it tetra-isogonic} if every four vertices of $S$ form an isogonic tetrahedron, i.e., a tetrahedron whose inspherical cevians are concurrent. Here, an {\it inspherical cevian} is the cevian that joins a vertex to the point where the insphere touches the opposite face. Other characterizations appear in \cite{RM-HHM}.
Let us denote the families of orthocentric, circumscriptible, isodynamic, and tetra-isogonic $n$-simplices and the families of $n$-kites and $n$-pre-kites by $F_o$, $F_c$, $F_d$, $F_g$, $F_k$, and $F_p$, respectively.
It is proved in \cite{kites} that the intersection of any two of the families \begin{eqnarray}\label{4four}
F_o, ~F_c, ~F_d, ~\mbox{and~} F_g
\end{eqnarray}
is the family $F_k$. In this section, we prove that this still holds if we enlarge the list in (\ref{4four}) to include our new family $F_p$. We prove this in Theorem \ref{5five}. The proof is a consequence of the following theorem, which is taken from \cite{kites}.
\begin{thm} \label{HHM}
Let $S = [A_1,\cdots,A_{d+1}]$, $d \ge 2$, be a $d$-simplex. Then
\begin{eqnarray*}
\mbox{$S$ is orthocentric} &\ifff&\mbox{there exist $\beta_1, \cdots, \beta_{d+1} \in \mathbb{R}$ such that} \nonumber \\
&&\mbox{$\|A_i-A_j\|^2=\beta_i + \beta_j$
for $1 \le i < j \le d+1$}\\
\mbox{$S$ is circumscriptible} &\ifff& \mbox{there exist $\beta_1, \cdots, \beta_{d+1} > 0$ such that} \nonumber \\ &&\mbox{$\|A_i-A_j\|=\beta_i + \beta_j$ for $1 \le i < j \le d+1$}\\
\mbox{$S$ is isodynamic} &\ifff&\mbox{there exist $\beta_1, \cdots, \beta_{d+1} > 0$ such that} \nonumber \\ &&\mbox{$\|A_i-A_j\|^2 =\beta_i \beta_j$ for $1 \le i < j \le d+1$}\\
\mbox{$S$ is tetra-isogonic} &\ifff&\mbox{there exist $\beta_1, \cdots, \beta_{n+1} > 0$ such that} \nonumber \\ &&\mbox{$\|A_i-A_j\|^2=\beta_i^2 + \beta_i \beta_j + \beta_j^2$ for $1 \le i < j \le d+1$}.
\end{eqnarray*}
Moreover, the numbers $\beta_i$, $1 \le i \le d+1$, appearing in the four equations are unique.
\end{thm}
\begin{thm} \label{5five}
For $n \ge 3$, the intersection of any two of the five families
\begin{eqnarray}\label{4}
F_o, ~F_c, ~F_d, ~F_g, ~\mbox{and~} F_p
\end{eqnarray}
is the family $F_k$.
\end{thm}
\vspace{.2cm} \noindent{\it Proof.}
In view of the fact, proved in \cite{kites}, that
the intersection of any two of the four families $F_o$, $F_c$, $F_d$, and $F_g$ is the family $F_k$, it remains to show that
\begin{eqnarray}\label{55five}
F_o \cap F_p = F_c \cap F_p = F_d \cap F_p = F_g \cap F_p = F_k.
\end{eqnarray}
For this, Theorem \ref{HHM} is very useful. Since the proofs of these statements are similar, we find it sufficient to prove the last statement only, i.e.,
\begin{eqnarray}\label{FgFp}
F_g \cap F_p = F_k.
\end{eqnarray}
Thus let $S = [A_0,\cdots,A_n]$, $n \ge 3$, be the $n$-pre-kite
$PK [n;u;v_1,\cdots,v_n]$ with apex $A_0$, and suppose that $S$ is
tetra-isogonic. By Theorem \ref{HHM}, there
exist $\beta_0, \cdots, \beta_{n} > 0$ such that
\begin{eqnarray}
\|A_i-A_j\|^2=\beta_i^2 + \beta_i \beta_j + \beta_j^2 \mbox{~~ for $0 \le i < j \le n$}.
\end{eqnarray}
By the definition of $PK[n;u;v_1,\cdots,v_n]$, we see that
\begin{eqnarray}
v_j &=& \beta_0^2 + \beta_0 \beta_j + \beta_j^2 \mbox{~~ for $1 \le j \le n$},\\
u &=& \beta_i^2 + \beta_i \beta_j + \beta_j^2 \mbox{~~ for $1 \le i < j \le n$}.
\end{eqnarray}
It follows from the second equation that
\begin{eqnarray}
u= \beta_1^2 + \beta_1 \beta_i + \beta_i^2
&=&
\beta_1^2 + \beta_1 \beta_j + \beta_j^2
\mbox{~~ for $2 \le i < j \le n$}.
\end{eqnarray}
Therefore $(\beta_i - \beta_j) (\beta_1 + \beta_i + \beta_j) = 0$
for $2 \le i < j \le n$.
Since $\beta_1 + \beta_i + \beta_j > 0$, it follows that
$\beta_i = \beta_j$ for $2 \le i < j \le n$. By symmetry, we conclude that
$\beta_i = \beta_j$ for $1 \le i < j \le n$. Letting $\beta$ be the common value of $\beta_1, \cdots, \beta_n$, we see that
\begin{eqnarray}
v_j &=& \beta_0^2 + \beta_0 \beta + \beta^2 \mbox{~~ for $1 \le j \le n$}.
\end{eqnarray}
This shows that $S$ is an $n$-kite. \hfill $\Box$
| {
"timestamp": "2017-02-01T02:01:27",
"yymm": "1701",
"arxiv_id": "1701.08833",
"language": "en",
"url": "https://arxiv.org/abs/1701.08833",
"abstract": "The investigation of the relation among the distances of an arbitrary point in the Euclidean space $\\mathbb{R}^n$ to the vertices of a regular $n$-simplex in that space has led us to the study of simplices having a regular facet. Calling an $n$-simplex with a regular facet an $n$-pre-kite, we investigate, in the spirit of [4], [10], [9], and [15], and using tools from linear algebra, the degree of regularity implied by the coincidence of any two of the classical centers of such simplices. We also prove that if $n \\ge 3$, then the intersection of the family of $n$-pre-kites with any of the four known special families is the family of $n$-kites, thus extending the result in [18]. A basic tool is a closed form of a determinant that arises in the context of a certain Cayley-Menger determinant, and that generalizes several determinants that appear in [9], [15], and [16]. Thus the paper is a further testimony to the special role that linear algebra plays in higher dimensional geometry.",
"subjects": "Metric Geometry (math.MG)",
"title": "Pre-kites: Simplices having a regular facet",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9728307716151471,
"lm_q2_score": 0.7279754607093178,
"lm_q1q2_score": 0.7081969291587379
} |
https://arxiv.org/abs/2102.12027 | The prelimit generator comparison approach of Stein's method | This paper uses the generator comparison approach of Stein's method to analyze the gap between steady-state distributions of Markov chains and diffusion processes. The "standard" generator comparison approach starts with the Poisson equation for the diffusion, and the main technical difficulty is to obtain bounds on the derivatives of the solution to the Poisson equation, also known as Stein factor bounds. In this paper we propose starting with the Poisson equation of the Markov chain; we term this the prelimit approach. Although one still needs Stein factor bounds, they now correspond to finite differences of the Markov chain Poisson equation solution rather than the derivatives of the solution to the diffusion Poisson equation. In certain cases, the former are easier to obtain. We use the $M/M/1$ model as a simple working example to illustrate our approach. | \section{Introduction.}\label{intro}
\section{Introduction}
\label{sec:introduction}
Recent years have seen growing use of the generator comparison approach of Stein's method to establish rates of convergence for steady-state diffusion approximations of Markov chains. One very active area has been the study of queueing and service systems, e.g.,\ \cite{Stol2015,Gurv2014, BravDai2017,BravDaiFeng2016,Ying2016, Ying2017, DaiShi2017, GurvHuan2018, FengShi2018, LiuYing2019, bravgurvhuan2020, Brav2020, BravDaiFang2020}. In the typical setup, one considers a parametric family of continuous-time Markov chains (CTMCs) $\{X(t)\}$ taking values in some discrete state space. This family is often termed the \emph{prelimit} sequence. As the parameters tend to some asymptotic limit, the prelimit sequence converges to a limiting diffusion process $\{Y(t)\}$. In queueing, for example, the CTMC parameters are usually the arrival rate, number of servers, and service rate, and one common asymptotic regime is where the system utilization approaches one, also known as the heavy-traffic regime. To allow for general CTMC families, we assume the CTMC takes values in $\delta \mathbb{Z}^{d} = \{\delta k :\ k \in \mathbb{Z}^{d}\}$ where $\delta > 0$ is a parameter of the CTMC and the asymptotic regime of interest has $\delta$ converging to zero. To simplify notation, we omit the dependence of the CTMC on $\delta$ (or any other parameters). Let $X$ and $Y$ denote vectors having the stationary distribution of the CTMC and diffusion, respectively. We emphasize that these refer to the stationary distributions and not the stochastic processes $\{X(t)\}$ and $\{Y(t)\}$. The generator approach of Stein's method has been used to study the rates of convergence of $X$ to $Y$ as $\delta \to 0$. The generator approach is attributed to \cite{Barb1988,Barb1990} and \cite{Gotz1991}, which were the first papers to connect Stein's method to generators of diffusions and CTMCs.
The limiting factor in the generator comparison approach is the curse of dimensionality, because the distance between $X$ and $Y$ depends on the derivatives of the solution to the Poisson equation of the diffusion. In the literature on Stein's method, the Poisson equation is also referred to as the Stein equation. When the diffusion is multidimensional, the Poisson equation is a second-order partial differential equation (PDE), and obtaining derivative bounds, also known as Stein factor bounds, becomes a challenge. The present paper is concerned with expanding the technical toolbox for getting multidimensional Stein factor bounds. Before discussing our contribution, let us examine this problem in detail.
Recall that $X \in \delta \mathbb{Z}^{d}$ and assume $Y \in \mathbb{R}^{d}$. Let $G_X$ and $G_Y $ be the infinitesimal generators of the CTMC and diffusion, respectively. Suppose $G_X$ has the form
\begin{align}
G_X f(\delta k) =&\ \sum_{k' \in \mathbb{Z}^{d} } q_{k,k'} (f(\delta k')-f(\delta k)), \quad k \in \mathbb{Z}^{d}, \label{eq:generalctmc}
\end{align}
where $q_{k,k'}$ are the transition rates from $\delta k$ to $\delta k'$. Further suppose the diffusion generator has the form
\begin{align*}
G_Y f(x) = \sum_{i=1}^{d} b_i(x) \frac{\partial}{\partial x_i} f(x) + \frac{1}{2} \sum_{i,j=1}^{d} a_{ij}(x) \frac{\partial^2}{\partial x_i \partial x_j} f(x), \quad x \in \mathbb{R}^{d},
\end{align*}
where $f: \mathbb{R}^{d}\to \mathbb{R}$ is a twice continuously differentiable function and $b(x) = (b_1(x), \ldots, b_d(x))$ and $a(x) = (a_{ij}(x))_{i,j=1}^{d}$ are known as the drift and diffusion coefficient, respectively.
The generator approach works as follows. First, we choose a test function $h^{*}: \mathbb{R}^{d} \to \mathbb{R}$ and consider the Poisson equation
\begin{align}
G_Y f_{h^{*}}(x) =&\ \mathbb{E} h^{*}(Y) - h^{*}(x), \quad x \in \mathbb{R}^{d}. \label{eq:poisson_diffusion}
\end{align}
We use the star superscript above to emphasize that the functions are defined on all of $\mathbb{R}^{d}$. Given an arbitrary random element $W \in \mathbb{R}^{d}$, one can compare $\mathbb{E} h^{*}(Y)$ to $\mathbb{E} h^{*}(W)$ by taking expected values with respect to $W$ above and attempting to bound the left-hand side. Choosing $W = X$ allows us to leverage the fact that $\mathbb{E} G_X f_{h^{*}}(X) = 0$ (under some mild conditions), and so
\begin{align}
\mathbb{E} h^{*}(Y) - \mathbb{E} h^{*}(X) = \mathbb{E} \big( G_Y f_{h^{*}}(X) - G_X f_{h^{*}}(X) \big). \label{eq:main_difference}
\end{align}
In practice, the chosen $h^{*}(x)$ frequently belongs to
\begin{align*}
\text{\rm Lip(1)} = \{h^{*}: \mathbb{R}^{d} \to \mathbb{R} : \abs{h(x)-h(y)} \leq \abs{x-y}, \text{ for all } x,y \in \mathbb{R}^{d} \}.
\end{align*}
This choice is made because functions in $\text{\rm Lip(1)}$ are simple to work with, and because the Wasserstein distance
\begin{align*}
d_{\text{\rm Lip(1)}}(X,Y) = \sup_{h^{*} \in \text{\rm Lip(1)}} \big| \mathbb{E} h^*(X) - \mathbb{E} h^*(Y) \big|
\end{align*}
is convergence determining; i.e.,\ convergence in the Wasserstein distance implies convergence in distribution (see, for instance, \cite{GibbSu2002}).
Bounding the error on the right hand side of \eqref{eq:main_difference} requires bounds on the derivatives of $f_{h^{*}}(x)$. We refer to these as ``derivative bounds.'' Depending on the transition structure of the CTMC, one may also need to bound certain moments of $X$. Usually, the approximation $Y$ is such that \eqref{eq:main_difference} converges to zero at a rate of $\delta$, and to prove this it suffices to bound the second and third derivatives of $f_{h^{*}}(x)$. However, when one seeks approximations $Y$ with convergence rates faster than $\delta$, as was done in \cite{BravDaiFang2020}, for example, one needs to bound fourth- and higher-order derivatives.
When $d = 1$, the explicit form of $f_{h^{*}}(x)$ is known and can be used to get the derivative bounds via a brute-force approach. When $d > 1$, the Poisson equation is a second-order PDE, and the same kind of brute-force analysis cannot be carried out. Instead, one has to rely on the fact that provided it is finite,
\begin{align}
f_{h^{*}}(x) = \int_{0}^{\infty} \big( \mathbb{E}_{Y(0)=x} h^{*}(Y(t)) - \mathbb{E} h^{*}(Y) \big) dt, \quad x \in \mathbb{R}^{d}, \label{eq:diffusionpoissonsolution}
\end{align}
solves the Poisson equation; see any one of \cite{Barb1990,Gotz1991, Gurv2014,GorhMack2016} for a proof. We then have that
\begin{align}
&\frac{\partial}{\partial x_i} f_{h^{*}}(x) \approx \frac{f_{h^{*}}(x + \varepsilon e^{(i)}) - f_{h^{*}}(x)}{\varepsilon} = \frac{1}{\varepsilon} \int_{0}^{\infty} \big( \mathbb{E}_{Y(0) = x + \varepsilon e^{(i)}} h^{*}(Y(t)) - \mathbb{E}_{Y(0) = x} h^{*}(Y(t)) \big) dt. \label{eq:first_deriv}
\end{align}
Higher-order derivatives can be accessed similarly.
There are a few ways to bound \eqref{eq:first_deriv}. In a handful of cases, the distribution of $Y(t)$ is known as a function of $Y(0)$, as in \cite{Barb1990,Gotz1991,GanRollRoss2017, GanRoss2019}, and \cite{Chenetal2019}, but one should not expect to be so lucky in general.
Another approach uses \textit{synchronous couplings}: one diffusion process is initialized at $x$, and another process sharing the same Brownian motion is started at $x + \delta e^{(i)}$. The bound then depends on the coupling time of the two diffusions. This idea was exploited heavily in \cite{GorhMack2016} for instance to study derivative bounds for overdamped Langevin diffusions. There are other approaches aside from synchronous couplings (we review them briefly in Section~\ref{sec:relatedwork}), each with its own merits and drawbacks. Ultimately, however, none are universally applicable to all problems, making derivative bounds a common bottleneck of the generator comparison approach. In this paper, we present a new way to bound the left-hand side of \eqref{eq:main_difference}. Let us illustrate the main steps.
Fix a test function $h: \delta \mathbb{Z}^{d} \to \mathbb{R}$, defined only on the lattice $\delta \mathbb{Z}^{d}$ as opposed to $\mathbb{R}^{d}$ as before. Now, instead of \eqref{eq:poisson_diffusion}, we consider the Poisson equation of the prelimit,
\begin{align}
G_X f_h(\delta k) = \mathbb{E} h(X) - h(\delta k), \quad k \in \mathbb{Z}^{d}. \label{eq:poisson_CTMC}
\end{align}
Proposition 7.1 of \cite{Asmu2003} can be adapted to show that a solution exists provided $\mathbb{E} \abs{h(X)} < \infty$. Furthermore, this solution is unique up to a constant. We are tempted to proceed analogously to \eqref{eq:main_difference} by taking expected values with respect to $Y$, but we cannot do so because $G_X f_h(\delta k)$ is not defined on $\mathbb{R}^{d} \setminus \delta \mathbb{Z}^{d}$. We get around this by interpolating the discrete Poisson equation. Namely, we introduce a spline $A$, which interpolates functions $f: \delta \mathbb{Z}^{d} \to \mathbb{R}$ and results in extended functions $A f: \mathbb{R}^{d} \to \mathbb{R}$. By applying $A$ to both sides of \eqref{eq:poisson_CTMC}, we obtain the interpolated Poisson equation
\begin{align*}
AG_X f_h(x) =&\ \mathbb{E} h(X) - A h (x) , \quad x \in \mathbb{R}^{d}.
\end{align*}
Under some mild conditions on $A f_h(x)$, It\^{o}'s lemma implies $\mathbb{E} G_Y A f_h(Y) = 0$, and so we take expected values with respect to $Y$ to arrive at
\begin{align}
\mathbb{E} h(X) - \mathbb{E} A h (Y) =&\ \mathbb{E} AG_X f_h(Y) \notag \\
=&\ \mathbb{E} AG_X f_h(Y) - \mathbb{E} G_Y A f_h(Y).\label{eq:main_difference_discrete}
\end{align}
To ensure that the convergence of \eqref{eq:main_difference_discrete} to zero implies the convergence of $X$ to $Y$, we again need to ensure that $h(\delta k)$ belongs to a rich-enough class of functions. We describe some convergence-determining classes of grid-restricted test functions in Section~\ref{sec:convdet}. Lastly, to make the right-hand side of \eqref{eq:main_difference_discrete} comparable to \eqref{eq:main_difference}, we want to interchange $A$ and $G_X$. This interchange is possible but results in some error; i.e.,\ $A G_X f_h(x) = G_X A f_h(x) + \text{ error}$.
After this interchange, the right-hand side of \eqref{eq:main_difference_discrete} becomes analogous to \eqref{eq:main_difference} in the sense that the derivatives of $f_{h^{*}}(x)$ that appear in \eqref{eq:main_difference} are replaced by corresponding derivatives of $A f_h(x)$. Our choice of $A$ is such that the derivatives of $A f_h(x) $ correspond to finite differences of $f_h(\delta k)$, thus replacing the problem of establishing derivative bounds by an analogous problem of bounding finite differences. The finite differences of $f_h(\delta k)$ are determined entirely by the Poisson equation, which itself is determined by the transition structure of the CTMC. As such, it is fitting to refer to the $k$th-order finite difference of $f_h(\delta k)$ as the \emph{$k$th-order Stein factor of the CTMC}. We can bound these Stein factors by relying on the fact that
\begin{align}
f_h(\delta k) = \int_{0}^{\infty} \big( \mathbb{E}_{X(0) = \delta k} h(X(t)) - \mathbb{E} h(X) \big) dt, \quad k \in \mathbb{Z}^{d}, \label{eq:generalpoisson}
\end{align}
solves the Poisson equation
and constructing synchronous couplings of the CTMC similar to the diffusion synchronous couplings. Ways to verify that \eqref{eq:generalpoisson} is well defined are discussed in Section~\ref{sec:diffbounds}.
For ease of reference, we refer to our approach as the prelimit generator comparison approach, or simply \emph{prelimit approach}, and to the traditional approach based on \eqref{eq:poisson_diffusion} as the \emph{diffusion approach}. The prelimit and diffusion approaches are in some sense parallel approaches with many conceptual similarities. If we choose $h^{*}(x)$ in \eqref{eq:main_difference} to equal $A h(x)$ from \eqref{eq:main_difference_discrete}, we see that the right-hand sides of \eqref{eq:main_difference} and \eqref{eq:main_difference_discrete} are equal. This means that any bound established via the diffusion approach should, in theory, be attainable via the prelimit approach, and vice versa.
In practice, technical differences can make the prelimit approach more attractive for some models.
First, when working with models that have state-space collapse i.e.,\ when the dimension of the CTMC is higher than that of the diffusion the prelimit approach does not require one to bound the so-called $\mathbb{E} \abs{X_{\perp}}$, which is the distance between the stationary distribution of the CTMC and its projection onto the state-space collapse manifold. This is illustrated in more detail in \cite{Brav2022}, a companion paper in which the prelimit approach is applied to the join-the-shortest-queue model. Second, the diffusion approach can suffer from what we call ``misalignment of synchronous couplings,'' which can complicate the process of getting derivative bounds via diffusion synchronous couplings. We illustrate this issue in Section~\ref{sec:compare_couplings} using a simple example.
Apart from showing how fast $X$ converges to $Y$, the prelimit Poisson equation \eqref{eq:poisson_CTMC} can also be used to establish tightness of a family of steady-state distributions. Tightness has become an important property since the seminal work of \cite{GamaZeev2006}, which initiated a wave of research into justifying steady-state diffusion approximations of queueing systems; see, for instance, \cite{DaiDiekGao2014, BudhLee2009, ZhanZwar2008, Kats2010, YaoYe2012, Tezc2008, GamaStol2012}, and \cite{Gurv2014a}. Roughly speaking, process-level convergence of the CTMC to a diffusion combined with tightness of the CTMC stationary distributions enables one to perform a limit-interchange argument to conclude convergence of steady-state distributions. The bottleneck is usually proving tightness, which has become synonymous with steady-state convergence.
We can use \eqref{eq:poisson_CTMC} to prove tightness as follows. Let $x_{\infty}$ be the fluid equilibrium of the CTMC, and assume for simplicity that $x_{\infty} \in \delta \mathbb{Z}^{d}$ (if not, consider the nearest point in $\delta \mathbb{Z}^{d}$). Pick $h(\delta k) = \abs{\delta k - x_{\infty}}$ and evaluate the Poisson equation at the point $\delta k = x_{\infty}$ to get
\begin{align*}
\mathbb{E} \abs{X - x_{\infty} } = G_{X} f_h(x_{\infty}).
\end{align*}
The right-hand side typically contains CTMC Stein factors up to the second order. We give an example in Section~\ref{sec:tight}. Proving tightness is therefore equivalent to bounding these factors at the \emph{single point} $x_{\infty}$. In contrast, bounding the approximation error of $Y$ requires third-order Stein factor bounds on the \emph{entire support} of $Y$. This highlights the extra work needed for convergence rates as opposed to convergence alone.
The idea of interpolating the discrete Poisson equation can be applied more broadly to the problem of comparing discrete and continuous distributions using Stein's method. To the author's knowledge, anytime Stein's method has been invoked for a discrete-versus-continuous random variable comparison, the starting point has always been the differential equation for the continuous random variable. Furthermore, in most applications of the method, the starting point has been the Stein/Poisson equation for the limiting distribution, whereas we start with the prelimit.
To summarize, our main contribution is the prelimit approach, which depends on two technical components. First, we establish the existence of an interpolator $A$ that satisfies certain convenient properties. Theorem~\ref{thm:1dinterpolant_def} contains the one-dimensional result, which is generalized to multiple dimensions in Theorem~\ref{thm:interpolant_def}. Second, we describe the error of interchanging $A$ with $G_{X}$. Proposition~\ref{lem:interchanged1} contains the one-dimensional result, while Proposition~\ref{lem:mdiminterchange} is the multidimensional generalization.
After illustrating the general framework, we apply it to the $M/M/1$ queueing system to showcase the prelimit approach. The steady-state customer count in the $M/M/1$ model is geometrically distributed, and is approximated by the exponential distribution; convergence rates are presented in Theorem~\ref{thm:mm1}. The $M/M/1$ system is chosen purely for illustrative purposes because of its simplicity, and our convergence rates actually have alternative derivations. For example, one can use existing results on Stein's method for the exponential distribution in Theorem 3.1 of \cite{PekoRoll2011} or Theorem 5.11 of \cite{Ross2011}. Furthermore, the Poisson equation for the $M/M/1$ system is the same as the Stein equation for the geometric distribution, which was first obtained in \cite{peko1996} and is also a special case of the Pascal Stein equation considered in \cite{Scho2001}. Bounds on geometric Stein factors have also been obtained in \cite{Daly2008}. We compare existing bounds with our own in Section~\ref{sec:diffbounds}.
It is important to add that using CTMC synchronous couplings dates back to \cite{Barb1988}, which was the first paper to connect Stein's method to Markov chains (the author of that paper did not use the language ``synchronous coupling''). In that work, the author viewed the Poisson distribution as the steady-state distribution of the infinite server queue. Later, the application of Stein's method to birth-death processes received a thorough treatment in \cite{BrowXia2001}. A more recent example of using CTMC synchronous couplings can be found in \cite{BarbLuczXia2018a,BarbLuczXia2018b}.
The remainder of the paper is structured as follows. In Section~\ref{sec:prelimapproach} we introduce the technical components of the prelimit approach. We then apply the prelimit approach to the $M/M/1$ model and illustrate the synchronous coupling idea in Section~\ref{sec:diffbounds}. We discuss the issue of misalignment of synchronous couplings in Section~\ref{sec:compare_couplings} and conclude in Section~\ref{sec:conclusion}.
\subsection{Related Work on Derivative Bounds}
\label{sec:relatedwork}
Let us briefly discuss several recent works on ways to obtain derivative bounds. In \cite{GorhMack2016} the authors used synchronous couplings to study derivative bounds for overdamped Langevin diffusions with strongly concave drifts. Later in \cite{Gorhetal2019}, the authors relaxed the strongly concave drift assumption to a dissipativity condition and used a combination of synchronous couplings and reflection couplings studied in \cite{Eber2016} and \cite{Wang2016} to establish derivative bounds for a class of fast-coupling diffusions. In \cite{ErdoMackSham2019} the authors establish derivative bounds for an even larger class of diffusions, but still require a dissipativity condition. The strong concavity and dissipativity conditions both imply that the diffusion generator $G_Y$ satisfies
\begin{align}
G_Y V(x) \leq - \alpha \abs{x}^2 + \beta, \quad x \in \mathbb{R}^{d}, \label{eq:dissip}
\end{align}
where $\abs{x}$ denotes the Euclidean norm, $V(x) = \abs{x}^2$, and $\alpha, \beta$ are some positive constants. Condition \eqref{eq:dissip} is also known as $V$-exponential ergodicity (with $V(x) = \abs{x}^2$), see \cite{MeynTwee1993b}.
While the aforementioned papers contain a large list of applications, their results are not directly applicable to many queueing settings because \eqref{eq:dissip} does not hold there. Even one of the most basic diffusion processes in queueing, the piecewise Ohrnstein-Uhlenbeck process used for approximating the many-server queue in \cite{BravDaiFeng2016}, does not satisfy \eqref{eq:dissip}.
Furthermore, the results in these papers hold only for diffusions on the entire space $\mathbb{R}^{d}$. This excludes diffusions with reflecting boundary conditions, such as reflecting Brownian motions that appear as heavy-traffic limits for networks of single-server queueing systems.
Another approach to getting derivative bounds was proposed in \cite{Gurv2014}, where the author used a priori Schauder estimates from PDE theory to bound the derivatives of $f_{h^{*}}(x)$ in terms of $f_{h^{*}}(x)$ and $h(x)$. He then bounded $f_{h^{*}}(x)$ by a Lyapunov function satisfying an exponential ergodicity condition for the diffusion. This approach requires finding a Lyapunov function satisfying an exponential ergodicity condition, which typically requires significant effort, e.g.\ \cite{DiekGao2013, Gurv2014}. Furthermore, in the case of a diffusion with a reflecting boundary, the complexity of the PDE machinery used makes it nontrivial to trace how the a priori Schauder estimates depend on the primitives of the diffusion process.
Most recently, another approach to getting derivative bounds based on Bismut's formula from Malliavin calculus was proposed in \cite{FangShaoXu2018}. The authors required the diffusion coefficient to be constant, and the assumptions imposed on the drift were similar to those in \cite{GorhMack2016}.
\subsection{Notation}
For any $B \subset \mathbb{R}^{d}$, let $\text{Conv}(B)$ denote its convex hull. We use $\mathbb{Z}$ to denote the set of integers and let $\mathbb{N} = \{0,1,2,\ldots\}$. For any $k \in \mathbb{N} $ and $B \subset \mathbb{R}^{d}$, we let $C^{k}(B)$ be the set of all $k$-times continuously differentiable functions $f: B \to \mathbb{R}$. Given a stochastic process $\{Z(t)\} \in D$ and a functional $f: D \to \mathbb{R}$, we write $\mathbb{E}_{x}(f(Z))$ to denote $\mathbb{E}(f(Z)\ |\ Z(0) = x)$. We let $e \in \mathbb{R}^{d}$ be the vector whose elements all equal $1$ and let $e^{(i)}$ be the element with $1$ in the $i$th entry and zeros otherwise. For any $\delta > 0$ and integer $d > 0$, we let $\delta \mathbb{Z}^{d} = \{\delta k :\ k \in \mathbb{Z}^{d}\}$ and define $\delta \mathbb{N}^{d}$ similarly. For any function $f: \delta \mathbb{Z}^{d} \to \mathbb{R}$, we define the forward difference operator in the $i$th direction as
\begin{align*}
\Delta_{i} f(\delta k) = f \big( \delta (k+e^{(i)}) \big) - f(\delta k), \quad k \in \mathbb{Z}^{d}, \ 1 \leq i \leq d,
\end{align*}
and for $j \geq 0$, we define
\begin{align}
\Delta_i^{j+1} f(\delta k) = \Delta_{i}^{j} f(\delta(k+e^{(i)})) - \Delta_{i}^{j} f(\delta k), \label{eq:diffdef}
\end{align}
with the convention that $\Delta_i^{0} f(\delta k) = f(\delta k)$. For a vector $a \in \mathbb{N}^{d}$, we also let
\begin{align*}
\Delta^{a} f(\delta k) =&\ \Delta_{1}^{a_1} \ldots \Delta_{d}^{a_d} f(\delta k),
\end{align*}
and if $f: \mathbb{R}^{d} \to \mathbb{R}$, then
\begin{align*}
\frac{\partial^{a}}{\partial x^{a}} f(x) =&\ \frac{\partial^{a_1}}{\partial x_1^{a_1}} \ldots \frac{\partial^{a_d}}{\partial x_d^{a_d}} f(x),
\end{align*}
and we adopt the convention that $\frac{\partial^{0}}{\partial x^{0}} f(x) = f(x)$. For any $x \in \mathbb{R}^{d}$, we define $\norm{x}_{1} = \sum_{i=1}^{d} \abs{x_i}$ and write $\abs{x}$ to denote the Euclidean norm. Throughout the paper we will often use $C$ to denote a generic positive constant that may change from line to line and that will be independent of any parameters not explicitly specified. For a random variable $X$, we write $\text{supp}(X)$ to denote the support of $X$.
\section{The Prelimit Generator Comparison Approach}
\label{sec:prelimapproach}
In this section, we work out the technical details of the prelimit approach. We begin by introducing the interpolation operator $A$ in Section~\ref{sec:interpol}. We follow this with a discussion of convergence-determining classes in Section~\ref{sec:convdet}. Then, we write the form of $A G_{X} f(x)$ in a manner that easily lends itself to analysis. Informally, we refer to this as interchanging $A$ with $G_{X}$. Bounded and unbounded domains require separate consideration. We treat unbounded domains in Section~\ref{sec:noref} and treat one example of a bounded domain in Section~\ref{sec:bounded}. To minimize notational burden, we restrict our discussion to one-dimensional CTMCs. In multiple dimensions, the results are analogous from a technical perspective, but may be harder to parse at first read. We therefore postpone the multidimensional discussion to the appendix, in which multidimensional interpolation is discussed in Appendix~\ref{sec:interpolation}, and multidimensional interchange is left to Appendix~\ref{sec:mdimunbound}.
\subsection{The Interpolator}
\label{sec:interpol}
The objective of this section is to state Theorem~\ref{thm:1dinterpolant_def}. Fix $\delta > 0$, and for $x \in \mathbb{R}$ define $k(x) = \lfloor x/\delta\rfloor$. Let $K \subset \mathbb{R}$ be a possibly unbounded interval and define
\begin{align*}
K_{4} = \{x \in K \cap \delta \mathbb{Z} : (x + 4\delta) \in K \cap \delta \mathbb{Z}\}.
\end{align*}
For example, if $K = (-\infty, \infty)$, then $K_4 = \delta \mathbb{Z}$. Let $f: K \cap \delta \mathbb{Z} \to \mathbb{R} $ be the function we want to extend to the continuum.
We interpolate the function using splines, which are standard tools in numerical analysis; see for instance Section 8 in \cite{Kres1998}. Instead of the popular cubic spline, which only results in a $C^1$ interpolant, we craft a degree-7 spline so that our extension is thrice continuously differentiable. Define
\begin{align*}
A f(x) =&\ P_{k(x)}(x), \quad \text{ where } \quad P_{k}(x) = \sum_{i=0}^{4} \alpha^{k}_{k+i}(x) f(\delta (k+i)), \quad x \in \mathbb{R}.
\end{align*}
Each $P_{k}(x)$ is a degree-7 polynomial and is best understood as a weighted sum of $f(\delta k), \ldots, f(\delta (k+4))$ with weights $\alpha^{k}_{k}(x), \ldots, \alpha^{k}_{k+4}(x)$. The precise form of $P_{k}(x)$ is distracting, so we state it in Appendix~\ref{sec:interpolation}. The following result summarizes the key properties we require of $A f(x)$ and the weights $\alpha^{k}_{k+i}(x)$.
\begin{theorem} \label{thm:1dinterpolant_def}
Given $f: K \cap \delta \mathbb{Z} \to \mathbb{R}$, the function
\begin{align}
Af(x) = \sum_{i=0}^{4} \alpha^{k(x)}_{k(x)+i}(x)f(\delta(k(x)+i)), \quad x \in \text{Conv}(K_4), \label{eq:af}
\end{align}
belongs to $ C^{3}(\text{Conv}(K_4))$ and is infinitely differentiable on $\text{Conv}(K_4) \setminus K_4$. Furthermore,
\begin{align}
A f(\delta k) = f(\delta k), \quad \delta k \in K_4, \label{eq:interpolates}
\end{align}
and the derivatives of $A f(x)$ are bounded by the corresponding finite differences of $f(\delta k)$. Namely, there exists $C > 0$ independent of $x$, $f(\cdot)$ and $\delta$ such that
\begin{align}
\Big| \frac{\partial^{a}}{\partial x^{a}} Af(x) \Big| \leq&\ C \delta^{-a} \max_{\substack{ 0 \leq i \leq 4-a }} \abs{\Delta^{a} f(\delta (k(x)+i))}, \quad x \in \text{Conv}(K_4),\ 0 \leq a \leq 3, \label{eq:multibound}
\end{align}
and \eqref{eq:multibound} also holds for $x \in \text{Conv}(K_4) \setminus K_4$ when $a = 4$. Additionally, the weights $\big\{\alpha_{k+i}^{k}: \mathbb{R} \to \mathbb{R} \ | \ k \in \mathbb{Z},\ i = 0,1,2,3,4\big\}$
are degree-$7$ polynomials in $(x-\delta k)/ \delta$ whose coefficients do not depend on $k$ or $\delta$. They satisfy
\begin{align}
&\alpha_{k}^{k}(\delta k) = 1, \quad \text{ and } \quad \alpha_{k+i}^{k} (\delta k) = 0, \quad &k \in \mathbb{Z},\ i = 1,2,3,4, \label{eq:alphas_interpolate} \\
&\sum_{i=0}^{4} \alpha^{k}_{k+i}(x) = 1, \quad &k \in \mathbb{Z},\ x \in \mathbb{R}, \label{eq:weights_sum_one}
\end{align}
and also the following translational invariance property:
\begin{align}
\alpha^{k+j}_{k+j+i}(x+ \delta j) = \alpha^{k}_{k+i}(x),\quad i,j,k \in \mathbb{Z}, \ x \in \mathbb{R}. \label{eq:weights}
\end{align}
\end{theorem}
Theorem~\ref{thm:1dinterpolant_def} is proved in Appendix~\ref{sec:interpolation} and follows directly from the form of $P_{k}(x)$ stated there. From \eqref{eq:multibound} we see that the reason $P_{k}(x)$ depends on $f(\delta k), \ldots, f(\delta (k+4))$, as opposed to also depending on $f(\delta (k+5))$, is that we want $\frac{\partial^{a}}{\partial x^{a}} Af(x)$ to be related to $\Delta^{a} f(\delta k(x))$ for $0 \leq a \leq 4$, and we do not care what happens beyond the fourth derivative. In theory, one can make $A f(x)$ as differentiable as is needed by using a higher degree polynomial $P_{k}(x)$.
\subsection{Convergence-Determining Classes}
\label{sec:convdet}
We mentioned in the introduction that when one uses the diffusion approach, $\text{\rm Lip(1)}$ is a commonly used convergence-determining class. In this section we discuss two convergence-determining classes of grid-valued functions that can be used with the prelimit approach. Lemma~\ref{lem:convdet} below presents the main result of this section.
Recall our convention of using a star superscript to emphasize that a function is defined on the continuum.
Given two random variables $U,V \in \mathbb{R}^{d}$ and a class of functions $\mathcal{H} = \{h^{*}: \mathbb{R}^{d} \to \mathbb{R}\}$, we define
\begin{align*}
d_{\mathcal{H}}(U,V) = \sup_{h^{*} \in \mathcal{H}} \Big| \mathbb{E} h^*(U) - \mathbb{E} h^*(V) \Big|.
\end{align*}
We already said that $\text{\rm Lip(1)}$ is a convergence-determining class because $d_{\text{\rm Lip(1)}}(U,V) \to 0$ implies $U$ converges to $V$ in distribution. There are, of course, other convergence-determining classes. For instance, it was shown in Lemma 2.2 of \cite{GorhMack2016} that if
\begin{align*}
\mathcal{H} = \mathcal{M} = \Big\{h^*: \mathbb{R}^{d} \to \mathbb{R} : \Big| \frac{\partial^{a}}{\partial x^a} h^{*}(x) \Big| \leq 1, \quad 1 \leq \norm{a}_{1} \leq 3 \Big\},
\end{align*}
then $d_{\mathcal{M}}(U,V) \to 0$ also implies convergence in distribution.
Both $\text{\rm Lip(1)}$ and $\mathcal{M}$ are classes of functions defined on $\mathbb{R}^{d}$, but the prelimit approach works with functions defined only on $\delta \mathbb{Z}^{d}$. To mimic the two classes, we define
\begin{align*}
\text{dLip}(1) =&\ \{h: \delta \mathbb{Z}^{d} \to \mathbb{R} : \abs{\Delta_{j} h(\delta k)} \leq \delta,\ 1 \leq j \leq d,\ k \in \delta \mathbb{Z}^{d} \},\\
\mathcal{M}_{disc}(C) =&\ \{h: \delta \mathbb{Z}^{d} \to \mathbb{R} : \abs{\Delta^{a} h(\delta k)} \leq C \delta^{\norm{a}_{1}} ,\ 1 \leq \norm{a}_{1} \leq 3,\ k \in \delta \mathbb{Z}^{d} \}.
\end{align*}
The following lemma relates $d_{\text{\rm Lip(1)}}(U,V)$ and $d_{\mathcal{M}}(U,V)$ to their grid-restricted counterparts. The lemma involves the multidimensional interpolator, which we have not yet formally introduced. However, that does not preclude an understanding of the lemma, which is proved in Section~\ref{sec:proofconvdet}.
\begin{lemma}
\label{lem:convdet}
Let $U \in \delta \mathbb{Z}^{d}$ and $V \in \mathbb{R}^{d}$ be two random vectors. For any $h^{*}: \mathbb{R}^{d} \to \mathbb{R}$, let $h: \delta \mathbb{Z}^{d} \to \mathbb{R}$ be the restriction of $h^{*}(x)$ to $\delta \mathbb{Z}^{d}$. Let $A h: \mathbb{R}^{d} \to \mathbb{R}$ be defined by \eqref{eq:af} when $d =1$, and by \eqref{eq:af2} when $d > 1$. Then there exists a constant $C > 0$ such that
\begin{align*}
\abs{\mathbb{E} h^{*}(U) - \mathbb{E} h^{*}(V)} \leq \abs{\mathbb{E} h(U) - \mathbb{E} A h(V)} + C \delta \max_{\substack{ 1 \leq j \leq d }} \sup_{\substack{ x \in \mathbb{R}^{d} }} \bigg| \frac{\partial}{\partial x_{j}} h^{*}(x) \bigg|.
\end{align*}
As a consequence, there exists a constant $C'>0$ such that
\begin{align*}
d_{ \text{\rm Lip(1)} }(U,V) \leq \sup_{h \in \text{\rm dLip(1)}} \abs{\mathbb{E} h(U) - \mathbb{E} A h(V)} + C \delta,\\
d_{\mathcal{M}}(U,V) \leq \sup_{h \in \mathcal{M}_{disc}(C' )} \abs{\mathbb{E} h(U) - \mathbb{E} A h(V)} + C \delta.
\end{align*}
\end{lemma}
\subsection{Interchange for Unbounded Domains}
\label{sec:noref}
Assume $G_{X} f(\delta k)$ is defined for all $k \in \mathbb{Z}$; i.e.,\ the CTMC lives on $\delta \mathbb{Z}$. The interchange result for $A$ and $G_X$ is given in Proposition~\ref{lem:interchanged1} below. We then apply this result to characterize the approximation error between $X$ and its diffusion approximation $Y$ in \eqref{eq:errordiff}.
Define $\beta_{\ell}(\delta k) = q_{\delta k,\delta (k+\ell)}$ for $k,\ell \in \mathbb{Z}$, where $q_{\delta k, \delta k'}$ are the CTMC transition rates.
Then
\begin{align*}
G_X f(\delta k) = \sum_{k' \in \mathbb{Z} } q_{\delta k,\delta k'} (f(\delta k')-f(\delta k)) =&\ \sum_{\ell \in \mathbb{Z}} \beta_{\ell}(\delta k) (f(\delta (k+\ell))-f(\delta k)), \quad k \in \mathbb{Z}.
\end{align*}
Fix $h : \delta \mathbb{Z} \to \mathbb{R}$ with $\mathbb{E} \abs{h(X)} < \infty$. Since $A$ is a linear operator, we apply to both sides of the CTMC Poisson equation to get
\begin{align*}
A G_{X} f_h(x) = A(\mathbb{E} h(X) - h) (x) = \mathbb{E} A h(X) - A h(x).
\end{align*}
The following result says $A G_{X} f(x) = G_{X} A f(x) + \text{error}(x)$ and characterizes the error term. We prove it in Section~\ref{sec:mdimunbound} by proving the multidimensional version, Proposition~\ref{lem:mdiminterchange}, there.
\begin{proposition} \label{lem:interchanged1}
Fix $f: \delta \mathbb{Z} \to \mathbb{R}$ and assume that $G_{X} f(\delta k)$ is defined on all of $\delta \mathbb{Z}$. Assume also that
\begin{align}
\sum_{\ell \in \mathbb{Z}} \abs{\beta_{\ell}(\delta k) (f(\delta (k+\ell))-f(\delta k))} < \infty, \quad k \in \mathbb{Z}, \label{eq:intergrab}
\end{align}
which is trivially satisfied when the number of transitions from each state is finite. Then
\begin{align}
A G_{X} f(x) =&\ \sum_{\ell \in \mathbb{Z}} A \beta_{\ell}(x) \big( A f(x+\delta \ell) - A f(x)\big) + \varepsilon(x), \quad x \in \mathbb{R}. \label{eq:intererror}
\end{align}
The error $\varepsilon(x)$ satisfies
\begin{align}
\varepsilon(x) =&\ \sum_{\ell \in \mathbb{Z}} \sum_{i=0}^{4} \alpha^{k(x)}_{k(x)+i}(x)\Big(\beta_{\ell}\big(\delta (k(x)+i)\big) - A \beta_{\ell}(x)\Big) \notag \\
& \times \Big(1(\ell > 0) \sum_{j=0}^{i-1} \sum_{m=0}^{\ell-1} \Delta^{2} f\big( \delta(k(x)+ m + i )\big) - 1(\ell < 0) \sum_{j=0}^{i-1} \sum_{m=\ell}^{-1} \Delta^{2} f\big( \delta(k(x)+ m + i )\big) \Big). \label{eq:vareps}
\end{align}
\end{proposition}
Let us now fix a CTMC and derive a diffusion approximation $Y$ for it. We also characterize the approximation error. Fix $h(\delta k)$ and consider \eqref{eq:intererror} with $f(\delta k) = f_h(\delta k)$ (we assume \eqref{eq:intergrab} holds). First, we apply Taylor expansion to $\big( A f_{h}(x+\delta \ell) - A f_{h}(x)\big)$ to get
\begin{align*}
A G_{X} f_h(x) =&\ (A f_h)'(x) \delta \sum_{\ell \in \mathbb{Z}} \ell A \beta_{\ell}(x) + \frac{1}{2} (A f_h)''(x) \delta^{2} \sum_{\ell \in \mathbb{Z}} \ell^{2} A \beta_{\ell}(x) \\
&+ \frac{1}{6} \delta^{3} \sum_{\ell \in \mathbb{Z}} \ell^{3} A \beta_{\ell}(x) (A f_h)'''(\xi_{\ell}(x)) + \varepsilon(x),
\end{align*}
where $\xi_{\ell}(x)$ is between $x$ and $x + \delta \ell$. To approximate $X$, we set
\begin{align*}
b(x) = \delta \sum_{\ell \in \mathbb{Z}} \ell A \beta_{\ell}(x), \quad \text{ and } \quad a(x) = \delta^2 \sum_{\ell \in \mathbb{Z}} \ell^2 A \beta_{\ell}(x), \quad x \in \mathbb{R},
\end{align*}
and consider the diffusion process
\begin{align}
Y(t) = Y(0) + \int_{0}^{t} b(Y(s)) ds + \int_{0}^{t} \sqrt{a(Y(s))} d W(s), \label{eq:sdediffusion}
\end{align}
where $\{W(t)\}$ is standard Brownian motion. The generator of this diffusion is $G_{Y} f(x) = b(x) f'(x) + \frac{1}{2} a(x) f''(x)$ and its stationary distribution has density $\frac{\kappa}{a(x)} \exp\big(\int_{0}^{x} \frac{2 b(y)}{a(y)} dy\big)$, where $\kappa$ is a normalizing constant that we assume to be finite. Let $Y$ be the random variable having this density. It\^{o}'s lemma tells us that for any $f \in C^{2}(\mathbb{R})$,
\begin{align*}
\mathbb{E}_{x} f(Y(t)) - \mathbb{E}_{x} f(Y(0)) =&\ \mathbb{E}_{x} \int_{0}^{t} G_Y f(Y(s)) ds, \quad t > 0.
\end{align*}
Provided $\mathbb{E} |f(Y)|<\infty$, we can initialize $Y(0) \stackrel{d}{=} Y $ to get
\begin{align*}
\mathbb{E} \Big[ \int_{0}^{t} G_Y f(Y(s)) ds \Big| Y(0) \stackrel{d}{=} Y \Big] = 0.
\end{align*}
If we further assume that $\mathbb{E} \abs{G_Y f(Y)} < \infty$, then we can apply the Fubini-Tonelli theorem to interchange the integral and expectation above and conclude that
\begin{align}
\mathbb{E} G_{Y} f(Y) = 0. \label{eq:ito}
\end{align}
Now, provided that \eqref{eq:ito} holds with $A f_h(x)$ in place of $f(x)$ there, we get
\begin{align}
\mathbb{E} A h(X) - \mathbb{E} A h(Y) =&\ \mathbb{E} A G_{X} f_h(Y) - \mathbb{E} G_{Y} A f_{h}(Y) \notag \\
=&\ \frac{1}{6} \delta^{3} \mathbb{E} \sum_{\ell \in \mathbb{Z}} \ell^{3} A \beta_{\ell}(Y) (A f_h)'''(\xi_{\ell}(Y)) + \mathbb{E} \varepsilon(Y). \label{eq:errordiff}
\end{align}
The bounds on $(A f_{h})'''(x)$ from Theorem~\ref{thm:1dinterpolant_def} imply
\begin{align*}
\frac{1}{6} \delta^{3} \Big| \mathbb{E} \sum_{\ell \in \mathbb{Z}} \ell^{3} A \beta_{\ell}(Y) (A f_h)'''(\xi_{\ell}(Y)) \Big| \leq&\ C \Big| \mathbb{E} \sum_{\ell \in \mathbb{Z}} \ell^{3} A \beta_{\ell}(Y) \max_{\substack{ 0 \leq i \leq 1 }} \abs{\Delta^{3} f_h(\delta (k(\xi_{\ell}(Y))+i))} \Big|.
\end{align*}
In other words, the term above depends on $\ell^{3} A \beta_{\ell}(Y)$ and third-order Stein factors. The second term in \eqref{eq:errordiff} is $\mathbb{E} \varepsilon(Y)$. We recall $\varepsilon(x)$ below for convenience:
\begin{align*}
\varepsilon(x) =&\ \sum_{\ell \in \mathbb{Z}} \sum_{i=0}^{4} \alpha^{k(x)}_{k(x)+i}(x)\Big(\beta_{\ell}\big(\delta (k(x)+i)\big) - A \beta_{\ell}(x)\Big) \notag \\
& \times \Big(1(\ell > 0) \sum_{j=0}^{i-1} \sum_{m=0}^{\ell-1} \Delta^{2} f\big( \delta(k(x)+ m + i )\big) - 1(\ell < 0) \sum_{j=0}^{i-1} \sum_{m=\ell}^{-1} \Delta^{2} f\big( \delta(k(x)+ m + i )\big) \Big).
\end{align*}
First, the fact that $\alpha^{k }_{k +i}(x)$ is a polynomial in $(x-\delta k)/\delta$ implies $\sup_{x \in \mathbb{R}} \big|\alpha^{k(x)}_{k(x)+i}(x)\big|$ is bounded by a constant independent of $\delta, x$, or any other parameters. Second, the fact that $0 \leq i \leq 4$ and the mean value theorem imply that
\begin{align*}
\abs{\beta_{\ell}\big( \delta(k(x) + i)\big) - A \beta_{\ell}(x)} = \abs{A \beta_{\ell}\big( \delta(k(x) + i)\big) - A \beta_{\ell}(x)} \leq 4\delta (A \beta_{\ell})'(\xi'_{\ell}(x)).
\end{align*}
Therefore, provided that the transition rates $\beta_{\ell}(\cdot)$ of the CTMC do not vary too much, e.g.,\ they are Lipschitz, the term above can be controlled, so bounding \eqref{eq:errordiff} comes down to bounding $\Delta^{2} f_h(\delta k)$ and $\Delta^{3} f_h(\delta k)$.
\subsection{Interchange for a Bounded Domain}
\label{sec:bounded}
When the domain of the CTMC is bounded, Proposition~\ref{lem:interchanged1} must be modified slightly to account for the boundary of the domain. In this section we illustrate this using the example of the birth-death process defined by the generator
\begin{align}
G_{X} f(\delta k) = \lambda \Delta f(\delta k) - \mu 1(k > 0) \Delta f(\delta (k-1)), \quad k \in \mathbb{N}. \label{eq:mm1gen}
\end{align}
This generator corresponds to the customer count, scaled by $\delta$, in a single-server queue where customers arrive according to a Poisson process with rate $\lambda$ and service times are exponentially distributed with rate $\mu$. Such a system is also known as the $M/M/1$ queueing system. The quantity $\rho = \lambda/\mu$ is the system utilization. In steady state, the customer count is geometrically distributed provided that $\rho < 1$. It is also well known that as $\rho \to 1$, the customer count can be approximated by an exponential random variable. A recent application of Stein's method in \cite{GaunWalt2020} establishes convergence rates of the waiting time distribution in the $M/G/1$ system (which is more general than the $M/M/1$ system) to the exponential distribution. Another example of a CTMC with a bounded domain can be found in \cite{Brav2022}.
The $M/M/1$ system is restricted to the non-negative integers. Let us see how this boundary affects the interchange of $A$ and $G_X$. Fix $f: \delta \mathbb{N} \to \mathbb{R}$ and consider $A G_X f(x)$, which is defined for $x \in [0,\infty)$. For $x \geq \delta$, the proof of Proposition~\ref{lem:interchanged1} can be repeated to see that
\begin{align*}
A G_{X} f(x) = \lambda \big(A f\big(x + \delta\big)-A f(x)\big) + \mu \big(A f\big(x - \delta\big)-A f(x)\big) + \varepsilon(x).
\end{align*}
In fact, $\varepsilon(x) = 0$ for $x \geq \delta$ because the birth and death rates $\lambda$ and $\mu$ are constant. However, the equality above does not hold when $x \in [0,\delta)$ because $A f\big(x - \delta\big)$ is not defined there. To see this, we recall that $A f(x) = \sum_{i=0}^{4} \alpha^{k(x)}_{k(x)+i}(x)f(\delta(k(x)+i)) $, and that $f(\delta k)$ is only defined for $k \geq 0$. Our restriction of $f(\delta k)$ to $k \geq 0$ is not artificial because instead of $f(\delta k)$ we intend to use the $M/M/1$ Poisson equation solution, which is defined only on $\delta \mathbb{N}$.
To resolve this, we extend the definition $f(\delta k)$ to $k = -1$. To motivate the extension, fix $x \in [0,\delta)$ and consider
\begin{align}
A G_{X} f(x) = \sum_{i=0}^{4} \alpha^{0}_{i}(x)\lambda \Big( f\big(\delta(i+1)\big) - f(i)\Big) + \sum_{i=1}^{4} \alpha^{0}_{i}(x)\mu \Big( f\big(\delta(i-1)\big) - f(i)\Big). \label{eq:mm1bdry}
\end{align}
Using the translational invariance property presented in \eqref{eq:weights} of Theorem~\ref{thm:1dinterpolant_def}, it follows that
\begin{align*}
\sum_{i=0}^{4} \alpha^{0}_{i}(x)\lambda \Big( f\big(\delta(i+1)\big) - f(i)\Big) =&\ \lambda \Big( \sum_{i=0}^{4} \alpha^{1}_{1+i}(x+\delta )f\big(\delta(i+1)\big) - \sum_{i=0}^{4} \alpha^{0}_{i}(x)f(i)\Big) \\
=&\ \lambda\big( A f(x+\delta) - A f(x)\big).
\end{align*}
We wish to do the same thing to the second term in \eqref{eq:mm1bdry}, but we cannot because the summation there starts from $i = 1$, not $i = 0$. Note that
\begin{align*}
\sum_{i=1}^{4} \alpha^{0}_{i}(x)\mu \Big( f\big(\delta(i-1)\big) - f(i)\Big) =&\ \alpha^{0}_{0}(x) \mu \big(f(0) - f(0)\big) + \sum_{i=1}^{4} \alpha^{0}_{i}(x)\mu \Big( f\big(\delta(i-1)\big) - f(i)\Big)\\
=&\ \mu \Big( \alpha^{0}_{0}(x)f(0) + \sum_{i=1}^{4} \alpha^{0}_{i}(x) f\big(\delta(i-1)\big) \Big) - \mu A f(x),
\end{align*}
which motivates us to define
\begin{align*}
\widehat f(\delta k) =
\begin{cases}
f(\delta k), \quad k \geq 0, \\
f(0), \quad k = -1.
\end{cases}
\end{align*}
It follows that $\mu \Big( \alpha^{0}_{0}(x)f(0) + \sum_{i=1}^{4} \alpha^{0}_{i}(x) f\big(\delta(i-1)\big) \Big) $ equals
\begin{align*}
\mu \sum_{i=0}^{4} \alpha^{0}_{i}(x) \widehat f\big(\delta(i-1)\big)
= \mu \sum_{i=0}^{4} \alpha^{-1}_{-1+ i}(x - \delta) \widehat f\big(\delta(i-1)\big) = \mu A \widehat f(x - \delta) .
\end{align*}
Since $A f(x) = A \widehat f(x)$ for $x \geq 0$, we conclude that
\begin{align}
A G_{X} f(x) = \lambda \big( A \widehat f(x + \delta) - A \widehat f(x)\big) + \mu \big( A \widehat f(x - \delta) - A \widehat f(x)\big), \quad x \geq 0. \label{eq:interchangemm1}
\end{align}
This result resembles Proposition~\ref{lem:interchanged1}, but uses the extension $\widehat f(\delta k)$ instead of $f(\delta k)$. We comment more on our choice of extension at the end of this section. To derive the diffusion approximation, we perform Taylor expansion. For $x \geq 0$,
\begin{align}
A G_{X} f_h(x) =&\ \lambda ( A \widehat f_h(x+\delta ) - A \widehat f_h(x)) + \mu ( A \widehat f_h(x-\delta ) - A \widehat f_h(x)) \notag \\
=&\ \delta (\lambda - \mu) (A f_h)'(x) + \frac{1}{2}\delta^{2} (\lambda + \mu) (A f_h)''(x) \notag \\
&+ \frac{1}{6} \delta^{3} \big( \lambda (A f_h)'''(\xi_{1}(x)) - \mu (A \widehat f_h)'''(\xi_{-1}(x))\big). \label{eq:rbmgx}
\end{align}
In the second equality, we used the fact that $A \widehat f_h(x) = A f_h(x)$ for $x \geq 0$. The diffusion approximation is driven by the first- and second-order terms above. Since the $M/M/1$ system lives on the non-negative integers, we add a reflection term at zero to our diffusion. To this end, let us define the reflected Brownian motion (RBM) satisfying
\begin{align}
Y(t) = Y(0) + \delta(\lambda - \mu) t + \delta \sqrt{\lambda + \mu} W(t) + R(t), \label{eq:rbm}
\end{align}
where $R(t)$ is the unique, continuous, and non-decreasing process such that $Y(t) \geq 0$, $R(0) = 0$ and $R(t)$ increases only at those times $t$ when $Y(t) = 0$. Let $Y$ be a random variable having the stationary distribution of this RBM. It is well known that $Y$ is exponentially distributed with mean $\frac{\delta (\lambda + \mu)}{2( \mu-\lambda) }$, and so using Lemma 5.2 of \cite{Ross2011} with $Z = Y/\mathbb{E} Y$ there implies
\begin{align}
\mathbb{E}\big( \delta(\lambda - \mu) f'(Y) + \frac{1}{2} \delta^{2} (\lambda + \mu) f''(Y) \big) + f'(0) \delta(\mu - \lambda) = 0 \label{eq:rbmbar}
\end{align}
for all $f \in C^{2}(\mathbb{R}_+)$ with $\mathbb{E} \abs{f''(Y)} < \infty$. One can also derive \eqref{eq:rbmbar} using Ito's lemma for RBMs from Theorem 2 in \cite{HarrReim1981}.
Assume we know \eqref{eq:rbmbar} is satisfied when $f(x) = A f_{h}(x)$, a fact that will be verified by Proposition~\ref{lem:mm1dif} of the following section. Taking expected values with respect to $Y$ in \eqref{eq:rbmgx}, and using the fact that $A G_{X} f_h(x) = \mathbb{E} h(X) - A h(x)$, we arrive at
\begin{align}
& \mathbb{E} h(X) - \mathbb{E} A h(Y) \notag \\
=&\ \mathbb{E} A G_{X} f_h(Y) - \Big( \mathbb{E}\big( \delta(\lambda - \mu) (A f_{h})'(Y) + \frac{1}{2} \delta^{2} (\lambda + \mu) (A f_{h})''(Y) \big) + (A f_{h})'(0) \delta(\mu - \lambda) \Big) \notag \\
=&\ \frac{1}{6} \delta^{3} \mathbb{E} \big( \lambda (A f_h)'''(\xi_{1}(Y)) - \mu (A \widehat f_h)'''(\xi_{-1}(Y))\big) - (A f_{h})'(0) \delta(\mu - \lambda). \label{eq:errorrbm}
\end{align}
Note that the only term that depends on how we chose our extension $\widehat f_h(\delta k)$ is $(A \widehat f_h)'''(\xi_{-1}(Y))$ when $\xi_{-1}(Y) \in [-\delta,0]$. Theorem~\ref{thm:1dinterpolant_def} tells us that the upper bound on this term depends on $\Delta^3 \widehat f_h(-\delta)$ and $\Delta^3 \widehat f_h(0) = \Delta^3 f_h(0)$. A straightforward calculation shows that $\Delta^3 \widehat f_h(-\delta) = \Delta^2 f_h(0) - \Delta f_h(0)$ because we chose $\widehat f_h(-\delta) = f_h(0)$. Although $\Delta^2 f_h(0) - \Delta f_h(0)$ looks like a second-order Stein factor, we will see in the next section that it is of the same order of magnitude as $\Delta^3 f_h(0)$, meaning it behaves likes a third-order Stein factor. This is important because as we will see in Proposition~\ref{lem:mm1dif}, third-order Stein factors are smaller than second-order factors by a factor of $\delta$. We also note that a different choice for $\widehat f_h(-\delta)$ would most likely make $\Delta^3 \widehat f_h(-\delta)$ larger than a third-order Stein factor.
\section{Stein Factor Bounds for the $M/M/1$ System} \label{sec:diffbounds}
In this section we show how to use synchronous couplings to bound the Stein factors of the $M/M/1$ system. We initialize several copies of the same CTMC, each of which have slightly perturbed initial conditions, and we observe how the CTMCs evolve jointly until the time they couple. As we will see in Section~\ref{sec:synchronous}, the magnitude of the Stein factors depends on a) the coupling time and b) on the distance of the coupled CTMCs relative to each other before coupling. As such, Stein factors simply measure the sensitivity of the CTMC to perturbations of its initial condition. As to the generality of this approach, one expects it to work well when there is insight into the joint evolution and coupling time of the perturbed and non-perturbed chains.
The main result of this section is Proposition~\ref{lem:mm1dif} below, which states the relevant Stein factor bounds. We use this lemma to close the loop on our $M/M/1$ example by proving Theorem~\ref{thm:mm1}. After discussing synchronous couplings, we also show how the Poisson equation can be used to establish tightness in Section~\ref{sec:tight}.
Our entire discussion relies on our yet-unproven claim that
\begin{align*}
f_h(\delta k) = \int_{0}^{\infty} \big( \mathbb{E}_{X(0) = \delta k} h(X(t)) - \mathbb{E} h(X) \big) dt
\end{align*}
solves the Poisson equation, so we now verify this fact.
\begin{lemma} \label{lem:genpoisson}
Consider a CTMC taking values on a set $E \subset \delta \mathbb{Z}^{d}$ with generator $G_{X}$ given in \eqref{eq:generalctmc}. For $\delta k \in E$, let $g(\delta k) = \int_{0}^{\infty} \big( \mathbb{E}_{\delta k} h(X(t)) - \mathbb{E} h(X) \big) dt$ and assume that
\begin{align}
g(\delta k) \text{ is finite for all } \delta k \in E, \label{eq:barbexponential}
\end{align}
and that $\sum_{\ell \in \mathbb{Z}^{d}} \abs{\beta_{\ell}(\delta k) (g(\delta (k+\ell))-g(\delta k))} < \infty$ for $ \delta k \in E$.
Then $G_X g(\delta k) = \mathbb{E} h(X) - h(\delta k)$ for all $\delta k \in E$.
\end{lemma}
Lemma~\ref{lem:genpoisson} is proved in Section~\ref{sec:proofgenpoisson} using an argument similar to one used in \cite{Barb1988}. In practice, there are several ways to verify that \eqref{eq:barbexponential} holds. One way is by showing that $\{X(t)\}$ is $h$-exponentially ergodic; i.e.,\ $\abs{\mathbb{E}_{\delta k} h(X(t)) - \mathbb{E} h(X)} \leq c_1 e^{-c_2 t} $ for some $c_1,c_2 > 0$. This is automatically true when the state space $E$ is finite but when $E$ is infinite, the usual way to prove this would be to find a Lyapunov function $V(\delta k)$ such that $G_{X} V(\delta k) \leq -c V(\delta k) + \bar c 1(k \in K)$ for some compact set $K$ and some constants $c,\bar c > 0$. We refer the reader to \cite{MeynTwee1993b} for more on exponential ergodicity.
We now discuss another way to verify \eqref{eq:barbexponential} using synchronous couplings. Note that
\begin{align*}
\Big| \int_{0}^{\infty} \big( \mathbb{E}_{\delta k} h(X(t)) - \mathbb{E} h(X) \big) dt\Big| \leq&\ \int_{0}^{\infty}\sum_{j\in \mathbb{Z}^{d}}\mathbb{P}(X = \delta j) \big| \mathbb{E}_{\delta k} h(X(t)) - \mathbb{E}_{\delta j} h(X(t)) \big| dt\\
=&\ \sum_{j\in \mathbb{Z}^{d} }\mathbb{P}(X = \delta j) \int_{0}^{\infty} \big| \mathbb{E}_{\delta k} h(X(t)) - \mathbb{E}_{\delta j} h(X(t)) \big| dt,
\end{align*}
where the last equality follows from by Fubini-Tonelli.
Let us use synchronous couplings to show the right-hand side is finite for the $M/M/1$ model.
Recall the $M/M/1$ generator $G_{X}$ introduced in \eqref{eq:mm1gen} of Section~\ref{sec:bounded}. For the remainder of the section, we let $\{X(t)\}$ and $X$ represent the corresponding CTMC and stationary distribution, respectively. Similarly, we let $Y$ have the stationary distribution of the RBM given by \eqref{eq:rbm} of the same section. Suppose we have proved that for $h \in \text{\rm dLip(1)}$,
\begin{align}
\int_{0}^{\infty} \big| \mathbb{E}_{\delta (k+1)} h(X(t)) - \mathbb{E}_{\delta k} h(X(t)) \big| \leq \frac{\delta(k+1)}{\mu - \lambda}, \quad k \in \mathbb{N}. \label{eq:diftover}
\end{align}
Using a telescoping sum and the triangle inequality,
\begin{align*}
& \sum_{j\in \mathbb{Z}}\mathbb{P}(X = \delta j) \int_{0}^{\infty} \big| \mathbb{E}_{\delta k} h(X(t)) - \mathbb{E}_{\delta j} h(X(t)) \big| dt \\
\leq&\ \sum_{j\in \mathbb{Z}}\mathbb{P}(X = \delta j) \sum_{i= k \wedge j}^{ k \vee j-1} \int_{0}^{\infty} \big| \mathbb{E}_{\delta (i+1)} h(X(t)) - \mathbb{E}_{\delta i} h(X(t)) \big| dt.
\end{align*}
Applying \eqref{eq:diftover}, we bound this by
\begin{align*}
\sum_{j\in \mathbb{Z}}\mathbb{P}(X = \delta j) \sum_{i= k \wedge j}^{ k \vee j-1} \frac{\delta(i+1)}{\mu - \lambda} \leq \sum_{j\in \mathbb{Z}}\mathbb{P}(X = \delta j) \frac{\delta(k + j +1)}{\mu - \lambda}(k + j).
\end{align*}
The right-hand side is finite because $\mathbb{E} X^{2} < \infty$, meaning \eqref{eq:barbexponential} is satisfied. The following result confirms \eqref{eq:barbexponential} and presents several Stein factor bounds. It is proved in Section~\ref{sec:synchronous}.
\begin{proposition} \label{lem:mm1dif}
For any $h \in \text{\rm dLip(1)}$, \eqref{eq:diftover} holds. Furthermore, for each $ 1 \leq a \leq 3$, if $\abs{\Delta^{v} h(\delta k)} \leq \delta^v$ for all $1 \leq v \leq a$, then
\begin{align}
\abs{\Delta^a f_h(\delta k)} \leq \frac{\delta^{a}(k+1)}{\mu - \lambda } + \delta^{a-1} \frac{1}{\mu - \lambda}, \quad k \in \mathbb{N}. \label{eq:d3}
\end{align}
Lastly, for any function $h : \delta \mathbb{N} \to \mathbb{R}$,
\begin{align*}
\big( \Delta^{2} f_h(0) - \Delta f_h(0) \big) = \frac{\lambda + \mu }{\mu } \Delta^{3} f_h(0) - \frac{1}{\mu } \Delta^{3} h(0) - \frac{\lambda}{ \mu } \Delta^{3} f_h(\delta).
\end{align*}
\end{proposition}
The last claim above says that $( \Delta^{2} f_h(0) - \Delta f_h(0) )$ behaves like a third-order Stein factor. Let us compare Proposition~\ref{lem:mm1dif} to existing Stein factor bounds for the geometric distribution.
As mentioned in the introduction, the first paper to bound Stein factors for the geometric distribution is \cite{peko1996}. That paper works with indicator test functions of the form $h(\delta k) = 1(k \in B)$ for $B \subset \mathbb{N}$ instead of allowing $h \in \text{\rm dLip(1)}$. Bounds for more general test functions can be found in Theorem 1.4 of \cite{Daly2008}. Setting $q$ and $h(k)$ there to equal $\rho = \lambda/\mu$ and $h(\delta k)/\mu$, respectively, we get the bound
\begin{align}
\abs{\Delta^3 f_h(\delta k)} \leq \frac{2\delta}{\lambda}, \quad k \in \mathbb{N}, h \in \text{\rm dLip(1)}. \label{eq:d3alt}
\end{align}
Compared to Proposition~\ref{lem:mm1dif}, the right-hand side of \eqref{eq:d3alt} is independent of $k$, and only requires $h \in \text{\rm dLip(1)}$, making it more convenient to work with. However, unlike synchronous couplings, the proof of \eqref{eq:d3alt} in \cite{Daly2008} is much harder to generalize to multidimensional settings. We now state and prove a bound on the approximation error between $Y$ and $X$.
\begin{theorem}
\label{thm:mm1}
There exists a constant $C > 0$ such that for $\rho < 1$ and $h^{*} \in \text{\rm Lip(1)}$,
\begin{align*}
\abs{\mathbb{E} h^{*}(X) - \mathbb{E} h^{*}(Y) } \leq C \delta \Big(1 + \frac{1}{\rho} \Big).
\end{align*}
\end{theorem}
To prove Theorem~\ref{thm:mm1} we will use the third-order bound in \eqref{eq:d3alt} instead of Proposition~\ref{lem:mm1dif}. After the proof, we comment on how to get a comparable bound using Proposition~\ref{lem:mm1dif}.
Before proving the theorem, let us say a few words on the possible choices of $\delta$. It is well known that $X \approx Y$ when $\rho \approx 1$, and that $\mathbb{E} X = \delta \rho/(1-\rho)$. Choosing $\delta = 1$, Theorem~\ref{thm:mm1} tells us that the approximation error $\abs{\mathbb{E} X - \mathbb{E} Y}$ does not grow even though $\mathbb{E} X \to \infty$ as $\rho \to 1$. However, since both $X$ and $Y$ diverge, we cannot conclude that $X$ converges to $Y$. To ensure convergence, we recall that $\mathbb{E} Y = \frac{\delta (\lambda + \mu)}{2( \mu-\lambda) }$. Choosing $\delta = (1-\rho)$ ensures that $\{X\}_{\rho < 1}$ and $\{Y\}_{\rho < 1}$ are tight, and Theorem~\ref{thm:mm1} then implies that $X$ converges to $Y$ in distribution as $\rho \to 1$. As discussed in the introduction, tightness of the prelimit sequence is a sought-after property because, when combined with process-level convergence to some diffusion limit, tightness implies convergence of stationary distributions as well. We discuss in Section~\ref{sec:tight} below how one can use the Poisson equation to establish tightness.
\proof{Proof of Theorem~\ref{thm:mm1}}
Let $h: \delta \mathbb{N} \to \mathbb{R}$ be the restriction of $h^{*}(x)$ to $\delta \mathbb{N}$.
Since $Y$ is exponentially distributed, Proposition~\ref{lem:mm1dif} implies \eqref{eq:rbmbar} holds with $f(x) = A f_h(x)$ there.
Consequently, \eqref{eq:errorrbm} holds, which we recall below:
\begin{align*}
\mathbb{E} h(X) - \mathbb{E} Ah(Y) =&\ \frac{1}{6} \delta^{3} \mathbb{E} \big( \lambda (A f_h)'''(\xi_{1}(Y)) - \mu (A \widehat f_h)'''(\xi_{-1}(Y))\big) - (A f_{h})'(0) \delta(\mu - \lambda).
\end{align*}
Once we bound the right-hand side above, Lemma~\ref{lem:convdet} will imply the theorem. Inequality \eqref{eq:multibound} from Theorem~\ref{thm:1dinterpolant_def} and the Stein factor bound in \eqref{eq:d3alt} imply
\begin{align*}
\delta^{3} \lambda \abs{(A f_h)'''(\xi_{1}(Y))} \leq&\ C \lambda \max_{0\leq i \leq 1} \abs{\Delta^{3} f_h\big(\delta (k(\xi_1(Y)) + i)\big) } \leq C \delta.
\end{align*}
Similarly, the bounds from Proposition~\ref{lem:mm1dif} imply $\abs{(A f_{h})'(0) \delta(\mu - \lambda)} \leq C\delta^2$. Lastly
\begin{align*}
\delta^{3} \mu \abs{(A \widehat f_h)'''(\xi_{-1}(Y))} \leq&\ C \mu \max_{0\leq i \leq 1} \abs{\Delta^{3} \widehat f_h\big(\delta (k(\xi_{-1}(Y)) + i)\big) }.
\end{align*}
As discussed at the end of Section~\ref{sec:bounded}, $\Delta^{3} \widehat f_h\big(\delta (k(\xi_{-1}(Y)) + i) = \Delta^{3} f_h\big(\delta (k(\xi_{-1}(Y)) + i)$ if $\xi_{-1}(Y) + i \geq 0$, and otherwise it equals $\Delta^{2} f_h(0) - \Delta f_h(0)$. Using the form of $\Delta^{2} f_h(0) - \Delta f_h(0)$ from Proposition~\ref{lem:mm1dif} together with \eqref{eq:d3alt}, we get
\begin{align*}
C \mu \max_{0\leq i \leq 1} \abs{\Delta^{3} \widehat f_h\big(\delta (k(\xi_{-1}(Y)) + i)\big) } \leq C \delta \frac{1}{\rho}.
\end{align*}
\hfill $\square$\endproof
The approximation error can also be bounded using only the bounds from Proposition~\ref{lem:mm1dif} instead of \eqref{eq:d3alt}. Consider the error term
\begin{align*}
\delta^{3} \lambda \mathbb{E} \abs{(A f_h)'''(\xi_{1}(Y))} \leq&\ C \lambda \mathbb{E} \max_{0\leq i \leq 1} \abs{\Delta^{3} f_h\big(\delta (k(\xi_1(Y)) + i)\big) }
\end{align*}
from the proof of the theorem above. If we apply Proposition~\ref{lem:mm1dif} and the fact that $\delta k(\xi_1(Y)) \leq Y+\delta $, we get
\begin{align*}
C \lambda \mathbb{E} \max_{0\leq i \leq 1} \abs{\Delta^{3} f_h\big(\delta (k(\xi_1(Y)) + i)\big) } \leq C \lambda \delta^{2}\frac{\mathbb{E} Y + 1}{\mu - \lambda} =&\ C \lambda \delta^{3}\frac{\lambda + \mu}{2(\mu - \lambda)^2} + C \lambda \delta^{2}\frac{1}{\mu - \lambda} \\
=&\ C \rho \delta^{3}\frac{\rho + 1}{2(1-\rho)^2} + C \rho \delta^{2}\frac{1}{1-\rho}.
\end{align*}
In the second-last equality we used $\mathbb{E} Y = \frac{\delta (\lambda + \mu)}{2( \mu-\lambda) }$. Choosing $\delta = (1-\rho)$ means the term on the right-hand side is bounded by $C \delta$, giving a comparable bound to the one in Theorem~\ref{thm:mm1}. Other error terms can be bounded similarly. As discussed previously, the choice of $\delta = (1-\rho)$ is natural because it ensures tightness of $\{X\}_{\rho < 1}$ and $\{Y\}_{\rho < 1}$.
\subsection{Synchronous Couplings}
\label{sec:synchronous}
We now use synchronous couplings to prove Proposition~\ref{lem:mm1dif}.
\proof{Proof of Proposition~\ref{lem:mm1dif}}
\textbf{First-order factors.}
Consider two $M/M/1$ systems, whose customer counts (scaled by $\delta$) are $\{X^{(0)}(t)\}$ and $\{X^{(1)}(t)\}$. We refer to these as system $0$ and system $1$, respectively. We couple the two systems by setting $X^{(1)}(0) = X^{(0)}(0)+1$ and defining their joint evolution via the following transition rate table.
\begin{table}[h!]
\caption{Transitions of the joint process $\{(X^{(1)}(t), X^{(0)}(t))\}$ in state $ (x^{(1)},x^{(0)}) $. }
\centering
\label{tab:mm1}
\begin{tabular}{|c|c|c|}
\hline
\# &Rate & Transition \\ \hline
1& $ \lambda $ & $(x^{(1)}+ \delta,x^{(0)}+ \delta)$ \\ \hline
2&$\mu 1(x^{(0)}>0) $ & $(x^{(1)}- \delta,x^{(0)}- \delta)$ \\ \hline
3&$\mu 1(x^{(1)}>0,x^{(0)}=0) $ & $(x^{(1)}- \delta,x^{(0)})$ \\ \hline
\end{tabular}
\end{table}
The transition table can be interpreted as follows. Both systems have the same customer arrival stream. The customers present in system 0 at time $t = 0$, and all newly arriving customers, have an identical counterpart in system 1. The only difference between the two is the extra initial customer in system 1, who behaves like a low-priority customer that only gets served when there are no other customers, and is preempted by new arrivals. The two systems couple once this extra customer is served. We refer to systems 1 and 0 as a synchronous coupling because the two systems are driven by the same underlying stochastic processes, i.e.,\ arrivals and services.
To bound $\Delta f_h(\delta k)$, define $\tau^{(i)}(\delta k) = \inf_{t \geq 0} \{X^{(i)}(t) = \delta k\}$. The discussion above implies
\begin{align}
\Delta f_h(\delta k) =&\ \int_{0}^{\infty} \big(\mathbb{E}_{\delta (k+1)} h(X (t)) - \mathbb{E}_{\delta k}h(X (t)) \big) dt \notag \\
=&\ \int_{0}^{\infty} \mathbb{E}_{X^{(0)}(0)=\delta k} \Big( h(X^{(1)}(t)) - h(X^{(0)}(t)) \Big) dt \notag \\
=&\ \int_{0}^{\infty} \mathbb{E}_{X^{(0)}(0)=\delta k} \bigg[ 1 (t \leq \tau^{(1)}(0) ) \Big( h(X^{(0)}(t)+\delta) - h(X^{(0)}(t)) \Big) \bigg] dt. \label{eq:dfk_mm1}
\end{align}
We emphasize that the last equality above is true because systems 0 and 1 always maintain a constant gap of a single customer until they couple. We bound $\mathbb{E}_{X^{(0)}(0)=\delta k} \tau^{(1)}(0)$ by combining the Lyapunov function $V(\delta k ) = k$ with Dynkin's formula. Observe that $V(\delta k)$ satisfies $G_X V(\delta k) = \lambda - \mu < 0$ whenever $k > 0$, which means that
\begin{align}
\mathbb{E}_{X^{(0)}(0)=\delta k} \tau^{(1)}(0) = \frac{\mathbb{E}_{X^{(1)}(0)=\delta (k+1)} \big(V(X^{(1)}(0) ) - V(X^{(1)}( \tau^{(1)}(0) ) )\big)}{\mu - \lambda} = \frac{k+1}{\mu - \lambda}. \label{eq:taubound}
\end{align}
To justify the above equality, we refer the reader to the proof of Theorem 4.3.i of \cite{MeynTwee1993b}, which is a direct application of Dynkin's formula.
Combining \eqref{eq:dfk_mm1}, \eqref{eq:taubound} and the fact that $h \in \text{\rm dLip(1)}$ proves
\begin{align*}
\abs{\Delta f_h(\delta k)} \leq \frac{\delta (k+1)}{\mu - \lambda}, \quad k \in \mathbb{N}.
\end{align*}
In fact, we have proved the stronger statement \eqref{eq:diftover}.
\textbf{Second- and third-order factors.}
We now prove the third-order bounds. Second-order bounds follow analogously. In addition to systems 0 and 1, we let $\{X^{(2)}(t)\}$ and $\{X^{(3)}(t)\}$ represent systems 2 and 3. System 2 is an identical copy of system 1 with one additional low-priority customer, and system 3 is a copy of system 2 with yet another low-priority customer. The relationship between the four systems is visualized in Figure~\ref{fig:starting_mm1_4}, where we note that $X^{(3)}(0) =X^{(2)}(0)+\delta = X^{(1)}(0)+2\delta = X^{(0)}(0)+3\delta$. The transitions of the joint chain are formally defined in Table~\ref{tab:mm1d2} below.
\begin{figure}[h!]
\centering
\includegraphics[scale=1]{Fig1.pdf}
\caption{The initial state of systems 0,1,2,3 when system 0 starts with 4 customers. The diamonds, squares, and stars represent the extra customers. }
\label{fig:starting_mm1_4}
\end{figure}
\begin{table}[h!]
\caption{Transitions of the joint process $\{(X^{(3)}(t),X^{(2)}(t),X^{(1)}(t), X^{(0)}(t))\}$ in state $ (x^{(3)},x^{(2)},x^{(1)},x^{(0)}) $. }
\centering
\label{tab:mm1d2}
\begin{tabular}{|c|c|c|}
\hline
\# &Rate & Transition \\ \hline
1& $ \lambda $ & $(x^{(3)}+ \delta,x^{(2)}+ \delta,x^{(1)}+ \delta,x^{(0)}+ \delta)$ \\ \hline
2&$\mu 1(x^{(0)}>0) $ & $(x^{(3)}- \delta,x^{(2)}- \delta,x^{(1)}- \delta,x^{(0)}- \delta)$ \\ \hline
3&$\mu 1(x^{(1)}>0,x^{(0)}=0) $ & $(x^{(3)}- \delta,x^{(2)}- \delta,x^{(1)}- \delta,x^{(0)})$ \\ \hline
4&$\mu 1(x^{(2)}>0,x^{(1)}=0) $ & $(x^{(3)}- \delta,x^{(2)}- \delta,x^{(1)},x^{(0)})$ \\ \hline
5&$\mu 1(x^{(3)}>0,x^{(2)}=0) $ & $(x^{(3)}- \delta,x^{(2)},x^{(1)},x^{(0)})$ \\ \hline
\end{tabular}
\end{table}
\noindent It follows that
\begin{align}
\Delta^3 f_h(\delta k) =&\ \int_{0}^{\infty} \mathbb{E}_{X^{(0)}(0)=\delta k} \Big( h(X^{(3)}(t)) - 3 h(X^{(2)}(t)) + 3 h(X^{(1)}(t)) - h(X^{(0)}(t)) \Big) dt \notag \\
=&\ \int_{0}^{\infty} \mathbb{E}_{X^{(0)}(0)=\delta k} \bigg[ 1 (t \leq \tau^{(1)}(0) ) \Delta^3 h(X^{(0)}(t)) \bigg] dt \notag \\
& + \int_{0}^{\infty} \mathbb{E}_{X^{(1)}(0)=0} \Big( h(X^{(3)}(t)) - 3 h(X^{(2)}(t)) + 2 h(X^{(1)}(t)) \Big) dt \notag \\
=&\ \int_{0}^{\infty} \mathbb{E}_{X^{(0)}(0)=\delta k} \bigg[ 1 (t \leq \tau^{(1)}(0) ) \Delta^3 h(X^{(0)}(t)) \bigg] dt + \Delta^{2} f_h(0) - \Delta f_h(0). \label{eq:d3calcs}
\end{align}
Similarly,
\begin{align*}
\Delta^2 f_h(\delta k) =&\ \int_{0}^{\infty} \mathbb{E}_{X^{(0)}(0)=\delta k} \Big( h(X^{(2)}(t)) - 2 h(X^{(1)}(t)) - h(X^{(0)}(t)) \Big) dt \notag \\
=&\ \int_{0}^{\infty} \mathbb{E}_{X^{(0)}(0)=\delta k} \bigg[ 1 (t \leq \tau^{(1)}(0) ) \Delta^2 h(X^{(0)}(t)) \bigg] dt + \Delta f_h(0).
\end{align*}
Combining the representations for $\Delta^3 f_h(\delta k) $ and $\Delta^2 f_h(\delta k) $ above, we conclude that
\begin{align*}
\Delta^3 f_h(\delta k) =&\ \int_{0}^{\infty} \mathbb{E}_{X^{(0)}(0)=\delta k} \bigg[ 1 (t \leq \tau^{(1)}(0) ) \Delta^3 h(X^{(0)}(t)) \bigg] dt \notag \\
&+ \int_{0}^{\infty} \mathbb{E}_{X^{(0)}(0)= 0} \bigg[ 1 (t \leq \tau^{(1)}(0) ) \Delta^2 h(X^{(0)}(t)) \bigg] dt.
\end{align*}
Assuming $h \in \text{\rm dLip(1)}$, $\abs{\Delta^{2} h(\delta k)} \leq \delta^2$, and $\abs{\Delta^{3} h(\delta k)} \leq \delta^3$, we apply \eqref{eq:taubound} to conclude that
\begin{align*}
\abs{\Delta^{3} f_h(\delta k)} \leq \frac{\delta^{3}(k+1)}{\mu - \lambda } + \delta^{2} \frac{1}{\mu - \lambda}.
\end{align*}
The bound on $\abs{\Delta^{2} f_h(\delta k)}$ is obtained similarly. Lastly, if we consider the first equality in \eqref{eq:d3calcs} with $k = 0$, and consider what happens after the first jump of the CTMC, we get
\begin{align*}
\Delta^{3} f_h(0) = \frac{1}{\lambda + \mu} \Delta^{3} h(0) + \frac{\mu }{\lambda + \mu }\big( \Delta^{2} f_h(0) - \Delta f_h(0) \big) + \frac{\lambda}{\lambda + \mu } \Delta^{3} f_h(\delta).
\end{align*}
\hfill $\square$\endproof
\subsection{Tightness via the Poisson Equation}
\label{sec:tight}
One way to show a sequence of CTMC stationary distributions is tight is by bounding $\mathbb{E} \abs{X}$. For the $M/M/1$ system this task is trivial because the stationary distribution is known. However, obtaining a useful upper bound on $\mathbb{E} \abs{X}$ is harder for more complicated systems and usually involves using some kind of Lyapunov function. We can also use the Poisson equation
\begin{align*}
G_X f_h(\delta k) = \mathbb{E} h(X) - h(\delta k), \quad k \geq 0
\end{align*}
as follows.
Pick $h(\delta k) = \abs{\delta k}$ and evaluating the above at $k = 0$ to get
\begin{align*}
\lambda \big( f_h(\delta) - f_h(0) \big) = \lambda \Delta f_h(0) = \mathbb{E} \abs{X} = \mathbb{E} X.
\end{align*}
Recall \eqref{eq:dfk_mm1} and \eqref{eq:taubound} from the previous section, which imply that $ \mathbb{E} X = \delta \lambda/(\mu -\lambda ) = \delta \rho /(1-\rho)$. Choosing $\delta$ to be any constant multiple of $1-\rho$ ensures that $\{X\}_{\rho < 1}$ is tight.
The main takeaway is that the problem of tightness is equivalent to bounding $G_X f_h(\delta k)$ at \emph{a single point}, usually the fluid equilibrium of the CTMC. At the fluid equilibrium, $G_{X} f_h(\delta k)$ typically consists of some combination of first- and second-order Stein factors.
\section{Misalignment of Diffusion Synchronous Couplings}
\label{sec:compare_couplings}
We have presented the prelimit approach as a parallel to the diffusion approach for the purposes of bounding $\abs{\mathbb{E} h(X) - \mathbb{E} h(Y)}$. As we have seen, the main challenge with either approach is bounding the differences/derivatives of the respective Poisson equation solution. In theory, any bound achievable using one approach should be achievable with the other. In practice, there are slight technical differences between working with a discrete-valued CTMC and a diffusion living on a continuum. In this section we illustrate one technical nuance that arises when we use synchronous couplings to bound the third derivatives of $f_{h^{*}}(x)$ in the diffusion approach. We term this the ``misalignment of synchronous couplings''. The main takeaway is that the misaligned synchronous couplings add extra complexity to the problem. In contrast, the analogous analysis using the prelimit approach in Section~\ref{sec:synchronous} is cleaner because the CTMC is restricted to the grid.
Recall the generic diffusion process $\{Y(t) \in \mathbb{R} \}$ defined in \eqref{eq:sdediffusion}. We assume for simplicity that the diffusion coefficient $a(x) = a$ for all $x \in \mathbb{R}$ and define the synchronous couplings
\begin{align*}
Y^{(i)}(t) = Y^{(0)}(0) + i \varepsilon + \int_{0}^{t} b(Y^{(i)}(s)) ds + \int_{0}^{t} \sqrt{a} d W(s), \quad i = 0,1,2,3.
\end{align*}
The four couplings start at different initial conditions but share the same Brownian motion. Since $f_{h^{*}}(x)$ is given by \eqref{eq:diffusionpoissonsolution}, it follows that
\begin{align}
&\frac{\partial^3}{\partial x^3} f_{h^{*}}(x) \notag \\
=&\ \lim_{\varepsilon \to 0} \frac{1}{\varepsilon^{3}} \int_{0}^{\infty} \mathbb{E}_{Y^{(0)}(0) = x } \Big(h^{*}(Y^{(3)}(t)) - 3 h^{*}(Y^{(2)}(t)) + 3 h^{*}(Y^{(1)}(t)) - h^{*}(Y^{(0)}(t)) \Big) dt. \label{eq:misalignstart}
\end{align}
To show the integral on the right-hand side is finite, one must characterize the speed at which the synchronous couplings converge to one another. Furthermore, the integral must be of order $\varepsilon^{3}$ for the limit to exist. Let us consider this last point further.
Given a sufficiently differentiable function $g: \mathbb{R} \to \mathbb{R}$, we know that its derivatives can be approximated by finite differences. For instance, Taylor expansion tells us that
\begin{align}
g'''(x) \approx \frac{\big( g(x''') - 3 g(x'') + 3 g(x') - g(x)\big) }{\varepsilon^{3}} \label{eq:stencil}
\end{align}
when $x''' = x+ 3\varepsilon$, $x'' = x + 2 \varepsilon$, and $x' = x + \varepsilon$. The precise spacing of $x,x',x'',x'''$ relative to each other is essential for the limit (as $\varepsilon \to 0$) of the right-hand side in \eqref{eq:stencil} to exist. For example, if $x''' = x + 4\varepsilon$, then the numerator is now of order $\varepsilon$ instead of $\varepsilon^3$, and the right-hand side diverges as $\varepsilon \to 0$. Therefore, one way to show that the integral in \eqref{eq:misalignstart} is of order $\varepsilon^{3}$ is to prove that the diffusion couplings maintain the appropriate spacing relative to each other so that the integrand is of order $\varepsilon^3$ for each $t \geq 0$.
Indeed, this is precisely the result of Lemma 3.3 of \cite{GorhMack2016}, which says that if $h(x)$ and $b(x)$ are smooth enough, and if $b(x)$ is $k$-strongly concave, then
\begin{align}
\abs{h^{*}(Y^{(3)}(t)) - 3 h^{*}(Y^{(2)}(t)) + 3 h^{*}(Y^{(1)}(t)) - h^{*}(Y^{(0)}(t)) } \leq \varepsilon^3 C e^{-kt/2} \label{eq:gormack}
\end{align}
almost surely, where the constant $C$ depends on $k$, $h(x)$ and $b(x)$. The above inequality then implies that
\begin{align}
\lim_{\varepsilon \to 0} \frac{1}{\varepsilon^{3}} \int_{0}^{\infty} \mathbb{E}_{Y^{(0)}(0) = x } \Big|h^{*}(Y^{(3)}(t)) - 3 h^{*}(Y^{(2)}(t)) + 3 h^{*}(Y^{(1)}(t)) - h^{*}(Y^{(0)}(t)) \Big| dt \leq 2C/k. \label{eq:gormakimply}
\end{align}
Similarly, \eqref{eq:gormack} also holds for $d$-dimensional diffusions with constant diffusion coefficients. Unfortunately, if the assumptions on the drift are violated, e.g.\ the drift is only Lipschitz-continuous or the diffusion has a reflecting boundary, then \eqref{eq:gormack} no longer holds because the diffusion couplings become misaligned. This misalignment complicates the problem of bounding \eqref{eq:misalignstart} because one cannot use \eqref{eq:gormack} anymore.
As an example, we now illustrate how this misalignment occurs in the RBM that approximates the $M/M/1$ system. In contrast, the discrete nature of the prelimit approach prevents this kind of misalignment from happening. Recall the RBM defined in \eqref{eq:rbm},
\begin{align*}
Y(t) = Y(0) + \delta (\lambda - \mu)t+ \delta\sqrt{(\lambda + \mu)}W(t) + R(t), \quad t \geq 0,
\end{align*}
and let $Y$ be the random variable having its stationary distribution. It was shown in \cite{HarrReim1981} that
\begin{align*}
R(t) = -\inf_{0 \leq s \leq t} \Big\{ Y(0) + \delta (\lambda - \mu)s+ \delta\sqrt{(\lambda + \mu)}W(s) \Big\}.
\end{align*}
We wish to bound the third derivative of $f_{h^{*}}(x)$.
For simplicity, we choose $h^{*}(x) = x$. Let us define the four coupled processes
\begin{align}
Y^{(i)}(t) =&\ Y^{(0)}(0) + i \varepsilon + \delta (\lambda - \mu)t+ \delta\sqrt{(\lambda + \mu)}W(t) + R^{(i)}(t), \notag \\
\text{ where } \quad R^{(i)}(t) =&\ -\inf_{0 \leq s \leq t} \Big\{ Y^{(i)}(0) + \delta (\lambda - \mu)s+ \delta\sqrt{(\lambda + \mu)}W(s) \Big\}, \quad i = 0,1,2,3. \label{eq:ridef}
\end{align}
We also define $D_3(t) =Y^{(3)}(t) - 3Y^{(2)}(t) + 3 Y^{(1)}(t) - Y^{(0)}(t)$. It follows that
\begin{align}
\frac{\partial^3}{\partial x^3} f_{h^{*}}(x) =&\ \lim_{\varepsilon \to 0} \frac{1}{\varepsilon^3} \int_{0}^{\infty} \mathbb{E}_{Y^{(0)}(0) = x}D_3(t) dt. \label{eq:d3_starting_point}
\end{align}
We define
\begin{align*}
\gamma_1 = \inf_{t \geq 0} \{Y^{(1)}(t) = 3\varepsilon/4 \}, \quad \gamma_2 = \inf_{t \geq 0} \{Y^{(1)}(t) = \varepsilon/4 \}.
\end{align*}
We will prove at the end of this section that
\begin{align}
D_3(t) \leq -\varepsilon/4, \quad \text{ for } t \in [\gamma_1,\gamma_2]. \label{eq:toproverbm}
\end{align}
We see that \eqref{eq:toproverbm} violates \eqref{eq:gormack}. Furthermore, the expected hitting time of a fixed level by a Brownian motion with drift is well known and implies that $\mathbb{E}(\gamma_2 - \gamma_1) = \varepsilon/(2\delta(\mu-\lambda)) $. Therefore, the integral in \eqref{eq:d3_starting_point} equals
\begin{align}
&\frac{1}{\varepsilon^3} \int_{0}^{\infty} \mathbb{E}_{Y^{(0)}(0) = x}(D_3(t) 1(t \in[\gamma_1,\gamma_2] )) dt +\frac{1}{\varepsilon^3} \int_{0}^{\infty} \mathbb{E}_{Y^{(0)}(0) = x}(D_3(t) 1(t \not\in[\gamma_1,\gamma_2] )) dt , \label{eq:int12}
\end{align}
and the first term is bounded from above by $ -(8\delta(\mu-\lambda) \varepsilon)^{-1}$, which diverges as $\varepsilon \to 0$. Therefore, unlike in \eqref{eq:gormakimply}, we cannot just bound $\abs{D_3(t)}$ and take the limit as $\varepsilon \to 0$.
In reality, $\frac{\partial^3}{\partial x^3} f_{h^{*}}(x)$ is well defined and the right-hand side of \eqref{eq:d3_starting_point} exists and it is possible to prove that the second integral in \eqref{eq:int12} contains a positive term of order $1/\varepsilon$ that cancels out the first integral. Indeed, Theorem 1.2 in \cite{Daly2008} presents bounds on $\frac{\partial^k}{\partial x^k} f_{h^{*}}(x)$ for any $k \geq 2$. We conclude by verifying \eqref{eq:toproverbm}. By definition,
\begin{align*}
Y^{(i+1)}(t) - Y^{(i)}(t) = \varepsilon + R^{(i+1)}(t) - R^{(i)}(t), \quad i = 0,1,2,
\end{align*}
for every $t \geq 0$. Since $R^{(i)}(t) = 0$ for $i = 1,2,3$ when $t < \inf_{s \geq 0} \{Y^{(1)}(s) = 0\}$, we have that
\begin{align*}
Y^{(3)}(t) - Y^{(2)}(t) =Y^{(2)}(t) - Y^{(1)}(t) = \varepsilon, \quad t \in [\gamma_1,\gamma_2]
\end{align*}
because $\gamma_2 < \inf_{s \geq 0} \{Y^{(1)}(s) = 0\}$. Thus,
\begin{align*}
D_3(t) =&\ -\varepsilon + Y^{(1)}(t) - Y^{(0)}(t)= R^{(1)}(t) - R^{(0)}(t) = - R^{(0)}(t), \quad t \in [\gamma_1,\gamma_2].
\end{align*}
One can check that $R^{(0)}(\gamma_1) = \varepsilon/4$ and $R^{(0)}(\gamma_2) = 3\varepsilon/4$ using the form of $R^{(i)}(t)$ in \eqref{eq:ridef}. Since $R^{(0)}(t)$ is non-decreasing, we have $R^{(0)}(t) \in [\varepsilon/4, 3\varepsilon/4]$ when $t \in [\gamma_1,\gamma_2]$, proving \eqref{eq:toproverbm}.
\section{Conclusion}
\label{sec:conclusion}
In this paper we introduced the prelimit generator comparison approach and used the $M/M/1$ model to illustrate it in practice. In applying the approach, we overcame two technical challenges. First, we used an interpolator to extend the prelimit Poisson equation to the continuum. Second, we interchanged the interpolator with the CTMC generator. Our solution to both challenges extends beyond the $M/M/1$ model. Appendices~\ref{sec:interpolation} and \ref{sec:mdimunbound} contain the general multidimensional interpolation and interchange results. These are intended to simplify as much as possible the tedious aspects of the prelimit approach, and to make it easy for readers to apply the approach to their own problem.
One direction we have not considered is working with the Kolmogorov distance. For two real-valued random variables $U,U'$, the Kolmogorov distance is defined as
\begin{align*}
d_{K}(U,U') = \sup_{z \in \mathbb{R}} \big| \mathbb{E} \big(1(U \geq z)\big) - \mathbb{E} \big(1(U' \geq z)\big)\big|.
\end{align*}
It is well known (e.g.,\ \cite{BravDaiFeng2016}) that the discontinuity in the test functions $1(\cdot \geq z)$ makes working with the Kolmogorov distance more difficult than the Wasserstein. Even though we deal with discrete functions and their interpolations, the issue with the discontinuity in $1(\cdot \geq z)$ will still come up in the difference bounds on $f_h(x)$. Although there are undoubtedly technical challenges to overcome, the author believes that the prelimit approach can be used as a tool to bound the Kolmogorov distance.
\begin{APPENDICES}
\section{Multidimensional Interpolation}
\label{sec:interpolation}
We now prove Theorem~\ref{thm:1dinterpolant_def}. We then state and prove the multidimensional version -- Theorem~\ref{thm:interpolant_def}.
\proof{Proof of Theorem~\ref{thm:1dinterpolant_def}}
Given $f: K \to \mathbb{R}$, for each $k \in \mathbb{Z}$ such that $\delta k \in K_{4}$ we define
\begin{align}
P_{k}(x) =&\ f(\delta k) + \Big(\frac{x-\delta k}{\delta} \Big)(\Delta - \frac{1}{2}\Delta^2 + \frac{1}{3}\Delta^3) f(\delta k) \notag \\
&+ \frac{1}{2} \Big(\frac{x-\delta k}{\delta} \Big)^2\big(\Delta^2 - \Delta^3\big) f(\delta k) + \frac{1}{6} \Big(\frac{x-\delta k}{\delta} \Big)^3 \Delta^3 f(\delta k)\notag \\
& -\frac{23}{3} \Big(\frac{x-\delta k}{\delta} \Big)^4 \Delta^4 f(\delta k) +\frac{41}{2} \Big(\frac{x-\delta k}{\delta} \Big)^5\Delta^4 f(\delta k) \notag \\
&- \frac{55}{3} \Big(\frac{x-\delta k}{\delta} \Big)^6\Delta^4 f(\delta k) +\frac{11}{2} \Big(\frac{x-\delta k}{\delta} \Big)^7\Delta^4 f(\delta k) , \quad x \in \mathbb{R}. \label{eq:pplus}
\end{align}
From \eqref{eq:pplus} we have $P_{k}(\delta k) = f(\delta k)$, implying \eqref{eq:interpolates}. Since $P_{k}(x) \in C^{\infty}(\mathbb{R})$, we know $A f(x)$ is infinitely differentiable on $\text{Conv}(K_4) \setminus K_4$. Furthermore, it is straightforward to verify that
\begin{align}
\frac{\partial^{a}}{\partial x^{a}} P_{k-1}(x)\Big|_{x =\delta k} = \frac{\partial^{a}}{\partial x^{a}} P_{k}(x)\Big|_{x =\delta k} , \quad \text{ for } a = 0,1,2,3. \label{eq:onedsmooth}
\end{align}
The property above implies $A f(x) = P_{k(x)}(x) \in C^{3}(\text{Conv}(K_4))$. The weights $\alpha^{k}_{k+i}(x)$ can be read off by combining the coefficients corresponding to $f(\delta (k+i))$ in \eqref{eq:pplus}. For example,
\begin{align*}
\alpha^{k}_k(x) =&\ 1 - \frac{11}{6} \Big(\frac{x-\delta k}{\delta} \Big) + \Big(\frac{x-\delta k}{\delta} \Big)^2 - \frac{1}{6}\Big(\frac{x-\delta k}{\delta} \Big)^3 - \frac{23}{3}\Big(\frac{x-\delta k}{\delta} \Big)^4 \notag \\
&+\frac{41}{2}\Big(\frac{x-\delta k}{\delta} \Big)^5 - \frac{55}{3}\Big(\frac{x-\delta k}{\delta} \Big)^6 + \frac{11}{2}\Big(\frac{x-\delta k}{\delta} \Big)^7.
\end{align*}
It is straightforward to check that
\begin{align*}
\sum_{i=0}^{4} \alpha^{k}_{k+i}(x) = 1, \quad \alpha_{k}^{k}(\delta k) = 1, \quad \text{ and } \quad \alpha_{k+i}^{k} (\delta k) = 0.
\end{align*}
The weights are degree-$7$ polynomials in $(x-\delta k)/ \delta$ whose coefficients do not depend on $k$ or $\delta$, i.e., $\alpha^{k}_{k+i}(x) = J_{i}\big((x-\delta k)/ \delta \big)$
for some polynomial $J_i(\cdot)$. Consequently, for any $x \in \mathbb{R}$,
\begin{align*}
\alpha^{k+j}_{k+j+i}(x+ \delta j) = J_{i }\Big(\frac{x + \delta j- \delta(k+j)}{\delta} \Big)= J_{i}\Big(\frac{x-\delta k}{\delta} \Big) = \alpha^{k}_{k+i}(x),\quad j,k \in \mathbb{Z},\ 0 \leq i \leq 4.
\end{align*}
\hfill $\square$\endproof
We now generalize Theorem~\ref{thm:1dinterpolant_def} and define an interpolation operator that can interpolate any function defined on $ K \cap \delta \mathbb{Z}^{d}$ where $K \subset \mathbb{R}^{d}$ is convex. The interpolator is based on forward differences, but one could also use central or backward differences to accommodate different domains shapes. The following theorem summarizes the key properties we want from it.
\begin{theorem}
\label{thm:interpolant_def}
Let the weights $\{\alpha_{k+i}^{k}: \mathbb{R} \to \mathbb{R} \ :\ k \in \mathbb{Z},\ i = 0,1,2,3,4 \}$ be as in Theorem~\ref{thm:1dinterpolant_def} and suppose we are given a convex set $K \subset \mathbb{R}^{d}$ and a function $f: K \cap \delta \mathbb{Z}^{d} \to \mathbb{R}$. Letting $i = (i_1, \ldots, i_d) \in \mathbb{Z}^{d}$, we use the weights to define
\begin{align}
A f(x) =&\ \sum_{i_d = 0}^{4} \alpha_{k_d(x)+i_d}^{k_d(x)}(x_d)\cdots \sum_{i_1 = 0}^{4} \alpha_{k_1(x)+i_1}^{k_1(x)}(x_1) f(\delta(k(x)+i)) \notag \\
=&\ \sum_{i_1, \ldots, i_d = 0}^{4} \bigg(\prod_{j=1}^{d} \alpha_{k_j(x) +i_j}^{k_j(x) }(x_j)\bigg) f(\delta k(x)+i) , \quad x \in \text{Conv}(K_4), \label{eq:af2}
\end{align}
where $k(x) \in \mathbb{Z}^{d}$ is defined by $k_{i}(x) = \lfloor x_i/\delta\rfloor$, and
\begin{align*}
K_{4} = \{x \in K \cap \delta \mathbb{Z}^{d} : \delta(k(x)+ i) \in K \cap \delta \mathbb{Z}^{d} \text{ for all } 0 \leq i \leq 4e\}.
\end{align*}
Then $A f(x) \in C^{3}(\text{Conv}(K_4))$ and is infinitely differentiable almost everywhere on $\text{Conv}(K_4) $. Also,
\begin{align}
A f(\delta k) = f(\delta k), \quad \delta k \in K \cap \delta \mathbb{Z}^{d}, \label{eq:interpolates2}
\end{align}
and there exists a constant $C > 0$ independent of $f(\cdot)$,$x$, and $\delta$, such that
\begin{align}
\Big| \frac{\partial^{a}}{\partial x^{a}} Af(x) \Big| \leq&\ C \delta^{-\norm{a}_{1}} \max_{\substack{ 0 \leq i_j \leq 4-a_j \\ j = 1,\ldots, d}} \abs{\Delta_{1}^{a_1}\ldots \Delta_{d}^{a_d} f(\delta (k(x)+i))}, \quad x \in \text{Conv}(K_4), \label{eq:multibound2}
\end{align}
for $0 \leq \norm{a}_{1} \leq 3$, and \eqref{eq:multibound2} also holds when $\norm{a}_{1} = 4$ for almost all $x \in \text{Conv}(K_4) $.
\end{theorem}
Note that for any $J \subset \{1, \ldots, d\}$ and $J^{c} = \{1,\ldots, d\} \setminus J$, we may rewrite \eqref{eq:af2} as
\begin{align}
Af(x) =&\ \sum_{\substack{i_j = 0 \\ j \in J^{c}}}^{4} \Bigg( \bigg(\prod_{j\in J^{c}} \alpha_{k_j+i_j}^{k_j}(x_j)\bigg) \sum_{\substack{i_j = 0 \\ j \in J}}^{4}\bigg(\prod_{j \in J} \alpha_{k_j+i_j}^{k_j}(x_j)\bigg) f(\delta(k+i))\Bigg). \label{eq:alpha2}
\end{align}
The representation in \eqref{eq:alpha2} will come in handy later on. We define
\begin{align}
F_k(x)=&\ \sum_{i_d = 0}^{4} \alpha_{k_d+i_d}^{k_d}(x_d)\cdots \sum_{i_1 = 0}^{4} \alpha_{k_1+i_1}^{k_1}(x_1) f(\delta(k+i)), \quad x \in \mathbb{R}^{d}, k \in K_4 \label{eq:alpha1}
\end{align}
to be the multidimensional analog of $P_{k}(x)$.
Note that $A f(x)$ defined in Theorem~\ref{thm:interpolant_def} satisfies $A f(x) = F_{k(x)}(x)$ for $x \in \text{Conv}(K_4)$. Furthermore, \eqref{eq:alphas_interpolate} of Theorem~\ref{thm:1dinterpolant_def} implies \eqref{eq:interpolates2}. To prove Theorem~\ref{thm:interpolant_def}, it remains to verify the smoothness of $A f(x)$ and \eqref{eq:multibound2}.
For any $x \in \mathbb{R}^{d}$ and $J \subset \{1, \ldots, d\}$, we write $x_{J}$ to denote the vector whose $i$th element equals $x_i 1(i \in J)$. The following result is the multidimensional analog of \eqref{eq:onedsmooth}. We prove it at the end of this section.
\begin{lemma}
\label{lem:multidinterp}
Fix $k \in K_4$. For $u \in [0,1]^{d}$, let $\Theta(u) = \{i: u_i = 1\}$ and $\Theta(u)^{c} = \{1,\ldots, d\}\setminus \Theta(u) $. Then
\begin{align}
& \frac{\partial^{a}}{\partial x^{a}} F_k(x) \Big|_{x = \delta(k+u)} = \frac{\partial^{a}}{\partial x^{a}}F_{k+e_{\Theta(u)}}(x) \Big|_{x=\delta(k+u)} \quad \text{ for } 0 \leq \norm{a}_{1} \leq 3. \label{eq:multipaste}
\end{align}
Furthermore, there exists a constant $C> 0$ independent of $f(\cdot)$, $k$, and $\delta$ such that
\begin{align}
\Big| \frac{\partial^{a}}{\partial x^{a}} F_k(x) \Big| \leq&\ C \delta^{-\norm{a}_{1}}\bigg(\prod_{j=1}^{d}\Big(1 + \Big|\frac{x_j-\delta k_j}{\delta} \Big| \Big)^{7-a_j} \bigg) \max_{\substack{ 0 \leq i_j \leq 4-a_j \\ j = 1,\ldots, d}} \abs{\Delta_{1}^{a_1}\ldots \Delta_{d}^{a_d} f(\delta (k+i))} \label{eq:premultibound}
\end{align}
for all $0 \leq \norm{a}_{1} \leq 4$ and all $x \in \text{Conv}(K_4)$ where the derivative above is well defined.
\end{lemma}
\noindent
Lemma~\ref{lem:multidinterp} implies Theorem~\ref{thm:interpolant_def}. Indeed, \eqref{eq:multipaste} implies $A f(x) \in C^{3}(\text{Conv}(K_4))$, and since $\alpha_{k+i}^{k}(x) \in C^{\infty}(\mathbb{R})$, we know $A f(x)$ is infinitely differentiable everywhere except at the points where the $F_k(x)$ are glued together, i.e., on the set $\{x \in \text{Conv}(K_4)\ |\ x_i \in \delta \mathbb{Z} \text{ for some } i \in \{1, \ldots, d\} \}$, which has Lebesgue measure zero. Furthermore, \eqref{eq:multibound2} follows directly from \eqref{eq:premultibound}. We now prove Lemma~\ref{lem:multidinterp}.
\proof{Proof of Lemma~\ref{lem:multidinterp}}
We first prove \eqref{eq:multipaste}. Fix $k \in K_{4}$ and let $j' \in \Theta(u)$. From \eqref{eq:alpha2} it follows that
\begin{align*}
\frac{\partial^{a}}{\partial x^{a}} F_k(x)\Big|_{x = \delta(k+u)} =&\ \sum_{\substack{i_j = 0 \\ j \in \{1, \ldots, d\}\setminus \{j'\}}}^{4} \Bigg( \bigg(\prod_{j \in \{1, \ldots, d\}\setminus \{j'\}} \frac{\partial^{a_j}}{\partial x_{j}^{a_j}} \alpha_{k_j+i_j}^{k_j}(x_j)\Big|_{x_j = \delta(k_j+u_j)}\bigg) \\
& \hspace{2cm} \times \sum_{ i_{j'} = 0 }^{4}\bigg( \frac{\partial^{a_{j'}}}{\partial x_{j'}^{a_{j'}}}\alpha_{k_{j'}+i_{j'}}^{k_{j'}}(x_{j'})\Big|_{x_{j'} = \delta(k_{j'}+1)} \bigg) f(\delta(k+i))\Bigg).
\end{align*}
For the inner sum, note that
\begin{align*}
& \sum_{ i_{j'} = 0 }^{4}\bigg( \frac{\partial^{a_{j'}}}{\partial x_{j'}^{a_{j'}}}\alpha_{k_{j'}+i_{j'}}^{k_{j'}}(x_{j'})\Big|_{x_{j'} = \delta(k_{j'}+1)} \bigg) f(\delta(k+i)) \\
=&\ \sum_{ i_{j'} = 0 }^{4}\bigg(\frac{\partial^{a_{j'}}}{\partial x_{j'}^{a_{j'}}}\alpha_{k_{j'}+1+i_{j'}}^{k_{j'}+1}(x_{j'})\Big|_{x_{j'} = \delta(k_{j'}+1)} \bigg) f(\delta(k+i + e_{j'})),
\end{align*}
which follows from \eqref{eq:onedsmooth}. Repeating the above procedure for all other elements of $\Theta(u)$, we see that
\begin{align*}
\frac{\partial^{a}}{\partial x^{a}} F_k(x)\Big|_{x = \delta(k+u)} =&\ \sum_{\substack{i_j = 0 \\ j \in \Theta(u)^{c}}}^{4} \Bigg( \bigg(\prod_{j\in \Theta(u)^{c}} \frac{\partial^{a_j}}{\partial x_{j}^{a_j}} \alpha_{k_j+i_j}^{k_j}(x_j)\Big|_{x_j = \delta(k_j+u_j)}\bigg) \\
& \hspace{1cm} \times \sum_{\substack{i_j = 0 \\ j \in \Theta(u)}}^{4}\bigg(\prod_{j \in \Theta(u)} \frac{\partial^{a_j}}{\partial x_{j}^{a_j}}\alpha_{k_j+1+i_j}^{k_j+1}(x_j)\Big|_{x_j = \delta(k_j+1)} \bigg) f(\delta(k+i+e_{\Theta(u)}))\Bigg)\\
=&\ \frac{\partial^{a}}{\partial x^{a}} F_{k+e_{\Theta(u)}}(x) \Big|_{x=\delta(k+u)},
\end{align*}
which proves \eqref{eq:multipaste}. It remains to prove the bound on $\big| \frac{\partial^{a}}{\partial x^{a}} F_k(x)\big| $ in \eqref{eq:premultibound}. We know
\begin{align*}
\frac{\partial^{a}}{\partial x^{a}}F_k(x) =&\ \sum_{i_1=0}^{4}\frac{\partial^{a_1}}{\partial x_{1}^{a_1}} \alpha_{k_1+i_1}^{k_1}(x_1) \cdots\sum_{i_d=0}^{4}\frac{\partial^{a_d}}{\partial x_{d}^{a_d}} \alpha_{k_d+i_d}^{k_d}(x_d) f(\delta(k+i)).
\end{align*}
By inspecting the form of the one-dimensional $P_k(\cdot)$ in \eqref{eq:pplus}, one can check that
\begin{align*}
\sum_{i_d=0}^{4}\frac{\partial^{a_d}}{\partial x_{d}^{a_d}} \alpha_{k_d+i_d}^{k_d}(x_d) f(\delta (k +i )) = \delta^{-a_{d}} Q^{(d)}\Big( \frac{x_d - \delta k_d}{\delta} \Big) ,
\end{align*}
where $Q^{(d)}(\cdot)$ is a degree-$(7-a_{d})$ polynomial whose coefficients depend on $ f(\cdot)$ only through
\begin{align*}
&\Delta_{d}^{a_d} f\big(\delta (k + (i_1,\ldots, i_{d-1},i_d))\big), \quad \text{ for } i_d = 0, \ldots, 4-a_{d},
\end{align*}
and are independent of $\delta$. This implies in particular that
\begin{align*}
\Big| \sum_{i_d=0}^{4}\frac{\partial^{a_d}}{\partial x_{d}^{a_d}} \alpha_{k_d+i_d}^{k_d}(x_d) f(\delta(k+i)) \Big| \leq&\ C \delta^{-a_{d}} \Big(1 + \Big|\frac{x_d-\delta k_d}{\delta} \Big| \Big)^{7-a_{d}} \max_{\substack{ 0 \leq i_d \leq 4-a_{d} }} \abs{\Delta_{d}^{a_d} f(\delta (k+i))}.
\end{align*}
We now consider
\begin{align*}
\sum_{i_{d-1}=0}^{4}\frac{\partial^{a_{d-1}}}{\partial x_{{d-1}}^{a_{d-1}}} \alpha_{k_{d-1}+i_{d-1}}^{k_{d-1}}(x_{d-1}) \bigg( \sum_{i_d=0}^{4}\frac{\partial^{a_d}}{\partial x_{d}^{a_d}} \alpha_{k_d+i_d}^{k_d}(x_d) f(\delta(k+i)) \bigg).
\end{align*}
When viewed as a one-dimensional function of $x_{d-1}$, the above is again a degree-$(7-a_{d-1})$ polynomial that depends on the quantity inside the parentheses only through
\begin{align*}
&\Delta_{d-1}^{a_{d-1}}\bigg( \sum_{i_d=0}^{4}\frac{\partial^{a_d}}{\partial x_{d}^{a_d}} \alpha_{k_d+i_d}^{k_d}(x_d) f(\delta(k+i)) \bigg), \quad \text{ for } i_{d-1} = 0, \ldots, 4-a_{d-1}.
\end{align*}
Hence,
\begin{align*}
&\bigg| \sum_{i_{d-1}=0}^{4}\frac{\partial^{a_{d-1}}}{\partial x_{{d-1}}^{a_{d-1}}} \alpha_{k_{d-1}+i_{d-1}}^{k_{d-1}}(x_{d-1}) \bigg( \sum_{i_d=0}^{4}\frac{\partial^{a_d}}{\partial x_{d}^{a_d}} \alpha_{k_d+i_d}^{k_d}(x_d) f(\delta(k+i)) \bigg) \bigg| \\
\leq&\ C \delta^{-a_{d-1}-a_{d}} \Big(1 + \Big|\frac{x_{d-1}-\delta k_{d-1}}{\delta} \Big| \Big)^{7-a_{d-1}} \Big(1 + \Big|\frac{x_d-\delta k_d}{\delta} \Big| \Big)^{7-a_{d}} \\
& \hspace{4.5cm} \times \max_{\substack{ 0 \leq i_d \leq 4-a_{d} \\ 0 \leq i_{d-1} \leq 4 - a_{d-1} }} \abs{\Delta_{d-1}^{a_{d-1}} \Delta_{d}^{a_d} f(\delta (k+i))}.
\end{align*}
Repeating this argument along each of the remaining $d-2$ dimensions proves \eqref{eq:premultibound}.
\hfill $\square$\endproof
\section{Interchange in Multiple Dimensions}
\label{sec:mdimunbound}
In this section we prove Proposition~\ref{lem:interchanged1} by proving the more general Proposition~\ref{lem:mdiminterchange} stated below. Consider a CTMC living on $\mathbb{Z}^{d}$ with generator
\begin{align*}
G_X f(\delta k) =&\ \sum_{\ell \in \mathbb{Z}^{d}} \beta_{\ell}(\delta k) (f(\delta (k+\ell))-f(\delta k)), \quad k \in \mathbb{Z}^{d}.
\end{align*}
\begin{proposition} \label{lem:mdiminterchange}
Fix $f: \delta \mathbb{Z}^{d} \to \mathbb{R}$ and assume that
\begin{align}
\sum_{\ell \in \mathbb{Z}^{d}} \abs{\beta_{\ell}(\delta k) (f(\delta (k+\ell))-f(\delta k))} < \infty, \quad k \in \mathbb{Z}^{d}, \label{eq:intergrab2}
\end{align}
which is trivially satisfied when the number of possible transitions from each state is finite. For $x \in \mathbb{R}^{d}$ define $k(x) \in \mathbb{Z}^{d}$ by $k_{i}(x) = \lfloor x_i/\delta\rfloor $. Then
\begin{align}
A G_{X} f(x) =&\ \sum_{\ell \in \mathbb{Z}} A \beta_{\ell}(x) \big( A f(x+\delta \ell) - A f(x)\big) + \varepsilon(x), \quad x \in \mathbb{R}^{d}, \label{eq:intererror2}
\end{align}
where
\begin{align}
\varepsilon(x) =&\ \sum_{\ell \in \mathbb{Z}} \sum_{i_1, \ldots, i_d = 0}^{4} \bigg(\prod_{j=1}^{d} \alpha_{k_j(x) +i_j}^{k_j(x) }(x_j)\bigg) \Big(\beta_{\ell}\big(\delta (k(x)+i)\big) - A \beta_{\ell}(x)\Big) \notag \\
& \hspace{3cm} \times \Big(f\big(\delta (k(x)+\ell+i)\big)-f\big(\delta (k(x)+i)\big) - \big(f\big(\delta (k(x)+\ell )\big)-f (\delta k(x) )\big)\Big). \label{eq:varepsmdim}
\end{align}
\end{proposition}
Before proving Proposition~\ref{lem:mdiminterchange}, let us reconcile the forms of $\varepsilon(x)$ in \eqref{eq:varepsmdim} above and in \eqref{eq:vareps} of Proposition~\ref{lem:interchanged1}. When $d = 1$, \eqref{eq:varepsmdim} equals
\begin{align*}
&\sum_{\ell \in \mathbb{Z}} \sum_{i=0}^{4} \alpha^{k(x)}_{k(x)+i}(x)\Big(\beta_{\ell}\big(\delta (k(x)+i)\big) - A \beta_{\ell}(x)\Big) \\
& \hspace{3cm} \times \Big(f\big(\delta (k(x)+\ell+i)\big)-f\big(\delta (k(x)+i)\big) - \big(f\big(\delta (k(x)+\ell )\big)-f\big(\delta k(x)\big)\big)\Big).
\end{align*}
Using a telescoping series, we see that if $\ell > 0$,
\begin{align*}
& f\big(\delta (k(x)+\ell+i)\big)-f\big(\delta (k(x)+i)\big) - \big(f\big(\delta (k(x)+\ell )\big)-f\big(\delta k(x)\big)\big)\\
=&\ \sum_{j=0}^{\ell-1} \big(\Delta f(\delta(k(x)+j+\ell)) - \Delta f(\delta(k(x) +j))\big) \\
=&\ \sum_{j=0}^{i-1} \sum_{m=0}^{\ell-1} \Delta^2 f(\delta(k(x)+m +j)).
\end{align*}
Similarly, when $\ell < 0$,
\begin{align*}
& f\big(\delta (k(x)+\ell+i)\big)-f\big(\delta (k(x)+i)\big) - \big(f\big(\delta (k(x)+\ell )\big)-f\big(\delta k(x)\big)\big)\\
=&\ - \sum_{j=0}^{i-1} \sum_{m=-\ell}^{ -1} \Delta^2 f(\delta(k(x)+m +j)).
\end{align*}
Therefore, Propositions~\ref{lem:interchanged1} and \ref{lem:mdiminterchange} are equivalent when $d = 1$. When $d > 1$, it is also possible to write \eqref{eq:varepsmdim} as a telescoping series of second-order differences of $f(\delta k)$. We leave this as an exercise to the interested reader.
\proof{Proof of Proposition~\ref{lem:mdiminterchange}}
Fix $x \in \mathbb{R}^{d}$. We will write $k$ instead of $k(x)$ for convenience. Recalling the form of $A$ from Theorem~\ref{thm:interpolant_def},
it follows that $A G_{X} f(x)$ equals
\begin{align}
& \sum_{i_1, \ldots, i_d = 0}^{4} \bigg(\prod_{j=1}^{d} \alpha_{k_j +i_j}^{k_j }(x_j)\bigg) \sum_{\ell \in \mathbb{Z}} \beta_{\ell}\big(\delta (k+i)\big) \Big(f\big(\delta (k+\ell+i)\big)-f\big(\delta (k+i)\big)\Big) \notag \\
=&\ \sum_{\ell \in \mathbb{Z}} A \beta_{\ell}(x) \sum_{i_1, \ldots, i_d = 0}^{4} \bigg(\prod_{j=1}^{d} \alpha_{k_j +i_j}^{k_j }(x_j)\bigg) \Big(f\big(\delta (k+\ell+i)\big)-f\big(\delta (k+i)\big)\Big) \label{eq:first} \\
&+ \sum_{\ell \in \mathbb{Z}} \sum_{i_1, \ldots, i_d = 0}^{4} \bigg(\prod_{j=1}^{d} \alpha_{k_j +i_j}^{k_j }(x_j)\bigg)\Big(\beta_{\ell}\big(\delta (k+i)\big) - A \beta_{\ell}(x)\Big) \Big(f\big(\delta (k+\ell+i)\big)-f\big(\delta (k+i)\big)\Big). \label{eq:second}
\end{align}
Interchanging the summations is allowed by the Fubini-Tonelli theorem due to assumption \eqref{eq:intergrab2}. Consider first the inner sum in \eqref{eq:first} and observe that for each $\ell \in \mathbb{Z}^{d}$,
\begin{align*}
& \sum_{i_1, \ldots, i_d = 0}^{4} \bigg(\prod_{j=1}^{d} \alpha_{k_j +i_j}^{k_j }(x_j)\bigg) f\big(\delta (k+\ell+i)\big)- \sum_{i_1, \ldots, i_d = 0}^{4} \bigg(\prod_{j=1}^{d} \alpha_{k_j +i_j}^{k_j }(x_j)\bigg) f\big(\delta (k+i)\big) \\
=&\ \sum_{i_1,\ldots,i_d = 0}^{4}\bigg(\prod_{j=1}^{d} \alpha_{k_j+\ell_{j} +i_j}^{k_j+\ell_{j}}(x_j+\delta \ell_{j})\bigg) f\big(\delta (k+\ell+i)\big)- A f(x)\\
=&\ A f(x+\delta \ell) - A f(x),
\end{align*}
where in the first equality we used the translation invariance property of the weights stated in \eqref{eq:weights} of Theorem~\ref{thm:1dinterpolant_def}. Moving on, we see that \eqref{eq:second} equals
\begin{align*}
& \sum_{\ell \in \mathbb{Z}} \sum_{i_1, \ldots, i_d = 0}^{4} \bigg(\prod_{j=1}^{d} \alpha_{k_j +i_j}^{k_j }(x_j)\bigg) \Big(\beta_{\ell}\big(\delta (k+i)\big) - A \beta_{\ell}(x)\Big) \\
& \hspace{3cm} \times \Big(f\big(\delta (k+\ell+i)\big)-f\big(\delta (k+i)\big) - \big(f\big(\delta (k+\ell )\big)-f (\delta k )\big)\Big)\\
&+ \sum_{\ell \in \mathbb{Z}} \Big(f\big(\delta (k+\ell )\big)-f (\delta k )\Big)\sum_{i_1, \ldots, i_d = 0}^{4} \bigg(\prod_{j=1}^{d} \alpha_{k_j +i_j}^{k_j }(x_j)\bigg) \Big(\beta_{\ell}\big(\delta (k+i)\big) - A \beta_{\ell}(x)\Big) .
\end{align*}
The second line equals zero because \eqref{eq:weights_sum_one} of Theorem~\ref{thm:1dinterpolant_def} implies $\sum_{i_1, \ldots, i_d = 0}^{4} \Big(\prod_{j=1}^{d} \alpha_{k_j +i_j}^{k_j }(x_j)\Big) = 1$ and because $A \beta_{\ell}(x) = \sum_{i_1, \ldots, i_d = 0}^{4} \Big(\prod_{j=1}^{d} \alpha_{k_j +i_j}^{k_j }(x_j)\Big) \beta_{\ell}\big(\delta (k+i)\big)$ by definition.
\hfill $\square$\endproof
\section{Proofs of Miscellaneous Technical Lemmas}
\subsection{Proof of Lemma~\ref{lem:convdet}}
\label{sec:proofconvdet}
We now state and prove an auxiliary result, and then prove Lemma~\ref{lem:convdet}.
\begin{lemma} \label{lem:auxconv}
Suppose $g^*:\mathbb{R} \to \mathbb{R}$ is three times continuously differentiable with an absolutely continuous third derivative, and let $g: \delta \mathbb{Z} \to \mathbb{R}$ be its restriction to $\delta \mathbb{Z} $. For $1 \leq v \leq 4$,
\begin{align}
\Delta^{v} g(\delta k) = \delta^{v} \int_{0}^{v \delta} c^{v}(u) \frac{\partial^{v}}{\partial x^{v}} g^*(\delta k + u) du, \label{eq:cver}
\end{align}
where $c^{v}(x)$ is a function such that $\sup_{x \in [0,\delta v]} \abs{c^{v}(x)} \leq C $ for some constant $C > 0$ independent of $k$, $g^*(x)$, and $\delta$.
\end{lemma}
\proof{Proof of Lemma~\ref{lem:auxconv}}
Suppose $v = 4$; the case $1 \leq v \leq 3$ is handled similarly. Using Taylor expansion, we have
\begin{align*}
\Delta^{4} g(\delta k) =&\ \Delta^{3} \big(g (\delta (k+1)) - g(\delta k) \big)\\
=&\ \Delta^{3} \Big( \delta (g^*)'(\delta k) + \frac{1}{2}\delta^{2} (g^*)''(\delta k) + \frac{1}{6} \delta^{3} (g^*)'''(\delta k) + \frac{1}{6}\int_{0}^{\delta} g^{(4)}(\delta k + u) (\delta - u)^{3} du \Big).
\end{align*}
We note that
\begin{align*}
&\Delta^{3} \Big( \frac{1}{6}\int_{0}^{\delta} g^{(4)}(\delta k + u) (\delta - u)^{3} du \Big) \\
=&\ \frac{1}{6}\Bigg( \int_{0}^{\delta} (g^*)^{(4)}(\delta (k+3) + u) (\delta - u)^{3} du - 3 \int_{0}^{\delta} (g^*)^{(4)}(\delta (k+1) + u) (\delta - u)^{3} du \\
& \qquad + 3\int_{0}^{\delta} (g^*)^{(4)}(\delta (k+2) + u) (\delta - u)^{3} du - \int_{0}^{\delta} (g^*)^{(4)}(\delta k + u) (\delta - u)^{3} du \Bigg).
\end{align*}
Applying a similar expansion to $\Delta^{3} \big( \delta (g^*)'(\delta k) + \frac{1}{2}\delta^{2} (g^*)''(\delta k) + \frac{1}{6} \delta^{3} (g^*)'''(\delta k) \big)$ proves \eqref{eq:cver}.
\hfill $\square$\endproof
\proof{Proof of Lemma~\ref{lem:convdet}}
Since $h^*(X) = h(X)$, the triangle inequality implies that
\begin{align*}
\big| \mathbb{E} h^*(X) - \mathbb{E} h^*(Y) \big| \leq&\ \big| \mathbb{E} h(X ) - \mathbb{E} Ah( Y) \big| + \big| \mathbb{E} Ah( Y) - \mathbb{E} h^*( Y) \big|.
\end{align*}
For $x \in \mathbb{R}^{d}$ let $k(x) \in \mathbb{Z}^{d}$ be defined by $k_{i}(x) = \lfloor x_i/\delta\rfloor$. Since $h^*(\delta k(x)) = h(\delta k(x)) = A h(\delta k(x))$ and $\abs{x_i - k_i(x)} \leq \delta$,
\begin{align*}
\big| \mathbb{E} Ah(Y) - \mathbb{E} h^*(Y) \big| \leq&\ \big| \mathbb{E} Ah(Y) - \mathbb{E} h^*(\delta k(Y)) \big| + \big| \mathbb{E} h^*(\delta k(Y)) - \mathbb{E} h^*(Y) \big|\\
=&\ \big| \mathbb{E} Ah(Y) - \mathbb{E} Ah(\delta k(Y)) \big| + \big| \mathbb{E} h^*(\delta k(Y)) - \mathbb{E} h^*(Y) \big|\\
\leq&\ C \delta \max_{\substack{ 1 \leq j \leq d }}\sup_{\substack{ x \in \mathbb{R}^{d} }} \bigg| \frac{\partial}{\partial x_{j}} Ah(x) \bigg| + C \delta \max_{\substack{ 1 \leq j \leq d }}\sup_{\substack{ x \in \mathbb{R}^{d} }} \bigg| \frac{\partial}{\partial x_{j}} h^{*}(x) \bigg|.
\end{align*}
Using the bound in \eqref{eq:multibound2} from Theorem~\ref{thm:interpolant_def}, it follows that
\begin{align*}
\max_{\substack{ 1 \leq j \leq d }}\sup_{\substack{ x \in \mathbb{R}^{d} }} \bigg| \frac{\partial}{\partial x_{j}} Ah(x)\bigg| \leq C \delta^{-1} \max_{\substack{ 1 \leq j \leq d }}\sup_{\substack{ x \in \mathbb{R}^{d} }} \abs{ \Delta_{j} h(\delta k(x))} =&\ C \delta^{-1}\max_{\substack{ 1 \leq j \leq d }}\sup_{\substack{ x \in \mathbb{R}^{d} }} \abs{ \Delta_{j} h^{*}(\delta k(x))} \\
\leq&\ C \max_{\substack{ 1 \leq j \leq d }}\sup_{\substack{ x \in \mathbb{R}^{d} }} \bigg| \frac{\partial}{\partial x_{j}} h^{*}(x) \bigg|.
\end{align*}
This proves the first claim. The other two claims follow by observing that if $h^{*} \in \text{\rm Lip(1)}$, then the mean-value theorem implies $h \in \text{\rm dLip(1)}$, and if $h^{*} \in \mathcal{M}$, then we can apply \eqref{eq:cver} along each dimension to show that $h \in \mathcal{M}_{disc}(C')$ for some $C' > 0$.
\hfill $\square$\endproof
\subsection{Proof of Lemma~\ref{lem:genpoisson}}
\label{sec:proofgenpoisson}
\proof{Proof of Lemma~\ref{lem:genpoisson}}
Let $\tau = \inf_{t \geq 0} \{X(t) \neq X(0)\}$ be the first jump time. For any $\varepsilon > 0$,
\begin{align*}
g(\delta k) =&\ \int_{0}^{\infty} \mathbb{E}_{X(0) = \delta k} \Big(\big(h(X(t)) - \mathbb{E} h(X) \big) 1(t < \tau \wedge \varepsilon)\Big) dt \\
&+ \int_{0}^{\infty} \mathbb{E}_{X(0) = \delta k} \Big(\big(h(X(t)) - \mathbb{E} h(X) \big) 1(t > \tau \wedge \varepsilon)\Big) dt.
\end{align*}
Set $r = \sum_{k' \neq k } q_{k,k'}$ and note that $\mathbb{P}(\tau > \varepsilon) = e^{-r \varepsilon}$. The strong Markov property implies
\begin{align*}
\int_{0}^{\infty} \mathbb{E}_{X(0) = \delta k} \Big(\big(h(X(t)) - \mathbb{E} h(X) \big) 1(t > \tau \wedge \varepsilon)\Big) dt =&\ e^{-r \varepsilon} g(\delta k) + (1- e^{-r \varepsilon}) \sum_{k' \neq k } \frac{q_{k,k'}}{r} g(\delta k').
\end{align*}
Furthermore, since $X(t) = X(0) $ for $t < \tau \wedge \varepsilon$,
\begin{align*}
\int_{0}^{\infty} \mathbb{E}_{X(0) = \delta k} \Big(\big(h(X(t)) - \mathbb{E} h(X) \big) 1(t < \tau \wedge \varepsilon)\Big) dt =&\ \big(h(\delta k) - \mathbb{E} h(X)\big) \int_{0}^{\infty} \mathbb{P}_{X(0) = \delta k} (\tau \wedge \varepsilon > t) dt\\
=&\ \big(h(\delta k) - \mathbb{E} h(X)\big) \mathbb{E} \big(\tau \wedge \varepsilon \big).
\end{align*}
Combining the three equations above and rearranging terms yields
\begin{align*}
0=&\ \big(h(\delta k) - \mathbb{E} h(X)\big) \mathbb{E} \big(\tau \wedge \varepsilon \big) + (1- e^{-r \varepsilon}) \sum_{k' \neq k } \frac{q_{k,k'}}{r} \big( g(\delta k') - g(\delta k)\big).
\end{align*}
We conclude by dividing both sides by $\varepsilon$ and taking $\varepsilon \to 0$, and by noting that
\begin{align*}
\frac{1}{\varepsilon} \mathbb{E} \big(\tau \wedge \varepsilon \big) = e^{-r \varepsilon} + (1 - e^{-r \varepsilon}) \frac{1}{\varepsilon}\mathbb{E} \big((\tau \wedge \varepsilon) 1(\tau < \varepsilon) \big) \to 1, \quad \text{ as } \varepsilon \to 0.
\end{align*}
\hfill $\square$\endproof
\end{APPENDICES}
\ACKNOWLEDGMENT{The author would like to thank Han Liang Gan for stimulating discussions during early stages of this work, as well as Robert Bray and Shane Henderson for providing feedback on early drafts. The author is also grateful to two anonymous referees for their numerous suggestions to improve the presentation of the material, as well as Zhe Su whose input helped significantly reduce the length of the manuscript. }
\bibliographystyle{informs2014}
| {
"timestamp": "2021-10-11T02:04:31",
"yymm": "2102",
"arxiv_id": "2102.12027",
"language": "en",
"url": "https://arxiv.org/abs/2102.12027",
"abstract": "This paper uses the generator comparison approach of Stein's method to analyze the gap between steady-state distributions of Markov chains and diffusion processes. The \"standard\" generator comparison approach starts with the Poisson equation for the diffusion, and the main technical difficulty is to obtain bounds on the derivatives of the solution to the Poisson equation, also known as Stein factor bounds. In this paper we propose starting with the Poisson equation of the Markov chain; we term this the prelimit approach. Although one still needs Stein factor bounds, they now correspond to finite differences of the Markov chain Poisson equation solution rather than the derivatives of the solution to the diffusion Poisson equation. In certain cases, the former are easier to obtain. We use the $M/M/1$ model as a simple working example to illustrate our approach.",
"subjects": "Probability (math.PR)",
"title": "The prelimit generator comparison approach of Stein's method",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9728307700397331,
"lm_q2_score": 0.7279754607093178,
"lm_q1q2_score": 0.7081969280118752
} |
https://arxiv.org/abs/2201.08606 | A test for fractal boundaries based on the basin entropy | In dynamical systems, basins of attraction connect a given set of initial conditions in phase space to their asymptotic states. The basin entropy and related tools quantify the unpredictability in the final state of a system when there is an initial perturbation or uncertainty in the initial state. Based on the basin entropy, the $\ln 2$ criterion allows for efficient testing of fractal basin boundaries at a fixed resolution. Here, we extend this criterion into a new test with improved sensitivity that we call the \textit{$S_{bb}$ fractality test}. Using the same single scale information, the $S_{bb}$ fractality test allows for the detection of fractal boundaries in many more cases than the $\ln 2$ criterion. The new test is illustrated with the paradigmatic driven Duffing oscillator, and the results are compared with the classical approach given by the uncertainty exponent. We believe that this work can prove particularly useful to study both high-dimensional systems and experimental basins of attraction. | \section{\label{sec:Introduction}Introduction}
Dynamical systems often show multistability with the coexistence of several asymptotic states, known as \emph{attractors}~\cite{feudel2008complex}. In dissipative systems, the set of initial conditions that asymptotically approach an attractor is called the \emph{basin of attraction}~\cite{nusse1996basinsSci}. In open Hamiltonian systems, instead of attractors we have exits, and consequently the initial conditions leaving the system by these exits are their \emph{escape basins}~\cite{aguirre2001wada}. In both cases, basin boundaries can either be smooth or fractal curves. Under repeated enlargement, fractal boundaries reveal new structures at arbitrarily small scales. This leads to non-integer dimensions and it is often considered as one of the hallmarks of chaos~\cite{mcdonald1985fractal,aguirre2009fractal}. The existence of fractal basin boundaries is profoundly intertwined with the unpredictability under uncertainty in the initial state of the orbit's attractor; this is the facet of chaos we aim to study here.
The classical method for studying the lack of predictability of a multistable system is via the \emph{uncertainty exponent} $\alpha$~\cite{grebogi1983final}. Basically, it measures the fractal dimension of the boundaries counting the number of boxes that lie between basins at different scales. The uncertainty exponent ranges from zero for the most unpredictable basins to one for smooth boundaries. Nonetheless, the uncertainty exponent presents some unavoidable numerical difficulties, such as accessing arbitrarily small scales or exploring multiple box sizes. Besides, it makes a poor use of the information obtained by sampling the phase space with the boxes, since it only classifies them as certain (lying in the interior) or uncertain (lying on a boundary).
Other tools able to quantify the attractors unpredictability given a finite uncertainty in the initial states are the \emph{basin entropy} and the \emph{boundary basin entropy}~\cite{daza2016basin}. Starting with a similar tiling of phase space, the idea is to compute the probabilities of going to each attractor within a box and exploit Shannon's information entropy. Since their formulation, the basin entropy and the boundary basin entropy have been applied to experiments with cold atoms~\cite{daza2017chaotic}, chaotic scattering~\cite{bernal2018uncertainty, nieto2018resonant}, biological systems~\cite{donepudi2018collective, mugnaine2019basin}, electronic micro/nanodevices~\cite{gusso2019nonlinear}, oscillators~\cite{kong2019special} and astrophysical models~\cite{zotos2018basins, zotos2018newton, zotos2018henon}, among others.
Based on the boundary basin entropy, the \emph{$\ln 2$ criterion}~\cite{daza2016basin} to detect fractal boundaries at a fixed resolution was developed. The use of a single scale to test for fractality makes it both computationally and experimentally convenient. The $\ln 2$ criterion applies to boundaries separating more than two different basins, like the Wada basins~\cite{kennedy1991basins}. The goal of this paper is to extend the $\ln 2$ criterion, presenting a test as a criterion to ascertain for fractal structures with an improved sensitivity that also applies for boundaries separating only two basins.
The article is organized as follows: we start in section~\ref{sec:Fractal structures: a perspective based on the basin entropy} with a quick revision of the basin entropy and its relation to fractal structures on phase space. We continue in section~\ref{sec:Testing for fractal structures with the basin entropy} introducing the fractal test and developing it for the case when basins are known exactly in two-dimensional phase spaces. Then, in sec.~\ref{sec:Effects of finite grids}, we study finite resolution effects in discrete phase spaces given by finite grids. In sec.~\ref{sec:Example}, we give a recipe of the test and illustrate it with an example. Moreover, in sec.~\ref{sec:Extension any dimension}, we extend the test to phase spaces in any dimension. Finally, we summarize and discuss the results in sec.~\ref{sec:conclusions}.
\section{Basin entropy and the definition of fractality\label{sec:Fractal structures: a perspective based on the basin entropy}}
Since our main goal is the identification of fractal boundaries, we first need to consider the definition of \emph{fractal}. Even though fractals are not defined in the literature in a precise and unambiguous manner, they typically share some of the following properties~\cite{falconer2004fractal}: (1) fine or detailed structure at arbitrarily small scales, (2) local and global irregularity non-describable by ordinary geometry, (3) some notion of self-similarity, (4) a `fractal dimension' greater than the topological dimension and (5) simple and perhaps recursive definitions. Here, we refer to fractal structures as those fulfilling at least properties (1), (2) and (4). Nonetheless, numerically we are always able to reach only up to a given scale. This connects with the experience that physical systems do not exhibit `true' fractal structures, in the sense that there are only some finite accessible or even defined scales~\cite{avnir1998geometry}.
The basin entropy provides a way to characterize phase space structures at a given scale. We consider a region of the phase space with $M$ distinct basins and we sample it using $N$ boxes of size $\varepsilon$ (see Fig.~\ref{fig:duf basins}). The box size $\varepsilon$ dictates the scale to study the boundary structures. These boxes can be thought as initial states either with uncertainty $\varepsilon$ or an initial perturbation $\varepsilon$. We can construct the discrete probability distribution of having a basin $k$ for each box, $p(k)$, as the ratio of the volume of the basin $k$ in the box, $v(k)$, to the phase space volume comprised by the box, $V$:
\begin{equation}
p(k) \equiv \frac{v(k)}{V},\label{eq:box prob as volume ratios}
\end{equation}
such that $\sum_{k=1}^M p(k ) = 1$. Its associated information entropy $s$,
\begin{equation}
s \equiv s(p(1), \dots, p(M)) = \sum_{k=1}^M -p(k ) \ln{p(k)},\label{eq:definition entropy s}
\end{equation}
measures the final state unpredictability of an initial condition chosen at random within the box defined by $\varepsilon$. It increases monotonically with the number of basins within the box and tends to a maximum value $s = \ln M$ as the probabilities $p(k)$ tend to the equiprobable conditions $p(k) = \frac{1}{M}$ for all $k$.
\begin{figure}
\centering
\subfloat[\label{fig:duf smooth}]{\includegraphics[width=0.32\textwidth]{FIG1a.eps}}
\hspace{0.01\textwidth}
\subfloat[\label{fig:duf fractal 2 basins}]
{\includegraphics[width=0.32\textwidth]{FIG1b.eps}}
\hspace{0.01\textwidth}
\subfloat[\label{fig:duf fractal multiple basins}]
{\includegraphics[width=0.32\textwidth]{FIG1c.eps}}
\caption{To compute the basin entropy, we draw boxes of size $\varepsilon$ from the phase space. For each box, we calculate the information entropy from the fractions of each basin within the box; then, averaging over all boxes we calculate the basin entropy $S_b$; and averaging only over boundary boxes (red dashed circles), the boundary basin entropy $S_{bb}$. These basins are from the periodically driven Duffing oscillator $\ddot{x} + 0.15 \dot{x} - x + x^3 = F \sin \omega t$, and (a) $F=0.100$, $\omega = 0.200$, (b) $F=0.395$, $\omega = 1.617$ and (c) $F=0.128$, $\omega = 1.106$. For $N_b = 10^4$ disks boxes in the boundary of radius $\varepsilon = 0.025$, the figures are ordered from left to right by the boundary basin entropy: $S_{bb} = 0.465 \pm 0.002 < 0.6323 \pm 0.0011 \leq \ln 2 < 0.760 \pm 0.004$. Using the $\ln 2$ criterion, only basins (c) are tested with fractal boundaries.\label{fig:duf basins}}
\end{figure}
Then, the \emph{basin entropy} $S_b$ is defined as the average of the entropy of the box $s$ for the total number of boxes $N$,
\begin{equation}
S_b \equiv \frac{1}{N} \sum_{i=1}^{N} s(i),\label{eq:basin entropy}
\end{equation}
where $i$ labels the boxes. Therefore, for an initial random position in the phase space region that is uncertain within a volume of size $\varepsilon$, the basin entropy quantifies the unpredictability of the orbit's attractor. Moreover, the \emph{boundary basin entropy} $ S_{bb}$ is defined as the average of the entropies $s(i)$ restricted only to the $N_b$ boxes falling on basin boundaries (in Fig.~\ref{fig:duf basins}, these correspond to the red dashed boxes):
\begin{equation}
S_{bb} \equiv \frac{1}{N_b} \sum_{i=1}^{N_b} s(i). \label{eq:boundary basin entropy}
\end{equation}
The boundary basin entropy quantifies the unpredictability focusing only on the unpredictable regions of the phase space: the basin boundaries.
Based on the boundary basin entropy, the $\ln 2$ criterion provides a sufficient condition to test for fractal boundaries. It is based on the fact that smooth boundaries separate only two basins, with the possible exception of a countable number of points that can separate three or more basins at a time. Therefore, for a sufficient large number of small boxes, for smooth boundaries $S_{bb} \in [0, \ln 2\simeq 0.693]$. This is the case of the basins in Fig.~\ref{fig:duf smooth}, where for a box scale $\varepsilon = 0.025$, $S_{bb} = 0.465 \pm 0.002$. Equivalently, if $S_{bb}$ is significantly larger than $\ln 2$, this implies the boundary will not be smooth (i.e., it will be fractal). This occurs for the basins in Fig.~\ref{fig:duf fractal multiple basins}, where $S_{bb} = 0.760 \pm 0.004$. Nonetheless, the criterion fails for the case of Fig.~\ref{fig:duf fractal 2 basins} with a manifested fractal boundaries between two basins and $S_{bb} = 0.6323 \pm 0.0011$. Following this idea, there is room to make a better test using a single scale with the basin entropy.
\section{\label{sec:Testing for fractal structures with the basin entropy} Beyond the ln 2 criterion: the $S_{bb}$ fractality test}
Simple smooth boundaries are those which are locally flat (as a straight line in 2D). According to our previous definition of a fractal, we consider as a fractal boundary any boundary that is not simple smooth. A simple smooth boundary could also include a few finite points where boundaries from several basins intersect. Nevertheless, in the infinitely fine scale these regions become negligible in front of the other locally flat regions.
Based on this idea and the boundary basin entropy, we can define a statistical test to identify fractal structures, which we name \textit{$S_{bb}$ fractality test}. It consists on comparing the value of the boundary basin entropy $S_{bb}$ of the boundary under study to the theoretical value $S_{bb}$ of a flat boundary, at a given small box scale $\varepsilon$. If they are deemed statistically significantly different, the boundary has a fractal structure. We will define it formally after studying the boundary basin entropy of a flat boundary. Since the boundary basin entropy is defined as an average of the box entropies $s$ and different configuration of the boxes' entropies can give rise to the same value, this test is a sufficient but not necessary condition. Nonetheless, it is more restrictive than the $\ln 2$ criterion and consequently it can detect fractal boundaries in many more cases, e.g. in regions with only two basins.
Assuming a perfect knowledge of the phase space structure for a flat boundary in a two-dimensional phase space, we can derive the theoretical value of the boundary basin entropy. We start by choosing a disk as box shape, because its rotational symmetry under any angle allows to compute the boundary basin entropy independently of the box orientation. Now, having disk boxes in a phase space with a flat boundary, we can obtain the boundary basin entropy by sliding the box from side to side along the boundary. If a disk box has a radius $\varepsilon$ and is centered at the coordinate $x_0 \in [-\varepsilon, \varepsilon]$ from the perpendicular direction to the boundary with origin on the boundary, the probability $p(x_0)$ to have a disk point in one of the basins is the fraction of the disk given by the circular segment
\begin{equation}
p(x_0) = \frac{1}{2} + \frac{1}{\pi} \left[ \frac{x_0}{\varepsilon}\sqrt{1-\left( \frac{x_0}{\varepsilon} \right)^2} + \arcsin{\frac{x_0}{\varepsilon}}\right].
\end{equation}
Then, $S_{bb}$ is given numerically by:
\begin{equation}
S_{bb} = \frac{1}{2 \varepsilon } \int_{-\varepsilon}^{\varepsilon} dx_0 \ s\left(p \left(x_0\right), 1 - p \left(x_0\right) \right) = 0.4395093(6).\label{eq:sb sliding box}
\end{equation}
This means that if we were able to compute exactly the boundary basin entropy of a smooth boundary using small disks we would get that result, and any other number would correspond to a fractal case. However, in practice we always have some errors induced by the use of a finite number of trajectories. The effects that this can introduce into our test are explored in the next section.
\section{\label{sec:Effects of finite grids}Effects of finite grids}
Both numerically and in experiments, basins are commonly calculated using a finite number of grid points. The number of grid points within a box determines the observed probabilities $\hat{p}(k)$ of the box. Ultimately, this lack of detail results in an observed boundary basin entropy $\hat{S}_{bb}$ with a systematic error or bias $\delta_{\hat{S}_{bb}}$. In addition, we are generally limited to draw a finite number of boxes too. This results in an observed boundary basin entropy with a statistical error $\sigma_{\hat{S}_{bb}}$. Here, we study these effects for a two-dimensional square grid with a flat boundary and disk boxes, which provides confidence intervals for our fractal basin boundary test.
The systematic error of the observed boundary basin entropy $\hat{S}_{bb}$ depends on two factors: the extension of the grid and the angular orientation of the grid. The effect of the grid extension decreases with the number of grid points per axis. Our numerical simulations showed that this factor is no longer relevant for larger values than $50 \varepsilon_g$ grid points per axis, where $\varepsilon_g$ is the disk radius in \textit{grid units}. The second factor, the angle between the grid and the boundary, is inherently unavoidable. We have investigated this effect in Fig.~\ref{fig:grid angular dependency}, representing in function of the disk radius $\varepsilon_g$ in grid units and for largely extended grids, $\hat{S}_{bb}$ for different angles (legend) and the exact $S_{bb}$ (magenta dashed line). The figure inset displays the absolute errors of $\hat{S}_{bb}$. Even in the worst case scenario (red line), the systematic error decreases approximately inversely proportional to the disk boxes radius $\varepsilon_g$. Indeed, a power law fit gives an upper bound for the systematic error:
\begin{equation}
\delta_{\hat{S}_{bb}}^\text{UB} \simeq A\varepsilon_g^B,\label{eq:Sbb UB systematic error}
\end{equation}
with $A=0.224 \pm 0.010$ and $B = -1.006 \pm 0.014$.
\begin{figure}[h]
\centering
\includegraphics[width=0.45\textwidth]{FIG2.eps}
\caption{For a grid with a flat boundary and disk boxes of radius $\varepsilon_g$ in grid units, the systematic error of the observed boundary basin entropy $\hat{S}_{bb}$ depends on the angle between the grid and the boundary (legend): (outset) $\hat{S}_{bb}$ in function of $\varepsilon_g$ and the exact $S_{bb}$ value (magenta dashed line); (inset) absolute errors of $\hat{S}_{bb}$ in function of $\varepsilon_g$. We considered large grid extensions of around $128 \varepsilon_g$ points per axis and $N_b = 10^5$ disk boxes in the boundary. The worst systematic error is given by the angle $0$ in red. In the infinite grid resolution ($\varepsilon_g \to \infty$) and for any angle $\alpha$, we recover the exact boundary basin entropy $S_{bb}$.\label{fig:grid angular dependency}}
\end{figure}
On the other hand, there is a statistical error of the observed boundary basin entropy $\hat{S}_{bb}$, which is due to the finite number $N_b$ of boxes in the boundary. Indeed, $\hat{S}_{bb}$ is the average of the boxes' entropies in the boundary. This means that, by the central limit theorem, the statistical error follows a Gaussian distribution with a standard deviation given by:
\begin{equation}
\sigma_{\hat{S}_{bb}} = a N_b^{-\frac{1}{2}},\label{eq: Sb statistical error}
\end{equation}
where $a \equiv \sqrt{\frac{1}{N_b - 1 } \sum_{i=1}^{N_b} \left (\hat{S}_{bb} - \hat{s}(i) \right )^2 }$ is the sampling standard deviation and $\hat{s}(i)$ the observed entropy for the box $i$.
Taking into account all the finite grid effects for 2D phase spaces, we can formulate our test for fractal boundaries as follows: \textit{under an infinitesimal disk box with radius $\varepsilon_g$ in grid units and a finite grid that is largely extended (with at least $50 \varepsilon_g$ grid points per axis), if the observed boundary basin entropy $\hat{S}_{bb}$ with a standard deviation $\sigma_{\hat{S}_{bb}}$ (Eq.~\ref{eq: Sb statistical error}) is deemed statistically significant away from a flat boundary, either below the exact boundary basin entropy value $S_{bb}$ (Eq.~\ref{eq:sb sliding box}) or above the upper-bound systematic error value $S_{bb} + \delta_{\hat{S}_{bb}}^\text{UB}$ ($\delta_{\hat{S}_{bb}}^\text{UB}$ is given by Eq.~\ref{eq:Sbb UB systematic error}), the boundary has a fractal structure.} Using one standard deviation $\sigma_{\hat{S}_{bb}}$ for the statistical error, we can express the $S_{bb}$ fractality test by the following sufficient conditions:
\begin{align}
\hat{S}_{bb} &< S_{bb} - \sigma_{\hat{S}_{bb}},\label{eq:1st-order fractal test eq 1} \\
\hat{S}_{bb} &> S_{bb} + \delta_{\hat{S}_{bb}}^\text{UB} + \sigma_{\hat{S}_{bb}}.\label{eq:1st-order fractal test eq 2}
\end{align}
\section{\label{sec:Example}Example of application}
The periodically driven Duffing oscillator is a paradigmatic model that accounts for nonlinear elastic effects in large displacements of a forced damped elastic structure. It is defined by:
\begin{equation}
\ddot{x} + \gamma \dot{x} - x + x^3 = F \sin \omega t,
\end{equation}
where $x$ is the displacement of the oscillator at time $t$, $\gamma$ is the damping coefficient, $F$ is the forcing amplitude and $\omega$ is the frequency of the driving. Depending on these parameters, the system exhibits a wide variety of dynamics. Here, we investigate a parameter space region given by $\gamma = 0.15$, $F \in [0.1, 0.5]$ and $\omega \in [0.2, 2.5]$. We search for fractal boundaries in the basins of attraction given by a finite grid in the phase space region $\Omega = \Omega_x \times \Omega_{\dot{x}} = [-2.5, 2.5] \times [-2.5, 2.5]$, with $10^3$ points per axis.
On the following, we use the example to illustrate an \textbf{easy-to-follow recipe for the \textit{$S_{bb}$ fractality test}} for finite grids in two-dimensional phase spaces:
\begin{enumerate}[\bfseries 1.]
\item \textbf{Choose the box disk radius $\varepsilon$ appropriately}. On the one hand, the smaller the box, the finer scales we can study. On the other hand, we want to have sufficient trajectories per box to get good estimates for the probabilities. For our example, we found that a value of $\varepsilon_g = 5$ / $\varepsilon = 0.025$ gave good results.
\item \textbf{Verify that we are in the largely extended grid limit, with at least $50 \varepsilon_g$ grid points per axis.} In our example, we had $10^3 = 200 \varepsilon_g$ points per axis.
\item \textbf{Draw $N_b$ disk boxes in the boundary, uniformly at random from the phase space (and not only at grid points).} In our example, we took $N_b = 10^4$ boxes in the boundary.
\item \textbf{For all boxes in the boundary, calculate both the observed probabilities of the basins and the observed box's entropy.}
\item \textbf{Compute the observed boundary basin entropy $\hat{S}_{bb}$ and its standard deviation $\sigma_{\hat{S}_{bb}}$ (Eq.~\ref{eq: Sb statistical error}).}
\item \textbf{Compute the upper-bound systematic error $\delta_{\hat{S}_{bb}}^\text{UB}$ (Eq.~\ref{eq:Sbb UB systematic error}) for the current value of $\varepsilon_g$.} In our example, we had $\delta_{\hat{S}_{bb}}^\text{UB} = 0.0474$.
\item \textbf{Check if the computed value of $\hat{S}_{bb}$ lies in the interval provided by the systematic and statistical errors.} If the value is outside the interval, we can affirm that according to our test the boundary is fractal. Otherwise, it is most likely to be a smooth boundary, although there could be pathological cases leading to wrong results. It is important to recall that this fractality test is a sufficient but not necessary condition.
\end{enumerate}
We display the results in the parameter space of Fig.~\ref{fig:duf Sb parameter set}. The color of each parameter value corresponds to its boundary basin entropy $\hat{S}_{bb}$ value: white is for regions with a single basin, cold colors are for regions compatible with simple smooth boundaries and hot colors are for fractal basin boundaries. In particular, red colors are for fractal basin boundaries detected by the $\ln 2$ criterion. We can observe that this just accounts for a $27\%$ fraction of all the fractal basin boundaries detected by the single-scale fractal basin boundary test.
\begin{figure}
\centering
\subfloat[\label{fig:duf Sb parameter set}]{\includegraphics[width=0.45\textwidth]{FIG3a.eps}}
\hspace{0.05\textwidth}
\subfloat[\label{fig:duf unc exp}]
{\includegraphics[width=0.45\textwidth]{FIG3b.eps}}
\caption{For the periodically driven Duffing oscillator $\ddot{x} + 0.15 \dot{x} - x + x^3 = F \sin \omega t$ in the parameter space $( F, \omega )$, we obtain similar regions with fractal boundaries (hot colors) and smooth boundaries (cold colors), both for (a) the \textit{$S_{bb}$ fractality test} and (b) the uncertainty exponent $\alpha$. In white, we have regions with a single basin. In (a), the color of each point corresponds to the boundary basin entropy $\hat{S}_{bb}$, for disk boxes with radius $\varepsilon = 0.025$ / $\varepsilon_g = 5$ and $N_b = 10^4$ boxes in the boundary: cold colors are compatible with simple smooth boundaries and hot colors for fractal basin boundaries (red colors are detected by the $\ln 2$ criterion). In (b), we represent the uncertainty exponent with a transition point at $\alpha = 0.8$ from fractal to smooth boundaries; this value is arbitrary and based on the observed basins.}
\end{figure}
To evaluate the performance of the method, we compare it to the uncertainty exponent $\alpha$ (Fig.~\ref{fig:duf unc exp}). From the observed basins, we have chosen the arbitrary value of $\alpha = 0.8$ as the transition point between smooth and fractal boundaries: $\alpha \geq 0.8$ correspond to smooth boundaries (cold colors) and $\alpha < 0.8$ to fractal basin boundaries (hot colors). Again, parameters with a single basin are represented in white. Indeed, both the uncertainty exponent and the \textit{$S_{bb}$ fractality test} give qualitatively similar regions with fractal boundaries. However, there are some quantitative differences due to numerical errors in both methods.
\section{\label{sec:Extension any dimension}Extension to any phase space dimension}
We can generalize the test to phase spaces for any dimension. The test again compares for small box scales $\varepsilon$ the boundary basin entropy $S_{bb}$ of the boundary under study to the theoretical value of that of a flat boundary of the corresponding dimension. Here, we only consider the case with an exact knowledge of the phase space. We believe that the study for basins with finite grids can be developed similarly to the two-dimensional case.
For a given dimension, we can derive the theoretical value of the boundary basin entropy of a phase space with a flat boundary. The box is now an hyperball of radius $\varepsilon$ centered at a coordinate $x_0$ in the direction that indicates the distance to the flat boundary. Furthermore, the probability $p(x_0)$ to have a hyperball point in one basin is the fraction of the hyperball given by the hyperspherical cap~\cite{li2011concise}:
\begin{equation}
p(x_0) = -\frac{1}{2} \text{sgn}\left(\frac{x_0}{\varepsilon} \right) I_{1-\left( \frac{x_0}{\varepsilon} \right)^2} \left( \frac{D+1}{2}, \frac{1}{2} \right) + \Theta \left(\frac{x_0}{\varepsilon}\right),\label{eq:D Dim px0}
\end{equation}
where $D$ is the dimension of the phase space, $\text{sgn}(x)$ is the sign function, $I_x \left(a,b \right)$ is the regularized incomplete beta function and $\Theta(x)$ is the Heaviside function. We have this quantity plotted for several dimensions (see colorbar) in Fig.~\ref{fig:D Dim px disk flat}. For $D=1$, the probability $p(x_0)$ is linear; for larger values of $D$, it deviates from this behavior; and in the limiting case $D \to \infty$, $p(x_0)$ becomes dominated by the $\Theta \left(\frac{x_0}{\varepsilon}\right)$ term and has a switching behavior. This large dimension behavior is understood because most of the volume of a high-dimensional hyperball lies within two parallel hyperplanes at a small distance from its center of order $\mathcal{O} \left( \frac{\varepsilon}{\sqrt{D-1}} \right)$~\cite{hopcroft2012computer}.
\begin{figure}[h]
\centering
\subfloat[\label{fig:D Dim px disk flat}]
{\includegraphics[width=0.4\textwidth]{FIG4a.eps}}
\hspace{0.1\textwidth}
\subfloat[\label{fig:D Dim Sbb}]{\includegraphics[width=0.4\textwidth]{FIG4b.eps}}
\caption{For a hyperball box and a flat boundary in the phase space, we plot (a) for several dimensions of the phase space (colorbar), the probability $p(x_0)$ to have a hyperball point in one basin in function of the box center coordinate $x_0$, which indicates the distance to the flat boundary; and (b) the boundary basin entropy $S_{bb}$ relationship with the dimension $D$ of the phase space (magenta circles). $S_{bb}$ decreases potentially fast with a power fit $S_{bb} = A D^B$ (purple line): $A=0.898 \pm 0.006$ and $B=-0.4995 \pm 0.0012$.}
\end{figure}
Moreover, we calculate the boundary basin entropy $S_{bb}$ analogously to the two-dimensional case, following Eq.~\ref{eq:sb sliding box}. We have the $S_{bb}$ relationship with the dimension $D$ in Fig.~\ref{fig:D Dim Sbb}; we can see how the boundary basin entropy goes to zero in the infinite dimension limit following the expression $S_{bb} \sim D^{-\frac{1}{2}}$. We have tabulated the boundary basin entropy $S_{bb}$ for the first five $D$ dimensions in table~\ref{tab:D dim Sb}.
\begin{table}[h]
\centering
\caption{Boundary basin entropy $S_{bb}$ for hyperball boxes in a flat boundary of a $D$ dimensional phase space, for the first five $D$ dimensions.}\label{tab:D dim Sb}
\begin{tabular}{ l @{\qquad} l }
\hline
$D$ & $S_{bb}$ \\
\hline
1 & 0.499999(9) \\
2 & 0.4395093(6) \\
3 & 0.39609176(4) \\
4 & 0.36319428(1) \\
5 &0.33722572 \\
\hline
\end{tabular}
\end{table}
\section{\label{sec:conclusions}Conclusions}
In this paper, we have applied the theory of the basin entropy and related tools to characterize fractal basin boundaries using a single scale. Our work consists on comparing, under a given box scale, the value of the boundary basin entropy $S_{bb}$ of the boundary to the $S_{bb}$ value of a smooth boundary. In contrast to the former basin-entropy-based test, the $\ln 2$ criterion, it achieves both an improved sensitivity and a capacity to identify fractal basin boundaries separating only two basins. We call it \textit{$S_{bb}$ fractality test}. Nonetheless, this test is still a sufficient but not necessary condition. One could think about an improvement by means of the probability density function (PDF) of the probabilities in the boundary boxes. The configuration of a PDF is more restrictive than an average (the boundary basin entropy), but it is also more demanding to implement.
On the other hand, testing for fractal basin boundaries with this line of work has important advantages compared to the classical approach given by the uncertainty exponent $\alpha$. While both methods hold numerical limitations in the infinitely fine scale, the uncertainty exponent requires accessing multiple scales of the system. This is not only numerically inconvenient but, for some physical systems, it may be impossible. Indeed, our methods only require accessing a single scale and have a natural formulation for experimental basins of attraction.
Finally, we believe the extension for basins in any phase space dimension could prove useful for studying the unpredictability in higher-dimensional systems, a generally unexplored area in the field; in particular, it could complement current studies for network systems~\cite{menck2013basin,menck2014dead,rakshit2017basin}.
\section*{Acknowledgements}
This work has been supported by the Spanish State Research Agency (AEI) and the European Regional Development Fund (ERDF, EU) under Projects No.~FIS2016-76883-P and No.~PID2019-105554GB-I00.
| {
"timestamp": "2022-01-24T02:10:24",
"yymm": "2201",
"arxiv_id": "2201.08606",
"language": "en",
"url": "https://arxiv.org/abs/2201.08606",
"abstract": "In dynamical systems, basins of attraction connect a given set of initial conditions in phase space to their asymptotic states. The basin entropy and related tools quantify the unpredictability in the final state of a system when there is an initial perturbation or uncertainty in the initial state. Based on the basin entropy, the $\\ln 2$ criterion allows for efficient testing of fractal basin boundaries at a fixed resolution. Here, we extend this criterion into a new test with improved sensitivity that we call the \\textit{$S_{bb}$ fractality test}. Using the same single scale information, the $S_{bb}$ fractality test allows for the detection of fractal boundaries in many more cases than the $\\ln 2$ criterion. The new test is illustrated with the paradigmatic driven Duffing oscillator, and the results are compared with the classical approach given by the uncertainty exponent. We believe that this work can prove particularly useful to study both high-dimensional systems and experimental basins of attraction.",
"subjects": "Chaotic Dynamics (nlin.CD)",
"title": "A test for fractal boundaries based on the basin entropy",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9728307700397332,
"lm_q2_score": 0.7279754548076477,
"lm_q1q2_score": 0.7081969222705489
} |
https://arxiv.org/abs/2009.06125 | A Qualitative Study of the Dynamic Behavior for Adaptive Gradient Algorithms | The dynamic behavior of RMSprop and Adam algorithms is studied through a combination of careful numerical experiments and theoretical explanations. Three types of qualitative features are observed in the training loss curve: fast initial convergence, oscillations, and large spikes in the late phase. The sign gradient descent (signGD) flow, which is the limit of Adam when taking the learning rate to 0 while keeping the momentum parameters fixed, is used to explain the fast initial convergence. For the late phase of Adam, three different types of qualitative patterns are observed depending on the choice of the hyper-parameters: oscillations, spikes, and divergence. In particular, Adam converges much smoother and even faster when the values of the two momentum factors are close to each other. This observation is particularly important for scientific computing tasks, for which the training process usually proceeds into the high precision regime. | \section{Introduction}
Adaptive gradient algorithms \cite{duchi2011adaptive, Tieleman2012, kingma2014adam,reddi2018convergence}, especially RMSprop \cite{Tieleman2012} and Adam \cite{kingma2014adam}, have demonstrated superior performance in training modern machine learning models, e.g. deep neural networks. Distinguished from the vanilla gradient descent (GD) or stochastic gradient descent (SGD), adaptive gradient algorithms use a coordinate-wise scaling of the update direction. The scaling factors are adaptively determined by using the past gradients \cite{duchi2011adaptive}, which makes the analysis of these algorithms much more challenging.
Recent theoretical efforts \cite{reddi2018convergence,zhou2018convergence,xie2020linear,li2019convergence, chen2018convergence} have focused on establishing the convergence of adaptive gradient algorithms. However, these results are still unsatisfactory, since they cannot explain
any of the particular features of these adaptive gradient algorithms. Moreover, all these results usually require
taking the limit that the step size $\eta_t$ goes to zero, e.g. $\eta_t = 1/\sqrt{t}$. However, in practice, one usually starts with a large step size and only decays the step size for several times during the training process. For the most of iterations, the step size is actually fixed. So It is interesting to see what happens for not only the convergence but also the behavior of whole training curve with a fixed step size.
Figure~\ref{fig: loss} shows an example of full-batch Adam with a fixed step size.
One can see that even with full batch, the loss curve does not decrease monotonically: Small oscillations and large spikes appear shortly after initialization. This complicated loss pattern, especially the large spikes, makes it difficult to pick a good stopping time.
\begin{figure}[!h]
\centering
\includegraphics[width=0.4\columnwidth]{loss_curve_nn_adam.png}
\caption{The training loss of full-batch Adam for a multi-layer neural network model on CIFAR-10. The network has $3$ hidden-layers with widths 256-256-128.
The step size is fixed to be $0.001$ and $(\beta_1, \beta_2)=(0.9, 0.999)$, the default values in PyTorch and TensorFlow.
$2$ classes are picked from CIFAR-10 with $1000$ images in each class. Square loss function is used.}
\label{fig: loss}
\end{figure}
In addition to the learning rate, adaptive gradient algorithms also use extra hyper-parameters such as the second-order momentum factor for RMSprop and the first and second-order momentum factor for Adam. Though default values are provided in mainstream packages (e.g. $\beta_1$=0.9, $\beta_2$=0.999 for Adam in PyTorch and TensorFlow), these default parameter are not necessarily optimal and changing these hyper-parameter values can drastically change the behavior and performance of the algorithms.
One objective of this paper is to carry out a comprehensive study of the influence of these hyper-parameters.
\paragraph{Contributions}
In this paper, we provide well-designed experiments to demystify the dynamic behavior of adaptive gradient algorithms. Specifically, our contributions are summarized as follows.
\begin{enumerate}
\item We identify three types of typical phenomena in the training process of these adaptive algorithms: initial fast convergence (sometimes even super-linear), small oscillations, and large spikes.
\item For RMSprop and Adam, if the learning rate decreases to zero while the momentum parameters are fixed, we show that the algorithms tends to signGD. For signGD, we prove finite-time convergence for strongly convex objective functions. These arguments together provide a partial explanation of the fast initial convergence of RMSprop and Adam, which could be the one of reasons behind the popularity of these algorithms.
\item We show that the spikes are caused by some instabilities of the algorithm at stationary points. For RMSprop on simple objective functions, we explicitly write down the limiting oscillating solution. For Adam, we classify the behavior into three different patterns in
the space of the two momentum factors: {the spike regime, the oscillation regime, and the divergence regime.} Empirical results show that training is most stable in the ``oscillation regime'', especially when $\beta_1\approx\beta_2$.
\end{enumerate}
\paragraph{Notations} To make the notations more consistent, from now on we use $\alpha$ to denote the second-order momentum factor in both Adam and RMSprop, and use $\beta$ to denote the first-order momentum in Adam. The conventional nations $\beta_1$ and $\beta_2$ above for Adam will become $\beta$ and $\alpha$, respectively. For vectors $\bm{u}$ and $\mathbf{v}$, operations such as $\bm{u}^2$, $\sqrt{\bm{u}}$ and $\bm{u}/\mathbf{v}$ are
understood to be element-wise.
\section{Preliminaries}\label{sec:2}
\subsection{Adaptive gradient algorithms}
Adaptive gradient algorithms are a family of optimization algorithms that use a coordinate-wise scaling of the update direction (gradient or gradient with momentum) according to the history of gradients. Many adaptive algorithms can be cast to the following form~\cite{da2018general},
\begin{equation}\label{eqn: adap_general}
\begin{aligned}
\mathbf{m}_{t+1} &= h_t\nabla f(\mathbf{x}_t) + r_t \mathbf{m}_t \\
\mathbf{v}_{t+1} &= p_t (\nabla f(\mathbf{x}_t))^2 + q_t \mathbf{v}_t\\
\mathbf{x}_{t+1} &= \mathbf{x}_t - \eta_t \frac{\mathbf{m}_{t+1}}{\sqrt{\mathbf{v}_{t+1}}+\epsilon},
\end{aligned}
\end{equation}
with different choice of $h, r, p, q$. In~\eqref{eqn: adap_general}, $h, r, p, q$ are scalar functions of $t$. For example, Adagrad~\cite{duchi2011adaptive} is recovered when $h, p, q=1$ and $r=0$, and RMSprop corresponds to the case when $h=1$, $r=0$, $p=1-\alpha$ and $q=\alpha$ for some constant $\alpha\in(0,1)$. { Viewed from the dynamics of $\mathbf{x}$ alone, adaptive gradient algorithms usually have a ``memory effect'' due to the momentum terms. The strength of the memory depends on the momentum factors ($h, r, p, q$) and the learning rate.}
Because of their efficiency in training neural network models, these algorithms are extensively used. Readers are referred to~\cite{ruder2016overview} for a more thorough review of existing adaptive algorithms.
In this paper, we focus on RMSprop and Adam --- the two algorithms that are most widely used by practitioners. The discrete update rules of these algorithms are
\begin{itemize}
\item {\bf RMSprop:}
\begin{equation} \label{eqn: rmsprop}
\begin{aligned}
\mathbf{v}_{t+1} &= \alpha \mathbf{v}_{t} + (1-\alpha)(\nabla f(\mathbf{x}_t))^2 \\
\mathbf{x}_{t+1} &= \mathbf{x}_t - \eta\frac{\nabla f(\mathbf{x}_t)}{\sqrt{\mathbf{v}_{t+1}}+\epsilon}
\end{aligned}
\end{equation}
\item {\bf Adam:}
\begin{equation} \label{eqn: adam}
\begin{aligned}
\mathbf{v}_{t+1} &= \alpha \mathbf{v}_{t} + (1-\alpha) (\nabla f(\mathbf{x}_t))^2\\
\mathbf{m}_{t+1} &= \beta \mathbf{m}_{t} + (1-\beta) \nabla f(\mathbf{x}_t)\\
\mathbf{x}_{t+1} &= \mathbf{x}_t - \eta\frac{\mathbf{m}_{t+1}/(1-\beta^{t+1})}{\sqrt{\mathbf{v}_{t+1}/(1-\alpha^{t+1})}+\epsilon}
\end{aligned}
\end{equation}
\end{itemize}
In~\eqref{eqn: rmsprop} and~\eqref{eqn: adam}, $\epsilon$ is a small constant used to avoid division by $0$. It is usually taken to be $10^{-8}$. In this paper, we always consider the full batch setting, i.e. $\nabla f(\mathbf{x}_t)$ is the gradient instead of some stochastic approximation.
\subsection{Continuous-time limits}
RMSProp and Adam can be studied by considering the limiting ordinary differential equations (ODE) obtained by taking the learning rate $\eta$ to $0$. However,
different limiting ODEs are obtained when the hyper-parameters are scaled differently. If the hyper-parameters are kept fixed, then as $\eta\rightarrow0$, the memory effect diminishes, because in each discrete iteration we loss the same amount of memory but one iteration
occupies shorter and shorter amount of time. In this case, the continuous-time limit for both RMSprop and Adam are the following dynamics
\begin{equation}\label{eqn: ode1}
\dot{\mathbf{x}} = -\frac{\nabla f(\mathbf{x})}{|\nabla f(\mathbf{x})|+\epsilon}.
\end{equation}
Since $\epsilon$ is a small value, this dynamics is close to the continuous-time signGD:
\begin{equation}\label{eqn: continuous-time-signGD}
\dot{\mathbf{x}} = -\textrm{sign}(\nabla f(\mathbf{x})).
\end{equation}
\begin{proposition}\label{thm: limit1}
Assume that $\nabla f$ is bounded and Lipschitz continuous, i.e. there exists constants $M$ and $L$ such that $\|\nabla f(\mathbf{x}_1)\|\leq M$ and $\|\nabla f(\mathbf{x}_1)-\nabla f(\mathbf{x}_2)\|\leq L\|\mathbf{x}_1-\mathbf{x}_2\|$ hold for any $\mathbf{x}_1$ and $\mathbf{x}_2$. Let $\{\mathbf{x}_k^\eta\}$, $k=0,1,2,\cdots$ be the solution given by algorithm~\eqref{eqn: rmsprop} or~\eqref{eqn: adam} starting from $\mathbf{x}_0$, $\mathbf{m}_0$ and $\mathbf{v}_0\geq0$, with learning rate $\eta$ and some fixed $\alpha, \beta\in(0,1)$ and $\epsilon>0$. Let $\bm{X}^\eta(\cdot)$ be a piece-wise constant function of $t\in[0,\infty)$ that satisfies
\begin{equation*}
\bm{X}^\eta(t) = \mathbf{x}_k^\eta,\quad for \,t \in[k\eta, (k+1)\eta).
\end{equation*}
In addition, let $\mathbf{x}(\cdot)$ be the solution of~\eqref{eqn: ode1} initialized from $\mathbf{x}_0$. Then, for any $T>0$, we have
\begin{equation}
\lim_{\eta\rightarrow0}\sup_{t\in[0,T]}\|\bm{X}^\eta(t)-\mathbf{x}(t)\| = 0.
\end{equation}
\end{proposition}
The proof of the Proposition is given in the appendix. Figure~\ref{fig: adap_signgd} provides numerical evidences that RMSprop and Adam are close to signGD in a finite time interval when $\eta$ is small while $\alpha$ and $\beta$ are fixed. The closeness between signGD and RMSprop is also shown in Figure~\ref{fig: signgd_rmsprop} for a synthetic objective function.
\begin{figure}[!h]
\centering
\includegraphics[width=0.4\columnwidth]{sign_adam_nn.png}
\hspace{-5mm}
\includegraphics[width=0.4\columnwidth]{sign_rmsprop_nn.png}
\caption{Early stage training loss curve of signGD and Adam/RMSprop with different learning rates. The x-axis is the time (learning rate$\times$number of iterations) and y-axis denotes the training loss.
For Adam, $\beta=0.9$ and $\alpha=0.999$; for RMSprop $\alpha=0.99$. Learning rate of signGD is $10^{-5}$. Experiments conducted on a fully-connected neural network with three hidden layers, with width $256$, $128$, and $64$, respectively. The training data is taken from $2$ classes of CIFAR10 with $1000$ data per class.}
\label{fig: adap_signgd}
\end{figure}
On the other hand, if we want to keep the strength of the memory effect fixed, we have to let $\alpha$ and $\beta$ go to $1$ when $\eta$ tends to $0$. Specifically, let $\alpha=1-a\eta$ and $\beta=1-b\eta$, with $a$ and $b$ being positive constants. Then, it is easy to show that the trajectories of~\eqref{eqn: rmsprop} and~\eqref{eqn: adam} converge to the following ODEs~\eqref{eqn: rmsprop_ode} and~\eqref{eqn: adam_ode}, respectively.
\begin{itemize}
\item {\bf ODE RMSprop:}
\begin{equation}
\begin{aligned}
\dot{\mathbf{v}} &= a(\nabla f(\mathbf{x})^2-\mathbf{v}) \\
\dot{\mathbf{x}} &= -\frac{\nabla f(\mathbf{x})}{\sqrt{\mathbf{v}}+\epsilon} \label{eqn: rmsprop_ode}
\end{aligned}
\end{equation}
\item {\bf ODE Adam:}
\vspace{-2mm}
\begin{equation}
\begin{aligned}
\dot{\mathbf{v}} &= a(\nabla f(\mathbf{x})^2-\mathbf{v}) \\
\dot{\mathbf{m}} &= b(\nabla f(\mathbf{x})-\mathbf{m}) \\
\dot{\mathbf{x}} &= -\frac{(1-e^{-bt})^{-1}\mathbf{m}}{\sqrt{(1-e^{-at})^{-1}\mathbf{v}}+\epsilon} \label{eqn: adam_ode}
\end{aligned}
\end{equation}
\end{itemize}
The following proposition is a simplification of Theorem 3.2 in~\cite{barakat2018convergence}.
More general results including the stochastic case for Adam are given in~\cite{barakat2018convergence}.
\begin{proposition}\label{thm: limit2}
Under the same condition of $f$ in Proposition~\ref{thm: limit1}, let $\{\mathbf{x}_k^\eta\}$, $k=0,1,2,\cdots$ be the solution given by algorithm~\eqref{eqn: rmsprop} starting from $\mathbf{x}_0$ and $\mathbf{v}_0=0$, with learning rate $\eta$ and $\alpha=1-a\eta$ for a fixed constant $a>0$. Let $\bm{X}^\eta(\cdot)$ be a piece-wise constant vector function of $t\in[0,\infty)$ that satisfies
\begin{equation*}
\bm{X}^\eta(t) = \mathbf{x}_k^\eta,\quad for t \, \in[k\eta, (k+1)\eta).
\end{equation*}
In addition, let $\mathbf{x}(\cdot)$ be the solution of~\eqref{eqn: rmsprop_ode} initialized from $\mathbf{x}_0$ and $\mathbf{v}_0\geq0$. Then, for any $T>0$, we have
\begin{equation}
\lim_{\eta\rightarrow0}\sup_{t\in[0,T]}\|\bm{X}^\eta(t)-\mathbf{x}(t)\| = 0.
\end{equation}
Similarly, if $\alpha=1-a\eta$ and $\beta=1-b\eta$ for some constants $a,b>0$, then the same convergence statements hold for
the solutions of~\eqref{eqn: adam} and~\eqref{eqn: adam_ode}.
\end{proposition}
In~\cite{barakat2018convergence} a Lyapunov function is found for the continuous version of Adam in the state space of $\bm{z}=(\mathbf{x}, \mathbf{m}, \mathbf{v})$:
\begin{equation}
V(t, \bm{z}) := f(\mathbf{x}) + \frac{1}{2}\|\mathbf{m}\|_{U(t,\mathbf{v})^{-1}}^2,
\end{equation}
where
\begin{equation}
U(t,\mathbf{v}) = b(1-e^{-bt})\left(\sqrt{\frac{\mathbf{v}}{1-e^{-at}}}+\epsilon\right),
\end{equation}
and $\|\mathbf{m}\|^2_{\bm{u}}$ is defined as $\sum_{i=1}^d \bm{u}_i\mathbf{m}_i^2$.
As can be seen from~\eqref{eqn: rmsprop_ode} and~\eqref{eqn: adam_ode},
the smaller the value of $a$ and $b$, the slower the dynamics of $\mathbf{v}$ (and $\mathbf{m}$), and consequently the slower the whole dynamics.
Numerical results in Figure~\ref{fig: speed} confirm this.
However, it is worth mentioning that this difference in convergence speed does not manifest at the very beginning of the training process. To understand this, consider the dynamics of Adam~\eqref{eqn: adam_ode}. Assume that at the beginning $\nabla f(\mathbf{x})=\nabla f(\mathbf{x}_0)$ is unchanged. Further assume that $\mathbf{v}_0=\mathbf{m}_0=0$ and $\epsilon=0$. Then,
\begin{align}
\mathbf{v}_t &= (1-e^{-at})\nabla f(\mathbf{x})^2, \nonumber\\
\mathbf{m}_t &= (1-e^{-bt})\nabla f(\mathbf{x}). \nonumber
\end{align}
Hence, we have
\begin{equation*}
\dot{\mathbf{x}}=-\frac{(1-e^{-bt})^{-1}(1-e^{-bt})\nabla f(\mathbf{x})}{\sqrt{(1-e^{-at})^{-1}(1-e^{-at})\nabla f(\mathbf{x})^2}} = -1,
\end{equation*}
which shows that the initial speed of $\mathbf{x}$ does not depend on $a$ and $b$.
\begin{figure*}[!h]
\centering
\includegraphics[width=0.4\columnwidth]{speed_adam_nn.png}
\hspace{-5mm}
\includegraphics[width=0.4\columnwidth]{speed_rmsprop_nn.png}
\caption{How the values of $a$ and $b$ affect the speed of dynamics. \textbf{Left:} Adam; \textbf{Right:} RMSprop. The learning rate is $0.001$ for all the experiments. The model and training data are the same as Figure~\ref{fig: adap_signgd}.
One can see that at the early stage of the training (after a very short period from initialization), optimizers with larger $a$ and $b$ converge faster. }
\label{fig: speed}
\end{figure*}
\section{RMSprop and signGD: Fast convergence and oscillation}
In this section we focus on RMSprop. Figure~\ref{fig: rmsprop} shows the loss curves and trajectories of RMSprop on a typical
multi-layer neural network model. There are three obvious features:
\begin{enumerate}
\item {\bf Fast initial convergence:} the loss curve decreases very fast, sometimes even super-linearly, at the early stage of the training.
\item {\bf Small oscillations:} The fast initial convergence is followed by oscillations around the minimum.
\item {\bf Large spikes:} spikes are sudden increase of the value of the loss. They are followed by an oscillating recovery. Different from small oscillations, spikes make the loss much larger and the interval between two spikes is longer.
\end{enumerate}
\begin{figure}[!h]
\centering
\includegraphics[width=0.4\columnwidth]{nn_rmsprop_loss1.png}
\hspace{-5mm}
\includegraphics[width=0.4\columnwidth]{nn_rmsprop_loss2.png}
\caption{The loss curves and trajectories of the RMSprop on a neural network model and CIFAR-10 data. Model and data the same as Figure~\ref{fig: loss}. The learning rate is 1e-3, and $\alpha=0.99$. $2000$ iterations are run. {\bf Left:} The whole training loss curves {\bf Right:} The training loss of the last $500$ iterations.}
\label{fig: rmsprop}
\end{figure}
\paragraph{Fast initial convergence.}
{ As discussed in the last section, when $\eta$ tends to $0$ while $\alpha$ stays fixed, RMSprop tends to signGD. So the loss curve of RMSprop and signGD align well during initial phase as shown in Figure \ref{fig: adap_signgd}. }
Figure~\ref{fig: signgd_rmsprop} shows the loss curves of both signGD and RMSprop on a quadratic objective function. Their behaviors are similar
---they both experience fast initial convergence and then the loss stops decreasing.
For this reason, we will study the fast initial convergence of RMSprop with the help of signGD.
In the (strongly) convex setting, the following proposition shows that continuous-time signGD can reach the global minimum in finite time.
\begin{figure}[!h]
\centering
\includegraphics[width=0.35\columnwidth]{compare_signgd.png}
\hspace{3mm}
\includegraphics[width=0.35\columnwidth]{compare_rmsprop.png}
\caption{The loss curves of signGD and RMSprop for a randomly generated quadratic function $f(\mathbf{x})=\frac{1}{2}\mathbf{x}^T A\mathbf{x}$ with different learning rates. { Here $A=UU^T$ with $U\in\mathbb{R}^{10\times 10}$ and the each entry of $U$ is randomly drawn from $\mathcal{N}(0,1)$.
For RMSprop $\alpha$ is fixed to be $0.9$.}}
\label{fig: signgd_rmsprop}
\end{figure}
\begin{proposition}
Assume that the objective function satisfies the Polyak-Lojasiewicz (PL) \cite{polyak1963gradient} condition: $\|\nabla f(\mathbf{x})\|_2^2 \geq \mu f(\mathbf{x})$ for any $\mathbf{x}$.
Assume that $\mathbf{x}(\cdot)$ is given by the continuous-time signGD dynamics \eqref{eqn: continuous-time-signGD},
then we have
\begin{equation*}
f(\mathbf{x}(t)) \leq \left(\sqrt{f(\mathbf{x}_0)} - \frac{\sqrt{\mu}}{2}t\right)^2.
\end{equation*}
\end{proposition}
\begin{proof}
We have
\begin{equation*}
\begin{aligned}
\frac{d}{dt}f(\mathbf{x}(t)) &= -\langle \textrm{sign}(\nabla f(\mathbf{x}(t))), \nabla f(\mathbf{x}(t)) \rangle = -\|\nabla f(\mathbf{x}(t))\|_1\\
&\leq -\|\nabla f(\mathbf{x}(t))\|_2 \leq -\sqrt{\mu f(\mathbf{x}(t))}.
\end{aligned}
\end{equation*}
Hence, we have
\begin{equation*}
\frac{d}{dt}\sqrt{f(\mathbf{x}(t))}\leq -\frac{\sqrt{\mu}}{2},
\end{equation*}
which implies
\begin{equation*}
f(\mathbf{x}(t)) \leq \left(\sqrt{f(\mathbf{x}_0)} - \frac{\sqrt{\mu}}{2}t\right)^2.
\end{equation*}
\end{proof}
\paragraph{Small oscillations}
For standard dynamical systems, small oscillations can be analyzed via linearization. Oscillations occur when the Jacobian has purely imaginary eigenvalues. However, in our case standard linearization cannot be applied easily. If we set $\epsilon=0$, then the dynamics is singular at the stationary point where $\mathbf{v}=0$. If $\epsilon$ is positive, all the eigenvalues of the Jacobian are negative real numbers, which means the linearized dynamics is strictly attractive and no oscillation will happen. However, since $\epsilon$ is usually small, the linearization approximates the original dynamics well only in a very small neighborhood of the stationary point, smaller than the range of the oscillations, hence cannot explain the oscillation.
For low dimensional strongly convex objective functions, RMSprop can converge to a $2$-periodic solution instead of the global minimum.
For example, if the objective function is $f(x)=\frac{1}{2}x^2$, then the $2$-periodic solution is an oscillation between $\frac{\eta}{2}$ and $-\frac{\eta}{2}$, where $\eta$ is the learning rate. Figure~\ref{fig: 2periodic} shows the convergence to this $2$-periodic solution.
\begin{figure}[!h]
\centering
\includegraphics[width=0.5\columnwidth]{rmsprop_2periodic.png}
\vspace{-2mm}
\caption{The trajectory of RMSprop for the $1$-dimensional quadratic function $f(x)=\frac{x^2}{2}$ for different values of $\alpha$. $\eta=0.01$. One sees that all the trajectories eventually converge to the $2$-periodic solution at $\frac{\eta}{2}$ and $-\frac{\eta}{2}$.}
\label{fig: 2periodic}
\end{figure}
For more complicated objective functions, such as high-dimensional quadratic function, or the loss function of neural network models, the RMSprop trajectories show more complicated oscillations patterns, such as the spikes. As we will see in the next section, Adam is more vulnerable to large spikes.
We will take a closer look of the large spikes in the next section.
\section{Adam: performances for different values of a and b}
The dynamic behavior of Adam is more complicated than RMSprop since it is influenced by $2$ hyper-parameters. Different combinations of $\alpha$ and $\beta$ (or $a$ and $b$) can lead to different dynamic patterns.
To rule out the influence of the learning rate, we will consider $a$ and $b$ instead of $\alpha$ and $\beta$. As is mentioned
before, $\alpha$ and $\beta$ are given by $a$ and $b$ through $\alpha=1-a\eta$ and $\beta=1-b\eta$.
{ As we have seen in Proposition \ref{thm: limit1} , when $a$ and $b$ are sufficiently large compared to $\eta$, Adam behaves like signGD. For relatively small $a$ and $b$,}
through extensive numerical experiments, we have found that there are roughly three different regimes of qualitative patterns in the parameter space (see Figure~\ref{fig: adam_patterns}):
\begin{enumerate}
\item {\bf The spike regime} happens when $b$ is sufficiently larger than $a$. In this regime large spikes appear in the loss curve, which makes the optimization process unstable.
\item {\bf The oscillation regime} happens when $a$ and $b$ have similar magnitude (or in the same order). In this regime the loss curve exhibits fast and small oscillations. Small loss and stable loss curve can be achieved.
\item {\bf The divergence regime} happens when $a$ is sufficiently larger than $b$. In this regime the loss curve is unstable and usually diverges after a period of training. This regime should be avoided in practice since the training loss stays large.
\end{enumerate}
In Figure~\ref{fig: adam_patterns} we show one typical loss curve for each regime for a typical neural network model. We also show typical trajectories in the state space of $(\|\mathbf{x}\|, \|\mathbf{m}\|, \|\sqrt{\mathbf{v}}\|)$ for the three regimes. These trajectories are also qualitatively different for different regimes.
\begin{figure*}
\centering
\includegraphics[width=0.32\textwidth]{spike_loss.png}
\includegraphics[width=0.32\textwidth]{oscillation_loss.png}
\includegraphics[width=0.32\textwidth]{divergence_loss.png}
\includegraphics[width=0.32\textwidth]{spike_loss2.png}
\includegraphics[width=0.32\textwidth]{oscillation_loss2.png}
\includegraphics[width=0.32\textwidth]{divergence_loss2.png}
\includegraphics[width=0.32\textwidth]{spike_traj.png}
\includegraphics[width=0.32\textwidth]{oscillation_traj.png}
\includegraphics[width=0.32\textwidth]{divergence_traj.png}
\caption{The three typical behavior patterns for Adam and the trajectories in the state space of $(\|\mathbf{x}\|, \|\mathbf{m}\|, \|\sqrt{\mathbf{v}}\|)$. $\eta=0.001$. The model and the training data are the same as Figure~\ref{fig: adap_signgd}. The first row shows the loss curve of totally $1000$ iterations, the second row shows part of the loss curve (the last $200$ iterations for oscillation and divergence regimes, and $400-800$ iterations for the spike regime), the bottom row shows the state space trajectory in the same period shown in the second row. {\bf Left:} $a=1$, $b=100$, large spikes appear in the loss curve; {\bf Middle:} $a=10$, $b=10$, the loss is small and oscillates very fast, and the amplitude of the oscillation is also small; {\bf Right:} $a=100$, $b=1$, the loss is large and blows up.}
\label{fig: adam_patterns}
\end{figure*}
Next we study the transition between the different regimes and the training loss behavior in different regimes.
To this end, we carried out experiments
for a multi-layer neural network model on the Fashion-MNIST dataset, with different values of $a$ and $b$ until the behavior of the training loss curve stabilizes. The left panel of Figure~\ref{fig: heatmap_nn} shows the heatmap of the average loss value of the last $1000$ iterations. The right panel of Figure~\ref{fig: heatmap_nn} shows the classification of the behavior of the training curve into three different categories (oscillations, spikes and divergence).
From these figures we see that in the divergence regime the training loss does not perform well (actually in some cases it may even blow up).
Hence this regime should be avoided in practice. In the oscillation regime the loss values are small and quite robust with respect to the change of hyper-parameters. Therefore this is the regime that should be preferred in practice. This is the regime when $a\approx b$.
\paragraph{Training ResNets on CIFAR10}
The above investigation suggests that Adam performs better when $\alpha\approx\beta$.
Here we provide further support by considering a more realistic problem: training a ResNet18 \cite{he2016deep} on CIFAR10 using stochastic Adam with large batch size. The results are shown in Figure \ref{fig: cifar}. We see that with the default parameters ($\beta=0.9, \alpha=0.999$), there are
are large spikes during the late phase of training. In contrast, when $a\approx b$, Adam converges very smoothly and is also faster than
using the default parameters.
\begin{figure}[!h]
\centering
\includegraphics[width=0.42\textwidth]{fmnist-loss-mean-lr1e-3.pdf}
\includegraphics[width=0.4\textwidth]{fmnist_spikes_lr1e-3.pdf}
\caption{ {\bf Left:} Heatmap of average training loss of Adam on a multi-layer neural network model. The loss is averaged over the last $1000$ iterations and is shown in logarithmic scale. $a$ and $b$ range from $0.1$ to $100$ and are also shown in logarithmic scale. {\bf Right:} The classification of the different training behavior. { $500$ data samples are taken from each class of Fashion-MNIST. The neural network model
is fully connected with $6$ hidden layers, with $500$ neurons per layer. The learning rate of is 1e-3. } }
\label{fig: heatmap_nn}
\end{figure}
\begin{figure}[!h]
\centering
\includegraphics[width=0.5\columnwidth]{adam_cifar_1.png}
\caption{Loss curves of stochastic Adam on a ResNet18 model and CIFAR-10 dataset. The learning rate is 1e-3. The red line shows the
results of using the default hyper-parameters setting ($\beta=0.9, \alpha=0.999$). $1000$ images are taken from each class to form the training dataset. The network is a ResNet18. The number of channels is half of the ones from the typical setting in~\cite{he2016deep}. The batch size is $1000$.}
\label{fig: cifar}
\end{figure}
\section{Discussion}
In this paper we reported the results of some systematical investigation on the dynamic behavior of adaptive gradient algorithms,
particularly RMSprop and Adam. Three typical phenomena---fast initial convergence, small oscillation and spikes---are observed and analyzed.
The influence of the choice of the hyper-parameters on the dominant training behavior is studied.
It is worth noting that the investigation in this paper focuses on the full-batch setting. However, the result in Figure \ref{fig: cifar} provides some evidence to show that the phenomena revealed here should also be of relevance for the stochastic setting when the batch size is large enough. The systematical study of the influence of batch size, especially in the small-batch regime, is left to future work.
There are still many other important open questions.
For example, learning rate decay is a common practice used in training large neural networks. When
performing learning rate decay, usually one does not change the values of $\alpha$ and $\beta$.
This makes the effective $a$ and $b$ larger, pushing the optimizer to the signGD-like regime. Another choice is to adaptively tune $\alpha, \beta$ such that $a$ and $b$ are kept fixed. It is interesting to see the comparison of the two strategies.
This paper focuses on optimization.
For machine learning problems, another important consideration when implementing optimization algorithms is the generalization performance. It has been reported that the solutions found by adaptive gradient algorithms usually perform a bit worse than those found by SGD
in terms of generalization (see~\cite{wilson2017marginal}).
The study of generalization performance of adaptive gradient algorithms is left for future work.
{\small
\bibliographystyle{plain}
| {
"timestamp": "2020-09-15T02:22:09",
"yymm": "2009",
"arxiv_id": "2009.06125",
"language": "en",
"url": "https://arxiv.org/abs/2009.06125",
"abstract": "The dynamic behavior of RMSprop and Adam algorithms is studied through a combination of careful numerical experiments and theoretical explanations. Three types of qualitative features are observed in the training loss curve: fast initial convergence, oscillations, and large spikes in the late phase. The sign gradient descent (signGD) flow, which is the limit of Adam when taking the learning rate to 0 while keeping the momentum parameters fixed, is used to explain the fast initial convergence. For the late phase of Adam, three different types of qualitative patterns are observed depending on the choice of the hyper-parameters: oscillations, spikes, and divergence. In particular, Adam converges much smoother and even faster when the values of the two momentum factors are close to each other. This observation is particularly important for scientific computing tasks, for which the training process usually proceeds into the high precision regime.",
"subjects": "Machine Learning (cs.LG); Machine Learning (stat.ML)",
"title": "A Qualitative Study of the Dynamic Behavior for Adaptive Gradient Algorithms",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.972830769252026,
"lm_q2_score": 0.7279754548076477,
"lm_q1q2_score": 0.7081969216971175
} |
https://arxiv.org/abs/0710.1994 | Metric Dichotomies | These are notes from talks given at ICMS, Edinburgh, 4/2007 ("Geometry and Algorithms workshop") and at Bernoulli Center, Lausanne 5/2007 ("Limits of graphs in group theory and computer science"). We survey the following type of dichotomies exhibited by certain classes X of finite metric spaces: For every host space H, either all metrics in X embed almost isometrically in H, or the distortion of embedding some metrics of X in H is unbounded. | \section{Problem statement and motivation}
\label{sec:intro}
In these notes we examine dichotomy phenomena exhibited by certain classes $\mathcal{X}$ of finite metric spaces. When attempting to embed the metrics in $\mathcal{X}$ in any given host spaces $H$, either all of them embed almost isometrically, or there are some metrics in $\mathcal{X}$ which are very poorly embedded in $H$. To make this statement precise we define the distortion of metric embeddings.
Given a mapping between metric spaces $f:X\to H$, define the Lipschitz norm of $f$ to be $\|f\|_{\mathrm{Lip}}=\sup_{x\ne y} d_H(f(x),f(y))/d_X(x,y)$. The distortion of injective mapping $f$ is defined as
$\dist(f)=\|f\|_{\mathrm{Lip}} \cdot \|f^{-1}\|_{\mathrm{Lip}}$, where
$f^{-1}$ is defined on $f(X)$.
The ``least distortion" in which $X$ can be embedded in $H$ is defined as
$c_H(X)=\inf\{\dist(f)|\; f:X\to H\}$. This is a measure of the faithfulness
possible when representing $X$ using a subset of $H$.
We formalize the discussion above as follows:
\begin{definition}[\textbf{Qualitative Dichotomy}]
\label{def:qual-dichotomy}
A class of \emph{finite} metric spaces $\mathcal{X}$ has the qualitative dichotomy property if
for any host space $H$, either
\begin{compactitem}
\item $\sup_{X\in\mathcal{X}} c_{H}(X)=1$; or
\item $\sup_{X\in\mathcal{X}}c_{H}(X)=\infty$.
\end{compactitem}
\end{definition}
\begin{remark}
As defined in Def.~\ref{def:qual-dichotomy}, the dichotomy is with respect to all metric spaces as hosts.
It is possible to extend the definition
to be with respect to all \emph{sets of metric spaces} as hosts. That is, for a set of metric spaces $\mathcal{H}$, define
$c_{\mathcal{H}}(X)=\inf _{H\in \mathcal{H}} c_H(X)$, and replace the use of ``$c_H(X)$" in Def.~\ref{def:qual-dichotomy} with
``$c_{\mathcal{H}}(X)$".
This extension, however, is inconsequential and the two definitions are equivalent. This follows from \emph{the proof} of Theorem~1.6 in~\cite{MN-cotype-full}, which implies that for any set of metric spaces $\mathcal{H}$, there exists a metric $\hat H$, such that for any finite metric space $X$, $c_{\hat H}(X)=c_{\mathcal{H}}(X)$.
\end{remark}
A dichotomy theorem for $\mathcal X$ can be interpreted as a
form of rigidity of $\mathcal X$: Small deformations of all the spaces in $\mathcal X$ is impossible.
We will also be interested in stronger dichotomies --- of a quantitative nature --- in which the
unboundedness condition of the
distortion is replaced with quantitative estimates on the rate in which it tends to infinity as a function of the size of the metric space.
I.e., by asymptotic lower bounds on the sequence
\[ D_N(H, \mathcal{X})=\sup\{c_{H}(X):\ X\in \mathcal{X}, \ |X|\le N\}. \]
The question of identifying such dichotomies was first explicitly raised by Arora \emph{et.~al.}~\cite{ALNRRV}.
They were motivated by a question from the theory of combinatorial approximation algorithms, where
bounded distortion embeddings have become a basic tool.
When dealing with algorithmically hard problem on a metric data
$X\in \mathcal{X}$, some algorithms first embed $X$ into
a better understood metric space $H$, $e:X\to H$, and then solve the algorithmic problem on $e(X)$.
This approach is used, for example,
in~\cite{LLR,Bartal-embed,GKR,Feige,FRT}.
For this approach to work:
\begin{compactenum}
\item $H$ should be simple enough to make the algorithmic problem tractable.
\item $e(X)$ should be close to $X$.
\end{compactenum}
Metric dichotomies draw limits on this approach when ``closeness" is measured in terms of the distortion. Dichotomy means that either $H$ already (essentially) contains $\mathcal{X}$, and therefore cannot be understood better than $\mathcal{X}$, or $H$ does not approximate some metrics in $\mathcal{X}$ very well. The algorithmic point of view also motivates the interest in
quantitative dichotomies: When dealing with finite objects, slowly growing approximation ratios
are also useful, and can be ruled out by quantitative dichotomies.
{Matou\v{s}ek}~\cite{Mat-BD} studied a closely related notion, which he called
\emph{bounded distortion (bd-) Ramsey}.
Simplifying his definitions a bit, a class of finite metric space $\mathcal{X}$ is called bd-Ramsey,
If for every $K>1$, $\varepsilon>0$, and $X\in\mathcal{X}$, there exists $Y\in\mathcal{X}$ such that
for any host space $H$, and any embedding $f:Y\to H$, if $\dist(f)\le K$,
then there exists $g:X\to Y$ such that
$\dist(g)\le 1+\varepsilon$, and $\dist(f|_{g(X)})\le 1+\varepsilon$.
As observed in~\cite{ALNRRV}, the bd-Ramsey property implies
qualitative dichotomy.
\begin{proposition}
If a class of finite metric spaces $\mathcal{X}$ is bd-Ramsey
then it has the qualitative dichotomy property.
\end{proposition}
\begin{proof}
Fix a host space $H$, and suppose that
\begin{equation} \label{eq:bounded}
\sup_{Y\in\mathcal{X}}c_H(Y)<\infty.
\end{equation}
Fix $X\in \mathcal{X}$, and $\varepsilon\in (0,1/2)$, and let $K=1+ \sup_{Y\in\mathcal{X}}c_H(Y)$. Pick
$Y\in\mathcal{X}$ that satisfies the bd-Ramsey condition.
By~\eqref{eq:bounded}, there exists $f:Y\to H$ such that $\dist(f)\le K$.
By the bd-Ramsey property, there exists $g:X\to Y$ such that
$\dist(g)\le 1+\varepsilon$, and $\dist(f|_{g(X)})\le 1+\varepsilon$, and so $c_H(X)\le
\dist(g)\cdot \dist(f|_{g(X)}) \le 1+3\varepsilon$. Since this is true for any $\varepsilon\in(0,1/2)$, we conclude that $c_H(X)=1$. As this is true for any $X\in\mathcal{X}$, we conclude that $\sup_{X\in\mathcal{X}}c_H(X)=1$.
\end{proof}
\begin{remark}
All the dichotomies results in this note
are actually bd-Ramsey results.
\end{remark}
{Matou\v{s}ek}'s study of bd-Ramsey phenomena~\cite{Mat-BD} is partially motivated by a general theme in the geometric theory of Banach spaces to translate notions and results from the linear theory of finite dimensional Banach spaces to finite metric spaces.
One such example is a theorem of Maurey, Pisier, and Krivine~\cite{MP-type-cotype,Krivine}
(see also~\cite{Maurey-survey} and~\cite[Ch.~12]{BL}) which implies that if a normed space $H$ contains linear images of $\ell_p^n$ for any $n$ with uniformly bounded distortion, then $H$ contains linear images of $\ell_p^n$ for any $n$ almost isometrically. More precisely, For every $t\in \mathbb N$, $\varepsilon>0$, $K\ge 1$, and $p\in [1,\infty]$, there exists $n=n(t,\varepsilon\,K,p)$ such that if there exists a linear mapping $f:\ell_p^n \to H$, with $\dist(f)\le K$, then there exists a linear mapping $g:\ell_p^t \to \ell_p^n$ such that both $\dist(g)\le 1+\varepsilon$, and
$\dist(f|_{g(\ell_p^t)})\le 1+\varepsilon$. The bd-Ramsey property is a similar property, without the linear structure.
\medskip
When studying metric dichotomy for a given class $\mathcal{X}$ of metric spaces, it is beneficial to work with a structured dense subclass $\mathcal{Y}\subset \mathcal{X}$.
\begin{proposition} \label{prop:dense}
Suppose that $\mathcal{Y} \subset \mathcal{X}$ and $\mathcal{Y}$ is dense in $\mathcal{X}$, i.e., for every
$X\in\mathcal{X}$, $c_\mathcal{Y}(X)=1$. Then if $\mathcal{Y}$ has a metric dichotomy (either qualitative or quantitative) then $\mathcal{X}$ has the same dichotomy.
\end{proposition}
\begin{proof} Since $\mathcal{Y}\subset \mathcal{X}$, for any host space $H$,
$D_N(H,\mathcal{Y})\ge D_N(H,\mathcal{X})$. On the other hand, if $\sup_{Y\in\mathcal{Y}}c_H(Y)=1$, then
\( 1\le \sup_{X\in\mathcal{X}}c_H(X) \le \sup_{Y\in\mathcal{Y}}c_H(Y) \cdot \sup_{X\in\mathcal{X}}c_\mathcal{Y}(X)= 1 . \)
\end{proof}
Table~\ref{tab:classes} lists the classes of finite metric spaces which we will deal with and their regular dense subclasses. The proofs of the
density are standard.
\begin{table}[ht]
\begin{center}
\begin{tabular}{l|ll}
Metric class & Dense structured & Shorthand\\
(Finite subsets of) & subclass ($n\in\mathbb N$) & ($n\in\mathbb N$) \\ \hline
$\mathbb R$ & $\{0,\ldots,n\}$ & $P_n$\\
$L_1$ & $(\{0,1\}^n,\|\cdot\|_1)$ & $\{0,1\}^n$ \\
$L_\infty$ ~ (i.e., $\mathcal{MET}$) & $(\{1,\ldots,n\}^n,\|\cdot\|_\infty)$ & $[n]^n_\infty$ \\
tree metrics & $(\{0,1\}^{\le n}, \text{tree distance})$ & $B_n$ \\ \hline
\end{tabular}
\end{center}
\caption{The classes of metric spaces considered in these notes, and their dense regular subclasses used in the proofs. $\mathcal{MET}$ is the class of all finite metric spaces. $\{0,1\}^{\le n}$ is the set of binary strings of length at most $n$. The tree distance on binary strings
$x,y\in \{0,1\}^{\le n}$ is defined as $|x|+|y|- 2 |\mathrm{lcp}(x,y)|$,
where $|x|$ is the length of $x$, and $\mathrm{lcp}(x,y)$ is the longest common prefix of $x$ and $y$.}
\label{tab:classes}
\end{table}
\section{Qualitative dichotomies} \label{sec:qualitative}
\begin{theorem}~\cite{Mat-BD} \label{thm:mat-bd}
The following classes of finite metric spaces have the qualitative dichotomy property:
\begin{enumerate}[1.]
\item Finite subsets of $\mathbb R$,
\item For any $p\in [1,\infty]$, the class of finite subsets of $L_p$.
\item Finite equilateral spaces.
\end{enumerate}
\end{theorem}
Here we just outline the proof for finite subsets of $L_p$, which is a nice demonstration of the linearization technique for Lipschitz mappings of normed spaces: Given a Lipschitz map, find a point of differentiability.
The differential is a linear map with the same Lipschitz norm. Now apply a result from the linear theory.
In our case, the linear result is the Maurey, Pisier, and Krivine theorem, and the differentiability argument
is due to Kirchheim~\cite{Kirchheim}.
\begin{proof}[Sketch of a proof of Theorem.~\ref{thm:mat-bd}, item 2]
Fix a host space $H$, and $p\ge 1$, and assume that there exists $K\in [1,\infty)$
such that any finite subset of $S\subset L_p$ embeds in $H$, $f_S:S\to H$, and $\dist(f_S)\le K$.
We fix a finite $S\subset L_p$, $\varepsilon>0$ and want to prove that $c_H(S)\le 1+\varepsilon$. It is known (see~\cite[Sec.~11.2]{DZ})
that $S$ can be isometrically embedded in $\ell_p^t$, for $t=\binom{|S|}{2}$. Let $n=n(p,t,K,\varepsilon)$ be chosen as in the Maurey-Pisier+Krivine theorem discussed in Section~\ref{sec:intro}.
The argument of the proof goes roughly as follows: Use a compactness argument to conclude that there exists an
embedding $\hat f:\ell_p^n \to H$ whose distortion at most $K$. Since this embedding is in particular Lipschitz, use a differentiation argument to find a point of ``differentiability". The differential is a ``linear mapping" of $\ell_p^n$ whose distortion is at most $K$,
and thus by the Maurey-Pisier+Krivine theorem alluded to above, there exists a $t$-dimensional subspace of
$\ell_p^n$ which is almost isometric to $\ell_p^t$ for which that mapping is almost isometry into $H$. This would finish the proof, since $S$ is isometrically embeddable in $\ell_p^t$.
The above argument has two major difficulties:
\begin{inparaenum}[1)]
\item Since $H$ is not compact, the required ``compactness argument" is false.
\item The notions of linear mapping and derivative when the target is general metric space, are not clear.
\end{inparaenum}
The first difficulty is addressed using a compactness argument, similar to Rado's Lemma (see \cite[Lemmas~3.4, \&~4.4]{Mat-BD}), which implies
there exists a metric space $\hat H$, and an embedding
$\hat f:\ell_p^n\to \hat H$ such that $\dist(\hat f)\le K$, and
moreover, for any finite $T\subset \ell_p^n$, and $\delta>0$, there exists $R$, $T\subset R\subset \ell_p^n$ such that $\hat f(R)$ distorts the distance in $f_R(R)$ by at most a factor of $1+\varepsilon$.
The second difficulty is addressed using
a metric differentiation theorem of Kirchheim~\cite{Kirchheim}. It implies that
there exists $x_0\in \mathbb R^n$, and a pseudo-norm on ${|\hspace{-0.9pt}|\hspace{-0.9pt}|}\cdot {|\hspace{-0.9pt}|\hspace{-0.9pt}|}$ on $\mathbb R^n$
such that for every $h,k\in \ell_p^n$,
$d_H(\hat f(x_0+h),\hat f(x_0+k))= {|\hspace{-0.9pt}|\hspace{-0.9pt}|} h-k{|\hspace{-0.9pt}|\hspace{-0.9pt}|} +o(\|h\|_p+\|k\|_p)$. We conclude that ${|\hspace{-0.9pt}|\hspace{-0.9pt}|} \cdot {|\hspace{-0.9pt}|\hspace{-0.9pt}|} $ is a norm on $\mathbb R^n$ whose Banach-Mazur distance from the $\ell_p$ norm is at most $K$.
Furthermore on a ball $B$ small enough around $x_0$ in $\ell_p^n$, ${|\hspace{-0.9pt}|\hspace{-0.9pt}|} \cdot {|\hspace{-0.9pt}|\hspace{-0.9pt}|} $ is $1+\varepsilon$ approximation to the metric on $\hat f(B)$.
Hence, by translating and rescaling $S$ we can assume it is inside $B$, and thus we can view $\hat f$ as an approximate mapping between $\ell_p^n$ and $(\mathbb R^n,{|\hspace{-0.9pt}|\hspace{-0.9pt}|} \cdot {|\hspace{-0.9pt}|\hspace{-0.9pt}|})$. At this point Maurey-Pisier+Krivine
theorem can be applied rigorously.
\end{proof}
\medskip
Finite subsets of finite (but larger than one) dimensional normed space is a natural class of metric spaces that does not have the dichotomy property:
\begin{proposition}
Fix $d>1$, and some norm, $\|\cdot \|$, on $\mathbb R^d$. Then the class of finite subsets of $(\mathbb R^d, \|\cdot\|)$ does not have qualitative dichotomy.
\end{proposition}
\begin{proof}[Sketch of a proof]
It is possible to construct another norm ${|\hspace{-0.9pt}|\hspace{-0.9pt}|} \cdot {|\hspace{-0.9pt}|\hspace{-0.9pt}|} $ on $\mathbb{R}^d$, whose Banach-Mazur
distance from $\|\cdot \|$ is some $B>1$.
I.e., for any
linear mapping $T:(\mathbb R^d, \|\cdot\|) \to (\mathbb R^d, {|\hspace{-0.9pt}|\hspace{-0.9pt}|} \cdot {|\hspace{-0.9pt}|\hspace{-0.9pt}|} )$, $\|T\|\cdot \|T^{-1}\|\ge B>1$,
and the inequality is tight for some $T$.%
\footnote{To see it, notice that the Banach-Mazur distance
between $\ell_2^d$, and $\ell_1^d$ is $\sqrt{d}>1$, and therefore by the triangle inequality any other $d$-dimensional norm must be at distance
at least $\sqrt[4]{d}>1$ from one of them. By John's Theorem (see e.g.~\cite[Sec.~13.4]{Mat-Discrete-Geometry}),
the distance is at most $d$.}
We take $H=(\mathbb R^d, {|\hspace{-0.9pt}|\hspace{-0.9pt}|} \cdot {|\hspace{-0.9pt}|\hspace{-0.9pt}|} )$, and so $c_H((\mathbb R^d,\|\cdot\|))\le B$.
On the other hand, assume for the sake of contradiction that there exists $A<B$ such that any finite subset of $(\mathbb R^d, \|\cdot\|)$ can be embedded in $H$ with distortion at most $A$. By a compactness argument there exists an embedding of the unit ball of
$(\mathbb R^d, \|\cdot \|)$ in $(\mathbb R^d, {|\hspace{-0.9pt}|\hspace{-0.9pt}|} \cdot {|\hspace{-0.9pt}|\hspace{-0.9pt}|} )$ with distortion at most $A$. Next, by Rademacher differentiation theorem
there exists a point of differentiability in this embedding. The differential is a linear mapping whose distortion is at most $A$, which is a contradiction.
\end{proof}
\section{Dichotomy for subsets of the line} \label{sec:line}
In the next two sections we discuss quantitative dichotomies and sketch direct proofs.
We begin with finite subsets of $\mathbb R$.
\begin{theorem} \label{thm:line-dich}
For every metric space $H$, either
\begin{compactitem}
\item $c_{ H}(A) =1$, for every finite $A\subset \mathbb R$; or
\item There exists $\beta>0$, such that $c_{ H}(P_n)\ge \Omega(n^\beta)$, where $P_n$ is the $n$-point path metric.
\end{compactitem}
\end{theorem}
As discussed above, {Matou\v{s}ek} showed a qualitative dichotomy for finite subsets of the line, based on
differentiation argument. His proof actually gives the same quantitative bounds as in Theorem~\ref{thm:line-dich}.
Here, following~\cite{MN-trees}, we sketch a somewhat different proof which conveys the approach to prove the more complicated quantitative dichotomies for finite subsets of $L_1$, and for all finite metric spaces.
The general approach in those proofs is to define an ``isomorphic" inequality, and prove sub-multiplicativity.
This approach --- originated in the work of Pisier~\cite{Pisier-type-1} --- is used in Banach space theory quite often.
We proceed to prove Theorem~\ref{thm:line-dich}. We first
choose an appropriate inequality that captures the distortion of embedding $P_n$ in $H$.
Let $\Psi_n(H)$ be the infimum over $\Psi>0$ such that for every $f:P_n \to H,$
\begin{equation} \label{eq:path}
d_H(f(0),f({n})) \le \Psi n\max_{i=0,\ldots n-1} d_H(f(i),f({i+1})) .
\end{equation}
\begin{lemma} \label{lem:path-ineq}
For every metric space $H$, and $m,n\in \mathbb N$,
\begin{compactenum}[1.]
\item $\Psi_n( H) \le 1$.
\item $c_{ H}(P_n)\ge 1/ \Psi_n( H)$.
\item If $\Psi_n( H)=1$, then $c_{ H}(P_n)=1$.
\item $\Psi_{mn}( H) \le \Psi_m( H) \cdot \Psi_n( H)$.
\end{compactenum}
\end{lemma}
Before proceeding with the proof of lemma~\ref{lem:path-ineq}, lets see how
Theorem~\ref{thm:line-dich} is derived.
\begin{proof}[Proof of Theorem~\ref{thm:line-dich}]
We will prove the dichotomy to $(P_n)_n$ (the path metrics), which by Prop.~\ref{prop:dense} is sufficient.
Fix a host space $H$.
\begin{itemize}
\item If for every $n\in\mathbb N$, $\Psi_n(H)=1$, then $c_{H}(P_n)=1$.
\item If there exists $n_0$ for which $\Psi_{n_0}(H)=\eta<1$, then let $\beta>0$ be such that $n_0^{-\beta}=\eta$,
and from the submultiplicativity, $\Psi_{n_0^k}(H)\le \eta^k= (n_0^k)^{-\beta}$, and so
$c_{H}(P_{n_0^k})\ge (n_0^k)^\beta$. \qedhere
\end{itemize}
\end{proof}
\begin{proof}[Proof of Lemma~\ref{lem:path-ineq}]
~
\begin{enumerate}[1.]
\item Follows from the triangle inequality.
\item
Fix $f:P_n\to H$, and $\Psi>\Psi_n(H)$. Plugging the Lipschitz norms into~\eqref{eq:path},
\begin{multline*} \frac{n}{\|f^{-1}\|_\mathrm{Lip}} \le d_H(f(0),f({n}))
\le \Psi n\max_{i=0,\ldots n-1} d_H(f(i),f({i+1})) \le \Psi n \|f\|_\mathrm{Lip} ,
\end{multline*}
So $\dist(f)=\|f\|_\mathrm{Lip} \cdot \|f^{-1}\|_\mathrm{Lip} \ge 1/\Psi$. Since this is true for any $\Psi>\Psi_n(H)$, $\dist(f)\ge 1/\Psi_n(H)$.
\item If $\Psi_n( H)=1$, then for any $\varepsilon\in(0,1/2n)$, there exists $f:P_n \to H$ for which
\begin{multline} \label{eq:path-2}
n\max_{i=0,\ldots n-1} d_H(f(i),f({i+1})) \ge d_H(f(0),f({n-1}))
\\ \ge (1-\varepsilon) n\max_{i=0,\ldots n-1} d_H(f(i),f({i+1})).
\end{multline}
Let $A= \max_{i=0,\ldots n-1} d_H(f_i,f_{i+1})$,
and for $i> j$,
\begin{multline*} (i-j)A \ge d_H(f(i),f(j)) \\\ge d_H(f(0),f({n})) - d_H(f(j),f(0)) - d_H(f(n),f(i)) \\
\ge (1-\varepsilon)n A - j A - (n-i)A= (i-j -\varepsilon n) A .
\end{multline*}
This means that $\dist(f)\le 1+2\varepsilon n$, which implies that $c_H(P_n)=1$.
\item Fix $f:P_{mn}\to H$. Define $g:P_n \to H$, by $g(i)=f(im)$.
Applying~\eqref{eq:path} to $g$, we obtain
\begin{equation}\label{eq:path-3}
d_H(f(0),f({mn})) \le (\Psi_n(H)+\varepsilon) n \max_{i=0\ldots n-1} d_H(f({im}),f({(i+1)m})).
\end{equation}
Next, define $h_i:P_m\to H$, $h_i(j)=f(im+j)$, and apply~\eqref{eq:path} for each $h_i$, and so
\begin{multline} \label{eq:path-4}
d_H(f({im}),f({(i+1)m})) \\ \le (\Psi_m(H)+\varepsilon) m \max_{j=0\ldots m-1} d_H(f({im+j}),f({im+j+1})) .
\end{multline}
Combining~\eqref{eq:path-3} with~\eqref{eq:path-4}, and we conclude the claim. \qedhere
\end{enumerate}
\end{proof}
The quantitative dichotomy in Theorem~\ref{thm:line-dich} is tight for finite subsets of the line:
For any $\beta\in(0,1]$, there exists $H_\beta$ such that $c_{H_\beta}(P_n)=\Theta(n^\beta)$. For $\beta\in(0,1)$, $H_\beta$ can be taken as the real line with the usual metric to the power of $1-\beta$.
For $\beta=1$, $H_1$ can be taken as the ultrametric defined on $\{0,1\}^{\mathbb N}$, with the distance function
$\rho(x,y)=2^{-|\mathrm{lcp}(x,y)|}$, where $\mathrm{lcp}$ is the longest common prefix of the two sequences.
\section{Dichotomies for finite subsets of $L_1$, and $L_\infty$}
The proofs of the quantitative dichotomies for subsets of $L_1$ and subsets of $L_\infty$
use the same general approach taken in Section~\ref{sec:line}:
we write inequalities for which we can prove a lemma similar to Lemma~\ref{lem:path-ineq}, but replacing paths with Hamming cubes (for subsets of $L_1$) and grids with the $\ell_\infty$ distance (for subsets of $L_\infty$).
In both cases the hard part in the proof seems to be coming up with the inequality. However, in contrast to
path metrics, the proofs of the lemmas analogous to Lemma~\ref{lem:path-ineq} (especially item~(3)) are technical and lengthy. We will therefore omit all these details and concentrate on the inequalities.
\subsection{Finite subsets of $L_1$}
The argument given here is essentially from a paper of Bourgain, Milman, and Wolfson~\cite{BMW} on metric type.%
\footnote{The paper~\cite{BMW} does not discuss dichotomy, but rather
a non-linear analogue for Pisier theorem for type-1. As we shall see in Section~\ref{sec:MP}, from that result it is easy to obtain the dichotomy.}
\begin{theorem}{\cite{BMW}} \label{thm:dich-cubes}
For every metric space $H$, either
\begin{compactitem}
\item $c_{H}(X) =1$, for every finite $X\subset L_1$; or
\item There exists $\beta>0$, such that $c_{H}((\{0,1\}^n,\|\cdot\|_1)\ge \Omega(n^\beta)$.
\end{compactitem}
\end{theorem}
Similarly to the dichotomy of subsets of $\mathbb R$, we use an inequality to guide the proof:
Let $(e_i)_{i=1}^n$ denote the standard basis of $\{0,1\}^n$, and $\mathbf 1=\sum_i e_i$.
Let $T_n(H)$ be the infimum over $T>0$ such that
for every $f:\{0,1\}^n \to H$,
\begin{equation} \label{eq:cube}
\Avg_{x\in\{0,1\}^n}
d_H(f(x),f(x+\mathbf{{1}}))^2 \le T^2 n\sum_{i=1}^n \Avg_{x\in\{0,1\}^n} d_H(f(x),f(x+e_i))^2 ,
\end{equation}
where the operator $\Avg$ means averaging.
Inequality~\eqref{eq:cube} was chosen to ``capture" the distortion of embeddings the Hamming cubes in $H$, and have
the sub-multiplicativity property (in $n$). It is a variant of the metric-type inequality from~\cite{BMW}. The connection (and motivation) to the type property is expanded upon in Section~\ref{sec:MP}.
Formally, we can prove a lemma analogous to Lemma~\ref{lem:path-ineq}:
\begin{lemma} \label{lem:cube-ineq}
For every metric space $H$, and $m,n\in \mathbb N$,
\begin{enumerate}[1.]
\item $T_m(H) \le 1$.
\item $c_{H}(\{0,1\}^n)\ge 1/ T_n( H)$.
\item If $T_n( H)=1$, then $c_{H}(\{0,1\}^n)=1$.
\item $T_{mn}( H) \le T_m(H) \cdot T_n(H)$.
\end{enumerate}
\end{lemma}
Using Lemma~\ref{lem:cube-ineq}, the proof of Theorem~\ref{thm:dich-cubes}
is the same as the proof of Theorem~\ref{thm:line-dich},
replacing references to $\Psi_n(H)$ with $T_n(H)$, and the path metric with the Hamming cube.
Regarding the quantitative tightness of Theorem~\ref{thm:dich-cubes}:
It is known~\cite{Enflo-cubes} that $c_{\ell_2}((\{0,1\}^n,\|\cdot\|_1))=\sqrt{n}$,
and that any $N$-point subset of $L_1$ is $O(\sqrt{\log N} \log\log N)$ embeddable in $\ell_2$~\cite{ARV,CKR,ALN}.
I do not know much more.
\begin{question}
Does there exist $\beta\in (0,1/2)$ and a metric space $H$
such that $1<D_N(H,2^{\ell_1})=O((\log N)^\beta)$? If so, is it true for every $\beta>0$?
\end{question}
\subsection{Finite metric spaces}
Next,
we consider $\mathcal{MET}$, the set of all finite metric spaces
which is equal to the set of finite subsets of $L_\infty$.
\begin{theorem}\cite{MN-cotype-full} \label{thm:dich-all}
For every metric space $H$, either
\begin{compactitem}
\item $\sup_{X\in \mathcal{MET}} c_{\mathcal H}(X) =1$; or
\item There exists $\beta>0$, such that $c_{H}((\{1,\ldots,n\}^n, \|\cdot\|_\infty)\ge \Omega(n^\beta)$,
where $[n]^n_\infty$ is the $\{1,\ldots,n\}^n$ grid with the $\ell_\infty$ distance.
\end{compactitem}
\end{theorem}
Again, we use an inequality (derived from the metric cotype inequality~\cite{MN-cotype-full}) to guide the proof.
Denote by $\Gamma_{n}(H)$ the infimum over $\Gamma>0$ such that for every $m\in \mathbb N$,
and every $f:\mathbb Z_m^n \to H$,
\begin{equation} \label{eq:grid}
\sum_{i=1}^n \Avg_{x\in \mathbb Z_m^n} d_H(f(x),f(x+n e_j))^2 \le
\Gamma^2\cdot n^2 \cdot n \Avg_{\varepsilon\in \{\pm 1\}^n} \Avg_{x\in \mathbb Z_m^n} d_H(f(x),f(x+\varepsilon))^2
\end{equation}
(the additions ``$x+n e_j$", and ``$x+\varepsilon$" are in $\mathbb Z_m^n$).
Inequality~\eqref{eq:grid} is designed to capture the distortion of embedding $\{1,\ldots,n\}^n$ with the $\ell_\infty$
distance in $H$. In this context, it seems more natural to average $\varepsilon$ over all $\{-1,0,1\}^n$ which are the distance 1 in the $\ell_\infty$ metric. However, this choice would complicate the proof of the submultiplicativity property. Note that the metric induced by the graph $\mathbb Z_n^n$ with the $\{\pm 1\}^n$ edges contains an isometric copy of $\{1,\ldots, n/4\}^n$
with the $\ell_\infty$ metric. Also, the design choice of the universal quantifier on $m$ (instead of say fixing $m=n$) was done to make the proof of the submultiplicativity easy. The connection to the cotype property of Banach spaces is expanded upon in Section~\ref{sec:MP}.
As before, we use a lemma analogous to Lemma~\ref{lem:path-ineq}:
\begin{lemma} \label{lem:torus-ineq}
For every metric space $H$, and $m,n\in \mathbb N$,
\begin{enumerate}[1.]
\item $\Gamma_n( H) \le 1$, for all even $n$.
\item $c_{ H}((\{1,\ldots,n/4\}^n, \|\cdot\|_\infty))\ge 1/ \Gamma_n(H)$.
\item If $\Gamma_n(H)=1$, then $c_{ H}((\{1,\ldots,n/4\}^n, \|\cdot\|_\infty))=1$.
\item $\Gamma_{m n}(H) \le \Gamma_{m}(H) \cdot \Gamma_{n}(H)$.
\end{enumerate}
\end{lemma}
Theorem~\ref{thm:dich-all} is deduced similarly to Theorems~\ref{thm:line-dich} and~\ref{thm:dich-cubes} but now using
Lemma~\ref{lem:torus-ineq}. A sketch of the proof of Lemma~\ref{lem:torus-ineq}
can be found in~\cite[Sec.~2]{MN-cotype}. The complete proof appears in~\cite[Sec.~6]{MN-cotype-full}.
I do not know much about the quantitative tightness of Theorem~\ref{thm:dich-all}.
\begin{question} \label{q:dich-all}
Does there exist $\beta\in(0,1)$ and a metric space $H$
for which $1<D_N(H,\mathcal{MET})=O((\log N)^\beta)$? If so, is it true for every $\beta\in(0,1)$?
\end{question}
Personally, the dichotomy of $\mathcal{MET}$ seems to me the most natural problem in thess notes, and Question~\ref{q:dich-all} the most fundamental
open problem.
Bourgain's embedding theorem~\cite{Bourgain-embed} and the matching lower bound~\cite{LLR} implies
that $D_N(\ell_2,\mathcal{MET})=\Theta(\log N)$.
Essentially, all examples of families of metrics having logarithmic distortion when embedded into Hilbert space, are families of expander graphs with a logarithmic diameter. {Matou\v{s}ek} proved that expanders have logarithmic distortion when embedded in $L_p$, for any $p\in [1,\infty)$. Recently, Lafforgue~\cite{Lafforgue} has exhibited classes of expanders
with logarithmic distortion when embedded in $B$-convex Banach spaces
--- spaces with type greater than~$1$.
In view of the seemingly surprising fact of the metric dichotomy, it is natural to ask which monotone
$f:\mathbb N\to\mathbb N$ has a metric space $H$ such that $D_N(H,\mathcal{MET})=\Theta^*(f(N))$ (here $\Theta^*$ can ``hide" polylogarithmic multiplicative factors).
From Bourgain embedding theorem we know that $f(N)=\log N$ is achievable using Hilbert space. {Matou\v{s}ek}~\cite{Mat-lowdim} showed that for every even $d$,
$D_N(\ell_2^d,\mathcal{MET})=\Theta^*(N^{2/d})$. Furthermore, in the full version of~\cite{MN-trees}
it is shown that for every $\varepsilon\in(0,1]$, there exists a metric space $H_\varepsilon$, for which $D_N(H_\varepsilon,\mathcal{MET})=\Theta(N^\varepsilon)$.
This leaves us with a concrete question:
\begin{question}
Does there exist $H$ for which $D_N(H,\mathcal{MET})\in \omega(\log N) \cap N^{o(1)}$?
\end{question}
Comparing the results in this section with those in Section~\ref{sec:qualitative}, we also ask:
\begin{question}
Is there a (substantial) quantitative dichotomy for finite subsets of
$L_p$, and in particular for $L_2$?
\end{question}
\subsection{Metric type and cotype and non-linear Maurey-Pisier theorems}
\label{sec:MP}
A normed space is said to have type $p$, $1\leq p\leq 2$, with constant $T$, if for every finite family $x_1,\dots,x_n\in X$,
\begin{equation}\label{eq:type}
\Big(\Avg_{\varepsilon\in\{-1,1\}^n}\Big\|\sum^n_{j=1}\varepsilon_j x_j\Big\|^p_X\Big)^{\frac1p}\leq T\Big(\sum^n_{j=1}\|x_j\|^p\Big)^{\frac 1p}
\end{equation}
and cotype $q$, $2\leq q\leq\infty$, with constant $C$,
if for every finite family $x_1,\dots,x_n\in X$,
\begin{equation} \label{eq:cotype}
C \Big(\Avg_{\varepsilon\in\{-1,1\}^n}\Big\|\sum^n_{j=1}\varepsilon_j x_j\Big\|^q_X\Big)^{\frac1q}\geq \Big(\sum^n_{j=1}\|x_j\|^q\Big)^{\frac 1q}
\end{equation}
The theory around these notions was developed since the 70's, with fascinating results. The interested reader may consult~\cite{MS,Maurey-survey} and references there in.
Here we are interested in one aspect of this theory, called
Maurey-Pisier Theorem.
A closely related conditions are \emph{equal norms type and cotype}: A normed space is said to have equal norm (en) type $p$, $1\leq p\leq 2$, with constant $\hat T$ if for every finite family $x_1,\dots,x_n\in X$,
\begin{equation}\label{eq:en-type}
\Avg_{\varepsilon\in\{-1,1\}^n} \Big\|\sum^n_{j=1}\varepsilon_j x_j\Big\|^2_X\leq \hat T^2 n^{\frac 2p-1}\sum^n_{j=1}\|x_j\|^2
\end{equation}
Similarly, a normed space is said to have equal norms (en) cotype $q$, $2\leq q\leq\infty$, with constant $\hat C$,
if for every finite family $x_1,\dots,x_n\in X$,
\begin{equation} \label{eq:en-cotype}
\hat C^2 n^{1-\frac2q} \Avg_{\varepsilon\in\{-1,1\}^n}\Big\|\sum^n_{j=1}\varepsilon_j x_j\Big\|^2_X\geq \sum^n_{j=1}\|x_j\|^2
\end{equation}
Type/cotype and en-type/cotype are closely related%
\footnote{The LHS are equivalent, up to a constant factor, by Kahane inequality. In the RHS, type/cotype condition implies the equal norms variant by H\"older inequality, and they are equal when all the $x_i$ has the same norm, hence the name.}:
Type $p$ implies en-type $p$ which implies type $p-\varepsilon$ for every $\varepsilon>0$.
Cotype $q$ implies en-cotype $q$ which implies cotype $q+\varepsilon$ for every $\varepsilon>0$ (see~\cite{T-J}).
We observe that any normed space has en-type 1 and en-cotype $\infty$, as these inequalities follow from the triangle inequality (and Cauchy-Schwarz).
A normed space has type $>1$ if and only if it has en-type $>1$,
and cotype $<\infty$ if and only if it has en-cotype $<\infty$.
Maurey and Pisier~\cite{MP-cotype-oo} proved:
\begin{theorem} \label{thm:MP-cotype-oo}
A normed space $X$ does not have (en-)cotype $<\infty$ if and only if for every $n\in \mathbb N$,
and $\eta>0$, $\ell_\infty^n$ can be linearly embedded in $X$ with distortion at most $1+\eta$.
\end{theorem}
Pisier~\cite{Pisier-type-1} proved an analogous result for type:
\begin{theorem} \label{thm:P-type-1}
A normed space $X$ does not have (en-)type $>1$ if and only if for every $n\in \mathbb N$,
and $\eta>0$, $\ell_1^n$ can be linearly embedded in $X$ with distortion at most $1+\eta$.
\end{theorem}
Those results imply dichotomy results for the class of finite dimensional subspaces.
\begin{proposition} \label{prop:linear-dichotomy}
For any normed space $H$, either
\begin{compactitem}
\item For every $\varepsilon>0$, any finite dimensional normed space $X$ can be linearly embedded in $H$ with distortion $1+\varepsilon$.
\item There exists $\beta>0$, such that for every $n\in\mathbb N$,
and linear embedding $f:\ell_\infty^n \to H$,
$\dist(f)= \Omega(n^{\beta})$.
\end{compactitem}
\end{proposition}
\begin{proof}
If $H$ does not have finite en-cotype, then by Theorem~\ref{thm:MP-cotype-oo} any $\ell_\infty^n$
can be linearly embedded in $H$ with distortion $1+\varepsilon$, for every $\varepsilon>0$. Combining this with the elementary observation that for any $\varepsilon>0$, any finite dimensional normed space can be $(1+\varepsilon)$-embedded in $\ell_\infty^n$, for some $n\in \mathbb N$ we obtain the first bullet.
If, on the other hand, $H$ has some finite en-cotype $q<\infty$, then for any linear embedding $f: \ell_\infty^n \to H$,
\begin{multline*}
\hat C \|f\|= \hat C\|f\| \Big(\Avg \Big\|\sum^n_{j=1}\varepsilon_j e_j\|^2_\infty \Big)^{\frac {1}{2}}\ge \hat C \Big(\Avg \Big\|\sum^n_{j=1}\varepsilon_j f(e_j)\Big\|^2_H\Big)^{\frac12} \\
\geq n^{\frac1q-\frac12} \Big(\sum^n_{j=1}\|f(e_j)\|_H^2\Big)^{\frac 12}
\ge \|f^{-1}\|n^{\frac1q-\frac12}\Big(\sum^n_{j=1}\|e_j\|_\infty^2\Big)^{\frac 12}= n^{\frac{1}{q}} \cdot \|f^{-1}\|,
\end{multline*}
which implies that $\dist(f)=\Omega(n^{1/q})$.
\end{proof}
A similar dichotomy can be proved for $\ell_1^n$, using Theorem~\ref{thm:P-type-1}.
Theorems~\ref{thm:MP-cotype-oo} and~\ref{thm:P-type-1} are proved using linear analogues of Lemmas~\ref{lem:cube-ineq}, and~\ref{lem:torus-ineq}.\footnote{The sub-multiplicativity argument originates in the work of Pisier~\cite{Pisier-type-1} on type $1$. The original proof
of the cotype $\infty$ in~\cite{MP-cotype-oo} uses a more complicated argument.}
Indeed,~\eqref{eq:cube} is a natural non-linear analogue of \eqref{eq:en-type}: One can view~\eqref{eq:en-type}
as~\eqref{eq:cube} restricted to \emph{linear} mappings of the cube (and $\hat T$ as a substitute to $Tn^{1-\frac 1p}$).
Variant of~\eqref{eq:cube} was suggested by Enflo~\cite{Enflo} as a non-linear version of type (Ineq.~\eqref{eq:type}), and the (almost) equivalence to \mbox{(en-)type} was proved in~\cite{BMW,Pisier-type,MN-type}.
The connection between en-cotype~\eqref{eq:en-cotype}, and~\eqref{eq:grid} is less apparent: It is shown
in~\cite{MN-cotype-full}%
\footnote{Actually, the paper~\cite{MN-cotype-full} proves a variant of the above statement: the equivalence between cotype and metric cotype. But the arguments are the same.}
that the following metric property is equivalent to en-cotype $q$ in Banach spaces:
There exists $\tilde \Gamma>0$ such that for every $n\in \mathbb N$ there exists $m\in \mathbb N$ such that for every
$f:\mathbb Z_m^n \to H$,
\begin{equation} \label{eq:metric-en-cotype}
\sum_{i=1}^n \Avg_{x\in \mathbb Z_m^n} d_H(f(x),f(x+\tfrac{m}{2} e_j))^2 \le
\tilde \Gamma^2 m^2 n^{1-\frac{2}{q}} \!\!\!\! \Avg_{\varepsilon\in \{0,\pm 1\}^n} \Avg_{x\in \mathbb Z_m^n} d_H(f(x),f(x+\varepsilon))^2.
\end{equation}
Inequality~\eqref{eq:metric-en-cotype} is ``close in spirit" to~\eqref{eq:grid}.
Rigorously, if we change~\eqref{eq:grid} by replacing the ``$n$" in the LHS with ``$n^3$", and the ``$n^2$" on the RHS with ``$n^6$", then it is proved in~\cite{MN-cotype-full} that \eqref{eq:metric-en-cotype} is satisfied for some $q<\infty$ if and only if $\limsup_{n\to \infty} \Gamma_n(H)<1$ (where we $\Gamma_n(H)$ is defined according to the modifications of~\eqref{eq:grid} we have just suggested).
\section{Tree metrics}
The class of finite tree metrics does not have the dichotomy property.
\begin{theorem} \cite{MN-trees} \label{thm:dich-trees}
For any $B> 4$, there exists a metric space $H$ such that
$\sup_T c_{H}(T)=B$, where $T$ ranges over the finite tree metrics.
\end{theorem}
\begin{comment}
From the perspective of Banach space theory this result is somewhat surprising, because when the
host space $H$ is a Banach space then there is a dichotomy:
\begin{itemize}
\item Either $H$ is isomorphic to a uniformly convex space, i.e., super-reflexive, and then by~\cite{Pisier??} this space has modulus of uniform convexity of the form $\varepsilon^p$, which implies by a theorem of Bourgain~\cite{Bourgain-trees} that
$c_H(B_n)\ge \Omega((\log n)^{1/p})$,
\item Or $H$ is not super-reflexive. In this case Bourgain has shown that $H$ contains bi-Lipschitz all finite trees, but his proof actually give distortion $1+\varepsilon$, for every $\varepsilon>0$. \mnote{TO CHECK, AND ADD REFERENCE TO JAMES}
\end{itemize}
\end{comment}
It is natural to ask whether there is a dichotomy between constant distortions and say $\log^* N$ distortions. We believe no such dichotomy exists. For complete binary trees we can prove:
\begin{theorem} \cite{MN-trees} \label{thm:dich-binary-trees}
For any $\delta\in(0,0.001)$, and
for any sequence $s(n)$ satisfying (i) $s(n)$ is non decreasing; (ii) $s(n)/n$ is non-increasing
(iii) $4<s(n) \le O(\delta \log n / \log \log n)$, there exists a metric space $H$ and $n_0$ such that for every $n\ge n_0$,
$(1-\delta) s(n) \le c_H(B_n) \le s(n)$. Here $B_n$ is the (metric on) unweighted complete binary tree of depth $n$.
\end{theorem}
\begin{question}
Is there an extension of Theorem~\ref{thm:dich-binary-trees} to all tree metrics, with dependence on the size of the metric? For example
``For every $\delta>0$, and $s(n)$ as in Theorem~\ref{thm:dich-binary-trees}, there exists a metric space $H$ such that for every tree metric $T$,
$(1-\delta) s(\log N) \le D_N(H,\text{trees}) \le s(\log N)$".
\end{question}
Theorem~\ref{thm:dich-trees} is a corollary of Theorem~\ref{thm:dich-binary-trees}, when substituting $s(n)=B$, and using
the fact that the complete binary trees are ``dense" in the finite tree metrics, in the sense of Prop.~\ref{prop:dense}.
The rest of the section is devoted to a non-quantitative sketch of the argument in the proof of Theorem~\ref{thm:dich-trees}.
The more complicated (and complete) proof of Theorem~\ref{thm:dich-binary-trees} can be found in~\cite{MN-trees}.
\begin{figure}[ht]
\centering
\includegraphics[scale=0.6]{d_e-metric}
\caption{The $d_\eta$ metric on $B_\infty$}
\label{fig:d_eta}
\end{figure}
Denote $\eta=1/B$. To define $H_\eta$, consider
the infinite binary tree $B_\infty$ with the tree metric on it, and contract the ``horizontal'' distances by a factor of $B$.
More precisely
Let $h(x)$ be the depth of $x\in B_\infty$, i.e. the distance from the root of $B_\infty$.
Assuming $h(y)\ge h(x)$, the distance
$d_\eta(x,y)$, for $x,y\in B_\infty$ is defined as
\begin{equation} d_\eta(x,y)=h(y)-h(x)+ 2(h(x)-h(\lca(x,y))\cdot \eta. \end{equation}
It is not hard to check that:
\begin{proposition}
\begin{enumerate}
\item $d_\eta$ is a metric on $B_\infty$.
\item $c_{d_\eta}(B_\infty)\le \eta^{-1}$. Indeed the identity mapping does not expand distances, and contracts
them by factor of at most $1/\eta$.
\end{enumerate}
\end{proposition}
Thus $H_\eta=(B_\infty,d_\eta)$ is our candidate host space for proving Theorem~\ref{thm:dich-trees}. We are left to show that
$\lim_{n\to \infty}c_{H_\eta}(B_n)=\eta^{-1}$.
Our approach follows {Matou\v{s}ek}'s proof~\cite{Mat-trees} of:
\begin{theorem} \label{thm:trees-in-l2}
$C_{L_2}(B_n)\ge \Omega(\sqrt{\log n})$.
\end{theorem}
Bourgain~\cite{Bourgain-trees} proved Theorem~\ref{thm:trees-in-l2} first,
and there are subsequent proofs~\cite{LS-trees,LNP-markov-convex}.
For our purpose, {Matou\v{s}ek}'s argument seems the most appropriate,
we therefore outline his proof of Theorem~\ref{thm:trees-in-l2}.
\begin{figure}[ht]
\centering
\includegraphics[scale=0.9]{fork}
\caption{A fork $(x,y,z,w)$}
\label{fig:fork}
\end{figure}
A $\delta$-fork is a quadruple $(x,y,z,w)$
such that both $(x,y,z)$,
and $(x,y,w)$ are $1+\delta$ equivalent to the metric $(0,1,2)$ (where $x$ is mapped to $0$ and $y$ is mapped to $1$).
It is not hard to see that in Hilbert space (and more generally, $2$-uniform convex spaces), if $(x,y,z,w)$ is a $\delta$-fork then $\|z-w\|\le O(\sqrt{\delta}) \|x-y\| $.
{Matou\v{s}ek}'s approach is to assume toward a contradiction that there exists a
Lipschitz embedding $f:B_n \to L_2$ such that $\dist(f)\le c \sqrt{\log n}$,
and use this assumption to find a 3-leaf star $(x,y,z,w)$ in $B_n$ whose center is $y$ such that
$(f(x),f(y),f(z),f(w))$ is a $\delta$ fork for $\delta\approx 1/\log n$.
This implies a large contraction of the distance between $z$ and $w$, which is a contradiction to the assumed upper bound on the distortion.
Consider the first part of {Matou\v{s}ek}'s proof: Finding a star in $B_n$ whose image is $\delta$-fork. It is proved along the following lines:
Call a metric embedding $f:B_n\to H$, $A$-vertically faithful if $\|f\|_{\mathrm{Lip}}\le 1$, and for every $x,y\in B_n$ in which $x$ is an ancestor of $y$, $d_X(f(x),f(y))\ge d_{B_n}(x,y)/A$.
It turns out (as proved by {Matou\v{s}ek}) that when considering only the vertical distances in $B_n$, this class has the BD-Ramsey property, or
the dichotomy property. In other words:
\begin{lemma} \label{lem:Bn-BD-Ramsey}
For every $t\in \mathbb N$, $\delta>0$, and $A>1$, there exists $n=n(t,\delta,A)$, such that
for any host space $H$, and $A$-vertically faithful embedding $f:B_n\to H$,
there exists a subset $C\subset B_n$ which is isometric to $B_t$, and $f(C)$ is $1+\delta$-vertically faithful to $B_t$.
\end{lemma}
Note that for $t=2$, $f(C)$ contains a copy of $\delta$-fork (actually, two copies). We should also mention that,
not surprisingly, the (simple) proof of Lemma~\ref{lem:Bn-BD-Ramsey} uses the BD-Ramsey property of the path metrics as proved in Section~\ref{sec:line}.
Since the part of finding a $\delta$-fork is independent of the range of the embedding, it makes sense to use it on embedding into $(B_\infty,d_\eta)$.
Examining possible $\delta$-forks $(x,y,z,w)$ inside $H_\eta$,
the two configurations in Fig.~\ref{fig:tip-contract} contracts the distance between $z$ and $w$ by at least $1/(O(\delta)+\eta)$ factor, which
is what we are looking for.
\begin{figure}[ht]
\centering
\includegraphics[scale=0.7]{tip-contract}
\caption{\emph{Forks $(x,y,z,w)$ with the distance between the prongs ($z$ and $w$) contracted}. On the right $x$ is an ancestor of the forking point $y$, which is an ancestor to the prongs $z$ and $w$.}
\label{fig:tip-contract}
\end{figure}
However this is not the whole story! There are other types of $\delta$ forks embedded in $H_\eta$. For example type~$II$ in
Fig.~\ref{fig:fork-types}, can be even made 0-fork, but with very small contraction of the tips. This means that the approach that attempts to show large contraction of $\delta$-forks in $H_\eta$ will not work.
There are also other configurations of ``bad" $\delta$-forks, such as type $I$, $III$, and $IV$ in
Fig.~\ref{fig:fork-types}.%
\footnote{A more careful examination reveals that the configurations labeled type $I$, $III$, $IV$ in Fig.~\ref{fig:fork-types} can be
$O(\eta)$-fork at best. Hence, by taking $\delta\ll \eta$, we can rule out their existence as $\delta$-forks. This approach, however, will fail to prove the more general result of Theorem~\ref{thm:dich-binary-trees}, in which $\eta$ is no longer a constant.}
\begin{figure}[ht]
\centering
\includegraphics[scale=0.7]{fork-types}
\caption{\emph{Forks in which the distance between the prongs ($z$ and $w$) do not contract}. In type $II$, $x$ is a descendant of the forking point $y$, which is deeper (in $B_\infty$) than the prongs $z$ and $w$.}
\label{fig:fork-types}
\end{figure}
It turns out that the situation is not that bad. The four types of
``bad forks" are the only ones that exist.
\begin{lemma} \label{lem:fork-types}
Every $\delta$-fork in $H_\eta$ is close to one of the 6 types of forks in Figures~\ref{fig:tip-contract}
and~\ref{fig:fork-types}, up to distortion of $1+O(\delta)$.
\end{lemma}
The proof of Lemma~\ref{lem:fork-types} is a tedious and contains long case analysises (not to mention the need to properly define
the configurations in Figures~\ref{fig:tip-contract}
and~\ref{fig:fork-types}). But having it,
it is reasonable to assume that a slight
generalization of the tip contraction argument for $\delta$-fork would be true in $H_\eta$. Indeed, we show that
\begin{lemma} \label{lem:B_4-nonembed}
Any $1+\delta$ vertically faithful embedding of $B_4$ in $H_\eta$, must
have distortion at least $1/(O(\delta)+\eta)$.
\end{lemma}
Notice that this lemma is sufficient to prove a lower bound
on the distortion of $B_n$ in $H_\eta$, by using
Lemma~\ref{lem:Bn-BD-Ramsey} with $t=4$.
In order to prove Lemma~\ref{lem:B_4-nonembed}, we view $1+\delta$ vertically faithful embedding of $B_4$
as a collection of $\delta$-forks ``glued" together in prong-to-handle fashion.
For this purpose,
it is helpful to analyze what are the possible configurations of $1+\delta$
embedding of 4-point paths, $\{0,1,2,3\}$, in $H_\eta$. There are
essentially only three different configurations, as depicted in Fig.~\ref{fig:3path-types}.
\begin{figure}[ht]
\centering
\includegraphics[scale=0.7]{3path-types}
\caption{\emph{The possible configurations of 4-point path $(x_0,x_1,x_2,x_3)$.} In type $B$, for example,
$x_1$ is an ancestor of $x_0$,
$x_3$ is an ancestor of $x_2$, and $x_1$ and $x_2$ have the same depth in $B_\infty$.}
\label{fig:3path-types}
\end{figure}
At this point we can do a ``syntactic" case analysis of how
the $\delta$-forks of Fig.~\ref{fig:fork-types} can be glued together into $1+\delta$ vertically faithful
embedding of $B_4$, using the ``rules" enforced by the configuration of 4-point paths
described in Fig.~\ref{fig:3path-types}. Doing this lead to the inevitable conclusion that a fork of a type described in Fig.~\ref{fig:tip-contract}
must appear in the embedding of $B_4$, leading to the conclusion that
the embedding of $B_4$ must have a large contraction, and hence a large distortion. \hfill \qed
\subsection*{Acknowledgments}
This work was supported by an Israel Science Foundation (ISF)
grant no.~221/07, and a US-Israel Bi-national Science Foundation (BSF) grant no.~2006009.
The author thanks Assaf Naor for his help in assimilating the subject while collaborating on the papers~\cite{MN-cotype-full,MN-trees}. He also thanks Assaf Naor and the anonymous referee for commenting
on an earlier version of these notes, which helped improving the presentation.
The figures in these notes are adapted from~\cite{MN-trees}.
Finally, the author wish to thank the organizers of
the ICMS ``Geometry and Algorithms workshop", (Edinburgh, 4/2007) and
the organizers of ``Limits of graphs in group theory and computer science semester" in the Bernoulli Center (Lausanne 5/2007) for inviting him to give a talk on which these notes are based.
\bibliographystyle{plain}
| {
"timestamp": "2008-04-10T06:19:24",
"yymm": "0710",
"arxiv_id": "0710.1994",
"language": "en",
"url": "https://arxiv.org/abs/0710.1994",
"abstract": "These are notes from talks given at ICMS, Edinburgh, 4/2007 (\"Geometry and Algorithms workshop\") and at Bernoulli Center, Lausanne 5/2007 (\"Limits of graphs in group theory and computer science\"). We survey the following type of dichotomies exhibited by certain classes X of finite metric spaces: For every host space H, either all metrics in X embed almost isometrically in H, or the distortion of embedding some metrics of X in H is unbounded.",
"subjects": "Metric Geometry (math.MG)",
"title": "Metric Dichotomies",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9728307676766118,
"lm_q2_score": 0.7279754548076477,
"lm_q1q2_score": 0.7081969205502545
} |
https://arxiv.org/abs/2206.04725 | Persistent Homology with k-nearest-neighbor Filtrations reveals Topological Convergence of PageRank | Graph-based representations of point-cloud data are widely used in data science and machine learning, including epsilon-graphs that contain edges between pairs of data points that are nearer than epsilon and kNN-graphs that connect each point to its k-nearest neighbors. Recently, topological data analysis has emerged as a family of mathematical and computational techniques to investigate topological features of data using simplicial complexes. These are a higher-order generalization of graphs and many techniques such as Vietoris-Rips (VR) filtrations are also parameterized by a distance epsilon. Here, we develop kNN complexes as a generalization of kNN graphs, leading to kNN-based persistent homology techniques for which we develop stability and convergence results. We apply this technique to characterize the convergence properties PageRank, highlighting how the perspective of discrete topology complements traditional geometrical-based analyses of convergence. Specifically, we show that convergence of relative positions (i.e., ranks) is captured by kNN persistent homology, whereas persistent homology with VR filtrations coincides with vector-norm convergence. Beyond PageRank, kNN-based persistent homology is expected to be useful to other data-science applications in which the relative positioning of data points is more important than their precise locations. | \section*{Acknowledgments} located above the reference section.
\thanks{The first author is supported by NSF grant xx-xxxx}
\begin{document}
\maketitle
\centerline{\scshape Minh Quang Le$^*$ and Dane Taylor$^*$}
\medskip
{\footnotesize
\centerline{Department of Mathematics}
\centerline{ University at Buffalo, State University of New York }
\centerline{USA}
}
\medskip
\bigskip
\centerline{(Communicated by the associate editor name)}
\begin{abstract}
Graph-based representations of point-cloud data are widely used in data science and machine learning, including $\epsilon$-graphs that contain edges between pairs of data points that are nearer than $\epsilon$ and kNN-graphs that connect each point to its $k$ nearest neighbors.
Recently, topological data analysis has emerged as a family of mathematical and computational techniques to investigate topological features of data using simplicial complexes. These are a higher-order generalization of graphs and many techniques such as Vietoris-Rips (VR) filtrations are also parameterized by a distance $\epsilon$.
Here, we develop kNN complexes as a generalization of kNN graphs, leading to kNN-based persistent homology techniques for which we develop stability and convergence results.
We apply this technique to characterize the convergence properties PageRank, highlighting how the perspective of discrete topology complements traditional geometrical-based analyses of convergence. Specifically, we show that convergence of relative positions (i.e., ranks) is captured by kNN persistent homology, whereas persistent homology with VR filtrations coincides with vector-norm convergence.
Beyond PageRank, kNN-based persistent homology is expected to be useful to other data-science applications in which the relative positioning of data points is more important than their precise locations.
\end{abstract}
\section{Introduction}
Topological data analysis (TDA) is a rapidly growing field of applied mathematics in which techniques from computational and applied topology are applied to extract structural information about the ``shape'' of data. TDA has been applied to numerous contexts ranging from
visualization and dimensionality reduction \cite{carlsson2009topology,kusano2016persistence} and time series analyses \cite{perea2015sliding} to applications in
cosmology \cite{sousbie2011persistent,weygaert2011alpha},
physical processes over networks
\cite{le2021persistent,taylor2015topological,kramar2013persistence,kondic2012topology}, neuroscience \cite{petri2014homological,giusti2015clique,chung2009persistence,nielson2017uncovering}, and systems biology \cite{rucco2014using,liang1998analytical,kasson2007persistent,ichinomiya2020protein}.
One of the main tools is the study of persistent homology, which can effectively reveal multiscale topological properties of data. This approach relies on examining how the homology of a topological space evolves as one applies a filtration, which is one of the basic notions in topology. There are many types of filtrations \cite{aktas2019persistence,rieck2017clique,huang2016persistent,chowdhury2018functorial,gasparovic2018relationship}, however we will focus on the widely used Vietoris-Rips (VR) filtration \cite{edelsbrunner2008persistent}.
In general, different filtrations reveal complementary insights, and it is important to develop additional filtrations that cater to different applications.
\begin{figure}[t]
\begin{center}
\includegraphics[width=\linewidth]{figures/minh_fig1.pdf}
\caption{
Visualizations of simplicial complexes and graphs resulting from (A) a Vietoris-Rips (VR) filtration and (B) our proposed k-nearest-neighbor (kNN) filtration. As shown in the second column, VR filtrations are parameterized by the radius $\epsilon$ of $\epsilon$-balls that are centered at the points, whereas kNN filtrations are parameterized by the number $k$ of nearest neighbors. (Dotted lines depict the nearest-neighbor orderings for node $i=4$.) The third column depicts simplicial complexes that are obtained at some $\epsilon$ and $k$. The fourth columns shows their 1-skeletons, which are graphs in which $k$-simplices of $k>1$ are discarded.
}
\label{fig:fig1}
\end{center}
\end{figure}
In the prototypical setting, one aims to construct and study empirical topological features for a set of data points, or \emph{point cloud}. The approach involves using points to construct a filtered topological space, which is often represented by a simplicial complex in which the data points are 0-simplices, and higher-dimensional $k$-simplices are constructed via some set of rules. Often, $k$-simplices are added according to the pairwise distances between 0-simplices.
As an example, in Fig.~~\ref{fig:fig1}(A), we visualize one of the most commonly studied point-cloud filtrations, the Vietoris-Rips (VR) filtration. As shown, VR filtrations are constructed by considering simplicial complexes in which k-simplices are added between 0-simplices that are less that $\epsilon>0$ distance apart.
In Fig.~~\ref{fig:fig1}(B), we illustrate a different filtration that we develop in this paper: \emph{kNN filtrations}, whereby $k$-simplices are created according to $k$-nearest-neighbor sets. We note that while $\epsilon$ and kNN graphs are both very prevalent in the data-science and machine-learning literatures, TDA methods largely focus on $\epsilon$-based simplicial complexes and filtrations, thereby limiting their potential utility to new applications.
We develop kNN-based simplicial complexes as a generalization of kNN graphs, allowing us to develop kNN-based TDA methods including kNN persistent homology.
We formulate and study persistent homology under kNN filtrations, constructing persistence diagrams and studying their robustness properties. We also define and study several local version of kNN complexes, filtrations, and persistent homology. By constructing filtrations using discrete sets, our approach relies on discrete topology, thereby contrasting approaches that are tied to continuous topological spaces \cite{kim2019homotopy}.
In addition, we further analyze and explore homological features for converging sequences of point sets, exploring how the convergence of persistent diagrams for kNN filtrations contrasts that for VR filtrations.
We apply kNN persistent homology to study the ranking of nodes in graphs, and in particular, we study the convergence of rankings given by approximate PageRank values that are obtained by the power iteration method.
Our work establishes a new connection between TDA and PageRank, and provides a topological approach for determining how many iterations are required for the node rankings to convergence.
We visualize this application and motivation in Fig.~\ref{Fig:graphPR}. In this application, approximate PageRank values $x_i(t)\to\pi_i$ asymptotically converge to their final values and the normed error $||{\bf x}(t) -\bm{\pi}||$ exponentially decays. However, as highlighted in Table \ref{table:student}, the orderings (i.e., node ranks) according to ${ x}_i(t)$ values converge after only $t=9$ iterations, even though $||{\bf x}(t) -\bm{\pi}||>0$ for all $t$.
We apply kNN-based persistent homology to study of convergence for these relative orderings, which is a property that is crucial to PageRank and which is not revealed through VR filtrations.
Although we focus here on PageRank, we expect kNN-based TDA to be widely useful to other applications in which the relative positioning of points---as opposed to the precise locations---is a property of main interest.
\begin{figure}[t]
\begin{center}
\includegraphics[width=.42\linewidth]{figures/grapht.pdf}
\includegraphics[width=.5\linewidth]{figures/thes.pdf}
\caption{
(left)~An example graph.
(right)~Convergence of approximate PageRank values $x_i(t)\to\pi_i$ with $t$ iterations.
}
\label{Fig:graphPR}
\end{center}
\end{figure}
\begin{table}[t]
\centering
{\tiny{
\begin{tabular}{ | l | r | r | r | r | r | r | r | r | r | r | r |}
\hline
Nodes & $R_i(0)$ & $R_i(1)$ & $R_i(2)$ & $R_i(3)$ & $R_i(4)$ & $R_i(5)$ &$R_i(6)$& $R_i(7)$&$R_i(8)$&$R_i(9)$& ${R}_i(\infty)$ \\ \hline \hline
$i=0$ & 5 & 1 & 4 & 4 & 4 & 4 & 4 & 4 & 4 &4 &4 \\ \hline
$i=1$ & 4 & 4 & 1 & 3 & 3 & 2 & 2 & 3 & 3 &2 &2 \\ \hline
$i=2$ & 3 & 3 & 3 & 1 & 2 & 3 & 3 & 2 & 2 &3 &3 \\ \hline
$i=3$ & 2 & 5 & 5 & 5 & 5 & 5 & 5 & 5 & 5 &5 &5 \\ \hline
$i=4$ & 1 & 2 & 2 & 2 & 1 & 1 & 1 & 1 & 1 &1 &1 \\ \hline
\end{tabular}
\caption{The relative orderings (i.e., node ranks $R_i(t)$) converge at $t\ge 9$.
For each $t$, we indicate the top-ranked node by $R_i(t)=1$.
}}}
\label{table:student}
\end{table}
This paper is organized as follows.
We provide background information in Sec.~\ref{sec:prior} and our main findings in Sec.~\ref{sec:main}.
In Sec.~\ref{sec:PagerankandRD}, we apply this approach to study convergence for the PageRank algorithm. We provide a discussion in Sec.~\ref{sec:discusion}.
\section{Background information}\label{sec:prior}
In Sec.~\ref{sec:simplical}, we discuss simplicial complexes that are derived from point clouds.
In Sec.~\ref{sec:holomology}, we define homology and persistent homology for simplicial complexes.
In Sec.~\ref{sec:stability}, we discuss the notions of stability and convergence for persistence diagrams.
\subsection{Simplicial complexes derived from point clouds} \label{sec:simplical}
\subsubsection{Simplicial complexes}
We will define simplicial complexes in a geometric way by considering a set of points, or ``point cloud'' \cite{schaub2018random}.
Consider a finite set $ \mathcal{Y}=\{y_i \} \in\mathbb{R}^p$ of $N$ points (which we enumerate $i\in\mathcal{V}=\{1,\dots,N\}$) in a $p$-dimensional space that we equip with the Euclidean metric.
\begin{definition}[Euclidean Metric Space]
\label{def:Euclidean}
Let $\mathbb{R}^p$ be the set of all ordered $p$-tuples, or vectors, over the real numbers and $d:\mathbb{R}^p\times \mathbb{R}^p$ be the Euclidean metric
$$d(x,y)=||x-y||_2=\sqrt{\sum_{i=1}^p(x_i-y_i)^2},$$
where $x=(x_1,..,x_p), y=(y_1,..,y_p) \in \mathbb{R}^p$.
Then the Euclidean metric space is given by $(\mathbb{R}^p,d)$.
\end{definition}
To facilitate later discussion, it is also helpful to define the following distance-related concepts.
\begin{definition}[Euclidean Ball, or $\epsilon$-ball]
\label{def:EucBal}
A $p$-dimensional Euclidean ball $\mathcal{B}_\epsilon(x)$ centered at $x$ with radius $\epsilon$ is defined by
$$\mathcal{B}_\epsilon(x)=\{y\in \mathbb{R}^p:d(x,y) \le \epsilon\}$$
\end{definition}
\begin{definition}[Pairwise-Distance Map]
\label{def:NNOF}
Let $\mathcal{Y} = \{y_i \}_{i\in\mathcal{V}} \subset \mathbb{R}^p$ be a finite set of $N$ points in a Euclidean metric space $(\mathbb{R}^p,d)$.
Then we define the ``pairwise-distance map''
$f:\mathcal{Y} \to \mathbb{R}_+^{N\times N}$
to encode the distances between all pairs of points so that
$f_{ij}(\mathcal{Y}) = d(y_i,y_j) $.
We similarly define a map $f_i:\mathcal{Y}\to \mathbb{R}_+^N$ for each row $i$ of matrix $[f_{ij}]$ by $f_i:\mathcal{Y}\to \mathbb{R}_+^N$ so that $[f_i(\mathcal{Y})]_j = f_{ij}(\mathcal{Y})$.
\end{definition}
Given a point-cloud, we define a simplicial complex as a collection of $k$-simplices involving a set $ \mathcal{V}=\{1,\dots,N\}$ of vertices with locations $\mathcal{Y}=\{y_i \}$.
A $k$-dimensional simplex---or simply \textbf{k-simplex}---$\mathcal{S}^k$ is defined by a subset of $\mathcal{V}$ having cardinality $k+1$. For example, each 0-simplex is a vertex, each 1-simplex is an edge, 2-simplices are informally ``triangles'' that must be defined using $k+1=3$ vertices. Because the vertices are spatially embedded in $\mathbb{R}^p$, each $k$-simplex is a $k$-dimensional geometrical object. It is then natural to consider their $(k-1)$-dimensional faces, which can be defined as follows. A face of a $k$-simplex $S^k$ is a subset of $S^k$ with cardinality $k$, i.e , with one of the elements of $S^k$ omitted. If $S^{k-1}_f$ is a face of simplex $S^k$, then $S^k$ is called $\textit{coface}$ of $S^{k-1}_f$. For example, the 0-simplices at the start and end of a 1-simplex are its faces, and the set of 1-simplices that are adjacent to a 0-simplex are its cofaces.
A simplicial complex $X$ is a collection of $k$-simplices with the property that if $S^k \in X$, then all the faces of $S^k$ are also members of $X$.
The notions of face and coface can also be extended to \textit{abstract simplicial complexes} that lack spatial coordinates, and one can also define connections between $k$-simplices of the same dimension by considering if their faces and cofaces overlap. Two $k$-simplices $S^k_i$ and $S^k_j$ are said to be lower adjacent if they have a common face, and they are upper adjacent if they are both faces of a common $(k + 1)$-simplex. For any $S^k \subset X$ we define its degree, denote by $deg(S^k)$, to be the number of cofaces of $S^k$. We use $X^k$ to denote the subset of $k$-simplices in $X$.
Note that a graph, while typically defined via two sets (vertices, edges), can also be interpreted as a 1-dimensional abstract simplicial complex, since it contains $k$-simplices of dimension $k \leq 1$. Moreover, for any simplicial complex, one can obtain an associated graph that is its 1-skeleton, whereby one discards any $k$-simplices of dimension $k>1$. More generally, a $\kappa$-skeleton of a simplicial complex $X$ can be obtained by discarding all $k$-simplices of dimension $k>\kappa$.
Thus, a simplicial complex can be understood as a generalization of a graph that allows for higher-order relationships between vertices. To emphasize this connection, we will make no distinction between 1-simplices in a simplical complex $X$ and the edges of a graph, and we will refer to them interchangeably.
\subsubsection{Two simplicial complexes based on $\epsilon$}
There are many ways in which one can construct a set of simplices involving the vertices $\mathcal{V}$ associated with a given a set of points $ \mathcal{Y}=\{y_i \} \in\mathbb{R}^p$. Most approaches stem from considering $\epsilon$-balls centered at the points $\mathcal{Y}$.
Here, we present two closely related simplicial complexes that are parameterized by a distance threshold $\epsilon$ and are often motivated by the assumption that the point cloud lies on a low-dimensional manifold.
\begin{definition}[\v Cech Complex \cite{bubenik2015statistical}]\label{def: Cech complex}
Given a collection of points $ \mathcal{Y}=\{y_i \}_{i=1}^N \in \mathbb{R}^p$, the {\v C}ech complex, $\mathcal{C}_\epsilon$, is the abstract simplicial complex whose $k$-simplices are determined by the unordered $(k + 1)$-tuples of points $\{y_i\}_{i=1}^{k+1}$ whose closed $(\epsilon/2)$-ball neighborhoods have a point of common intersection.
\end{definition}
The $\textit{Vietoris-Rips (VR) complex}$ is closely related and is defined as follows.
\begin{definition}[Vietoris-Rips Complex {\cite{attali2013vietoris}}]
\label{def:Vietoris-Rips complex}
Given a collection of points $\mathcal{Y}=\{y_i \}_{i=1}^N\in \mathbb{R}^p$, the Rips complex, $\mathcal{R}_\epsilon$, is the abstract simplicial complex whose $k$-simplices correspond to unordered $(k + 1)$-tuples of points $C = \{y_i\}^{k+1}_{i=1} \in \mathcal{R}_\epsilon$ whose pairwise distance satisfy $|| y_i-y_j||_2 \le \epsilon $ for any $y_i,y_j\in C$.
\end{definition}
Notably, \v Cech complexes are sometimes preferred over VR complexes because of their relation to the a continuous topological space that is associated with unions of $\epsilon$-balls, as established in the following Theorem.
\begin{theorem}[\v Cech theorem {\cite{kim2019homotopy}}]
The $\textbf{\v Cech theorem}$ (or, equivalently, the “Nerve theorem”) states that a {\v Cech} complex $\mathcal{C}_\epsilon$ has the homotopy type of the union of closed $(\epsilon/2)$-balls about the point set $\mathcal{Y}=\{y_i \}$.
\end{theorem}
This result is somewhat surprising since a {\v Cech complex} $\mathcal{C}_\epsilon$ is an abstract simplicial complex, which is a discrete topological space of potentially high dimension. In contrast, the union of closed $(\epsilon/2)$-balls is a continuous topological space (i.e., subset of $\mathbb{R}^p$), and so it is interesting that these two spaces have the same homotopy type. While an identical relation has not been identified for VR complexes, it has been shown that they are very closely related {\v Cech complexes} through the following interleaving result.
\begin{lemma}[{Relation between \v Cech complex and Vietoris-Rips complex \cite{kim2019homotopy} }]
For any $\epsilon > 0$, there is a chain of inclusion maps for {\v Cech} complexes and VR complexes
\[
\mathcal{R}_\epsilon \subseteq \mathcal{C}_{\epsilon \sqrt{2}} \subseteq \mathcal{R}_{\epsilon \sqrt{2}}.
\]
\end{lemma}
From a practical perspective, applications involving TDA often focus on VR complexes because they are more computationally efficient to compute than
{\v Cech complexes}. That is, it is easier to compute whether the distance between two points is less than $\epsilon$ versus check whether the intersection of $\epsilon$-balls is nonempty. As such, TDA methods based on VR complexes are very popular in applied settings, and we will later conduct numerical experiments that use them as a baseline for comparison. Given the relations between VR and {\v Cech complexes} and the unions of $\epsilon$-balls, these all may approximate the structure of a manifold, assuming all data points lie on the manifold. Such an assumption, however, is not appropriate for every data set.
Before continuing, we note that simplicial complexes can be constructed in ways that do not require a set of points. For example, one can construct abstract simplicial complexes based on a given graph.
\begin{definition}[Clique Complex \cite{zomorodian2010fast}] \label{def:Clique_complex}
The $\textbf{clique complex}$ $Cl(G)$ of an undirected graph $G=(\mathcal{V},\mathcal{E})$ is a simplicial complex where $\mathcal{V}$ are vertices of $G$ and each $k$-clique (i.e. a complete subgraph with $k$ vertices) in $G$ corresponds to a $(k-1)$-simplex in $Cl (G)$.
More precisely, it is the simplicial complex
\[
Cl(G) =\mathcal{V} \cup \mathcal{E} \cup \left\{\sigma | \binom{\sigma}{2} \subseteq \mathcal{E}\right\},
\]
where $\sigma \subset \mathcal{V} $ denotes a simplex
and $\binom{\sigma}{2} \in \mathcal{V}\times \mathcal{V}$ denotes the set of pairs that can be obtained by selecting two 0-simplices from $\sigma$ (which must be an edge in the graph).
\end{definition}
It is worth noting that the \emph{1-skeleton} (i.e., 0-simplices and 1-simplices) of any clique complex recovers its associated graph $G=(\mathcal{V},\mathcal{E})$. That is, the construction of a graph's clique complex is an invertible transformation.
\subsection{Homology and persistent homology}
\label{sec:holomology}
After representing a data set by a simplicial complex, one can study its topological structure via its homology. To obtain multiscale insights, it is also useful to study how that homology changes and persists as one varies a parameter, such as a distance threshold $\epsilon$.
\subsubsection{Homology}
We formulate homology by considering functions defined over $k$-simplices within a simplicial complex.
A chain complex over a field $\mathbf{F} $ is a tuple $(\mathcal{C}; \partial)$ where $\mathcal{C}$ is a collection $\{\mathcal{C}_{k}\}_{k\in \mathbf{N}}$ of vector spaces together with a collection $\partial$ of $\textit{F}$-linear maps $\{\partial_k: C_k \longrightarrow C_{k-1}\}_{k \in \mathbf{N}}$ such that $\partial_{k-1} \circ \partial_k =0$ for all integers $k$. The maps $\partial_k$ are called boundary maps. The $k$-cycles of the complex are the elements that are sent to zero by the map $\partial_k$; the $k$-boundaries are the elements in the image of $\partial_{k+1}$. A map of chain complexes $f : (C; \partial) \longrightarrow (C^\prime; \partial^\prime)$ is a collection $\{f_k: C_k \longrightarrow C_{k^\prime}\}_{k \in \mathbf{N}}$ of $\mathbf{F}$-linear map such that $f_{k-1} \circ \partial_k = \partial
_k^{\prime} \circ f_k$ for all natural numbers $k$ .
The $k$-cycles form a vector space,
and so do the $k$-boundaries; we denote these vector spaces by $Z_k$ and $B_k$, respectively. The $k$th homology of a chain complex $(C, \partial)$ over a field $\mathbf{F}$ is the quotient vector space
$$H_k((C, \partial)) = Z_k/B_k.$$
The number
\[
\beta_k (C) \equiv \textsf{dim}(H_k(C,\partial))=\textsf{dim} Z_k- \textsf{dim} B_k
\]
is called the $k$-th Betti number. For a geometric intuition, the dimension of $H_k(X)$ can be thought of as the number of `$k$-dimensional holes' of X:
\begin{itemize}
\item The 0th Betti number $\beta_0$ is the number of connected components.
\item The 1st Betti number $\beta_1$ counts the number of loops, or 1-cycles.
\item The 2nd Betti number $\beta_2$ counts the number of voids, or 2-cycles.
\end{itemize}
The dimension of a simplicial complex is the maximum over the dimensions of its simplices. If $X$ is a simplicial complex of dimension $d$ then $H_k(X) = 0$ for all $k \geq d.$
Any map of chain complexes $\Xi : (C, \partial) \longrightarrow (C^\prime; \partial^\prime)$ induces a linear map on homology:
\[
H(\Xi): H_k((C,\partial)) \longrightarrow H_k((C^\prime,\partial^\prime))
\]
To simplify notation, we later use $\Xi$ in place of $H(\Xi)$.
\subsubsection{Filtrations}
To discuss how homology changes and persists, it is necessary to define a filtration for simplicial complexes.
Given a simplicial complex $X$, a $\textit{filtration}$ is a totally ordered set of subcomplexes $X_i$ of $X$, indexed by the nonnegative integers, such that if $i \leq j$ then $X_i \subset X_j$ \cite{wang2012basic} (or equivalently a sequence of simplicial complexes $ X_1,...,X_l$ such that $X_1 \subseteq \dots \subseteq X_l = X$). The totally ordering itself is called a $\textit{filter}$. We call a simplicial complex together with a filtration a $ \textit{filtered simplicial complex}$.
There are different ways to define a filtration \cite{aktas2019persistence}, and we will focus herein on
the popular $\textit{Vietoris-Rips filtration}$.
\begin{definition}[Vietoris-Rips Filtration \cite{aktas2019persistence}]\label{VR filtration}
Let $G=({V},{E})$ be an undirected graph
in which each vertex $i\in\mathcal{V}$ has a position $x_i\in \mathcal{R}^p$ in a metric space with metric $d$.
Consider a weight function $W: \mathcal{V} \times \mathcal{V} \longrightarrow \mathbb{R}$ defined on the edges $\mathcal{E}$ so that $W(i,j)=d(x_i,x_j)$ encodes the distance between $x_i$ and $x_j$. Let $Cl(G)$ be the associated clique complex for $G(\mathcal{V},\mathcal{E})$. For any $\delta \in \mathbb{R}$, the 1-skeleton $G_\delta = (V,E_\delta)$ is defined as the subgraph of $G$ where $E_\delta \subseteq E$ includes only the edges such that $d(x_i,x_j)\le \delta$. Then, for any $\delta \in \mathbb{R}$, we define the $\textit{Vietoris-Rips filtration}$ by
\[
\{Cl(G_\delta) \longrightarrow Cl(G_{\delta^ \prime})\}_{0\leq \delta \leq \delta^\prime} .
\]
\end{definition}
In this work, we will only consider when the distance function is the Euclidean metric, although one can in principle use other metrics. That said, we note in passing that one can also construct filtrations in a variety of ways including functional metric filtrations \cite{aktas2019persistence}, vertex-based clique filtrations \cite{rieck2017clique}, $k$-clique filtrations \cite{rieck2017clique}, weighted simplex filtrations \cite{huang2016persistent}, vertex function based filtrations \cite{aktas2019persistence}, Dowker sink and source filtration \cite{chowdhury2018functorial}, and the intrinsic \v Cech filtrations \cite{gasparovic2018relationship}.
\subsubsection{Persistent homology}
Persistent homology involves studying how homology changes (or more precisely, identifies when it does not change) during a filtration.
Consider a simplicial complex, for all $i \leq j$ the inclusion maps $X_i \hookrightarrow {X_j}$ induce $\mathbf{F}$-linear maps $\Xi_{i,j} : H_k(X_i) \longrightarrow H_k(X_j)$ on simplicial homology. For a generator $x \in H_k(X_i)$ with $x \neq 0$, we say that $x$ dies in $H_k(X_j)$ if $j > i$ is the smallest index for which $\Xi_{i,j}(x) = 0$.
We similarly say that $ x \in H_k(X_i)$ is born in $H_k(X_i)$ if $\Xi^{-1}_{t,i} (x) = 0$ for all $t < i$. We can represent the lifetime of $x$ by the half open interval $[i, j)$. If $\Xi_{i,j}(x) \neq 0$ for all $i < j \leq l$, then we say that $x$ lives forever and we represent its lifetime by the interval $[i;\infty)$.
The $k$-th persistent homology vector spaces of a filtered simplicial complex $X$ are defined as $H_k^{i,j}= \mathbf{img}(\Xi_{i,j})$, and the total $k$th persistent homology of $X$ is defined as $\oplus _{i=1}^l H_k(X_i)$. By the Correspondence Theorem of Persistent Homology \cite{zomorodian2005computing}, for each $k$ we can assign to the total $k$-th persistent homology vector space a finite well-defined collection of half open intervals, i.e., its so called \emph{barcode}. An alternative way to represent persistent homology graphically is given by persistence diagrams, in which case each open interval $[i, j)$ is represented by the point $(i, j)$ in $\mathbb{R}^2$. See Fig.~\ref{fig:SPD} for an example persistence diagram, which we will more formally define in the next section.
\subsection{Stability and convergence of persistence diagrams}\label{sec:stability}
Persistence diagrams and barcodes are widely used as a concise representation for the multiscale homological features of a point-cloud data set. As such, it is important to understand their robustness to perturbations, which can represent data error or noise \cite{cohen2007stability}. One can also study for a converging sequence of point clouds whether their associated persistence diagrams also converge. In this section, we present basic results related to stability and convergence.
\begin{figure}[t!]
\centering
\includegraphics[width=1\linewidth]{figures/stability_1.pdf}
\vspace{-.1cm}
\caption{Example of stability for persistence diagrams resulting from Vietoris-Rips filtrations.
(left) Two point-cloud sets $\mathcal{Y}$ and $\mathcal{Z}$ are close with respect to the $\mathsf{L}_\infty$ norm. The center and right panels depict their associated persistence diagrams $D_{\mathcal{Y}}$ and $D_{\mathcal{Z}}$, which are also close with respect to the bottleneck distance.
}
\label{fig:SPD}
\end{figure}
\subsubsection{Stability}
We first discuss stability in a general setting, which requires several definitions. Let $X$ be a simplicial complex and $g : X \longrightarrow \mathbb{R}$ be a tame function.
\begin{definition}[Tame \cite{cohen2007stability}] \label{def:tame}
A function $g : X \longrightarrow \mathbb{R} $ is tame if it has a finite number of homological critical values and the homology groups $H_k (g^{-1}(-\infty,a))$ are finite-dimensional for all $k \in \mathbb{Z}$ and $a \in \mathbb{R}$.
\end{definition}
In particular, Morse functions on compact manifolds are tame, as well as piece-wise linear functions on finite simplicial complexes and, more generally, Morse functions on compact Whitney-stratified spaces \cite{goresky1988stratified}.
Assuming a fixed integer $k$, we define $G_x = H_k(g^{-1}(-\infty,x])$, and for any $x<y$, we let $g_x^y:G_x \longrightarrow G_y$ be the map induced by inclusion of the sub-level set of x in that of y.
\begin{lemma}[Critical Value Lemma \cite{cohen2007stability}]\label{lemma:critical}
If some closed interval $[x, y]$ contains no homological critical value of $g$, then $g^y_x$ is an isomorphism for every integer $k$.
\end{lemma}
We now formally define a $\textbf{persistence diagram}.$ Using the same notation as above, we write $G_x^y= {\bf img}( g_x^y) $ for the image of $G_x$ in $G_y$. By convention, we set $G_x^y=\{0\}$ whenever $x$ or $y$ is infinite. The group $G_x^y$ is called the persistent homology group.
Let $g:X \longrightarrow \mathbb{R}$ be a tame function and denote $(a_i)_{i=1,...,n}$ as its homological critical values, and let $(b_i)_{i=0..n}$ be an interleaved sequence in which $b_{i-1} < a_i < b_i$ for all $i$. We set $b_{-1} = a_0 = -\infty$ and $b_{n+1} = a_{n+1} = +\infty$. For two integers $0 \leq i < j \leq n + 1$, we define the multiplicity of the pair $(a_i, a_j)$ by:
\[
\mu_i^j=\beta_{b_{i-1}}^{b_j}-\beta_{b_{i}}^{b_j}+\beta_{b_{i}}^{b_{j-1}}-\beta_{b_{i-1}}^{b_{j-1}}
\]
where $\beta_x^y = {\bf{dim}}( F_x^y)$ denotes the persistent Betti numbers for all $-\infty \leq x\leq y \leq +\infty$.
\begin{definition}[Persistence Diagram \cite{cohen2007stability}]\label{def:Persistence diagram}
The persistence diagram $D(\mathcal{A}) \subset \overline{\mathbb{R}}^2$ of a filtered topological space $\mathcal{A}$ is the set of points $(a_i, a_j)$, which are counted with multiplicity $\mu_i^j$ for $0\leq i < j \leq n+1$, along with all points on the diagonal, which are counted with infinite multiplicity.
\end{definition}
Given two persistence diagrams, we can quantify their dissimilarity using various metrics. In particular, we will consider the following metric.
\begin{definition}[Bottleneck Distance \cite{cohen2007stability}] \label{def:Hausdorff and bottleneck}
The
bottleneck distance between two point sets $\mathcal{Y}$ and $\mathcal{Z}$ is given by
\begin{align}
d_B(\mathcal{Y},\mathcal{Z}) &= \inf_{\gamma} \sup_{y} \left\lVert{y-\gamma(y)} \right\rVert_\infty ~,
\end{align}
where $y \in \mathcal{Y}$ and $z \in \mathcal{Z}$ range over all points and $\gamma$ ranges over all bijections from $\mathcal{Y}$ to $\mathcal{Z}$. Here, we interpret each point with multiplicity $k$ as $k$ individual points and the bijection is between the resulting sets.
\end{definition}
Next, we present
triangulable spaces. As in \cite{edelsbrunner1997triangulating}, an \textit{underlying space} of a simplicial complex $K$ is defined as $\bigcup K=\bigcup_{\sigma \in K}\sigma$. If there exists a simplicial complex $K$ such that $\bigcup K$ is homeomorphic to $X$, then $X$ is triangulable.
In Fig.~\ref{fig:SPD}, we show that it two point clouds are similar, then their associated persistence diagrams are similar. That is, persistence diagrams are `stable' with respect to small changes, which can be quantified through the bottleneck distance.
\begin{theorem}[Stability Theorem \cite{bukkuri2021applications} ]\label{thm:global_stability}
Let $X$ be a triangulable space with continuous tame functions $g, h : X \longrightarrow \mathbb{R}$, and let $D_g$ and $D_h$ denote their associated persistence diagrams obtained using a height filtration. Then the bottleneck distance between these persistence diagrams satisfies a global uniform bound
$$d_B(D_g,D_h)\leq \left\lVert {g-h} \right\rVert_\infty .$$
\end{theorem}
While Thm.~\ref{thm:global_stability} describes the stability of persistence diagrams with respect to changes to the functions to which filtrations are applied, one can also use it to equivalently prove the stability of persistence diagrams for when the function changes due to perturbations of points. We present the following example.
\begin{corollary}[Stability of VR Persistence Diagrams to Point Perturbations]\label{cor:point_stability}
Let $\mathcal{Y}=\{y_i\}_{i \in \mathcal{V}}\in\mathbb{R}^d$ with $\mathcal{V}=\{1,...,N\}$ denote a set of $N$ points, and let $\mathcal{Y'}=\{y_i'\}_{i \in \mathcal{V}}\in\mathbb{R}^d$ be a set of perturbed points
such that $||y_i-y_i'||_2 \le \epsilon$ for all $i$.
If
$g: \mathcal{Y} \longrightarrow \mathbb{R}^{n \times n}_+$
denotes the pairwise distance function in which $g_{ij}(\mathcal{Y})=||y_i-y_j||_2$ and $h: \mathcal{Y'} \longrightarrow \mathbb{R}^{n \times n}_+$
with $h_{ij}(\mathcal{Y'})= ||y'_i-y'_j||_2$. It then follows that
$d_B(D_g,D_h) \le 2\epsilon$.
\begin{proof}
\begin{align}
d_B(D_g,D_h) \le ||g-h||_\infty&=\max_{i,j\in\mathcal{V}} |g(y_i,y_j)-h(y'_i,y'_j)|\nonumber\\
&=\max_{i,j\in\mathcal{V}} \left| ||y_i-y_j||_2-||y'_i-y'_j||_2 \right|\nonumber .
\end{align}
However, if we define $y'_i = y_i + e_i$, then $||e_i||_2\le\epsilon$ for any $i\in\mathcal{V}$ and
\begin{align}
||y'_i-y'_j||_2 = ||(y_i-y_j) + (e_i-e_j) ||_2 \le ||(y_i-y_j)||_2 + || (e_i-e_j) ||_2.\nonumber
\end{align}
It follows that
\begin{align}
\left| ||y_i-y_j||_2-||y'_i-y'_j||_2 \right| \le || (e_i-e_j) ||_2 \le 2\epsilon \nonumber ,
\end{align}
which subsequently also bounds $d_B(D_g,D_h) $.
\end{proof}
\end{corollary}
\subsubsection{Convergence}\label{sec:convergence}
The stability of persistence diagrams also has important consequences for the convergence of a sequence of persistence diagrams that is associated with convergent sequence of point clouds.
Focusing on VR filtrations, consider a sequence of point clouds $\mathcal{Y}^m = \{y^m_i\}_{i=1}^N $ of fixed size $N = |\mathcal{Y}^m |$ for each $ m \in \mathbb{N}_{+}$ with $y_i^m \in \mathbb{R}^p$. Assume for each point $i$ that the sequence converges $y_i^m \to y_i$ such that $\lVert{y^m_i-y_i}\rVert_2 \le 1/m$.
Note that such a convergence criterion can be ensured by considering a subsequence in which each subsequent element is chosen so that the bound is true for all $i\in\{1,\dots,N\}$.
Let $D_{f_m}$ and $D_f$ be the persistent diagrams for VR filtrations using the pairwise distance functions $f_m$ and $f$, which are respectively defined using the respective point clouds $\mathcal{Y}^m$ and $\mathcal{Y}$.
The Stability Theorem and Corollary~\ref{cor:point_stability} imply convergence of the associated persistence diagrams since
$$ d_B(D_{f_m},D_f)\leq \left\lVert{f_m-f} \right\rVert_\infty \le 2 \lVert{y^m_i-y_i}\rVert_2 \le \frac{2}{m} ,$$ which converges to $0$ as $m\to\infty$.
\section{Topological data analysis using kNN orderings}\label{sec:main}
We now present our main findings: an approach for persistent homology that is based on the relative positioning of points according to their $k$-nearest-neighbor (kNN) sets. Our approach
introduces kNN complexes and filtrations using a filtration parameter $k\in\mathbb{N}_+$, thereby contrasting VR and other filtrations that use a distance threshold $\epsilon$ as the filtration parameter. We will show that kNN-based persistent homology has certain advantages that can benefit applications for which the relative positioning of data points is important, i.e., as opposed to their precise locations.
This section is organized as follows. In Sec.~\ref{sec:NNO}, we define kNN orderings of points. In Sec.~\ref{sec:NNO_ph}, we develop kNN filtrations, use them to study kNN persistent homology, and compare them to the study of VR filtrations. In Sec.~\ref{sec:NNO_stab}, we analyze the stability and convergence of persistence diagrams resulting from kNN filtrations.
\subsection{kNN orderings, neighborhoods, and symmetrization}\label{sec:NNO}
We begin with a definition for kNN orderings. One complication is that the ordering of nearest neighbors is not necessarily symmetric, and so we will also describe additional transformations that we will use to symmetrize orderings prior to constructing filtrations.
\begin{definition}[$k$-Nearest-Neighbor Orderings] \label{def:NNO}
Let $\mathcal{V} = \{1,\dots,N\}$ enumerate a set of points $\mathcal{Y}=\{y_i\}_{i\in\mathcal{V}}\in\mathbb{R}^p$ in a normed metric space and $\{f_{ij} = || y_i-y_j||_2 \}$ be their pairwise distances resulting from the pairwise distance map given in Definition~\ref{def:NNOF}. For each $i$, let $k_{ij}$ denote the \textbf{nearest-neighbor order} of $y_j$ with respect to $y_i$ so that $y_j$ is the ($k_{ij}$)-th nearest neighbor of $y_i$ ($k_{ii}\equiv 0$). The ordering is formally defined for each fixed $i$ by $\{k_{ij}\}_{j=1}^N = \phi(\{f_{ij}\}_{j=1}^N)$, where
$\phi: \mathbb{R}^{N} \longrightarrow \mathbb{N}^{N}$ is the ``argsort function''.
\end{definition}
In the above, we assume that the orderings are well-defined in that for a given $i\in\mathcal{V}$, the entries in the set $\{f_{ij}\}_{j=1}^N$ are unique. In practice, if two distances $f_{ij}$ and $f_{ij'}$ are the same and they correspond or order $k$, then we assign an ordering at random so that either $k_{ij} = k$ and $k_{ij'} = k+1$, or vice versa. If even more distances are the same, then we similarly assign them a relative ordering uniformly at random. In principle, the argsort function $\phi$ can be defined in a variety ways to handle the ordering of repeated entries. Given the above definition of kNN orderings, we now introduce an associated map between a set of points and an associated matrix with entries $k_{ij}$.
\begin{definition}[kNN Ordering Function] \label{def:NOf}
Let $\mathcal{Y}=\{y_i\}_{i \in \mathcal{V}}$ denote a set of $N\ge 2$ points with $\mathcal{V}=\{1,2,...,N\}$ in a Euclidean metric space and recall the row-defined pairwise distance function $f_i: \mathcal{Y} \longrightarrow \mathbb{R}^{N }_{+}$ given in Definition~\ref{def:NNOF}.
Letting $\phi$ be the argsort function, we define the map
\textbf{k-nearest-neighbor (kNN) ordering function} $F:\mathcal{Y}\longrightarrow \mathbb{N}^{N\times N }_{+}$ as the matrix-valued function $F=[F_1,\dots,F_N]^T$ in which each row $F_i$ is defined by $F_i = \phi \circ f_i: \mathcal{Y} \longrightarrow \mathbb{N}^{N }_{+}$.
\end{definition}
We will use kNN orderings of points to define local neighborhoods $\mathcal{N}_{ik}$ for each point $i\in\mathcal{V}$, noting that any nested sequence of neighborhoods defines a local filtration of the vertices $\mathcal{V}$.
\begin{definition}[kNN Neighborhoods] \label{def:local_NNO_filtration}
Given a set of kNN orderings $\{k_{ij}\}$ for $i,j \in\mathcal{V}$, we define the \textbf{kNN neighborhoods} as the sets $\mathcal{N}_{ik} = \{j | k_{ij} \le k \}$ for each $i\in\mathcal{V}$ and $k \in \{0\}\cup \mathcal{V}\setminus \{N\}$.
\end{definition}
Importantly, the kNN orderings and kNN neighborhoods are not symmetric---that is, $j\in\mathcal{N}_{ik}$ does not imply $i\in\mathcal{N}_{jk}$. Because we would like to use the kNN neighborhood sets to construct filtrations of simplicial complexes that contain undirected simplices, we will now describe how to construct symmetric kNN neighborhoods.
\begin{definition}[Symmetrized kNN Orderings and Neighborhoods] \label{def:local_NNO_symm}
We define three types of \textbf{symmetrized kNN neighborhoods}---${\mathcal{N}}_{ik}^{min}$, ${\mathcal{N}}_{ik}^{trans}$ and ${\mathcal{N}}_{ik}^{max}$---that use, respectively, the following three symmetrizations of kNN orderings $\{k_{ij}\}$:
\begin{align}\label{eq:sym}
\tilde{k}_{ij} &= \min\{ k_{ij} , k_{ji} \} \nonumber\\
\overline{k}_{ij} &= (k_{ij} + k_{ji} )/2 \nonumber\\
\hat{k}_{ij} &= \max\{ k_{ij} , k_{ji} \}.
\end{align}
\end{definition}
Notably, these kNN neighborhoods satisfy the following nestedness relation.
\begin{lemma}[Nestedness of Symmetrized kNN Neighborhoods] \label{lemma:monotonic2}
Consider fixed $k$ and $i$. Then the neighborhood sets $ \mathcal{N}_{ik}^{min}, \mathcal{N}_{ik}^{trans}, \mathcal{N}_{ik}^{max} $ satisfy the following nestedness relationships:
\[
\mathcal{N}^{max}_{ik} \subseteq \mathcal{N}^{trans}_{ik} \subseteq \mathcal{N}^{min}_{ik}.
\]
\begin{proof}
For any $k_{ij},k_{ji}\in\mathbb{R}$, one has $\tilde{k}_{ij} \le \overline{k}_{ij} \le \hat{k}_{ij}$ by definition of the min and max functions.
If $j \in \mathcal{N}^{max}_{ik}$, then $\hat{k}_{ij} \leq k$. This implies $\tilde{k}_{ij} \le \overline{k}_{ij} \le \hat{k}_{ij} \le k$, and so $j \in \mathcal{N}^{min}_{ik} \subseteq \mathcal{N}^{trans}_{ik} \subseteq \mathcal{N}^{max}_{ik}$.
\end{proof}
\end{lemma}
We now define graphs and clique complexes using symmetrized kNN orderings and neighborhoods.
\begin{definition}[Symmetrized kNN Graph]
\label{def:NNO_graph}
Given a set of points enumerated by $\mathcal{V}$, let $\mathcal{E}_k = \{(i,j)| i\in \mathcal{V}, j\in \mathcal{N}_{ik}\}$ be a set of undirected edges connecting pairs of nearest neighbors, which are symmetrically defined by choosing $\mathcal{N}_{ik}\in\{ \mathcal{N}_{ik}^{min}, \mathcal{N}_{ik}^{trans}, \mathcal{N}_{ik}^{max} \}$. We denote the respective graphs $\mathcal{G}_k(\mathcal{V},\mathcal{E}_k)$, depending on symmetrization method, by
$\mathcal{G}_{k}^{min}$, $\mathcal{G}_{k}^{trans}$, and $\mathcal{G}_{k}^{max}$.
\end{definition}
\begin{definition}[Symmetrized kNN Clique Complex] \label{def:NNO_global}
Given a symmetrized kNN graph $\mathcal{G}_k \in\{\mathcal{G}_{k}^{min}, \mathcal{G}_{k}^{trans}, \mathcal{G}_{k}^{max}\}$ with vertices $\mathcal{V}$ and edges $\mathcal{E}_k$, we construct its associated clique complex $X_k = Cl(\mathcal{G}_{k} )$, and we similarly use a superscript to denote the method of symmetrization, i.e., $X_k^{min}$, $X_k^{trans}$, and $X_k^{max}$.
\end{definition}
\begin{corollary}
[Nestedness of Symmetrized kNN Graphs and kNN Complexes] \label{lemma:monotonic3}
Symmetrized kNN graphs and kNN complexes satisfy the following nestedness relations
\begin{align}
\mathcal{G}^{max}_{k} &\hookrightarrow \mathcal{G}^{trans}_{k} \hookrightarrow \mathcal{G}^{min}_{k}\nonumber\\
X^{max}_{k} &\hookrightarrow X^{trans}_{k} \hookrightarrow X^{min}_{k}.
\end{align}
\begin{proof}
The results follow immediately from Lemma~\ref{lemma:monotonic2}, which proved the nestedness of kNN neighborhoods.
\end{proof}
\end{corollary}
Corollary~\ref{lemma:monotonic3} implies that $\mathcal{G}^{max}_{k}$ is a subgraph of $\mathcal{G}^{trans}_{k}$, which is a subgraph of $\mathcal{G}^{min}_{k}$. Similarly, $X^{max}_{k}$ is a subcomplex of $X^{trans}_{k}$, which is a subcomplex of $X^{min}_{k}$.
\subsection{Filtrations and persistent homology using kNN complexes}\label{sec:NNO_ph}
We now formulate filtrations and persistent homology using symmetrized kNN complexes. Varying $k$ gives rise to a sequence of nested sets called a filtration, and we will define several types based on the different symmetrization methods.
We will also define local and global versions of filtrations.
\begin{figure}[htp]
\begin{center}
\includegraphics[width=\linewidth]{figures/NNOmin.pdf}
\includegraphics[width=\linewidth]{figures/NNOtrans.pdf}
\includegraphics[width=\linewidth]{figures/NNOmax.pdf}
\caption{Visualization of kNN-filtered simplicial complexes for a point cloud using the three types of symmetrization given by Definition~\ref{def:local_NNO_symm}.
Comparing across the columns, observe the nestedness property given by Corollary~\ref{lemma:monotonic3}.
}\label{Fig:NNOs_sym}
\end{center}
\end{figure}
\begin{definition}[Global kNN Filtration]
\label{def:NNO_globa}
Let $X_k,X_{k'}\in \{ X_k^{min},X_k^{trans},X_k^{max}\}$ be kNN complexes with the same symmetrization method.
Then we define a \textbf{kNN-filtered simplicial complex}
\[
\{X_k \longrightarrow X_{k'} \}_{0\leq k \leq k^\prime\le N-1} .
\]
\end{definition}
In Fig.~\ref{Fig:NNOs_sym}, we illustrate global kNN filtrations with the three different symmetrization methods given by Definition~\ref{def:local_NNO_symm}. For simplicity, we only visualize 0-simplices and 1-simplices. By comparing the kNN complexes across a given column, one can observe the nestedness relations defined by Corollary~\ref{lemma:monotonic3}.
We formulate persistent homology for kNN filtrations of a simplicial complex anaologous to that defined for a VR filtrations (recall Sec.~\ref{sec:holomology}), and in Fig.~\ref{Fig:VR_filtration}, we visualize persistence diagrams for both (left) a kNN filtration and (right) a VR filtration for an example point cloud. The red and blue persistence barcodes indicate 0-dimensional and 1-dimensional cycles, respectively. Observe that the kNN filtration reveals a 1-cycle, whereas the VR filtration does not. Hence, filtrations of simplicial complexes (specifically clique complexes) according to pairwise distances and kNN provide complementary homological information.
\begin{figure}[t]
\begin{center}
\includegraphics[width=.45\linewidth]{figures/NO_filtration.pdf}~~~
\includegraphics[width=.45\linewidth]{figures/VR_filtration.pdf}
\includegraphics[width=.45\linewidth]{figures/RankBar.pdf}
\includegraphics[width=.45\linewidth]{figures/VRBar.pdf}
\caption{Comparison of persistent homology for an example point cloud using two different filtrations: (left) our proposed kNN filtration; and (right) a VR filtration. The red and blue persistence barcodes indicate 0-dimensional and 1-dimensional cycles, respectively.
Observe that the kNN filtration reveals a 1-cycle, whereas the VR filtration does not.
}\label{Fig:VR_filtration}
\end{center}
\end{figure}
In the next sections, we will further compare persistence diagrams resulting from kNN and VR filtrations;
however, it's worth highlighting several differences here.
First, one benefit of using kNN sets versus a distance threshold $\epsilon$ is that filtrations are standardized. That is, the filtration parameter range $k\in[0,N]$ for kNN filtrations is always the same for a set of $N$ points. In contrast, different point clouds have different lengths scales, and so the range of a distance-based filtration parameter is in general not standardized. One could seek to standardize the ranges of VR filtration parameters by standardizing distances; however, there are many ways to normalize a point cloud, such as dividing by the mean distance or by the distances' standard deviation. Given that one could implement a variety of normalization approaches, comparing VR filtrations across different point clouds that have different length scales or different dimensions is not straightforward. In contrast, kNN filtrations are standardized by definition.
Second, the filtration parameter $\epsilon\in\mathbb{R}_+$ for VR filtrations can in principle take on any positive number. In contrast, $k\in\{0,\dots, N-1\}$ can only take on integer values (or in the case of the symmetrization method of $trans$, half integers as well). Potentially, this reduced space of possible filtration parameters could benefit the computationally efficiency of implementing kNN filtrations, although we don't explore that pursuit herein. That said, our experiments generally find kNN filtrations to have higher computational complexity, since they require both computing pairwise distances as well as their orderings.
\subsection{Stability and convergence of kNN homology}\label{sec:NNO_stab}
Here, we will study two key properties for the space of persistence diagrams that follow from kNN filtrations: stability and convergence.
\subsubsection{kNN-preserving transformations}\label{sec:NNO-preserve}
We observe that if points undergo perturbations that are sufficiently small, then it is possible to move points without changing any of the neighbor orderings. To make this more precise, we define a family of point-cloud transformations that have this property.
\begin{definition}[Local and Global kNN-Preserving Transformations] \label{def:NNO_transformation}
Let $\mathcal{Y}=\{y_i\}_{i \in \mathcal{V}}$ with $ y_i \in \mathbb{R}^p$ be a set of points that are enumerated $\mathcal{V}=\{1,...,N\}$, $f: \mathcal{Y} \longrightarrow \mathbb{R}^{N \times N}_{+}$ be the pairwise distance function given by Def.~\ref{def:NNOF}, and $F: \mathcal{Y} \longrightarrow \mathbb{N}^{N \times N}_{+}$ be the neighbor-ordering function given by Def.~\ref{def:NOf}.
We define a transformation $h: \mathcal{Y} \longrightarrow \mathcal{Y'}$ with $h(x)=(h_1(y_1),...,h_n(y_n))=(y'_1,...,y'_n)$ and define two types of preservation properties:
\begin{itemize}
\item $h$ is local kNN-preserving for some $i \in \mathcal{V}$ if and only if $(F \circ h)_{ij}(\mathcal{Y}) = F_{ij}(\mathcal{Y})$ for fixed $i$ and $j \in \mathcal{V}$.
\item $h$ is global kNN-preserving if and only if it is local kNN-preserving for all $i\in\mathcal{V}$.
\end{itemize}
\end{definition}
Note that local kNN-preservation implies that the neighbor orderings are preserved for a particular point $x_i$, while global-kNN preservation implies that all neighbor orderings are preserved for all points. We also note that local and global kNN-preserving transformations can also be similarly defined for the three symmetrized versions of kNN orderings.
We also define a slight variation in which the preservation of orderings is only required for the nearest neighbor orderings up to a finite size $k\le K$, which we refer to as ``$K$-bounded'' local and global kNN-preserving transformations.
As one example, consider a set of three points $x_i\in \{0,2,3\}\subset \mathbb{R}$ and the transformation $h(\{0,2,3\}) = \{0,2.1,3\} $. It is both local and global kNN-preserving, since the perturbation $ 2\to2.1$ is sufficiently small such that none of the neighbor orderings change. In fact, all isometric transformations including (i.e., translation, rotation, reflection and glide reflection) are global kNN-preserving transformations because they leave the pairwise distances unchanged.
\begin{figure}[htp]
\begin{center}
\includegraphics[width=1\linewidth]{figures/instability_1.pdf}
\caption{Example with 3 points $\mathcal{Y} = \{x_a,x_b,x_c\}$ with $x_a=-1$, $x_c =1$ and either (A) $x_b=-\epsilon$ or (B) $b=\epsilon$. Note that the perturbation can be made arbitrarily small for any $\epsilon>0$, and the nearest-neighbor orderings are different. (See Tables \ref{table:Rank line origin} and \ref{table:Rank line transform}.) }
\label{fig:stab}
\end{center}
\end{figure}
However, there are many transformations that are not kNN preserving. In Fig.~\ref{fig:stab}, we illustrate an example where a small perturbation to one point can change the neighbor ordering in a discontinuous way. Consider a set of points $\mathcal{V}=\{a,b,c\}$ with locations $\mathcal{Y}=\{x_a,x_b,x_c\}\subset \mathbb{R}$ where $x_a=-1$, $x_c=1$ and $x_b=-\epsilon$ (for some small value $0<\epsilon \ll 1$). Then the neighbor ordering matrix is given in Table~\ref{table:Rank line origin}.
Now consider the transformation $h_1(x_a)=x_a$, $h_2(x_b)=x_b+2\epsilon$, and $h_3(x_c)=x_c$, then the neighbor ordering matrix changes, as shown in Table~\ref{table:Rank line transform}.
\begin{table}[ht]
\begin{minipage}[b]{0.4\linewidth}
\centering
{\small{\begin{tabular}{ | l | r | r | r | r |}
\hline
& point a & point b & point c\\ \hline \hline
point a & 0 & 1&2 \\ \hline
point b & 1 & 0&2 \\ \hline
point c & 2 & 1&0 \\ \hline
\end{tabular}}}
\caption{kNN orderings before transformation $h$.}
\label{table:Rank line origin}
\end{minipage}\hfill
\begin{minipage}[b]{0.4\linewidth}
\centering
{\small{\begin{tabular}{ | l | r | r | r | r |}
\hline
& point a & point b & point c \\ \hline \hline
point a & 0 & 1 &2 \\ \hline
point b & 2 & 0& 1 \\ \hline
point c & 2 & 1&0 \\ \hline
\end{tabular}}}
\caption{kNN orderings after transformation $h$.}
\label{table:Rank line transform}
\end{minipage}
\end{table}
Given the definitions of local and global kNN-preserving transformations, we
can also define equivalence classes for any finite set of points $ \mathcal{Y}= \{y_i\} \subset \mathbb{R}^p$.
\begin{definition}[kNN Equivalence of Point Sets] \label{def:NNO_equiv}
Let $F$ denote the neighbor-ordering function given by Def.~\ref{def:NOf}.
Any two point clouds $\mathcal{Y}=\{y_i\}_{i \in \mathcal{V}}$ and $\mathcal{Y'}=\{y_i'\}_{i \in \mathcal{V}}$ with $ y_i,y'_i \in \mathbb{R}^p$ are said to be kNN-equivalent if $F(\mathcal{Y}) = F(\mathcal{Y'})$. We denote the equivalence by $\mathcal{Y}\sim \mathcal{Y'}$.
Furthermore, we have $[\mathcal{Y}] = \{ \mathcal{Y'} | \mathcal{Y'}\sim \mathcal{Y} \}$ as the kNN-equivalence class of $\mathcal{Y}$.
\end{definition}
Indeed, one can show
\begin{itemize}
\item {\bf Reflexivity: $\mathcal{Y} \sim \mathcal{Y}'$.} \\
It is trivial. Take $h(t)= \mathbf{id}$ as the kNN-preserving transformation.
\item {\bf Symmetry: if $\mathcal{Y} \sim \mathcal{Y'}$ then $\mathcal{Y'} \sim \mathcal{Y}$ .} \\
Let $h: \mathcal{Y} \longrightarrow \mathcal{Y'}$ be a global kNN-preserving transformation. Then $(F \circ h)_{ij}(\mathcal{Y})=(F_{ij})(\mathcal{Y})$ and $h$ is a one-to-one function. Let $h^{-1}$ be the inverse function of $h$. So, we have: $(F \circ h^{-1})_{ij}(\mathcal{Y'})=(F_{ij})(\mathcal{Y'})$, thus $h^{-1}$ is the global kNN-preserving transformation from $\mathcal{Y'}$ to $\mathcal{Y}$.
\item {\bf Transitivity: if $\mathcal{Y} \sim \mathcal{Y'}$ and $\mathcal{Y'} \sim \mathcal{Y''}$ then $\mathcal{Y} \sim \mathcal{Y''}$.}\\
Let $h(t)$ be kNN-preserving transformation between $\mathcal{Y}$ and $\mathcal{Y'}$, $g(t)$ be kNN-preserving transformation between $\mathcal{Y'}$ and $\mathcal{Y''}$, then $g \circ h$ be kNN-preserving transformation between $\mathcal{Y}$ and $\mathcal{Y''}$.\\
\end{itemize}
The equivalence of kNN orderings implies that kNN filtrations and kNN-based persistent homology are identical for any two point clouds within a given equivalence class. Thus, persistence diagrams for kNN persistent homology are robust to (and in fact, unaffected by) any kNN-preserving transformation. That said, such persistence diagrams can also be unstable to other types of perturbations, as we explore below.
\subsubsection{Stability of kNN persistence diagrams} \label{sec: stability1}
In analogy to the stability theorem for the space $\mathcal{H}_{VR}$ of persistence diagrams obtained using VR filtrations, we consider whether the space $\mathcal{H}_{kNN}$ of persistence diagrams using kNN filtrations is also stable with respect to perturbations of the associated point cloud.
We fist show that persistence diagrams resulting from kNN filtrations are do not satisfy the same stability condition as that for VR filtrations.
\begin{theorem}[kNN Persistence Diagrams are not Uniformly Globally Stable to Point Changes]\label{thm:kNN_sta}
Let $\mathcal{Y}=\{y_i\}_{i \in \mathcal{V}}$ with $\mathcal{V}=\{1,...,N\}$ be a point cloud with a neighbor-ordering function $F=\phi \circ f$, where
$f: \mathcal{Y} \longrightarrow \mathbb{R}^{N \times N}_{+}$ is the pairwise distance function and
$\phi: \mathbb{R}^{N \times N}_{+} \longrightarrow \mathbb{N}^{N \times N}_{+}$ is the `argsort' function. Further let $\mathcal{Y'}=\{y'_i\}_{i \in \mathcal{V}}$ be a second point cloud with the neighbor-ordering function $G=\phi \circ g$.
Further let $D_F$ and $D_G$ denote the persistence diagrams of kNN filtrations applied to $\mathcal{Y}$ and $\mathcal{Y'}$.
Further, define the metric for two enumerated point clouds $\left\lVert{\mathcal{Y}-\mathcal{Y'}} \right\rVert_\infty= \max_{i\in\mathcal{V}} ||y_i-y_i'||_2$.
Then there does not exist a uniform global bound on the bottleneck distance between persistence diagrams of the form:
$$
d_B(D_F,D_G) \leq \lVert{F-G}\rVert_\infty
=\left\lVert{\phi \circ f- \phi \circ g} \right\rVert_\infty \not\leq L \left\lVert{\mathcal{Y}-\mathcal{Y'}} \right\rVert_\infty.
$$
\begin{proof}
The first inequality is satisfied by applying the original statement of the stability theorem: persistence diagrams are uniformly bound by the maximal difference between the functions that are filtered. By definition, the neighbor-ordering function yields a matrix of k nearest neighbors, $[k_{ij}] = F(\mathcal{Y})$ and $[k'_{ij}] = G(\mathcal{Y'})$, implying
$$
\lVert{F-G}\rVert_\infty \equiv \max_{ij} |k_{ij}-k'_{ij}|.
$$
That is, the difference in persistence diagrams is bound by the maximum difference in neighbor orderings. However, the stability of persistent homology with respect to neighbor-ordering changes does not imply stability with respect to point-location changes.
To disprove the last inequality, we provide a counter example. Recall the sets of $N=3$ points in Fig.~\ref{fig:stab} with $\mathcal{V}=\mathcal{V}'=\{a,b,c\}$ and locations $\mathcal{Y} = \{1,-\epsilon,1\} \in\mathbb{R}$ and $\mathcal{Y}' = \{1,\epsilon,1\} \in\mathbb{R}$. By construction, $||F-G||_\infty=1$ for any small $\epsilon$, but $\left\lVert{\mathcal{Y}-\mathcal{Y'}} \right\rVert_\infty=2\epsilon$. Suppose there did exist a uniform bound with Lipschitz constant $L>1$, then the points $\mathcal{Y}$ and $\mathcal{Y'}$ with any $\epsilon<1/2L$ yields a contradiction since $||F-G||_\infty=1$ and $||\mathcal{Y}-\mathcal{Y'}||_\infty<1$.
\end{proof}
\end{theorem}
\subsubsection{Topological convergence of kNN orderings} \label{sec:convergence1}
In this section, we will use the neighbor-ordering function to define a discrete-topological notion of convergence for a sequence of point sets using kNN persistent homology.
\begin{definition}
Consider a sequence of point sets $\mathcal{Y}^{(t)}=\{y^{(t)}_i| i \in \mathcal{V}\}$ with $t\in \mathbb{N}$ and $ \mathcal{V}= \{1,...,N\}$ and $y^{(t)}_i \in \mathbb{R}^p$. We say that the sequence $\{\mathcal{Y}^{(t)}\}$ has {\bf global convergence in kNN topology} to the limit $\mathcal{Y}$ iff $\lim_{t\to \infty} F(\mathcal{Y}^{(t)}) = F(\mathcal{Y})$, where $F$ is the neighbor-ordering function. More precisely, for any $\epsilon$, there exists a $t^*$ such that $\max_{ij}|F_{ij}(\mathcal{Y}) - F_{ij}(\mathcal{Y}^{(t)})|<\epsilon$ for all $t>t^*$.
\end{definition}
\begin{definition}
Consider a sequence of point sets $\mathcal{Y}^{(t)}=\{y^{(t)}_i| i \in \mathcal{V}\}$ with $t\in \mathbb{N}$ and $\mathcal{V}= \{1,...,N\}$ and $y^{(t)}_i \in \mathbb{R}^p$. We say that the sequence $\{\mathcal{Y}^{(t)}\}$ has {\bf $\kappa$-bounded convergence in kNN topology} to a limit $\mathcal{Y}$ iff $\lim_{t\to \infty} F_{ij}(\mathcal{Y}^{(t)})=F_{ij}(\mathcal{Y})$
for all $i \in \mathcal{V}$ and $j \in \mathcal{N}_{ik}$, where $k\le \kappa$ and $\mathcal{N}_k \in\{\mathcal{N}_{ik}^{min}, \mathcal{N}_{ik}^{trans}, \mathcal{N}_{ik}^{max}\}$ are symmetrized neighborhoods that are defined in Def.~\ref{def:local_NNO_filtration}.
More precisely, for any $\epsilon$, there exists a $t^*$ such that $\max_{i\in\mathcal{V},j\in \mathcal{N}_{ik}}|F_{ij}(\mathcal{Y}) - F_{ij}(\mathcal{Y}^{(t)})|<\epsilon$ for all $t>t^*$.
\end{definition}
\begin{remark}
Note that $\kappa$-bounded convergence is inclusive so that convergence for a given $\kappa$ also implies convergence for any $\kappa'\le \kappa$. In particular, global convergence in kNN topology implies $\kappa$-bounded convergence for any $\kappa$.
\end{remark}
We now define local variants of kNN topological convergence.
\begin{definition}
Consider a sequence of point sets $\mathcal{Y}^{(t)}=\{y^{(t)}_i| i \in \mathcal{V}\}$ with $t\in \mathbb{N}$ and $ \mathcal{V}= \{1,...,N\}$ and $y^{(t)}_i \in \mathbb{R}^p$. We say that the sequence $\{\mathcal{Y}^{(t)}\}$ has {\bf $\mathcal{U}$-local convergence in kNN topology} to a limit $\mathcal{Y}$ iff $\lim_{t\to \infty} F_{ij}(\mathcal{Y}^{(t)})=F_{ij}(\mathcal{Y})$
for all $i,j \in \mathcal{U}\subseteq \mathcal{V}$ .
More precisely, for any $\epsilon$, there exists a $t^*$ such that $\max_{i,j\in\mathcal{U}}|F_{ij}(\mathcal{Y}) - F_{ij}(\mathcal{Y}^{(t)})|<\epsilon$ for all $t>t^*$.
\end{definition}
\begin{definition}
Consider a sequence of point sets $\mathcal{Y}^{(t)}=\{y^{(t)}_i| i \in \mathcal{V}\}$ with $t\in \mathbb{N}$ and $ \mathcal{V}= \{1,...,N\}$ and $y^{(t)}_i \in \mathbb{R}^p$. We say that the sequence $\{\mathcal{Y}^{(t)}\}$ has {\bf $\kappa$-bounded $\mathcal{U}$-local convergence in kNN topology} to a limit $\mathcal{Y}$ iff $\lim_{t\to \infty} F_{ij}(\mathcal{Y}^{(t)})=F_{ij}(\mathcal{Y})$
for all $i \in \mathcal{U}\subseteq \mathcal{V}$ and $j \in \mathcal{N}_{ik}$, where the symmetrized neighborhoods
$\mathcal{N}_k \in\{\mathcal{N}_{ik}^{min}, \mathcal{N}_{ik}^{trans}, \mathcal{N}_{ik}^{max}\}$
are given in Def.~\ref{def:local_NNO_filtration} and $k\le \kappa$.
More precisely, for any $\epsilon$, there exists a $t^*$ such that $\max_{i\in\mathcal{U},j\in \mathcal{V}}|F_{ij}(\mathcal{Y}) - F_{ij}(\mathcal{Y}^{(t)})|<\epsilon$ for all $t>t^*$.
\end{definition}
Note for any point set that convergences in global kNN topology (or k-bounded convergence in kNN topology), that there exists a $t^*$ such that $F_{ij}(\mathcal{Y}) = F_{ij}(\mathcal{Y}^{(t)})$ for all $t>t^*$ and $i,j \in\mathcal{V}$ (or $i \in\mathcal{V}$ and $j\in N_{ik}$).
Specifically, we can choose $\epsilon\in(0,1)$. Since $F_{ij}(\mathcal{Y})\in\mathbb{N}$ for any point set $\mathcal{Y}$, the condition $|F_{ij}(\mathcal{Y})-F_{ij}(\mathcal{Y}^{(t)})|<\epsilon$ implies $F_{ij}(\mathcal{Y})-F_{ij}(\mathcal{Y}^{(t)})=0$ for all $t>t^*$.
That is, kNN convergence is a discrete property that is exactly obtained for sufficiently large $t>t^*$.
This significantly contrasts the notion of convergence in norm, which is often asymptotically approached rather than exactly obtained.
The next theorem more precisely establishes a relation between convergence in a normed metric space and convergence in kNN topology.
\begin{theorem}[Convergence in Norm Implies Convergence in kNN Topology]\label{thm: convergence_NNO}
Let $\mathcal{Y}^{(t)} = \{y^{(t)}_i| i \in \mathcal{V}\}\subset\mathbb{R}^p$ be an element of a sequence of point clouds in which each point converges in norm to some limit,
$\lim_{t\to\infty}||y_i^{(t)} - y_i||=0$ for each $i\in \mathcal{V}$.
Define $\mathcal{Y} = \{y_i | i\in\mathcal{V} \} \subset \mathbb{R}^p$ as that limit.
Then $\mathcal{Y}^{(t)} \to \mathcal{Y}$ in kNN topology, and there exists a $t^*$ such that $F(\mathcal{Y}^{(t)})=F(\mathcal{Y})$ for any $t>t^*$, where $F$ is the neighbor-ordering given by Def.~\ref{def:NOf}.
\begin{proof}
To prove this theorem, we use a contradiction.
If $\mathcal{Y}^{(t)}$ does not converge to $\mathcal{Y}$ in kNN-topology, then for any $t$, there exists at least one neighbor ordering that differs, i.e., $F_{ij}(\mathcal{Y}^{(t)}) \neq F_{ij}(\mathcal{Y})$ for some $i,j \in\mathcal{V}$. Letting $d_{ij}^{(t)} = ||y_i^{(t)} - y_j^{(t)}||_2 $ and $d_{ij}^{(\infty)} = ||y_i - y_j ||_2 $, this implies $d_{ij}^{(\infty)} - d_{ik}^{(\infty)}>0$ but $d_{ij}^{(t)} - d_{ik}^{(t)} \le 0$ for some $i,j, k \in\mathcal{V}$. (This must be true of the distances if the neighbor orderings differ.) Let $\delta^*= \max_{i,j,k} |d_{ij}^{(\infty)} - d_{ik}^{(\infty)}|$. However, because $\lim_{ t\to\infty} || y_i^{(t)} -y_i|| $ for each $i$, there exists a $t^*$ such that $|| y_i^{(t)} -y_i||_\infty < \epsilon/4$ for all $i$ when $t >t^*$. It then follows that $ |d_{ij}^{(t)} - d_{ij}^{(\infty)}| < \epsilon/2$ and $|d_{ik}^{(t)} - d_{ik}^{(\infty)}| < \epsilon/2$.
Now suppose $d_{ij}^{(\infty)} - d_{ik}^{(\infty)} >0$.
Then
\begin{align}
d_{ij}^{(t)}-d_{ik}^{(t)} &> d_{ij}^{(t)} - (d_{ik}^{(\infty)} + \epsilon/2) \nonumber\\
&> (d_{ij}^{(\infty)} - \epsilon/2) - (d_{ik}^{(\infty)} + \epsilon/2) \nonumber\\
&= d_{ij}^{(\infty)} - d_{ik}^{(\infty)} - \epsilon\nonumber\\
&= \delta^* - \epsilon.
\end{align}
This is true for any $\epsilon$, so we may choose $\epsilon<\delta^*$, which implies $d_{ij}^{(t)}-d_{ik}^{(t)}>0$, contradicting the above statement.
\end{proof}
\end{theorem}
Finally, we prove that the reverse property does not necessarily hold.
\begin{theorem}[Convergence in kNN Topology Does Not Imply Convergence in Norm] \label{thm: convergence NNO1}
Let $\mathcal{Y}^{(t)} = \{y^{(t)}_i| i \in \mathcal{V} \}\subset\mathbb{R}^p$ be an element of a sequence of point sets that converges in kNN topology.
Then each $y_i^{(t)}$ may or may not converge to some limit $y_i$ as $t\to \infty$.
\begin{proof}
See the proof to Thm.~\ref{thm:kNN_sta} for a counter example in which points converge as $\epsilon\to 0$, but not for kNN topology, which remains bounded below by 1.
\end{proof}
\end{theorem}
\section{kNN persistent homology reveals topological convergence of pageRank algorithm} \label{sec:PagerankandRD}
In this section, we study the convergence of an iterative method for approximating Google's PageRank an use kNN persistent homology to develop a perspective from discrete topology---that is, as opposed to the typical geometric perspective of convergence under a vector norm.
In Sec.~\ref{sec:PR1}, we present the PageRank algorithm.
In Sec.~\ref{sec:Dolph}, we present numerical experiments for an empirical network: a dolphin social network.
In Sec.~\ref{sec:Dolpu}, we study PageRank for the same network using local versions of kNN topological convergence.
\subsection{PageRank}
\label{sec:PR1}
Google developed \emph{PageRank} to solve the problem of web search and ranking for the World Wide Web \cite{brin1998anatomy}. Their aim was to create an importance measure for each webpage distinguish highly recognizable, relevant pages from those that are less known. There are many derivations of PageRank \cite{langville2006updating,higham2005google}, all of which stem from modeling `websurfing' (i.e., how people navigate the web) as a Markov chain. In this analogy, the fraction of random websurfers at a particular web page is given by the stationary distribution of a Markov chain.
A main challenge for this formulation is that in practice, a network connecting webpages via their hyperlinks does not usually consist of a single connected component. Instead, there are isolated webpages that cannot be navigated to, or navigated from. To address this issue, the Google founders introduced `teleportation' so that with probability $\alpha$, websurfers click a hyperlink to move between webpages, and with probability $1-\alpha$, websurfers randomly jump to webpage $i$ with probability $v_i$. Parameter $\alpha$ is called as \emph{teleportation parameter} or \emph{damping factor}.
In the original formulation, a walker would jump uniformly at random to another webpage so that the transition probability to each webpage is the same: $v_i=v_j=1/N$, where $N$ is the number of webpages. It is also beneficial to allow the $v_i$ values to be heterogeneous to bias the random websurfing to remain near a particular set of webpages. Such dynamics is called `personalized' PageRank, and ${\bf v}$ is called the \emph{personalization vector}.
Both PageRank and personalized PageRank can be formulated as a discrete-time Markov chain in which the transition matrix is given by the Google matrix:
\begin{align}
{\bf G} = \alpha {\bf P} + (1-\alpha) {\bf e}{\bf v}^T,
\end{align}
where ${\bf e}$ is a vector of ones and each entry $P_{ij}$ gives the probability of transitioning from webpage $i$ to webpage $j$ following a hyperlink. The matrices $P$, ${\bf e}{\bf v}^T$ and $G$ are all row-stochastic transition matrices.
%
The stationary distribution of the Markov chain with transition matrix ${\bf G}$ is called the PageRank vector $\pi\in\mathbb{R}^N$, which is the limit of the iterative equation
\begin{align}\label{eq:it}
{\bf x}(t+1)^T = {\bf x}(t)^T {\bf G} ,
\end{align}
for which the fixed point is the solution to the eigenvalue problem $ {\bf G}^T\bm{\pi} = \bm{\pi}$.
The PageRank algorithm has contributed toward Google's rise as a leading technology company, but it's worth noting that it has also been applied in a wide variety of domains beyond web search \cite{gleich2015pagerank}. In any context, the practical usage of PageRank comes with various challenges. For example, what $\alpha$ value should be used? The PageRank vector can be integrated with a machine learning framework for web search \cite{najork2007hits}, used to help fit functions over directed graphs \cite{zhou2005learning}, and can be used to predict missing genes and protein functions \cite{ichinomiya2020protein,morrison2005generank}. In these various cases, $\alpha$ must be separately chosen as appropriate. As in \cite{pan2004automatic}, the author suggests choosing $\alpha = 0.15$ for correlated discovery in a multimedia database. Google historically set $\alpha = 0.85$, which often remains as the default choice in the literature.
\begin{figure}[t]
\begin{center}
\includegraphics[width=\linewidth]{figures/Dolphins2.pdf}
\caption{(A) A social network of interactions among $N=62$ dolphins in New Zealand.
(B) Node colors indicate the nodes' respective PageRank values. In both panels, nodes with larger/smaller size have more/fewer connecting edges.
}
\label{Fig:Dol1}
\end{center}
\end{figure}
In our paper, we explore a different challenge that arises when considering how many times to iterate Eq.~\eqref{eq:it}. The iterated values ${\bf x}(t)\to\bm{\pi}$ converge as $t\to\infty$, but a more practical questions involves studying how many iterations are required for the associated ranks to converge. That is, if one ranks webpages from top to bottom based on their $\pi_i$ values, then one only needs to iterate Eq.~\eqref{eq:it} until those ranks converge. Understanding the asymptotic convergence rate and asymptotic error is not immediately relevant when considering this practical question, and we propose kNN persistent homology as a mathematically principled technique to study the convergence of rankings (and relative orderings more generally).
\subsection{Ranking nodes in a dolphin social network}
\label{sec:Dolph}
In this section, we consider the convergence of the iterative method for computing PageRank. We will study the convergence of persistence diagrams associated with converging approximate PageRank values ${\bf x}(t) \to \bm{ \pi}$, comparing the result for kNN filtrations to those of VR filtrations. We will show that the convergence of persistence diagrams for VR filtrations closely relates to the convergence in vector norm (i.e., due to the stability theorem). In contrast, we the convergence of persistence diagrams for kNN filtrations more closely resembles the convergence of rank orderings. In other words, convergence of kNN persistent homology can be used to predict how many iterations are required for ${\bf x}(t)$ to be sufficiently close to $\bm{\pi}$ such that the ranking of nodes---i.e., from 1 to N---has converged.
\begin{figure}[t]
\begin{center}
\includegraphics[width=\linewidth]{figures/Dolphins_iterations.pdf}
\caption{Visualization of the approximate PageRank values $x_i(t)\approx \pi_i$ for the iterative method with different $t\in\{3,6,9,12\}$. Node colors indicate the $x_i(t)$ values. The black circles around nodes indicate the ranks $R_i({\bf x}(t))$ that have already converged to their limiting rank $R_i(\bm{\pi})$. For each node $i$, we define $t^*_i$ to be the time step for which $R_i({\bf x}(t))={R}_i(\bm{\pi})$ for $t\ge t^*$. Observe that most ranks converge after 12 time steps, even though $||{\bf x}-\bm{\pi}|| >0$.
}
\label{Fig:Dol2}
\end{center}
\end{figure}
We present our results for an empirical network in which undirected, unweighted edges encode social interactions among $N=62$ bottlenose dolphins living near Doubtful Sound in New Zealand
\cite{lusseau2003bottlenose}.
We apply the iterative PageRank algorithm with teleportation parameter $\alpha=0.85$ to the network with an initial condition given by $x_i(0) = i / (\sum_{j=1}^N j)$.
In the Figure ~\ref{Fig:Dol1}, we illustrate the dolphin network and indicate the nodes' converged PageRank values by node color.
We formally define the converged rank order according to PageRank by
$$
[R_i(\bm{\pi)}] = \phi(-\bm{\pi}),
$$
where $\phi:\mathbb{R} \to \{1,\dots,N\}$ is the `argsort' function.
Recall that the function $\phi$ was previously defined to sort the pairwise distances in ascending order. By multiplying $\bm{\pi}$ by negative one, we now sort the $\pi_i$ values in descending order so that $R_i(\bm{\pi})=1$ for the top-ranked node $i=\text{argmax}_j \pi_j$, which is considered to be the most important dolphin in the social network. Similarly, $R_i(\bm{\pi})=N$ for the lowest-ranked node $i=\text{argmin}_j \pi_j$, which is the least important dolphin. (
For each time $t$, we similarly define the approximate rank orderings
$$
[R_i({\bf x}(t))] = \phi(-{\bf x}(t)).
$$
We note that the rank orderings satisfy $R_i({\bf x}(t)),{R}_i(\bm{\pi}) \in \{1,\dots, N\}$.
Because ${\bf x}(t)\to\bm{\pi}$, the approximate rank orderings $R_i({\bf x}(t))$ converge to their final rank orderings ${R}_i(\bm{\pi})$. Moreover, for each node $i$, the rank $R_i({\bf x}(t))$ can converge to $R_i(\bm{\pi})$ at a different time step $t$. Therefore, we define $t^*_i$ to be the {\bf iteration of rank convergence}, which is the value of $t$ at which $R_i({\bf x}(t))={R}_i(\bm{\pi})$ for all $t\ge t^*_i$.
In Fig.~\ref{Fig:Dol2}, we illustrate the convergence for the approximate PageRank values $x_i(t)\approx \pi_i$ for different time steps. That is, for the time steps $t\in\{3,6,9,12\}$, we use node color to indicate the values $x_i(t)$. Moreover, we use black circles to indicate which nodes have already obtained their limiting rank, i.e., $R_i({\bf x}(t))={R(\bm{\pi})}_i$ for $t\ge t^*_i$. Most ranks converge after 12 time steps.
\begin{figure}[t]
\begin{center}
\includegraphics[width=\linewidth]{figures/dolphins_agh.pdf}
\caption{(left) Convergence of nodes' rank orderings $R_i({\bf x}(t))\to {R}_i(\bm{\pi})$ versus time step $t$.
(right) Scatter plot comparing $t^*_i$ and $\pi_i$ across the nodes $i$.
}
\label{Fig:BR}
\end{center}
\end{figure}
In Fig.~\ref{Fig:BR}, we further study the convergence of the rank orderings $R_i({\bf x}(t))\to {R}_i(\bm{\pi})$ for the dolphin network. In Fig.~\ref{Fig:BR}(left), observe that the $R_i({\bf x}(t))$ values change up until a time step $t^*_i$, which is potentially different for each node $i$. In Fig.~\ref{Fig:BR}(left), we depict a scatter plot comparing $t^*_i$ and $\pi_i$, noting that we do not see any strong correlation. From a practical perspective, one is often most interested in the top-ranked nodes, and so one is primarily interested in how many iterations are required for the top-ranks to converge. However, there is no guarantee that the top-rank nodes converge before the lower-ranked ones do, or vice versa.
We now study the convergence of persistence diagrams for the converging $x_i(t)\to\pi_i$ values, comparing persistence diagrams resulting from kNN filtrations to those of VR filtrations. More precisely, we define $D_{VR}({\bf x})$ and $D_{kNN}({\bf x})$ to be the persistence diagrams according to VR and kNN filtrations, respectively. We will also study kNN filtrations with two types of symmetrization for k-nearest neighbor sets: the minimum and maximum approaches. Given the persistence diagrams for $\bm{\pi}$ and ${\bf x}(t)$, we study homological convergence through the bottleneck distance, e.g., $d(D_{VR}({\bf x}(t))-D_{VR}(\bm{\pi}))$.
In Fig.~\ref{Fig:Dolconv}, we illustrate the convergence of persistence diagrams for (left) VR filtrations and (right) kNN filtrations. We compare these two converging topological spaces, respectively, with the normed approximation error, $||{\bf x}(t)-\bm{\pi}||$, and the total difference in rank orderings, $\sum_i |{R}_i(\bm{\pi}) - R_i({\bf x}(t))|$. Observe in Fig.~\ref{Fig:Dolconv}(left) that VR persistent homology converges similarly to $||{\bf x}(t)-\bm{\pi}||$, whereas kNN persistent homology converges similarly to $\sum_i |{R}_i(\bm{\pi}) - R_i({\bf x}(t))|$. That is, kNN topological convergence can be used as a proxy to estimate the number of iterations required for the rank orderings to converge to exactly their final values.
\begin{figure}[t]
\begin{center}
\includegraphics[width=\linewidth]{figures/Dolphins_convergence.pdf}
\caption{Homological convergence of an iterative algorithm for PageRank for the dolphin social network.
(left) Convergence of persistent homology for VR filtrations coincides with a geometrical notion of convergence $||{\bf x}(t)-\bm{\pi}|| \to 0$ due to the stability theorem. Both asymptotically approach 0 with exponential decay.
(right) In contrast, convergence of persistent homology for kNN filtrations more closely resembles the convergence of the rank ordering, which exactly converges after $t=14$ time steps in this case.
Observe the kNN persistence diagrams for the max and min methods of symmetrization for kNN sets converge at around the same number of iterations.
}\label{Fig:Dolconv}
\end{center}
\end{figure}
\subsection{U-local convergence of dolphin social network} \label{sec:Dolpu}
Before concluding, we further study the homological convergence of PageRank for the dolphin network, except we now focus on
our notion of $\mathcal{U}$-local topological convergence, which we presented in Sec.~\ref{sec:convergence1}. In this context, we focus on convergence for a subset $\mathcal{U}\subset \mathcal{V}$ of the nodes, and we will consider two subsets: a randomly selected set of nodes---$\mathcal{U}_1= \{$Ripplefluke, Zig, Feather, Gallatin, SN90, DN16, Wave, DN21, Web, Upbang$\}$ and $\mathcal{U}_2$ is the set of 10 nodes that have the top PageRank values. Note that we still compute the iterative approximation to PageRank in the usual way, except that we only consider the values $\pi_i$ and $x_i(t)$ for which $i\in\mathcal{U}$.
\begin{figure}[t]
\begin{center}
\includegraphics[width=\linewidth]{figures/loc.pdf}
\caption{$\mathcal{U}$-local topological convergence for the following subset of nodes: $\mathcal{U}_1=\{$Ripplefluke, Zig, Feather, Gallatin, SN90, DN16, Wave, DN21, Web, Upbang$\}$ . Similar to Fig.~\ref{Fig:Dolconv}, the left and right panels depict convergence of persistence diagrams for VR and kNN filtrations, respectively.
}\label{Fig:locd}
\end{center}
\end{figure}
\begin{figure}[t]
\begin{center}
\includegraphics[width=\linewidth]{figures/top10.pdf}
\caption{Same information as in Fig.~\ref{Fig:locd}, except we consider the subset $\mathcal{U}_1$ of nodes that have the largest PageRank values.
}\label{Fig:top10d}
\end{center}
\end{figure}
In Fig.~\ref{Fig:locd}(left) and (right), we depict convergence of VR and kNN persistence homology, respectively for the subset of nodes $\mathcal{U}_1$. Observer that convergence for VR persistent homology is similar to that for the full system. In contrast, observe in Fig.~\ref{Fig:locd}(right) that the kNN homology and rank orderings converge after approximately $t=5$ iterations for this subset.
In Fig.~\ref{Fig:top10d}, we depict the same information, except that we now consider the subset $\mathcal{U}_2$ of nodes with top PageRank values.
In this case, the VR convergence more closely aligns with $||{\bf x}(t)-\bm{\pi}||$ than when we considered the full set $\mathcal{V}$ of nodes or the subset $\mathcal{U}_1$. Moreover, the kNN homology and the rank orderings converge after approximately $t=9$ iterations in this case.
\section{Discussion}\label{sec:discusion}
In this paper, we developed an approach for topological data analysis (TDA) that utilizes k-nearest neighbor sets to define kNN complexes, kNN filtrations, and kNN persistent homology. Our approach was developed
in the spirit of discrete topology and by examining the relative ordering of points, as opposed to the precise distances between points.
Although kNN orderings are related to pairwise distances, we provided theory and many experiments highlighting important differences between persistent homology that is based on kNN filtrations versus Vietoris-Rips (VR) filtrations.
To gain theoretical insights into kNN-based TDA,
we investigated stability properties for the resulting persistence diagrams
in Sec.~\ref{sec: stability1} and convergence properties in
Sec.~\ref{sec:convergence1}.
While persistence diagrams resulting from kNN filtrations do not satisfy a stability theorem involving a universal bound on perturbed point sets (see Theorem~\ref{thm:kNN_sta}), we identified and characterized different notions of stability and convergence by identifying equivalence classes as well as bounds on the bottleneck distance between persistence diagrams for kNN homology that are based on the maximum difference for a kNN ordering. Our formulation of convergence also led to several types including global convergence, $\kappa$-bounded convergence and $\mathcal{U}$-local convergence. Moreover, we identified some relations among these types as in Theorem~\ref{thm: convergence_NNO} and Theorem~\ref{thm: convergence NNO1}.
As a concrete application, we applied persistent homology to study the topological convergence of an iterative method for solving PageRank. We showed that the convergence of persistence diagrams for VR filtrations closely relates to the convergence in vector norm (i.e., due to the stability theorem). In contrast, the convergence of persistence diagrams for kNN filtrations more closely resembles the convergence of rank ordering. In other words, convergence of kNN persistent homology can be used to predict how many iterations are required for ${\bf x}(t)$ to be sufficiently close to $\bm{\pi}$ such that the ranking of nodes---i.e., from 1 to N---has converged. Although we have focused on the PageRank algorithm, iterative algorithms for solving systems (e.g., root finding) are some of the most widely used numerical algorithms. We have shown that the existing TDA approach of VR filtrations coincides with geometrical (i.e., normed) convergence and thereby provides one perspective for the convergence of numerical algorithms. In contrast, kNN filtrations provide complementary insights from the perspective of a discrete topological space that is associated with the relative positioning of points. As such, we expect kNN complexes, filtrations, and persistent homology to have many applications for converging numerical algorithms beyond PageRank.
\section*{Acknowledgments} M.Q.L. and D.T. were supported in part by the Simons Foundation (Grant No. 578333). D.T also acknowledges the National Science Foundation (Grants No. DMS-2052720 and No. EDT-1551069).
\bibliographystyle{plain}
| {
"timestamp": "2022-06-13T02:01:21",
"yymm": "2206",
"arxiv_id": "2206.04725",
"language": "en",
"url": "https://arxiv.org/abs/2206.04725",
"abstract": "Graph-based representations of point-cloud data are widely used in data science and machine learning, including epsilon-graphs that contain edges between pairs of data points that are nearer than epsilon and kNN-graphs that connect each point to its k-nearest neighbors. Recently, topological data analysis has emerged as a family of mathematical and computational techniques to investigate topological features of data using simplicial complexes. These are a higher-order generalization of graphs and many techniques such as Vietoris-Rips (VR) filtrations are also parameterized by a distance epsilon. Here, we develop kNN complexes as a generalization of kNN graphs, leading to kNN-based persistent homology techniques for which we develop stability and convergence results. We apply this technique to characterize the convergence properties PageRank, highlighting how the perspective of discrete topology complements traditional geometrical-based analyses of convergence. Specifically, we show that convergence of relative positions (i.e., ranks) is captured by kNN persistent homology, whereas persistent homology with VR filtrations coincides with vector-norm convergence. Beyond PageRank, kNN-based persistent homology is expected to be useful to other data-science applications in which the relative positioning of data points is more important than their precise locations.",
"subjects": "Algebraic Topology (math.AT)",
"title": "Persistent Homology with k-nearest-neighbor Filtrations reveals Topological Convergence of PageRank",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9728307661011976,
"lm_q2_score": 0.7279754548076478,
"lm_q1q2_score": 0.7081969194033917
} |
https://arxiv.org/abs/math/9612217 | Subspace Arrangements over Finite Fields: Cohomological and Enumerative Properties | The enumeration of points on (or off) the union of some linear or affine subspaces over a finite field is dealt with in combinatorics via the characteristic polynomial and in algebraic geometry via the zeta function. We discuss the basic relations between these two points of view. Counting points is also related to the $\ell$-adic cohomology of the arrangement (as a variety). We describe the eigenvalues of the Frobenius map acting on this cohomology, which corresponds to a finer decomposition of the zeta function. The $\ell$-adic cohomology groups and their decomposition into eigenspaces are shown to be fully determined by combinatorial data. Finally, it is shown that the zeta function is determined by the topology of the corresponding complex variety in some important cases. | \section{Integral cohomology}
\let\refitem\DHrefitem
\newcommand\hodli{\mathop{\hbox{\vtop{\setbox0=\hbox{holim}%
\dimen0\wd0\box0
\nointerlineskip\hbox to\dimen0{\rightarrowfill}}}}}
\newcommand\Topos{\symb{{\Cal T}opos}}
\newcommand\Top{\symb{{\Cal T}op}}
\newcommand\Sets{\symb{{\Cal S}ets}}
\newcommand\Sh{\symb{Sh}}
\def\hbox{\tiny \'et}{\hbox{\tiny \'et}}
\candef{HW}
\begin{section}{The ${\Bbb Q}_\ell$-cohomology of subspace arrangements}
We will now consider the computation of the cohomology of a subspace
arrangement, and in particular its \'etale cohomology. Most of the results that we
will prove are well-known in the case of an arrangement over the real or complex
numbers, and are at least partially to be found in the literature (cf.~\cite{Ya})
over a general field (including positive characteristic). The new contribution
is that we keep track of the action of the Galois group of the base field, which
has important arithmetic significance. As the general ``philosophy of weights''
(cf.~\[D3]) would predict, we can use the same argument to get the mixed
Hodge structure in the complex case, a result which seems to be new (except for
the case of hyperplane configurations which is due to Kim \cite{Ki}). In this
paper we will also be concerned only with results on rational cohomology. In
that case one can use the action of the Galois group (or the rational mixed
Hodge structure) on cohomology to very quickly get to the desired result.
As we are dealing with varieties over arbitrary fields (our main interest being
the case of finite fields) we are forced to deal with \'etale cohomology of
algebraic varieties instead of classical cohomology, since the latter make sense
only over the real or complex numbers. Its construction is based on the
realisation that to define the usual cohomology one needs access not to the
topological space itself but only the category of sheaves on it. Though neither
the topological space underlying a complex algebraic variety nor its category of
sheaves can be constructed algebraically, a category with properties very
similar to this category of sheaves can be constructed in a purely algebraic
fashion. In general this category is most definitely not equivalent to the
category of sheaves on a topological space, and Grothendieck and his
collaborators \[G4] introduced an axiomatisation under the name of
\Definition{topos} that covers both these new
categories and the category of sheaves on a topological space. The category
associated to an algebraic variety (or more
generally a scheme) then goes under the name \Definition{\'etale topos}. Our
technical results would be most naturally formulated in terms of toposes and
diagrams of them, but in the interest of concreteness we will confine ourselves
to algebraic varieties (and implicitly their \'etale toposes).
\begin{remark}
It should be noted that in the case of a reasonable topological space the
topological space itself can be recovered from the category of sheaves on it,
hence not only is knowledge of the category of sheaves on a (reasonable)
topological space enough to be able to compute its cohomology, it actually is
equivalent to knowledge about the topological space itself. For the reader
interested in details we can add that ``reasonable'' in this context means that
every \Definition{irreducible} (not the union of two non-empty closed subsets)
closed subset is the closure of a unique point -- a condition almost always
fulfilled in practice.
\end{remark}
The construction of \'etale cohomology is quite involved. The original work of
Grothendieck and his collaborators \cite{D4,G4,G4a,G5} is still the only place
where a detailed treatment of its technical properties can be found. The
monographs \cite{FK} and \cite{Mi} are easier to approach, but deal mainly
with the case of smooth varieties. We will make a thumbnail sketch of how
the $\ell$-adic \'etale cohomology groups $H^i_{\hbox{\tiny \'et}}(X,{\Bbb Q}_\ell)$
that we shall use are constructed.
The analogy between the \'etale topos and the category of sheaves on the space
underlying a complex algebraic variety is a very close one, though there are
some definite differences. The most important is that the ``\'etale topology''
is not fine enough to capture the ordinary cohomology with integer coefficients;
one has to use finite coefficients. This is not an artifact of the \'etale topos
but depends on the fact that one wants an algebraically defined
cohomology. Consider for instance the fact that the first cohomology group, with
integer coefficients, of ${\Bbb C}^*:={\Bbb C}\setminus \{0\}$ is {{\Bbb Z}}. This reflects the
fact that there is a non-trivial covering space of ${\Bbb C}^*$ with structure group
{{\Bbb Z}}, given by $\symb{exp}\co{\Bbb C} \to {\Bbb C}^*$. As the exponential function is
transcendental this makes no algebraic sense, whereas the first cohomology group
with ${\Bbb Z}/n{\Bbb Z}$-coefficients describes covering spaces with structure group
${\Bbb Z}/n{\Bbb Z}$. In the case of ${\Bbb C}^*$ these are described using $n$'th roots, which
are eminently algebraic functions. One is therefore forced, to begin with, to
work with cohomology with finite coefficients. In that case one obtains a theory
very close to the classical topological one. In fact, a basic theorem
\cite[Exp.~XVI, Thm.~4.2]{G4a} says that for any algebraic variety $X$ over
the complex numbers and any finite group $A$ we have a natural isomorphism of
abelian groups $H^i_{\hbox{\tiny \'et}}(X,A)\cong H^i_{{\rm cl}}(X,A)$, where the subscripts cl (as in ``classical'') and $\hbox{\tiny \'et}$ denote respectively the
ordinary cohomology of the topological space
underlying $X$ and the \'etale cohomology. (This isomorphism is in fact induced
by a map from the topos of sheaves on the topological space of $X$ to the
\'etale topos and hence preserves supplementary structures such as cup
products.)
Having \'etale cohomology for finite coefficients one then {\em
defines}, for an algebraic variety $X$ over an algebraically closed field,
$H^i_{\hbox{\tiny \'et}}(X,{\Bbb Z}_\ell)$, $\ell$ a prime, to be the inverse limit
$\ili_nH^i_{\hbox{\tiny \'et}}(X,{\Bbb Z}/\ell^n{\Bbb Z})$. Using the result above on equality of \'etale and
classical cohomology for complex varieties $X$ and the universal coefficient
theorem,
one then gets $H^i_{\hbox{\tiny \'et}}(X,{\Bbb Z}_\ell)\cong H^i_{{\rm cl}}(X,{\Bbb Z})\bigotimes {\Bbb Z}_l$.
If $X$ is defined over a
field ${\Bbb F}$ which is not algebraically closed, then (using a not quite standard
notation) we define $H^i_{\hbox{\tiny \'et}}(X,{\Bbb Z}_\ell)$ to be the \'etale cohomology of $X$
considered as an algebraic variety over some algebraic closure of the base
field. The fact that $X$ is defined over ${\Bbb F}$ is then reflected in the fact that
we have a natural action of the Galois group of ${\Bbb F}$ on $H^i_{\hbox{\tiny \'et}}(X,{\Bbb Z}_\ell)$,
of which we will see examples later on. Finally, we put
$H^i_{\hbox{\tiny \'et}}(X,{\Bbb Q}_\ell):=H^i_{\hbox{\tiny \'et}}(X,{\Bbb Z}_\ell)\bigotimes_{{\Bbb Z}_\ell}{\Bbb Q}_\ell$, which
then is a finite dimensional vector space over the field ${\Bbb Q}_\ell$.
We will also normally
dispense with the $\hbox{\tiny \'et}$-subscript (the comparison theorem guarantees that confusion
should only rarely result.)
\begin{remark}
When the base field is not algebraically closed one may also consider the
\'etale cohomology of $X$ as a variety over the base field. That cohomology will
be an appropriate mixture of the \'etale cohomology of $X$ as an algebraic variety
over an algebraic closure and the Galois cohomology of the base field. As we
will not be interested in it we have chosen to use $H^i_{\hbox{\tiny \'et}}(X,{\Bbb Z}_\ell)$ to
denote the object which interests us.
\end{remark}
If we consider an arrangement $\Aa$ of subspaces, $V_\Aa$ is by definition their
union. If there are only two of them we would have the Mayer-Vietoris long exact
sequence relating the cohomology of the arrangement, the two linear spaces
covering it and their intersection. In the general case one gets a
Mayer-Vietoris spectral sequence.
The closest analogue of the Mayer-Vietoris long exact sequence would be a
spectral sequence starting with an $E_1$-term. We prefer to start at the
$E_2$-term which, as usual, is more intrinsic. Starting from a covering of an
algebraic variety by closed subvarieties, one may consider the cohomology of
these subvarieties and their intersections. It forms a diagram of abelian groups
over the ordered set of intersections of covering subvarieties. (We will follow
the convention of \[ZZ] in that an ordered set is considered as a category with
morphisms $p \to q$ iff
$p \ge q$, so that a diagram $\{X_p\}$ over the poset has morphisms $X_p \to X_q$
when $p \ge q$). The $E_2$-term will involve the inverse limit and its right
derived functors of this diagram, and we begin by recalling a standard way of
computing such limits.
\begin{lemma}\label{std}
Let \Cal C be a category and $F_.$ a diagram of abelian groups. Then the groups
$\ili_{\Cal C}^*F_.$ are the cohomology groups of the complex $S^*(F)$ whose
$i$'th component is the product
\begin{displaymath}
\prod_{X_0\mapright{f_0}X_1\mapright{f_1} \dots \mapright{f_{i-1}}X_i} F_{X_0}
\end{displaymath}
and whose differential is the alternating sum of the maps obtained by
composing two subsequent morphisms and from the structure map
\pil{f_0^*}{F_{X_0}}{F_{X_1}}.
\pro This is shown in \[JER].
\end{lemma}
\begin{lemma}\label{ss}
Let $\{X_p\}$ be a covering, closed under intersections, of an algebraic variety $X$ by
closed subvarieties. Let $P$ be the poset of these subvarieties ordered
by reverse inclusion. Then there is the Mayer-Vietoris spectral sequence
\begin{equation}
E_2^{i,j} = \ili_{P}{}^j{H^i(X_p,A)} \Rightarrow H^{i+j}(X,A)
\end{equation}
for any finite abelian group $A$.
\pro Let \pil{i_p}{X_p}X be the inclusion map. We may consider the complex
\begin{displaymath}
0 \to A \to \prod_p i_{p*}A \to \prod_{p \ge q}i_{p*}A \to \dots
\end{displaymath}
of sheaves on $X$, where $A$ is considered as the constant sheaf on $X$ and on
the subvarieties $X_p$. To show that this is
an exact sequence it is enough to show that it is exact
on all fibres. For a given point $x \in X$ the fibre at $x$ of this complex is
the (extended) cochain complex with values in A of the (abstract) simplex with
vertices the set of those $p$ for which $x \in X_p$. The simplex being
contractible, this is exact. Now, using that
$H^i(X,i_{p*}A)=H^i(X_p,A)$, as $i_p$ is a closed embedding, and the
spectral sequence of a resolution, we arrive at the Mayer-Vietoris spectral
sequence.
\end{lemma}
We will now apply this result to the rational cohomology of a subspace
arrangement. For this we need to recall some known facts about the action of the
Galois group on \'etale cohomology. The best control on the Galois action is
obtained when one ignores torsion, so we will look only at \'etale cohomology with
${\Bbb Q}_\ell$-coefficients (as defined above). It turns out that in positive
characteristic the properties of this cohomology is quite pathological when
$\ell$ is equal to the characteristic of the base field,
so from now on we will assume that {\em
the prime $\ell$ is invertible in the base field}.
Furthermore, the properties of the Galois
action on cohomology is simplest to formulate when the base field is finite,
so for the moment we will make that assumption and let $q$ denote its
cardinality. Then the Galois group is topologically cyclic (meaning that it has
a dense subgroup generated by one element) with a canonical generator called the
\Definition{Frobenius element}. It is the inverse of the map which raises an
element of an algebraic closure to its $q$'th power.
\begin{remark}
Often it is this map itself rather than its inverse that is called the Frobenius
element, in matters cohomological the present choice is the more suitable
however. The situation is somewhat confusing since the definition of the
Frobenius map in cohomology could appear to give the opposite impression.
However, there is a subtle distinction between the $q$'th power as a generator
of the Galois group of ${\Bbb F}_q$, and the $q$'th power as an algebraic map on, for
instance, affine space. More precisely, both induce actions on the cohomology of
a variety defined over ${\Bbb F}_q$ and these actions are each other's inverses. For
a more thorough discussion of this relation see \cite[pp.~76--81]{D4}.
\end{remark}
We have seen that the action of the Galois group on ${\Bbb Q}_\ell$-cohomology
of a variety defined over ${\Bbb F}_q$ is
given by a single linear map, usually called the \Definition{Frobenius map}. As
a first invariant of such a map one may look at the eigenvalues (defined over
some algebraically closed overfield and counted with the multiplicity in which
it appears as zeros of the characteristic polynomial). The following definition
may look very strong.
\begin{definition}
Let $F$ be a linear map of a finite dimensional vector space $V$ over a field of
characteristic zero, and let $q$ be a positive integer.
\item $F$ is said to be \Definition{pure of weight $n$ (wrt to $q$)} if all of its
eigenvalues are algebraic numbers, all of whose algebraic conjugates have
(complex) absolute value $q^{n/2}$.
\item $F$ is said to be \Definition{mixed} if $V$ has a filtration by $F$-stable
subspaces such that $F$ is pure of some weight on each successive subquotient of
the filtration (where the weight may depend on the subquotient). The set of the
weights of these subquotients will be called the \Definition{weights of $F$}.
\end{definition}
\begin{remark}
\item The condition that all the algebraic conjugates of an algebraic number
have the same absolute value is very strong. For instance, if one bounds
the degree and the absolute value then there are only finitely many such
numbers. This is seen by bounding the coefficients of the minimal polynomial,
coefficients that are also integers.
\item It is implicit in the definition that the set of weights of a mixed linear
operator is independent of the choice of filtration. This is obvious as the set
of weights can be immediately read off from the eigenvalues of $F$.
\end{remark}
A deep result of Deligne (\cite[3.3.8]{D1a}) says that if $X$ is smooth and
proper (over a finite field of cardinality $q$) then its degree $n$
${\Bbb Q}_\ell$-cohomology is pure of weight $n$, and without any assumptions the
cohomology is mixed.
\begin{example}
\item Affine space is the simplest example, we have
$H^0(\A^n,{\Bbb Q}_\ell)={\Bbb Q}_\ell$ and the rest of the cohomology groups are equal to
zero. The Frobenius map acts as the identity on this vector space and is hence
pure of weight zero.
\item For the projective line we have $H^1({\Bbb P}^1,{\Bbb Q}_\ell)=0$, and
$H^0({\Bbb P}^1,{\Bbb Q}_\ell)={\Bbb Q}_\ell$, with $F$ acting again as the identity, whereas
$H^2({\Bbb P}^1,{\Bbb Q}_\ell)$ is more interesting. As a ${\Bbb Q}_\ell$-vector space it is
1-dimensional. The Galois group of the base field acts by the inverse of the
\Definition {cyclotomic character}. The cyclotomic character is the character of
the Galois group for which an element {\gsi} acts by multiplication by the
$\ell$-adic number by which {\gsi} acts on roots of unity of order any power of
$\ell$. In particular, when the base field is finite (of cardinality
$q$) the Frobenius element acts by multiplication by $q$ on
$H^2({\Bbb P}^1,{\Bbb Q}_\ell)$. Therefore the weight is indeed 2.
In general we denote by
${\Bbb Q}_\ell(i)$ the 1-dimensional ${\Bbb Q}_\ell$-vector space on which the Galois group
acts by multiplication by the $i$'th power of the cyclotomic character. Then
from the fact (which can be proved by some of the methods used in the classical
case) that the cohomology of $N$-dimensional projective space is a truncated
polynomial ring and the fact that the Frobenius map preserves multiplication,
one sees that $H^{2i}({\Bbb P}^N,{\Bbb Q}_\ell)={\Bbb Q}_\ell(-i)$ when $i \le N$ (and all other
cohomology groups are zero), which confirms that it is indeed pure of weight $2i$.
\item Let us consider the multiplicative group $\mul:=\A^1\setminus \{0\}$. Then
$H^1(\mul,{\Bbb Q}_\ell)={\Bbb Q}_\ell(-1)$ which is not pure of weight 1 but rather of
weight 2. This does not contradict Deligne's theorem as $\mul$, though smooth, is
not proper.
More generally, punctured affine space $\A^n\setminus \{0\}$ behaves
cohomologically as an
odd-dimensional sphere with $H^0(\A^n\setminus \{0\},{\Bbb Q}_\ell)={\Bbb Q}_\ell$,
$H^{2n-1}(\A^n\setminus \{0\},{\Bbb Q}_\ell)={\Bbb Q}_\ell(-n)$, and the other
cohomology groups equal to zero.
\item One can also introduce \'etale cohomology with compact support, see below.
Intuitively this
is the reduced cohomology of the one-point compactification --- only that this
doesn't make sense in our setting since the one-point compactification,
even if defined, is not usually an algebraic
variety. For the affine spaces one has that there is only a single non-zero
cohomology group for cohomology with compact support:
$H_c^{2n}(\A^n,{\Bbb Q}_\ell)={\Bbb Q}_\ell(-n)$.
\end{example}
In all of these examples each weight was associated to only one cohomology
group. In the case of projective space that follows from Deligne's theorem,
in the others it seems to be more of an accident. In any case, they are all
situations to which the following theorem applies. Before going into its
formulation we want to comment on the cohomology of 1-point compactifications, which
occurs prominently in the study of subspace arrangements.
The 1-point compactification of a complex algebraic variety $X$ need not be an
algebraic variety. If one is interested only in its cohomology a substitute may
be found, the \Definition{cohomology with compact support}. For this one chooses
some realisation of $X$ as an open subset of some proper variety $j\co
X\hookrightarrow{\bar X}$ and then one considers the \'etale cohomology
$H^i(\bar X,j_!A)$, where $j_!A$ is the sheaf on $\bar X$ that is equal to $A$
on $X$ and has fibre 0 at all points of $\bar X$ outside of $X$ (``the extension
by zero''). This turns out to be independent of the choice of $j$ and computes
in the case that the base field is the complex numbers the reduced cohomology of
the 1-point compactification. For these cohomology groups the notation
$H^i_c(X,A)$ is used. (Properly speaking we should also add the subscript
$\hbox{\tiny \'et}$, as the cohomology with compact support makes excellent sense also in the
classical case. In the interest of readability we will dispense with that.)
There is now an analogue of the spectral sequence of Lemma \ref{ss} for
cohomology with compact support, the proof is the same (the essential point is
that when one embeds $X$ as an open subset of a proper variety one gets at the
same time a compactification of all the $X_p$ by taking their closures in the
compactification of $X$).
\begin{theorem}\label{degen}
Let $X$ be an algebraic variety that is the union of a family of closed subvarieties
$\{X_p\}$, closed under intersection. Suppose that there is a function {\gph} from
{{\Bbb N}}, the natural numbers, to subsets of the integers such that different numbers are
taken to disjoint sets, and that the degree $i$ cohomology
of each $X_p$ is mixed with weights in $\phi(i)$. Then, with the notations of Lemma
\ref{ss}, the spectral sequence of (loc.~cit.) degenerates at the
$E_2$-term. The same is true if instead cohomology with compact support is
considered.
\pro
Let us first assume that the base field is finite. If we can prove that
$E_2^{i,j}$ is mixed with weights in $\phi(i)$ we are finished, since then all
the differentials $d^{i,j}_k$ at the $E_k$-term, for $k \ge 2$, will be between
spaces of disjoint weights. However, Lemma \ref{std} presents $E_2^{i,j}$ as a
subquotient of spaces with weights in $\phi(i)$.
For the case of a general base field there are standard techniques for reducing
to the case of a finite base field, for which we refer to for instance
\cite[6.1]{BBD} rather than repeating them here. Very quickly described, one
first uses that base extension from one algebraically closed field to an
algebraically closed overfield does not change cohomology to reduce to the case
where the base field is finitely generated over the prime field. Then there is a
specialisation to a finite field, which again does not change cohomology.
\end{theorem}
\begin{remark}
\item The idea that one could use weights to show that spectral sequences
degenerate is not new. One of its first uses can be found in \[D2], where
it is applied to the study of the cohomology of the complement of a divisor with
normal crossings in a smooth and projective variety. However, there one is
using the mixed Hodge structure on classical cohomology rather than the Galois
action on \'etale cohomology. The arguments of (loc.~cit.) were one of the major
inspirations for our theorem.
\item The theorem applies to the cohomology (including cohomology with compact
support) of an affine subspace arrangement, as there only the cohomology of
affine spaces are involved and we have seen that they fulfill the required
condition. It also applies to projective subspace arrangements, again the
cohomology of projective spaces fulfills the condition. Another case is a
punctured central arrangement, where one considers the arrangement minus a
central point (this is the algebraic analogue of the spherical arrangement
associated to a central arrangement over the reals).
\end{remark}
To apply this result to the various cases of subspace arrangements
we need to compute the cohomology of some diagrams of abelian groups.
Recall from Section 2 the definition of the order complex $\Delta(P)$
of a poset $P$, and of subposets of type $P_{<p}$ and $P_{\le p}$.
\begin{proposition}\label{cohdiag}
Let $P$ be a finite poset and let $A$ be an abelian group.
\item[ii] \label{cohdiag:ii}Let $Q$ be an order ideal (i.e.,
a subset of $P$ such that any element of $P$ less than an
element of $Q$ is also in $Q$), and let $\Cal F_{A,Q}$ be the diagram which is 0
outside of $Q$ and constant with value $A$ on $Q$. Then we have
a natural isomorphism
\begin{displaymath}
\ili{}^j\Cal F_{A,Q} \cong H^j(\Delta(Q),A).
\end{displaymath}
\item[iii] Let $p \in P$ and let $\Cal F_{A,p}$ be the diagram with value $A$ on $p$
and 0 elsewhere. Then we have a natural isomorphism
\begin{displaymath}
\ili{}^j\Cal F_{A,p} \cong \widetilde H^{j-1}(\Delta(P_{<p}),A).
\end{displaymath}
(For this formula, recall that the reduced cohomology of the empty complex is $A$ in
degree $-1$ and 0 otherwise.)
\pro For part \refitem{ii} we simply use Lemma \ref{std}, which shows that the
higher inverse limits can be computed using a complex which is also the cochain
complex of $\Delta(Q)$ with values in $A$. As for \refitem{iii}, we have a
natural inclusion $\Cal F_{A,p}\hookrightarrow \Cal F_{A, P_{\le p}}$,
whose quotient is $\Cal F_{A, P_{<p}}$. Using the long exact sequence of
higher inverse limits, part \refitem{ii} and the fact that
$\Delta(P_{\le p})$ is contractible, we immediately reach the desired conclusion.
\end{proposition}
\begin{example}
\item Consider the cohomology of an affine arrangement $\Cal A$. The only
non-trivial cohomology group of affine spaces is $H^0(-,A)=A$, so the spectral
sequence degenerates to the isomorphism
$H^i(V_\Aa,A) \cong H^i(\Delta(L_{\Aa}\setminus\{\hat0\}),A)$.
\item If $\Cal A$ is central we may remove the central point to get an
arrangement of punctured affine spaces (in the real or complex case it is
homotopic to the associated spherical arrangement). Again the condition of Theorem
\ref{degen} is fulfilled. Furthermore, $H^{2i}(-,{\Bbb Q}_\ell)$ is zero for $i>0$ and
${\Bbb Q}_\ell$ for $i=0$, and $H^{2i-1}(-,{\Bbb Q}_\ell)$ is ${\Bbb Q}_\ell(-i)$ on
$i$-dimensional elements of the
intersection lattice and zero otherwise. This then is a direct sum
of diagrams of the type considered in the proposition. Hence,
letting $P=L_{\Aa}\setminus \{\hat0\}$ we get
\begin{equation}\label{centrallim}
\begin{array}{lcl}
\ili{}^j H^0(-,{\Bbb Q}_\ell)&=&H^j(\Delta(P),{\Bbb Q}_\ell)\\
\ili{}^j H^{2i-1}(-,{\Bbb Q}_\ell)&=&\Dsum_{\dim p = i}\widetilde
H^{j-1}(\Delta(P_{<p}),{\Bbb Z})\bigotimes {\Bbb Q}_\ell(-i).
\end{array}
\end{equation}
\item If $\Cal A$ is a projective arrangement, then $H^{2*+1}(-,A)=0$ and
$H^{2i}(-,A)$ is the constant diagram on the elements of dimension greater than
or equal to $i$. Thus, we may again use the proposition to compute the higher
inverse limits and get the result
\begin{equation}\label{projlim}
\ili{}^j H^{2i}(-,{\Bbb Q}_\ell)=H^j(\Delta(P^{\ge i}),{\Bbb Z})\bigotimes {\Bbb Q}_\ell(-i),
\end{equation}
where $P=L_{\Aa}\setminus \{\hat0\}$ and $P^{\ge i}
:=\set{p \in P}{dim(p) \ge i}$.
\item Once more let $\Aa$ be an affine arrangement, but this time consider
cohomology with compact support. As has been noted, we get a spectral sequence
also in that case, and from the computation of the cohomology with compact
support of affine space we get that $H^{2*+1}_c(-,{\Bbb Q}_\ell)=0$ and that
$H^{2i}_c(-,{\Bbb Q}_\ell)$ is ${\Bbb Q}_\ell(-i)$ on $i$-dimensional elements of the
intersection lattice and zero otherwise. As in the central affine case we get
\begin{equation}\label{complim}
\ili{}^j H^{2i}_c(-,{\Bbb Q}_\ell)=\Dsum_{\dim p = i}\widetilde
H^{j-1}(\Delta(P_{<p}),{\Bbb Z})\bigotimes {\Bbb Q}_\ell(-i).
\end{equation}
\end{example}
Even if one sticks to the case of the base field being the complex
numbers there are advantages to considering diagrams of algebraic varieties. For
algebraic varieties over the complex numbers there is an additional structure on
its cohomology alluded to previously --- its {\it mixed Hodge structure}.
To give the definition of this notion we first recall that a \Definition{Hodge
structure of weight $n$} consists of a finitely generated abelian group $H_{\Bbb Z}$
and a decreasing finite filtration $F^m$ of $H_{\Bbb Z}\bigotimes{\Bbb C}$ by complex
sub-vector spaces such that $\bar F^m$, the complex conjugate of of $F^m$ (the
complex conjugation being induced by that of the second factor in the tensor
product), is a complementary subspace to $F^{n-m+1}$. We then recall
\cite[2.3.1]{D2} that a \Definition{mixed Hodge structure} is a finitely
generated abelian group $H_{\Bbb Z}$ together with one increasing finite filtration
$W_p$ of sub-vector spaces of $H_{\Bbb Q}:=H_{\Bbb Z}\bigotimes_{\Bbb Z}{\Bbb Q}$ and one
decreasing finite filtration $F^m$ by {\Bbb C}-sub-vector spaces of
$H_{\Bbb C}:=H_{\Bbb Z}\bigotimes_{\Bbb Z}{\Bbb C}$, such that for every $i$ the filtration induced by $F^m$ on
$W_i/W_{i-1}\bigotimes{\Bbb C}$ forms a Hodge structure of weight $i$. The
class (with the obvious morphisms) of mixed Hodge structures form an abelian
category. We also use the term \Definition{set of weights} of a mixed Hodge
structure for the set of integers for which $W_i\ne W_{i-1}$. If instead one
looks at only a rational vector space $H_{\Bbb Q}$ without a choice of $H_{\Bbb Z}$ one
speaks about a \Definition{rational Hodge structure}.
\begin{example}
Let ${\Bbb Z}(i)$ be the mixed Hodge structure with $H_{\Bbb Z}={\Bbb Z}$, $0=W_{2i-1} \subset
W_{2i}={\Bbb Q}$, and $0=F^{i+1} \subset F^i={\Bbb C}$\,; and similarly for ${\Bbb Q}(i)$, a
rational Hodge structure. This notation will be used to
describe the mixed Hodge structures relevant to subspace arrangements
after the next theorem.
\end{example}
There is a very strong analogy between mixed Hodge structure and the action of
the Galois group on \'etale cohomology. Parts of the analogy can actually be
proven --- for instance, if one considers the cohomology of a complex algebraic
variety, then the filtration on rational cohomology induced from the Hodge
structure coincides with the weight filtration with respect to the Galois
action. We illustrate this analogy by giving another proof of the degeneration
of the Mayer-Vietoris spectral sequence when the base field is the complex
numbers.
\begin{theorem}\label{deg}
Let $X$ be a complex algebraic variety that is the union of a family of closed subvarieties
$\{X_p\}$, closed under intersection. Suppose that there is a function {\gph} from
{{\Bbb N}}, the natural numbers, to subsets of the integers such that different numbers are
taken to disjoint sets, and that the degree $i$
cohomology of each $X_p$
has weights, with respect to its mixed Hodge structure, in
$\phi(i)$. Then, with the notation of Lemma \ref{ss}, the spectral sequence of
(loc.~cit.) with ${\Bbb Q}$-coefficients degenerates at the $E_2$-term.
\pro
We first need to prove that the spectral sequence is a spectral sequence of
rational mixed Hodge structures. For this we note another way of constructing
it. Namely, we consider the simplicial complex variety $sX_.$ for which $sX_j$
is the disjoint union of the $X_{i_0}$ over the index set $\{i_0 \ge i_1 \ge
\dots \ge i_j\}$ with the obvious structure maps. The spectral sequence
(cf.~\cite[5.3.3.3]{D2a}) applied to the constant sheaf {{\Bbb Z}} of this simplicial
variety converges to the cohomology of $X$ and has an $E_1$-term which is the
standard complex for computing $\ili_P\{H^i(-,{\Bbb Z})\}$, and hence gives our
spectral sequence from the $E_2$-term on. According to \cite[8.3.5]{D2a} this is
a spectral sequence of mixed Hodge structures which becomes a spectral sequence
of rational mixed Hodge structures when tensored with ${\Bbb Q}$.
If we can prove that $E_2^{i,j}$ is mixed with weights in $\phi(i)$ we are
finished, since then all the differentials $d^{i,j}_k$, for $k \ge 2$, will be
between rational mixed Hodge structures of disjoint weights. However,
Lemma \ref{std} presents
$E_2^{i,j}$ as a subquotient of spaces with weights in $\phi(i)$.
\end{theorem}
\begin{remark}
We have the following computations of the mixed Hodge structure on the
cohomology of affine space, puctured affine space and projective space,
completely analogous to the action of the Frobenius on \'etale cohomology:
\begin{displaymath}
\begin{array}{lcl}
H^0(\A^n,{\Bbb Z})&=&{\Bbb Z}(0),\\
H^0(\A^n\setminus\{0\},{\Bbb Z})&=&{\Bbb Z}(0),\\
H^{2n-1}(\A^n\setminus\{0\},{\Bbb Z})&=&{\Bbb Z}(n),\\
H^{2i}({\Bbb P}^n,{\Bbb Z})&=&{\Bbb Z}(i),\;i \le n.
\end{array}
\end{displaymath}
Hence the theorem may be applied to subspace arrangements.
\end{remark}
Having developed the necessary general tools, we now want to collect our results
as applied to subspace arrangements over finite fields. In that case one extra
refinement is possible which is given in the following lemma.
\begin{lemma}\label{caniso}
Let assumptions be as in Theorem \ref{degen} and assume that the base field is
finite. Then there is a canonical isomorphism between $H^*(X,{\Bbb Q}_\ell)$
and the $E_2$-term of the spectral sequence. This isomorphism preserves
the action of the fundamental group of the base field.
\pro We may use the action of the Frobenius map to split up $H^*(X,{\Bbb Q}_\ell)$ as a
sum of generalised eigenspaces under it. Since each such eigenspace occurs in
just one row of the $E_2$-term we get the canonical isomorphism.
\end{lemma}
\begin{remark}
It is not possible to conclude from what we have proven so far that this result
remains true for a general field or has an analogue for mixed Hodge
structures. The reason for this is that it would be possible for the extensions
provided by the spectral sequence to be non-trivial, there are indeed
non-trivial extensions between the Galois representations (resp.~mixed Hodge
structures) involved. It will be proved elsewhere that in the case of subspace
arrangements these possibilities are not realised and in fact the isomorphisms
of the theorem exist for ${\Bbb Z}_\ell$-cohomology (resp.~for cohomology with its mixed
Hodge structure).
\end{remark}
We now collect the various results obtained so far about the cohomology of
unions $V_{\Aa}$ of subspace arrangements over finite fields. To simplify
statements of formulas in this and the following theorem we introduce,
just as in the classical case, reduced $\ell$-adic cohomology
$\ti H^*(X,{\Bbb Q}_\ell)$. This differs from ordinary cohomology
for all varieties $X$ only in one dimension, namely
$\ti H^0(X,{\Bbb Q}_\ell)=H^0(X,{\Bbb Q}_\ell)/{\Bbb Q}_\ell$ when $X$ is non-empty, and
$\ti H^{-1}(X,{\Bbb Q}_\ell)={\Bbb Q}_\ell$ when $X$ is empty.
\newpage
\begin{theorem}\label{coharr}
Let $\Aa$ be a subspace arrangement over a finite field, \pil d{L_\Aa}{{\Bbb Z}}
the dimension function of its intersection semilattice and $P := L_{\Aa}\setminus
\{\hat0\}$.
\item If $\Cal A$ is an affine arrangement then we have a canonical isomorphism
\begin{displaymath}
H^*(V_\Aa,{\Bbb Z}_\ell)\cong H^*(\Delta(P),{\Bbb Z}_\ell),
\end{displaymath}
which respects the action of the Frobenius map if it is assumed to
act trivially on the right hand side.
\item\label{affcpct}
If $\Aa$ is an affine arrangement then we have a canonical isomorphism
\begin{displaymath}
H^*_{c}(V_\Aa,{\Bbb Q}_\ell)\cong \bigoplus_{p \in P}
\widetilde H^{*-2d(p)-1}(\Delta(P_{<p}))\bigotimes{\Bbb Q}_\ell(-d(p)),
\end{displaymath}
which respects the Frobenius action if it is assumed to act trivially on the
cohomology of the order complexes $\Delta(P_{<p})$.
\item If $\Aa$ is a central arrangement then we have a canonical isomorphism
\begin{displaymath}
\ti H^*(V_\Aa\setminus \{0\},{\Bbb Q}_\ell)\cong\bigoplus_{p \in P}
\widetilde H^{*-2d(p)}(\Delta(P_{<p}))\bigotimes{\Bbb Q}_\ell(-d(p)),
\end{displaymath}
which respects the Frobenius action if it is assumed to act trivially on the
cohomology of the $\Delta(P_{<p})$.
\item\label{projarr} If $\Aa$ is a projective arrangement then we have a
canonical isomorphism
\begin{displaymath}
H^*(V_\Aa,{\Bbb Q}_\ell)\cong
\bigoplus_{0 \le j} H^{*-2j}(\Delta(P^{\ge j}))\bigotimes{\Bbb Q}_\ell(-j),
\end{displaymath}
where $P^{\ge j}:=\set{p \in P}{d(p) \ge j}$, which respects the Frobenius action if
it is assumed to act trivially on the cohomology of the $\Delta(P^{\ge j})$.
\begin{proof}
This follows from the results \ref{ss}, \ref{degen}, \ref{cohdiag} and
\ref{caniso}, together with (\ref{centrallim}), (\ref{projlim}) and
(\ref{complim}).
\end{proof}
\end{theorem}
\medskip
Finally we also collect the consequences of our results for the cohomology of the complements of subspace arrangements over finite fields.
\begin{theorem}\label{cohcomp}
Let $\Aa$ be a subspace arrangement in a space of $n$ dimensions over a
finite field, \pil d{L_\Aa}{{\Bbb Z}} the dimension function of its intersection
semilattice, and $M_\Aa$ the complement of the union $V_\Aa$. Furthermore,
let $P=L_{\Aa}\setminus \{\hat0\}$.
\item If $\Aa$ is affine we have canonical isomorphisms
\begin{displaymath}
H^*_c(M_\Aa,{\Bbb Q}_\ell)\cong \bigoplus_{p \in P}\widetilde H^{*-2d(p)-2}
(\Delta(P_{<p}))\bigotimes{\Bbb Q}_\ell(-d(p)),
\end{displaymath}
when $* \ne 2n$ and $H^{2n}_c(M_\Aa,{\Bbb Q}_\ell)\cong {\Bbb Q}_\ell(-n)$,
and
\begin{displaymath}
\widetilde H^*(M_\Aa,{\Bbb Q}_\ell)\cong \bigoplus_{p \in P}\widetilde H_{2n-*-2d(p)-2}
(\Delta(P_{<p}))\bigotimes{\Bbb Q}_\ell(d(p)-n).
\end{displaymath}
\item If $\Aa$ is projective we have canonical isomorphisms
\begin{displaymath}
H^*_c(M_\Aa,{\Bbb Q}_\ell)\cong \bigoplus_{0 \le j \le n}\widetilde H^{*-2j-1}
(\Delta(P^{\ge j}))\bigotimes{\Bbb Q}_\ell(-j),
\end{displaymath}
when $* \ne 2n$ and $H^{2n}_c(M_\Aa,{\Bbb Q}_\ell)\cong {\Bbb Q}_\ell(-n)$,
and
\begin{displaymath}
\widetilde H^*(M_\Aa,{\Bbb Q}_\ell)\cong \bigoplus_{0 \le j \le n}\widetilde H_{2n-*-2j-1}
(\Delta(P^{\ge j}))\bigotimes{\Bbb Q}_\ell(j-n).
\end{displaymath}
All these isomorphisms respect the action of Frobenius if it is assumed
to act trivially on the order complexes occurring in the right hand sides.
\pro One enjoyable property of the cohomology with compact support is
``additivity'' for a closed subvariety and its complement. More precisely, for a
variety $Y$, a closed subvariety $F$ and its complement $U$ we have
(cf.~\cite[Exp.~XVII, 5.1.16.3]{G4a}) a long exact sequence
\begin{eqnarray*}
0 \to H^0_c(U,{\Bbb Q}_\ell) \rightarrow &H^0_c(Y,{\Bbb Q}_\ell) \rightarrow& H^0_c(F,{\Bbb Q}_\ell) \rightarrow \\
H^1_c(U,{\Bbb Q}_\ell) \rightarrow &H^1_c(Y,{\Bbb Q}_\ell) \rightarrow& H^1_c(F,{\Bbb Q}_\ell) \rightarrow \dots,
\end{eqnarray*}
where the maps $H^i_c(Y,{\Bbb Q}_\ell) \to H^i_c(F,{\Bbb Q}_\ell)$ are the restriction
maps. If we apply this to an affine arrangement, using the computation of the
cohomology of affine space as well as that of the union $V_\Aa$, we get that
$H^{2n}_c(M_\Aa,{\Bbb Q}_\ell)=H^{2n}_c(\A^n,{\Bbb Q}_\ell)={\Bbb Q}_\ell(-n)$ and that
$H^i_c(M_\Aa,{\Bbb Q}_\ell)=H^{i-1}_c(V_\Aa,{\Bbb Q}_\ell)$ for $i \ne 2n$. Now, $M_\Aa$ is
a smooth variety and so we may apply the Poincar\'e duality theorem
\cite[Exp. XVIII,3.2.6.1]{G4a}, which says that the cup-product
$H^i(M_\Aa,{\Bbb Q}_\ell)\bigotimes_{{\Bbb Q}_\ell} H^{2n-i}_c(M_\Aa,{\Bbb Q}_\ell)\to
H^{2n}_c(M_\Aa,{\Bbb Q}_\ell)$ composed with the trace map $H^{2n}_c(M_\Aa,{\Bbb Q}_\ell)\to
{\Bbb Q}_\ell(-n)$ gives a perfect pairing. This gives that $H^i(M_\Aa,{\Bbb Q}_\ell)$ is
canonically isomorphic to $H^{2n-i}_c(M_\Aa,{\Bbb Q}_\ell)^*\bigotimes
{\Bbb Q}_\ell(-n)$. Using this formula, the relation
$H^i_c(M_\Aa,{\Bbb Q}_\ell)=H^{i-1}_c(V_\Aa,{\Bbb Q}_\ell)$ for $i \ne 2n$,
Theorem \ref{affcpct} and the universal coefficient formula applied to the
cohomology of the $\Delta(P_{<p})$, we get the first part of the theorem.
As for the second, we consider again the long exact sequence of cohomology with
compact support, using that for a proper variety it is equal to cohomology
without compact support, so that we can use Theorem \ref{projarr}. Now, it is
clear that the restriction map $H^{2i}({\Bbb P}^n,{\Bbb Q}_\ell) \to H^{2i}(V_\Aa,{\Bbb Q}_\ell)$
maps ${\Bbb Q}_\ell(-i)$ to $1\otimes {\Bbb Q}_\ell(-i)\subseteq H^0(\Delta(P^{\ge
i}))\otimes {\Bbb Q}_\ell(-i)$. This is evidently an injection when $i \le m$, where
$m$ is the maximal dimension of subspaces in $\Cal A$, so the long exact
sequence splits up into the desired isomorphisms for cohomology with compact
support. Using duality gives the formula for cohomology without compact support.
\end{theorem}
\begin{remark}
Analogs of the formulas in Theorems \ref{coharr} and \ref{cohcomp} for arrangements
over the real and complex numbers were proved by Goresky and MacPherson \cite{GM},
Ziegler and \v{Z}ivaljevi\'c \cite{ZZ} and others. Some of these formulas in \'etale
cohomology version appear in the paper by Yan \cite{Ya}, however without the
decomposition into eigenspaces under Frobenius.
\end{remark}
\end{section}
\begin{section}{Arrangements over the integers}
In this section we shall be concerned with arrangements specified by
integer forms. Let a ${\Bbb Z}$-{\em arrangement} ({\em affine} resp.~{\em
projective}) mean an arrangement $\Aa=\{K_1,\dots,K_t\}$ where each
subspace is specified by a certain collection of linear forms (general
resp.~homogenous) with integer coefficients. Thus, a ${\Bbb Z}$-arrangement
is really a list of linear forms over ${\Bbb Z}$ partitioned into $t$
groups. With a ${\Bbb Z}$-arrangement $\Aa$ we associate on the one hand the
complex subspace arrangement $\Aa_{\Bbb C}$ (affine or projective, as the
case may be) obtained by interpreting the given ${\Bbb Z}$-forms over ${\Bbb C}$;
and on the other hand the subspace arrangement $\Aa_q$ over the finite
field ${\Bbb F}_q$ obtained from the ${\Bbb Z}$-forms by reduction modulo $p$, for
arbitrary prime powers $q=p^\alpha$.
\begin{remark}
We could here equally well replace {{\Bbb Z}} with an arbitrary number
ring. Except for trivial notational changes nothing in the arguments to follow
would need to be modified.
\end{remark}
\begin{Lem}\label{A} Let $\Aa$ be a ${\Bbb Z}$-arrangement and $p$ a
prime. Let $\varepsilon$ be the identity map on the set of subspaces of
$\Aa$. Then the following conditions are equivalent:
\begin{enumerate}
\item{$\varepsilon$ extends to a dimension-preserving isomorphism
$L_{\Aa_{\Bbb C}}\cong L_{\Aa_p}$;}
\item{$\varepsilon$ extends to a dimension-preserving isomorphism
$L_{\Aa_{\Bbb C}}\cong L_{\Aa_{p^\alpha}}$, for all $\alpha\ge1$;}
\item{$\text{\em{rank}}_{\Bbb C}\{\ell_1,\dots,\ell_g\}=\text{\em{rank}}_{{\Bbb F}_p}\{\ell_1,\dots,\ell_g\}$
for any collection $\ell_1,\dots,\ell_g$ of linear forms from $\Aa$,
containing for each subspace either all of its defining forms or none
of them.}
\end{enumerate}
\end{Lem}
\begin{proof}
The implications (ii)$\Rightarrow$(i)$\Rightarrow$(iii) are immediate. For
(iii)$\Rightarrow$(ii) one checks that the linear algebra in ${\Bbb F}_{p^\alpha}$ of
the given forms (reduced modulo $p$) takes place in the subfield
${\Bbb F}_p$.
\end{proof}
We shall call a prime $p$ {\em good} with respect to a
${\Bbb Z}$-arrangement $\Aa$ if it satisfies the conditions of the lemma,
otherwise {\em bad}. Part (iii) shows that for a given $\Aa$ there is
only a finite number of bad primes (these being the divisors of a
finite collection of determinants in the $\ell_i$'s). In the special
case when $\Aa$ is a hyperplane arrangement condition (iii) can be
expressed by saying that $\Aa$ determines the same matroid over ${\Bbb C}$
and over ${\Bbb F}_p$.
\begin{example} The $k$-equal arrangements defined in Section 2 are
${\Bbb Z}$-arrangements, and $\Aa_{n,k}$ has no bad primes, while
${\E{B}}_{n,k}$ and ${\E{D}}_{n,k}$ have the bad prime 2.
\end {example}
\medskip
Let $\Aa$ be a $d$-dimensional projective ${\Bbb Z}$-arrangement and
$q=p^\alpha$, where $p$ is a good prime. Let $L_\Aa=L_{\Aa_{\Bbb C}}\cong
L_{\Aa_q}$ and $L^{\ge j}_{\Aa}=\{x\in L_\Aa\mid\dim(x)\ge
j\}\smallsetminus\{\hat0\}$. Define
\begin{equation}\label{m}
\beta^{\ge j}_i=\dim_{\Bbb Q} H_i \left(L^{\ge j}_\Aa, {\Bbb Q}\right).
\end{equation}
These order homology Betti numbers of the $j$-truncated intersection
lattices are possibly nontrivial only in the range $0\le i\le d-j\le
d$. We will call the triangular array $(\beta^{\ge j}_i)$ the {\em
beta triangle} of $\Aa$.
A formula of Ziegler and \v{Z}ivaljevi\'c \cite[Prop. 2.15]{ZZ}
\cite[Coroll. 6.7]{WZZ}, which is the complex analog of
Theorem \ref{projarr}, shows that
\begin{equation}\label{o}
\beta^{\Bbb C}_i:=\dim_{\Bbb Q} H_i \left(V_{\Aa_{\Bbb C}}, {\Bbb Q}\right)=\sum^d_{j=0} \beta^{\ge
j}_{i-2j},
\end{equation}
and from formula \eqref{i} we have that
\begin{equation}\label{p}
Z\left(V_{\Aa_q};t\right)=\prod^d_{j=0}\left(1-q^jt\right)^{\sum(-1)^{i+1}
\beta^{\ge j}_i}.
\end{equation}
Thus both the rational Betti numbers of the union of the complex arrangement
$\Aa_{\Bbb C}$ and the zeta function of the discrete arrangement $\Aa_q$ are
governed by the same primitive combinatorial data, namely the beta triangle of
$\Aa$.
\begin{example}
Here is the beta triangle $(\beta^{\ge j}_i)$ of
$\Aa_{6,3}$ in the $(i,j)$ Cartesian plane:
\begin{equation*}
\begin{matrix}
20 \\
1 & 26\\
1 & 10 & 10\\
1 & 0 & 0 & 0
\end{matrix}
\end{equation*}
It follows that $\Aa_{6,3}$ has Betti numbers
$\beta^{\Bbb C}=(1,\,0,\,1,\,10,\,11,\,26,\,20)$ as a complex variety,
and zeta function
$$
Z\left(V_{\Aa_{6,3}};t\right)=\frac{\left(1-q^2t\right)^{25}}
{\left(1-t\right)\left(1-qt\right)\left(1-q^3t\right)^{20}}.
$$
as a variety over ${\Bbb F}_q$. Furthermore, from Theorem 1.1 we have that
\begin{align*}
P_0(t) &= 1-t\\
P_1(t) &=1\\
P_2(t) &=1-qt\\
P_3(t) &=(1-qt)^{10}\\
P_4(t) &=(1-qt)^{10}\left(1-q^2t\right)\\
P_5(t) &= \left(1-q^2t\right)^{26}\\
P_6(t) &=\left(1-q^3t\right)^{20}
\end{align*}
\end{example}
We will now show that for an important class of ${\Bbb Z}$-arrangements the
Betti numbers $\beta^{\Bbb C}_i$ of the complex variety and the zeta function
of the ${\Bbb F}_q$-variety determine each other. This is clearly not true
in general.
The intersection semilattice $L_\Aa$ is said to be {\em
rationally Cohen-Macaulay} if for all $x<y$ in
$\widehat{L}_\Aa=L_\Aa\cup\{\hat1\}$, where $\hat1$ is a new top element,
we have
$$
\widetilde{H}_i\left(\Delta(x,y),{\Bbb Q}\right)=
,\qquad\text{for all\,\,}i<\dim\Delta(x,y).
$$
This definition is via a theorem of Reisner equivalent to the
Cohen-Macaulayness of the Stanley-Reisner ring of $L_\Aa$. See
Stanley \cite{S1} for more about this concept.
We will consider ${\Bbb Z}$-arrangements $\Aa$ whose semilattice $L_\Aa$ is
both Cohen-Macaul\-ay and hereditary (defined in connection with Theorem
3.8). Then every maximal chain in $L_\Aa$ has the form
$x_0>x_1>\dots>x_d>\hat0$ with $\dim(x_i)=i$ for $0\le i\le
d$. Examples are all hyperplane arrangements, many of the orbit
arrangements $\Aa_\lambda$ shown to the shellable by Kozlov \cite{Ko}, and
the arrangements corresponding to Cohen-Macaulay simplicial complexes
considered in Bj\"orner and Sarkaria \cite{BSar}. The following
generalises the main result of \cite{BSar}.
\begin{Thm}\label{L}
Let $\Aa$ be a $d$-dimensional projective ${\Bbb Z}$-arrangement such that
$L_\Aa$ is Cohen-Macaulay and hereditary, and let $q$ be a power of a good
prime. Then
$$
Z\left(V_{\Aa_q};t\right)=\prod^d_{j=0}\left(1-q^{d-j}t\right)^{(-1)^{j+1}
\beta^{\Bbb C}_{2d-j}-\delta_j},
$$
where $\delta_j=1$ if $j$ is odd and = 0 otherwise.
\end{Thm}
\begin{proof}
We will use the fact \cite[Theorem III.4.5]{S1} that the truncated
posets $L^{\ge j}_\Aa$ are Cohen-Macaulay for all $0\le j\le d$. Thus,
the beta triangle $(\beta^{\ge j}_i)$ has internal zeros, and ones along
the $i=0$ boundary:
\begin{equation*}
\beta^{\ge j}_i =
\begin{cases}
1, &\quad\text{if\,\,} i=0, \,0\le j<d\\
0, &\quad\text{if\,\,} 0<i<d-j
\end{cases}.
\end{equation*}
Therefore formula \eqref{o} simplifies as follows for $0\le j\le d$:
\begin{equation*}
\beta^{\Bbb C}_{2d-j} =
\begin{cases}
\beta^{\ge d-j}_j, &\quad\text{if $j=0$ or $j$ is odd}\\
\beta^{\ge d-j}_j+1,&\quad\text{otherwise}
\end{cases}.
\end{equation*}
These two formulas imply
\begin{equation*}
\sum_i(-1)^{i+1} \beta^{\ge d-j}_i =
\begin{cases}
\beta^{\Bbb C}_{2d-j}-1, &\quad\text{if $j$ is odd}\\
-\beta^{\Bbb C}_{2d-j}, &\quad\text{otherwise}
\end{cases},
\end{equation*}
which because of formula \eqref{p} is equivalent to the theorem.
\end{proof}
Note that the rational Betti numbers $\beta^{\Bbb C}_{2d-j}$ for $0\le j\le d$
appearing in the theorem are the only essential ones, since the structure
of the beta triangle in the Cohen-Macaulay case shows that for
$0\le j< d$:
\begin{equation*}
\beta^{\Bbb C}_{j} =
\begin{cases}
1, &\quad\text{if $j$ is even}\\
0, &\quad\text{otherwise}
\end{cases}.
\end{equation*}
The preceding proof hinges on the very simple structure of the beta
triangle $(\beta^{\ge j}_i)$ given by the almost total vanishing of
Betti numbers in the Cohen-Macaulay case. The beta triangle has
simplified structure also for some other arrangements, including the
$k$-equal arrangements $\Aa_{n,k}$ and ${\E{B}}_{n,k}$, as we will now show.
Let us say that an intersection semilattice $L_\Aa$ is {\em
mod-$m$-pure} if the lengths of all maximal chains are congruent mod
$m$.
\begin{Thm}\label{M}
Suppose that $L_\Aa$ is hereditary, mod-$m$-pure and
CL-shellable.
\newline Then $\widetilde\beta^{\ge j}_i=0$, unless $i+j\equiv d$ (mod $m$).
\end{Thm}
\begin{proof}
Let $0\le j\le d$. As in the proof of Theorem 3.8 we conclude that
$L^{\ge j}_\Aa$ is CL-shellable and mod-$m$-pure. Furthermore, since
$L_\Aa$ is hereditary and $\dim \Aa=d$ there is in $L^{\ge j}_\Aa$ a
maximal chain $x_j>x_{j+1}>\dots>x_d$ with $\dim(x_i)=i$ for all $j\le
i\le d$. Hence, all maximal chains of $L^{\ge j}_\Aa$ have lengths
congruent to $d-j$ (mod $m$), and by \cite[Theorem 5.9]{BW1}
$\widetilde\beta^{\ge j}_i\no=0$ is possible only if $i\equiv d-j$ (mod $m$).
\end{proof}
The intersection lattices of $\Aa_{n,k}$ and ${\E{B}}_{n,k}$ satisfy
these conditions
with $m=k-2$. The nontrivial part here is the CL-shellability, which
was shown in \cite{BW1} and \cite{BSag} respectively.
The material of this section has parallels in the affine case. The
results come out in
essentially the same way, and we will not repeat the arguments.
\end{section}
\vfill\eject
| {
"timestamp": "1998-06-13T17:49:57",
"yymm": "9612",
"arxiv_id": "math/9612217",
"language": "en",
"url": "https://arxiv.org/abs/math/9612217",
"abstract": "The enumeration of points on (or off) the union of some linear or affine subspaces over a finite field is dealt with in combinatorics via the characteristic polynomial and in algebraic geometry via the zeta function. We discuss the basic relations between these two points of view. Counting points is also related to the $\\ell$-adic cohomology of the arrangement (as a variety). We describe the eigenvalues of the Frobenius map acting on this cohomology, which corresponds to a finer decomposition of the zeta function. The $\\ell$-adic cohomology groups and their decomposition into eigenspaces are shown to be fully determined by combinatorial data. Finally, it is shown that the zeta function is determined by the topology of the corresponding complex variety in some important cases.",
"subjects": "Algebraic Geometry (math.AG)",
"title": "Subspace Arrangements over Finite Fields: Cohomological and Enumerative Properties",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9728307661011976,
"lm_q2_score": 0.7279754548076478,
"lm_q1q2_score": 0.7081969194033917
} |
https://arxiv.org/abs/1606.03336 | Solution for the nonlinear relativistic harmonic oscillator via Laplace-Adomian decomposition method | Far as we know there are not exact solutions to the equation of motion for a relativistic harmonic oscillator. In this paper, the relativistic harmonic oscillator equation which is a nonlinear ordinary differential equation is studied by means of a combined use of the Adomian Decomposition Method and the Laplace Transform (LADM). The results that we have obtained, a series of powers of functions, have never been reported and show a very good match when compared with other approximate solutions, obtained by different methods. The method here proposed works with high degree of accuracy and because it requires less computational effort, it is very convenient to solve this kind of nonlinear differential equations. | \section{Introduction}
\label{Intro}
\noindent Many of the phenomena that arise in the real world can be described by means of nonlinear partial and ordinary differential equations and, in some cases, by integral or differo-integral equations. However, most of the mathematical methods developed so far, are only capable to solve linear differential equations. In the 1980's, George Adomian (1923-1996) introduced a powerful method to solve nonlinear differential equations. Since then, this method is known as the Adomian decomposition method (ADM) \cite{ADM-0,ADM-1}. The technique is based on a decomposition of a solution of a nonlinear differential equation in a series of functions. Each term of the series is obtained from a polynomial generated by a power series expansion of an analytic function. The Adomian method is very simple in an abstract formulation but the difficulty arises in calculating the polynomials that becomes a non-trivial task. This method has widely been used to solve equations that come from nonlinear models as well as to solve fractional differential equations \cite{Das-1,Das-2,Das-3}. The advantage of this method is that, it solves the problem directly without the need of linearization, perturbation, or any other transformation, and also, reduces the massive computation works required by most other methods.
\noindent The relativistic nonlinear harmonic oscillator, was studied by first time in the middle of the last century \cite{Pen,Gold}. In spite of its importance in several models of physics,
exact solutions of its equation of motion have not been obtained.
In the present work we will use the Adomian decomposition method in combination with the Laplace transform (LADM) \cite{Waz-Lap} to determine the relativistic oscillator
solutions. This equation is a nonlinear ordinary differential equation that, in physics, is used to model a simple one dimensional harmonic oscillator with relativistic velocities.
We will decomposed the nonlinear terms of this equation using the Adomian polynomials and then, in combination with the use of the Laplace transform, we will obtain an algorithm to solve the problem subject to initial conditions. Finally, we will illustrate our procedure and the quality of the obtained algorithm by solving two examples in which the nonlinear differential equation is solved for different initial conditions .
\noindent Our work is divided in several sections. In ``The Adomian Decomposition Method Combined With Laplace Transform'' section, we present, in a brief and self-contained manner, the LADM. Several references are given to delve deeper into the subject and to study its mathematical foundation that is beyond the scope of the present work. In ``The Relativistic Harmonic Oscillator'' section, we also give a brief introduction to the model described by the relativistic harmonic oscillator and we compare our results with the previous one obtained in this respect.
In ``Solution of the Relativistic Harmonic Oscillator Equation Through LADM'' section,
we will establish that LADM can be used to solve this equation in a very simple way. In ``Application to the Relativistic Harmonic Oscillator'' section, we will show by means of two examples, the quality and precision of our method, comparing the obtained results with existing approximate solutions available in the literature and obtained by other methods.
Finally, in the ``Conclusion and Summary" section, we present our conclusions.
\section{The Adomian Decomposition Method Combined with Laplace Transform}
\label{H-O-0}
\noindent The ADM is a method to solve ordinary and nonlinear differential equations. Using this method is possible to express analytic solutions in terms of a series \cite{ADM-1}.
In a nutshell, the method identifies and separates the linear and nonlinear parts of a differential equation. Inverting and applying the highest order differential operator that is contained in the linear part of the equation, it is possible to express the solution in terms of the rest of the equation affected by the inverse operator. At this point, the solution is proposed by means of a series
with terms that will be determined and that give rise to the Adomian Polynomials \cite{Waz-0}. The nonlinear part can also be expressed in terms of these polynomials.
The initial (or the border conditions) and the terms that contain the independent variables will be considered as the initial approximation. In this way and by means of a recurrence relations, it is possible to find the terms of the series that give the approximate solution of the differential equation. In the next paragraph we will see how to use the Adomian decomposition method in combination with the Laplace transform (LADM).\\
\noindent Let us consider the homogeneous differential equation of second order:
\begin{equation}
\frac{d^{2}x}{dt^2}+N(x)=0\label{eq:y1}
\end{equation}
with initial conditions
\begin{equation}
x(0)=\alpha,\quad x'(0)=\beta\label{eq:y2}
\end{equation}
where $\alpha$, $\beta$ are real constants and $N$ is a nonlinear operator acting on the dependent variable $x$ and some of its derivatives.\\
In general, if we consider the second-order differential operator $L_{tt}=\frac{\partial^2}{\partial t^2}$, then the equation (\ref{eq:y1}) could be written as
\begin{equation}
L_{tt}x(t)+N(x(t))=0.\label{eq:y3}
\end{equation}
Solving for $L_{tt}x(t)$, we have
\begin{equation}
L_{tt}x(t)=-N(x(t))\label{eq:y4}.
\end{equation}
The LADM consists of applying Laplace transform (denoted throughout this paper by $\mathcal{L}$) first on both sides of Eq. (\ref{eq:y4}), obtaining
\begin{equation}
\mathcal{L}\{L_{tt}x(t)\}= -\mathcal{L}\{N(x(t))\}.\label{eq:y5}
\end{equation}
An equivalent expression to (\ref{eq:y5}) is
\begin{equation}
s^{2}x(s)-sx(0)-x'(0)= -\mathcal{L}\{Nx(t)\},\label{eq:y6}
\end{equation}
using the initial conditions (\ref{eq:y2}), we have
\begin{equation}
x(s)=\frac{\alpha}{s}+\frac{\beta}{s^2}-\frac{1}{s^2}\mathcal{L}\{N(x(t))\}\label{eq:y7}
\end{equation}
now, applying the inverse Laplace transform to equation (\ref{eq:y7})
\begin{equation}
x(t)=\alpha+\beta t-\mathcal{L}^{-1}\big[\frac{1}{s^2}\mathcal{L}\{N(x(t))\}\big]. \label{eq:y8}
\end{equation}
\noindent The ADM method proposes a series solution $x(t)$ given by,
\begin{equation}
x(t)= \sum_{n=0}^{\infty}x_{n}(t).\label{eq:y7-1}
\end{equation}
The nonlinear term $N(x)$ is given by
\begin{equation}
N(x)= \sum_{n=0}^{\infty}A_{n}(x_{0},x_{1},\ldots, x_{n})\label{eq:y8-1}
\end{equation}
where $\{A_{n}\}_{n=0}^{\infty}$ is the so-called Adomian polynomials sequence established in \cite{Waz-0} and \cite{Ba} and, in general, give us term to term:\\
$A_{0}=N(x_0)$\\
$A_{1}=x_{1}N'(x_0)$\\
$A_{2}=x_{2}N'(x_{0})+\frac{1}{2}x_{1}^{2}N''(x_0)$\\
$A_{3}=x_{3}N'(x_{0})+x_{1}x_{2}N''(x_0)+\frac{1}{3!}x_{1}^{3}N^{(3)}(x_{0})$\\
$A_{4}=x_{4}N'(x_{0})+(\frac{1}{2}x_{2}^{2}+x_{1}x_{3})N''(x_0)+\frac{1}{2!}x_{1}^{2}x_{2}N^{(3)}(x_{0})+\frac{1}{4!}x_{1}^{4}N^{(4)}(x_0)$\\
$ \vdots$.\\
\noindent Other polynomials can be generated in a similar way. Some other approaches to obtain Adomian's polynomials can be found in \cite{Duan,Duan1}.\\
\noindent Using (\ref{eq:y7-1}) and (\ref{eq:y8-1}) into equation (\ref{eq:y8}), we obtain,
\begin{equation}
\sum_{n=0}^{\infty}x_{n}(t)= \alpha+\beta t-\mathcal{L}^{-1}\Big[\frac{1}{s^2}\mathcal{L}\{\sum_{n=0}^{\infty}A_{n}(x_{0},x_{1},\ldots, x_{n})\}\Big].\label{eq:y10}
\end{equation}
From the equation (\ref{eq:y10}) we deduce the recurrence formula:
\begin{equation}
\left\{
\begin{array}{ll}
x_{0}(t)=\alpha+\beta t,\\
x_{n+1}(t)=-\mathcal{L}^{-1}\Big[\frac{1}{s^2}\mathcal{L}\{A_{n}(x_{0},x_{1},\ldots, x_{n})\}\Big],\;\; n=0,1,2,\ldots
\end{array}
\right.\label{eq:y11}
\end{equation}
Using (\ref{eq:y11}) we can obtain an approximate solution of (\ref{eq:y1}), (\ref{eq:y2}) using
\begin{equation}
x(t)\approx \sum_{n=0}^{k}x_{n}(t),\;\; \mbox{where} \;\; \lim_{k\to\infty}\sum_{n=0}^{k}x_{n}(t)=x(t).\label{eq:y12}
\end{equation}
It becomes clear that, the Adomian decomposition method, combined with the Laplace transform needs less work in comparison with the traditional Adomian decomposition method. This method decreases considerably the volume of calculations. The decomposition procedure of Adomian will be easily set, without linearising the problem. With this approach, the solution is found in the form of a convergent series with easily computed components; in many cases, the convergence of this series is very fast and only a few terms are needed in order to have an idea of how the solutions behave. Convergence conditions of this series are examined by several authors, mainly in \cite{Y3,Y4,Y1,Y2}. Additional references related to the use of the Adomian Decomposition Method, combined with the Laplace transform, can be found in \cite{Waz-Lap,Khu,Y} and references therein.
\section{The Relativistic Harmonic Oscillator}
\label{H-O-1}
\noindent The equation of motion of the relativistic harmonic oscillator is given by the nonlinear differential equation \cite{Bia}:
\begin{equation}\label{Osc-1}
\frac{d^2x}{dt^2}+\Big[1-\Big(\frac{dx}{dt}\Big)^2\Big]^{\frac{3}{2}}x=0,\quad x(0)=0,\quad \frac{dx}{dt}(0)=\beta.
\end{equation}
This normalized, dimensionless form of the equation is based on taking the rest mass $m$ to be unity and the speed of light $c$ to also be unity \cite{Mic1}. It is easy to verify that the dimensionless length $x$ and the dimensionless time $t$ are related to the dimensional variables $\bar{x}$ and $\bar{t}$ through $x=\omega_{0}\bar{x}/c$ and $t=\omega_{0}\bar{t}$, respectively, where $\omega_{0}=\sqrt{k/m}$ is the angular frequency for the non-relativistic oscillator.\\
\noindent As far as we know, no exact solution of the nonlinear equation (\ref{Osc-1}) has yet been published and therefore the research work about equation
(\ref{Osc-1}) has been intense; a fundamental result reported in \cite{Mic1}
is that all the solutions of (\ref{Osc-1}) are periodic functions with the period dependent of the initial velocity
$\beta$.
In the same work, an approximation solution of (\ref{Osc-1}) was found using the harmonic balance method (HBM), it is given by
\begin{equation} \label{hbm}
\begin{split}
x_{\mbox{\tiny HBM}}(t) & = \frac{\beta}{\omega}\Big(\frac{3\beta^{4}+8\beta^{2}+64}{64}\Big)\sin(\omega t)-\frac{\beta^3}{24\omega}\Big(\frac{3\beta^{2}+128}{128}\Big)\sin(3\omega t) \\
& +\Big(\frac{3\beta^{5}}{640\omega}\Big)\sin(5\omega t),\;\; \mbox{where}\;\; \omega=\sqrt[4]{\frac{2-2\beta^2}{2-\beta^2}}\;\; \mbox{and}\;\; 0<\beta<1.
\end{split}
\end{equation}
Some more detailed work in the same direction was reported ten years later in \cite{Bel-1,Bel-2}.
After that, in \cite{Eb}, using the differential transformation method (DTM), some periodic solutions were obtained and more recently the relativistic harmonic oscillator is
studied by using the homotopy perturbation method (HPM) \cite{Bia}, where a good approximation is obtained using the fact that the solutions are periodic functions.\\
In the following section we will develop an algorithm using the method described in ``The
Adomian Decomposition Method Combined with Laplace Transform'' section in order to
solve the nonlinear differential equation (\ref{Osc-1}) without resort to any truncation or linearization and not assuming {\it a priori} that the solutions are periodic functions.
\section{\bf Solution of the Relativistic Harmonic Oscillator Equation Through LADM}
\label{NLC-ADM}
\noindent Comparing (\ref{Osc-1}) with equation (\ref{eq:y4}) we have that $L_{tt}$ and $N$ becomes:
\begin{equation}
L_{tt}x=\frac{d^2}{dt^2}x,\;\; Nx=\Big[1-\Big(\frac{dx}{dt}\Big)^2\Big]^{\frac{3}{2}}x .\;\; \label{Oper-1}
\end{equation}
\noindent By using now equation (\ref{eq:y11}) through the LADM method we obtain recursively
\begin{equation}
\left\{
\begin{array}{ll}
x_{0}(t)=\beta t,\\
x_{n+1}(t)=-\mathcal{L}^{-1}\Big[\frac{1}{s^2}\mathcal{L}\{A_{n}(x_{0},x_{1},\ldots, x_{n})\}\Big],\;\; n=0,1,2,\ldots
\end{array}
\right.\label{eq:ADM1}
\end{equation}
Also the nonlinear term is decomposed as
\begin{equation}
Nx=\Big[1-\Big(\frac{dx}{dt}\Big)^2\Big]^{\frac{3}{2}}x=\sum_{n=0}^{\infty}A_{n}(x_{0},x_{1},\ldots, x_{n})
\label{eq:N-1}
\end{equation}
where $\{A_{n}\}_{n=0}^{\infty}$ is the so-called Adomian polynomials sequence, the terms will be calculated according to \cite{Duan} and \cite{Duan1}. The first few polynomials are given by\\
\noindent $A_{0}(x_0)=x_{0}(1-x_{0}'^{2})^{\frac{3}{2}}, $\\
$A_{1}(x_0,x_1)=x_{1}(1-x_{0}'^{2})^{\frac{3}{2}},$\\
$A_{2}(x_0,x_1, x_2)=x_{2}(1-x_{0}'^{2})^{\frac{3}{2}}, $\\
$A_{3}(x_0,x_1, x_2, x_3)=x_{3}(1-x_{0}'^{2})^{\frac{3}{2}}, $\\
$A_{4}(x_0,x_1, x_2, x_3, x_4)=x_{4}(1-x_{0}'^{2})^{\frac{3}{2}}, $\\
$ \vdots $\\
$A_{m}(x_0,x_1,\ldots, x_m)=x_{m}(1-x_{0}'^{2})^{\frac{3}{2}} $ for every $m\geq 0$.\\
Now, recursively using (\ref{eq:ADM1}) with the Adomian polynomials given by the later sequence $\{A_{n}\}_{n=0}^{\infty}$, we obtain, for a given initial velocity
$\beta$:\\
\begin{equation} \label{s-0}
x_{0}(t)=\beta t,
\end{equation}
\begin{equation} \label{s-1}
\begin{split}
x_{1}(t)&=-\mathcal{L}^{-1}\Big[\frac{1}{s^2}\mathcal{L}\{\beta(1-\beta^2)^{\frac{3}{2}}t\}\Big]=-\mathcal{L}^{-1}\Big[\frac{1}{s^4}\beta(1-\beta^2)^{\frac{3}{2}}\Big]\\ &=-\beta(1-\beta^2)^{\frac{3}{2}}\frac{t^3}{3!},
\end{split}
\end{equation}
\begin{equation} \label{s-2}
\begin{split}
x_{2}(t)&=-\mathcal{L}^{-1}\Big[\frac{1}{s^2}\mathcal{L}\{-\beta(1-\beta^2)^{3}\frac{t^3}{3!}\}\Big]=\mathcal{L}^{-1}\Big[\frac{1}{s^6}\beta(1-\beta^2)^{3}\Big]\\ &=\beta(1-\beta^2)^{3}\frac{t^5}{5!},
\end{split}
\end{equation}
\begin{equation} \label{s-3}
\begin{split}
x_{3}(t)&=-\mathcal{L}^{-1}\Big[\frac{1}{s^2}\mathcal{L}\{\beta(1-\beta^2)^{\frac{9}{2}}\frac{t^5}{5!}\}\Big]=-\mathcal{L}^{-1}\Big[\frac{1}{s^8}\beta(1-\beta^2)^{\frac{9}{2}}\Big]\\ &=-\beta(1-\beta^2)^{\frac{9}{2}}\frac{t^7}{7!},
\end{split}
\end{equation}
\begin{equation} \label{s-4}
\begin{split}
x_{4}(t)&=-\mathcal{L}^{-1}\Big[\frac{1}{s^2}\mathcal{L}\{-\beta(1-\beta^2)^{6}\frac{t^7}{7!}\}\Big]=\mathcal{L}^{-1}\Big[\frac{1}{s^{10}}\beta(1-\beta^2)^{6}\Big]\\ &=\beta(1-\beta^2)^{6}\frac{t^9}{9!},
\end{split}
\end{equation}
\begin{equation} \label{s-5}
\begin{split}
x_{5}(t)&=-\mathcal{L}^{-1}\Big[\frac{1}{s^2}\mathcal{L}\{\beta(1-\beta^2)^{\frac{15}{2}}\frac{t^9}{9!}\}\Big]=-\mathcal{L}^{-1}\Big[\frac{1}{s^{12}}\beta(1-\beta^2)^{\frac{15}{2}}\Big]\\ &=-\beta(1-\beta^2)^{\frac{15}{2}}\frac{t^{11}}{11!},
\end{split}
\end{equation}
$$\vdots . $$
In view of equations (\ref{s-0})-(\ref{s-5}), the series solution is
\begin{equation} \label{solser}
\begin{split}
x(t)&=\beta t-\beta(1-\beta^2)^{\frac{3}{2}}\frac{t^3}{3!}+\beta(1-\beta^2)^{3}\frac{t^5}{5!}-\beta(1-\beta^2)^{\frac{9}{2}}\frac{t^7}{7!}\\ & +\beta(1-\beta^2)^{6}\frac{t^9}{9!}-\beta(1-\beta^2)^{\frac{15}{2}}\frac{t^{11}}{11!}+\beta(1-\beta^2)^{9}\frac{t^{13}}{13!}\cdots\\
\end{split}
\end{equation}
$$=\beta\Big(t-(1-\beta^2)^{\frac{3}{2}}\frac{t^3}{3!}+(1-\beta^2)^{3}\frac{t^5}{5!}-(1-\beta^2)^{\frac{9}{2}}\frac{t^7}{7!}+(1-\beta^2)^{6}\frac{t^9}{9!}-+\cdots\Big)
$$
\begin{eqnarray}\label{SOL}
=\beta\sum_{n=0}^{\infty}\Big((1-\beta^2)^{\frac{3}{2}}\Big)^{n}Ç(-1)^{n}\frac{t^{2n+1}}{(2n+1)!}.
\end{eqnarray}
From (\ref{SOL}) we conclude that the solution of the equation (\ref{Osc-1}), that is, the position of the relativistic harmonic oscillator is given by
the series of power of functions with $0<\beta<1$
\begin{equation}\label{Ser}
x(t)=\beta\sum_{n=0}^{\infty}\Big((1-\beta^2)^{\frac{3}{2}}\Big)^{n}Ç(-1)^{n}\frac{t^{2n+1}}{(2n+1)!}.
\end{equation}
According to \cite{Bart}, we easily see that the power series (\ref{Ser}) converges in all $\mathbb{R}$
and it also converges uniformly in any compact subinterval of $\mathbb{R}$.\\
Using the expressions obtained above for the solution of equation (\ref{Osc-1}), we will illustrate, with two examples, the
efectiveness of LADM to solve the nonlinear relativistic harmonic oscillator.
\section{\bf Application to the Relativistic Harmonic Oscillator}
\noindent {\bf Example 1}\\
In this first example, we consider the particular case of (\ref{Osc-1}) such that $\beta =0.1$;
this case was studied in \cite{Eb} via differential transformation method (DTM) and also in \cite{Bia} through the homotopy perturbation method (HPM).
Good approximations were obtained in both works in comparison with the first known approximation solution of
(\ref{Osc-1}) obtained in \cite{Mic1} by the harmonic balance method (HBM). We will use the formula
(\ref{Ser}) taking only the first fourteen terms (since the next one will be very small)
\begin{equation} \label{s1}
\begin{split}
x(t)&=0.1\sum_{n=0}^{13}(0.9850375)^{n}Ç(-1)^{n}\frac{t^{2n+1}}{(2n+1)!}=0.1t-0.0985037\frac{t^3}{3!}+0.0970299\frac{t^5}{5!}\\
& -0.095578\frac{t^7}{7!}+0.094148\frac{t^9}{9!}-0.0927393\frac{t^{11}}{11!}+\cdots
-0.0822027\frac{t^{27}}{27!}
\end{split}
\end{equation}
The approximations obtained for $\beta=0.1$ v\'ia DTM in \cite{Eb} by using HPM in \cite{Bia} are respectively:
\begin{equation}\label{n1}
x_{\mbox{\tiny DTM}}(t)=0.10033\sin(0.998t)-0.000047097\sin(2.997t)+0.00000008254\sin(4.841t)
\end{equation}
\begin{equation}\label{n2}
x_{\mbox{\tiny HPM}}(t)=0.10010\sin(0.999t)-0.00004689\sin(2.997t)+0.00000005062\sin(4.995t)
\end{equation}
Moreover, using $\beta=0.1$ in (\ref{hbm}) we find
\begin{equation}\label{n3}
x_{\mbox{\tiny HBM}}(t)=0.10025\sin(0.998t)-0.00004173\sin(2.996t)+0.00000004369\sin(4.944t)
\end{equation}
\noindent The results obtained are shown in Table \ref{tab1} in which the comparison with the ones obtained in \cite{Eb}, \cite{Bia} and \cite{Mic1} using DTM, HPM and HBM respectively has been done. We also display in figures \ref{fi1}, \ref{fi2} and \ref{fi3} this comparison.
All the numerical work was accomplished with the Mathematica software package.
\begin{figure}[h!]
\begin{center}
\includegraphics[width=110mm, height=65mm, scale=1.0]{tab-1.pdf}
\end{center}
\caption{Table for $\beta=0.1$ \label{tab1}}
\end{figure}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=80mm, height=50mm, scale=1.0]{G-1-1.pdf}
\end{center}
\caption{Graph of the values of $x_{\mbox{\tiny our}}$ and $x_{\mbox{\tiny DTM}}$ for $\beta=0.1$ \label{fi1}}
\end{figure}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=80mm, height=50mm, scale=1.0]{G-1-2.pdf}
\end{center}
\caption{Graph of the values of $x_{\mbox{\tiny our}}$ and $x_{\mbox{\tiny HPM}}$ for $\beta=0.1$ \label{fi2}}
\end{figure}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=80mm, height=50mm, scale=1.0]{G-1-3.pdf}
\end{center}
\caption{Graph of the values of $x_{\mbox{\tiny our}}$ and $x_{\mbox{\tiny HBM}}$ for $\beta=0.1$ \label{fi3}}
\end{figure}
\noindent {\bf Example 2}\\
In this second example, we consider the particular case of (\ref{Osc-1}) such that $\beta =0.2$; this case was studied in \cite{Eb} via DTM and also in \cite{Bia} using
HPM. Once again, in both works, good approximations were found in comparison with the first obtained in (\ref{Osc-1}) and the one obtained in \cite{Mic1} by HBM.
As before, using the formula (\ref{Ser}) taking the first fourteen terms we obtain
\begin{equation} \label{s1}
\begin{split}
x(t)&=0.2\sum_{n=0}^{13}(0.940604)^{n}Ç(-1)^{n}\frac{t^{2n+1}}{(2n+1)!}=0.2t-0.1881208\frac{t^3}{3!}+0.1769472\frac{t^5}{5!}\\
& -0.1664372\frac{t^7}{7!}+0.1565515\frac{t^9}{9!}-0.1472253\frac{t^{11}}{11!}+\cdots
-0.0902233\frac{t^{27}}{27!}
\end{split}
\end{equation}
The approximations obtained in the case of $\beta=0.2$ v\'ia DTM in \cite{Eb} through HPM in \cite{Bia} are respectively:
\begin{equation}\label{p1}
x_{\mbox{\tiny DTM}}(t)=0.203\sin(0.992t)-0.0003695\sin(3.051t)+0.000009257\sin(4.29t)
\end{equation}
\begin{equation}\label{p2}
x_{\mbox{\tiny HPM}}(t)=0.201\sin(0.995t)-0.0003768\sin(2.985t)+0.000001652\sin(4.974t)
\end{equation}
And also using $\beta=0.2$ in (\ref{hbm}) we obtain
\begin{equation}\label{p3}
x_{\mbox{\tiny HBM}}(t)=0.202\sin(0.995t)-0.0003354\sin(2.985t)+0.000001508\sin(4.974t)
\end{equation}
\noindent Comparison of our results with the ones obtained in \cite{Eb}, \cite{Bia} and \cite{Mic1} using DTM, HPM and HBM are showed in Table \ref{tab2} and displayed
in figures \ref{fi4}, \ref{fi5} and \ref{fi6}. In this example we can also see that the approximation accuracy depends of the initial velocity of the oscillator.
All the numerical work was accomplished with the Mathematica software package.
\begin{figure}[h!]
\begin{center}
\includegraphics[width=110mm, height=65mm, scale=1.0]{tab-2.pdf}
\end{center}
\caption{Table for $\beta=0.2$ \label{tab2}}
\end{figure}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=80mm, height=50mm, scale=1.0]{G-2-1.pdf}
\end{center}
\caption{Graph of the values of $x_{\mbox{\tiny our}}$ and $x_{\mbox{\tiny DTM}}$ for $\beta=0.2$ \label{fi4}}
\end{figure}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=80mm, height=50mm, scale=1.0]{G-2-2.pdf}
\end{center}
\caption{Graph of the values of $x_{\mbox{\tiny our}}$ and $x_{\mbox{\tiny HPM}}$ for $\beta=0.2$ \label{fi5}}
\end{figure}
\begin{figure}[h!]
\begin{center}
\includegraphics[width=80mm, height=50mm, scale=1.0]{G-2-3.pdf}
\end{center}
\caption{Graph of the values of $x_{\mbox{\tiny our}}$ and $x_{\mbox{\tiny HBM}}$ for $\beta=0.2$ \label{fi6}}
\end{figure}
\noindent As we seen from the last examples, the solutions we have obtained are periodic functions and the amplitude depends of the initial velocity
as found by the author in \cite{Mic1}. The main difference of our results with the reported ones is that the final series is uniformly convergent in any compact subset
of the real line and therefore we can obtain the results with the required accuracy.
\section{Summary and Conclusions}
\noindent Far as we know there is no exact solutions to the equation of motion for
a relativistic harmonic oscillator. In this work, we have obtained the solution of the problem without
the {\it a priori} assumption that the solutions are periodic functions;
the solution that we have obtained is a series of powers of functions which uniformly converge on compact subsets of $\mathbb{R}$, never before reported.
The problem of find the limit function of the series solution is an open question that we are currently work.\\
\noindent In order to show the accuracy and efficiency of our method, we have solved two examples and comparing our results with the ones obtained with three different methods \cite{Mic1,Eb,Bia}. Our results show that LADM produces highly accurate solutions in complicated nonlinear problems. We therefore, conclude that the Laplace-Adomian decomposition method is a notable non-sophisticated powerful tool that produces high quality approximate solutions for nonlinear ordinary differential equations using simple calculations and that reaches convergence with only a few terms. Finally, the Laplace-Adomian decomposition method would be a powerful mathematical tool for solving other nonlinear differential equations
related with mathematical physics models. All the numerical work and the graphics was accomplished with the Mathematica software package.
| {
"timestamp": "2016-06-13T02:11:02",
"yymm": "1606",
"arxiv_id": "1606.03336",
"language": "en",
"url": "https://arxiv.org/abs/1606.03336",
"abstract": "Far as we know there are not exact solutions to the equation of motion for a relativistic harmonic oscillator. In this paper, the relativistic harmonic oscillator equation which is a nonlinear ordinary differential equation is studied by means of a combined use of the Adomian Decomposition Method and the Laplace Transform (LADM). The results that we have obtained, a series of powers of functions, have never been reported and show a very good match when compared with other approximate solutions, obtained by different methods. The method here proposed works with high degree of accuracy and because it requires less computational effort, it is very convenient to solve this kind of nonlinear differential equations.",
"subjects": "Classical Analysis and ODEs (math.CA); Mathematical Physics (math-ph)",
"title": "Solution for the nonlinear relativistic harmonic oscillator via Laplace-Adomian decomposition method",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9728307653134903,
"lm_q2_score": 0.7279754548076477,
"lm_q1q2_score": 0.70819691882996
} |
https://arxiv.org/abs/2302.02334 | Revisiting Discriminative vs. Generative Classifiers: Theory and Implications | A large-scale deep model pre-trained on massive labeled or unlabeled data transfers well to downstream tasks. Linear evaluation freezes parameters in the pre-trained model and trains a linear classifier separately, which is efficient and attractive for transfer. However, little work has investigated the classifier in linear evaluation except for the default logistic regression. Inspired by the statistical efficiency of naive Bayes, the paper revisits the classical topic on discriminative vs. generative classifiers. Theoretically, the paper considers the surrogate loss instead of the zero-one loss in analyses and generalizes the classical results from binary cases to multiclass ones. We show that, under mild assumptions, multiclass naive Bayes requires $O(\log n)$ samples to approach its asymptotic error while the corresponding multiclass logistic regression requires $O(n)$ samples, where $n$ is the feature dimension. To establish it, we present a multiclass $\mathcal{H}$-consistency bound framework and an explicit bound for logistic loss, which are of independent interests. Simulation results on a mixture of Gaussian validate our theoretical findings. Experiments on various pre-trained deep vision models show that naive Bayes consistently converges faster as the number of data increases. Besides, naive Bayes shows promise in few-shot cases and we observe the "two regimes" phenomenon in pre-trained supervised models. Our code is available atthis https URL. | \section{Introduction}
\label{introduction}
Deep representation learning has achieved great success in many fields such as computer vision \cite{ren2015faster,he2017mask,chen2020generative,DBLP:conf/cvpr/He0WXG20Moco, DBLP:conf/icml/ChenK0H20SimCLR, DBLP:conf/cvpr/ChenH21SimSiam,grill2020bootstrap,DBLP:conf/cvpr/HeCXLDG22MAE}, natural language processing \cite{DBLP:conf/naacl/DevlinCLT19Bert, DBLP:conf/nips/BrownMRSKDNSSAA20GPT3,raffel2020exploring} and cross-modal learning~\cite{CLIP} over the past few years.
The common paradigm behind them is to (pre-)train a large-scale model on an enormous amount of labeled or unlabeled data and transfer it to downstream tasks. During the transfer, \emph{linear evaluation}~\cite{chen2020generative,DBLP:conf/cvpr/He0WXG20Moco, DBLP:conf/icml/ChenK0H20SimCLR,grill2020bootstrap,CLIP}
freezes all parameters in the pre-trained model and learns a linear classifier separately. Theoretically, it is validated by the (approximate) linear separability of the representations extracted by pre-trained models~\cite{DBLP:conf/icml/SaunshiPAKK19, DBLP:conf/nips/LeeLSZ21, DBLP:conf/alt/ToshK021, DBLP:conf/nips/HaoChenWGM21}.
Practically,
linear evaluation is an efficient and attractive alternative to fine-tuning, considering the extremely large and continually growing size of modern pre-trained models.
Although new algorithms and models for deep pre-training emerge in endlessly, little work has investigated the classifier except for the default logistic regression. Directly inspired by the classical work~\cite{efron1975efficiency,DBLP:conf/nips/NgJ01} (detailed in Section~\ref{sec: Preliminaries}) on the statistical efficiency of generative linear classifiers (e.g. na\"ive Bayes), we revisit the discriminative vs. generative linear classifiers in the context of deep representation learning.
In Section~\ref{sec: Theory}, we improve the classical theory~\cite{DBLP:conf/nips/NgJ01} in two aspects for subsequent analysis in deep representation learning. First, we characterize asymptotic behaviors of both multiclass na\"ive Bayes and logistic regression, generalizing the results in binary classification~\cite{DBLP:conf/nips/NgJ01}. Second, in logistic regression, we consider the practically used surrogate loss in our analysis instead of directly optimizing the zero-one loss as assumed in~\cite{DBLP:conf/nips/NgJ01}. To establish it, we introduce a general \emph{multiclass $\mathcal{H}$-consistency bound} framework upon recent advances~\cite{DBLP:conf/icml/AwasthiMM022} and a nontrivial explicit bound for multiclass logistic regression, which are of independent interests. We prove that for a fixed number of classes, the number of samples required to approach the corresponding optimal classifier is $O(\log n)$ and $O(n)$ for na\"ive Bayes and logistic regression respectively, where $n$ is the feature dimension. We conduct synthetic experiments with tractable $\mathcal{H}$-optimal classifiers to validate our theory.
In Section~\ref{sec: implications}, we discuss the implications of our theory in the linear evaluation of pre-trained deep models.
We first analyze the main assumptions in our theory upon deep representations.
We then perform extensive experiments on CIFAR10 and CIFAR100 datasets with various representative pre-trained vision models~\cite{resnet,dosovitskiy2020image,DBLP:journals/corr/abs-2003-04297MocoV2,DBLP:conf/nips/simclrv2,CLIP, DBLP:conf/cvpr/simmim, DBLP:conf/cvpr/HeCXLDG22MAE}, which are trained in supervised or self-supervised manners.
The results show that na\"ive Bayes consistently converges faster as the number of data increases in all settings, which agrees with our theory. Besides, na\"ive Bayes shows promise in few-shot cases and we observe the ``two regimes'' phonomenan~\cite{DBLP:conf/nips/NgJ01} in models
pre-trained in a supervised manner, suggesting a distinction between the representations learned by supervised and self-supervised approaches.
\section{Preliminaries}
\label{sec: Preliminaries}
In this section, we present notations and preliminaries on discriminative vs. generative classifiers and $\mathcal{H}$-consistency.
Let lower, boldface lower and capital case letters denote scalers (e.g., a), vectors (e.g., $\boldsymbol{a}$), and matrices (e.g., $\boldsymbol{A}$), respectively. For a matrix $\boldsymbol{A}$, $\boldsymbol{A}_i$ and $A_{ij}$ denote its $i$-th row and $(i, j)$-th element. For a vector $\boldsymbol{a}$, $a_i$ denotes its $i$-th element. Similarly, for a vector function $\boldsymbol{f}$, $f_i({\boldsymbol x})$ denotes the $i$-th element of $\boldsymbol{f}({\boldsymbol x})$. We do not distinguish constants and random variables in notations if there is no confusion. We denote the KL Divergence between two distributions $p$ and $q$ by $D(p \Vert q)$. We use $\mathbb{E}$, $\mathbb{V}$, $\Delta_k$ to represent expectation, variance, and $k$-dimensional possibility simplex, respectively.
Let $\mathcal{X}$ denote the domain set and $\mathcal{Y}= \{1,\dots, K\}$
denote the label set, where $K$ is the number of classes. For simplicity, we assume $\mathcal{X} = \{0,1\}^n$ when inputs are discrete and $\mathcal{X} = [0,1]^n$ otherwise, where $n$ is the feature dimension. Note that our analysis can be easily extended to the general case with bounded features.
Let $\mathcal{H}$ be a hypothesis set of functions mapping from $\mathcal{X} \times \mathcal{Y}$ to $\mathbb{R}^K$. The prediction associated by a hypothesis ${\boldsymbol h} \in \mathcal{H}$ and ${\boldsymbol x} \in \mathcal{X}$ is $\mathop{\mathrm{argmax}}_{y \in \mathcal{Y}} h_y({\boldsymbol x})$. In the main paper, we focus on the family of constrained linear hypotheses $\mathcal{H}_{lin} = \{{\boldsymbol x} \to \boldsymbol{h}({\boldsymbol x}): h_y({\boldsymbol x}) = \langle {\boldsymbol w}_y, {\boldsymbol x}\rangle + b_y, \Vert {\boldsymbol w}_y \Vert_2 \leq W, \vert b_y\vert \leq B, y \in \mathcal{Y}\}$, where $W, B \in \mathbb{R}^+$. We also denote the hypothesis set of all measurable functions by $\mathcal{H}_{all}$. Given a hypothesis set $\mathcal{H}$ and distribution $\mathcal{D}$, the generalization error and minimal generalization error of a hypothesis ${\boldsymbol h}$ with respect to the loss function $\ell: \mathbb{R}^K \times \mathcal{Y} \rightarrow \mathbb{R}$ are defined as $R_{\ell}({\boldsymbol h}) = \mathbb{E}_{({\boldsymbol x},y) \sim \mathcal{D}} [\ell({\boldsymbol h}({\boldsymbol x}), y)]$ and $R_{\ell, \mathcal{H}}^* = \inf_{{\boldsymbol h} \in \mathcal{H}} R_{\ell}({\boldsymbol h})$.
\subsection{Discriminative vs. Generative Classifiers}
$K$-class logistic regression is parameterized by $[{\boldsymbol w}_1,\dots,{\boldsymbol w}_K, {\boldsymbol b}]$, where ${\boldsymbol w}_i \in \mathbb{R}^n$ and ${\boldsymbol b} \in \mathbb{R}^K$. Its prediction is given by $\mathop{\mathrm{argmax}}_{y \in \mathcal{Y}} (\langle {\boldsymbol w}_y, {\boldsymbol x}\rangle + b_y)$.
It's well known that the generative counterpart of the logistic regression is na\"ive Bayes (with some constraints presented later)~\cite{DBLP:conf/nips/NgJ01, DBLP:conf/kdd/RubinsteinH97}. When inputs are discrete, a na\"ive Bayes classifier uses a training set with $m$ i.i.d examples to calculate the empirical conditional distributions $\hat{p}(x_i\vert y)$ and empirical marginal distribution $\hat{p}(y)$ as follows:
\begin{align}
\hat{p}(x_i = 1\vert y = k) &= \frac{\#\{x_i = 1, y = k\} + \alpha}{\#\{y = k\} + K\alpha}, \label{eq:nb estimation 1}\\
\hat{p}(y = k) &= \frac{\#\{y = k\} + \alpha}{m + K\alpha}, \label{eq:nb estimation 2}
\end{align}
where $\#\{\cdot\}$ is the counting function and $\alpha$ is a positive Laplace smoothing parameter. Corresponding population versions are denoted by $p(x_i\vert y)$ and $p(y)$ respectively. In case of continuous inputs, we let $\hat{p}(x_i \vert y = k)$ be a univariate Gaussian distribution with parameters $\hat{\mu}_{ki}$ and $\hat{\sigma}_i^2$. We note that $\hat{\sigma}^2_i$s do not depend on $y$ to keep the linearity of its decision boundary, otherwise logistic regression and na\"ive Bayes are no longer a fair discriminative-generative pair~\cite{DBLP:journals/npl/XueT08}. They are calculated as the empirical version of ${\mu}_{ki} =\mathbb{E}[x_i \vert y = k]$ and ${\sigma}_i^2 = \mathbb{E}_y[\mathbb{V}(x_i\vert y)]$.
\citet{DBLP:conf/nips/NgJ01} proved that in binary classification, logistic regression enjoys a lower asymptotic error but approaches it much slower (w.r.t. the sample size) than na\"ive Bayes. The theory explains the \emph{``two regimes''}~\cite{DBLP:conf/nips/NgJ01} phenomenon in practice. In particular, na\"ive Bayes generalizes better with limited data. However, the multiclass case has not been investigated yet, which is the main focus of this paper. Besides, prior work~\cite{DBLP:conf/nips/NgJ01} assumes that the zero-one loss can be directly optimized in logistic regression, which is impractical. To weaken the assumption, we introduce tools from \emph{$\mathcal{H}$-consistency}.
\subsection{$\mathcal{H}$-consistency}
$\mathcal{H}$-consistency~\cite{long2013consistency} analyzes the relationship between the estimation error of zero-one loss w.r.t. a hypothesis class $\mathcal{H}$ and that of a surrogate loss. It includes the classical Bayes consistency~\cite{zhang2004statistical,bartlett2006convexity,tewari2007consistency} as a special case by setting $\mathcal{H}$ to $\mathcal{H}_{all}$. In this paper, we analyze the linear discriminative vs. generative classifiers upon recent advances on $\mathcal{H}$-consistency bounds~\cite{, DBLP:conf/icml/AwasthiMM022}.
We first introduce some notations. We denote by ${\boldsymbol p}({\boldsymbol x})$ the conditional distribution of $Y$ given ${\boldsymbol x}$, i.e., $p_y({\boldsymbol x}) = \mathbb{P}(Y=y\vert X={\boldsymbol x})$. We define the conditional risk as $\mathscr{C}_{\ell}(\boldsymbol{h}, {\boldsymbol x}) = \sum_{y=1}^K p_y({\boldsymbol x}) \ell({\boldsymbol h}({\boldsymbol x}), y)$, and note that generalization error $R_{\ell}({\boldsymbol h})$ can be rewritten as $\mathbb{E}_{{\boldsymbol x}} [\mathscr{C}_{\ell} ({\boldsymbol h}, {\boldsymbol x})]$. We also define its infimum $\mathscr{C}_{\ell, \mathcal{H}}^*({\boldsymbol x}) = \inf_{\boldsymbol{h} \in \mathcal{H}} \mathscr{C}_{\ell}(\boldsymbol{h}, {\boldsymbol x})$ and the gap between them $\Delta \mathscr{C}_{\ell, \mathcal{H}}(\boldsymbol{h}, {\boldsymbol x}) = \mathscr{C}_{\ell}(\boldsymbol{h}, {\boldsymbol x}) - \mathscr{C}_{\ell, \mathcal{H}}^*({\boldsymbol x})$. A key quantity appears in our bounds is $M_{\ell, \mathcal{H}} = R_{\ell, \mathcal{H}}^* - \mathbb{E}_{{\boldsymbol x}}(\mathscr{C}_{\ell, \mathcal{H}}^* ({\boldsymbol x}))$, which is difficult to estimate~\cite{DBLP:conf/icml/AwasthiMM022}, but can be bounded by approximate error. Furthermore, for any ${\boldsymbol p}$ in probability simplex $\Delta_K$, we can define $\mathscr{C}_{\ell}(\boldsymbol{h}, {\boldsymbol x}, {\boldsymbol p}) = \sum_{y=1}^K p_y \ell(\boldsymbol{h}({\boldsymbol x}), y)$ and $\Delta \mathscr{C}_{\ell, \mathcal{H}}(\boldsymbol{h}, {\boldsymbol x}, {\boldsymbol p}) = \mathscr{C}_{\ell}(\boldsymbol{h}, {\boldsymbol x}, {\boldsymbol p}) - \inf_{\boldsymbol{h} \in \mathcal{H}} \mathscr{C}_{\ell}(\boldsymbol{h}, {\boldsymbol x}, {\boldsymbol p})$. We define $\epsilon$-regret of $t$ as $\langle t \rangle_\epsilon = t \mathbbm{1}_{t > \epsilon}$.
The general $\mathcal{H}$-consistency bound~\cite{DBLP:conf/icml/AwasthiMM022} for two loss functions $\ell_1$ and $\ell_2$ is defined as follows.
\begin{mydef}[$\mathcal{H}$-consistency bound]
\label{Def :h consistency bound}
$\mathcal{H}$-consistency bound is in the following form that holds for all ${\boldsymbol h} \in \mathcal{H}$, $\mathcal{D} \in \mathcal{P}$ and some non-decreasing function $f: \mathbb{R}_+ \to \mathbb{R}_+$:
\begin{align}
R_{\ell_2}({\boldsymbol h}) - R_{\ell_2, \mathcal{H}}^* \leq f(R_{\ell_1}({\boldsymbol h}) - R_{\ell_1, \mathcal{H}}^*).
\end{align}
If $\mathcal{P}$ is composed of all distributions over $\mathcal{X} \times \mathcal{Y}$, we call it a distribution-independent bound.
\end{mydef}
Note that it covers the classical Bayes consistency bounds~\cite{bartlett2006convexity} by setting $\mathcal{H}=\mathcal{H}_{all}$. When $\ell_1$ is logistic loss $\ell_{log}$ and $\ell_2$ is zero-one loss $\ell_{0-1}$,~\citet{DBLP:conf/icml/AwasthiMM022} proved the following $\mathcal{H}$-consistency bound w.r.t. the bounded linear hypotheses.
\begin{theorem}[$\mathcal{H}$-consistency bound for binary logistic loss and zero-one loss, Appendix K.1.2~\cite{DBLP:conf/icml/AwasthiMM022}]
\label{lemma: binary H consistency bounds}
Given binary linear hypothesis set $\mathcal{H} = \{{\boldsymbol x} \to \langle {\boldsymbol w}, {\boldsymbol x}\rangle + b: \Vert {\boldsymbol w} \Vert_2 \leq W, \vert b\vert \leq B\}$, if $R_{\ell_{log}}(h) - R^*_{\ell_{log}, \mathcal{H}} + M_{\ell_{log}, \mathcal{H}} \leq \frac{1}{2}({\frac{e^{B}-1}{e^{B}+ 1}})^2$, then it holds for any distribution that $R_{\ell_{0-1}}(h) - R^*_{\ell_{0-1}, \mathcal{H}} + M_{\ell_{0-1}, \mathcal{H}} \leq \sqrt{2}(R_{\ell_{log}}(h) - R^*_{\ell_{log}, \mathcal{H}} + M_{\ell_{log}, \mathcal{H}})^{\frac{1}{2}}$.
\end{theorem}
To the best of our knowledge, there is no $\mathcal{H}$-consistency bound for logistic loss and zero-one loss in multiclass classification. In this paper, we extend the binary framework~\cite{DBLP:conf/icml/AwasthiMM022} to multiclass cases and derive an explicit bound for logistic loss.
\section{Theory}
\label{sec: Theory}
In this section, we present our main theoretical results in Section~\ref{sec: Discriminative vs. Generative: Multiclass Classification}: Under some mild assumptions, for any fixed $K$, the number of training samples required by na\"ive Bayes to approach its asymptotic error is $O(\log(n))$ (Theorem~\ref{cor: multiclass NB sample complexity}), and that of logistic regression is $O(n)$ (Theorem~\ref{cor: sample complexity of multiclass lr}).
To establish it, we propose a general multiclass $\mathcal{H}$-consistency framework (Theorem~\ref{cor: Distribution-independent convex Psi bound}) and a nontrivial multiclass $\mathcal{H}$-consistency bound for logistic loss and zero-one loss (Theorem~\ref{thm: H-consistency bound for log}) in Section~\ref{sec: multiclass H-consistency framework}. Notably, our theory includes the analysis for $K = 2$ in Appendix~\ref{sec: Discriminative vs. Generative: Binary Classification} as a special case.
\subsection{On Multiclass Discriminative vs. Generative Linear Classifiers}
\label{sec: Discriminative vs. Generative: Multiclass Classification}
Let $\boldsymbol{h}_{Gen, m}$ and $\boldsymbol{h}_{Dis, m}$ denote the hypothesis returned by multiclass logistic regression and na\"ive Bayes with $m$ $i.i.d$ samples, respectively. Let $\boldsymbol{h}_{Gen, \infty}$ and $\boldsymbol{h}_{Dis, \infty}$ be the corresponding asymptotic version. We are interested in comparing the statistical efficiency of na\"ive Bayes and logistic regression~\cite{DBLP:conf/nips/NgJ01}. Formally, we need to bound $R_{\ell_{0-1}} (\boldsymbol{h}_{Gen, m}) - R_{\ell_{0-1}} (\boldsymbol{h}_{Gen, \infty})$ and $R_{\ell_{0-1}} (\boldsymbol{h}_{Dis, m}) - R_{\ell_{0-1}} (\boldsymbol{h}_{Dis, \infty})$ respectively.
\textbf{Na\"ive Bayes.} Notably, the solution of Na\"ive Bayes is in a closed-form, as presented in Eq.~(\ref{eq:nb estimation 1}\&\ref{eq:nb estimation 2}). Therefore, we can characterize the gap between parameters in $\boldsymbol{h}_{Gen, m}$ and $\boldsymbol{h}_{Gen, \infty}$ to bound $R_{\ell_{0-1}} (\boldsymbol{h}_{Gen, m}) - R_{\ell_{0-1}} (\boldsymbol{h}_{Gen, \infty})$, similarly to the binary case~\cite{DBLP:conf/nips/NgJ01}.
We make two mild assumptions about the data distribution similar to~\citet{DBLP:conf/nips/NgJ01}. We avoid trivial cases where $p(y = k) =1$ or $p(y = k) =0$ for some $k$ in Assumption~\ref{Assumption: p(y=k)} and assume that the conditional distribution of ${\boldsymbol x}$ given $y$ can not be too concentrated in Assumption~\ref{Assumption: parmeters bounded}.
\begin{assumption}
\label{Assumption: p(y=k)}
For some fixed $\rho_1 \in (0, \frac{1}{2}]$, we have that $\rho_1 \leq p(y = k) \leq 1 - \rho_1$ for all $k \in \mathcal{Y}$.
\end{assumption}
\begin{assumption}
\label{Assumption: parmeters bounded}
For some fixed $\rho_2 \in (0, \frac{1}{2}]$, $\rho_2 \leq p(x_i = 1\vert y = k) \leq 1 - \rho_2$ for all $i, k$ in the discrete case, and $\sigma^2_i \ge \rho_2$ for all $i$ in the continuous case.
\end{assumption}
In practice, most deep learning work considers the balanced case where $\rho_1 = \frac{1}{K}$~\cite{imagenet}.
Empirically, we found that $\rho_2 \in [10^{-5}, 10^{-2}]$ on the features extracted by representative pre-trained vision models in Section~\ref{sec: implications}. For clarity, we denote $\rho_0 = \min\{\rho_1, \rho_2\}$ throughout the paper. We now define two key quantities in our proof as follows.
\begin{mydef}[Pair activation function of na\"ive Bayes]
\label{Def :multiclass delta a}
For every $k_1, k_2 \in \mathcal{Y}$, we define the pair activation function $\Delta a_{Gen}({\boldsymbol x}, k_1, k_2)$ as
\begin{align}
\Delta a_{Gen}({\boldsymbol x}, k_1, k_2) = a_{Gen}({\boldsymbol x}, k_1) - a_{Gen}({\boldsymbol x}, k_2),
\end{align}
where $a_{Gen}({\boldsymbol x}, k) = \sum_{i=1}^n \log \hat{p}(x_i\vert y=k) + \log\hat{p}(y=k)$.
\end{mydef}
The paired activation function is important because it connects the estimated parameters and predictions of the hypothesis. For instance, $\Delta a_{Gen}({\boldsymbol x}, k_1, k_2) > 0$ means that ${\boldsymbol x}$ is more likely to be predicted as an instance of class $k_1$ than class $k_2$. We can easily bound the gap between the parameters in $\boldsymbol{h}_{Gen, m}$ and $\boldsymbol{h}_{Gen, \infty}$ by standard concentration inequalities. To bound $R_{\ell_{0-1}} (\boldsymbol{h}_{Gen, m}) - R_{\ell_{0-1}} (\boldsymbol{h}_{Gen, \infty})$ as presented in Theorem~\ref{Thm: multiclass generalization bound}, we further upper bound the probability of getting ``bad training samples'', which are predicted as different classes with high probability by $\boldsymbol{h}_{Gen, m}$ and $\boldsymbol{h}_{Gen, \infty}$, via the following
$\widetilde{G}(\tau)$.
\begin{mydef}
\label{Def :multiclass G}
We define the function $\widetilde{G}(\tau)$ as follows:
\begin{align*}
\widetilde{G}(\tau) = \max_{k_1, k_2} \mathbb{P}_{({\boldsymbol x},y) \sim \mathcal{D}}(\vert \Delta a_{Gen, \infty}({\boldsymbol x}, k_1, k_2)\vert \leq \tau n).
\end{align*}
\end{mydef}
\begin{theorem}[Proof in Appendix~\ref{proof: Proof of Theorem Thm: multiclass generalization bound}]
\label{Thm: multiclass generalization bound}
Suppose that Assumption~\ref{Assumption: p(y=k)} and~\ref{Assumption: parmeters bounded} are valid. Then with probability at least $1-\delta$:
\begin{align*}
R_{\ell_{0-1}}(\boldsymbol{h}_{Gen, m}) &\leq R_{\ell_{0-1}}(\boldsymbol{h}_{Gen, \infty}) \\
&+ \frac{K(K-1)}{2} \biggl(\widetilde{G}\bigl(O(\sqrt{\frac{1}{m} \log(\frac{n}{\delta})})\bigr) + \delta\biggr).
\end{align*}
\end{theorem}
The core of Theorem~\ref{Thm: multiclass generalization bound} is the $\widetilde{G}(\tau)$, which must be small when $\tau$ is small in order to obtain meaningful bound about $R_{\ell_{0-1}}(\boldsymbol{h}_{Gen, m}) - R_{\ell_{0-1}}(\boldsymbol{h}_{Gen, \infty})$. It holds under the following assumptions, similarly to~\citet{DBLP:conf/nips/NgJ01}.
\begin{assumption}
\label{Assumption: multiclass KL}
For all $k_1, k_2 (k_1 \ne k_2)$ and $k \in \mathcal{Y}$, it holds that $\lvert \sum_{i=1}^n (D(p(x_i \vert y=k) \Vert p(x_i \vert y=k_1)) - D(p(x_i \vert y=k) \Vert p(x_i \vert y=k_2))) \rvert = \beta_{k_1, k_2, k}n = \Omega(n)$.
\end{assumption}
\begin{assumption}
\label{Assumption: multiclass likelihood ratio var}
For all $k_1, k_2 (k_1 \ne k_2)$ and $k \in \mathcal{Y}$, it holds that $\mathbb{V}_{{\boldsymbol x}}[\sum_{i=1}^n \log \frac{{p}(x_i\vert y=k_1) }{{p}(x_i\vert y=k_2) } \vert y = k] = \alpha_{k_1, k_2, k}n = O(n^r)$ for any $r \in [1,2)$.
\end{assumption}
Intuitively, Assumption~\ref{Assumption: multiclass KL} requires that $\Omega(1)$ fraction of features distinct for any two different classes. Assumption~\ref{Assumption: multiclass likelihood ratio var} is more technical. In fact, it is derived when we attempt to bound $\widetilde{G}(\tau)$ via Chebyshev's inequality\footnote{Indeed, if the na\"ive Bayes assumption really holds, we can obtain a stronger guarantee for $\widetilde{G}(\tau)$ by using Chernoff's bound. We put the result in Proposition~\ref{Prop: multiclass exp -n}.}. We empirically analyze both assumptions in Section~\ref{sec: implications}. Proposition~\ref{Prop: multiclass ploy -n} presents a meaningful bound for $\widetilde{G}(\tau)$, which is followed by the main result of na\"ive Bayes in Theorem~\ref{cor: multiclass NB sample complexity}.
\begin{prop}[Proof in Appendix~\ref{proof: Proposition Prop: multiclass poly -n}]
\label{Prop: multiclass ploy -n}
Suppose that Assumption~\ref{Assumption: p(y=k)},~\ref{Assumption: multiclass KL} and~\ref{Assumption: multiclass likelihood ratio var} hold, then $\widetilde{G}(\tau)$ is polynomially small in $n$:
\begin{equation*}
\widetilde{G}(\tau) \leq \frac{\alpha}{(\tau - \zeta)^2 n},
\end{equation*}
where $\alpha = \max_{k_1, k_2, k} \alpha_{k_1,k_2,k} = O(n^{r-1})$, $\mathbb{E}_{{\boldsymbol x}}[\Delta a_{Gen, \infty}({\boldsymbol x}, k_1, k_2)\vert y=k] = \zeta_{k_1,k_2,k} n$, $\zeta = \min_{k_1, k_2, k} \vert\zeta_{k_1,k_2,k}\vert = \Omega(1)$ and $\tau < \zeta$.
\end{prop}
\begin{theorem}[Results for na\"ive Bayes, proof in Appendix~\ref{proof: Proof of Corollary cor: multiclass NB sample complexity}]
\label{cor: multiclass NB sample complexity}
Suppose the precondition of Proposition~\ref{Prop: multiclass ploy -n} holds. Then, it suffices to pick $m = O(\log (n))$ training samples such that $R_{\ell_{0-1}}(\boldsymbol{h}_{Gen, m}) \leq R_{\ell_{0-1}}(\boldsymbol{h}_{Gen, \infty}) +\epsilon_0$ hold with probability $1 - \delta_0$, for any $\epsilon_0 \in (0,1)$ and $\delta_0 \in (0, \frac{\epsilon_0}{K^2}]$.
\end{theorem}
\textbf{Logistic Regression.} To directly compare with na\"ive Bayes, we aim to bound $R_{\ell_{0-1}} (\boldsymbol{h}_{Dis, m}) - R_{\ell_{0-1}} (\boldsymbol{h}_{Dis, \infty})$. However, the optimization of logistic regression does not have an analytic form, making the proof idea of na\"ive Bayes infeasible. Besides, \citet{DBLP:conf/nips/NgJ01} proves the bound by directly optimizing the zero-one loss, which is impractical. Instead, we present a bound considering the surrogate logistic loss in this paper. To establish it, we exploit recent advances on $\mathcal{H}$-consistency bound~\cite{DBLP:conf/icml/AwasthiMM022} as detailed in Defition~\ref{Def :h consistency bound}. It is worth discussing an alternative approach based on \emph{Bayes consistency bounds}~\cite{bartlett2006convexity}. For a direct comparison with na\"ive Bayes, we care about the asymptotic error in $\mathcal{H}_{lin}$ instead of $\mathcal{H}_{all}$. Therefore, a $\mathcal{H}$-consistency bound is more natural and potentially tighter than a Bayes consistency bound. In fact, existing Bayes consistency bounds~\cite{bartlett2006convexity} are special cases of the $\mathcal{H}$-consistency bounds~\cite{DBLP:conf/icml/AwasthiMM022}.
Note that the binary $\mathcal{H}$-consistency bound~\cite{DBLP:conf/icml/AwasthiMM022} in Theorem~\ref{lemma: binary H consistency bounds} does not directly apply to multiclass cases. We generalize the binary framework~\cite{DBLP:conf/icml/AwasthiMM022} to multiclass cases and prove an explicit $\mathcal{H}$-consistency bound for logistic loss. We present the bound in Theorem~\ref{thm: H-consistency bound for log} and defer the establishment to Section~\ref{sec: multiclass H-consistency framework}.
\begin{theorem}
[$\mathcal{H}$-consistency bound for multiclass logistic loss and zero-one loss, proof in Appendix~\ref{proof: thm: H-consistency bound for log}]
\label{thm: H-consistency bound for log}
If $R_{\ell_{log}}({\boldsymbol h}) - R^*_{\ell_{log}, \mathcal{H}_{lin}} + M_{\ell_{log}, \mathcal{H}_{lin}} \leq \frac{1}{2}({\frac{e^{2B}-1}{e^{2B}+ K - 1}})^2$, then for any distribution satisfiying $\max_y p_y({\boldsymbol x}) - \min_y p_y({\boldsymbol x}) \leq \frac{e^{2B} - 1}{e^{2B} + K - 1}$ for all ${\boldsymbol x}$, it holds that $R_{\ell_{0-1}}(\boldsymbol{h}) - R^*_{\ell_{0-1}, \mathcal{H}_{lin}} + M_{\ell_{0-1}, \mathcal{H}_{lin}} \leq \sqrt{2}(R_{\ell_{log}}(\boldsymbol{h}) - R^*_{\ell_{log}, \mathcal{H}_{lin}} + M_{\ell_{log}, \mathcal{H}_{lin}})^{\frac{1}{2}}$.
\end{theorem}
Note that $R_{\ell_{0-1}} (\boldsymbol{h}_{Dis, \infty}) = R^*_{\ell_{log}, \mathcal{H}_{lin}}$ by definition and when $B \to +\infty$, we have $ \frac{e^{2B} - 1}{e^{2B} + K - 1} \to 1$, then Theorem~\ref{thm: H-consistency bound for log} holds for all distribution. Theorem~\ref{thm: H-consistency bound for log} provides a tool to analyze the asymptotic behavior of multiclass logistic regression considering the surrogate loss. According to it, we need to bound the gap $R_{\ell_{log}}(\boldsymbol{h}_{Dis, m})-R_{\ell_{log}}(\boldsymbol{h}_{Dis, \infty})$ and $M_{\ell_{log}, \mathcal{H}_{lin}}$ to guarantee a small $R_{\ell_{0-1}}(\boldsymbol{h}_{Dis, m})-R_{\ell_{0-1}}(\boldsymbol{h}_{Dis, \infty})$. The following Proposition characterizes $R_{\ell_{log}}(\boldsymbol{h}_{Dis, m})-R_{\ell_{log}}(\boldsymbol{h}_{Dis, \infty})$ by Radmancher complexity~\cite{DBLP:conf/colt/BartlettBM02, mohri2018foundations} and a contraction lemma~\cite{maurer2016vector}.
\begin{prop}[Proof in appendix~\ref{proof: Prop: multiclass logisticbound}]
\label{Prop: multiclass logisticbound}
With probability at least $1 - \delta_0$, the following holds:
\begin{equation*}
\label{Eq: multiclass logisticboundO}
R_{\ell_{log}}(\boldsymbol{h}_{Dis, m}) \leq R_{\ell_{0-1}} (\boldsymbol{h}_{Dis, \infty}) + O(\sqrt{\frac{K^3n}{m}}).
\end{equation*}
\end{prop}
$M_{\ell, \mathcal{H}}$ is a constant determined by the hypothesis set $\mathcal{H}$, loss function $\ell$, and data distribution $\mathcal{D}$. Its value is difficult to estimate directly~\cite{DBLP:conf/icml/AwasthiMM022}. However, according to the definition, $M_{\ell, \mathcal{H}}$ can be bounded by the corresponding approximate error. Prior work~\cite{DBLP:conf/icml/SaunshiPAKK19, DBLP:conf/nips/LeeLSZ21, DBLP:conf/alt/ToshK021, DBLP:conf/nips/HaoChenWGM21} proves the (approximate) linear separability of the representations extracted by deep pre-trained models, suggesting a small approximation error for the logistic loss. Therefore, we make the following assumption, which is validatable in the context of linear evaluation of deep models.
\begin{assumption}
\label{Assumption: Optimal classifier has finite empirical loss}
The approximate error of the logistic loss is bounded by a small constant $\nu < \frac{1}{2}({\frac{e^{2B}-1}{e^{2B}+ K - 1}})^2$. Namely, $\mathop{\mathrm{argmin}}_{\boldsymbol{h} \in \mathcal{H}_{lin}} R_{\ell_{log}}(\boldsymbol{h}) - \mathop{\mathrm{argmin}}_{\boldsymbol{h} \in \mathcal{H}_{all}} R_{\ell_{log}}(\boldsymbol{h}) \leq \nu$, which implies that $M_{\ell_{log}, \mathcal{H}_{lin}} \le \nu$.
\end{assumption}
We characterize the number of samples required to approach the asymptotic error for logistic regression in Theorem~\ref{cor: sample complexity of multiclass lr} by combining Proposition~\ref{Prop: multiclass logisticbound} and Theorem~\ref{thm: H-consistency bound for log}.
\begin{theorem}[Results for multiclass logistic regression, proof in appendix~\ref{proof: Proof of Corollary cor: sample complexity of multiclass lr}]
\label{cor: sample complexity of multiclass lr}
Suppose that Assumption~\ref{Assumption: Optimal classifier has finite empirical loss} holds. Then, it suffices to pick $m = O(n)$ training samples such that $R_{\ell_{0-1}}(\boldsymbol{h}_{Dis, m}) \leq R_{\ell_{0-1}}(\boldsymbol{h}_{Dis, \infty}) +\epsilon_0$ hold with probability $1 - \delta_0$, for any $\epsilon_0 \in [\sqrt{2\nu}, {\frac{e^{2B}-1}{e^{2B}+ K - 1}}]$ and $\delta_0 \in (0, 1)$.
\end{theorem}
Notably, according to the multiclass fundamental theorem from ~\cite{shalev2014understanding} (Theorem 29.3), the sample complexity of $\mathcal{H}_{lin}$ for any algorithm is $\Omega(n)$ because the Natarajan dimension for $\mathcal{H}_{lin}$ is $\Omega(Kn)$, indicating the upper bound in Thereom~\ref{Prop: multiclass logisticbound} is tight.
Theorem~\ref{cor: multiclass NB sample complexity} and Theorem~\ref{cor: sample complexity of multiclass lr} show that the $O(n)$ vs. $O(\log(n))$ result~\cite{DBLP:conf/nips/NgJ01} still holds in multiclass cases, suggesting that na\"ive Bayes is possibly better than logistic regression when the sample size is limited. We validate our theory on a mixture of Gaussian, as presented in Figuire~\ref{figures: multiclass simulation}. For a fixed feature dimension $n$, we increase the number of samples $m$ until the two models approach the corresponding asymptotic error, which is tractable in the experiment. Detailed configurations of the experiments and more results are presented in Appendix~\ref{app: Configurations of Simulation Experiment}.
\subsection{Multiclass $\mathcal{H}$-consistency Framework}
\label{sec: multiclass H-consistency framework}
We now present the general multiclass $\mathcal{H}$-consistency bound framework and prove the explicit bound for the logistic loss in Theorem~\ref{thm: H-consistency bound for log}, which are of independent interest. Similarly to the binary case~\cite{DBLP:conf/icml/AwasthiMM022}, we first introduce the following general multiclass $\mathcal{H}$-consistency bound between any target loss $\ell_2$ and surrogate loss $\ell_1$.
\begin{prop}[Distribution-dependent convex bound, proof in Appendix~\ref{proof: Thm: Distribution-dependent convex bound}]
\label{Thm: Distribution-dependent convex bound}
For a fixed distribution, if there exists a convex function $g: \mathbb{R}_+ \to \mathbb{R}$ with $g(0) \ge 0$ and $\epsilon \ge 0$, and the following holds for any $\boldsymbol{h} \in \mathcal{H}$ and ${\boldsymbol x} \in \mathcal{X}$:
\begin{equation}
\label{eqn: Thm: Distribution-dependent convex bound 1}
g(\langle \Delta \mathscr{C}_{\ell_2, \mathcal{H}}(\boldsymbol{h}, {\boldsymbol x}) \rangle_\epsilon) \leq \Delta \mathscr{C}_{\ell_1, \mathcal{H}}(\boldsymbol{h}, {\boldsymbol x}).
\end{equation}
Then it holds for all $\boldsymbol{h} \in \mathcal{H}$ that
\begin{align}
&g(R_{\ell_2}(\boldsymbol{h}) - R^*_{\ell_2, \mathcal{H}} + M_{\ell_2, \mathcal{H}}) \nonumber \\
&\leq R_{\ell_1}(\boldsymbol{h}) - R^*_{\ell_1, \mathcal{H}} + M_{\ell_1, \mathcal{H}} + \max(g(0), g(\epsilon)).\label{eq:results-of-dependent}
\end{align}
\end{prop}
We present the concave counterpart of it as Proposition~\ref{Thm: Distribution-dependent concave bound} of Appendix~\ref{sec: Deferred Results: Multiclass H-consistency}. For simplicity, we fix the target loss $\ell_2$ as the zero-one loss in the following. Note that
Proposition~\ref{Thm: Distribution-dependent convex bound} is distribution-dependent while an asymptotically distribution-independent version is necessary for our analysis in Section~\ref{sec: Discriminative vs. Generative: Multiclass Classification}.
To this end, we introduce a tool called \emph{multiclass $\mathcal{H}$-estimation error transformation}.
\begin{mydef}[Multiclass $\mathcal{H}$-estimation error transformation] \label{def:trans}
The multiclass $\mathcal{H}$-estimation error transformation of a surrogate loss $\ell$ is defined on $t \in [0,1]$ as $\mathcal{J}_{\ell}(t) = \inf_{\hat{y} \in \mathcal{Y}, {\boldsymbol p} \in \mathcal{P}_{\hat{y}}(t), {\boldsymbol x} \in \mathcal{X}, \boldsymbol{h} \in \mathcal{H}_{\hat{y}}({\boldsymbol x}) }\Delta \mathscr{C}_{\ell, \mathcal{H}}(\boldsymbol{h}, {\boldsymbol x}, {\boldsymbol p})$. Here $\mathcal{H}_{\hat{y}}({\boldsymbol x}) \coloneqq \{\boldsymbol{h} \in \mathcal{H}: \mathop{\mathrm{argmax}}_{y \in \mathcal{Y}} h_y({\boldsymbol x}) = \hat{y}\}$ is a collection of hypotheses that predicts ${\boldsymbol x}$ as class $\hat{y}$. $\mathcal{P}_{\hat{y}}(t) \coloneqq \{{\boldsymbol p} \in \Delta_K: \max_y p_y - p_{\hat{y}} = t\}$ is a subset of $K$-dimensional simplex indexed by classes and the gap between the max component and class-indexed component of ${\boldsymbol p}$.
\end{mydef}
$\mathcal{J}_{\ell}(t)$ in Defition~\ref{def:trans} is carefully derived such that
plugging it to the right-hand side of Eq.~(\ref{eqn: Thm: Distribution-dependent convex bound 1}) provides a sufficient condition such that Eq.~(\ref{eq:results-of-dependent}) holds for any ${\boldsymbol h}, {\boldsymbol x}$, and ${\boldsymbol p}$ (i.e., distribution-independent). It is worth noting that the condition is actually necessary as well under further assumptions, as presented later in Theorem~\ref{thm:tightness}. Defition~\ref{def:trans} generalizes the binary freamwork~\cite{DBLP:conf/icml/AwasthiMM022} by optimizing ${\boldsymbol p}$ in a collection of subsets $\mathcal{P}_{\hat{y}}(t)$ to handle multiclass cases. Built upon Defition~\ref{def:trans}, we establish the multiclass distribution-independent bound for zero-one loss as follows.
\begin{theorem}
[Distribution-independent convex $\ell_{0-1}$ bound, proof in Appendix~\ref{proof: cor: Distribution-independent convex Psi bound}]
\label{cor: Distribution-independent convex Psi bound}
Suppose that $\mathcal{H}$ satisfies that $\{\mathop{\mathrm{argmax}}_{y \in \mathcal{Y}} h_y({\boldsymbol x}) : \boldsymbol{h} \in \mathcal{H}\} = \{1, \dots, K\}$ for any ${\boldsymbol x} \in \mathcal{X}$. If there exists a convex function $g: \mathbb{R}_+ \to \mathbb{R}$ with $g(0) = 0$ and $g(t) \leq \mathcal{J}_{\ell}(t)$. Then it holds for any $\boldsymbol{h} \in \mathcal{H}$ and any distribution $\mathcal{D}$ that
\begin{equation*}
g(R_{\ell_{0-1}}({\boldsymbol h}) - R^*_{\ell_{0-1}, \mathcal{H}} + M_{\ell_{0-1}, \mathcal{H}}) \leq R_{\ell}({\boldsymbol h}) - R^*_{\ell, \mathcal{H}} + M_{\ell, \mathcal{H}}.
\end{equation*}
\end{theorem}
\begin{figure}[t]
\centering
\includegraphics[width=1. \columnwidth]{figures/simulation.jpg}
\caption{Multiclass ($K = 5$) simulation results. Empirically, logistic regression and na\"ive Bayes require $O(n)$ and $O(\log n)$ samples to approach the corresponding asymptotic error respectively. Error bars show the variance estimated by 5 runs.}
\label{figures: multiclass simulation}
\end{figure}
\begin{table*}[t!]
\centering
\caption{Analysis of assumptions on CIFAR10 training dataset.}
\vskip 0.15in
\label{tab: assumptions}
\begin{tabular}{lccccc}
\toprule
Method & Backbone & Pre-training data & $\rho_0$ & $\beta$ & $\alpha$ \\
\midrule
ViT~\cite{dosovitskiy2020image} & ViT-B/16 & Image-label & 2.80E-3 & 0.004 & 690 \\
ResNet~\cite{resnet} & ResNet50 & Image-label & 1.70E-3 & 0.06 & 11516 \\
CLIP~\cite{CLIP} & ResNet50 & Image-text & 4.78E-3 & 0.203 & 6383 \\
MoCov2~\cite{DBLP:journals/corr/abs-2003-04297MocoV2} & ResNet50 & Image & 5.03E-5 & 0.005 & 26640 \\
SimCLRv2~\cite{DBLP:conf/nips/simclrv2} & ResNet50 & Image & 3.74E-5 & 0.01 & 2490 \\
MAE~\cite{DBLP:conf/cvpr/HeCXLDG22MAE} & ViT-B/16 & Image & 6.37E-3 & 0.032 & 6919 \\
SimMIM~\cite{DBLP:conf/cvpr/simmim} & ViT-B/16 & Image & 7.86E-3 & 0.002 & 5201 \\
\bottomrule
\end{tabular}
\end{table*}
We present the concave counterpart of it as Theorem~\ref{cor: Distribution-independent concave bound} in Appendix~\ref{sec: Deferred Results: Multiclass H-consistency}. This theorem holds for any hypothesis set $\mathcal{H}$ that can divide any sample ${\boldsymbol x}$ into any category, including the linear hypothesis set and hypotheses of neural network. Notably, our multiclass $\mathcal{H}$-consistency result degenerates to the binary one exactly~\cite{DBLP:conf/icml/AwasthiMM022} with $K=2$. In addition, we note that if $\mathcal{J}_{\ell}(t)$ is convex and $\mathcal{J}_{\ell}(0) = 0$, then $\mathcal{J}_{\ell}$ satisfies the condition of $g$ in Theorem~\ref{cor: Distribution-independent convex Psi bound}. In fact, it leads to the tightest multiclass $\mathcal{H}$-consistency bound.
\begin{theorem}[Tightness, proof in Appendix~\ref{Proofs of thm:tightness}]
\label{thm:tightness}
If $\mathcal{J}_{\ell}(t)$ is convex with $\mathcal{J}_{\ell}(0)=0$, then for any $t\in[0,1]$ and $\delta>0$, there exist a distribution $\mathcal{D}$ and a hypothesis $h\in \mathcal{H}$ such that $R_{\ell_{0-1}}(\boldsymbol{h}) - R^*_{\ell_{0-1}, \mathcal{H}} + M_{\ell_{0-1}, \mathcal{H}} = t$ and $\mathcal{J}_{\ell}(t) \leq R_{\ell}(\boldsymbol{h}) - R^*_{\ell, \mathcal{H}} + M_{\ell, \mathcal{H}} \leq \mathcal{J}_{\ell}(t) + \delta$.
\end{theorem}
To establish our main result in Section~\ref{sec: Discriminative vs. Generative: Multiclass Classification}, we have presented an asymptotically distribution-independent multiclass $\mathcal{H}$-consistency bound for the logistic loss in an explicit form in Theorem~\ref{thm: H-consistency bound for log}. We mention that the proof of Theorem~\ref{thm: H-consistency bound for log} is nontrivial because $\mathcal{J}_{\ell}(t)$ in the multiclass case involves a much more complex optimization problem than that in the binary case~\cite{DBLP:conf/icml/AwasthiMM022}.
The proposed framework is not limited to the linear hypothesis class and the logistic loss. In particular, we present a similar result for the hypothesis class of one-hidden-layer neural networks in Theorem~\ref{thm: H-consistency bound for log neural network} of Appendix~\ref{sec: Deferred Results: Multiclass H-consistency}. Besides, the general bound in Theorem~\ref{cor: Distribution-independent convex Psi bound} and the proof idea of Theorem~\ref{thm: H-consistency bound for log} are applicable to hinge loss, exponential loss, $\rho$-margin loss, and so on, which are left for future work. Furthermore, the analysis idea can be used to obtain multiclass Bayes consistency bounds by setting $\mathcal{H}$ to $\mathcal{H}_{all}$.
\section{Implications in Deep Learning}
\label{sec: implications}
In this section, we discuss the implications of our theoretical results in the linear evaluation of pre-trained deep neural networks.
First, as presented in Section~\ref{sec: Validate the assumptions}, we empirically analyze the main assumptions of our theory in various deep vision models~\cite{dosovitskiy2020image, resnet, CLIP, DBLP:journals/corr/abs-2003-04297MocoV2, DBLP:conf/nips/simclrv2, DBLP:conf/cvpr/HeCXLDG22MAE, DBLP:conf/cvpr/simmim}. Second, we systematically compare logistic regression and na\"ive Bayes on CIFAR10 and CIFAR100 datasets~\cite{cifar} with various models and sample sizes in Section~\ref{sec: Deep Learning Results}. Na\"ive Bayes always converges much faster, which agrees with our theory. The ``two regimes'' phenomenon~\cite{DBLP:conf/nips/NgJ01} almost happens with models pre-trained in a supervised manner~\cite{dosovitskiy2020image, resnet}, which is analyzed in detail in Section~\ref{sec: On the difference between supervised pertaining and self-supervised}. Details of experiments can be found in Appendix~\ref{app: details of DL Experiments}.
\subsection{Analyzing the Assumptions}
\label{sec: Validate the assumptions}
We empirically analyze and discuss the main assumptions made in Section~\ref{sec: Theory} on the CIFAR10 dataset. The results are summarized in Table~\ref{tab: assumptions}. We emphasize that the concrete values of the quantities in the table won't affect the asymptotic analyses in Section~\ref{sec: Theory}, i.e., $O(\log n)$ results for na\"ive Bayes, but may affect its performance given a fixed data size.
We consider linear evaluation for transfer learning on top of pre-trained models, whose parameters are frozen. Therefore, it is valid to assume that the features extracted on the target dataset satisfy the $i.i.d.$ assumption.
\subsubsection{Assumption~\ref{Assumption: p(y=k)} and~\ref{Assumption: parmeters bounded}}
Assumption~\ref{Assumption: p(y=k)} holds naturally because the CIFAR10 dataset is class-balanced. For Assumption~\ref{Assumption: parmeters bounded}, we calculate the $\hat{\sigma_i}^2$ for each dimension of the training representations as approximations for $\sigma_i^2$. We present $\rho_0 = \min(\min_i \hat{\sigma_i}^2, \frac{1}{10})$ in Table~\ref{tab: assumptions}, and Figure~\ref{figures: sigmas} in Appendix~\ref{app: Additional Results of Validating the Assumptions} plots the histogram of $\hat{\sigma_i}^2$. Assumption~\ref{Assumption: parmeters bounded}
holds for all models.
\subsubsection{Assumption~\ref{Assumption: multiclass KL} and~\ref{Assumption: multiclass likelihood ratio var}}
It is hard to directly validate the two assumptions in practice. Nevertheless, we estimate $\beta_{k_1,k_2,k}$ and $\alpha_{k_1,k_2,k}$ for all $k_1, k_2 (k_1 \ne k_2)$ and $k \in \mathcal{Y}$ in different models for a comparison. We note that $\beta_{k_1,k_2,k} = \zeta_{k_1,k_2,k}$ in our experiments, because the CIFAR10 dataset is class-balanced. We report the $\beta = \zeta = \min_{k1,k_2,k} \vert\beta_{k_1,k_2,k}\vert$ and $\alpha = \max_{k1,k_2,k} \alpha_{k_1,k_2,k}$ in Table~\ref{tab: assumptions}. We also present the histograms of $\vert\beta_{k_1,k_2,k}\vert$ and $\alpha_{k_1,k_2,k}$ in Figure~\ref{figures: beta} and Figure~\ref{figures: alpha} of Appendix~\ref{app: Additional Results of Validating the Assumptions}, respectively.
\subsubsection{Assumption~\ref{Assumption: Optimal classifier has finite empirical loss}}
Assumption~\ref{Assumption: Optimal classifier has finite empirical loss} is hard to validate in practice because the Bayes-optimal classifier is unknown. However, recent theoretical results in prior works~\cite{DBLP:conf/icml/SaunshiPAKK19, DBLP:conf/nips/LeeLSZ21, DBLP:conf/alt/ToshK021, DBLP:conf/nips/HaoChenWGM21} suggest that it holds when the number of samples for pre-training is sufficiently large.
\subsection{Empirical Results in Deep Learning}
\label{sec: Deep Learning Results}
We systematically compare logistic regression and naïve Bayes on CIFAR10 and CIFAR100 datasets in various models, which are trained on image-label pairs~\cite{dosovitskiy2020image, resnet}, image-text pairs~\cite{CLIP}, or pure images~\cite{DBLP:journals/corr/abs-2003-04297MocoV2, DBLP:conf/nips/simclrv2,DBLP:conf/cvpr/HeCXLDG22MAE, DBLP:conf/cvpr/simmim}.
For a fair comparison, we keep the linear evaluation setting in~\cite{CLIP} throughout the experiments. Specially, we train the logistic regression using scikit-learn's~\cite{scikit-learn} L-BFGS implementation, with a maximum of 1000 iterations. We adjust the weight of $\ell_2$ regularization of logistic regression carefully to reproduce the results reported in~\cite{CLIP} on both datasets with full training data. We then adjust the number of training samples $m$ gradually. For each $m$, we obtain training samples randomly 5 times and record the mean test error of two models.
We plot the convergence curves in all settings in Appendix~\ref{app: Additional Deep Learning Results}, which are linked in Table~\ref{tab: visual results sum}. Notably, na\"ive Bayes approaches its asymptotic error much faster than logistic regression in all settings, like that presented in Figure~\ref{figures: resnet cifar100 long}, which is consistent with our theoretical results.
\begin{figure}[t!]
\centering
\includegraphics[width=0.9\columnwidth]{figures/resnet_long.jpg}
\caption{Comparison between na\"ive Bayes and logistic regression with the features extracted by ResNet on the CIFAR100 dataset. Na\"ive Bayes approaches its asymptotic error much
faster.}
\label{figures: resnet cifar100 long}
\end{figure}
\begin{figure}[t]
\centering
\includegraphics[width=0.9\columnwidth]{figures/vit_short.jpg}
\caption{Comparison between na\"ive Bayes and logistic regression with the features extracted by ViT on the CIFAR100 dataset. The ``two regimes'' phenomenon is observed.}
\label{figures: vit cifar100 short}
\end{figure}
\begin{table}[t]
\centering
\caption{Convergence comparison between multiclass logistic regression and na\"ive Bayes. ``NB faster'' means na\"ive Bayes approaches its asymptotic error faster.}
\vskip 0.15in
\label{tab: visual results sum}
\begin{tabular}{lccc}
\toprule
Method & Visual results & \multicolumn{2}{c}{NB faster/ Two regimes} \\
& & CIFAR10 & CIFAR100 \\
\midrule
ViT & Figure~\ref{figures: vit} & $\surd$ / $\surd$ & $\surd$ / $\surd$ \\
ResNet & Figure~\ref{figures: resnet} & $\surd$ / $\surd$ & $\surd$ / $\surd$ \\
CLIP & Figure~\ref{figures: clip} & $\surd$ / $\surd$ & $\surd$ / $\surd$ \\
MoCov2 & Figure~\ref{figures: mocov2} & $\surd$ / $\times$ & $\surd$ / $\times$ \\
SimCLRv2 & Figure~\ref{figures: simclrv2} & $\surd$ / $\times$ & $\surd$ / $\surd$ \\
MAE & Figure~\ref{figures: mae} & $\surd$ / $\surd$ &$\surd$ / $\times$ \\
SimMIM & Figure~\ref{figures: simmim} & $\surd$ / $\times$ & $\surd$ / $\times$ \\
\bottomrule
\end{tabular}
\end{table}
\subsection{On the ``Two Regimes'' Phenomenon}
\label{sec: On the difference between supervised pertaining and self-supervised}
\citet{DBLP:conf/nips/NgJ01} suggests that there can often be two regimes of performance between na\"ive Bayes and logistic regression, that is, though logistic regression enjoys lower asymptotic error, na\"ive Bayes performs better with smaller
training sets because of its fast convergence rate. They observed this phenomenon on many datasets from the UCI Machine Learning repository~\cite{UCI}. These classical datasets are small and the features are mostly low-dimensional. However, nowadays, people prefer to obtain representations by using deep neural networks pre-trained by massive data. The occurrence of the ``two regimes'' phenomenon in this new setting has not been investigated yet.
We summarize the occurrence of the ``two regimes'' phenomenon in Table~\ref{tab: visual results sum}. The ``two regimes'' phenomenon occurs in half of our experiments, which suggests that na\"ive Bayes still shows promise when the training data is limited. We present a typical case in Figure~\ref{figures: vit cifar100 short} and see Appendix~\ref{app: Additional Deep Learning Results} for complete results. Interestingly, the ``two regimes'' phenomenon almost happens when the deep vision model is pre-trained in a supervised manner (ViT, ResNet, and CLIP), which suggests a distinction between representations learned by supervised learning and self-supervised learning.
We conjecture that representations learned by supervised methods could have some better properties to make na\"ive Bayes converges faster than that learned by self-supervised methods. As validated in Section~\ref{sec: Deep Learning Results}, though our theory could only prove the fast convergence rate of na\"ive Bayes, it does help us to understand this distinction to some extent. Combining the values presented in Table~\ref{tab: assumptions}, we can get some preliminary results.
\emph{Representations learned by supervised methods could be more robust for each dimension.} As shown in Table~\ref{tab: assumptions}, features learned by supervised methods (ViT, ResNet, CLIP) tend to have larger $\rho_0$. In other words, these representations tend to have larger in-class variance $\sigma_i^2$ than others. Intuitively, it suggests that data in each dimension could be more robust to relieve the over-fitting and boost na\"ive Bayes learning better in the few-shot case. Besides, according to Eq.~(\ref{eq: discrete delata a}-\ref{eq: conti delata a}) in Appendix~\ref{proof: Proof of Theorem Thm: multiclass generalization bound} and the derivation in Appendix~\ref{proof: Proof of Corollary cor: multiclass NB sample complexity}, a larger $\rho_0$ implies faster convergence in a $1/\rho_0^2$ order, which explains it in a certain sense.
\emph{Representations learned by supervised methods could be more separable between different categories.} From Table~\ref{tab: assumptions}, representations learned by supervised methods (ResNet, CLIP) are inclined to have larger $\beta$ than others. Namely, there exists more distinction between the distributions of samples in different classes, which are easier to predict. In addition, by our derivation in Appendix~\ref{proof: Proof of Corollary cor: multiclass NB sample complexity}, a larger $\beta$ implies faster convergence in a $1/\beta^2$ order, which agrees with our observation.
\section{Related Work}
\subsection{Deep Representative Learning}
Deep representation learning aims to learn representations on the raw unlabeled data and transfer them to the downstream tasks. It has made remarkable progress in various machine learning fields~\cite{ren2015faster,he2017mask,chen2020generative,DBLP:conf/cvpr/He0WXG20Moco, DBLP:conf/icml/ChenK0H20SimCLR, DBLP:conf/cvpr/ChenH21SimSiam,grill2020bootstrap,DBLP:conf/cvpr/HeCXLDG22MAE, DBLP:conf/cvpr/simmim, DBLP:conf/naacl/DevlinCLT19Bert, DBLP:conf/nips/BrownMRSKDNSSAA20GPT3,raffel2020exploring}.
In particular, the promise of \emph{linear evaluation}~\cite{chen2020generative,DBLP:conf/cvpr/He0WXG20Moco, DBLP:conf/icml/ChenK0H20SimCLR,grill2020bootstrap,CLIP} suggests that representations extracted by pre-trained models are near to linear separable.
Besides, the performance of such representations in linear evaluation is
guaranteed in recent theoretical works~\cite{DBLP:conf/icml/SaunshiPAKK19, DBLP:conf/nips/LeeLSZ21, DBLP:conf/alt/ToshK021, DBLP:conf/nips/HaoChenWGM21}.
All of these empirical and theoretical works encourage us to rethink the role of linear classifiers.
\subsection{Discriminative vs. Generative Learning}
Comparing discriminative with generative classifiers has long been an interesting topic~\cite{efron1975efficiency, DBLP:conf/kdd/RubinsteinH97, DBLP:conf/nips/NgJ01}. \citet{efron1975efficiency} compared the logistic regression and normal discriminant analysis and claimed that the latter is only slightly more efficient. \citet{DBLP:conf/nips/NgJ01} simplified the normal discriminant analysis to na\"ive Bayes and concluded that the discriminative model has lower asymptotic error while the generative classifier may approach its higher asymptotic error much faster.
\citet{DBLP:conf/nips/NgJ01} assume that one can directly optimize on zero-one loss. Instead, we weaken the assumption and introduce the theoretical tools from $\mathcal{H}$-consistency to obtain more reliable results.
\subsection{$\mathcal{H}$-consistency}
Most machine learning algorithms depend on optimizing a surrogate loss function rather than the target loss function. To find the favorable property of surrogate loss, consistency has been studied broadly in the last two decades. Classical Bayes consistency~\cite{DBLP:journals/jmlr/Zhang04a, zhang2004statistical,bartlett2006convexity,tewari2007consistency} analyzes the relationship between the excess error of zero-one loss and that of a surrogate loss. Instead, $\mathcal{H}$-consistency~\cite{long2013consistency} considers the estimation error w.r.t. a hypothesis set $\mathcal{H}$. It includes the classical Bayes consistency as a special case by setting $\mathcal{H}$ to $\mathcal{H}_{all}$. Most recently, \citet{DBLP:conf/icml/AwasthiMM022} proposed a novel and solid framework named $\mathcal{H}$-consistency bounds, which consider the upper bounds on the target estimation error expressed by surrogate estimation error.
We proposed a novel multiclass $\mathcal{H}$-consistency framework, which includes the framework in~\cite{DBLP:conf/icml/AwasthiMM022} as a special case. We notice that the independent work of~\cite{awasthimulti} also proposed a multiclass $\mathcal{H}$-consistency framework from the same general theorem (Proposition~\ref{Thm: Distribution-dependent convex bound}). We highlight the following comparison that distinguishes our work.
First, the proof ideas are totally different. In particular, we directly generalize the binary framework in~\cite{DBLP:conf/icml/AwasthiMM022} to the multiclass case in Theorem~\ref{cor: Distribution-independent convex Psi bound}, which is general and tight (Theorem~\ref{thm:tightness}). In contrast, \citet{awasthimulti} argues that generalizing the binary framework is nontrivial and instead provides a case-by-case analysis for different losses, which does not enjoy the tightness guarantee. Second, we provide an explicit bound for logistic loss (Theorem~\ref{thm: H-consistency bound for log}), which is necessary for our subsequent analysis, while it is unclear how to derive such a bound by the prior work~\cite{awasthimulti}.
\section{Conclusion}
We revisit the classical topic of discriminative vs. generative classifiers~\cite{DBLP:conf/nips/NgJ01}. Specially, we weaken the assumption in the previous work and extend the analysis to multiclass cases. As result, under some assumptions, we prove that multiclass na\"ive Bayes requires $O(\log n)$ samples to approach its asymptotic error while the logistic regression needs $O(n)$ samples. Technically, we proposed a multiclass $\mathcal{H}$-consistency framework, which is of independent interest. Experiments with various pre-trained deep vision models verify our theory and show the potential of the generative linear head in the few-shot cases. Finally, our experiments suggest differences between representations learned by supervised and self-supervised methods.
\textbf{Social Impact:} This is mainly theoretical work and we do not see a direct social impact of our theory. The experiments on Na\"ive Bayes may benefit applications with a few training data such as medical analysis.
\ignore{
\section*{Acknowledgements}
}
| {
"timestamp": "2023-02-07T02:13:49",
"yymm": "2302",
"arxiv_id": "2302.02334",
"language": "en",
"url": "https://arxiv.org/abs/2302.02334",
"abstract": "A large-scale deep model pre-trained on massive labeled or unlabeled data transfers well to downstream tasks. Linear evaluation freezes parameters in the pre-trained model and trains a linear classifier separately, which is efficient and attractive for transfer. However, little work has investigated the classifier in linear evaluation except for the default logistic regression. Inspired by the statistical efficiency of naive Bayes, the paper revisits the classical topic on discriminative vs. generative classifiers. Theoretically, the paper considers the surrogate loss instead of the zero-one loss in analyses and generalizes the classical results from binary cases to multiclass ones. We show that, under mild assumptions, multiclass naive Bayes requires $O(\\log n)$ samples to approach its asymptotic error while the corresponding multiclass logistic regression requires $O(n)$ samples, where $n$ is the feature dimension. To establish it, we present a multiclass $\\mathcal{H}$-consistency bound framework and an explicit bound for logistic loss, which are of independent interests. Simulation results on a mixture of Gaussian validate our theoretical findings. Experiments on various pre-trained deep vision models show that naive Bayes consistently converges faster as the number of data increases. Besides, naive Bayes shows promise in few-shot cases and we observe the \"two regimes\" phenomenon in pre-trained supervised models. Our code is available atthis https URL.",
"subjects": "Machine Learning (cs.LG); Artificial Intelligence (cs.AI); Machine Learning (stat.ML)",
"title": "Revisiting Discriminative vs. Generative Classifiers: Theory and Implications",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9728307708274402,
"lm_q2_score": 0.7279754489059774,
"lm_q1q2_score": 0.7081969171026539
} |
https://arxiv.org/abs/0903.3828 | On the derivation of the Dirac Equation | We point out that the anticommutation properties of the Dirac matrices can be derived without squaring the Dirac hamiltonian, that is, without explicit reference to the Klein-Gordon equation. We only require the Dirac equation to admit two linearly independent plane wave solutions with positive energy for all momenta. The necessity of negative energies as well as the trace and determinant properties of the Dirac matrices are also a direct consequence of this simple and minimal requirement. | \section{Introduction}
Many textbooks \cite{bjorken,messiah,roman,scadron,schiff,thaller}
derive the Dirac equation
for a free particle of mass $m$ following
the method used by Dirac himself in his 1928 paper \cite{dirac}. This
method involves two steps. First, one admits that the
wave function should be a multi-component object, as in the
non-relativistic theory of spin, and that its time
evolution is ruled by a partial differential equation of first order
in both the time and space derivatives. Working in a system of
units where $\hbar=c=1$, we have
\begin{equation}
i{\partial \Psi\over\partial t}=H_D\Psi,
\label{d}
\end{equation}
where the Dirac hamiltonian $H_D$ is defined by
\begin{eqnarray}
H_D&=&\mbox{\boldmath$\alpha$}\cdot(-i\nabla)+\beta m\nonumber\\
&=&\sum_{k=1}^3\alpha_k(-i{\partial\over\partial x^k})+\beta m.
\end{eqnarray}
In this equation, the $\alpha_k$'s
and $\beta$ are constant hermitian matrices. In the second step, one
'squares' Eq. (\ref{d}) by acting on both sides of it with the operator
$i{\partial\over\partial t}$. This yields
\begin{eqnarray}
-{\partial^2 \Psi\over\partial t^2}
&=&H_D(i{\partial\over\partial t}\Psi)\nonumber\\
&=&H_D^2\Psi.
\end{eqnarray}
Then one requires $H_D^2$ to be identical to the operator $-\Delta+m^2$,
thereby ensuring that each of the components of $\Psi$ satisfies the
Klein-Gordon equation.This implies the anticommutation relations
\begin{eqnarray}
\alpha_i\alpha_j+\alpha_j\alpha_i&=&2\delta_{ij},\nonumber\\
\alpha_i\beta+\beta\alpha_i&=&0,\nonumber\\
\beta^2=1.
\label{rel}
\end{eqnarray}
Starting from these results, one usually proceeds
by showing that such matrices indeed exist when $\Psi$
is a four-component object and then one 'finds' that Eq. (\ref{d})
admits both positive and negative energy plane wave solutions. So
the Dirac equation does not solve the 'problem of negative energies'
which appears when studying the Klein-Gordon equation.
However, it is difficult to be immediately convinced that this fact
is not a mere consequence of the requirement appearing in the second
step of the above derivation. Thus, to discard any doubts about
the necessity of negative energies, it would be more satisfactory
to avoid the squaring of the hamiltonian $H_D$. In this
paper, we show that this is indeed feasible.
\section{Necessity of negative energies}
In order to implement the program outlined at the end of the previous
section, we require Eq. (\ref{d}) to admit two linearly independent
plane wave solutions of the form
\begin{equation}
\Psi({\bf x},t)=u({\bf p})e^{i({\bf p}\,\cdot\,{\bf x}-E_{\bf p}\,t)},
\label{pw}
\end{equation}
where $E_{\bf p}$ is the positive energy associated with a free
particle of momentum ${\bf p}$, that is,
\begin{equation}
E_{\bf p}=+({\bf p}^2+m^2)^{1/2}.
\label{pose}
\end{equation}
Since our aim is to describe spin $1/2$ particles such as
electrons, this requirement is both natural and minimal. By
inserting Eq. (\ref{pw}) into Eq. (\ref{d}), we obtain
\begin{equation}
E_{\bf p}\,u({\bf p})=h_D({\bf p})\,u({\bf p}),
\end{equation}
with
\begin{equation}
h_D({\bf p})=\mbox{\boldmath$\alpha$}\cdot{\bf p}+\beta m.
\end{equation}
Note that $h_D({\bf p})$ is a matrix of numbers
whereas $H_D$ is a matrix of differential operators. Since
we obviously discard the solution $u({\bf p})=0$, we see, from
the above requirement, that $E_{\bf p}$ should
be a double root of the eigenvalue equation pertaining to
the matrix $h_D({\bf p})$. Thus, if we introduce the characteristic
polynomial of $h_D({\bf p})$
\begin{equation}
P_n(E)=dtm[E-h_D({\bf p})],
\label{char}
\end{equation}
we should have
\begin{equation}
P_n(E_{\bf p})=0,
\label{p}
\end{equation}
and
\begin{equation}
P_n'(E_{\bf p})=0,
\label{pp}
\end{equation}
where $P_n'$ is the derivative of $P_n$ with respect to $E$. The index $n$
in these equations stands for the degree of $P_n(E)$
or, equivalently, for the number of components of the wave
function $\Psi$.
Let us now try to satisfy Eqs. (\ref{p}) and (\ref{pp}) within a
two-component theory ($n=2$). We have
\begin{equation}
P_n(E)\equiv P_2(E)=E^2+c_1({\bf p})E+c_0({\bf p}),
\end{equation}
where the coefficients $c_1$ and $c_0$ are polynomials homogeneous
in $m$ and the components $p_1$, $p_2$, $p_3$ of the
momentum ${\bf p}$. Eq. (\ref{pp}) yields
\begin{equation}
2E_{\bf p}+c_1({\bf p})=0.
\end{equation}
It is not possible to satisfy this equation for all momenta
since the square root $E_{\bf p}$ cannot be expressed as a polynomial.
Thus, a two-component theory is immediately ruled out. So, let us try
a Dirac equation with three components. Now, we have
\begin{equation}
P_n(E)\equiv P_3(E)=E^3+c_2({\bf p})E^2+c_1({\bf p})E+c_0({\bf p}),
\end{equation}
where the coefficients $c_2$, $c_1$ and $c_0$ are again polynomials homogeneous
in $m$ and the components of the momentum ${\bf p}$. Eqs. (\ref{p})
and (\ref{pp}) yield
\begin{equation}
E_{\bf p}^3+c_2({\bf p})E_{\bf p}^2+c_1({\bf p})E_{\bf p}+c_0({\bf p})=0,
\end{equation}
\begin{equation}
3E_{\bf p}^2+2c_2({\bf p})E_{\bf p}+c_1({\bf p})=0.
\end{equation}
Again using the fact that $E_{\bf p}$ cannot be expressed as a polynomial, we
see that these equations imply
\begin{equation}
E_{\bf p}^2+c_1({\bf p})=0,
\label{p3}
\end{equation}
\begin{equation}
c_2({\bf p})E_{\bf p}^2+c_0({\bf p})=0,
\end{equation}
\begin{equation}
3E_{\bf p}^2+c_1({\bf p})=0,
\label{pp3}
\end{equation}
\begin{equation}
c_2({\bf p})=0.
\end{equation}
Eqs. (\ref{p3}) and (\ref{pp3}) lead to $E_{\bf p}=0$ for all momenta. This
is not possible and, as a consequence, a three-component Dirac theory is
also ruled out. Finally, let us turn to a four-component
theory. Now,
\begin{equation}
P_n(E)\equiv P_4(E)=E^4+c_3({\bf p})E^3+c_2({\bf p})E^2+c_1({\bf p})E
+c_0({\bf p}),
\label{pc4}
\end{equation}
where our notations are similar to those used above in the
two- and three-component cases. Eqs. (\ref{p}) and (\ref{pp}) yield
\begin{equation}
E_{\bf p}^4+c_3({\bf p})E_{\bf p}^3+c_2({\bf p})E_{\bf p}^2
+c_1({\bf p})E_{\bf p}+c_0({\bf p})=0,
\end{equation}
\begin{equation}
4E_{\bf p}^3+3c_3({\bf p})E_{\bf p}^2+2c_2({\bf p})E_{\bf p}
+c_1({\bf p})=0.
\end{equation}
These equations imply
\begin{equation}
E_{\bf p}^4+c_2({\bf p})E_{\bf p}^2+c_0({\bf p})=0,
\label{p4a}
\end{equation}
\begin{equation}
c_3({\bf p})E_{\bf p}^2+c_1({\bf p})=0,
\label{p4b}
\end{equation}
\begin{equation}
2E_{\bf p}^2+c_2({\bf p})=0,
\label{pp4a}
\end{equation}
\begin{equation}
3c_3({\bf p})E_{\bf p}^2+c_1({\bf p})=0.
\label{pp4b}
\end{equation}
From Eq. (\ref{pp4a}), we obtain
\begin{equation}
c_2({\bf p})=-2E_{\bf p}^2.
\label{c2}
\end{equation}
Inserting this expression into Eq. (\ref{p4a}) yields
\begin{equation}
c_0({\bf p})=E_{\bf p}^4.
\label{c0}
\end{equation}
Finally, comparing Eqs. (\ref{p4b}) and (\ref{pp4b}) leads to
\begin{equation}
c_1({\bf p})=0
\label{c1}
\end{equation}
and
\begin{equation}
c_3({\bf p})=0.
\label{c3}
\end{equation}
If we insert these results back into Eq. (\ref{pc4}), we see
that the eigenvalue equation for $h_D({\bf p})$ reads
\begin{equation}
(E-E_{\bf p})^2(E+E_{\bf p})^2=0.
\label{eve}
\end{equation}
This shows that the positive energy solutions to the Dirac equation
will always be accompanied by
solutions with negative energy. To prove that the approach adopted in this
paper is self-contained, we still have to derive the anticommutation
relations (\ref{rel}). This is performed in the next section.
\section{Derivation of the anticommutation relations}
We now show that Eqs. (\ref{c2}), (\ref{c0}), (\ref{c1})
and (\ref{c3}) do indeed
imply Eqs. (\ref{rel}). We remark that once we have replaced $E_{\bf p}$
by its expression (\ref{pose}), all of these equations require
some polynomial homogeneous in $m$ and the components of ${\bf p}$ to
vanish identically, that is for all momenta. This is possible only if
all the polynomial coefficients are zero. We shall
rely repeatedly on this remark in what follows.
From Eqs. (\ref{char}) and (\ref{pc4}), we obtain
\begin{eqnarray}
c_3({\bf p})&=&-Tr(h_D({\bf p}))\nonumber\\
&=&\sum_{k=1}^3p_kTr(\alpha_k)+m Tr(\beta),
\end{eqnarray}
where the symbol $Tr$ denotes the trace. Thus, Eq.(\ref{c3}) implies
\begin{equation}
Tr(\alpha_1)=Tr(\alpha_2)=Tr(\alpha_3)=Tr(\beta)=0.
\label{trace}
\end{equation}
Eqs.(\ref{char}) and (\ref{pc4}) also yield
\begin{equation}
c_0({\bf p})=dtm(h_D({\bf p})).
\end{equation}
Inserting this expression into Eq.(\ref{c0}) and considering the
terms in $p_1^4$, $p_2^4$, $p_3^4$ and $m^4$ leads to
\begin{equation}
dtm(\alpha_1)=dtm(\alpha_2)=dtm(\alpha_3)=dtm(\beta)=1.
\label{dtm}
\end{equation}
To make things simpler, it is convenient to work in a
representation where the matrix $\beta$ is
diagonal. Note that Eqs.(\ref{trace})
and (\ref{dtm}) are representation independent. Consider the terms
in $m^3$ in Eq.(\ref{c1}) and in $m^2$ in Eq.(\ref{c2}). They yield
\begin{equation}
\beta_{11}\beta_{22}\beta_{33}+
\beta_{11}\beta_{22}\beta_{44}+
\beta_{11}\beta_{33}\beta_{44}+
\beta_{22}\beta_{33}\beta_{44}=0
\label{prod3}
\end{equation}
and
\begin{equation}
\beta_{11}\beta_{22}+\beta_{11}\beta_{33}+\beta_{11}\beta_{44}+
\beta_{22}\beta_{33}+\beta_{22}\beta_{44}+\beta_{33}\beta_{44}
=-2,
\label{prod2}
\end{equation}
respectively. Combining these equations with
\begin{equation}
dtm(\beta)=\beta_{11}\beta_{22}\beta_{33}\beta_{44}=1,
\end{equation}
(see Eq.(\ref{dtm})), we obtain
\begin{equation}
(1+\beta_{11})(1+\beta_{22})(1+\beta_{33})(1+\beta_{44})=0
\end{equation}
and
\begin{equation}
(1-\beta_{11})(1-\beta_{22})(1-\beta_{33})(1-\beta_{44})=0.
\end{equation}
These equations show that one of the eigenvalues of $\beta$ is
equal to $+1$ and another to $-1$. Let us assume that $\beta_{11}=+1$
and $\beta_{33}=-1$. Taking Eqs.(\ref{trace}) and (\ref{dtm})
into account, this implies
$\beta_{44}=-\beta_{22}$ and $\beta_{22}=\pm 1$. We shall assume
that $\beta_{22}=+1$ and thus $\beta_{44}=-1$. We do not have to consider
other choices for the
diagonal elements to be put equal to $+1$ or $-1$ since this would
correspond to a mere rearrangement of
the lines and columns of $\beta$. Thus, we have
\begin{equation}
\beta=\left(
\begin{array}{cccc}
1&0&0&0\\
0&1&0&0\\
0&0&-1&0\\
0&0&0&-1
\end{array}
\right).
\label{bstruc}
\end{equation}
Obviously,
\begin{equation}
\beta^2=1,
\label{b2}
\end{equation}
and it is easy to show that this equation implies
\begin{equation}
\alpha_i^2=1\hspace{5mm}(i=1,2,3).
\label{a2}
\end{equation}
Indeed, let us just imagine that we perform, on all the Dirac matrices, a
unitary transformation which brings $\alpha_1$, say, into diagonal form. We
expect that the matrix $\beta$ will no longer be diagonal but
Eq.(\ref{b2}) will remain true because it is representation
independent. We now proceed for $\alpha_1$ as we did above for $\beta$, that
is, we concentate on the terms in $p_1^3$ in Eq.(\ref{c1}) and in
$p_1^2$ in Eq.(\ref{c2}). This will lead us to $\alpha_i^2=1$ which is
also representation independent. Proceeding in this way for
$\alpha_2$ and $\alpha_3$, we prove the
other identities in Eq.(\ref{a2}). This trick can be used each time
we establish a representation independent identity even if we arrived at
that identity within a particular representation. In what follows, we
go back to the representation in which $\beta$ is given by Eq.(\ref{bstruc})
and we derive the structure of $\alpha_1$ in that representation. For the
moment, we drop the index $1$ to simplify our notations. Thus, $\alpha$
stands for $\alpha_1$. Consider Eq. (\ref{c2}). The terms in $m p_1$
and in $p_1^2$ give
\begin{equation}
-\alpha_{11}-\alpha_{22}+\alpha_{33}+\alpha_{44}=0
\label{mp}
\end{equation}
and
\begin{eqnarray}
&&\alpha_{11}\alpha_{22}+\alpha_{11}\alpha_{33}+\alpha_{11}\alpha_{44}+
\alpha_{22}\alpha_{33}+\alpha_{22}\alpha_{44}+\alpha_{33}\alpha_{44}
\nonumber\\
&&-|\alpha_{12}|^2-|\alpha_{13}|^2-|\alpha_{14}|^2-
|\alpha_{23}|^2-|\alpha_{24}|^2-|\alpha_{34}|^2
=-2,
\label{p2}
\end{eqnarray}
respectively. Consider now the terms in $m^2p_1^2$ in Eq.(\ref{c0}),
they give
\begin{eqnarray}
&&\alpha_{11}\alpha_{22}-\alpha_{11}\alpha_{33}-\alpha_{11}\alpha_{44}-
\alpha_{22}\alpha_{33}-\alpha_{22}\alpha_{44}+\alpha_{33}\alpha_{44}
\nonumber\\
&&-|\alpha_{12}|^2+|\alpha_{13}|^2+|\alpha_{14}|^2+
|\alpha_{23}|^2+|\alpha_{24}|^2-|\alpha_{34}|^2
=2.
\label{m2p2}
\end{eqnarray}
Adding Eqs.(\ref{p2}) and (\ref{m2p2}) yields
\begin{equation}
\alpha_{11}\alpha_{22}+\alpha_{33}\alpha_{44}
-|\alpha_{12}|^2-|\alpha_{34}|^2=0.
\label{p2m2p2}
\end{equation}
On the other hand, comparing Eq.(\ref{mp}) with Eq.(\ref{trace}) yields
\begin{equation}
\alpha_{22}=-\alpha_{11}
\end{equation}
and
\begin{equation}
\alpha_{44}=-\alpha_{33}.
\end{equation}
If we insert these results back into Eq.(\ref{p2m2p2}), we obtain
\begin{equation}
\alpha_{11}^2+\alpha_{33}^2
+|\alpha_{12}|^2+|\alpha_{34}|^2=0.
\end{equation}
Thus, we have
\begin{equation}
\alpha_{11}=\alpha_{22}=\alpha_{33}=\alpha_{44}
=\alpha_{12}=\alpha_{34}=0,
\end{equation}
and, restoring the index $1$, we see that the matrix $\alpha_1$
has the following structure:
\begin{equation}
\alpha_1=\left(
\begin{array}{cccc}
0&0&(\alpha_1)_{13}&(\alpha_1)_{14}\\
0&0&(\alpha_1)_{23}&(\alpha_1)_{24}\\
(\alpha_1)_{13}^*&(\alpha_1)_{23}^*&0&0\\
(\alpha_1)_{14}^*&(\alpha_1)_{24}^*&0&0
\end{array}
\right),
\label{astruc}
\end{equation}
where the non-vanishing elements are restricted by the condition
\begin{equation}
|(\alpha_1)_{13}|^2+|(\alpha_1)_{14}|^2+|(\alpha_1)_{23}|^2
+|(\alpha_1)_{24}|^2=2.
\end{equation}
Actually, this equation tells us nothing new since it can be
derived from Eq.(\ref{a2}).
An analogous proof shows that the matrices $\alpha_2$ and
$\alpha_3$ have also this structure. We note that the
structure of the $\alpha_i$'s and of $\beta$
(see Eq.(\ref{bstruc})) imply
\begin{equation}
\alpha_i\beta+\beta\alpha_i=0 \hspace{5mm}(i=1,2,3),
\label{ab}
\end{equation}
as can be checked simply by performing matrix multiplications. Since
these equations are representation independent, we conclude, using
the trick described after Eq.(\ref{a2}), that
we should also require
\begin{equation}
\alpha_i\alpha_j+\alpha_j\alpha_i=0 \hspace{5mm}(i\neq j=1,2,3).
\label{aa}
\end{equation}
Eqs.(\ref{b2}), (\ref{a2}), (\ref{ab}) and (\ref{aa}) are
the anticommutation relations we were looking for. It is easy to
check that Eqs.(\ref{c2}), (\ref{c0}) and (\ref{c1}) do not
give rise to additional restrictions on the Dirac matrices. As an
example, consider the terms in $p_1p_2$ in Eq.(\ref{c2}). They
impose
\begin{eqnarray}
&&(\alpha_1)_{13}^*(\alpha_2)_{13}+(\alpha_1)_{13}(\alpha_2)_{13}^*
+(\alpha_1)_{14}^*(\alpha_2)_{14}+(\alpha_1)_{14}(\alpha_2)_{14}^*
\nonumber\\
&+&(\alpha_1)_{23}^*(\alpha_2)_{23}+(\alpha_1)_{23}(\alpha_2)_{23}^*
+(\alpha_1)_{24}^*(\alpha_2)_{24}+(\alpha_1)_{24}(\alpha_2)_{24}^*
=0.
\end{eqnarray}
This equation can be written as
\begin{equation}
(\alpha_1\alpha_2+\alpha_2\alpha_1)_{11}+
(\alpha_1\alpha_2+\alpha_2\alpha_1)_{22}=0
\end{equation}
and, indeed, requires nothing new.
\section{Summary and comments}
In this paper, we have provided a method to derive the anticommutation
properties of the Dirac matrices without relying on the squaring
of the Dirac hamiltonian. We have only required Eq.(\ref{d}) to admit
two linearly independent plane wave solutions with positive energy for all
momenta. At an early stage in the derivation, we have seen that, despite this
conservative requirement, it was not possible to rule out negative energy
solutions, thereby establishing that these
are not an artefact of
the standard derivation. It might also be interesting to note
that, within the method described in this
paper, the trace and determinant properties of the Dirac matrices
appear in the course of the derivation and not as by-products
of the anticommutation relations. Finally, a few comments are appropriate
concerning our proof of the impossibility of a two-component Dirac
equation. As is well known, such an equation appears in a space-time
with less than three space dimensions \cite{thaller} or in the study of
massless fermions, where it is known as the Weyl
equation \cite{bjorken,scadron}. This by no means
contradicts our assertions. Indeed, no
impossibility arises in a two-component theory if one only requires
the Dirac equation to admit a single plane wave solution with positive
energy for all momenta. In that case, only Eq.(\ref{p}) with $n=2$
has to be imposed and this yields
\begin{equation}
E_{\bf p}^2+c_1({\bf p})E_{\bf p}+c_0({\bf p})=0.
\end{equation}
This equation implies
\begin{equation}
E_{\bf p}^2+c_0({\bf p})=0,
\end{equation}
and
\begin{equation}
c_1({\bf p})=0.
\end{equation}
From these equations, we see that the eigenvalue equation for
$h_D({\bf p})$ reads
\begin{equation}
(E-E_{\bf p})(E+E_{\bf p})=0.
\end{equation}
Thus, we have a plane wave solution with positive
energy and another with negative energy. As a consequence, in a two-component
theory, the 'twofold degeneracy' only corresponds to the existence
of antiparticles. The derivation of the properties of the
Dirac matrices (actually, of the Pauli matrices, since we are now in a
two-component theory) can be performed as in the previous section and
will not be repeated here.
| {
"timestamp": "2009-03-23T11:30:37",
"yymm": "0903",
"arxiv_id": "0903.3828",
"language": "en",
"url": "https://arxiv.org/abs/0903.3828",
"abstract": "We point out that the anticommutation properties of the Dirac matrices can be derived without squaring the Dirac hamiltonian, that is, without explicit reference to the Klein-Gordon equation. We only require the Dirac equation to admit two linearly independent plane wave solutions with positive energy for all momenta. The necessity of negative energies as well as the trace and determinant properties of the Dirac matrices are also a direct consequence of this simple and minimal requirement.",
"subjects": "Quantum Physics (quant-ph)",
"title": "On the derivation of the Dirac Equation",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9728307700397331,
"lm_q2_score": 0.7279754489059774,
"lm_q1q2_score": 0.7081969165292223
} |
https://arxiv.org/abs/1803.09235 | An improved Popoviciu-type inequality for a new Bernstein-type operator | Recently we introduced a new Bernstein-type operator using Pólya's urn model with negative replacement, and we showed that it satisfies a Popoviciu-type inequality with a constant slightly larger than that of the corresponding inequality for the classical Bernstein operator.In the present paper we prove an inequality for the rising factorial (of independent interest), and we use it in order to show that the constant in the Popoviciu inequality for the new operator is in fact smaller than the corresponding constant for the Bernstein operator. | \section{Introduction}
It is known that the classical Bernstein operator (\cite{Berstein 1912}) defined by
\begin{equation}\label{Probabilistic repr of Bernstein polynomial}
B_{n}\left( f;x\right) =\sum_{k=0}^{n}f\left( \frac{k}{n}\right)
C_{n}^{k}x^{k}\left( 1-x\right) ^{n-k}
\end{equation
satisfies the inequality
\begin{equation} \label{Popoviciu's error estimate for Bernstein}
\left\vert B_{n}\left( f;x\right) -f\left( x\right) \right\vert \leq C\omega
\left( n^{-1/2}\right) ,\qquad x\in \left[ 0,1\right] , \; n=1,2,\ldots,
\end{equation
where $f:[0,1]\rightarrow \mathbb R$ is an arbitrary continuous function, $\omega(\cdot)$ denotes the modulus of continuity of $f$.
T. Popoviciu (\cite{Popoviciu}) proved the above inequality for the value of the constant $C=\frac32$. Lorentz (\cite{Lorentz}, pp. 20 --21) improved the value of the constant to $C=\frac{5}{4}$, and also showed that the constant $C$ cannot be less than one. The optimal value of the constant $C$ for which the above inequality holds true for any continuous function was obtained by Sikkema (\cite{Sikkema}), who obtained the value
\begin{equation}
C_{opt}=\frac{4306+837\sqrt{6}}{5932}\approx 1.0898873...,
\label{Sikkema optimal constant}
\end{equation
attained in the case $n=6$ for a particular function.
Recently (\cite{PPT1}), we introduced the Bernstein-type operator $R_{n}$ defined by (\ref{Rational Bernstein operator}) and we showed that its also satisfies a Popoviciu-type inequality, with the constant $C=\frac{31}{27}\approx 1.14815$, smaller than Popoviciu's and Lorentz's constants, but slightly larger than Sikkema's optimal constant.
In the present paper we refine this result, by showing that the constant in the Popoviciu-type estimate is smaller than Sikkema's constant (Theorem \ref{thm for upper bound for optimal constant}). The proof is based on a certain inequality for the rising factorial (Lemma \ref{Monotonicity of F^c}) which is of independent interest.
\section{Preliminaries}\label{Preliminaries}
For $x,h\in \mathbb{R}$ and $n\in \mathbb{N}$ we set
\begin{equation}
x^{\left( n,h\right) }=x\left( x+h\right) \left( x+2h\right) \cdot \ldots
\cdot \left( x+\left( n-1\right) h\right) \label{rising factorial}
\end{equation
for the generalized (rising) factorial with increment $h$. When $n=0$ we are using the convention $x^{\left( 0,h\right) }=1$ for any $x,h\in \mathbb{R}$.
A random variable $X_{n}^{a,b,c}$ has a P\'{o}lya's urn distribution (also known as P\'{o}lya-Eggenberger distribution, see \cite{Eggenberger-Polya}, \cite{Polya}) with parameters $ n\geq 1$, $a,b\in \mathbb{R}_{+}$, and $c\in\mathbb{R}$ satisfying \begin{equation}\label{Hypothesis on c}
a+\left( n-1\right) c\geq 0\qquad \text{and }\qquad b+\left( n-1\right)
c\geq 0,
\end{equation}
if it is given by (see for example \cite{Johnson-Kotz})
\begin{equation}
P\left(X_n^{a,b,c}=k\right)=p_{n,k}^{a,b,c}=C_{n}^{k}\frac{\left( a\right) ^{\left( k,c\right) }\left(
b\right) ^{\left( n-k,c\right) }}{\left( a+b\right) ^{\left( n,c\right) }
,\qquad k\in \left\{ 0,1,\ldots ,n\right\} . \label{Polya urn probabilities}
\end{equation}
In the case when $a,b\in \mathbb N$ and $c\in \mathbb Z$, the physical interpretation of the random variable $X_n^{a,b,c}$ is the total number of white balls obtained in $n$ extractions from an urn containing initially $a$ white balls and $b$ black balls, when the extractions are made with $c$ replacements (the extracted ball is returned to the urn together with $c$ balls of the same color, a negative value of $c$ being interpreted as removing $\vert c\vert$ balls from the urn).
It is known (e.g. \cite{Johnson-Kotz}) that the mean and variance of $X_n^{a,b,c}$ are given by
\begin{equation}
E\left( X_{n}^{a,b,c}\right) =\frac{na}{a+b}\qquad \text{and} \qquad \sigma
^{2}\left( X_{n}^{a,b,c}\right) =\frac{nab}{\left( a+b\right) ^{2}}\left( 1+\frac
\left( n-1\right) c}{a+b+c}\right) . \label{Polya mean and variance}
\end{equation}
In \cite{PPT1} we considered the operator $R_n$ defined on the space of real-valued functions on $[0,1]$ by
\begin{eqnarray} \label{Rational Bernstein operator}
R_{n}\left( f;x\right) &=&Ef\left( \frac{1}{n}X_{n}^{x,1-x,-\min \left\{
x,1-x\right\} /(n-1)}\right) \\
&=&\sum_{k=0}^{n}C_{n}^{k}\frac{x^{\left( k,-\min
\left\{ x,1-x\right\} /(n-1)\right) }\left( 1-x\right) ^{\left( n-k,-\min
\left\{ x,1-x\right\} /(n-1)\right) }}{1^{\left( n,-\min \left\{
x,1-x\right\} /(n-1)\right) }}f\left( \frac{k}{n}\right),\notag
\end{eqnarray}
and we showed that it satisfies pointwise estimates (in terms of the moduli of continuity of the function, of its first or second derivative) which improve the corresponding estimates for the classical Bernstein operator. We also showed that the operator $R_n$ satisfies a global Popoviciu-type inequality of the form (\ref{Popoviciu's error estimate for Bernstein}), with a constant slightly larger than the optimal constant found by Sikkema in the case of Bernstein operator.
In the next section we will show that the constant in the Popoviciu-type inequality for the operator $R_n$ is in fact strictly smaller than Sikkema's optimal constant for the Bernstein operator $B_n$. This suggests (although we do not have a proof) that the operator $R_n$ provides a better approximation than the Bernstein operator $B_n$, claim which is also supported by the numerical and theoretical results obtained in \cite{PPT1}.
\section{Main results}
The proof of our main result rests on the following inequality for the rising factorial (or equivalently, Pochammer symbol), which may be of independent interest.
\begin{lemma}
\label{Monotonicity of F^c}For any $x\in \left[ 0,1\right] $ and any
non-negative integers $n>1$ and $r\leq nx-\sqrt{n}$, we hav
\begin{equation}
\frac{x^{\left( r+1,c\right) }\left( 1-x\right) ^{\left( n-r,c\right) }}
1^{\left( n,c\right) }}\leq x^{r+1}\left( 1-x\right) ^{n-r} \label{claim}
\end{equation
for any $c\in \left[ -\min \left\{ x,1-x\right\} /\left( n-1\right) ,0\right]
$.
Moreover, the above inequality is strict except for the case $c=0$.
\end{lemma}
\begin{proof}
For $c=0$ and the claim becomes an identity, so we may assume $c\ne0$. The claim also holds true for $r=0$, sinc
\[
\frac{x\left( 1-x\right) ^{\left( n,c\right) }}{1^{\left( n,c\right) }
=x\prod_{i=0}^{n-1}\frac{1-x+ic}{1+ic} < x\prod_{i=0}^{n-1}\left(
1-x\right) =x\left( 1-x\right) ^{n}.
\]
Since $r\leq nx-\sqrt{n}<n-1$, we may assume that $0<r<n-1$, which in
particular shows that all the factors appearing in the rising factorials on
the left of (\ref{claim}) are positive.
Taking logarithms and rearranging the terms, the claim (\ref{claim}) is
equivalent t
\begin{equation}
\sum_{i=1}^{n-1}\ln \frac{1}{1+ic}\leq \sum_{i=1}^{r}\ln \frac{x}{x+ic
+\sum_{i=1}^{n-r-1}\ln \frac{1-x}{1-x+ic}. \label{claim equivalent}
\end{equation}
It is easy to see that the function $\varphi :\left\{ \left(
u,t\right) \in \mathbb{R}^{2}:u,u+ct>0\right\} \rightarrow \mathbb{R}$
defined b
\[
\varphi \left( u,t\right) =\ln \frac{u}{u+ct},
\
is convex and increasing in the variable $t$ (holding $u>0$ fixed), and it
is also satisfies $\varphi \left( \alpha u,\alpha t\right) =\varphi \left(
u,t\right) $ for any $\alpha >0$. Using this and an area comparison, we
obtai
\begin{equation}
\sum_{i=1}^{r}\ln \frac{x}{x+ic}=\sum_{i=1}^{r}\varphi \left( x,i\right)
=r\sum_{i=1}^{r}\frac{1}{r}\varphi \left( \frac{x}{r},\frac{i}{r}\right)
> r\int_{0}^{1}\varphi \left( \frac{x}{r},t\right)
dt=\int_{0}^{1}r\varphi \left( 1,\frac{rt}{x}\right) dt, \label{ineq1}
\end{equation
and similarl
\begin{equation}
\sum_{i=1}^{n-r-1}\ln \frac{1-x}{1-x+ic}> \int_{0}^{1}\left( n-r-1\right)
\varphi \left( 1,\frac{\left( n-r-1\right) t}{1-x}\right) dt, \label{ineq2}
\end{equation
an
\begin{equation}
\sum_{i=1}^{n-1}\ln \frac{1}{1+ic}=n\sum_{i=1}^{n-1}\frac{1}{n}\varphi
\left( \frac{1}{n},\frac{i}{n}\right) < n\int_{1/n}^{1}\varphi \left(
\frac{1}{n},t\right) dt. \label{ineq3}
\end{equation}
Using a substitution, we can simplify the last term above as follow
\begin{eqnarray*}
&&n\int_{1/n}^{1}\varphi \left( \frac{1}{n},t\right) dt=n\left(
\int_{0}^{1}\varphi \left( \frac{1}{n},t\right) dt-\int_{0}^{1/n}\varphi
\left( \frac{1}{n},t\right) dt\right) \\
&=&n\left( \int_{0}^{1}\varphi \left( \frac{1}{n},t\right) dt-\frac{1}{n
\int_{0}^{1}\varphi \left( \frac{1}{n},\frac{t}{n}\right) dt\right)
=\int_{0}^{1}n\varphi \left( 1,nt\right) dt-\int_{0}^{1}\varphi \left(
1,t\right) dt.
\end{eqnarray*}
Combining the above with (\ref{ineq1}) -- (\ref{ineq3}), it follows that
\ref{claim equivalent}) holds if we prove the inequalit
\begin{equation}
\int_{0}^{1}\varphi \left( 1,nt\right) dt\leq \int_{0}^{1}\frac{1}{n}\varphi
\left( 1,t\right) dt+\frac{r}{n}\varphi \left( 1,\frac{rt}{x}\right) +\frac
n-r-1}{n}\varphi \left( 1,\frac{\left( n-r-1\right) t}{1-x}\right) dt.
\label{claim equivalent biss}
\end{equation}
The convexity of the function $\varphi $ in the second variable shows that
the right side of the above inequality is larger tha
\[
\int_{0}^{1}\varphi \left( 1,\frac{t}{n}\left( 1+\frac{r^{2}}{x}+\frac
\left( n-r-1\right) ^{2}}{1-x}\right) \right) dt,
\]
and using the monotonicity of the function $\varphi $ in the second variable
it follows that the inequality (\ref{claim equivalent biss}) holds provided
we show tha
\begin{equation}
nt\leq \frac{t}{n}\left( 1+\frac{r^{2}}{x}+\frac{\left( n-r-1\right) ^{2}}
1-x}\right) \label{claim final}
\end{equation
for any positive integer $r<nx-\sqrt{n}$.
Extending the above inequality to positive real values $r\leq nx-\sqrt{n}$,
we hav
\[
\frac{d}{dr}\left( 1+\frac{r^{2}}{x}+\frac{\left( n-r-1\right) ^{2}}{1-x
\right) =\frac{2r}{x}-\frac{2(n-r-1)}{1-x} =\frac{2\left(
r-nx+x\right) }{x\left( 1-x\right) }<\frac{2\left( r-nx+\sqrt{n}\right) }
x\left( 1-x\right) }\leq 0,
\
and therefore the right hand side of (\ref{claim final}) is decreasing in
r\leq nx-\sqrt{n}$.
It follows that (\ref{claim final}) holds, provided we sho
\[
n^{2}\leq 1+\frac{\left( nx-\sqrt{n}\right) ^{2}}{x}+\frac{\left( n\left(
1-x\right) +\sqrt{n}-1\right) ^{2}}{1-x},
\
or equivalen
\[
n^{2}\leq 1+n^{2}x-2n\sqrt{n}+\frac{n}{x}+n^{2}\left( 1-x\right) +2n\left(
\sqrt{n}-1\right) +\frac{n-2\sqrt{n}+1}{1-x}.
\]
Rearranging terms, we have equivalen
\[
\frac{\left( 2n-1\right) x^{2}-2\left( n+\sqrt{n}-1\right) x+n}{x\left(
1-x\right) }\geq 0,
\
which holds true, since the numerator is a quadratic function of $x$ with a
negative dis\-cri\-mi\-nant $\Delta =-4\left( n-1\right) \left( \sqrt{n}-1\right)
^{2}<0$ for all $n>1$, thus concluding the proof.
\end{proof}
Using the above lemma, we can now prove the following.
\begin{theorem}
\label{thm for upper bound for optimal constant} There exists a constant
C\leq 1.08970< C_{opt}=1.0898873...$ such that for any conti\-nu\-ous function $f:\left[
0,1\right] \rightarrow \mathbb{R}$ and any $n>1$ we hav
\begin{equation}
\left\vert R_{n}\left( f;x\right) -f\left( x\right) \right\vert \leq C\omega
\left( n^{-1/2}\right) ,\qquad x\in \left[ 0,1\right] ,
\end{equation
where $\omega \left( \delta \right) =\omega ^{f}\left( \delta \right) $
denotes the modulus of continuity of $f$.
\end{theorem}
\begin{proof}
For $a\in \mathbb{R}$ denote by $]a[$ the largest integer strictly smaller
than $a$, that is $]a[=k\in \mathbb{Z}$ if $k<a\leq k+1$.
Using the definition of the modulus of continuity of $f$, it can be seen
that $\omega \left( \lambda \delta \right) \leq \left( 1+]\lambda \lbrack
\right) \omega \left( \delta \right) $ for any $\lambda \geq 0$ and $\delta
>0$ (see \cite{Sikkema}). Using this, with $\delta =n^{-1/2}$ and $\lambda
=\left\vert x-\frac{k}{n}\right\vert $, $k\in \left\{ 0,1,\ldots ,n\right\} $, and $c\in \left[ -\frac{\min \left\{ x,1-x\right\} }{n-1},0\right] $, we obtai
\begin{eqnarray}
\left\vert Ef\left( \frac{1}{n}X_{n}^{x,1-x,c}\right) -f\left( x\right)
\right\vert &\leq& \left\vert \sum_{k=0}^{n}\left( f\left( \frac{k}{n
\right) -f\left( x\right) \right) p_{k,n}^{x,1-x,c}\right\vert \label{aux 1}
\\
&\leq & \omega \left( n^{-1/2}\right) \left( 1+\sum_{k=0}^{n}\left] \frac
\left\vert x-\frac{k}{n}\right\vert }{n^{-1/2}}\right[ p_{n,k}^{x,1-x,c
\right) \nonumber \\
&\leq &\omega \left( n^{-1/2}\right) \left( 1+\sqrt{n}\sum_{\substack{k\in \left\{
0,1,\ldots ,n\right\} :\\\left\vert x-\frac{k}{n}\right\vert
>n^{-1/2}}}\left\vert x-\frac{k}{n}\right\vert p_{n,k}^{x,1-x,c}\right) .
\nonumber
\end{eqnarray
For $x\in (\frac{1}{\sqrt{n}},1]$, denoting by $r=r\left( x\right) $ the
largest integer for which $x-\frac{r}{n}>n^{-1/2}$, or equivalent $r=r\left(
x\right) =]nx-\sqrt{n}[\in \left\{ 0,1,\ldots ,n-1\right\} $, and using
Lemma 3 in \cite{Kozniewska}, we have
\begin{eqnarray}
&&\sum_{\substack{k\in \left\{ 0,1,\ldots ,n\right\} :\\x-\frac{k}{n}>n^{-1/2}}}\lef
\vert x-\frac{k}{n}\right\vert p_{n,k}^{x,1-x,c}=\sum_{k=0}^{r}\left( x
\frac{k}{n}\right) p_{n,k}^{x,1-x,c}=\frac{1}{n}\sum_{k=0}^{r}\left(
nx-k\right) p_{n,k}^{x,1-x,c} \quad\label{aux 2} \\
&&=\frac{1}{n}(r+1)p_{n,r+1}^{x,1-x,c}\left( 1-x+\left( n-r-1\right)
c\right) =C_{n-1}^{r}\frac{x^{\left( r+1,c\right) }\left( 1-x\right)
^{\left( n-r,c\right) }}{1^{\left( n,c\right) }}. \nonumber
\end{eqnarray}
The above sum equal zero for $x\in \left[ 0,\frac{1}{\sqrt{n}}\right] $, and therefore we hav
\[
\sum_{\substack{k\in \left\{ 0,1,\ldots ,n\right\} :\\x-\frac{k}{n}>n^{-1/2}}}\left( x
\frac{k}{n}\right) p_{n,k}^{x,1-x,c}=F_{n}^{c}\left( x\right) ,\qquad x\in
\left[ 0,1\right] ,
\
where $F_{n}^{c}:\left[ 0,1\right] \mathbb{\rightarrow R}$ is the function
defined b
\begin{equation}
F_{n}^{c}\left( x\right) =\left\{
\begin{tabular}{ll}
$0,$ & $x\in \left[ 0,\frac{1}{\sqrt{n}}\right] $ \\
$C_{n-1}^{r}\frac{x^{\left( r+1,c\right) }\left( 1-x\right) ^{\left(
n-r,c\right) }}{1^{\left( n,c\right) }},$ & $x\in (\frac{1}{\sqrt{n}},1]
\end{tabular
\right. \text{, where }r=r\left( x\right) =]nx-\sqrt{n}[. \label{definition of the function F^c}
\end{equation
For $x\in \lbrack 0,1-\frac{1}{\sqrt{n}})$, denoting $s=s\left( x\right) $
the smallest integer for which $x-\frac{s}{n}<-\frac{1}{\sqrt{n}}$, or
equivalent $s=\left[ nx+\sqrt{n}+1\right] \in \left\{ 1,\ldots ,n\right\} $,
and using again Lemma 3 in \cite{Kozniewska} and (\ref{Polya mean and
variance}), we hav
\begin{eqnarray}
&&\sum_{\substack{k\in \left\{ 0,1,\ldots ,n\right\} :\\x-\frac{k}{n}<-n^{-1/2}}}\lef
\vert x-\frac{k}{n}\right\vert p_{n,k}^{x,1-x,c}=\frac{1}{n
\sum_{k=s}^{n}\left( k-nx\right) p_{n,k}^{x,1-x,c} =\frac{1}{n}\sum_{k=0}^{s-1}\left( nx-k\right)
p_{n,k}^{x,1-x,c}\quad \quad\\
&=& C_{n-1}^{s-1}\frac{x^{\left( s,c\right) }\left( 1-x\right)
^{\left( n-s+1,c\right) }}{1^{\left( n,c\right) }}.\notag
\end{eqnarray}
Denoting by $r^{\prime }=n-s$ and $x^{\prime }=1-x$, and using the property
-\left[ a+1\right] =]-a[$, we hav
\[
r^{\prime }=n-s=n-\left[ n\left( 1-x\right) ^{\prime }+\sqrt{n}+1\right] =
\left[ -nx^{\prime }+\sqrt{n}+1\right] =]nx^{\prime }-\sqrt{n}[,
\
and using the definition (\ref{definition of the function F^c}) of
F_{n}^{c} $, we can rewrite the sum above as follow
\begin{equation}
\sum_{\substack{k\in \left\{ 0,1,\ldots ,n\right\} :\\x-\frac{k}{n}<-n^{-1/2}}}\left\vert
x-\frac{k}{n}\right\vert p_{n,k}^{x,1-x,c}=C_{n-1}^{r^{\prime }}\frac{\left(
x^{\prime }\right) ^{\left( r^{\prime }+1,c\right) }\left( 1-x^{\prime
}\right) ^{\left( n-r^{\prime },c\right) }}{1^{\left( n,c\right) }
=F_{n}^{c}\left( x^{\prime }\right) =F_{n}^{c}\left( 1-x\right) ,\label{aux 4}
\end{equation
for any $x\in \left[ 0,1\right]$ (note that for $x\in \left[ 1-\frac{1}{\sqrt{n}},1\right] $ the left
hand-side of the above equality also equals the right hand-side, both sides
being equal to zero).
Combining (\ref{aux 1}) -- (\ref{aux 4}) above, we obtai
\[
\left\vert Ef\left( \frac{1}{n}X_{n}^{x,1-x,c}\right) -f\left( x\right)
\right\vert \leq \omega \left( n^{-1/2}\right) \left( 1+\sqrt{n}\left(
F_{n}^{c}\left( x\right) +F_{n}^{c}\left( 1-x\right) \right) \right) ,
\
for any $x\in \left[ 0,1\right]$ and $c\in \left[ -\frac{\min \left\{ x,1-x\right\} }{n-1},0\right] $. Considering in particular $c=-\frac{\min \left\{ x,1-x\right\} }{n-1}$, and using Lemma \ref{Monotonicity of F^c} (which shows that $F_{n}^{c}\left( x\right)\leq F_{n}^{0}\left( x\right)$ for any $x\in[0,1]$), we obtai
\begin{equation}\label{estimate using Sikkema's function}
\left\vert R_{n}\left( f;x\right) -f\left( x\right) \right\vert \leq \omega \left( n^{-1/2}\right) \left( 1+\sqrt{n}\left( F_{n}^{0}\left( x\right) +F_{n}^{0}\left( 1-x\right) \right) \right),\qquad x\in[0,1] .
\end{equation}
In \cite{Sikkema} (Section 4), the author obtained the following estimate
\begin{equation}\label{Sikkema estimate}
1+\sqrt{n}\left( F_{n}^{0}\left( x\right) +F_{n}^{0}\left( 1-x\right) \right) \leq 1.0897, \qquad x\in[0,1],
\end{equation}
valid any positive integer $n\neq6$. Combining this with (\ref{estimate using Sikkema's function}) proves the claim of the theorem for any positive integer $n\neq 6$.
To conclude the proof, we have left to consider the case $n=6$. First note that $c=-\frac{\min\{x,1-x\}}{5}=-\frac{x}{5}$ or $\frac{x-1}{5}$, depending whether $x\leq\frac12$ or $x>\frac12$, and the function $r=r\left( x\right) =]6x-\sqrt{6}[$ in the definition $F^c_6$ takes the value $k\in\{0,1,2,3\}$ for $x\in\left( \frac{1}{\sqrt{6}}+\frac{k}{6},\frac{1}{\sqrt{6}}+\frac{k+1}{6}\right]$, thus in order to estimate $F_6^c$ there are several cases to consider. For example, in the case $x\in\left( \frac{1}{\sqrt{6}},\frac12\right]$ we have $r=0$ and $c=-\frac{x}{5}$, and from (\ref{definition of the function F^c}) we obtain
\begin{equation*}
F_6^c(x)=\frac{x(1-x)^{(6,-x/5)}}{1^{(6,-x/5)}}<\frac{\frac12(1-\frac{1}{\sqrt{6}})^{(6,-\frac{1}{5\sqrt{6}})}}{1^{(6,-1/10)}}=\frac{193282 - 78887 \sqrt{6}}{6804}\approx 0.00721673,
\end{equation*}
and for $x\in\left( \frac12 , \frac{1}{\sqrt{6}}+\frac16 \right]$ we have $r=0$ and $c=\frac{x-1}{5}$, and we obtain
\begin{equation*}
F_6^c(x)=\frac{x(1-x)^{(6,\frac{x-1}{5})}}{1^{(6,\frac{x-1}{5})}}\equiv 0.
\end{equation*}
In the remaining three cases we have $c=\frac{1-x}{5}$, and proceeding similarly we obtain
\begin{equation*}
F_6^c(x)\leq\left\{
\begin{tabular}{ll}
$C_5^1 \frac{x^{\left(2,\frac{1-x}{5}\right)} (1-x)^{\left(5,\frac{1-x}{5}\right)}}{1^{\left(6,\frac{1-x}{5}\right)}}$, &$x\in \left(\frac{1}{\sqrt{6}}+\frac16,\frac{1}{\sqrt{6}}+\frac26\right ]$\\
$C_5^2 \frac{x^{\left(3,\frac{1-x}{5}\right)}(1-x)^{4,\frac{1-x}{4}}}{1^{6,\frac{1-x}{5}}}$, & $x\in\left( \frac{1}{\sqrt{6}}+\frac26, \frac{1}{\sqrt{6}}+\frac36\right]$ \\
$C_5^3 \frac{x^{\left(3,\frac{1-x}{5}\right)}(1-x)^{3,\frac{1-x}{5}}}{1^{6,\frac{1-x}{5}}}$, & $x\in\left( \frac{1}{\sqrt{6}}+\frac36, 1\right]$ \\
\end{tabular}
\right.
\leq 0.014271.
\end{equation*}
In all cases above we obtained $F_6^c(x)\leq 0.014271$ for $x\in[0,1]$, and therefore
\begin{equation*}
1+\sqrt{6}\left(F_6^c(x)+F_6^c(1-x)\right) \leq 1+2 \sqrt{6}\cdot 0.014271 \approx 1.0699134 < 1.0897, \quad x\in[0,1],
\end{equation*}
concluding the proof of the theorem.
\end{proof}
The above result leaves open the problem of finding the value of the optimal constant $C$ in the above theorem. Although we do not have an answer here, we believe that the optimal constant is much smaller than the value hinted by the above theorem.
A second remark is that we believe that Lemma \ref{Monotonicity of F^c}, on which the above proof rests, can be improved to show that the left hand-side of the inequality (\ref{claim}) is in fact an increasing function of $c\geq -\frac{\min \left\{ x,1-x\right\} }{n-1}$. If this conjecture is correct, an argument similar to the one used in
the above proof would show that
\[
\frac{\left\vert Ef\left( \frac{1}{n}X_{n}^{x,1-x,c}\right) -f\left(
x\right) \right\vert }{\omega \left( n^{-1/2}\right) }
\
is in fact a monotone increasing function of $c\geq -\min \left\{ x,1-x\right\}/(n-1)$, hence among all P\'{o}lya-Bernstein type operators of the form
$$P_{n}^{x,1-x,c}(f;x) = Ef\left( \tfrac{1}{n}X_{n}^{x,1-x,c}\right),\qquad x\in[0,1],$$
the one that provides the best approximation in the class of continuous functions is the operator $R_{n}$ given by (\ref{Rational Bernstein operator}), which corresponds to $c=-\min \left\{ x,1-x\right\} /(n-1)$.
\section*{Acknowledgements}
The first author kindly acknowledges the support by a grant of the Romanian National Authority for Scientific Research, CNCS - UEFISCDI, project number PNII-ID-PCCE-2011-2-0015.
| {
"timestamp": "2018-03-28T02:10:54",
"yymm": "1803",
"arxiv_id": "1803.09235",
"language": "en",
"url": "https://arxiv.org/abs/1803.09235",
"abstract": "Recently we introduced a new Bernstein-type operator using Pólya's urn model with negative replacement, and we showed that it satisfies a Popoviciu-type inequality with a constant slightly larger than that of the corresponding inequality for the classical Bernstein operator.In the present paper we prove an inequality for the rising factorial (of independent interest), and we use it in order to show that the constant in the Popoviciu inequality for the new operator is in fact smaller than the corresponding constant for the Bernstein operator.",
"subjects": "Classical Analysis and ODEs (math.CA)",
"title": "An improved Popoviciu-type inequality for a new Bernstein-type operator",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9728307676766119,
"lm_q2_score": 0.7279754489059774,
"lm_q1q2_score": 0.7081969148089282
} |
https://arxiv.org/abs/1202.2840 | Geometric Pricing: How Low Dimensionality Helps in Approximability | Consider the following toy problem. There are $m$ rectangles and $n$ points on the plane. Each rectangle $R$ is a consumer with budget $B_R$, who is interested in purchasing the cheapest item (point) inside R, given that she has enough budget. Our job is to price the items to maximize the revenue. This problem can also be defined on higher dimensions. We call this problem the geometric pricing problem.In this paper, we study a new class of problems arising from a geometric aspect of the pricing problem. It intuitively captures typical real-world assumptions that have been widely studied in marketing research, healthcare economics, etc. It also helps classify other well-known pricing problems, such as the highway pricing problem and the graph vertex pricing problem on planar and bipartite graphs. Moreover, this problem turns out to have close connections to other natural geometric problems such as the geometric versions of the unique coverage and maximum feasible subsystem problems.We show that the low dimensionality arising in this pricing problem does lead to improved approximation ratios, by presenting sublinear-approximation algorithms for two central versions of the problem: unit-demand uniform-budget min-buying and single-minded pricing problems. Our algorithm is obtained by combining algorithmic pricing and geometric techniques. These results suggest that considering geometric aspect might be a promising research direction in obtaining improved approximation algorithms for such pricing problems. To the best of our knowledge, this is one of very few problems in the intersection between geometry and algorithmic pricing areas. Thus its study may lead to new algorithmic techniques that could benefit both areas. |
\section{Open Problems}\label{sec:conclusion}
Several interesting problems are open. The most important problem is whether we can obtain better approximation factors for $d$-{\sf UUDP-MIN} and $d$-{\sf SMP}.
We tend to believe that there is an $f(d)$-approximation algorithm for $d$-{\sf UUDP-MIN} and $d$-{\sf SMP} where $f(d)$ is a function that depends on $d$ only. However,
it seems to be a very challenging task to obtain approximation ratio like $\log^{O(d)} n$ or $O_d(\log^{1-\epsilon(d)} m)$, for some constant $\epsilon(d)>0$ depending on $d$.
\fullonly{
\begin{openproblem}
\label{conjecture: main one}
For any integer $d >0$, are there $\log^{O(d)} n$-approximation algorithms for $d$-{\sf UUDP-MIN} and $d$-{\sf SMP}?
\end{openproblem}
\begin{openproblem}
\label{conjecture: main two}
For any integer $d >0$, are there $O_d(\log^{1-\epsilon_d} m)$-approximation algorithms for $d$-{\sf UUDP-MIN} and $d$-{\sf SMP}?
\end{openproblem}
}
One promising direction in attacking the above problems is to improve Theorem~\ref{thm: dimension reduction for UDP}, e.g., getting $O_d(\rho\cdot\mathrm{polylog}(n))$ for $d$-{\sf UUDP-MIN} using a $\rho$-approximation algorithm of $(d-1)$-{\sf UUDP-MIN} as a blackbox.
\fullonly{
\begin{openproblem}
For any constant $d >0$, given a $\rho$-approximation algorithm for $(d-1)$-{\sf UUDP-MIN} (and $(d-1)$-{\sf SMP}), is it possible to get an $O_d(\rho\cdot\polylog n)$ approximation algorithm for $d$-{\sf UUDP-MIN} (and $d$-{\sf SMP})?
\end{openproblem}
}
A positive resolution to this problem would imply $(\log^{O(d)} n)$-approximation algorithm for $d$-{\sf UUDP-MIN}. We believe that, even resolving this problem would require some new insights on geometric and poset structures.
There are two special cases that can be thought of as barriers in dealing with standard versions of {\sf SMP} and {\sf UUDP-MIN}, and we believe that these two special cases serve as good starting points in attacking our problems. The first problem is the geometric version of the Maximum Expanding Subsequence ({\sc Mes}) problem which is the key problem to show the hardness of {\sf UUDP-MIN}~\cite{BriestK11}. The second problem is the Unique Coverage problem \cite{DemaineFHS08} when the sets have constant VC-dimension. Another interesting problem is to obtain {\sf PTAS}s for $2$-{\sf UUDP-MIN} and $2$-{\sf SMP} (e.g., by extending the techniques in \cite{GrandoniR10}).
\fullonly{
There are two special cases that can be thought of as barriers in dealing with standard versions of {\sf SMP} and {\sf UUDP-MIN}, and we believe that these two special cases serve as good starting points in attacking our problems. First, in order to get an $O_d(\polylog n)$-approximation algorithm for $d$-{\sf UUDP-MIN}, we need to deal with the Maximum Expanding Subsequence ({\sc Mes}) problem which is the key problem to show the hardness of {\sf UUDP-MIN}~\cite{Briest08}.
\fullonly{
\begin{definition}[Maximum Expanding Sub-sequence ({\sc Mes})]
We are given a set of $n$ ground elements $U$ and a set system $S_1,\ldots, S_m$ where $S_i \subseteq U$ for all $i$. We say that $S_{\phi(1)},\dots, S_{\phi(\ell)}$ is {\em expanding sequence} if for all $j$, we have $S_{\phi(j)} \not\subseteq \bigcup_{j' < j} S_{\phi(j')}$. The objective is to find an expanding sequence of maximum length.
\end{definition}
Briest \cite{Briest08} introduces this problem as an intermediate problem to prove hardness of approximation results for {\sf UUDP-MIN} and {\sf SMP}. Roughly speaking, Briest used $n^{\epsilon}$-hardness for this problem to show $n^{\delta}$-hardness of approximating {\sf UUDP-MIN}. His results showed a very strong evidence that {\sc Mes} is very closely related to {\sf UUDP-MIN} and {\sf SMP}. Now we ask the following question.
\begin{openproblem}
Suppose that the underlying set system of {\sc Mes} is defined by our framework in $\ensuremath{\mathbb R}^d$. That is, ground elements are points in $\ensuremath{\mathbb R}^d$, and each set is defined by an unbounded rectangle in $\ensuremath{\mathbb R}^d$. Can we get, says, an $O(\operatorname{poly} \log n)$ approximation algorithm for {\sc Mes}?
\end{openproblem}
}
\sodaonly{This motivates us to propose the following open problem, called $d$-{\sc Mes}: Suppose that the underlying set system of {\sc Mes} is defined by our framework in $\ensuremath{\mathbb R}^d$. That is, ground elements are points in $\ensuremath{\mathbb R}^d$, and each set is defined by an unbounded rectangle in $\ensuremath{\mathbb R}^d$. Can we get, says, $O(\operatorname{poly} \log n)$ approximation algorithm for {\sc Mes}?} It can be observed that our algorithm implies an $\tilde O_d(n^{1-\epsilon(d)})$-approximation for $d$-{\sc Mes}. Solving this problem can be considered the first step in getting $\log^{O(d)} n$ approximation.
Getting $O_d(\log^{1-\epsilon(d)} m)$ approximation for $d$-{\sf SMP} also has the following barrier. Previous results suggest that {\sf SMP} has inherited its intractibility from the problem called {\sf Unique Coverage} \cite{DemaineFHS08}. Roughly speaking, {\sf Unique Coverage} is equivalent to {\sf SMP} in which all consumers have budget one. An $O(\log n)$ approximation is known for this problem \cite{DemaineFHS08}. The question is whether we can improve this ratio in the case of $d$-{\sf SMP}.
\fullonly{
\begin{openproblem}
Is there a sub-logarithmic approximation algorithm for $d$-{\sf SMP} with unit budgets.
\end{openproblem}
}
Solving this problem is a first step in breaking another barrier of $O(\log m)$ for $d$-{\sf SMP}. Moreover, it would give a convincing evidence that our model may not inherit intractibility from {\sf Unique Coverage} problem. In fact, this problem itself can be seen as {\sf Unique Coverage} when the sets have {\em constant VC-dimension} and may be of independent interest.
Another interesting direction is to further investigate $2$-{\sf UUDP-MIN} and $2$-{\sf SMP}. We know that {\sf PTAS}s are likely to exist but do not even have an $O(\log n)$-approximation algorithm running in polynomial time. Getting {\sf PTAS} would be very interesting, and we believe that it will require novel ideas and structural properties. Even weaker approximation guarantees, such as $n^{\epsilon}$-approximation in time $n^{O(1/\epsilon)}$, would still be interesting.
\fullonly{
\begin{openproblem}
Can we construct $O(\log n)$ approximation algorithms for $2$-{\sf UUDP-MIN} or $2$-{\sf SMP}?
\end{openproblem}
}
}
\iffalse
\squishlist
\item Are there polylogarithmic approximation algorithms for $d$-{\sf UUDP} and $d$-{\sf SMP}, for any constant $d$? We note that, no explicit $O(\log n)$-approximation algorithms are known even for the case of $d=2$, although constant approximation factors are the real hope for a constant $d$ and we know that a {\sf PTAS} is likely to exist for the case of $d=2$.
\item In this paper, a {\sf PTAS} is shown to likely to exist in many cases but it is interesting to have explicit algorithms.\danupon{I'm not sure if we should list this problem.} (A {\sf PTAS} for $2$-{\sf SMP} is especially interesting since it will generalize the recent {\sf PTAS} result for the highway problem.)
\item
Another very interesting problem is the case where there are some items already priced by other sellers (i.e., Stackelberg problem). This case is still not much explored although it is usually the case in reality.
\end{itemize
\fi
\section{Hardness}\label{sec:hardness}
We provide hardness results in both scenarios when the number of attributes $d$ is small and when $d$ is large. We sketch our results here. More details can be found in Appendix~\ref{sec:omitted_hardness}.
\paragraph{Few attributes} First we discuss the \mbox{\sf NP}-hardness of $3$-{\sf UUDP-MIN} and \mbox{\sf APX}-hardness of $4$-{\sf UUDP-MIN}. These hardness results hold even when the consumer budgets are either 1 or 2. We perform a reduction from Vertex Cover~\cite{DBLP:journals/tcs/GareyJS76,DBLP:conf/ciac/AlimontiK97}, essentially using the same ideas as in~\cite{GuruswamiHKKKM05}, except for the fact that we use Schnyder's result~\cite{Schnyder89,DBLP:conf/soda/Schnyder90} to ``embed'' the instance into posets of low order dimensions.
First, let us recall the reduction in \cite{GuruswamiHKKKM05}. We start
from a graph $G=(V,E)$, which is an input instance of Vertex Cover. We create two types of consumers: (i) poor
consumer ${\bf C}_e$ for each edge $e$ with budget $1$ and (ii) rich
consumer ${\bf C}_v$ for each vertex $v$ with budget $2$. The items
are ${\mathcal{I}} = \set{{\bf I}_v: v\in V}$. Each poor consumer ${\bf C}_e$ has a
consideration set containing two items ${\bf I}_u$ and ${\bf I}_v$
where $e=(u,v)$ and each rich consumer ${\bf C}_v$ considers only
one item ${\bf I}_v$. Using the analysis essentially the same as
\cite{GuruswamiHKKKM05}, one can show that the problem is {\sf NP}-hard
if we start from Vertex Cover on planar graphs and {\sf APX}-hard if
we start from Vertex Cover on cubic graphs.
Therefore, it only remains to map consumers and items to points in ${\mathbb R}_{\geq 0}^d$ (where $d=3,4$) such that for each consumer
${\bf C}$, the set of items that pass her criteria (i.e., $\{{\bf I}\in {\mathcal{I}} \mid{\bf I}[i]\geq {\bf C}[i]~~\mbox{for all $1\leq i\leq d$}\}$) is exactly her consideration set.
The main idea is to first embed the problem into an {\em adjacency poset} of the input graph. Then, we invoke Schnyder's theorem~\cite{Schnyder89,DBLP:conf/soda/Schnyder90} to again embed this poset into a Euclidean space.
An adjacency poset of a graph can be constructed as follows. First we construct a $2$-layer poset with minimal elements in the first layer and maximal elements in the second layer. For each edge $e \in E$, we have a minimal element in the poset corresponding to $e$ (for convenience, we also denote the poset element by $e$). For each vertex $v \in V$, we have a maximal poset element corresponding to $v$. There is a relation $e\preceq v$ if and only if vertex $v$ is an endpoint of $e$. \fullonly{This is called an {\em adjacency poset} of the graph.}
The last task is to ``embed'' poset elements into points in the Euclidean space in such a way that, for any poset elements $e_1$ and $e_2$, $e_1\preceq e_2$ if and only if $q_{e_1}[i]\geq q_{e_2}[i]$ for all $i$ where $q_{e_1}$ and $q_{e_2}$ are points that $e_1$ and $e_2$ are mapped to, respectively. If we can do this, we would be done, simply by defining the coordinates of each consumer ${\bf C}_e$ to be $q_{e}$, and the coordinates of each consumer ${\bf C}_v$ to be $q_v$. Similarly, we define the coordinates of each item ${\bf I}_v$ as $q_v$. In order to obtain such an embedding, we use part of Schnyder's theorem~\cite{Schnyder89} which states that any planar graph has an adjacency poset of dimension three, and any $4$-colorable graph (including cubic graphs) has an adjacency poset of dimension four. Moreover, embedding these graphs into Euclidean spaces can be done in polynomial time~\cite{DBLP:conf/soda/Schnyder90}.
Finally we note that $2$-{\sf SMP} is strongly $\mbox{\sf NP}$-hard and $4$-{\sf SMP} is $\mbox{\sf APX}$-hard. The proof follows from the fact that these problems generalize Highway pricing and graph vertex pricing on bipartite graphs, respectively, and can be found in \sodaonly{the full version}\fullonly{Appendix~\ref{sec:omitted_hardness}}.\danupon{TO DO: Write the proof for $4$-{\sf SMP}}
\paragraph{Many attributes}
We establish a connection between the {\sf UUDP-MIN} with bounded-size consideration sets and our problem. This connection immediately implies hardness results for $d$-{\sf UUDP-MIN} when $d$ is at least poly-logarithmic in $n$. Our main result in this section is the following:
\begin{theorem}(Informal)
\label{theorem: higher dimension}
Let $A =({\mathcal{C}}, {\mathcal{I}}, \{S_{\bf C}\}_{ {\bf C} \in {\mathcal{C}}} )$ be an instance of {\sf UUDP-MIN} where $B=\max_{{\bf C} \in {\mathcal{C}}} \size{S_{\bf C}}$. We can (with high probability of success) create an instance $A'=({\mathcal{C}}', {\mathcal{I}}')$ of $d$-{\sf UUDP-MIN}, where $d= O(B^2 \log n)$, that is ``equivalent'' to $A$.
\end{theorem}
In other words, the above theorem shows that any {\sf UUDP-MIN} instance with consideration sets of size bounded by $B$, can be realized by a $d$-{\sf UUDP-MIN} instance for $d= O(B^2 \log n)$.
\sodaonly{
Combining this with the result in \cite{ChalermsookPricing}, we have a hardness of $\Omega(d^{1/4-\epsilon)}$ for any $\epsilon>0$.
}
\fullonly{
This implies that $d$-{\sf UUDP-MIN} is at least as hard
as the original problem when consideration sets have size at most $B$.
When
$B$ is at least logarithmic in the number of items, our
reduction yields the following corollary, assuming the hardness of the balanced bipartite independent set problem in constant degree graphs or refuting random 3{\sf CNF} formulas \cite{BriestThesis}. \danupon{I shortened the last sentence.}
\begin{corollary}
There is a constant $\epsilon$ such that for every $d \geq \log^2
n$, it is hard to approximate $d$-{\sf UUDP-MIN} to
within a factor of $d^{\epsilon}$.
\end{corollary}
}
We remark that our reduction here in fact works independently of the decision model, so this result works for {\sf SMP} and {\sf UDP-Util} as well.
\subsection{Hardness Results in Higher Dimensions}\label{sec:higher
dimensions}
In this section, we present the proof of Theorem~\ref{theorem: higher dimension}. Let $A = ({\mathcal{I}}, {\mathcal{C}})$ be an instance of {\sf UUDP-MIN} where every consumer ${\bf C}$ has its consideration set $S_{\bf C}$ of size at most $B$. Let ${\mathcal{I}} = \set{{\bf I}_1,\ldots, {\bf I}_n}$. For each $i \in [d]$, we pick a random permutation $\pi_i: [n] \rightarrow [n]$, so we have $d$ permutations $\pi_1,\ldots, \pi_d$. The function $\phi$ on items ${\mathcal{I}}$ can be defined as $\phi({\bf I}_j)[i] = \pi_i(j)$, and we extend the function to the set of consumers as follows: $\phi({\bf C})[i] = \min_{j \in S_C} \pi_i(j)$. Now we have a well-defined function $\phi$.
\begin{lemma}
With probability at least $1- 1/n$, for all consumer ${\bf C} \in {\mathcal{C}}$, the consideration set $S'_{\bf C}$ defined by $S'_{\bf C} = \set{{\bf I}_j: \phi({\bf I}_j) \mbox{ dominates } \phi({\bf C})}$ is exactly $S_{\bf C}$.
\end{lemma}
\begin{proof}
Since we define $\phi({\bf C})$ to be the minimum of $\phi({\bf I}_j)$ over all items in $S_{\bf C}$, we have $S_{\bf C} \subseteq S'_{\bf C}$. Let $k$ be the index of an item that does not belong to $S_{\bf C}$. We show the following claim.
\begin{claim}
\label{claim: tiny prob}
The probability that $\phi({\bf I}_k)$ dominates $\phi({\bf C})$ is at most $1/n^{B+2}$.
\end{claim}
\begin{proof
Fix some $i \in [d]$. The bad event that $\pi_i(k)\geq \min_{j \in
S_C} \pi_i(j)$ happens only if $\pi_i$ does not put $k$ in the last
position among $S_C \cup \set{k}$. This probability is exactly
$(1-1/(B+1))$. Therefore, the bad event happens for all values of
$i$ with probability at most $(1-1/(B+1))^d \leq 1/n^{B+2}$ for $d=
O(B^2 \log n)$.
\end{proof}
This claim immediately implies the lemma: By the union bound, the
probability that $\phi({\bf I}_k)$ dominates $\phi({\bf C})$ is at
most $1/n^{B+1}$. So we have that $\pr{}{S_{\bf C} \neq S'_{\bf C}} \leq
1/n^{B+2}$. There are at most $n^B$ possible consideration sets of
size at most $B$, so by union bounds, the probability that a bad
event $S_C \neq S'_{\bf C}$ happens for some consumer ${\bf C}$ is at most
$1/n$.
\end{proof}
\paragraph{$n$ attributes capture general problem} Finally, we end this section with the proof that $n$-{\sf UUDP-MIN} captures the whole generality of {\sf UUDP-MIN}: Consider an instance $({\mathcal{C}}, {\mathcal{I}}, \set{S_{\bf C}}_{{\bf C}\in {\mathcal{C}}})$ of {\sf UUDP-MIN}. Denote the set of items by ${\mathcal{I}} = \set{{\bf I}_1,\ldots, {\bf I}_n}$. Notice that we can define the coordinates of each consumer by ${\bf C}[i] = 0$ if ${\bf I}_i \in S_{\bf C}$, and ${\bf C}[i] = 1$ otherwise. We define the coordinates of each item as ${\bf I}_i[i] = 0$ and ${\bf I}_i[j] =1$ for all $j \neq i$. It is easy to check that the consideration sets are preserved by this reduction.
\section{Introduction}
This paper studies a geometric version of two central {\em unlimited-supply pricing} problems. We are given a set ${\mathcal{I}}$ of $n$ consumers and a set ${\mathcal{C}}$ of $m$ items.
Every item ${\bf I}\in{\mathcal{I}}$ is represented by a point ${\bf I} = ({\bf I}[1],\ldots, {\bf I}[d]) \in \ensuremath{\mathbb R}^d_{\geq 0}$, where $\ensuremath{\mathbb R}_{\geq 0}$ denotes the set of non-negative reals and ${\bf I}[j]$ expresses the quality of item ${\bf I}$ in the $j$-th attribute. Every consumer ${\bf C}\in{\mathcal{C}}$ is represented by a point ${\bf C}= ({\bf C}[1],\ldots, {\bf C}[d]) \in \ensuremath{\mathbb R}^d_{\geq 0}$, where ${\bf C}[j]$ is the criterion of consumer ${\bf C}\in\mathcal{C}$ in the $j$-th attribute. Each consumer ${\bf C}$ is additionally equipped with budget $B_{{\bf C}}\in\mathbb{R}_{\geq 0}$ and a consideration set
\begin{equation}\label{eq:SC for UDP-MIN}
S_{\bf C}=\set{{\bf I}: {\bf I}[j] \geq {\bf C}[j], \mbox{for all } 1\leq j \leq d}.
\end{equation}
\begin{wrapfigure}{r}{0.3\textwidth}
\center
\vspace{-.9 cm}
\includegraphics[width=1.1\linewidth, clip=true, trim= 0.5cm 1cm 1cm 0.8cm]{problem_visualization.pdf}\\
\caption{Problem visualization}\label{fig:visualization}\vspace{-.3cm}
\end{wrapfigure}
In the {\em $d$-dimensional uniform-budget unit-demand min-buying pricing problem} ($d$-{\sf UUDP-MIN}), once we assign prices to items, each consumer ${\bf C}$ will buy the cheapest item ${\bf I}$ in $S_{{\bf C}}$ if the price of item ${\bf I}$ is at most $B_{{\bf C}}$. In the {\em $d$-dimensional single-minded pricing problem} ($d$-{\sf SMP}), consumer ${\bf C}$ will buy the {\em all} items in $S_{{\bf C}}$ if the total price of those items is at most $B_{{\bf C}}$.
The objective is to set the price of items in ${\mathcal{I}}$ in order to maximize the revenue. That is, we want to find $p: {\mathcal{I}}\rightarrow {\mathbb R}_{\geq 0}$ that maximizes $\sum_{{\bf C}\in {\mathcal{C}}, \min_{{\bf I}\in S_{\bf C}} p({\bf I})\leq B_{\bf C}} \min_{{\bf I}\in S_{\bf C}} p({\bf I})$ in the case of $d$-{\sf UUDP-MIN} and $\sum_{{\bf C}\in {\mathcal{C}}, \sum_{{\bf I}\in S_{\bf C}} p({\bf I})\leq B_{\bf C}} \sum_{{\bf I}\in S_{\bf C}} p({\bf I})$ in the case of $d$-{\sf SMP}.
Fig.~\ref{fig:visualization} illustrates the problem: Each item corresponds to a point in the plane. The consideration set of each consumer ${\bf C}$ is represented by an (unbounded) axis-parallel rectangle with point ${\bf C}$ as a lower-left corner.
The above problems when $d$ is unbounded (called {\sf UUDP-MIN} and {\sf SMP}) have been widely studied recently
(e.g.,
\cite{Rusmevichientong03,GuruswamiHKKKM05,RusmevichientongRG06,BriestK11,ChalermsookPricing,AggarwalFMZ04,BalcanB07})
and are known to be $O(\log m)$-approximable~\cite{AggarwalFMZ04}; so we have a reasonable approximation guarantee when there are not many consumers.
However, in many cases, one would expect the number of consumers to be much larger than the number of items~$n$. In this case, we are still stuck at the trivial $O(n)$ approximation ratio, and there are evidences that suggest that getting a sub-linear approximation ratios might be impossible: Unless $\mbox{\sf NP} \subseteq \mbox{\sf DTIME}(n^{\operatorname{poly} \log n})$, these problem are hard to approximate within a $2^{\log^{1-\epsilon} n}$ for any constant $\epsilon>0$~\cite{ChalermsookPricing}. Moreover, assuming a stronger (but still plausible) assumption, these problems are hard to approximate to within a factor of $n^{\epsilon}$ for some $\epsilon >0$ \cite{BriestK11}.
Motivated by various types of assumptions, the pricing problems with special structures have been studied (e.g., when there is a {\em price-ladder constraint}~\cite{Rusmevichientong03,AggarwalFMZ04,RusmevichientongRG06,RusmevichientongRG06-2,BriestK11}, consideration sets are small~\cite{BalcanB07,BriestK11} or consideration sets correspond to paths on graphs~\cite{BalcanB07,ElbassioniSZ07,GrandoniR10,ElbassioniRRS09,DBLP:conf/icalp/GamzuS10}). In these cases, better approximation ratios are usually possible.
In this paper we consider the geometric structure of pricing problems arising naturally from real-world scenarios, which turns out to be quite general. Our motivation is two-fold: We hope that the geometric structures will lead to better approximation algorithms, and we found these problems interesting on their own as they have connections to other pricing and geometric problems.
Our problems are motivated by the following simple observation on the consumers' behavior.
Consider a setting where we sell cars. If a consumer has car $A$ with horse power $130$HP in her consideration set, she would not mind buying car $B$ with horse power $150$HP. Maybe she does not want $B$ because it is less energy-efficient or has lower reputation. But, if we list {\em all} attributes of the cars that people care about and it happens that $B$ is not worse than $A$ in all other aspects, then $B$ should also be in the list.
In particular, instead of looking at a full generality where each consumer ${\bf C}$ considers any set of items $S_{\bf C}$, it is reasonable to assume that each consumer has some criterion in mind for each attribute of the cars, and her consideration set consists of any car that passes all her criteria, i.e. consumers judge items according to their attributes.
This natural assumption has been a model of study in other fields such as marketing research, healthcare economics and urban planning. It is referred to as the {\em attribute-based screening process}. In particular, using criteria to define consideration sets as in Eq.~\eqref{eq:SC for UDP-MIN} is called {\em conjunctive screening rule}. Besides being natural, this assumption has been supported by a number of studies where it is concluded that consumers typically use a conjunctive screening rule in obtaining their consideration sets (see further detail in Section~\ref{sec:relatedwork}).
It is also interesting that $d$-{\sf SMP} captures many previously studied problems as special cases. For example, $2$-{\sf SMP} generalizes the highway pricing problem~\cite{GuruswamiHKKKM05,BalcanB07,ElbassioniSZ07,GrandoniR10} and thus our algorithmic results on $2$-{\sf SMP} can immediately be applied to this problem. Moreover, $3$-{\sf SMP} generalizes the upward case of the tollbooth pricing problem \cite{ElbassioniRRS09,KortsarzRR11} as well as the graph vertex pricing problem on planar graphs \cite{BalcanB07,ChalermsookKLN11}. $4$-{\sf SMP} generalizes the unlimited-supply version of the {\em exhibition} problem \cite{ChristodoulouEF10}, the graph vertex pricing problem on bipartite graphs \cite{BalcanB07,KhandekarKMS09}, and the ``rectangle version'' of the {\em unique coverage problem} ({\sc UC}) \cite{DemaineFHS08}, which are the geometric variants of {\sc UC} studied recently in~\cite{ErlebachL08,ItoNOOUUU12}.
Moreover, {\sf SMP} is a special case of the {\em maximum feasible subsystem with 0/1 coefficients} problem ({\sc Mrfs}) \cite{ElbassioniRRS09}. Elbassioni et~al. \cite{ElbassioniRRS09} showed that a very special geometric version of {\sc Mrfs} (the ``interval version'') admits much better approximation ratios than the general one. A geometric {\sc Mrfs} can be seen as a special case of ``$2$-{\sc Mrfs}'' in our terminologies, and it is thus interesting whether ``$d$-{\sc Mrfs}'' is easier than general {\sc Mrfs} for other values of $d$. Our geometric {\sf SMP} is a special case of $d$-{\sc Mrfs}. Thus, solving $d$-{\sf SMP} serves as the first step towards solving $d$-{\sc Mrfs}.
\subsection{Our Results and Techniques}
We show that geometric structures lead to breaking the linear-approximation barrier: While the pricing problems are likely to be hard to approximate within a factor of $n^{1-\epsilon}$ in the general cases, we obtain an $o(n)$-approximation algorithms in the geometric setting, as follows.
\begin{theorem}\label{thm:intro thm udp smp}
For any $d>0$, there is an $\tilde O_d\left(n^{1-\epsilon(d)}\right)$-approximation algorithm for $d$-{\sf UUDP-MIN} and $d$-{\sf SMP} where function $\epsilon(d): = \frac{1}{4^{d-1}}$ and $\tilde O_d$ treats $d$ as a constant and hides a $\mathrm{polylog}(n)$ factor.
\end{theorem}
The essential idea behind our algorithm is to partition the problem instance into sub-instances without decreasing the optimal revenue (we call this {\em consideration-preserving decomposition}). This is done by using Dilworth's Theorem (partitioning items into chains and anti-chains) and epsilon-nets to find subsets of items satisfying certain structural properties. Subsequently, we show that the dimensions of these sub-instances can be reduced through the notion of {\em consideration-preserving embedding}. In the end of our algorithm, we are left with a sub-linear number of sub-instances, each of which can be solved almost optimally in polynomial time. Returning the best solution among the solutions of these sub-instances guarantees a sub-linear approximation ratio.
The spirit of our technique is in some sense in a similar flavor to Chan's algorithm \cite{Chan12} which computes a {\em conflict-free} coloring of $d$-dimensional points (w.r.t. rectangle ranges) using $O(n^{1-0.632/(2^{d-3}-0.368)})$ colors. In particular, in the 2-dimensional cases of both our geometric pricing and Chan's conflict-free coloring problems, the upper bounds of $O(\sqrt{n})$ can be obtained by a simple application of Dilworth's theorem (Ajwani et al. \cite{AjwaniEGR12} obtained a better bound in this case for the latter problem). However, the techniques of the two results are different in higher dimensions.
\paragraph{QPTASs} We also obtain {\sf QPTAS}s for $2$-{\sf UUDP-MIN} and $2$-{\sf SMP}. We present this in Appendix~\ref{sec: qptas 2 udp} and \ref{sec:2-SMP}. These results, together with a widely-believed assumption that the existence of a {\sf QPTAS} for any problem implies that {\sf PTAS} exists for the same problem (e.g., \cite{BansalCES06,ElbassioniSZ07}), imply that the value of $\epsilon(d)$ in Theorem~\ref{thm:intro thm udp smp} could be improved slightly to $1/4^{d-2}$.
As a by-product of these results, we show a {\sf QPTAS} for $2$-{\sf SMP} which subsumes the previous {\sf QPTAS} for highway pricing \cite{ElbassioniSZ07}.
\begin{comment}
\paragraph{Further results} We note that the notion of attribute and our techniques are useful not only for attacking {\sf UUDP-MIN} but also for several other models.
For instance we consider another well-known setting called {\em single-minded pricing} ({\sf SMP})~\cite{GuruswamiHKKKM05}, in which each consumer buys the entire bundle $S_{{\bf C}}$ if she can afford to and buys nothing otherwise.
We prove that the $d$-attribute version of {\sf SMP}, called $d$-{\sf SMP}, admits the same approximation ratio as in Theorem~\ref{thm:intro thm udp smp}.
More importantly, we obtain {\sf QPTAS}s for important subroutines which are crucial building blocks in our main theorem. These results, together with a widely-believed assumption that the existence of a {\sf QPTAS} for any problem implies that {\sf PTAS} exists for the same problem (e.g., \cite{BansalCES06,ElbassioniSZ07}), imply that the approximation ratios of the aforementioned problems can be improved. (We emphasize that this claim relies on the fact that the subroutines that we obtain {\sf QPTAS}s have {\sf PTAS}s.) They also imply that obtaining sublinear-approximation algorithms is possible even for more general problems. Here, we consider two broader models of the pricing problem. \fullonly{(see their full definitions in Section~\ref{sec: algorithms for small d})}\sodaonly{Due to the page limitation, most of the proof details are omitted in this extended abstract and can be found in the full version of our paper \cite{Fullversion}}.
\squishlist
\item The {\em unit-demand utility-maximizing pricing problem} ({\sf UDP-Util}) is a model that generalizes {\sf UUDP-MIN} in that the consumers may have different {\em valuations} on different items and want to maximize the difference between the valuations and the price of the purchased items. (This problem is sometimes called {\em envy-free pricing problem} \cite{BriestK07} and {\em max-gain-buying pricing problem} \cite{AggarwalFMZ04}. We name this problem {\sf UDP-Util} to distinguish it from other general pricing problems studied in this paper.) We show that there is an $\tilde O_d\left(n^{1-1/4^{d-1}}\right)$-approximation algorithm for a naturally defined $d$-attribute version of {\sf UDP-Util} when valuations are monotone functions of $d$ attributes (i.e., each consumer values an item not less than the inferior ones). This result also holds for the $d$-attribute version of the (non-uniform) unit-demand min-buying pricing problem ({\sf UDP-MIN}) where consumers can have different budgets on different items.
\item The pricing problem with {\em symmetric valuations} and {\em subadditive revenues} is a model that includes both {\sf UUDP-MIN} and {\sf SMP} as well as many other natural problems. Informally, symmetric valuation means that each consumer's valuation only depends on the number of items she gets, and subadditive revenue implies that the amount that each consumer pays to get an item bundle $X$ is at most the amount she pays to get the items in $X$ separately. We show that there is an $\tilde O_d\left(n^{1-1/4^{d-1}}\right)$-approximation algorithm for the $d$-attribute version of this problem.
\end{itemize
\fullonly{
We summarize our results in the following theorem.
\begin{theorem}\label{thm:intro all}
Unless there is a problem that admits {\sf QPTAS} but no {\sf PTAS}, there exist $\tilde O_d(n^{1-1/4^{d-1}})$-approximation algorithms for (1) {\em unit-demand utility-maximizing} $d$-attribute pricing problem where valuations depend only on attributes and (2) $d$-attribute pricing problem with {\em symmetric valuations} and {\em subadditive revenues}. Moreover, there exist $\tilde O_d(n^{1-1/4^{d-2}})$-approximation algorithms for $d$-{\sf UUDP-MIN} and $d$-{\sf SMP}.
\end{theorem}
}
As a by-product of our proof, we show a {\sf QPTAS} for $2$-{\sf SMP} which subsumes the previous {\sf QPTAS} for the highway pricing problem \cite{ElbassioniSZ07}. \fullonly{We discuss this in the next subsection.}
\end{comment}
\paragraph{Hardness} We also study the hardness of approximation of our problems. We show that $3$-{\sf UUDP-MIN} and $2$-{\sf SMP} are {\sf NP}-hard, and $4$-{\sf UUDP-MIN} and $4$-{\sf SMP} are {\sf APX}-hard. Hence, our problem is already non-trivial for small $d$.
Our hardness proofs establish a cute connection between our problem and the vertex cover problem on graphs of low {\em order dimensions}~\cite{Schnyder89,DBLP:conf/soda/Schnyder90}.
Moreover, we show that the hardness of our problem tends to increase as we increase $d$, and the whole generality is captured when $d=n$. In particular, we show that when the dimension is sufficiently high (i.e. $d \geq \log^2 n$), the problems are hard to approximate to within a factor of $d^{1/4-\epsilon}$ for any $\epsilon>0$.
Table~\ref{table:current status} concludes our results for $d$-{\sf UUDP-MIN} and $d$-{\sf SMP}.
\begin{table}
\begin{center}
\footnotesize
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline
{\bf Problem} & & $\mathbf{d=1}$ & $\mathbf{d=2}$ & $\mathbf{d=3}$ & $\mathbf{d=4}$ & {\bf large $\mathbf{d}$ \{range \} }\\
\hline
\multirow{2}{*}{$d$-{\sf UUDP-MIN}} & Upper bound & Polytime & {\sf QPTAS} & & & $n^{1-\frac{1}{4^{d-1}}}$ \{constant $d$\}\\
& Lower bound & & & {\sf NP}-hard & {\sf APX}-hard & $d^{\frac{1}{4}-\epsilon}$ \{$d=\omega(\log n)$\} \\
\hline
\multirow{2}{*}{$d$-{\sf SMP}} & Upper bound & Polytime & {\sf QPTAS} & & & $n^{1-\frac{1}{4^{d-1}}}$ \{constant $d$\}\\
& Lower bound & & {\sf NP}-hard & & {\sf APX}-hard & $d^{\frac{1}{4}-\epsilon}$ \{$d=\omega(\log n)$\} \\
\hline
\end{tabular}
\end{center}
\vspace{-.5cm}
\caption{Results of $d$-{\sf UUDP-MIN} and $d$-{\sf SMP} for small values of $d$.}
\label{table:current status}
\vspace{-.3cm}
\end{table}
\fullonly{
\subsection{General Framework}\label{sec:framework}
We explain the general framework of the proofs in this subsection, and we will solve $d$-{\sf UUDP-MIN} to give the key ideas of our techniques in Section~\ref{sec:d udp min}. The proofs of our results can be divided into two parts.
In the first part, we essentially show that a large class of $d$-attribute pricing problems is sublinear-approximable {\em if} it is sublinear-approximable for some small $d$. This class is the class of pricing problems with {\em subadditive revenue} (informally described in the previous subsection), which includes all the aforementioned problems. Thus, in one shot we reduce our task to solving the pricing problems on a very simple input! The following informal theorem shows the essence of the first part (detail in Section~\ref{sec:dim reduction statement} and \ref{sec:dim reduction proof}).
\begin{theorem}[Dimension Reduction Theorem (Informal)] Let $\mathcal{P}$ be any pricing problem with subadditive revenue. For any $d$ and $d'<d$, if there is an $\tilde O_d(1)$ approximation algorithm for the $d'$-attribute version of $\mathcal{P}$ then there is an $\tilde O_d(n^{1-\varepsilon(d,d')})$-approximation algorithm for its $d$-attribute version, where $\varepsilon(\cdot)$ is a function defined as $\varepsilon(t,t') = 1/4^{t-t'}$.
\end{theorem}
In the second part we show that the aforementioned problems can be solved in the case of one and two attributes. First, the cases of $1$-{\sf UUDP-MIN} and $1$-{\sf SMP} can be solved optimally by simple dynamic programs and sublinear-approximation algorithms of both problems thus follow. Furthermore, we show quasi-polynomial time approximation schemes ({\sf QPTAS}s) for $2$-{\sf UUDP-MIN} and $2$-{\sf SMP} as well as (1) {\em unit-demand utility-maximizing} $1$-attribute pricing problem where valuations depend only on attributes and (2) $1$-attribute pricing problem with {\em symmetric valuations} and {\em subadditive revenues}. These results rule out the possibility of these problems being {\sf APX}-hard unless $\mbox{\sf NP} \subseteq \mbox{\sf DTIME}(n^{\operatorname{poly} \log n})$. Thus {\sf PTAS}s for these problems are likely to exist. This, along with the Dimension Reduction Theorem, implies Theorem~\ref{thm:intro all}.
On a technical side, we note that our {\sf QPTAS} for $2$-{\sf SMP} generalizes the {\sf QPTAS} result in \cite{ElbassioniSZ07} for the Highway pricing problem as the Highway pricing problem is a special case of $2$-{\sf SMP}. However, $2$-{\sf SMP} seems to have a more complicated structure and is harder to handle. A good evidence of this is that while the Highway problem has a very simple $O(\log n)$-approximation algorithm \cite{BalcanB07}, getting a polynomial-time algorithm with $o(\sqrt{n})$ approximation guarantee for $2$-{\sf SMP} without assuming anything is already a challenging task. Obtaining $O(\log n)$-approximation algorithm or extending the {\sf PTAS} technique in \cite{GrandoniR10} to $2$-{\sf SMP} (or $2$-{\sf UUDP-MIN}) is an interesting open problem.
}
\subsection{Related Work}\label{sec:relatedwork}
Rusmevichientong et al.~\cite{Rusmevichientong03,RusmevichientongRG06,RusmevichientongRG06-2} defined the {\em non-parametric multi-product
pricing problem}, motivated by the possibility that the data about consumers' preferences and budgets can be predicted based on previous data, which can be gathered and mined by web sites designed for this purpose, e.g., \cite{HaublTrifts00,RusmevichientongRG06-2}. This problem is what we call uniform-budget unit-demand pricing problem ({\sf UUDP}). Rusmevichientong et al. proposed many decision rules such as min-buying, max-buying and rank-buying and showed that {\sf UUDP-MIN} allows a polynomial-time algorithm, assuming the {\em price-ladder constraint}, i.e., a predefined total order on the prices of all products. Aggarwal et al.~\cite{AggarwalFMZ04} later showed that the price ladder constraint also leads to a $4$-approximation algorithm for the max-buying case, even in the case of limited supply.
We note that the price ladder constraint is closely related to our notion of attributes in the following sense. It can be shown that $1$-{\sf UUDP-MIN} satisfies the price ladder constraint (this is the reason we can solve it in polynomial time). Moreover, although $2$-{\sf UUDP-MIN} does not satisfy this constraint, it {\em partially} satisfies the constraint in the sense that if one item is better than another item in all attributes then we can assume that it has a higher price. This property plays an important role in obtaining {\sf QPTAS} for $2$-{\sf UUDP-MIN} and also holds for general $d$.
Other variants defined later include non-uniform and utility-maximizing unit-demand, single-minded ({\sf SMP}), tollbooth and highway models~\cite{AggarwalFMZ04,GuruswamiHKKKM05}.
These problems were later found to have important connections to
algorithmic mechanism design~\cite{AggarwalH06,BalcanB05,GuruswamiHKKKM05} and online pricing
problems~\cite{BalcanB07,BlumH05}.
As we mentioned in the introduction, many problems can be approximated within the factor of $O(\log m + \log n)$ and $O(n)$, and these seem to be tight.
The observation that consumers make decisions based on attributes has been used in other areas outside computer science. For example, most pricing models are captured by the {\em two-stage consider-then-choose} model (e.g., \cite{Gensch87,Payne82,PayneBJ88,GilbrideAllenby,HaublTrifts00,JedidiK05,HauserTEBS10,LiuArora2011}) in marketing research: Each consumer first screens out some undesirable items ({\em screening process}) and is left with the consideration set which is used to make a final decision. Pricing problems such as {\sf UUDP-MIN} are the case where consideration sets are arbitrary (as defined in, e.g. \cite{Shocker91,HauserW90}) while the final decision is simplified to, e.g., buying the cheapest item.
The idea of using the consideration sets defined from attributes is called {\em attribute-based screening process}~\cite{GilbrideAllenby} in marketing research where it was shown to be a rational choice for trading off between accuracy and cognitive effort~\cite{Bettman79,BettmanJP90,BettmanPark80,Shugan80}. Our model is equivalent to the attribute-based screening process with {\em conjunctive screening rules} (e.g., \cite{GilbrideAllenby,LiuArora2011}). This type of rules was justified by many studies that it is what consumers typically use when making decisions (e.g., \cite{Bettman79,GilbrideAllenby,HauserTEBS10}).
\section*{Appendix}
\input{proof_algo_ec}
\input{udp}
\input{single-minded-3}
\input{omitted_hardness}
\input{higherdimension}
\end{document}
\section{Omitted hardness results}\label{sec:omitted_hardness}
\subsection{Hardness of 3-{\sf UUDP-MIN} and 4-{\sf UUDP-MIN}}
In this section we show that $3$-{\sf UUDP-MIN} is \mbox{\sf NP}-hard, and
$4$-{\sf UUDP-MIN} is \mbox{\sf APX}-hard by a reduction from {\sf Vertex Cover}. Our
reduction relies on the concepts of adjacency poset and its
embedding into Euclidean space. We describe basic terminologies
here. Given a graph $G=(V,E)$, an adjacency poset $(V \cup E,
\preceq_G)$ of graph $G$ can be constructed as follows: First we
define a poset with its maximal elements corresponding to vertices in $V$ and its minimal
elements corresponding to edges $E$. For each vertex $v$ and each edge $e$, we
have the relation $e \preceq_G v$ if and only if vertex $v$ is an endpoint of
$e$. We say that a map $\phi: V \cup E \rightarrow \ensuremath{\mathbb R}^d$ is an {\em
embedding} of adjacency poset $(V \cup E, \preceq_G)$ into $\ensuremath{\mathbb R}^d$
if and only if it preserves the relations $\preceq_G$, i.e., for any two
elements $a,b \in V \cup E$, we have that $a \preceq_G b$ iff $\phi(a)[i] \leq \phi(b)[i]$ for all coordinates $i \in [d]$.
Now we describe our reductions. Since two reductions are essentially the same, we give a general procedure which will imply both results. Given an instance $G=(V,E)$ of {\sf Vertex Cover}, we first construct an adjacency poset $(V \cup E, \preceq_G)$ for $G$,
and then we compute the embedding $\phi$ of this poset into Euclidean space $\ensuremath{\mathbb R}^d$. We will use the graph $G$, as well as the embedding $\phi$, to define the instance of $d$-{\sf UUDP-MIN} as follows:
\begin{itemize}
\item {\bf Consumers:} We have two types of consumers, i.e. the rich consumers and the poor ones. For each vertex $v \in V$, we create a {\em rich} consumer $C_v$ with budget $2$ at coordinates $\phi(v)$. For each edge $e \in E$, we create a {\em poor} consumer $C_e$ with budget $1$ at coordinates $\phi(e)$.
\item {\bf Items:} For each vertex $v \in V$, we create item ${\bf I_v}$ at coordinates $\phi(v)$.
\end{itemize}
Note that for each $e=(u,v)$, each poor consumer $C_e$ has $S_{C_{e}} = \set{{\bf I}_v, {\bf I}_u}$, while each rich consumer $C_v$ has $S_{C_v} = \set{{\bf I}_v}$. We denote the resulting instance by $({\mathcal{C}}, {\mathcal{I}})$.
The following lemma gives a characterization of the optimal solution for $({\mathcal{C}}, {\mathcal{I}})$. It says that we may assume without loss of generality that every poor consumer gets some item.
\begin{lemma}
For any price $p$ that is a solution for $({\mathcal{C}}, {\mathcal{I}})$ constructed above, we can transform $p$ to $p'$ such that every poor consumer buys some item with respect to $p'$, and the revenue of $p'$ is at least as much as the revenue of $p$.
\end{lemma}
\begin{proof}
Consider edge $e=(u,v)$. Suppose poor consumer $C_e$ does not get
any item, so it implies that both items ${\bf I}_u$ and ${\bf I}_v$
have price $p({\bf I}_u)= p({\bf I}_v) =2$ (recall that, since budgets are $1$ or $2$, the optimal prices would never set prices that are not in $\set{1,2}$). We define the price function $p'$ by setting
$p'({\bf I}_u) = 1$ while $p'({\bf I}_w) = p({\bf I}_w)$ for all
other vertices $w \in V \setminus \set{u}$. The only rich consumer that gets affected is $C_u$, whose payment may decrease by one. However, we earn the revenue of one back from poor consumer $C_e$. For $e' \in E: e'\neq e$, poor consumer $C_{e'}$ is never affected because his budget is one. Overall, changing the price from $p$ to $p'$ never decreases revenue.
\end{proof}
Let $p^*$ be the optimal price for $({\mathcal{C}}, {\mathcal{I}})$ and ${\sf VC}(G)$ denote the size of
minimum vertex cover of $G$. We show the following connection
between the size of minimum vertex cover and the optimal revenue
collected by $p^*$.
\begin{theorem}
The optimal revenue collected by $p^*$ is exactly $2n-{\sf VC}(G)
+m$.
\end{theorem}
\begin{proof}
From the previous lemma, we can assume that the pricing $p^*$ sells
items to every poor consumer. In other words, if $V' = \set{v:
p^*({\bf I}_v)=1}$, it must be the case that $V'$ is a vertex cover:
otherwise, let $e=(u,w)$ be an edge which is not covered by any
vertex in $V'$, so $C_e$ is only interested in items with price $2$,
which he cannot afford. This contradicts the assumption that $p^*$
sells items to every poor consumer.
The revenue collected from poor consumers is exactly $m$. Each rich
consumer $C_v$ in the vertex cover gets the item with price $1$ while
others get the items with price $2$, so the total revenue is $m + {\sf
VC}(G) + 2(n - {\sf VC}(G))$.
\end{proof}
This theorem immediately implies the gap between {\sc Yes-Instance} and {\sc No-Instance} for $d$-{\sf UUDP-MIN}. The only detail we left out is the computation of the embedding $\phi$, and this is where the hardness proofs
of $3$-{\sf UUDP-MIN} and $4$-{\sf UUDP-MIN} depart (other steps are
exactly the same). For $3$-dimensional case, we start from planar
graphs whose adjacency poset can be embedded into $\ensuremath{\mathbb R}^3$. Since
planar vertex cover has a polynomial-time approximation scheme, we
only get \mbox{\sf NP}-hardness here. For $4$-dimensional case, we start from
vertex cover in cubic graphs, which is known to be \mbox{\sf APX}-hard, but
unfortunately we can only embed its adjacency poset into $\ensuremath{\mathbb R}^4$,
thus obtaining the hardness of $4$-{\sf UUDP-MIN}.
\paragraph{\mbox{\sf NP}-Hardness of $3$-{\sf UUDP-MIN}}
To show the \mbox{\sf NP}-hardness, we start from {\sf Vertex Cover} in planar graphs, which is known to be \mbox{\sf NP}-complete~\cite{DBLP:journals/tcs/GareyJS76}. We will use the following theorem,
due to Schnyder~\cite{Schnyder89}.
\begin{theorem}
Let $(V \cup E, \preceq_G)$ be an incident poset of planar graph
$G$. Then there exists an embedding $\phi$ from the poset into
$\ensuremath{\mathbb R}^3$.
\end{theorem}
Schnyder shows later that the crucial step in the theorem
can be computed in polynomial time \cite{DBLP:conf/soda/Schnyder90}, which
immediately implies the following theorem.
\begin{theorem}
$3$-{\sf UUDP-MIN} is \mbox{\sf NP}-hard even when the consumer budgets are
either $1$ or $2$.
\end{theorem}
\paragraph{\mbox{\sf APX}-Hardness of $4$-{\sf UUDP-MIN}}
We will be using the fact that {\sf Vertex Cover} in cubic graphs
is \mbox{\sf APX}-hard \cite{DBLP:conf/ciac/AlimontiK97}, stated in the
language convenient for our use below.
\begin{theorem}
For some $0 < \alpha < \beta < 1$, it is \mbox{\sf NP}-hard to distinguish
between (i) the graph that has a vertex cover of size at most
$\alpha n$, and (ii) the graph whose minimum vertex cover is at
least $\beta n$.
\end{theorem}
Now we assume that our input graph $G$ is a cubic graph and use the
following theorem to embed the adjacency poset of $G$ into $\ensuremath{\mathbb R}^4$.
\begin{theorem}[Schnyder]
An adjacency poset of any $4$-colorable graph can be embedded into
$\ensuremath{\mathbb R}^4$. Moreover, the embedding is computable in polynomial time.
\end{theorem}
It only requires a straightforward computation to prove the
following theorem.
\begin{theorem}
$4$-{\sf UUDP-MIN} is \mbox{\sf APX}-hard even when the consumer budgets are
either $1$ or $2$.
\end{theorem}
\begin{proof}
In the {\sc Yes-Instance}\xspace, we can collect the revenue of $(2-\alpha) n +m$.
However, in the \ni, the revenue is at most $(2-\beta)n +m$. Since
the graph is cubic, we may assume that $m =\gamma n$ for some $1\leq
\gamma <2$. Hence we have a gap of
$(2-\alpha+\gamma)/(2-\beta+\gamma)$.
\end{proof}
\subsection{\mbox{\sf NP}-hardness of $2$-{\sf SMP}}
Highway problem can be defined as follows: We are given a line $P = (v_0,\ldots, v_n)$ consisting of $n$ edges and $n+1$ vertices and a set of consumers ${\mathcal{C}}$ where each consumer ${\bf C}$ corresponds to a subpath $P_{{\bf C}}$ of $P$ and is equipped with budget $B_{\bf C}$. Our goal is to set price to edges so as to maximize the revenue, where each consumer ${\bf C}$ buys a path $P_{{\bf C}}$ if she can afford the whole path; otherwise she buys nothing.
\begin{lemma}
There is a polynomial-time algorithm that transforms an instance of Highway problem to an instance of $2$-{\sf SMP}.
\end{lemma}
\begin{proof}
For each $i=1,\ldots, n$, each edge $(v_{i-1}, v_i)$, we create an item ${\bf I}_i$ at coordinates $(i, n+1-i)$. Then for each consumer ${\bf C}$ whose path is $P_{\bf C} = (v_s,\ldots, v_t)$, we create a consumer point at $(s+1, n+1-t)$. It is easy to see that the consideration set remains unchanged.
\end{proof}
\subsection{\mbox{\sf APX}-hardness of $4$-{\sf SMP}}
We perform a reduction from {\sf Graph Vertex Pricing} on bipartite graphs. In this problem, we are given a graph $G=(V,E)$, where each vertex corresponds to item and each edge $e \in E$ corresponds to a consumer, additionally equipped with budget $B_e$. Each consumer edge is interested in items that correspond to her incident vertices. Our goal is to set a price $p: V \rightarrow \ensuremath{\mathbb R}$ so as to maximize our revenue.
Given an instance $(G, \set{B_e}_{e \in E})$ of {\sf Graph Vertex Pricing} where graph $G$ is a bipartite graph $(U \cup W, E)$, we create an instance of $4$-{\sf SMP} as follows. Suppose we have $U = \set{u_1,\ldots, u_{|U|}}$ and $W = \set{w_1,\ldots, w_{|W|}}$. For each vertex $u_i \in U$, we have a corresponding item ${\bf I}^u_{i}$ with coordinates $(i, |U|+1-i, \infty,\infty)$. Similarly, for each vertex $w_j \in W$, we have a corresponding item ${\bf I}^w_{j}$ with coordinates $(\infty,\infty,j, |W|+1-j)$. Finally, for each edge $(u_i, w_j) \in E$, we have a consumer ${\bf C}_{ij} = (i, |U|+1-i, j, |W|+1-j)$, whose budget is the same as the budget of edge $(u_i,w_j)$. The following claim is almost immediate.
\begin{claim}
For each consumer ${\bf C}_{ij}$, we have that $S_{{\bf C}_{ij}} = \set{{\bf I}^u_i, {\bf I}^w_j}$
\end{claim}
\begin{proof}
It is easy to see that $\set{{\bf I}^u_i, {\bf I}^w_j} \subseteq S_{{\bf C}_{ij}}$. Notice that for $i'<i$, we have ${\bf C}_{ij}[1] > {\bf I}^u_{i'}[1]$, so any such item ${\bf I}^u_{i'}$ cannot belong to $S_{{\bf C}_{ij}}$. Similarly for $i' >i$, we have ${\bf C}_{ij}[2] > {\bf I}^u_{i'}[2]$, so such an item cannot belong to $S_{{\bf C}_{ij}}$. By using similar arguments for items of the form ${\bf I}^w_{j'}$ for $j' \neq j$, we reach the conclusion that $S_{{\bf C}_{ij}} = \set{{\bf I}^u_i, {\bf I}^w_j}$.
\end{proof}
Since the $4$-{\sf SMP} instance is equivalent to the instance of {\sf Graph Vertex Pricing}, the maximum revenue is preserved. Using the \mbox{\sf APX}-hardness result of {\sf Graph Vertex Pricing} on bipartite graphs~\cite{KhandekarKMS09}, we conclude that $4$-{\sf SMP} is \mbox{\sf APX}-hard.
\section{Sub-linear Approximation Algorithm (Proof of Theorem~\ref{thm:intro thm udp smp})}\label{sec:d udp min}
To simplify the presentation, we present the algorithm for $d$-{\sf UUDP-MIN} in this section. The algorithm for $d$-{\sf SMP} is almost identical. Let ${\mathcal{C}}$ and ${\mathcal{I}}$ be the set of points in $\mathbb{R}^d$, where every consumer ${\bf C}\in{\cal C}$ has budget $B_{{\bf C}}$ and consideration set $S_{{\bf C}}$ which is specified by coordinates of the input point. For any subset $\mathcal{C}'\subseteq{\mathcal{C}}$ and ${\cal I}'\subseteq {\mathcal{I}}$, let
$\mathcal{P}({\mathcal{C}}',{\mathcal{I}}')$ be the $d$-{\sf UUDP-MIN} problem with input ${\mathcal{C}}'$ and ${\mathcal{I}}'$. Moreover, for any ${\mathcal{C}}'$ and ${\mathcal{I}}'$, we use $\mbox{\sf OPT}({\mathcal{C}}', {\mathcal{I}}')$ to express the optimal revenue of the instance $({\mathcal{C}}', {\mathcal{I}}')$. At a high level, our algorithm proceeds in an inductive manner and obtains a solution of $d$-{\sf UUDP-MIN} problem by invoking the algorithms for $(d-1)$-{\sf UUDP-MIN} and $1$-{\sf UUDP-MIN} as a subroutine. Our result is summarized in the following theorem.
\begin{theorem}
\label{thm: dimension reduction for UDP}
For any $\epsilon \in (0,1]$, if there is an $\tilde O_d(n^{1-\epsilon})$-approximation algorithm for $(d-1)$-{\sf UUDP-MIN} then there is an $\tilde O_d(n^{1-\epsilon/4})$-approximation algorithm for $d$-{\sf UUDP-MIN} as well.
\end{theorem}
Theorem~\ref{thm:intro thm udp smp} then follows from the fact that $1$-{\sf UUDP-MIN} can be solved optimally in polynomial time (see Appendix~\ref{sec:one udp min}\fullonly{ and its generalization in Section~\ref{sec: algorithms for small d}}). As we noted earlier, it can be improved slightly since $2$-{\sf UUDP-MIN} admits {\sf QPTAS} \sodaonly{(see Appendix~\ref{sec: qptas 2 udp})}\fullonly{(see Section~\ref{subsection: qptas minbuying})}.
\subsection{Consideration-preserving Decomposition}
Our algorithm partitions the input instance into many subinstances and tries to collect the profit from some of them. The notion of consideration-preserving decomposition, defined below, allows us to do so without losing revenue.
\begin{definition}\label{def:decomposition} We call a collection $\set{({\mathcal{C}}'_1,{\mathcal{I}}'_1),\ldots, ({\mathcal{C}}'_k, {\mathcal{I}}'_k)}$ a {\em consideration-preserving decomposition} of the problem $(\mathcal{C}, \mathcal{I})$ if and only if for any ${\bf C} \in {\mathcal{C}}$ and ${\bf I} \in S_{{\bf C}}$, there exists (not necessarily unique) $i$ such that ${\bf C} \in {\mathcal{C}}'_i$ and ${\bf I} \in {\mathcal{I}}'_i$.
\end{definition}
By definition, for any consumer ${\bf C}$ and item ${\bf I}$ the fact that consumer ${\bf C}$ considers item ${\bf I}$ is preserved by at least one instance $({\mathcal{C}}'_i, {\mathcal{I}}'_i)$. The following lemma says that this decomposition preserves the total revenue.
\begin{lemma}
\label{lemma: UDP decomposition}
For any consideration-preserving decomposition $\left\{({\mathcal{C}}'_1,{\mathcal{I}}'_1), \ldots, ({\mathcal{C}}'_k, {\mathcal{I}}'_k)\right\}$ of $({\mathcal{C}}, {\mathcal{I}})$, it holds that
$\sum_{i=1}^k \mbox{\sf OPT}({\mathcal{C}}'_i, {\mathcal{I}}'_i)\geq \mbox{\sf OPT}({\mathcal{C}}, {\mathcal{I}})
\,.$
Moreover, any price function for ${\mathcal{P}}({\mathcal{C}}'_i, {\mathcal{I}}'_i)$ can be extended to a price function for the original problem ${\mathcal{P}}({\mathcal{C}}, {\mathcal{I}})$ that gives revenue at least $\mbox{\sf OPT}({\mathcal{C}}'_i, {\mathcal{I}}'_i)$.
\end{lemma}
\iffalse
\sodaonly{\begin{proof}[Proof Idea]}
\fullonly{\begin{proof}[Proof idea]}
(See Appendix~\ref{sec: proof of UDP decomposition lemma} for the full proof.)
Let $p^{\star}:{\mathcal{I}}\rightarrow\mathbb{R}$ be the optimal price function for ${\mathcal{P}}({\mathcal{C}}, {\mathcal{I}})$. It is straightforward to see that, for any consumer ${\bf C}$, the optimal revenue that ${\bf C}$ gets from any subset of ${\mathcal{I}}$ is at least the optimal revenue ${\bf C}$ gets when considering the set ${\mathcal{I}}$. This proves the first statement.
For the second one, let $p$ be a price function for ${\mathcal{P}}({\mathcal{C}}_i, {\mathcal{I}}_i)$. We extend $p$ to $p':{\mathcal{I}}\rightarrow\ensuremath{\mathbb R}$ defined by $p'({\bf I})=p({\bf I})$ if ${\bf I}\in{\mathcal{I}}_i$, and $p'({\bf I})=\infty $ otherwise.
\end{proof}
\fi
This is simply by applying the optimal price function of one problem to the other (see Appendix~\ref{sec: proof of UDP decomposition lemma} for the full proof). In the rest of our discussion, we mainly use two different types of consideration-preservation decomposition, as explained in the following observation.
\begin{observation}\label{observation: conseration preserving decomposition}
Given an input instance $({\mathcal{C}}',{\mathcal{I}}')$, let ${\mathcal{C}}' = \bigcup_{i=1}^k {\mathcal{C}}'_i$. Then $\{({\mathcal{C}}'_1,{\mathcal{I}}')$, $\ldots$, $({\mathcal{C}}'_k, {\mathcal{I}}')\}$ is a consideration-preserving decomposition of $({\mathcal{C}}', {\mathcal{I}}')$. Similarly, if ${\mathcal{I}}' = \bigcup_{i=1}^k {\mathcal{I}}'_i$, then we have that $\set{({\mathcal{C}}',{\mathcal{I}}'_1), \ldots, ({\mathcal{C}}', {\mathcal{I}}'_k)}$ is a consideration-preserving decomposition of $({\mathcal{C}}', {\mathcal{I}}')$.
\end{observation}
\subsection{Algorithm}
\begin{wrapfigure}{r}{0.5\textwidth}
\vspace{-1.5cm}
\center\includegraphics[width=\linewidth]{overview.pdf}
\vspace{-.5cm}
\caption{Decomposition overview}\label{fig:overview}
\vspace{-.5cm}
\end{wrapfigure}
At a high level, the algorithm proceeds in four steps where each step involves consideration-preserving decomposition (see Fig.~\ref{fig:overview} for an overview). In Step 1, we partition ${\mathcal{I}}$ into different subsets where
every subset satisfies certain properties, i.e. the elements in each subset either form a chain or an antichain. The problem on those subsets
in which elements form a chain can be solved easily, and we deal with the antichains in later steps. In Step 2, we partition consumers in ${\mathcal{C}}$ into two types, those with large and small consideration sets. We use the algorithm of \cite{BalcanB07,BriestK11} to deal with consumers with small consideration sets and handle the rest consumers in later steps. In Step 3, we find a subset of items, i.e. a ``hitting set'', and partition consumers further into several sets. Each set of consumers has the following property: There is some item desired by all consumers in the set. Using this property, we show in Step 4 that the problem can be further partitioned into a few problems where each of them can be viewed as a $(d-1)$-{\sf UUDP-MIN} problem. (We call this a ``consideration-preserving embedding''.)
\paragraph{\underline{Step 1:} Partitioning items into chains and antichains}
Let $({\mathcal{C}}, {\mathcal{I}})$ be an input of $d$-{\sf UUDP-MIN}. First we define a partially ordered set $({\mathcal{I}}, \leq)$ on the item set as follows. We say that ${\bf I}_1 \leq {\bf I}_2$ if and only if ${\bf I}_1$ has a lower quality than ${\bf I}_2$ in every attribute, i.e. ${\bf I}_1[d'] \leq {\bf I}_2[d']$ for all $d' \in [d]$.
We say that a subset ${\mathcal{I}}' \subseteq {\mathcal{I}}$ is a chain if ${\mathcal{I}}'$ can be written as ${\mathcal{I}}'= \set{{\bf I}_1,\ldots, {\bf I}_z}$ such that ${\bf I}_j \leq {\bf I}_{j+1}$ for all $j\in[z-1]$. We say that ${\mathcal{I}}' \subseteq {\mathcal{I}}$ is an antichain if and only if for any pair of items ${\bf I}, {\bf I}' \in {\mathcal{I}}'$, neither ${\bf I} \leq {\bf I}'$ nor ${\bf I}' \leq {\bf I}$.
\begin{lemma}
\label{lemma: chain decomposition}
For any $\epsilon >0$ and any $s = n^{\epsilon/4}, t= n^{1-\epsilon/4}$, we can partition ${\mathcal{I}}$ into $A_1, \ldots, A_s$ and $B_1, \ldots, B_t$ in polynomial-time. Moreover, each $A_i$ is an antichain and each $B_j$ is a chain.
\end{lemma}
\sodaonly{\begin{proof}[Proof Idea]}
\fullonly{\begin{proof}[Proof idea]}
(See Section~\ref{sec:detail one} for detailed definitions and proofs.)
By Dilworth's theorem \cite{Dilworth,Fulkerson-Dilworth}, the minimum chain decomposition equals to the maximum antichain size. We will use the fact that both minimum chain decomposition and maximum-size antichain can be computed in polynomial time as follows:
As long as the maximum-size antichain is bigger than $n^{\epsilon/4}$, we repeatedly extract such an antichain out of the input; otherwise, we would have the decomposition into at most $n^{\epsilon/4}$ chains, so we stop.
\end{proof}
By Observation~\ref{observation: conseration preserving decomposition}, the collection
$\{({\mathcal{C}}, A_1), \ldots, ({\mathcal{C}}, A_s), ({\mathcal{C}}, B_1), \ldots, ({\mathcal{C}}, B_t)\}$
is a consideration-preserving decomposition of $({\mathcal{C}}, {\mathcal{I}})$. It follows by Lemma~\ref{lemma: UDP decomposition} that
$\sum_{i=1}^s\mbox{\sf OPT}({\mathcal{C}}, A_i)+\sum_{j=1}^t \mbox{\sf OPT}({\mathcal{C}}, B_j) \geq \mbox{\sf OPT}({\mathcal{C}}, {\mathcal{I}}).$
Further, observe that if there exists $j$ such that $\mbox{\sf OPT}({\mathcal{C}}, B_j)\geq\mbox{\sf OPT}({\mathcal{C}}, {\mathcal{I}})/(2n^{1-\epsilon/4})$, then we would be done: the $d$-{\sf UUDP-MIN} problem ${\mathcal{P}}({\mathcal{C}}, B_j)$ can be seen as a $1$-{\sf UUDP-MIN} problem (since $B_j$ is a chain) and hence can be solved optimally! (See Lemma~\ref{lem:chain to one dim} for detailed analysis)
Otherwise $\mbox{\sf OPT}({\mathcal{C}}, B_j)\leq \mbox{\sf OPT}({\mathcal{C}}, {\mathcal{I}})/ (2n^{1-\epsilon/4})$ for every $j$. Therefore
$\sum_{j=1}^t \mbox{\sf OPT}({\mathcal{C}}, B_j) \leq n^{1-\varepsilon/4}\cdot\mbox{\sf OPT}({\mathcal{C}},{\mathcal{I}})/ (2n^{1-\varepsilon/4})< \mbox{\sf OPT}({\mathcal{C}},{\mathcal{I}})/2.$
If this is not the case then we know that there must be an antichain $A_i$ such that
$\mbox{\sf OPT}({\mathcal{C}}, A_i)\geq \mbox{\sf OPT}({\mathcal{C}},{\mathcal{I}})/2n^{\epsilon/4}\,.$
\begin{wrapfigure}{r}{.5\textwidth}
\vspace{-.9cm}
\center\includegraphics[width=\linewidth]{step1.pdf}\\
\vspace{-.3cm}\caption{Example of Step 1}\label{fig:step1}\vspace{-.2cm}
\end{wrapfigure}
\paragraph{\underline{Step 2:} Dealing with small consideration sets}
For simplicity, let us assume that we know $i$ such that $\mbox{\sf OPT}({\mathcal{C}}, A_i)\geq \mbox{\sf OPT}({\mathcal{C}}, {\mathcal{I}})/(2n^{\epsilon/4})$. Now we focus on collecting revenue from the subproblem ${\mathcal{P}}({\mathcal{C}}, A_i)$. Let ${\mathcal{C}}_1 \subseteq {\mathcal{C}}$ be the set of consumers who are interested in at most $n^{1-2\epsilon/4}$ items in $A_i$, i.e. ${\mathcal{C}}_1=\left\{{\bf C}\in {\mathcal{C}}: |S_{\bf C}\cap A_{i}|\leq n^{1-2\epsilon/4}\right\}$, and define ${\mathcal{C}}_2={\mathcal{C}}\setminus {\mathcal{C}}_1$.
Since $\{({\mathcal{C}}_1, A_i), ({\mathcal{C}}_2, A_i)\}$ is a consideration-preserving decomposition of $({\mathcal{C}}, A_i)$, we have
$\mbox{\sf OPT}({\mathcal{C}}_1, A_i)+\mbox{\sf OPT}({\mathcal{C}}_2, A_i) \geq \mbox{\sf OPT}({\mathcal{C}}, A_i) \geq \frac{\mbox{\sf OPT}({\mathcal{C}}, {\mathcal{I}})}{2n^{\epsilon/4}}.$
Now we need an algorithm of \cite{BalcanB07,BriestK11}. Balcan and Blum give an approximation algorithm for {\sf SMP} whose approximation guarantee depends on the sizes of consideration sets. Briest and Krysta, by using a slight modification of this algorithm, give an approximation algorithm with the same guarantee for {\sf UDP-MIN}. Their result, stated in terms of {\sf UUDP-MIN}, is summarized in the following theorem. (For completeness, we provide the proof in Appendix~\ref{sec:balcanblum for udp min}.)
\begin{theorem}\label{theorem:BalcanBlum}\cite{BalcanB07,BriestK11}
Given a {\sf UUDP-MIN} instance $({\mathcal{C}}, {\mathcal{I}}, \set{S_{{\bf C}}}_{{\bf C} \in {\mathcal{C}}})$, there is a deterministic $O(k)$-approximation algorithm of {\sf UUDP-MIN}, where $k: = \max_{{\bf C} \in {\mathcal{C}}} \size{S_{{\bf C}}}$.
\end{theorem}
We remark that we extend this technique to deal with any pricing problem with subadditive revenue in the full version of this paper.
If $\mbox{\sf OPT}({\mathcal{C}}_1, A_i)\geq \mbox{\sf OPT}({\mathcal{C}}, {\mathcal{I}})/(4n^{\epsilon/4})$, then we could invoke the algorithm in Theorem~\ref{theorem:BalcanBlum} on $({\mathcal{C}}_1, A_i)$ to get a solution with approximation ratio
$O\left(\max_{{\bf C} \in {\mathcal{C}}_1} \size{S_{{\bf C}}\cap A_i}\right)=O(n^{1-2\epsilon/4}).$
This yields a solution that gives a desired revenue of
$\Omega\left(\mbox{\sf OPT}({\mathcal{C}}_1, A_i)/n^{1-2\epsilon/4}\right)=\Omega\left(\mbox{\sf OPT}({\mathcal{C}}, {\mathcal{I}})/n^{1-\epsilon/4}\right)\,.$
Otherwise we have $\mbox{\sf OPT}({\mathcal{C}}_1, A_i)< \mbox{\sf OPT}({\mathcal{C}}, {\mathcal{I}})/4n^{\epsilon/4}$. Then
$\mbox{\sf OPT}({\mathcal{C}}_2, A_i)=\Omega\left(\mbox{\sf OPT}({\mathcal{C}}, {\mathcal{I}})/n^{\epsilon/4}\right).$
We will deal with this case in the next steps.
\paragraph{\underline{Step 3:} Partitioning consumers using a small hitting set}
First, we apply the epsilon net theorem~\cite{Chazelle-book,Epsilon-net} to derive the following lemma.
\begin{lemma}\label{lem:hitting set}
We can find a set $H\subseteq A_i$ of size $\tilde O(n^{2\epsilon})$ in randomized polynomial time such that for any ${\bf C}\in {\mathcal{C}}_2$, there exists ${\bf I}\in H$ such that ${\bf I}\geq {\bf C}$.
\end{lemma}
\begin{proof}
\danupon{We should consider providing a bit more detail (``a primer'') in the Appendix.}
The instance $(\mathcal{C}_2, A_i)$ defines a set system $\{S_{\bf C}\}_{{\bf C}\in \mathcal{C}_2}$ over $A_i$, where $S_{\bf C}=\{{\bf I}\in A_i\mid {\bf I}\geq {\bf C}\}$. We note that each set $S_{{\bf C}}$ has {\em descriptive complexity} at most $d$, i.e. set $S_{{\bf C}}$ can be described by $d$ linear inequalities of the form $S_{{\bf C}} = \bigcap_{d'=1}^d \set{{\bf I} \in {\mathcal{I}}: {\bf I}[d'] \geq {\bf C}[d']}$. In this case, this set system has VC dimension $O(d)$, c.f. \cite{Sharir-book}. More specifically, it is well known (e.g., \cite{AronovES10}) that any collection of $d$-dimensional axis-parallel boxes has VC dimension $O(d)$. We will not formally define VC-dimension here. The following theorem is all we need.
\begin{theorem}(\cite{Chazelle-book,Epsilon-net}; Epsilon net theorem)
Let ${\mathcal{X}}$ be a set system of VC-dimension at most $d'$ over $N$. Then for any $\delta \in (0,1)$, we can find a set $H \subseteq N$ with $\size{H} = O(\frac{d'}{\delta}\log \frac{d'}{\delta})$ in randomized polynomial time such that, for all $X_i \in {\mathcal{X}}$ with $\size{X_i} \geq \delta \size{N}$, it holds that $H \cap X_i \neq \emptyset$.
\end{theorem}
Using the theorem with $\delta = n^{-2\epsilon/4}$, we can find a set $H \subseteq A_i$ of size at most $\tilde{O}(n^{2\epsilon/4})$, and since we have $|S_{{\bf C}} \cap A_i| \geq \delta n$ for all ${\bf C} \in {\mathcal{C}}_2$, we are guaranteed that $H \cap S_{{\bf C}} \neq \emptyset$ for all ${\bf C} \in {\mathcal{C}}_2$.
\end{proof}
We call $H$ a {\em hitting set} of $\mathcal{C}_2$ since $H$ intersects $S_{\bf C}$ for all ${\bf C}\in \mathcal{C}_2$. We use $H$ to decompose $(\mathcal{C}_2, A_i)$ into a small number of subproblems and show in Step 4 that each of these problems can be viewed as a $(d-1)$-{\sf UUDP-MIN} problem.
For each ${\bf I}\in H$, let ${\mathcal{C}}_{\bf I}=\{{\bf C}\in {\mathcal{C}}_2\mid {\bf I} \in S_{\bf C}\}$, i.e., ${\mathcal{C}}_{\bf I}$ consists of all consumers in ${\mathcal{C}}_2$ that consider item ${\bf I}$. Observe that $\bigcup_{{\bf I} \in H} {\mathcal{C}}_{{\bf I}} = {\mathcal{C}}_2$, and therefore by Lemma~\ref{lemma: UDP decomposition}, we have
$\sum_{{\bf I} \in H} \mbox{\sf OPT}({\mathcal{C}}_{\bf I}, A_i)\geq \mbox{\sf OPT}({\mathcal{C}}_2, A_i)
\geq \Omega\left(\mbox{\sf OPT}({\mathcal{C}}, {\mathcal{I}})/n^{\epsilon/4}\right).$
Since $\size{H}=O(n^{2\epsilon/4})$, there exists ${\bf I}^*\in H$ such that
\begin{align*}
\mbox{\sf OPT}({\mathcal{C}}_{{\bf I}^*}, A_i) &=\tilde\Omega\left(\mbox{\sf OPT}(\mathcal{C}, \mathcal{I})\cdot n^{-\epsilon/4}/|H|\right)
=\tilde\Omega\left(\mbox{\sf OPT}(\mathcal{C}, \mathcal{I})/n^{3\epsilon/4}\right).
\end{align*}
Now we, again, assume that we know ${\bf I}^*$ and turn our focus to the subproblem ${\mathcal{P}}({\mathcal{C}}_{{\bf I}^*}, A_i)$.
\paragraph{\underline{Step 4:} Reducing the dimension} We have now reached the most crucial step.
We will (crucially) rely on the fact that all consumers in ${\mathcal{C}}_{{\bf I}^*}$ consider item ${\bf I}^*$, and that $A_i$ is an antichain.
For each $j \leq d$, define $A_i^j$ as the set of items in $A_i$ that are at least as good as ${\bf I}^*$ in the $j$-th coordinate, i.e., $A_i^j=\{{\bf I}\in A_i \mid {\bf I}[j]\geq {\bf I}^*[j]\}$. See Fig.~\ref{fig:step4_1} for an example in the case of $2$-{\sf UUDP-MIN}.
\begin{lemma}\label{lem:union of d dimensions}
$A_i = \bigcup_{j=1}^d A_i^j$.
\end{lemma}
\begin{figure}
\begin{center}
\subfigure[]{
\includegraphics[height=0.12\textheight, clip=true, trim=1.5cm .3cm 1cm .3cm]{step3.pdf}
\label{fig:step4_1}
}
\subfigure[]{
\includegraphics[height=0.12\textheight, clip=true, trim=3cm .75cm 7cm 1cm]{step4.pdf}
\label{fig:step4}
}
\end{center}
\vspace{-.5cm}
\caption{\subref{fig:step4_1} Example of Step 4. \subref{fig:step4} Example of Step 4 when we view the instance $\mathcal{C}_{{\bf I}^*}, A_i^j$ as a $(d-1)$-{\sf UUDP-MIN} instance.}
\vspace{-.5cm}
\end{figure}
This lemma holds simply because $A_i$ is an antichain (in any antichain, no item can completely dominate the others, so at least one coordinate of any ${\bf I}\in \mathcal{I}_{{\bf I}^*}$ has to be at least as good as ${\bf I}^*$; see detailed proof in Appendix~\ref{proof:union of d dimensions}). Then $\{(\mathcal{C}_{{\bf I}^*}, A_i^1), \ldots, (\mathcal{C}_{{\bf I}^*}, A_i^d)\}$ is a consideration-preserving decomposition of $(\mathcal{C}_{{\bf I}^*}, A_i)$ and thus there exists $j$ such that
$\mbox{\sf OPT}(\mathcal{C}_{{\bf I}^*}, A_i^j) \geq \mbox{\sf OPT}({\mathcal{C}}_{{\bf I}^*}, A_i)/d = \tilde\Omega_d(\mbox{\sf OPT}(\mathcal{C}, \mathcal{I})/n^{3\epsilon/4}).$
Observe that, for all ${\bf C}\in {\mathcal{C}}_{{\bf I}^*}$ and ${\bf I} \in A_i^j$, ${\bf C}[j]\leq {\bf I}^*[j]\leq {\bf I}[j]$. This implies that we can ignore the $j$-th coordinate when we solve ${\mathcal{P}}({\mathcal{C}}_{{\bf I}^*}, A_i^j)$. (In particular, for any ${\bf C}\in\mathcal{C}_{{\bf I}^*}$, the consideration set $S_{\bf C}=\left\{{\bf I}\geq {\bf C}\mid {\bf I}\in A_i^j\right\}$ remains the same even when we drop the $j$-th coordinate of all points.) In other words, the problem can be viewed as a $(d-1)$-{\sf UUDP-MIN} problem (see Fig.~\ref{fig:step4} for an idea).
We defer the formal statement and proof of this claim to Section~\ref{sec:detail four}.
Finally, we can invoke the $\tilde O_d(n^{1-\epsilon})$-approximation algorithm for $(d-1)$-{\sf UUDP-MIN} to collect the revenue of
$\tilde\Omega_d\left(\mbox{\sf OPT}({\mathcal{C}}, {\mathcal{I}}) n^{-3\epsilon/4}/n^{1-\epsilon}\right)$ $=\tilde\Omega_d\left(\mbox{\sf OPT}({\mathcal{C}}, {\mathcal{I}})/n^{1-\epsilon/4}\right).$ Therefore we obtain an approximation ratio of $\tilde O_d(n^{1-\epsilon/4})$ in all cases. Algorithm~\ref{algo:udp} summaries our algorithm for solving $d$-{\sf UUDP-MIN}.
\begin{algorithm}
\caption{\footnotesize {\sf UUDP-MIN-APPROX}($d$)}\label{algo:udp}
\begin{algorithmic}[1]
\footnotesize
\IF{$d=1$}
\STATE Solve the problem $\mathcal{P}(\mathcal{C}, \mathcal{I})$ optimally using an algorithm for $1$-{\sf UUDP-MIN} (cf. Appendix~\ref{sec:one udp min})
\ELSE
\STATE Partition $\mathcal{I}$ into antichains $A_1, \ldots, A_s$ and chains $B_1, \ldots, B_t$ where $s\leq n^{\epsilon/4}$ and $t\leq n^{1-\epsilon/4}$ as in Step 1.
\STATE We claim that the problems $\mathcal{P}(\mathcal{C}, B_1), \ldots, \mathcal{P}(\mathcal{C}, B_t)$ are equivalent to $1$-{\sf UUDP-MIN} problems (cf. Section~\ref{sec:detail one}). Solve them optimally using an algorithm for $1$-{\sf UUDP-MIN} (cf. Appendix~\ref{sec:one udp min}).
\FOR{$i=1, \ldots, s$}
\STATE Partition $\mathcal{C}$ into $\mathcal{C}_1$ and $\mathcal{C}_2$ as in Step 2. Find an $O(\max_{{\bf C} \in {\mathcal{C}}_1} \size{S_{{\bf C}}\cap A_i})$=$O(n^{1-2\epsilon/4})$ approximate solution of problem $\mathcal{P}(\mathcal{C}_1, A_i)$.
\STATE Find a hitting set $H$ of $(\mathcal{C}_2, A_i)$ as in Step 3
\FOR{each ${\bf I}\in H$}
\STATE Define $\mathcal{C}_{\bf I}$ as in Step 3
\STATE Define $A_i^1, \ldots, A_i^d$ as in Step 4
\STATE Solve problem $\mathcal{P}(\mathcal{C}_{\bf I}, A_i^1), \ldots, \mathcal{P}(\mathcal{C}_{\bf I}, A_i^d)$ using an $O(n^{1-\epsilon})$-approximation algorithm for $(d-1)$-{\sf UUDP-MIN}
\ENDFOR
\ENDFOR
\ENDIF
\RETURN the solution with highest revenue among the solutions of all solved problems
\end{algorithmic}
\end{algorithm}
\subsection{Consideration-preserving Embedding}\label{sec:detail four}
To formally discuss the reduction of dimensions, we introduce the notion of consideration-preserving embedding. For any $d$, let $(\mathcal{C}, \mathcal{I})$ be any instance of $d$-{\sf UUDP-MIN}. For any $d'$, consider one-to-one functions $f$ and $g$ that map points in $\ensuremath{\mathbb R}^d$ to the ones in $\ensuremath{\mathbb R}^{d'}$. We say that $(f, g)$ is a {\em consideration-preserving embedding} if, for any item ${\bf I} \in {\mathcal{I}}$ and consumer ${\bf C} \in {\mathcal{C}}$, we have that ${\bf I} \geq {\bf C}$ if and only if $g({\bf I}) \geq f({\bf C})$. That is, the fact that consumer ${\bf C}$ is considering or not considering item ${\bf I}$ must be preserved in $f({\bf C})$ and $g({\bf I})$.
Given a consideration-preserving embedding $(f,g)$, we can naturally define a $d'$-{\sf UUDP-MIN} problem ${\mathcal{P}}(f({\mathcal{C}}), g({\mathcal{I}}))$ where $f({\mathcal{C}})=\{f({\bf C})\mid {\bf C}\in {\mathcal{C}}\}$, $g({\mathcal{I}})=\{g({\bf I})\mid {\bf I} \in {\mathcal{I}}\}$ \danupon{Should we say this? ``(note that we allow the sets to contain identical points)''} and the budget $B_{f({\bf C})}$ is $B_{\bf C}$ for any ${\bf C}\in{\mathcal{C}}$.
Observe that, although $(\mathcal{C}, \mathcal{I})$ and $(f(\mathcal{C}), g(\mathcal{I}))$ correspond to points on different spaces, they represent the same pricing problem (i.e., the consumers' consideration sets and budgets are exactly the same). Thus, we sometimes say that $(\mathcal{C}, \mathcal{I})$ and $(f(\mathcal{C}), g(\mathcal{I}))$ are {\em equivalent}. The following observation follows trivially.
\begin{observation}
\label{observation: consideration preserving embedding}
For any instance $({\mathcal{C}},{\mathcal{I}})$, let $(f,g)$ be a consideration-preserving embedding of $({\mathcal{C}}, {\mathcal{I}})$ into $\ensuremath{\mathbb R}^{d'}$. Then we have that
$\mbox{\sf OPT}({\mathcal{C}}, {\mathcal{I}}) = \mbox{\sf OPT}(f({\mathcal{C}}), g({\mathcal{I}})).$
Moreover, if $f$ and $g$ are polynomial-time computable then a solution for $\mathcal{P}(f({\mathcal{C}}), g({\mathcal{I}}))$ can be efficiently transformed into one for $\mathcal{P}({\mathcal{C}}, {\mathcal{I}})$ that gives the same revenue.
\end{observation}
The transformation in the above lemma is trivial: For any price function $p$ for $(f({\mathcal{C}}), g({\mathcal{I}}))$, we simply price item ${\bf I}\in {\mathcal{I}}$ to $p(g({\bf I}))$. Observe that we will receive the same revenue from both problems using this pricing strategy.
In Step 1, we claimed that when the items form a chain, our instance would be equivalent to $1$-{\sf UUDP-MIN}. Now we prove this fact formally below.
\begin{lemma}\label{lem:chain to one dim}
Let $({\mathcal{C}}, {\mathcal{I}})$ be a $d$-{\sf UUDP-MIN} instance where $({\mathcal{I}}, \leq)$ is a chain. Then $(\mathcal{C}, \mathcal{I})$ is equivalent to a $1$-{\sf UUDP-MIN} instance. Moreover, the corresponding consideration-preserving embedding $(f, g)$ can be computed in polynomial time.
\end{lemma}
\begin{proof}
Order items in $\mathcal{I}$ by ${\bf I}_1\leq {\bf I}_2\leq \ldots$. Now map each item into a one-dimensional point: $g({\bf I}_i)=(i)$. Moreover, map each consumer according to $f({\bf C})=g({\bf I}_i)$, where $i$ is the minimum number such that ${\bf I}_i\geq {\bf C}$. Observe that $(f, g)$ is a consideration-preserving embedding since $S_{\bf C}=\{{\bf I}_i, {\bf I}_{i+1}, \ldots\}$ while $S_{f({\bf C})}=\{g({\bf I}_i), g({\bf I}_{i+1}), \ldots\}$ for any ${\bf C}\in {\mathcal{C}}$. (Note that this embedding might create redundancy since it is possible that $f({\bf C})=f({\bf C}')$ for some ${\bf C}\neq {\bf C}'$. This can be fixed easily by slightly perturbing the points.\danupon{I'm not sure if this is necessary.})
\end{proof}
In Step 4, we also claimed the dimension reduction of sub-instances $({\mathcal{C}}_{{\bf I}^*}, A_i^j)$, and we now prove the claim formally. Recall that the item ${\bf I}^*\in A_i^j$ has the property that ${\bf I}^*\geq {\bf C}$ for all ${\bf C} \in \mathcal{C}_{{\bf I}^*}$ and ${\bf I}^*[j]\leq {\bf I}[j]$ for all ${\bf I}\in A_i^j$.
\begin{lemma}\label{lem:dim reduction}
The instance
$(\mathcal{C}_{{\bf I}^*}, A_i^j)$ is equivalent to a $(d-1)$-{\sf UUDP-MIN} instance. Moreover, the corresponding consideration-preserving embedding $(f, g)$ can be computed in polynomial time.
\end{lemma}
\begin{proof}
Consider ``ignoring'' the $j$-th coordinate as follows. For any ${\bf C}\in \mathcal{C}_{{\bf I}^*}$ and ${\bf I}\in A_i^j$, let $f({\bf C})=({\bf C}[1], {\bf C}[2], \ldots, {\bf C}[j-1], {\bf C}[j+1], \ldots, {\bf C}[d])$
and
$g({\bf I})=({\bf I}[1], {\bf I}[2], \ldots, {\bf I}[j-1], {\bf I}[j+1], \ldots, {\bf I}[d]).$
Observe that for any ${\bf C}\in \mathcal{C}_{{\bf I}^*}$ and ${\bf I}\in A_i^j$, ${\bf I}\geq {\bf C}$ trivially implies that $g({\bf I})\geq f({\bf C})$. Conversely, if $g({\bf I})\geq f({\bf C})$ then ${\bf I}\geq {\bf C}$ since ${\bf I}[j]\geq {\bf I}^*[j]\geq {\bf C}[j]$. Thus, $(f, g)$ is a consideration-preserving embedding.
\end{proof}
\section{Proof Omitted from Section~\ref{sec:d udp min}}
\subsection{Proof of Lemma~\ref{lemma: UDP decomposition}}
\label{sec: proof of UDP decomposition lemma}
Let $p^*$ be the optimal price function for ${\mathcal{P}}({\mathcal{C}}, {\mathcal{I}})$. For each $i=1,\ldots, k$, we define $p^*_i: {\mathcal{I}}'_i \rightarrow \ensuremath{\mathbb R}$ by
$$p^*_i({\bf I}) = p^*({\bf I})~~\mbox{if ${\bf I} \in {\mathcal{I}}'_i$, and $p^*_i({\bf I}) = \infty$ otherwise.}$$
Let $r_i$ be the total revenue made by $p^*_i$ in ${\mathcal{P}}({\mathcal{C}}'_i, {\mathcal{I}}'_i)$. We argue below that
\begin{align}\label{eq:one}
\sum_{i=1}^k r_i \geq \mbox{\sf OPT}({\mathcal{C}}, {\mathcal{I}}).
\end{align}
Let ${\mathcal{C}}^* \subseteq {\mathcal{C}}$ be the set of consumers who make a positive payment with respect to $p^*$. For each consumer ${\bf C} \in {\mathcal{C}}^*$, denote by $\phi({\bf C}) \in {\mathcal{I}}$ the item that consumer ${\bf C}$ buys with respect to the price $p^*$.
So we can write $\mbox{\sf OPT}({\mathcal{C}}, {\mathcal{I}})$ as
\begin{align}\label{eq:two}
\mbox{\sf OPT}({\mathcal{C}}, {\mathcal{I}}) = \sum_{{\bf C} \in {\mathcal{C}}^*} p^*(\phi({\bf C})).
\end{align}
For each $i=1,\ldots, k$, let ${\mathcal{C}}^*_i \subseteq {\mathcal{C}}'_i$ be the set of consumers ${\bf C} \in {\mathcal{C}}'_i$ such that $\phi({\bf C}) \in {\mathcal{I}}'_i$. That is, ${\mathcal{C}}^*_i$ is a set of consumers whose item she bought in $\mbox{\sf OPT}({\mathcal{C}}, {\mathcal{I}})$ is in ${\mathcal{I}}'_i$.
Notice that
\begin{align}\label{eq:three}
r_i \geq \sum_{{\bf C} \in {\mathcal{C}}^*_i} p^*(\phi({\bf C})).
\end{align}
Since $\set{({\mathcal{C}}'_i, {\mathcal{I}}'_i)}_{i=1}^k$ is a consideration-preserving decomposition, we have that
\begin{align}\label{eq:four}
\bigcup_{i=1}^k {\mathcal{C}}^*_i \supseteq {\mathcal{C}}^*,
\end{align}
since for any ${\bf C} \in {\mathcal{C}}^*$, we must have $\phi({\bf C})\in {\mathcal{I}}_i$ for some $i$. By summing Eq.\eqref{eq:three} over all $i=1,\ldots, k$, we have
\begin{align*}
\sum_{i=1}^k r_i &\geq \sum_{i=1}^k \sum_{{\bf C}\in {\mathcal{C}}^*_i} p^*(\phi({\bf C})) &\mbox{(by Eq.\eqref{eq:three})}\\
&\geq \sum_{{\bf C}\in {\mathcal{C}}^*} p^*(\phi({\bf C})) &\mbox{(by Eq.\eqref{eq:four})}\\
&=\mbox{\sf OPT}({\mathcal{C}}, {\mathcal{I}})&\mbox{(by Eq.\eqref{eq:two})}
\end{align*}
This proves Eq.\eqref{eq:one} and thus the first claim.
Now suppose we have a price $p': {\mathcal{I}}_i \rightarrow \ensuremath{\mathbb R}$ that collects revenue $r'$ in ${\mathcal{P}}({\mathcal{C}}'_i, {\mathcal{I}}'_i)$. We define a function $p: {\mathcal{I}} \rightarrow \ensuremath{\mathbb R}$ by $p({\bf I}) = p'({\bf I})$ for ${\bf I} \in {\mathcal{I}}'_i$ and $p({\bf I}) = \infty$ otherwise. We can use $p'$ to obtain a revenue of $r'$ from $\mathcal{P}({\mathcal{C}}, {\mathcal{I}})$. This proves the second claim.
\subsection{Decomposing items into small number of chains and antichains}\label{sec:detail one}
We will use the following theorem, first proved by Dilworth \cite{Dilworth}, and its polynomial computability follows from the equivalence between Dilworth's theorem and K\"onig's theorem~\cite{Fulkerson-Dilworth}.
\begin{theorem}\label{theorem:dilworth}
Let $(S, \leq)$ be a partially ordered set, and $Z$ be the maximum number of elements in any antichain of $S$. Then there is a polynomial-time algorithm that produces a partition of $S$ into $Z$ chains $S_1,\ldots, S_Z$.
\end{theorem}
We now use the theorem to prove Lemma~\ref{lemma: chain decomposition}.
\begin{proof}[of Lemma~\ref{lemma: chain decomposition}]
Initially, let $i=1$. In iteration $i$, we check if the size of maximum antichain in ${\mathcal{I}}$ is at least $t=n^{1-\epsilon/4}$. If so, we find the maximum antichain $A_i$, update ${\mathcal{I}} = {\mathcal{I}} \setminus A_i$, and proceed to the next iteration; otherwise, we stop the iterations. Notice that the number of iterations is at most $s = n^{\epsilon/4}$, and when the iteration stops, the size of maximum-size antichain is at most $t\leq n^{1-\epsilon/4}$. We apply the above theorem to compute a decomposition of ${\mathcal{I}}$ into $t$ chains, denoted by $B_1,\ldots, B_t$. This concludes the proof of Lemma~\ref{lemma: chain decomposition}.
\end{proof}
\subsection{Proof of Balcan-Blum Theorem for {\sf UUDP-MIN} (cf. Theorem~\ref{theorem:BalcanBlum})}\label{sec:balcanblum for udp min}
We first explain a randomized algorithm, and then we discuss how to derandomize it. This part is essentially the same as \cite{BalcanB07,BriestK11}. First, we randomly construct a set ${\mathcal{I}}^* \subseteq {\mathcal{I}}'$ where each item ${\bf I}$ is independently added to ${\mathcal{I}}^*$ with probability $1/k$ (recall that $k=\max_{{\bf C}\in {\mathcal{C}}} |S_{\bf C}|$). Then let ${\mathcal{C}}^*$ be a set of consumer ${\bf C}$ such that $\size{S_{\bf C}\cap {\mathcal{I}}^*}=1$
(i.e. consumers who care about exactly one item in ${\mathcal{I}}^*$). We show that the problem $\mathcal{P}({\mathcal{C}}^*, {\mathcal{I}}^*)$ has expected revenue at least $\Omega(\mbox{\sf OPT}({\mathcal{C}}, {\mathcal{I}})/k)$.
Let $p$ be the optimal price function for $({\mathcal{C}}, {\mathcal{I}})$ and $\phi: {\mathcal{C}} \rightarrow {\mathcal{I}}\cup \{\perp\}$ be a function that maps each consumer to the item she buys with respect to $p$ (let $\phi({\bf C})=\perp$ if consumer ${\bf C}$ buys nothing and $p(\perp)=0$). Therefore, we have that $\mbox{\sf OPT}({\mathcal{C}}, {\mathcal{I}}) = \sum_{{\bf C}} p(\phi({\bf C}))$. We denote by $p^*$ the price function $p$ restricted to ${\mathcal{I}}^*$. For each ${\bf C}$, if ${\bf C} \in {\mathcal{C}}^*$ and $\phi({\bf C}) \in {\mathcal{I}}^*$, the revenue created by $p^*$ in $({\mathcal{C}}^*, {\mathcal{I}}^*)$ would be at least $p(\phi({\bf C}))$. Therefore,
$$\expect{}{\mbox{\sf OPT}({\mathcal{C}}^*, {\mathcal{I}}^*)} \geq \sum_{{\bf C} \in {\mathcal{C}}} \Pr[\mbox{$\phi({\bf C})\in {\mathcal{I}}^*$ and ${\bf C}\in {\mathcal{C}}^*$}] \times p(\phi({\bf C}))\,.$$
Notice that, for any ${\bf C}\in{\mathcal{C}}$ and ${\bf I}\in S_{{\bf C} }$,
\[\Pr[\mbox{${\bf I}\in {\mathcal{I}}^*$ and ${\bf C}\in {\mathcal{C}}^*$}] \geq \frac{1}{k}\left(1-\frac{1}{k}\right)^{k-1}\geq \frac{1}{k\mathrm{e}},\]
which implies that $\expect{}{\mbox{\sf OPT}({\mathcal{C}}^*, {\mathcal{I}}^*)} \geq \frac{1}{k \mathrm{e}}\cdot \mbox{\sf OPT}({\mathcal{C}}, {\mathcal{I}})$.
{\bf Derandomization:} First, note that we can assume that $k=O(\log m+\log n)$. Otherwise, we can use the result of \cite{AggarwalFMZ04,GuruswamiHKKKM05,BalcanBM08} (see \cite[Section 4]{BalcanBM08} for the result in a general setting) to obtain $O(\log m+ \log n)$ approximation algorithm for {\sf UUDP-MIN}, which will also be $O(k)$-approximation.
Now, assuming that $k=O(\log m+\log n)$, we follow the argument of Balcan and Blum~\cite{BalcanB07}. In particular, we observe that we need only $k$-wise independence among the events of the form ``${\bf I}\in {\mathcal{I}}^*$ and ${\bf C}\in {\mathcal{C}}^*$'', for any ${\bf I}$ and ${\bf C}$, in order to get the above expectation result. In this case, we can use the tools from Even et al \cite{EvenGLNV98} to derandomize the above algorithm while blowing up the running time by a factor of $2^{O(k)}=\operatorname{poly}(m, n)$. For more details, we refer the readers to \cite{BalcanB07}.
\subsection{Proof of Lemma \ref{lem:union of d dimensions}}\label{proof:union of d dimensions}
Recall that each $A_i$ is an antichain, i.e., for any distinct ${\bf I}_1, {\bf I}_2\in A_i$, there exists $1\leq d_1, d_2\leq d$ such that ${\bf I}_1[d_1]<{\bf I}_2[d_1]$ and ${\bf I}_1[d_2]>{\bf I}_2[d_2]$. In particular, if ${\bf I}_1={\bf I}^*$, then we have that for any ${\bf I}\in A_i$, there exists coordinate $j$ such that ${\bf I}[j]\geq {\bf I}^*$. This means that ${\bf I}\in A_i^j$. The lemma follows.
\subsection{Polynomial-Time Algorithm for $1$-{\sf UUDP-MIN}}\label{sec:one udp min}
We provide a polynomial-time algorithm for solving $1$-{\sf UUDP-MIN}. Let ${\bf I}_1,\ldots, {\bf I}_n$ be a sequence of items ordered non-increasingly by their coordinates. We can assume without loss of generality that their coordinates are different (by slightly perturbing their values), and we say that consumer ${\bf C}$ is at {\em level $j$} if her coordinate lies between ${\bf I}_{j-1}$ and ${\bf I}_j$. Notice that, for any consumer ${\bf C}$ at level $j$, we have $S_{{\bf C}} = \set{{\bf I}_1,\ldots, {\bf I}_j}$.
\begin{claim}
Let $p^*$ be an optimal price. Then we can assume that $p^*({\bf I}_1) \geq p^*({\bf I}_2) \geq \ldots \geq p^*({\bf I}_n)$.
\end{claim}
\begin{proof}
Suppose that $p^*({\bf I}_i) < p^*({\bf I}_j)$ for some $i <j$. Recall that ${\bf I}_i \geq {\bf I}_j$, so for each consumer ${\bf C}$ such that ${\bf C} \leq {\bf I}_j$, we know that ${\bf C}$ does not buy item ${\bf I}_j$ with respect to this solution. Thus, we can reduce $p^*({\bf I}_i)$ slightly, while maintaining the same revenue.
\end{proof}
The claim will ensure that consumers at level $j$ only buy item ${\bf I}_j$ but not any other items in $\set{{\bf I}_1,\ldots, {\bf I}_{j-1}}$, and this allows us to solve the problem by dynamic programming. For each $j=1,\ldots, n$, for each price $P \in \ensuremath{\mathbb R}$ we have a table entry $T[j,P]$ that keeps the maximum revenue achievable from consumers at levels $1,\ldots, j$ and items $\set{{\bf I}_1,\ldots, {\bf I}_j}$ where the price of ${\bf I}_j$ is set to $P$. Notice that it is easy to compute the profit from consumers at level $j$ if we know $p({\bf I}_j) = P$. Denote such value by $\gamma$. Then we have that $T[j,P] = \gamma + \max_{P' \geq P} T[j-1,P']$. Finally, we note that there are at most $|{\mathcal{C}}|$ possibilities of prices $P$ because one can assume without loss of generality that, for {\sf UUDP-MIN}, the prices always belong to $\set{B_{{\bf C}}}_{{\bf C} \in {\mathcal{C}}}$.
\section{{\sf QPTAS} for $2$-{\sf SMP}}\label{sec:2-SMP}
In this section, we show that {\sf QPTAS} for $2$-{\sf SMP}.
\subsection{Overview}
\danupon{This problem is harder than the highway pricing problem since it doesn't have a separator. For example, the log n approx of Balcan-Bum for Highway heavily relies on the separator. Similarly, Khaled's QPTAS also relies on the separator (once you remember the profile in the middle, you can solve two sides separately).}
We sketch the key ideas here and leave the details in next sections. First, consider the special case where each consumer has
budget $1$ or $2$ and each item must be priced either $0$ or $1$.
The exact optimal solution of this case can be found in
$n^{O(\log^2 m n)}$ time. We later
show how to extend the idea to the general cases, which turns out to
be easy for the case of highway problem but need a few more ideas
for the case of $2$-{\sf SMP}.
\paragraph{Algorithm for highway pricing problem reviewed:}
Let us first start with the highway pricing problem which can be
casted as a special case of $2$-{\sf SMP} where items are in the
form $(1,n), (2,n-1), \ldots, (n,1)$.
The main idea used in \cite{ElbassioniSZ07}, casted in our language
of ``partition tree'' (for convenience in explaining our $2$-{\sf SMP} algorithm later) is the following.\danupon{I removed [htb!] from the figure.}
\begin{figure
\centering \scalebox{1.2}[1.2]{\includegraphics{pic1}}
\caption{A partition tree}\label{fig:Partition_Tree}
\end{figure}
We first construct a balanced binary tree called a {\em partition tree} and denoted by
${\cal T}$. We define the vertical gridline in the middle to be a level-$0$ line, denoted by $\ell_r$, dividing the items equally to left and right sides. This line corresponds to the root node $r$ of the tree. We also assign the consumers whose consideration set contains items on both sides to the root node. Then we recursively define the subtrees on the subproblems on the two sides of line $\ell_r$ as shown in Figure~\ref{fig:Partition_Tree} until we reach the subproblem containing only one item. For any node $v \in {\mathcal T}$, let ${\mathcal{C}}_v$ be the set of consumers assigned to $v$, and $\ell_v$ be the line associated with node $v$.
Now we show a top-down recursive algorithm to solve this problem.
This algorithm can be converted to a dynamic program by working
bottom-up instead. At the root node $r$ of ${\cal T}$, we would like
to compute $f_r({\bf I}_{L, 1}, {\bf I}_{L, 2}, {\bf I}_{L, 3}, {\bf I}_{R,
1}, {\bf I}_{R, 2}, {\bf I}_{R, 3})$ which is defined to be the optimal
revenue that we can collect from consumers in ${\cal C}\setminus
{\cal C}_r$ when we price the items in such a way that ${\bf I}_{L,
1}$, ${\bf I}_{L, 2}$ and ${\bf I}_{L, 3}$ (${\bf I}_{R, 1}$, ${\bf I}_{R,
2}$ and ${\bf I}_{R, 3}$, respectively) are the first, second, and
third closest items on the left (respectively, right) of $\ell_r$
that have price $1$. To avoid long notation, let us denote
$\{{\bf I}_{L, 1}, {\bf I}_{L, 2}, {\bf I}_{L, 3}, {\bf I}_{R, 1}, {\bf I}_{R,
2}, {\bf I}_{R, 3}\}$ by $\Gamma_r$ and $f_r({\bf I}_{L, 1}, {\bf I}_{L,
2}, {\bf I}_{L, 3}, {\bf I}_{R, 1}, {\bf I}_{R, 2}, {\bf I}_{R, 3})$ by
$f_r(\Gamma_r)$.
If we can compute $f_r(\Gamma_r)$ for all $\Gamma_r$ then the
optimal revenue can be obtained via the following formula.
\begin{align}
\text{Optimal revenue} &=
\max_{\Gamma_r} f_r(\Gamma_r) + m_1(\Gamma_r) + 2m_2(\Gamma_r)\label{eq:highway}
\end{align}
where, for any node $v$, $m_1(\Gamma_v)$ is the number of consumers
in ${\cal C}_v$ whose consideration sets contain exactly one item in
$\Gamma_v$, and $m_2(\Gamma_v)$ is the number of consumers in ${\cal
C}_v$ with budget $2$ whose consideration sets contain exactly two
items in $\Gamma_v$. The point is that we can calculate the revenue
from consumers in ${\cal C}_r$ as $m_1(\Gamma_r) + 2m_2(\Gamma_r)$
and use $f_r(\Gamma_r)$ to compute the revenue obtained from the
rest of the consumers.
It is left to compute $f_r(\Gamma_r)$. Let $u$ and $v$ be the left
and right children of $r$, respectively. In order to compute
$f_r(\Gamma_r)$, we will compute $f_u(\Gamma_r, \Gamma_u)$ which is
the maximum revenue we can collect from consumers assigned to the
descendants of $u$ (excluding $u$) where $\Gamma_r$ is the set of
six items of price $1$ that are closest to $\ell_r$ as defined
earlier. And, similarly, $\Gamma_u=\{{\bf I}'_{L, 1}, {\bf I}'_{L, 2},
{\bf I}'_{L, 3}, {\bf I}'_{R, 1}, {\bf I}'_{R, 2}, {\bf I}'_{R, 3}\}$ is the
set of six items of price $1$ that are closest to $\ell_u$.
Moreover, we require that $\Gamma_u$ must be {\em consistent} with
$\Gamma_r$ in the sense that there is some price assignment such
that items in $\Gamma_u$ are the items closest to $\ell_u$ of price $1$
and items in $\Gamma_r$ are the items closest to $\ell_u$ of price $1$ as
well.
(For example, if we let $\Gamma_r=\{{\bf I}_{L, 1}, {\bf I}_{L, 2},
{\bf I}_{L, 3}, {\bf I}_{R, 1}, {\bf I}_{R, 2}, {\bf I}_{R, 3}\}$ then an
item ${\bf I}$ with property ${\bf I}_{L, 3}[1]<{\bf I}[1]<{\bf I}_{L,
2}[1]$ cannot be in $\Gamma_u$ since this item must have price $0$.)
We use $\Gamma_u\bowtie\Gamma_r$ to denote ``$\Gamma_u$ is
consistent with $\Gamma_r$''. We define $f_v(\Gamma_r, \Gamma_v)$ in
a similar way.
Once we have $f_u(\Gamma_r, \Gamma_u)$ and $f_v(\Gamma_r, \Gamma_v)$
for all $\Gamma_u\bowtie\Gamma_r$ and $\Gamma_v\bowtie\Gamma_r$, we
can compute $f_r(\Gamma_r)$:
\begin{align}
f_r(\Gamma_r) & = \max_{\Gamma_u\bowtie\Gamma_r} \left\{f_u(\Gamma_r, \Gamma_u) + m_1(\Gamma_u) + 2m_2(\Gamma_u)\right\} + \max_{\Gamma_v\bowtie\Gamma_r} \left\{f_v(\Gamma_r, \Gamma_v) + m_1(\Gamma_v) + 2m_2(\Gamma_v)\right\} \,.\label{eq:highway_recurse}
\end{align}
The main point here is that there is no consistency requirement between $\Gamma_u$ and $\Gamma_v$ so we have two independent subproblems. We define the function $f_z$, for all nodes $z$ in ${\cal T}$ similarly: Let $r=v_0$, $v_1$, $v_2$, ..., $v_{q-1}$ be the ancestors of $z$ and $v_q=z$. We have to compute $f_z(\Gamma_{v_0}, \Gamma_{v_1}, \ldots, \Gamma_{v_q})$ for all $\Gamma_{v_0}, \Gamma_{v_1}, \ldots, \Gamma_{v_q}$ such that $\Gamma_{v_i}\bowtie \Gamma_{v_j}$ for all $i\neq j$.
The computation of $f_z(\Gamma_{v_0}, \Gamma_{v_1}, \ldots,
\Gamma_{v_q})$ is done in the same way as Eq.\eqref{eq:highway_recurse} for every non-leaf node $z$. At leaf node $z$, $f_z(\Gamma_{v_0}, \Gamma_{v_1}, \ldots,
\Gamma_{v_q})$ can also be easily computed: $f_z(\Gamma_{v_0},
\Gamma_{v_1}, \ldots, \Gamma_{v_q})=m_1(\Gamma_z)+2m_2(\Gamma_z)$.
Observe that $q=O(\log m + \log n)$ and there are $n^{6}$
choices for each $\Gamma_{v_i}$. Therefore, we can precompute
$f_z(\Gamma_{v_0}, \Gamma_{v_1}, \ldots, \Gamma_{v_q})$ for all
$n^{O(\log m + \log n)}$ combinations of $\Gamma_{v_0},
\Gamma_{v_1}, \ldots, \Gamma_{v_q}$. By working bottom-up from the
leaf nodes, the running time becomes $\operatorname{poly}(m)n^{O(\log m + \log n)}$.
\paragraph{Algorithm for $2$-{\sf SMP} (special case):} To solve the special case of $2$-{\sf SMP} defined above, we need to modify a few definitions in a right way.
Let us again consider the top-down algorithm and start at the root node $r$ of the partition tree ${\cal T}$. (Recall that we can assume that there is at most one item in each row and column so we can still define the paritition tree by drawing the vertical line through the point in the middle when sorted by the first dimension.)
One problem immediately appears: $f_r(\Gamma_r)$ cannot be used to compute the optimal revenue as we did in Eq.\eqref{eq:highway}. The reason is that we cannot compute the revenue from ${\cal C}_r$ using $m_1(\Gamma_r) + 2m_2(\Gamma_r)$ anymore.
To fix this, we have to redefine ${\cal C}_r$ in the following way:
We assign all consumers lying on the left (respectively, right) of
${\bf I}_r$ to the left (respective, right) child and keep only those
consumers lying exactly on the vertical line going through ${\bf I}_r$
in ${\cal C}_r$.
Now we can compute the revenue from the newly defined ${\cal C}_r$
and a function that computes the total revenue. To do this, we
define $f_r({\bf I}_1, {\bf I}_2, {\bf I}_3)$ to be the total revenue we
can get from consumers in ${\cal C}\setminus{\cal C}_r$ by pricing
the items in such a way that, among the items on the right side of
${\bf I}_r$, items ${\bf I}_1$, ${\bf I}_2$, and ${\bf I}_3$ are the items
with price $1$ that have the highest, second highest, and third
highest values in the second dimension, respectively. Again, let
$\Gamma_r$ denote a possible choice of $\{{\bf I}_1, {\bf I}_2,
{\bf I}_3\}$ and write $f_r(\Gamma_r)$ instead of $f_r({\bf I}_1,
{\bf I}_2, {\bf I}_3)$. If we can compute $f_r(\Gamma_r)$ then we can
get the optimal revenue by Eq.\eqref{eq:highway},
where $m_1(\Gamma_r)$ and $m_2(\Gamma_r)$ is as defined earlier (with the new definition of ${\cal C}_r$).
Some more complications lie in computing $f_r(\Gamma_r)$, for any
$\Gamma_r$. As before, we will compute $f_u(\Gamma_r, \Gamma_u)$ and
$f_v(\Gamma_r, \Gamma_v)$ where $u$ and $v$ are the left and right
children of $r$, respectively. Howerver, we have to carefully define
$f_u(\Gamma_r, \Gamma_u)$ and $f_v(\Gamma_r, \Gamma_v)$, in a
different way.
We define $f_u(\Gamma_r, \Gamma_u)$, for any $\Gamma_u=\{{\bf I}_1,
{\bf I}_2, {\bf I}_3\}$, to be the maximum revenue from the consumers
assigned to the descendants of $u$ when we price the items in such a
way that, among the items lying on the right side of ${\bf I}_u$ and
left side of ${\bf I}_r$, items ${\bf I}_1$, ${\bf I}_2$, and ${\bf I}_3$
are the items with price $1$ that have the highest, second highest,
and third highest values in the second dimension, respectively. Note
that we do not need to check any consistency between $\Gamma_r$ and
$\Gamma_u$: For any choice of $\Gamma_r$ and $\Gamma_u$, there is
always a price assignment such that items in $\Gamma_r$ and
$\Gamma_u$ are the items of price $1$ that have the highest values
in the second dimension in their respective regions. In this case,
we say that $\Gamma_r\bowtie \Gamma_u$ is always true for any
$\Gamma_r$ and $\Gamma_u$.
On the other hand, we define $f_v(\Gamma_r, \Gamma_v)$, for any
$\Gamma_v=\{{\bf I}_1, {\bf I}_2, {\bf I}_3\}$, to be the maximum revenue
from the consumers assigned to the descendants of $v$ when we price
the items in such a way that, among the items lying on the right
side of ${\bf I}_v$, items ${\bf I}_1$, ${\bf I}_2$, and ${\bf I}_3$ are the
items with price $1$ that have the highest, second highest, and
third highest values in the second dimension, respectively. In this
case, we have to make sure that $\Gamma_v$ is consistent with
$\Gamma_r$, i.e., there is some price assignment such that items in
$\Gamma_r$ and $\Gamma_u$ are the items of price $1$ that have the
highest values in the second dimension in their respective regions.
Now we have defined $f_u(\Gamma_r, \Gamma_u)$ and $f_v(\Gamma_r,
\Gamma_v)$, we compute $f_r(\Gamma_r)$ using
Eq.\eqref{eq:highway_recurse}.
As in the case of the highway pricing problem, we can extend the
definition to other nodes. In particular, at a leaf node $z$ we have
to compute $f_z(\Gamma_{v_0}, \Gamma_{v_1}, \ldots, \Gamma_{v_q})$
where $q=O(\log m + \log n)$. Hence, this case can be solved in
$\operatorname{poly}(|{\cal C}|)\cdot\size{\cal I}^{\polylog{|{\cal I}|}}$ time.
\danupon{To do: try not to use $m$ and $n$.}
\paragraph{Algorithm for general $2$-{\sf SMP}:}
We now remove the restrictions that each item must be priced $0$ or
$1$ and each budget must be $1$ or $2$. The removal of the
restriction on item price does not affect the case of highway
pricing problem since this can be easily assumed (see, e.g.,
\cite{GrandoniR10}).\danupon{Actually, can we assume this? In
\cite{GrandoniR10}, they also assume this for tollbooth problem on
trees (Section 4.1).}
Moreover, we can still assume that the maximum budget is $O(m n)$. Now we can deal with the general highway problem by
redefining $f_r(\Gamma_r)$: Let $\Gamma_r=\{{\bf I}_{L,0}, {\bf I}_{L,
1}, \ldots, {\bf I}_{L, q}, {\bf I}_{R,0}, {\bf I}_{R, 1},
\ldots,{\bf I}_{R, q}\}$ where $q=O(\log m n)$.
For any $i\leq q$, we want to price in such a way that ${\bf I}_{L,
i}$ is the item closest to ${\bf I}_r$ on the left such that the sum
of the price of all items between ${\bf I}_r$ and ${\bf I}_{L, i}$ is at
least $(1+\epsilon)^i$. Computing $f_r(\Gamma_r)$ can be done in the
same manner as before and consistency checking is easy to deal with.
Function $f_{v_q}(\Gamma_{v_0}, \Gamma_{v_1}, ..., \Gamma_{v_q})$,
for any node $v_q$ at level $q$ in $\cal T$, can be defined in a
similar manner.
For $2$-{\sf SMP}, we may not in general assume the item prices to
be $0/1$. Instead, we show that it can be assumed that each item
must have price $0$, or $(1+\epsilon)^j$, for any $j=0, 1, \ldots,
O(\log m)$. A natural extension of the above idea is to
define the notion of ``volume of regions'': For each item ${\bf I}$,
let $H_{{\bf I}}$ and $V_{{\bf I}}$ denote the horizontal and vertical
line cutting through item ${\bf I}$, respectively. Any rectangle
resulting from drawing some horizontal and vertical lines through
some items are called {\em regions} and the regions that do not
contain other regions are called {\em minimal regions}. For any
price assignment, we define the {\em volume} of a region to be the
sum of the price of all items within the region.
\begin{figure
\centering \scalebox{0.7}[0.7]{\includegraphics{pic2}}
\caption{Approximating the revenue from consumer ${\bf C}$ assigned to node $z$ in $\mathcal{T}$.}\label{fig:area-idea}
\end{figure}
Now, similar to the highway problem, we define $\Gamma_r=\{{\bf I}_0,
{\bf I}_1, ..., {\bf I}_k\}$ (note that $k=O(\log m)$) as the
``region guess'': We define $f_r(\Gamma_r)$ to be the maximum
revenue from ${\cal C}\setminus {\cal C}_r$ when we price in such a
way that, for any $i$, item ${\bf I}_i$ is the highest item (in the
second dimension) such that the volume of the region on the right of
the vertical line $V_{{\bf I}_r}$ and above the horizontal line
$H_{{\bf I}_i}$ (including ${\bf I}_i$) is at least $(1+\epsilon)^i$.
Using these volume guesses, we can approximate the upper and lower
bounds of the revenue from each consumer ${\bf C}$ at node $z$ by
looking at $\Gamma_v$ for all ancestors $v$ of $z$. This is because
each consumer's consideration set will contain some set of regions
$B_1, B_2, ...$ with volume guesses $(1+\epsilon)^{i_1},
(1+\epsilon)^{i_2}, ...$, respectively (such as the blue regions in
Figure~\ref{fig:area-idea}). Also, this consideration set will also
be contained in some set of regions $R_1, R_2, ...$ with volume
guesses $(1+\epsilon)^{i_1+1}, (1+\epsilon)^{i_2+1}, ...$ (such as
the blue and red regions together in Figure~\ref{fig:area-idea}).
However, in contrast to the case of highway problem, the consistency
between the guesses (e.g., between $\Gamma_r$ and its children
$\Gamma_u$ and $\Gamma_v$) is harder to guarantee. In order to
guarantee the consistency, we add another parameter, denoted by
$\Delta_r=\{\delta_0, \delta_1, \ldots, \delta_{O(k^2)}\}\subseteq
R_{\geq 0}^{O(k^2)}$ (recall that $|\Gamma_r|=k+1$). $\Delta_r$ is
used as a ``volume guess''. That is, we define $f_r(\Gamma_r,
\Delta_r)$ to be the maximum revenue from ${\cal C}\setminus {\cal
C}_r$ when we price in such a way that the restriction on $\Gamma_r$
is as before and, additionally, the volumn of the $i$-th minimal
region is exactly $\delta_i$ (where we make any order of the minimal
regions). We can now guarantee the consistency by making sure that
the sum of the volume guesses in smaller regions defined by
$\Gamma_u$ and $\Delta_u$ (as well as $\Gamma_v$ and $\Delta_v$) is
exactly the volume guesses in the bigger regions defined by
$\Gamma_r$ and $\Delta_r$. \danupon{Need to make this part much more
precise later. Also, we have to make it very clear why we need the
volume guesses, not region guesses alone.}
For any node $z$, we also define a function $f_z(\Gamma_{v_0},
\Gamma_{v_1}, \ldots, \Gamma_{v_q}, \Delta_z)$ where $v_0, v_1,
\ldots, v_{i-1}$ are ancestors of $z$ and $v_q=z$. In this case, we
consider the minimal regions obtained by drawing vertical lines
$V_{{\bf I}_{v_0}}, V_{{\bf I}_{v_1}}, \ldots, V_{{\bf I}_{v_q}}$ and
horizontal lines $H_{{\bf I}}$ for ${\bf I}\in \Gamma_{v_i}$, for all
$i$. We use $\Delta_z$ to store the numbers that are the ``volume
guesses'' of all these regions. We also check the consistency in
terms of volume, i.e., $\Pi=\{\Gamma_{v_0}, \Gamma_{v_1}, \ldots,
\Gamma_{v_q}, \Delta_{v_q}\}$ is consistent with
$\Pi'=\{\Gamma_{v_0}, \Gamma_{v_1}, \ldots, \Gamma_{v_{q-1}},
\Delta_{v_{q-1}}\}$ if the volume guesses of the smaller regions
defined by lines in $\Pi$ add up to the volume guesses of the bigger
regions defined by lines in $\Pi'$.
\subsection{Preprocessing}
Fix some $\epsilon >0$. Given an instance $({\mathcal{I}}, {\mathcal{C}})$, our goal
is to compute a price that collects a revenue of at least
$(1-O(\epsilon)) \mbox{\sf OPT}$. Recall that we can assume that the consumers
are on the intersection of grid lines, and the items are in the grid
cells (cf. Lemma~\ref{lem:perturb}). First we process the input so
that the budgets and prices are polynomially bounded. Moreover, the
optimal solution only assigns prices of the form $(1+\epsilon)^j$
for some $j \leq O(\log m)$. The proof of this fact only uses
standard arguments (along the same line as in \cite{BalcanB07}).
\begin{lemma}\label{lem:preprocess}
Let $M= O(mn /\epsilon)$. The input instance ${\cal P}$ can be
reduced to ${\cal P'}$ with the following properties.
\squishlist
\item For each consumer ${\bf C}$, the budget of ${\bf C}$ in ${\cal P}'$ is between $1$ and $M$.
\item Any price $p'$ that $\alpha$-approximates the optimal pricing of ${\mathcal{P}}'$ can be transformed in polynomial time into another price $p$ that gives $(1+3\epsilon) \alpha$-approximation for ${\mathcal{P}}$.
\item There is a $(1+\epsilon)$-approximate solution $\tilde{p}$ satisfying the following: For all ${\bf I} \in {\mathcal{I}}$, $1 \leq \tilde{p}({\bf I}) \leq M$, and $\tilde{p}({\bf I})$ is in the form $(1+\epsilon)^j$ for some $j \leq O(\log m)$.
\end{itemize
\end{lemma}
\begin{proof
Let $B_{\max}$ be the maximum budget among all consumers. We first
remove all consumers whose budgets are less than $\epsilon
B_{\max}/mn$. Notice that we only lose the revenue of at most
$\epsilon B_{\max} \leq \epsilon \mbox{\sf OPT}$ by this removal. We denote
the new set of consumers by ${\mathcal{C}}'$. Now look at the optimal price
$p^*$ for the resulting instance. If for some ${\bf I} \in {\mathcal{I}}$,
the price $p^*({\bf I})$ is less than $\epsilon B_{\max}/mn$, we
change its price to $p'({\bf I}) = 0$ and remove item ${\bf I}$
completely from the instance. Again, since each such item can only
be sold to at most $m$ consumers, discarding it only decreases the
revenue by $\epsilon B_{\max}/n$. There are at most $n$ such items,
so we lose a revenue of at most $\epsilon \mbox{\sf OPT}$ in total. Let
${\mathcal{I}}'$ denote the resulting set of items.
Next we scale each consumer budget by $M'=mn/\epsilon B_{\max}$ to
get a new budget, i.e. $B'_C = M'B_C$. Now we have a complete
description of the instance ${\mathcal{P}}'$ in which consumer budgets are
between $1$ and $M$. Let $\mbox{\sf OPT}'$ be the optimal value of the new
instance. First we try to lower bound the value of $\mbox{\sf OPT}'$. Consider
the same price $p^*: {\mathcal{I}}' \rightarrow \ensuremath{\mathbb R}$ scaled up by a factor of
$M'$. The revenue from this price is at least $(1 - 2\epsilon)
M'\mbox{\sf OPT}$, so we have that $\mbox{\sf OPT}' \geq (1-2\epsilon)M'\mbox{\sf OPT}$.
We are now ready to prove the second part. Assume that we have a
price $p'$ that gives $\alpha$-approximation for ${\mathcal{P}}'$, so the
revenue collected by $p'$ is at least $\mbox{\sf OPT}'/\alpha$. We construct
the price $p$ by scaling down the price of $p'$ by $M'$. Notice that
for each consumer ${\bf C}$ who can afford his consideration set in
${\mathcal{P}}'$ with price $p'(S_C)$, he can also afford his set in ${\mathcal{P}}$
with price $p'(S_C) = p(S_C)/M'$. Therefore, the revenue collected
by $p$ is at least $\mbox{\sf OPT}'/\alpha M' \geq (1-2\epsilon)\mbox{\sf OPT}/ \alpha$.
This argument also implies that $\mbox{\sf OPT} \geq \mbox{\sf OPT}'/M'$.
Finally we show that there is a good solution $\tilde p$ that only
assigns prices in the form $(1+\epsilon)^j$, as follows. We round
down the price of $p^*$ to the nearest scale of $(1+\epsilon)^j$ for
some $j$. For each consumer ${\bf C}$ who purchases item ${\bf I}$
w.r.t. price $p^*$, by scaling down every item price, she can still
afford her consideration set $S_{{\bf C}}$, whose new price is at
least $p^*(S_{Consumer})/(1+\epsilon) \geq (1-\epsilon)
p^*(S_{{\bf C}})$.
\end{proof}
From now on, we assume that our input instance and its optimal price
are in such format. Our goal is to devise a {\sf QPTAS} for this
instance. We note here that in some special cases of single-minded
pricing problems, especially the Highway problem, an even stronger
statement can be assumed; namely, that the optimal price is
integral~\cite{GuruswamiHKKKM05}. It seems that such a nice property
may not hold in our case, and we anyway do not need it.
\subsection{Partition tree}
We first construct a (almost balanced) binary tree ${\mathcal T}$ where
each node in ${\mathcal T}$ is associated with a rectangular region in the
plane (from now on, whenever we talk about region, we always mean a
rectangular one). We call this tree the {\em partition tree}. It can
be constructed recursively as follows. In the beginning, we have
${\mathcal T}=\set{r}$ where $r$ is the root of the tree whose region $A_r$
is the whole grid. We repeat the following process: For each leaf $v
\in {\mathcal T}$, if the region $A_v$ of $v$ contains at least two items,
we choose a vertical grid line $\ell_v$ dividing the items in a
balanced manner to the left and right side. We then add the left
child $v'$ of $v$ with the region $A_{v'}$ being the region of $A_v$
on the left side of $\ell_v$. We also add the right child $v''$ of
$v$ associated with the region $A_{v''}$ on the right side of
$\ell_v$. We repeat the process until every leaf is associated with
a region containing only one item; see Figure~\ref{figure: tree}.
\danupon{It's better to define ``volume'' later because we need to
define ``region'' with horizontal lines first.} For each node $v \in
{\mathcal T}$, we define the item set ${\mathcal{I}}_v$ to be the set of all items
in the region $A_v$. Fix a price $p: {\mathcal{I}} \rightarrow \ensuremath{\mathbb R}$. For any
region $A$, we define the ``volume'' $\mathsf{vol}_p(A)$ to be the total
price among all items in the region, i.e. $\mathsf{vol}_p(A) = \sum_{{\bf I}
\in A} p({\bf I})$. The following simple claim
is crucial in designing our algorithm.
\begin{claim}\label{claim:bound_sum}
Let $p^*$ be an optimal price. Then for any region $A$, there are
only $n^{O(\log m)}$ possible values of $\mathsf{vol}_{p^*}(A)$.
\end{claim}
\begin{proof
Let $x_j$ denote the number of items ${\bf I}$ in $A$ with price
$p^*({\mathbf I}) = (1+\epsilon)^j$. Notice that we can write the
volume of $A$ as $\sum_{j=1}^{q} x_j (1+\epsilon)^j$ where $x_j$
only takes non-negative integer values at most $n$. So we have at
most $n^{O(\log m)}$ possibilities for the volume.
\end{proof}
\subsection{Horizontal partition and local profile}
From the construction, each node $v$ of the partition tree, is
associated with a vertical line $\ell_v$ which divides the plane
into two region. We further partition the right region using
vertical line, as follows.
Consider a non-leaf node $v \in {\mathcal T}$ with left child $v'$ and
right child $v''$. A {\em horizontal partition} for node $v$,
denoted by $H_v$, is a collection of (not-necessarily distinct)
horizontal lines $\ell^v_1,\ldots, \ell^v_q$, partitioning the
region of $A_{v''}$ into many pieces; note that the left endpoints
of these lines are on $\ell_v$.\danupon{Need a picture here.} The
line $\ell^v_j$ is supposed to mark the highest $y$-coordinate such
that the volume inside $A_{v''}$ above $\ell^v_j$ is at least
$(1+\epsilon)^j$. Notice that each node $v$ has at most $n^{O(\log
m)}$ feasible partitions since there are at most $n$ possibilities
for the choice of each $\ell^v_j$.
Now if we fix a horizontal partition of every non-leaf node in the
partition tree, we can define {\em minimal} regions for each
non-leaf node $v$ as follows. For each node $v$, we consider all
vertical and horizontal lines associated with $v$ and all its
ancestors (i.e., all lines in $\ell_u$ and $H_u$ where $u=v$ or $u$
is an ancestor of $v$). Let ${\mathcal{L}}_v$ denote the set of these lines.
${\mathcal{L}}_v$ naturally defines minimal regions: We say that a region
$A$ is minimal with respect to ${\mathcal{L}}_v$ if and only if $A$ is a
rectangle whose four boundaries are the lines in ${\mathcal{L}}_v$, and
there is no line in ${\mathcal{L}}_v$ that intersects with the interior of
$A$.
Now, we define a {\em local profile} of a node $v$. It consists of
(i) horizontal partitions for $v$ and for all its ancestors, and
(ii) numbers on every minimal region resulting from vertical and
horizonal lines. The numbers are supposedly the ``volume guesses''
of every minimal region of $v$.
Now we try to guess the ``right'' local profile of every node in the
partition tree. We show that if this guess is right, then we get a
good approximation of the optimal solution. Moreover, we can use
dynamic programming to make the right guess.
\subsection{Dynamic Programming Solution}
A {\em global profile} (or just {\em profile} in short) of a node
$v$ consists of the local profile of $v$ and all its ancestors in
such a way that the volumes of minimal regions of $v$ is consistent
with its ancestors. More formally, fix a node $v$. A profile $\Pi_v$
for $v$ consists of, for any ancestor $v'$ of $v$, $\Pi_{v, v'}$
which is the local profile that node $v$ wants its ancestor $v'$ to
have (we also think of $v$ has an ancestor of itself for
convenience).
As a reminder, for each ancestor $v'$ of $v$, local profile $\Pi_{v,
v'}$ some horizontal partition $H_{v'}$ and the ``volume guess''of
each minimal region of $v$.
In addition, we restrict that these local profiles $\Pi_{v, v'}$
have to be consistent in themselves in the following sense. For each
vertex $v'$, for any minimal region $A'$ of $\Pi_{v,v'}$ that is
further partitioned into minimal regions $A'_1, A'_2,\ldots,
A'_{\gamma}$ of $\Pi_{v,v''}$ for some descendant $v''$ of $v'$, the
number $z_{A'}$ at $\Pi_{v,v'}$ is equal to the sum of the numbers
$z'_{A_j}$ of $\Pi_{v,v''}$.\danupon{This is still not clear and a
bit informal.}
We argue that the number of global profiles for each node is not too
large, i.e. only $n^{\operatorname{poly} \log m}$. There are
$n^{O(\log m)}$ horizontal partitions for each ancestor $v'$ of $v$,
making a total of $n^{O(\log m \log n)}$ possibilities of the lines
$\ell^{v'}_j$. Now fix a choice of such horizontal partitions. If we
draw all lines $\ell^{v'}_j$ involved in the global profiles, we
will see a number of regions formed by intersections between these
lines and the vertical lines $\ell_{v''}$. Since there are $O(\log m
\log n)$ such horizontal lines and $O(\log n)$ vertical lines
involved, we have at most $O(\log m \log^2 n)$ minimal rectangular
regions. Each region has at most $n^{O(\log m)}$ possible volumes,
so there are at most $n^{O(\log^2 m \log^2 n)}$ global profiles for
each node $v$\parinya{We implicitly used the fact that $m \leq O(n)$. Have to say
it somewhere}.
Now we define a {\em valid tree profile} $\Pi$ for ${\mathcal T}$ as the
set of global profiles $\set{\Pi_v}_{v \in {\mathcal T}}$ such that $\Pi_v$
is a global profile for node $v$. Moreover, for every parent-child
pair $v, v'$ where $v$ is a parent of $v'$ in ${\mathcal T}$, the profile
$\Pi_{v'}$ agrees with $\Pi_v$. That is, all profiles about
ancestors of $v$ in $\Pi_v$ and $\Pi_{v'}$ are exactly the same.
Given a valid tree profile $\Pi$, we have the notion of cost of the
profile $\Pi$ (denoted by $\mbox{\sf Cost}(\Pi)$) which is supposed
to approximate the total revenue we can collect by a price function
consistent with $\Pi$. The cost of a profile can be computed as
follows. For each node $v \in {\mathcal T}$, let ${\mathcal{C}}_v$ be the set of
all consumers on line $\ell_v$. For each consumer ${\bf C} \in
{\mathcal{C}}_v$, the rectangular region enclosed by horizontal line ${\bf
C}[2]$ and vertical line $\ell_v$
is the actual amount the consumer needs to pay. This is the amount
we do not know, but we can approximate: We let $v_0, v_1,\ldots,
v_{\alpha}$ be a sequence of ancestors of $v$ such that $v$ is on
the left subtree of $v_i$ (in the order from $v$ to the root), where
$v_0= v$. And we let for each $i$, $j_i$ be the maximum number such
that $\ell^{v_i}_{j_i}$ does not lie below ${\bf C}[2]$. The cost of
consumer ${\bf C}$ is just the sum $\sum_{i=0}^{\alpha}
(1+\epsilon)^{j_i}$ if $B_C \leq \sum_{i=0}^{\alpha}
(1+\epsilon)^{j_i}$ and zero otherwise. The cost at node $v$ is just
the total cost of all consumers in ${\mathcal{C}}_v$, and the cost of the
profile is the sum of the cost over all nodes $v \in {\mathcal T}$.
\begin{figure}
\centering
\subfigure[Tree Decomposition up to depth $2$ where the shaded
region denotes $A_v$]{
\includegraphics[height=5cm]{pic3}\label{figure:
tree} }
\hspace{.05\textwidth}
\subfigure[The actual volume]{
\includegraphics[height=5cm]{pic4}
\label{fig:area1} }
\hspace{.05\textwidth}
\subfigure[The approximate volume]{
\includegraphics[height=5cm]{pic5}
\label{fig:area2}}
\caption{Computing the cost of consumer ${\bf C}$}\label{fig:area}
\end{figure}
\begin{lemma}\label{lem:exist_good_tree}
There is a valid tree profile $\Pi^*$ such that the cost is at least
$(1-\epsilon)\mbox{\sf OPT}$.
\end{lemma}
\begin{proof
We start from the optimal price $p^*$ and construct the valid
profile as follows. For each node $v$, we define a feasible
partition of $v$ by choosing the line $\ell^v_j$ to be at the
highest $y$-coordinate such that the total volume enclosed is at
least $(1+\epsilon)^j$. Then we create a profile $\Pi^*_v$ for each
node $v$ according to the actual volume of each minimal region.
Notice that this gives a valid tree profile.
\end{proof}
Our goal now is to compute the valid profile $\Pi$ of maximum cost
by dynamic programming, and the profile will automatically suggest a
near-optimal pricing.
\paragraph{Computing the Solution:} Let $v \in {\mathcal T}$.
We say that a price $p: {\mathcal{I}}_v \rightarrow \ensuremath{\mathbb R}$ is consistent with
global profile $\Pi_v$ if and only if for every minimal region $A$
of $\Pi_v$ that is completely contained in $A_v$, we have $\mathsf{vol}_p(A)
= z_A$. The minimum cost profile can be computed in a bottom-up
fashion, as follows. For a leaf node $v$, a global profile for $v$
automatically determines the price of the only item in $A_v$;
discard a profile which does not have consistent price.
The following lemma shows that a price $p$ consistent with a valid
tree profile $\Pi$ can be computed from $\Pi$.
\begin{lemma}\label{lem:consistent}
For each node $v$ with left child $v'$ and right child $v''$, let
$p':{\mathcal{I}}_{v'}\rightarrow \ensuremath{\mathbb R}$ and $p'':{\mathcal{I}}_{v''}\rightarrow \ensuremath{\mathbb R}$ be
the prices that are consistent with the profile $\Pi_{v'}$ and
$\Pi_{v''}$ respectively. Then the price $p: {\mathcal{I}}_v \rightarrow \ensuremath{\mathbb R}$
defined to agree with $p'$ on ${\mathcal{I}}_{v'}$ and with $p''$ on
${\mathcal{I}}_{v''}$, is consistent with $\Pi_v$.
\end{lemma}
\begin{proof
Consider a minimal region $A \subseteq A_v$ and a volume guess $z_A$ in
$\Pi_v$. If $A \subseteq A_{v'}$ where $A$ is the union of minimal
regions $A'_1,\ldots A'_{\gamma}$ of $\Pi_{v'}$ (similar argument
can be made in case $A \subseteq A_{v''}$), then by assumption that
$\Pi_v$ is consistent with $\Pi_{v'}$, we know that the total value
$z_A = \sum_{j=1}^{\gamma} z'_{A'_j}$. Since $p'$ is consistent with
the profile $\Pi_{v'}$, we have that $\mathsf{vol}_{p}(A) = \mathsf{vol}_{p'}(A) =
\sum_{j} \mathsf{vol}_{p'}(A'_j) = \sum_j z'_{A'_j} = z_A$ as desired.
\end{proof}
We have shown that a valid tree profile $\Pi$ always has a price $p$
consistent with it. The following lemma
basically says that this price $p$ gives a revenue close to the cost
of the profile, which will in turn imply that the maximum cost
profile gives the revenue of at least $(1-O(\epsilon)) \mbox{\sf OPT}$.
\begin{lemma}\label{lem:tree-revenue}
For any valid tree profile $\Pi$, let $p$ be a price consistent with
$\Pi$ and let $p' = p/(1+\epsilon)$. Then $p'$ collects revenue at
least $(1-\epsilon)$ fraction of the profile cost.
\end{lemma}
\danupon{Still need to prove this lemma.}
\section{{\sf QPTAS} for $2$-{\sf UUDP-MIN}}
\label{sec: bicriteria}\label{sec: qptas 2 udp}
We note that we will write $O(\log m)$ instead of $O(\log n+\log m)$ since we assume that $n\leq m$ in this paper. (Otherwise, we already have approximation ratio of $O(\log m)=O(\log n)$.)
\sodaonly{
We explain the main idea first. The intuition can be realized by solving the following simple case: Assume for now that we have $\Theta(n^2)$ items, which form a set $\set{(2i-1,2j-1): 1 \leq i,j \leq n}$. In this case it is
possible to have two different consumers at the same coordinate, i.e. ${\bf C}={\bf C}'$, while there is exactly one item at each point $(2i-1,2j-1)$. Assume further that each consumer has budget either $1$
or $2$. We show below how to solve this case in polynomial time.
Note that there is an optimal solution such that each item is
priced either $1$ or $2$: otherwise we could increase the price by small amount to collect more revenue.
Now, for any item point $(2i-1, 2j-1)$ and any price assignment
$p$, define\danupon{(to polish after submission) $r_p(i, j)$ may be confused with $r_{\bf C}(p)$.}
\[r_p(i, j):=\min_{\substack{{\bf I}[1]\ge 2i-1, {\bf I}[2]\geq 2j-1 \\ {\bf I}\in{\mathcal{I}}}}\{p({\bf I})\}
\]
to be the minimum price among the items
dominating $(2i-1, 2j-1)$. This quantity immediately tells us how much
revenue we will get from consumers at point $(2i-2, 2j-2)$: each consumer
will buy an item at price $r_p(i, j)$ if and only if she has budget
at least $r_p(i, j)$.
By the definition of $r_p$, we know that for any fixed value $j$, $r_p(i,j)$ is non-decreasing in terms of $i$.
In other words, for any pricing $p$ and integer $j$, there exists
a ``threshold'' $\gamma(p, j)$ such that $r_p(i', j)=1$ for all $i'\leq
\gamma(p, j)$ and $r_p(i',j)=2$ for all $i'> \gamma(p, x)$.
Additionally, for any $j$, $\gamma(p, j)\geq \gamma(p, j+1)$.
Using these observations, we are ready to define the dynamic programming table. The table entry $T[i,j]$ is defined to be the maximum revenue we can get among the price assignment $p$ such that $r_p(i', j)=1$ for all
$i'\leq i$ and $r_p(i', j)=2$ for all $i'>i$. The table $T$ can be
computed as follows.
\begin{align}
T[i, j]&=\max_{i'\leq i} \{T[i',j+1]+m_1(i',j) + 2m_2(i',j)\}
\label{eq:udp-min-table}
\end{align}
where $m_1(i',j)$ is the number of consumers of the form
$(2i''-2,2j-2)$ for $i''\leq i'$ with budget $1$ and $m_2(i',j)$ is the number of
consumers of the form $(2i''-2,2j-2)$ for $i'' > i'$ with budget $2$. Moreover, let $T[i,n+1]=0$ for all $i$.
The optimal solution is then $\max_i T[i,1]$.
The above discussion captures almost all the key ideas for solving the general $2$-{\sf UUDP-MIN} problem. To get a {\sf QPTAS} in the general case,
we extend these ideas in the following way.
\squishlist
\item Consider a slight generalization when there is only one item in each column and row of grid cells (cf.
Lemma~\ref{lem:perturb}) while each budget is still $1$ and $2$. In
this case, we cannot pick arbitrary value of $i'$ when we compute
$T[i, j]$ as in Eq.\eqref{eq:udp-min-table} since it might not
correspond to any pricing. Through some additional observations,
table $T$ can be computed as follows: Let ${\bf I}_j$ be the item whose $y$-coordinate is $j$. If $i={\bf I}_j[1]$ then we can use Eq.\eqref{eq:udp-min-table}; otherwise, $T[i,j]=T[i,j+1]+m_1(i,j)+2m_2(i,j)$. This algorithm runs in $O(n^3)$ time.
\item When there are $q$ different budgets, say $B_1, B_2, \ldots, B_q$,
we can solve the problem in $n^{O(q)}$ time. This is done
by defining $T[i_1,\ldots, i_{q-1}, j]$ to be the maximum
revenue we can get among the price assignment $p$ such that, for all
$q': 1 \leq q' \leq q$, $r_p(i',j)=B_{q'}$ for all $i_{q'-1}<i'\leq i_{q'}$ (where we let
$i_0=-1$ and $i_q=n$).
\item Finally, we obtain a {\sf QPTAS} by ``discretizing'' the prices so that there are not many choices of item prices (cf. Lemma~\ref{lem:uudp-discretize}). This enables us to assume that the prices are in $\Gamma=\{0, (1+\epsilon)^0, (1+\epsilon)^1, ..., (1+\epsilon)^q\}$ where $q=O(\log_{1+\epsilon} m)$, and we can get the algorithm running in time $n^{O(\size{\Gamma})}=n^{O(\log m n)}$.\danupon{I changed this slightly: I point to Lemma~\ref{lemma:discretization}.}
\end{itemize
}
\subsection{Preprocessing}
\fullonly{We need to do a preprocessing, which will be used in the next two sections to design {\sf QPTAS} for $2$-{\sf UUDP-MIN} and $2$-{\sf SMP}.}
The following lemma says that we can assume the input lies on the grid where each row and column of the grid contains exactly one item.
\begin{lemma}\label{lem:perturb}
We are given an instance $({\mathcal{C}},{\mathcal{I}})$ of $2$-{\sf UUDP-MIN}\fullonly{ (or $2$-{\sf SMP})}. Then we can, in polynomial time, transform $({\mathcal{C}}, {\mathcal{I}})$ into an ``equivalent'' instance $({\mathcal{C}}', {\mathcal{I}}')$ such that
\begin{itemize}
\item Each consumer ${\bf C}' \in {\mathcal{C}}'$ has even coordinates $(2i, 2j)$ for $0 \leq i,j \leq n$.
\item Each item ${\bf I}' \in {\mathcal{I}}'$ has odd coordinate $(2i-1, 2j-1)$ for $1 \leq i,j \leq n$.
\item For each odd number $2i-1$, $1 \leq i \leq n$, there is exactly one item ${\bf I}' \in {\mathcal{I}}'$ with ${\bf I}'[1]=2i-1$ and exactly one item ${\bf I}'$ with ${\bf I}'[2]=2i-1$.
\end{itemize}
\end{lemma}
\begin{proof}
We sweep the horizontal line from top to bottom, and whenever the
line meets the items ${\bf I'}_1,\ldots, {\bf I'}_z$ such that ${\bf
I}'_1[1] < {\mathbf I}'_2[1] < \ldots < {\mathbf I}'_z[1]$ with the
same $y$-coordinate $y'$, we break ties as follows. Let $\delta$ be
the vertical distance from the line to the next item point below the
line. We set the new $y$-coordinates of these items to ${\bf
I'}_j[2] = y'-(z-j)\delta/2 z$. Notice that some consumers whose
$y$-coordinates lie in $[y', y'-\delta)$ get affected by this move.
We also change the $y$-coordinates of those consumers to
$y'-\delta/2$. Then we add the horizontal grid lines between the
space of every consecutive items, while making sure that consumer
points are on the line passing $y-\delta/2$. It is easy to see that
this process preserves the consideration set of every consumer. We
repeat the above steps until the sweeping line passes the bottommost
item.
We do a similar sweep of vertical line from right to left, inserting
the grid lines along the way. In the end, each consumer lies on the
intersection of the grid lines and each item in its cell, which guarantees
that no two items appear in the same row or column of the grid.
\end{proof}
\subsection{Detail of {\sf QPTAS} for {\sf UUDP-MIN}}
First, we can make the following simple assumption.
\begin{lemma
\label{lem:uudp-discretize}
We can assume that the prices are in the form $(1+\epsilon)^0,
(1+\epsilon)^1, ..., (1+\epsilon)^q$ or zero where
$q=O(\log_{1+\epsilon} m)$ by sacrificing $(1+\epsilon)$ in the approximation factor.
\end{lemma}
\begin{proof
We use the following standard arguments. Consider an optimal price
$p^*$. For each item ${\bf I}_j$, if the price is non-zero, we round
down the price $p^*({\bf I}_j)$ to the nearest scale of
$(1+\epsilon)^{q'}$, so the price of each item gets decreased by at
most a factor of $(1+\epsilon)$. Consider a consumer ${\bf C}$ who
bought ${\bf I}_{j}$ with price $p^*$. After the rounding, she can
still afford ${\bf I}_j$, so we can still collect at least
$(1-\epsilon)p^*({\bf I}_j)$ from $C$.\danupon{Missing: We have to
first show that we can bound the maximum budget by $O(m)$.}
\end{proof}
Now, assuming that the optimal price $p^*$ has the above structure,
we show how to solve the problem in quasi-polynomial time. First, we
reorder the items based on their $y$-coordinates in descending
order, so we have ${\bf I}_1[2] > {\bf I}_2[2] > \ldots > {\bf
I}_n[2]$. A consumer ${\bf C}$ is said to belong to {\em level
$j$} if it lies between the row\danupon{We didn't define ``row'' before.} of ${\bf I}_j$ and
that of ${\bf I}_{j+1}$, so each consumer belongs to exactly one
level. Moreover, observe that a consumer ${\bf C}$ at level $j$ is
only interested in (a subset of) items in $\set{{\bf I}_1,\ldots,
{\bf I}_j}$ (since ${\bf I}_{j'}[2]<{\bf C}[2]$ for any $j'>j$).
We define a subproblem ${\mathcal{P}}_j$ as the pricing problem with items
$\set{{\bf I}_1,\ldots, {\bf I}_j}$ and consumers at levels
$1,\ldots, j$. We use the dynamic programming technique to solve this
problem.
\paragraph{Profiles} We will remember the profile for each
subproblem ${\mathcal{P}}_j$. A profile $\Pi$ of ${\mathcal{P}}_j$ consists of
$O(\log m)$ item indices $\pi_1,\ldots, \pi_q \in \set{1,\ldots,
j}$. Each value $\pi_i$ is supposed to tell us the index of the item ${\bf I}$
of price $(1+\epsilon)^i$ with maximum value ${\bf I}[1]$. That is, we say
that a price $p$ for ${\mathcal{P}}_j$ is {\em consistent} with profile
$\Pi =(\pi_1,\ldots, \pi_q)$ if, for each $i$, the item ${\bf I}_{\pi_i}$ has the highest value in the first coordinate among the items with price at most $(1+\epsilon)^i$, i.e., for all
$i$,
$$\pi_i=\arg\max_{j'} \{ {\bf I}_{j'}[1]\ |\ p({\bf I}_{j'})\leq (1+\epsilon)^i\}\,.$$
Since $\{ {\bf I}_{j'}\ |\ p({\bf I}_{j'})\leq (1+\epsilon)^i\}\subseteq
\{ {\bf I}_{j'}\ |\ p({\bf I}_{j'})\leq (1+\epsilon)^{i+1}\}$ for any
$i$,
$${\bf I}_{\pi_1}[1] \leq {\bf I}_{\pi_2}[1] \leq \ldots \leq {\bf
I}_{\pi_q}[1]\,.$$
Observe that if two prices $p'$ and $p''$ have the
same ${\mathcal{P}}_j$ profile, then consumers at level $j$ see no
difference between these two prices, as shown formally by the
following lemma. We say that an item ${\bf I}_k$ is a profile item for profile $\Pi=(\pi_1,\ldots, \pi_q)$ if and only if $k = \pi_{q'}$ for some $q' \in [q]$.
\begin{lemma}
\label{lemma:reconstruction}
Let $\Pi$ be a profile for subproblem ${\mathcal{P}}_j$, and let $p$ be any
price function for ${\mathcal{P}}_j$ that is consistent with profile $\Pi$. Then we
can assume without loss of generality that every consumer at level $j$ only purchases
profile items.
\end{lemma}
\begin{proof
Suppose that a consumer ${\bf C}$ buys an item ${\bf I}$ in ${\mathcal{I}}$
with $p({\bf I}) =(1+\epsilon)^{q'}$ which is not a profile item.
Then consider the profile item ${\bf I}_{\pi_{q'}}$, which satisfies ${\bf I'}[1] \geq {\bf I}[1]$, so we must have ${\bf I}_{\pi_{q'}} \in S_{{\bf C}}$. We can
therefore assume that consumer ${\bf C}$ buys ${\bf I}_{\pi_{q'}}$ instead
of ${\bf I}$.
\end{proof}
Let $\Pi=(\pi_1,\ldots, \pi_q)$ be a profile for ${\mathcal{P}}_{j}$ and
$\Pi'= (\pi'_1,\ldots, \pi'_q)$ be a profile for ${\mathcal{P}}_{j-1}$. We
say that $\Pi$ is {\em consistent} with $\Pi'$ if for any price
$p':\set{{\bf I}_1,\ldots, {\bf I}_{j-1}} \rightarrow \ensuremath{\mathbb R}$ that is
consistent with $\Pi'$, we can extend $p'$ to $p$ by assigning value
$p({\bf I}_j)$ such that $p$ is consistent with $\Pi$. Notice that
consistency between any two profiles can be checked in polynomial
time by trying all $q$ possibilities of prices.
We recall that we use $p^*$ to denote the optimal price.
\begin{lemma}\label{lem:udp-consistency}
There are profiles $\Pi^1,\ldots, \Pi^n$ for ${\mathcal{P}}_1,\ldots,
{\mathcal{P}}_n$ respectively such that for each $j\in\{1,\cdots, n-1\}$,
$\Pi^j$ is consistent with $\Pi^{j+1}$. Moreover, all such profiles
are consistent with price $p^*$.
\end{lemma}
\begin{proof
For each subproblem ${\mathcal{P}}_j$, we define the profile
$\Pi^j=(\pi^j_1,\ldots, \pi^j_q)$ based on the price $p^*$ (there is
only one possible profile consistent with $p^*$). It is clear that
$\Pi^j$ is always consistent with $\Pi^{j+1}$.
\end{proof}
\paragraph{Dynamic Programming Table} For each $j=1,\ldots, n$ and for each profile $\Pi$ of ${\mathcal{P}}_j$, we use a table entry $T(j,\Pi)$ to store the maximum revenue achievable among the price function for ${\mathcal{P}}_j$ that is consistent with the profile $\Pi$. Since there are $n^{O(\log m)}$ possibilities for the profile $\Pi$, the table size is $n^{O(\log m)}$. We now show the computation of the table. To compute $T(j,\Pi)$, we recall that given the profile $\Pi$, the revenue from consumers at level $j$ can be computed efficiently. Denote such revenue by $r_j(\Pi)$. The following equation holds:
\[T(j,\Pi) = r_j(\Pi) + \max_{\Pi' \mbox{ consistent with } \Pi} T(j-1, \Pi') \]
\paragraph{Computing the Solution} For each table entry $T(j,\Pi)$, we can keep track of the profile $\Pi'$ such that $T(j-1, \Pi')$ is the entry that is used to compute $T(j,\Pi)$. Let $T(n, \Pi)$ be the entry that contains the maximum value over all $\Pi$. The value in this entry represents the revenue we can get from the optimal pricing $p^*$, so it is enough to reconstruct the price function $p^*$. We first obtain a sequence of profiles $\Pi^1,\ldots, \Pi^n = \Pi$ such that $\Pi^j$ is a profile for ${\mathcal{P}}_j$ and that $\Pi^j$ is consistent with $\Pi^{j-1}$ for any $j =1,\ldots, n$. This sequence allows us to reconstruct a price function that is consistent with all the profiles in polynomial time.
\section{Open Problems}\label{sec:conclusion}
Several interesting problems are open. The most important problem is whether we can obtain better approximation factors for $d$-{\sf UUDP-MIN} and $d$-{\sf SMP}.
We tend to believe that there is an $f(d)$-approximation algorithm for $d$-{\sf UUDP-MIN} and $d$-{\sf SMP} where $f(d)$ is a function that depends on $d$ only. However,
it seems to be a very challenging task to obtain approximation ratio like $\log^{O(d)} n$ or $O_d(\log^{1-\epsilon(d)} m)$, for some constant $\epsilon(d)>0$ depending on $d$.
\fullonly{
\begin{openproblem}
\label{conjecture: main one}
For any integer $d >0$, are there $\log^{O(d)} n$-approximation algorithms for $d$-{\sf UUDP-MIN} and $d$-{\sf SMP}?
\end{openproblem}
\begin{openproblem}
\label{conjecture: main two}
For any integer $d >0$, are there $O_d(\log^{1-\epsilon_d} m)$-approximation algorithms for $d$-{\sf UUDP-MIN} and $d$-{\sf SMP}?
\end{openproblem}
}
One promising direction in attacking the above problems is to improve Theorem~\ref{thm: dimension reduction for UDP}, e.g., getting $O_d(\rho\cdot\mathrm{polylog}(n))$ for $d$-{\sf UUDP-MIN} using a $\rho$-approximation algorithm of $(d-1)$-{\sf UUDP-MIN} as a blackbox.
\fullonly{
\begin{openproblem}
For any constant $d >0$, given a $\rho$-approximation algorithm for $(d-1)$-{\sf UUDP-MIN} (and $(d-1)$-{\sf SMP}), is it possible to get an $O_d(\rho\cdot\polylog n)$ approximation algorithm for $d$-{\sf UUDP-MIN} (and $d$-{\sf SMP})?
\end{openproblem}
}
A positive resolution to this problem would imply $(\log^{O(d)} n)$-approximation algorithm for $d$-{\sf UUDP-MIN}. We believe that, even resolving this problem would require some new insights on geometric and poset structures.
There are two special cases that can be thought of as barriers in dealing with standard versions of {\sf SMP} and {\sf UUDP-MIN}, and we believe that these two special cases serve as good starting points in attacking our problems. The first problem is the geometric version of the Maximum Expanding Subsequence ({\sc Mes}) problem which is the key problem to show the hardness of {\sf UUDP-MIN}~\cite{BriestK11}. The second problem is the Unique Coverage problem \cite{DemaineFHS08} when the sets have constant VC-dimension. Another interesting problem is to obtain {\sf PTAS}s for $2$-{\sf UUDP-MIN} and $2$-{\sf SMP} (e.g., by extending the techniques in \cite{GrandoniR10}).
\fullonly{
There are two special cases that can be thought of as barriers in dealing with standard versions of {\sf SMP} and {\sf UUDP-MIN}, and we believe that these two special cases serve as good starting points in attacking our problems. First, in order to get an $O_d(\polylog n)$-approximation algorithm for $d$-{\sf UUDP-MIN}, we need to deal with the Maximum Expanding Subsequence ({\sc Mes}) problem which is the key problem to show the hardness of {\sf UUDP-MIN}~\cite{Briest08}.
\fullonly{
\begin{definition}[Maximum Expanding Sub-sequence ({\sc Mes})]
We are given a set of $n$ ground elements $U$ and a set system $S_1,\ldots, S_m$ where $S_i \subseteq U$ for all $i$. We say that $S_{\phi(1)},\dots, S_{\phi(\ell)}$ is {\em expanding sequence} if for all $j$, we have $S_{\phi(j)} \not\subseteq \bigcup_{j' < j} S_{\phi(j')}$. The objective is to find an expanding sequence of maximum length.
\end{definition}
Briest \cite{Briest08} introduces this problem as an intermediate problem to prove hardness of approximation results for {\sf UUDP-MIN} and {\sf SMP}. Roughly speaking, Briest used $n^{\epsilon}$-hardness for this problem to show $n^{\delta}$-hardness of approximating {\sf UUDP-MIN}. His results showed a very strong evidence that {\sc Mes} is very closely related to {\sf UUDP-MIN} and {\sf SMP}. Now we ask the following question.
\begin{openproblem}
Suppose that the underlying set system of {\sc Mes} is defined by our framework in $\ensuremath{\mathbb R}^d$. That is, ground elements are points in $\ensuremath{\mathbb R}^d$, and each set is defined by an unbounded rectangle in $\ensuremath{\mathbb R}^d$. Can we get, says, an $O(\operatorname{poly} \log n)$ approximation algorithm for {\sc Mes}?
\end{openproblem}
}
\sodaonly{This motivates us to propose the following open problem, called $d$-{\sc Mes}: Suppose that the underlying set system of {\sc Mes} is defined by our framework in $\ensuremath{\mathbb R}^d$. That is, ground elements are points in $\ensuremath{\mathbb R}^d$, and each set is defined by an unbounded rectangle in $\ensuremath{\mathbb R}^d$. Can we get, says, $O(\operatorname{poly} \log n)$ approximation algorithm for {\sc Mes}?} It can be observed that our algorithm implies an $\tilde O_d(n^{1-\epsilon(d)})$-approximation for $d$-{\sc Mes}. Solving this problem can be considered the first step in getting $\log^{O(d)} n$ approximation.
Getting $O_d(\log^{1-\epsilon(d)} m)$ approximation for $d$-{\sf SMP} also has the following barrier. Previous results suggest that {\sf SMP} has inherited its intractibility from the problem called {\sf Unique Coverage} \cite{DemaineFHS08}. Roughly speaking, {\sf Unique Coverage} is equivalent to {\sf SMP} in which all consumers have budget one. An $O(\log n)$ approximation is known for this problem \cite{DemaineFHS08}. The question is whether we can improve this ratio in the case of $d$-{\sf SMP}.
\fullonly{
\begin{openproblem}
Is there a sub-logarithmic approximation algorithm for $d$-{\sf SMP} with unit budgets.
\end{openproblem}
}
Solving this problem is a first step in breaking another barrier of $O(\log m)$ for $d$-{\sf SMP}. Moreover, it would give a convincing evidence that our model may not inherit intractibility from {\sf Unique Coverage} problem. In fact, this problem itself can be seen as {\sf Unique Coverage} when the sets have {\em constant VC-dimension} and may be of independent interest.
Another interesting direction is to further investigate $2$-{\sf UUDP-MIN} and $2$-{\sf SMP}. We know that {\sf PTAS}s are likely to exist but do not even have an $O(\log n)$-approximation algorithm running in polynomial time. Getting {\sf PTAS} would be very interesting, and we believe that it will require novel ideas and structural properties. Even weaker approximation guarantees, such as $n^{\epsilon}$-approximation in time $n^{O(1/\epsilon)}$, would still be interesting.
\fullonly{
\begin{openproblem}
Can we construct $O(\log n)$ approximation algorithms for $2$-{\sf UUDP-MIN} or $2$-{\sf SMP}?
\end{openproblem}
}
}
\iffalse
\squishlist
\item Are there polylogarithmic approximation algorithms for $d$-{\sf UUDP} and $d$-{\sf SMP}, for any constant $d$? We note that, no explicit $O(\log n)$-approximation algorithms are known even for the case of $d=2$, although constant approximation factors are the real hope for a constant $d$ and we know that a {\sf PTAS} is likely to exist for the case of $d=2$.
\item In this paper, a {\sf PTAS} is shown to likely to exist in many cases but it is interesting to have explicit algorithms.\danupon{I'm not sure if we should list this problem.} (A {\sf PTAS} for $2$-{\sf SMP} is especially interesting since it will generalize the recent {\sf PTAS} result for the highway problem.)
\item
Another very interesting problem is the case where there are some items already priced by other sellers (i.e., Stackelberg problem). This case is still not much explored although it is usually the case in reality.
\end{itemize
\fi
\section{Hardness}\label{sec:hardness}
We provide hardness results in both scenarios when the number of attributes $d$ is small and when $d$ is large. We sketch our results here. More details can be found in Appendix~\ref{sec:omitted_hardness}.
\paragraph{Few attributes} First we discuss the \mbox{\sf NP}-hardness of $3$-{\sf UUDP-MIN} and \mbox{\sf APX}-hardness of $4$-{\sf UUDP-MIN}. These hardness results hold even when the consumer budgets are either 1 or 2. We perform a reduction from Vertex Cover~\cite{DBLP:journals/tcs/GareyJS76,DBLP:conf/ciac/AlimontiK97}, essentially using the same ideas as in~\cite{GuruswamiHKKKM05}, except for the fact that we use Schnyder's result~\cite{Schnyder89,DBLP:conf/soda/Schnyder90} to ``embed'' the instance into posets of low order dimensions.
First, let us recall the reduction in \cite{GuruswamiHKKKM05}. We start
from a graph $G=(V,E)$, which is an input instance of Vertex Cover. We create two types of consumers: (i) poor
consumer ${\bf C}_e$ for each edge $e$ with budget $1$ and (ii) rich
consumer ${\bf C}_v$ for each vertex $v$ with budget $2$. The items
are ${\mathcal{I}} = \set{{\bf I}_v: v\in V}$. Each poor consumer ${\bf C}_e$ has a
consideration set containing two items ${\bf I}_u$ and ${\bf I}_v$
where $e=(u,v)$ and each rich consumer ${\bf C}_v$ considers only
one item ${\bf I}_v$. Using the analysis essentially the same as
\cite{GuruswamiHKKKM05}, one can show that the problem is {\sf NP}-hard
if we start from Vertex Cover on planar graphs and {\sf APX}-hard if
we start from Vertex Cover on cubic graphs.
Therefore, it only remains to map consumers and items to points in ${\mathbb R}_{\geq 0}^d$ (where $d=3,4$) such that for each consumer
${\bf C}$, the set of items that pass her criteria (i.e., $\{{\bf I}\in {\mathcal{I}} \mid{\bf I}[i]\geq {\bf C}[i]~~\mbox{for all $1\leq i\leq d$}\}$) is exactly her consideration set.
The main idea is to first embed the problem into an {\em adjacency poset} of the input graph. Then, we invoke Schnyder's theorem~\cite{Schnyder89,DBLP:conf/soda/Schnyder90} to again embed this poset into a Euclidean space.
An adjacency poset of a graph can be constructed as follows. First we construct a $2$-layer poset with minimal elements in the first layer and maximal elements in the second layer. For each edge $e \in E$, we have a minimal element in the poset corresponding to $e$ (for convenience, we also denote the poset element by $e$). For each vertex $v \in V$, we have a maximal poset element corresponding to $v$. There is a relation $e\preceq v$ if and only if vertex $v$ is an endpoint of $e$. \fullonly{This is called an {\em adjacency poset} of the graph.}
The last task is to ``embed'' poset elements into points in the Euclidean space in such a way that, for any poset elements $e_1$ and $e_2$, $e_1\preceq e_2$ if and only if $q_{e_1}[i]\geq q_{e_2}[i]$ for all $i$ where $q_{e_1}$ and $q_{e_2}$ are points that $e_1$ and $e_2$ are mapped to, respectively. If we can do this, we would be done, simply by defining the coordinates of each consumer ${\bf C}_e$ to be $q_{e}$, and the coordinates of each consumer ${\bf C}_v$ to be $q_v$. Similarly, we define the coordinates of each item ${\bf I}_v$ as $q_v$. In order to obtain such an embedding, we use part of Schnyder's theorem~\cite{Schnyder89} which states that any planar graph has an adjacency poset of dimension three, and any $4$-colorable graph (including cubic graphs) has an adjacency poset of dimension four. Moreover, embedding these graphs into Euclidean spaces can be done in polynomial time~\cite{DBLP:conf/soda/Schnyder90}.
Finally we note that $2$-{\sf SMP} is strongly $\mbox{\sf NP}$-hard and $4$-{\sf SMP} is $\mbox{\sf APX}$-hard. The proof follows from the fact that these problems generalize Highway pricing and graph vertex pricing on bipartite graphs, respectively, and can be found in \sodaonly{the full version}\fullonly{Appendix~\ref{sec:omitted_hardness}}.\danupon{TO DO: Write the proof for $4$-{\sf SMP}}
\paragraph{Many attributes}
We establish a connection between the {\sf UUDP-MIN} with bounded-size consideration sets and our problem. This connection immediately implies hardness results for $d$-{\sf UUDP-MIN} when $d$ is at least poly-logarithmic in $n$. Our main result in this section is the following:
\begin{theorem}(Informal)
\label{theorem: higher dimension}
Let $A =({\mathcal{C}}, {\mathcal{I}}, \{S_{\bf C}\}_{ {\bf C} \in {\mathcal{C}}} )$ be an instance of {\sf UUDP-MIN} where $B=\max_{{\bf C} \in {\mathcal{C}}} \size{S_{\bf C}}$. We can (with high probability of success) create an instance $A'=({\mathcal{C}}', {\mathcal{I}}')$ of $d$-{\sf UUDP-MIN}, where $d= O(B^2 \log n)$, that is ``equivalent'' to $A$.
\end{theorem}
In other words, the above theorem shows that any {\sf UUDP-MIN} instance with consideration sets of size bounded by $B$, can be realized by a $d$-{\sf UUDP-MIN} instance for $d= O(B^2 \log n)$.
\sodaonly{
Combining this with the result in \cite{ChalermsookPricing}, we have a hardness of $\Omega(d^{1/4-\epsilon)}$ for any $\epsilon>0$.
}
\fullonly{
This implies that $d$-{\sf UUDP-MIN} is at least as hard
as the original problem when consideration sets have size at most $B$.
When
$B$ is at least logarithmic in the number of items, our
reduction yields the following corollary, assuming the hardness of the balanced bipartite independent set problem in constant degree graphs or refuting random 3{\sf CNF} formulas \cite{BriestThesis}. \danupon{I shortened the last sentence.}
\begin{corollary}
There is a constant $\epsilon$ such that for every $d \geq \log^2
n$, it is hard to approximate $d$-{\sf UUDP-MIN} to
within a factor of $d^{\epsilon}$.
\end{corollary}
}
We remark that our reduction here in fact works independently of the decision model, so this result works for {\sf SMP} and {\sf UDP-Util} as well.
\subsection{Hardness Results in Higher Dimensions}\label{sec:higher
dimensions}
In this section, we present the proof of Theorem~\ref{theorem: higher dimension}. Let $A = ({\mathcal{I}}, {\mathcal{C}})$ be an instance of {\sf UUDP-MIN} where every consumer ${\bf C}$ has its consideration set $S_{\bf C}$ of size at most $B$. Let ${\mathcal{I}} = \set{{\bf I}_1,\ldots, {\bf I}_n}$. For each $i \in [d]$, we pick a random permutation $\pi_i: [n] \rightarrow [n]$, so we have $d$ permutations $\pi_1,\ldots, \pi_d$. The function $\phi$ on items ${\mathcal{I}}$ can be defined as $\phi({\bf I}_j)[i] = \pi_i(j)$, and we extend the function to the set of consumers as follows: $\phi({\bf C})[i] = \min_{j \in S_C} \pi_i(j)$. Now we have a well-defined function $\phi$.
\begin{lemma}
With probability at least $1- 1/n$, for all consumer ${\bf C} \in {\mathcal{C}}$, the consideration set $S'_{\bf C}$ defined by $S'_{\bf C} = \set{{\bf I}_j: \phi({\bf I}_j) \mbox{ dominates } \phi({\bf C})}$ is exactly $S_{\bf C}$.
\end{lemma}
\begin{proof}
Since we define $\phi({\bf C})$ to be the minimum of $\phi({\bf I}_j)$ over all items in $S_{\bf C}$, we have $S_{\bf C} \subseteq S'_{\bf C}$. Let $k$ be the index of an item that does not belong to $S_{\bf C}$. We show the following claim.
\begin{claim}
\label{claim: tiny prob}
The probability that $\phi({\bf I}_k)$ dominates $\phi({\bf C})$ is at most $1/n^{B+2}$.
\end{claim}
\begin{proof
Fix some $i \in [d]$. The bad event that $\pi_i(k)\geq \min_{j \in
S_C} \pi_i(j)$ happens only if $\pi_i$ does not put $k$ in the last
position among $S_C \cup \set{k}$. This probability is exactly
$(1-1/(B+1))$. Therefore, the bad event happens for all values of
$i$ with probability at most $(1-1/(B+1))^d \leq 1/n^{B+2}$ for $d=
O(B^2 \log n)$.
\end{proof}
This claim immediately implies the lemma: By the union bound, the
probability that $\phi({\bf I}_k)$ dominates $\phi({\bf C})$ is at
most $1/n^{B+1}$. So we have that $\pr{}{S_{\bf C} \neq S'_{\bf C}} \leq
1/n^{B+2}$. There are at most $n^B$ possible consideration sets of
size at most $B$, so by union bounds, the probability that a bad
event $S_C \neq S'_{\bf C}$ happens for some consumer ${\bf C}$ is at most
$1/n$.
\end{proof}
\paragraph{$n$ attributes capture general problem} Finally, we end this section with the proof that $n$-{\sf UUDP-MIN} captures the whole generality of {\sf UUDP-MIN}: Consider an instance $({\mathcal{C}}, {\mathcal{I}}, \set{S_{\bf C}}_{{\bf C}\in {\mathcal{C}}})$ of {\sf UUDP-MIN}. Denote the set of items by ${\mathcal{I}} = \set{{\bf I}_1,\ldots, {\bf I}_n}$. Notice that we can define the coordinates of each consumer by ${\bf C}[i] = 0$ if ${\bf I}_i \in S_{\bf C}$, and ${\bf C}[i] = 1$ otherwise. We define the coordinates of each item as ${\bf I}_i[i] = 0$ and ${\bf I}_i[j] =1$ for all $j \neq i$. It is easy to check that the consideration sets are preserved by this reduction.
\section{Introduction}
This paper studies a geometric version of two central {\em unlimited-supply pricing} problems. We are given a set ${\mathcal{I}}$ of $n$ consumers and a set ${\mathcal{C}}$ of $m$ items.
Every item ${\bf I}\in{\mathcal{I}}$ is represented by a point ${\bf I} = ({\bf I}[1],\ldots, {\bf I}[d]) \in \ensuremath{\mathbb R}^d_{\geq 0}$, where $\ensuremath{\mathbb R}_{\geq 0}$ denotes the set of non-negative reals and ${\bf I}[j]$ expresses the quality of item ${\bf I}$ in the $j$-th attribute. Every consumer ${\bf C}\in{\mathcal{C}}$ is represented by a point ${\bf C}= ({\bf C}[1],\ldots, {\bf C}[d]) \in \ensuremath{\mathbb R}^d_{\geq 0}$, where ${\bf C}[j]$ is the criterion of consumer ${\bf C}\in\mathcal{C}$ in the $j$-th attribute. Each consumer ${\bf C}$ is additionally equipped with budget $B_{{\bf C}}\in\mathbb{R}_{\geq 0}$ and a consideration set
\begin{equation}\label{eq:SC for UDP-MIN}
S_{\bf C}=\set{{\bf I}: {\bf I}[j] \geq {\bf C}[j], \mbox{for all } 1\leq j \leq d}.
\end{equation}
\begin{wrapfigure}{r}{0.3\textwidth}
\center
\vspace{-.9 cm}
\includegraphics[width=1.1\linewidth, clip=true, trim= 0.5cm 1cm 1cm 0.8cm]{problem_visualization.pdf}\\
\caption{Problem visualization}\label{fig:visualization}\vspace{-.3cm}
\end{wrapfigure}
In the {\em $d$-dimensional uniform-budget unit-demand min-buying pricing problem} ($d$-{\sf UUDP-MIN}), once we assign prices to items, each consumer ${\bf C}$ will buy the cheapest item ${\bf I}$ in $S_{{\bf C}}$ if the price of item ${\bf I}$ is at most $B_{{\bf C}}$. In the {\em $d$-dimensional single-minded pricing problem} ($d$-{\sf SMP}), consumer ${\bf C}$ will buy the {\em all} items in $S_{{\bf C}}$ if the total price of those items is at most $B_{{\bf C}}$.
The objective is to set the price of items in ${\mathcal{I}}$ in order to maximize the revenue. That is, we want to find $p: {\mathcal{I}}\rightarrow {\mathbb R}_{\geq 0}$ that maximizes $\sum_{{\bf C}\in {\mathcal{C}}, \min_{{\bf I}\in S_{\bf C}} p({\bf I})\leq B_{\bf C}} \min_{{\bf I}\in S_{\bf C}} p({\bf I})$ in the case of $d$-{\sf UUDP-MIN} and $\sum_{{\bf C}\in {\mathcal{C}}, \sum_{{\bf I}\in S_{\bf C}} p({\bf I})\leq B_{\bf C}} \sum_{{\bf I}\in S_{\bf C}} p({\bf I})$ in the case of $d$-{\sf SMP}.
Fig.~\ref{fig:visualization} illustrates the problem: Each item corresponds to a point in the plane. The consideration set of each consumer ${\bf C}$ is represented by an (unbounded) axis-parallel rectangle with point ${\bf C}$ as a lower-left corner.
The above problems when $d$ is unbounded (called {\sf UUDP-MIN} and {\sf SMP}) have been widely studied recently
(e.g.,
\cite{Rusmevichientong03,GuruswamiHKKKM05,RusmevichientongRG06,BriestK11,ChalermsookPricing,AggarwalFMZ04,BalcanB07})
and are known to be $O(\log m)$-approximable~\cite{AggarwalFMZ04}; so we have a reasonable approximation guarantee when there are not many consumers.
However, in many cases, one would expect the number of consumers to be much larger than the number of items~$n$. In this case, we are still stuck at the trivial $O(n)$ approximation ratio, and there are evidences that suggest that getting a sub-linear approximation ratios might be impossible: Unless $\mbox{\sf NP} \subseteq \mbox{\sf DTIME}(n^{\operatorname{poly} \log n})$, these problem are hard to approximate within a $2^{\log^{1-\epsilon} n}$ for any constant $\epsilon>0$~\cite{ChalermsookPricing}. Moreover, assuming a stronger (but still plausible) assumption, these problems are hard to approximate to within a factor of $n^{\epsilon}$ for some $\epsilon >0$ \cite{BriestK11}.
Motivated by various types of assumptions, the pricing problems with special structures have been studied (e.g., when there is a {\em price-ladder constraint}~\cite{Rusmevichientong03,AggarwalFMZ04,RusmevichientongRG06,RusmevichientongRG06-2,BriestK11}, consideration sets are small~\cite{BalcanB07,BriestK11} or consideration sets correspond to paths on graphs~\cite{BalcanB07,ElbassioniSZ07,GrandoniR10,ElbassioniRRS09,DBLP:conf/icalp/GamzuS10}). In these cases, better approximation ratios are usually possible.
In this paper we consider the geometric structure of pricing problems arising naturally from real-world scenarios, which turns out to be quite general. Our motivation is two-fold: We hope that the geometric structures will lead to better approximation algorithms, and we found these problems interesting on their own as they have connections to other pricing and geometric problems.
Our problems are motivated by the following simple observation on the consumers' behavior.
Consider a setting where we sell cars. If a consumer has car $A$ with horse power $130$HP in her consideration set, she would not mind buying car $B$ with horse power $150$HP. Maybe she does not want $B$ because it is less energy-efficient or has lower reputation. But, if we list {\em all} attributes of the cars that people care about and it happens that $B$ is not worse than $A$ in all other aspects, then $B$ should also be in the list.
In particular, instead of looking at a full generality where each consumer ${\bf C}$ considers any set of items $S_{\bf C}$, it is reasonable to assume that each consumer has some criterion in mind for each attribute of the cars, and her consideration set consists of any car that passes all her criteria, i.e. consumers judge items according to their attributes.
This natural assumption has been a model of study in other fields such as marketing research, healthcare economics and urban planning. It is referred to as the {\em attribute-based screening process}. In particular, using criteria to define consideration sets as in Eq.~\eqref{eq:SC for UDP-MIN} is called {\em conjunctive screening rule}. Besides being natural, this assumption has been supported by a number of studies where it is concluded that consumers typically use a conjunctive screening rule in obtaining their consideration sets (see further detail in Section~\ref{sec:relatedwork}).
It is also interesting that $d$-{\sf SMP} captures many previously studied problems as special cases. For example, $2$-{\sf SMP} generalizes the highway pricing problem~\cite{GuruswamiHKKKM05,BalcanB07,ElbassioniSZ07,GrandoniR10} and thus our algorithmic results on $2$-{\sf SMP} can immediately be applied to this problem. Moreover, $3$-{\sf SMP} generalizes the upward case of the tollbooth pricing problem \cite{ElbassioniRRS09,KortsarzRR11} as well as the graph vertex pricing problem on planar graphs \cite{BalcanB07,ChalermsookKLN11}. $4$-{\sf SMP} generalizes the unlimited-supply version of the {\em exhibition} problem \cite{ChristodoulouEF10}, the graph vertex pricing problem on bipartite graphs \cite{BalcanB07,KhandekarKMS09}, and the ``rectangle version'' of the {\em unique coverage problem} ({\sc UC}) \cite{DemaineFHS08}, which are the geometric variants of {\sc UC} studied recently in~\cite{ErlebachL08,ItoNOOUUU12}.
Moreover, {\sf SMP} is a special case of the {\em maximum feasible subsystem with 0/1 coefficients} problem ({\sc Mrfs}) \cite{ElbassioniRRS09}. Elbassioni et~al. \cite{ElbassioniRRS09} showed that a very special geometric version of {\sc Mrfs} (the ``interval version'') admits much better approximation ratios than the general one. A geometric {\sc Mrfs} can be seen as a special case of ``$2$-{\sc Mrfs}'' in our terminologies, and it is thus interesting whether ``$d$-{\sc Mrfs}'' is easier than general {\sc Mrfs} for other values of $d$. Our geometric {\sf SMP} is a special case of $d$-{\sc Mrfs}. Thus, solving $d$-{\sf SMP} serves as the first step towards solving $d$-{\sc Mrfs}.
\subsection{Our Results and Techniques}
We show that geometric structures lead to breaking the linear-approximation barrier: While the pricing problems are likely to be hard to approximate within a factor of $n^{1-\epsilon}$ in the general cases, we obtain an $o(n)$-approximation algorithms in the geometric setting, as follows.
\begin{theorem}\label{thm:intro thm udp smp}
For any $d>0$, there is an $\tilde O_d\left(n^{1-\epsilon(d)}\right)$-approximation algorithm for $d$-{\sf UUDP-MIN} and $d$-{\sf SMP} where function $\epsilon(d): = \frac{1}{4^{d-1}}$ and $\tilde O_d$ treats $d$ as a constant and hides a $\mathrm{polylog}(n)$ factor.
\end{theorem}
The essential idea behind our algorithm is to partition the problem instance into sub-instances without decreasing the optimal revenue (we call this {\em consideration-preserving decomposition}). This is done by using Dilworth's Theorem (partitioning items into chains and anti-chains) and epsilon-nets to find subsets of items satisfying certain structural properties. Subsequently, we show that the dimensions of these sub-instances can be reduced through the notion of {\em consideration-preserving embedding}. In the end of our algorithm, we are left with a sub-linear number of sub-instances, each of which can be solved almost optimally in polynomial time. Returning the best solution among the solutions of these sub-instances guarantees a sub-linear approximation ratio.
The spirit of our technique is in some sense in a similar flavor to Chan's algorithm \cite{Chan12} which computes a {\em conflict-free} coloring of $d$-dimensional points (w.r.t. rectangle ranges) using $O(n^{1-0.632/(2^{d-3}-0.368)})$ colors. In particular, in the 2-dimensional cases of both our geometric pricing and Chan's conflict-free coloring problems, the upper bounds of $O(\sqrt{n})$ can be obtained by a simple application of Dilworth's theorem (Ajwani et al. \cite{AjwaniEGR12} obtained a better bound in this case for the latter problem). However, the techniques of the two results are different in higher dimensions.
\paragraph{QPTASs} We also obtain {\sf QPTAS}s for $2$-{\sf UUDP-MIN} and $2$-{\sf SMP}. We present this in Appendix~\ref{sec: qptas 2 udp} and \ref{sec:2-SMP}. These results, together with a widely-believed assumption that the existence of a {\sf QPTAS} for any problem implies that {\sf PTAS} exists for the same problem (e.g., \cite{BansalCES06,ElbassioniSZ07}), imply that the value of $\epsilon(d)$ in Theorem~\ref{thm:intro thm udp smp} could be improved slightly to $1/4^{d-2}$.
As a by-product of these results, we show a {\sf QPTAS} for $2$-{\sf SMP} which subsumes the previous {\sf QPTAS} for highway pricing \cite{ElbassioniSZ07}.
\begin{comment}
\paragraph{Further results} We note that the notion of attribute and our techniques are useful not only for attacking {\sf UUDP-MIN} but also for several other models.
For instance we consider another well-known setting called {\em single-minded pricing} ({\sf SMP})~\cite{GuruswamiHKKKM05}, in which each consumer buys the entire bundle $S_{{\bf C}}$ if she can afford to and buys nothing otherwise.
We prove that the $d$-attribute version of {\sf SMP}, called $d$-{\sf SMP}, admits the same approximation ratio as in Theorem~\ref{thm:intro thm udp smp}.
More importantly, we obtain {\sf QPTAS}s for important subroutines which are crucial building blocks in our main theorem. These results, together with a widely-believed assumption that the existence of a {\sf QPTAS} for any problem implies that {\sf PTAS} exists for the same problem (e.g., \cite{BansalCES06,ElbassioniSZ07}), imply that the approximation ratios of the aforementioned problems can be improved. (We emphasize that this claim relies on the fact that the subroutines that we obtain {\sf QPTAS}s have {\sf PTAS}s.) They also imply that obtaining sublinear-approximation algorithms is possible even for more general problems. Here, we consider two broader models of the pricing problem. \fullonly{(see their full definitions in Section~\ref{sec: algorithms for small d})}\sodaonly{Due to the page limitation, most of the proof details are omitted in this extended abstract and can be found in the full version of our paper \cite{Fullversion}}.
\squishlist
\item The {\em unit-demand utility-maximizing pricing problem} ({\sf UDP-Util}) is a model that generalizes {\sf UUDP-MIN} in that the consumers may have different {\em valuations} on different items and want to maximize the difference between the valuations and the price of the purchased items. (This problem is sometimes called {\em envy-free pricing problem} \cite{BriestK07} and {\em max-gain-buying pricing problem} \cite{AggarwalFMZ04}. We name this problem {\sf UDP-Util} to distinguish it from other general pricing problems studied in this paper.) We show that there is an $\tilde O_d\left(n^{1-1/4^{d-1}}\right)$-approximation algorithm for a naturally defined $d$-attribute version of {\sf UDP-Util} when valuations are monotone functions of $d$ attributes (i.e., each consumer values an item not less than the inferior ones). This result also holds for the $d$-attribute version of the (non-uniform) unit-demand min-buying pricing problem ({\sf UDP-MIN}) where consumers can have different budgets on different items.
\item The pricing problem with {\em symmetric valuations} and {\em subadditive revenues} is a model that includes both {\sf UUDP-MIN} and {\sf SMP} as well as many other natural problems. Informally, symmetric valuation means that each consumer's valuation only depends on the number of items she gets, and subadditive revenue implies that the amount that each consumer pays to get an item bundle $X$ is at most the amount she pays to get the items in $X$ separately. We show that there is an $\tilde O_d\left(n^{1-1/4^{d-1}}\right)$-approximation algorithm for the $d$-attribute version of this problem.
\end{itemize
\fullonly{
We summarize our results in the following theorem.
\begin{theorem}\label{thm:intro all}
Unless there is a problem that admits {\sf QPTAS} but no {\sf PTAS}, there exist $\tilde O_d(n^{1-1/4^{d-1}})$-approximation algorithms for (1) {\em unit-demand utility-maximizing} $d$-attribute pricing problem where valuations depend only on attributes and (2) $d$-attribute pricing problem with {\em symmetric valuations} and {\em subadditive revenues}. Moreover, there exist $\tilde O_d(n^{1-1/4^{d-2}})$-approximation algorithms for $d$-{\sf UUDP-MIN} and $d$-{\sf SMP}.
\end{theorem}
}
As a by-product of our proof, we show a {\sf QPTAS} for $2$-{\sf SMP} which subsumes the previous {\sf QPTAS} for the highway pricing problem \cite{ElbassioniSZ07}. \fullonly{We discuss this in the next subsection.}
\end{comment}
\paragraph{Hardness} We also study the hardness of approximation of our problems. We show that $3$-{\sf UUDP-MIN} and $2$-{\sf SMP} are {\sf NP}-hard, and $4$-{\sf UUDP-MIN} and $4$-{\sf SMP} are {\sf APX}-hard. Hence, our problem is already non-trivial for small $d$.
Our hardness proofs establish a cute connection between our problem and the vertex cover problem on graphs of low {\em order dimensions}~\cite{Schnyder89,DBLP:conf/soda/Schnyder90}.
Moreover, we show that the hardness of our problem tends to increase as we increase $d$, and the whole generality is captured when $d=n$. In particular, we show that when the dimension is sufficiently high (i.e. $d \geq \log^2 n$), the problems are hard to approximate to within a factor of $d^{1/4-\epsilon}$ for any $\epsilon>0$.
Table~\ref{table:current status} concludes our results for $d$-{\sf UUDP-MIN} and $d$-{\sf SMP}.
\begin{table}
\begin{center}
\footnotesize
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline
{\bf Problem} & & $\mathbf{d=1}$ & $\mathbf{d=2}$ & $\mathbf{d=3}$ & $\mathbf{d=4}$ & {\bf large $\mathbf{d}$ \{range \} }\\
\hline
\multirow{2}{*}{$d$-{\sf UUDP-MIN}} & Upper bound & Polytime & {\sf QPTAS} & & & $n^{1-\frac{1}{4^{d-1}}}$ \{constant $d$\}\\
& Lower bound & & & {\sf NP}-hard & {\sf APX}-hard & $d^{\frac{1}{4}-\epsilon}$ \{$d=\omega(\log n)$\} \\
\hline
\multirow{2}{*}{$d$-{\sf SMP}} & Upper bound & Polytime & {\sf QPTAS} & & & $n^{1-\frac{1}{4^{d-1}}}$ \{constant $d$\}\\
& Lower bound & & {\sf NP}-hard & & {\sf APX}-hard & $d^{\frac{1}{4}-\epsilon}$ \{$d=\omega(\log n)$\} \\
\hline
\end{tabular}
\end{center}
\vspace{-.5cm}
\caption{Results of $d$-{\sf UUDP-MIN} and $d$-{\sf SMP} for small values of $d$.}
\label{table:current status}
\vspace{-.3cm}
\end{table}
\fullonly{
\subsection{General Framework}\label{sec:framework}
We explain the general framework of the proofs in this subsection, and we will solve $d$-{\sf UUDP-MIN} to give the key ideas of our techniques in Section~\ref{sec:d udp min}. The proofs of our results can be divided into two parts.
In the first part, we essentially show that a large class of $d$-attribute pricing problems is sublinear-approximable {\em if} it is sublinear-approximable for some small $d$. This class is the class of pricing problems with {\em subadditive revenue} (informally described in the previous subsection), which includes all the aforementioned problems. Thus, in one shot we reduce our task to solving the pricing problems on a very simple input! The following informal theorem shows the essence of the first part (detail in Section~\ref{sec:dim reduction statement} and \ref{sec:dim reduction proof}).
\begin{theorem}[Dimension Reduction Theorem (Informal)] Let $\mathcal{P}$ be any pricing problem with subadditive revenue. For any $d$ and $d'<d$, if there is an $\tilde O_d(1)$ approximation algorithm for the $d'$-attribute version of $\mathcal{P}$ then there is an $\tilde O_d(n^{1-\varepsilon(d,d')})$-approximation algorithm for its $d$-attribute version, where $\varepsilon(\cdot)$ is a function defined as $\varepsilon(t,t') = 1/4^{t-t'}$.
\end{theorem}
In the second part we show that the aforementioned problems can be solved in the case of one and two attributes. First, the cases of $1$-{\sf UUDP-MIN} and $1$-{\sf SMP} can be solved optimally by simple dynamic programs and sublinear-approximation algorithms of both problems thus follow. Furthermore, we show quasi-polynomial time approximation schemes ({\sf QPTAS}s) for $2$-{\sf UUDP-MIN} and $2$-{\sf SMP} as well as (1) {\em unit-demand utility-maximizing} $1$-attribute pricing problem where valuations depend only on attributes and (2) $1$-attribute pricing problem with {\em symmetric valuations} and {\em subadditive revenues}. These results rule out the possibility of these problems being {\sf APX}-hard unless $\mbox{\sf NP} \subseteq \mbox{\sf DTIME}(n^{\operatorname{poly} \log n})$. Thus {\sf PTAS}s for these problems are likely to exist. This, along with the Dimension Reduction Theorem, implies Theorem~\ref{thm:intro all}.
On a technical side, we note that our {\sf QPTAS} for $2$-{\sf SMP} generalizes the {\sf QPTAS} result in \cite{ElbassioniSZ07} for the Highway pricing problem as the Highway pricing problem is a special case of $2$-{\sf SMP}. However, $2$-{\sf SMP} seems to have a more complicated structure and is harder to handle. A good evidence of this is that while the Highway problem has a very simple $O(\log n)$-approximation algorithm \cite{BalcanB07}, getting a polynomial-time algorithm with $o(\sqrt{n})$ approximation guarantee for $2$-{\sf SMP} without assuming anything is already a challenging task. Obtaining $O(\log n)$-approximation algorithm or extending the {\sf PTAS} technique in \cite{GrandoniR10} to $2$-{\sf SMP} (or $2$-{\sf UUDP-MIN}) is an interesting open problem.
}
\subsection{Related Work}\label{sec:relatedwork}
Rusmevichientong et al.~\cite{Rusmevichientong03,RusmevichientongRG06,RusmevichientongRG06-2} defined the {\em non-parametric multi-product
pricing problem}, motivated by the possibility that the data about consumers' preferences and budgets can be predicted based on previous data, which can be gathered and mined by web sites designed for this purpose, e.g., \cite{HaublTrifts00,RusmevichientongRG06-2}. This problem is what we call uniform-budget unit-demand pricing problem ({\sf UUDP}). Rusmevichientong et al. proposed many decision rules such as min-buying, max-buying and rank-buying and showed that {\sf UUDP-MIN} allows a polynomial-time algorithm, assuming the {\em price-ladder constraint}, i.e., a predefined total order on the prices of all products. Aggarwal et al.~\cite{AggarwalFMZ04} later showed that the price ladder constraint also leads to a $4$-approximation algorithm for the max-buying case, even in the case of limited supply.
We note that the price ladder constraint is closely related to our notion of attributes in the following sense. It can be shown that $1$-{\sf UUDP-MIN} satisfies the price ladder constraint (this is the reason we can solve it in polynomial time). Moreover, although $2$-{\sf UUDP-MIN} does not satisfy this constraint, it {\em partially} satisfies the constraint in the sense that if one item is better than another item in all attributes then we can assume that it has a higher price. This property plays an important role in obtaining {\sf QPTAS} for $2$-{\sf UUDP-MIN} and also holds for general $d$.
Other variants defined later include non-uniform and utility-maximizing unit-demand, single-minded ({\sf SMP}), tollbooth and highway models~\cite{AggarwalFMZ04,GuruswamiHKKKM05}.
These problems were later found to have important connections to
algorithmic mechanism design~\cite{AggarwalH06,BalcanB05,GuruswamiHKKKM05} and online pricing
problems~\cite{BalcanB07,BlumH05}.
As we mentioned in the introduction, many problems can be approximated within the factor of $O(\log m + \log n)$ and $O(n)$, and these seem to be tight.
The observation that consumers make decisions based on attributes has been used in other areas outside computer science. For example, most pricing models are captured by the {\em two-stage consider-then-choose} model (e.g., \cite{Gensch87,Payne82,PayneBJ88,GilbrideAllenby,HaublTrifts00,JedidiK05,HauserTEBS10,LiuArora2011}) in marketing research: Each consumer first screens out some undesirable items ({\em screening process}) and is left with the consideration set which is used to make a final decision. Pricing problems such as {\sf UUDP-MIN} are the case where consideration sets are arbitrary (as defined in, e.g. \cite{Shocker91,HauserW90}) while the final decision is simplified to, e.g., buying the cheapest item.
The idea of using the consideration sets defined from attributes is called {\em attribute-based screening process}~\cite{GilbrideAllenby} in marketing research where it was shown to be a rational choice for trading off between accuracy and cognitive effort~\cite{Bettman79,BettmanJP90,BettmanPark80,Shugan80}. Our model is equivalent to the attribute-based screening process with {\em conjunctive screening rules} (e.g., \cite{GilbrideAllenby,LiuArora2011}). This type of rules was justified by many studies that it is what consumers typically use when making decisions (e.g., \cite{Bettman79,GilbrideAllenby,HauserTEBS10}).
\section*{Appendix}
\input{proof_algo_ec}
\input{udp}
\input{single-minded-3}
\input{omitted_hardness}
\input{higherdimension}
\end{document}
\section{Omitted hardness results}\label{sec:omitted_hardness}
\subsection{Hardness of 3-{\sf UUDP-MIN} and 4-{\sf UUDP-MIN}}
In this section we show that $3$-{\sf UUDP-MIN} is \mbox{\sf NP}-hard, and
$4$-{\sf UUDP-MIN} is \mbox{\sf APX}-hard by a reduction from {\sf Vertex Cover}. Our
reduction relies on the concepts of adjacency poset and its
embedding into Euclidean space. We describe basic terminologies
here. Given a graph $G=(V,E)$, an adjacency poset $(V \cup E,
\preceq_G)$ of graph $G$ can be constructed as follows: First we
define a poset with its maximal elements corresponding to vertices in $V$ and its minimal
elements corresponding to edges $E$. For each vertex $v$ and each edge $e$, we
have the relation $e \preceq_G v$ if and only if vertex $v$ is an endpoint of
$e$. We say that a map $\phi: V \cup E \rightarrow \ensuremath{\mathbb R}^d$ is an {\em
embedding} of adjacency poset $(V \cup E, \preceq_G)$ into $\ensuremath{\mathbb R}^d$
if and only if it preserves the relations $\preceq_G$, i.e., for any two
elements $a,b \in V \cup E$, we have that $a \preceq_G b$ iff $\phi(a)[i] \leq \phi(b)[i]$ for all coordinates $i \in [d]$.
Now we describe our reductions. Since two reductions are essentially the same, we give a general procedure which will imply both results. Given an instance $G=(V,E)$ of {\sf Vertex Cover}, we first construct an adjacency poset $(V \cup E, \preceq_G)$ for $G$,
and then we compute the embedding $\phi$ of this poset into Euclidean space $\ensuremath{\mathbb R}^d$. We will use the graph $G$, as well as the embedding $\phi$, to define the instance of $d$-{\sf UUDP-MIN} as follows:
\begin{itemize}
\item {\bf Consumers:} We have two types of consumers, i.e. the rich consumers and the poor ones. For each vertex $v \in V$, we create a {\em rich} consumer $C_v$ with budget $2$ at coordinates $\phi(v)$. For each edge $e \in E$, we create a {\em poor} consumer $C_e$ with budget $1$ at coordinates $\phi(e)$.
\item {\bf Items:} For each vertex $v \in V$, we create item ${\bf I_v}$ at coordinates $\phi(v)$.
\end{itemize}
Note that for each $e=(u,v)$, each poor consumer $C_e$ has $S_{C_{e}} = \set{{\bf I}_v, {\bf I}_u}$, while each rich consumer $C_v$ has $S_{C_v} = \set{{\bf I}_v}$. We denote the resulting instance by $({\mathcal{C}}, {\mathcal{I}})$.
The following lemma gives a characterization of the optimal solution for $({\mathcal{C}}, {\mathcal{I}})$. It says that we may assume without loss of generality that every poor consumer gets some item.
\begin{lemma}
For any price $p$ that is a solution for $({\mathcal{C}}, {\mathcal{I}})$ constructed above, we can transform $p$ to $p'$ such that every poor consumer buys some item with respect to $p'$, and the revenue of $p'$ is at least as much as the revenue of $p$.
\end{lemma}
\begin{proof}
Consider edge $e=(u,v)$. Suppose poor consumer $C_e$ does not get
any item, so it implies that both items ${\bf I}_u$ and ${\bf I}_v$
have price $p({\bf I}_u)= p({\bf I}_v) =2$ (recall that, since budgets are $1$ or $2$, the optimal prices would never set prices that are not in $\set{1,2}$). We define the price function $p'$ by setting
$p'({\bf I}_u) = 1$ while $p'({\bf I}_w) = p({\bf I}_w)$ for all
other vertices $w \in V \setminus \set{u}$. The only rich consumer that gets affected is $C_u$, whose payment may decrease by one. However, we earn the revenue of one back from poor consumer $C_e$. For $e' \in E: e'\neq e$, poor consumer $C_{e'}$ is never affected because his budget is one. Overall, changing the price from $p$ to $p'$ never decreases revenue.
\end{proof}
Let $p^*$ be the optimal price for $({\mathcal{C}}, {\mathcal{I}})$ and ${\sf VC}(G)$ denote the size of
minimum vertex cover of $G$. We show the following connection
between the size of minimum vertex cover and the optimal revenue
collected by $p^*$.
\begin{theorem}
The optimal revenue collected by $p^*$ is exactly $2n-{\sf VC}(G)
+m$.
\end{theorem}
\begin{proof}
From the previous lemma, we can assume that the pricing $p^*$ sells
items to every poor consumer. In other words, if $V' = \set{v:
p^*({\bf I}_v)=1}$, it must be the case that $V'$ is a vertex cover:
otherwise, let $e=(u,w)$ be an edge which is not covered by any
vertex in $V'$, so $C_e$ is only interested in items with price $2$,
which he cannot afford. This contradicts the assumption that $p^*$
sells items to every poor consumer.
The revenue collected from poor consumers is exactly $m$. Each rich
consumer $C_v$ in the vertex cover gets the item with price $1$ while
others get the items with price $2$, so the total revenue is $m + {\sf
VC}(G) + 2(n - {\sf VC}(G))$.
\end{proof}
This theorem immediately implies the gap between {\sc Yes-Instance} and {\sc No-Instance} for $d$-{\sf UUDP-MIN}. The only detail we left out is the computation of the embedding $\phi$, and this is where the hardness proofs
of $3$-{\sf UUDP-MIN} and $4$-{\sf UUDP-MIN} depart (other steps are
exactly the same). For $3$-dimensional case, we start from planar
graphs whose adjacency poset can be embedded into $\ensuremath{\mathbb R}^3$. Since
planar vertex cover has a polynomial-time approximation scheme, we
only get \mbox{\sf NP}-hardness here. For $4$-dimensional case, we start from
vertex cover in cubic graphs, which is known to be \mbox{\sf APX}-hard, but
unfortunately we can only embed its adjacency poset into $\ensuremath{\mathbb R}^4$,
thus obtaining the hardness of $4$-{\sf UUDP-MIN}.
\paragraph{\mbox{\sf NP}-Hardness of $3$-{\sf UUDP-MIN}}
To show the \mbox{\sf NP}-hardness, we start from {\sf Vertex Cover} in planar graphs, which is known to be \mbox{\sf NP}-complete~\cite{DBLP:journals/tcs/GareyJS76}. We will use the following theorem,
due to Schnyder~\cite{Schnyder89}.
\begin{theorem}
Let $(V \cup E, \preceq_G)$ be an incident poset of planar graph
$G$. Then there exists an embedding $\phi$ from the poset into
$\ensuremath{\mathbb R}^3$.
\end{theorem}
Schnyder shows later that the crucial step in the theorem
can be computed in polynomial time \cite{DBLP:conf/soda/Schnyder90}, which
immediately implies the following theorem.
\begin{theorem}
$3$-{\sf UUDP-MIN} is \mbox{\sf NP}-hard even when the consumer budgets are
either $1$ or $2$.
\end{theorem}
\paragraph{\mbox{\sf APX}-Hardness of $4$-{\sf UUDP-MIN}}
We will be using the fact that {\sf Vertex Cover} in cubic graphs
is \mbox{\sf APX}-hard \cite{DBLP:conf/ciac/AlimontiK97}, stated in the
language convenient for our use below.
\begin{theorem}
For some $0 < \alpha < \beta < 1$, it is \mbox{\sf NP}-hard to distinguish
between (i) the graph that has a vertex cover of size at most
$\alpha n$, and (ii) the graph whose minimum vertex cover is at
least $\beta n$.
\end{theorem}
Now we assume that our input graph $G$ is a cubic graph and use the
following theorem to embed the adjacency poset of $G$ into $\ensuremath{\mathbb R}^4$.
\begin{theorem}[Schnyder]
An adjacency poset of any $4$-colorable graph can be embedded into
$\ensuremath{\mathbb R}^4$. Moreover, the embedding is computable in polynomial time.
\end{theorem}
It only requires a straightforward computation to prove the
following theorem.
\begin{theorem}
$4$-{\sf UUDP-MIN} is \mbox{\sf APX}-hard even when the consumer budgets are
either $1$ or $2$.
\end{theorem}
\begin{proof}
In the {\sc Yes-Instance}\xspace, we can collect the revenue of $(2-\alpha) n +m$.
However, in the \ni, the revenue is at most $(2-\beta)n +m$. Since
the graph is cubic, we may assume that $m =\gamma n$ for some $1\leq
\gamma <2$. Hence we have a gap of
$(2-\alpha+\gamma)/(2-\beta+\gamma)$.
\end{proof}
\subsection{\mbox{\sf NP}-hardness of $2$-{\sf SMP}}
Highway problem can be defined as follows: We are given a line $P = (v_0,\ldots, v_n)$ consisting of $n$ edges and $n+1$ vertices and a set of consumers ${\mathcal{C}}$ where each consumer ${\bf C}$ corresponds to a subpath $P_{{\bf C}}$ of $P$ and is equipped with budget $B_{\bf C}$. Our goal is to set price to edges so as to maximize the revenue, where each consumer ${\bf C}$ buys a path $P_{{\bf C}}$ if she can afford the whole path; otherwise she buys nothing.
\begin{lemma}
There is a polynomial-time algorithm that transforms an instance of Highway problem to an instance of $2$-{\sf SMP}.
\end{lemma}
\begin{proof}
For each $i=1,\ldots, n$, each edge $(v_{i-1}, v_i)$, we create an item ${\bf I}_i$ at coordinates $(i, n+1-i)$. Then for each consumer ${\bf C}$ whose path is $P_{\bf C} = (v_s,\ldots, v_t)$, we create a consumer point at $(s+1, n+1-t)$. It is easy to see that the consideration set remains unchanged.
\end{proof}
\subsection{\mbox{\sf APX}-hardness of $4$-{\sf SMP}}
We perform a reduction from {\sf Graph Vertex Pricing} on bipartite graphs. In this problem, we are given a graph $G=(V,E)$, where each vertex corresponds to item and each edge $e \in E$ corresponds to a consumer, additionally equipped with budget $B_e$. Each consumer edge is interested in items that correspond to her incident vertices. Our goal is to set a price $p: V \rightarrow \ensuremath{\mathbb R}$ so as to maximize our revenue.
Given an instance $(G, \set{B_e}_{e \in E})$ of {\sf Graph Vertex Pricing} where graph $G$ is a bipartite graph $(U \cup W, E)$, we create an instance of $4$-{\sf SMP} as follows. Suppose we have $U = \set{u_1,\ldots, u_{|U|}}$ and $W = \set{w_1,\ldots, w_{|W|}}$. For each vertex $u_i \in U$, we have a corresponding item ${\bf I}^u_{i}$ with coordinates $(i, |U|+1-i, \infty,\infty)$. Similarly, for each vertex $w_j \in W$, we have a corresponding item ${\bf I}^w_{j}$ with coordinates $(\infty,\infty,j, |W|+1-j)$. Finally, for each edge $(u_i, w_j) \in E$, we have a consumer ${\bf C}_{ij} = (i, |U|+1-i, j, |W|+1-j)$, whose budget is the same as the budget of edge $(u_i,w_j)$. The following claim is almost immediate.
\begin{claim}
For each consumer ${\bf C}_{ij}$, we have that $S_{{\bf C}_{ij}} = \set{{\bf I}^u_i, {\bf I}^w_j}$
\end{claim}
\begin{proof}
It is easy to see that $\set{{\bf I}^u_i, {\bf I}^w_j} \subseteq S_{{\bf C}_{ij}}$. Notice that for $i'<i$, we have ${\bf C}_{ij}[1] > {\bf I}^u_{i'}[1]$, so any such item ${\bf I}^u_{i'}$ cannot belong to $S_{{\bf C}_{ij}}$. Similarly for $i' >i$, we have ${\bf C}_{ij}[2] > {\bf I}^u_{i'}[2]$, so such an item cannot belong to $S_{{\bf C}_{ij}}$. By using similar arguments for items of the form ${\bf I}^w_{j'}$ for $j' \neq j$, we reach the conclusion that $S_{{\bf C}_{ij}} = \set{{\bf I}^u_i, {\bf I}^w_j}$.
\end{proof}
Since the $4$-{\sf SMP} instance is equivalent to the instance of {\sf Graph Vertex Pricing}, the maximum revenue is preserved. Using the \mbox{\sf APX}-hardness result of {\sf Graph Vertex Pricing} on bipartite graphs~\cite{KhandekarKMS09}, we conclude that $4$-{\sf SMP} is \mbox{\sf APX}-hard.
\section{Sub-linear Approximation Algorithm (Proof of Theorem~\ref{thm:intro thm udp smp})}\label{sec:d udp min}
To simplify the presentation, we present the algorithm for $d$-{\sf UUDP-MIN} in this section. The algorithm for $d$-{\sf SMP} is almost identical. Let ${\mathcal{C}}$ and ${\mathcal{I}}$ be the set of points in $\mathbb{R}^d$, where every consumer ${\bf C}\in{\cal C}$ has budget $B_{{\bf C}}$ and consideration set $S_{{\bf C}}$ which is specified by coordinates of the input point. For any subset $\mathcal{C}'\subseteq{\mathcal{C}}$ and ${\cal I}'\subseteq {\mathcal{I}}$, let
$\mathcal{P}({\mathcal{C}}',{\mathcal{I}}')$ be the $d$-{\sf UUDP-MIN} problem with input ${\mathcal{C}}'$ and ${\mathcal{I}}'$. Moreover, for any ${\mathcal{C}}'$ and ${\mathcal{I}}'$, we use $\mbox{\sf OPT}({\mathcal{C}}', {\mathcal{I}}')$ to express the optimal revenue of the instance $({\mathcal{C}}', {\mathcal{I}}')$. At a high level, our algorithm proceeds in an inductive manner and obtains a solution of $d$-{\sf UUDP-MIN} problem by invoking the algorithms for $(d-1)$-{\sf UUDP-MIN} and $1$-{\sf UUDP-MIN} as a subroutine. Our result is summarized in the following theorem.
\begin{theorem}
\label{thm: dimension reduction for UDP}
For any $\epsilon \in (0,1]$, if there is an $\tilde O_d(n^{1-\epsilon})$-approximation algorithm for $(d-1)$-{\sf UUDP-MIN} then there is an $\tilde O_d(n^{1-\epsilon/4})$-approximation algorithm for $d$-{\sf UUDP-MIN} as well.
\end{theorem}
Theorem~\ref{thm:intro thm udp smp} then follows from the fact that $1$-{\sf UUDP-MIN} can be solved optimally in polynomial time (see Appendix~\ref{sec:one udp min}\fullonly{ and its generalization in Section~\ref{sec: algorithms for small d}}). As we noted earlier, it can be improved slightly since $2$-{\sf UUDP-MIN} admits {\sf QPTAS} \sodaonly{(see Appendix~\ref{sec: qptas 2 udp})}\fullonly{(see Section~\ref{subsection: qptas minbuying})}.
\subsection{Consideration-preserving Decomposition}
Our algorithm partitions the input instance into many subinstances and tries to collect the profit from some of them. The notion of consideration-preserving decomposition, defined below, allows us to do so without losing revenue.
\begin{definition}\label{def:decomposition} We call a collection $\set{({\mathcal{C}}'_1,{\mathcal{I}}'_1),\ldots, ({\mathcal{C}}'_k, {\mathcal{I}}'_k)}$ a {\em consideration-preserving decomposition} of the problem $(\mathcal{C}, \mathcal{I})$ if and only if for any ${\bf C} \in {\mathcal{C}}$ and ${\bf I} \in S_{{\bf C}}$, there exists (not necessarily unique) $i$ such that ${\bf C} \in {\mathcal{C}}'_i$ and ${\bf I} \in {\mathcal{I}}'_i$.
\end{definition}
By definition, for any consumer ${\bf C}$ and item ${\bf I}$ the fact that consumer ${\bf C}$ considers item ${\bf I}$ is preserved by at least one instance $({\mathcal{C}}'_i, {\mathcal{I}}'_i)$. The following lemma says that this decomposition preserves the total revenue.
\begin{lemma}
\label{lemma: UDP decomposition}
For any consideration-preserving decomposition $\left\{({\mathcal{C}}'_1,{\mathcal{I}}'_1), \ldots, ({\mathcal{C}}'_k, {\mathcal{I}}'_k)\right\}$ of $({\mathcal{C}}, {\mathcal{I}})$, it holds that
$\sum_{i=1}^k \mbox{\sf OPT}({\mathcal{C}}'_i, {\mathcal{I}}'_i)\geq \mbox{\sf OPT}({\mathcal{C}}, {\mathcal{I}})
\,.$
Moreover, any price function for ${\mathcal{P}}({\mathcal{C}}'_i, {\mathcal{I}}'_i)$ can be extended to a price function for the original problem ${\mathcal{P}}({\mathcal{C}}, {\mathcal{I}})$ that gives revenue at least $\mbox{\sf OPT}({\mathcal{C}}'_i, {\mathcal{I}}'_i)$.
\end{lemma}
\iffalse
\sodaonly{\begin{proof}[Proof Idea]}
\fullonly{\begin{proof}[Proof idea]}
(See Appendix~\ref{sec: proof of UDP decomposition lemma} for the full proof.)
Let $p^{\star}:{\mathcal{I}}\rightarrow\mathbb{R}$ be the optimal price function for ${\mathcal{P}}({\mathcal{C}}, {\mathcal{I}})$. It is straightforward to see that, for any consumer ${\bf C}$, the optimal revenue that ${\bf C}$ gets from any subset of ${\mathcal{I}}$ is at least the optimal revenue ${\bf C}$ gets when considering the set ${\mathcal{I}}$. This proves the first statement.
For the second one, let $p$ be a price function for ${\mathcal{P}}({\mathcal{C}}_i, {\mathcal{I}}_i)$. We extend $p$ to $p':{\mathcal{I}}\rightarrow\ensuremath{\mathbb R}$ defined by $p'({\bf I})=p({\bf I})$ if ${\bf I}\in{\mathcal{I}}_i$, and $p'({\bf I})=\infty $ otherwise.
\end{proof}
\fi
This is simply by applying the optimal price function of one problem to the other (see Appendix~\ref{sec: proof of UDP decomposition lemma} for the full proof). In the rest of our discussion, we mainly use two different types of consideration-preservation decomposition, as explained in the following observation.
\begin{observation}\label{observation: conseration preserving decomposition}
Given an input instance $({\mathcal{C}}',{\mathcal{I}}')$, let ${\mathcal{C}}' = \bigcup_{i=1}^k {\mathcal{C}}'_i$. Then $\{({\mathcal{C}}'_1,{\mathcal{I}}')$, $\ldots$, $({\mathcal{C}}'_k, {\mathcal{I}}')\}$ is a consideration-preserving decomposition of $({\mathcal{C}}', {\mathcal{I}}')$. Similarly, if ${\mathcal{I}}' = \bigcup_{i=1}^k {\mathcal{I}}'_i$, then we have that $\set{({\mathcal{C}}',{\mathcal{I}}'_1), \ldots, ({\mathcal{C}}', {\mathcal{I}}'_k)}$ is a consideration-preserving decomposition of $({\mathcal{C}}', {\mathcal{I}}')$.
\end{observation}
\subsection{Algorithm}
\begin{wrapfigure}{r}{0.5\textwidth}
\vspace{-1.5cm}
\center\includegraphics[width=\linewidth]{overview.pdf}
\vspace{-.5cm}
\caption{Decomposition overview}\label{fig:overview}
\vspace{-.5cm}
\end{wrapfigure}
At a high level, the algorithm proceeds in four steps where each step involves consideration-preserving decomposition (see Fig.~\ref{fig:overview} for an overview). In Step 1, we partition ${\mathcal{I}}$ into different subsets where
every subset satisfies certain properties, i.e. the elements in each subset either form a chain or an antichain. The problem on those subsets
in which elements form a chain can be solved easily, and we deal with the antichains in later steps. In Step 2, we partition consumers in ${\mathcal{C}}$ into two types, those with large and small consideration sets. We use the algorithm of \cite{BalcanB07,BriestK11} to deal with consumers with small consideration sets and handle the rest consumers in later steps. In Step 3, we find a subset of items, i.e. a ``hitting set'', and partition consumers further into several sets. Each set of consumers has the following property: There is some item desired by all consumers in the set. Using this property, we show in Step 4 that the problem can be further partitioned into a few problems where each of them can be viewed as a $(d-1)$-{\sf UUDP-MIN} problem. (We call this a ``consideration-preserving embedding''.)
\paragraph{\underline{Step 1:} Partitioning items into chains and antichains}
Let $({\mathcal{C}}, {\mathcal{I}})$ be an input of $d$-{\sf UUDP-MIN}. First we define a partially ordered set $({\mathcal{I}}, \leq)$ on the item set as follows. We say that ${\bf I}_1 \leq {\bf I}_2$ if and only if ${\bf I}_1$ has a lower quality than ${\bf I}_2$ in every attribute, i.e. ${\bf I}_1[d'] \leq {\bf I}_2[d']$ for all $d' \in [d]$.
We say that a subset ${\mathcal{I}}' \subseteq {\mathcal{I}}$ is a chain if ${\mathcal{I}}'$ can be written as ${\mathcal{I}}'= \set{{\bf I}_1,\ldots, {\bf I}_z}$ such that ${\bf I}_j \leq {\bf I}_{j+1}$ for all $j\in[z-1]$. We say that ${\mathcal{I}}' \subseteq {\mathcal{I}}$ is an antichain if and only if for any pair of items ${\bf I}, {\bf I}' \in {\mathcal{I}}'$, neither ${\bf I} \leq {\bf I}'$ nor ${\bf I}' \leq {\bf I}$.
\begin{lemma}
\label{lemma: chain decomposition}
For any $\epsilon >0$ and any $s = n^{\epsilon/4}, t= n^{1-\epsilon/4}$, we can partition ${\mathcal{I}}$ into $A_1, \ldots, A_s$ and $B_1, \ldots, B_t$ in polynomial-time. Moreover, each $A_i$ is an antichain and each $B_j$ is a chain.
\end{lemma}
\sodaonly{\begin{proof}[Proof Idea]}
\fullonly{\begin{proof}[Proof idea]}
(See Section~\ref{sec:detail one} for detailed definitions and proofs.)
By Dilworth's theorem \cite{Dilworth,Fulkerson-Dilworth}, the minimum chain decomposition equals to the maximum antichain size. We will use the fact that both minimum chain decomposition and maximum-size antichain can be computed in polynomial time as follows:
As long as the maximum-size antichain is bigger than $n^{\epsilon/4}$, we repeatedly extract such an antichain out of the input; otherwise, we would have the decomposition into at most $n^{\epsilon/4}$ chains, so we stop.
\end{proof}
By Observation~\ref{observation: conseration preserving decomposition}, the collection
$\{({\mathcal{C}}, A_1), \ldots, ({\mathcal{C}}, A_s), ({\mathcal{C}}, B_1), \ldots, ({\mathcal{C}}, B_t)\}$
is a consideration-preserving decomposition of $({\mathcal{C}}, {\mathcal{I}})$. It follows by Lemma~\ref{lemma: UDP decomposition} that
$\sum_{i=1}^s\mbox{\sf OPT}({\mathcal{C}}, A_i)+\sum_{j=1}^t \mbox{\sf OPT}({\mathcal{C}}, B_j) \geq \mbox{\sf OPT}({\mathcal{C}}, {\mathcal{I}}).$
Further, observe that if there exists $j$ such that $\mbox{\sf OPT}({\mathcal{C}}, B_j)\geq\mbox{\sf OPT}({\mathcal{C}}, {\mathcal{I}})/(2n^{1-\epsilon/4})$, then we would be done: the $d$-{\sf UUDP-MIN} problem ${\mathcal{P}}({\mathcal{C}}, B_j)$ can be seen as a $1$-{\sf UUDP-MIN} problem (since $B_j$ is a chain) and hence can be solved optimally! (See Lemma~\ref{lem:chain to one dim} for detailed analysis)
Otherwise $\mbox{\sf OPT}({\mathcal{C}}, B_j)\leq \mbox{\sf OPT}({\mathcal{C}}, {\mathcal{I}})/ (2n^{1-\epsilon/4})$ for every $j$. Therefore
$\sum_{j=1}^t \mbox{\sf OPT}({\mathcal{C}}, B_j) \leq n^{1-\varepsilon/4}\cdot\mbox{\sf OPT}({\mathcal{C}},{\mathcal{I}})/ (2n^{1-\varepsilon/4})< \mbox{\sf OPT}({\mathcal{C}},{\mathcal{I}})/2.$
If this is not the case then we know that there must be an antichain $A_i$ such that
$\mbox{\sf OPT}({\mathcal{C}}, A_i)\geq \mbox{\sf OPT}({\mathcal{C}},{\mathcal{I}})/2n^{\epsilon/4}\,.$
\begin{wrapfigure}{r}{.5\textwidth}
\vspace{-.9cm}
\center\includegraphics[width=\linewidth]{step1.pdf}\\
\vspace{-.3cm}\caption{Example of Step 1}\label{fig:step1}\vspace{-.2cm}
\end{wrapfigure}
\paragraph{\underline{Step 2:} Dealing with small consideration sets}
For simplicity, let us assume that we know $i$ such that $\mbox{\sf OPT}({\mathcal{C}}, A_i)\geq \mbox{\sf OPT}({\mathcal{C}}, {\mathcal{I}})/(2n^{\epsilon/4})$. Now we focus on collecting revenue from the subproblem ${\mathcal{P}}({\mathcal{C}}, A_i)$. Let ${\mathcal{C}}_1 \subseteq {\mathcal{C}}$ be the set of consumers who are interested in at most $n^{1-2\epsilon/4}$ items in $A_i$, i.e. ${\mathcal{C}}_1=\left\{{\bf C}\in {\mathcal{C}}: |S_{\bf C}\cap A_{i}|\leq n^{1-2\epsilon/4}\right\}$, and define ${\mathcal{C}}_2={\mathcal{C}}\setminus {\mathcal{C}}_1$.
Since $\{({\mathcal{C}}_1, A_i), ({\mathcal{C}}_2, A_i)\}$ is a consideration-preserving decomposition of $({\mathcal{C}}, A_i)$, we have
$\mbox{\sf OPT}({\mathcal{C}}_1, A_i)+\mbox{\sf OPT}({\mathcal{C}}_2, A_i) \geq \mbox{\sf OPT}({\mathcal{C}}, A_i) \geq \frac{\mbox{\sf OPT}({\mathcal{C}}, {\mathcal{I}})}{2n^{\epsilon/4}}.$
Now we need an algorithm of \cite{BalcanB07,BriestK11}. Balcan and Blum give an approximation algorithm for {\sf SMP} whose approximation guarantee depends on the sizes of consideration sets. Briest and Krysta, by using a slight modification of this algorithm, give an approximation algorithm with the same guarantee for {\sf UDP-MIN}. Their result, stated in terms of {\sf UUDP-MIN}, is summarized in the following theorem. (For completeness, we provide the proof in Appendix~\ref{sec:balcanblum for udp min}.)
\begin{theorem}\label{theorem:BalcanBlum}\cite{BalcanB07,BriestK11}
Given a {\sf UUDP-MIN} instance $({\mathcal{C}}, {\mathcal{I}}, \set{S_{{\bf C}}}_{{\bf C} \in {\mathcal{C}}})$, there is a deterministic $O(k)$-approximation algorithm of {\sf UUDP-MIN}, where $k: = \max_{{\bf C} \in {\mathcal{C}}} \size{S_{{\bf C}}}$.
\end{theorem}
We remark that we extend this technique to deal with any pricing problem with subadditive revenue in the full version of this paper.
If $\mbox{\sf OPT}({\mathcal{C}}_1, A_i)\geq \mbox{\sf OPT}({\mathcal{C}}, {\mathcal{I}})/(4n^{\epsilon/4})$, then we could invoke the algorithm in Theorem~\ref{theorem:BalcanBlum} on $({\mathcal{C}}_1, A_i)$ to get a solution with approximation ratio
$O\left(\max_{{\bf C} \in {\mathcal{C}}_1} \size{S_{{\bf C}}\cap A_i}\right)=O(n^{1-2\epsilon/4}).$
This yields a solution that gives a desired revenue of
$\Omega\left(\mbox{\sf OPT}({\mathcal{C}}_1, A_i)/n^{1-2\epsilon/4}\right)=\Omega\left(\mbox{\sf OPT}({\mathcal{C}}, {\mathcal{I}})/n^{1-\epsilon/4}\right)\,.$
Otherwise we have $\mbox{\sf OPT}({\mathcal{C}}_1, A_i)< \mbox{\sf OPT}({\mathcal{C}}, {\mathcal{I}})/4n^{\epsilon/4}$. Then
$\mbox{\sf OPT}({\mathcal{C}}_2, A_i)=\Omega\left(\mbox{\sf OPT}({\mathcal{C}}, {\mathcal{I}})/n^{\epsilon/4}\right).$
We will deal with this case in the next steps.
\paragraph{\underline{Step 3:} Partitioning consumers using a small hitting set}
First, we apply the epsilon net theorem~\cite{Chazelle-book,Epsilon-net} to derive the following lemma.
\begin{lemma}\label{lem:hitting set}
We can find a set $H\subseteq A_i$ of size $\tilde O(n^{2\epsilon})$ in randomized polynomial time such that for any ${\bf C}\in {\mathcal{C}}_2$, there exists ${\bf I}\in H$ such that ${\bf I}\geq {\bf C}$.
\end{lemma}
\begin{proof}
\danupon{We should consider providing a bit more detail (``a primer'') in the Appendix.}
The instance $(\mathcal{C}_2, A_i)$ defines a set system $\{S_{\bf C}\}_{{\bf C}\in \mathcal{C}_2}$ over $A_i$, where $S_{\bf C}=\{{\bf I}\in A_i\mid {\bf I}\geq {\bf C}\}$. We note that each set $S_{{\bf C}}$ has {\em descriptive complexity} at most $d$, i.e. set $S_{{\bf C}}$ can be described by $d$ linear inequalities of the form $S_{{\bf C}} = \bigcap_{d'=1}^d \set{{\bf I} \in {\mathcal{I}}: {\bf I}[d'] \geq {\bf C}[d']}$. In this case, this set system has VC dimension $O(d)$, c.f. \cite{Sharir-book}. More specifically, it is well known (e.g., \cite{AronovES10}) that any collection of $d$-dimensional axis-parallel boxes has VC dimension $O(d)$. We will not formally define VC-dimension here. The following theorem is all we need.
\begin{theorem}(\cite{Chazelle-book,Epsilon-net}; Epsilon net theorem)
Let ${\mathcal{X}}$ be a set system of VC-dimension at most $d'$ over $N$. Then for any $\delta \in (0,1)$, we can find a set $H \subseteq N$ with $\size{H} = O(\frac{d'}{\delta}\log \frac{d'}{\delta})$ in randomized polynomial time such that, for all $X_i \in {\mathcal{X}}$ with $\size{X_i} \geq \delta \size{N}$, it holds that $H \cap X_i \neq \emptyset$.
\end{theorem}
Using the theorem with $\delta = n^{-2\epsilon/4}$, we can find a set $H \subseteq A_i$ of size at most $\tilde{O}(n^{2\epsilon/4})$, and since we have $|S_{{\bf C}} \cap A_i| \geq \delta n$ for all ${\bf C} \in {\mathcal{C}}_2$, we are guaranteed that $H \cap S_{{\bf C}} \neq \emptyset$ for all ${\bf C} \in {\mathcal{C}}_2$.
\end{proof}
We call $H$ a {\em hitting set} of $\mathcal{C}_2$ since $H$ intersects $S_{\bf C}$ for all ${\bf C}\in \mathcal{C}_2$. We use $H$ to decompose $(\mathcal{C}_2, A_i)$ into a small number of subproblems and show in Step 4 that each of these problems can be viewed as a $(d-1)$-{\sf UUDP-MIN} problem.
For each ${\bf I}\in H$, let ${\mathcal{C}}_{\bf I}=\{{\bf C}\in {\mathcal{C}}_2\mid {\bf I} \in S_{\bf C}\}$, i.e., ${\mathcal{C}}_{\bf I}$ consists of all consumers in ${\mathcal{C}}_2$ that consider item ${\bf I}$. Observe that $\bigcup_{{\bf I} \in H} {\mathcal{C}}_{{\bf I}} = {\mathcal{C}}_2$, and therefore by Lemma~\ref{lemma: UDP decomposition}, we have
$\sum_{{\bf I} \in H} \mbox{\sf OPT}({\mathcal{C}}_{\bf I}, A_i)\geq \mbox{\sf OPT}({\mathcal{C}}_2, A_i)
\geq \Omega\left(\mbox{\sf OPT}({\mathcal{C}}, {\mathcal{I}})/n^{\epsilon/4}\right).$
Since $\size{H}=O(n^{2\epsilon/4})$, there exists ${\bf I}^*\in H$ such that
\begin{align*}
\mbox{\sf OPT}({\mathcal{C}}_{{\bf I}^*}, A_i) &=\tilde\Omega\left(\mbox{\sf OPT}(\mathcal{C}, \mathcal{I})\cdot n^{-\epsilon/4}/|H|\right)
=\tilde\Omega\left(\mbox{\sf OPT}(\mathcal{C}, \mathcal{I})/n^{3\epsilon/4}\right).
\end{align*}
Now we, again, assume that we know ${\bf I}^*$ and turn our focus to the subproblem ${\mathcal{P}}({\mathcal{C}}_{{\bf I}^*}, A_i)$.
\paragraph{\underline{Step 4:} Reducing the dimension} We have now reached the most crucial step.
We will (crucially) rely on the fact that all consumers in ${\mathcal{C}}_{{\bf I}^*}$ consider item ${\bf I}^*$, and that $A_i$ is an antichain.
For each $j \leq d$, define $A_i^j$ as the set of items in $A_i$ that are at least as good as ${\bf I}^*$ in the $j$-th coordinate, i.e., $A_i^j=\{{\bf I}\in A_i \mid {\bf I}[j]\geq {\bf I}^*[j]\}$. See Fig.~\ref{fig:step4_1} for an example in the case of $2$-{\sf UUDP-MIN}.
\begin{lemma}\label{lem:union of d dimensions}
$A_i = \bigcup_{j=1}^d A_i^j$.
\end{lemma}
\begin{figure}
\begin{center}
\subfigure[]{
\includegraphics[height=0.12\textheight, clip=true, trim=1.5cm .3cm 1cm .3cm]{step3.pdf}
\label{fig:step4_1}
}
\subfigure[]{
\includegraphics[height=0.12\textheight, clip=true, trim=3cm .75cm 7cm 1cm]{step4.pdf}
\label{fig:step4}
}
\end{center}
\vspace{-.5cm}
\caption{\subref{fig:step4_1} Example of Step 4. \subref{fig:step4} Example of Step 4 when we view the instance $\mathcal{C}_{{\bf I}^*}, A_i^j$ as a $(d-1)$-{\sf UUDP-MIN} instance.}
\vspace{-.5cm}
\end{figure}
This lemma holds simply because $A_i$ is an antichain (in any antichain, no item can completely dominate the others, so at least one coordinate of any ${\bf I}\in \mathcal{I}_{{\bf I}^*}$ has to be at least as good as ${\bf I}^*$; see detailed proof in Appendix~\ref{proof:union of d dimensions}). Then $\{(\mathcal{C}_{{\bf I}^*}, A_i^1), \ldots, (\mathcal{C}_{{\bf I}^*}, A_i^d)\}$ is a consideration-preserving decomposition of $(\mathcal{C}_{{\bf I}^*}, A_i)$ and thus there exists $j$ such that
$\mbox{\sf OPT}(\mathcal{C}_{{\bf I}^*}, A_i^j) \geq \mbox{\sf OPT}({\mathcal{C}}_{{\bf I}^*}, A_i)/d = \tilde\Omega_d(\mbox{\sf OPT}(\mathcal{C}, \mathcal{I})/n^{3\epsilon/4}).$
Observe that, for all ${\bf C}\in {\mathcal{C}}_{{\bf I}^*}$ and ${\bf I} \in A_i^j$, ${\bf C}[j]\leq {\bf I}^*[j]\leq {\bf I}[j]$. This implies that we can ignore the $j$-th coordinate when we solve ${\mathcal{P}}({\mathcal{C}}_{{\bf I}^*}, A_i^j)$. (In particular, for any ${\bf C}\in\mathcal{C}_{{\bf I}^*}$, the consideration set $S_{\bf C}=\left\{{\bf I}\geq {\bf C}\mid {\bf I}\in A_i^j\right\}$ remains the same even when we drop the $j$-th coordinate of all points.) In other words, the problem can be viewed as a $(d-1)$-{\sf UUDP-MIN} problem (see Fig.~\ref{fig:step4} for an idea).
We defer the formal statement and proof of this claim to Section~\ref{sec:detail four}.
Finally, we can invoke the $\tilde O_d(n^{1-\epsilon})$-approximation algorithm for $(d-1)$-{\sf UUDP-MIN} to collect the revenue of
$\tilde\Omega_d\left(\mbox{\sf OPT}({\mathcal{C}}, {\mathcal{I}}) n^{-3\epsilon/4}/n^{1-\epsilon}\right)$ $=\tilde\Omega_d\left(\mbox{\sf OPT}({\mathcal{C}}, {\mathcal{I}})/n^{1-\epsilon/4}\right).$ Therefore we obtain an approximation ratio of $\tilde O_d(n^{1-\epsilon/4})$ in all cases. Algorithm~\ref{algo:udp} summaries our algorithm for solving $d$-{\sf UUDP-MIN}.
\begin{algorithm}
\caption{\footnotesize {\sf UUDP-MIN-APPROX}($d$)}\label{algo:udp}
\begin{algorithmic}[1]
\footnotesize
\IF{$d=1$}
\STATE Solve the problem $\mathcal{P}(\mathcal{C}, \mathcal{I})$ optimally using an algorithm for $1$-{\sf UUDP-MIN} (cf. Appendix~\ref{sec:one udp min})
\ELSE
\STATE Partition $\mathcal{I}$ into antichains $A_1, \ldots, A_s$ and chains $B_1, \ldots, B_t$ where $s\leq n^{\epsilon/4}$ and $t\leq n^{1-\epsilon/4}$ as in Step 1.
\STATE We claim that the problems $\mathcal{P}(\mathcal{C}, B_1), \ldots, \mathcal{P}(\mathcal{C}, B_t)$ are equivalent to $1$-{\sf UUDP-MIN} problems (cf. Section~\ref{sec:detail one}). Solve them optimally using an algorithm for $1$-{\sf UUDP-MIN} (cf. Appendix~\ref{sec:one udp min}).
\FOR{$i=1, \ldots, s$}
\STATE Partition $\mathcal{C}$ into $\mathcal{C}_1$ and $\mathcal{C}_2$ as in Step 2. Find an $O(\max_{{\bf C} \in {\mathcal{C}}_1} \size{S_{{\bf C}}\cap A_i})$=$O(n^{1-2\epsilon/4})$ approximate solution of problem $\mathcal{P}(\mathcal{C}_1, A_i)$.
\STATE Find a hitting set $H$ of $(\mathcal{C}_2, A_i)$ as in Step 3
\FOR{each ${\bf I}\in H$}
\STATE Define $\mathcal{C}_{\bf I}$ as in Step 3
\STATE Define $A_i^1, \ldots, A_i^d$ as in Step 4
\STATE Solve problem $\mathcal{P}(\mathcal{C}_{\bf I}, A_i^1), \ldots, \mathcal{P}(\mathcal{C}_{\bf I}, A_i^d)$ using an $O(n^{1-\epsilon})$-approximation algorithm for $(d-1)$-{\sf UUDP-MIN}
\ENDFOR
\ENDFOR
\ENDIF
\RETURN the solution with highest revenue among the solutions of all solved problems
\end{algorithmic}
\end{algorithm}
\subsection{Consideration-preserving Embedding}\label{sec:detail four}
To formally discuss the reduction of dimensions, we introduce the notion of consideration-preserving embedding. For any $d$, let $(\mathcal{C}, \mathcal{I})$ be any instance of $d$-{\sf UUDP-MIN}. For any $d'$, consider one-to-one functions $f$ and $g$ that map points in $\ensuremath{\mathbb R}^d$ to the ones in $\ensuremath{\mathbb R}^{d'}$. We say that $(f, g)$ is a {\em consideration-preserving embedding} if, for any item ${\bf I} \in {\mathcal{I}}$ and consumer ${\bf C} \in {\mathcal{C}}$, we have that ${\bf I} \geq {\bf C}$ if and only if $g({\bf I}) \geq f({\bf C})$. That is, the fact that consumer ${\bf C}$ is considering or not considering item ${\bf I}$ must be preserved in $f({\bf C})$ and $g({\bf I})$.
Given a consideration-preserving embedding $(f,g)$, we can naturally define a $d'$-{\sf UUDP-MIN} problem ${\mathcal{P}}(f({\mathcal{C}}), g({\mathcal{I}}))$ where $f({\mathcal{C}})=\{f({\bf C})\mid {\bf C}\in {\mathcal{C}}\}$, $g({\mathcal{I}})=\{g({\bf I})\mid {\bf I} \in {\mathcal{I}}\}$ \danupon{Should we say this? ``(note that we allow the sets to contain identical points)''} and the budget $B_{f({\bf C})}$ is $B_{\bf C}$ for any ${\bf C}\in{\mathcal{C}}$.
Observe that, although $(\mathcal{C}, \mathcal{I})$ and $(f(\mathcal{C}), g(\mathcal{I}))$ correspond to points on different spaces, they represent the same pricing problem (i.e., the consumers' consideration sets and budgets are exactly the same). Thus, we sometimes say that $(\mathcal{C}, \mathcal{I})$ and $(f(\mathcal{C}), g(\mathcal{I}))$ are {\em equivalent}. The following observation follows trivially.
\begin{observation}
\label{observation: consideration preserving embedding}
For any instance $({\mathcal{C}},{\mathcal{I}})$, let $(f,g)$ be a consideration-preserving embedding of $({\mathcal{C}}, {\mathcal{I}})$ into $\ensuremath{\mathbb R}^{d'}$. Then we have that
$\mbox{\sf OPT}({\mathcal{C}}, {\mathcal{I}}) = \mbox{\sf OPT}(f({\mathcal{C}}), g({\mathcal{I}})).$
Moreover, if $f$ and $g$ are polynomial-time computable then a solution for $\mathcal{P}(f({\mathcal{C}}), g({\mathcal{I}}))$ can be efficiently transformed into one for $\mathcal{P}({\mathcal{C}}, {\mathcal{I}})$ that gives the same revenue.
\end{observation}
The transformation in the above lemma is trivial: For any price function $p$ for $(f({\mathcal{C}}), g({\mathcal{I}}))$, we simply price item ${\bf I}\in {\mathcal{I}}$ to $p(g({\bf I}))$. Observe that we will receive the same revenue from both problems using this pricing strategy.
In Step 1, we claimed that when the items form a chain, our instance would be equivalent to $1$-{\sf UUDP-MIN}. Now we prove this fact formally below.
\begin{lemma}\label{lem:chain to one dim}
Let $({\mathcal{C}}, {\mathcal{I}})$ be a $d$-{\sf UUDP-MIN} instance where $({\mathcal{I}}, \leq)$ is a chain. Then $(\mathcal{C}, \mathcal{I})$ is equivalent to a $1$-{\sf UUDP-MIN} instance. Moreover, the corresponding consideration-preserving embedding $(f, g)$ can be computed in polynomial time.
\end{lemma}
\begin{proof}
Order items in $\mathcal{I}$ by ${\bf I}_1\leq {\bf I}_2\leq \ldots$. Now map each item into a one-dimensional point: $g({\bf I}_i)=(i)$. Moreover, map each consumer according to $f({\bf C})=g({\bf I}_i)$, where $i$ is the minimum number such that ${\bf I}_i\geq {\bf C}$. Observe that $(f, g)$ is a consideration-preserving embedding since $S_{\bf C}=\{{\bf I}_i, {\bf I}_{i+1}, \ldots\}$ while $S_{f({\bf C})}=\{g({\bf I}_i), g({\bf I}_{i+1}), \ldots\}$ for any ${\bf C}\in {\mathcal{C}}$. (Note that this embedding might create redundancy since it is possible that $f({\bf C})=f({\bf C}')$ for some ${\bf C}\neq {\bf C}'$. This can be fixed easily by slightly perturbing the points.\danupon{I'm not sure if this is necessary.})
\end{proof}
In Step 4, we also claimed the dimension reduction of sub-instances $({\mathcal{C}}_{{\bf I}^*}, A_i^j)$, and we now prove the claim formally. Recall that the item ${\bf I}^*\in A_i^j$ has the property that ${\bf I}^*\geq {\bf C}$ for all ${\bf C} \in \mathcal{C}_{{\bf I}^*}$ and ${\bf I}^*[j]\leq {\bf I}[j]$ for all ${\bf I}\in A_i^j$.
\begin{lemma}\label{lem:dim reduction}
The instance
$(\mathcal{C}_{{\bf I}^*}, A_i^j)$ is equivalent to a $(d-1)$-{\sf UUDP-MIN} instance. Moreover, the corresponding consideration-preserving embedding $(f, g)$ can be computed in polynomial time.
\end{lemma}
\begin{proof}
Consider ``ignoring'' the $j$-th coordinate as follows. For any ${\bf C}\in \mathcal{C}_{{\bf I}^*}$ and ${\bf I}\in A_i^j$, let $f({\bf C})=({\bf C}[1], {\bf C}[2], \ldots, {\bf C}[j-1], {\bf C}[j+1], \ldots, {\bf C}[d])$
and
$g({\bf I})=({\bf I}[1], {\bf I}[2], \ldots, {\bf I}[j-1], {\bf I}[j+1], \ldots, {\bf I}[d]).$
Observe that for any ${\bf C}\in \mathcal{C}_{{\bf I}^*}$ and ${\bf I}\in A_i^j$, ${\bf I}\geq {\bf C}$ trivially implies that $g({\bf I})\geq f({\bf C})$. Conversely, if $g({\bf I})\geq f({\bf C})$ then ${\bf I}\geq {\bf C}$ since ${\bf I}[j]\geq {\bf I}^*[j]\geq {\bf C}[j]$. Thus, $(f, g)$ is a consideration-preserving embedding.
\end{proof}
\section{Proof Omitted from Section~\ref{sec:d udp min}}
\subsection{Proof of Lemma~\ref{lemma: UDP decomposition}}
\label{sec: proof of UDP decomposition lemma}
Let $p^*$ be the optimal price function for ${\mathcal{P}}({\mathcal{C}}, {\mathcal{I}})$. For each $i=1,\ldots, k$, we define $p^*_i: {\mathcal{I}}'_i \rightarrow \ensuremath{\mathbb R}$ by
$$p^*_i({\bf I}) = p^*({\bf I})~~\mbox{if ${\bf I} \in {\mathcal{I}}'_i$, and $p^*_i({\bf I}) = \infty$ otherwise.}$$
Let $r_i$ be the total revenue made by $p^*_i$ in ${\mathcal{P}}({\mathcal{C}}'_i, {\mathcal{I}}'_i)$. We argue below that
\begin{align}\label{eq:one}
\sum_{i=1}^k r_i \geq \mbox{\sf OPT}({\mathcal{C}}, {\mathcal{I}}).
\end{align}
Let ${\mathcal{C}}^* \subseteq {\mathcal{C}}$ be the set of consumers who make a positive payment with respect to $p^*$. For each consumer ${\bf C} \in {\mathcal{C}}^*$, denote by $\phi({\bf C}) \in {\mathcal{I}}$ the item that consumer ${\bf C}$ buys with respect to the price $p^*$.
So we can write $\mbox{\sf OPT}({\mathcal{C}}, {\mathcal{I}})$ as
\begin{align}\label{eq:two}
\mbox{\sf OPT}({\mathcal{C}}, {\mathcal{I}}) = \sum_{{\bf C} \in {\mathcal{C}}^*} p^*(\phi({\bf C})).
\end{align}
For each $i=1,\ldots, k$, let ${\mathcal{C}}^*_i \subseteq {\mathcal{C}}'_i$ be the set of consumers ${\bf C} \in {\mathcal{C}}'_i$ such that $\phi({\bf C}) \in {\mathcal{I}}'_i$. That is, ${\mathcal{C}}^*_i$ is a set of consumers whose item she bought in $\mbox{\sf OPT}({\mathcal{C}}, {\mathcal{I}})$ is in ${\mathcal{I}}'_i$.
Notice that
\begin{align}\label{eq:three}
r_i \geq \sum_{{\bf C} \in {\mathcal{C}}^*_i} p^*(\phi({\bf C})).
\end{align}
Since $\set{({\mathcal{C}}'_i, {\mathcal{I}}'_i)}_{i=1}^k$ is a consideration-preserving decomposition, we have that
\begin{align}\label{eq:four}
\bigcup_{i=1}^k {\mathcal{C}}^*_i \supseteq {\mathcal{C}}^*,
\end{align}
since for any ${\bf C} \in {\mathcal{C}}^*$, we must have $\phi({\bf C})\in {\mathcal{I}}_i$ for some $i$. By summing Eq.\eqref{eq:three} over all $i=1,\ldots, k$, we have
\begin{align*}
\sum_{i=1}^k r_i &\geq \sum_{i=1}^k \sum_{{\bf C}\in {\mathcal{C}}^*_i} p^*(\phi({\bf C})) &\mbox{(by Eq.\eqref{eq:three})}\\
&\geq \sum_{{\bf C}\in {\mathcal{C}}^*} p^*(\phi({\bf C})) &\mbox{(by Eq.\eqref{eq:four})}\\
&=\mbox{\sf OPT}({\mathcal{C}}, {\mathcal{I}})&\mbox{(by Eq.\eqref{eq:two})}
\end{align*}
This proves Eq.\eqref{eq:one} and thus the first claim.
Now suppose we have a price $p': {\mathcal{I}}_i \rightarrow \ensuremath{\mathbb R}$ that collects revenue $r'$ in ${\mathcal{P}}({\mathcal{C}}'_i, {\mathcal{I}}'_i)$. We define a function $p: {\mathcal{I}} \rightarrow \ensuremath{\mathbb R}$ by $p({\bf I}) = p'({\bf I})$ for ${\bf I} \in {\mathcal{I}}'_i$ and $p({\bf I}) = \infty$ otherwise. We can use $p'$ to obtain a revenue of $r'$ from $\mathcal{P}({\mathcal{C}}, {\mathcal{I}})$. This proves the second claim.
\subsection{Decomposing items into small number of chains and antichains}\label{sec:detail one}
We will use the following theorem, first proved by Dilworth \cite{Dilworth}, and its polynomial computability follows from the equivalence between Dilworth's theorem and K\"onig's theorem~\cite{Fulkerson-Dilworth}.
\begin{theorem}\label{theorem:dilworth}
Let $(S, \leq)$ be a partially ordered set, and $Z$ be the maximum number of elements in any antichain of $S$. Then there is a polynomial-time algorithm that produces a partition of $S$ into $Z$ chains $S_1,\ldots, S_Z$.
\end{theorem}
We now use the theorem to prove Lemma~\ref{lemma: chain decomposition}.
\begin{proof}[of Lemma~\ref{lemma: chain decomposition}]
Initially, let $i=1$. In iteration $i$, we check if the size of maximum antichain in ${\mathcal{I}}$ is at least $t=n^{1-\epsilon/4}$. If so, we find the maximum antichain $A_i$, update ${\mathcal{I}} = {\mathcal{I}} \setminus A_i$, and proceed to the next iteration; otherwise, we stop the iterations. Notice that the number of iterations is at most $s = n^{\epsilon/4}$, and when the iteration stops, the size of maximum-size antichain is at most $t\leq n^{1-\epsilon/4}$. We apply the above theorem to compute a decomposition of ${\mathcal{I}}$ into $t$ chains, denoted by $B_1,\ldots, B_t$. This concludes the proof of Lemma~\ref{lemma: chain decomposition}.
\end{proof}
\subsection{Proof of Balcan-Blum Theorem for {\sf UUDP-MIN} (cf. Theorem~\ref{theorem:BalcanBlum})}\label{sec:balcanblum for udp min}
We first explain a randomized algorithm, and then we discuss how to derandomize it. This part is essentially the same as \cite{BalcanB07,BriestK11}. First, we randomly construct a set ${\mathcal{I}}^* \subseteq {\mathcal{I}}'$ where each item ${\bf I}$ is independently added to ${\mathcal{I}}^*$ with probability $1/k$ (recall that $k=\max_{{\bf C}\in {\mathcal{C}}} |S_{\bf C}|$). Then let ${\mathcal{C}}^*$ be a set of consumer ${\bf C}$ such that $\size{S_{\bf C}\cap {\mathcal{I}}^*}=1$
(i.e. consumers who care about exactly one item in ${\mathcal{I}}^*$). We show that the problem $\mathcal{P}({\mathcal{C}}^*, {\mathcal{I}}^*)$ has expected revenue at least $\Omega(\mbox{\sf OPT}({\mathcal{C}}, {\mathcal{I}})/k)$.
Let $p$ be the optimal price function for $({\mathcal{C}}, {\mathcal{I}})$ and $\phi: {\mathcal{C}} \rightarrow {\mathcal{I}}\cup \{\perp\}$ be a function that maps each consumer to the item she buys with respect to $p$ (let $\phi({\bf C})=\perp$ if consumer ${\bf C}$ buys nothing and $p(\perp)=0$). Therefore, we have that $\mbox{\sf OPT}({\mathcal{C}}, {\mathcal{I}}) = \sum_{{\bf C}} p(\phi({\bf C}))$. We denote by $p^*$ the price function $p$ restricted to ${\mathcal{I}}^*$. For each ${\bf C}$, if ${\bf C} \in {\mathcal{C}}^*$ and $\phi({\bf C}) \in {\mathcal{I}}^*$, the revenue created by $p^*$ in $({\mathcal{C}}^*, {\mathcal{I}}^*)$ would be at least $p(\phi({\bf C}))$. Therefore,
$$\expect{}{\mbox{\sf OPT}({\mathcal{C}}^*, {\mathcal{I}}^*)} \geq \sum_{{\bf C} \in {\mathcal{C}}} \Pr[\mbox{$\phi({\bf C})\in {\mathcal{I}}^*$ and ${\bf C}\in {\mathcal{C}}^*$}] \times p(\phi({\bf C}))\,.$$
Notice that, for any ${\bf C}\in{\mathcal{C}}$ and ${\bf I}\in S_{{\bf C} }$,
\[\Pr[\mbox{${\bf I}\in {\mathcal{I}}^*$ and ${\bf C}\in {\mathcal{C}}^*$}] \geq \frac{1}{k}\left(1-\frac{1}{k}\right)^{k-1}\geq \frac{1}{k\mathrm{e}},\]
which implies that $\expect{}{\mbox{\sf OPT}({\mathcal{C}}^*, {\mathcal{I}}^*)} \geq \frac{1}{k \mathrm{e}}\cdot \mbox{\sf OPT}({\mathcal{C}}, {\mathcal{I}})$.
{\bf Derandomization:} First, note that we can assume that $k=O(\log m+\log n)$. Otherwise, we can use the result of \cite{AggarwalFMZ04,GuruswamiHKKKM05,BalcanBM08} (see \cite[Section 4]{BalcanBM08} for the result in a general setting) to obtain $O(\log m+ \log n)$ approximation algorithm for {\sf UUDP-MIN}, which will also be $O(k)$-approximation.
Now, assuming that $k=O(\log m+\log n)$, we follow the argument of Balcan and Blum~\cite{BalcanB07}. In particular, we observe that we need only $k$-wise independence among the events of the form ``${\bf I}\in {\mathcal{I}}^*$ and ${\bf C}\in {\mathcal{C}}^*$'', for any ${\bf I}$ and ${\bf C}$, in order to get the above expectation result. In this case, we can use the tools from Even et al \cite{EvenGLNV98} to derandomize the above algorithm while blowing up the running time by a factor of $2^{O(k)}=\operatorname{poly}(m, n)$. For more details, we refer the readers to \cite{BalcanB07}.
\subsection{Proof of Lemma \ref{lem:union of d dimensions}}\label{proof:union of d dimensions}
Recall that each $A_i$ is an antichain, i.e., for any distinct ${\bf I}_1, {\bf I}_2\in A_i$, there exists $1\leq d_1, d_2\leq d$ such that ${\bf I}_1[d_1]<{\bf I}_2[d_1]$ and ${\bf I}_1[d_2]>{\bf I}_2[d_2]$. In particular, if ${\bf I}_1={\bf I}^*$, then we have that for any ${\bf I}\in A_i$, there exists coordinate $j$ such that ${\bf I}[j]\geq {\bf I}^*$. This means that ${\bf I}\in A_i^j$. The lemma follows.
\subsection{Polynomial-Time Algorithm for $1$-{\sf UUDP-MIN}}\label{sec:one udp min}
We provide a polynomial-time algorithm for solving $1$-{\sf UUDP-MIN}. Let ${\bf I}_1,\ldots, {\bf I}_n$ be a sequence of items ordered non-increasingly by their coordinates. We can assume without loss of generality that their coordinates are different (by slightly perturbing their values), and we say that consumer ${\bf C}$ is at {\em level $j$} if her coordinate lies between ${\bf I}_{j-1}$ and ${\bf I}_j$. Notice that, for any consumer ${\bf C}$ at level $j$, we have $S_{{\bf C}} = \set{{\bf I}_1,\ldots, {\bf I}_j}$.
\begin{claim}
Let $p^*$ be an optimal price. Then we can assume that $p^*({\bf I}_1) \geq p^*({\bf I}_2) \geq \ldots \geq p^*({\bf I}_n)$.
\end{claim}
\begin{proof}
Suppose that $p^*({\bf I}_i) < p^*({\bf I}_j)$ for some $i <j$. Recall that ${\bf I}_i \geq {\bf I}_j$, so for each consumer ${\bf C}$ such that ${\bf C} \leq {\bf I}_j$, we know that ${\bf C}$ does not buy item ${\bf I}_j$ with respect to this solution. Thus, we can reduce $p^*({\bf I}_i)$ slightly, while maintaining the same revenue.
\end{proof}
The claim will ensure that consumers at level $j$ only buy item ${\bf I}_j$ but not any other items in $\set{{\bf I}_1,\ldots, {\bf I}_{j-1}}$, and this allows us to solve the problem by dynamic programming. For each $j=1,\ldots, n$, for each price $P \in \ensuremath{\mathbb R}$ we have a table entry $T[j,P]$ that keeps the maximum revenue achievable from consumers at levels $1,\ldots, j$ and items $\set{{\bf I}_1,\ldots, {\bf I}_j}$ where the price of ${\bf I}_j$ is set to $P$. Notice that it is easy to compute the profit from consumers at level $j$ if we know $p({\bf I}_j) = P$. Denote such value by $\gamma$. Then we have that $T[j,P] = \gamma + \max_{P' \geq P} T[j-1,P']$. Finally, we note that there are at most $|{\mathcal{C}}|$ possibilities of prices $P$ because one can assume without loss of generality that, for {\sf UUDP-MIN}, the prices always belong to $\set{B_{{\bf C}}}_{{\bf C} \in {\mathcal{C}}}$.
\section{{\sf QPTAS} for $2$-{\sf SMP}}\label{sec:2-SMP}
In this section, we show that {\sf QPTAS} for $2$-{\sf SMP}.
\subsection{Overview}
\danupon{This problem is harder than the highway pricing problem since it doesn't have a separator. For example, the log n approx of Balcan-Bum for Highway heavily relies on the separator. Similarly, Khaled's QPTAS also relies on the separator (once you remember the profile in the middle, you can solve two sides separately).}
We sketch the key ideas here and leave the details in next sections. First, consider the special case where each consumer has
budget $1$ or $2$ and each item must be priced either $0$ or $1$.
The exact optimal solution of this case can be found in
$n^{O(\log^2 m n)}$ time. We later
show how to extend the idea to the general cases, which turns out to
be easy for the case of highway problem but need a few more ideas
for the case of $2$-{\sf SMP}.
\paragraph{Algorithm for highway pricing problem reviewed:}
Let us first start with the highway pricing problem which can be
casted as a special case of $2$-{\sf SMP} where items are in the
form $(1,n), (2,n-1), \ldots, (n,1)$.
The main idea used in \cite{ElbassioniSZ07}, casted in our language
of ``partition tree'' (for convenience in explaining our $2$-{\sf SMP} algorithm later) is the following.\danupon{I removed [htb!] from the figure.}
\begin{figure
\centering \scalebox{1.2}[1.2]{\includegraphics{pic1}}
\caption{A partition tree}\label{fig:Partition_Tree}
\end{figure}
We first construct a balanced binary tree called a {\em partition tree} and denoted by
${\cal T}$. We define the vertical gridline in the middle to be a level-$0$ line, denoted by $\ell_r$, dividing the items equally to left and right sides. This line corresponds to the root node $r$ of the tree. We also assign the consumers whose consideration set contains items on both sides to the root node. Then we recursively define the subtrees on the subproblems on the two sides of line $\ell_r$ as shown in Figure~\ref{fig:Partition_Tree} until we reach the subproblem containing only one item. For any node $v \in {\mathcal T}$, let ${\mathcal{C}}_v$ be the set of consumers assigned to $v$, and $\ell_v$ be the line associated with node $v$.
Now we show a top-down recursive algorithm to solve this problem.
This algorithm can be converted to a dynamic program by working
bottom-up instead. At the root node $r$ of ${\cal T}$, we would like
to compute $f_r({\bf I}_{L, 1}, {\bf I}_{L, 2}, {\bf I}_{L, 3}, {\bf I}_{R,
1}, {\bf I}_{R, 2}, {\bf I}_{R, 3})$ which is defined to be the optimal
revenue that we can collect from consumers in ${\cal C}\setminus
{\cal C}_r$ when we price the items in such a way that ${\bf I}_{L,
1}$, ${\bf I}_{L, 2}$ and ${\bf I}_{L, 3}$ (${\bf I}_{R, 1}$, ${\bf I}_{R,
2}$ and ${\bf I}_{R, 3}$, respectively) are the first, second, and
third closest items on the left (respectively, right) of $\ell_r$
that have price $1$. To avoid long notation, let us denote
$\{{\bf I}_{L, 1}, {\bf I}_{L, 2}, {\bf I}_{L, 3}, {\bf I}_{R, 1}, {\bf I}_{R,
2}, {\bf I}_{R, 3}\}$ by $\Gamma_r$ and $f_r({\bf I}_{L, 1}, {\bf I}_{L,
2}, {\bf I}_{L, 3}, {\bf I}_{R, 1}, {\bf I}_{R, 2}, {\bf I}_{R, 3})$ by
$f_r(\Gamma_r)$.
If we can compute $f_r(\Gamma_r)$ for all $\Gamma_r$ then the
optimal revenue can be obtained via the following formula.
\begin{align}
\text{Optimal revenue} &=
\max_{\Gamma_r} f_r(\Gamma_r) + m_1(\Gamma_r) + 2m_2(\Gamma_r)\label{eq:highway}
\end{align}
where, for any node $v$, $m_1(\Gamma_v)$ is the number of consumers
in ${\cal C}_v$ whose consideration sets contain exactly one item in
$\Gamma_v$, and $m_2(\Gamma_v)$ is the number of consumers in ${\cal
C}_v$ with budget $2$ whose consideration sets contain exactly two
items in $\Gamma_v$. The point is that we can calculate the revenue
from consumers in ${\cal C}_r$ as $m_1(\Gamma_r) + 2m_2(\Gamma_r)$
and use $f_r(\Gamma_r)$ to compute the revenue obtained from the
rest of the consumers.
It is left to compute $f_r(\Gamma_r)$. Let $u$ and $v$ be the left
and right children of $r$, respectively. In order to compute
$f_r(\Gamma_r)$, we will compute $f_u(\Gamma_r, \Gamma_u)$ which is
the maximum revenue we can collect from consumers assigned to the
descendants of $u$ (excluding $u$) where $\Gamma_r$ is the set of
six items of price $1$ that are closest to $\ell_r$ as defined
earlier. And, similarly, $\Gamma_u=\{{\bf I}'_{L, 1}, {\bf I}'_{L, 2},
{\bf I}'_{L, 3}, {\bf I}'_{R, 1}, {\bf I}'_{R, 2}, {\bf I}'_{R, 3}\}$ is the
set of six items of price $1$ that are closest to $\ell_u$.
Moreover, we require that $\Gamma_u$ must be {\em consistent} with
$\Gamma_r$ in the sense that there is some price assignment such
that items in $\Gamma_u$ are the items closest to $\ell_u$ of price $1$
and items in $\Gamma_r$ are the items closest to $\ell_u$ of price $1$ as
well.
(For example, if we let $\Gamma_r=\{{\bf I}_{L, 1}, {\bf I}_{L, 2},
{\bf I}_{L, 3}, {\bf I}_{R, 1}, {\bf I}_{R, 2}, {\bf I}_{R, 3}\}$ then an
item ${\bf I}$ with property ${\bf I}_{L, 3}[1]<{\bf I}[1]<{\bf I}_{L,
2}[1]$ cannot be in $\Gamma_u$ since this item must have price $0$.)
We use $\Gamma_u\bowtie\Gamma_r$ to denote ``$\Gamma_u$ is
consistent with $\Gamma_r$''. We define $f_v(\Gamma_r, \Gamma_v)$ in
a similar way.
Once we have $f_u(\Gamma_r, \Gamma_u)$ and $f_v(\Gamma_r, \Gamma_v)$
for all $\Gamma_u\bowtie\Gamma_r$ and $\Gamma_v\bowtie\Gamma_r$, we
can compute $f_r(\Gamma_r)$:
\begin{align}
f_r(\Gamma_r) & = \max_{\Gamma_u\bowtie\Gamma_r} \left\{f_u(\Gamma_r, \Gamma_u) + m_1(\Gamma_u) + 2m_2(\Gamma_u)\right\} + \max_{\Gamma_v\bowtie\Gamma_r} \left\{f_v(\Gamma_r, \Gamma_v) + m_1(\Gamma_v) + 2m_2(\Gamma_v)\right\} \,.\label{eq:highway_recurse}
\end{align}
The main point here is that there is no consistency requirement between $\Gamma_u$ and $\Gamma_v$ so we have two independent subproblems. We define the function $f_z$, for all nodes $z$ in ${\cal T}$ similarly: Let $r=v_0$, $v_1$, $v_2$, ..., $v_{q-1}$ be the ancestors of $z$ and $v_q=z$. We have to compute $f_z(\Gamma_{v_0}, \Gamma_{v_1}, \ldots, \Gamma_{v_q})$ for all $\Gamma_{v_0}, \Gamma_{v_1}, \ldots, \Gamma_{v_q}$ such that $\Gamma_{v_i}\bowtie \Gamma_{v_j}$ for all $i\neq j$.
The computation of $f_z(\Gamma_{v_0}, \Gamma_{v_1}, \ldots,
\Gamma_{v_q})$ is done in the same way as Eq.\eqref{eq:highway_recurse} for every non-leaf node $z$. At leaf node $z$, $f_z(\Gamma_{v_0}, \Gamma_{v_1}, \ldots,
\Gamma_{v_q})$ can also be easily computed: $f_z(\Gamma_{v_0},
\Gamma_{v_1}, \ldots, \Gamma_{v_q})=m_1(\Gamma_z)+2m_2(\Gamma_z)$.
Observe that $q=O(\log m + \log n)$ and there are $n^{6}$
choices for each $\Gamma_{v_i}$. Therefore, we can precompute
$f_z(\Gamma_{v_0}, \Gamma_{v_1}, \ldots, \Gamma_{v_q})$ for all
$n^{O(\log m + \log n)}$ combinations of $\Gamma_{v_0},
\Gamma_{v_1}, \ldots, \Gamma_{v_q}$. By working bottom-up from the
leaf nodes, the running time becomes $\operatorname{poly}(m)n^{O(\log m + \log n)}$.
\paragraph{Algorithm for $2$-{\sf SMP} (special case):} To solve the special case of $2$-{\sf SMP} defined above, we need to modify a few definitions in a right way.
Let us again consider the top-down algorithm and start at the root node $r$ of the partition tree ${\cal T}$. (Recall that we can assume that there is at most one item in each row and column so we can still define the paritition tree by drawing the vertical line through the point in the middle when sorted by the first dimension.)
One problem immediately appears: $f_r(\Gamma_r)$ cannot be used to compute the optimal revenue as we did in Eq.\eqref{eq:highway}. The reason is that we cannot compute the revenue from ${\cal C}_r$ using $m_1(\Gamma_r) + 2m_2(\Gamma_r)$ anymore.
To fix this, we have to redefine ${\cal C}_r$ in the following way:
We assign all consumers lying on the left (respectively, right) of
${\bf I}_r$ to the left (respective, right) child and keep only those
consumers lying exactly on the vertical line going through ${\bf I}_r$
in ${\cal C}_r$.
Now we can compute the revenue from the newly defined ${\cal C}_r$
and a function that computes the total revenue. To do this, we
define $f_r({\bf I}_1, {\bf I}_2, {\bf I}_3)$ to be the total revenue we
can get from consumers in ${\cal C}\setminus{\cal C}_r$ by pricing
the items in such a way that, among the items on the right side of
${\bf I}_r$, items ${\bf I}_1$, ${\bf I}_2$, and ${\bf I}_3$ are the items
with price $1$ that have the highest, second highest, and third
highest values in the second dimension, respectively. Again, let
$\Gamma_r$ denote a possible choice of $\{{\bf I}_1, {\bf I}_2,
{\bf I}_3\}$ and write $f_r(\Gamma_r)$ instead of $f_r({\bf I}_1,
{\bf I}_2, {\bf I}_3)$. If we can compute $f_r(\Gamma_r)$ then we can
get the optimal revenue by Eq.\eqref{eq:highway},
where $m_1(\Gamma_r)$ and $m_2(\Gamma_r)$ is as defined earlier (with the new definition of ${\cal C}_r$).
Some more complications lie in computing $f_r(\Gamma_r)$, for any
$\Gamma_r$. As before, we will compute $f_u(\Gamma_r, \Gamma_u)$ and
$f_v(\Gamma_r, \Gamma_v)$ where $u$ and $v$ are the left and right
children of $r$, respectively. Howerver, we have to carefully define
$f_u(\Gamma_r, \Gamma_u)$ and $f_v(\Gamma_r, \Gamma_v)$, in a
different way.
We define $f_u(\Gamma_r, \Gamma_u)$, for any $\Gamma_u=\{{\bf I}_1,
{\bf I}_2, {\bf I}_3\}$, to be the maximum revenue from the consumers
assigned to the descendants of $u$ when we price the items in such a
way that, among the items lying on the right side of ${\bf I}_u$ and
left side of ${\bf I}_r$, items ${\bf I}_1$, ${\bf I}_2$, and ${\bf I}_3$
are the items with price $1$ that have the highest, second highest,
and third highest values in the second dimension, respectively. Note
that we do not need to check any consistency between $\Gamma_r$ and
$\Gamma_u$: For any choice of $\Gamma_r$ and $\Gamma_u$, there is
always a price assignment such that items in $\Gamma_r$ and
$\Gamma_u$ are the items of price $1$ that have the highest values
in the second dimension in their respective regions. In this case,
we say that $\Gamma_r\bowtie \Gamma_u$ is always true for any
$\Gamma_r$ and $\Gamma_u$.
On the other hand, we define $f_v(\Gamma_r, \Gamma_v)$, for any
$\Gamma_v=\{{\bf I}_1, {\bf I}_2, {\bf I}_3\}$, to be the maximum revenue
from the consumers assigned to the descendants of $v$ when we price
the items in such a way that, among the items lying on the right
side of ${\bf I}_v$, items ${\bf I}_1$, ${\bf I}_2$, and ${\bf I}_3$ are the
items with price $1$ that have the highest, second highest, and
third highest values in the second dimension, respectively. In this
case, we have to make sure that $\Gamma_v$ is consistent with
$\Gamma_r$, i.e., there is some price assignment such that items in
$\Gamma_r$ and $\Gamma_u$ are the items of price $1$ that have the
highest values in the second dimension in their respective regions.
Now we have defined $f_u(\Gamma_r, \Gamma_u)$ and $f_v(\Gamma_r,
\Gamma_v)$, we compute $f_r(\Gamma_r)$ using
Eq.\eqref{eq:highway_recurse}.
As in the case of the highway pricing problem, we can extend the
definition to other nodes. In particular, at a leaf node $z$ we have
to compute $f_z(\Gamma_{v_0}, \Gamma_{v_1}, \ldots, \Gamma_{v_q})$
where $q=O(\log m + \log n)$. Hence, this case can be solved in
$\operatorname{poly}(|{\cal C}|)\cdot\size{\cal I}^{\polylog{|{\cal I}|}}$ time.
\danupon{To do: try not to use $m$ and $n$.}
\paragraph{Algorithm for general $2$-{\sf SMP}:}
We now remove the restrictions that each item must be priced $0$ or
$1$ and each budget must be $1$ or $2$. The removal of the
restriction on item price does not affect the case of highway
pricing problem since this can be easily assumed (see, e.g.,
\cite{GrandoniR10}).\danupon{Actually, can we assume this? In
\cite{GrandoniR10}, they also assume this for tollbooth problem on
trees (Section 4.1).}
Moreover, we can still assume that the maximum budget is $O(m n)$. Now we can deal with the general highway problem by
redefining $f_r(\Gamma_r)$: Let $\Gamma_r=\{{\bf I}_{L,0}, {\bf I}_{L,
1}, \ldots, {\bf I}_{L, q}, {\bf I}_{R,0}, {\bf I}_{R, 1},
\ldots,{\bf I}_{R, q}\}$ where $q=O(\log m n)$.
For any $i\leq q$, we want to price in such a way that ${\bf I}_{L,
i}$ is the item closest to ${\bf I}_r$ on the left such that the sum
of the price of all items between ${\bf I}_r$ and ${\bf I}_{L, i}$ is at
least $(1+\epsilon)^i$. Computing $f_r(\Gamma_r)$ can be done in the
same manner as before and consistency checking is easy to deal with.
Function $f_{v_q}(\Gamma_{v_0}, \Gamma_{v_1}, ..., \Gamma_{v_q})$,
for any node $v_q$ at level $q$ in $\cal T$, can be defined in a
similar manner.
For $2$-{\sf SMP}, we may not in general assume the item prices to
be $0/1$. Instead, we show that it can be assumed that each item
must have price $0$, or $(1+\epsilon)^j$, for any $j=0, 1, \ldots,
O(\log m)$. A natural extension of the above idea is to
define the notion of ``volume of regions'': For each item ${\bf I}$,
let $H_{{\bf I}}$ and $V_{{\bf I}}$ denote the horizontal and vertical
line cutting through item ${\bf I}$, respectively. Any rectangle
resulting from drawing some horizontal and vertical lines through
some items are called {\em regions} and the regions that do not
contain other regions are called {\em minimal regions}. For any
price assignment, we define the {\em volume} of a region to be the
sum of the price of all items within the region.
\begin{figure
\centering \scalebox{0.7}[0.7]{\includegraphics{pic2}}
\caption{Approximating the revenue from consumer ${\bf C}$ assigned to node $z$ in $\mathcal{T}$.}\label{fig:area-idea}
\end{figure}
Now, similar to the highway problem, we define $\Gamma_r=\{{\bf I}_0,
{\bf I}_1, ..., {\bf I}_k\}$ (note that $k=O(\log m)$) as the
``region guess'': We define $f_r(\Gamma_r)$ to be the maximum
revenue from ${\cal C}\setminus {\cal C}_r$ when we price in such a
way that, for any $i$, item ${\bf I}_i$ is the highest item (in the
second dimension) such that the volume of the region on the right of
the vertical line $V_{{\bf I}_r}$ and above the horizontal line
$H_{{\bf I}_i}$ (including ${\bf I}_i$) is at least $(1+\epsilon)^i$.
Using these volume guesses, we can approximate the upper and lower
bounds of the revenue from each consumer ${\bf C}$ at node $z$ by
looking at $\Gamma_v$ for all ancestors $v$ of $z$. This is because
each consumer's consideration set will contain some set of regions
$B_1, B_2, ...$ with volume guesses $(1+\epsilon)^{i_1},
(1+\epsilon)^{i_2}, ...$, respectively (such as the blue regions in
Figure~\ref{fig:area-idea}). Also, this consideration set will also
be contained in some set of regions $R_1, R_2, ...$ with volume
guesses $(1+\epsilon)^{i_1+1}, (1+\epsilon)^{i_2+1}, ...$ (such as
the blue and red regions together in Figure~\ref{fig:area-idea}).
However, in contrast to the case of highway problem, the consistency
between the guesses (e.g., between $\Gamma_r$ and its children
$\Gamma_u$ and $\Gamma_v$) is harder to guarantee. In order to
guarantee the consistency, we add another parameter, denoted by
$\Delta_r=\{\delta_0, \delta_1, \ldots, \delta_{O(k^2)}\}\subseteq
R_{\geq 0}^{O(k^2)}$ (recall that $|\Gamma_r|=k+1$). $\Delta_r$ is
used as a ``volume guess''. That is, we define $f_r(\Gamma_r,
\Delta_r)$ to be the maximum revenue from ${\cal C}\setminus {\cal
C}_r$ when we price in such a way that the restriction on $\Gamma_r$
is as before and, additionally, the volumn of the $i$-th minimal
region is exactly $\delta_i$ (where we make any order of the minimal
regions). We can now guarantee the consistency by making sure that
the sum of the volume guesses in smaller regions defined by
$\Gamma_u$ and $\Delta_u$ (as well as $\Gamma_v$ and $\Delta_v$) is
exactly the volume guesses in the bigger regions defined by
$\Gamma_r$ and $\Delta_r$. \danupon{Need to make this part much more
precise later. Also, we have to make it very clear why we need the
volume guesses, not region guesses alone.}
For any node $z$, we also define a function $f_z(\Gamma_{v_0},
\Gamma_{v_1}, \ldots, \Gamma_{v_q}, \Delta_z)$ where $v_0, v_1,
\ldots, v_{i-1}$ are ancestors of $z$ and $v_q=z$. In this case, we
consider the minimal regions obtained by drawing vertical lines
$V_{{\bf I}_{v_0}}, V_{{\bf I}_{v_1}}, \ldots, V_{{\bf I}_{v_q}}$ and
horizontal lines $H_{{\bf I}}$ for ${\bf I}\in \Gamma_{v_i}$, for all
$i$. We use $\Delta_z$ to store the numbers that are the ``volume
guesses'' of all these regions. We also check the consistency in
terms of volume, i.e., $\Pi=\{\Gamma_{v_0}, \Gamma_{v_1}, \ldots,
\Gamma_{v_q}, \Delta_{v_q}\}$ is consistent with
$\Pi'=\{\Gamma_{v_0}, \Gamma_{v_1}, \ldots, \Gamma_{v_{q-1}},
\Delta_{v_{q-1}}\}$ if the volume guesses of the smaller regions
defined by lines in $\Pi$ add up to the volume guesses of the bigger
regions defined by lines in $\Pi'$.
\subsection{Preprocessing}
Fix some $\epsilon >0$. Given an instance $({\mathcal{I}}, {\mathcal{C}})$, our goal
is to compute a price that collects a revenue of at least
$(1-O(\epsilon)) \mbox{\sf OPT}$. Recall that we can assume that the consumers
are on the intersection of grid lines, and the items are in the grid
cells (cf. Lemma~\ref{lem:perturb}). First we process the input so
that the budgets and prices are polynomially bounded. Moreover, the
optimal solution only assigns prices of the form $(1+\epsilon)^j$
for some $j \leq O(\log m)$. The proof of this fact only uses
standard arguments (along the same line as in \cite{BalcanB07}).
\begin{lemma}\label{lem:preprocess}
Let $M= O(mn /\epsilon)$. The input instance ${\cal P}$ can be
reduced to ${\cal P'}$ with the following properties.
\squishlist
\item For each consumer ${\bf C}$, the budget of ${\bf C}$ in ${\cal P}'$ is between $1$ and $M$.
\item Any price $p'$ that $\alpha$-approximates the optimal pricing of ${\mathcal{P}}'$ can be transformed in polynomial time into another price $p$ that gives $(1+3\epsilon) \alpha$-approximation for ${\mathcal{P}}$.
\item There is a $(1+\epsilon)$-approximate solution $\tilde{p}$ satisfying the following: For all ${\bf I} \in {\mathcal{I}}$, $1 \leq \tilde{p}({\bf I}) \leq M$, and $\tilde{p}({\bf I})$ is in the form $(1+\epsilon)^j$ for some $j \leq O(\log m)$.
\end{itemize
\end{lemma}
\begin{proof
Let $B_{\max}$ be the maximum budget among all consumers. We first
remove all consumers whose budgets are less than $\epsilon
B_{\max}/mn$. Notice that we only lose the revenue of at most
$\epsilon B_{\max} \leq \epsilon \mbox{\sf OPT}$ by this removal. We denote
the new set of consumers by ${\mathcal{C}}'$. Now look at the optimal price
$p^*$ for the resulting instance. If for some ${\bf I} \in {\mathcal{I}}$,
the price $p^*({\bf I})$ is less than $\epsilon B_{\max}/mn$, we
change its price to $p'({\bf I}) = 0$ and remove item ${\bf I}$
completely from the instance. Again, since each such item can only
be sold to at most $m$ consumers, discarding it only decreases the
revenue by $\epsilon B_{\max}/n$. There are at most $n$ such items,
so we lose a revenue of at most $\epsilon \mbox{\sf OPT}$ in total. Let
${\mathcal{I}}'$ denote the resulting set of items.
Next we scale each consumer budget by $M'=mn/\epsilon B_{\max}$ to
get a new budget, i.e. $B'_C = M'B_C$. Now we have a complete
description of the instance ${\mathcal{P}}'$ in which consumer budgets are
between $1$ and $M$. Let $\mbox{\sf OPT}'$ be the optimal value of the new
instance. First we try to lower bound the value of $\mbox{\sf OPT}'$. Consider
the same price $p^*: {\mathcal{I}}' \rightarrow \ensuremath{\mathbb R}$ scaled up by a factor of
$M'$. The revenue from this price is at least $(1 - 2\epsilon)
M'\mbox{\sf OPT}$, so we have that $\mbox{\sf OPT}' \geq (1-2\epsilon)M'\mbox{\sf OPT}$.
We are now ready to prove the second part. Assume that we have a
price $p'$ that gives $\alpha$-approximation for ${\mathcal{P}}'$, so the
revenue collected by $p'$ is at least $\mbox{\sf OPT}'/\alpha$. We construct
the price $p$ by scaling down the price of $p'$ by $M'$. Notice that
for each consumer ${\bf C}$ who can afford his consideration set in
${\mathcal{P}}'$ with price $p'(S_C)$, he can also afford his set in ${\mathcal{P}}$
with price $p'(S_C) = p(S_C)/M'$. Therefore, the revenue collected
by $p$ is at least $\mbox{\sf OPT}'/\alpha M' \geq (1-2\epsilon)\mbox{\sf OPT}/ \alpha$.
This argument also implies that $\mbox{\sf OPT} \geq \mbox{\sf OPT}'/M'$.
Finally we show that there is a good solution $\tilde p$ that only
assigns prices in the form $(1+\epsilon)^j$, as follows. We round
down the price of $p^*$ to the nearest scale of $(1+\epsilon)^j$ for
some $j$. For each consumer ${\bf C}$ who purchases item ${\bf I}$
w.r.t. price $p^*$, by scaling down every item price, she can still
afford her consideration set $S_{{\bf C}}$, whose new price is at
least $p^*(S_{Consumer})/(1+\epsilon) \geq (1-\epsilon)
p^*(S_{{\bf C}})$.
\end{proof}
From now on, we assume that our input instance and its optimal price
are in such format. Our goal is to devise a {\sf QPTAS} for this
instance. We note here that in some special cases of single-minded
pricing problems, especially the Highway problem, an even stronger
statement can be assumed; namely, that the optimal price is
integral~\cite{GuruswamiHKKKM05}. It seems that such a nice property
may not hold in our case, and we anyway do not need it.
\subsection{Partition tree}
We first construct a (almost balanced) binary tree ${\mathcal T}$ where
each node in ${\mathcal T}$ is associated with a rectangular region in the
plane (from now on, whenever we talk about region, we always mean a
rectangular one). We call this tree the {\em partition tree}. It can
be constructed recursively as follows. In the beginning, we have
${\mathcal T}=\set{r}$ where $r$ is the root of the tree whose region $A_r$
is the whole grid. We repeat the following process: For each leaf $v
\in {\mathcal T}$, if the region $A_v$ of $v$ contains at least two items,
we choose a vertical grid line $\ell_v$ dividing the items in a
balanced manner to the left and right side. We then add the left
child $v'$ of $v$ with the region $A_{v'}$ being the region of $A_v$
on the left side of $\ell_v$. We also add the right child $v''$ of
$v$ associated with the region $A_{v''}$ on the right side of
$\ell_v$. We repeat the process until every leaf is associated with
a region containing only one item; see Figure~\ref{figure: tree}.
\danupon{It's better to define ``volume'' later because we need to
define ``region'' with horizontal lines first.} For each node $v \in
{\mathcal T}$, we define the item set ${\mathcal{I}}_v$ to be the set of all items
in the region $A_v$. Fix a price $p: {\mathcal{I}} \rightarrow \ensuremath{\mathbb R}$. For any
region $A$, we define the ``volume'' $\mathsf{vol}_p(A)$ to be the total
price among all items in the region, i.e. $\mathsf{vol}_p(A) = \sum_{{\bf I}
\in A} p({\bf I})$. The following simple claim
is crucial in designing our algorithm.
\begin{claim}\label{claim:bound_sum}
Let $p^*$ be an optimal price. Then for any region $A$, there are
only $n^{O(\log m)}$ possible values of $\mathsf{vol}_{p^*}(A)$.
\end{claim}
\begin{proof
Let $x_j$ denote the number of items ${\bf I}$ in $A$ with price
$p^*({\mathbf I}) = (1+\epsilon)^j$. Notice that we can write the
volume of $A$ as $\sum_{j=1}^{q} x_j (1+\epsilon)^j$ where $x_j$
only takes non-negative integer values at most $n$. So we have at
most $n^{O(\log m)}$ possibilities for the volume.
\end{proof}
\subsection{Horizontal partition and local profile}
From the construction, each node $v$ of the partition tree, is
associated with a vertical line $\ell_v$ which divides the plane
into two region. We further partition the right region using
vertical line, as follows.
Consider a non-leaf node $v \in {\mathcal T}$ with left child $v'$ and
right child $v''$. A {\em horizontal partition} for node $v$,
denoted by $H_v$, is a collection of (not-necessarily distinct)
horizontal lines $\ell^v_1,\ldots, \ell^v_q$, partitioning the
region of $A_{v''}$ into many pieces; note that the left endpoints
of these lines are on $\ell_v$.\danupon{Need a picture here.} The
line $\ell^v_j$ is supposed to mark the highest $y$-coordinate such
that the volume inside $A_{v''}$ above $\ell^v_j$ is at least
$(1+\epsilon)^j$. Notice that each node $v$ has at most $n^{O(\log
m)}$ feasible partitions since there are at most $n$ possibilities
for the choice of each $\ell^v_j$.
Now if we fix a horizontal partition of every non-leaf node in the
partition tree, we can define {\em minimal} regions for each
non-leaf node $v$ as follows. For each node $v$, we consider all
vertical and horizontal lines associated with $v$ and all its
ancestors (i.e., all lines in $\ell_u$ and $H_u$ where $u=v$ or $u$
is an ancestor of $v$). Let ${\mathcal{L}}_v$ denote the set of these lines.
${\mathcal{L}}_v$ naturally defines minimal regions: We say that a region
$A$ is minimal with respect to ${\mathcal{L}}_v$ if and only if $A$ is a
rectangle whose four boundaries are the lines in ${\mathcal{L}}_v$, and
there is no line in ${\mathcal{L}}_v$ that intersects with the interior of
$A$.
Now, we define a {\em local profile} of a node $v$. It consists of
(i) horizontal partitions for $v$ and for all its ancestors, and
(ii) numbers on every minimal region resulting from vertical and
horizonal lines. The numbers are supposedly the ``volume guesses''
of every minimal region of $v$.
Now we try to guess the ``right'' local profile of every node in the
partition tree. We show that if this guess is right, then we get a
good approximation of the optimal solution. Moreover, we can use
dynamic programming to make the right guess.
\subsection{Dynamic Programming Solution}
A {\em global profile} (or just {\em profile} in short) of a node
$v$ consists of the local profile of $v$ and all its ancestors in
such a way that the volumes of minimal regions of $v$ is consistent
with its ancestors. More formally, fix a node $v$. A profile $\Pi_v$
for $v$ consists of, for any ancestor $v'$ of $v$, $\Pi_{v, v'}$
which is the local profile that node $v$ wants its ancestor $v'$ to
have (we also think of $v$ has an ancestor of itself for
convenience).
As a reminder, for each ancestor $v'$ of $v$, local profile $\Pi_{v,
v'}$ some horizontal partition $H_{v'}$ and the ``volume guess''of
each minimal region of $v$.
In addition, we restrict that these local profiles $\Pi_{v, v'}$
have to be consistent in themselves in the following sense. For each
vertex $v'$, for any minimal region $A'$ of $\Pi_{v,v'}$ that is
further partitioned into minimal regions $A'_1, A'_2,\ldots,
A'_{\gamma}$ of $\Pi_{v,v''}$ for some descendant $v''$ of $v'$, the
number $z_{A'}$ at $\Pi_{v,v'}$ is equal to the sum of the numbers
$z'_{A_j}$ of $\Pi_{v,v''}$.\danupon{This is still not clear and a
bit informal.}
We argue that the number of global profiles for each node is not too
large, i.e. only $n^{\operatorname{poly} \log m}$. There are
$n^{O(\log m)}$ horizontal partitions for each ancestor $v'$ of $v$,
making a total of $n^{O(\log m \log n)}$ possibilities of the lines
$\ell^{v'}_j$. Now fix a choice of such horizontal partitions. If we
draw all lines $\ell^{v'}_j$ involved in the global profiles, we
will see a number of regions formed by intersections between these
lines and the vertical lines $\ell_{v''}$. Since there are $O(\log m
\log n)$ such horizontal lines and $O(\log n)$ vertical lines
involved, we have at most $O(\log m \log^2 n)$ minimal rectangular
regions. Each region has at most $n^{O(\log m)}$ possible volumes,
so there are at most $n^{O(\log^2 m \log^2 n)}$ global profiles for
each node $v$\parinya{We implicitly used the fact that $m \leq O(n)$. Have to say
it somewhere}.
Now we define a {\em valid tree profile} $\Pi$ for ${\mathcal T}$ as the
set of global profiles $\set{\Pi_v}_{v \in {\mathcal T}}$ such that $\Pi_v$
is a global profile for node $v$. Moreover, for every parent-child
pair $v, v'$ where $v$ is a parent of $v'$ in ${\mathcal T}$, the profile
$\Pi_{v'}$ agrees with $\Pi_v$. That is, all profiles about
ancestors of $v$ in $\Pi_v$ and $\Pi_{v'}$ are exactly the same.
Given a valid tree profile $\Pi$, we have the notion of cost of the
profile $\Pi$ (denoted by $\mbox{\sf Cost}(\Pi)$) which is supposed
to approximate the total revenue we can collect by a price function
consistent with $\Pi$. The cost of a profile can be computed as
follows. For each node $v \in {\mathcal T}$, let ${\mathcal{C}}_v$ be the set of
all consumers on line $\ell_v$. For each consumer ${\bf C} \in
{\mathcal{C}}_v$, the rectangular region enclosed by horizontal line ${\bf
C}[2]$ and vertical line $\ell_v$
is the actual amount the consumer needs to pay. This is the amount
we do not know, but we can approximate: We let $v_0, v_1,\ldots,
v_{\alpha}$ be a sequence of ancestors of $v$ such that $v$ is on
the left subtree of $v_i$ (in the order from $v$ to the root), where
$v_0= v$. And we let for each $i$, $j_i$ be the maximum number such
that $\ell^{v_i}_{j_i}$ does not lie below ${\bf C}[2]$. The cost of
consumer ${\bf C}$ is just the sum $\sum_{i=0}^{\alpha}
(1+\epsilon)^{j_i}$ if $B_C \leq \sum_{i=0}^{\alpha}
(1+\epsilon)^{j_i}$ and zero otherwise. The cost at node $v$ is just
the total cost of all consumers in ${\mathcal{C}}_v$, and the cost of the
profile is the sum of the cost over all nodes $v \in {\mathcal T}$.
\begin{figure}
\centering
\subfigure[Tree Decomposition up to depth $2$ where the shaded
region denotes $A_v$]{
\includegraphics[height=5cm]{pic3}\label{figure:
tree} }
\hspace{.05\textwidth}
\subfigure[The actual volume]{
\includegraphics[height=5cm]{pic4}
\label{fig:area1} }
\hspace{.05\textwidth}
\subfigure[The approximate volume]{
\includegraphics[height=5cm]{pic5}
\label{fig:area2}}
\caption{Computing the cost of consumer ${\bf C}$}\label{fig:area}
\end{figure}
\begin{lemma}\label{lem:exist_good_tree}
There is a valid tree profile $\Pi^*$ such that the cost is at least
$(1-\epsilon)\mbox{\sf OPT}$.
\end{lemma}
\begin{proof
We start from the optimal price $p^*$ and construct the valid
profile as follows. For each node $v$, we define a feasible
partition of $v$ by choosing the line $\ell^v_j$ to be at the
highest $y$-coordinate such that the total volume enclosed is at
least $(1+\epsilon)^j$. Then we create a profile $\Pi^*_v$ for each
node $v$ according to the actual volume of each minimal region.
Notice that this gives a valid tree profile.
\end{proof}
Our goal now is to compute the valid profile $\Pi$ of maximum cost
by dynamic programming, and the profile will automatically suggest a
near-optimal pricing.
\paragraph{Computing the Solution:} Let $v \in {\mathcal T}$.
We say that a price $p: {\mathcal{I}}_v \rightarrow \ensuremath{\mathbb R}$ is consistent with
global profile $\Pi_v$ if and only if for every minimal region $A$
of $\Pi_v$ that is completely contained in $A_v$, we have $\mathsf{vol}_p(A)
= z_A$. The minimum cost profile can be computed in a bottom-up
fashion, as follows. For a leaf node $v$, a global profile for $v$
automatically determines the price of the only item in $A_v$;
discard a profile which does not have consistent price.
The following lemma shows that a price $p$ consistent with a valid
tree profile $\Pi$ can be computed from $\Pi$.
\begin{lemma}\label{lem:consistent}
For each node $v$ with left child $v'$ and right child $v''$, let
$p':{\mathcal{I}}_{v'}\rightarrow \ensuremath{\mathbb R}$ and $p'':{\mathcal{I}}_{v''}\rightarrow \ensuremath{\mathbb R}$ be
the prices that are consistent with the profile $\Pi_{v'}$ and
$\Pi_{v''}$ respectively. Then the price $p: {\mathcal{I}}_v \rightarrow \ensuremath{\mathbb R}$
defined to agree with $p'$ on ${\mathcal{I}}_{v'}$ and with $p''$ on
${\mathcal{I}}_{v''}$, is consistent with $\Pi_v$.
\end{lemma}
\begin{proof
Consider a minimal region $A \subseteq A_v$ and a volume guess $z_A$ in
$\Pi_v$. If $A \subseteq A_{v'}$ where $A$ is the union of minimal
regions $A'_1,\ldots A'_{\gamma}$ of $\Pi_{v'}$ (similar argument
can be made in case $A \subseteq A_{v''}$), then by assumption that
$\Pi_v$ is consistent with $\Pi_{v'}$, we know that the total value
$z_A = \sum_{j=1}^{\gamma} z'_{A'_j}$. Since $p'$ is consistent with
the profile $\Pi_{v'}$, we have that $\mathsf{vol}_{p}(A) = \mathsf{vol}_{p'}(A) =
\sum_{j} \mathsf{vol}_{p'}(A'_j) = \sum_j z'_{A'_j} = z_A$ as desired.
\end{proof}
We have shown that a valid tree profile $\Pi$ always has a price $p$
consistent with it. The following lemma
basically says that this price $p$ gives a revenue close to the cost
of the profile, which will in turn imply that the maximum cost
profile gives the revenue of at least $(1-O(\epsilon)) \mbox{\sf OPT}$.
\begin{lemma}\label{lem:tree-revenue}
For any valid tree profile $\Pi$, let $p$ be a price consistent with
$\Pi$ and let $p' = p/(1+\epsilon)$. Then $p'$ collects revenue at
least $(1-\epsilon)$ fraction of the profile cost.
\end{lemma}
\danupon{Still need to prove this lemma.}
\section{{\sf QPTAS} for $2$-{\sf UUDP-MIN}}
\label{sec: bicriteria}\label{sec: qptas 2 udp}
We note that we will write $O(\log m)$ instead of $O(\log n+\log m)$ since we assume that $n\leq m$ in this paper. (Otherwise, we already have approximation ratio of $O(\log m)=O(\log n)$.)
\sodaonly{
We explain the main idea first. The intuition can be realized by solving the following simple case: Assume for now that we have $\Theta(n^2)$ items, which form a set $\set{(2i-1,2j-1): 1 \leq i,j \leq n}$. In this case it is
possible to have two different consumers at the same coordinate, i.e. ${\bf C}={\bf C}'$, while there is exactly one item at each point $(2i-1,2j-1)$. Assume further that each consumer has budget either $1$
or $2$. We show below how to solve this case in polynomial time.
Note that there is an optimal solution such that each item is
priced either $1$ or $2$: otherwise we could increase the price by small amount to collect more revenue.
Now, for any item point $(2i-1, 2j-1)$ and any price assignment
$p$, define\danupon{(to polish after submission) $r_p(i, j)$ may be confused with $r_{\bf C}(p)$.}
\[r_p(i, j):=\min_{\substack{{\bf I}[1]\ge 2i-1, {\bf I}[2]\geq 2j-1 \\ {\bf I}\in{\mathcal{I}}}}\{p({\bf I})\}
\]
to be the minimum price among the items
dominating $(2i-1, 2j-1)$. This quantity immediately tells us how much
revenue we will get from consumers at point $(2i-2, 2j-2)$: each consumer
will buy an item at price $r_p(i, j)$ if and only if she has budget
at least $r_p(i, j)$.
By the definition of $r_p$, we know that for any fixed value $j$, $r_p(i,j)$ is non-decreasing in terms of $i$.
In other words, for any pricing $p$ and integer $j$, there exists
a ``threshold'' $\gamma(p, j)$ such that $r_p(i', j)=1$ for all $i'\leq
\gamma(p, j)$ and $r_p(i',j)=2$ for all $i'> \gamma(p, x)$.
Additionally, for any $j$, $\gamma(p, j)\geq \gamma(p, j+1)$.
Using these observations, we are ready to define the dynamic programming table. The table entry $T[i,j]$ is defined to be the maximum revenue we can get among the price assignment $p$ such that $r_p(i', j)=1$ for all
$i'\leq i$ and $r_p(i', j)=2$ for all $i'>i$. The table $T$ can be
computed as follows.
\begin{align}
T[i, j]&=\max_{i'\leq i} \{T[i',j+1]+m_1(i',j) + 2m_2(i',j)\}
\label{eq:udp-min-table}
\end{align}
where $m_1(i',j)$ is the number of consumers of the form
$(2i''-2,2j-2)$ for $i''\leq i'$ with budget $1$ and $m_2(i',j)$ is the number of
consumers of the form $(2i''-2,2j-2)$ for $i'' > i'$ with budget $2$. Moreover, let $T[i,n+1]=0$ for all $i$.
The optimal solution is then $\max_i T[i,1]$.
The above discussion captures almost all the key ideas for solving the general $2$-{\sf UUDP-MIN} problem. To get a {\sf QPTAS} in the general case,
we extend these ideas in the following way.
\squishlist
\item Consider a slight generalization when there is only one item in each column and row of grid cells (cf.
Lemma~\ref{lem:perturb}) while each budget is still $1$ and $2$. In
this case, we cannot pick arbitrary value of $i'$ when we compute
$T[i, j]$ as in Eq.\eqref{eq:udp-min-table} since it might not
correspond to any pricing. Through some additional observations,
table $T$ can be computed as follows: Let ${\bf I}_j$ be the item whose $y$-coordinate is $j$. If $i={\bf I}_j[1]$ then we can use Eq.\eqref{eq:udp-min-table}; otherwise, $T[i,j]=T[i,j+1]+m_1(i,j)+2m_2(i,j)$. This algorithm runs in $O(n^3)$ time.
\item When there are $q$ different budgets, say $B_1, B_2, \ldots, B_q$,
we can solve the problem in $n^{O(q)}$ time. This is done
by defining $T[i_1,\ldots, i_{q-1}, j]$ to be the maximum
revenue we can get among the price assignment $p$ such that, for all
$q': 1 \leq q' \leq q$, $r_p(i',j)=B_{q'}$ for all $i_{q'-1}<i'\leq i_{q'}$ (where we let
$i_0=-1$ and $i_q=n$).
\item Finally, we obtain a {\sf QPTAS} by ``discretizing'' the prices so that there are not many choices of item prices (cf. Lemma~\ref{lem:uudp-discretize}). This enables us to assume that the prices are in $\Gamma=\{0, (1+\epsilon)^0, (1+\epsilon)^1, ..., (1+\epsilon)^q\}$ where $q=O(\log_{1+\epsilon} m)$, and we can get the algorithm running in time $n^{O(\size{\Gamma})}=n^{O(\log m n)}$.\danupon{I changed this slightly: I point to Lemma~\ref{lemma:discretization}.}
\end{itemize
}
\subsection{Preprocessing}
\fullonly{We need to do a preprocessing, which will be used in the next two sections to design {\sf QPTAS} for $2$-{\sf UUDP-MIN} and $2$-{\sf SMP}.}
The following lemma says that we can assume the input lies on the grid where each row and column of the grid contains exactly one item.
\begin{lemma}\label{lem:perturb}
We are given an instance $({\mathcal{C}},{\mathcal{I}})$ of $2$-{\sf UUDP-MIN}\fullonly{ (or $2$-{\sf SMP})}. Then we can, in polynomial time, transform $({\mathcal{C}}, {\mathcal{I}})$ into an ``equivalent'' instance $({\mathcal{C}}', {\mathcal{I}}')$ such that
\begin{itemize}
\item Each consumer ${\bf C}' \in {\mathcal{C}}'$ has even coordinates $(2i, 2j)$ for $0 \leq i,j \leq n$.
\item Each item ${\bf I}' \in {\mathcal{I}}'$ has odd coordinate $(2i-1, 2j-1)$ for $1 \leq i,j \leq n$.
\item For each odd number $2i-1$, $1 \leq i \leq n$, there is exactly one item ${\bf I}' \in {\mathcal{I}}'$ with ${\bf I}'[1]=2i-1$ and exactly one item ${\bf I}'$ with ${\bf I}'[2]=2i-1$.
\end{itemize}
\end{lemma}
\begin{proof}
We sweep the horizontal line from top to bottom, and whenever the
line meets the items ${\bf I'}_1,\ldots, {\bf I'}_z$ such that ${\bf
I}'_1[1] < {\mathbf I}'_2[1] < \ldots < {\mathbf I}'_z[1]$ with the
same $y$-coordinate $y'$, we break ties as follows. Let $\delta$ be
the vertical distance from the line to the next item point below the
line. We set the new $y$-coordinates of these items to ${\bf
I'}_j[2] = y'-(z-j)\delta/2 z$. Notice that some consumers whose
$y$-coordinates lie in $[y', y'-\delta)$ get affected by this move.
We also change the $y$-coordinates of those consumers to
$y'-\delta/2$. Then we add the horizontal grid lines between the
space of every consecutive items, while making sure that consumer
points are on the line passing $y-\delta/2$. It is easy to see that
this process preserves the consideration set of every consumer. We
repeat the above steps until the sweeping line passes the bottommost
item.
We do a similar sweep of vertical line from right to left, inserting
the grid lines along the way. In the end, each consumer lies on the
intersection of the grid lines and each item in its cell, which guarantees
that no two items appear in the same row or column of the grid.
\end{proof}
\subsection{Detail of {\sf QPTAS} for {\sf UUDP-MIN}}
First, we can make the following simple assumption.
\begin{lemma
\label{lem:uudp-discretize}
We can assume that the prices are in the form $(1+\epsilon)^0,
(1+\epsilon)^1, ..., (1+\epsilon)^q$ or zero where
$q=O(\log_{1+\epsilon} m)$ by sacrificing $(1+\epsilon)$ in the approximation factor.
\end{lemma}
\begin{proof
We use the following standard arguments. Consider an optimal price
$p^*$. For each item ${\bf I}_j$, if the price is non-zero, we round
down the price $p^*({\bf I}_j)$ to the nearest scale of
$(1+\epsilon)^{q'}$, so the price of each item gets decreased by at
most a factor of $(1+\epsilon)$. Consider a consumer ${\bf C}$ who
bought ${\bf I}_{j}$ with price $p^*$. After the rounding, she can
still afford ${\bf I}_j$, so we can still collect at least
$(1-\epsilon)p^*({\bf I}_j)$ from $C$.\danupon{Missing: We have to
first show that we can bound the maximum budget by $O(m)$.}
\end{proof}
Now, assuming that the optimal price $p^*$ has the above structure,
we show how to solve the problem in quasi-polynomial time. First, we
reorder the items based on their $y$-coordinates in descending
order, so we have ${\bf I}_1[2] > {\bf I}_2[2] > \ldots > {\bf
I}_n[2]$. A consumer ${\bf C}$ is said to belong to {\em level
$j$} if it lies between the row\danupon{We didn't define ``row'' before.} of ${\bf I}_j$ and
that of ${\bf I}_{j+1}$, so each consumer belongs to exactly one
level. Moreover, observe that a consumer ${\bf C}$ at level $j$ is
only interested in (a subset of) items in $\set{{\bf I}_1,\ldots,
{\bf I}_j}$ (since ${\bf I}_{j'}[2]<{\bf C}[2]$ for any $j'>j$).
We define a subproblem ${\mathcal{P}}_j$ as the pricing problem with items
$\set{{\bf I}_1,\ldots, {\bf I}_j}$ and consumers at levels
$1,\ldots, j$. We use the dynamic programming technique to solve this
problem.
\paragraph{Profiles} We will remember the profile for each
subproblem ${\mathcal{P}}_j$. A profile $\Pi$ of ${\mathcal{P}}_j$ consists of
$O(\log m)$ item indices $\pi_1,\ldots, \pi_q \in \set{1,\ldots,
j}$. Each value $\pi_i$ is supposed to tell us the index of the item ${\bf I}$
of price $(1+\epsilon)^i$ with maximum value ${\bf I}[1]$. That is, we say
that a price $p$ for ${\mathcal{P}}_j$ is {\em consistent} with profile
$\Pi =(\pi_1,\ldots, \pi_q)$ if, for each $i$, the item ${\bf I}_{\pi_i}$ has the highest value in the first coordinate among the items with price at most $(1+\epsilon)^i$, i.e., for all
$i$,
$$\pi_i=\arg\max_{j'} \{ {\bf I}_{j'}[1]\ |\ p({\bf I}_{j'})\leq (1+\epsilon)^i\}\,.$$
Since $\{ {\bf I}_{j'}\ |\ p({\bf I}_{j'})\leq (1+\epsilon)^i\}\subseteq
\{ {\bf I}_{j'}\ |\ p({\bf I}_{j'})\leq (1+\epsilon)^{i+1}\}$ for any
$i$,
$${\bf I}_{\pi_1}[1] \leq {\bf I}_{\pi_2}[1] \leq \ldots \leq {\bf
I}_{\pi_q}[1]\,.$$
Observe that if two prices $p'$ and $p''$ have the
same ${\mathcal{P}}_j$ profile, then consumers at level $j$ see no
difference between these two prices, as shown formally by the
following lemma. We say that an item ${\bf I}_k$ is a profile item for profile $\Pi=(\pi_1,\ldots, \pi_q)$ if and only if $k = \pi_{q'}$ for some $q' \in [q]$.
\begin{lemma}
\label{lemma:reconstruction}
Let $\Pi$ be a profile for subproblem ${\mathcal{P}}_j$, and let $p$ be any
price function for ${\mathcal{P}}_j$ that is consistent with profile $\Pi$. Then we
can assume without loss of generality that every consumer at level $j$ only purchases
profile items.
\end{lemma}
\begin{proof
Suppose that a consumer ${\bf C}$ buys an item ${\bf I}$ in ${\mathcal{I}}$
with $p({\bf I}) =(1+\epsilon)^{q'}$ which is not a profile item.
Then consider the profile item ${\bf I}_{\pi_{q'}}$, which satisfies ${\bf I'}[1] \geq {\bf I}[1]$, so we must have ${\bf I}_{\pi_{q'}} \in S_{{\bf C}}$. We can
therefore assume that consumer ${\bf C}$ buys ${\bf I}_{\pi_{q'}}$ instead
of ${\bf I}$.
\end{proof}
Let $\Pi=(\pi_1,\ldots, \pi_q)$ be a profile for ${\mathcal{P}}_{j}$ and
$\Pi'= (\pi'_1,\ldots, \pi'_q)$ be a profile for ${\mathcal{P}}_{j-1}$. We
say that $\Pi$ is {\em consistent} with $\Pi'$ if for any price
$p':\set{{\bf I}_1,\ldots, {\bf I}_{j-1}} \rightarrow \ensuremath{\mathbb R}$ that is
consistent with $\Pi'$, we can extend $p'$ to $p$ by assigning value
$p({\bf I}_j)$ such that $p$ is consistent with $\Pi$. Notice that
consistency between any two profiles can be checked in polynomial
time by trying all $q$ possibilities of prices.
We recall that we use $p^*$ to denote the optimal price.
\begin{lemma}\label{lem:udp-consistency}
There are profiles $\Pi^1,\ldots, \Pi^n$ for ${\mathcal{P}}_1,\ldots,
{\mathcal{P}}_n$ respectively such that for each $j\in\{1,\cdots, n-1\}$,
$\Pi^j$ is consistent with $\Pi^{j+1}$. Moreover, all such profiles
are consistent with price $p^*$.
\end{lemma}
\begin{proof
For each subproblem ${\mathcal{P}}_j$, we define the profile
$\Pi^j=(\pi^j_1,\ldots, \pi^j_q)$ based on the price $p^*$ (there is
only one possible profile consistent with $p^*$). It is clear that
$\Pi^j$ is always consistent with $\Pi^{j+1}$.
\end{proof}
\paragraph{Dynamic Programming Table} For each $j=1,\ldots, n$ and for each profile $\Pi$ of ${\mathcal{P}}_j$, we use a table entry $T(j,\Pi)$ to store the maximum revenue achievable among the price function for ${\mathcal{P}}_j$ that is consistent with the profile $\Pi$. Since there are $n^{O(\log m)}$ possibilities for the profile $\Pi$, the table size is $n^{O(\log m)}$. We now show the computation of the table. To compute $T(j,\Pi)$, we recall that given the profile $\Pi$, the revenue from consumers at level $j$ can be computed efficiently. Denote such revenue by $r_j(\Pi)$. The following equation holds:
\[T(j,\Pi) = r_j(\Pi) + \max_{\Pi' \mbox{ consistent with } \Pi} T(j-1, \Pi') \]
\paragraph{Computing the Solution} For each table entry $T(j,\Pi)$, we can keep track of the profile $\Pi'$ such that $T(j-1, \Pi')$ is the entry that is used to compute $T(j,\Pi)$. Let $T(n, \Pi)$ be the entry that contains the maximum value over all $\Pi$. The value in this entry represents the revenue we can get from the optimal pricing $p^*$, so it is enough to reconstruct the price function $p^*$. We first obtain a sequence of profiles $\Pi^1,\ldots, \Pi^n = \Pi$ such that $\Pi^j$ is a profile for ${\mathcal{P}}_j$ and that $\Pi^j$ is consistent with $\Pi^{j-1}$ for any $j =1,\ldots, n$. This sequence allows us to reconstruct a price function that is consistent with all the profiles in polynomial time.
| {
"timestamp": "2012-07-25T02:02:44",
"yymm": "1202",
"arxiv_id": "1202.2840",
"language": "en",
"url": "https://arxiv.org/abs/1202.2840",
"abstract": "Consider the following toy problem. There are $m$ rectangles and $n$ points on the plane. Each rectangle $R$ is a consumer with budget $B_R$, who is interested in purchasing the cheapest item (point) inside R, given that she has enough budget. Our job is to price the items to maximize the revenue. This problem can also be defined on higher dimensions. We call this problem the geometric pricing problem.In this paper, we study a new class of problems arising from a geometric aspect of the pricing problem. It intuitively captures typical real-world assumptions that have been widely studied in marketing research, healthcare economics, etc. It also helps classify other well-known pricing problems, such as the highway pricing problem and the graph vertex pricing problem on planar and bipartite graphs. Moreover, this problem turns out to have close connections to other natural geometric problems such as the geometric versions of the unique coverage and maximum feasible subsystem problems.We show that the low dimensionality arising in this pricing problem does lead to improved approximation ratios, by presenting sublinear-approximation algorithms for two central versions of the problem: unit-demand uniform-budget min-buying and single-minded pricing problems. Our algorithm is obtained by combining algorithmic pricing and geometric techniques. These results suggest that considering geometric aspect might be a promising research direction in obtaining improved approximation algorithms for such pricing problems. To the best of our knowledge, this is one of very few problems in the intersection between geometry and algorithmic pricing areas. Thus its study may lead to new algorithmic techniques that could benefit both areas.",
"subjects": "Computer Science and Game Theory (cs.GT); Computational Geometry (cs.CG); Data Structures and Algorithms (cs.DS)",
"title": "Geometric Pricing: How Low Dimensionality Helps in Approximability",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9728307668889047,
"lm_q2_score": 0.7279754489059774,
"lm_q1q2_score": 0.7081969142354967
} |
https://arxiv.org/abs/2105.02827 | Tight Approximation Algorithms for Geometric Bin Packing with Skewed Items | In the Two-dimensional Bin Packing (2BP) problem, we are given a set of rectangles of height and width at most one and our goal is to find an axis-aligned nonoverlapping packing of these rectangles into the minimum number of unit square bins. The problem admits no APTAS and the current best approximation ratio is $1.406$ by Bansal and Khan [SODA'14]. A well-studied variant of the problem is Guillotine Two-dimensional Bin Packing (G2BP), where all rectangles must be packed in such a way that every rectangle in the packing can be obtained by recursively applying a sequence of end-to-end axis-parallel cuts, also called guillotine cuts. Bansal, Lodi, and Sviridenko [FOCS'05] obtained an APTAS for this problem. Let $\lambda$ be the smallest constant such that for every set $I$ of items, the number of bins in the optimal solution to G2BP for $I$ is upper bounded by $\lambda\operatorname{opt}(I) + c$, where $\operatorname{opt}(I)$ is the number of bins in the optimal solution to 2BP for $I$ and $c$ is a constant. It is known that $4/3 \le \lambda \le 1.692$. Bansal and Khan [SODA'14] conjectured that $\lambda = 4/3$. The conjecture, if true, will imply a $(4/3+\varepsilon)$-approximation algorithm for 2BP. According to convention, for a given constant $\delta>0$, a rectangle is large if both its height and width are at least $\delta$, and otherwise it is called skewed. We make progress towards the conjecture by showing $\lambda = 4/3$ for skewed instance, i.e., when all input rectangles are skewed. Even for this case, the previous best upper bound on $\lambda$ was roughly 1.692. We also give an APTAS for 2BP for skewed instance, though general 2BP does not admit an APTAS. | \section{Introduction}
Two-dimensional Bin Packing (2BP) is a well-studied problem in combinatorial optimization.
It finds numerous applications in logistics, databases, and cutting stock.
In 2BP, we are given a set of $n$ rectangular items and square bins of side length 1.
The $i^{\textrm{th}}$ item is characterized by its width $w(i) \in (0,1]$ and height $h(i) \in (0,1]$.
Our goal is to find an axis-aligned nonoverlapping packing of
these items into the minimum number of square bins of side length 1.
There are two well-studied variants: (i) where the items cannot be rotated, and
(ii) they can be rotated by 90 degrees.
As is conventional in bin packing, we focus on asymptotic approximation algorithm.
For any optimization problem, the asymptotic approximation ratio (AAR)
of algorithm $\mathcal{A}$ is defined as $\lim_{m \to \infty} \sup_{I: \opt(I) = m} ({\mathcal{A}(I)}/{\opt(I)})$,
where $\opt(I)$ is the optimal objective value
and $\mathcal{A}(I)$ is the objective value of the solution output by algorithm $\mathcal{A}$, respectively, on input $I$.
Intuitively, AAR captures the algorithm's behavior
when $\opt(I)$ is large.
We call a bin packing algorithm $\alpha$-asymptotic-approximate iff its AAR is at most $\alpha$.
An Asymptotic Polynomial-Time Approximation Scheme (APTAS) is an algorithm
that accepts a parameter $\eps$ and has AAR of $(1+\eps)$.
2BP is a generalization of classical 1-D bin packing problem \cite{HobergR17, bp-aptas}.
However, unlike 1-D bin packing, 2BP does not admit an APTAS unless P=NP \cite{bansal2006bin}.
In 1982, Chung, Garey, and Johnson~\cite{chung1982packing} gave an approximation algorithm
with AAR 2.125 for 2BP. Caprara~\cite{caprara2008} obtained
a $T_{\infty}(\approx 1.691)$-asymptotic-approximation algorithm.
Bansal, Caprara, and Sviridenko~\cite{rna} introduced the Round and Approx framework
to obtain an AAR of $1+\ln(T_{\infty})$ ($\approx 1.525$).
Then Jansen and Praedel~\cite{JansenP2013} obtained an AAR of 1.5.
The present best AAR is $1+\ln(1.5)$ ($\approx 1.405$),
due to Bansal and Khan~\cite{bansal2014binpacking},
and works for both the cases with and without rotations.
The best lower bounds on the AAR for 2BP are
1 + 1/3792 and 1 + 1/2196 \cite{chlebik2009hardness},
for the versions with and without rotations, respectively.
In the context of geometric packing, guillotine cuts are well-studied and heavily used in practice \cite{sweeney1992cutting}.
The notions of {\em guillotine cuts} and {\em $k$-stage packing} were introduced
by Gilmore and Gomory in their seminal paper \cite{gilmore1965multistage} on cutting stock problem.
In $k$-stage packing each stage consists of either vertical or horizontal (but not both) axis-parallel end-to-end cuts, also called guillotine cuts.
In each stage, each of the rectangular regions obtained on the previous stage is considered separately
and can be cut again by using guillotine cuts. In $k$-stage packing,
the minimum number of cuts to obtain each rectangle from the initial packing is at most $k$, plus an
additional cut to trim (i.e., separate the rectangles themselves from waste area).
Note that in the cutting process we change the orientation (vertical or horizontal) of the cuts $k-1$ times.
2-stage packing, also called {\em shelf packing}, has been studied extensively.
In {\em guillotine packing}, the packing of items in each bin should be \emph{guillotinable},
i.e., items have to be packed in alternate horizontal and vertical stages
but there is no limit on the number of stages that can be used.
See \cref{sec:guill-examples} for examples.
Caprara et al.~\cite{caprara2005fast} gave an APTAS for 2-stage 2BP.
Bansal et al.~\cite{bansal2005tale} showed an APTAS for guillotine 2BP.
The presence of an APTAS for guillotine 2BP raises an important question:
can the optimal solution to guillotine 2BP be used as a good approximate solution to 2BP?
Formally, let $\opt(I)$ and $\opt_g(I)$ be the minimum number of bins and the
minimum number of guillotinable bins, respectively, needed to pack items $I$.
Let $\lambda$ be the smallest constant such that for some constant $c$
and for every set $I$ of items, we get $\opt_g(I) \le \lambda\opt(I) + c$.
Then $\lambda$ is called the Asymptotic Price of Guillotinability (APoG).
It is easy to show that $\mathrm{APoG} \ge 4/3$\footnote{Consider
a set $I$ of items containing $2m$ rectangles of width 0.6 and height 0.4 and
$2m$ rectangles of width 0.4 and height 0.6.
Then $\opt(I) = m$ and $\opt_g(I) = \ceil{4m/3}$.}.
Bansal and Khan~\cite{bansal2014binpacking} conjectured that $\mathrm{APoG} = 4/3$.
If true, this would imply a $(4/3+\eps)$-asymptotic-approximation algorithm for 2BP.
However, the present upper bound on APoG is only $T_\infty$ ($\approx1.691$),
due to Caprara's HDH algorithm~\cite{caprara2008} for 2BP, which produces a 2-stage packing.
APTASes are known for some special cases for 2BP,
such as when all items are squares~\cite{bansal2006bin} or
when all rectangles are small in both dimensions
\cite{coffman1980performance} (see \cref{thm:nfdh-small-2} in \cref{sec:nfdh}).
Another important class is {\em skewed} rectangles.
We say that a rectangle is $\delta$-large if,
for some constant $\delta>0$, its width and height are more than $\delta$;
otherwise, the rectangle is $\delta$-skewed.
We just say that a rectangle is large or skewed when $\delta$ is clear from the context.
An instance of 2BP is skewed if all the rectangles in the input are skewed.
Skewed instances are important in geometric packing (see Section \ref{subs:prior}).
This special case is practically relevant~\cite{galvez2020tight}:
e.g., in scheduling, it captures scenarios where
no job can consume a significant amount of a shared resource
(energy, memory space, etc.) for a significant amount of time.
Even for skewed instance for 2BP, the best known AAR is 1.406 \cite{bansal2014binpacking}.
Also, for skewed instance, the best known upper bound on APoG is $T_\infty \approx 1.691$.
\subsection{Related Works}
\label{subs:prior}
Multidimensional packing and covering problems are fundamental in combinatorial optimization \cite{CKPT17}.
Vector packing (VP) is another variant of bin packing, where the input is a set of vectors in $[0, 1]^d$ and the
goal is to partition the vectors into the minimum number of parts (bins) such that in each part, the sum of
vectors is at most 1 in every coordinate.
The present best approximation algorithm attains an AAR of $(0.807+\ln(d+1))$ \cite{bansal2016improved}
and there is a matching $\Omega(\ln d)$-hardness \cite{sandeep2021optimal}. Generalized multidimensional
packing \cite{aco-gvbp, aco-gvks} generalizes both geometric and vector packing.
In two-dimensional strip packing (2SP) \cite{coffman1980performance, steinberg1997strip}, we are given a
set of rectangles and a bounded width strip. The goal is to obtain an axis-aligned nonoverlapping packing
of all rectangles such that the height of the packing is minimized.
The best-known approximation ratio for SP is $5/3+\eps$ \cite{harren20115} and it is NP-hard to obtain
better than 3/2-approximation.
However, there exist APTASes for the problem, for both the cases
with and without rotations \cite{kenyon1996strip, jansen2005strip}.
In two-dimensional knapsack (2GK) \cite{jansen2004rectangle}, the rectangles have associated profits and
our goal is to pack the maximum profit subset into a unit square knapsack.
The present best polynomial-time (resp.~pseudopolynomial-time) approximation ratio
for 2GK is 1.809~\cite{galvez2017approximating} (resp.\ 4/3~\cite{GalSocg21}).
These geometric packing problem are studied for $d$-dimensions ($d\ge 2$) \cite{eku-hdhk} as well.
Both 2SP and 2GK are also well-studied under guillotine packing.
Seiden and Woeginger~\cite{seiden2005two} gave an APTAS for guillotine 2SP.
Khan et al.~\cite{KhanSocg21} have recently given a pseudopolynomial-time
approximation scheme for guillotine 2GK.
Recently, guillotine cuts \cite{pach2000cutting} have received attention due to
their connection with the maximum independent set of rectangles (MISR) problem~\cite{AdamaszekHW19}.
In MISR, we are given a set of possibly overlapping rectangles and the goal is to find the maximum cardinality set of rectangles so that there is
no pairwise overlap. It was noted in \cite{khan2020guillotine, abed2015guillotine} that for any set of $n$ non-overlapping axis-parallel rectangles, if there is a guillotine cutting sequence
separating $\alpha n$ of them, then it implies a $1/\alpha$-approximation for MISR.
Skewed instance is an important special case in these problems.
In some problems, such as MISR and 2GK, if all items are $\delta$-large then we can solve them exactly in polynomial time.
So, the inherent difficulty of these problems lies in skewed instances.
For VP, hard instances are again skewed, e.g.,
Bansal, Eli\'a\v{s} and Khan~\cite{bansal2016improved} showed that
hard instances for 2-D VP (for a class of algorithms called {\em rounding based algorithms})
are skewed instances, where one dimension is $1-\eps$ and the other dimension is $\eps$.
Galvez el al.~\cite{galvez2020tight} recently studied strip packing when all items are skewed.
For skewed instances, they showed $(3/2-\eps)$ hardness of approximation and a matching $(3/2+\eps)$-approximation algorithm.
For 2GK, when the height of each item is at most $\eps^3$,
a $(1-72\eps)$-approximation algorithm is known~\cite{fishkin2005efficient}.
\subsection{Our Contributions}
We study 2BP for the special case of $\delta$-skewed{} rectangles,
where $\delta \in (0, 1/2]$ is a constant.
First, we make progress towards the conjecture \cite{bansal2014binpacking} that $\mathrm{APoG} = 4/3$.
Even for skewed{} rectangles, we only knew $4/3 \le \mathrm{APoG} \le T_{\infty}(\approx 1.691)$.
We resolve the conjecture for skewed{} rectangles, by giving lower and upper bounds of roughly $4/3$
when $\delta$ is a small constant.
Specifically, we give an algorithm for 2BP, called $\thinGPack_{\eps}$,
that takes a parameter $\eps \in (0, 1)$ as input.
For a set $I$ of $\delta$-skewed{} rectangles, we show that when
$\delta$ and $\eps$ are close to 0, $\thinGPack_{\eps}(I)$ outputs a
4-stage packing of $I$ into roughly $4\opt(I)/3 + O(1)$ bins.
\begin{restatable}{theorem}{rthmThinGPack}
\label{thm:thin-gpack}
Let $I$ be a set of $\delta$-skewed items, where $\delta \in (0, 1/2]$.
Then $\thinGPack_{\eps}(I)$ outputs a 4-stage packing of $I$
in time $O((1/\eps)^{O(1/\eps)} + n\log n)$.
Furthermore, the number of bins used is at most
$({4}/{3})(1+8\delta)(1+7\eps)\opt(I) + ({8}/{\eps^2}) + 30$.
\end{restatable}
A tighter analysis shows that
when $\delta \le 1/16$ and $\eps \le 10^{-4}$,
then $\thinGPack$ has AAR $(76/45)(1+7\eps) < T_{\infty}$,
which improves upon the best-known bound on APoG for the general case.
The lower bound of $4/3$ on APoG can be extended to skewed{} items.
We formally prove this in \cref{sec:apog-lb}.
Hence, our bounds on APoG are tight for skewed{} items.
Our result indicates that to improve the bounds for APoG in the general case,
we should focus on $\delta$-large items.
Our bounds on APoG also hold when items can be rotated.
See \cref{sec:guill-rot} for details.
Our other main result is an APTAS for 2BP for skewed{} items.
Formally, we give an algorithm for 2BP, called $\thinCPack$,
and we show that for every constant $\eps \in (0, 1)$,
there exists a constant $\delta \in (0, \eps)$ such that the algorithm has an AAR of $1+\eps$
when all items in the input are $\delta$-skewed{} rectangles.
$\thinCPack$ can also be extended to the case where items can be rotated. %
The best-known AAR for 2BP is $1 + \ln(1.5) + \eps$.
Our result indicates that to improve upon algorithms for 2BP,
one should focus on $\delta$-large items.
In \cref{sec:guill-thin}, we describe the $\thinGPack$ algorithm
and prove \cref{thm:thin-gpack}.
In \cref{sec:thin-bp}, we describe the $\thinCPack$ algorithm
and prove that it has an AAR of $1+\eps$.
\section{Preliminaries}
\label{sec:preliminaries}
Let $[n] := \{1, 2, \ldots, n\}$, for $n \in \mathbb{N}$.
For a rectangle $i$, its area $a(i) := w(i)h(i)$.
For a set $I$ of rectangles, let $a(I) := \sum_{i \in I} a(i)$.
An \emph{axis-aligned packing} of an item $i$ in a bin
is specified by a pair $(x(i), y(i))$, where $x(i), y(i) \in [0,1]$,
so that $i$ is placed in the region
$[x(i), x(i)+w(i)] \times [y(i), y(i)+h(i)]$.
A packing of rectangles in a bin is called \emph{nonoverlapping} iff for any two
distinct items $i$ and $j$, the rectangles
$(x(i), x(i)+w(i)) \times (y(i), y(i)+h(i))$ and
$(x(j), x(j)+w(j)) \times (y(j), y(j)+h(j))$ are disjoint.
Equivalently, items may only intersect at their boundaries.\\
\noindent \textbf{Next-Fit Decreasing Height (NFDH):}
The NFDH algorithm~\cite{coffman1980performance}
is a simple algorithm for 2SP and 2BP. We will use the following results on NFDH.
We defer the proofs to \cref{sec:nfdh}.
\begin{restatable}{lemma}{rthmNfdhSmall}
\label{thm:nfdh-small}
Let $I$ be a set of items where each item $i$ has $w(i) \le \delta_W$
and $h(i) \le \delta_H$. Let there be a bin of width $W$ and height $H$.
If $a(I) \le (W - \delta_W)(H - \delta_H)$, then NFDH can pack $I$ into the bin.
\end{restatable}
\begin{lemma}
\label{thm:nfdh-wide-tall}
\label{thm:nfdh-tall}
\label{thm:nfdh-wide}
Let $I$ be a set of rectangular items. Then NFDH uses less than
$(2a(I)+1)/(1-\delta)$ bins to pack $I$ when $h(i) \le \delta$ for each item $i$
and $2a(I)/(1-\delta) + 3$ bins when $w(i) \le \delta$ for each item $i$.
\end{lemma}
If we swap the coordinate axes in NFDH, we get the
Next-Fit Decreasing Width (NFDW) algorithm.
Analog{}s of the above results hold for NFDW.\\
\noindent \textbf{Slicing Items:}
We will consider variants of 2BP where some items can be \emph{sliced}.
Formally, slicing a rectangular item $i$ using a horizontal cut is the operation of
replacing $i$ by two items $i_1$ and $i_2$ such that
$w(i) = w(i_1) = w(i_2)$ and $h(i) = h(i_1) + h(i_2)$.
Slicing using vertical cut is defined analogously.
Allowing some items to be sliced may reduce the number of bins required to pack.
See \cref{fig:frac-pack} for an example.
\begin{figure}[htb]
\centering
\begin{tikzpicture}[
myarrow/.style = {->,>={Stealth},semithick},
mybrace/.style = {decoration={amplitude=3pt,brace,mirror,raise=1pt},semithick,decorate},
every node/.style = {scale=0.8},
scale=0.8
]
\draw (0,0) rectangle +(3,3);
\draw[fill={black!30}] (0,0) rectangle +(1.2,3);
\draw[fill={black!10}] (3,0) rectangle +(-1.5,1.2);
\draw[fill={black!10}] (3,1.2) rectangle +(-1.5,1.2);
\draw[mybrace] (0,0) -- node[below=1pt] {0.4} +(1.2,0);
\draw[mybrace] (1.5,0) -- node[below=1pt] {0.5} +(1.5,0);
\draw[mybrace] (3,0) -- node[right=2pt] {0.4} +(0,1.2);
\draw[mybrace] (3,1.2) -- node[right=2pt] {0.4} +(0,1.2);
\draw[fill={black!30}] (-8,0) rectangle +(1.2,3);
\path (-8,0) -- node[below=0pt] {0.4} +(1.2,0);
\path (-8,0) -- node[left=0pt] {1} +(0,3);
\node at (-6.2,1.5) {+};
\draw[fill={black!10}] (-5,1) rectangle +(3,1.2);
\path (-5,1) -- node[left=0pt] {0.4} +(0,1.2);
\draw[dashed] (-3.5,2.5) -- (-3.5,0.5);
\node[rotate=90,transform shape] at (-3.5,0.3) {\large\ding{34}};
\draw[mybrace] (-5,1) -- node[below=1pt] {0.5} +(1.5,0);
\draw[mybrace] (-3.5,1) -- node[below=1pt] {0.5} +(1.5,0);
\draw[myarrow] (-1.5,1.5) -- (-0.5,1.5);
\end{tikzpicture}
\caption{Packing two items into a bin, where one item is sliced using a vertical cut.
If slicing were forbidden, two bins would be required.}
\label{fig:frac-pack}
\end{figure}
Alamdari et al.~\cite{alamdari2013smart} explored algorithms for
a variant of 2SP where items can be sliced using vertical cuts,
which has applications in smart-grid electricity allocation.
Many packing algorithms \cite{kenyon1996strip,JansenP2013,bansal2005tale}
solve the sliceable version of the problem as a subroutine.
\section{Guillotinable Packing of Skewed{} Rectangles}
\label{sec:guill-thin}
An item is called $(\delta_W, \delta_H)$-skewed{} iff its width is at most $\delta_W$
or its height is at most $\delta_H$.
In this section, we consider the problem of obtaining tight upper and lower bounds
on APoG for $(\delta_W, \delta_H)$-skewed{} items.
We will describe the $\thinGPack$ algorithm and prove \cref{thm:thin-gpack}.
\subsection{Packing With Slicing}
Before describing $\thinGPack$,
let us first look at a closely-related variant of this problem,
called the \emph{sliceable 2D bin packing problem}, denoted as S2BP.
In this problem, we are given two sets of rectangular items, $\widetilde{W}$ and $\widetilde{H}$, where
items in $\widetilde{W}$ have width more than $1/2$, and items in $\widetilde{H}$ have height more than $1/2$.
$\widetilde{W}$ is called the set of wide items and $\widetilde{H}$ is called the set of tall items.
We are allowed to \emph{slice} items in $\widetilde{W}$ using horizontal cuts
and slice items in $\widetilde{H}$ using vertical cuts, and our task is to pack
$\widetilde{W} \cup \widetilde{H}$ into the minimum number of bins without rotating the items.
See \cref{fig:bp-vs-sbp} for an example that illustrates the difference
between 2BP and S2BP.
\begin{figure}[htb]
\begin{subfigure}{0.45\textwidth}
\centering
\begin{tikzpicture}[
witem/.style={draw,fill={black!30}},
hitem/.style={draw,fill={black!10}},
bin/.style={draw,thick},
myarrow/.style={->,>={Stealth},thick},
scale=0.7,
]
\begin{scope}
\node at (0.3, 2.0) {$\widetilde{W}$:};
\node at (0.3, 0.6) {$\widetilde{H}$:};
\path[hitem] (1.0, 0.0) rectangle +(1.2, 1.2);
\path[hitem] (2.4, 0.0) rectangle +(1.2, 1.2);
\path[witem] (1.0, 1.4) rectangle +(1.2, 1.2);
\path[witem] (2.4, 1.4) rectangle +(1.2, 1.2);
\end{scope}
\draw[myarrow] (3.9, 1.3) -- (5.2, 1.3);
\begin{scope}[xshift={5.5cm},yshift={-0.8cm}]
\path[hitem] (0.0, 0.0) rectangle +(1.2, 1.2);
\path[hitem] (2.2, 0.0) rectangle +(1.2, 1.2);
\path[witem] (0.0, 2.2) rectangle +(1.2, 1.2);
\path[witem] (2.2, 2.2) rectangle +(1.2, 1.2);
\path[bin] (0.0, 0.0) rectangle +(2, 2);
\path[bin] (2.2, 0.0) rectangle +(2, 2);
\path[bin] (0.0, 2.2) rectangle +(2, 2);
\path[bin] (2.2, 2.2) rectangle +(2, 2);
\end{scope}
\end{tikzpicture}
\caption{Packing items into 4 bins without slicing.}
\end{subfigure}
\hfil
\begin{subfigure}{0.5\textwidth}
\centering
\begin{tikzpicture}[
witem/.style={draw,fill={black!30}},
hitem/.style={draw,fill={black!10}},
bin/.style={draw,thick},
myarrow/.style={->,>={Stealth},thick},
cutline/.style={draw={black!50!red},dashed,semithick},
scale=0.7,
]
\begin{scope}
\node at (0.3, 2.0) {$\widetilde{W}$:};
\node at (0.3, 0.6) {$\widetilde{H}$:};
\path[hitem] (1.0, 0.0) rectangle +(1.2, 1.2);
\path[hitem] (2.4, 0.0) rectangle +(1.2, 1.2);
\path[witem] (1.0, 1.4) rectangle +(1.2, 1.2);
\path[witem] (2.4, 1.4) rectangle +(1.2, 1.2);
\path[cutline] (2.3, 2.0) -- (3.7, 2.0);
\node[xscale=-1,transform shape] at (3.9, 1.98) {\large\ding{34}};
\path[cutline] (3.0, -0.1) -- (3.0, 1.3);
\node[rotate=90,transform shape] at (3.02, -0.3) {\large\ding{34}};
\end{scope}
\draw[myarrow] (3.9, 1.3) -- (5.2, 1.3);
\begin{scope}[xshift={5.5cm},yshift={0.3cm}]
\path[witem] (0.0, 0.0) rectangle +(1.2, 1.2);
\path[hitem] (2.2, 0.0) rectangle +(1.2, 1.2);
\path[witem] (0.0, 1.2) rectangle +(1.2, 0.6);
\path[witem] (2.2, 1.2) rectangle +(1.2, 0.6);
\path[hitem] (1.2, 0.0) rectangle +(0.6, 1.2);
\path[hitem] (3.4, 0.0) rectangle +(0.6, 1.2);
\path[bin] (0.0, 0.0) rectangle +(2, 2);
\path[bin] (2.2, 0.0) rectangle +(2, 2);
\end{scope}
\end{tikzpicture}
\caption{Packing items into 2 bins by horizontally slicing an item in $\widetilde{W}$
and vertically slicing an item in $\widetilde{H}$.}
\end{subfigure}
\caption[2BP vs.~S2BP]{Example illustrating 2BP vs.~S2BP.
There are 2 wide items ($\widetilde{W}$) and 2 tall items ($\widetilde{H}$).
The items are squares of side length 0.6 and the bins are squares of side length 1.}
\label{fig:bp-vs-sbp}
\end{figure}
We first describe a simple $4/3$-asymptotic-approximation algorithm
for S2BP, called $\greedyPack$, that outputs a 2-stage packing.
Later, we will show how to use $\greedyPack$ to design $\thinGPack$.
We assume that the bin is a square of side length 1. Since we can slice items,
we allow items in $\widetilde{W}$ to have height more than 1
and items in $\widetilde{H}$ to have width more than 1.
For $X \subseteq \widetilde{W}$, $Y \subseteq \widetilde{H}$, define $\hsum(X) := \sum_{i \in X} h(i)$;
$\wsum(Y) := \sum_{i \in Y} w(i)$;
$\wmax(X) := \max_{i \in X} w(i) \textrm{ if } X \neq \emptyset$, and $0 \textrm{ if } X = \emptyset$;
$\hmax(Y) := \max_{i \in Y} h(i) \textrm{ if } Y \neq \emptyset$, and $0 \textrm{ if } Y = \emptyset$.
In the algorithm $\greedyPack(\widetilde{W}, \widetilde{H})$, we first sort items $\widetilde{W}$ in decreasing order
of width and sort items $\widetilde{H}$ in decreasing order of height.
Suppose $\hsum(\widetilde{W}) \ge \wsum(\widetilde{H})$. Let $X$ be the largest prefix of $\widetilde{W}$
of total height at most 1, i.e., if $\hsum(\widetilde{W}) > 1$,
then $X$ is a prefix of $\widetilde{W}$ such that $\hsum(X) = 1$ (slice items if needed),
and $X = \widetilde{W}$ otherwise.
Pack $X$ into a bin such that the items touch the right edge of the bin.
Then we pack the largest possible prefix of $\widetilde{H}$
into the empty rectangular region of width $1 - \wmax(X)$ in the left side of the bin.
We call this a type-1 bin. See \cref{fig:greedy-pack:1} for an example.
If $\hsum(\widetilde{W}) < \wsum(\widetilde{H})$, we proceed analogously in a coordinate-swapped way,
i.e., we first pack tall items in the bin and then pack wide items in the remaining space.
Call this bin a type-2 bin.
We pack the rest of the items into bins in the same way.
\begin{figure}[htb]
\begin{subfigure}{0.45\textwidth}
\centering
\tikzset{mytransform/.style={scale=0.7}}
\tikzset{wItem/.style={draw,fill={black!30}}}
\tikzset{hItem/.style={draw,fill={black!10}}}
\ifcsname pGameL\endcsname\else\newlength{\pGameL}\fi
\setlength{\pGameL}{0.2cm}
\tikzset{bin/.style={draw,thick}}
\begin{tikzpicture}[mytransform]
\path[wItem] (5\pGameL, 0\pGameL) rectangle +(15\pGameL, 2\pGameL);
\path[wItem] (6\pGameL, 2\pGameL) rectangle +(14\pGameL, 3\pGameL);
\path[wItem] (6\pGameL, 5\pGameL) rectangle +(14\pGameL, 2\pGameL);
\path[wItem] (7\pGameL, 7\pGameL) rectangle +(13\pGameL, 3\pGameL);
\path[wItem] (8\pGameL, 10\pGameL) rectangle +(12\pGameL, 4\pGameL);
\path[wItem] (8\pGameL, 14\pGameL) rectangle +(12\pGameL, 2\pGameL);
\path[wItem] (9\pGameL, 16\pGameL) rectangle +(11\pGameL, 2\pGameL);
\path[wItem] (10\pGameL, 18\pGameL) rectangle +(10\pGameL, 2\pGameL);
\path[hItem] (1\pGameL, 4\pGameL) rectangle +(2\pGameL, 16\pGameL);
\path[hItem] (0\pGameL, 2\pGameL) rectangle +(1\pGameL, 18\pGameL);
\path[hItem] (3\pGameL, 5\pGameL) rectangle +(2\pGameL, 15\pGameL);
\draw[semithick,dashed] (5\pGameL, 0\pGameL) -- +(0\pGameL, 20\pGameL);
\path[bin] (0\pGameL, 0\pGameL) rectangle (20\pGameL, 20\pGameL);
\end{tikzpicture}
\caption{A type-1 bin produced by $\greedyPack$.
Wide items are packed on the right. Tall items are packed on the left.}%
\label{fig:greedy-pack:1}
\end{subfigure}
\hfill
\begin{subfigure}{0.45\textwidth}
\centering
\tikzset{mytransform/.style={xscale=-0.7,yscale=0.7,rotate=90}}
\tikzset{wItem/.style={draw,fill={black!10}}}
\tikzset{hItem/.style={draw,fill={black!30}}}
\ifcsname pGameL\endcsname\else\newlength{\pGameL}\fi
\setlength{\pGameL}{0.2cm}
\tikzset{bin/.style={draw,thick}}
\begin{tikzpicture}[mytransform]
\path[wItem] (5\pGameL, 0\pGameL) rectangle +(15\pGameL, 2\pGameL);
\path[wItem] (6\pGameL, 2\pGameL) rectangle +(14\pGameL, 3\pGameL);
\path[wItem] (6\pGameL, 5\pGameL) rectangle +(14\pGameL, 2\pGameL);
\path[wItem] (7\pGameL, 7\pGameL) rectangle +(13\pGameL, 3\pGameL);
\path[wItem] (8\pGameL, 10\pGameL) rectangle +(12\pGameL, 4\pGameL);
\path[wItem] (8\pGameL, 14\pGameL) rectangle +(12\pGameL, 2\pGameL);
\path[wItem] (9\pGameL, 16\pGameL) rectangle +(11\pGameL, 2\pGameL);
\path[wItem] (10\pGameL, 18\pGameL) rectangle +(10\pGameL, 2\pGameL);
\path[hItem] (1\pGameL, 4\pGameL) rectangle +(2\pGameL, 16\pGameL);
\path[hItem] (0\pGameL, 2\pGameL) rectangle +(1\pGameL, 18\pGameL);
\path[hItem] (3\pGameL, 5\pGameL) rectangle +(2\pGameL, 15\pGameL);
\draw[semithick,dashed] (5\pGameL, 0\pGameL) -- +(0\pGameL, 20\pGameL);
\path[bin] (0\pGameL, 0\pGameL) rectangle (20\pGameL, 20\pGameL);
\end{tikzpicture}
\caption{A type-2 bin produced by $\greedyPack$.
Tall items are packed above. Wide items are packed below.}%
\label{fig:greedy-pack:2}
\end{subfigure}
\caption{Examples of type-1 and type-2 bins produced by $\greedyPack$.}
\label{fig:greedy-pack}
\end{figure}
\begin{claim}
\label{thm:greedy-pack}
$\greedyPack(\widetilde{W}, \widetilde{H})$ outputs a 2-stage packing of $\widetilde{W} \cup \widetilde{H}$.
It runs in $O(m + |\widetilde{W}|\log|\widetilde{W}| + |\widetilde{H}|\log|\widetilde{H}|)$ time,
where $m$ is the number of bins used.
Furthermore, it slices items in $\widetilde{W}$ by making at most $m-1$ horizontal cuts
and slices items in $\widetilde{H}$ by making at most $m-1$ vertical cuts.
\end{claim}
Since items in $\widetilde{W}$ have width more than $1/2$,
no two items can be placed side-by-side.
Hence, $\smallceil{\hsum(\widetilde{W})} = \opt(\widetilde{W}) \le \opt(\widetilde{W} \cup \widetilde{H})$.
Similarly, $\smallceil{\wsum(\widetilde{H})} \le \opt(\widetilde{W} \cup \widetilde{H})$.
So, if all bins have the same type, $\greedyPack$ uses
$\max(\smallceil{\hsum(\widetilde{W})}, \smallceil{\wsum(\widetilde{H})}) = \opt(\widetilde{W} \cup \widetilde{H})$ bins.
We will now focus on the case where
some bins have type 1 and some have type 2.
\begin{definition}
In a type-1 bin, let $X$ and $Y$ be the wide and tall items, respectively.
The bin is called \emph{full} iff $\hsum(X) = 1$ and $\wsum(Y) = 1 - \wmax(X)$.
Define fullness for type-2 bins analogously.
\end{definition}
We first show that full bins pack items of a large total area,
and then we show that if some bins have type 1 and some bins have type 2,
then there can be at most 2 non-full bins.
This will help us get an upper-bound on the number of bins used by $\greedyPack(\widetilde{W}, \widetilde{H})$
in terms of $a(\widetilde{W} \cup \widetilde{H})$.
\begin{lemma}
\label{thm:area-bound}
Let there be $m_1$ type-1 full bins.
Let $J_1$ be the items in them.
Then $m_1 \le 4a(J_1)/3 + 1/3$.
\end{lemma}
\begin{proof}
In the $j^{\textrm{th}}$ full bin of type 1, let $X_j$ be the items from $\widetilde{W}$
and $Y_j$ be the items from $\widetilde{H}$. Let
$\ell_j := \wmax(X_j) \textrm{ if } j \le m_1$
and $\ell_{m_1+1} := 1/2$.
Since all items have their larger dimension more than $1/2$,
$\ell_j \ge 1/2$ and $\hmax(Y_j) > 1/2$, for any $j \in [m_1]$.
$a(X_j) \ge \ell_{j+1}$, since $X_j$ has height 1 and width at least $\ell_{j+1}$.
$a(Y_j) \ge (1-\ell_j)/2$, since $Y_j$ has width $1 - \ell_j$ and height more than $1/2$.
Therefore,
$a(J_1) = \sum_{j=1}^{m_1} (a(X_j) + a(Y_j))
\ge \sum_{j=1}^{m_1} (\ell_{j+1} + (1-\ell_j)/2)
\ge \sum_{j=1}^{m_1} \left(({\ell_{j+1}}/{2}) + ({1}/{4})
+ ({1}/{2}) - ({\ell_j}/{2})\right)
= ({3m_1}/{4}) + ({1}/{4}) - ({\ell_1}/{2})
\ge ({3m_1-1}/{4})$.\\
In the above inequalities, we used that $\ell_{j+1} \ge 1/2$
and $\ell_1 \le 1$.
Therefore, $m_1 \le 4a(J_1)/3 + 1/3$.
\end{proof}
An analog{} of \cref{thm:area-bound} can be proven for type-2 bins.
Note that \cref{thm:area-bound} implies that the average area of full bins is close to $3/4$.
It is possible for an individual full bin to have area close to 1/2,
but the number of such bins is small, due to the telescopic sum in \cref{thm:area-bound}.
Let $m$ be the number of bins used by $\greedyPack(\widetilde{W}, \widetilde{H})$.
After $j$ bins have been packed, let $A_j$ be the height of the remaining items in $\widetilde{W}$
and $B_j$ be the width of the remaining items in $\widetilde{H}$.
Let $t_j$ be the type of the $j^{\textrm{th}}$ bin (1 for type-1 bin and 2 for type-2 bin).
So $t_j = 1 \iff A_{j-1} \ge B_{j-1}$.
We first show that $|A_{j-1} - B_{j-1}| \le 1 \implies |A_j - B_j| \le 1$.
This means that once the difference between $\hsum(\widetilde{W})$ and $\wsum(\widetilde{H})$ becomes at most 1,
it continues to stay at most 1.
Next, we show that $t_j \neq t_{j+1} \implies |A_{j-1} - B_{j-1}| \le 1$.
This means that if some bins have type 1 and some have type 2,
then the difference between $\hsum(\widetilde{W})$ and $\wsum(\widetilde{H})$ will eventually become at most 1.
In the first non-full bin, we will use up all the wide items or the tall items.
We will show that the remaining items have total height or total width at most 1,
so we can have at most 1 more non-full bin.
Hence, there can be at most 2 non-full bins when we have both type-1 and type-2 bins.
In the $j^{\textrm{th}}$ bin, let $a_j$ be the height of items from $\widetilde{W}$
and $b_j$ be the width of items from $\widetilde{H}$.
Hence, for all $j \in [m]$,
$A_{j-1} = A_j + a_j$ and $B_{j-1} = B_j + b_j$.
\begin{lemma}
\label{thm:diff-capture}
$|A_{j-1} - B_{j-1}| \le 1 \implies |A_j - B_j| \le 1$.
\end{lemma}
\begin{proof}
W.l.o.g.{}, assume $A_{j-1} \ge B_{j-1}$. So, $t_j = 1$. Suppose $a_j < b_j$.
Then $a_j < 1$, so we used up $\widetilde{W}$ in the $j^{\textrm{th}}$ bin. Therefore,
$A_j = 0 \implies A_{j-1} = a_j < b_j \le b_j + B_j = B_{j-1}$,
which contradicts. Hence, $a_j \ge b_j$.
As $0 \le (A_{j-1} - B_{j-1}), (a_j - b_j) \le 1$, we get
$A_j - B_j = (A_{j-1} - B_{j-1}) - (a_j - b_j) \in [-1, 1]$.
\end{proof}
\begin{lemma}
\label{thm:tdiff-implies-adiff}
$t_j \neq t_{j+1} \implies |A_{j-1} - B_{j-1}| \le 1$.
\end{lemma}
\begin{proof}
W.l.o.g.{}, assume $t_j = 1$ and $t_{j+1} = 2$. Then
\[ A_{j-1} \ge B_{j-1} \textrm{ and } A_j < B_j
\implies B_{j-1} \le A_{j-1} < B_{j-1} + a_j - b_j
\implies A_{j-1} - B_{j-1} \in \ropenInterval{0, 1}. \qedhere \]
\end{proof}
\begin{lemma}
\label{thm:non-full-ub}
If all bins don't have the same type, then there can be at most 2 non-full bins.
\end{lemma}
\begin{proof}
Let there be $p$ full bins. %
Assume w.l.o.g.{} that in the $(p+1)^{\textrm{th}}$ bin, we used up all items from $\widetilde{W}$ but not $\widetilde{H}$.
Hence, $A_{p+1} = 0$ and $\forall i \ge p+2$, $t_i = 2$.
Since all bins don't have the same type, $\exists k \le p+1$ such that
$t_k = 1$ and $t_{k+1} = 2$.
By \cref{thm:tdiff-implies-adiff,thm:diff-capture}, we get $|A_{p+1} - B_{p+1}| \le 1$,
implying $B_{p+1} \le 1$.
Hence, the $(p+1)^{\textrm{th}}$ bin will use up all tall items,
implying at most 2 non-full bins.
\end{proof}
\begin{theorem}
\label{thm:greedy-pack-bins}
The number of bins $m$ used by $\greedyPack$ is at most
\\ $\max\left(\smallceil{\hsum(\widetilde{W})}, \smallceil{\wsum(\widetilde{H})},
\frac{4}{3}a(\widetilde{W} \cup \widetilde{H}) + \frac{8}{3}\right)$.
\end{theorem}
\begin{proof}
If all bins have the same type, then $m \le \max(\smallceil{\hsum(\widetilde{W})}, \smallceil{\wsum(\widetilde{H})})$.
Let there be $m_1$ (resp.~$m_2$) full bins of type 1 (resp.~type 2)
and let $J_1$ (resp.~$J_2$) be the items inside those bins.
Then by \cref{thm:area-bound}, we get $m_1 \le 4a(J_1)/3 + 1/3$ and $m_2 \le 4a(J_2)/3 + 1/3$.
Hence, $m_1 + m_2 \le 4a(\widetilde{W} \cup \widetilde{H})/3 + 2/3$.
If all bins don't have the same type, then by \cref{thm:non-full-ub},
there can be at most 2 non-full bins, so $\greedyPack(\widetilde{W}, \widetilde{H})$
uses at most $4a(\widetilde{W} \cup \widetilde{H})/3 + 8/3$ bins.
\end{proof}
\subsection{The \texorpdfstring{$\thinGPack$}{skewed4Pack} Algorithm}
\label{sec:thin-gpack}
We now return to the 2BP problem.
$\thinGPack$ is an algorithm for 2BP takes as input a set $I$ of rectangular items
and a parameter $\eps \in (0, 1)$ where $\eps^{-1} \in \mathbb{Z}$.
It outputs a 4-stage bin packing of $I$.
$\thinGPack$ has the following outline:
\begin{enumerate}[A.]
\item Use linear grouping \cite{bp-aptas,kenyon1996strip} to round up the width or height of each item in $I$.
This gives us a new instance $\widehat{I}$.
\item Pack $\widehat{I}$ into $1/\eps^2 + 1$ shelves,
after possibly \emph{slicing} some items.
Each shelf is a rectangular region with width or height more than $1/2$ and is fully packed, i.e.,
the total area of items in a shelf equals the area of the shelf.
If we treat each shelf as an item, we get a new instance $\widetilde{I}$.
\item Compute a packing of $\widetilde{I}$ into bins, after possibly slicing some items,
using $\greedyPack$.
\item Pack most of the items of $I$ into the shelves in the bins. We will prove that
the remaining items have very small area, so they can be packed separately using NFDH.
\end{enumerate}
\noindent \textbf{A. Item Classification and Rounding.}
Define $W := \{i \in I: h(i) \le \delta_H \}$ and $H := I - W$.
Items in $W$ are called \emph{wide} and items in $H$ are called \emph{tall}.
Let $W^{(L)} := \{i \in W: w(i) > \eps \}$ and $W^{(S)} := W - W^{(L)}$.
Similarly, let $H^{(L)} := \{i \in H: h(i) > \eps \}$ and $H^{(S)} := H - H^{(L)}$.
We will now use \emph{linear grouping}~\cite{bp-aptas,kenyon1996strip}
to round up the widths of items $W^{(L)}$ and the heights of items $H^{(L)}$
to get items $\What^{(L)}$ and $\Hhat^{(L)}$, respectively.
By \cref{thm:lingroup-n} in \cref{sec:lingroup},
items in $\What^{(L)}$ have at most $1/\eps^2$ distinct widths
and items in $\Hhat^{(L)}$ have at most $1/\eps^2$ distinct heights.
Let $\widehat{W} := \What^{(L)} \cup W^{(S)}$,
$\widehat{H} := \Hhat^{(L)} \cup H^{(S)}$,
and $\widehat{I} := \widehat{W} \cup \widehat{H}$.
Let $\fopt(\widehat{I})$ be the minimum number of bins needed to pack $\widehat{I}$,
where items in $\What^{(L)}$ can be sliced using horizontal cuts,
items in $\Hhat^{(L)}$ can be sliced using vertical cuts,
and items in $W^{(S)} \cup H^{(S)}$ can be sliced using both vertical and horizontal cuts.
Then the following lemma follows from \cref{thm:lingroup-repack} in \cref{sec:lingroup}.
\begin{lemma}
\label{thm:lingroup-opt-compare}
$\fopt(\widehat{I}) < (1+\eps)\opt(I) + 2$.
\end{lemma}
\noindent \textbf{B. Creating Shelves.}
We will use ideas from Kenyon and R\'emila's 2SP algorithm~\cite{kenyon1996strip}
to pack $\widehat{I}$ into \emph{shelves}.
Roughly, we solve a linear program to compute an optimal strip packing of $\widehat{W}$,
where the packing is 3-stage. The first stage of cuts gives us shelves
and the second stage gives us containers.
From each shelf, we trim off space that doesn't belong to any container.
See \cref{sec:guill-thin-extra:shelves} for details.
Let $\widetilde{W}$ be the shelves thus obtained.
Analogously, we can pack items $\widehat{H}$ into shelves $\widetilde{H}$.
Shelves in $\widetilde{W}$ are called \emph{wide shelves}
and shelves in $\widetilde{H}$ are called \emph{tall shelves}.
Let $\widetilde{I} := \widetilde{W} \cup \widetilde{H}$.
We can interpret each shelf in $\widetilde{I}$ as a rectangular item.
We allow slicing $\widetilde{W}$ and $\widetilde{H}$ using horizontal cuts and vertical cuts, respectively.
In \cref{sec:guill-thin-extra:shelves}, we prove the following facts.%
\begin{restatable}{lemma}{rthmCreateShelves}
\label{thm:shelves}
$\widetilde{I}$ has the following properties:
(a) $|\widetilde{W}| \le 1+1/\eps^2$ and $|\widetilde{H}| \le 1+1/\eps^2$;
(b) Items in $\widetilde{W}$ have width more than $1/2$
and items in $\widetilde{H}$ have height more than $1/2$;
(c) $a(\widetilde{I}) = a(\widehat{I})$;
(d) $\max(\smallceil{\hsum(\widetilde{W})}, \smallceil{\wsum(\widetilde{H})}) \le \fopt(\widehat{I})$.
\end{restatable}
\noindent \textbf{C. Packing Shelves Into Bins.}
So far, we have packed $\widehat{I}$ into shelves $\widetilde{W}$ and $\widetilde{H}$.
We will now use $\greedyPack(\widetilde{W}, \widetilde{H})$ to pack the shelves into bins.
By \cref{thm:greedy-pack}, we get a 2-stage packing of $\widetilde{W} \cup \widetilde{H}$
into $m$ bins, where we make at most $m-1$ horizontal cuts in $\widetilde{W}$
and at most $m-1$ vertical cuts in $\widetilde{H}$.
The horizontal cuts (resp.~vertical cuts) increase the number of wide shelves
(resp.~tall shelves) from at most $1 + 1/\eps^2$
to at most $m + 1/\eps^2$.
By \cref{thm:greedy-pack-bins}, \cref{thm:shelves}(d)
and \cref{thm:shelves}(c), we get
$m \le \max\left(\smallceil{\hsum(\widetilde{W})}, \smallceil{\wsum(\widetilde{H})},
\frac{4}{3}a(\widetilde{I}) + \frac{8}{3}\right)
\le \frac{4}{3}\fopt(\widehat{I}) + \frac{8}{3}$.
\noindent \textbf{D. Packing Items Into Containers.}
So far, we have a packing of shelves into $m$ bins,
where the shelves contain slices of items $\widehat{I}$.
We will now repack a large subset of the items $\widehat{I}$ into the shelves
without slicing them. See \cref{fig:thin-gpack-output} for an example output.
We will do this using a standard greedy algorithm.
See \cref{sec:guill-thin-extra:pack-into-containers} for details of the
algorithm and proof of the following lemma.
\begin{figure}[htb]
\centering
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\end{tikzpicture}
\caption[A type-1 bin in the packing of $\widehat{I}$ computed by $\thinGPack$.]%
{A type-1 bin in the packing of $\widehat{I}$ computed by $\thinGPack$.
The packing contains 5 tall containers in 2 tall shelves
and 18 wide containers in 8 wide shelves.}%
\label{fig:thin-gpack-output}
\end{figure}
\begin{restatable}{lemma}{rthmDiscardAreaUb}
\label{thm:discard-area-ub}
Let $P$ be a packing of $\widetilde{I}$ into $m$ bins, where we sliced wide shelves by making
at most $m-1$ horizontal cuts and sliced tall shelves by making at most $m-1$ vertical cuts.
Then we can (non-fractionally) pack a large subset of items $\widehat{I}$
into the shelves in $P$ such that
the unpacked items (also called \emph{discarded items}) from $\widehat{W}$ have area less than
$\eps \hsum(\widetilde{W}) + \delta_H(1 + \eps)(m + 1/\eps^2)$,
and the unpacked items from $\widehat{H}$ have area less than
$\eps \wsum(\widetilde{H}) + \delta_W(1 + \eps)(m + 1/\eps^2)$.
\end{restatable}
We will pack the wide discarded items into new bins using NFDH
and pack the tall discarded items into new bins using NFDW.
Finally, we prove the performance guarantee of $\thinGPack_{\eps}(I)$.
\begin{lemma}
\label{thm:thin-gpack-strong}
Let $I$ be a set of $(\delta_W, \delta_H)$-skewed items.
Then $\thinGPack_{\eps}(I)$ outputs a 4-stage packing of $I$
in time $O((1/\eps)^{O(1/\eps)} + n\log n)$
and uses less than $\alpha(1+\eps)\opt(I) + 2\beta$ bins, where
$\Delta := \frac{1}{2}\left(\frac{\delta_H}{1-\delta_H}
+ \frac{\delta_W}{1-\delta_W}\right),\;
\alpha := \frac{4}{3}(1+4\Delta)(1+3\eps),\;
\beta := \frac{2\Delta(1+\eps)}{\eps^2} + \frac{10}{3}
+ \frac{19\Delta}{3} + \frac{16\Delta\eps}{3}$.
\end{lemma}
\begin{proof}
The discarded items are packed using NFDH or NFDW, which output a 2-stage packing.
Since $\greedyPack$ outputs a 2-stage packing of the shelves
and the packing of items into the shelves is a 2-stage packing,
the bin packing of non-discarded items is a 4-stage packing.
The time taken by $\thinGPack$ is at most $O((1/\eps)^{O(1/\eps)} + n\log n)$.
Suppose $\greedyPack$ uses at most $m$ bins. Then by \cref{thm:greedy-pack-bins},
$m \le 4\fopt(\widehat{I})/3 + 8/3$.
Let $W^d$ and $H^d$ be the items discarded from $W$ and $H$, respectively.
By \cref{thm:discard-area-ub} and \cref{thm:shelves}(d),
$a(W^d) < \eps\fopt(\widehat{I}) + \delta_H(1 + \eps)(m + 1/\eps^2)$
and $a(H^d) < \eps\fopt(\widehat{I}) + \delta_W(1 + \eps)(m + 1/\eps^2)$.
By \cref{thm:nfdh-wide}, the number of bins used by $\thinGPack_{\eps}(I)$ is less than
$m + \frac{2a(W^d)+1}{1-\delta_H} + \frac{2a(H^d)+1}{1-\delta_W}
\le (1 + 4\Delta(1+\eps))m + 4\eps(1+\Delta)\fopt(\widehat{I})
+ 2(1+\Delta) + 4\Delta(1+\eps)/\eps^2
\le \alpha\fopt(\widehat{I}) + 2(\beta - 1)
< \alpha(1+\eps)\opt(I) + 2\beta.$
The last inequality follows from \cref{thm:lingroup-opt-compare}.
\end{proof}
Now we conclude with the proof of \cref{thm:thin-gpack}.
\begin{proof}[Proof of \cref{thm:thin-gpack}]
This is a simple corollary of \cref{thm:thin-gpack-strong}, where
$\delta \le 1/2$ gives us $\Delta \le 2\delta$,
$\alpha(1+\eps) \le (4/3)(1+8\delta)(1+7\eps)$,
and $\beta \le 4/\eps^2 + 15$.
\end{proof}
\section{Almost-Optimal Bin Packing of Skewed{} Rectangles}
\label{sec:thin-bp}
In this section, we will describe the algorithm $\thinCPack$. $\thinCPack$ takes as input
a set $I$ of items and a parameter $\eps \in (0, 1/2]$, where $\eps^{-1} \in \mathbb{Z}$.
We will prove that $\thinCPack$ has AAR $1+20\eps$ when $\delta$ is sufficiently small.
$\thinCPack$ works roughly as follows:
\begin{enumerate}
\item Invoke the subroutine $\round(I)$ (described in \cref{sec:thin-bp:round}).
$\round(I)$ returns a pair $(\widetilde{I}, I_{\mathrm{med}})$.
Here $I_{\mathrm{med}}$, called the set of \emph{medium items}, has low total area,
so we can pack it in a small number of bins.
$\widetilde{I}$, called the set of \emph{rounded items}, is obtained by
rounding up the width or height of each item in $I - I_{\mathrm{med}}$,
so that $\widetilde{I}$ has special properties that help us pack it easily.
\item Compute the optimal \emph{fractional compartmental} bin packing of $\widetilde{I}$
(we will define \emph{compartmental} and \emph{fractional} later).
\item Use this packing of $\widetilde{I}$ to obtain a packing of $I$
that uses slightly more number of bins.
\end{enumerate}
To bound the AAR of $\thinCPack$, we will prove a structural theorem
in \cref{sec:thin-bp:struct}, i.e., we will prove that
the optimal fractional compartmental packing of $\widetilde{I}$ uses close to $\opt(I)$ bins.
\subsection{Classifying and Rounding Items}
\label{sec:thin-bp:round}
\label{sec:thin-bp:remmed}
We now describe the algorithm $\round$ and
show that its output satisfies important properties.
First, we will find a set $I_{\mathrm{med}} \subseteq I$
and positive constants $\eps_1$ and $\eps_2$
such that $a(I_{\mathrm{med}}) \le \eps a(I)$, $\eps_2 \ll \eps_1$,
and $I - I_{\mathrm{med}}$ is $(\eps_2, \eps_1]$-free, i.e.,
no item in $I - I_{\mathrm{med}}$ has its width or height in the interval $(\eps_2, \eps_1]$.
Then we can remove $I_{\mathrm{med}}$ from $I$ and pack it separately
into a small number of bins using NFDH. We will see that
the $(\eps_2, \eps_1]$-freeness of $I - I_{\mathrm{med}}$
will help us pack $I - I_{\mathrm{med}}$ efficiently.
Specifically, we require $\eps_1 \le \eps$, $\eps_1^{-1} \in \mathbb{Z}$,
and $\eps_2 = f(\eps_1)$, where $f(x) := \frac{\eps x}{104(1+1/(\eps x))^{2/x-2}}$.
We explain this choice of $f$ in \cref{sec:thinCPack}.
Intuitively, such an $f$ ensures that $\eps_2 \ll \eps_1$
and $\eps_2^{-1} \in \mathbb{Z}$.
For $\thinCPack$ to work, we require $\delta \le \eps_2$.
Finding such an $I_{\mathrm{med}}$ and $\eps_1$ is a standard technique \cite{JansenP2013, bansal2014binpacking},
so we defer the details to \cref{sec:thin-bp-extra:remmed}.
Next, we classify the items in $I - I_{\mathrm{med}}$ into three disjoint classes:
\begin{itemize}[noitemsep]
\item Wide items: $W := \{i \in I: w(i) > \eps_1 \textrm{ and } h(i) \le \eps_2 \}$.
\item Tall items: $H := \{i \in I: w(i) \le \eps_2 \textrm{ and } h(i) > \eps_1 \}$.
\item Small items: $S := \{i \in I: w(i) \le \eps_2 \textrm{ and } h(i) \le \eps_2 \}$.
\end{itemize}
We will now use \emph{linear grouping}~\cite{bp-aptas,kenyon1996strip}
to round up the widths of items $W$ and the heights of items $H$
to get items $\widetilde{W}$ and $\widetilde{H}$, respectively.
By \cref{thm:lingroup-n} in \cref{sec:lingroup},
items in $\widetilde{W}$ have at most $1/(\eps\eps_1)$ distinct widths
and items in $\widetilde{H}$ have at most $1/(\eps\eps_1)$ distinct heights.
Let $\widetilde{I} := \widetilde{W} \cup \widetilde{H} \cup S$.
\begin{definition}[Fractional packing]
Suppose we are allowed to slice wide items in $\widetilde{I}$ using horizontal cuts,
slice tall items in $\widetilde{I}$ using vertical cuts and slice
small items in $\widetilde{I}$ using both horizontal and vertical cuts.
For any $\widetilde{X} \subseteq \widetilde{I}$, a bin packing of the slices of $\widetilde{X}$
is called a \emph{fractional packing} of $\widetilde{X}$.
The optimal fractional packing of $\widetilde{X}$ is denoted by $\fopt(\widetilde{X})$.
\end{definition}
\begin{lemma}
\label{thm:thin-bp:lingroup-opt-compare}
$\fopt(\widetilde{I}) < (1+\eps)\opt(I) + 2$.
\end{lemma}
\begin{proof}
Directly follows from \cref{thm:lingroup-repack} in \cref{sec:lingroup}.
\end{proof}
\subsection{Structural Theorem}
\label{sec:thin-bp:struct}
We will now define compartmental packing
and prove the structural theorem, which says that
the number of bins in the optimal fractional compartmental packing of $\widetilde{I}$
is roughly equal to $\fopt(\widetilde{I})$.
We first show how to \emph{discretize} a packing, i.e.,
we show that given a fractional packing of items in a bin,
we can remove a small fraction of tall and small items
and shift the remaining items leftwards so that the left and right edges
of each wide item belong to a constant-sized set $\mathcal{T}$,
where $|\mathcal{T}| \le (1+1/\eps\eps_1)^{2/\eps_1 - 2}$.
Next, we define \emph{compartmental} packing and show how to convert
a discretized packing to a compartmental packing.
For any rectangle $i$ packed in a bin, let $x_1(i)$ and $x_2(i)$ denote the $x$-coordinates
of its left and right edges, respectively, and let $y_1(i)$ and $y_2(i)$
denote the $y$-coordinates of its bottom and top edges, respectively.
Let $R$ be the set of distinct widths of items in $\widetilde{W}$.
Given the way we rounded items, $|R| \le 1/\eps\eps_1$.
Recall that $\eps_1 \le \eps \le 1/2$ and $\eps_1^{-1}, \eps^{-1} \in \mathbb{Z}$.
\begin{theorem}
\label{thm:disc-hor-pos}
Given a fractional packing of items $\widetilde{J} \subseteq \widetilde{I}$ into a bin,
we can remove tall and small items of total area less than $\eps$
and shift some of the remaining items to the left such that for every wide item $i$,
we get $x_1(i), x_2(i) \in \mathcal{T}$.
\end{theorem}
\begin{proof}
For wide items $u$ and $v$ in the bin, we say that $u \prec v$ iff
the right edge of $u$ is to the left of the left edge of $v$.
Formally $u \prec v \iff x_2(u) \le x_1(v)$.
We call $u$ a \emph{predecessor} of $v$.
A sequence $[i_1, i_2, \ldots, i_k]$ such that $i_1 \prec i_2 \prec \ldots \prec i_k$
is called a \emph{chain} ending at $i_k$.
For a wide item $i$, define $\level(i)$ as the number of items in the longest chain
ending at $i$. Formally, $\level(i) := 1$ if $i$ has no predecessors,
and $\left(1 + \max_{j \prec i} \level(j)\right)$ otherwise.
Let $W_j$ be the items at level $j$, i.e., $W_j := \{i: \level(i) = j\}$.
Note that the level of an item can be at most $1/\eps_1-1$,
since each wide item has width more than $\eps_1$.
\begin{figure}[htb]
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\tikzset{cutline/.style={draw,dashed}}
\begin{tikzpicture}
\path[cutline] (2\pGameL, 0\pGameL) -- +(0, 20\pGameL);
\path[cutline] (5\pGameL, 0\pGameL) -- +(0, 20\pGameL);
\path[cutline] (7\pGameL, 0\pGameL) -- +(0, 20\pGameL);
\path[cutline] (9\pGameL, 0\pGameL) -- +(0, 20\pGameL);
\path[cutline] (14\pGameL, 0\pGameL) -- +(0, 20\pGameL);
\path[cutline] (16\pGameL, 0\pGameL) -- +(0, 20\pGameL);
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\path[item] (5\pGameL, 11\pGameL) rectangle +(9\pGameL, 2\pGameL) node[mynode] (wb) {$b$};
\path[item] (16\pGameL, 10\pGameL) rectangle +(4\pGameL, 2\pGameL) node[mynode] (wc) {$c$};
\path[item] (9\pGameL, 18\pGameL) rectangle +(5\pGameL, 2\pGameL) node[mynode] (wd) {$d$};
\path[item] (2\pGameL, 6\pGameL) rectangle +(7\pGameL, 2\pGameL) node[mynode] (we) {$e$};
\path[item] (9\pGameL, 2\pGameL) rectangle +(11\pGameL, 2\pGameL) node[mynode] (wf) {$f$};
\path[bin] (0\pGameL, 0\pGameL) rectangle (20\pGameL, 20\pGameL);
\path[myarrow] (wa) -- (wc);
\path[myarrow] (wa) -- (wd);
\path[myarrow] (wa) -- (wf);
\path[myarrow] (wb) -- (wc);
\path[myarrow] (wd) -- (wc);
\path[myarrow] (we) -- (wc);
\path[myarrow] (we) -- (wd);
\path[myarrow] (we) -- (wf);
\end{tikzpicture}
\caption[Relation $\prec$ among items in a bin]%
{Example illustrating the $\prec$ relationship between wide items in a bin.
An edge is drawn from $u$ to $v$ iff $u \prec v$.
Here $W_1 = \{a, e, b\}$, $W_2 = \{d, f\}$ and $W_3 = \{c\}$.}
\label{fig:precedence-graph}
\end{figure}
We will describe an algorithm for discretization.
But first, we need to introduce two recursively-defined set families
$(S_1, S_2, \ldots)$ and $(T_0, T_1, \ldots)$.
Let $T_0 := \{0\}$ and $t_0 := 1$. For any $j > 0$, define
$t_j := (1+1/\eps\eps_1)^{2j},\, \delta_j := \eps\eps_1/t_{j-1},\,
S_j := T_{j-1} \cup \{k\delta_j: k \in \mathbb{Z}, 0 \le k < 1/\delta_j\},\,
T_j := \{x + y: x \in S_j, y \in R \cup \{0\}\}$.
Note that $\forall j > 0$, we have $T_{j-1} \subseteq S_j \subseteq T_j$
and $\delta_j^{-1} \in \mathbb{Z}$.
Define $\mathcal{T} := T_{1/\eps_1-1}$.
Our discretization algorithm proceeds in stages, where in the $j^{\textrm{th}}$ stage,
we apply two transformations to the items in the bin,
called \emph{strip-removal} and \emph{compaction}.
\\ \textbf{Strip-removal}: For each $x \in T_{j-1}$,
consider a strip of width $\delta_j$ and height 1 in the bin
whose left edge has coordinate $x$.
Discard the slices of tall and small items inside the strips.
\\ \textbf{Compaction}: Move all tall and small items as much towards the left as possible
(imagine a gravitational force acting leftwards on the tall and small items)
while keeping the wide items fixed.
Then move each wide item $i \in W_j$ leftwards till $x_1(i) \in S_j$.
Observe that the algorithm maintains the following invariant:
\textsl{after $k$ stages, for each $j \in [k]$,
each item $i \in W_j$ has $x_1(i) \in S_j$ (and hence $x_2(i) \in T_j$).}
This ensures that after the algorithm ends, $x_1(i), x_2(i) \in \mathcal{T}$.
All that remains to prove is that the total area of items discarded
during strip-removal is at most $\eps$ and that compaction is always possible.
\begin{lemma}
For all $j \ge 0$, $|T_j| \le t_j$.
\end{lemma}
\begin{proof}
We will prove this by induction. The base case holds because $|T_0| = t_0 = 1$.
Now assume $|T_{j-1}| \le t_{j-1}$. Then
$|T_j| \le (|R|+1)|S_j|
\le \left(\frac{1}{\eps\eps_1}+1\right)\left(|T_{j-1}| + \frac{1}{\delta_j}\right)
\le \left(\frac{1}{\eps\eps_1}+1\right)^2 t_{j-1}
= t_j.$
\end{proof}
Therefore, $|\mathcal{T}| \le t_{1/\eps_1-1} = (1+1/\eps\eps_1)^{2/\eps_1 - 2}$.
\begin{lemma}
Items discarded by strip-removal (across all stages)
have total area less than $\eps$.
\end{lemma}
\begin{proof}
In the $j^{\textrm{th}}$ stage, we create $|T_{j-1}|$ strips,
and each strip has total area at most $\delta_j$.
Therefore, the area discarded in the $j^{\textrm{th}}$ stage is at most
$|T_{j-1}|\delta_j \le t_{j-1}\delta_j = \eps\eps_1$.
Since there can be at most $1/\eps_1-1$ stages,
we discard an area of less than $\eps$ across all stages.
\end{proof}
\begin{lemma}
Compaction always succeeds, i.e., in the $j^{\textrm{th}}$ stage, while moving
item $i \in W_j$ leftwards, no other item will block its movement.
\end{lemma}
\begin{proof}
Let $i \in W_j$. Let $z$ be the $x$-coordinate of the left edge of the strip
immediately to the left of item $i$, i.e.,
$z := \max(\{x \in T_{j-1}: x \le x_1(i)\})$.
For any wide item $i'$, we have $x_2(i') \le x_1(i) \iff i' \prec i \iff \level(i') \le j-1$.
By our invariant, we get
$\level(i') \le j-1 \implies x_2(i') \in T_{j-1} \implies x_2(i') \le z$.
Therefore, for every wide item $i'$, $x_2(i') \not\in (z, x_1(i)]$.
In the $j^{\textrm{th}}$ strip-removal, we cleared the strip $[z, z+\delta_j] \times [0, 1]$.
If $x_1(i) \in [z, z+\delta_j]$, then $i$ can freely move to $z$,
and $z \in T_{j-1} \subseteq S_j$.
Since no wide item has its right edge in $(z, x_1(i)]$, if $x_1(i) > z + \delta_j$,
all the tall and small items whose left edge lies in $[z+\delta_j, x_1(i)]$
will move leftwards by at least $\delta_j$ during compaction.
Hence, there would be an empty space of width at least $\delta_j$
to the left of item $i$
(see \cref{fig:compaction-zoom}).
Therefore, we can move $i$ leftwards to make
$x_1(i)$ a multiple of $\delta_j$, and then $x_1(i)$ would belong to $S_j$.
\end{proof}
\begin{figure}[!htb]
\centering
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\draw[mybrace] (0\myu, 0\myu) -- node[below=5pt] {$\delta_j$} (1\myu, 0\myu);
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\draw[myarrow] (9\myu, 3\myu) -- (13\myu, 3\myu)
node[pos=0.5,anchor=north,align=center,text width=3cm]
{shift tall and small items leftwards by $\delta_j$};
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(2\myu, 1\myu) rectangle (3\myu, 1.5\myu)
(3.8\myu, 2\myu) rectangle (4.8\myu, 2.5\myu)
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\draw[mybrace] (2\myu, 1\myu) -- node[below=5pt] {$\delta_j$} (3\myu, 1\myu);
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\end{tikzpicture}
\caption[Creation of empty space during compaction.]%
{This figure shows a region in the bin in the vicinity of item $i \in W_j$.
It illustrates how shifting tall and small items during compaction in the $j^{\textrm{th}}$ stage
creates a free space of width $\delta$ to the left of some wide items, including $i$.
Wide items are shaded dark and the lightly shaded region
potentially contains tall and small items.
Note that some tall and small items in the region $C$
may be unable to shift left because item $k$ is blocking them.
All other tall and small items in this figure to the right of $z$
can shift left by $\delta_j$.}
\label{fig:compaction-zoom}
\end{figure}
Hence, compaction always succeeds and we get $x_1(i), x_2(i) \in \mathcal{T}$
for each wide item $i$.
\end{proof}
\begin{definition}[Compartmental packing]
\label{defn:thin-bp:compartmental}
Consider a bin with some items packed into it. A \emph{compartment} $C$
is defined as a rectangular region in the bin satisfying the following properties:
\begin{itemize}[noitemsep]
\item $x_1(C), x_2(C) \in \mathcal{T}$.
\item $y_1(C), y_2(C)$ are multiples of $\eps_{\mathrm{cont}} := \eps\eps_1/6|\mathcal{T}|$.
\item $C$ does not contain both wide items and tall items.
\item If $C$ contains tall items, then $x_1(C)$ and $x_2(C)$
are consecutive values in $\mathcal{T}$.
\end{itemize}
If a compartment contains a wide item, it is called a \emph{wide compartment}.
Otherwise it is called a \emph{tall compartment}.
A packing of items into a bin is called \emph{compartmental}
iff there is a set of non-overlapping \emph{compartments} in the bin
such that each wide or tall item lies completely inside some compartment,
and there are at most $n_W := 3(1/\eps_1-1)|\mathcal{T}| + 1$ wide compartments
and at most $n_H := (1/\eps_1-1)|\mathcal{T}|$ tall compartments in the bin.
A packing of items into multiple bins is called compartmental iff
each bin is compartmental.
\end{definition}
Note that small items can be packed both inside and outside compartments.
The following two results are proved in \cref{sec:thin-bp-extra:compartmentalize}
using standard techniques.
\begin{restatable}{lemma}{rthmCompartmentalize}
\label{thm:thin-bp:compartmentalize}
Suppose $x_1(i), x_2(i) \in \mathcal{T}$ for each wide item $i$ in a bin.
Then by removing wide and small items of area less than $\eps$,
we can get a compartmental packing of the remaining items.
\end{restatable}
\begin{restatable}{theorem}{rthmStruct}
\label{thm:struct}
For a set $\widetilde{I}$ of $\delta$-skewed{} rounded items,
define $\fcopt(\widetilde{I})$ as the number of bins in
the optimal fractional compartmental packing%
\footnote{A \emph{fractional compartmental} packing of $\widetilde{I}$ is a fractional packing
of $\widetilde{I}$ that is also compartmental.}\!
of $\widetilde{I}$.
Then $\fcopt(\widetilde{I}) < (1+4\eps)\fopt(\widetilde{I}) + 2$.
\end{restatable}
\subsection{Packing Algorithm}
\label{sec:thin-bp:algo}
We now describe the $\thinCPack$ algorithm for packing a set $I$
of $\delta$-skewed{} items.
Roughly, $\thinCPack$ first computes $(\widetilde{I}, I_{\mathrm{med}}) := \round(I)$.
It then computes the optimal fractional compartmental packing of $\widetilde{I}$
by first guessing a packing of empty compartments into bins
and then fractionally packing the wide and tall items into the compartments
using a linear program.
It then converts the fractional packing of $\widetilde{I}$ to a non-fractional packing of $I$
with only a tiny increase in the number of bins.
See \cref{fig:thincpack} for a visual overview of $\thinCPack$.
We defer the details to \cref{sec:enum-configs,sec:feas-lp,sec:greedy-cont,sec:thinCPack}
and simply state the final result.
\begin{restatable}{theorem}{rthmThinCPack}
The number of bins used by $\thinCPack_{\eps}(\widetilde{I})$ is less than
\[ (1+20\eps)\opt(I) +
\frac{1}{13}\left(1 + \frac{1}{\eps\eps_1}\right)^{2/\eps_1 - 2} + 23. \]
\end{restatable}
\begin{figure}[htb]
\hbadness=10000
\begin{subfigure}[t]{0.3\textwidth}
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0.10/0.65,0.09/0.75,0.08/0.84,0.07/0.92}{
foreach \hdiff/\w/\x in {0.000/0.15/0.00,0.001/0.14/0.15,0.002/0.13/0.29,0.003/0.12/0.42,
0.004/0.11/0.54,0.005/0.10/0.65,0.006/0.09/0.75,0.007/0.08/0.84,0.008/0.07/0.92}{
(\x\myu, \y\myu) rectangle +(\w\myu, \h\myu-\hdiff\myu)
}};
}}
\begin{tikzpicture}
\pic at (2\myu, 0\myu) {pic4};
\pic[xscale=-3,rotate=90] at (2\myu, 2\myu) {pic1};
\pic[xscale=-3,rotate=90] at (1\myu, 4\myu) {pic2};
\pic[xscale=-2,rotate=90] at (1\myu, 1\myu) {pic3};
\pic[xscale=-2,rotate=90] at (2\myu, 3\myu) {pic1};
\pic[xscale=-2,rotate=90] at (0\myu, 0\myu) {pic2};
\pic[yscale=2] at (1\myu, 2\myu) {pic1};
\pic[yscale=2] at (3\myu, 0\myu) {pic2};
\pic[yscale=2] at (4\myu, 0\myu) {pic3};
\pic[yscale=2] at (4\myu, 3\myu) {pic1};
\pic[yscale=4] at (0\myu, 1\myu) {pic2};
\draw[ultra thick] (0\myu, 0\myu) rectangle (5\myu, 5\myu);
\end{tikzpicture}
\caption{Guess the packing of empty compartments in each bin (\cref{sec:enum-configs}).}
\end{subfigure}
\hfill
\begin{subfigure}[t]{0.3\textwidth}
\centering
\tikzset{compartment/.style={draw,thick},
container/.style={draw,fill={black!25}},
item/.style={}}
\ifcsname myu\endcsname\else\newlength{\myu}\fi
\setlength{\myu}{0.8cm}
\tikzset{pic1/.pic={
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(0.7\myu, 0.7\myu) rectangle +(0.3\myu, 0.3\myu);
\path[compartment] (0\myu, 0\myu) rectangle (1\myu, 1\myu);
}}
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\path[compartment] (0\myu, 0\myu) rectangle (1\myu, 1\myu);
}}
\tikzset{pic3/.pic={
\path[item]
(0.0\myu, 0\myu) rectangle +(0.1\myu, 0.4\myu)
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(0.2\myu, 0\myu) rectangle +(0.1\myu, 0.4\myu)
(0.3\myu, 0\myu) rectangle +(0.1\myu, 0.4\myu);
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(0.27\myu, 0.4\myu) rectangle +(0.09\myu, 0.3\myu);
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(0.88\myu, 0.5\myu) rectangle +(0.08\myu, 0.5\myu);
\path[container]
(0\myu, 0\myu) rectangle +(0.4\myu, 0.4\myu)
(0\myu, 0.4\myu) rectangle +(0.4\myu, 0.3\myu)
(0\myu, 0.7\myu) rectangle +(0.4\myu, 0.3\myu)
(0.4\myu, 0\myu) rectangle +(0.6\myu, 0.5\myu)
(0.4\myu, 0.5\myu) rectangle +(0.6\myu, 0.5\myu);
\path[compartment] (0\myu, 0\myu) rectangle (1\myu, 1\myu);
}}
\tikzset{pic4/.pic={
\path[item] foreach \h/\y in {0.15/0.00,0.14/0.15,0.13/0.29,0.12/0.42,0.11/0.54,
0.10/0.65,0.09/0.75,0.08/0.84,0.07/0.92}{
foreach \hdiff/\w/\x in {0.000/0.15/0.00,0.001/0.14/0.15,0.002/0.13/0.29,0.003/0.12/0.42,
0.004/0.11/0.54,0.005/0.10/0.65,0.006/0.09/0.75,0.007/0.08/0.84,0.008/0.07/0.92}{
(\x\myu, \y\myu) rectangle +(\w\myu, \h\myu-\hdiff\myu)
}};
}}
\begin{tikzpicture}
\pic at (2\myu, 0\myu) {pic4};
\pic[xscale=-3,rotate=90] at (2\myu, 2\myu) {pic1};
\pic[xscale=-3,rotate=90] at (1\myu, 4\myu) {pic2};
\pic[xscale=-2,rotate=90] at (1\myu, 1\myu) {pic3};
\pic[xscale=-2,rotate=90] at (2\myu, 3\myu) {pic1};
\pic[xscale=-2,rotate=90] at (0\myu, 0\myu) {pic2};
\pic[yscale=2] at (1\myu, 2\myu) {pic1};
\pic[yscale=2] at (3\myu, 0\myu) {pic2};
\pic[yscale=2] at (4\myu, 0\myu) {pic3};
\pic[yscale=2] at (4\myu, 3\myu) {pic1};
\pic[yscale=4] at (0\myu, 1\myu) {pic2};
\draw[ultra thick] (0\myu, 0\myu) rectangle (5\myu, 5\myu);
\end{tikzpicture}
\caption{Fractionally pack wide and tall items into compartments.
This partitions each compartment into \emph{containers} (\cref{sec:feas-lp}).}
\end{subfigure}
\hfill
\begin{subfigure}[t]{0.3\textwidth}
\centering
\tikzset{compartment/.style={draw,ultra thick},
container/.style={draw,thick},
item/.style={draw,very thin,fill={black!25}}}
\ifcsname myu\endcsname\else\newlength{\myu}\fi
\setlength{\myu}{0.8cm}
\tikzset{pic1/.pic={
\path[item]
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}}
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(0.88\myu, 0.5\myu) rectangle +(0.08\myu, 0.5\myu);
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(0\myu, 0.4\myu) rectangle +(0.4\myu, 0.3\myu)
(0\myu, 0.7\myu) rectangle +(0.4\myu, 0.3\myu)
(0.4\myu, 0\myu) rectangle +(0.6\myu, 0.5\myu)
(0.4\myu, 0.5\myu) rectangle +(0.6\myu, 0.5\myu);
\path[compartment] (0\myu, 0\myu) rectangle (1\myu, 1\myu);
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\tikzset{pic4/.pic={
\path[item] foreach \h/\y in {0.15/0.00,0.14/0.15,0.13/0.29,0.12/0.42,0.11/0.54,
0.10/0.65,0.09/0.75,0.08/0.84,0.07/0.92}{
foreach \hdiff/\w/\x in {0.000/0.15/0.00,0.001/0.14/0.15,0.002/0.13/0.29,0.003/0.12/0.42,
0.004/0.11/0.54,0.005/0.10/0.65,0.006/0.09/0.75,0.007/0.08/0.84,0.008/0.07/0.92}{
(\x\myu, \y\myu) rectangle +(\w\myu, \h\myu-\hdiff\myu)
}};
}}
\begin{tikzpicture}
\pic at (2\myu, 0\myu) {pic4};
\pic[xscale=-3,rotate=90] at (2\myu, 2\myu) {pic1};
\pic[xscale=-3,rotate=90] at (1\myu, 4\myu) {pic2};
\pic[xscale=-2,rotate=90] at (1\myu, 1\myu) {pic3};
\pic[xscale=-2,rotate=90] at (2\myu, 3\myu) {pic1};
\pic[xscale=-2,rotate=90] at (0\myu, 0\myu) {pic2};
\pic[yscale=2] at (1\myu, 2\myu) {pic1};
\pic[yscale=2] at (3\myu, 0\myu) {pic2};
\pic[yscale=2] at (4\myu, 0\myu) {pic3};
\pic[yscale=2] at (4\myu, 3\myu) {pic1};
\pic[yscale=4] at (0\myu, 1\myu) {pic2};
\draw[ultra thick] (0\myu, 0\myu) rectangle (5\myu, 5\myu);
\end{tikzpicture}
\caption{Pack the items non-fractionally (\cref{sec:greedy-cont}).}
\end{subfigure}
\caption{Major steps of $\thinCPack$ after $\round$ing $I$.}
\label{fig:thincpack}
\end{figure}
\input{skewedbp.bbl}
| {
"timestamp": "2021-05-07T02:25:05",
"yymm": "2105",
"arxiv_id": "2105.02827",
"language": "en",
"url": "https://arxiv.org/abs/2105.02827",
"abstract": "In the Two-dimensional Bin Packing (2BP) problem, we are given a set of rectangles of height and width at most one and our goal is to find an axis-aligned nonoverlapping packing of these rectangles into the minimum number of unit square bins. The problem admits no APTAS and the current best approximation ratio is $1.406$ by Bansal and Khan [SODA'14]. A well-studied variant of the problem is Guillotine Two-dimensional Bin Packing (G2BP), where all rectangles must be packed in such a way that every rectangle in the packing can be obtained by recursively applying a sequence of end-to-end axis-parallel cuts, also called guillotine cuts. Bansal, Lodi, and Sviridenko [FOCS'05] obtained an APTAS for this problem. Let $\\lambda$ be the smallest constant such that for every set $I$ of items, the number of bins in the optimal solution to G2BP for $I$ is upper bounded by $\\lambda\\operatorname{opt}(I) + c$, where $\\operatorname{opt}(I)$ is the number of bins in the optimal solution to 2BP for $I$ and $c$ is a constant. It is known that $4/3 \\le \\lambda \\le 1.692$. Bansal and Khan [SODA'14] conjectured that $\\lambda = 4/3$. The conjecture, if true, will imply a $(4/3+\\varepsilon)$-approximation algorithm for 2BP. According to convention, for a given constant $\\delta>0$, a rectangle is large if both its height and width are at least $\\delta$, and otherwise it is called skewed. We make progress towards the conjecture by showing $\\lambda = 4/3$ for skewed instance, i.e., when all input rectangles are skewed. Even for this case, the previous best upper bound on $\\lambda$ was roughly 1.692. We also give an APTAS for 2BP for skewed instance, though general 2BP does not admit an APTAS.",
"subjects": "Computational Geometry (cs.CG); Data Structures and Algorithms (cs.DS)",
"title": "Tight Approximation Algorithms for Geometric Bin Packing with Skewed Items",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9728307645257831,
"lm_q2_score": 0.7279754489059774,
"lm_q1q2_score": 0.7081969125152022
} |
https://arxiv.org/abs/2103.02338 | A survey of the noise-correcting tools for Dynamic Mode Decomposition | Dynamic Mode Decomposition (DMD) is a data-driven modeling tool that generates a model from spatio-temporal data. The data needs to be as clean as possible for DMD to come up with a faithful model. We review a few data-filtering methods to be integrated with DMD and test them on datasets of varying complexity. The impact of SNR on these methods and the error variation in the DMD model due to each method are observed and discussed. | \section{Introduction}
There is a lot of data relevant to what is going on around us, be it from the ocean-atmosphere interaction or the turbulent blood flow in the veins. Problems vary from being simple (linear) to complex (highly non-linear). But, one common aspect in the data is the noise. Noise can be of two types: white and colored. The difference between white noise and colored noise is that the power spectral density for the latter varies with frequency. Noise in the data is an issue when making a reduced order model based on that dataset. Reduced order models (ROM) are approximate models of a given full order model (FOM). ROMS appear to be very effective in terms of time and storage. A few notable reduced order models are Proper Orthogonal Decomposition \cite{chatterjee2000introduction}, Balanced Proper Orthogonal Decomposition \cite{willcox2002balanced}, Eigensystem Realization Algorithm \cite{juang1985eigensystem,murshed2020towards}, and Dynamic Mode Decomposition (DMD) \cite{jtu,schmid2010dynamic,9038561,murshed2020projection}.\\ \\
DMD is a data-driven method of creating a model solely from time-series data whose performance usually declines when dealing with corrupted data. There are two types of noise: process noise and sensor noise. The former relates to the system dynamics and latter to the measurements. There are several variants of DMD, namely Total least square DMD \cite{hemati2017}, Forward-Backward DMD \cite{dawson2016characterizing}, and Optimized DMD \cite{askham2018variable}, which are developed to tackle the effect of stochastic sensor disturbance on the dataset. \\ \\
The rest of the paper is organized as follows: Section \ref{BACK} is a brief discussion of the basics of the DMD and some of the remarkable data-filtering recipes. The results are shown in Section \ref{NR} and a summary provided in Section \ref{CFW}.
\section{Background}
\label{BACK}
This section comprises a review of the algorithm behind Dynamic Mode Decomposition and a precise discussion about the data-filtering methods that would be considered as a pre-processing step in DMD.
\subsection{Dynamic Mode Decomposition}
Given time-series data, Dynamic Mode Decomposition approximates the given data and predicts approximate states in future. The idea of extraction of dynamic information from snapshots is closely related to the Arnoldi algorithm \cite{arnoldi1951principle}. If we have available \(P\) number of snapshots, then, we consider those snapshots in a matrix
\begin{equation}
\textbf{X}_{1}^{P} = [\textbf{x}_{1}\ \textbf{x}_{2}\ ...\ \textbf{x}_{P}],
\end{equation}
where each snapshot contains \(Q\) pixels. It is assumed that each snapshot is separated by some uniform time interval, \(\Delta t\). DMD is thought to be the approximation of \emph{Koopman operator} (a linear, infinite-dimensional operator that represents nonlinear, infinite-dimensional dynamics). DMD algorithm considers a best fit linear operator \textbf{A} to approximate the dynamics,
\begin{equation}
\textbf{x}_{k+1} \approx \textbf{A}\textbf{x}_k,
\label{equation_3}
\end{equation}
where \(\textbf{x}_k\) is the snapshot at time \(t_k\) and \(\textbf{x}_{k+1}\) the snapshot one time step ahead in future. The data in the matrix can also be written, in terms of the operator, as
\begin{equation}
\textbf{X}_{2}^{P} \approx \textbf{A}\textbf{X}_{1}^{P-1}.
\end{equation}
The eigendecomposition of \textbf{A} facilitates analysis of the data despite the large size of the operator matrix.\\ \\
\textbf{Algorithm of DMD}\\ \\
(1) DMD takes the singular value decomposition (SVD) of \textbf{X} \cite{trefethen1997numerical}:
\begin{equation}
\textbf{X}_{1}^{P-1} = \textbf{U}\Sigma \textbf{V}^\ast
\Rightarrow \textbf{X}_{2}^{P} = \textbf{A} \textbf{U}\Sigma \textbf{V}^\ast,
\end{equation}
here matrix \(\textbf{U}\) is \(q \times r\), \(\Sigma\) is \(r \times r\) and \(\textbf{V}\) is \(m \times r\). \(\textbf{V}^\ast\) denotes the conjugate transpose of \textbf{V} and \(r\) is the rank for truncation after SVD. Note that the columns in \(\textbf{U}\) represent the POD modes. \\ \\
(2) \(\tilde{\textbf{A}}\) is computed as the \(K \times K\) projected version of matrix \(\textbf{A}\) in POD modes:
\begin{equation}
\textbf{A} = \textbf{X}_{2}^{P} \textbf{V}\Sigma^{-1} \textbf{U}^\ast
\Rightarrow \tilde{\textbf{A}} = \textbf{U}^\ast \textbf{A} \textbf{U} = \textbf{U}^\ast \textbf{X}_{2}^{P} \textbf{V}\Sigma^{-1}.
\end{equation}
(3) The eigendecomposition of \(\tilde{\textbf{A}}\) leads to,
\begin{equation}
\textbf{A}\textbf{W} = \textbf{W}\Lambda,
\end{equation}
where the columns of \(\textbf{W}\) represents the eigenvectors and \(\Lambda\) is a diagonal matrix that contains the corresponding eigenvalues.\\ \\
(4) DMD model reads,
\begin{equation}
\textbf{X}_{DMD} (t) = \bm{\Phi} \text{exp}(\bm{\Omega} t))\textbf{b},
\label{equation_9}
\end{equation}
where \(\bm{\Phi} = \textbf{X}_{2}^{P} \textbf{V}\Sigma^{-1} \textbf{W}\) and \(\textbf{b} = \bm{\phi}^{\dagger}\textbf{x}_1\). \(\bm{\Phi}\) is a matrix containing the eigenvectors, and \textbf{b} the vector where each entry is the initial amplitude of each mode. Note that the dagger symbol used in the definition of \textbf{b} denotes the Moore-Penrose pseudoinverse.
\subsection{Noise in data}
We have just seen how DMD begins working with the data matrix, \textbf{X}, to form a model for the dynamics of the data. In practice, the dataset would pick up noise from many sources. In case of numerical data, improper step size in time and space may introduce error in the simulation. For experimental data, uncertainties arise due to poor quality of the instrument and the way the technician is using it. We define a corrupted snapshot as,
$$ \textbf{x}_{c}(t) = \textbf{x}(t) + \bm{\eta},$$
where \(\bm{\eta}\) is a vector containing white Gaussian noise. The first subset of data can be written as,
$$ \textbf{X}_{c} =\textbf{X} + \textbf{C}_{X},$$
where \(\textbf{X}_{c}\) is the corrupted data, \(\textbf{X}\) the noise-free data, and \(\textbf{C}_{X}\) the noise. In the same manner, the second subset would appear as,
$$ \textbf{Y}_{c} =\textbf{Y} + \textbf{C}_{Y}.$$
Taking into account the error in the subsets, the linear mapping that takes place in DMD then becomes,
$$ \textbf{A} = (\textbf{Y}_{c} + \textbf{C}_{Y}) (\textbf{X}_{c} + \textbf{C}_{X})^{\dagger}, $$
$$ \textbf{A} = (\textbf{Y}_{c} + \textbf{C}_{Y}) (\textbf{X}_{c} + \textbf{C}_{X})^{*} \{(\textbf{X}_{c} + \textbf{C}_{X})(\textbf{X}_{c} + \textbf{C}_{X})^{*}\}^{\dagger}. $$
\subsection{Data-filtering methods}
Many algorithms have been proposed to remove noise from corrupted data. Possible ways are Principal Component Analysis (PCA) and Robust Principal Component Analysis (RPCA)\cite{wright2009}. RPCA handles noise of large magnitude much better than PCA. RPCA is based on the idea that the data, \textbf{X}, can be decomposed into two matrices: \(\textbf{X} = \textbf{L} \ + \ \textbf{S}\), where \textbf{L} is a low-rank matrix and \textbf{S} a sparse matrix. The idea of calculating \textbf{L} and \textbf{S}, hence, becomes a minimization problem. Previously, this has been done by Iterative Threshold\cite{wright2009}, Accelerated Proximal Gradient \cite{lin2009}, Dual Approach\cite{lin2009}, Singular Value Thresholding \cite{cai2010}, Alternating Direction Method \cite{yuan2009}, Exact Augmented Lagrange Multiplier \cite{lin2010}, and Inexact Augmented Lagrange Multiplier \cite{lin2010}. Variants of DMD can also be used to clean the data. Forward-Backward DMD and Total Least Squares DMD (tlsDMD) \cite{hemati2017} are a few notable ones that complete such task. In what follows, we will review the noise reduction by
\begin{itemize}
\item RPCA with ADM,
\item RPCA with inexact ALM,
\item Total least-squares DMD.
\end{itemize}
\textbf{RPCA with ADM}\\ \\
RPCA with ADM algorithm uses two helper functions: Shrink and Singular Value Threshold (SVT).\\ \\
\textbf{Shrink}: it takes two parameters as input: \textbf{X} and \(\tau\), where \textbf{X} can be any matrix and \(\tau\) is the threshold. It first checks each element in \textbf{X} and compares its absolute value with \(\tau\). If the absolute value of the element is greater \(\tau\), then the element is replaced with the difference between its absolute value and \(\tau\), otherwise, it is replaced with zero. The resulting matrix is then multiplied by the sign of the original matrix \textbf{X}. The final product is then returned as an output.\\ \\
\textbf{SVT}: this function takes as inputs \textbf{X} and \(\tau\) (a threshold for the consideration of the singular values). The matrix \textbf{X} is first decomposed into its component using SVD. We will then use shrink operator on the singular value matrix. This will cause many small values in the singular value matrix to become zero. The product of the \textbf{U}, \textbf{V}, and the modified \textbf{S} matrix is returned. The implication of this is that the \textbf{X} matrix will become more low-ranked.\\ \\
\noindent \textbf{RPCA with inexact ALM}\\ \\
The augmented Lagrange multiplier (ALM) is a class of algorithm for solving constrained optimization problems. ALM is used for problems where we have to minimize $f(X)$ such that $h(X)=0$. The ALM can be then defined as:
\begin{center}
$L(\textbf{X},\textbf{Y},\mu)=f(\textbf{X})+<\textbf{Y},h(\textbf{X})>+\frac{\mu}{2}{\lvert\lvert h(\textbf{X}) \rvert\rvert}^2_F. $
\end{center}
This algorithm is very fast in solving RPCA compared to other methods such as the Accelerated Proximal Gradient approach.
For the RPCA problem,
\begin{center}
$ \bm{X} = (\textbf{A,E})$,\\
$f(\textbf{X}) = \textbf{A}_* + \lambda{\lvert\lvert \textbf{E} \rvert\rvert}_1$,\\
$h(\textbf{X}) = \textbf{D}-\textbf{A}-\textbf{E}$
\end{center}
such that the lagrangian becomes,
\begin{center}
$L(\textbf{A,E,Y},\mu) = \textbf{A}_* + \lambda{\lvert\lvert \textbf{E} \rvert\rvert}_1+<\textbf{Y},\textbf{D}-\textbf{A}-\textbf{E}>+\frac{\mu}{2}{\lvert\lvert \textbf{D}-\textbf{A}-\textbf{E}} {\rvert\rvert}^2_F , $
\end{center}
where \(\lambda = \frac{\lambda_0}{\sqrt{max(n,m)}}\) with \(\lambda_0\) optimally being 1 and \(\mu\) is a hyperparameter. The inexact ALM can then be calculated as prescribed in \cite{lin2010}.\\ \\
\textbf{Total least-squares DMD}\\ \\
Data is collected and separated into \(\textbf{X}_1\) and \(\textbf{X}_2\). We define \textbf{Z} as,
\begin{equation}
\textbf{Z} = \textbf{X}_1^{'}\textbf{X}_{1} + \textbf{X}_{2}^{'}\textbf{X}_{2}.
\end{equation}
Then, SVD is performed on \textbf{Z},
\begin{equation}
\textbf{Z}=\textbf{U}\textbf{S}\textbf{V}^*,
\end{equation}
to identify the number of significant modes, \(r\).
$\textbf{V}$ is truncated to $\textbf{V}_n$,
$$ [\textbf{V}_{n}, \textbf{D}] = eig(\textbf{Z}) $$
$$ \textbf{d} = diag(\textbf{D}) $$
$$ [ \sim , idx] = sort(\textbf{d}, 'descend') $$
$$ \textbf{V}_{n} = \textbf{V}_{n}(:,idx(1:r)),$$
to get projected \(\textbf{X}_1\) and \(\textbf{X}_2\):
\begin{center}
$\tilde{\textbf{X}_1}=\textbf{X}_{1} \textbf{V}_{n} \textbf{V}_n^*,$\\
$\tilde{\textbf{X}_2}=\textbf{X}_{2} \textbf{V}_{n} \textbf{V}_{n}^*.$
\end{center}
Note that \textbf{eig} means the eigendecomposition of a matrix, \textbf{diag} picks out the elements of a matrix from the main diagonal, and \textbf{sort} arranges the entries in the given vector in either ascending or descending order.\\ \\
SVD on $\tilde{\textbf{X}_1}$ yields,
\begin{center}
$\tilde{\textbf{X}_{1}}=\tilde{\textbf{U}}\tilde{\textbf{S}}\tilde{\textbf{V}}.$
\end{center}
$\tilde{\textbf{A}}$ is then computed as,
\begin{center}
$\tilde{\textbf{A}} = \tilde{\textbf{U}}\tilde{\textbf{X}_2}\tilde{\textbf{V}}\tilde{\textbf{S}}^{-1}.$
\end{center}
\section{Numerical Results}
\label{NR}
The discussed filtering techniques will be implemented on different dataset. Gaussian white noise is added to the raw data to mimic the corrupted data. The experiments are run on MATLAB 2019b on a laptop with configuration of Intel Core i7-7700HQ CPU @ 2.80 GHz 2.81 GHz with 16 GB RAM.\\ \\
The performance of the model is to be evaluated as per two measures: Root Mean Squared Error (RMSE) and Correlation Co-efficient (CC). These two metrics are defined below, where, \(\textbf{X}_{DMD}\) is the prediction, \textbf{X} the ground truth, and \(n_s\) the number of samples or recordings.
\begin{itemize}
\item RMSE measures the standard deviation of the prediction error (residuals) as,
\begin{equation}\label{rmse}
RMSE = \sqrt{\frac{\sum {\lvert \textbf{X}_{DMD} - \textbf{X} \rvert}^2}{n_s}}
\end{equation}
The smaller the spread, the better the model.
\item CC indicates how closely two variables (prediction and ground truth) are related and is computed as,
\begin{equation}\label{cc}
CC = \sqrt{1 - \frac{\frac{\sum \lvert \textbf{X}_{DMD} - mean(\textbf{X}_{DMD}) \rvert}{n_s}}{\frac{\sum \lvert \textbf{X} - mean(\textbf{X}) \rvert}{n_s}}}.
\end{equation}
CC allows us to understand if the general structure of the signal/data is retained.
\end{itemize}
The error in the DMD approximation over time will be computed as,
$$ \epsilon = \frac{|\textbf{x}_{DMD} - \textbf{x}|_{2}}{|\textbf{x}|_{2}}, $$
where \textbf{x} is the approximation at a certain time instant.
\subsection{Non-linear Schrodinger Equation}
The data for this example, Figure \ref{nlse_odat}, comes from the solution to the non-linear Schrodinger's equation (NLSE),
$$ \frac{\partial p}{\partial t} = \frac{i}{2} \frac{\partial^{2} p}{\partial w^2} + i |p|^{2} p ,$$
where \(w\) and \(t\) are space and time, respectively. The domain are defined as:
$$ -15 < w < 15, $$
$$ 0 < t < 8 \pi. $$
The corrupted data (SNR = 20) is shown in Figure \ref{nlse_ndat}. The effect of RPCA via ADM, RPCA via inexact ALM and TLS-DMD on the noisy data is investigated and displayed in Figure \ref{nlse_adm}, Figure \ref{nlse_alm}, and Figure \ref{nlse_tls}, respectively. All of them tend to deal well with the noise in the data. The impact of the amount of the signal compared to that of noise is studied for each method based on RMSE and CC. It appears that RPCA via ADM performs much better that the other two methods for some given level of SNR, Figure \ref{RMSE_snr_nlse} and Figure \ref{CC_snr_nlse}. \\ \\
The approximation error in DMD is plotted in Figure \ref{nlse_error}. RPCA via inexact ALM sees a gradual increase in the error over time, whereas error in RPCA via ADM grows early on and then gets roughly constant. In case of the TLS-DMD, there is a sharp increase in the error in the very beginning after which a drop is witnessed.
\begin{figure}[H]
\centering
\begin{subfigure}{0.47\textwidth}
\begin{adjustbox}{max width=1\textwidth,center}
\includegraphics{spdat_nlse.png}
\end{adjustbox}
\caption{Original Data}
\label{nlse_odat}
\end{subfigure}
\hspace{0.04\textwidth}
\begin{subfigure}{0.47\textwidth}
\begin{adjustbox}{max width=1\textwidth,center}
\includegraphics{spdat_snr20_nlse.png}
\end{adjustbox}
\caption{Noisy Data}
\label{nlse_ndat}
\end{subfigure}
\begin{subfigure}{0.47\textwidth}
\begin{adjustbox}{max width=1\textwidth,center}
\includegraphics{adm_nlse.png}
\end{adjustbox}
\caption{ADM}
\label{nlse_adm}
\end{subfigure}
\hspace{0.04\textwidth}
\begin{subfigure}{0.47\textwidth}
\begin{adjustbox}{max width=1\textwidth,center}
\includegraphics{rpca_inex_nlse.png}
\end{adjustbox}
\caption{Inexact ALM}
\label{nlse_alm}
\end{subfigure}
\begin{subfigure}{0.47\textwidth}
\begin{adjustbox}{max width=1\textwidth,center}
\includegraphics{rpca_tls_nlse.png}
\end{adjustbox}
\caption{TLS}
\label{nlse_tls}
\end{subfigure}
\caption{NLSE}
\label{nlse adm surf}
\end{figure}
\begin{figure}[H]
\centering
\begin{subfigure}{0.47\textwidth}
\begin{adjustbox}{max width=1\textwidth,center}
\includegraphics{RMSE_nlse.png}
\end{adjustbox}
\caption{RMSE}
\label{RMSE_snr_nlse}
\end{subfigure}
\hspace{0.04\textwidth}
\begin{subfigure}{0.47\textwidth}
\begin{adjustbox}{max width=1\textwidth,center}
\includegraphics{CC_nlse.png}
\end{adjustbox}
\caption{CC}
\label{CC_snr_nlse}
\end{subfigure}
\begin{subfigure}{0.47\textwidth}
\begin{adjustbox}{max width=1\textwidth,center}
\includegraphics{error_evo_nlse.png}
\end{adjustbox}
\caption{Error variation with time}
\end{subfigure}
\begin{subfigure}{0.47\textwidth}
\begin{adjustbox}{max width=1\textwidth,center}
\includegraphics{rp_nlse.png}
\end{adjustbox}
\caption{Rank for each data-filtering method}
\label{rank_nlse}
\end{subfigure}
\caption{NLSE}
\label{nlse_error}
\end{figure}
\noindent \textbf{Fitzhugh-Nagumo Equation}\\ \\
The Fitzhugh-Nagumo equation (FNE) models the voltage spikes in neurons. FNE consists of the following system of partial differential equations,
\begin{equation}\label{FNE 1}
\frac{\delta V}{\delta t} = D\frac{\delta^2 V}{\delta x^2} + V(a-V)(V-1) - W
\end{equation}
\begin{equation}\label{FNE 2}
\frac{\delta W}{\delta t} = bV - cW
\end{equation}
where $V(x,t)$ measures the voltage in neuron. The initial conditions are as,
\begin{equation}\label{FNE 3}
V(x,0) = exp(-x^2),
\end{equation}
\begin{equation}\label{FNE 4}
W(x,0) = 0.2exp(-(x+2)^2).
\end{equation}
The boundary condition used on the partial differential equation system is,
\begin{equation} \label{FNE 5}
\frac{\delta V}{\delta x} = 0 \;\; and \;\; \frac{\delta W}{\delta x} = 0 \;\; at\; x = a,b.
\end{equation}
The raw data and the noisy data (SNR of 20) are shown in Figure \ref{FNE_odat} and Figure \ref{FNE_ndat}, respectively. The noise correction is somewhat done by RPCA via ADM (Figure \ref{FNE_adm}), but it is poorly done by RPCA via inexact ALM Figure \ref{FNE_alm}. Among the three techniques, TLS-DMD happens to be the best in denoising the data-set, Figure \ref{FNE_tls}.\\ \\
RMSE is found to go down as SNR is increased for each method. The effect of the SNR is more pronounced on TLS-DMD than on RPCA via ADM or RPCA via inexact ALM, Figure \ref{fne_rmse} and Figure \ref{fne_cc}. The error in the DMD approximation goes up in the beginning and then stabilizes for both TLS-DMD and RPCA via ADM, whereas it is a slowly increasing trend for RPCA via inexact ALM, Figure \ref{error_fne}.
\begin{figure}[H]
\centering
\begin{subfigure}{0.47\textwidth}
\begin{adjustbox}{max width=1\textwidth,center}
\includegraphics{spdat_fhn.png}
\end{adjustbox}
\caption{Original Data}
\label{FNE_odat}
\end{subfigure}
\hspace{0.04\textwidth}
\begin{subfigure}{0.47\textwidth}
\begin{adjustbox}{max width=1\textwidth,center}
\includegraphics{spdat_snr20_fhn.png}
\end{adjustbox}
\caption{Noisy Data}
\label{FNE_ndat}
\end{subfigure}
\begin{subfigure}{0.47\textwidth}
\begin{adjustbox}{max width=1\textwidth,center}
\includegraphics{adm_fhn.png}
\end{adjustbox}
\caption{ADM}
\label{FNE_adm}
\end{subfigure}
\hspace{0.04\textwidth}
\begin{subfigure}{0.47\textwidth}
\begin{adjustbox}{max width=1\textwidth,center}
\includegraphics{rpca_inex_fhn.png}
\end{adjustbox}
\caption{Inexact ALM}
\label{FNE_alm}
\end{subfigure}
\begin{subfigure}{0.47\textwidth}
\begin{adjustbox}{max width=1\textwidth,center}
\includegraphics{rpca_tls_fhn.png}
\end{adjustbox}
\caption{TLS}
\label{FNE_tls}
\end{subfigure}
\caption{FNE}
\label{nlse adm surf}
\end{figure}
\begin{figure}[H]
\centering
\begin{subfigure}{0.47\textwidth}
\begin{adjustbox}{max width=1\textwidth,center}
\includegraphics{RMSE_fhn.png}
\end{adjustbox}
\caption{RMSE}
\label{fne_rmse}
\end{subfigure}
\hspace{0.04\textwidth}
\begin{subfigure}{0.47\textwidth}
\begin{adjustbox}{max width=1\textwidth,center}
\includegraphics{CC_fhn.png}
\end{adjustbox}
\caption{CC}
\label{fne_cc}
\end{subfigure}
\begin{subfigure}{0.47\textwidth}
\begin{adjustbox}{max width=1\textwidth,center}
\includegraphics{error_evo_fhn.png}
\end{adjustbox}
\caption{Error variation with time}
\label{error_fne}
\end{subfigure}
\begin{subfigure}{0.47\textwidth}
\begin{adjustbox}{max width=1\textwidth,center}
\includegraphics{rp_fhn.png}
\end{adjustbox}
\caption{Rank for each data-filtering method}
\label{rank_fne}
\end{subfigure}
\caption{FNE}
\label{nlse adm surf}
\end{figure}
\subsection{Shallow water equations}
The motion of water in rivers and channels are often referred to as the shallow water equations (SWE) given by,
\begin{equation}
\frac{\partial(\rho \kappa)}{\partial t} + \frac{\partial(\rho \kappa u)}{\partial x} + \frac{\partial(\rho \kappa v)}{\partial y} = 0,
\end{equation}
\begin{equation}
\frac{\partial(\rho \kappa u)}{\partial t} + \frac{\partial}{\partial x} (\rho \kappa u^2 + \frac{1}{2} \rho g \kappa^2) + \frac{\partial (\rho \kappa u v)}{\partial y} = 0,
\end{equation}
\begin{equation}
\frac{\partial(\rho \kappa v)}{\partial t} + \frac{\partial (\rho \kappa u v)}{\partial x} + \frac{\partial}{\partial y} (\rho \kappa v^2 + \frac{1}{2} \rho g \kappa^2) = 0,
\end{equation}
to model the dynamics of fluid column height (\(\kappa\)) of a fluid of constant density \(\rho\). Here, \((x,y)\) defines the horizontal 2D space, and \(u\) and \(v\) denote the flow velocity in the 2D space. The solution to SWE via finite differencing produces the data for this example. The raw data and the noisy data are illustrated in Figure \ref{sw_odat} and Figure \ref{sw_ndat}, respectively for a certain time instant (the tenth temporal node in this case). Much like the previous example, the uncertainty in the DMD approximation is similar for both RPCA via ADM and TLS-DMD, whereas it moderately increases with time for RPCA via ALM, Figure \ref{sw_error}.
\begin{figure}[H]
\centering
\begin{subfigure}{0.47\textwidth}
\begin{adjustbox}{max width=1\textwidth,center}
\includegraphics{sw_dat.png}
\end{adjustbox}
\caption{Original Data}
\label{sw_odat}
\end{subfigure}
\hspace{0.04\textwidth}
\begin{subfigure}{0.47\textwidth}
\begin{adjustbox}{max width=1\textwidth,center}
\includegraphics{sw_noi_dat.png}
\end{adjustbox}
\caption{Noisy Data}
\label{sw_ndat}
\end{subfigure}
\begin{subfigure}{0.47\textwidth}
\begin{adjustbox}{max width=1\textwidth,center}
\includegraphics{error_evo_SW.png}
\end{adjustbox}
\caption{Error variation with time}
\label{sw_error}
\end{subfigure}
\begin{subfigure}{0.47\textwidth}
\begin{adjustbox}{max width=1\textwidth,center}
\includegraphics{rp_sw.png}
\end{adjustbox}
\caption{Rank for each data-filtering method}
\label{rank_sw}
\end{subfigure}
\hspace{0.04\textwidth}
\caption{SWE}
\end{figure}
\section{Conclusion and Future Work}
This work is concerned with three major noise-correction tools that can be integrated with DMD. These tools were previously used in other applications like detecting fake moustache or glasses in an image. In this work, we exploit them to remove noise from the dataset generated from PDEs and eventually process the data via DMD to come up with reduced order models. The rank of the filtered data from each method is plotted in Figure \ref{rank_nlse}, \ref{rank_fne}, \ref{rank_sw}. It is high for both RPCA via ADM and TLS-DMD, but remains very low for RPCA via inexact ALM in all three examples. The effect of SNR on each method is examined and no absolute pattern is found. But, there exists a trend in the way the error in the DMD model accumulates over time. The uncertainty in the DMD model increases very early in time and then stabilizes to a certain value with time for RPCA via ADM and RPCA via inexact ALM method, whereas the error for TLS-DMD sees a gradual increase in time. This observations are made based on the results from testing the methods on three PDEs: the Non-linear Schrodinger Equation, the Fitzhugh-Nagumo Equation, and the Shallow water equations. In future, we would like to design a denoising scheme that can outperform the ones that are used in this work.
\label{CFW}
\section{Acknowledgement}
This project is partially funded by the Office of Research, North South University, under grant \textbf{CTRG-20/SEPS/08}.
\bibliographystyle{ieeetr}
| {
"timestamp": "2021-03-04T02:23:39",
"yymm": "2103",
"arxiv_id": "2103.02338",
"language": "en",
"url": "https://arxiv.org/abs/2103.02338",
"abstract": "Dynamic Mode Decomposition (DMD) is a data-driven modeling tool that generates a model from spatio-temporal data. The data needs to be as clean as possible for DMD to come up with a faithful model. We review a few data-filtering methods to be integrated with DMD and test them on datasets of varying complexity. The impact of SNR on these methods and the error variation in the DMD model due to each method are observed and discussed.",
"subjects": "Optimization and Control (math.OC)",
"title": "A survey of the noise-correcting tools for Dynamic Mode Decomposition",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9728307700397331,
"lm_q2_score": 0.7279754430043072,
"lm_q1q2_score": 0.7081969107878959
} |
https://arxiv.org/abs/2112.06674 | Recalibration of Predictive Models as Approximate Probabilistic Updates | The output of predictive models is routinely recalibrated by reconciling low-level predictions with known derived quantities defined at higher levels of aggregation. For example, models predicting turnout probabilities at the individual level in U.S. elections can be adjusted so that their aggregation matches the observed vote totals in each state, thus producing better calibrated predictions. In this research note, we provide theoretical grounding for one of the most commonly used recalibration strategies, known colloquially as the "logit shift." Typically cast as a heuristic optimization problem (whereby an adjustment is found such that it minimizes the difference between aggregated predictions and the target totals), we show that the logit shift in fact offers a fast and accurate approximation to a principled, but often computationally impractical adjustment strategy: computing the posterior prediction probabilities, conditional on the target totals. After deriving analytical bounds on the quality of the approximation, we illustrate the accuracy of the approach using Monte Carlo simulations. The simulations also confirm analytical results regarding scenarios in which users of the simple logit shift can expect it to perform best -- namely, when the aggregated targets are comprised of many individual predictions, and when the distribution of true probabilities is symmetric and tight around 0.5. | \section{Problem Description}
A common problem in predictive modeling is that of calibrating probabilities to observed totals. For example, an analyst may obtain individual-level scores $p_i \in (0, 1), i = 1, \dots, N,$ to estimate the probability that each of the $N$ registered voters in a particular voting precinct will support the Democratic candidate in an upcoming election.
After the election occurs, the analyst can observe the total number of Democratic votes, $D$, cast among the subset $\mathcal{V} \subset \{1, \dots, N\}$ of registered voters who cast a ballot. But she cannot observe individual-level outcomes due to the secret ballot. In the absence of perfect prediction, the analyst will find that $\sum_{i \in \mathcal{V}} p_i \neq D$. She must then decide how to compute recalibrated scores, $\tilde p_i$, to better reflect the realized electoral outcome.
This practical problem has direct implications for public opinion research. For example, \cite{ghitza2020voter} recalibrate their MRP estimates of voter support levels after an election to match county-level totals, while \cite{schwenzfeierPolMeth} proposes using the magnitude of the calibration to estimate non-response bias in public opinion polls. The problem is also of great importance in campaign work. Campaigns frequently seek to target voters who are most likely to have supported their party in the prior presidential election. Estimates of prior party support may also serve as predictor variables in models estimating support in successive elections. Recalibrating the scores to match observed outcomes is thus a crucial step to improve the scores' accuracy and bolster future electioneering.
A common heuristic solution to the recalibration problem is the use of the ``uniform swing'' \citep{butler1951appendix} on the logit scale. This approach is simple: first, one defines the function
\[ h(\alpha) = \sum_{i \in \mathcal{V}} \frac{1}{1 + \frac{1-p_i}{p_i} \alpha} \,,\]
and, having observed a total $D$, one finds the $\alpha$ that satisfies the equation
\begin{equation}\label{eq:basicSwing}
h(\alpha) = D\,.
\end{equation}
The function $h(\cdot)$ is monotonic in $\alpha$, so Equation \ref{eq:basicSwing} can be solved in logarithmic time using binary search. The updated scores are then computed as
\[ \tilde p_i = \frac{1}{1 + \frac{1-p_i}{p_i} \alpha} \,. \]
Solving Eq.~\ref{eq:basicSwing} is equivalent to finding the set of probabilities $\tilde p_i$ which sum to $D$ and minimize the Kullback–Leibler divergence \citep{kullback1951information} with the distribution induced by the original scores $p_i$.
Moreover, if the $p_i$ are defined based on a logistic regression, then this update is equivalent to shifting the intercept in the model by $\log(\alpha)$. For more details on these characterizations, see the Appendix, Section \ref{sec:char}.
Examples of this simple recalibration strategy are given by \cite{ghitza2013deep}, \cite{hanrettyetal2016}, and \cite{ghitza2020voter}. This procedure is also familiarly referred to by campaign workers as the ``logit shift".\footnote{The term ``logit swing" is also commonly used.}
In this research note, we provide analytical justification for the logit shift. First, we introduce an alternative procedure for score updating, which simply computes the updated scores as posterior probabilities, conditional on the target totals. In this procedure, we assume the original scores $p_i$ capture a kind of prior Democratic support probability, while the updated scores $\tilde p_i$ reflect the conditional Democratic voting probability given observed outcomes. Next, we show that this second, more principled approach is well approximated by the logit shift in large samples. We demonstrate this result analytically and illustrate it in a small simulation study. Finally, we discuss potential extensions to cases where a uniform swing is insufficient to capture observed electoral dynamics.
\section{Recalibration as a posterior update}
To motivate the posterior update approach, we introduce some additional notation. We define each voter's choice as a binary variable $W_i \in \{0, 1\}$, where $W_i = 1$ signifies a Democratic vote and $W_i = 0$ signifies a Republican vote (we suppose a two-candidate election for simplicity). The $W_i$ are modeled as independent Bernoulli random variables, where $W_i \sim \text{Bern}(p_i)$. In other words, the $p_i = P(W_i = 1)$ can be thought of as the prior, unconditional probability of casting a Democratic vote.
In this model, it is straightforward to approach score recalibration by defining a new set of updated scores, $\{p_i^{\star}\}$, using the following conditional probability (which automatically sum to $D$ over voters $i$):
\begin{align}
\begin{split}
p_i^{\star} &= \mathbb{P} \left( W_i = 1 \,\middle\vert\, \sum_{j \in \mathcal{V}} W_j = D \right)\\
&=\frac{\mathbb{P} \left(W_i = 1, \sum_{j \in \mathcal{V}} W_j = D\right)}{\mathbb{P}\left(\sum_{j \in \mathcal{V}} W_j = D\right)}\\
&= p_i \times \xi_i
\end{split}
\label{eq:posterior}
\end{align}
where $\xi_i=\frac{\mathbb{P}\left( \sum_{j \neq i} W_j = D - 1\right)}{\mathbb{P}\left(\sum_j W_{j \in \mathcal{V}} = D\right)}$
is a ratio of two Poisson-Binomial probabilities --- that is, probabilities over the sum of independent \emph{but not identically distributed} Bernoulli random variables \citep{chen1997statistical}. Explicit computation of the $p_i^{\star}$ is quite challenging, as efficient computation of Poisson-Binomial probabilities is extremely computationally demanding at even moderate sample sizes, despite substantial recent advances in the literature \citep{olivella2017poisbinom, junge2020package}. To compute the $p_i^{\star}$, we would need to compute one unique Poisson-Binomial probability per unit in the population. Hence, if the number of actual voters $|\mathcal{V}|$ were even modestly large, it would be computationally infeasible to obtain these exact posterior probabilities.
\section{Logit shift approximates the correct posterior}
\subsection{Preliminaries}
In this section, we show analytically why the logit shift is a good approximation to the general posterior update in Eq.~\ref{eq:posterior}.
To do so, we begin by defining two terms, the ratio $\phi_i = \frac{\mathbb{P}\left(\sum_{j \neq i} W_j = D\right)}{\mathbb{P}\left(\sum_{j \neq i} W_j = D-1\right)}$, and the function $f(x, s) = \frac{1}{1 + \frac{1-x}{x}(s)} \,. $
Simple substitution, along with a useful recursive property of the Poisson-Binomial distribution,\footnote{\label{fn:recursion}Namely,
\begin{equation*}
\mathbb{P}\left(\sum_j W_j = D \right) = p_i\times\mathbb{P}\left(\sum_{j \neq i} W_j = D-1\right) + (1-p_i)\times\mathbb{P}\left(\sum_{j \neq i} W_j = D\right) \,.
\end{equation*} } makes it clear that
\begin{align}
\begin{split}
\sum_if(p_i, \phi_i) &= \sum_i\frac{1}{1 + \frac{1-p_i}{p_i} \phi_i}\\
& = \sum_i\frac{1}{1 + \frac{1-p_i}{p_i} \frac{\mathbb{P}\left(\sum_{j \neq i} W_j = D\right)}{\mathbb{P}\left(\sum_{j \neq i} W_j = D-1\right)} } \\
&= \sum_i\frac{p_i\times\mathbb{P}\left(\sum_{j \neq i} W_j = D-1\right) }{ p_i\times\mathbb{P}\left(\sum_{j \neq i} W_j = D-1\right) + (1-p_i)\times\mathbb{P}\left(\sum_{j \neq i} W_j = D\right) } \\
&= \sum_i\frac{\mathbb{P} \left(W_i = 1, \sum_i W_i = D\right)}{\mathbb{P}\left(\sum_i W_i = D\right)} \\
& =\sum_i p_i^{\star}\\
&=D
\end{split}
\label{eq:indswing}
\end{align}
In words, Eq.~\ref{eq:indswing} shows that $\phi_i$ is precisely the ``shift" that turns each $p_i$ into the desired, recalibrated posterior probability $p_i^{\star}$. The logit shift, however, uses a constant $\alpha$ to approximate the vector of recalibrating shifts $\{\phi_i\}_{i \in \mathcal{V}}$.
What remains, therefore, is to show that the value of $\alpha$ that solves Eq.~\ref{eq:basicSwing} is a very good approximation of $\phi_i$ for all values of $i$.
To do so, we establish a couple of facts: that the value of $\alpha$ is bounded by the range of $\{\phi_i\}_{i \in \mathcal{V}}$, and that each $\phi_i$ in turn has well-defined bounds:
\begin{theorem}\label{thm:boundsOnAlpha}
The value of $\alpha$ which solves Equation \ref{eq:basicSwing} satisfies:
\[ \min_i \frac{\mathbb{P}\left(\sum_{j \neq i} W_j = D\right)}{\mathbb{P}\left(\sum_{j \neq i} W_j = D-1\right)} \leq \alpha \leq \max_i \frac{\mathbb{P}\left(\sum_{j \neq i} W_j = D\right)}{\mathbb{P}\left(\sum_{j \neq i} W_j = D-1\right)}\,. \]
\end{theorem}
\begin{proof}
The proof can be found in the Appendix, Section \ref{sec:prfThm1}.
\end{proof}
\begin{theorem}\label{thm:boundsOnPhi}
For any choice of $i \in \mathcal{V}$, we have
\[ \frac{\mathbb{P}\left(\sum_{j \in \mathcal{V}} W_j = D + 1\right)}{\mathbb{P}\left(\sum_{j \in \mathcal{V}} W_j = D\right)}\leq \frac{\mathbb{P}\left(\sum_{j \neq i} W_j = D\right)}{\mathbb{P}\left(\sum_{j \neq i} W_j = D-1\right)}\leq \frac{\mathbb{P}\left(\sum_{j \in \mathcal{V}} W_j = D\right)}{\mathbb{P}\left(\sum_{j \in \mathcal{V}} W_j = D-1\right)} \]
\end{theorem}
\begin{proof}
The proof can be found in the Appendix, Section \ref{sec:prfThm2}.
\end{proof}
\subsection{Main Results}
The bounds from Theorem \ref{thm:boundsOnPhi} apply regardless of the choice of $i$, so we can combine the two theorems to find that
\begin{equation}\label{eq:double-bounds}
\begin{aligned}
\frac{\mathbb{P}\left(\sum_{j \in \mathcal{V}} W_j = D + 1\right)}{\mathbb{P}\left(\sum_{j \in \mathcal{V}} W_j = D\right)} &\leq \min_i \frac{\mathbb{P}\left(\sum_{j \neq i} W_j = D\right)}{\mathbb{P}\left(\sum_{j \neq i} W_j = D-1\right)} \leq \alpha \\ & \leq \max_i \frac{\mathbb{P}\left(\sum_{j \neq i} W_j = D\right)}{\mathbb{P}\left(\sum_{j \neq i} W_j = D-1\right)} \leq \frac{\mathbb{P}\left(\sum_{j \in \mathcal{V}} W_j = D\right)}{\mathbb{P}\left(\sum_{j \in \mathcal{V}} W_j = D-1\right)}.
\end{aligned}
\end{equation}
This is useful, because we can now use the outer bounds in Eq.~\ref{eq:double-bounds} to obtain a bound on the approximation error when estimating recalibrated scores $p_i^{\star}$ (obtained from the posterior update approach) via $\tilde p_i$ (obtained from the logit shift):
\begin{theorem}\label{thm:boundsOnError}
For large sample sizes, we obtain
\[ \tilde p_i = p_i^{\star} + \mathcal{O} \left( \frac{1}{\sum_{j \in \mathcal{V}} p_j (1-p_j)} \right)\,. \]
\end{theorem}
\begin{proof}
The proof can be found in the Appendix, Section \ref{sec:prfThm3}.
\end{proof}
Theorem~\ref{thm:boundsOnError} states that the error in using the logit shift approach to approximate the posterior recalibration update depends on the precision of the Poisson-Binomial distribution over sums of binary outcomes being aggregated (votes for the Democratic candidate, in our running example). As the variance of Poisson-Binomial deviates is maximal when all underlying probabilities are equal to 0.5, it follows that, in our running example, the approximation will perform best when voters in $\mathcal{V}$ are
equally likely to vote for either party. As voters become more heterogenous, or as their support becomes more lopsided (or both, as would be the case in heavily polarized electorates), the quality of the approximation is expected to suffer. Fortunately, the binding bounds in Eq.~\ref{eq:double-bounds} are extremely tight for large enough samples, so that even in the worst case-scenarios, the approximation can be expected to perform well. We now briefly illustrate our analytical results with a small Monte-Carlo simulation.
\begin{comment}
\begin{theorem}
For large $N$, we have that the lower and upper bounds from Theorem \ref{thm:boundsOnPhi} differ by a multiplicative factor no larger than approximately $\left(\sum_j p_j (1- p_j) \right)^{-1}$.
\end{theorem}
\begin{proof}
It is a well-established result that, for large $N$, the Poisson-Binomial behaves approximately as a Normal random variable with the same mean and variance \citep[see e.g.][]{rosenman2018using}, namely
\[ \mu = \sum_j p_j \hspace{5mm} \text{ and } \hspace{5mm} \sigma^2 = \sum_j p_j(1-p_j). \]
Denote as $\psi(d)$ the density of a Normal distribution $\mathcal{N}(\mu, \sigma^2)$ with this mean and variance. We can simply compute
\begin{align*}
\frac{\mathbb{P}\left(\sum_j W_j = D\right)}{\mathbb{P}\left(\sum_j W_j = D-1\right)}\bigg/ \frac{\mathbb{P}\left(\sum_j W_j = D + 1\right)}{\mathbb{P}\left(\sum_j W_j = D\right)} \approx \frac{\psi(D^2)}{\psi(D-1)\psi(D+1)} = \exp\left(\frac{1}{\sigma^2} \right) = \exp\left(\frac{1}{\sum_j p_j (1-p_j)} \right)\,.
\end{align*}
For large $N$, $\sum_j p_j(1-p_j)$ will be large, and hence we can use a Taylor approximation:
\[ \frac{\mathbb{P}\left(\sum_j W_j = D\right)}{\mathbb{P}\left(\sum_j W_j = D-1\right)}\bigg/ \frac{\mathbb{P}\left(\sum_j W_j = D + 1\right)}{\mathbb{P}\left(\sum_j W_j = D\right)} \approx 1 + \frac{1}{\sum_j p_j(1-p_j)}\,. \]
\end{proof}
We have now established that the $\phi_i$ vary minimally across choices of $i$ for large $N$; it follows that $\alpha$ is a very good approximation for all of the $\phi_i$ and hence that our proposed procedure will approximate the true $p_i^{\star}$ very well.
\end{comment}
\section{Simulations}
We conduct a brief simulation study to empirically demonstrate the efficacy of this approach. We simulate using $1,000$ units, a sample size at which computations of the true $p_i^{\star}$ are possible. We draw the initial probabilities $p_i$ according to the six distributions discussed in \cite{biscarri2018simple}. We then consider the cases in which the observed $D$ is either 20\% above or 20\% below the expectation, $\sum_i p_i$.
We compute the true probabilities using Biscarri's algorithm as implemented in the \textit{PoissonBinomial} package \citep{junge2020package}, and compare it against the estimates obtained using our heuristic method. We report the RMSE as well as the proportion of variance in the true $p_i^{\star}$ that is \emph{not} explained by our method. Results are given in Table \ref{tab:simulations}. Across all settings, our approximations perform extremely well.
\begin{table}[bt]
\label{tab:simulations}
\centering
\begin{tabular}{llrccc}
\toprule
\textbf{$\boldsymbol{p_i}$ Setting} & \textbf{Sampling Distribution} & \textbf{Observed D} & \textbf{RMSE} & \textbf{1 $-$ R$\boldsymbol{^2}$ } \\ \midrule
Uniform & Uniform(0, 1) & $-$20\% & 0.00022 & 5.58$\times 10^{-7}$ \\
Uniform & Uniform(0, 1) & 20\% & 0.00021 & 5.46$\times 10^{-7}$ \\
Close to Zero & Beta(0.1, 3) & $-$20\% & 0.00047 & 3.41$\times 10^{-5}$ \\
Close to Zero & Beta(0.1, 3) & 20\% & 0.00039 & 1.85$\times 10^{-5}$ \\
Close to One & Beta(3, 0.1) & $-$20\% & 0.00036 & 1.22$\times 10^{-6}$ \\
Close to One & Beta(3, 0.1) & 20\% & -- & -- \\
Extremal & 0.5*Beta(0.1, 3) + 0.5*Beta(3, 0.1) & $-$20\% & 0.00048 & 1.14$\times 10^{-6}$ \\
Extremal & 0.5*Beta(0.1, 3) + 0.5*Beta(3, 0.1) & 20\% & 0.00048 & 1.12$\times 10^{-6}$ \\
Central & Beta(3, 3) & $-$20\% & 0.00015 & 6.86$\times 10^{-7}$ \\
Central & Beta(3, 3) & 20\% & 0.00015 & 6.86$\times 10^{-7}$ \\
Bimodal & 0.5*Beta(3, 10) + 0.5*Beta(10, 3) & $-$20\% & 0.00023 & 6.92$\times 10^{-7}$ \\
Bimodal & 0.5*Beta(3, 10) + 0.5*Beta(10, 3) & 20\% & 0.00023 & 6.86$\times 10^{-7}$ \\ \bottomrule
\end{tabular}
\caption{Approximation error, as measured by RMSE and $1 - R^2$, using our heuristic method vs. the true Poisson-Binomial probabilities, under various settings. No results are reported in the row in which $1.2 \times \sum_i p_i$ would exceed the sample size of $1,000$.}
\end{table}
\section{Discussion}
In this paper, we have considered the problem of updating voter scores to match observed vote totals from an election. We have shown that the relatively simple ``logit shift" algorithm is a very good approximation to computing the true conditional probability. This is an especially useful insight for campaign analysts and researchers alike, because the logit shift is significantly more efficient computationally than the calculation of the exact posterior recalibration update.
It is worth being explicit about the limitations of this approach. Under the posterior update model, we treat the original scores $p_i$ as a prior over Democratic vote probability. In turn, the updated scores $p_i^{\star}$ deviate from the initial scores only by assuming the observed vote tallies deviate from the expectation of $\sum_i p_i$ due to random error. Crucially, the updated probabilities retain the same ordering as the prior probabilities, which implies the original scoring model must discriminate positive and negative (but unobservable, in the case of voting) individual cases well. It is also important to note that the realization of $\mathcal{V}$ over which values of $D$ are defined can have an impact on the quality of the approximation: the approximation will be better when the number of Democratic votes $D$ tallies the choices of voters in very competitive districts than when it tallies votes in landslide ones, and choosing a level of aggregation with too few voters in it could render the error bounds too loose. In most practical instances, however, the logit shift can be expected to perform very well.
Hence, this approach represents a useful -- albeit crude -- method of updating individual-level scores to incorporate information from a completed election. More complex insights about the electorate, such as the marked underperformance of Democrats among Hispanics voters in the 2020 election, cannot be directly incorporated by computing the posterior probabilities (or their approximation via the logit shift). Methods based on ecological inference \citep[e.g.][]{king2004ecological} would be necessary to capture this structure. Such methods represent a promising potential extension of the insights provided in this manuscript.
\bibliographystyle{apalike}
| {
"timestamp": "2021-12-14T02:37:06",
"yymm": "2112",
"arxiv_id": "2112.06674",
"language": "en",
"url": "https://arxiv.org/abs/2112.06674",
"abstract": "The output of predictive models is routinely recalibrated by reconciling low-level predictions with known derived quantities defined at higher levels of aggregation. For example, models predicting turnout probabilities at the individual level in U.S. elections can be adjusted so that their aggregation matches the observed vote totals in each state, thus producing better calibrated predictions. In this research note, we provide theoretical grounding for one of the most commonly used recalibration strategies, known colloquially as the \"logit shift.\" Typically cast as a heuristic optimization problem (whereby an adjustment is found such that it minimizes the difference between aggregated predictions and the target totals), we show that the logit shift in fact offers a fast and accurate approximation to a principled, but often computationally impractical adjustment strategy: computing the posterior prediction probabilities, conditional on the target totals. After deriving analytical bounds on the quality of the approximation, we illustrate the accuracy of the approach using Monte Carlo simulations. The simulations also confirm analytical results regarding scenarios in which users of the simple logit shift can expect it to perform best -- namely, when the aggregated targets are comprised of many individual predictions, and when the distribution of true probabilities is symmetric and tight around 0.5.",
"subjects": "Methodology (stat.ME); Statistics Theory (math.ST); Applications (stat.AP)",
"title": "Recalibration of Predictive Models as Approximate Probabilistic Updates",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9728307700397332,
"lm_q2_score": 0.727975443004307,
"lm_q1q2_score": 0.7081969107878959
} |
https://arxiv.org/abs/1512.02137 | On orienting edges of unstructured two- and three-dimensional meshes | Finite element codes typically use data structures that represent unstructured meshes as collections of cells, faces, and edges, each of which require associated coordinate systems. One then needs to store how the coordinate system of each edge relates to that of neighboring cells. On the other hand, we can simplify data structures and algorithms if we can a priori orient coordinate systems in such a way that the coordinate systems on the edges follows uniquely from those on the cells \textit{by rule}.Such rules require that \textit{every} unstructured mesh allows assigning directions to edges that satisfy the convention in adjacent cells. We show that the convention chosen for unstructured quadrilateral meshes in the \texttt{deal.II} library always allows to orient meshes. It can therefore be used to make codes simpler, faster, and less bug prone. We present an algorithm that orients meshes in $O(N)$ operations. We then show that consistent orientations are not always possible for 3d hexahedral meshes. Thus, cells generally need to store the direction of adjacent edges, but our approach also allows the characterization of cases where this is not necessary. The 3d extension of our algorithm either orients edges consistently, or aborts, both within $O(N)$ steps. | \section{Introduction}
\label{sec:introduction}
In most of the common numerical methods for the solution of partial
differential equations on bounded domains $\Omega\subset {\mathbb R}^d, d=2,3$,
one defines approximate solutions by first subdividing $\Omega$ into a
finite number of cells, and then setting up a system of equations on
these cells. Usually, cells are either triangular or quadrilateral
(for $d=2$), or tetrahedral, prismatic, pyramidal, or hexahedral (for
$d=3$). Because certain aspects of the solution may be associated with
cells or edges (in 2d), or cells, faces and edges (in 3d),
essentially all sufficiently general codes use data structures for such meshes that
explicitly or implicitly store not only cells and vertex locations,
but also faces and edges and allow associating data with these objects.
In many cases, the data that is associated with a cell, face, edge, or
vertex may have a physical location somewhere on this object. For
example, when using a ${\mathbb Q}_3$ bicubic finite element on a
rectangular cell, we need to store the index (and possibly the value)
of one degree of freedom per vertex, two along the edge (typically at
1/3 and 2/3 along its extent), and 4 inside the cell. In order to define where
a distance of 1/3 or 2/3 along the edge is, we need to define a coordinate
system on the edge. The same is true when implementing N\'ed\'elec elements
that define degrees of freedom as tangential vectors along edges, and
therefore need to define a direction on each edge. For similar
reasons, we typically also need coordinate systems within each cell.
We then need to define how the coordinate
system defined on the edge relates to that of the adjacent cells. We can
either do this by letting every
quadrilateral cell store pointers to the four edges along with one bit per
edge that indicates how the direction of the edge embeds in the coordinate system of the
cell. Or we could seek a convention by which we orient all edges of the mesh once at the
beginning so that the orientation of each cell implies the orientation of its
bounding edges. In the latter case, we would not need to store direction
flags, and algorithms built on this convention would not need to provision for
different possible directions, thereby greatly simplifying code construction
and maintenance.
This paper is concerned with the following two questions:
\begin{itemize}
\item Is it possible to
find such a convention for quadrilateral meshes
(an example of such a mesh is shown in Fig.~\ref{fig:wing})? We will
constructively show that this is indeed possible in 2d when adopting
the convention that the edges that bound opposite sides of each cell
point in the same direction; see the left panel of
Fig.~\ref{fig:convention}.%
\footnote{On the other hand, the only two other reasonable conventions for edge
directions, namely either requiring them to be (i) oriented in clockwise or
counter-clockwise direction around the cell, or (ii) oriented away
from two, oppositely located vertices, both do not always allow for
globally unique directions of all edges of a mesh; see
Remark~\ref{rem:circular-edges}.}
\item Is it possible to assign directions to all edges of a mesh that satisfy
this convention in a computationally efficient manner? We will demonstrate
that this is in fact so: Our construction of a proof for the answer to the
first question (in
Section~\ref{sec:2d}) also implies an algorithm that we show to be order
optimal, i.e., it runs in a time proportional to the number of edges
in the mesh.
\end{itemize}
\begin{figure}[tbp]
\begin{center}
\phantom{.}
\hfill
\includegraphics[height=.3\textwidth]{graphics/wing}
\hfill
\includegraphics[height=.225\textwidth]{graphics/mesh_3d}
\hfill
\phantom{.}
\caption{\it A typical two-dimensional quadrilateral mesh around an
airfoil with 29,490 cells (left). Surface of a
three-dimensional mesh with 2,304 cells (right).}
\label{fig:wing}
\end{center}
\end{figure}
\begin{figure}[tbp]
\begin{center}
\phantom{.}
\hfill
\includegraphics[width=.2\textwidth]{graphics/convention_2d}
\hfill
\includegraphics[width=.26\textwidth]{graphics/convention_3d}
\hfill
\phantom{.}
\caption{\it Choice of directions of edges when seen with regard
to one particular orientation of a ``coordinate system'' on a
cell. Note in particular that this choice implies that opposite
edges in a cell must have parallel directions. Left: For quadrilaterals in $d=2$.
Right: For hexahedra in $d=3$.}
\label{fig:convention}
\end{center}
\end{figure}
On the other hand, we will show in Section~\ref{sec:3d} that the
corresponding convention in 3d (see the right
panel of Fig.~\ref{fig:convention}) allows for examples
where it is not possible to orient edges so that coordinate systems of
adjacent cells are implied. However, we will show that the
extension of the 2d algorithm to 3d either produces a
consistent set of edge orientations or fails, both within order optimal
complexity. There are, however, important classes of 3d meshes that
always allow such edge directions, and we will discuss these in
Section~\ref{sec:3d-always-orientable}.
\begin{includefaces}
On the other hand, in 3d one may want to
not only have consistent directions for edges of cells, but also for
faces, and the corresponding case is discussed in
Section~\ref{sec:3d-faces} where we show that there are cases where it is
possible to find either consistent edge or face orientations, but not
both.
\end{includefaces}
The paper is complemented by Section~\ref{sec:notation} defining
notation, conclusions in Section~\ref{sec:conclusions}, and an
appendix in which we we prove a generalization of the main
statement of the paper to general manifolds.
\paragraph*{Related literature}
Finite element software packages have traditionally taken different routes to
dealing with the problem of relative orientations of cells and their edges and
faces. In many cases, software has been developed to only support linear or
quadratic $H^1$-type elements, in which case edge and face orientations are not
in fact of any concern at all. Others use triangular or tetrahedral meshes for
which it is necessary to explicitly store edge orientations; see, for example,
\cite{BM02,AC03}. For quadrilaterals and hexahedra,
strategies for implementation in specific packages are discussed in \cite{RKL09} for the FEniCS
library and \cite{TL08} for the KARDOS package. We have found that DUNE \cite{PBDEKKOS08} and Nektar++ \cite{Can15} appear to
explicitly store edge directions as part of their mesh data structures or
finite element implementations, but we could not find written elaborations of
their strategies in publications or overview documents. Finally, descriptions
of finite
elements that use \textit{global} conventions for edge orientations
(i.e., based on vertex coordinates or indices, instead of in relation
to locally adjacent cells)
can be found in \cite{Zag06} and are used,
for example, in libMesh
\cite{KPSC07}.
Constructions similar to those in this paper have previously been discussed in
the discrete geometry literature, see \cite{Het95,AZ04,HW08} for examples. However,
this part of the literature is not typically concerned with algorithms and
their complexity (such as our discussions in Sections~\ref{sec:2d} and
\ref{sec:3d}), nor with the particular application of these ideas to finite
element meshes (such as our discussion of specific types of meshes
in Section~\ref{sec:3d-always-orientable}). Our contribution therefore provides a
relevant extension of what is available in the literature.
\paragraph*{A historical note} The algorithms discussed herein were
implemented in the \textsc{deal.II}{} library in 2003, with an incomplete
discussion of the topic available in the documentation of
\textsc{deal.II}{}'s \texttt{GridReordering} class. A more formal description of these
algorithms has recently appeared in \cite{HH15}; it extends our 2d algorithms to
meshes stored on distributed memory, parallel machines, but it lacks the
complexity analysis that we provide here, as well as much of the discussion of
the 3d case.
The introduction in \textsc{deal.II}{} of the convention discussed above predates 2003.
In its earliest days, the library was almost always used on small
problems for which edge orientations could be determined by hand on a
piece of paper, and little consideration was given to the
question whether it always exists and if so whether there is an efficient
algorithm to generate it automatically. However, as the project grew
and more applications used the 3d part, these questions became more
important. Initially, an algorithm that generated edge orientations using a
backtracking algorithm was implemented. This works for meshes
with up to a few hundred cells, but fails due to excessive run times for
larger ones. In particular, it is easy to construct meshes for which it had
exponential run time. Therefore, more efficient algorithms were needed,
leading to the results reported here.
\paragraph*{Availability of implementations} We have implemented the
algorithms outlined here in the \textsc{deal.II}{} library (see
\url{http://www.dealii.org/} and \cite{BHK07,dealII82}), and they are
available as part of the \texttt{GridReordering} class under the LGPL open source
license.
\section{Notation and conventions}
\label{sec:notation}
Throughout this paper, we will consider triangulations $\tria$ such as
those shown in Fig.~\ref{fig:wing}, as a collection
$\tria=\{K_1,\ldots,K_{N_K}\}$ of quadrilateral or hexahedral cells
$K_i$. These cells can be considered as open geometric
objects $K_i\subset {\mathbb R}^d, d=2,3$ so that (i)~$K_i\cap K_j=\emptyset$
if $i\neq j$, (ii)~the intersection of the closure of two cells, $\bar
K_i \cap \bar K_j$, is either empty, a vertex of the mesh, or a
complete edge or face of both cells, and (iii)~$\bigcup_i \bar K_i =
\bar\Omega$ where $\Omega\subset {\mathbb R}^d$ is the bounded, open domain that is
subdivided into the triangulation. We assume that the triangulation
has only finitely many cells. For the purposes of this paper, we do
not require that the union of mesh cells corresponds to a simply
connected domain. We will rely on the very practical assumption that
the volume of each cell is positive and that cells are convex.
That said, the bulk of our arguments will not make use of this
geometric view of a triangulation. Rather,
it is convenient to reformulate the problem under consideration using
the language of graphs. When viewing a finite element mesh as an undirected
graph, we consider it as a pair $G_\tria=(V,E)$ with the vertices
$V=\{v_1,\ldots,v_{N_v}\}$ being the vertices of the mesh, and
edges $E=\{e_1,\ldots,e_{N_e}\}\subset \{\{v_a,v_b\}: v_a, v_b\in
V, v_a\neq v_b\}$ being the four ($d=2$) or twelve ($d=3$) edges of the
cells. Given the construction of this graph as a representation of a
mesh, there is then a collection of cells $\tria=\{K_1,\ldots,K_{N_K}\}$
where we can alternatively see each $K_i$ as either an ordered
collection of 4 vertices or an ordered collection of 4 edges (in 2d;
in 3d it is 8 vertices and 12 edges). The index $\tria$ on $G_\tria$
indicates that we are not considering general graphs but indeed only
those graphs that originate from a triangulation of a domain
$\Omega\subset{\mathbb R}^d$.
For a given cell $K$, let $v(K)\subseteq V$ be the set of its vertices
and $e(K)\subseteq E$ the set of its edges. For a given edge, let
$K(e)\subseteq{\tria}$ be the set of adjacent cells. In 2d,
$|K(e)|$ is either one or two (depending on whether the edge is at the
boundary or not); in 3d, $|K(e)|\ge 1$ since arbitrarily many cells may be
adjacent to a single edge.
As discussed above, we are interested in assigning a direction to
each edge in such a way that the direction of the edge is implied from
the orientation of a cell. That is, for a mesh with associated graph
$G_\tria$, we would like to have a directed graph $G_\tria^+=(V,E^+)$ with the same vertex
set and edges, but where each edge is now considered directed (i.e.,
represented by an ordered pair of vertices). To be precise, this
graph is in fact \textit{oriented} since we never have both $(v_i, v_j) \in E^+$ and
$(v_j, v_i) \in E^+$ as may occur in directed graphs.
\subsection{Goals for orienting meshes}
As discussed in the introduction, practical implementations of the
finite element method need to define a coordinate system on both cells
and edges of a mesh. This is typically done by \textit{mapping} a
reference cell $\hat K=[0,1]^d$ and edge $\hat e=[0,1]$ to each cell
$K$ and edge $e$, along with the coordinate systems. The details of
this are of no importance here other than the following two
statements: (i) On each cell, we need to designate one of the four
(or eight) vertices as the ``origin''; each of the two (three) edges adjacent to the
origin then form the first and second (and third) ``coordinate
axis''. If we insist that the mapping from the reference cell $\hat K$
to $K$ has positive volume, then the choice of the origin fixes the
coordinate system in 2d; in 3d, each of the edges adjacent to the
origin can be chosen as the first coordinate axis, with the other two
then fixed. In other words, each cell allows for 4 possible choices of
coordinate systems in 2d, and $8\times 3=24$ in 3d. (ii) On each edge,
we define a coordinate system by choosing a ``first'' vertex.
With this, there are a total of $4^{|N_K|}$ choices in 2d for the coordinate systems
on the cells (and $24^{|N_K|}$ in 3d), and $2^{|N_e|}$ for the
coordinate systems of the edges. The question is now whether it is
possible to choose them in such a way that if the coordinate systems
of all cells are specified, we can infer the coordinate system on each
edge unambiguously, regardless of which cell adjacent to the edge we
consider. Or conversely: is it possible to specify directions for all
edges in such a way that
this implies a unique choice of coordinate systems for all cells?
For reasons that will become clear later, we will prescribe directions
of the edges of a cell when seen in the ``coordinate system'' of the
cell as shown in Fig.~\ref{fig:convention}. Clearly, it will be
possible to choose the coordinate systems of two neighboring cells in
such a way that they do not agree on the direction of the common
edge. Such a choice would require an implementation to store for each
cell whether or not an edge's (global) direction does or does not match the
direction that results from the (local) convention.
On the other hand, let us assume that we can find an orientation for
all edges of the mesh so that in each cell, ``opposite'' edges are
``parallel''. Then each cell has one vertex from which all oriented
edges originate, and we can choose this as the ``origin'' of the
cell. Using this choice of cell coordinate system, edge directions are
then again uniquely implied and the problem is solved. In other words,
we have reduced the problem of orienting all cells and edges to the
problem of finding one particular set of edge orientations that
satisfy the \textit{``opposite'' edges are
``parallel''} property.
This property is easy to understand intuitively. It is significantly
more cumbersome to describe it rigorously in the graph
theoretical language, and so we will only do this for the 2d case. (The
3d case follows the same approach but requires lengthy notation
despite the fact that the situation is relatively easy to understand.)
In order to formulate the convention,
we need to fix the order in which we consider
vertices and edges as part of a cell. We do so using a lexicographic
order for vertices, as shown in
Fig.~\ref{fig:vertex-edge-convention}. Edges are numbered so that we
first number the edge from vertex 0 to 2, then its translation in
the perpendicular direction (i.e., from vertex 1 to 3), and then the
edges connecting the vertices of the first two edges. Both of these orders
reflect the tensor product structure of quadrilaterals and are easily
generalized to hexahedra (or higher dimensions, if desired). The
choice of edge directions within each cell, as shown in
Fig.~\ref{fig:convention}, then
ensures that the coordinate system of the edge is simply the
restriction of the cell's coordinate system to the edge.
\begin{figure}[tbp]
\begin{center}
\phantom{.}
\hfill
\includegraphics[width=.3\textwidth]{graphics/numbering_2d}
\hfill
\phantom{.}
\caption{\it Numbering convention for vertices and edges of
two-dimensional cells. Here, $v(K)^+$ is the ordered set of
vertices that bound the cell, and $e(K)^+$ the ordered set of
edges.}
\label{fig:vertex-edge-convention}
\end{center}
\end{figure}
With this definition, each cell is described by an ordered tuple of
its vertices where we will assume that the first element of this tuple
corresponds to the ``origin''. Equivalently, we can describe each cell
as an ordered tuple of four (unordered) edges, where the ``origin'' of
the cell is now the common vertex of edges 0 and 2. Because there are
4 possible choices for the origin of the cell, there are four
ways to describe a cell that are equivalent up to rotation.
With these preparations, we can finally define what it means for the
edges of a directed graph $G_\tria^+$ that represents a quadrilateral mesh
to be consistently oriented:
\begin{convention}
\label{conv:1}
We call a graph $G_\tria^+=(V,E^+)$ \textit{consistently oriented with respect to cell $K\in\tria$}
if among the four equivalent choices of vertex tuples of $K$, there is one so that the
following directed edges are all elements of $E^+$:
$e(K)_0=(v(K)_0,v(K)_2)$,
$e(K)_1=(v(K)_1,v(K)_3)$,
$e(K)_2=(v(K)_0,v(K)_1)$,
$e(K)_3=(v(K)_2,v(K)_3)$.
\end{convention}
\begin{convention}
\label{conv:2}
We call a graph $G_\tria^+=(V,E^+)$ \textit{consistently oriented} if it is
consistently oriented with respect to all cells in $\tria$.
\end{convention}
As discussed above, a consistently oriented graph has edges that allow
us to choose a coordinate system on each cell so that the edge
orientations follow immediately from the cell orientations.
Similar definitions can be given for the 3d case. The purpose of this
paper is to ask the question whether it is always possible to
consistently orient the edges of a given mesh $\tria$, and if this is the case,
whether it can efficiently be done by an algorithm.
\subsection{Reformulation of Conventions \ref{conv:1} and \ref{conv:2}}
The developments
in the following sections all depend on the fact that
Convention~\ref{conv:1} can equivalently be stated by only looking at
sets of ``parallel edges'' of quadrilaterals:%
\footnote{The definition of whether edges are parallel given here only
uses the graph theoretic context. In the language of finite element
methods, it could equivalently be defined in geometric terms by
using the coordinates of the vertices of the original mesh
$\tria$. We can then view each edge of the
graph as a (not necessarily straight) line connecting the adjacent
vertices. Each cell $K\subseteq \tria$ occupies a
subset of ${\mathbb R}^d$ that is the image of the reference square or cube
$[0,1]^d$ under a homeomorphic mapping $\phi_K$. We can then
call two edges $e_1,e_2$ parallel in $K$ if their preimages,
$\phi_K^{-1}(e_1)$ and $\phi_K^{-1}(e_2)$, i.e., the corresponding
edges of the reference cell, are parallel line segments in the geometric sense. This may be more
intuitive, but we have no further use for mappings and
transformations in this paper and will therefore not further explore
the geometric setting.}
\begin{definition}
\label{def:parallel}
Two edges $e',e''\in E$ are called \textit{parallel edges of $K$} if $e'=e''$
or if they bound $K$ but do not share a vertex. If $e',e''\in E$ are parallel
edges of $K$, then we denote $e' \CellPar{K}
e''$. We say that $e',e''\in E$ are \textit{locally parallel} and denote $e'
\LocPar e''$ if there exists a cell $K \in {\cal K}$ so that $e' \CellPar{K} e''$.
\end{definition}
The four edges of a quadrilateral then fall into two classes of two edges each that are
parallel. (In 3d, a corresponding but more notationally cumbersome definition would yield three classes of
four parallel edges each.) In practice, one does not usually have to verify
if edges are parallel, but only enumerate classes of parallel edges;
this does not require testing all equivalent edge sets: in 2d, edges
$e(K)_0$ and $e(K)_1$ are parallel, as are $e(K)_2$ and $e(K)_3$, for
any arbitrarily chosen equivalent edge set.
We can then define consistent orientations via
these classes of parallel edges:
\begin{convention}
\label{conv:3}
Two oriented, parallel edges $e', e''\in E^+$ are called
\textit{consistently oriented with respect to cell $K$} if
$e'=(v(K)_0,v(K)_2)$,
$e''=(v(K)_1,v(K)_3)$ with regard to the one equivalent vertex set
within which $e'=e(K)_0$ and $e''=e(K)_1$.
\end{convention}
\begin{convention}
\label{conv:4}
We call a graph $G_\tria^+=(V,E^+)$ \textit{consistently oriented with respect to cell $K$}
if all sets of parallel edges of $K$ are consistently oriented
with regard to $K$.
\end{convention}
In other words, consistent orientation on a cell can be tested by only
verifying consistent orientation of all the edges in all sets of
parallel edges. Consequently, \textit{testing} that a graph $G_\tria^+$ is
consistently oriented is relatively easy and can be done by verifying
the condition on every cell separately. A verification algorithm can
therefore easily be written with $O(N_K)$ complexity and,
consequently, with $O(N_e)$ because $\frac 12 N_K \le N_e \le 4N_K$.
On the other hand, \textit{generating}
a consistent orientation requires a \textit{global} algorithm because
the orientation of one edge implies that of all of its parallel edges
on all of its adjacent cells, which itself implies the orientation of
edges parallel on cells twice removed, etc. Because of this property,
it is not a priori clear that one can find a linear-time algorithm
that can find a consistently oriented graph $G_\tria^+$ given the graph $G_\tria$
associated with a mesh. However, as we will show below, this is indeed
possible.
As a final note in this section, let us state that for all algorithms that
follow, we assume that we have methods to generate the vertex and edge sets
$v(K),e(K)$ for a given cell $K$ with ${\cal O}(1)$ complexity. This can
easily be achieved by storing this information as the rows of $N_K\times 4$
matrices as is commonly done in all widely used software (in 3d, the vertex
adjacency matrix is of size $N_K\times 8$ and the edge adjacency
matrix is of size $N_K\times 12$). Furthermore, we will assume that finding
the cell neighbors of a given edge, $K(e)$, requires ${\cal O}(1)$ time. It is
obvious that in 2d, this can be achieved by storing an $N_e\times 2$ matrix
storing the indices of the one or two cells that are adjacent to each edge;
populating this matrix from $e(K)$ requires only a single loop over all
cells. In 3d, the number of cells adjacent to each cell can in principle be
equal to $N_K$, requiring data structures that can either not be queried in
${\cal O}(1)$ complexity, or can not be built with ${\cal O}(N_K)$ complexity;
however, in actual finite element practice, the meshes we consider will never
have more than, say, a dozen or so elements joined at any one edge, so that we
can consider $|K(e)|$ to be bounded by a constant, thereby allowing
storing information in tables of fixed width and ensuring that $K(e)$ can be queried
with ${\cal O}(1)$ complexity. These assumptions will be important to
guarantee the overall complexity of the algorithms we will consider in the
following.
As stated above, we assume that $K(e)$ can be \textit{evaluated} in ${\cal
O}(1)$ time. To make this possible requires \textit{building} appropriate data structures, and
depending on what information is available this may require more than ${\cal
O}(N)$ time. For example, if one only knew the vertices of each cell, i.e.,
$v(K)$, then building tables that can evaluate $e(K)$ in ${\cal O}(1)$ time
requires ${\cal O}(N\,\log N)$ time; this is the typical case with file
formats that store mesh information. On the other hand, if each cell already
knows all of its edge neighbors, as is often the case inside mesh generators
or finite element codes,
then building the tables to evaluate $e(K)$ in ${\cal O}(1)$ only costs ${\cal
O}(N)$ time and is consequently of the same complexity as the algorithms we
will discuss in the following.
\section{The two-dimensional case}
\label{sec:2d}
In this section, we will consider the two-dimensional problem, which can be
stated as follows: \textit{Given a graph $G_\tria$ that originates
from a mesh $\tria$ composed of
quadrilaterals, find a consistently oriented graph $G_\tria^+$.}
As mentioned above, consistency of edge directions only requires us to ensure
consistency within sets of parallel edges. To this end, we will in the
following describe methods that first find all edges that are, in some sense
that goes beyond that defined in Definition~\ref{def:parallel}, parallel to
each other (Section~\ref{sec:2d-parallel}). We will then show how we can
consistently orient all edges that are in this sense parallel to each other
(Section~\ref{sec:2d-orienting-ribbon}) and finally how we can orient all
edges in the graph (Section~\ref{sec:2d-orienting-graph}).
\subsection{Decomposing $E$ into sets of parallel edges}
\label{sec:2d-parallel}
As we will show next, the set of edges $E$ of $G_\tria$ can be decomposed into
a collection of mutually exclusive sets of edges, where the edges in
each set are all parallel to each other in some global sense.
This follows from the fact that the relation \emph{locally parallel}
is reflexive (i.e., $e \LocPar e$ for all edges $e$) and symmetric (i.e. $e' \LocPar e''$
implies $e'' \LocPar e'$ for all edges $e',e''$) by construction. Let us define two
edges $e'$, $e''$ to be globally parallel if there is a finite, possibly empty sequence of
edges $e_0, e_1, \ldots, e_s$ such that $e' \LocPar e_0 \LocPar e_1 \LocPar \ldots \LocPar
e_s \LocPar e''$ and denote this by $e' \GlobPar e''$. One immediately checks that this
relation is reflexive, symmetric and transitive (i.e. $e' \GlobPar e''$ and $e'' \GlobPar
e'''$ imply $e' \GlobPar e'''$ for all edges $e',e'',e'''$) and hence forms an equivalence
relation.%
\footnote{In an abstract sense, we have constructed the relation $\GlobPar$ as
the \emph{transitive hull} of the relation $\LocPar$.}
This relation then partitions $E$ into disjoint sets of (globally) parallel edges
\cite[Chapter I, \S 3]{Kur63} or \cite[Corollary 28.19]{LP98}.
Algorithmically, we can
recursively construct each of these sets by starting with an edge $e$ and build
the set $\Pi(e)\subseteq E$ of all edges
that are globally parallel to $e$ by first
adding the edges opposite $e$ in the cells adjacent to $e$, then those
edges that are opposite to the ones added previously, and so on. Sets
of parallel edges are central to the rest of the paper, since our edge
direction convention requires that they will all have parallel
directions. An intuitive interpretation of the importance of parallel edges is as
follows: assume we had already found the directed graph $G_\tria^+$. Then,
flipping the direction of an edge $e$ would make the triangulation
non-consistent, and to make it consistent again we would have to flip
the directions of a number of other edges as well; the entire set of
edges that needs to be flipped, including $e$, is precisely $\Pi(e)$.
With this knowledge, let us concisely define an algorithm to find all
elements of a parallel set for an edge $e$ as a first step:
\begin{algorithm}[(Construct one set of parallel edges)]
\label{alg:parallel-edges}
Let $e\in E$ be a given edge and generate the set $\Pi(e)$ of edges
parallel to $e$ recursively as follows, where the set $\Delta_k$ consists
of those edges that we add to $\Pi(e)$ in each step as we grow it away from $e$:
\begin{enumerate}
\item Set $\Pi_0(e)=\emptyset$, $\Pi_1(e)=\{e\}$,
$\Delta_0=\{e\}$, $k=1$.
\item While $\Delta_{k-1}\neq \emptyset$, do:
\begin{enumerate}
\item Set $\Delta_k=\emptyset$.
\item For each $\delta\in\Delta_{k-1}$:
\begin{itemize}
\item Set ${\cal E}(\delta)=\emptyset$.
\item For each $K\in K(\delta)$:
\begin{enumerate}
\item Set ${\cal N}_K(\delta):=\{\varepsilon \in e(K):
\varepsilon \CellPar{K} \delta \} \backslash \{\delta\}$.
\item Set ${\cal E}(\delta) = {\cal E}(\delta) \cup {\cal N}_K(\delta)$.
\end{enumerate}
\item Set $\Delta_k = \Delta_k \cup \left({\cal E}(\delta) \backslash\Pi_k(e))\right)$.
\end{itemize}
\item $\Pi_{k+1}(e) = \Pi_k(e) \cup \Delta_k$.
\item Set $k:=k+1$.
\end{enumerate}
\item Set $\Pi(e)=\Pi_k(e)$.
\end{enumerate}
\end{algorithm}
Fig.~\ref{fig:parallel-sets} gives a graphical depiction of the construction
of one such set. Starting from a given edge $e$, the set
$\Pi(e)$ grows in each step by
at most the two edges in $\Delta_k$ along a line that always intersects cells from
one side to the opposite one, and connects these parallel edges.
Growth of the set in one direction stops whenever this line hits a boundary
edge (in which case the set of opposite edges for this boundary edge $\delta$,
${\cal E}_K(\delta)$, has only one member, which furthermore is already in
$\Pi_{k-1}(e)$), or if both ends of the line meet ``head-on'' (in which case all
elements of ${\cal E}(\delta)$ for all $\delta\in\Delta_{k-1}$ are already in
$\Pi_{k-1}(e)$ and thus $\Delta_k=\emptyset$, upon which the
iteration terminates).
\begin{figure}[tbp]
\begin{center}
\includegraphics[width=.2\textwidth]{graphics/sample_grid_1}
\hfill
\includegraphics[width=.2\textwidth]{graphics/sample_grid_2}
\hfill
\includegraphics[width=.2\textwidth]{graphics/sample_grid_3}
\hfill
\includegraphics[width=.2\textwidth]{graphics/sample_grid_4}
\caption{\it Construction of the parallel sets $\Pi_k(e)$,
$k=1,\ldots, 4$, indicated by bold edges. The red line connects all of
them and grows in both directions.}
\label{fig:parallel-sets}
\end{center}
\end{figure}
To assess the overall run time of this algorithm, we note that in 2d, each
edge has exactly two neighboring cells unless it is at the boundary,
i.e., $|K(\delta)|\le 2$. Furthermore, within each cell, there is
exactly one other parallel edge to a given edge $\delta$, i.e.,
$|{\cal N}_K(\delta)|=1$. Consequently, in step (2)(b) we have
$|{\cal E}(\delta)|\le 2$ and it follows that $|\Delta_k|\le 2$. With the
appropriate data structures -- for example, by representing sets of
known maximal cardinality through fixed-sized arrays --, all
operations in steps (2)(a)--(d) can then be executed in
${\cal O}(1)$ operations. Furthermore, since $\Pi_k(e)$ grows by
one or two elements per iteration, the loop represented by step (2)
executes at most $|\Pi(e)|$ times. The total cost of the
algorithm is therefore ${\cal O}(|\Pi(e)|)$.
The next step is based on the realization that every edge $e$ in the
graph $G_\tria$ can be uniquely sorted into one of a collection of mutually
exclusive sets
$\pi=\{\Pi_1,\ldots,\Pi_{N_\Pi}\}$.
Each class $\Pi_i$ is constructed as above. Because the connecting
line for each parallel set is either closed or ends on both sides at
the boundary, the number of distinct sets of parallel edges, $|\pi|$, is at least
half the number of boundary edges, but of course at most half the number of edges
in $G_\tria$.
Algorithmically, we can construct the collection $\pi$ of parallel sets in the following way:
\begin{algorithm}[(Construct the set of parallel edge sets)]
\label{alg:all-parallel-edges}
Construct the set of parallel edge sets as follows:
\begin{enumerate}
\item Set $\pi=\emptyset, {\cal E}=E$.
\item While ${\cal E} \neq \emptyset$, do:
\begin{enumerate}
\item Choose any $e \in {\cal E}$.
\item Compute $\Pi=\Pi(e)$ using Algorithm~\ref{alg:parallel-edges}.
\item Set $\pi = \pi \cup \{\Pi\}$.
\item Set ${\cal E} = {\cal E} \backslash \Pi$.
\end{enumerate}
\end{enumerate}
\end{algorithm}
Here, the set of not-yet-classified edges ${\cal E}$ is reduced one
equivalence class -- i.e., by one set of globally parallel edges -- at a
time. Because the decomposition of edges into
equivalence classes is unique, it is clear that in each iteration,
$\Pi\subseteq{\cal E}$. Furthermore,
$\Pi(e)\supseteq\{e\}$ and so $|\Pi|\ge 1$; thus,
the iteration is guaranteed to terminate.
More concretely, the cost of each iteration is given by
Algorithm~\ref{alg:parallel-edges}, i.e., ${\cal O}(|\Pi(e)|)$. The
overall cost is therefore $\sum_{\Pi\in\pi} {\cal
O}(|\Pi|)$. On the other hand, because edges can be uniquely sorted
into their equivalence classes, we know that
$\bigcup_{\Pi\in\pi} \Pi = E$. Thus, the cost of
Algorithm~\ref{alg:all-parallel-edges} is ${\cal O}(|E|)$, i.e., of
optimal complexity.
\subsection{Orienting the elements of a set of parallel edges}
\label{sec:2d-orienting-ribbon}
Our convention was only that opposite edges in a cell have parallel
directions, but there was no requirement on the relative directions of
adjacent (non-opposite) edges within a cell. In fact, that was the
basis for restating Conventions~\ref{conv:1} and \ref{conv:2} in terms of
Convention~\ref{conv:3} and \ref{conv:4}. It is thus easy to see that we
only have to make sure that we have consistent directions of all edges within
each set of parallel edges, and that consistency of edges within each such set
is independent of the directions of edges in all other parallel sets. The
following lemma proves that within each such equivalence class a
consistent set of directions exists:
\begin{lemma}
\label{lemma:existence-for-one-parallel-set}
Let $e\in E$. Then there exists a choice of orientations for the elements of
$\Pi(e)$ that is consistent; i.e., for all
$e',e''\in\Pi(e)$ so that $e' \CellPar{K} e''$ for some
cell $K$, then the
orientations we associate with $e'$ and $e''$ are consistent in $K$.
\end{lemma}
This statement can be proven in a variety of ways, both constructively
and in indirect ways. The most intuitive way uses the
fact that a curve in the plane, closed or not and possibly
self-intersecting, such as the dotted line in
Fig.~\ref{fig:parallel-sets}, allows for the definition of a unique direction
``from one side of the curve to the other'', and we can then orient
each edge it crosses according to this direction. This statement appears
obvious on its face. For closed, non-intersecting curves, it
follows from the Jordan Curve Theorem that states that such curves
partition the plane into an ``inside'' and ``outside'' area and we can
then, for example, choose the direction from the inside to the outside to orient
edges. Proving the existence of such a direction field for
self-intersecting curves requires more work. The history of the Jordan
Curve Theorem teaches us that care is necessary, and our variations
on a proof typically required a page or more of geometry, even when
taking into account that we only need to show the statement for the
piecewise linear curves that connect edge midpoints.
Thus, rather than providing such a proof here, let us for now
simply consider the lemma to be true. A proof will later follow from
Remark~\ref{remark:extension-manifold} and a statement in the appendix
where we show a more general statement of which
Lemma~\ref{lemma:existence-for-one-parallel-set} is simply a special case.
Given this statement of feasibility, we can now ask for an
algorithm that assigns directions to all edges in a set
$\Pi(e)$ and that can be implemented with ${\cal
O}(|\Pi(e)|)$ complexity. We do so by extending
Algorithm~\ref{alg:parallel-edges} to include edge orientation
assignment:
\begin{algorithm}[(Orient edges for consistency)]
\label{alg:orient-edges}
Let $\pi=\{\Pi_1,\ldots,\Pi_{N_\Pi}\}$ be
given. Then perform the following operations for each $i=1,\ldots,N_\Pi$:
\begin{enumerate}
\item Assign an ``unassigned'' orientation to each edge $e\in \Pi_i$.
\item Choose some $e\in \Pi_i$ and set $\Delta_0=\{e\}, k=1$.
\item Assign an arbitrary orientation to $e$.
\item Set $\Delta_k=\emptyset$.
\item While $\Delta_{k-1}\neq \emptyset$, do:
\begin{enumerate}
\item For each $\delta\in\Delta_{k-1}$:
\begin{itemize}
\item Set ${\cal E}(\delta)=\emptyset$.
\item For each $K\in K(\delta)$:
\begin{enumerate}
\item Set ${\cal N}_K(\delta):=\{\varepsilon \in e(K):
\varepsilon \CellPar{K} \delta \} \backslash \{\delta\}$.
\item Assign an orientation to the elements of ${\cal N}_K(\delta)$
that is consistent in $K$ with that of $\delta$.
\item Set ${\cal E}(\delta) = {\cal E}(\delta) \cup {\cal N}_K(\delta)$.
\end{enumerate}
\item Set $\Delta_k = \Delta_k \cup \left({\cal E}(\delta) \backslash\Pi_k(e))\right)$.
\end{itemize}
\item $\Pi_{k+1}(e) = \Pi_k(e) \cup \Delta_k$.
\item Set $k:=k+1$.
\end{enumerate}
\end{enumerate}
\end{algorithm}
It is important to note that in step (5)(a)(ii), we assign directions only to
the cell neighbors ${\cal N}_K(\delta)$ of $\delta$, all of which either do
not yet have an orientation, or that have already been assigned
that very same orientation when coming ``from the other side of the parallel set''
in case the connecting line for this set is a closed line through our
mesh. The step therefore never changes an already assigned orientation.
The algorithm above would, in practice, simply be implemented as part of
computing the parallel sets $\Pi_i$. In that case, it isn't even
necessary to explicitly build $\Pi_k(e)$ in step (5)(b) as this set
is only used in the last part of step (5)(a) where we could simply exclude all
elements from ${\cal E}(\delta)$ that had previously already been assigned an
orientation.
\subsection{Orienting all edges in a graph}
\label{sec:2d-orienting-graph}
With these results, we can state the main result for the case $d=2$:
\begin{theorem}
\label{theorem:2d}
For every planar, undirected graph $G_\tria$ generated by subdividing a bounded
domain in ${\mathbb R}^2$ into finitely many quadrilaterals, there exists a
corresponding, consistently oriented directed graph $G_\tria^+$.
\end{theorem}
\begin{proof}
The proof follows from the preceding subsections: first, we can uniquely sort
all edges into equivalence classes, for which we can choose edge directions
independently; second, we can find a consistent choice of edge
directions within each such set.
\end{proof}
Since both sorting edges into equivalence classes and assigning
directions to edges of all equivalence classes are linear in
the number of edges in the equivalence set, the overall algorithm is
${\cal O}(|E|)$ and therefore order optimal in the number of edges.
Furthermore, because each edge is shared by no more than two cells and
because each cell is bounded by exactly four edges, it is obvious that
$\frac 14 |E| \le |{\tria}| \le 2|E|$. It
follows directly that the algorithm is not only linear in the number of
edges, but also in the number of cells (which is the more important estimate
in practice).%
\footnote{In practice, not only the complexity class matters, but also
the constant in front of it. Our reference implementation in
\textsc{deal.II}{} orients the edges of the roughly 30,000 cells of the mesh
shown in the left panel of
Fig.~\ref{fig:wing} in 0.035 seconds on a current laptop -- far
faster than solving any equation on this mesh would take.}
\begin{remark}
\label{remark:extension-manifold}
The arguments above showing that 2d meshes are always orientable can
be carried over to meshes on two-dimensional, orientable surfaces,
and we will prove so in the appendix. In particular, this
holds for the practically important case of
2d meshes covering (part of) the surface of a 3d domain. Indeed,
this is true whether the domain is homeomorphic to the
unit ball or not -- i.e., meshes on the surface of 3d domains with
handles are still orientable.
The proof of this more general statement then also covers
Lemma~\ref{lemma:existence-for-one-parallel-set}: the lemma states
the orientability of edges of a mesh in the plane ${\mathbb R}^2$, which is
obviously an orientable, two-dimensional manifold. The proof of
Lemma~\ref{lemma:existence-for-one-parallel-set} thus follows from
the proof given in the appendix.
On the other hand, meshes on non-orientable surfaces,
for example on a M{\"o}bius strip, are not necessarily
orientable. This is because some of the connecting lines of parallel
edges (see Fig.~\ref{fig:parallel-sets}) may wrap around the strip
and return what we thought to be a vector from ``one side to the
other side'' in its reverse orientation. It is
therefore not possible to define a unique ``right'' and ``left'' of
a curve on a non-orientable manifold, and not all sets of edges of a
mesh defined on it may have a consistent orientation.
\end{remark}
\begin{remark}
\label{rem:circular-edges}
As mentioned in a footnote to Section~\ref{sec:introduction} and
Fig.~\ref{fig:convention}, one can imagine other conventions for the
relative orientations of edges and cell. For example, for triangles one
often assumes a circular orientation. Such
a convention could also be adopted for quadrilaterals. However, it has two
problems: (i)~It is not obvious how to generalize it to
hexahedra. (ii) It is easy to construct meshes for which no set of
globally consistent edge orientations can be found. To see this, note that a
circular choice for edge directions in each cell implies that opposite edges
have anti-parallel directions. Then imagine a closed string of $n$
cells. One of the sets of parallel edges then contains all of the $n$ edges
that separate the $n$ cells forming a circle. The convention requires
us to orient them alternatingly -- something that is only possible
if $n$ happens to be even.
\end{remark}
\section{The three-dimensional case}
\label{sec:3d}
Let us now also look at the case of three spatial dimensions, and a
subdivision of the domain into hexahedral cells. The right panel of
Fig.~\ref{fig:convention} shows the convention for $d=3$, where again we will
want to have all parallel edges point in the same direction. Note that now we
have three sets of four edges within each cell.
We will have to investigate both whether the algorithms discussed in
the previous section always yield a consistent edge orientation, and
what their complexity is given the changed circumstances.
\subsection{Are the edges of 3d meshes always orientable?}
\label{sec:3d-always}
The first step is to construct the sets of parallel edges,
$\pi$. Algorithm~\ref{alg:parallel-edges} again finds all
elements of an equivalence set $\Pi(e)$ for a given starting
edge $e$. Instead of following a line intersecting
opposite edges starting at $e$ in both directions, we now have to
follow a sheet going through all four parallel edges of a hexahedron.
It is easy to see that again there is a unique classification of all
edges into equivalence sets of parallel edges.
The second step was to assign a consistent orientation to the edges in a set
of parallel edges. This was possible for $d=2$ since the line that connects
all these edges is always orientable. However, that is not the case for the
sheet connecting the edges in $d=3$: It may not be orientable, and in this
case we will not be able to find a consistent orientation for the edges in this
equivalence class because we can no longer choose their directions to
be from ``one side'' to the ``other side'' of the sheet. The following
two sections show examples of meshes whose edges are not orientable according
to our conventions.
\subsubsection*{A first counterexample}
\begin{figure}[tbp]
\begin{center}
\includegraphics[width=.65\textwidth]{graphics/torus_0}
\\[6pt]
\phantom{.}
\hfill
\includegraphics[width=.4\textwidth]{graphics_raw/downsampled-downsampled-NonTwistedMesh}
\hfill
\includegraphics[width=.4\textwidth]{graphics_raw/downsampled-downsampled-TwistedMesh}
\hfill
\phantom{.}
\\[6pt]
\phantom{.}
\hfill
\includegraphics[width=0.4\textwidth]{graphics_raw/downsampled-downsampled-NonTwistedSurfaces}
\hfill
\includegraphics[width=0.4\textwidth]{graphics_raw/downsampled-downsampled-moebius04}
\hfill
\phantom{.}
\caption{\it First counterexample for $d=3$. Taking a string of cells
(top) and bending them into a torus yields a graph for which a
consistent orientation of edges exists (center left). However, twisting
the cells by $180^\circ$ before re-attaching ends fails to yield such a
graph (center right): The sheets connecting the radial and axial
parallel edges form two intersecting,
non-orientable M\"obius strips (bottom).}
\label{fig:torus}
\end{center}
\end{figure}
A first non-orientable example is shown in Fig.~\ref{fig:torus},
demonstrating a subdivision of a toroidal domain for which no
consistent edge
orientation exists. If we take a string of cells (top) and bend it into a
torus (bottom left), then all edges can be grouped into one radial, one axial,
and $|{\tria}|$ tangential classes (the surfaces connecting the edges of
the first two classes are shown in the bottom row of Fig.~\ref{fig:torus}).
One possible consistent orientation would be radially inward and axially into
positive $z$-direction. The tangential edges can be oriented arbitrarily for
each cell separately. However, such a consistent choice of direction for edges
no longer exists if we twist the string of cells by $180^\circ$ before
attaching ends (bottom right). In that case, there must be at least one cell
with radial edges that both point inward and outward, in violation of our
convention. The same holds for a twist of $90^\circ$, in which case the sheet
connecting parallel edges has to circle the torus twice, before meeting itself
in the wrong orientation again. The sheet that passes through parallel edges
is, in these cases, a M\"obius strip with either a twist of $180^\circ$ or
$90^\circ$, and it is well known that this surface is not orientable.
\subsubsection*{A second counterexample}
Conjecturing from the first example that triangulations into hexahedra with no
consistent orientation must have a hole or be multiply connected, is wrong,
though. Fig.~\ref{fig:rainers-example} shows an example of 14 hexahedra
subdividing a simply connected domain without holes for which no consistent
orientation exists. The bottom row of the figure shows the top face of the seven
lower hexahedra (left) and the bottom face of the upper seven (right). Only
faces $A$ and $B$ match and are connected.
\begin{figure}[tbp]
\begin{center}
\phantom{.}
\hfill
\includegraphics[width=.44\textwidth]{graphics_raw/downsampled-downsampled-RainerMeshNotHollow}
\hfill
\includegraphics[width=.44\textwidth]{graphics_raw/downsampled-downsampled-RainerSurface}
\hfill
\phantom{.}
\\
\phantom{.}
\hfill
\includegraphics[height=.17\textwidth]{graphics/counterexample_2}
\hfill
\includegraphics[height=.17\textwidth]{graphics/counterexample_1}
\hfill
\phantom{.}
\caption{\it Second counterexample for $d=3$ with a simply connected
domain. Top row: Hexahedralization and one of the sheets passing through
parallel edges. Bottom row: Top face of lower half of domain and
bottom face of upper part of domain, both showing consistent directions of
edges if one oriented the edge $(v_4,v_3)$ as shown. This leads
to conflicting directions for $(v_4,v_1)$, and the same happens
if one had oriented $(v_4,v_3)$ in the opposite direction. Note that the
two parts shown are connected
only on faces $A$ and $B$.}
\label{fig:rainers-example}
\end{center}
\end{figure}
Let us consider the orientation of the radial edges of the lower left
picture: starting, for example, at edge $e$, then all radial edges
must either point inward or outward due to the opposite edge
rule. Also, the direction of the other two edges of face $A$ is then
fixed. One of the two possibilities for directions of these edges is shown in
the bottom left part of Fig.~\ref{fig:rainers-example}. Independent of these
two possible choices, we have the \textit{invariant} that $v_2$ and
$v_4$ are vertices to which both lines in $A$ either converge, or from
which they emanate, and $v_1$ and $v_3$ are vertices into which one
line enters and from which one line emerges.
Now let us consider the underside of the upper seven hexahedra, given by
the lower right figure. It is connected to the lower part along faces
$A$ and $B$. By the same argument, starting for example at
edge $e'$, the directions of the radial edges and the other two edges
of face $A$ are fixed. However, this time vertices $v_1$ and $v_3$
are the vertices from which both adjacent edges in $A$ emerge and into
which they vanish, and $v_2$ and $v_4$ are the ones through which
they pass, i.e.~the character of the vertices is changed. Since in the
joint domain vertices and edges of the upper- and undersides are
identified, this creates a conflict: whatever direction we choose for
edges $e$ and $e'$, no consistent edge orientation of the face $A$
is possible.
This is easily understood by looking at the sheet that connects the
set of parallel edges, $\Pi(e)=\Pi(e')$,
shown in the upper right part of the figure. It is a split,
self-intersecting sheet that is not orientable.
\subsection{Algorithm complexity}
The examples in the previous subsection show that not all 3d meshes
allow for a consistent edge orientation. In a sense, the complexity of
our algorithm may therefore be a moot point: codes do necessarily have
to store explicit edge orientation flags for each cell because the
orientation of edges can no longer be inferred from the orientation of
the cell.
On the other hand, it may still be of interest to orient edges
consistently for those meshes for which this is possible, for example
to ensure that the code paths for default edge orientation are always
chosen. We may also be interested in seeing whether our algorithm can
at least detect in optimal complexity whether a mesh is orientable.
To investigate these questions, we note that
Algorithms~\ref{alg:parallel-edges} and \ref{alg:all-parallel-edges}
separating edges into the set $\pi$ of equivalence classes continue to
work. Algorithm~\ref{alg:parallel-edges} determines the overall
complexity. In 3d, we need to note that in step (2)(b)(i), ${\cal
N}_K(\delta)$, the set of edges parallel to $\delta$ in cell $K$,
now has cardinality 3. Furthermore, the loop in step (2)(b) now
iterates over all cell neighbors $K(\delta)$, of which in 3d there may
in fact be up to $|{\tria}|$. Consequently, we can no longer bound
$|{\cal E}(\delta)|$ by a constant independent of the number of edges
or cells, and this then applies to the cost of the entire step
(2)(b). This would not be a problem if we added at least a fixed
fraction of the elements of ${\cal E}(\delta)$ to $\Delta_k$ (i.e., if
we could guarantee that $\frac{|{\cal E}(\delta)\backslash
\Pi_k(e)|}{|{\cal E}(\delta)|} \ge c >0$), but we are not
aware of any theoretical argument that this would be so.
Consequently, it is conceivable that there exist sequences of meshes
for which $|{\cal E}(\delta)|={\cal
O}(|E|)$ but for which step (2)(c) adds only a fixed number of
elements to $\Pi_k$ (or at least a number that grows less
quickly than $|E|$). This would destroy the linear complexity
of the overall algorithm.
From a practical perspective, however, this question is not terribly
interesting as it requires meshes in which the number of cells
adjacent to individual edges becomes very large. While the optimal
number of cells adjacent to each edge would be four (for example in a
cubic lattice), practical meshes generated by mesh generators rarely
have more than 8 or 10 such adjacent cells per edge, and this number
is independent of the number of cells. Consequently, for
such meshes, $|{\cal E}(\delta)|={\cal O}(1)$ and the algorithm is
then again guaranteed to run with optimal complexity. We believe that
it would also be possible to reformulate the algorithms in ways that
allow for optimal complexity even for these pathological cases, though
this is of no interest in the current paper.
Finally, we note that when a mesh is not orientable, step (5)(a)(ii) of
Algorithm~\ref{alg:orient-edges} will eventually try to assign a direction to
an edge that already has an orientation that is inconsistent with the one that
we want to assign to it (a case that cannot happen in 2d). Thus, the algorithm
will be able to detect non-orientable meshes with the same run time complexity
as it can orient meshes.
\section{Meshes in $d=3$ that are always orientable}
\label{sec:3d-always-orientable}
The fact that not all 3d meshes can be consistently oriented of course does
not rule out that there may be important subclasses of 3d meshes that can in
fact be oriented. Indeed, we will show these statements in the
following subsections:
\begin{enumerate}
\item Refining a non-orientable mesh uniformly yields a mesh that is
consistently orientable.
\item Three-dimensional meshes generated by extruding a
two-dimensional mesh into a third direction are always orientable.
\item Hexahedral meshes that result from subdividing each tetrahedron of a
tetrahedral mesh into hexahedra are always orientable.
\end{enumerate}
We will discuss these three cases in the following.
\subsection{Meshes originating from refinement of non-orientable meshes}
If one is given a three-dimensional mesh with a sheet of parallel edges that
are not consistently orientable, then it turns out that we can generate
an orientable mesh by subdividing all the cells along this sheet. To
understand this intuitively, remember that a
non-orientable surface somehow connects to itself ``with a twist of
$180^\circ$''. If we refine all the cells along its way, we duplicate this
sheet and replace it by one that ``goes around twice before connecting to
itself again''. The twist is thus $360^\circ$ and the resulting sheet is
orientable. This is illustrated in Fig.~\ref{fig:doubling} for the two
examples discussed in Section~\ref{sec:3d-always}.
\begin{figure}[tbp]
\begin{center}
\phantom{.}
\hfill
\includegraphics[width=.4\textwidth]{graphics_raw/downsampled-downsampled-SplitMoebius}
\hfill
\includegraphics[width=.4\textwidth]{graphics_raw/downsampled-downsampled-TwistedRefinedMesh}\\
\hfill
\phantom{.}
\\
\includegraphics[width=.4\textwidth]{graphics_raw/downsampled-downsampled-SplitRainer}
\caption{\it Illustration that splitting a non-orientable sheet by
subdividing each edge along the sheet into two, yields a single
orientable sheet. Top left: The sheet that results from subdividing
one of the two M{\"o}bius strips in Fig.~\ref{fig:torus}. Top right: The mesh one obtains by refining all edges along both of
the non-orientable sheets in Fig.~\ref{fig:torus}. This mesh is
orientable. Bottom: The
same for the example shown in Fig.~\ref{fig:rainers-example}.}
\label{fig:doubling}
\end{center}
\end{figure}
To make this more formal, let us consider an equivalence class
$\Pi_{G_\tria}(e)$ of parallel edges which is not consistently orientable.
Here, we use the index to indicate that this is a set of edges in the graph
$G_\tria$. Now split each cell that the sheet associated with $\Pi_{G_\tria}(e)$
intersects, into 2, 4, or 8 new cells (depending on whether the sheet
intersects the cell once, twice, or three times, i.e., whether one, two, or
all three of the sets of parallel edges within this cell are part of
$\Pi_{G_\tria}(e)$). We subdivide in the
directions orthogonal to the sheet, see
Figure~\ref{fig:refinement-cases}. This removes the affected edges
from the graph, and replaces them by 8, 24, or 54 new ``child'' edges. With this
refined mesh is associated a new graph $G'$. Then the following results hold:
\begin{figure}[tbp]
\begin{center}
\phantom{.}
\hfill
\includegraphics[width=.24\textwidth]{graphics_raw/downsampled-downsampled-cell_div0_new}
\hfill
\includegraphics[width=.24\textwidth]{graphics_raw/downsampled-downsampled-cell_div1_new}
\hfill
\includegraphics[width=.24\textwidth]{graphics_raw/downsampled-downsampled-cell_div2_new}
\hfill
\includegraphics[width=.24\textwidth]{graphics_raw/downsampled-downsampled-cell_div3_new}
\hfill
\phantom{.}
\caption{\it Illustration of splitting a cell (left) along the purple
connecting sheet (left center), as well as along the purple
sheet in cases where it runs through the same cell twice (center right) or
three times (right).}
\label{fig:refinement-cases}
\end{center}
\end{figure}
\begin{lemma}
\label{lemma:refinement}
Let $\Pi_{G_\tria}(e)$ be a set of parallel edges that are not consistently
orientable, and $\Pi_{G'}(e')$, $\Pi_{G'}(e'')$ the sets of parallel edges
in the refined graph $G'$ associated with the ``children'' $e',e''$ of
$e$. Then $\Pi_{G'}(e')=\Pi_{G'}(e'')=:\Pi$. Furthermore, $\Pi$ is orientable.
\end{lemma}
\begin{proof}
By refining each cell with edges in $\Pi_{G_\tria}(e)$ along the non-orientable
sheet, we replace each edge $\varepsilon_i\in\Pi_{G_\tria}(e)$ by two edges
$\varepsilon_i',\varepsilon_i''$. Let us select an arbitrarily chosen
element from $\Pi_{G_\tria}(e)$, and denote it for simplicity by the symbol $e$. Its
children, after refinement, $e',e''$ are edges in the refined graph $G'$,
and their respective sets of parallel edges are
$\Pi_{G'}(e'),\Pi_{G'}(e'')$. It is easy to see that each of the two
children $\varepsilon',\varepsilon''$ of each edge $\varepsilon\in\Pi_{G_\tria}(e)$
must be in either $\Pi_{G'}(e')$ or $\Pi_{G'}(e'')$.
We first show that these sets are equal. Assume that
$\Pi_{G'}(e')\neq\Pi_{G'}(e'')$. Then we could find a unique direction for
each edge $\varepsilon\in\Pi_{G_\tria}(e)$ by assigning it the direction
``from the $\Pi_{G'}(e')$ side to the $\Pi_{G'}(e'')$ side.'' This, however,
would be a consistent orientation of the edges in $\Pi_{G_\tria}(e)$, in
contradiction to the assumption that this set is non-orientable. Thus,
$\Pi_{G'}(e')=\Pi_{G'}(e'')$. Intuitively, this means that splitting the
single sheet associated with $\Pi_{G_\tria}(e)$ still yields a single sheet.
The second step is to show that the single set
$\Pi=\Pi_{G'}(e')=\Pi_{G'}(e'')$ is orientable. We do this purely locally:
choose for the two children $\varepsilon',\varepsilon''$ of an edge
$\varepsilon\in\Pi_{G_\tria}(e)$ the direction away from their common node
(``centrifugal direction'': the direction vectors of the child edges are
``rooted'' in the original sheet through $\Pi_{G_\tria}(e)$). This orientation of the children of
$\Pi_{G_\tria}(e)$ is consistent with our convention for every one of the affected child cells,
and, consequently, for all cells which the sheet for $\Pi_{G_\tria}(e)$
intersects. Intuitively, this can be interpreted as follows: while
the sheet has no associated normal direction, the direction ``away from the
sheet'' exists on both sides.
\end{proof}
As can be seen in Fig.~\ref{fig:refinement-cases}, refinement of a cell along one sheet
also adds new edges to other sets of parallel edges. Depending on whether the
sheet intersects a cell once, twice, or three times, refinement adds 4, 5, or
no new edges to other parallel sets. However, we can show that refining the
cells along one non-orientable sheet does not render a previously orientable,
other sheet non-orientable:
\begin{lemma}
\label{lemma:other-sets}
Let $\Pi_{G_\tria}(e)$ be a set of parallel edges that are refined to make it
orientable. Let $\Pi_{G_\tria}(e')$ be a set of parallel edges to which edges are
added by this refinement step. Then, if $\Pi_{G_\tria}(e')$ was orientable before
the refinement step, then it is also after the refinement step.
\end{lemma}
\begin{proof}
If the edges in $\Pi_{G_\tria}(e')$ were already orientable in $G_\tria$, then they must
have parallel directions in the cell shown in Fig.~\ref{fig:refinement-cases}. If
we assign the same, parallel direction to the newly added edges in $G'$ that
belong to this set of parallel edges in $G'$, then the resulting directions
are consistent in the child cells as well. Because the remainder of the
cells intersected by $\Pi_{G_\tria}(e')$ are not affected, the locally consistent
edge orientations in the child cells implies that $\Pi_{G'}(e')$ remains
orientable.
On the other hand, if $\Pi_{G_\tria}(e')$ was not orientable, then this fact is
unaltered by the addition of new edges.
\end{proof}
With these lemmas, we can state the final result of this section:
\begin{theorem}
Let $G_\tria$ be the graph associated with a subdivision of a domain into
hexahedra. If it is not orientable, then we can generate a new graph $G'$ by
refining all cells exactly once along the sheets associated with those sets
of parallel edges that are not orientable.
\end{theorem}
\begin{proof}
The theorem follows from the previous two lemmas. As was shown in
Lemma~\ref{lemma:refinement}, we can convert a non-orientable set of
parallel edges into an orientable one. In Lemma~\ref{lemma:other-sets}, we
showed that this does not create new non-orientable sets. Since the original
graph can only have finitely many non-orientable sets of parallel edges, we decrease this
number by one in each refinement step and thus obtain an orientable graph in
a finite number of steps.
\end{proof}
Thus, even though there are three-dimensional meshes that cannot be
oriented according to our convention, above theorem shows that there is a
simple and inexpensive remedy for these cases. While the meshes resulting from
such anisotropic refinement have a worse aspect ratio, one can
simply refine \textit{all} hexahedra uniformly into eight
children to obtain an orientable mesh with the same aspect ratio of cells as
the original one. This corresponds to refining the cells intersected by \textit{all}
sheets associated with sets of parallel edges, not only those corresponding to
non-orientable ones. Obviously, this mesh is also orientable: in addition to
the originally non-orientable parallel sets, we now also refine the originally
orientable parallel sets, which yields two distinct sets of parallel edges
$\Pi_{G'}(e')$ and $\Pi_{G'}(e'')$ that can independently be oriented, one on
``this'' side of the original sheet and one on ``that'' side of it (these
sides are distinct because the sheet associated with $\Pi_{G_\tria}(e)$ was
orientable).
\begin{remark}
The observations of this section also point out an important
optimization in practice for codes that create finer
meshes by subdividing the cells of an existing mesh. In such cases, it is not
necessary to re-run the mesh orientation algorithm on the finer mesh
with four (2d) or eight (3d) times as many cells. Rather, if the
original mesh was already consistently oriented, then the new mesh
will be consistently oriented by simply choosing the directions of
the refined edges to be the same as those of their parent edge.
\end{remark}
\subsection{Extruded meshes}
\label{sec:3d-extrude}
An important class of meshes consists of those that start with a
two-dimensional quadrilateral mesh and then ``extrude'' it into a third direction by
replicating it one or more times and connecting the vertices of the
original mesh and its replicas in this third direction. Indeed, the
mesh shown at the top of Fig.~\ref{fig:torus} is such a mesh: the
sequence of quadrilaterals at the bottom has been replicated to the
top, and each pair of original and replicated vertices are connected
by a new edge.
Extruded meshes are often used for ``thin'' domains. The
technique is obviously also applicable if the original two-dimensional
mesh lived on an (orientable) manifold such as the bottom surface of
the object we want to mesh. The extrusion also need not necessarily be
in a perpendicular direction, nor do the replicas have to be parallel
to the original mesh. That said, for the purposes of orienting edges,
such geometric considerations are immaterial.
For extruded meshes, we can state the following result:
\begin{theorem}
\label{theorem:extruded-always}
Let $\{K\}$ be a hexahedral mesh obtained by extruding a
quadrilateral mesh defined on a two-dimensional, orientable manifold
in a third direction. Then the
edges of this mesh are consistently orientable.
\end{theorem}
\begin{proof}
To understand why this is the case, recall that \textit{all}
two-dimensional quadrilateral meshes on such manifolds are orientable. In other words,
in the original mesh, opposite edges can already be chosen to be
parallel, and this is also the case for its replicas.
Next, consider the sets of parallel edges we may consider. Obviously, the edges
of the original mesh and its replicas are parallel, and we can
consistently orient them if we choose edge directions of the
original mesh and its replicas the same. An alternative viewpoint is
that the sheets that connects these sets of parallel edges are the
``extruded'' version of the line that connected these parallel edges
in the two-dimensional mesh, and the orientability of the
one-dimensional line then extends to the orientability of the
corresponding sheet to which it gives rise.
The only additional sets of parallel edges are the ones that
connect the original mesh and its first replica, and then each
replica with the next. The corresponding sheets can be thought of as
copies of the two-dimensional domain spanned by the original mesh,
located halfway between the replicas. Each of these sheets is
obviously orientable: The edges of each of these new sets are
independent of each other, and each class can be consistently
oriented, for example by always choosing the direction from the
original to the first replica, and from each replica to the next
(i.e., an ``upward'' direction).
\end{proof}
\subsection{Meshes originating from tetrahedralizations}
The examples in Section~\ref{sec:3d-always} show that there is no
\textit{topological} characterization of those
domains for which we can always find a consistent orientation of
edges. Rather, it is a question of \textit{how} that domain
was subdivided into cells. Indeed, we can show that for the very
general class of domains that can be subdivided into
tetrahedra, there also exists a subdivision into hexahedra with a consistent
orientation of edges:
\begin{theorem}
\label{theorem:tet-to-hex}
Let $\{T\}$ be a given subdivision of a domain $\Omega$ into a set of
tetrahedra so that two distinct tetrahedra either have no intersection,
share a common vertex, a complete edge, or a complete face. Divide each
tetrahedron $T$ into four hexahedra, $K(T)_0,\ldots, K(T)_3$, by using the
vertices, edge midpoints, face midpoints, and cell midpoint of $T$ as
vertices for the hexahedra. Then the edges of the mesh consisting of the
union of these hexahedra, $\bigcup_{T, 0\le i< 4} K_i(T)$, are consistently
orientable.
\end{theorem}
Before we prove this claim, we note that the resulting hexahedra are
in almost all cases not close to the optimal cube shape. Also, almost
all vertices in the resulting mesh have a
number of cells meeting at this vertex that deviates from the optimal
number of eight (encountered, for example, in a regular subdivision into
cubes). These meshes are therefore hardly optimal for finite element
computations. However, given the complexity of generating hexahedral
meshes, until recently many mesh generators used this approach to
generate an initial hexahedral mesh. (For example, the widely used
open source mesh generator gmsh \cite{gmsh} can use this
approach to generate hexahedral meshes.)
\begin{figure}[tbp]
\begin{center}
\phantom{.}
\hfill
\includegraphics[width=.22\textwidth]{graphics/tri_to_quad_1}
\hfill
\includegraphics[width=.22\textwidth]{graphics/tri_to_quad_2}
\hfill
\phantom{.}
\\[12pt]
\includegraphics[width=.22\textwidth]{graphics/tri_to_quad_3}
\hfill
\includegraphics[width=.22\textwidth]{graphics/tri_to_quad_4}
\hfill
\includegraphics[width=.22\textwidth]{graphics/tri_to_quad_5}
\caption{\it Construction in $d=2$ for the proof of
Theorem~\ref{theorem:tet-to-hex}. Top: Part of a subdivision into
triangles before and after cutting each triangle into three
quadrilaterals (top row). Bottom: Three sets of parallel edges and their
connecting line (bottom row).}
\label{fig:theorem-2-2d}
\end{center}
\end{figure}
\begin{proof}[of Theorem \ref{theorem:tet-to-hex}]
In order to explain the proof in $d=3$, it is instructive to first
consider a similar case in $d=2$, where we would cut all cells of a triangular mesh
into three quadrilaterals, see
Fig.~\ref{fig:theorem-2-2d}. From the edges of these
quadrilaterals, we then generate independent sets of parallel
edges; the bottom row of the figure shows three of these sets
along with their connecting line.
Importantly,
each of these lines forms a closed, non-self-intersecting loop
around one of the original vertices of the triangles, cutting
through all the cells in the second layer of quadrilaterals around
this vertex (unless, of course, the vertex is at the boundary of the
domain, in which case the line is not closed but starts and ends at
the boundary). Indeed, the edges of each quadrilateral are
part of the loops for two vertices of the triangle
from which it arose. Each
vertex thus gives rise to at least one set of parallel
edges in the subdivision into quadrilaterals. These can, of course, all be
oriented in $d=2$. (It may give rise to more than one such set if
two parts of the domain touch at a vertex, i.e., if there are two
sets of cells adjacent to a vertex that are not mutual face neighbors.)
It is easy to generalize these considerations to the case $d=3$ (though we
have not found good ways to visualize hexahedral meshes resulting from
even a small collection of tetrahedra forming an unstructured mesh).
First, we note that each original interior vertex
now induces a set of parallel edges that is connected by a closed
non-self-intersecting sheet around the vertex. Since it is homeomorphic to the
surface of the unit sphere, it is of course orientable, and so are the edges
of this parallel set. For original vertices on the boundary, the sheet is not
closed, but it is obviously still orientable.
\end{proof}
\begin{includefaces}
\section{Extension to face orientations for 3d meshes}
\label{sec:3d-faces}
In 3d, one not only has the problem of aligning the coordinate system of
one-dimensional edges with that of cells, but a similar problem also appears
with the coordinate system of two-dimensional faces to which one would like to
associate data (such as degrees of freedom or their numeric values). This then
raises similar questions as in the edge case: can we assign an orientation --
in the simplest case a normal vector -- to each face that can uniquely be determined
from each of the adjacent cells simply from its position within this cell?
The answer to this first question is an unconditional ``yes'', by a similar argument
as the one we used for edge directions in 2d. Here, we first have to collect
the set of parallel \textit{faces}, denoted by $\Pi(f)$, of a
particular face $f$. These faces are constructed by always hopping from one
face to the one opposite of it in a cell, so we can draw a line through all of
them. The second step is to give all of them a certain direction. In 2d, we
used that each line is orientable, so we could use the direction pointing from
one side of the line to the other as the direction for the edges it
intersects. Here, we simply use the ``forward'' (or ``backward'') direction as
we move along the line connecting the faces in $\Pi(f)$, to provide
each face with a normal direction that matches that of all parallel faces
within its two adjacent cells. By construction, this line cannot split, so
even if it is a closed or self-intersecting line, we can always define such a
direction; thus there is a consistent orientation of all faces in
$\Pi(f)$ for all subdivisions of domains into hexahedra, and an
algorithm is easily derived that does this in a time linear in the number of
faces or cells.
In reality, however, the orientation of the face's coordinate system is not
only described by a single bit such as whether its normal vector points into
or out of a cell. Rather, the coordinate system of the face may also be
rotated by $0^\circ, 90^\circ, 180^\circ$, or $270^\circ$ against the
coordinate system of the cell when restricted to that face. An alternative way
of describing these four rotations is to state the vertex at which the
coordinate system originates. Defining properties such as these does
not naturally fit into the graph theoretical context we have so far
used, and we will therefore use some concepts of discrete geometry in
the following.
To concisely define what properties we would like to have in the
finite element context, let us introduce two
definitions that assume that a mesh is given by a collection of
vertices $V$, linear edges $E$, quadrilateral faces $F$, and
hexahedral cells ${\tria}$ (where obviously each of faces bound one
or two cells, and each edge bounds at least one face). We then want to
associate with each cell $K\in {\tria}$ a right handed coordinate
system and denote $K^+$, and similarly for faces and edges which we
will then denote $f^+$ and $e^+$ for $f\in F, e\in E$. For each edge,
there are two possible ways of defining $e^+$, namely the two edge
orientations. For each face $f$, the previous paragraphs have shown that
there are 8 possibilities to define $f^+$. One can easily verify that
for each cell $K$, there are 24 possible $K^+$. One can then define
consistent orientations as follows:
\begin{convention}
An oriented cell $K^+$ and its bounding faces $f^+(K)$ and bounding
edges $e^+(K)$ are called \textit{consistently edge and face oriented} if the
coordinate systems defined on the edges and faces bounding $K$ are
simply the restrictions of the cell's coordinate system to the edge
or face.
\end{convention}
\begin{figure}[tbp]
\begin{center}
\phantom{.}
\hfill
\includegraphics[width=.3\textwidth]{graphics/convention_3d_faces}
\hfill
\hfill
\includegraphics[width=.3\textwidth]{graphics/convention_3d_faces_misaligned_1}
\hfill
\includegraphics[width=.3\textwidth]{graphics/convention_3d_faces_misaligned_2}
\hfill
\phantom{.}
\caption{\it Left: Illustration of consistently face oriented cell
in 3d. For each face, a filled arrow indicates the local $x$-axis
and an open arrow the local $y$ axis. The coordinate system of the
cell is shown in red to the left, with the local $z$ axis using an
open box as arrow. The cell would also be
consistently edge oriented if edges were oriented as shown in the
right panel of Fig.~\ref{fig:convention}.
Center: A cell that could still be consistently edge oriented but
is not consistently face oriented because the right face has an
inverted coordinate system.
Right: A cell where the right face's coordinate system has its
origin at the wrong location. For this cell, edge orientations can
not be chosen in such a way that they would be consistent with
both the top and right face, for example.}
\label{fig:3d-face-edge-orientations}
\end{center}
\end{figure}
An example of a consistently oriented cell, and two cells that are
not consistently oriented are shown in
Fig.~\ref{fig:3d-face-edge-orientations}. Note that for a consistently
edge and face oriented cell, the coordinate systems defined on the
edges are naturally not only the restrictions of the cell coordinate
systems to this edge, but also the restrictions of the face coordinate
systems to this edge.
This convention is a natural
generalization of the two-dimensional edge orientations we have used
in Section~\ref{sec:2d}. There, we only considered the coordinate
systems on the edge (described by the edge orientation) and did not
mention a coordinate system on cells, but it is easy to see that we
can choose a unique right-handed coordinate system originating at the
one vertex with two outgoing edges, as long as opposite edges are
parallel. Given this local consistency, we can define globally
consistently oriented meshes:
\begin{convention}
A mesh with associated coordinate systems,
$(V,E^+,F^+,{\tria}^+)$, is called \textit{consistently edge and
face oriented} if each cell and its bounding faces and edges are
consistently oriented.
\end{convention}
Clearly, this is a stronger condition than just the edge orientations
considered in Section~\ref{sec:3d} and every mesh that is consistently
edge and face oriented is also consistently edge oriented following
Definition~\ref{conv:2} (as well as consistently face oriented, using
the definition earlier in this section). It is therefore clear that at least those
meshes that can not be consistently edge oriented can also not be
consistently edge and face oriented. On the other hand, we have shown
in Section~\ref{sec:3d-always-orientable} that important classes of
meshes are always edge orientable, and one may hope
that these are in fact also face orientable.
Unfortunately, this turns out to be not true, even for the otherwise
relatively well-behaved class of extruded meshes for which we showed in
Section~\ref{sec:3d-extrude} that their edges can always be oriented.
This can be shown for a simple counter-example involving only three
cells and that is shown in Fig.~\ref{fig:3d-face-edge-counter}.
\textbf{ACTUALLY, THIS APPEARS TO REQUIRE MORE THOUGHT. I'M NO LONGER
CLEAR ON WHAT IS INTENDED OR IMPLEMENTED, AND THEY MAY BE DIFFERENT.}
\begin{figure}[tbp]
\begin{center}
\phantom{.}
\hfill
\includegraphics[width=.3\textwidth]{graphics/convention_3d_faces_counter}
\hfill
\phantom{.}
\caption{\it xxx.}
\label{fig:3d-face-edge-counter}
\end{center}
\end{figure}
This example shows that there are simple examples of meshes that
cannot be consistently edge and face oriented at the same time. It is
conceivable that there is a simple characterization for which meshes
consistent orientations can be found, in the same way as we showed in
Section~\ref{sec:3d} that meshes can be consistently oriented if no
set of parallel edges is connected by a non-orientable surface. We
have not attempted to find such a characterization here since the
counterexample showed that the subset of consistently edge and face
orientable meshes is too small for practical purposes.
\end{includefaces}
\section{Conclusions}
\label{sec:conclusions}
Finite element codes can be made significantly simpler if they can
make assumptions about the relative orientations of coordinate
systems defined on cells, edges, and faces. If such assumptions always hold,
then this reduces the number
of cases one has to implement and, consequently, the potential for
bugs. In this paper, we have described a way to orient edges and the cells
they bound, and shown that not only
is it possible to choose edge directions consistently with regard to
this convention for two-dimensional quadrilateral meshes, but also
that there is an efficient algorithm to find such edge orientations.
On the other hand, it is not always possible to orient edges of
three-dimensional hexahedral meshes according to the three dimensional
generalization of this convention. The obvious generalization of
our two-dimensional algorithm is able to detect these cases, again in
optimal complexity, but the result implies that codes dealing
with hexahedral meshes necessarily have to store flags for each of the
edges of each cell that indicate the orientation of that edge relative
to the coordinate system of the cell. This is not a significant
overhead in terms of memory and possibly not in terms of algorithmic
complexity. Nevertheless, in actual practice, this has turned out to be an endless
source of frustration and bugs in \textsc{deal.II}{} as the cases where edge
orientations are relevant are restricted to the use of higher order
elements, as well as complex and three-dimensional geometries. In case of bugs,
methods generally converge but at suboptimal orders. Consequently,
debugging such cases and detecting where in the interplay of geometry,
mappings, degrees of freedoms, shape functions, and quadrature the bug
resides has proven to be a very significant challenge. This experience also
supports our claim that being able to enforce a convention in the
two-dimensional case almost certainly saved a great deal of development time.
At the same time, this paper at least identifies broad classes of
three-dimensional meshes for which one can always consistently orient
edges, and for which no special treatment of edges is
necessary. Through counter-examples, we have shown that there is no
topological description for which domains do or do not allow for
consistent orientations, but that it is indeed a property of how the
domain is subdivided into cells, and our analysis demonstrates ways by which
meshes can be constructed in ways so that edges can always be oriented
consistently. This analysis can therefore also provide constructive feedback
for the design of mesh generation algorithms.
\begin{includefaces}
In an ideal world, finite element codes would use meshes for which not
only edges but also faces are consistently orientable. As shown in
Section~\ref{sec:3d-faces}, this turns out to be a much more
restrictive condition, and a simple counterexample demonstrates that
we cannot hope for such a property for meshes that arise in realistic
situations.
\end{includefaces}
\paragraph*{Acknowledgments} WB would like to thank J.~M.~Landsberg,
J.-L.~Guermond, and A.~Ern for illuminating discussions.
| {
"timestamp": "2016-10-20T02:01:08",
"yymm": "1512",
"arxiv_id": "1512.02137",
"language": "en",
"url": "https://arxiv.org/abs/1512.02137",
"abstract": "Finite element codes typically use data structures that represent unstructured meshes as collections of cells, faces, and edges, each of which require associated coordinate systems. One then needs to store how the coordinate system of each edge relates to that of neighboring cells. On the other hand, we can simplify data structures and algorithms if we can a priori orient coordinate systems in such a way that the coordinate systems on the edges follows uniquely from those on the cells \\textit{by rule}.Such rules require that \\textit{every} unstructured mesh allows assigning directions to edges that satisfy the convention in adjacent cells. We show that the convention chosen for unstructured quadrilateral meshes in the \\texttt{deal.II} library always allows to orient meshes. It can therefore be used to make codes simpler, faster, and less bug prone. We present an algorithm that orients meshes in $O(N)$ operations. We then show that consistent orientations are not always possible for 3d hexahedral meshes. Thus, cells generally need to store the direction of adjacent edges, but our approach also allows the characterization of cases where this is not necessary. The 3d extension of our algorithm either orients edges consistently, or aborts, both within $O(N)$ steps.",
"subjects": "Numerical Analysis (math.NA)",
"title": "On orienting edges of unstructured two- and three-dimensional meshes",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.972830769252026,
"lm_q2_score": 0.7279754430043072,
"lm_q1q2_score": 0.7081969102144646
} |
https://arxiv.org/abs/1909.12107 | Critical properties of the Ising model in hyperbolic space | The Ising model exhibits qualitatively different properties in hyperbolic space in comparison to its flat space counterpart. Due to the negative curvature, a finite fraction of the total number of spins reside at the boundary of a volume in hyperbolic space. As a result, boundary conditions play an important role even when taking the thermodynamic limit. We investigate the bulk thermodynamic properties of the Ising model in two and three dimensional hyperbolic spaces using Monte Carlo and high and low-temperature series expansion techniques. To extract the true bulk properties of the model in the Monte Carlo computations, we consider the Ising model in hyperbolic space with periodic boundary conditions. We compute the critical exponents and critical temperatures for different tilings of the hyperbolic plane and show that the results are of mean-field nature. We compare our results to the 'asymptotic' limit of tilings of the hyperbolic plane: the Bethe lattice, explaining the relationship between the critical properties of the Ising model on Bethe and hyperbolic lattices. Finally, we analyze the Ising model on three dimensional hyperbolic space using Monte Carlo and high-temperature series expansion. In contrast to recent field theory calculations, which predict a non-mean-field fixed point for the ferromagnetic-paramagnetic phase-transition of the Ising model on three-dimensional hyperbolic space, our computations reveal a mean-field behavior. | \section{Introduction}
The critical properties of a statistical mechanics model on curved space can be drastically different from its flat-space counterpart~\cite{Callan1990}. In particular, statistical mechanics on negatively curved hyperbolic space has attracted much attention. First, they are relevant for quantum field theories in curved space-time~\cite{Wald1994} and serve as means to separate the roles of geometric curvature and topological defects~\cite{Callan1990}. Second, they arise in condensed matter settings as toy models for amorphous solids or exotic liquid crystalline structures~\cite{Kleman1982, Rubinstein1983, Nelson2002}. Third, hyperbolic spaces arise in quantum information theory as natural candidates for encoding quantum information with toric codes. This is because a toric code on hyperbolic space encodes a larger number of logical qubits than a flat-space counterpart for the same number of physical qubits~\cite{Freedman2002, Breuckmann2017}.
An essential consequence of negative curvature in a hyperbolic space is the exponential growth of the volume with an associated linear dimension. This leads to the expectation that the space is effectively infinite-dimensional and the critical properties of any statistical mechanical model will be effectively mean-field. At first sight, this might lead to the impression that the thermodynamical properties of the hyperbolic-space statistical model can be straightforwardly inferred from its flat-space counterpart. However, this is not true and the physics is qualitatively different due to the curvature of the embedding space. For instance, in contrast to the euclidean-space counterpart, the order-disorder phase-transition of the XY model on a 2D hyperbolic plane is driven by the fluctuations of topological defects, while the infrared properties of the spin-wave fluctuations are well-behaved~\cite{Callan1990}. Critical statistical mechanical systems show exponential decay of correlations~\cite{Mnasri2015} and new-phases with broken translational invariance can arise, which are absent in the flat-space counterparts. Mathematical proofs for the existence of such phases exist for models describing percolation~\cite{Benjamini1999, Benjamini2001, Baek2009} and ferromagnetic Ising models~\cite{Wu1996, Wu2000}.
For the ferromagnetic Ising model on a hyperbolic plane, high and low-temperature expansion had obtained mean-field exponents for susceptibility and magnetization~\cite{Rietman1992}. Interestingly, the transition temperature, obtained from the series expansion calculations for a self-dual lattice was different from that of the 2D square-lattice Ising model. Since the Kramers-Wannier duality~\cite{Krammers1941} also holds for self-dual hyperbolic lattices, assuming convergence of the series up to the critical point, this indicates the existence of a second phase-transition at a temperature related by the duality relation [see below, Eq. \eqref{eqn:KWduality}]. Indeed, the existence of this second phase-transition for a hyperbolic plane with free boundary condition was later proved~\cite{Wu1996, Wu2000,Jiang2018}. This phase-transition separates the low temperature, purely ferromagnetic phase from an intermediate phase. In this intermediate phase, the system has broken translational invariance, where there are infinitely many, infinitely large clusters of magnetized spins. These clusters survive in the thermodynamic limit purely due to the negative curvature of the embedding space and cannot arise in ordinary euclidean space~\cite{Aizenman1980}. Further increase of temperature eventually causes the system to transition from the intermediate phase to the high temperature, disordered phase. It is this latter phase-transition that was found using series expansion methods. Additional evidence of mean-field behavior of magnetization was shown with corner-transfer-matrix renormalization group calculations~\cite{Krcmar2008}. Monte Carlo calculations of bulk thermodynamics (throughout the paper, by bulk thermodynamics, we mean thermodynamics of a model on a vertex-transitive graph) have so far only been done on open boundary lattices. To eliminate the effects of the large fraction of total spins being on the boundary arising due to the negative curvature, outer layers of these open lattices are removed, focussing the analysis to the central region~\cite{Shima2005}. While the boundary effects can be interesting on their own~\cite{d_Auriac2001}, they prevent the analysis of bulk behavior of the hyperbolic plane Ising model.
In the first part of this work, we investigate the bulk critical properties of the Ising model on a hyperbolic plane. To remove boundary effects in the Monte Carlo simulations, we perform these simulations on a hyperbolic plane with periodic boundary conditions~\cite{Sausset2007}. We concentrate on the self-dual $\{5,5\}$ lattice, which is a tiling of the hyperbolic plane with regular pentagons (see Fig.~\ref{fig_ising}). We concentrate on this lattice since it captures all the qualitatively different physics of curved space. We compute the critical exponents and critical temperature as the system transitions to the ferromagnetic phase. A hyperbolic plane with periodic boundary conditions is a manifold with genus $\gg 1$. As a result, the finite size scaling is nontrivial and provides additional insight into the fundamental differences between statistical mechanical models defined on spaces with zero and negative curvature. We find that the universal critical exponents for the different thermodynamic quantities are close to the mean-field predictions. To lend additional credence to the above findings, we perform high and low temperature series expansion of various thermodynamic quantities. Performing this analysis, we find results that are close to our Monte Carlo predictions and confirm the mean-field nature of the phase-transition. Our results for the $\{5,5\}$ lattice are close to the results for the critical temperature recently obtained by Mone Carlo simulations of the same model~\cite{Jiang2018}.
Subsequently, we also compute the critical temperature and the critical exponents using Monte Carlo and series expansion for lattices with different curvatures and compare our results for the different hyperbolic lattices to the Bethe lattice. We explain the relation between ferromagnetic-paramagnetic phase-transitions of the Ising model on hyperbolic lattices and those on the Bethe lattice, the latter being the limit where the number of edges of the polygons tiling the hyperbolic lattice is taken to infinity. Although not the point of the paper, we mention that we provide the high temperature series for several lattices not analyzed before, going up to ${\cal O} (v^{24})$, where $v = \tanh(\beta J)$. Here, $\beta$ is inverse temperature and~$J$ is the ferromagnetic coupling.
The last part of this work is devoted to the analysis of the Ising model in 3D hyperbolic space. Recent field theory calculations~\cite{Doyon2004, Benedetti2015} using 1/N expansion predict deviations from mean-field exponents for the Ising model in 3D hyperbolic space~\cite{Mnasri2015}. This result is remarkable since the basic intuition for critical properties of a statistical mechanical model in hyperbolic space being mean-field -- that the hyperbolic space acts like an infinitely connected lattice -- is even more valid in 3D compared to 2D since the number of neighbors of each spin is higher. Previous calculations using series expansion or corner-transfer-matrix renormalization group have all been done for 2D hyperbolic Ising models. In this part of the work, we calculate the bulk critical properties of the model in 3D hyperbolic space with Monte Carlo and high-temperature series expansion technique. Our results, in contrast to the field theory calculations, reveal a mean-field nature of the transition.
We perform the Monte Carlo analysis of the Ising model on the $\{5,3,5\}$ lattice with periodic boundary condition, the latter being a tiling of 3D space with pentagonal dodecahedrons. Then, we compute the susceptibility exponents for the $\{5,3,4\}$ and $\{5,3,5\}$ lattices using high-temperature series expansion. We also consider a 3D analogue of the Bethe lattice which has Schl\"afli symbol $\{5,3,6\}$.
\begin{figure}
\includegraphics[width = 0.30\textwidth]{55.png}
\caption{\label{fig_ising} The self-dual $\{5,5\}$ tiling of the hyperbolic plane in the Poincar\'e disc model. Ising degrees of freedom reside on the nodes and any two spins connected by an edge interact. }
\end{figure}
The article is organized as follows. In Sec.~\ref{sec:model}, we analyze the Ising model on a 2D hyperbolic plane. After a brief summary of the well-known properties of the model, in Sec.~\ref{sec:monte_carlo}, we perform Monte Carlo analysis to compute the universal critical exponents and infer the phase-transition point for the $\{5,5\}$ lattice. Then, we compute with high and low-temperature series expansions the susceptibility and the magnetization in Sec.~\ref{sec:series_expansion}. We find the results of the series-expansion computations to be close to those obtained by Monte Carlo. In Sec. \ref{sec:variation_with_curvature}, we describe the variation of the critical properties with curvature for self-dual lattices and explain the relationship between the critical properties of the Ising model in hyperbolic lattices and in the Bethe lattice. In Sec.~\ref{3Dmc}, we analyze the properties of the Ising model in 3D hyperbolic space with $\{5,3,5\}$ tiling using Monte Carlo. The high-temperature expansion calculations for the $\{5,3,k\}$ lattices, $k=4,5,6$ are done in Sec.~\ref{3dtemp}. Finally, in Sec.~\ref{sec:conclusion}, we provide a concluding summary and outlook. In Appendix~\ref{hyperbolic_tiling}, we summarize the relevant properties of hyperbolic planes and describe how to tessellate them in the presence of periodic boundary condition. Appendix~\ref{ap:series_derivation} provides additional details on the high-temperature series expansion. Appendix~\ref{sec:kramers_wannier} provides a general derivation of the Kramers-Wannier duality.
\section{The Ising Model on a 2D Hyperbolic Plane}
\label{sec:model}
Consider a regular tiling of the hyperbolic plane, denoted by $\{r,s\}$. Here, $r$ denotes the number of sides of the polygon and $s$ the coordination number of each site (more details in Appendix~\ref{hyperbolic_tiling}).
Note that for $r=s$ the lattice is self-dual; the case $r=s=5$ is shown in Fig.~\ref{fig_ising}. The Hamiltonian for the ferromagnetic Ising model on the hyperbolic plane is defined by
\begin{align}
\label{ham_ising}
H = -J \sum_{\langle i, j\rangle} \sigma_i\sigma_j,
\end{align}
where the spins ($\sigma_i = \pm1$) reside on the vertices of the lattice and any two spins connected by an edge interact with each other with a coupling $J>0$.
For either free, fixed or periodic boundary conditions, for temperatures $T\gg J$, the system is in a disordered, paramagnetic phase. Upon lowering the temperature, the system undergoes a phase-transition at a critical temperature, denoted by $T_c$, to a magnetically ordered phase. For free boundary conditions, in contrast to the euclidean space Ising model, this phase has broken translational invariance and is characterized by {\it infinitely many} clusters of magnetized spins. The survival of these clusters in the thermodynamic limit is due to the negative curvature of the hyperbolic plane, which allows an infinite number of these clusters to be squeezed within the plane. For free boundary conditions, upon further lowering of the temperature, at a temperature $\bar{T}_c$, the system undergoes another phase-transition, where a {\it single} cluster of magnetized spins covers the entire hyperbolic plane and translational invariance is restored. The two temperatures, $T_c, \bar{T}_c$, are related to one another by the Kramers-Wanner duality relation:
\begin{align}\label{eqn:KWduality}
\sinh(2J/T_c)\sinh(2J/\bar{T}_c) = 1,
\end{align}
(see Appendix \ref{sec:kramers_wannier} for a proof of the duality for all self-dual lattices).
This qualitative difference between Ising models in absence of external magnetic field for planes with and without curvature can be mathematically formulated in a unified framework using the cluster model, which describes both percolation and Ising models as special cases.
The proof for free boundary condition is formulated in terms of the (non-)uniqueness of the Gibbs states~\cite{Wu1996, Wu2000} and the intuitive explanation is given below. Consider the Gibbs states $\nu_\pm, \nu_f$ for the Ising model with fixed, free boundary conditions. Here, $+(-)$ refers to the fixed boundary condition case when the boundary spins are fixed to $+1(-1)$ states. First, consider planes without curvature. In the high-temperature phase ($T\gg J$), the Gibbs state is unique: $\nu_+ = \nu_- = \nu_f$. The average magnetization ($m$) is zero in this phase and the correlation function $\langle\sigma_i \sigma_j\rangle_{f,\pm}\rightarrow0$ for $|i-j|\rightarrow \infty$, the decay to zero being exponential due the presence of a gapped spectrum. On the other hand, in the low-temperature phase ($T\ll J$), the Gibbs-state is non-unique $\nu_+ \neq \nu_-$; However, these two states remain extremal, which means that the Gibbs state with free boundary condition is a symmetric mixture of the two:~$\nu_f = (\nu_+ + \nu_-)/2$. In this phase, the absolute-value of the magnetization is nonzero and $\langle\sigma_i \sigma_j\rangle_{f,\pm}\geq m^2$. The two phases are separated by a second-order phase-transition. Now, consider the Ising model on the hyperbolic plane. The high-temperature phase ($T\gg J$) again has unique Gibbs states. The average internal magnetization is zero and correlations are exponentially damped. Upon lowering the temperature, at $T_c$, the system magnetizes. However, the Gibbs states, $\nu_\pm$, are no-longer extremal, which leads to~$\nu_f \neq (\nu_+ + \nu_-)/2$. As a result, there exists a finite fraction of uncorrelated spins, which survive in the thermodynamic limit:~$\langle\sigma_i \sigma_j\rangle_{f}\rightarrow0$, despite there being a finite overall magnetization. Finally, upon further lowering of temperature, at $\bar{T}_c$, the system is magnetized as a whole and the extremal nature of the Gibbs states~$\nu_\pm$ is restored. This leads to~$\nu_f = (\nu_+ + \nu_-)/2$ and~$\langle\sigma_i \sigma_j\rangle_{f,\pm}\geq m^2$. This low-temperature phase is similar to the low-temperature phase of the euclidean space Ising model.
We emphasize that the phase-diagram of the model depends on the different choice of boundary conditions. The existence of the intermediate phase has been proven for free boundary conditions. We do not know if a similar intermediate phase exists for the case of periodic boundary conditions; see Sec. \ref{intermphase} for further discussion. Note that, despite the presence of two phase-transitions, the magnetic susceptibility diverges and the magnetization vanishes only at the higher phase-transition temperature Tc. These critical properties of these thermodynamic quantities at this higher temperature phase-transition are analyzed using Monte Carlo and series expansion methods below.
\subsection{Monte Carlo analysis of the critical exponents and temperature}
\label{sec:monte_carlo}
In this subsection, we perform Monte Carlo analysis of the Ising model on the $\{5,5\}$ lattice using the Metropolis algorithm.
In order to perform finite-size scaling we perform the numerical analysis for perfiodic lattices of different size.
We compute the average magnetization per spin $m = M/N$ and the energy per spin $e = E/N$, where~$M$ is the total magnetization,~$E$ the total energy and~$N$ the number of spins in the lattice.
Then, we compute the average absolute susceptibility per spin $\chi$ and the specific heat per spin $c$, defined below~\cite{Newman1999}
\begin{align}
\chi &= \frac{\beta}{N}(\langle M^2\rangle - \langle |M|\rangle^2), \\
\ c &= \frac{\beta^2}{N}(\langle E^2\rangle - \langle E\rangle^2),
\end{align}
where $\beta=1/T$ is the inverse temperature and $k_B$ is the Boltzmann constant. The phase-transition point is inferred form the fourth Binder cumulant~\cite{Landau2009}, given by
\begin{align}
U_4 = 1-\frac{\langle m^4\rangle}{3\langle m^2\rangle^2}.
\end{align}
Each Monte Carlo simulation was equilibriated with $10^5$ sweeps of the lattice, where one sweep corresponds to the number of Metropolis updates equal to the number of spins in the lattice. The measurements were performed over $10^6$ sweeps. The simulations were done for lattices with the following number of spins: $N = 360$, $1024$, $1920$, $2304$, $6888$ and $11760$. Fig. \ref{fig:mc_1} shows the absolute magnetization per spin, the average energy per spin, the average absolute susceptibility per spin and the specific heat per spin. The error analysis was performed using the bootstrap method~\cite{Newman1999}. The computed errors in the measured quantities are too small to show in the plots. The maximum of the absolute susceptibility occurs at the same temperature at which the magnetization goes to zero, in this case indicating a transition from a magnetically ordered phase to a disordered, high temperature phase. A crude estimate of the location of this phase-transition point can be obtained with help of the fourth Binder cumulant $U_4$, plotted in the top left panel of Fig. \ref{fig_mc_2}.
\begin{figure}
\centering
\includegraphics[width = 0.5\textwidth]{fig_mc_1.png}
\caption{Results of the Monte Carlo simulations for the hyperbolic space with $\{5,5\}$ tiling. The absolute magnetization per spin ($|m|$), the energy per spin ($\langle e\rangle$), average absolute susceptibility per spin ($\chi$) and the specific heat per spin ($c$) are plotted in the top left, top right, bottom left and bottom right panels respectively. We only show a zoom of the different quantities near the phase-transition region; but simulations were done from deep in the ferromagnetic regime to deep in the disordered region.}
\label{fig:mc_1}
\end{figure}
Accurate estimates of the phase-transition temperature and the critical exponents require finite system size analysis. In simulations on a finite plane with zero curvature, the divergence of the correlation length is cut off by the linear dimension of the system. From standard homogeneity arguments~\cite{Newman1999, Landau2009}, one can then obtain the ratio of the critical exponents: {\it e.g.}, from a linear fit of $\ln\chi$ versus $\ln L$, one gets the ratio $\gamma/\nu$, where $\gamma$ and $\nu$ are the critical exponents for the susceptibility and the correlation length and $L$ is the linear system size. Similarly, the the ratio $\beta/\nu$ is obtained from the linear fit of $\ln \langle |m|\rangle$ versus $\ln L$, where $\beta$ is the critical exponent for magnetization. The exact numerical values of the exponents and the phase-transition point are extracted by collapsing the data. This situation is different for a hyperbolic plane for following reason. In order to remove boundary effects, we perform the Monte Carlo simulations on a compactified hyperbolic plane, which is a manifold with genus larger than one.
The size of the smallest non-contractible loop along different handles this manifold may vary.\footnote{Note that this is not in contradiction to the space being isotropic, as several non-contractible loops my run through the same vertex (or the same edge) and these loops may have different lengths.}
As a result, the choice of a suitable linear dimension is ambiguous.
\begin{figure}
\centering
\includegraphics[width = 0.5\textwidth]{fig_mc_2.png}
\caption{\label{fig_mc_2} Results of the Monte Carlo simulations: (top left) the fourth Binder cumulant, (top right) data collapse for average absolute susceptibility and the linear fits are system size scaling for average absolute magnetization (bottom left) and average absolute susceptibility per spin (bottom right). The vertical line in the top left panel $T/J = 3.93$ (see main text for error estimate) indicates the location of phase-transition temperature obtained from the data collapse. The linear fits in the bottom panels denote are obtained from the finite size analysis with respect to the number of spins. The slope for the linear fit for average absolute susceptibility (magnetization) per spin provides an estimate of the ratio $\gamma/\mu$ ($\beta/\mu$), where $\gamma$ ($\beta$) denote the exponents for susceptibility (magnetization) and $\mu$ is the scaling of the coherence number. In the bottom panels, the error indicated is only the fit error, the actual value and the error in the exponent is provided in the main text. }
\end{figure}
To combat these difficulties of finite size scaling, we choose to perform finite size scaling with respect to the number of spins, which was initially proposed for an infinitely connected lattice~\cite{Botet1982} and has been used for hyperbolic space Ising models with open boundaries \cite{Shima2005}. We define a correlation number $N_c$, which plays the role of the correlation length in flat space, with an assumed scaling: $N_c\propto |T-T_c|^{-\mu}$, where $\mu$ is the associated critical exponent and $T_c$ is the critical temperature. The exponent $\mu$ is given by $\nu_{\rm{MF}}d_c$~\cite{Botet1982}, where $\nu_{MF}$ is the mean-field exponent of the divergence of correlation length for the Ising model ($=1/2$) and $d_c$ is the upper critical dimension ($=4$). Thus, $\mu = 2$. As explained in Ref.~\cite{Botet1982}, the correlation number $N_c \sim \xi^{d_c}$, where $\xi$ is the correlation length of the Euclidean space model. The use of an effective correlation number, which scales with the overall volume of the space, avoids the ambiguities associated with the linear dimension. The rest of the finite size scaling is similar to that of euclidean space, with the linear dimension replaced by the number of spins and the correlation length exponent replaced by $\mu$. Physically, due to the exponential growth of hyperbolic space volume with linear dimension, we expect the finite-size scaling analysis used for infinitely connected lattices to be applicable in our case. Since there is no proof that such an analysis is indeed valid, in Sec. \ref{sec:series_expansion}, we compute the critical exponents and critical temperature using high and low-temperature expansions. We do this since the series expansion computations, by design, do not suffer from finite size effects (but have different sources of error, see below). We find the results of these two independent computations to be close to each other, which lends additional credence to our Monte Carlo findings.
Performing this analysis, we find that the critical transition temperature occurs at $T_c / J = 3.93\pm0.03$. The critical exponents are $\beta = 0.46\pm 0.05$ and $\gamma = 1.03\pm 0.02$. The critical temperature, $T_c$, is close to the two possible candidates for critical temperature given in Ref. \cite{Jiang2018}. The exponents obtained are close to those for the Ising model within the mean-field approximation. The divergence of the specific heat does not develop as the system size is increased and we expect the critical exponent $\alpha$ to be 0, as expected for a mean-field theory. We note our Monte Carlo results are not exactly the mean-field predictions, despite being close to them. We attribute the deviations to the finite-size scaling done by considering a divergence of a correlation number. Additional support for the mean-field behavior is obtained by high and low temperature series expansion computations, described below.
\subsection{High-temperature expansion analysis of the critical exponents and temperature}
\label{sec:series_expansion}
In this section, we perform high-temperature series expansion of the susceptibility on the infinite lattice.
We chose to concentrate on the susceptibility rather than the free energy density as we found the latter harder to analyze.
It turns out that it is favourable to perform the high-temperature expansion in the {\em inverse} susceptibility.
The reason for this is that it can be shown~\cite{singh87} that the only non-trivial contributions come from graphs which have the property that they stay connected if any of their vertices (and the edges attached to it) are being removed.
Such graphs are called {\em biconnected} graphs and since it is a more restricted class there are fewer of these graphs which simplifies the expansion and allows us to go to much higher orders.
The inverse susceptibility can be expanded in terms of these graphs as
\begin{align}\label{eqn:sus_series}
\tilde{\chi}^{-1} = 1\, + \, \sum_{g}\, c(g)\, W(g)
\end{align}
where the sum is over all graphs, $c(g)$ is the coefficient of~$N$ of the number of embeddings of the graph~$g$ into the lattice and~$W$ is a weight function. For more details and a derivation of Eq.~(\ref{eqn:sus_series}) see Appendix~\ref{ap:series_derivation}.
\begin{figure}
\centering
\includegraphics[width = 0.45\textwidth]{55bicon.png}
\caption{Some small biconnected subgraphs of the $\{5,5\}$-tiling. Removing a vertex and all its incident edges will leave the graphs connected. Only biconnected graphs contribute to the series expansion.}\label{fig:55bicon}
\end{figure}
\subsubsection{Analysis of the Series}
The coefficients of the susceptibility on the $\{5,5\}$-lattices are given in Table ~\ref{tab:susseries}.
Our results for the high-temperature series expansion agree with \cite{Rietman1992} who obtained the series for~$\{5,5\}$ up to order~10.
We analyze the series $\tilde{\chi}(v)$ using \emph{first-order homogeneous integrated differential approximants (FO-IDAs)}.
One reason to choose FO-IDAs over simpler methods is that they are known to be less biased towards the lower-order coefficients of the expansion~\cite{singh2}.
This is important here, as the non-trivial contributions come from graphs with at least~$r$ edges.
The analysis using FO-IDAs proceeds as follows:
We assume that the series $\tilde{\chi}$ is the solution of a first-order differential equation of the form
\begin{equation}\label{eqn:IDAdef}
Q_L(v) \frac{d \tilde{\chi}(v)}{d v} + R_M(v)\, \tilde{\chi}(v) + S_T(v) = 0
\end{equation}
where $Q_L$, $R_M$ and $S_T$ are polynomials of degree $L$, $M$, $T$, respectively.
By equating the series order-by-order with the coefficients of Eq.~\ref{eqn:IDAdef} we obtain a linear system of equations in the coefficients of the polynomials $Q_L$, $R_M$ and $S_T$.
It can be shown that for any root $v_c$ of the polynomial~$Q_L$, a solution of Eq.~\ref{eqn:IDAdef} has an algebraic singularity of the form $(v-v_c)^{-\gamma}$ \cite{oitmaa2006}.
The exponent of the singularity is given by
\begin{align}\label{eqn:crit_exp}
\gamma = \frac{R_{M}(v_c)}{Q'_L(v_c)} .
\end{align}
Generally, the results for $v_c$ and $\gamma$ will depend on the choice of degrees $L$, $M$ and $T$.
If we have the series up to order $N$ then we can choose all possible values satisfying $L+M+T \leq N-2$.
Following~\cite{singh2} we exclude approximants if
\begin{itemize}
\item a root of $R_M$ is close to $v_c$, giving rise to a small estimate of $\gamma$
\item a complex root of $Q_L$ with small absolute value smaller than $v_c$ is close to the real axis
\end{itemize}
We observe that the convergence of the series is very good, since the approximants for different choices of $L$, $M$ and $T$ are all close.
The mean critical value of $v$ obtained by averaging over 45 approximants is $v_c = 0.25200759 \pm 0.00000006$.
This is in agreement with the result obtained by Monte Carlo [$\tanh(1/3.93) \approx 0.249$].
The critical exponent $\gamma$ is found via Eq. \ref{eqn:crit_exp} and averaged over all approximants. The result of $\gamma = 1.000001 \pm 0.000005$ is again close to the Monte Carlo result and the mean field value $\gamma = 1$.
\subsubsection{Comparison to low-temperature series}
\begin{figure}
\includegraphics[width = 0.45\textwidth]{55magn.png}
\caption{All the graphs contributing up to order $u^{19}$ of the magnetization of the model on the $\{5,5\}$-tiling.}\label{fig:55all}
\end{figure}
To verify that thermodynamic properties only diverge at the higher critical temperature $T_c$, we perform also a low temperature series of the magnetization of the model on the $\{5,5\}$ lattice. In the case of the low temperature expansion, we use an elementary expansion technique. We write the free energy in the presence of a field as the sum of configurations of flipped spins with respect to the ground state, to obtain an expansion in the temperature $u = \exp(-2\beta J)$ and the field $\mu = \exp(-2 \beta h)$
\begin{equation}\label{eqn:low_temp_Z}
\frac{1}{N} \ln Z = \frac{\ln(2)}{N} + \frac{q}{2} \beta J + \beta h + \sum_{\{g\}} c(g) u^{q n_v-2n_l} \mu^{n_v},
\end{equation}
where $q$ is the coordination number, $h$ is the magnetic field and $g$ is any graph on the lattice (including articulated and disconnected graphs). $c(g)$ here is the part of the number of embeddings of the graph $g$ into the lattice that is proportional to $N$, and $n_v$ and $n_l$ are the number of vertices and edges of the graph respectively.
We note that setting $h=0$ ($\mu=1$) we obtain the exact same coefficients as for the high-temperature expansion by exchanging $u \leftrightarrow v$, as predicted by the Kramers-Wannier duality (cf. Appendix~\ref{sec:kramers_wannier}).
From Eq.~\ref{eqn:low_temp_Z} the magnetization is obtained via
\begin{equation}
M = -2 \mu \frac{\partial}{\partial \mu} \frac{1}{N} \ln Z.
\end{equation}
All graphs contributing to the magnetization on the $\{5,5\}$ lattice up to order $u^{19}$ are shown in Fig. \ref{fig:55all}. Summing up their contributions yields:
\begin{align}
M = 1 &- 2u^5 \mu - 10 u^8 \mu^2 + 12 u^{10} \mu^2 - 60 u^{11} \mu^3 \nonumber\\
&+ 150 u^{13} \mu^3 - 400 u^{14} \mu^4 - (92\mu^3+10\mu^5)u^{15}\nonumber\\
&+ 1530\mu^4 u^{16} - 2800\mu^5u^{17} - (1920\mu^4+180\mu^6)u^{18}\nonumber\\
&+14600\mu^5u^{19}+ \mathcal{O}(u^{20})
\end{align}
The magnetization is expected to vanish at the critical temperature with a power law
\begin{equation}
M \sim (T_c - T)^\beta.
\end{equation}
We can thus analyze the inverse magnetization using FO-IDAs as before. This yields a critical point
\begin{equation}
u_c = 0.608 \pm 0.014
\end{equation}
which corresponds to a upper critical temperature of $T_c = 4.01\pm0.19$. This is in agreement with both, the result from the high-temperature series and the Monte Carlo simulation. Also, in agreement with the Monte Carlo simulation, we see that the magnetization does not vanish at the lower critical temperature $\bar{T}_c$. Finally, we note that our results are commensurate with those obtained in Ref. \onlinecite{Rietman1992}.
\section{Variation of critical properties with curvature}\label{sec:variation_with_curvature}
In this section, we perform Monte Carlo and series expansion analysis of Ising models on hyperbolic planes with different (but uniform) curvatures.
Before we do so we explain how the lattice type $\{r,s\}$ is connected to the magnitude of curvature of the underlying space.
\subsection{Curvature and lattices}
One way to quantify curvature is by considering a circle centered around a point $p$ with radius $r$.
Let~$C_p(r)$ be the circumference of this circle.
The curvature around the point $p$ is the difference between the circumference~$C_p(r)$ and the circumference of a circle with the same radius in the euclidean plane
\begin{align}
\kappa_p = \lim_{r \rightarrow 0^+} 3\; \frac{2\pi r - C_p(r)}{\pi r^3}.
\end{align}
Intuitively, the curvature at $p$ is negative if there is an excess of space in a local neighborhood around it.
This excess of space allows for a large variety of lattices in hyperbolic space.
\begin{figure}
\centering
\begin{minipage}{.6\columnwidth}
\centering
\includegraphics[width=0.45\columnwidth]{37.png}
\includegraphics[width=0.45\columnwidth]{73.png} \\
\includegraphics[width=0.45\columnwidth]{66.png}
\includegraphics[width=0.45\columnwidth]{77.png}
\end{minipage}%
\begin{minipage}{.4\columnwidth}
\centering
\includegraphics[width=0.7\columnwidth]{angles.eps}
\end{minipage}
\caption{Left: More tilings of the hyperbolic plane. Their Schl\"afli symbols from top to bottom and left to right: $\{3,7\}$, $\{7,3\}$, $\{6,6\}$, $\{7,7\}$. The tilings in the upper row are dual to one another, while the tilings in the bottom row are self-dual. Right: Angles of a triangle in the lattice.}
\label{fig:hyptilings}
\end{figure}
In fact, curvature and the type of regular tiling are intimately connected.
This can be seen as follows.
If we normalize the edge-length to be $1$, the hyperbolic law of cosines states that for any triangle in the hyperbolic plane with internal angles $(\alpha, \beta, \gamma)$ and $a$ the side-length opposing $\alpha$ we have
\begin{align}\label{eqn:hypcosinelaw}
\cos(\alpha) = -\cos(\beta)\, \cos(\gamma) + \sin(\beta)\, \sin(\gamma)\, \cosh(\kappa a) .
\end{align}
Triangulating a face of the tessellation by drawing lines from the center of the face to a vertex and the mid-point of an edge gives a triangle with angles $(\alpha,\beta,\gamma) = (\pi/r,\pi/2,\pi/s)$ and $a=1/2$ (see Fig.~\ref{fig:hyptilings}).
It now follows that
\begin{align}\label{eqn:schlafli_curvature}
\kappa = -4\, \cosh^{-1}\left(\frac{\cos(\pi/r)}{\sin(\pi/s)}\right) .
\end{align}
\subsection{Monte Carlo and series expansion analysis}
Next, we perform Monte Carlo and high-temperature series expansion analysis of the Ising model for the following lattices: $\{r,r\}$, $r = 5,6,7,8$. As the coordination number, $r$, increases, the curvature of the plane also increases [cf. Eq.~\eqref{eqn:schlafli_curvature}]. This causes the phase-transition temperature to increase as well. This is because increase in the coordination number increases the effective strength of the interaction seen by each spin.
Fig. \ref{fig:mc_variation} shows the variation of the transition temperature, $T_c$ (in units of $J$), and the exponents for susceptibility and magnetization, $\gamma$ and $\beta$, for the different self-dual lattices (since we perform only high-temperature expansion for these lattices, we obtain only the susceptibility exponent). We see that as the coordination number is increased by 1, the $T_c$ also increases by approximately the same amount (within errors in estimating the critical temperature). Intuitively, this can be understood as a further indication of the mean-field nature of the phase-transition since in a mean-field setting, the transition temperature is proportional to the coordination number~\cite{Chaikin2000}: $T_c\propto r$.
Note that the critical exponents obtained by the series expansion are extremely close to the mean-field estimate of 1. This indicates that the series expansions are extremely well-behaved and there are no appreciable corrections to scaling. On the other hand, the Monte Carlo estimates are close to the mean-field value. We believe the deviations from mean-field predictions for the Monte Carlo are due to finite size effects, not all of which is taken into account by our correlation number scaling. The analysis of the Monte Carlo data is done as in Sec. \ref{sec:monte_carlo} and the detailed plots are not shown here for brevity.
\begin{figure}
\centering
\includegraphics[width = 0.5\textwidth]{varn.png}
\caption{\label{fig:mc_variation} Results for self-dual $\{r,r\}$ lattices from Monte Carlo (filled blue diamonds) and high-temperature series expansion (filled orange circles) computations. (Left panel) The obtained transition temperatures, $T_c$ (in units of $J$), with the coordination numbers, $r$, subtracted to ensure the visibility of the error bars.
We note that the transition temperatures are increase by the same amount as the coordination number is increased by 1. This behavior is further indication that the nature of the transition is indeed mean-field (see main text for further details). (Right panel) The critical exponents for the susceptibility ($\gamma$) and magnetization ($\beta$). We see that the high-temperature series expansion gives extremely precise mean-field results, indicating the lack of corrections to scaling behavior. We believe the deviations from mean-field predictions for the Monte Carlo are due to finite size effects, not all of which is taken into account by our correlation number scaling (see Sec. \ref{sec:monte_carlo} for more details). Note that high-temperature expansion only gives $\gamma$ and not~$\beta$, for which only Monte Carlo data is provided.
}
\end{figure}
Table ~\ref{tab:susseries} shows the results of the high-temperature series expansion for susceptibility for self-dual lattices $\{r,r\}$, $r = 5,6,7,8$. In addition, we have also provided the series for the $\{3,7\}$ and the $\{7,3\}$ lattices, the latter two dual of one-another. The last two lattices will be needed when comparing our results to the Bethe lattice (see below). The exact values of $T_c$ and $\gamma$ obtained by the series expansions are shown in Table ~\ref{tab:susIDA}.
{
\renewcommand{\arraystretch}{1.2
\begin{table}
\centering
\bgroup
\def\arraystretch{1.5}%
\resizebox{1\columnwidth}{!}{%
\begin{tabular}{c r r r r r r}
\hline
\hline
$n$ & $\{3,7\}$ & $\{7,3\}$ & $\{5,5\}$ & $\{6,6\}$ & $\{7,7\}$ & $\{8,8\}$ \\
\hline
1 & 3 & 7 & 5 & 6 & 7 & 8 \\
2 & 6 & 42 & 20 & 30 & 42 & 56 \\
3 & 12 & 238 & 80 & 150 & 252 & 392 \\
4 & 24 & 1316 & 320 & 750 & 1512 & 2744 \\
5 & 48 & 7196 & 1270 & 3750 & 9072 & 19208 \\
6 & 96 & 39144 & 5040 & 18738 & 54432 & 134456 \\
7 & 186 & 212394 & 20010 & 93630 & 326578 & 941192 \\
8 & 360 & 1150968 & 79400 & 467862 & 1959384 & 6588328 \\
9 & 702 & 6233150 & 315060 & 2337870 & 11755814 & 46118184 \\
10 & 1368 & 33745698 & 1250260 & 11682090 & 70531944 & 322826520 \\
11 & 2664 & 182669074 & 4961180 & 58374174 & 423174024 & 2259780264 \\
12 & 5148 & 988735958 & 19686500 & 291689754 & 2538938220 & 15818424216 \\
13 & 9948 & 5351558814 & 78119090 & 1457543742 & 15232993804 & 110728706088 \\
14 & 19308 & 28964952422 & 309987000 & 7283195826 & 91394150092 & 775099098536 \\
15 & 37434 & 156769556314 & 1230068820 & & 548342025112 & 5425680781224 \\
16 & 72504 & 848494238298 & 4881081760 & & 3289914904549 & 37979675110288 \\
17 & 140238 & & 19368790490 & & & 265857093267968 \\
18 & 271242 & & & & & 1860995225360608 \\
19 & 525528 & & & & & 13026935584994368 \\
20 & 1017726 & & & & & \\
21 & 1969458 & & & & & \\
22 & 3811128 & & & & & \\
23 & 7375278 & & & & & \\
24 & 14279604 & & & & & \\
\hline
\hline
\end{tabular}
}
\egroup
\caption{The coefficients $x_n$ of the susceptibility series $\tilde{\chi} = 1 + \sum_{n=1}^{\infty} x_n\, v^n$. Our results for the high-temperature series expansion agree with \cite{Rietman1992} who obtained the series for~$\{5,5\}$ up to order~10 and for and for~$\{3,7\}$ up to order~11. }\label{tab:susseries}
\end{table}
}
\begin{table}
\centering
\bgroup
\def\arraystretch{1.5}%
\begin{tabular}{c c c}
\hline
\hline
lattice & $v_c$ & $\gamma$ \\
\hline
$\{3,7\}$ & $0.184764\pm 0.000004$ & $0.9999\pm 0.0004$ \\
$\{7,3\}$ & $0.51\pm 0.04$ & $1.00\pm 0.02$ \\
$\{5,5\}$ & $0.25200759 \pm 0.00000006$ & $1.000001 \pm 0.000005$ \\
$\{6,6\}$ & $0.200125\pm 0.000001$ & $1.00002\pm 0.00004$ \\
$\{7,7\}$ & $0.166673621\pm 4\cdot10^{-9}$ & $1.0000001\pm 2\cdot10^{-7}$ \\
$\{8,8\}$ & $0.142857482725 \pm 7\cdot 10^{-12}$ & $0.9999999993 \pm 9\cdot 10^{-10}$ \\
\hline
\hline
\end{tabular}
\egroup
\caption{Estimation of $v_c = \tanh(J/T_c)$ and the critical exponent~$\gamma$ obtained from the high-temperature series expansion of~$\tilde{\chi}$. The series analysis was done via first-order IDAs.
}\label{tab:susIDA}
\end{table}
\subsection{Comparison to the Bethe lattice}
In this section, we compare the critical properties of the above-analyzed hyperbolic lattices to the Bethe lattice. The lattice is the infinite $s$-regular tree, i.e. every vertex has $s$ neighbours and there are no cycles (closed loops) in the graph. It can be understood as a hyperbolic tiling where all faces have an infinite number of sides (see Fig.~\ref{fig:bethe}) which means that we can formally assign it the Schl\"afli symbol $\{\infty,s\}$.
Intuitively, for hyperbolic tilings $\{r,s\}$, where the number of edges around each face(~$r$) is large the Bethe lattice should be a good approximation.
\begin{figure}
\centering
\includegraphics[width=0.5\columnwidth]{bethe.png}
\caption{The Bethe lattice with coordination number $s=3$. It can be interpreted as a hyperbolic tiling with Schl\"afli symbol $\{\infty, s\}$.}
\label{fig:bethe}
\end{figure}
Due to its tree-structure it is straightforward to solve the Ising model defined on the Bethe lattice~\cite{Baxter2013}.
The critical temperature is given by
\begin{align}\label{eqn:betheexacttemp}
T_c^\text{B} = \frac{2}{\ln \frac{s}{s-2}}
\end{align}
and hence
\begin{align}\label{eqn:betheexact}
v_c^\text{B} = \frac{1}{s-1}.
\end{align}
In Fig.~\ref{fig:betheapprox} the exact solution of the Bethe lattice is plotted together with the results of the high-temperature series expansion (see Table ~\ref{tab:susIDA}).
Evidently, Eq. ~\eqref{eqn:betheexact} provides a good approximation to the results that we obtained for the hyperbolic tilings.
The relative error between the critical values of the hyperbolic lattice and the Bethe lattice $|v_c - v_c^\text{B}| / v_c$ is shown in Fig.~\ref{fig:betheapprox}. The relative error decreases exponentially in the number of sides of a face~$r$. Note that the Ising model on 2D square-lattice ($\{4,4\}$ tiling) shows critical properties that are furthest from the mean-field behavior of the Bethe lattice.
\begin{figure}
\centering
\includegraphics[width=0.49\columnwidth]{bethe_comparison.pdf}
\hfil
\includegraphics[width=0.49\columnwidth]{bethe_comparison_error.pdf}
\caption{(a) Comparison of the results from Table ~\ref{tab:susIDA} to the exact solution of the Ising model on the Bethe lattice~$\{\infty,s\}$ given by Eq.~\eqref{eqn:betheexact}. The upper bound is obtained in~\cite{hypSAW} where the authors derive a bound on the coefficients of the series expansion of $\chi$ for self-dual lattices. (b) The relative error when comparing the critical value $v_c$ of the hyperbolic Ising model and the Bethe lattice with the same coordination number. The tessellations here are $\{3,7\}$, $\{4,4\}$ (euclidean square lattice), $\{5,5\}$, $\{6,6\}$ and $\{7,7\}$.}\label{fig:betheapprox}
\end{figure}
\subsection{The intermediate phase and the second phase-transition}
\label{intermphase}
Recall that in addition to the phase-transition at $T_c$, the Ising model on the self-dual hyperbolic lattices undergoes a second phase-transition at a lower temperature~$\bar{T}_c$, where~$T_c, \bar{T}_c$ are related by the Kramers-Wannier duality relation [see discussion below Eq. \eqref{eqn:KWduality}]. Since the critical properties of the self-dual hyperbolic lattices are quite close to the Bethe lattice, in what follows, we describe the fate of this second phase-transition at $\bar{T}_c$ in the case of the Bethe lattice.
In the range of temperatures $\bar{T}_c< T < T_c$, the correlation function of two far away spins approaches $m^2$ instead of going to zero (that happens at temperatures $T>T_c$). Since the magnetization changes from zero to nonzero as $T$ changes from $T_c +\epsilon$ to $T_c - \epsilon$, $\epsilon\rightarrow0$, it is clear that the critical temperature $T^\text{B}_c$ in Eq.~\eqref{eqn:betheexacttemp} (obtained by analyzing the change in magnetization~\cite{Baxter2013}), corresponds to the critical temperature $T_c$. This implies that the low-temperature phase of the Bethe lattice Ising model, in fact, corresponds to the intermediate phase of the hyperbolic plane Ising models. This begs the question: what is $\bar{T}_c$ for the Bethe lattice when the system transitions to the pure ferromagnetic phase?
To answer this question, consider the correlation function of the Ising model on the Bethe lattice. It can be obtained exactly~\cite{kumar1976two} [cf.~Eq.~\eqref{eqn:high_temp_exp}]:
\begin{align}
\begin{split}
\langle \sigma_{0}\, \sigma_{k} \rangle &= \frac{\sum_\sigma e^{-\beta H(\sigma)} \sigma_{0}\sigma_{k}}{\sum_\sigma e^{-\beta H(\sigma)}}\\
&= \frac{\sum_\sigma \prod_{(i,j)\in E} (1+\sigma_i \sigma_j v) \sigma_{0}\sigma_{k}}{\sum_\sigma \prod_{(i,j)\in E} (1+\sigma_i \sigma_j v)}
\end{split}
\end{align}
Observe that those terms in the product which contain an odd number of spin variables will cancel when summing over all spin configurations.
As taking the product can be interpreted as picking subsets of edges, we can interpret the numerator and the denominator as summing over subgraphs of the lattice (cf.~Appendix~\ref{sec:kramers_wannier}). Together with the previous observation we see that in the numerator the sum is taken over all subgraphs for which vertices 0 and $k$ have odd degree and all other vertices have even degree, while in the denominator all contributing graphs must have even degree.
For the Bethe lattice, the only contribution in the numerator is the line graph connecting $0$ and $k$. This graph has $\text{dist}(0,k)$ edges and hence, the numerator is exactly equal to $v^{\text{dist}(0,k)}$. The denominator has only the empty graph as a non-trivial contribution as the Bethe lattice has no finite subgraphs with all vertices of even degree.
Thus, for the Bethe lattice,
\begin{align}\label{eqn:bethe_correlation}
\langle \sigma_{0} \sigma_{k} \rangle = v^{\text{dist}(0,k)}.
\end{align}
The correlation function is exponentially decaying {\em to zero} for $k \rightarrow \infty$ at any fixed non-zero temperature (just as for the hyperbolic Ising model in the intermediate phase) and is non-analytic at temperature~$0$ where it jumps to~$m^2 = 1$. Thus, for the Bethe lattice, $\bar{T}_c = 0$. Note that we assuming the convergence of the sum to arrive at this conclusion.
We tried to obtain signatures of this intermediate phase for the different hyperbolic lattices. However, with the system-sizes that we explored, the signal-to-noise ratio was not good enough to make any conclusive predictions. We hope to return to this problem in the future.
\section{The Ising model in 3D hyperbolic space}
\label{3DIsing}
So far, in this work, we analyzed the Ising model in 2D hyperbolic plane with Monte Carlo and high-temperature series expansion techniques.
While our analysis confirms the mean-field nature of the phase-transition, our results are not compatible with the conjectured formulas for critical exponents given by the field theory calculations~\cite{Mnasri2015}.
In this section, we analyze the Ising model in 3D hyperbolic space with periodic boundary condition, for which explicit $1/N$ computations revealed non-mean-field behavior.
For Monte Carlo simulations, we consider the $\{5,3,5\}$ lattice where the 3D hyperbolic space is tiled with dodecahedra in a way that there are 5 dodecahedra around each edge. The high-temperature analyses of the next section are done for $\{5,3,k\}$ lattices, where $k = 4,5,6$.
\subsection{Monte Carlo Analysis}
\label{3Dmc}
The Monte Carlo simulations were done for three different lattice sizes of the $\{5,3,5\}$ lattice with number of nodes in the vertices given by $N = 4428$, $14762$ and $390963$. We were able to perform simulations on only three lattice sizes since finding compactifications of 3D hyperbolic space is computationally even more challenging than for their 2D counterparts.
The results of the Monte Carlo simulations for the absolute magnetization per spin, the energy per spin, average absolute susceptibility per spin and the specific heat per spin are shown in Fig. \ref{fig:mc_3D_1}. The smaller two lattices were equilibrated with $10^4$ sweeps of the lattice and the Monte Carlo measurements were done over $10^5$ sweeps of the lattice. The finite size scaling analysis was done as in Sec. \ref{sec:monte_carlo} using the correlation number exponent $\mu = 2$, thereby avoiding difficulties associated with the multiple linear dimensions. Since the connectivity of the 3D lattice is higher (in the case considered, the number of nearest neighbors is 12 for any given spin), we expect the correlation number scaling to yield better results than that obtained for the 2D lattices analyzed in Sec. \ref{sec:variation_with_curvature}.
\begin{figure}
\centering
\includegraphics[width = 0.5\textwidth]{fig_mc_3D_1.png}
\caption{\label{fig:mc_3D_1}Results of the Monte Carlo simulations for the hyperbolic space with $\{5,3,5\}$ tiling. The absolute magnetization per spin ($|m|$), the energy per spin ($\langle e\rangle$), average absolute susceptibility per spin ($\chi$) and the specific heat per spin ($c$) are plotted in the top left, top right, bottom left and bottom right panels respectively.}
\end{figure}
From the finite size scaling analysis, we infer that the critical temperature is $T_c = 10.96\pm 0.01$ and $\gamma = 0.97\pm 0.02$, $\beta = 0.51\pm 0.04$, which are close to the mean-field predictions. Comparing our results to those obtained by field theory ($1/N$) computations~\cite{Mnasri2015}, we see that the susceptibility exponent is not compatible with the field theory computations, who obtain $\gamma = 2$. On the other hand, the magnetization exponent, inferred from scaling relations, together with the field theory computations, agree. As in the 2D case, the peak in the specific heat did not develop upon increase of system size, which indicates that the specific heat does not diverge and we expect $\alpha = 0$ in this case as well.
\begin{figure}
\centering
\includegraphics[width = 0.5\textwidth]{fig_mc_3D_2.png}
\caption{\label{fig:mc_3D_2} Results of the Monte Carlo simulations: (top left) the fourth Binder cumulant, (top right) data collapse for average absolute susceptibility and the linear fits are system size scaling for average absolute magnetization (bottom left) and average absolute susceptibility per spin (bottom right). The vertical line in the top left panel at $T/J = 10.96$ (see main text for error estimate) indicates the location of phase-transition temperature obtained from the data collapse. The linear fits in the bottom panels denote are obtained from the finite size analysis with respect to the number of spins. The slope for the linear fit for average absolute susceptibility (magnetization) per spin provides the ratio $\gamma/\mu$ ($\beta/\mu$), where $\gamma$ ($\beta$) denote the exponents for susceptibility (magnetization) and $\mu$ is the scaling of the coherence number (see Sec. \ref{sec:monte_carlo} for details). In the bottom panels, the error indicated is only the fit error, the actual value and the error in the exponent is provided in the main text. }
\end{figure}
\subsection{High-Temperature Series Analysis}
\label{3dtemp}
To verify the results obtained by Monte Carlo simulation, we also compute the high-temperature series expansion of the susceptibility for both the $\{5, 3, 5\}$ and $\{5, 3, 4\}$ lattices.
We also consider the $\{5, 3, 6\}$ lattice which is a 3D generalization of the dual of the Bethe lattice.
The Bethe lattice can be interpreted as a tessellation where the faces are $\infty$-gons, i.e. they have the (only possible) tessellation of the infinite line~$\mathbb{R}$ at their boundary.
The $\{5, 3, 6\}$ tessellation is dual to the $\{6, 3, 5\}$ tessellation which is a space tessellated by non-compact polyhedra which have a hexagonal tiling~$\{6, 3\}$ at their boundary.
The first four graphs contributing to those series, together with their embeddig numbers per site, are given in Tab.~\ref{fig:3d-series}.
The weights $W(g)$ are the same as in Sec.~\ref{sec:series_expansion}.
Summing the contributions yields the inverse susceptibility $\bar\chi^{-1}$.
Inverting that series gives the coefficients in Tab.~\ref{tab:susseries3d}.
For analysis of the series, we use FO-IDAs. Averaging over 8 different approximants using a minimum number of 8 terms yields an estimate of the critical properties tabulated in Tab.~\ref{tab:susIDA3d}.
The results are very close to the mean field predictions and the $\{5, 3, 5\}$ result is close to the result of the Monte~Carlo simulation. The exponents $\gamma$ are not compatible with the field-theory~($1/N$) prediction of $\gamma=2$~\cite{Mnasri2015}.
\begin{table}
\centering
\bgroup
\def\arraystretch{1.5}%
\begin{tabular}{c c c c c}
\hline
\hline
$g$ &
\begin{minipage}[c]{1.5cm}\vspace{.2cm}\includegraphics[width=1.3cm]{g2.png}\end{minipage}&
\begin{minipage}[c]{1.5cm}\vspace{.2cm}\includegraphics[width=1.3cm]{0.png}\end{minipage}&
\begin{minipage}[c]{1.5cm}\vspace{.2cm}\includegraphics[width=1.3cm]{1.png}\end{minipage}&
\begin{minipage}[c]{1.5cm}\vspace{.2cm}\includegraphics[width=1.3cm]{2.png}\end{minipage}\\[0.7cm]
\hline
$c_{\{5, 3, 4\}}(g)$ & 6 & 24/5 & 36 & 36\\[0.1cm]
$c_{\{5, 3, 5\}}(g)$ & 6 & 6 & 60 & 60\\[0.1cm]
$c_{\{5, 3, 6\}}(g)$ & 6 & 36/5 & 90 & 90\\[0.1cm]
\hline
\hline
\end{tabular}
\egroup
\caption{\label{fig:3d-series}The four smallest graphs (exlcuding the single bond) contributing to the susceptibility series of the Ising model in hyperbolic space with $\{5, 3, 4\}$, $\{5, 3, 5\}$ and $\{5, 3, 6\}$ tiling. The number of embeddings per site $c(g)$ for each graph~$g$ is given explicitly for all three tilings}
\end{table}
\begin{table}
\centering
\bgroup
\def\arraystretch{1.5}%
\begin{tabular}{c r r r }
\hline
\hline
$n$ & $\{5, 3, 4\}$ & $\{5, 3, 5\}$ & $\{5, 3, 6\}$\\
\hline
1 & 12 & 12 & 12\\
2 & 132 & 132 & 132\\
3 & 1452 & 1452 & 1452\\
4 & 15972 & 15972 & 15792\\
5 & 175644 & 175632 & 175620\\
6 & 1931556 & 1931292 & 1931028\\
7 & 21241356 & 21237012 & 21232668\\
8 & 233590980 & 233526972 & 233462868\\
9 & 2568797772 & 2567915412 & 2567030988\\
10 & 28249045956 & 28237381152 & 28225683060\\
\hline
\hline
\end{tabular}
\egroup
\caption{The coefficients $x_n$ of the susceptibility series $\tilde{\chi} = 1 + \sum_{n=1}^{\infty} x_n\, v^n$. For the Ising model on hyperbolic tilings in three dimensions}\label{tab:susseries3d}
\end{table}
\begin{table}
\centering
\bgroup
\def\arraystretch{1.5}%
\begin{tabular}{c c c}
\hline
\hline
lattice & $v_c$ & $\gamma$ \\
\hline
$\{5,3,4\}$ & $0.09093417 \pm 0.00000008$ & $1.000012 \pm 0.000003$ \\
$\{5,3,5\}$ & $0.09094066 \pm 0.00000013$ & $1.000023 \pm 0.000005$ \\
$\{5,3,6\}$ & $0.09094723 \pm 0.00000018$ & $1.000037 \pm 0.000007$ \\
\hline
\hline
\end{tabular}
\egroup
\caption{\label{tab:susIDA3d}Estimation of $v_c = \tanh(J/T_c)$ and the critical exponent~$\gamma$ obtained from the high-temperature series expansion of~$\tilde{\chi}$ for hyperbolic tilings in three dimensions. The series analysis was done via first-order IDAs.
}
\end{table}
\section{Conclusion and Perspectives}
\label{sec:conclusion}
To summarize, we have analyzed the critical properties of the Ising model in hyperbolic space using Monte Carlo and series expansion methods. The negative curvature of hyperbolic space leads to a comparable number of spins on the boundary as in the bulk of an open hyperbolic space, which leads to large boundary effects. While these boundary effects can be interesting in themselves, they obscure the bulk properties of the model. We analyze the bulk properties of the Ising model in hyperbolic spaces with periodic boundary condition. First, we find compactifications of the hyperbolic manifolds, which are manifolds with genus larger than 1, in contrast to the euclidean plane with periodic boundary condition, a torus, with genus 1. Performing Monte Carlo simulations on these compactified hyperbolic manifolds, we infer the critical temperature and critical exponents for several two and three dimensional lattices. We obtain results that are close to mean-field predictions. Subsequently, we perform high-temperature series analysis for the different lattices which confirm the mean-field nature of the critical behavior and are close to our Monte Carlo findings. We analyze the variation of the critical temperature as a function of curvature and explain how the properties of the Ising model on the Bethe lattice can be viewed as an asymptotic case of that on the different hyperbolic lattices. Recently, 2D mesoscopic superconducting circuit lattices have been engineered which implement spin-systems in hyperbolic lattices~\cite{Kollar2019}. With this remarkable experimental progress, we are optimistic that statistical mechanical models in hyperbolic space will be experimentally realized in the near future.
Before concluding, we outline several possible research directions that are of interest.
First, the 2D Ising model on the hyperbolic plane with free boundary conditions is expected to exhibit an intermediate phase, between the ferromagnetic and the disordered phases. This intermediate phase, absent in the euclidean space Ising model, shows broken translational invariance, where the lattice is covered with infinitely large, infinitely many magnetized domains. We were unable to find conclusive evidence of this phase in our Monte Carlo simulations for the case of periodic boundary conditions. This could be due to the fact that the sizes of the systems analyzed were not large enough to ensure several magnetized clusters to form in addition to domains of randomized spins. We emphasize that the main limitation is not the Monte Carlo aspect of the simulation, but finding the compactifcations of the hyperbolic space itself, which is computationally costly. Larger scale Monte Carlo simulations on compactified hyperbolic planes may be able to find evidence for this intermediate phase.
Second, the found compactifications of hyperbolic space are valuable for analyzing bulk properties of different gauge and matter spin-systems in these spaces --- an interesting problem, much less explored than its flat-space counterpart.
Third, the high genus compactified hyperbolic manifolds can be used to implement quantum codes such as toric codes, which are promising for quantum information processing. In contrast to their euclidean space counterparts, for toric codes on these manifolds, the number of encoded logical qubits scales proportionally with the number of actual physical qubits \cite{Freedman2002, Breuckmann2017} .
Interestingly, the decoding of these hyperbolic space toric codes can be related to the random-bond Ising model in hyperbolic space \cite{dklp}, which, to the best of our knowledge, has not been analyzed before. The ferromagnetic to paramagnetic phase-transition points provide the threshold for successful decoding of the error syndrome for the different decoders of the quantum code. We hope to report on properties of the random-bond Ising model on hyperbolic manifolds in the near future.
\section*{Acknowledgments}
We thank Leonid Pryadko for encouraging discussions.
N.P.B. is supported by the UCLQ Fellowship.
A.R. acknowledges the support of the Alexander von Humboldt foundation.
| {
"timestamp": "2020-02-21T02:15:26",
"yymm": "1909",
"arxiv_id": "1909.12107",
"language": "en",
"url": "https://arxiv.org/abs/1909.12107",
"abstract": "The Ising model exhibits qualitatively different properties in hyperbolic space in comparison to its flat space counterpart. Due to the negative curvature, a finite fraction of the total number of spins reside at the boundary of a volume in hyperbolic space. As a result, boundary conditions play an important role even when taking the thermodynamic limit. We investigate the bulk thermodynamic properties of the Ising model in two and three dimensional hyperbolic spaces using Monte Carlo and high and low-temperature series expansion techniques. To extract the true bulk properties of the model in the Monte Carlo computations, we consider the Ising model in hyperbolic space with periodic boundary conditions. We compute the critical exponents and critical temperatures for different tilings of the hyperbolic plane and show that the results are of mean-field nature. We compare our results to the 'asymptotic' limit of tilings of the hyperbolic plane: the Bethe lattice, explaining the relationship between the critical properties of the Ising model on Bethe and hyperbolic lattices. Finally, we analyze the Ising model on three dimensional hyperbolic space using Monte Carlo and high-temperature series expansion. In contrast to recent field theory calculations, which predict a non-mean-field fixed point for the ferromagnetic-paramagnetic phase-transition of the Ising model on three-dimensional hyperbolic space, our computations reveal a mean-field behavior.",
"subjects": "Statistical Mechanics (cond-mat.stat-mech)",
"title": "Critical properties of the Ising model in hyperbolic space",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9728307684643189,
"lm_q2_score": 0.7279754430043072,
"lm_q1q2_score": 0.7081969096410331
} |
https://arxiv.org/abs/1103.0992 | Associated primes of powers of edge ideals | Let G be a graph and let I be its edge ideal. Our main result shows that the sets of associated primes of the powers of I form an ascending chain. It is known that the sets of associated primes of I(i) and intcl(I(i)) stabilize for large i, where "intcl" denotes integral closure and I(i) denotes the i-th power of I. We show that for edge ideals their corresponding stable sets are equal. To show our main result we use a classical result of Berge from matching theory and certain notions from combinatorial optimization. | \section{Introduction}
Let $G$ be a simple {\it graph\/} with finite vertex set
$X=\{x_1,\ldots,x_n\}$, i.e., $G$ is the set $X$ together with a family of
subsets of $X$ of cardinality $2$, called edges, none
of which is included in another. The sets of
vertices and edges of $G$ are denoted by $V(G)$
and $E(G)$ respectively. We shall always assume that $G$ has no
isolated vertices, i.e., every vertex of $G$ has to occur in at least
one edge. Let $R=K[x_1,\ldots,x_n]$ be a polynomial ring
over a field $K$. The {\it edge ideal\/} of $G$,
denoted by $I=I(G)$, is the ideal of $R$
generated by all square-free monomials $x_ix_j$ such that
$\{x_i,x_j\}\in E(G)$.
The assignment $G\mapsto
I(G)$ gives a natural one to one
correspondence between the family of graphs and the family
of monomial ideals generated by square-free monomials of degree $2$.
Let $G$ be a graph and let $I=I(G)$ be its edge ideal. In this paper
we will examine the sets of
associated primes of the powers of $I$, that is, the sets
$${\rm Ass}(R/I^k)=\{{\mathfrak p}\subset R \, | \, {\mathfrak p} \
{\mbox{\rm is prime and }}
{\mathfrak p}=(I^k:c)\
{\mbox{\rm for some }} c\in R\},\ \ k\geq 1.$$
Since $I$ is a monomial ideal of a polynomial ring $R$, the
associated primes
will be monomial primes, which are primes that are
generated by subsets of the variables, see
\cite[Proposition~5.1.3]{monalg}.
The associated primes of $I$
correspond to
minimal vertex covers of the graph $G$ and ${\rm Min}(R/I)={\rm
Ass}(R/I)$, where ${\rm Min}(R/I)$ denotes the
set of minimal primes of $I$, see \cite{Vi2}. For edge ideals, ${\rm
Ass}(R/I)\subset {\rm Ass}(R/I^k)$ for all positive integers $k$. In
the case where
equality holds for all $k$, the ideal $I$ is said to
be {\em normally torsion-free}.
In \cite{brod}, Brodmann showed that the sets ${\rm Ass}(R/I^k)$ stabilize
for large $k$. That is, there exists a positive integer $N_1$ such that
${\rm Ass}(R/I^k)={\rm Ass}(R/I^{N_1})$ for all $k \geq N_1$. A minimal
such $N_1$ is called the {\em index of stability} of $I$. One important
result in this area establishes that $N_1=1$ if and only if $G$ is a
bipartite graph \cite[Theorem~5.9]{ITG}. A useful upper bound for
$N_1$ was
shown in \cite[Corollary~4.3]{AJ}, namely that if $G$ is a connected
non-bipartite graph with $n$ vertices, $s$ leaves, and the smallest odd
cycle of $G$ has length $2k+1$, then $N_1 \leq n-k-s$. We make use of
this upper bound in Example~\ref{intcl1.m2}.
Although the sets ${\rm Ass}(R/I^k)$ are known to stabilize for large
$k$, their behavior for small $k$ can be erratic. Finding the stable
set ${\rm Ass}(R/I^{N_1})$ is complicated by the fact that a prime
ideal $\mathfrak{p}$ that is associated to a low power of an ideal
$I$ need not be associated
to higher powers. For example, \cite[Example,~p.~2]{McAdam} gives an example,
due to A. Sathaye, of
an ideal $I$ in a ring $R$ and a prime $\mathfrak{p}$ for which
$\mathfrak{p} \in {\rm Ass}(R/I^k)$ for $k$
even and $\mathfrak{p} \not\in {\rm Ass}(R/I^k)$ for $k$ odd for all
$k$ below a stated bound. When,
for an ideal $I$, $\mathfrak{p} \in {\rm Ass}(R/I^k)$ implies
$\mathfrak{p}\in {\rm
Ass}(R/I^{k+1})$ for all $k \geq 1$, one says that the sets ${\rm
Ass}(R/I^k)$ form an {\it ascending chain\/}. Although this property is
highly desirable, few classes of ideals are known to possess it. Examples
of monomial ideals for which the sets ${\rm Ass}(R/I^k)$ do not form
ascending chains can be found in \cite[Section~4]{HH} (stated in terms
of depths), and \cite[Example~4.18]{edge-ideals}.
Let $\overline{I^k}$ denote the integral closure of $I^k$. An ideal
$I$ is called {\it normal} if $I^k=\overline{I^k}$ for all
$k\geq 1$. By results of
Ratliff \cite{ratliff,ratliff-increasing}, one has that the sets ${\rm
Ass}(R/{\overline{I^k}})$ form an ascending chain which stabilizes for
large $k$. Thus, there exists $N_2$ such that ${\rm Ass}(R/\overline{I^k})={\rm
Ass}(R/\overline{I^{N_2}})$ for $k\geq N_2$. The set ${\rm
Ass}(R/\overline{I^{N_2}})$ is nicely described in \cite{mcadam}, and
for edge ideals of graphs the set ${\rm Ass}(R/I^{N_1})$ is
described in \cite{AJ}.
Our main result is:
\noindent {\bf Theorem~\ref{persistence-edge-ideals}}{\it\ If $I$ is
the edge ideal of a graph, then ${\rm
Ass}(R/I^k) \subset{\rm Ass}(R/I^{k+1})$ for all $k$. That
is, the sets of associated primes of the powers of $I$ form an
ascending chain.
}
There are two cases where the sets of associated primes of a
square-free monomial ideal are known to form an ascending chain. The
first case is the family of normal ideals (as was pointed out above),
which includes, for instance, ideals of vertex covers of perfect graphs
\cite{persistence-dm,FHV,perfect}. The second case
is the family of graphs with at
least one leaf \cite{edge-ideals}, which is now a particular case of
our main result.
In a more general setting, i.e., when $I\neq(0)$ is an ideal of a
commutative
Noetherian domain, Ratliff showed that $(I^{k+1}\colon I)=I^k$ for all
large $k$ \cite[Corollary~4.2]{ratliff} and that equality holds for
all $k$ when $I$ is normal \cite[Proposition~4.7]{ratliff}. We show
that equality holds for
all $k$ when $I$ is an edge ideal.
\noindent {\bf Lemma~\ref{jan29-2011}}{\it\ $(I^{k+1}\colon I)=I^k$
for $k\geq 1$.}
This lemma is central to the proof of our main result.
To show this lemma, we need to link the algebraic and
combinatorial data. This is achieved using matching theory
and basic notions from combinatorial optimization.
Given an edge $f$, we denote by $G^f$ the
graph obtained from $G$ by duplicating the two vertices of $f$ (see
Definition~\ref{parallelization-def}). The {\it deficiency} of $G$,
denoted by ${\rm def}(G)$, is the number of vertices left uncovered
by any maximum matching of $G$. The matching number of $G$ is denoted
by $\nu(G)$ (see Definition~\ref{matching-number}). Using a formula of Berge (see
Theorem~\ref{berge-formula}), we compare the deficiencies of $G$ and $G^f$.
Our main combinatorial result is:
\noindent {\bf Theorem~\ref{pepe-vila-berge}}{\it\ ${\rm
def}(G^f)=\delta$ for all $f\in E(G)$
if and only if ${\rm
def}(G)=\delta$ and $\nu(G^f)=\nu(G)+1$ for all $f\in E(G)$.}
As a byproduct, we present the following
characterization of graphs with a perfect matching.
\noindent {\bf Corollary~\ref{pepe-vila}}{\it\ $G$ has a perfect matching if
and only if $G^f$ has a perfect matching for every edge $f$ of $G$.}
In general, for ideals in commutative Noetherian rings,
${\rm Ass}(R/\overline{I^{N_2}})$ is a subset of
${\rm Ass}(R/I^{N_1})$, see \cite[Proposition~3.17]{McAdam}. We show that
for edge ideals these stable sets are equal.
\noindent {\bf Theorem~\ref{ass=assic}}{\it\ ${\rm Ass}(R/I^k)={\rm
Ass}(R/\overline{I^k})$ for $k\geq
\max\{N_1,N_2\}$.}
As an application we show that an edge ideal $I$ is normally
torsion-free if and only if
${\rm Ass}(R/\overline{I^i})={\rm Ass}(R/I)$ for $i\geq 1$ (see
Corollary~\ref{ntf-ass-corollary}).
Throughout the paper we introduce most of the
notions that are relevant for our purposes. For unexplained
terminology we refer to \cite{diestel,Mats,bookthree}. Two excellent
references for the
general theory of asymptotic prime divisors in commutative Noetherian
rings are \cite{huneke-swanson-book} and \cite{McAdam}.
\section{Perfect matchings and persistence of associated primes}
In this section we give a characterization of graphs with a perfect
matching and show that the sets of associated primes of powers of an
edge ideal form an ascending chain. We continue using the
definitions and terms from the introduction.
Let $G$ be a graph with vertex set $X=\{x_1,\ldots,x_n\}$ and let
$I=I(G)\subset R$ be its edge ideal. In what follows
$F=\{f_1,\ldots,f_q\}$
denotes the set of all monomials $x_ix_j$ such that $\{x_i,x_j\}\in
E(G)$. As usual, we use
$x^a$ as an
abbreviation for $x_1^{a_1} \cdots x_n^{a_n}$ and we set
$|a|=a_1+\cdots+a_n$, where $a=(a_i)\in \mathbb{N}^n$. For convenience
we will consider $0$ to be an element of $\mathbb{N}$. We also use
$f^c$ as an abbreviation
for $f_1^{c_1}\cdots f_q^{c_q}$, where $c=(c_i)\in\mathbb{N}^q$.
\begin{definition}\label{parallelization-def}\rm Following Schrijver
\cite{Schr2}, the {\it
duplication\/} of a vertex $x_i$
of a graph $G$ means extending its vertex set $X$ by a
new vertex $x_i'$
and replacing
$E(G)$ by
$$
E(G)\cup\{(e\setminus\{x_i\})\cup\{x_i'\}\vert\, x_i\in e\in
E(G)\}.
$$
The {\it deletion\/} of $x_i$, denoted by
$G\setminus\{x_i\}$, is the graph formed from
$G$ by deleting the vertex $x_i$ and all edges containing
$x_i$. A graph obtained from $G$ by a sequence of
deletions and
duplications of vertices is called a {\it parallelization\/} of $G$.
\end{definition}
It is not difficult to verify that these two operations commute.
If $a=(a_i)$ is a vector in $\mathbb{N}^n$, we denote by $G^a$
the graph obtained from
$G$ by successively deleting any vertex $x_i$ with $a_i=0$ and
duplicating $a_i-1$ times any vertex $x_i$ if $a_i\geq 1$
(cf. \cite[p.~53]{golumbic}). The notion of a parallelization was used
in \cite{cm-mfmc,symboli} to describe the symbolic Rees algebra of an
edge ideal. This notion has its origin in combinatorial optimization
and has been used to describe the max-flow min-cut property of
clutters \cite{covers,Schr2}.
\begin{example}\label{april9-09} Let $G$ be the graph of Fig. 1
and let $a=(3,3)$. We set $x_i^1=x_i$ for $i=1,2$. The parallelization
$G^a$ is a complete bipartite graph
with bipartition $V_1=\{x_1^1,x_1^2,x_1^3\}$ and
$V_2=\{x_2^1,x_2^2,x_2^3\}$. Note that $x_i^k$ is a vertex, i.e., $k$
is an index not an exponent.
\vspace{0.8cm}
$$
\begin{array}{cccc}
\setlength{\unitlength}{.04cm}
\thicklines
\begin{picture}(80,35)
\put(0,10){\circle*{3.1}}
\put(0,40){\circle*{3.1}}
\put(-15,42){$x_1$}
\put(-15,3){$x_2$}
\put(0,10){\line(0,1){30}}
\put(10,15){$G$}
\put(-20,-15){\mbox{Fig. 1. Graph}}
\end{picture}
&
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\thicklines
\begin{picture}(80,35)
\put(30,40){\circle*{3.1}}
\put(60,40){\circle*{3.1}}
\put(0,40){\circle*{3.1}}
\put(-20,-15){\mbox{Fig. 2. Duplications of $x_1$}}
\put(0,10){\circle*{3.1}}
\put(-15,42){$x_1^1$}
\put(18,42){$x_1^2$}
\put(45,42){$x_1^3$}
\put(-15,3){$x_2^1$}
\put(0,10){\line(0,1){30}}
\put(0,10){\line(1,1){30}}
\put(0,10){\line(2,1){60}}
\put(35,15){$G^{(3,1)}$}
\end{picture}
&
\ \ \ \ \ \ \ \ \ \
&
\setlength{\unitlength}{.04cm}
\thicklines
\begin{picture}(80,35)
\put(0,10){\circle*{3.1}}
\put(-15,42){$x_1^1$}
\put(18,42){$x_1^2$}
\put(45,42){$x_1^3$}
\put(-15,3){$x_2^1$}
\put(18,3){$x_2^2$}
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\put(30,10){\line(1,1){30}}
\put(30,10){\line(-1,1){30}}
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\put(60,10){\line(0,1){30}}
\put(60,10){\line(-2,1){60}}
\put(60,10){\line(-1,1){30}}
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\put(30,40){\circle*{3.1}}
\put(60,40){\circle*{3.1}}
\put(70,15){$G^{(3,3)}$}
\put(-25,-15){\mbox{Fig. 3. Duplications of $x_1$ and $x_2$}}
\end{picture}
\end{array}
$$
\end{example}
\bigskip
\begin{definition}\label{matching-number} Two edges of $G$ are {\it
independent} if they do not intersect. A {\it matching\/} of $G$ is a
set of pairwise
independent edges. The {\it matching number\/} of
$G$, denoted by $\nu(G)$, is the size of any maximum matching of $G$.
A matching that covers all the vertices of
$V(G)$ is called a {\it perfect matching} of $G$.
\end{definition}
A very readable and comprehensive reference about matchings in finite
graphs is the book of Lov\'asz and Plummer \cite{matching-theory}.
Given a graph $G$, the {\it edge-subring\/} of $G$ is the
subring $K[G]=K[x_ix_j\vert\, \{x_i,x_j\}\in E(G)]$.
\begin{lemma}\label{multiset-perfect-matching} Let $G$ be a graph
with vertex set
$X=\{x_1,\ldots,x_n\}$ and let $a=(a_1,\ldots,a_n)\in\mathbb{N}^n$.
Then $G^a$ has a perfect matching if and only if $x^a\in K[G]$.
\end{lemma}
\begin{proof} We may assume that $a_i\geq 1$ for all $i$, because if
$a$ has zero entries we can use the induced subgraph on the vertex
set $\{x_i\vert\, a_i>0\}$. The vertex set of $G^a$ is
$$
X^a=\{x_1^1,\ldots,x_1^{a_1},\ldots,x_i^1,\ldots,x_i^{a_i},
\ldots,x_n^1,\ldots,x_n^{a_n}\}
$$
and the edges of $G^a$ are exactly those pairs of the form
$\{x_{i}^{k_{i}},x_{j}^{k_{j}}\}$ with $i\neq j$, $k_{i}\leq
a_{i}$, $k_j\leq a_j$, for some edge $\{x_i,x_j\}$ of $G$. We can
regard $x^a$ as an ordered multiset
$$
x^a=x_1^{a_1}\cdots x_n^{a_n}=(\underbrace{x_1\cdots x_1}_{a_1})\cdots
(\underbrace{x_n\cdots x_n}_{a_n})
$$
on the set $X$, that is, we can identify the monomial $x^a$ with the multiset
$$X_a=\{\underbrace{x_1,\ldots,x_1}_{a_1},\ldots,
\underbrace{x_n,\ldots,x_n}_{a_n}\}
$$
on $X$ in which each variable is uniquely identified with an integer
between $1$ and $|a|$. This
integer is the position, from left to right, of $x_i$ in $X_a$. There
is a bijective map
$$
\begin{array}{ccccccccccc}
1&2&\cdots&a_1&a_1+1&\cdots&a_1+a_2&\cdots&a_1+\cdots+a_{n-1}+1&\cdots&
a_1+\cdots+a_n\\
\downarrow&\downarrow&\cdots&\downarrow&\downarrow&\cdots&\downarrow
&\cdots&\downarrow&\cdots&\downarrow\\
x_1&x_1&\cdots&x_1&x_2&\cdots&x_2&\cdots&x_n&\cdots&x_n\\
\downarrow&\downarrow&\cdots&\downarrow&\downarrow&\cdots&\downarrow
&\cdots&\downarrow&\cdots&\downarrow\\
x_1^1&x_1^2&\cdots&x_1^{a_1}&x_2^1&\cdots&x_2^{a_2}&\cdots&x_n^1&\cdots&x_n^{a_n}.\\
\end{array}
$$
Hence if $G^a$ has a perfect matching, then the perfect matching
induces a
factorization of $x^a$ in which each factor corresponds to an
edge of $G$, i.e., $x^a\in K[G]$. Conversely, if $x^a\in K[G]$
we can factor $x^a$ as a product of monomials corresponding to edges
of $G$ and this factorization induces a perfect matching of $G^a$.
\end{proof}
Note that the process of passing from the vertex set $X$ to the
set $X^a$ and the multiset $X_a$ used in the lemma above can also be
used to view a
general monomial as a square-free monomial in a polynomial ring with
additional variables. This is referred to as {\em polarization} in
the literature. The copies of $x_i$ that are used are called shadows
of $x_i$. Conversely, a square-free monomial $M$ in the ring $K[X^a]$
can be viewed as a monomial in the ring $K[X]$ by setting the
exponent of $x_i$ to be the number of shadows of $x_i$ that divide
$M$. This process is called {\em depolarization}.
Given an edge $f=\{x_i,x_j\}$ of a graph $G$, we denote by $G^f$ or
$G^{\{x_i,x_j\}}$ the graph obtained
from $G$ by successively duplicating the vertices $x_i$ and $x_j$,
i.e., $G^f:=G^{\mathbf{1}+e_i+e_j}$, where $e_i$ is the $i${\it th} unit
vector in $\mathbb{R}^n$ and $\mathbf{1}=(1,\ldots,1)$.
\begin{example} Consider the graph $G$ of Fig. 4, where vertices
are labeled with $i$ instead of $x_i$. The duplication of the
vertices $x_1$ and
$x_2$ of $G$ is shown in Fig. 6.
$$
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\put(64,28){\lb{2}}\put(24,28){\lb{6}} \put(24,15){\lb{7}}
\put(8,-20){Fig. 4.
$G$}
\end{picture}
& \ \ &
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\begin{picture}(0,50)(-60,20)
\put(0,0){\circle*{4.2}} \put(60,0){\circle*{4.2}} \put(0,30){\circle*{4.2}}
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$$
\end{example}
\vspace{1.5cm}
Recall that ${\rm def}(G)$, the {\it
deficiency\/} of $G$, is given by ${\rm def}(G)=|V(G)|-2\nu(G)$,
where $\nu(G)$ is the matching number of $G$. Hence ${\rm def}(G)$ is
the number of vertices left uncovered by any maximum matching.
\begin{lemma}\label{feb9-2011} Let $G$ be a graph and
let $a\in\mathbb{N}^n$ and $c\in\mathbb{N}^q$. Then
\begin{itemize}
\item[{(a)}] $x^a=x^\delta f^c$,
where $|\delta|={\rm def}(G^a)$ and $|c|=\nu(G^a)$.
\item[{(b)}] $x^a$ belongs to $I(G)^k\setminus
I(G)^{k+1}$ if and only if $k=\nu(G^a)$.
\item[{(c)}] $(G^a)^f=(G^a)^{\{x_i,x_j\}}$
for any edge $f=\{x_i^{k_i},x_j^{k_j}\}$ of $G^a$.
\end{itemize}
\end{lemma}
\begin{proof} Parts (a) and (b) follow using the bijective map used
in the proof of
Lemma~\ref{multiset-perfect-matching}. To show (c) we use the
notation used in the proof of Lemma~\ref{multiset-perfect-matching}.
We now prove the inclusion $E((G^a)^f)\subset
E((G^a)^{\{x_i,x_j\}})$. Let $y_i$
and $y_j$ be the duplications
of $x_i^{k_i}$ and $x_j^{k_j}$ respectively. We also denote the
duplications of $x_i$ and $x_j$ by $y_i$ and $y_j$ respectively. The common
vertex set of $(G^a)^f$ and $(G^a)^{\{x_i,x_j\}}$ is
$V(G^a)\cup\{y_i,y_j\}$. Let $e$ be an edge of $(G^a)^f$. If
$e=\{y_i,y_j\}$ or $e\cap\{y_i,y_j\}=\emptyset$, then clearly $e$ is
an edge of $(G^a)^{\{x_i,x_j\}}$. Thus, we may assume that
$e=\{y_i,x_\ell^{k_\ell}\}$. Then $\{x_i^{k_i},x_\ell^{k_\ell}\}\in
E(G^a)$, so $\{x_i,x_\ell\}\in E(G)$. Hence $\{x_i,x_\ell^{k_\ell}\}$
is in $E(G^a)$, so $e=\{y_i,x_\ell^{k_\ell}\}$ is an edge of
$(G^a)^{\{x_i,x_j\}}$. This proves the inclusion ``$\subset$''. The
other inclusion follows using similar arguments (arguing backwards).
\end{proof}
\begin{theorem}{\rm (Berge; see
\cite[Theorem~3.1.14]{matching-theory})}\label{berge-formula} Let $G$
be a graph. Then
$${\rm def}(G)=\max\{c_0(G\setminus S)-|S|\, \vert\, S\subset V(G)\},$$
where $c_0(G)$ denotes the number of odd components $($components
with an odd number of vertices$)$ of a graph $G$.
\end{theorem}
We come to the main combinatorial result of this section.
\begin{theorem}\label{pepe-vila-berge} Let $G$ be a graph. Then
${\rm def}(G^f)=\delta$ for all $f\in E(G)$ if and only if ${\rm
def}(G)=\delta$ and $\nu(G^f)=\nu(G)+1$ for all $f\in E(G)$.
\end{theorem}
\begin{proof} Assume that ${\rm def}(G^f)=\delta$ for all $f\in
E(G)$. In general, ${\rm def}(G)\geq {\rm
def}(G^f)$ for any $f\in E(G)$. We proceed by contradiction. Assume
that ${\rm def}(G)>\delta$. Then, by Berge's theorem, there is an $S\subset V(G)$
such that $c_0(G\setminus S)-|S|>\delta$. We set $r=c_0(G\setminus S)$ and
$s=|S|$. Let $H_1,\ldots,H_r$ be the odd
components of $G\setminus S$.
Case (I): $|V(H_k)|\geq 2$ for some $1\leq k\leq r$. Pick an edge
$f=\{x_i,x_j\}$ of $H_k$. Consider the
parallelization $H_k'$ obtained from $H_k$ by duplicating the vertices $x_i$
and $x_j$, i.e., $H_k'=H_k^f$. The odd connected components of
$G^f\setminus S$ are $H_1,H_2,\ldots,H_{k-1},H_k',H_{k+1}\ldots,H_r$. Thus
$$
c_0(G^f\setminus S)-|S|>\delta={\rm def}(G^f).
$$
This contradicts
Berge's theorem when applied to $G^f$.
Case (II): $|V(H_k)|=1$ for $1\leq k\leq r$. Notice that in this case
$S\neq\emptyset$ because $G$ has no isolated vertices. Pick
$f=\{x_i,x_j\}$ an edge of $G$ with $\{x_i\}=V(H_1)$ and $x_j\in S$.
Let $y_i$ and $y_j$ be the duplications of $x_i$
and $x_j$ respectively. The odd components of
$G^f\setminus(S\cup\{y_j\})$ are $H_1,\ldots,H_r,\{y_i\}$.
Thus
$$
c_0(G^f\setminus(S\cup\{y_j\}))-|S\cup\{y_j\}|=c_0(G\setminus
S)-|S|>\delta={\rm def}(G^f).
$$
This again contradicts Berge's theorem when applied to $G^f$. Therefore
${\rm def}(G)={\rm def}(G^f)$ for all $f\in E(G)$. Consequently
$\nu(G^f)=\nu(G)+1$ for all $f\in E(G)$. The converse follows readily
using the definition of ${\rm def}(G)$ and ${\rm def}(G^f)$.
\end{proof}
The result of Theorem~\ref{pepe-vila-berge} depends upon the
deficiency of $G^f$ being constant for all $f$. In general,
the deficiencies of $G$ and $G^f$ need not be equal.
\begin{example} Consider the graph $G$ of Fig. 7, where vertices
are labeled with $i$ instead of $x_i$. The duplication of the vertices $x_3$ and
$x_4$ of $G$ is shown in Fig. 8.
$$
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$\mbox{def}(G)=2$}
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\put(-40,-40){Fig. 8. $\mbox{def}(G^{(1,1,2,2,1,1)})=0$}
\end{picture}
\end{array}
$$
\end{example}
\vspace{1.5cm}
The theorem of Berge is equivalent to the following classical result of Tutte
describing perfect matchings \cite{matching-theory}.
\begin{theorem}{\rm (Tutte; see
\cite[Theorem~2.2.1]{diestel})}\label{tutte-theorem}
A graph $G$ has a perfect matching if and only if
$c_0(G\setminus S) \leq | S | $ for all $S\subset V(G)$.
\end{theorem}
We give the following characterization of perfect matchings in terms
of duplications of edges.
\begin{corollary}\label{pepe-vila} Let $G$ be a graph.
Then $G$ has a perfect matching if
and only if $G^f$ has a perfect matching for every edge $f$ of $G$.
\end{corollary}
\begin{proof} Assume that $G$ has a perfect matching. Let
$f_1,\ldots,f_{n/2}$ be a set of
edges of $G$ that form a perfect matching of $V(G)$,
where $n$ is the number of vertices of $G$. If $f=\{x_i,x_j\}$ is any
edge of $G$ and $y_i$, $y_j$ are the duplications of the vertices
$x_i$ and $x_j$ respectively, then clearly
$f_1,\ldots,f_{n/2},\{y_i,y_j\}$ form a perfect matching of $V(G^f)$.
Conversely, if $G^f$ has a perfect matching for all $f\in E(G)$, then
${\rm def}(G^f)=0$ for all $f\in E(G)$. Hence, by
Theorem~\ref{pepe-vila-berge}, we get that ${\rm def}(G)=0$, so $G$
has a perfect matching.
\end{proof}
The following lemma will play an important role in the proof of the
main theorem. It uses the preceding combinatorial results about
matchings to prove an algebraic equality.
\begin{lemma}\label{jan29-2011} Let $I$ be the edge ideal of a graph $G$.
Then $(I^{k+1}\colon I)=I^k$ for $k\geq 1$.
\end{lemma}
\begin{proof} Let $F=\{f_1,\ldots,f_q\}$
be the set of all monomials $x_ix_j$ such that $\{x_i,x_j\}\in E(G)$. Given
$c=(c_i)\in\mathbb{N}^q$, we set $f^c=f_1^{c_1}\cdots f_q^{c_q}$. It
is well known that the colon ideal of two monomial ideals is a
monomial ideal, see for instance \cite[p.~137]{monalg}. In particular
$(I^{k+1}\colon I)$ is a monomial
ideal. Clearly $I^k\subset(I^{k+1}\colon I)$. To show the
reverse inclusion it suffices to show that any monomial of
$(I^{k+1}\colon I)$ is in $I^k$. Take $x^a\in(I^{k+1}\colon I)$. Then $f_ix^a\in I^{k+1}$ for
$i=1,\ldots,q$. We may assume that $f_ix^a\notin I^{k+2}$, otherwise
$x^a\in I^k$ as required. Thus $x^{a+e_i+e_j}\in I^{k+1}\setminus
I^{k+2}$ for any $e_i+e_j$ such that
$\{x_i,x_j\}\in E(G)$. Hence, by Lemma~\ref{feb9-2011}(b),
$\nu(G^{a+e_i+e_j})=k+1$ for any $\{x_i,x_j\}\in E(G)$, that is,
$(G^a)^{\{x_i,x_j\}}$ has a maximum matching of size $k+1$ for any edge
$\{x_i,x_j\}$ of $G$. With the notation used in the proof of
Lemma~\ref{multiset-perfect-matching}, for any edge $\{x_i^{k_i},x_j^{k_j}\}$ of $G^a$ we have
$$
(G^a)^{\{x_i^{k_i},x_j^{k_j}\}}=(G^a)^{\{x_i,x_j\}},
$$
see Lemma~\ref{feb9-2011}(c). Then, $(G^a)^f$ has a maximum matching
of size $k+1$ for any edge $f$
of $G^a$. As a consequence
$$
{\rm def}((G^a)^f)=(|a|+2)-2(k+1)=|a|-2k
$$
for any edge $f$ of $G^a$. Therefore, by
Theorem~\ref{pepe-vila-berge}, ${\rm def}(G^a)=|a|-2k$. Using
Lemma~\ref{feb9-2011}(a), we can write $x^a=x^\delta f^c$,
where $|\delta|={\rm def}(G^a)$ and $|c|=\nu(G^a)$. Taking degrees in
the equality $x^a=x^\delta f^c$ gives
$|a|=|\delta|+2|c|=(|a|-2k)+2|c|$, that is, $|c|=k$. Then $x^a\in I^k$
and the proof is complete.
\end{proof}
\begin{proposition}\label{jan29-2011-1}
Let $I=I(G)$ be the edge ideal of a graph $G$ and let
$\mathfrak{m}=(x_1,\ldots,x_n)$. If $\mathfrak{m}\in {\rm
Ass}(R/I^k)$, then $\mathfrak{m}\in {\rm
Ass}(R/I^{k+1})$.
\end{proposition}
\begin{proof} As $\mathfrak{m}$ is an associated prime of $R/I^{k}$,
there is $x^a\notin I^k$ such that $\mathfrak{m}x^a\subset I^k$.
By Lemma~\ref{jan29-2011} there is an edge $\{x_i,x_j\}$ of $G$ such
that $x_ix_jx^a\notin I^{k+1}$. Then, $x_\ell(x_ix_jx^a)\in I^{k+1}$
for $\ell=1,\ldots,n$, that is, $\mathfrak{m}$ is an associated prime
of $R/I^{k+1}$.
\end{proof}
To generalize from the maximal ideal to arbitrary associated primes,
we will use localization. Since this process frequently results in
disjoint graphs, we first recall the following fact about associated
primes.
\begin{lemma}\label{disjoint}{\rm (\cite[Lemma~3.4]{HaM}, see also
\cite[Lemma~2.1]{AJ})} Let $I$ be a square-free monomial ideal in $S = K[x_1,\dots,x_m,
x_{m+1}, \dots, x_r]$ such that
$I=I_1S + I_2S$, where $I_1 \subset S_1 = K[x_1, \dots, x_m]$
and $I_2 \subset S_2 = K[x_{m+1}, \dots, x_r]$. Then ${\mathfrak p}\in
{\rm Ass} (S/I^k)$
if and only if
${\mathfrak p}={\mathfrak p}_1S + {\mathfrak p}_2S$, where ${\mathfrak
p}_1 \in {\rm Ass} (S_1/I_1^{k_1})$ and ${\mathfrak p}_2 \in
{\rm Ass} (S_2/I_2^{k_2})$ with $(k_1-1) + (k_2-1) = k-1$.
\end{lemma}
Note that this lemma easily generalizes to an ideal $I=(I_1,\ldots,I_s)$ where the
$I_i$ are square-free monomial ideals in disjoint sets of variables.
Then ${\mathfrak p} \in {\rm Ass}(R/I^k)$ if and only if
${\mathfrak p}=({\mathfrak p}_1,\ldots , {\mathfrak p}_s)$, where
${\mathfrak p}_i \in {\rm Ass}(R/I_i^{k_i})$ with $(k_1-1)+\cdots
+(k_s-1)=k-1$.
Note that although $\mathfrak{p}_i$ is an ideal of $R$, the
generators of $\mathfrak{p}_i$ will generate a prime ideal in
any ring that contains those variables. We will abuse notation in the sequel by
denoting the ideal generated by the generators of $\mathfrak{p}_i$ in
any other ring by $ \mathfrak{p}_i$ as
well.
\medskip
We come to the main algebraic result of this paper.
\begin{theorem}\label{persistence-edge-ideals}
Let $G$ be a graph and let $I=I(G)$ be its edge
ideal. Then
$${\rm Ass}(R/I^k) \subset{\rm Ass}(R/I^{k+1})$$
for all $k$. That
is, the sets of associated primes of the powers of $I$ form an
ascending chain.
\end{theorem}
\begin{proof} Recall that we are assuming that $G$ has no isolated vertices.
Let $\mathfrak{p}$ be an associated prime of
$R/I^k$ and let $\mathfrak{m}=(x_1,\ldots,x_n)$ be the irrelevant maximal ideal
of $R$. For simplicity of notation we may assume that
$\mathfrak{p}=(x_1,\ldots,x_r)$. Then, the set $C=\{x_1,\ldots,x_r\}$ is a vertex cover
of $G$. By Proposition~\ref{jan29-2011-1}, we may assume that
$\mathfrak{p}\subsetneq\mathfrak{m}$. Write
$I_{\mathfrak p}=(I_2,I_1)_\mathfrak{p}$, where $I_2$ is the ideal of
$R$ generated by all square-free monomials of degree two $x_ix_j$ whose image,
under the canonical map $R\rightarrow R_\mathfrak{p}$,
is a minimal generator of $I_{\mathfrak p}$, and $I_1$ is the prime ideal
of $R$ generated by all variables $x_i$ whose image is a minimal generator of
$I_{\mathfrak p}$, which correspond to the isolated vertices
of the graph associated to $I_{\mathfrak p}$. The minimal generators
of $I_2$ and $I_1$ lie in $S=K[x_1,\ldots,x_r]$, and the two sets of
variables occurring in
the minimal generating sets of $I_1$ and $I_2$ (respectively) are
disjoint and their union
is $C=\{x_1,\ldots,x_r\}$. If $I_2=(0)$, then
${\mathfrak p}$ is a minimal prime of $I$ so it is an associated
prime of $R/I^{k+1}$. Thus, we may assume $I_2\neq(0)$. An important fact is that
localization
preserves associated primes, that is $\mathfrak{p}\in{\rm Ass}(R/I^k)$ if and only if
$\mathfrak{p}R_\mathfrak{p}\in{\rm
Ass}(R_\mathfrak{p}/(I_\mathfrak{p}R_\mathfrak{p})^k)$, see
\cite[p.~38]{Mats}. Hence,
$\mathfrak{p}$ is in ${\rm Ass}(R/I^k)$ if and only if $\mathfrak{p}$
is in ${\rm Ass}(R/(I_1,I_2)^k)$ if and only if $\mathfrak{p}$ is
in ${\rm Ass}(S/(I_1,I_2)^k)$. By Proposition~\ref{jan29-2011-1} and
Lemma~\ref{disjoint}, $\mathfrak{p}$ is an associated prime of
$S/(I_1,I_2)^{k+1}$. Hence, we can argue backwards to conclude that
$\mathfrak{p}$ is an associated prime of $R/I^{k+1}$.
\end{proof}
\begin{remark} Using Proposition~\ref{jan29-2011-1} and
Lemma~\ref{disjoint}, this result can also be shown by induction
on the number of variables because
localizing at $\mathfrak{p}\subsetneq\mathfrak{m}$ yields the ideal
$(I_1,I_2)$ in a polynomial ring with fewer than $n$ variables. Using
induction may be useful to extend
Theorem~\ref{persistence-edge-ideals} to other classes of monomial
ideals (for instance to edge ideals of clutters, see \cite{symboli}). In the case where
the conclusion of Lemma~\ref{jan29-2011} holds for a class of
square-free monomial ideals, this result immediately extends.
\end{remark}
\begin{corollary}\label{square-free-chain}
Let $I$ be a square-free monomial ideal and suppose $(I^{k+1} \colon
I)=I^k$ for $k \geq 1$. Then the sets of associated primes of the
powers of $I$ form an ascending chain.
\end{corollary}
\begin{proof}
As in Proposition~\ref{jan29-2011-1}, we first show that $\mf \in
{\rm Ass}(R/I^k)$ implies $\mf \in {\rm Ass}(R/I^{k+1})$. Assume $\mf
\in {\rm Ass}(R/I^k)$. Then there is a monomial $x^a \not\in I^k$
with $x_ix^a \in I^{k+1}$ for all $i$. By the hypothesis, $x^a
\not\in (I^{k+1} \colon I)$, so there is a square-free monomial
generator $e$ of $I$ (which can be viewed as the edge of a clutter
associated to $I$) with $ex^a \not\in I^{k+1}$. But
$x_iex^a=e(x_ix^a) \in I^{k+1}$ for all $i$, so $\mf \in {\rm
Ass}(R/I^{k+1})$.
Recall that since $I$ is finitely generated, $(I^{k+1} \colon
I)_{\mathfrak p}= (I^{k+1}_{\mathfrak p} \colon
I_{\mathfrak p})$. Thus $(I^{k+1}_{\mathfrak p} \colon
I_{\mathfrak p})=I^k_{\mathfrak p}$.
The remainder of the argument now follows from localization, as in the
proof of Theorem~\ref{persistence-edge-ideals}, after noting that
Lemma~\ref{disjoint} applies to an arbitrary square-free monomial
ideal.
\end{proof}
In \cite[Question~4.16]{edge-ideals} it was asked if the sets ${\rm
Ass}(R/I^k)$ form an ascending chain for all square-free monomial
ideals $I$. Corollary~\ref{square-free-chain} provides one possible
approach for answering this question for some classes of square-free
monomial ideals. However, this approach will not work for all
square-free monomial ideals, as can be seen by the following example.
\begin{example}\label{ass-powers-ce} Let
$R=\mathbb{Q}[x_1,\ldots,x_6]$ and let
$I$ be the square-free monomial ideal
$$
I=(x_1x_2x_5,\, x_1x_3x_4,\, x_1x_2x_6,\, x_1x_3x_6,\, x_1x_4x_5,\,
x_2x_3x_4,\, x_2x_3x_5,\, x_2x_4x_6,\, x_3x_5x_6,\, x_4x_5x_6).
$$
Using {\it Normaliz\/} \cite{normaliz2} together with {\em Macaulay\/}$2$
\cite{mac2}, it is seen that $I$ is a non-normal ideal such that
$(I^2 : I)=I$ and $(I^3 : I)\neq I^2$. Nevertheless, it is not hard to see that the
sets of associated primes of the powers of $I$ form an ascending
chain and that the index of stability of $I$ is equal to $3$.
\end{example}
It is also of interest to note that
for square-free monomial ideals, knowing that the sets ${\rm
Ass}(R/I^k)$ form an ascending chain immediately implies that the
sets ${\rm Ass}(I^{k-1}/I^{k})$ form an ascending chain as well. Thus
we get the following corollary of
Theorem~\ref{persistence-edge-ideals}. A similar corollary would
follow from Corollary~\ref{square-free-chain} as well.
\begin{corollary} Let $I=I(G)$ be the edge ideal of a graph $G$, then
${\rm Ass}(I^{k-1}/I^{k})$ form an ascending chain for $k\geq 1$.
\end{corollary}
\begin{proof} It follows from \cite[Lemma~4.4 ]{edge-ideals} and
Theorem~\ref{persistence-edge-ideals}.
\end{proof}
\section{Integral Closures and Stable Sets}
As mentioned in the Introduction, the sets of associated primes of the
integral closures of the powers of $I$ are also known to form an
ascending chain that stabilizes. In order to compare the stable sets
of the two chains ${\rm Ass}(R/I^k)$ and ${\rm
Ass}(R/\overline{I^k})$, we recall the following definition and
lemma.
\begin{definition}\rm Let $I=(x^{v_1},\ldots,x^{v_q})$ be a monomial
ideal of $R$. The {\it Rees algebra\/} of $I$, denoted by $R[It]$, is
the monomial subring
$$
R[It]=R[x^{v_1}t,\ldots,x^{v_q}t]\subset R[t].
$$
The ring ${\mathcal F}(I)=R[It]/{\mathfrak
m}R[It]$ is called the
{\it special fiber ring\/} of $I$. The Krull dimension of ${\mathcal
F}(I)$, denoted by $\ell(I)$, is called the {\it analytic spread\/}
of $I$.
\end{definition}
\begin{lemma}{\cite[Proposition~7.1.17,
Exercise~7.4.10]{monalg}}\label{analytic-spread-formula}
Let $I=(x^{v_1},\ldots,x^{v_q})$ be a monomial ideal and let $A$ be
the matrix with column vectors $v_1,\ldots,v_q$. If $\deg(x^{v_i})=d$ for all $i$, then
$$
\mathcal{F}(I)\simeq K[x^{v_1}t,\ldots,x^{v_q}t]\simeq K[x^{v_1},\ldots,x^{v_q}]\
\mbox{ and }\ \ell(I)=\dim\,
K[x^{v_1},\ldots,x^{v_q}]={\rm rank}(A).
$$
\end{lemma}
Once again, localization will allow us to reduce to the case of the
maximal ideal. To that end, we prove a result that characterizes when
$\mathfrak{m}$ is in the stable sets of ${\rm Ass}(R/I^k)$ and ${\rm
Ass}(R/\overline{I^k})$ .
\begin{proposition}\label{feb1-2011} Let $G$ be a graph. The
following are equivalent\/{\rm :}
\begin{itemize}
\item[(a)] $\mathfrak{m}\in {\rm Ass}(R/I(G)^k)$ for some $k$.
\item[(b)] The connected components of $G$ are non-bipartite.
\item[(c)] $\mathfrak{m}\in{\rm Ass}(R/\overline{I(G)^{t}})$ for some $t$.
\item[(d)] ${\rm rank}(A)=n$, where $A$ is the incidence matrix of $G$
and $n=|V(G)|$.
\end{itemize}
\end{proposition}
\begin{proof} The equivalence between (c) and (d) follows from
\cite[Theorem~3]{mcadam} because the analytic spread of $I$ is equal
to the rank of $A$, see Lemma~\ref{analytic-spread-formula}. The
equivalence between (b) and (d) follows from the fact that
${\rm rank}(A)=|V(G)|$ if $G$ is a connected non-bipartite graph and
${\rm rank}(A)=|V(G)|-1$ if $G$ is a connected bipartite graph, see
\cite[Lemma~8.3.2]{monalg}.
Let $G_1,\ldots,G_r$ be the connected components of $G$. We set
$S_i=K[V(G_i)]$ and $\mathfrak{m}_i=(V(G_i))$. Assume that
$\mathfrak{m}=(\mathfrak{m}_1,\ldots,\mathfrak{m}_r)$ is an associated
prime of $R/I(G)^k$ for some $k$. Then, by Lemma~\ref{disjoint},
there are positive integers $k_i$ such that $\mathfrak{m}_i$ is an
associated prime of $S_i/I(G_i)^{k_i}$. Therefore $G_i$ is
non-bipartite for all $i$ because edge ideals of bipartite graphs are
normally torsion-free \cite{ITG}. This proves that (a) implies (b).
Finally we prove that (b) implies (a).
Assume that $G_i$ is non-bipartite for all $i$. Then, by
\cite[Corollary~3.4]{AJ}, $\mathfrak{m}_i\in {\rm
Ass}(S_i/I(G_i)^{k_i})$ for $k_i\gg 0$. Then,
again by Lemma~\ref{disjoint}, it follows that $\mathfrak{m}$ is an
associated prime of $R/I(G)^k$ for some $k$.
\end{proof}
Combining Lemma~\ref{analytic-spread-formula} with
Proposition~\ref{feb1-2011}(d) illustrates the importance of the
analytic spread in determining associated primes. Localizing will
allow the use of these results for primes other than $\mathfrak{m}$,
but this will require control over the analytic spread of
the edge ideal of a disconnected graph.
\begin{lemma}\label{analytic-spread-additive} Let $L_1,L_2$ be
monomial ideals with disjoint sets of
variables. If $L_1,L_2$ are gene\-rated by monomials of degrees $d_1$
and $d_2$ respectively, then $\ell(L_1+L_2)=\ell(L_1)+\ell(L_2)$.
\end{lemma}
\begin{proof} We set $L=L_1+L_2$. Let $g_1,\ldots,g_r$ and
$h_1,\ldots,h_s$ be the minimal generating sets of $L_1$ and $L_2$
respectively that consist of monomials. By hypothesis $L_1$ (resp.
$L_2$) lives in a polynomial ring $K[\mathbf{x}]$ (resp.
$K[\mathbf{y}])$, where $\mathbf{x}=\{x_1,\ldots,x_q\}$ and
$\mathbf{y}=\{y_1,\ldots,y_m\}$. We set $R=K[\mathbf{x},\mathbf{y}]$.
The special fiber ring of $L$ can be written
as
$$
\mathcal{F}(L)\simeq
K[\mathbf{x},\mathbf{y},u_1,\ldots,u_r,z_1,\ldots,z_s]/(\mathbf{x},\mathbf{y},J),
$$
where $J$ is the presentation ideal of the Rees algebra $R[Lt]$ and
$u_1,\ldots,u_r,z_1,\ldots,z_s$ is a set of new indeterminates. The
ideal $J$ is the kernel of the map
$$
K[\mathbf{x},\mathbf{y},u_1,\ldots,u_r,z_1,\ldots,z_s]\rightarrow
R[Lt],\ \ \ \ \ x_i\mapsto x_i,\, y_j\mapsto y_j,\, u_i\mapsto g_it,\,
z_j\mapsto h_jt.
$$
Since $J$ is a toric ideal, there is a generating set of $J$
consisting of binomials of the form
$$
x^{\alpha}y^{\beta}u^{\gamma}z^\delta-x^{\alpha'}y^{\beta'}u^{\gamma'}z^{\delta'}
$$
such that $x^{\alpha}y^{\beta}g^{\gamma}h^{\delta}
t^{|\gamma|+|\delta|}=
x^{\alpha'}y^{\beta'}g^{\gamma'}h^{\delta'}t^{|\gamma'|+|\delta'|}$.
From this equation we get $x^\alpha g^\gamma=x^{\alpha'}g^{\gamma'}$, $y^\beta
h^\delta=y^{\beta'}h^{\delta'}$ and
$t^{|\gamma|+|\delta|}=t^{|\gamma'|+|\delta'|}$. Hence
$$
|\alpha|+d_1|\gamma|=|\alpha'|+d_1|\gamma'|,\
|\beta|+d_2|\delta|=|\beta'|+d_2|\delta'|,\
{|\gamma|+|\delta|}={|\gamma'|+|\delta'|}.
$$
We claim that $\deg(x^{\alpha}y^{\beta})=0$ if and only if
$\deg(x^{\alpha'}y^{\beta'})=0$. Assume that
$\deg(x^{\alpha}y^{\beta})=0$, i.e., $\alpha=\beta=0$. From the first
equality we have $|\alpha'|=d_1(|\gamma|-|\gamma'|)$. From the second
and third equality we get
$$
|\beta'|+d_2|\delta'|=d_2|\delta|=d_2({|\gamma'|+|\delta'|}-|\gamma|)\
\Rightarrow\ |\beta'|=d_2(|\gamma'|-|\gamma|).
$$
As $|\alpha'|\geq 0$ and $|\beta'|\geq 0$, we get $\gamma-\gamma'=0$.
Thus $\alpha'=\beta'=0$. This proves the claim. Therefore one has the
following simpler expression for the special fiber ring of $L$
\begin{equation}\label{feb13-11}
\mathcal{F}(L)\simeq
K[u_1,\ldots,u_r,z_1,\ldots,z_s]/P\simeq
K[g_1t,\ldots,g_rt,h_1t,\ldots, h_st],
\end{equation}
where $P$ is the toric ideal of $K[g_1t,\ldots,g_rt,h_1t,\ldots,
h_st]$. Let $A_1$ (resp. $A_2$) be the matrix whose columns are the
exponent vectors of the monomials $g_1t,\ldots,g_rt$ (resp.
$h_1t,\ldots,h_st$) and let $A$ be the matrix whose columns are the
exponent vectors of $g_1t,\ldots,g_rt,h_1t,\ldots,h_st$. The sets of
variables $\mathbf{x}$ and $\mathbf{y}$ are disjoint. Therefore
${\rm rank}(A)={\rm rank}(A_1)+{\rm rank}(A_2)$. Since
$$
\mathcal{F}(L_1)=K[g_1t,\ldots,g_rt]\ \mbox{ and }\
\mathcal{F}(L_2)=K[h_1t,\ldots,h_st],
$$
using Lemma~\ref{analytic-spread-formula} and Eq.~(\ref{feb13-11}), it follows that
$\ell(L)=\ell(L_1)+\ell(L_2)$.
\end{proof}
\begin{remark} When $L_2$ is generated by a set of variables, which is
the case that we really need, the
lemma follows at once from \cite[Corollary~6.2, p.~43]{McAdam} because in this case the
set of variables form an asymptotic sequence over $L_1$ in the sense of
\cite{McAdam}.
\end{remark}
The sets ${\rm Ass}(R/I^i)$ and ${\rm Ass}(R/\overline{I^i})$
stabilize for large $i$. The next result shows that, for edge ideals, their
corresponding stable sets are equal.
\begin{theorem}\label{ass=assic} Let $I$ be the edge ideal of a graph
$G$. There exists
a positive integer $N$ such that ${\rm Ass}(R/I^k)={\rm
Ass}(R/\overline{I^k})$ for
$k\geq N$.
\end{theorem}
\begin{proof} Recall that we are assuming that $G$ has no isolated vertices. By
\cite{brod}
there is a positive integer $N_1$ such that ${\rm Ass}(R/I^{N_1})={\rm
Ass}(R/I^k)$ for $k \geq N_1$, and by \cite{ratliff,ratliff-increasing},
there is a positive integer $N_2$ such that ${\rm
Ass}(R/\overline{I^{N_2}})={\rm Ass}(R/\overline{I^{k}})$
for $k\geq N_2$. Let $N={\rm max}\{N_1,N_2\}$, and assume that $k\geq
N$. First we show the
inclusion ``$\subset$''. Take $\mathfrak{p}$ in
${\rm Ass}(R/I^k)$.
Case (I): $\mathfrak{p}=\mathfrak{m}$. By
Proposition~\ref{feb1-2011}, $\mathfrak{p}\in{\rm
Ass}(R/\overline{I^i})$ for some $i$. Hence $\mathfrak{p}\in{\rm
Ass}(R/\overline{I^k})$ because the sets ${\rm
Ass}(R/\overline{I^j})$ form an ascending sequence, see
\cite[Proposition~3.4, p.~13]{McAdam}.
Case (II): $\mathfrak{p}=(x_1,\ldots,x_r)\subsetneq\mathfrak{m}$. Let
$I_1$, $I_2$, and $S$ be as in the proof of
Theorem~\ref{persistence-edge-ideals} and let $X_i$ be the set of
variables that occur in the minimal generating set of $I_i$. Notice
that $\mathfrak{p}=(X_1,X_2)$. As $\mathfrak{p}$ is an associated
prime of $S/(I_1+I_2)^{k}$, applying Lemma~\ref{disjoint} to
$I_1S+I_2S$, where we regard $I_i$
as an ideal of $S_i=K[X_i]$, we can write ${\mathfrak p}={\mathfrak
p}_1S + {\mathfrak p}_2S$, where ${\mathfrak
p}_1 \in {\rm Ass} (S_1/I_1^{k_1})$ and ${\mathfrak p}_2 \in
{\rm Ass} (S_2/I_2^{k_2})$, with $(k_1-1) + (k_2-1) = k-1$. Notice
that $\mathfrak{p}_i=(X_i)$. Thus, applying
Proposition~\ref{feb1-2011} to the graph $G_2$ associated to
$I_2$, we get that the rank of the incidence matrix $A_{G_2}$ of
$G_2$ is $|X_2|$. On the other hand $\ell(I_2)$, the analytic spread
of $I_2$, is equal to the Krull dimension of the edge subring
$K[G_2]$,
which is equal to the rank of
$A_{G_2}$ (see Lemma~\ref{analytic-spread-formula}). Since
$I_1$ is generated by $|X_1|$ variables, one has
$\ell(I_1)=|X_1|$. By Lemma~\ref{analytic-spread-additive}, the
analytic spread $\ell(I_1+I_2)$ is equal to
$|X_1|+|X_2|={\rm ht}(\mathfrak{p})$. Thus, using
\cite[Theorem~3]{mcadam}, we conclude that
$\mathfrak{p}\in S/\overline{(I_1+I_2)^i}$ for $i\gg 0$. Then,
$\mathfrak{p}R_\mathfrak{p}\in
R_\mathfrak{p}/\overline{(I_1+I_2)_\mathfrak{p}^i}=
R_\mathfrak{p}/\overline{I_\mathfrak{p}^i}$. Consequently, by
\cite[Corollary, p.~38]{Mats} and the fact that the
integral closure of ideals commute with localizations, we get
$\mathfrak{p}\in R/\overline{I^k}$.
The inclusion ``$\supset$'' holds for any ideal $I$ of a commutative
Noetherian ring $R$ by a result of Ratliff
\cite{ratliff,ratliff-increasing}, see \cite[Proposition~3.17]{McAdam}
for additional details.
\end{proof}
\begin{corollary}\label{ntf-ass-corollary} Let $G$ be a graph and let $I$ be its edge ideal.
Then $I$ is normally torsion-free if and only if
${\rm Ass}(R/\overline{I^i})={\rm Ass}(R/I)$ for $i\geq 1$.
\end{corollary}
\begin{proof} $\Rightarrow$) This implication follows at once by
noticing that $I$ is normal, i.e., $\overline{I^i}=I^i$ for $i\geq 1$.
$\Leftarrow$) Since ${\rm Ass}(R/I)\subset{\rm
Ass}(R/{I^i})$ for $i\geq 1$, it suffices to show that
${\rm Ass}(R/{I^i})\subset{\rm Ass}(R/I)$ for $i\geq 1$.
Let $\mathfrak{p}$
be an associated prime of $R/{I^i}$ and let $N$ be the index
of stability of $I$. Then, by Theorem~\ref{persistence-edge-ideals},
$\mathfrak{p}$ is an associated prime of $R/I^{N}$. Hence, by
Theorem~\ref{ass=assic}, $\mathfrak{p}$ is an associated prime of
$R/\overline{I^k}$ for $k\gg 0$. Thus by hypothesis $\mathfrak{p}$ is
an associated prime of $I$.
\end{proof}
\begin{procedure}\label{mac-norm-proc} The following simple procedure for {\it
Macaulay\/}$2$ (version 1.4) decides
whether ${\rm Ass}(R/I^3)$ is contained in ${\rm Ass}(R/I^4)$ and
whether we have the equality ${\rm Ass}(R/I^3)={\rm Ass}(R/I^4)$. It
also computes $\overline{I^4}$ and
decides whether ${\rm Ass}(R/I^4)$ is equal to ${\rm
Ass}(R/\overline{I^4})$.
\begin{verbatim}
R=QQ[x1,x2,x3,x4,x5,x6,x7,x8,x9];
load "normaliz.m2";
I=monomialIdeal(x1*x2,x2*x3,x1*x3,x3*x4,x4*x5,x5*x6,x6*x7,x7*x8,x8*x9,x5*x9);
isSubset(ass(I^3),ass(I^4))
ass(I^3)==ass(I^4)
(intCl4,normRees4)=intclMonIdeal I^4;
intCl4'=substitute(intCl4,R);
ass(monomialIdeal(intCl4'))==ass(I^4)
\end{verbatim}
\end{procedure}
The next example was computed using version 1.4 of {\em
Macaulay\/}$2$ \cite{mac2}. This version allows the use of {\it Normaliz\/}
\cite{normaliz2} inside {\em Macaulay\/}$2$ in order
to compute the integral closure of a monomial ideal and the normalization of the Rees
algebra of a monomial ideal. Example~\ref{intcl1.m2} shows that
although the stable sets of ${\rm Ass}(R/{I}^{i})$ and ${\rm
Ass}(R/\overline{I^{i}})$ are equal, they do not
need to be reached at the same power.
\begin{example}\label{intcl1.m2} Let $R=\mathbb{Q}[x_1,\ldots,x_9]$ and let
$I=I(G)$ be the edge ideal of the graph below (Fig. 9). Notice that
this example was computed without using Theorem~\ref{ass=assic}.
$$
\setlength{\unitlength}{.04cm} \thicklines
\begin{picture}(0,40)(40,-5)
\put(0,0){\circle*{4.2}} \put(30,0){\circle*{4.2}} \put(60,0){\circle*{4.2}}
\put(-20,20){\circle*{4.2}} \put(-20,-20){\circle*{4.2}}
\put(80,20){\circle*{4.2}}\put(80,-20){\circle*{4.2}}
\put(110,20){\circle*{4.2}}\put(110,-20){\circle*{4.2}}
\put(0,0){\line(-1,-1){20}}\put(0,0){\line(-1,1){20}}\put(0,0){\line(1,0){60}}
\put(60,0){\line(1,1){20}}
\put(60,0){\line(1,-1){20}}\put(-20,-20){\line(0,1){40}}
\put(80,20){\line(1,0){30}}\put(80,-20){\line(1,0){30}}
\put(110,-20){\line(0,1){40}}
\newcommand{\lb}[1]{\tiny $#1$}
\put(-27,20){\lb{1}} \put(-27,-20){\lb{2}} \put(-7,-2){\lb{3}}\put(30,3){\lb{4}}
\put(64,-2){\lb{5}} \put(73,20){\lb{6}}\put(115,20){\lb{7}}\put(115,-20){\lb{8}}
\put(73,-21){\lb{9}}
\put(-35,-40){Fig. 9. Graph $G$ with non-normal $I(G)$}
\end{picture}
$$
\vspace{1.5cm}
Using {\em Macaulay\/}$2$ (see Procedure~\ref{mac-norm-proc}),
together with the fact that the index of stability of $I$ is at most
$8$ \cite[Corollary~4.3]{AJ} and
the fact that the stable set of ${\rm Ass}(R/\overline{I^i})$ is
contained in the stable set
of ${\rm Ass}(R/{I^i})$ \cite[Proposition~3.17]{McAdam}, we get
\begin{eqnarray*}
I^i=\overline{I^i},\ i=1,2,3,\
\overline{I^4}=I^4+(x^a),\, \overline{I^5}=I^5+I(x^a), \mbox{ where }
x^a=x_1x_2x_3x_5x_6x_7x_8x_9,&&\\
{\rm Ass}(R/I^i)={\rm Ass}(R/\overline{I^i})\ \mbox{for }\ i\neq 4\
\mbox{ and }\ {\rm Ass}(R/\overline{I^4})\subsetneq{\rm Ass}(R/I^4),& &\\
{\rm Ass}(R/I)\subsetneq{\rm Ass}(R/I^2)\subsetneq{\rm
Ass}(R/I^3)\subsetneq{\rm Ass}(R/I^4)={\rm
Ass}(R/I^i)\ \mbox{for }\ i\geq 4,& &\\
{\rm
Ass}(R/\overline{I^3})\subsetneq{\rm
Ass}(R/\overline{I^4})\subsetneq{\rm Ass}(R/\overline{I^5})={\rm
Ass}(R/\overline{I^i})\ \mbox{ for }\ i\geq 5.& &
\end{eqnarray*}
\end{example}
\medskip
\begin{center}
ACKNOWLEDGMENT
\end{center}
\noindent The authors would like to thank an anonymous referee for
providing us
with useful comments and suggestions.
| {
"timestamp": "2011-05-23T02:02:52",
"yymm": "1103",
"arxiv_id": "1103.0992",
"language": "en",
"url": "https://arxiv.org/abs/1103.0992",
"abstract": "Let G be a graph and let I be its edge ideal. Our main result shows that the sets of associated primes of the powers of I form an ascending chain. It is known that the sets of associated primes of I(i) and intcl(I(i)) stabilize for large i, where \"intcl\" denotes integral closure and I(i) denotes the i-th power of I. We show that for edge ideals their corresponding stable sets are equal. To show our main result we use a classical result of Berge from matching theory and certain notions from combinatorial optimization.",
"subjects": "Commutative Algebra (math.AC); Combinatorics (math.CO)",
"title": "Associated primes of powers of edge ideals",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9728307684643189,
"lm_q2_score": 0.7279754430043072,
"lm_q1q2_score": 0.7081969096410331
} |
https://arxiv.org/abs/2107.14799 | Generating Boolean Functions on Totalistic Automata Networks | We consider the problem of studying the simulation capabilities of the dynamics of arbitrary networks of finite states machines. In these models, each node of the network takes two states 0 (passive) and 1 (active). The states of the nodes are updated in parallel following a local totalistic rule, i.e., depending only on the sum of active states. Four families of totalistic rules are considered: linear or matrix defined rules (a node takes state 1 if each of its neighbours is in state 1), threshold rules (a node takes state 1 if the sum of its neighbours exceed a threshold), isolated rules (a node takes state 1 if the sum of its neighbours equals to some single number) and interval rule (a node takes state 1 if the sum of its neighbours belong to some discrete interval). We focus in studying the simulation capabilities of the dynamics of each of the latter classes. In particular, we show that totalistic automata networks governed by matrix defined rules can only implement constant functions and other matrix defined functions. In addition, we show that t by threshold rules can generate any monotone Boolean functions. Finally, we show that networks driven by isolated and the interval rules exhibit a very rich spectrum of boolean functions as they can, in fact, implement any arbitrary Boolean functions. We complement this results by studying experimentally the set of different Boolean functions generated by totalistic rules on random graphs. | \section{Introduction}
Consider an automata network $ \mathcal{A}= (G=(V,E), Q=\{0,1\},\{f_v: v \in V\})$ where $G=(V,E)$ such that $|V|=n$ is an undirected finite graph, $Q$ a finite set of states and $\{f_v: v \in V\}$ is a collection of functions called \textit{local functions} such that each local function $f_v$ depends only in neighbourhood of $v$ in $G$ given by $N_v = \{u\in V/ (u,v)\in E\}$ i.e. $f_v:N_v \to Q$. We define $F:Q^n \to Q^n$ as the \textit{global transition function} of the automata network given by $F(x)_v = f_v(x|_{N_v})$ for all $x \in Q^n$ and for all $v \in V$ where $x|_{N_v}$ are the coordinates of $x$ that are representing the neighbours of $v$. Formally, $x|_{N_v} \in Q^{N_v}$ and for all $u \in N_v$ we have $(x|_{N_v})_u = x_u.$ In this paper we consider only the case in which every local function $f_v$ is \textit{totalistic}, i.e., those which depends of the sum of the active values (states $1$'s ) in its neighbourhood. Suppose that the maximum degree $\Delta(G)$ on $G$ is such that $\Delta(G) = \Delta$, so the sum may take values in the set
$\{0,1, ..., \Delta\} $, i.e, given a configuration $x\in Q^{|V|}$ we have $f_v(x|_{N_v})= 1$ if and only if $S= \sum \limits_{u\in N_v} x_u \in \mathcal{I}_v = \{a_1, ..,a_s\}$ where $\{a_1, ..,a_s\}\subseteq \{0,1, ..., \Delta\}$
During the rest of the paper, we will refer to each local rule by the digit associated to $\mathcal{I}_v $, so if $\mathcal{I}_v = \{a_1,\hdots, a_s\}$ then, the totalistic rule number will be $a_1a_2, \cdots, a_s$. For instance, rule $25$ means that the associated vertices become $1$ if and only if the sum if either $2$ or $5$.
\section{The problem}
Given a totalistic automata $A$ assume that $\vec{0}$ is a fixed point, i.e. if $S=0$ then $f_v(x_u/u \in V_v)= 0$. Consider the set $X= \{ \vec{0}=X^1, ..., X^p\}$ of fixed point of $A$ and an arbitrary set of vertices $\{i_1, ...i_l,i_{l+1}\}\subseteq V$, the first $l$ being ``inputs'' and the last one $i_{l+1}$ the output.
Consider now a fixed point $X^k$. Over it consider every assignment of values $(z_1, ..., z_l) \in \{0,1\}^{|V|}$ to the inputs $i_1, ...i_l$. From that one consider the configuration $y_i = x_i^k if i \not \in \{i_1, ...i_l\}.$ Else $y_i= z_i$. Apply now $F^T(y)$ and inspect the value at site $i_{l+1}$, that is to say, $(F^T(y))_{i_{l+1}}$ which we consider as the output site. By doing so for every $z\in\{0,1\}^{|V|}$ we obtain a set of Boolean functions. In this context we associate the complexity of the automaton $A$ (over the graph $G$ with the totalistic rule $F$) to the distribution of different Boolean functionы found by previous procedure. More different Boolean functions aere generated by the procedure more complex we will consider the automaton. In this context we will say the complexity of $A$ is given by the number of different Boolean functions that could be generated over any fixed point we consider. Of course we have to make this definition precise.
\textcolor{red}{As a matter of fact, to our knowledge this complexity notion was introduced by AA in several papers ( we have to include the bibliography and discussion).}
In this context there are at least two different things. First, the distribution of Boolean functions determined for a given $A$ over a family of random graphs. Second, is the following problem:
to measure the complexity of the totalistic functions. Namely, given a family of graphs and a totalistic function we will say that it is complex if the probability to generate an arbitrary Boolean function is positive. This notion is completely related with the P-Completeness, because if the automaton $A$, for a prediction problem , say PRE, is P-complete, since we deduce usually that from the possibility to construct arbitrary circuits. So, let's consider that a automaton is COMPLEX if and only if the probability to generate an arbitrary Boolean function is positive. If not we will say that is simple. We will develop this approach with two points of view, theoretically, by proving when an automaton is, COMPLEX; and experimentally,by studying really in a family of graph what is, for a given totalistic automaton the distribution of its Boolean functions.
\section{Results}
First, we will characterize those totalistiuc aurtomata which are ``simple''. Apart fromthe obvious constant functions 0 and 1 (i.e. for every value of $S$, the next state is 0; or for every value $S\not =0$ next value is 1) we have the following result: \\
\begin{prop}
The OR function, say, $f_v(x_u/u\in V_v)= 1$ if and only if $S \geq 1$, only generates by previous procedure OR functions, so it is a simple network. Similarily, the AND function, i.e., $f_v(x_u/u\in V_v)= 1$ if and only if $S=|V_v|$, only generate AND functions.
\end{prop}
\begin{proof} Since we assume $G$ connected, for the OR the only fixed points are $\vec{0}$ or $\vec{1}$. So, taking the fixed point $\vec{0}$ for instance for any perturbation we take (changing a finite number of $0$s for $1$s, and a selected site as output, say $ k= i_{l+1}$ after $T$ steps, we have $F^T(y)=A^Tu$ where $A$ is the incidence matrix of the graph $G$, so $g(z)= (A^Tu)_{i–{l+1}}$ is an OR function. Actually if $A^T= B =(b_ij)$ we have
$ g(z)= b_k1u_1\lor ....\lor kn$
where $n$ is the matrix dimension, i.e. $|V|= n$.
Since the AND is the dual function of the OR, we get a similar result.
\end{proof}
The second result is related with totalistic threshold automata, that is to say,
$f_v(x_u/u \in V_v)= 1$ if and only if $S\ge\theta_v $, where $2\leq \theta_v \leq |V_v|-1$. As an important particular case we have the majority functions i.e., by considering for every $v\in V$ $\theta_v =\frac{|V_v|}{2}$
\begin{prop} The class of threshold automata may generate every monotone Boolean function.
\end{prop}
\begin{proof} Recall a monotone Boolean function is a disjuntion of conjuctions. So it is enough to demonstrate wires, the AND and the OR functions and duplication of signals
\textcolor{red}{
ACA VA MONO para $\theta$ arbitrario.
}
Since we are dealing with an arbitrary graph it is enough to consider non planar graphs in order to cross wires. On the other hand, any Boolean function can be written as a circuit so, previous results are enough to built any Boolean function.
\end{proof}
\textcolor{red}{
Remark: The particular case for the majority is MONO
}
Let us analyse other classes of totalistic automata. We will define the class of``isolated'' totallistic automata when there exist $k \in < \{1,2, ...,q\}$, the domain of the sum $S$, such that $f_v(x_u/u \in V_v)= 1$ if $S=k$ and $f_v(x_u/u \in V_v)= 0$ for $k-1$ and $k+1$. For instance, $F=256$ has the isolated value $S=2$.
\begin{prop}Every totalistic isolated automaton implements an arbitrary Boolean functions.
\end{prop}
\begin{proof} Let us consider case $k\geq 2$. One may construct AND and XOR, so by using negation and by the de Morgan the OR.
\textcolor{red}{
juntar MONO x FOTO
}
For the case $k=1$ (recall $S=0$ implies $f_v(x_u/u \in V_v)= 0$), similarly to previous case we get the XOR and from the XOR the negation.
\textcolor{red}{MONO
}
To build the OR it is enough to desynchronise signals, i.e. one of the OR inputs arrives later than the other (after that we resynchornise signals by shortening selected wires).
\end{proof}
The remaining cases to study deal with non-monotone (threshold) and non-isolated functions, i.e. those that accepts an ``isolated interval'' (\textcolor{red}{see figure MONOO}); i.e. there exist $k_1$ and $k_2$, $1\leq k_1\leq k_2\leq q $ such that
$f_v(x_u/u \in V_v)= 0$ for $ S\in\{k_1,k_2\} $ and $F_v(x_u/u \in V_v)= 1$ for
$k\in\{k_1, ...,k_2\}$. For instance, for $q \ge 8$ the function $F$ 2356 has two isolated intervals, [23] and [56]. From this definition we have the proposition:
\begin{prop} The class of totalistic functions which at least one isolated interval may generate every Boolean function.
\end{prop}
\begin{proof} One may construct wires, OR, AND and XOR
\textcolor{red}{
SEE MONOS
}
\end{proof}
So, from that we have proved the following Theorem
\begin{prop}.
Every other totalistic network
1) Networks governed by constant functions, OR and AND, only may generates trivial functions.
2) Networks governed by threshold functions generate every monotone Boolean function.
3) All the others totalistic networks may generate any Boolean function (not only monotone).
\end{prop}
\section{Discussion}
Since the circuits are constructed from perturbations of quiescent configurations, in order for some some vertices to be active (in the state 1) they must have the certain number of its neighbours be active. In.fact, if for instance a vertex to remain active needs the sum $S=p$ each one of them has to be connected with the other so the graph admits as a sub-graph a complete graph of $p+1$ vertices, $K_{p+1}$. In this context it is not difficult to see that for $q\leq 5$ one may construct planar sets of vertices such that every one has at least $q$ vertices actives but $q \geq 6$ those graph are non-planar. \textcolor{red}{VER MNONOS PARA CONSTRUCCIONES HASTA 5.}
For every totalistic rule where we have to put some fixed active vertices essentially we may do that in as planar graph when the maximum degree is less than 6. Thus we have the following result.
\begin{prop} Every totalistic rule with isolated points or intervals such that maximum degree $\leq 6 $ one may construct every Boolean function in a planar graph.
\end{prop}
\end{document}
\section{Introduction}
\section{Preliminaries}
An automata network is a tuple $ \mathcal{A}= (G=(V,E), Q=\{0,1\},\mathcal{F} = \{f_v: v \in V\})$ where $G=(V,E)$ is an undirected finite graph such that $|V|=n,$ $Q$ a finite set of states called $\textit{alphabet}$ and $\mathcal{F} = \{f_v: v \in V\}$ is a collection of functions called \textit{local functions} such that each local function $f_v:N_v \to Q$ depends only in neighbourhood of $v$ in $G$ given by $N_v = \{u\in V: uv \in E\}.$ We define $F:Q^n \to Q^n$ as the \textit{global transition function} of the automata network defined by $F(x)_v = f_v(x|_{N_v})$ for all $x \in Q^n$ and for all $v \in V$ where $x|_{N_v}$ are the coordinates of $x$ that are representing the neighbours of $v$. Formally, $x|_{N_v} \in Q^{N_v}$ and for all $u \in N_v$ we have $(x|_{N_v})_u = x_u.$ In this paper we consider only the case in which every local function $f_v$ is \textit{totalistic}, i.e., those which depend on the sum of the active values (states $1$'s ) in its neighbourhood. Suppose that the maximun degree $\Delta(G)$ on $G$ is such that $\Delta(G) = \Delta$, so the sum may take values in the set
$\{1, ..., \Delta\} $, i.e, given a configuration $x\in Q^{n}$ we have $f_v(x|_{N_v})= 1$ if and only if $\sum \limits_{u\in N_v} x_u \in \mathcal{I}_v = \{a_1, ..,a_s\}$ where $\{a_1, ..,a_s\}\subseteq \{1, ..., \Delta\}.$ We will call the set $\mathcal{I}_v$ the activation set of $f_v.$
During the rest of the paper, we will refer to each local rule by the digit associated to $\mathcal{I}_v $, so if $\mathcal{I}_v = \{a_1,\hdots, a_s\}$ then, the totalistic rule number will be $a_1a_2, \cdots, a_s$. For instance, rule $25$ means that the associated vertices become $1$ if and only if the sum if either $2$ or $5$. Note that, as each of these rules depend also in the set of values that each node in some neighborhood will take, we can consider that totalistic rules are defined over the set $S_{\Delta} = \{1,\hdots, \Delta\}$. This will be useful as, in the next sections, as we would like to work with a fixed set of totalistic rules and use them to define automata networks over different graphs. In this regard, as we will always work over some class of graphs $\mathcal{G}$ in which every graph have at most degree $\Delta$, we note that each totalistic rule having an activation set $I \subseteq \{1,\hdots,\Delta\}$ will be well defined over any graph in $\mathcal{G}$. For example, if $\Delta = 10$ rule $25$ will be well-define over any graph in $\mathcal{G}.$ More precisely, for any totalistic function $f$ such that $I_f \subseteq S_{\Delta}$ we can asume that $f: S_{\Delta} \to \{0,1\}$ and that $f(s) = 1$ if and only if $s \in I_f$.
\subsection{The problem}
Let $\mathcal{A} = (G, \mathcal{F})$ be a totalistic automata network with global transition function $F$. We say that $\overline{x} \in Q^n$ is a fixed point of $\mathcal{A}$ if $F(\overline{x}) = \overline{x}.$ Note that, by definition, $\vec{0}$ is always a fixed point of $\mathcal{A}$. Consider the set $\text{Fix}(\mathcal{A})$ of fixed point of $\mathcal{A}$. Note that particularly, as $\vec{0}$ is a fixed point, $\text{Fix}(\mathcal{A}) \not = \emptyset$. Now we consider two arbitrary disjoint sets of vertices $I = \{i_1, \hdots, i_l\}\subseteq V$ and $O = \{o_1,\hdots,o_s\} \subseteq V$ for $s,l \geq 1$. We call these sets the \textit{input} set of $\mathcal{A}$ and the \textit{output} set of $\mathcal{A}$ respectively. As we are interested in the simulation of logic gates, during the rest of the paper, we will focus in the case in which at most two outputs, that is to say $O \subseteq \{o_1,o_2\},$ with special emphasis in the case $O = \{o\}$. In fact, we will only use the case with two outputs for our theoretical results, in order to simplify our reasoning.
Let $t \geq 0$, we call a tuple $(I,o,t ) \in 2^{V} \times V \times \mathbb{N}$ an $I/O$ setting for $\mathcal{A}$. The variable $t$ represents some specific time step in which we are interested to analyze the output of the automata network in $o$. We call $t$ an \textit{observation} time.
Consider now a fixed point $\overline{x} \in \text{Fix} (\mathcal{A})$ with $I/O$ setting $(I,o,t) \in 2^{V} \times V \times \mathbb{N}.$ Let $z = (z_1, ..., z_l) \in \{0,1\}^{l}$ be an assignation of values for the input set $I = \{i_1, ...i_l\}$. We say that a configuration $y \in Q^n$ is a perturbation of $\overline{x}$ by $z$ if $y_u = \overline{x}_u$ for all $u \in V \setminus I$ and $y_u= z_u$ for $u \in I$. Now, given some observation time $t\geq 0$ we are interested in studying all possible state of the output $o$ after $t$ time steps given different assignation of variables for the inputs. More precisely, given a perturbation $y$ of $\overline{x}$ for the last $I/O$ setting we define a realization of $\overline{x}$ as a boolean function $g^{(I,o,t)}_{\overline{x}}: \{0,1\}^l \to \{0,1\}$ such that $g^{(I,o,t)}_{\overline{x}}(y) = F^t(y)_o.$ Let $\binom{V}{l}$ the collection of subsets of $V$ with size $l$. Considering all possible combinations of inputs of certain size $l$ and possible outputs for a fixed time, we define the spectrum of Boolean functions with $l$ inputs generated by $\overline{x}$ at time $t$ as the set of realizations of $\overline{x}$ given by $\mathbb{F}^{t,l}_{\overline{x}} = \{g^{(I,o,t)}_{\overline{x}}: (I,o) \in 2^V \times V\}$ and the spectrum of functions with $l$ inputs of $\mathcal{A}$ at time $t$ as the set $\mathbb{F}^{t,l}_{\mathcal{A}} = \bigcup \limits_{\overline{x} \in \text{Fix}(\mathcal{A})} \mathbb{F}_{\overline{x}}$. Note that the set of realizable boolean functions depends on the observation time that we are considering and in the interaction graph of the network. Thus, it is strictly related to classical dynamical properties such as transient length and the attractor landscape of the automata network.
\subsection{Totalistic classes of functions}
We recall that $S_{\Delta}$ is the set of natural numbers $\{1,\hdots,\Delta\}$ for some bound $\Delta \leq 2$. One of the advantages of working with totalistic rules is that we can fix a collection of these rules and change the underlying interaction graph in order to define different automata networks. For example, if we fix $n \in \mathbb{N}$ and we consider the set of local functions $\{f_k: S_{\Delta} \to \{0,1\}, k=1,\hdots,n\}$ in which every rule is the rule $1$, i.e., $f_k(s) = 1$ if and only if $s = 1$ for every $k \in V$, we can, for every graph $G = (V,E)$ with $n$ nodes, define an automata network $\mathcal{A}_G = (G,\{\tilde{f}_k: N(k) \to \{0,1\}\})$ where $\tilde{f}(x|_{N(k)}) = f(\sum_{v \in N(k)} x_v)$. In the next sections, we will write simply $f$ for refering both to $\tilde{f}$ and $f$ in the previous context in order to simplify the notation. Note now that if $G$ and $G'$ are two different graphs with $n$ nodes then, their corresponding automata networks $\mathcal{A}_G$ and $\mathcal{A}_{G'}$ will have the same global rule. Of course, the degree of the graph $G$ plays a fundamental role in this definition. For example, suppose that some rule $f_k$ needs $l$ active neighbors in order to activate itself, i.e. $f_k(x|_S) = 1$ if and only if $\sum \limits_{v \in S} x_v = l$ for some, $l \in \mathbb{N}$, for every $S\subseteq V$ and $x \in \{0,1\}^n.$ In addition, suppose that the degree of node $k$ is less than $l$. In that case, $f_k$ will be fixed in the initial configuration. Roughly this will not have an important effect in our results (nor theoretical or numerical) as we will work with connected graphs with bounded maximum degree and, in addition, it will allow us to keep the latter set-up simple. In addition, this way to generated different automata networks from the same class of local functions is more practical in order to study the dynamical behavioir of different totalistic rules for a fixed large collections of randomly generated graphs.
Roughly, in this context we will say that the complexity of $\mathcal{A}$ is given by the number of different Boolean functions that could be generated over any fixed point we consider. Formally, we define the simulation complexity for a $\mathcal{A}$ by $\rho(\mathcal{A},t,l) = |\mathbb{F}^{t,l}_{\mathcal{A}}|.$ More different Boolean functions are generated by the latter procedure, more complex we will consider the automaton. In this context we will say the complexity of $\mathcal{A}$ is given by the number of different Boolean functions that could be generated over any fixed point we consider. In addition, we can define the complexity of a class of totalistic functions $\mathcal{F}$ related to some class of graph $\mathcal{G}$ as $\rho(\mathcal{F},\mathcal{G},t,l) =|\mathbb{F}^{G,t,l}_{\mathcal{F}}|$
\subsection{The spectrum as a measure of complexity}
Generally speaking, the spectrum of a set or a class of totalistic rules is a measure of how many different types of a boolean gates it can simulate. Nevertheless, it is well known that every boolean function $f: \{0,1\}^r \to \{0,1\}^s$ can be represented by a directed graph $C$ in which every node is a boolean gate. An asynchronous evaluation of every boolean gate in $C$ performs the evaluation of the function $f$. In this regard, we are interested in the study sets of totalistic rules which not only are capable of simulating different boolean gates, but to organize them in way that, they can simulate the evaluation of a boolean circuit.
Roughly, our main idea is to show that some class $\mathcal{F}$ is able to simulate a complete set of logical gates, for example, $\textsc{AND}, \textsc{NOT}, \textsc{OR}$. Let $\mathcal{A}_{\textsc{AND}},\mathcal{A}_{\textsc{NOT}}$ and $\mathcal{A}_{\textsc{OR}}$ be the automata networks that simulates each of this gates. Then, we will try to combine its different underlying graphs in order to simulate an arbitrary circuit $C$, by considering for each gate $g$ in c one of the latter graphs and then, try to connect them somehow. Note that this process is not straightforward as every automata network simulates a logical gate through its dynamics and so, it is not trivial how we should glue them in order to generate coherent global dynamical behavior. In simple words, what we want to achieve is, exhibit a big automata network that has a set of small subgraphs simulating logic gates. This big network will simulate the evaluation of $C$ through its dynamics, in the sense that, by identifying a group of nodes as "input nodes" we can read the same output we would have read after the evaluation of $C$ by reading the state of another group of nodes labelled as "output nodes", after some time steps. More precisely, we introduce the following definition:
\begin{definition}
Let $\Delta \in \mathbb{N}$ and let $\mathcal{G}$ be a collection of graphs with maximum degree at most $\Delta$. Let $\mathcal{F}$ be a set of totalistic rules and $f:\{0,1\}^r \to \{0,1\}^s$ an arbitrary boolean function. We say that $\mathcal{F}$ simulates $f$ in $\mathcal{G}$ if there exist $n \in \mathbb{N}$ such that $n = r^{\mathcal{O}(1)}$, a graph $G_n = (V,E) \in \mathcal{G}$ with $|V| = n$ with global rule $F_n$ and $t = n^{\mathcal{O}(1)}$ such that $f(y) = (F_n^t(x|_I))|_O$ for every $y \in \{0,1\}^r$, for some $x \in \{0,1\}^n$ and for some sets $I,O \subseteq V$ such that $|I|=r$ and $|O|=s$.
\end{definition}
\begin{remark}
Note that, in the latter definition, the sets $I$ and $O$ are one-to-one related to the input and outputs of the graph. This means the simulation is very strong in the sense that inputs and outputs are represented by one node in the network.
\end{remark}
On the other hand, we remark that the latter definition is intrinsically related to the computation complexity of some decision problems that have been studied in order to measure the complexity of the dynamics of automata networks. In particular, it is closely related to \textit{prediction problem}. Given an automata network $\mathcal{A} = (G,\mathcal{F})$, this problem is roughly defined by a given configuration $x \in Q^n$ and a node $v \in V$ for which we would want to know if the state of $v$ will change at some point in the orbit of $x$. More precisely, we ask if there exist $t$ such that $F^t(x)_v \not = x_v.$ Depending on $\mathcal{A}$, it can be shown that the complexity of this decision problem is closely related to the capability of $\mathcal{A}$ of simulating the evaluation of an arbitrary boolean circuit \cite{bibid}. This is because, roughly, depending on the rules defining $\mathcal{A}$ this problem can be verified or solved in polynomial time, and thus the complexity bounds are deduced through a reduction to canonical problems such as \textsc{Circuit value problem} or \textsc{SAT}.
\section{Results}
In this section we present different results on the complexity of different totalistic rules in the sense of its spectrum. In particular, we focus in exhibiting for different classes of totalistic rules, small automata networks, that we call gadgets, that can simulate logic gates. Then, we show how we can combine them in order to simulate arbitrary boolean functions. In this regard, we present a classification based in the structure of the activation sets $\mathcal{I}_v$ of the different totalistic rules defining a given class of rules $\mathcal{F}$. In fact, we start studying the simple case in which $\mathcal{I}_v = \{1\}$ which lead us to the study of disjunctive and conjunctive networks, and , in a more general way, to study rules which dynamics are defined by a some sort of matrix product. Then, we study the classic case in which $\mathcal{I}_v$ is given by some interval $[\theta_v,\delta_v] \subseteq \{0,\hdots, \Delta\}.$ This class includes the well-known threshold networks which, as we show in this section, have the capability of simulating any monotone boolean network. Finally, we explore the case in which isolated activation values are considered. Roughly we explore the case in which if $a \in \mathcal{I}_v$ then $a-2,a-1,a+1 \not \in \mathcal{I}_v$ and we find that this class is also capable of simulating arbitrary boolean networks.
\subsection{Matrix-defined rules}
We start by studying of canonical cases of totalistic rules such as disjunctive (conjunctive) networks. Let $Q = \{0,1\}$ and let $\mathcal{G}$ be a family of graphs. We say that some totalistic rule $f:S_{\Delta}\to Q$ is disjunctive if it takes the value $1$ if and only if there exist at least one $1$ in its assignment, i.e., $\mathcal{I}_f = \{1\}$. We say that a set or class of totalistic rules $\mathcal{F}$ is disjuntive if every $f \in \mathcal{F}$ is disjuntive. Analogously, we can define a conjunctive totalistic function $f$ over a graph $G$ in some node $v \in V(G)$ by defining the transition to $1$ only in the case in which every neighbour of $v$ is in state $1$, i.e $\mathcal{I}_v= \{\delta_v\}$. Note that in this case we cannot define the rule independently of the interaction graph. Nevertheless, both rules are completely analogous as it suffices to change the role of $1$ and $0$ in order to change from disjunctive to conjunctive and vice versa. As a consequence of these, and in order to simplify following reasoning we focus on disjunctive rules but of course all of the next results are valid also for conjunctive rules.
\begin{lem}
Let $\mathcal{G}$ an arbitrary family of graphs and take some graph $G \in \mathcal{G}$. Let $\mathcal{D} = (G,\mathcal{F})$ be an automata network where $\mathcal{F}$ is disjuntive then, the spectrum of $\mathcal{D}$ are only constant gates (everything goes to $1$ or $0$) and disjunctive gates (OR gates). More precisely, for any $t,l\geq 1$ we have $\mathbb{F}^{(t,l)}_{\mathcal{D}} \subseteq \{0,1\} \cup \{\vee_J\}_{J \subseteq I} $ where $\vee :\{0,1\}^l \to \{0,1\}$ is such that $\vee (z_1,\hdots,z_l) = \bigvee \limits^l_{k=1} z_l$ for $J \subseteq I.$
\end{lem}
\begin{proof}
Let $G \in \mathcal{G}$, $t,l\geq 1$ and define $\mathcal{A} = (G,\mathcal{D}(G))$. Let $F$ its global transition function. Fix an input $I \subset V.$ Note that $\text{Fix}(\mathcal{A}) = \{\vec{0},\vec{1}\}.$ Also note that there exist a matrix $A \in M_n(\{0,1\})$ such that $F^t(x) = A^t \oplus x = \bigvee \limits_{i \in N_v(G^t) \cap I} x_i $ for all $t \geq 0$. In particular, $A$ is the adjacency matrix of $G$. Note that for every $i \in I$ there exist a path between $i$ and $o$ of length $t$ if and only if $(A^t)_{io} = 1$ and thus, the $t$-th power of $A$ define the power graph $G^t$. By definition we have that
\begin{equation}
F^t(x)_o = (A^tx)_o = \bigvee \limits_{i \in N_v(G^t)} x_i = \left( \bigvee \limits_{i \in N_v(G^t) \cap I} x_i \right) \vee \left( \bigvee \limits_{i \in N_v(G^t) \cap V \setminus I} x_i \right).
\label{eq:or}
\end{equation}
Now, note that, if we start perturbing $\vec{1}$ then we have that $\mathbb{F}^{(G,t,l)}_{\vec{1}} \subseteq \{1,\vee\}$ as the only case in which we can do something different than $1$ is when $l = \delta_o$, $I = N_o$ and $t = 1$.
On the other hand, as $G$ is connected, consider $P_1, \hdots, P_l$ all the minimum length paths connecting each node in $I$ to $o$. Let $d_1,\hdots, d_l$ the lengths of each of the latter paths and let $d = \min \limits_{i \in \{1,\hdots,l\}} d_i$ and $D = \max \limits_{i \in \{1,\hdots,l\}} d_i.$ If we perturb $\vec{0}$, we have that, for $t\leq d$ where we have $g^{I,o,t} \equiv 0.$ For $t \geq d$ we can have that not all the nodes in $I$ are connected in $G^t$ with $o$ and thus by (\ref{eq:or}) we have that $g^{I,o,t} \equiv \vee_{J}$ for some $J \subseteq I.$ Finally if $t \geq D$ then, we can have influence of external nodes in $N_v(G^t) \cap V\setminus.$ but because the influence is given by an OR function, we have two possible cases: a) one external have state $1$ in time $t$ and then $ g^{I,o,t} \equiv 1$ or all stay in state $0$ and then there is no influence. In every case we conclude that $\mathbb{F}^{(t,l)}_{\vec{0}} \subseteq \{0,1,\vee\} \cup \{\vee_J\}_{J \subseteq I}$ and thus the lemma holds.
\label{lemma:OR}
\end{proof}
\begin{theo}
For every class of graphs $\mathcal{G}$ and for every $l,t \geq 1$, we have that $\rho(\mathcal{G},\mathcal{D},l,t) \leq 2(2^{l-1}+1)$.
\end{theo}
\begin{proof}
The bound on the spectrum of $\mathcal{D}$ follows directly from Lemma \ref{lemma:OR}. The theorem holds.
\end{proof}
\begin{remark}
Note that if $G= (V,E)$ is such that every totalistic function takes the value $1$ when the sum of the states of each neighbour of certain vertex is odd, then, we have the known XOR rule. More precisely, we have the XOR rule if for every $v \in V$ we have that $\mathcal{I}_v = \{a \in \{0,\hdots,\delta_v\}: a \text{ is odd.} \}$ Also the global rule of that automata network in that case can be seen as a matrix product, i.e. $F^t(x) = A^t x$ where the product is the usual product in $\mathbb{F}_2.$ Thus, the previous result holds for XOR rules.
\end{remark}
\subsection{Threshold networks}
In this section we introduce a class of totalistic functions called threshold functions. Roughly, in this family we have that a function takes the value $1$ if the sum of the states of the neighbours of the corresponding vertex is in some interval $[\theta_v,\delta_v]$ where $\delta_v$ is the degree of the vertex $v$ that we are considering and $\theta_v$ is some positive threshold. We present this notion in the following definition:
\begin{definition}
A totalistic function $f: S_{\Delta} \to \{0,1\}$ is threshold if there exist some positive integer $\theta$ such that $ [\theta,\Delta] \subseteq \mathcal{I}_f $, where $I_f$ is the activation set of $f$.
\end{definition}
We denote by $\mathcal{T}$ the class of all totalistic functions that are threshold i.e. $f \in \mathcal{T}$ if and only if $f$ is threshold. We will show that there exist a class of graphs $\mathcal{G}$ for which $\mathcal{T}$ simulates any monotone boolean function.
\begin{lem}
There exist two automata networks $\mathcal{A}_2 = (G_1=(V_1,E_1),\mathcal{F}_1)$ and $\mathcal{A}_2 = (G_2=(V_2,E_2),\mathcal{F}_2)$ with global rules $F_1$ and $F_2 $ respectively, such that $\mathcal{F}_i \in \mathcal{T}$ and that :
\begin{enumerate}
\item $\wedge(x,y) = F^2_1(z_1)_{o} = F^2_1(z_1)_{o'}.$
\item $\vee(x,y) = F^2_2(z_2)_{o} = F^2_2(z_2)_{o'},$
\end{enumerate}
for some $z_i \in \{0,1\}^{|V_i|},$ $i=1,2.$
\label{lem:andorthreshold}
\end{lem}
\begin{proof}
Consider the graph $G_1$ and $G_2$ given in Figure \ref{fig:andthrehsold} and Figure \ref{fig:orthrehsold}. Observe that $\overline{z}_1 = (0,0,0,0,0)$ and $\overline{z}_2 = (0,0,0,0,0)$ are fixed points for $F_1$ and $F_2$ respectively. Then, we can define $z_1$ and $z_2$ as a perturbation of these fixed points as it is shown in Figures \ref{fig:andthrehsold} and \ref{fig:orthrehsold}. As we stated in the last section, it suffices to define $\theta_v \in \{1, \delta_v\}$ in order to define an AND or an OR function. Observe that this is exactly the threshold defined for each vertex in Figure \ref{fig:andthrehsold} and Figure \ref{fig:orthrehsold}. The result follows from the calculations in latter figures.
\end{proof}
\begin{figure}
\includegraphics[scale=0.85]{imagenes/ANDthreshold}
\caption{AND gadget for the class of threshold totalistic functions.}
\label{fig:andthrehsold}
\end{figure}
\begin{figure}
\includegraphics[scale=0.85]{imagenes/ORthreshold}
\caption{OR gadget for the class of threshold totalistic functions.}
\label{fig:orthrehsold}
\end{figure}
\begin{theo}
Let $r,s \in \mathbb{N}$ and $f:\{0,1\}^r \to \{0,1\}^s$ be a monotone boolean function. There exist a collection of graphs $\mathcal{G}$ such that $\mathcal{T}$ simulates f in $\mathcal{G}.$
\label{teo:thershold}
\end{theo}
\begin{proof}
Fix $r,,s \in \mathbb{N}$ and $f: \{0,1\}^r \to \{0,1\}^s$ an arbitrary function. It is well known that $f$ can be represented by a boolean circuit $C_f:\{0,1\}^{l} \to \{0,1\}.$ More precisely, for every variable assignation $y \in \{0,1\}^r$ the evaluation of the circuit computes $f(z)$. We are going to show that there exist some $t \in \mathbb{N}$, a graph $G=(V,E)$ and a set of threshold rules $\mathcal{F} = \{f_v:\{0,1\}^{N(v)} \to \{0,1\}\}$ defining an automata network $\mathcal{A}_f$ such that its associated global rule $F$ is such that $f(y) = F(x|_I)|_O$ for some sets nodes $I,O \subseteq V$ and some $x$ depending on $y$. Let $D_f$ the digraph defining circuit $C_f$. Without loss of generality, we can assume that $C_f$ is monotone, i.e., any gate computes only an AND or an OR gate. In other words, any node $v \in V(D_f)$ is labelled by a symbol $l(d) \in \{\wedge, \vee\}$ which represent the corresponding gate in the circuit. We define $G$ in the following way: for each $v \in V(D_f)$ that is not an input we assign one of the gadgets $\varphi(v)$ in Figure \ref{fig:andthrehsold} or Figure \ref{fig:orthrehsold} according to $l(v)$. For input gates we consider input nodes of gadgets representing gates in the first layer. Note that in order to represent output gates it suffices to consider output nodes in some gadget given by Figure \ref{fig:andthrehsold} or Figure \ref{fig:orthrehsold}. In addition, we can assume that $\delta^{+}_v = \delta^{-}_v = 2.$ Note also that $\varphi(v)$ has two possible outputs that we have called $o$ and $o'$ in Figure \ref{fig:andthrehsold}. We are going to define edges in $G$ locally by the connections in each gadget $\varphi(v)$ for each $v \in V(D_f)$ and also we identify the output of gadget $\varphi(v)$ with one of the inputs of gadget $\varphi(v')$ if $v' \in N^{-}(v)$. Note that $|V(G)| \leq \sum \limits_{v \in V(D_f)} |\varphi(v)| = 5 |V(D_f)| = r^{\mathcal{O}(1)}.$ From previous lemma we now that $\varphi(v)$ computes $\wedge(x,y)$ or $\vee(x,y)$ where $x,y$ are its inputs. We know also that it is done in uniform time $t=2$ and there is also two possible choices for the output $o$ and $o'$ which receive the signal carrying this computation at the same time. We define now an automata network $(G,\mathcal{F})$ where $\mathcal{F}$ contains all the rules defined for each gadget $\varphi(v)$ for each $v \in V(D_f).$ We define sets $I$ and $O$ as the nodes in $G$ corresponding to input gates in $D_f$ and the output nodes of gadgets representing output gates in $D_f$. Now, we locally set every gadget $\varphi(v)$ to its fixed point configuration and we call it $x$. We assign $z =(z_1,z_2,\hdots, z_r)$ to each of the inputs of corresponding input gadgets. We claim that in time $t = 2 \text{deph}(D_f) = r^{\mathcal{O}(n)}$ global function satisfies $f(z)|_I = F^t(x)|_{O}.$ In fact, it is not difficult to see that inductively, in $t_1 = 2$ all the gadgets in the first layer compute the assignation $(z_1,z_2,\hdots, z_r)$ and each of the outputs that are associated inputs in the first layer have now this information as a perturbation of their fixed point configuration $x$. Now assume that in some time $t = 2k$, gadgets in the $k$-th layer are computing the information received from layer $k-1.$ Again, because of Lemma \ref{lem:andorthreshold} we know that each gadget $\varphi(v)$ produces consistently an AND or an OR computation of its inputs in uniform time $t = 2$ and thus, $k+1$-th layer computes information of $k$-th layer in time $2k+2 = 2(k+1)$. Then, the claim holds. As a consequence of the claim we have that $(G,\mathcal{F})$ simulates $f$. The result holds.
\end{proof}
\subsubsection{Majority rules}
An important example is the case in which each local rule will change to $1$ when the majority of the nodes in the neighbourhood of its associated node $v$ is in state $1$. More precisely, when $\theta_v = \frac{\delta_v}{2}.$ When this happens, we say that local rule $f_v$ is a majority rule. Of course this depend on the graph. Now we will show that we can simulate any monotone boolean network by using only majority rules. Analogously to the previous result, we show first that we can find AND and OR functions as a part of some automata networks defined by majority rules.
\begin{lem}
There exist two automata networks $\mathcal{A}_2 = (G_1=(V_1,E_1),\mathcal{F}_1)$ and $\mathcal{A}_2 = (G_2=(V_2,E_2),\mathcal{F}_2)$ with global rules $F_1$ and $F_2 $ respectively, such that $\mathcal{F}_i$ are majority rules and
\begin{enumerate}
\item $\wedge(x,y) = F^2_1(z_1)_{o} = F^2_1(z_1)_{o'}.$
\item $\vee(x,y) = F^2_2(z_2)_{o} = F^2_2(z_2)_{o'},$
\end{enumerate} for some $z_i \in \{0,1\}^{|V_i|},$ $i=1,2.$
\label{lem:andormaj}
\end{lem}
\begin{proof}
Consider the graph $G_1$ and $G_2$ given in Figure \ref{fig:andmaj} and Figure \ref{fig:ormaj}. We define $\overline{z}_1 = (0,0,0,0,0)$ and $\overline{z}_2 = (0,0,0,0,0)$. The result follows from the calculations in latter figures.
\end{proof}
\begin{figure}
\includegraphics[scale=0.85]{imagenes/ANDmaj}
\caption{AND gadget for the class of majority totalistic functions.}
\label{fig:andmaj}
\end{figure}
\begin{figure}
\includegraphics[scale=0.85]{imagenes/ORmaj}
\caption{OR gadget for the class of majority totalistic functions.}
\label{fig:ormaj}
\end{figure}
\begin{theo}
Let $r,s \in \mathbb{N}$ and $f:\{0,1\}^r \to \{0,1\}^s$ be a monotone boolean function. There exist an automata network $\mathcal{A}_f = (G,\mathcal{F})$ with global rule $F$ such that $\mathcal{F}$ are majority rules such that there exist $t= r^{\mathcal{O}(1)}$ satisfying $f(y) = F(x|_I)|_O$ for all $y \in \{0,1\}^r$, some sets $I,0 \subseteq V$ and some $x$ depending on $y$.
\end{theo}
\begin{proof}
The proof of this result is completely analogous to Theorem \ref{teo:thershold}.
\end{proof}
\subsection{Isolated totalistic rules}
Now we introduce a new class of totalistic rules that we call \textit{isoladed}. In general, the class of isolated totalistic rules are rules that are activated by a precise level of activation in the neighbourhood of a given node. In fact, these rules will be activated if and only if the amount of active neighbours is exactly some value $\alpha$ and will be $0$ for any other value in sufficiently large enough interval containing $\alpha$. We precise this notion in the following definition:
\begin{definition}
A totalistic rule $f:S_{\Delta} \to \{0,1\}$ is isolated if there exist some positive integer $\alpha\geq3$ such that $[\alpha-2,\alpha+1] = \{\alpha-2,\alpha-1, \hdots, \alpha+1\} \cap \mathcal{I}_f = \{\alpha\}.$
\end{definition}
For example the rule $3$ is isolated because configurations that have an amount of $1$s in the interval $[1,4]$ will only produce $1$ as image if they have exactly $3$ ones. Note that, for example, any other totalistic rule of the form $3a$ with $a\geq 5$ will be isolated with $\alpha = 3$. In the next section, we will call the value $\alpha$ an isolated value for some fixed rule. For example $3$ is an isolated value for rule $35$ and so for rule $3$. Note also that one fixed isolated rule can have multiple isolated values. For example, rule $36$ has $3$ as isolated value and also $6$.
\begin{lem}
For each $\alpha \geq 3$ there exists an automata network $\mathcal{A}_\alpha= (G_\alpha,\mathcal{F}_\alpha)$ such that every $f \in \mathcal{F}_\alpha$ is a totalistic function with isolated value $\alpha$ and such that its global rule $F_{\alpha}$ satisfies that: $\exists i_1, i_2 ,o_1,o_2\in V(G): F_{\alpha}^3(x)_{o_j} = \textbf{NAND}(x|_{i_1},x|_{i_2}), j=1,2$ for any $x$ that is a perturbation in $i_0$ and $i_1$ of some $z \in \text{Fix}(\mathcal{A}_\alpha)$, i.e. $x_v = z_v$ for all $v \not \in \{i_1,i_2\}.$ In particular, NAND gate is in the spectrum of $\mathcal{A}_{\alpha}$ for $t=3$ and $l = 2$.
\label{lemma:NAND}
\end{lem}
\begin{proof}
Let $ \alpha \geq 3.$ We exhibit explicitly the structure and dynamics of $\mathcal{A}_\alpha$ in Figure \ref{fig:NANDiso}. Note that graph structure strongly depends on the fact that complete graphs $K_{\alpha +1}$ are stable connected components for state $1$ in the sense that nodes inside this clique will be always in state $1.$ From the figure it is clear that fixed point $z$ is given by the state in which $z_v = 0$ for any $v$ which is not part of one the two cliques in the graph and that computation of NAND is a consequence of a perturbation of $z$.
\end{proof}
\begin{figure}
\includegraphics[scale=0.5]{imagenes/NANDiso0.pdf}
\includegraphics[scale=0.5]{imagenes/NANDiso2.pdf}
\includegraphics[scale=0.5]{imagenes/NANDiso1.pdf}
\includegraphics[scale=0.5]{imagenes/NANDiso3.pdf}
\caption{NAND gadget for $\alpha$-uniform isolated totalistic rules with $\alpha \geq 3.$ Grey $K_{\alpha+1}$ components are fixed in state $1$.}
\label{fig:NANDiso}
\end{figure}
During the rest of this section we will call the automata network in Figure \ref{fig:NANDiso} a NAND gadget. We will now show that the class of $\alpha$-uniform totalistic isolated rules is complete for some class of graphs $\mathcal{G}$ that we will exhibit. Nevertheless, in order to do that, we need to define two more gadgets: a clock gadget and a clocked NAND gadget.
\begin{figure}
\includegraphics[scale=0.8]{imagenes/clockiso.pdf}
\caption{Clock gadget for $\alpha$-uniform isolated totalistic rules with $\alpha \geq 3.$ Grey $K_{\alpha+1}$ components are fixed in state $1$.}
\label{fig:clockiso}
\end{figure}
\begin{lem}
For each $\alpha \geq 3$ and $d \geq 1$ there exists an automata network $\mathcal{A}_{\alpha,d}= (G_{\alpha,d},\mathcal{F}_{\alpha,d})$ such that $\mathcal{F}_{\alpha,d} \in \mathcal{S}_\alpha$ and such that its global rule $F_C$ satisfies that there exists $o \in V(G)$ such that $F_C^s(x)_{o} = 1$ for $0\leq s \leq d-1$ and $F_C^{d}(x)_{o} = 0$ for some $x \in \{0,1\}^n$.
\end{lem}
\begin{proof}
See Figure \ref{fig:clockiso} for the structure of the gadget and the definition of $x$. Note that the gadget works as a wire defined by its path structure that carries the $1$ signal which perturbs one node in one of the complete graphs $K_{\alpha+1}$ at the end of the path after $d$ time steps. This is because each node in the path has exactly $\alpha-1$ neighbours in state $1$ and thus, incoming signal allow them to change its state.
\end{proof}
Finally we introduce the clocked NAND gadget in the following lemma.
\begin{lem}
For each $\alpha \geq 3$ and $d \geq 1$ there exists an automata network $\mathcal{A}_{\alpha,d}= (G_{\alpha,d},\mathcal{F}_{\alpha,d})$ such that $\mathcal{F}_{\alpha,d} \in \mathcal{S}_\alpha$ and such that its global rule $F_{CN}$ satisfies that there exist $i_1,i_2,o_1,o_2 \in V(G)$ such that $F_{CN}^s(w)_{o_j} = 1, j=1,2$ for $0\leq s \leq d-1$ and $F_{CN}^{d+3}(x)_{o_j} = \textbf{NAND}(F^d(w)|_{i_1},F^d(w)|_{i_2}), j=1,2$ for some $w \in \{0,1\}^n$
\end{lem}
\begin{proof}
In Figure \ref{fig:clockedNANDiso} we show the structure of a clocked NAND gadget. We define $i_1 $and $i_2$ as the nodes that are labeled by $x$ and $y$ in Figure \ref{fig:clockedNANDiso}. We define $w' = z \cup r$ the concatenation of the configuration $z$ in Lemma \ref{lemma:NAND} and $r$ in the previous lemma. Finally we define $w_{i_1} = w_{i_2} = 1$ and $w_v = w'_v$ for all $v \not \in \{i_1,i_2\}$. Note that central node (which is conected to nodes labelled as $x$ and $y$) is in state $0$ and has exactly $\alpha + 1$ active networks: $\alpha - 3$ active networks from the clique in the upper part of the gadget, one from the clique in the right, one from the clock gadget and two from the inputs. As $\alpha$ is an isolated value for the local rules in the gadget then, we have $F_{CN}^s(w)_{o_j} = 1, j=1,2$ for $0\leq s \leq d-1.$ Now note that in $t=d$ the neighbour of the central node located in the clock gadget will change to $0$ and then we will recover the same situation showed in Figure \ref{fig:clockiso} with nodes labelled $x$ and $y$ assuming the values $F(x)|_{i_1}$ and $F(x)|_{i_2}$ and thus as a consequence of latter lemma, the result holds.
\end{proof}
\begin{figure}
\includegraphics{imagenes/clockedNAND.pdf}
\caption{Clocked NAND gadget for $\alpha$-uniform isolated totalistic rules with $\alpha \geq 3$ and delay $d.$ Clock picture represents gadget in Figure \ref{fig:clockiso}.}
\label{fig:clockedNANDiso}
\end{figure}
Note that as inputs in clocked NAND gadget in Figure \ref{fig:clockedNANDiso} will be fixed in $1$ and thus, the second part of the latter lemma may seem trivial. However, as now we would want to connect different clocked NAND gadgets in order to simulate a circuit, nodes $i_1$ and $i_2$ will be identify with some outputs of some other gadget, allowing us to make simulate the calculations of a NAND gate at the wright time. We will precise this in hereunder. Now we are prepared to show the main result of this section.
\begin{theo}
Let $r,s \in \mathbb{N}$ and $f: \{0,1\}^r \to \{0,1\}^s$ a boolean function. For each $\alpha \geq 3$ there exist a set of isolated functions $\mathcal{F}$ with isolated value $\alpha$ and a bounded degree class of graphs $\mathcal{G}$ such that $\mathcal{F}$ simulates $f$ in $\mathcal{G}$.
\end{theo}
\begin{proof}
Let $C_f$ a circuit representing $f$. Without loss of generalization we can assume that any gate is a NAND gate. We will use again the same reasoning we used for Theorem \ref{teo:thershold}. The only difference here is that we have to be very careful in order to initialize clocked NAND gadgets in each layer of the circuit as it is shown in Figure \ref{fig:circuitiso}. In order to do that, we set locally each clocked NAND get to the local configuration $w$ given in the latter lemma with exception of the nodes representing input gates (those nodes will be identify with inputs of gadgets in first layer which do not have any clock ). Nodes in the second layer will wait time $d=3$ as a consequence of their clocks. When the calculations of the first layer arrives to the second one, their clocks will be off (the neighbour located in the clock will be inactive) and thus they will be able to compute a NAND from the values computed in the first layer. Then, third layer has to wait $6$ in order to coherently simulate the evaluation of the circuit so we choose $d=3$ as it is shown in Figure \ref{fig:circuitiso}. In general the $k$-th layer will have a clock with delay $d = 3k$. Note that, as a consequence of last lemma, any gadget with exception of the first layer will stay fixed in configuration $w$ and will calculate NAND when its clock gets inactive. Then, after $t = 3 \text{depth}(C_f)$ and thus, the result holds.
\end{proof}
\begin{figure}
\includegraphics{imagenes/circuit}
\caption{Scheme of the underlying graph of an automata network simulating a circuit defined over a class of $\alpha$-uniform isolated totalistic rules with $\alpha \geq 3.$ $\text{NAND}_d$ represents clocked NAND gadget in Figure \ref{fig:clockedNANDiso} }
\label{fig:circuitiso}
\end{figure}
\subsubsection{Planarity of previous gadgets}.
In this section, we make a remark concerning the planarity of the previous graphs. In the context of this work this is very interesting as it refers to the regularity of the structures that are able to make some type of calculations. In the context of the study of automata networks it is also interesting as one of the most studied examples, cellular automata, are automata networks defined over planar graphs, i.e. grid graphs ($2$-dimensions) or cycle graphs ($1$-dimensions). In order to study planarity we use a classic result that we present in the following proposition:
\begin{prop}[Kuratowski 1930, Wagner 1937]
The following assertions are equivalent for any graph $G$
\begin{enumerate}
\item $G$ is planar;
\item $G$ contains neither $K_{5}$ and $K_{3,3}$ as a minor;
\item $G$ contains neither $K_{5}$ and $K_{3,3}$ as a topological minor;
\end{enumerate}
\end{prop}
Using this last proposition we directly derive the following corollary:
\begin{cor}
For any $\alpha \geq 4$, NAND gadgets and clocked NAND gadgets are not planar.
\end{cor}
\begin{proof}
We have from Figure \ref{fig:NANDiso} and Figure \ref{fig:clockiso} that each gadget contains at least $K_{\alpha+1}$ as a subgraph. Thus, the result holds.
\end{proof}
\section{Simulations}
In this section, we show some results on simulations of totalistic automata networks over random graphs. More precisely, we generate a collection of $1000$ random graphs using the well known Erdös-Renyi model and we study all the totalistic rules with a maximum of $4$ active neighbors (i.e. $\mathcal{I}_v \subseteq
\{0,\hdots,4\}$ for each node $v$ in the network).
In general, in this section we study all the possible boolean gates that each rule in the latter class can calculate in some randomly generated graph, by trying any possible combination of three nodes as a set of two inputs and one output. We are also interested in identify a set of small graphs, that we call, gadgets, which exhibit a richer spectrum of boolean gates in the latter simulation.
Specifically, this section is organize as follows:
\begin{enumerate}
\item First we show a general collection of results with the aim of describe the landscape of simulation capabilities of each totalistic automata network with at most $4$ active neighbors: we exhibit the relative frequency in which each graph is capable of simulation all the possible $2^{2^2} = 16$ boolean gates and we choose the graph that has the greater spectrum from the collection of randomly generated graph (that is to say that exhibit more logic gates from the set of $16$ possible boolean gates with $2$ inputs and $1$ output). In this regard, we study the impact of the connectivity of each random generated graph by changing the probability of two given nodes to be connected. We call this probability $p \in [0,1]$.
\item From the latter simulations, we choose some rules from the set of totalistic rules that we have considered in the latter subsection and we exhibit the most representative gadget (i.e. the smallest graph with greatest spectrum)). Then, we study its dynamics, with emphasis in understanding how it simulates a given a boolean gate.
\end{enumerate}
\subsection{General landscape in random graphs}
We start by remarking that as we use the Erdös-Renyi model \cite{}, two parameters must be chosen in order to define a random graph: the number of nodes $n$ and the probability $p$ of two arbitrary nodes to be connected. We also recall that we are considering only logic gates with two inputs and one output. As we can encode each of these boolean functions as a sequence of $4$ bits, we can represent each function by a number $i \in \{0,\hdots,15\}.$ For example, if $i=8$ then we have the AND rule since $8 = 0001$ and we are considering the following coding: $00 \to 0$, $01 \to 0$ $10 \to 0$ and $11 \to 1$. Analogously, OR gate is $i=14$, XOR gate is $i=6$, NAND gate is $i=7$ and NOR gate is $i=1.$ Finally, we start all the simulations from the fixed point $\overline{x} = \vec{0}$ and we choose perturbations in all the possible assignations for two inputs and one output over a randomly generated graph.
We are now in condition of summarize the set-up of parameters that we used for the simulations.
\subsection{Simulations set-up}
We start by describing the parameters of the following simulations:
\begin{enumerate}
\item Probability of adjacency of two given nodes ($p$): 0.1, 0.5, 0.8
\item Number of generated graphs ($N$): 100
\item Number of nodes per graph ($n$): 10
\item Simulation time $t$: 100
\end{enumerate}
\subsubsection{Results}
\paragraph{The impact of connectivity.}
One of the most straightforwards observations from the results we show in Figures \ref{} is that there is an effect of the connectivity of the different random graphs in the diversity of boolean gates that certain rules can simulate. More precisely, if we see Table \ref{tab:p01} we observe that rules that need at least $2$ neighbors in order to activate one node (i.e. $2,23,24,3,34,$ etc ) can only simulate the trivial function (i.e. all inputs goes to $0$). Of course, this is intrinsically related to the definition of the dynamics which depends on the number of active neighbors in order to produce any dynamic behavior different from $\vec{0}$. In particular, in Table \ref{tab:p01} and \ref{tab:p05} we can observe that most of the gates are simulated by rules that have the possibility to change to state $1$ when they have at least one active neighbor. \\
\begin{table}[H]
\resizebox{\textwidth}{!}{%
\begin{tabular}{|l|l|l|l|l|l|l|l|l|l|l|l|l|l|l|l|l|}
\hline
& 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 & 14 & 15 \\ \hline
1 & 100 & 0 & 6 & 0 & 10 & 0 & 18 & 0 & 7 & 0 & 42 & 0 & 44 & 0 & 12 & 0 \\ \hline
2 & 100 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \hline
3 & 100 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \hline
4 & 100 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \hline
12 & 100 & 0 & 6 & 0 & 8 & 0 & 1 & 0 & 0 & 0 & 79 & 0 & 75 & 0 & 38 & 0 \\ \hline
13 & 100 & 0 & 1 & 0 & 1 & 0 & 33 & 0 & 1 & 0 & 51 & 0 & 53 & 0 & 1 & 0 \\ \hline
14 & 100 & 0 & 6 & 0 & 9 & 0 & 20 & 0 & 7 & 0 & 43 & 0 & 45 & 0 & 12 & 0 \\ \hline
23 & 100 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \hline
24 & 100 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \hline
34 & 100 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \hline
123 & 100 & 0 & 1 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 83 & 0 & 82 & 0 & 50 & 0 \\ \hline
124 & 100 & 0 & 8 & 0 & 8 & 0 & 2 & 0 & 1 & 0 & 80 & 0 & 76 & 0 & 39 & 0 \\ \hline
134 & 100 & 0 & 3 & 0 & 2 & 0 & 34 & 0 & 3 & 0 & 52 & 0 & 54 & 0 & 4 & 0 \\ \hline
234 & 100 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \hline
1234 & 100 & 0 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 84 & 0 & 83 & 0 & 51 & 0 \\ \hline
\end{tabular}%
}
\caption{Spectrum of boolean gates by totalistic rule for $p=0.1$. Each column contains the number of graphs that are capable of simulate the corresponding boolean gate.}
\label{tab:p01}
\end{table}
\begin{figure}[H]
\centering
\includegraphics[scale=0.20]{imagenes/01p_10nodes_100graphs_graph_list.pdf}
\caption{Graphs with the greatest spectrum by rule for $p=0.1.$}
\label{plot:p01}
\end{figure}
On the other hand, if we observe Table \ref{tab:p05} we can see that rules with higher threshold (that is to say that needs at least 2 active neighbors in order to change to state $1$) start to show some simulation capabilities. In particular, rules that change with $2$ active neighbors exhibit the possibility of calculate AND gates supporting the remark that higher connectivity implies higher simulation capabilities for some rule. \\
Finally, if we observe Figures \ref{plot:p01}, \ref{plot:p05} and \ref{plot:p08} we have that if $p \geq 0.5$ we have that every gadget is connected. Contrarily, in the case $p=0.1$ we observe that rules with higher threshold (those who need more active neighbors to active its nodes) tend to exhibit gadgets with bigger connected components. We remark the case of rule $1$ which seems to exhibit a minimal modular structure which is able to calculate several different logic gates.
\begin{table}[H]
\resizebox{\textwidth}{!}{%
\begin{tabular}{|l|l|l|l|l|l|l|l|l|l|l|l|l|l|l|l|l|}
\hline
& 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 & 14 & 15 \\ \hline
1 & 100 & 0 & 74 & 0 & 74 & 0 & 63 & 0 & 82 & 0 & 60 & 0 & 56 & 0 & 44 & 0 \\ \hline
2 & 100 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 58 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \hline
3 & 100 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \hline
4 & 100 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \hline
12 & 100 & 0 & 86 & 0 & 88 & 0 & 86 & 0 & 79 & 0 & 83 & 0 & 80 & 0 & 91 & 0 \\ \hline
13 & 100 & 0 & 94 & 0 & 94 & 0 & 91 & 0 & 96 & 0 & 89 & 0 & 90 & 0 & 83 & 0 \\ \hline
14 & 100 & 0 & 83 & 0 & 85 & 0 & 78 & 0 & 90 & 0 & 81 & 0 & 81 & 0 & 71 & 0 \\ \hline
23 & 100 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 83 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \hline
24 & 100 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 70 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \hline
34 & 100 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \hline
123 & 99 & 0 & 78 & 0 & 77 & 0 & 82 & 0 & 71 & 0 & 75 & 0 & 73 & 0 & 100 & 0 \\ \hline
124 & 100 & 0 & 93 & 0 & 96 & 0 & 94 & 0 & 95 & 0 & 93 & 0 & 95 & 0 & 97 & 0 \\ \hline
134 & 100 & 0 & 92 & 0 & 92 & 0 & 95 & 0 & 97 & 0 & 92 & 0 & 94 & 0 & 97 & 0 \\ \hline
234 & 100 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 100 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \hline
1234 & 94 & 0 & 53 & 0 & 50 & 0 & 56 & 0 & 40 & 0 & 46 & 0 & 46 & 0 & 100 & 0 \\ \hline
\end{tabular}%
}
\caption{Spectrum of boolean gates by totalistic rule for $p=0.5$. Each column contains the number of graphs that are capable of simulate the corresponding boolean gate.\}}
\label{tab:p05}
\end{table}
\begin{figure}[H]
\centering
\includegraphics[scale=0.20]{imagenes/05p_10nodes_100graphs_graph_list}
\caption{Graphs with the greatest spectrum by rule for $p=0.5$}
\label{plot:p05}
\end{figure}
\paragraph{More connectivity does not imply more diversity.}
If we study now rules that exhibit strong simulation capabilities even for low connectivity such as rule $1$, we observe that they do not show a significant change in their spectrum for higher values of $p$ (even if we can observe an increase in the frequencies of each boolean gate in their spectrum, which implies that more graphs of the random sample are being capable of simulating one specific gate). Roughly, this observation may suggest that probably we need to study graphs with more nodes in order to observe a significant impact of connectivity in the spectrum of different totalistic rules. This is also coherent with the theoretical results regarding isolated totalistic networks in which we have theoretically constructed gadgets requiring big cliques subgraphs fixed in the state $1$ in order to perform calculations. \\
\paragraph{Complete boolean gates set are not modular.}
Finally, we observe that gates NOR and NAND ($i=7$ and $i=1$ respectively in our notation) do not seem to be calculable by totalistic rules with at most $4$ active neighbors (see Table \ref{tab:p01}, Table \ref{tab:p05} and Table \ref{tab:p08}). It might suggest that we need a more complex structure in order to simulate complete sets of boolean gates. We conjecture that maybe we can simulate them by "glueing" different gadgets in a coherent way. Nevertheless, in order to do achieve this task, we need a better understanding of the particular dynamics of certain gadgets (see next subsection).
\begin{table}[H]
\resizebox{\textwidth}{!}{%
\begin{tabular}{|l|l|l|l|l|l|l|l|l|l|l|l|l|l|l|l|l|}
\hline
& 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 & 14 & 15 \\ \hline
1 & 100 & 0 & 1 & 0 & 1 & 0 & 0 & 0 & 39 & 0 & 1 & 0 & 1 & 0 & 0 & 0 \\ \hline
2 & 100 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 27 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \hline
3 & 100 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \hline
4 & 100 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \hline
12 & 100 & 0 & 11 & 0 & 10 & 0 & 8 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 \\ \hline
13 & 100 & 0 & 40 & 0 & 41 & 0 & 28 & 0 & 81 & 0 & 34 & 0 & 32 & 0 & 18 & 0 \\ \hline
14 & 100 & 0 & 61 & 0 & 61 & 0 & 48 & 0 & 91 & 0 & 54 & 0 & 53 & 0 & 38 & 0 \\ \hline
23 & 100 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 66 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \hline
24 & 100 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 83 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \hline
34 & 100 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \hline
123 & 99 & 0 & 29 & 0 & 27 & 0 & 18 & 0 & 11 & 0 & 14 & 0 & 13 & 0 & 14 & 0 \\ \hline
124 & 100 & 0 & 54 & 0 & 54 & 0 & 39 & 0 & 26 & 0 & 29 & 0 & 24 & 0 & 22 & 0 \\ \hline
134 & 99 & 0 & 66 & 0 & 63 & 0 & 52 & 0 & 90 & 0 & 58 & 0 & 53 & 0 & 41 & 0 \\ \hline
234 & 100 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 91 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \hline
1234 & 96 & 0 & 63 & 0 & 65 & 0 & 48 & 0 & 38 & 0 & 33 & 0 & 32 & 0 & 37 & 0 \\ \hline
\end{tabular}%
}
\caption{Spectrum of boolean gates by totalistic rule for $p=0.8$. Each column contains the number of graphs that are capable of simulate the corresponding boolean gate.\}}
\label{tab:p08}
\end{table}
\begin{figure}[H]
\centering
\includegraphics[scale=0.20]{imagenes/08p_10nodes_100graphs_graph_list.pdf}
\caption{Graphs with the greatest spectrum by rule for $p=0.8$}
\label{plot:p08}
\end{figure}
\subsection{Gadget dynamics: a rule $1$ automata network example }
In this section we focus in take a deeper look in the structures obtained for rule $1$ in latter simulations. In particular, we study the dynamics of a gadget obtained in the latter simulation which is capable of calculating different logic gates by changing the assignation of inputs and output. This gadget exhibit the maximum value of the spectrum for the rule $1$ in the simulations for $p=0.1$ (i.e. it is able to simulate the maximum amount of different logic gates observed in the simulations). In addition, this gadget has only $6$ nodes as it is shown in Figure \ref{plot:gadget} \\
\begin{figure}[H]
\centering
\includegraphics[scale=0.75]{grafoG1.pdf}
\caption{Gadget obtained for rule $1$ in the previous simulations with $p=0.1$ (see Figure \ref{plot:p01})}
\label{plot:gadget}
\end{figure}
This section is orgnaized in the following way: in the first subsection we study the dependency of the spectrum in the simulation time, showing that there is a critical time in which the gadget starts to simulate some gates such as AND and OR gates. In the second subsection, we explore the dynamics behind the simulation of the AND gate.
\subsubsection{Frequency of simulated gates v/s simulation time}
We now study the dependency of the spectrum of the latter gadget automata network that we have found for rule $1$ on the simulation time. Remember that the frequency here is a measure of how many different assignation of two input and one output produces the same logic gate. In this case, we focus in the studying the AND and OR gates. In Figure \ref{plot:gadget} we can see that the frequency for these two logic gates v/s simulation time. As we can observe in the latter Figure, there is no monotonic behavior on the frequency. More over it appears to have a periodic behavior. We conjecture that it is related to the maximal period of some attractor of the gadget automata network.
\begin{figure}[H]
\centering
\includegraphics[scale=0.45]{ejemploANDt1.pdf}
\caption{Frequency of AND and OR gates v/s simulation time for the gadget in Figure \ref{plot:gadget}}
\label{plot:exampleANDt1}
\end{figure}
On the other hand, we can observe that a critical time is necessary for the gadget in order to simulate AND and OR gates. More precisely, the system is capable of simulating both logic gates starting from $t = 3$ as it is shown in Figure \ref{plot:exampleANDt2}. Finally, note that the time steps in which the system is capable of simulating AND gates are not necessarily the same for simulating OR gates. For example, as it is shown in Figure \ref{plot:exampleANDt2}, the gadget is not capable of simulating OR gates in time $t=4$ but it can simulate AND gates with $6$ different assignations of input and output at this same time step.
\begin{figure}[H]
\centering
\includegraphics[scale=0.45]{ejemploANDt2.pdf}
\caption{Frequency of AND and OR gates v/s simulation time for the gadget in Figure \ref{plot:gadget}). The system is capable of simulating both gates starting from $t=3$}
\label{plot:exampleANDt2}
\end{figure}
\subsubsection{AND gate dynamics}
In the section we study in the detail the dynamics behind the simulation of an AND gate by the gadget automata network which graph is shown in Figure \ref{plot:gadget}. In order to do that, we choose the minimum simulation time required for the system to compute an AND gate. As it is shown in Figure \ref{plot:exampleANDt2} this time step is $t=3$. In Figure \ref{plot:exampleAND1} we show the dynamics of the network starting from the input $11$. In this case, nodes marked as $2$ and $3$ are the inputs and $4$ is the output. Active nodes are colored in yellow and inactive nodes are colored blue. Note that AND gate is computed after $t=3$ time steps. \\
\begin{figure}[H]
\centering
\includegraphics[scale=0.35]{ejemploAND1.pdf}
\caption{Dynamics behind simulation of an AND gate for an input $11$. Nodes $2$ and $3$ act as input nodes and node $4$ operates as the output. }
\label{plot:exampleAND1}
\end{figure}
On the other hand, Figure \ref{plot:exampleAND2} shows the evolution of the input $00$ using the same input-out assignation. We see here how the paths that are connected to the central triangle of the graph play an essential role in blocking a single $1$ signal. \\
\begin{figure}[H]
\centering
\includegraphics[scale=0.35]{ejemploAND2.pdf}
\caption{Dynamics behind simulation of an AND gate for an input $01$. Nodes $2$ and $3$ act as input nodes and node $4$ operates as the output. }
\label{plot:exampleAND2}
\end{figure}
Contrarily to the case of the gadgets that we have theoretically proposed in the latter sections, we observe that, in both Figure \ref{plot:exampleAND1} and \ref{plot:exampleAND2}, the simulation of the AND gate takes more than a straightforward sequence of calculations performed in some linear way. We conjecture that the system uses different attractors in order to simulate the output of one logic gate for different input assignments.
\end{document}
\section{Introduction}
Unconventional computing aims to uncover principles of information processing in chemical, physical and living computing substrates~\cite{adamatzky2016advances}. A predominant majority of living systems are comprised of two key networks: vascular system (metabolites transfer and processing) and nervous or other signalling systems (information transfer and processing). Both types of networks are conducive to propagation of electrical~\cite{hodgkin1952propagation,fromm2007electrical}, mechanical/sound~\cite{heimburg2005soliton,shrivastava2014evidence,fichtl2016protons} and optical~\cite{contreras2013non,appali2012comparison} signals. When these electrical, mechanical or optical solitons interact with each other while travelling and colliding on the networks they change velocity vectors, states or existence. Thus by encoding Boolean values in presence and absence of solitons one can implement logical circuits on the networks.
An idea to implement a computation by using collisions of signals travelling along one-dimensional non-linear geometries can be traced back to the mid 1960s when Atrubin developed a chain of finite-state machines executing multiplication~\cite{atrubin1965one}, Fisher designed prime numbers generators in cellular automata~\cite{fischer1965generation} and Waksman proposed the eight-state solution for a firing squad synchronisation problem~\cite{waksman1966optimum}.
In 1986, Park, Steiglitz, and Thurston~\cite{park1986soliton} designed a parity filter in cellular automata with soliton-like dynamics of localisations. Their design led to a construction of a one-dimensional particle machine, which performs the computation by colliding particles in one-dimensional cellular automata, i.e. the computing is embedded in a bulk media~\cite{squier1994programmable}. In 1990s Goles and colleagues demonstrated that sand pile and chip firing game are universal computers. That is by representing logical truth by presence of a sand grain or a chip and logical false by absence of the grain/chip one route information as avalanches and implement logical gates via interaction of avalanches in an appropriate geometrical structure~\cite{goles1996sand,gajardo2006crossing,goles1997universality}.
Most close to biophysical reality models of computing on biological networks have been implemented with molecular structures of verotoxin protein~\cite{adamatzky2017computing} and actin monomer~\cite{
adamatzky2017logical}, actin bundles networks derived from experimental laboratory data~\cite{adamatzky2019computing}, plant leaf vascular system~\cite{adamatzky2019plant} and microscopic images of three-dimensional fungal colonies~\cite{adamatzky2020boolean}. These approaches employed the following method. Two loci (atoms, molecules, parts of the network) are considered to be input and all other loci outputs. Sequences (01), (10), (11), represented by solitons or impulses are sent to the inputs and solitons/impulses are recorded on the outputs. Each of the outputs implements a two-input-one-output logical gate. In the computational experiments with the molecular, polymer, vascular or mycelial networks \cite{adamatzky2017computing,adamatzky2019computing,adamatzky2019plant,adamatzky2020boolean} we did not analyse where exactly in the networks the computation takes place. In this context, and in order to fill the gap in our knowledge, we study the generation of Boolean functions on totalistic automaton networks, i.e., each site changes state according to specific values of the sum of active sites in its neighbourhood. From a mathematical point of view, totalistic rules are well-known models in the context of the study of cellular automata and automata networks as dynamical systems \cite{wolfram1984universality,marr2009outer}. In addition, the approach of studying the computational complexity of some specific decision problems that are some how related to the dynamical behaviour of totalisitc automata network has been also broadly studied, particularly in regards to how difficult is to predict the dynamical behaviour of some specific entity in the network \cite{goles2014computational,goles2018complexity,goles2020complexity}. Although, as from a theoretical viewpoint, computational complexity has been proposed as an approach for somehow measure the complexity of the dynamics of an specific automata network model, the simulation capabilities of the network, in the sense of the experiments performed in the previous context, has remained unexplored. In this paper, we characterise the complexity of totalistic automata networks according to its capabilities in order to simulate Boolean gates by the automaton's dynamics. Particularly, we explore this approach from a mathematical point of view, by systematizing and formalizing the spectrum of an automata network as a measure of the different boolean networks that the system is able to implement. In this context we proved that linear or matrix rules generate few different Boolean networks:only constant or matrix defined ones. Threshold totalistic can generate any monotone Boolean Function and the isolated or interval ones, every Boolean function. Further, we study by computational experiments the generation of Boolean functions capabilities in random totalistic networks.
\section{Preliminaries}
\label{Preliminaries}
An automata network is a tuple $ \mathcal{A}= (G,Q,\mathcal{F})$ where $G=(V,E)$ is an undirected finite graph such that $|V|=n,$ $Q=\{0,1\}$ is the finite set of states called $\textit{alphabet}$ and $\mathcal{F} = \{f_v: v \in V\}$ is a collection of functions called \textit{local functions} such that each local function $f_v:N_v \to Q$ takes it arguments as a neighbourhood of $v$ in $G$ given by $N_v = \{u\in V: uv \in E\}.$ We define $F:Q^n \to Q^n$ as the \textit{global transition function} of the automata network defined by $F(x)_v = f_v(x|_{N_v})$ for all $x \in Q^n$ and for all $v \in V$ where $x|_{N_v}$ are the coordinates of $x$ that are representing the neighbours of $v$. Formally, $x|_{N_v} \in Q^{N_v}$ and for all $u \in N_v$ we have $(x|_{N_v})_u = x_u.$ We consider only the case where every local function $f_v$ is \textit{totalistic}, i.e., a result of the function depends on the sum of the active values (states $1$'s ) as its arguments. Suppose that the maximun degree $\Delta(G)$ on $G$ is such that $\Delta(G) = \Delta$, so the sum may take values in the set
$\{1, ..., \Delta\} $, i.e, given a configuration $x\in Q^{n}$ we have $f_v(x|_{N_v})= 1$ if and only if $\sum \limits_{u\in N_v} x_u \in \mathcal{I}_v = \{a_1, ..,a_s\}$, where $\{a_1, ..,a_s\}\subseteq \{1, ..., \Delta\}.$ We will call the set $\mathcal{I}_v$ the activation set of $f_v$.
We refer to each local rule by the digit associated to $\mathcal{I}_v $, so if $\mathcal{I}_v = \{a_1,\hdots, a_s\}$ then, the totalistic rule number will be $a_1a_2 \cdots a_s$. For instance, rule $25$ means that the associated vertices become $1$ if and only if the sum if either $2$ or $5$. Note that, as each of these rules depends also on the set of values that each node in some neighborhood will take, we can consider that totalistic rules are defined over the set $S_{\Delta} = \{1,\hdots, \Delta\}$. This is required because in further sections we will work with a fixed set of totalistic rules and use them to define automata networks over different graphs. In this regard, as we will always work over some class of graphs $\mathcal{G}$ in which every graph have at most degree $\Delta$, we note that each totalistic rule having an activation set $I \subseteq \{1,\hdots,\Delta\}$ will be well defined over any graph in $\mathcal{G}$. For example, if $\Delta = 10$ rule $25$ will be well-define over any graph in $\mathcal{G}$. More precisely, for any totalistic function $f$ such that $I_f \subseteq S_{\Delta}$ we can assume that $f: S_{\Delta} \to \{0,1\}$ and that $f(s) = 1$ if and only if $s \in I_f$.
\subsection{The problem}
Let $\mathcal{A} = (G, \mathcal{F})$ be a totalistic automata network with global transition function $F$. A configuration $\overline{x} \in Q^n$ is a fixed point of $\mathcal{A}$ if $F(\overline{x}) = \overline{x}.$ Note that, by definition, $\vec{0}$ is always a fixed point of $\mathcal{A}$. Consider the set $\text{Fix}(\mathcal{A})$ of fixed points of $\mathcal{A}$. Since $\vec{0}$ is a fixed point, $\text{Fix}(\mathcal{A}) \not = \emptyset$. Now we consider two arbitrary disjoint sets of vertices $I = \{i_1, \hdots, i_l\}\subseteq V$ and $O = \{o_1,\hdots,o_s\} \subseteq V$ for $s,l \geq 1$. We call these sets the \textit{input} set of $\mathcal{A}$ and the \textit{output} set of $\mathcal{A}$ respectively. As we are interested in the simulation of logic gates, during the rest of the paper, we will focus on the case of at most two outputs, that is to say $O \subseteq \{o_1,o_2\},$ with special emphasis in the case $O = \{o\}$.
Let $t \geq 0$, we call a tuple $(I,o,t ) \in 2^{V} \times V \times \mathbb{N}$ an $I/O$ setting for $\mathcal{A}$. The variable $t$ represents some specific time step in which we are interested to analyze the output of the automata network in $o$. We call $t$ an \textit{observation} time.
Consider now a fixed point $\overline{x} \in \text{Fix} (\mathcal{A})$ with $I/O$ setting $(I,o,t) \in 2^{V} \times V \times \mathbb{N}.$ Let $z = (z_1, ..., z_l) \in \{0,1\}^{l}$ be an assignation of values for the input set $I = \{i_1, ...i_l\}$. We say that a configuration $y \in Q^n$ is a perturbation of $\overline{x}$ by $z$ if $y_u = \overline{x}_u$ for all $u \in V \setminus I$ and $y_u= z_u$ for $u \in I$. Now, given some observation time $t\geq 0$ we are interested in studying all possible states of the output $o$ after $t$ time steps given different assigments of variables to the inputs. More precisely, given a perturbation $y$ of $\overline{x}$ for the last $I/O$ setting we define a realisation of $\overline{x}$ as a Boolean function $g^{(I,o,t)}_{\overline{x}}: \{0,1\}^l \to \{0,1\}$ such that $g^{(I,o,t)}_{\overline{x}}(y) = F^t(y)_o.$ Let $\binom{V}{l}$ be a collection of subsets of $V$ with size $l$. Considering all possible combinations of inputs of certain size $l$ and possible outputs for a fixed time, we define the spectrum of Boolean functions with $l$ inputs generated by $\overline{x}$ at time $t$ as the set of realisations of $\overline{x}$ given by $\mathbb{F}^{t,l}_{\overline{x}} = \{g^{(I,o,t)}_{\overline{x}}: (I,o) \in 2^V \times V\}$ and the spectrum of functions with $l$ inputs of $\mathcal{A}$ at time $t$ as the set $\mathbb{F}^{t,l}_{\mathcal{A}} = \bigcup \limits_{\overline{x} \in \text{Fix}(\mathcal{A})} \mathbb{F}_{\overline{x}}$. Previous set may consider the "same" Boolean function but only by changing the choosing input-output indexes of variables. To avoid that, in previous set we only consider different Boolean functions, i.e., that is its difer in al least one set of Boolean inputs. On the other hand, the set of realisable Boolean functions depends on the observation time that we are considering and in the interaction graph of the network. Thus, it is strictly related to classical dynamical properties such as transient length and the attractor landscape of the automata network.
\subsection{Totalistic classes of functions}
We recall that $S_{\Delta}$ is the set of natural numbers $\{1,\hdots,\Delta\}$ for some bound $\Delta \geq 2$. One of the advantages of working with totalistic rules is that we can fix a collection of these rules and change the underlying interaction graph in order to define different automata networks. For example, if we fix $n \in \mathbb{N}$ and we consider the set of local functions $\{f_k: S_{\Delta} \to \{0,1\}, k=1,\hdots,n\}$ in which every rule is the rule $1$, i.e., $f_k(s) = 1$ if and only if $s = 1$ for every $k \in V$, we can, for every graph $G = (V,E)$ with $n$ nodes, define an automata network $\mathcal{A}_G = (G,\{\tilde{f}_k: N(k) \to \{0,1\}\})$ where $\tilde{f}(x|_{N(k)}) = f(\sum_{v \in N(k)} x_v)$. In further sections, we will write simply $f$ while referring both to $\tilde{f}$ and $f$ to simplify the notation. Note now that if $G$ and $G'$ are two different graphs with $n$ nodes then their corresponding automata networks $\mathcal{A}_G$ and $\mathcal{A}_{G'}$ will have the same kind of global rule (each vertex has the same totalistic function). The degree of the graph $G$ plays a fundamental role in this definition. For example, suppose that some rule $f_k$ needs $l$ active neighbors in order to activate the node, i.e. $f_k(x|_S) = 1$ if and only if $\sum \limits_{v \in S} x_v = l$ for some, $l \in \mathbb{N}$, for every $S\subseteq V$ and $x \in \{0,1\}^n$. In addition, suppose that the degree of node $k$ is less than $l$. In that case, $f_k$ will be fixed in the initial configuration. Roughly this will not have an important effect in our results (nor theoretical or numerical) as we will work with connected graphs with bounded maximum degree and, in addition, it will allow us to keep the latter set-up simple. In addition, this way to generated different automata networks from the same class of local functions is more practical in order to study the dynamical behaviour of different totalistic rules for a fixed large collections of randomly generated graphs.
Roughly, in this context we will say that the complexity of $\mathcal{A}$ is given by the number of different Boolean functions that could be generated over any fixed point we consider. Formally, we define the simulation complexity for a $\mathcal{A}$ by $\rho(\mathcal{A},t,l) = |\mathbb{F}^{t,l}_{\mathcal{A}}|$. The large is variery of Boolean functions generated by the latter procedure, the more complex the automaton will be considered. In this context we will say the complexity of $\mathcal{A}$ is given by the number of different Boolean functions that could be generated over any fixed point we consider. In addition, we can define the complexity of a class of totalistic functions $\mathcal{F}$ related to some class of graph $\mathcal{G}$ as $\rho(\mathcal{F},\mathcal{G},t,l) =|\mathbb{F}^{G,t,l}_{\mathcal{F}}|$
\subsection{The spectrum as a measure of complexity}
\label{spectrum}
Generally speaking, the spectrum of a set or a class of totalistic rules is a measure of how many different types of a Boolean gates it can simulate. Nevertheless, it is well known that every Boolean function $f: \{0,1\}^r \to \{0,1\}^s$ can be represented by a directed graph $C$ in which every node is a Boolean gate. An asynchronous evaluation of every Boolean gate in $C$ performs the evaluation of the function $f$. In this regard, we are interested in the study sets of totalistic rules which not only are capable of simulating different Boolean gates, but to organize them in way that, they can simulate the evaluation of a Boolean circuit.
Roughly, our main idea is to show that some class $\mathcal{F}$ is able to simulate a complete set of logic gates, for example, $\textsc{AND}, \textsc{NOT}, \textsc{OR}$. Let $\mathcal{A}_{\textsc{AND}},\mathcal{A}_{\textsc{NOT}}$ and $\mathcal{A}_{\textsc{OR}}$ be the automata networks that simulates each of this gates. Then, we will try to combine its different underlying graphs in order to simulate an arbitrary circuit $C$, by considering for each gate $g$ in c one of the latter graphs and then, try to connect them somehow. Note that this process is not straightforward as every automata network simulates a logic gate through its dynamics and so, it is not trivial how we should glue them in order to generate coherent global dynamical behaviour. In simple words, what we want to achieve is, exhibit a large automata network that has a set of small subgraphs simulating logic gates. This big network will simulate the evaluation of $C$ through its dynamics, in the sense that, by identifying a group of nodes as ``input nodes'' we can read the same output we would have read after the evaluation of $C$ by reading the state of another group of nodes labelled as ``output nodes'', after some time steps. More precisely, we introduce the following definition:
\begin{definition}
Let $\Delta \in \mathbb{N}$ and let $\mathcal{G}$ be a collection of graphs with maximum degree at most $\Delta$. Let $\mathcal{F}$ be a set of totalistic rules and $f:\{0,1\}^r \to \{0,1\}^s$ an arbitrary Boolean function. We say that $\mathcal{F}$ simulates $f$ in $\mathcal{G}$ if there exists such $n \in \mathbb{N}$ such that $n = r^{\mathcal{O}(1)}$, a graph $G_n = (V,E) \in \mathcal{G}$ with $|V| = n$ with global rule $F_n$ and $t = n^{\mathcal{O}(1)}$ such that $f(y) = (F_n^t(x|_I))|_O$ for every $y \in \{0,1\}^r$, for some $x \in \{0,1\}^n$ and for some sets $I,O \subseteq V$ such that $|I|=r$ and $|O|=s$.
\end{definition}
\begin{remark}
Note that, in the latter definition, the sets $I$ and $O$ are one-to-one related to the input and outputs of the graph. This means the simulation is very strong in the sense that inputs and outputs are represented by one node in the network.
\end{remark}
On the other hand, we note that the latter definition is intrinsically related to the computation complexity of some decision problems that have been studied in order to measure the complexity of the dynamics of automata networks. In particular, it is closely related to \textit{prediction problem}. Given an automata network $\mathcal{A} = (G,\mathcal{F})$, this problem is roughly defined by a given configuration $x \in Q^n$ and a node $v \in V$ for which we would want to know if the state of $v$ will change at some point in the orbit of $x$. More precisely, we ask if there exist $t$ such that $F^t(x)_v \not = x_v.$ Depending on $\mathcal{A}$, it can be shown that the complexity of this decision problem is closely related to the capability of $\mathcal{A}$ of simulating the evaluation of an arbitrary Boolean circuit. This is because, depending on the rules defining $\mathcal{A}$ this problem can be verified or solved in a polynomial time, and thus the complexity bounds are deduced through a reduction to canonical problems such as \textsc{Circuit value problem} or \textsc{SAT}.
\section{Results}
In this section we present different results on the complexity of different totalistic rules in the sense of its spectrum. In particular, we focus in exhibiting for different classes of totalistic rules, small automata networks, that we call gadgets, that can simulate logic gates. Then, we show how we can combine them in order to simulate arbitrary Boolean functions. In this regard, we present a classification based in the structure of the activation sets $\mathcal{I}_v$ of the different totalistic rules in $\mathcal{F}$. We start by studying the simple case where $\mathcal{I}_v = \{1\}$. This relates to disjunctive and conjunctive networks, and, in a more general way, rules which dynamics are defined by a some sort of matrix product. Then, we study the classic case in which $\mathcal{I}_v$ is given by some interval $[\theta_v,\Delta] \subseteq \{0,\hdots, \Delta\}.$ This class includes the well-known threshold networks which, as we show in this section, have the capability of simulating any monotone Boolean network. In particular, we explore the case in which isolated activation values are considered. Roughly we explore the case in which if $a \in \mathcal{I}_v$ then $a-2,a-1,a+1 \not \in \mathcal{I}_v$ and we find that this class is also capable of simulating arbitrary Boolean networks. Finally we study the intermediate case in which $\mathcal{I}_v = \{\alpha,\hdots, \beta\}$ with $\alpha \leq\beta \leq \Delta$. Particularly, we are interested in studying automata networks $\mathcal{A} = (G,\mathcal{F})$ in which for any $v \in V(G)$ its local function is such that $\mathcal{I}_v = [\alpha,\beta]$ with $\beta < \delta_v$, i.e. there exist a threshold that deactivates the local function for each node. We illustrate the dynamics of each subclass of threshold networks in Figure \ref{fig:threshold}.
\begin{figure}[!tbp]
\centering
\begin{subfigure}[t]{3in}
\centering
\includegraphics[scale=0.15]{plots/threshold1}
\caption{Threshold rules}\label{fig:1a}
\end{subfigure}
\quad
\begin{subfigure}[t]{3in}
\centering
\includegraphics[scale=0.15]{plots/threshold3}
\caption{Isolated rules}\label{fig:1b}
\end{subfigure}
\begin{subfigure}[t]{3in}
\centering
\includegraphics[scale=0.15]{plots/threshold2}
\caption{Interval rules}\label{fig:3b}
\end{subfigure}
\caption{Different classes of threshold totalistic rules}\label{fig:threshold}
\end{figure}
We resume our main results in the following table:
\begin{table}[H]
\resizebox{\textwidth}{!}{%
\begin{tabular}{|l|l|}
\hline
\textbf{Class of Totalistic Rules} &\textbf{Simulation capabilities} \\ \hline
Matrix-defined rules & Constant functions and other matrix-defined functions. \\ \hline
Threshold rules & Arbitrary monotone Boolean functions. \\ \hline
Isolated rules & Arbitrary Boolean functions. \\ \hline
Interval rules & Arbitrary Boolean functions. \\ \hline
\end{tabular}%
}
\caption{Classification of totalistic rules according to their simulation capabilities.}
\label{tab:summary}
\end{table}
\subsection{Matrix-defined rules}
We start by studying canonical cases of totalistic rules such as disjunctive (conjunctive) networks. Let $Q = \{0,1\}$ and let $\mathcal{G}$ be a family of graphs. We say that some totalistic rule $f:S_{\Delta}\to Q$ is disjunctive if it takes the value $1$ if and only if there exist at least one $1$ in its assignment, i.e., $\mathcal{I}_f = \{1\}$. We say that a set or a class of totalistic rules $\mathcal{F}$ is disjunctive if every $f \in \mathcal{F}$ is disjunctive. Analogously, we can define a conjunctive totalistic function $f$ over a graph $G$ in some node $v \in V(G)$ by defining the transition to $1$ only in the case in which every neighbour of $v$ is in state $1$, i.e $\mathcal{I}_v= \{\delta_v\}$. Note that in this case we cannot define the rule independently of the interaction graph. Nevertheless, both rules are completely analogous as it suffices to change the role of $1$ and $0$ in order to change from disjunctive to conjunctive and vice versa. As a consequence of these, and in order to simplify following reasoning we focus on disjunctive rules but of course all of the next results are valid also for conjunctive rules.
\begin{lem}
Let $\mathcal{G}$ be an arbitrary family of graphs and take some graph $G \in \mathcal{G}$. Let $\mathcal{D} = (G,\mathcal{F})$ be an automata network where $\mathcal{F}$ is disjunctive then the spectrum of $\mathcal{D}$ contains only constant gates (everything goes to $1$ or $0$) and disjunctive gates (OR gates). For any $t,l\geq 1$ we have $\mathbb{F}^{(t,l)}_{\mathcal{D}} \subseteq \{0,1\} \cup \{\vee_J\}_{J \subseteq I} $ where $\vee :\{0,1\}^l \to \{0,1\}$ is such that $\vee (z_1,\hdots,z_l) = \bigvee \limits^l_{k=1} z_l$ for $J \subseteq I.$
\end{lem}
\begin{proof}
Let $G \in \mathcal{G}$, $t,l\geq 1$ and $\mathcal{A} = (G,\mathcal{D}(G))$. Let $F$ be a global transition function of $G$. Fix an input $I \subset V.$ Note that $\text{Fix}(\mathcal{A}) = \{\vec{0},\vec{1}\}.$ Also note that there exists a matrix $A \in M_n(\{0,1\})$ such that $F^t(x) = A^t \vee x = \bigvee \limits_{i \in N_v(G^t) \cap I} x_i $ for all $t \geq 0$. In particular, $A$ is the adjacency matrix of $G$. Note that for every $i \in I$ there exists a path between $i$ and $o$ of length $t$ if and only if $(A^t)_{io} = 1$ and thus, the $t$-th power of $A$ define the power graph $G^t$. By definition we have that
\begin{equation}
F^t(x)_o = (A^tx)_o = \bigvee \limits_{i \in N_v(G^t)} x_i = \left( \bigvee \limits_{i \in N_v(G^t) \cap I} x_i \right) \vee \left( \bigvee \limits_{i \in N_v(G^t) \cap V \setminus I} x_i \right).
\label{eq:or}
\end{equation}
Now, note that, if we start perturbing $\vec{1}$ then we have that $\mathbb{F}^{(G,t,l)}_{\vec{1}} \subseteq \{1,\vee\}$ as the only case in which we can do something different than $1$ is when $l = \delta_o$, $I = N_o$ and $t = 1$.
On the other hand, as $G$ is connected, consider $P_1, \hdots, P_l$ as all the minimum length paths connecting each node in $I$ to $o$. Let $d_1,\hdots, d_l$ be the lengths of each of the path and let $d = \min \limits_{i \in \{1,\hdots,l\}} d_i$ and $D = \max \limits_{i \in \{1,\hdots,l\}} d_i.$ If we perturb $\vec{0}$, we have that, for $t\leq d$ where we have $g^{I,o,t} \equiv 0.$ For $t \geq d$ we can have that not all the nodes in $I$ are connected in $G^t$ with $o$ and thus by (\ref{eq:or}) we have that $g^{I,o,t} \equiv \vee_{J}$ for some $J \subseteq I$. Finally if $t \geq D$ then, we can have influence of external nodes in $N_v(G^t) \cap V\setminus$. The influence is given by an OR function. Therefore we have two possible cases: a) one external have state $1$ at time step $t$ and then $ g^{I,o,t} \equiv 1$ or all stay in state $0$ and then there is no influence. In every case we conclude that $\mathbb{F}^{(t,l)}_{\vec{0}} \subseteq \{0,1,\vee\} \cup \{\vee_J\}_{J \subseteq I}$ and thus the lemma holds.
\label{lemma:OR}
\end{proof}
\begin{remark}
Note that if $G= (V,E)$ is such that every totalistic function takes the value $1$ when the sum of the states of all neighbours of certain vertex is odd, then, we have the known XOR rule. More precisely, we have the XOR rule if for every $v \in V$ we have that $\mathcal{I}_v = \{a \in \{0,\hdots,\delta_v\}: a \text{ is odd} \}$. Also the global rule of that automata network in that case can be seen as a matrix product, i.e. $F^t(x) = A^t x$ where the product is the usual product in $\mathbb{F}_2.$ Thus, the previous result holds for XOR rules.
\end{remark}
\subsection{Threshold networks}
In this section we introduce a class of totalistic functions called threshold functions. Roughly, in this family we have that a function takes the value $1$ if the sum of the states of the neighbours of the corresponding vertex is in some interval $[\theta_v,\delta_v]$ where $\delta_v$ is the degree of the vertex $v$ that we are considering and $\theta_v$ is some positive threshold. We present this notion in the following definition:
\begin{definition}
A totalistic function $f: S_{\Delta} \to \{0,1\}$ is a threshold if there exists some positive integer $\theta$ such that $ [\theta,\Delta] \subseteq \mathcal{I}_f $, where $I_f$ is the activation set of $f$.
\end{definition}
We denote by $\mathcal{T}$ the class of all totalistic functions that are threshold, i.e. $f \in \mathcal{T}$ if and only if $f$ is threshold. We will show that there exist a class of graphs $\mathcal{G}$ for which $\mathcal{T}$ simulates any monotone Boolean function.
\begin{lem}
There are two automata networks $\mathcal{A}_2 = (G_1=(V_1,E_1),\mathcal{F}_1)$ and $\mathcal{A}_2 = (G_2=(V_2,E_2),\mathcal{F}_2)$ with global rules $F_1$ and $F_2 $ respectively, such that $\mathcal{F}_i \in \mathcal{T}$ and that :
\begin{enumerate}
\item $\wedge(x,y) = F^2_1(z_1)_{o} = F^2_1(z_1)_{o'}.$
\item $\vee(x,y) = F^2_2(z_2)_{o} = F^2_2(z_2)_{o'},$
\end{enumerate}
for some $z_i \in \{0,1\}^{|V_i|},$ $i=1,2.$
\label{lem:andorthreshold}
\end{lem}
\begin{proof}
Consider the graph $G_1$ and $G_2$ given in Figure \ref{fig:andthrehsold} and Figure \ref{fig:orthrehsold}. Observe that $\overline{z}_1 = (0,0,0,0,0)$ and $\overline{z}_2 = (0,0,0,0,0)$ are fixed points for $F_1$ and $F_2$ respectively. Then, we can define $z_1$ and $z_2$ as a perturbation of these fixed points as it is shown in Figures \ref{fig:andthrehsold} and \ref{fig:orthrehsold}. As we stated in the last section, it suffices to define $\theta_v \in \{1, \delta_v\}$ in order to define an AND or an OR function. Observe that this is exactly the threshold defined for each vertex in Figure \ref{fig:andthrehsold} and Figure \ref{fig:orthrehsold}. The result follows from the calculations in latter figures.
\end{proof}
\begin{figure}[!tbp]
\centering
\includegraphics[scale=0.5]{plots/ANDthreshold}
\caption{AND gadget for the class of threshold totalistic functions.}
\label{fig:andthrehsold}
\end{figure}
\begin{figure}[!tbp]
\centering
\includegraphics[scale=0.5]{plots/ORthreshold}
\caption{OR gadget for the class of threshold totalistic functions.}
\label{fig:orthrehsold}
\end{figure}
\begin{theo}
Let $r,s \in \mathbb{N}$ and $f:\{0,1\}^r \to \{0,1\}^s$ be a monotone Boolean function. There exist a collection of graphs $\mathcal{G}$ such that $\mathcal{T}$ simulates $f$ in $\mathcal{G}$.
\label{teo:thershold}
\end{theo}
\begin{proof}
Fix $r,s \in \mathbb{N}$ and $f: \{0,1\}^r \to \{0,1\}^s$ an arbitrary function. It is well known that $f$ can be represented by a Boolean circuit $C_f:\{0,1\}^{r} \to \{0,1\}^{s}$. More precisely, for every variable assignment $y \in \{0,1\}^r$ the evaluation of the circuit computes $f(z)$. In addition, it suffices to consider bounded fanin and fanout circuits (more specifically we can always assume fanin and fanout $2$ for all gates, with the exception of input and output gates) and vertex set can always be considered as partitioned in layers (see \cite[Section 6.2]{greenlaw1995limits}). Each layer is defined by the length of longest path connecting a gate to an input gate. We are going to show that there exist some $t \in \mathbb{N}$, a graph $G=(V,E)$ and a set of threshold rules $\mathcal{F} = \{f_v:\{0,1\}^{N(v)} \to \{0,1\}\}$ defining an automata network $\mathcal{A}_f$ such that its associated global rule $F$ is such that $f(y) = F(x|_I)|_O$ for some sets nodes $I,O \subseteq V$ and some $x$ depending on $y$. Let $D_f$ the digraph defining circuit $C_f$. Without loss of generality, we can assume that $C_f$ is monotone, i.e., any gate computes only an AND or an OR gate. In other words, any node $v \in V(D_f)$ is labelled by a symbol $l(d) \in \{\wedge, \vee\}$ which represents the corresponding gate in the circuit. We define $G$ in the following way: for each $v \in V(D_f)$ that is not an input we assign one of the gadgets $\varphi(v)$ in Figure \ref{fig:andthrehsold} or Figure \ref{fig:orthrehsold} according to $l(v)$. For input gates we consider input nodes of gadgets representing gates in the first layer. Note that in order to represent output gates it is sufficient to consider output nodes in some gadget given by Figure \ref{fig:andthrehsold} or Figure \ref{fig:orthrehsold}. In addition, we can assume that $\delta^{+}_v = \delta^{-}_v = 2$. Note also that $\varphi(v)$ has two possible outputs $o$ and $o'$ in Figure \ref{fig:andthrehsold}. We define edges in $G$ locally by the connections in each gadget $\varphi(v)$ for each $v \in V(D_f)$ and also we identify the output of gadget $\varphi(v)$ with one of the inputs of gadget $\varphi(v')$ if $v' \in N^{-}(v)$. Note that $|V(G)| \leq \sum \limits_{v \in V(D_f)} |\varphi(v)| = 5 |V(D_f)| = r^{\mathcal{O}(1)}.$ From previous lemma we now that $\varphi(v)$ computes $\wedge(x,y)$ or $\vee(x,y)$ where $x,y$ are its inputs. We know also that it is done in a uniform time $t=2$ and there is also two possible choices for the outputs $o$ and $o'$ which receive the signal carrying the result of the computation at the same time. We define now an automata network $(G,\mathcal{F})$ where $\mathcal{F}$ contains all the rules defined for each gadget $\varphi(v)$ for each $v \in V(D_f)$. We define sets $I$ and $O$ as the nodes in $G$ corresponding to input gates in $D_f$ and the output nodes of gadgets representing output gates in $D_f$. Now, we locally set every gadget $\varphi(v)$ to its fixed point configuration and we call it $x$. We assign $z =(z_1,z_2,\hdots, z_r)$ to each of the inputs of corresponding input gadgets. We claim that at time $t = 2 \text{deph}(D_f) = r^{\mathcal{O}(n)}$ the global function satisfies $f(z)|_I = F^t(x)|_{O}.$ In fact, it is not difficult to see that inductively, in $t_1 = 2$ all the gadgets in the first layer compute the assignment $(z_1,z_2,\hdots, z_r)$ and each of the outputs that are associated inputs in the first layer have now this information as a perturbation of their fixed point configuration $x$. Now assume that in some time $t = 2k$, gadgets in the $k$-th layer are computing the information received from layer $k-1.$ Again, because of Lemma \ref{lem:andorthreshold} we know that each gadget $\varphi(v)$ produces consistently an AND or an OR computation of its inputs in uniform time $t = 2$ and thus, $k+1$-th layer computes information of $k$-th layer in time $2k+2 = 2(k+1)$. Then, the claim holds. As a consequence of the claim we have that $(G,\mathcal{F})$ simulates $f$. The result holds.
\end{proof}
\subsubsection{Majority rules}
An important example is the case in which each local rule will change to $1$ when the majority of the nodes in the neighbourhood of its associated node $v$ is in state $1$. More precisely, when $\theta_v = \frac{\delta_v}{2}.$ When this happens, we say that local rule $f_v$ is a majority rule. Of course this depend on the graph. Now we will show that we can simulate any monotone Boolean network by using only majority rules. Analogously to the previous result, we show first that we can find AND and OR functions as a part of some automata networks defined by majority rules.
\begin{lem}
There exist two automata networks $\mathcal{A}_2 = (G_1=(V_1,E_1),\mathcal{F}_1)$ and $\mathcal{A}_2 = (G_2=(V_2,E_2),\mathcal{F}_2)$ with global rules $F_1$ and $F_2 $ respectively, such that $\mathcal{F}_i$ are majority rules and
\begin{enumerate}
\item $\wedge(x,y) = F^2_1(z_1)_{o} = F^2_1(z_1)_{o'}.$
\item $\vee(x,y) = F^2_2(z_2)_{o} = F^2_2(z_2)_{o'},$
\end{enumerate} for some $z_i \in \{0,1\}^{|V_i|},$ $i=1,2.$
\label{lem:andormaj}
\end{lem}
\begin{proof}
Consider the graph $G_1$ and $G_2$ given in Figure \ref{fig:andmaj} and Figure \ref{fig:ormaj}. We define $\overline{z}_1 = (0,0,0,0,0)$ and $\overline{z}_2 = (0,0,0,0,0)$. The result follows from the calculations in latter figures.
\end{proof}
\begin{figure}[!tbp]
\centering
\includegraphics[scale=0.5]{plots/ANDmaj}
\caption{AND gadget for the class of majority totalistic functions.}
\label{fig:andmaj}
\end{figure}
\begin{figure}
\centering
\includegraphics[scale=0.5]{plots/ORmaj}
\caption{OR gadget for the class of majority totalistic functions.}
\label{fig:ormaj}
\end{figure}
\begin{theo}
Let $r,s \in \mathbb{N}$ and $f:\{0,1\}^r \to \{0,1\}^s$ be a monotone Boolean function. There exist an automata network $\mathcal{A}_f = (G,\mathcal{F})$ with global rule $F$ such that $\mathcal{F}$ are majority rules such that there exist $t= r^{\mathcal{O}(1)}$ satisfying $f(y) = F(x|_I)|_O$ for all $y \in \{0,1\}^r$, some sets $I,0 \subseteq V$ and some $x$ depending on $y$.
\end{theo}
\begin{proof}
The proof of this result is completely analogous to Theorem \ref{teo:thershold}.
\end{proof}
\begin{remark}
It is also possible to simulate the evaluation of an arbitrary Boolean circuit by a monotone circuit. The construction duplicates the gates and uses De Morgan's laws in order to simulate the evaluation of NOT gates. Roughly, for each gate $v$ we work with a two duplicates $v_+$ and $v_{-}$ such that $v_+$ is true if and only if $v$ is true and $v_-$ is true if and only if $v$ is false. By duplicating in this way every gate in the original circuit we use $v_{-}$ any time we need to evaluate a NOT gate. However, we mention this only as a remark because we consider it goes quite far away from our definition of simulation (though, one could adapt things to make it work).
\end{remark}
\subsection{Isolated totalistic rules}
Now we introduce another class of totalistic rules that we call \textit{isolated}. In general, the class of isolated totalistic rules are rules that are activated by a precise level of activation in the neighbourhood of a given node. In fact, these rules will be activated if and only if the amount of active neighbours is exactly some value $\alpha$ and will be $0$ for any other value in sufficiently large enough interval containing $\alpha$. We detail this as following.
\begin{definition}
A totalistic rule $f:S_{\Delta} \to \{0,1\}$ is isolated if there is a positive integer $\alpha\geq3$ such that $[\alpha-2,\alpha+1] = \{\alpha-2,\alpha-1, \hdots, \alpha+1\} \cap \mathcal{I}_f = \{\alpha\}.$
\end{definition}
For example the rule $3$ is isolated because configurations that have an amount of $1$s in the interval $[1,4]$ will only produce $1$ as image if they have exactly $3$ ones. Note that, for example, any other totalistic rule of the form $3a$ with $a\geq 5$ will be isolated with $\alpha = 3$. In the next section, we will call the value $\alpha$ an isolated value for some fixed rule. For example $3$ is an isolated value for rule $35$ and so for rule $3$. Note also that one fixed isolated rule can have multiple isolated values. For example, rule $36$ has $3$ as isolated value and also $6$.
\begin{remark}
Note that in the latter definition, taking $\alpha -2$ as a non active value for the rules is necessary in order to avoid considering matrix-defined functions that we have already studied. In fact, if we assume $\alpha=3$ and we allow $1$ to be active, then rule $135$ is the XOR rule for a neighbourhood with $6$ nodes.
\end{remark}
\begin{lem}
For each $\alpha \geq 3$ there is an automata network $\mathcal{A}_\alpha= (G_\alpha,\mathcal{F}_\alpha)$ such that every $f \in \mathcal{F}_\alpha$ is a totalistic function with isolated value $\alpha$ and such that its global rule $F_{\alpha}$ satisfies that: $\exists i_1, i_2 ,o_1,o_2\in V(G): F_{\alpha}^3(x)_{o_j} = \textbf{NAND}(x|_{i_1},x|_{i_2}), j=1,2$ for any $x$ that is a perturbation in $i_0$ and $i_1$ of some $z \in \text{Fix}(\mathcal{A}_\alpha)$, i.e. $x_v = z_v$ for all $v \not \in \{i_1,i_2\}.$ In particular, NAND gate is in the spectrum of $\mathcal{A}_{\alpha}$ for $t=3$ and $l = 2$.
\label{lemma:NAND}
\end{lem}
\begin{proof}
Let $ \alpha \geq 3$. We show explicitly the structure and dynamics of $\mathcal{A}_\alpha$ in Figure \ref{fig:NANDiso}. Note that the graph structure strongly depends on the fact that complete graphs $K_{\alpha +1}$ are stable connected components for state $1$ in the sense that nodes inside this clique will be always in state $1.$ From the Figure \ref{fig:NANDiso} it is evident that fixed point $z$ is given by the state in which $z_v = 0$ for any $v$ which is not part of one the two cliques in the graph and that computation of NAND is a consequence of a perturbation of $z$.
\end{proof}
\begin{figure}[!tbp]
\includegraphics[scale=0.45]{plots/NANDiso0.pdf}
\includegraphics[scale=0.45]{plots/NANDiso2.pdf}
\includegraphics[scale=0.45]{plots/NANDiso1.pdf}
\includegraphics[scale=0.45]{plots/NANDiso3.pdf}
\caption{NAND gadget for $\alpha$-uniform isolated totalistic rules with $\alpha \geq 3.$ Grey $K_{\alpha+1}$ components are fixed in state $1$.}
\label{fig:NANDiso}
\end{figure}
During the rest of this section we will call the automata network in Figure \ref{fig:NANDiso} a NAND gadget. We will now show that the class of $\alpha$-uniform totalistic isolated rules is complete for some class of graphs $\mathcal{G}$ that we will show. To do that, we need to define two more gadgets: a clock gadget and a clocked NAND gadget.
\begin{figure}[!tbp]
\centering
\includegraphics[scale=0.5]{plots/clockiso.pdf}
\caption{Clock gadget for $\alpha$-uniform isolated totalistic rules with $\alpha \geq 3.$ Grey $K_{\alpha+1}$ components are fixed in state $1$.}
\label{fig:clockiso}
\end{figure}
\begin{lem}
For each $\alpha \geq 3$ and $d \geq 1$ there is an automata network $\mathcal{A}_{\alpha,d}= (G_{\alpha,d},\mathcal{F}_{\alpha,d})$ such that every $f \in \mathcal{F}_{\alpha,d}$ is isolated with isolated value $\alpha$ and such that its global rule $F_C$ satisfies that there exists $o \in V(G)$: $F_C^s(x)_{o} = 1$ for $0\leq s \leq d-1$ and $F_C^{d}(x)_{o} = 0$ for some $x \in \{0,1\}^n$.
\end{lem}
\begin{proof}
See Figure \ref{fig:clockiso} for the structure of the gadget and the definition of $x$. Note that the gadget works as a wire defined by its path structure that carries the $1$ signal which perturbs one node in one of the complete graphs $K_{\alpha+1}$ at the end of the path after $d$ time steps. This is because each node in the path has exactly $\alpha-1$ neighbours in state $1$ and thus, incoming signal allow them to change its state.
\end{proof}
Finally we introduce the clocked NAND gadget in the following lemma.
\begin{lem}
For each $\alpha \geq 3$ and $d \geq 1$ there is an automata network $\mathcal{A}_{\alpha,d}= (G_{\alpha,d},\mathcal{F}_{\alpha,d})$ such that $\mathcal{F}_{\alpha,d} \in \mathcal{S}_\alpha$ and such that its global rule $F_{CN}$ satisfies that there exist $i_1,i_2,o_1,o_2 \in V(G)$ such that $F_{CN}^s(w)_{o_j} = 1, j=1,2$ for $0\leq s \leq d-1$ and $F_{CN}^{d+3}(x)_{o_j} = \textbf{NAND}(F^d(w)|_{i_1},F^d(w)|_{i_2}), j=1,2$ for some $w \in \{0,1\}^n$
\end{lem}
\begin{proof}
In Figure \ref{fig:clockedNANDiso} we show the structure of a clocked NAND gadget. We define $i_1 $and $i_2$ as the nodes that are labelled by $x$ and $y$ in Figure \ref{fig:clockedNANDiso}. We define $w' = z \cup r$ the concatenation of the configuration $z$ in Lemma \ref{lemma:NAND} and $r$ in the previous lemma. Finally we define $w_{i_1} = w_{i_2} = 1$ and $w_v = w'_v$ for all $v \not \in \{i_1,i_2\}$. Note that central node (which is connected to nodes labelled as $x$ and $y$) is in state $0$ and has exactly $\alpha + 1$ active networks: $\alpha - 3$ active networks from the clique in the upper part of the gadget, one from the clique in the right, one from the clock gadget and two from the inputs. As $\alpha$ is an isolated value for the local rules in the gadget then, we have $F_{CN}^s(w)_{o_j} = 1, j=1,2$ for $0\leq s \leq d-1.$ Now note that in $t=d$ the neighbour of the central node located in the clock gadget will change to $0$ and then we will recover the same scenario shown in Figure \ref{fig:clockiso} with nodes labelled $x$ and $y$ assuming the values $F(x)|_{i_1}$ and $F(x)|_{i_2}$ and thus as a consequence of latter lemma, the result holds.
\end{proof}
\begin{figure}[!tbp]
\centering
\includegraphics[scale=0.5]{plots/clockedNAND.pdf}
\caption{Clocked NAND gadget for $\alpha$-uniform isolated totalistic rules with $\alpha \geq 3$ and delay $d.$ Clock picture represents gadget in Figure \ref{fig:clockiso}.}
\label{fig:clockedNANDiso}
\end{figure}
Note that as inputs in clocked NAND gadget in Figure \ref{fig:clockedNANDiso} will be fixed in $1$ and thus, the second part of the latter lemma may seem trivial. However, as now we would want to connect different clocked NAND gadgets in order to simulate a circuit, nodes $i_1$ and $i_2$ will be identified with some outputs of some other gadget, allowing us to make simulate the calculations of a NAND gate at the wright time. We will detail this below. Now we are prepared to show the main result of this section.
\begin{theo}
Let $r,s \in \mathbb{N}$ and $f: \{0,1\}^r \to \{0,1\}^s$ a Boolean function. For each $\alpha \geq 3$ there is a set of isolated functions $\mathcal{F}$ with isolated value $\alpha$ and a bounded degree class of graphs $\mathcal{G}$ such that $\mathcal{F}$ simulates $f$ in $\mathcal{G}$.
\label{teo:isolated}
\end{theo}
\begin{proof}
Let $C_f$ be a circuit representing $f$. Without loss of generalisation we can assume that any gate is a NAND gate. We will use again the same reasoning we used for Theorem \ref{teo:thershold}. The only difference here is that we have to be very careful in order to initialise clocked NAND gadgets in each layer of the circuit as it is shown in Figure \ref{fig:circuitiso}. In order to do that, we set locally each clocked NAND get to the local configuration $w$ given in the latter lemma with exception of the nodes representing input gates (those nodes will be identify with inputs of gadgets in first layer which do not have any clock ). Nodes in the second layer will wait time $d=3$ as a consequence of their clocks. When the calculations of the first layer arrives to the second one, their clocks will be off (the neighbour located in the clock will be inactive) and thus they will be able to compute a NAND from the values computed in the first layer. Then, third layer has to wait $6$ in order to coherently simulate the evaluation of the circuit so we choose $d=3$ as it is shown in Figure \ref{fig:circuitiso}. In general the $k$-th layer will have a clock with delay $d = 3k$. Note that, as a consequence of last lemma, any gadget with exception of the first layer will stay fixed in configuration $w$ and will calculate NAND when its clock gets inactive. Then, after $t = 3 \text{depth}(C_f)$ and thus, the result holds.
\end{proof}
\begin{figure}[!tbp]
\centering
\includegraphics[scale=0.56]{plots/circuit}
\caption{Scheme of the underlying graph of an automata network simulating a circuit defined over a class of $\alpha$-uniform isolated totalistic rules with $\alpha \geq 3.$ $\text{NAND}_d$ represents clocked NAND gadget in Figure \ref{fig:clockedNANDiso} }
\label{fig:circuitiso}
\end{figure}
\subsubsection{Special cases: Rule $1$ and Rule $2$.}
In this section, we study two special cases of totalistic rules that are not strictly included in the last formalism. These are the rule $1$ and rule $2$ that change its values to active (change to state $1$) if and only if there is $1$ (respectively $2$) active neighbours in some given time step. The case of rule $1$ is particularly interesting considering that the study of the dynamics (particularly related to decision problems such as prediction) is still open in the case in which the network is a two-dimensional cellular automata (the underlying interaction graph of the network is a grid).
\paragraph{Rule $1$}
We start that showing that the XOR gates is in the spectrum of some automata network in which every rule is given by the rule $1$
\begin{prop}
There is a Rule $1$ automata network $\mathcal{A} = (G =(V,E),\mathcal{F})$ such that for some $i_1, i_2, o_1, o_2 \in V$ we have that $F^2(z)_{o_j} = \textbf{XOR}(z|_{i_1},z|_{i_2}) = z|_{i_1} \oplus z|_{i_2} $ for $j=1,2$ and some perturbation $z$ of fixed point $\vec{0}$ in $i_1$ and $i_2$. In particular, XOR gate is in the spectrum of $\mathcal{A}.$
\end{prop}
\begin{proof}
We show the gadget that defines $\mathcal{A} = (G =(V,E),\mathcal{F})$ in Figure \ref{fig:XOR1}. We identify the nodes at the left labelled by $x$ and $y$ as inputs $i_1$ and $i_2$ and the ones in the wright in state $0$ as $o_1$ and $o_2$. Note that because of the local rule we will read in the output a $1$ if exclusively one of the inputs is in state $1$. Thus, gadget in Figure \ref{fig:XOR1} computes a XOR gate. The result holds.
\end{proof}
\begin{figure}[!tbp]
\centering
\includegraphics[scale=0.28]{plots/XOR10.pdf}
\includegraphics[scale=0.28]{plots/XOR11.pdf}
\includegraphics[scale=0.28]{plots/XOR12.pdf}
\caption{XOR gadget for Rule $1$. $x$ and $y$ nodes represent the inputs and the nodes in the left represent the output. Computation takes place in central node and output is read after $2$ time-steps.}
\label{fig:XOR1}
\end{figure}
Now, we slightly modify this gadget in order to compute a NOR gate. This will allow us to simulate an arbitrary Boolean circuit.
\begin{prop}
There is a Rule $1$ automata network $\mathcal{A} = (G =(V,E),\mathcal{F})$ such that for some $i_1, i_2, o_1, o_2 \in V$ and a configuration $y \in \{0,1\}^{|V|}$, we have that $F^2(z)_{o_j} = \textbf{NOR}(z|_{i_1},z|_{i_2})$ for $j=1,2$ and some perturbation $z$ of $y$ in $i_1$ and $i_2$.
\end{prop}
\begin{proof}
Figure \ref{fig:NOR1} shows the gadget that defines $\mathcal{A} = (G =(V,E),\mathcal{F})$. We identify the nodes on the left labelled by $x$ and $y$ as inputs $i_1$ and $i_2$ and the ones on the right in state $0$ as $o_1$ and $o_2$. Note that the configuration $y$ sets all the nodes in state $0$ with exception of the node in the upper part, connected to the node in the central part of the graph, which is in state $1$. Note that this node blocks every signal coming from the inputs (every assignation for $x$ and $y$), forcing the outputs to change to $1$ in $2$ times steps only in the eventuality in which $x=y=0$. Thus, gadget in Figure \ref{fig:NOR1} computes a NOR gate. The result holds.
\end{proof}
\begin{figure}[!tbp]
\centering
\includegraphics[scale=0.28]{plots/NOR10.pdf}
\includegraphics[scale=0.28]{plots/NOR11.pdf}
\includegraphics[scale=0.28]{plots/NOR12.pdf}
\caption{NOR gadget for Rule $1$. $x$ and $y$ nodes represent the inputs and the nodes in the left represent the output. Computation takes place in central node and output is read after $2$ time-steps.}
\label{fig:NOR1}
\end{figure}
Now, we need to synchronise different NOR gadgets in order to connect them. This will allow us to simulate arbitrary Boolean circuits. In order to fulfil this task, we need first to introduce the $d$-wire gadget. This latter gadget is completely analogous to the clock gadget developed for isolated rules in Figure \ref{fig:clockiso}.
\begin{prop}
There is a Rule $1$ automata network $\mathcal{A} = (G =(V,E),\mathcal{F})$ such that for some nodes $i$ and $o$ we have that $F^d(y)|_{o} = y|_i$ for any perturbation in $i$ of the steady-state $\vec{0}$.
\end{prop}
\begin{proof}
$d$-wire gadget is shown in Figure \ref{fig:wire1}. The result holds directly from the definition of rule $1$.
\end{proof}
\begin{figure}[!tbp]
\centering
\includegraphics[scale=0.5]{plots/wire1.pdf}
\caption{$d$-wire gadget for Rule $1$.}
\label{fig:wire1}
\end{figure}
Now, we show a general gadget which computes a NOR gate in some given time $t = d$ taking as output an arbitrary state defined for the dynamics of an arbitrary Rule $1$ automata network, provided that, the dynamics of these automata networks must not locally perturb the computation of the NOR gadget, i.e. nodes connected to this latter gadget must remain in state $0$ until time of computation, more precisely, for $t =0,\hdots, d-1$ time steps.
\begin{prop}
Let $\mathcal{A} = (G,\mathcal{F})$ and $\mathcal{A'} = (G',\mathcal{F}')$ be two rule $1$ automata networks such that for some $i \in V(G)$ and $i' \in V(G')$ the respective global rules $F$ and $F'$ are such that $F^s(x)_i = F'^s(x')_{i'} = 0$ for $s = 0,\hdots, d-1$. Then, there exist a rule $1$ automata network $\mathcal{A}_d = (G_d,\mathcal{F}_d)$ such that its global rule $F_d$ is satisfies that $(F_d^{d+2}(z))_{o_1} = \textbf{NOR}(F^{d}(z)|_i,F'^{d}(z)|_{i'}).$
\end{prop}
\begin{figure}[!tbp]
\centering
\includegraphics[scale=0.5]{plots/NOR1general}
\caption{$d$-delayed NOR gadget for rule $1$.}
\label{fig:NORgen1}
\end{figure}
\begin{proof}
We show the gadget $\mathcal{A}_d$ in Figure \ref{fig:NORgen1}. The result is the direct consequence of the last proposition and the fact that dynamics of $\mathcal{A}$ and $\mathcal{A'}$ do not perturb the dynamics of NOR gadget. The system remains in state $0$ with exception of the signal that travels through the $d$-wire gadget and arrives to the terminal node connected to central node of the gadget in time $t=d$. Computation takes place as shown in Figure \ref{fig:NOR1}. The result holds.
\end{proof}
We call the gadget in Figure \ref{fig:NORgen1} a delayed NOR gadget. Finally, as a direct consequence of an analogous construction of the one we exhibited for isolated rules, we can show that we can simulate an arbitrary Boolean by correctly synchronizing different delayed NOR gates.
\begin{theo}
Let $r,s \in \mathbb{N}$ and $f: \{0,1\}^r \to \{0,1\}^s$ a Boolean function. There exist a a bounded degree class of graphs $\mathcal{G}$ such that rule $1$ simulates $f$ in $\mathcal{G}$.
\end{theo}
\begin{proof}
Proof is analogous to the proof of Theorem \ref{teo:isolated}.
\end{proof}
\paragraph{Rule 2}
For rule $2$ we show in Figure \ref{fig:NAND2} that it is capable of computing a NAND gate and we note that it is possible to use an analogous reasoning of the one of Theorem \ref{teo:isolated} in order to deduce that this rule can also simulate arbitrary Boolean networks. Particularly, we will need a clocked version of this gate that can be built by attaching three clock gadgets (see Figure \ref{fig:clockiso}) to the node that computes an AND gate from inputs (labelled by $x$ and $y$) in Figure \ref{fig:NAND2}.
\begin{figure}[!tbp]
\includegraphics[scale=0.45]{plots/NAND20.pdf}
\includegraphics[scale=0.45]{plots/NAND21.pdf}
\includegraphics[scale=0.45]{plots/NAND22.pdf}
\includegraphics[scale=0.45]{plots/NAND23.pdf}
\caption{NAND gadget for Rule $2$. $x$ and $y$ nodes represent the inputs and the nodes in the left represent the output. Gray nodes are part of a clique (triangle) fixed in state $1$. Computation takes place in central node and output is read after $4$ time-steps.}
\label{fig:NAND2}
\end{figure}
\subsection{Interval rules}
In this section we study the case in which the class of totalistic rules is defined by interval rules. In this particular sub-class of automata networks, active values are achieve by reaching an amount of active neighbours in a fixed interval $[\alpha,\beta]$. Of course we are assuming that $\beta$ is strictly less than the degree of each node in the interaction graph. We do this to avoid the case in which all the rules are simply threshold. Results of this section are completely analogous to the results related to isolated rules. In fact, we start by showing a NAND gadget, then we show a clock gadget and finally we show that we can use it to coordinate the evaluation of an arbitrary Boolean circuit.
First, we present a NOT gadget:
\begin{figure}[!tbp]
\includegraphics[scale=0.35]{plots/NOTinter0.pdf}
\includegraphics[scale=0.35]{plots/NOTinter1.pdf}
\includegraphics[scale=0.35]{plots/NOTinter2.pdf}
\caption{NOT gadget for Interval rules. $x$ and $y$ nodes represent the inputs and the nodes in the left represent the output.}
\label{fig:NOTinter}
\end{figure}
\begin{lem}
For each $\alpha, \beta$, $2\leq\alpha\leq \beta$, there is an automata network $\mathcal{A} =(G, \mathcal{F})$ in which every rule is interval with threshold $\alpha$ and $\beta$ and such its global rule $F$ satisfies $F^(x)_o = \overline{x_i}$ for some configuration $x \in Q^{|V(G)|}$.
\end{lem}
\begin{proof}
See Figure \ref{fig:NOTinter}.
\end{proof}
Then, we combine the latter gadget with other structures in order to generate a NAND gadget.
\begin{lem}
For each $\alpha, \beta$, $2\leq\alpha\leq \beta$, there is an automata network $\mathcal{A} =(G, \mathcal{F})$ in which every rule is interval with threshold $\alpha$ and $\beta$ and such its global rule $F$ satisfies $F^3(y)_{o_j} = \textbf{NAND}(y|_{i_1},y|_{i_2})$
\end{lem}
\begin{figure}[!tbp]
\includegraphics[scale=0.45]{plots/NANDinter0.pdf}
\includegraphics[scale=0.45]{plots/NANDinter1.pdf}
\includegraphics[scale=0.45]{plots/NANDinter2.pdf}
\includegraphics[scale=0.45]{plots/NANDinter3.pdf}
\caption{NAND gadget for Interval rules. $x$ and $y$ nodes represent the inputs and the nodes in the left represent the output.}
\label{fig:NANDinter}
\end{figure}
We show now that we can generate a clock gadget from a wire gadget.
\begin{lem}
For each $2\geq\alpha\geq\beta$ and $d \geq 1$ there is an automata network $\mathcal{A}_{\alpha,d}= (G_{\alpha,d},\mathcal{F}_{\alpha,d})$ such that every $f \in \mathcal{F}_{\alpha,d}$ is an interval rule with threshold $\alpha$ and $\beta$ and such that its global rule $F_C$ satisfies that there exists $o \in V(G)$: $F_C^s(x)_{o} = 1$ for $0\leq s \leq d-1$ and $F_C^{d}(x)_{o} = 0$ for some $x \in \{0,1\}^n$.
\end{lem}
\begin{proof}
We use the gadget from Figure \ref{fig:wireinter} to build a clock gadget. This is analogous to Figure \ref{fig:clockiso}.
\end{proof}
\begin{figure}[!tbp]
\centering
\includegraphics[scale=0.5]{plots/wireinter.pdf}
\caption{$d$-wire gadget for Interval rules.}
\label{fig:wireinter}
\end{figure}
\begin{figure}[!tbp]
\centering
\includegraphics[scale=0.35]{plots/clockinter.pdf}
\caption{$d$-clock gadget for Interval rules.}
\label{fig:clock}
\end{figure}
We now introduce a clocked-NAND gadget.
\begin{lem}
For each $2\geq\alpha\geq\beta$ and $d \geq 1$ there exists an automata network $\mathcal{A}_{\alpha,d}= (G_{\alpha,d},\mathcal{F}_{\alpha,d})$ such that every $f \in \mathcal{F}_{\alpha,d}$ is an interval rule with threshold $\alpha$ and $\beta$ and such that its global rule $F_{CN}$ satisfies that there exist $i_1,i_2,o_1,o_2 \in V(G)$ such that $F_{CN}^s(w)_{o_j} = 1, j=1,2$ for $0\leq s \leq d-1$ and $F_{CN}^{d+3}(x)_{o_j} = \textbf{NAND}(F^d(w)|_{i_1},F^d(w)|_{i_2}), j=1,2$ for some $w \in \{0,1\}^n$
\end{lem}
\begin{figure}[!tbp]
\centering
\includegraphics[scale=0.35]{plots/clockNANDinter.pdf}
\caption{A clocked NAND gadget with delay $d$ for interval rules.}
\label{fig:clockedNANDinter}
\end{figure}
\begin{proof}
Gadget is shown in Figure \ref{fig:clockedNANDinter}. This latter gadget is analogous to clocked NAND gadget for isolated rules (see Figure \ref{fig:clockedNANDiso}.)
\end{proof}
Now we can introduce the main result of the section:
\begin{theo}
Let $r,s \in \mathbb{N}$ and $f: \{0,1\}^r \to \{0,1\}^s$ a Boolean function. For each $2\leq \alpha \leq \beta$ there exist a set of interval functions $\mathcal{F}$ with interval values [$\alpha$,$\beta$] and a bounded degree class of graphs $\mathcal{G}$ such that $\mathcal{F}$ simulates $f$ in $\mathcal{G}$.
\label{teo:interval}
\end{theo}
\begin{proof}
Proof is analogous to the proof o Theorem \ref{teo:interval}. See Figure \ref{fig:circuitiso} for a scheme of the network simulating an arbitrary Boolean circuit.
\end{proof}
\begin{remark}
The case in which $\alpha = 1$ is analogous to Rule $1$. In fact, the same gadgets can be use to simulate arbitrary Boolean functions with the exception of delayed NOR gadget which is given in Figure \ref{fig:1intervalNOR}
\begin{figure}[!tbp]
\centering
\includegraphics[scale=0.5]{plots/1interNOR.pdf}
\caption{Delayed NOR gadget for $\alpha = 1$.}
\label{fig:1intervalNOR}
\end{figure}
\end{remark}
\begin{remark}[Planarity of the gadgets.]
It is important to point out that in most of our gadgets we have to consider a huge amount of fixed states (1), which implies that those subset are in general complete graphs (i.e., the vertex are fully connected) which are for most than 4 vertex non-planar. Actually, by considering the Kuratowski characterisation \cite[Theorem 4.4.6]{diestelgraph} in our constructions we have that for any $\alpha \geq 4$, the NAND gadgets and clocked NAND gadgets are not planar, because from Figure \ref{fig:NANDiso} and Figure \ref{fig:clockiso} we realise that each gadget contains at least $K_{\alpha+1}$ as a subgraph, which is a forbidden subgraph in the Kuratowski characterisation.
\end{remark}
\section{Numerical Experiments}
In this section, first we show some results on simulations of totalistic automata networks over random graphs. More precisely, we generate a collection of $1000$ random graphs using the well known Erdös-Renyi model and we study all the totalistic rules with a maximum of $4$ active neighbours (i.e. $\mathcal{I}_v \subseteq
\{0,\hdots,4\}$ for each node $v$ in the network).
In general, in this section we study all the possible Boolean gates that each rule in the latter class can calculate in some randomly generated graph, by trying any possible combination of three nodes as a set of two inputs and one output. We are also interested in identify a set of small graphs, that we call, gadgets, which exhibit a richer spectrum of Boolean gates in the latter simulation. We remark that in this section, in order to simplify the simulations, we observe the system capabilities to simulate boolean gates starting from perturbations of the quiescent state $\vec{0}.$ This latter choice could introduce some bias in some of our results because rules which need more than $2$ active neighbours in order to change to state $1$ (e.g. rules $3$, $34$, etc) will probably not show any interesting behaviour (we are introducing changes in only two nodes and the rest of the system stays in state $0$). In order to avoid this misleading effect, we consider, in the second part of this section, a sample of different fixed points and we repeat the previous simulations. However, as we consider that changing the interaction graph and the fixed point might add some extra complexity to the analysis of our results, instead we fix the underlying interaction graph as a small two dimensional grid and we study the simulation capabilities of the system starting from different fixed points. This is also interesting considering that for some of the totalistic rules that we are considering here, as rule $1$, the study of the capabilities of the system in order to simulate boolean circuits is still open in the case in which the interaction graph is the two dimensional grid.
Specifically, this section is organised as follows:
\begin{enumerate}
\item First we show a general collection of results to describe the landscape of simulation capabilities of each totalistic automata network with at most $4$ active neighbours: we exhibit the relative frequency in which each graph can simulat all the possible $2^{2^2} = 16$ Boolean gates and we choose the graph that has the greater spectrum from the collection of randomly generated graph (that is to say that exhibits more logic gates from the set of $16$ possible Boolean gates with $2$ inputs and $1$ output). In this regard, we study the impact of the connectivity of each random generated graph by changing the probability of two given nodes to be connected. We call this probability $p \in [0,1]$.
\item From the latter simulations, we choose one rule from the set of totalistic rules (rule $1$) that we have considered in the latter subsection and we exhibit the most representative gadget (i.e. the smallest graph with greatest spectrum)). Then, we study its dynamics, with emphasis in understanding how it simulates a given a Boolean gate.
\item We extend the study of this specific gadget into the study of a particular one dimensional celullar automaton inspired in its topology.
\item We fix a small two dimensional grid (4x4) and we generated a sample of fixed points for each rule. Then, we repeat previous simulations but now different fixed points play the role of random graphs of the latter sections.
\end{enumerate}
\subsection{General landscape in random graphs}
We start by remarking that as we use the Erdös-Renyi model, two parameters must be chosen in order to define a random graph: the number of nodes $n$ and the probability $p$ of two arbitrary nodes to be connected. We also recall that we are considering only logic gates with two inputs and one output. As we can encode each of these Boolean functions as a sequence of $4$ bits, we can represent each function by a number $i \in \{0,\hdots,15\}.$ For example, if $i=8$ then we have the AND rule since $8 = 0001$ and we are considering the following coding: $00 \to 0$, $01 \to 0$ $10 \to 0$ and $11 \to 1$. Analogously, OR gate is $i=14$, XOR gate is $i=6$, NAND gate is $i=7$ and NOR gate is $i=1.$ Finally, we start all the simulations from the fixed point $\overline{x} = \vec{0}$ and we choose perturbations in all the possible assignations for two inputs and one output over a randomly generated graph.
We are now in condition of summarize the set-up of parameters that we used for the simulations.
\subsection{Simulations set-up}
We start by describing the parameters of the following simulations:
\begin{enumerate}
\item Probability of adjacency of two given nodes ($p$): 0.1, 0.5, 0.8
\item Number of generated graphs ($N$): 100
\item Number of nodes per graph ($n$): 10
\item Simulation time $t$: 100
\end{enumerate}
\subsubsection{Results}
\paragraph{The impact of connectivity.}
One of the most straightforwards observations from the results we show in Figures \ref{plot:p01}, \ref{plot:p05} and \ref{plot:p08} is that there is an effect of the connectivity of the different random graphs in the diversity of Boolean gates that certain rules can simulate. More precisely, if we see Table \ref{tab:p01} we observe that rules that need at least $2$ neighbours in order to activate one node (i.e. $2,23,24,3,34,$ etc ) can only simulate the trivial function (i.e. all inputs goes to $0$). Of course, this is intrinsically related to the definition of the dynamics which depends on the number of active neighbours to produce any dynamic behavior different from $\vec{0}$. So, in those cases, in order to study the spectrum we have, if possible, to consider other fixed points. In this context, in Table \ref{tab:p01} and \ref{tab:p05} we can observe that most of the gates are simulated by rules that have the possibility to change to state $1$ when they have at least one active neighbour. \\
\begin{table}[H]
\resizebox{\textwidth}{!}{%
\begin{tabular}{|l|l|l|l|l|l|l|l|l|l|l|l|l|l|l|l|l|}
\hline
& 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 & 14 & 15 \\ \hline
1 & 100 & 0 & 6 & 0 & 10 & 0 & 18 & 0 & 7 & 0 & 42 & 0 & 44 & 0 & 12 & 0 \\ \hline
2 & 100 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \hline
3 & 100 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \hline
4 & 100 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \hline
12 & 100 & 0 & 6 & 0 & 8 & 0 & 1 & 0 & 0 & 0 & 79 & 0 & 75 & 0 & 38 & 0 \\ \hline
13 & 100 & 0 & 1 & 0 & 1 & 0 & 33 & 0 & 1 & 0 & 51 & 0 & 53 & 0 & 1 & 0 \\ \hline
14 & 100 & 0 & 6 & 0 & 9 & 0 & 20 & 0 & 7 & 0 & 43 & 0 & 45 & 0 & 12 & 0 \\ \hline
23 & 100 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \hline
24 & 100 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \hline
34 & 100 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \hline
123 & 100 & 0 & 1 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 83 & 0 & 82 & 0 & 50 & 0 \\ \hline
124 & 100 & 0 & 8 & 0 & 8 & 0 & 2 & 0 & 1 & 0 & 80 & 0 & 76 & 0 & 39 & 0 \\ \hline
134 & 100 & 0 & 3 & 0 & 2 & 0 & 34 & 0 & 3 & 0 & 52 & 0 & 54 & 0 & 4 & 0 \\ \hline
234 & 100 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \hline
1234 & 100 & 0 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 84 & 0 & 83 & 0 & 51 & 0 \\ \hline
\end{tabular}%
}
\caption{Spectrum of Boolean gates by totalistic rule for $p=0.1$. Each column contains the number of graphs that are capable of simulate the corresponding Boolean gate.}
\label{tab:p01}
\end{table}
\begin{figure}[!tbp]
\centering
\includegraphics[scale=0.20]{plots/01p_10nodes_100graphs_graph_list.pdf}
\caption{Graphs with the greatest spectrum by rule for $p=0.1.$}
\label{plot:p01}
\end{figure}
On the other hand, if we observe Table \ref{tab:p05} we can see that rules with higher threshold (that is to say that needs at least 2 active neighbours in order to change to state $1$) start to show some simulation capabilities. In particular, rules that change with $2$ active neighbours exhibit the possibility of calculate AND gates supporting the remark that higher connectivity implies higher simulation capabilities for some rule. \\
Finally, if we observe Figures \ref{plot:p01}, \ref{plot:p05} and \ref{plot:p08} we see that if $p \geq 0.5$ we have that every gadget is connected. Contrarily, in the case $p=0.1$ we observe that rules with higher threshold (those who need more active neighbours to active its nodes) tend to exhibit gadgets with bigger connected components. We remark the case of rule $1$ which seems to exhibit a minimal modular structure which is able to calculate several different logic gates.
\begin{table}[H]
\resizebox{\textwidth}{!}{%
\begin{tabular}{|l|l|l|l|l|l|l|l|l|l|l|l|l|l|l|l|l|}
\hline
& 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 & 14 & 15 \\ \hline
1 & 100 & 0 & 74 & 0 & 74 & 0 & 63 & 0 & 82 & 0 & 60 & 0 & 56 & 0 & 44 & 0 \\ \hline
2 & 100 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 58 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \hline
3 & 100 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \hline
4 & 100 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \hline
12 & 100 & 0 & 86 & 0 & 88 & 0 & 86 & 0 & 79 & 0 & 83 & 0 & 80 & 0 & 91 & 0 \\ \hline
13 & 100 & 0 & 94 & 0 & 94 & 0 & 91 & 0 & 96 & 0 & 89 & 0 & 90 & 0 & 83 & 0 \\ \hline
14 & 100 & 0 & 83 & 0 & 85 & 0 & 78 & 0 & 90 & 0 & 81 & 0 & 81 & 0 & 71 & 0 \\ \hline
23 & 100 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 83 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \hline
24 & 100 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 70 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \hline
34 & 100 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \hline
123 & 99 & 0 & 78 & 0 & 77 & 0 & 82 & 0 & 71 & 0 & 75 & 0 & 73 & 0 & 100 & 0 \\ \hline
124 & 100 & 0 & 93 & 0 & 96 & 0 & 94 & 0 & 95 & 0 & 93 & 0 & 95 & 0 & 97 & 0 \\ \hline
134 & 100 & 0 & 92 & 0 & 92 & 0 & 95 & 0 & 97 & 0 & 92 & 0 & 94 & 0 & 97 & 0 \\ \hline
234 & 100 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 100 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \hline
1234 & 94 & 0 & 53 & 0 & 50 & 0 & 56 & 0 & 40 & 0 & 46 & 0 & 46 & 0 & 100 & 0 \\ \hline
\end{tabular}%
}
\caption{Spectrum of Boolean gates by totalistic rule for $p=0.5$. Each column contains the number of graphs that can simulate the corresponding Boolean gate.\}}
\label{tab:p05}
\end{table}
\begin{figure}[!tbp]
\centering
\includegraphics[scale=0.20]{plots/05p_10nodes_100graphs_graph_list}
\caption{Graphs with the greatest spectrum by rule for $p=0.5$}
\label{plot:p05}
\end{figure}
\paragraph{More connectivity does not imply more diversity.}
If we study now rules that exhibit strong simulation capabilities even for low connectivity such as rule $1$, we observe that they do not show a significant change in their spectrum for higher values of $p$ (even if we can observe an increase in the frequencies of each Boolean gate in their spectrum, which implies that more graphs of the random sample are being capable of simulating one specific gate). Roughly, this observation may suggest that probably we need to study graphs with more nodes to observe a significant impact of connectivity in the spectrum of different totalistic rules. This is also coherent with the theoretical results regarding isolated totalistic networks in which we have theoretically constructed gadgets requiring big cliques subgraphs fixed in the state $1$ in order to perform calculations. \\
\paragraph{Complete Boolean gates set are not modular.}
Finally, we observe that gates NOR and NAND ($i=7$ and $i=1$ respectively in our notation) do not seem to be calculable by totalistic rules with at most $4$ active neighbours (see Table \ref{tab:p01}, Table \ref{tab:p05} and Table \ref{tab:p08}). It might suggest that we need a more complex structure in order to simulate complete sets of Boolean gates. We conjecture that maybe we can simulate them by ``glueing'' different gadgets in a coherent way. Nevertheless, in order to achieve this task, we need a better understanding of the particular dynamics of certain gadgets (see next subsection).
\begin{table}[H]
\resizebox{\textwidth}{!}{%
\begin{tabular}{|l|l|l|l|l|l|l|l|l|l|l|l|l|l|l|l|l|}
\hline
& 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 & 14 & 15 \\ \hline
1 & 100 & 0 & 1 & 0 & 1 & 0 & 0 & 0 & 39 & 0 & 1 & 0 & 1 & 0 & 0 & 0 \\ \hline
2 & 100 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 27 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \hline
3 & 100 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \hline
4 & 100 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \hline
12 & 100 & 0 & 11 & 0 & 10 & 0 & 8 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 \\ \hline
13 & 100 & 0 & 40 & 0 & 41 & 0 & 28 & 0 & 81 & 0 & 34 & 0 & 32 & 0 & 18 & 0 \\ \hline
14 & 100 & 0 & 61 & 0 & 61 & 0 & 48 & 0 & 91 & 0 & 54 & 0 & 53 & 0 & 38 & 0 \\ \hline
23 & 100 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 66 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \hline
24 & 100 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 83 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \hline
34 & 100 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \hline
123 & 99 & 0 & 29 & 0 & 27 & 0 & 18 & 0 & 11 & 0 & 14 & 0 & 13 & 0 & 14 & 0 \\ \hline
124 & 100 & 0 & 54 & 0 & 54 & 0 & 39 & 0 & 26 & 0 & 29 & 0 & 24 & 0 & 22 & 0 \\ \hline
134 & 99 & 0 & 66 & 0 & 63 & 0 & 52 & 0 & 90 & 0 & 58 & 0 & 53 & 0 & 41 & 0 \\ \hline
234 & 100 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 91 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \hline
1234 & 96 & 0 & 63 & 0 & 65 & 0 & 48 & 0 & 38 & 0 & 33 & 0 & 32 & 0 & 37 & 0 \\ \hline
\end{tabular}%
}
\caption{Spectrum of Boolean gates by totalistic rule for $p=0.8$. Each column contains the number of graphs that can simulate the corresponding Boolean gate.\}}
\label{tab:p08}
\end{table}
\begin{figure}[!tbp]
\centering
\includegraphics[scale=0.20]{plots/08p_10nodes_100graphs_graph_list.pdf}
\caption{Graphs with the greatest spectrum by rule for $p=0.8$}
\label{plot:p08}
\end{figure}
\subsection{Generating boolean gates in the two dimensional grid.}
In this subsection, in order to consider fixed points different from $\vec{0}$ (if there exists)we repeat the same computational experiments that we did before with different random graphs but this time, we fix the graph and we change the fixed point we are perturbing in order to generate boolean gates. More precisely we consider a two dimensional grid and for each rule we consider a sample of fixed points that we previously generated by simply simulating the system and waiting for it to attain a fixed point. This simulation time was previously established as $t=100$. Once the fixed point sample is obtained for each rule, we try any combination of input and output in order to observe the boolean gates that system is capable of generating as a result of perturbing given inputs. Then, as same as we did in the latter section, we show the fixed points that have shown the greater spectrum (that have exhibit the greater amount of boolean gates) and we discuss a possible link between its structure and its simulation capabilities.
\subsection{Simulations set-up}
\begin{enumerate}
\item Grid size: $4 \times 4$
\item Simulation time: $100$
\item Rule set: any totalistic rule up to a maximum activation value (number of neighbours in state $1$) of $\Delta = 4.$
\end{enumerate}
\subsection{Results}
\begin{table}[H]
\centering
\begin{tabular}{|l|l|}
\hline
Rules & \#Fixed Points \\ \hline
1 & 41 \\ \hline
2 & 57 \\ \hline
3 & 9 \\ \hline
4 & 2 \\ \hline
12 & 9 \\ \hline
13 & 1 \\ \hline
14 & 58 \\ \hline
23 & 57 \\ \hline
24 & 74 \\ \hline
34 & 34 \\ \hline
123 & 25 \\ \hline
124 & 58 \\ \hline
134 & 74 \\ \hline
234 & 34 \\ \hline
1234 & 2 \\ \hline
\end{tabular
\caption{Number of fixed points found for each totalistic rule after $t=100$ time steps of simulation.}
\label{tab:nfixedpoints}
\end{table}
\begin{table}[]
\resizebox{\textwidth}{!}{%
\begin{tabular}{|l|l|l|l|l|l|l|l|l|l|l|l|l|l|l|l|l|}
\hline
& 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 & 14 & 15 \\ \hline
1 & 100 & 98 & 76 & 98 & 59 & 98 & 78 & 0 & 78 & 0 & 78 & 0 & 78 & 0 & 83 & 98 \\ \hline
2 & 100 & 70 & 21 & 88 & 11 & 86 & 21 & 95 & 56 & 0 & 42 & 0 & 42 & 0 & 96 & 98 \\ \hline
3 & 100 & 78 & 78 & 78 & 56 & 78 & 0 & 0 & 89 & 0 & 89 & 0 & 89 & 0 & 0 & 89 \\ \hline
4 & 50 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 50 & 0 & 50 & 0 & 50 & 0 & 0 & 50 \\ \hline
12 & 100 & 0 & 89 & 0 & 67 & 0 & 0 & 0 & 0 & 89 & 89 & 0 & 89 & 0 & 0 & 89 \\ \hline
13 & 100 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ \hline
14 & 98 & 64 & 76 & 79 & 62 & 78 & 95 & 38 & 100 & 76 & 98 & 90 & 98 & 88 & 97 & 98 \\ \hline
23 & 100 & 63 & 96 & 86 & 68 & 86 & 63 & 42 & 84 & 54 & 84 & 32 & 84 & 28 & 40 & 98 \\ \hline
24 & 99 & 0 & 0 & 62 & 0 & 58 & 0 & 88 & 95 & 0 & 97 & 46 & 97 & 57 & 99 & 99 \\ \hline
34 & 97 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 94 & 0 & 94 & 0 & 94 & 0 & 68 & 97 \\ \hline
123 & 100 & 96 & 96 & 96 & 88 & 96 & 96 & 0 & 100 & 0 & 96 & 0 & 96 & 0 & 0 & 96 \\ \hline
124 & 98 & 38 & 90 & 79 & 88 & 78 & 76 & 64 & 97 & 95 & 98 & 76 & 98 & 62 & 100 & 98 \\ \hline
134 & 99 & 88 & 46 & 62 & 57 & 58 & 0 & 0 & 99 & 0 & 97 & 0 & 97 & 0 & 95 & 99 \\ \hline
234 & 97 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 68 & 0 & 94 & 0 & 94 & 0 & 94 & 97 \\ \hline
1234 & 50 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 50 & 0 & 50 & 0 & 50 & 50 \\ \hline
\end{tabular}
}
\caption{Spectrum of different totalistic rules considering a sample of fixed points in a $4x4$ grid. Each entry of the matrix represent the percent frequency of each boolean gate (enumerated from $0$ to $15$) related to total amount of fixed points}
\label{tab:spectrumfixedpoint}
\end{table}
\begin{figure}[!tbp]
\centering
\includegraphics[scale=0.5]{plots/bestfixedpoints2D.pdf}
\caption{Fixed points exhibiting the greatest spectrum of boolean gates for each totalistic rule.}
\label{plot:fixedpoint2D}
\end{figure}
\paragraph{Uniformity and stability.}
Roughly, as it is shown in Figure \ref{plot:fixedpoint2D} we can identify two types of fixed points according to two criteria: uniformity and stability. In the first group we observe those rules which have fixed point $\vec{0}$ as the one who has greater spectrum of simulated Boolean gates. As it is shown in Figure \ref{plot:fixedpoint2D} this is the case of rule $4,1234$ and $13$. Note that one can easily deduce that rules $4$ and $1234$ only have $\vec{1}$ and $\vec{0}$ as fixed points. In addition rule $13$ is the XOR rule. We have already shown in Theorem \ref{teo:linear} that matrix-defined rules can only generate other matrix-defined rules on every possible subset of inputs defined by fixing some of the input variables. As a consequence of the latter result, in this case, we can only expect the XOR function (rule $13$) to generate the XOR gate (gate $6$), its complementary gate by conjugation (gate $9$), constant gates ($0$ and $15$), projection gates ($10$ and $12$) and their conjugated gates ($3$ and $5$ respectively). Nevertheless, we observe in Table \ref{tab:nfixedpoints} a that $\vec{0}$ is the only fixed point of $13$ and one can easy show this in general for a $4 \times 4$ grid. In order to eliminate any misleading effect of the small dimension chosen for the experiment, we exhibit a fixed point for this rule in a $6\times 6$ grid in Figure \ref{plot:13fix}. For this particular fixed point, we have repeated latter experiment and, and we have obtain as result that it is capable to implement any of the previously described gates that XOR function can simulate.
\begin{figure}[!tbp]
\centering
\includegraphics[scale=0.25]{plots/fixed13.pdf}
\caption{Fixed point for Rule $13$ in a $6 \times 6$ grid.}
\label{plot:13fix}
\end{figure}
On the other hand, some of the non-uniform rules, i.e. those who have a fixed point different from $\vec{1}$ or $\vec{0}$ contributing the most to its spectrum, have actually complete spectrum (they can implement any possible Boolean gate). Then, ignoring the obvious misleading effects produced by taking a small grid as a interaction graph, we can roughly deduce that uniformity by itself plays a role on how complex is the spectrum of some given rule. Nevertheless, as the results on spectrum of non-uniform rules suggest, it is not the only element that seems to impact in the richness of the spectrum of some rules. In fact, taking the example of rule $14$ and $4$ (the first produces a complete spectrum while the second one only produce a reduced amount of Boolean gates), we conjecture that diffusion properties induced by having $1$ in the active set might play an interesting role explaining the difference between the spectrum of both rules. We observe this effect also in the other way: rule $23$ has complete spectrum but rule $123$ doesn't.
In addition, according to Table \ref{tab:spectrumfixedpoint} we can see that there are only three rules exhibiting only $2$ or $1$ fixed point.
On the other hand, we observe that there are fixed points in which, if we change only two cells, roughly, we do not produce any different dynamical behaviour compared to the initial condition. More precisely, dynamics tend to the original fixed point. It is interesting to observe that most of the rules having greatest spectrum exhibit a non-uniform unstable fixed point such us rules $1,23,123,2$. However, here we can also roughly identify two types of fixed points: those who are uniformly unstable, that is to say, they produce different dynamical behaviour independently from the position of he cell we perturb. This is the case of rule $1$ and $12$ according to igure \ref{plot:fixedpoint2D}. Contrarily, some rules (such as rule) $34$, $24$ and $124$ have a fixed point in which there are some stable areas and unstable areas. Again, the effect of having this type of distribution of $1$s and $0$s in the fixed points of some rules might play some role in the richness of their spectrum. Nevertheless it is not possible to directly deduce that from the results as there some rules such as rule $1$ having an uniformly unstable fixed point and having at the same time a considerable amount of richness in its spectrum while $124$ have some stable zones and produces a complete spectrum.
\subsection{Gadget dynamics: a rule $1$ automata network example.}
In this section we take a deeper look in the structures obtained for rule $1$ in latter simulations. We study the dynamics of a gadget obtained in the latter simulation which is capable of calculating different logic gates by changing the assignation of inputs and output. This gadget exhibits the maximum value of the spectrum for the rule $1$ in the simulations for $p=0.1$ (i.e. it is able to simulate the maximum amount of different logic gates observed in the simulations). In addition, this gadget has only $6$ nodes as it is shown in Figure \ref{plot:gadget} \\
\begin{figure}[!tbp]
\centering
\includegraphics[scale=0.5]{plots/graphG1.pdf}
\caption{Gadget obtained for rule $1$ in the previous simulations with $p=0.1$ (see Figure \ref{plot:p01})}
\label{plot:gadget}
\end{figure}
This section is organised in the following way: in the first subsection we study the dependency of the spectrum in the simulation time, showing that there is a critical time in which the gadget starts to simulate some gates such as AND and OR gates. In the second subsection, we explore the dynamics behind the simulation of the AND gate.
\subsubsection{Frequency of simulated gates v/s simulation time}
We now study the dependency of the spectrum of the latter gadget automata network that we have found for rule $1$ on the simulation time. Remember that the frequency here is a measure of how many different assignation of two input and one output produces the same logic gate. In this case, we focus in the studying the AND and OR gates. In Figure \ref{plot:gadget} we can see that the frequency for these two logic gates v/s simulation time. As we can observe in Figure \ref{plot:gadget}, there is no monotonic behaviour on the frequency. More over it appears to have a periodic behaviour. We conjecture that it is related to the maximal period of some attractor of the gadget automata network.
\begin{figure}[!tbp]
\centering
\includegraphics[scale=0.25]{plots/exampleANDt1.pdf}
\caption{Frequency of AND and OR gates v/s simulation time for the gadget in Figure \ref{plot:gadget}}
\label{plot:exampleANDt1}
\end{figure}
On the other hand, we can observe that a critical time is necessary for the gadget in order to simulate AND and OR gates. More precisely, the system is capable of simulating both logic gates starting from $t = 3$ as it is shown in Figure \ref{plot:exampleANDt2}. Finally, note that the time steps in which the system is capable of simulating AND gates are not necessarily the same for simulating OR gates. For example, as it is shown in Figure \ref{plot:exampleANDt2}, the gadget is not capable of simulating OR gates in time $t=4$ but it can simulate AND gates with $6$ different assignations of input and output at this same time step.
\begin{figure}[!tbp]
\centering
\includegraphics[scale=0.25]{plots/exampleANDt2.pdf}
\caption{Frequency of AND and OR gates v/s simulation time for the gadget in Figure \ref{plot:gadget}). The system is capable of simulating both gates starting from $t=3$.}
\label{plot:exampleANDt2}
\end{figure}
\subsubsection{AND gate dynamics}
In the section we study in the detail the dynamics behind the simulation of an AND gate by the gadget automata network which graph is shown in Figure \ref{plot:gadget}. We choose the minimum simulation time required for the system to compute an AND gate. As it is shown in Figure \ref{plot:exampleANDt2} this time step is $t=3$. In Figure \ref{plot:exampleAND1} we show the dynamics of the network starting from the input $11$. In this case, nodes marked as $2$ and $3$ are the inputs and $4$ is the output. Active nodes are coloured in yellow and inactive nodes are coloured blue. Note that AND gate is computed after $t=3$ time steps. \\
\begin{figure}[!tbp]
\centering
\includegraphics[scale=0.25]{plots/exampleAND1.pdf}
\caption{Dynamics behind simulation of an AND gate for an input $11$. Nodes $2$ and $3$ act as input nodes and node $4$ operates as the output. }
\label{plot:exampleAND1}
\end{figure}
On the other hand, Figure \ref{plot:exampleAND2} shows the evolution of the input $00$ using the same input-out assignment. We see here how the paths that are connected to the central triangle of the graph play an essential role in blocking a single $1$ signal. \\
\begin{figure}[!tbp]
\centering
\includegraphics[scale=0.25]{plots/exampleAND2.pdf}
\caption{Dynamics behind simulation of AND gate for an input $01$. Nodes $2$ and $3$ act as input nodes and node $4$ operates as the output. }
\label{plot:exampleAND2}
\end{figure}
Contrarily to the case of the gadgets that we have theoretically proposed in the latter sections, we observe that, in both Figure \ref{plot:exampleAND1} and \ref{plot:exampleAND2}, the simulation of the AND gate takes more than a straightforward sequence of calculations performed in some linear way. We conjecture that the system uses different attractors in order to simulate the output of one logic gate for different input assignments.
\subsection{Rule $1$ Bull Cellular Automata}
In this section, we introduce a non-uniform one dimensional cellular automata based in the gadget shown in Figure \ref{plot:gadget}. Considering the topology of latter gadget we introduce a cellular automaton with uniform radius $1$ (as in the definition of elementary cellular automata) but we modify the neighbourhood of certain cells in order to allow them to consider not only the value of their adjacent neighbours but to consider the values of the cells at distance $2$ to the right (right radius equal $2$) and also, to mantain symmetry, to the left. More precisely, we define this cellular automaton as a function $F:\{0,1,2,3,4,5\}^n: \to \{0,1,2,3,4,5\}^n.$ defined by:
$$F(x)_i\begin{cases}
1 & \text{ if } \overline{x}_{i+1} + \overline{x}_{i-1} = 1 \wedge x_i \in \{0,1\} \\
0 & \text{ if } \overline{x}_{i+1} + \overline{x}_{i-1} \not = 1 \wedge \overline{x}_i \in \{0,1\} \\
3 & \text{ if } \overline{x}_{i+2} + \overline{x}_{i+1} + \overline{x}_{i-1} = 1 \wedge \overline{x}_i \in \{2,3\} \\
2 & \text{ if } \overline{x}_{i+2} + \overline{x}_{i+1} + \overline{x}_{i-1} \not = 1 \wedge \overline{x}_i \in \{2,3\} \\
5 & \text{ if } \overline{x}_{i-2} + \overline{x}_{i+1} + \overline{x}_{i-1} = 1 \wedge \overline{x}_i \in \{4,5\} \\
4 & \text{ if } \overline{x}_{i-2} + \overline{x}_{i+1} + \overline{x}_{i-1} \not = 1 \wedge \overline{x}_i \in \{4,5\}
\end{cases}$$
where $\overline{x}_i = \begin{cases}
1 & \text{ if } x \in \{1,3,5\} \\
0 & \sim
\end{cases}$ for $i = 1, \hdots, n$ and taking $n+k$ and $0-k$ modulo $n$ for any positive integer $k$ and where $n$ is some positive integer. In simple words, we are non uniformly repeating several copies of gadget in Figure \ref{plot:gadget}. Based on our lasts results, our main aim here is to study simulation capabilities of this system. We remark that, for example, it is not clear wheter we can simulate an evaluation of an arbitrary boolen circuit in two dimensional rule $1$ cellular automata. We remark also that in the case in which there are no cell with special neighbourhood (so the cellular automaton is an elementary cellular automaton) the function is the well known rule $90$ which is a class $4$ rule in Wolfram classification \cite{wolfram1984universality}.
In order to illustrate the dynamics of the system, we show a simulation starting from a random initial condition in Figure \ref{fig:dynbull}. In the latter simulation, we have generated randomly a few bulls in three different areas of length $8$ cells (at the begining, in the middle and at the end of the ring) and the rest of the states of the rest of the cells were randomly generated with uniform probability.
\begin{figure}[!tbp]
\centering
\includegraphics[scale=0.6]{plots/dynamcisBull.pdf}
\caption{Dynamics of Rule $1$ Bull cellular automaton starting from a random initial condition. Green cells are regular cells in state $1$; blue cells are regular cells in state $0$; purple cells are right-modified neighbourhood cells in state $0$; pink cells are modified right-neighbourhood cells in state $1$; yellow cells are left-modified neighbourhood cells in state $0$ and light blue cells are modified left-neighbourhood cells in state $1$ }
\label{fig:dynbull}
\end{figure}
In this subsection, we have repeated the same experiments we did for totalistic two dimensional cellular automata in previous sections, i.e., we start by searching for a set of fixed points by simulating for small values of $n$ ($n=12$) the dynamics of the system starting from any initial condition. Note that in this case, the state values also code the type of neighbourhood of the cell so, we are also taking into account any possible assignation for different types of neighbourhoods in any cell. More precisely, given a initial condition we simulate the system for $t=100$ steps and we verify if the system has reach a fixed point. Then, we search for boolean gates by perturbing this sample of fixed points in the same way we did in previous sections. We show the fixed point which attains the great spectrum (the one which simulates the greatest amount of different boolean gates) and we exhibit the its dynamics in order to understand how it simulate certain boolean gates.
\subsubsection{Simulation set-up}
We start by describing the parameters of the following simulations:
\begin{enumerate}
\item Number of cells ($n$) : 12
\item Simulation time $t$: 100
\item Number of fixed points considered: 608
\end{enumerate}
\subsubsection{Results}
\begin{table}[]
\resizebox{\textwidth}{!}{%
\begin{tabular}{|l|l|l|l|l|l|l|l|l|l|l|l|l|l|l|l|l|}
\hline
& 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 & 13 & 14 & 15 \\ \hline
1Bull & 100 & 67 & 96 & 70 & 98 & 71 & 98 & 69 & 98 & 67 & 100 & 69 & 99 & 69 & 96 & 71 \\ \hline
\end{tabular}%
}
\caption{Spectrum for rule $1$ bull cellular automaton. Numbers are percent frequency of each boolean gate related to the total amount of gates simulated by a sample of 608 fixed points obtained by simulating the system starting from different initial conditions.}
\label{tab:spectrumbull}
\end{table}
\begin{figure}[H]
\centering
\includegraphics[scale=0.55]{plots/fixedpoint13graph.pdf}
\caption{Fixed point for rule 1 bull cellular automaton with the greatest spectrum value. Gray nodes are considered active and white nodes inactive. Coded state is shown inside each node.}
\label{fig:fpbull}
\end{figure}
\paragraph{Non-uniformity produces all possible boolean network}
From Table \ref{tab:spectrumbull} we deduce that system can simulate all two input-one output boolean gates. That is very interesting as it confirms our initial insight from previous result on gadget shown in Figure \ref{plot:gadget}. In order to precise what role plays these modified neighbourhoods in terms of how many of them we need to simulate a large amount of boolean gates, we show in Figure \ref{fig:fpbull} the most representative fixed point of the sample needs roughly $33\%$ of cells with a extended neighborhood (two bull graphs) to produce the greatest spectrum. Moreover, we have analysed its particular spectrum of generated boolean gates and we observe it can simulate all of them.
Finally, we are interested in exhibit the dynamical behaviour of the sistem starting from different perturbations of this fixed point in some given input-output assignation. This experiment will simulate a NAND gate. As it is shown in Figure \ref{fig:NANDBull}, we fixed the first and the fourth cells as a inputs (they are marked with red dots in the figure) and the fifth cell as output.
We studied the whole dynamics for all the simulation time $t=100$ however, we have found that, actually, at different time steps system is capable of simulate a NAND gate. In fact, one can deduce from in Figure \ref{fig:NANDBull} (which shows $t=27$ time steps) , system is exhibiting a periodic behaviour for perturbations $(1,0)$ and $(0,1)$ and it remains in a fixed point for the rest of them. We observe that depending of the period of both attractors we can periodically see the simulation of the NAND gate for different simulation times.
\begin{figure}
\centering
\includegraphics[scale=0.5]{plots/FixPoint1Bull.pdf}
\caption{Dynamics of Rule $1$ Bull cellular automaton starting from fixed point shown in Figure \ref{fig:fpbull}. Title of plots indicates a perturbation of inputs. Inputs are marked by red dots and output is marked by a blue dot. Green cells are regular cells in state $1$; blue cells are regular cells in state $0$; purple cells are right-modified neighbourhood cells in state $0$; pink cells are modified right-neighbourhood cells in state $1$; yellow cells are left-modified neighbourhood cells in state $0$ and light blue cells are modified left-neighbourhood cells in state $1$ }
\label{fig:NANDBull}
\end{figure}
\begin{figure}[!tbp]
\centering
\includegraphics[scale=0.6]{plots/RatioOfGatesComparative}
\caption{Comparative ratios of Boolean gates discovered in
mycelium network of real fungal colony~\cite{adamatzky2020booleanFungal}, black disc and solid line;
slime mould \emph{Physarum polycephalum}~\cite{harding2018discovering}, black circle and dotted line;
succulent plant~\cite{adamatzky2018computers}, red snowflake and dashed line;
single molecule of protein verotoxin~\cite{adamatzky2017computing}, light blue `+' and dash-dot line;
actin bundles network~\cite{adamatzky2019computing}, green triangle pointing right and dash-dot-dot line;
actin monomer~\cite{adamatzky2017logical}, magenta triangle pointing left and dashed line. Area of {\sc xor} gate is magnified in the insert. Lines are to guide eye only.}
\label{fig:gatesDistribution}
\end{figure}
In numerical experiments we calculated spectra of Boolean gates implementable by totalistic automata network with various probabilities of connectivity (Tabs.~\ref{tab:p01}, \ref{tab:p05}, \ref{tab:p08}). Let us compare these with our previous experimental laboratory and numerical modelling results shown in Fig.~\ref{fig:gatesDistribution}. The ratios of the gate discovered are obtained in experimental laboratory reservoir computing with slime mould \emph{Physarum polycephalum}~\cite{harding2018discovering}, succulent plant~\cite{adamatzky2018computers} and numerical modelling experiments on computing with
protein verotoxin~\cite{adamatzky2017computing},actin bundles network~\cite{adamatzky2019computing}, actin monomer~\cite{adamatzky2017logical}, and fungal colony~\cite{adamatzky2020boolean}. The distributions are quite similar. The gates selecting one of the inputs are in majority, followed by {\sc or} gate, {\sc not-and} an {\sc and-not} gates. The gate {\sc and} is usually underrepresented in experimental and modelling experiments. The gate {\sc xor} is a rare find. The similarity of the distributions give us a hint on universality of all types of biological networks and the totalistic automata network echoing them.
The fact that only automata networks governed by totalistic rules with at least one isolated interval can generate any Boolean function echoes our previous numerical experiments on Boolean gates in protein molecules~\cite{adamatzky2017computing,adamatzky2017logical}: a variety of logical gates realisable on excitable automata networks, where functions rely on excitation interval is much higher than that of the excitable automata networks with threshold excitation functions.
\section{Discussion}
In this article we have studied theoretically and computationally the complexity of totalistic networks, according to their ability to simulate different Boolean functions (or, equivalently, Boolean circuits). It was shown that using networks of isolated and interval totalistic rules any Boolean function is generated. Using threshold networks it is possible to construct any monotonic function and with those defined by a matrix (linear: with local disjunction or the XOR) the generation of Boolean functions is poor: only constants or those of their own type (OR or XOR ). Furthermore, by means of computational experiments, the spectrum of Boolean functions generated by totalistic networks on random graphs and in a two-dimensional grid was established.
It is important to point out that for the generation of Boolean functions not only the totalistic rule and the considered graph are important, but also the fixed points. In this sense, there is a relationship between the totalistic function and the balance of active (1) and inactive (0) states of the fixed point considered. For example, if the local function reaches the state 1 with at least one active state, then the fixed point may have very few nodes in active state and still its disturbance can produce a wide spectrum of Boolean functions. In particular, for this type of functions we established that the fixed point $\vec{0}$ is enough to determine a complete spectrum of Boolean functions. However, if the local totalistic rule requires two or more active states to be activated, then the fixed point $\vec{0}$ is not a sufficient support to generate an interesting set of Boolean functions and, if they exist, fixed points with active states should be considered. In this perspective, a future work could be the study of the relationship between the perturbations of fixed points, the dynamics generated (how many nodes of the network change state before reaching a steady state again) and the Boolean functions obtained. In some way, this problem seems to be related to the amplitude of the “avalanche” produced by the disturbance (cells that change their value) and the latter could be studied from the point of view of the sand pile model proposed by Per Bak and the self-organized criticality paradigm \cite{bak1988self}.
On the other hand, in this work we have associated Boolean functions with circuits implemented in the network, but apart from proving that it is possible to generate the universal base functions (AND, OR, NAND, XOR) that allow us to construct arbitrary Boolean functions. However, this is a result, we would say, extreme: we are the ones who, based on the local functions considered, build these gadgets, so the network. In the biological problem, the network is given. Therefore, we also did numerical experiments to observe the type of functions that appear according to the type of totalistic function and the topology of the network. This led us to looking at the problem from a different angle: searching for “wild” Boolean functions, that is, setting a totalistic function and discovering subgraphs with active and inactive states that emulate interesting Boolean functions. Specifically, we study one of them, the bull gadget, inserting it into a one-dimensional totalistic automaton. Clearly, this path: the search for “wild gadgets” and its applications, is a promising line of future work.
\section*{Acknowledgement}
AA has received funding from the European Union's Horizon 2020 research and innovation programme FET OPEN ``Challenging current thinking'' under grant agreement No 858132. EG residency in UWE has been supported by funding from the Leverhulme Trust under the Visiting Research Professorship grant VP2-2018-001 and from the project the project 1200006, FONDECYT-Chile. MRW has recieved funding from ANID via PFCHA/DOCTORADO NACIONAL/2018 – 21180910 and was also supported by PIA AFB 170001.
\bibliographystyle{plain}
| {
"timestamp": "2021-08-02T02:24:04",
"yymm": "2107",
"arxiv_id": "2107.14799",
"language": "en",
"url": "https://arxiv.org/abs/2107.14799",
"abstract": "We consider the problem of studying the simulation capabilities of the dynamics of arbitrary networks of finite states machines. In these models, each node of the network takes two states 0 (passive) and 1 (active). The states of the nodes are updated in parallel following a local totalistic rule, i.e., depending only on the sum of active states. Four families of totalistic rules are considered: linear or matrix defined rules (a node takes state 1 if each of its neighbours is in state 1), threshold rules (a node takes state 1 if the sum of its neighbours exceed a threshold), isolated rules (a node takes state 1 if the sum of its neighbours equals to some single number) and interval rule (a node takes state 1 if the sum of its neighbours belong to some discrete interval). We focus in studying the simulation capabilities of the dynamics of each of the latter classes. In particular, we show that totalistic automata networks governed by matrix defined rules can only implement constant functions and other matrix defined functions. In addition, we show that t by threshold rules can generate any monotone Boolean functions. Finally, we show that networks driven by isolated and the interval rules exhibit a very rich spectrum of boolean functions as they can, in fact, implement any arbitrary Boolean functions. We complement this results by studying experimentally the set of different Boolean functions generated by totalistic rules on random graphs.",
"subjects": "Computational Complexity (cs.CC); Formal Languages and Automata Theory (cs.FL); Cellular Automata and Lattice Gases (nlin.CG)",
"title": "Generating Boolean Functions on Totalistic Automata Networks",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9728307684643189,
"lm_q2_score": 0.7279754430043072,
"lm_q1q2_score": 0.7081969096410331
} |
https://arxiv.org/abs/math/9809159 | A review of Hardy inequalities | We review the literature concerning the Hardy inequality for regions in Euclidean space and in manifolds, concentrating on the best constants. We also give applications of these inequalities to boundary decay and spectral approximation. | \section{Introduction}
\par
Let $H$ be a non-negative second order elliptic operator
acting in $L^{2}(U)$ subject to Dirichlet boundary
conditions, where $U$ is a region in ${\bf R}^{N}$ or
in a Riemannian manifold. Also let $d$ be a positive
function on $U$ which is continuous and satisfies $|\nabla
d |\leq 1$. Traditionally one takes $d(x)$ to be the distance
of $x\in U$ from the boundary $\partial
U$, but another possibility is that $d(x)$ is the distance
from any closed subset of $M\backslash U$ if $U$ is embedded
in some larger Riemannian manifold $M$.
We say that $H$ satisfies a weak Hardy inequality with
respect to $d$ if there exists a
constant $c>0$ and a constant
$a\geq 0$ such that
\begin{equation}
\int_{U}\frac{|f|^{2}}{d^{2}}\leq c^{2}\left( Q(f) +a\Vert f
\Vert^{2 }\right)
\label{1}
\end{equation}
is valid for all $f\in C_{c}^{\infty}(U)$, and hence for all $f$ in
the domain of the quadratic
form $Q$ of $H$. The infimum of all possible $c$ in
(\ref{1}) is then
called the weak Hardy constant. We say that $H$ satisfies a
strong Hardy inequality if (\ref{1}) holds with $a=0$, in
which case the minimum possible $c$ is called the strong
Hardy constant.
There are also $L^{p}$ and higher order analogues of the above notion, which
we mention briefly later in this review.
In section 2 we describe the method of geodesic integrals
for proving Hardy inequalities in higher dimensions.
Section 3 describes a method ultimately due to Jacobi,
while Section 4 gives various miscellaneous results.
We then turn to the applications of the HI to the
proof of boundary decay. It was shown
in \cite{D1} that Hardy's inequality can be used to prove
the $L^{2}$ boundary decay of eigenfunctions without any
further assumptions. This in turn leads to the possibility
of controlling the rate of convergence of the eigenvalues
when the region $U$ is approximated by a family $U_{\varepsilon}$
of slightly smaller regions. Very recently progress has
been made on this problem, \cite{D4}, and we are able to announce
bounds on the rate of convergence which are sharp
in a certain sense.
Our main results on boundary decay, Theorems 11 and 12, may
be regarded as $L^{2}$ analogues of much stronger pointwise bounds on
eigenfunctions given in \cite{Ba2,CZ,LP}. Note however that our
bounds depend only on the validity of (\ref{1}), hold for
all functions in the domains of
the operators, not just for eigenfunctions, and have rather
precise constants.
If we abandon interest in the
precise value of the constant, and choose $d$ to be the
Euclidean distance from an arbitrary point of $U$, then it
may be seen that our results are related to Morrey space
estimates. These have been of considerable importance in the
theory of elliptic operators, and recently in the proof of
heat kernel bounds, and we refer the reader to
\cite{Au,AT,Gi} for further details.
\section{Geodesic integrals}
The first method which we describe depends upon
the one-dimensional case, which is the only one Hardy actually
studied. We refer to \cite{OK} for an exhaustive study, which
involves generalizations to the variable
coefficient case of the original formula
\[
\int_{0}^{\infty}\frac{|f(x)|^{2}}{x^{2}} {\rm d} x \leq
4\int_{0}^{\infty}|f^{\prime}(x)|^{2}{\rm d} x
\]
valid for all $f\in C_{c}^{\infty}(0,\infty)$, and hence for
all $f\in W^{1,2}_{0}(0,\infty)$.
Let $H:= -{\Delta}_{DIR}$
in the Hilbert space $L^{2}(U)$ where $U$ is a bounded
region in ${\bf R}^{N}$. For every unit vector $u\in S^{N-1}$
and $x\in U$ let
\[
d_{u}(x):=\min \{ |t| :x+tu\notin U\}
\]
if the set of such $t$ is non-empty, and put $d_{u}(x):=+\infty$
otherwise.
We define the (harmonic) mean distance of $x$ from $\partial
U$ by
\begin{equation}
m(x)^{-2}:=|S^{N-1}|^{-1}\int_{S^{N-1}}
d_{u}(x)^{-2} {\rm d} S(u). \label{mdist}
\end{equation}
It is easy to prove that $d(x)\leq m(x)$ for all $x\in U$.
\begin{lemma}
We have
\[
\frac{N}{4 m^{2}} \leq H
\]
in the sense of quadratic forms. If $\lambda_{1}$ is the
smallest eigenvalue of $H$ then
\[
\lambda_{1}\geq \frac{N}{4\mu^{2}}
\]
where the quasi-inradius $\mu$ of $U$ is defined by
\[
\mu:=\sup \{ m(x):x\in U\} .
\]
\end{lemma}
\underbar{Proof}{\hskip 0.1in} See \cite{D3} or \cite[Th. 1.5.3]{HKST}.
Applications of the above lemma depend on making
assumptions on $U$ which enable one to bound $m(x)$ above
by some multiple of $d(x)$. The first of these is folklore
and seems not to
have been written down explicitly until very recently; see
\cite{MMP,MS} and the next section for alternative proofs.
\begin{theorem}
If $U$ is a convex subset of ${\bf R}^{N}$ then
\[
\frac{1}{d^{2}} \leq 4 H
\]
in the sense of quadratic forms.
\end{theorem}
\underbar{Proof}{\hskip 0.1in} If $a$ is the point of $\partial U$ closest to
$x$ then we can obtain the relevant upper bound of $m(x)$ by
computing an appropriate integral over the supporting hyperplane
at $a$. See \cite[Exercise 5.7]{STDO}.
The following lemma is typical of a variety of methods of
obtaining crude upper bounds on $m(x)$. The hypothesis is
valid not only for regions with Lipschitz boundaries, but
also for a variety of regions with fractal boundaries,
such as the Koch snowflake region in ${\bf R}^{2}$.
\begin{lemma}
Suppose that there is a constant $k$ such that
for each $a\in \partial U$ and each $\alpha >0$
there exists a ball $B$ disjoint from $U$ with centre $b$
and radius $\beta\geq k\alpha$, where $|b-a|=\alpha$. Then there
exists constants $c_{0}, c_{1}$ such that $m(x)\leq c_{0}d(x)$ and
hence
\[
\frac{1}{ d^{2}} \leq c_{1}H
\]
in the sense of quadratic forms.
\end{lemma}
\underbar{Proof}{\hskip 0.1in} See \cite{D3}, \cite[Th. 1.5.4]{HKST} and \cite[Th.
3]{A}.
The condition of Lemma 3 is not satisfied for regions
satisfying a uniform exterior power-like cusp condition. In
such cases one may prove a modified Hardy inequality using
Lemma 1, namely
\begin{equation}
\int_{U}\frac{|f|^{2}}{d^{\gamma}}\leq c^{2}\left( Q(f) +a\Vert f
\Vert^{2 }\right) \nonumber
\end{equation}
for some $0<\gamma <2$; see \cite[p 369]{DS}. See also
\cite[Th. 3.2, 3.3]{DL} where a similar situation arises
for locally Euclidean manifolds with fractal boundaries.
A procedure closely related to the idea of this section
was developed for regions in
Riemannian manifolds independently by Croke and Derdzinski, \cite{CD}, and
Donnelly, \cite{Don}. The integrals over straight
lines were replaced by integrals over geodesics, so the
formulation involves the geodesic flow on the unit
sphere bundle of the manifold. However, both papers are
concerned with obtaining lower bounds on the
bottom eigenvalue, much as in Lemma 1, rather than Hardy's
inequality.
We mention in passing that there is no requirement that one
should assign equal weights to every direction in Euclidean
space. In some cases one obtains a better constant in the
Hardy inequality by taking
an average over a few directions which are well adapted to
the region in question.
\section{The Classical Method}
The following method goes back to Jacobi,
and was used by Barta and Kasue to obtain lower bounds on
the first eigenvalue, \cite{Ba,Ka}. It is the easy half of a
theorem of Allegretto, Moss and Piepenbrink
characterising the bottom of the spectrum of a Schr\"odinger
operator in terms of the existence of positive
distributional solutions of
the eigenvalue equation, \cite[p. 23]{CFKS}. Assume that
\[
Hf(x):=-\sum\frac{\partial}{\partial x_{i}}\left\{
a_{i,j}(x)\frac{\partial f}{\partial x_{j}} \right\}
\]
where $a(x)$ is a non-negative $C^{1}$ real symmetric matrix-valued
function and $f\in C_{c}^{2}(U)$. Then $H$ is a
non-negative symmetric operator and we can use the same symbol
to denote its Friedrichs extension.
\begin{lemma}
Let $\phi$ be a positive $C^{2}$ function on $U$ and let $V$
be a continuous function on $U$ such that
\[
-\sum\frac{\partial}{\partial x_{i}}\left\{
a_{i,j}(x)\frac{\partial \phi}{\partial x_{j}} \right\}
\geq V\phi .
\]
Then we have
\[
H\geq V
\]
in the sense of quadratic forms.
\end{lemma}
\underbar{Proof}{\hskip 0.1in} See \cite[Th. 4.2.1]{HKST}.
The conditions of the above lemma can be weakened to allow a
distributional inequality.
\underbar{Second proof of Theorem 2} If we put
$\phi:=d^{1/2}$ and use the fact that ${\Delta} d \leq 0$ for
any convex set $U$ then the result follows immediately from
the last lemma.
The method of this section can be extended to
Riemannian manifolds without difficulty. We refer to \cite{Ca,DH,Ow}
for a variety of Hardy and Rellich type inequalities with
explicit constants in Riemannian manifolds obtained in this
manner. The following theorem is only one of a range of related
results due to Brezis and Marcus, \cite{BM}. In particular they find explicit
bounds on the minimum possible negative value of $a$ in the theorem
when $U$ is convex.
\begin{theorem}
If $\, U\subseteq {\bf R}^{N}$ is bounded
with a $C^{2}$ boundary and $H:=-{\Delta}_{DIR}$ in
$L^{2}(U)$ then there exists $a\in{\bf R}$ such that
\begin{equation}
d^{-2}\leq 4(H+a) \label{bm}
\end{equation}
in the sense of quadratic forms. If $U$ is convex
then (\ref{bm}) holds for certain $a<0$.
\end{theorem}
\underbar{Proof}{\hskip 0.1in} Let $\phi$ be a positive $C^{2}$ function on $U$
such that $\phi(x)=d(x)^{1/2} -d(x)$
for all $x$ close enough to $\partial U$. The first
statement of the theorem
follows by applying Lemma 4 to $\phi$.
There are various other improvements of the strong Hardy
inequality of which we mention just two. For a definitive
treatment of the one-dimensional theory see \cite{OK}.
\begin{theorem}
If $U:=\{ x\in{\bf R}^{N}:x_{N}>0\}$ where $N>1$ then
\[
\int_{U}\left\{
\frac{1}{x_{N}^{2}}
+\frac{1}{4x_{N}(x_{N}^{2}+x_{N-1}^{2})^{1/2}}
\right\} |f|^{2} {\rm d} x\leq
4\int_{U}|\nabla f|^{2}{\rm d} x
\]
for all $f\in C_{c}^{\infty}(U)$.
\end{theorem}
\underbar{Proof}{\hskip 0.1in} See \cite[Sect. 2.1.6]{Maz}.
\begin{theorem}
If $U:=(0,a)$ then
\[
\int_{U} \frac{a^{2}|f|^{2}}{x^{2}(a-x)^{2}}{\rm d} x\leq
4\int_{U}| f^{\prime}|^{2}{\rm d} x
\]
for all $f\in C_{c}^{\infty}(U)$.
\end{theorem}
\underbar{Proof}{\hskip 0.1in} Put $\phi(x):=x^{1/2}(a-x)^{1/2}$ in Lemma 4.
\section{Capacity-based methods}
In this section we mention a few of the very general theorems
which involve the use of capacity arguments. These have
been developed in an $L^{p}$ context, but we only treat the
case $p=2$. If $K$ is a compact subset of $U\subseteq
{\bf R}^{N}$ we define its relative capacity by
\[
{\rm cap} (K,U):=\inf\left\{
\int_{U}|\nabla f|^{2}: f\in C_{c}^{\infty}(U) \mbox{ {\rm
and} }
f\vert_{K}\geq 1
\right\}.
\]
It is particularly appropriate in this conference
to mention one version of the most quantitatively precise
theorems of this type, due to Professor Maz'ya.
\begin{theorem}
If $\mu$ is a positive measure on $U$ and
\[
\mu(K)\leq \beta \,{\rm cap}(K,U)
\]
for all compact subsets $K$ of $U$, then
\[
\int_{U}|f|^{2}{\rm d} \mu \leq 4\beta \int_{U}|\nabla f|^{2}
\]
for all $f\in C_{c}^{\infty}(U)$. Conversely the second
inequality implies
\[
\mu(K)\leq 4\beta \,{\rm cap}(K,U)
\]
for all compact subsets $K$ of $U$.
\end{theorem}
\underbar{Proof}{\hskip 0.1in} See \cite[p.113]{Maz}.
Our next results are taken from a paper of Ancona,
\cite{A}. We say that $U$ is uniformly ${\Delta}$-regular if
for all $x\in\partial U$ and all $r>0$
the harmonic measure $w$ of $U\cap \partial B(x,r)$ in
$U\cap B(x,r)$ satisfies
$w\leq 1-\beta$ on $U\cap \partial B(x,r/2)$, for some
constant $\beta\in(0,1)$ independent of $x,r$. If $N\geq 3$
this is equivalent to the uniform capacitary density
condition that there
exists a constant $\alpha>0$ such that
\[
{\rm cap}(B(x,r)\backslash U) \geq \alpha r^{N-2}
\]
for all $x\in\partial U$ and all $r>0$.
\begin{theorem} \label{CapTh}
If $N\geq 2$ and $U\subseteq {\bf R}^{N}$
is uniformly ${\Delta}$-regular then $U$
satisfies a strong Hardy inequality with respect to the
Laplace operator. If $N=2$ then the converse is also true.
\end{theorem}
Although \cite{A} does not provide sharp information
about the size of the strong Hardy constant, it contains many
more results than we have indicated above. An $L^{p}$
converse of Theorem \ref{CapTh} for $N=p>2$
may be found in \cite{Lew},
using an appropriate $L^{p}$ Riesz capacity.
\section{Miscellaneous results}
The weak Hardy constant $c$ as defined in Section 1
was proved in \cite{D2} to be local in the sense that it is
the maximum value of a certain upper semi-continuous function on
the boundary, whose value at each point depends only on the
geometry of the boundary around that point. Various methods
of evaluating this function at different types of boundary
point are described in \cite{D2}.
For the remainder of this section we
assume that $H:=-{\Delta}_{DIR}$. The strong Hardy constant is a global
invariant of $U$. It equals $2$ for any convex set,
but the condition of convexity is not necessary for
this conclusion. Let
\[
U_{\beta}:=\{ r{\rm e}^{i\theta}:0<r<1 \,\,{\rm and}\,\, 0<\theta <\beta\}.
\]
Then $U_{\beta}$ has strong Hardy constant $2$ if and only if
the internal angle $\beta$ is less than or equal to a certain critical
value $\beta_{c}\sim 4.856$ radians, \cite{D2}. For larger
$\beta$ the strong and weak Hardy constants are larger than
$2$. Similar conclusions hold for other plane regions with
piecewise smooth boundaries.
If $U$ is a simply connected region in
${\bf R}^{2}$ then $U$ has strong Hardy constant at most $4$ by \cite{A},
\cite[Th. 1.5.10]{HKST}. The proof of this result depends
upon a fact from analytic function theory, namely Koebe's
one-quarter theorem.
There is an interesting connection between the possible
constants in the strong Hardy inequality and the Minkowski
dimension of the boundary, \cite{DM}. In two dimensions there is also a
relationship with hyperbolic geometry, which we do not pursue.
We say that the boundary $\partial U$ has interior Minkowski dimension
$\alpha$ if there exist positive constants $k_{1}$ and $k_{2}$
such that
\[
k_{1}\varepsilon^{N-\alpha}\leq
|\{ x\in U:{\rm dist} (x,\partial U)<\varepsilon\}| \leq
k_{2}\varepsilon^{N-\alpha}
\]
for all $\varepsilon >0$. The following theorem is adapted from
\cite[Th. 3.3]{DM}. We allow $\alpha <N-1$ because the theorem
is applicable in manifolds, for example if $U$ is obtained
by removing a compact set $K$ from a sphere endowed with the
standard metric.
\begin{theorem}
If $\partial U$ has interior Minkowski dimension
$\alpha>N-2$, then the strong Hardy
constant of $U$ with respect to the Laplacian satisfies
\[
c(2+\alpha -N)\geq 2.
\]
\end{theorem}
In most of the above lemmas we have restricted attention to
Hardy inequalities in $L^{2}$. In fact many of the results
have been extended to $L^{p}$ with sharp constants; see
\cite{MMP,MS} for the proofs of the following two theorems.
\begin{theorem}
Let\[
c^{-p}:=\inf\left\{ \frac{ \int_{U}|\nabla f|^{p}}{\int_{U}|f/d|^{p}}
:f\in W^{1,p}_{0}(U)
\right\}
\]
where $1<p<\infty$. If $\partial U$ is smooth then
$c\geq p/(p-1)$. If in addition $p=2$ then $c>2$ if and only
if the infimum is achieved by some $f\in W^{1,2}_{0}(U)$.
\end{theorem}
\begin{theorem}
If $U$ is a convex set in ${\bf R}^{N}$ and $1<p<\infty$ then
\[
\int_{U} \frac{|f|^{p}}{d^{p}} \leq \left(
\frac{p}{p-1}\right)^{p}\int_{U}|\nabla f |^{p}
\]
for all $f\in W_{0}^{1,p}(U)$.
\end{theorem}
We refer to \cite{A,BM,Ca,Lew,MMP,Maz,OK,W1,W2} for further $L^{p}$
results, since they do not yet have such direct
consequences for spectral theory. We refer to
\cite{DH, Ow} for the analogues for
higher order operators, known as Rellich inequalities, and
to \cite{Cia} for analogues in Orlicz spaces.
We describe some trace inequalities in
\cite{D5} which may be proved using Hardy's
inequality. We assume that $H:=-{\Delta}_{DIR}$ acting in
$L^{2}(U)$ where $U$ is a region in ${\bf R}^{N}$. The theorems
are only of interest when $U$ has infinite volume.
\begin{theorem}
We have
\[
{\rm tr} [{\rm e}^{-Ht}] \leq (2\pi t)^{-N/2}
\int_{U}{\rm e}^{-Nt/8m(x)^{2}}{\rm d}^{N}x
\]
for all $t>0$, where $m$ is defined by (\ref{mdist}).
\end{theorem}
\begin{theorem}
If $U$ satisfies the regularity condition
\[
d(x)\leq m(x)\leq bd(x)
\]
for all $x\in U$ then
\[
2^{-N}(2\pi t)^{-N/2}
\int_{U}{\rm e}^{-8\pi^{2}N^{2}t/d(x)^{2}}{\rm d}^{N}x
\leq{\rm tr} [{\rm e}^{-Ht}]
\leq (2\pi t)^{-N/2}
\int_{U}{\rm e}^{-Nt/8b^{2}d(x)^{2}}{\rm d}^{N}x
\]
for all $t>0$. Hence
\[
{\rm tr} [{\rm e}^{-Ht}]<\infty
\]
for all $t>0$ if and only if
\[
\int_{U}{\rm e}^{-t/d(x)^{2}}{\rm d}^{N}x <\infty
\]
for all $t>0$.
\end{theorem}
\section{Boundary estimates}
The size of the constant $s$ in an inequality of the form
\begin{equation}
\int_{U}\frac{|f|^{2}}{d^{s}} <\infty \label{power}
\end{equation}
conveys information about the behaviour of the function $f$
near the boundary of $U$.
We conjecture that it is not possible to have $s>2$ in the inequality
for any region $U$
if we only assume that $f\in{\rm Dom}(Q)$, where $Q$ is the
quadratic form associated with a uniformly elliptic second
order operator $H$ acting in $L^{2}(U)$ subject to
Dirichlet boundary conditions. However, if
we make stronger assumptions on $f$ then one may be able to
prove (\ref{power}) for a larger value of
$s$. The first paper with results of
this type was \cite{EHK}, where it was assumed that $f$ was
an eigenfunction of $H$. Subsequently \cite{D1} obtained
better bounds for all $f\in{\rm Dom}(H)$, assuming only the Hardy
inequality.
Although we have concentrated on $L^{2}$ boundary
estimates, there is a substantial literature on pointwise
decay of eigenfunctions and their gradients at the boundary.
Bounds of the type
\begin{equation}
|\phi_{n}(x)| \leq c_{n} \phi_{1}(x) \label{phi1}
\end{equation}
are immediate consequences of intrinsic ultracontractivity
(IU),
\cite{DS,HKST}, in which a major ingredient of the proof
is the existence of an inequality
\begin{equation}
\phi_{1}(x) \geq ad(x)^{\alpha} \label{lower}
\end{equation}
for some positive constants $a$ and $\alpha$. The proof of
(\ref{lower}) depends in turn upon the Harnack inequality
and a boundary accessibility
property. The BAP was proved in \cite{DS,HKST} for
Lipschitz domains, but Ancona and Simon commented
that it holds under a suitable
twisted interior cone condition, i.e. for John domains,
\cite[p 98]{Green}. Finally Banuelos gave a
detailed analysis of the relationship between (\ref{phi1}),
IU, John domains, Holder domains, NTA domains. etc. in \cite{Ba1}.
Pointwise bounds on the gradients of the
eigenfunctions $\phi_{n}$ of $-{\Delta}_{DIR}$ and of
Schr\"odinger operators with potentials in restricted Kato
classes acting in
$L^{2}(U)$ are proved in
\cite{Caf,CZ,LP,Ba2,BP} in steadily increasing generality. The
best upper bound is for IU domains and is in \cite{Ba2},
while the best lower bound is for Lipschitz domains and is
in \cite{BP}. The inequalities are of the form
\begin{eqnarray*}
|\nabla\phi_{n}(x)| & \leq & c_{n} \phi_{1}(x) /d(x)\\
|\nabla\phi_{1}(x)| & \geq & c_{1} \phi_{1}(x) /d(x),
\end{eqnarray*}
the latter being for $x$ close enough to the boundary.
One may also obtain upper bounds of the form
\[
|\phi_{n}(x)|\leq c_{n}d(x)^{\beta}
\]
for explicit but non-optimal constants $c_{n},\beta$ which
depend only on the eigenvalue $\lambda_{n}$, the dimension and
the constant $\alpha$ in the uniform capacitary density
inequality, \cite{BB}. For an open simply connected region
in ${\bf R}^{2}$ the bound
\[
|\phi_{n}(x)|\leq c_{n}d(x)^{1/2}
\]
is proved in \cite{Ba2,BB,P2}; in this case the power $1/2$ is
sharp.
We finally present some new results on $L^{2}$ boundary
decay, taken from \cite{D4}. Let $U$ be
a bounded region in ${\bf R}^{N}$ and let
$H:=-{\Delta}_{DIR}$ acting in $L^{2}(U)$. Let $d(x)$ denote the
distance of $x$ from some closed subset of ${\bf R}^{N}\backslash
U$. We make no assumptions on the boundary $\partial U$
apart from the validity of (\ref{1}) for certain values
of $c\geq 2$ and $a\geq 0$. We are then able to draw the
following conclusions
about the boundary decay of functions in the domain of $H$.
We have proved in \cite{D4} that the powers of $\varepsilon$ in these theorems
are sharp, and conjecture that the constant $c_{0}$ is also
sharp.
Analogues of the theorems for uniformly elliptic operators
in divergence form are proved in \cite{D4}.
\begin{theorem}\label{5A}
If $f\in{\rm Dom}(H)$ and $\varepsilon >0$ then
\[
\int_{\{x:d(x) <\varepsilon \}}|f|^{2}\leq
c_{0}\varepsilon^{2+2/c}\Vert (H+a)f\Vert_{2}
\Vert(H+a)^{1/c}f\Vert_{2}
\]
where
\[
c_{0}:=c^{2+2/c}.
\]
\end{theorem}
\begin{theorem} \label{5B}
If $f\in{\rm Dom} (H)$ and $\varepsilon >0$ then
\[
\int_{\{x:d(x) <\varepsilon\}}|\nabla f|^{2}
\leq c_{1}\varepsilon^{2/c}\Vert
(H+a)f\Vert_{2}\Vert(H+a)^{1/c}f\Vert_{2}.
\]
where
\[
c_{1}:=c^{2/c}+c^{2/c}(1+c)^{2+2/c}.
\]
\end{theorem}
\begin{corollary}
If $Hf=\lambda f$ for some $\lambda > 0$, $\Vert f \Vert_{2}=1$ and $\varepsilon >0$ then
\[
\int_{\{x:d(x) <\varepsilon \}}|f|^{2}\leq
c_{0}\varepsilon^{2+2/c}(\lambda +a)^{1+1/c}
\]
and
\[
\int_{\{x:d(x) <\varepsilon\}}|\nabla f|^{2}
\leq c_{1}\varepsilon^{2/c}(\lambda +a)^{1+1/c}.
\]
\end{corollary}
\begin{corollary}
If $U$ is a
simply connected proper subregion of ${\bf R}^{2}$ then
\[
\int_{\{x:d(x) <\varepsilon \}}|f|^{2}\leq
32\varepsilon^{5/2}\Vert Hf\Vert_{2}
\Vert H^{1/4}f\Vert_{2}
\]
and
\[
\int_{\{x:d(x) <\varepsilon \}}|\nabla f|^{2}\leq
114\varepsilon^{1/2}\Vert Hf\Vert_{2}
\Vert H^{1/4}f\Vert_{2}
\]
for all $f\in{\rm Dom}(H)$ and $\varepsilon >0$.
\end{corollary}
\underbar{Proof}{\hskip 0.1in} We put $c=4$ and $a=0$ in Theorems \ref{5A} and
\ref{5B}.
We use the results above to consider the
effect on the spectrum of $H:=-{\Delta}_{DIR}$ of replacing the
bounded region $U$
by a slightly smaller region $U_{\varepsilon}$ such that
\[
\{ x\in U:d(x) >\varepsilon\} \subseteq U_{\varepsilon}\subseteq U.
\]
If $\lambda_{n}(U_{\varepsilon})$ denote the eigenvalues of the
operator $H_{\varepsilon}$ defined by restricting $H$ to $L^{2}(U_{\varepsilon})$
where we again impose Dirichlet boundary conditions, then
variational arguments imply that $\lambda_{n}(U)\leq
\lambda_{n}(U_{\varepsilon})$
for all $n$ and $\varepsilon >0$. Our theorem below provides
quantitative estimates of the difference, again only
assuming (\ref{1}). The first version in \cite{D1} did not
obtain what we believe to be the sharp power of $\varepsilon$ given
below. Pang, \cite{P2}, obtained the result of Theorem
\ref{5E} for
$n=1$ for simply connected
plane regions by a method involving conformal mappings,
improving his own earlier
results in \cite{P1}. See \cite{D4} for a more general version
of the theorem below, and its proof.
\begin{theorem}\label{5E}
There exist constants $c_{n}$ for all positive integers
$n$ such that
\[
\lambda_{n}(U)\leq \lambda_{n}(U_{\varepsilon})\leq
\lambda_{n}(U)+c_{n}\varepsilon^{2/c}.
\]
\end{theorem}
We finally mention that there is extensive literature which
compares $\lambda_{n}(U)$ with $\lambda_{n}(U\backslash K)$,
where $K$ is a compact subset of $U$ which has a small capacity
in a suitable sense; we believe that \cite{RT} is one of the earliest
contributions to this subject, often known as the crushed
ice problem. See \cite{DMN} for a survey, including an
explicit asymptotic formula for the difference of the
eigenvalues in the limit
of small ${\rm Cap}(K)$ and also estimates of the
difference for $n=1$, both proved in the abstract setting of
regular Dirichlet forms. See also \cite{No}, where
estimates of the difference for $n=1$ in terms of an appropriate
definition of capacity are obtained in an abstract contect
applicable to higher order elliptic operators.
\par
\vskip 0.3in
{\bf Acknowledgments } I should like to thank R Banuelos,
H Brezis, V G Maz'ya and M Pang for comments on an
early draft of this paper.
\par
\vfil
\eject
| {
"timestamp": "1998-09-28T13:57:58",
"yymm": "9809",
"arxiv_id": "math/9809159",
"language": "en",
"url": "https://arxiv.org/abs/math/9809159",
"abstract": "We review the literature concerning the Hardy inequality for regions in Euclidean space and in manifolds, concentrating on the best constants. We also give applications of these inequalities to boundary decay and spectral approximation.",
"subjects": "Spectral Theory (math.SP); Mathematical Physics (math-ph)",
"title": "A review of Hardy inequalities",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9728307661011976,
"lm_q2_score": 0.727975443004307,
"lm_q1q2_score": 0.7081969079207387
} |
https://arxiv.org/abs/2002.11582 | Proximal Gradient Algorithm with Momentum and Flexible Parameter Restart for Nonconvex Optimization | Various types of parameter restart schemes have been proposed for accelerated gradient algorithms to facilitate their practical convergence in convex optimization. However, the convergence properties of accelerated gradient algorithms under parameter restart remain obscure in nonconvex optimization. In this paper, we propose a novel accelerated proximal gradient algorithm with parameter restart (named APG-restart) for solving nonconvex and nonsmooth problems. Our APG-restart is designed to 1) allow for adopting flexible parameter restart schemes that cover many existing ones; 2) have a global sub-linear convergence rate in nonconvex and nonsmooth optimization; and 3) have guaranteed convergence to a critical point and have various types of asymptotic convergence rates depending on the parameterization of local geometry in nonconvex and nonsmooth optimization. Numerical experiments demonstrate the effectiveness of our proposed algorithm. | \section{Experiments}\label{sec:exp}
In this section, we implement the APG-restart algorithm with different restart schemes listed in \Cref{table: 1} to corroborate our theory that APG-restart has guaranteed convergence with any restart scheme. In specific, for the fixed restart scheme we set the restart period to be $q=10,30,50$, respectively.
We first solve two smooth nonconvex problems, i.e., the logistic regression problem with a nonconvex regularizer (i.e., $g(x):=\alpha \sum_{i=1}^{d} \frac{x_i^2}{1+x_i^2}$) and the robust linear regression problem. For the logistic regression problem, we adopt the cross-entropy loss and set $\alpha=0.01$, and for the robust linear regression problem, we adopt the robust nonconvex loss $\ell(s):= \log(\frac{s^2}{2}+1)$. We test both problems on two LIBSVM datasets: a9a and w8a \cite{Chang_2011}. We use stepsizes $\beta_k = 1, \lambda_{k}=(1+\alpha_{k+1})\beta_{k}$ for the APG-restart as suggested by our theorems. We note that in these experiments, we plot the loss gap versus number of iterations for all algorithms. The comparison of running time is similar as all the algorithms require the same computation per iteration.
\begin{figure}
\begin{subfigure}{0.48\linewidth}
\includegraphics[width=\linewidth]{figs/E1_a9a_Loss.jpg}
\caption{logistic regression \\ a9a}
\end{subfigure}
\begin{subfigure}{0.48\linewidth}
\includegraphics[width=\linewidth]{figs/E1_w8a_Loss.jpg}
\caption{logistic regression \\ w8a}
\end{subfigure}%
\\
\begin{subfigure}{0.49\linewidth}
\includegraphics[width=\linewidth]{figs/E2_a9a_Loss.jpg}
\caption{robust regression \\ a9a}
\end{subfigure}
\begin{subfigure}{0.49\linewidth}
\includegraphics[width=\linewidth]{figs/E2_w8a_Loss.jpg}
\caption{robust regression \\ w8a}
\end{subfigure}%
\caption{Comparison of different restart schemes in smooth nonconvex optimization.} \label{Experment_1}
\end{figure}
\Cref{Experment_1} shows the experiment results of APG-restart with fixed scheme (constant $q$), function value scheme (FS), gradient mapping scheme (GS) and non-monotone scheme (NS). It can be seen that APG-restart under the function scheme performs the best among all restart schemes. In fact, the function scheme restarts the APG algorithm the most often in these experiments. The gradient mapping scheme and the non-monotone scheme have very similar performance, and both of them perform slightly worse than the function scheme. Moreover, the fixed restart schemes have the worst performance. In particular, the performance of fixed scheme gets better as the restart period $q$ decreases (i.e., more restarts take place).
Next, we further add a nonsmooth $\ell_1$ norm regularizer to the objective functions of all the problems mentioned above, and apply APG-restart with different restart schemes to solve them. The results are shown in \Cref{Experment_3}. One can see that for the nonsmooth logistic regression, all the non-fixed restart schemes have comparable performances and they perform better than the fixed restart schemes. For the nonsmooth robust linear regression, both the gradient mapping scheme and the non-monotone scheme outperform the other schemes. In this experiment, the function scheme has a degraded performance that is comparable to the fixed restart schemes. This is possibly due to the highly nonconvexity of the loss landscape.
\begin{figure}
\centering
\begin{subfigure}{0.48\linewidth}
\includegraphics[width=\linewidth]{figs/E3_a9a_Loss.jpg}
\caption{logistic regression \\a9a}
\end{subfigure}
\begin{subfigure}{0.48\linewidth}
\includegraphics[width=\linewidth]{figs/E3_w8a_Loss.jpg}
\caption{logistic regression \\ w8a}
\end{subfigure}%
\\
\begin{subfigure}{0.48\linewidth}
\includegraphics[width=\linewidth]{figs/E4_a9a_Loss.jpg}
\caption{robust regression \\ a9a}
\end{subfigure}
\begin{subfigure}{0.48\linewidth}
\includegraphics[width=\linewidth]{figs/E4_w8a_Loss.jpg}
\caption{robust regression \\ w8a}
\end{subfigure}%
\caption{Comparison of different restart schemes in {\em nonsmooth} nonconvex optimization.} \label{Experment_3}
\end{figure}
\section{Introduction}
Training modern machine learning models in real applications typically involves highly nonconvex optimization, and some effective interesting examples include deep learning \cite{RELU}, nature language processing and computer vision, etc. To solve these nonconvex optimization problems, gradient-based algorithms \cite{Nesterov2014} are popular choices due to their simplicity, effectiveness as well as well-understood convergence guarantees.
In practical training of machine learning models, momentum has been a successful and widely applied optimization trick that facilitates the convergence of gradient-based algorithms. Various types of momentum schemes have been developed, e.g., \cite{Nesterov2014,Beck2009,Tseng2010,Ghadimi2016b,Li:2015}, and have been shown to improve the order of convergence rates of gradient-based algorithms in solving convex and strongly convex optimization problems. In specific, gradient descent algorithms with momentum have been shown to achieve the complexity lower bound for convex optimization \cite{Nesterov2014,Beck2009} and have guaranteed convergence in nonconvex optimization \cite{Ghadimi2016b,Li:2015}.
Despite the superior theoretical advantages of momentum acceleration schemes, they do not fully exploit the potential for acceleration. For example, the basic momentum scheme \cite{Nesterov2014,Beck2009} adopts a diminishing momentum coefficient for accelerating smooth convex optimization, and it does not provide much momentum acceleration after a large number of iterations. Also, for accelerating strongly convex optimization, the choice of momentum coefficient requires the knowledge of condition number of the Hessian matrix, which is typically unknown a priori. To resolve these issues and further facilitate the practical convergence of gradient algorithms with momentum, various types of {\em parameter restart} techniques have been proposed, e.g., \cite{Donoghue2015,Fercoq2016,Fercoq2017,Giselsson2014,Kim2018,Liang2017,Lin2015,Liu2017,Renegar2018,Roulet2017}.
In these works, it has been demonstrated that restarting algorithm parameters (i.e., variables and momentum coefficient) periodically can suppress the oscillations of the training loss induced by the extrapolation step and improve the practical convergence in {\em convex} optimization. In specific, parameter restart is typically triggered by certain occurrences that may slow down the convergence, such as function value divergence \cite{Donoghue2015,Renegar2018} and gradient mismatch \cite{Donoghue2015}, etc. Therefore, parameter restart can reduce the instability and oscillations caused by momentum. However, in {\em nonconvex} optimization, the applications of parameter restart to gradient algorithms with momentum require to deal with the following open issues.
{\em (a)} While the convergence of gradient algorithms with momentum and parameter restart have been well explored in {\em convex} optimization, they are of lack of theoretical understandings in {\em nonconvex} optimization, which are important for modern machine learning purpose. {\em (b)} Previous works on gradient algorithms with momentum and restart for convex optimization are based on very specific restart schemes in order to have convergence guarantee, but practically the best restart scheme can be problem dependent. Therefore, it is much desired to design a momentum scheme that allows to adopt flexible parameter restart schemes with theoretical convergence guarantee.
{\em (c)} The existing gradient algorithms with momentum for nonconvex optimization have convergence guarantees at the cost of either introducing extra computation steps \cite{Li:2015,Li2017} or imposing restrictions on the objective function \cite{Ghadimi2016b}. It is important to explore whether parameter restart can help alleviate these costs or restrictions.
Considering all the issues above, we are motivated to design a gradient algorithm with momentum and parameter restart that {\em (a)} has convergence guarantee in nonconvex optimization, {\em (b)} allows to apply flexible restart schemes in practice and {\em (c)} avoids the existing weakness and restrictions in design of accelerated methods for nonconvex optimization. We summarize our contributions as follows.
\subsection{Our Contributions}
We consider the problem of minimizing a smooth nonconvex function plus a (non)smooth regularizer. To solve such a class of problems, we propose APG-restart: a momentum-accelerated proximal gradient algorithm with parameter restart (see \Cref{alg: Acc-PGD}) and show that APG-restart satisfies the following properties.
\begin{itemize}[leftmargin=*,topsep=0pt]
\item APG-restart allows for adopting any parameter restart scheme (hence covers many existing ones). In particular, it guarantees to make monotonic progress on function value between successive restart periods of iterations.
\item The design of the proximal momentum component in APG-restart leverages the notion of generalized gradient mapping (see \cref{eq: grad_map}), which leads to convergence guarantee in nonconvex optimization. Also, APG-restart does not require extra computation steps compared to other accelerated algorithms for nonconvex optimization \cite{Li2017,Li:2015}, and removes the restriction of bounded domain on the regularizer function in existing works \cite{Ghadimi2016b}.
\item APG-restart achieves the stationary condition at a global sublinear convergence rate (see \Cref{lemma: Acc-PGD dynamic}).
\item Under the Kurdyka-{\L}ojasiewicz (K{\L}) property of nonconvex functions (see \Cref{def: KL}), the variable sequence generated by APG-restart is guaranteed to converge to a critical point. Moreover, the asymptotic convergence rates of function value and variable sequences generated by APG-restart are fully characterized by the parameterization of the K{\L}~ property of the objective function. This work is the first study of gradient methods with momentum and parameter restart under the K{\L}~ property.
\end{itemize}
\subsection{Related Works}
\paragraph{Gradient algorithms with momentum and parameter restart:} Various types of parameter restart schemes have been proposed for accelerated gradient-based algorithms for convex optimization. Specifically, \cite{Donoghue2015} proposed to restart the accelerated gradient descent algorithm whenever certain function value-based criterion or gradient-based criterion is violated. These restart schemes were shown to achieve the optimal convergence rate without prior knowledge of the condition number of the function. \cite{Giselsson2014} further proposed an accelerated gradient algorithm with restart and established formal convergence rate analysis for smooth convex optimization. \cite{Lin2015} proposed a restart scheme that automatically estimates the strong convexity parameter and achieves a near-optimal iteration complexity. \cite{Fercoq2016,Fercoq2017} proposed a restart scheme for accelerated algorithms that achieves a linear convergence in convex optimization under the quadratic growth condition. \cite{Liu2017,Roulet2017} studied convergence rate of accelerated algorithms with restart in convex optimization under the error bound condition and the {\L}ojasiewicz condition, respectively. \cite{Renegar2018} proposed a restart scheme that is based on achieving a specified amount of decrease in function value. All these works studied accelerated gradient algorithms with restart in convex optimization, whereas this work focuses on nonconvex optimization.
\paragraph{Nonconvex optimization under K{\L}~ property:} The Kurdyka-{\L}ojasiewicz property is a generalization of the {\L}ojasiewicz gradient inequality for smooth analytic functions to nonsmooth sub-analytic functions. Such a local property was then widely applied to study the asymptotic convergence behavior of various gradient-based algorithms in nonconvex optimization \cite{Attouch2009,Bolte2014,Zhou2016,Zhou_2017a}. The K{\L}~ property has also been applied to study convergence properties of accelerated gradient algorithms \cite{Li2017,Li:2015} and heavy-ball algorithms \cite{Ochs2018,Liang2016} in nonconvex optimization. Some other works exploited the K{\L}~ property to study the convergence of second-order algorithms in nonconvex optimization, e.g., \cite{Yi2018}.
\section{Conclusion}
In this paper, we propose a novel accelerated proximal gradient algorithm with parameter restart for nonconvex optimization. Our proposed APG-restart allows for adopting any parameter restart schemes and have guaranteed convergence. We establish both the global convergence rate and various types of asymptotic convergence rates of the algorithm, and we demonstrate the effectiveness of the proposed algorithm via numerical experiments. We expect that such a parameter restart algorithm framework can inspire new design of optimization algorithms with faster convergence for solving nonconvex machine learning problems.
{\small
\section*{Acknowledgment}
The work of Z. Wang, K. Ji and Y. Liang was supported in part by the U.S. National Science Foundation under the grants CCF-1761506, CCF-1909291 and CCF-1900145.
\bibliographystyle{named}
\section{Preliminaries}
In this section, we introduce some definitions that are useful in our analysis later.
Consider a proper\footnote{An extended real-valued function $h$ is proper if its domain $\mathop{\mathrm{dom}} h := \{ x: h(x) < \infty \}$ is nonempty.} and lower-semicontinuous function $h:\mathbb{R}^d \to \mathbb{R}$ which is {\em not} necessarily smooth nor convex. We introduce the following generalized notion of derivative for the function $h$.
\begin{definition}(Subdifferential and critical point, \cite{vari_ana})\label{def:sub}
The Frech\'et subdifferential $\widehat\partial h$ of function $h$ at $x\in \mathop{\mathrm{dom}} h$ is the set of $u\in \mathbb{R}^d$ defined as
\begin{align*}
\widehat\partial h(x) := \bigg\{u: \liminf_{z\neq x, z\to x} \frac{h(z) - h(x) - u^\intercal(z-x)}{\|z-x\|} \ge 0 \bigg\},
\end{align*}
and the limiting subdifferential $\partial h$ at $x\in\mathop{\mathrm{dom}} h$ is the graphical closure of $\widehat\partial h$ defined as:
\begin{align*}
\partial h(x) := \{ u: \exists x_k \to x, h(x_k) \to h(x), u_k \in \widehat{\partial} h(x_k) \to u \}.
\end{align*}
The set of critical points of $h$ is defined as $\mathbf{\mathop{\mathrm{crit}}}\!~h := \{ x: \mathbf{0}\in\partial h(x) \}$.
\end{definition}
Note that when the function $h$ is continuously differentiable, the limiting sub-differential $\partial h$ reduces to the usual notion of gradient $\nabla h$.
Next, we introduce the Kurdyka-{\L}ojasiewicz (K{\L}) property of a function $h$. Throughout, we define the distance between a point $x\in \mathbb{R}^d$ and a set $\Omega \subseteq \mathbb{R}^d$ as $\mathrm{dist}_\Omega(x) := \inf_{w\in \Omega} \|x - w\|$.
\begin{definition}(K{\L}~ property, \cite{Bolte2014})\label{def: KL}
A proper and lower-semicontinuous function $h$ is said to satisfy the K{\L}~ property if for every compact set $\Omega\subset \mathop{\mathrm{dom}} h$ on which $h$ takes a constant value $h_\Omega \in \mathbb{R}$, there exist $\varepsilon, \lambda >0$ such that for all $x \in \{z\in \mathbb{R}^d : \mathrm{dist}_\Omega(z)<\varepsilon, h_\Omega < h(z) <h_\Omega + \lambda\}$, the following inequality is satisfied
\begin{align}\label{eq: KL}
\varphi' \left(h(x) - h_\Omega\right) \mathrm{dist}_{\partial h(x)}(\mathbf{0}) \ge 1,
\end{align}
where $\varphi'$ is the derivative of function $\varphi: [0,\lambda) \to \mathbb{R}_+$, which takes the form $\varphi(t) = \frac{c}{\theta} t^\theta$ for some $c>0, \theta\in (0,1]$.
\end{definition}
To elaborate, consider the case where $h$ is differentiable. Then, the K{\L}~ property in \cref{eq: KL} can be rewritten as
\begin{align}
h(x) - h_\Omega \le C \|\nabla h(x)\|^{p} \label{eq: KLsimple}
\end{align}
for some constant $C>0$ and $p\in (1, +\infty)$. In fact, \Cref{eq: KLsimple} can be viewed as a generalization of the gradient dominance condition
that corresponds to the special case of $p = 2$. A large class of functions have been shown to satisfy the K{\L}~ property, e.g., sub-analytic functions, logarithm and exponential functions, etc \cite{Bolte2007}. These function classes cover most of nonconvex objective functions encountered in practical applications, e.g., logistic loss, vector and matrix norms, rank, and polynomial functions, etc. Please refer to \cite[Section 5]{Bolte2014} and \cite[Section 4]{Attouch2010} for more example functions.
To handle non-smooth objective functions, we introduce the following notion of proximal mapping.
\begin{definition}(Proximal mapping)\label{def:prox}
For a proper and lower-semicontinuous function $h$, its proximal mapping at $x\in \mathbb{R}^d$ with parameter $\eta > 0$ is defined as:
\begin{align}
\prox{\eta h}(x) := \mathop{\mathrm{argmin}}_{z\in \mathbb{R}^d} \bigg\{h(z) + \frac{1}{2\eta}\|z - x\|^2\bigg\}.
\end{align}
\end{definition}
\section{APG-restart for Nonsmooth \& Nonconvex Optimization}
In this section, we propose a novel momentum-accelerated proximal gradient with parameter restart (referred to as APG-restart) for solving nonsmooth and nonconvex problems.
Consider the composite optimization problem of minimizing a smooth and nonconvex function $f:\mathbb{R}^d \to \mathbb{R}$ plus a possibly nonsmooth and convex function $g:\mathbb{R}^d \to \mathbb{R}$, which is written as
\begin{align*}
\min_{x\in \mathbb{R}^d} F(x):= f(x) + g(x). \tag{P}
\end{align*}
We adopt the following standard assumptions on the objective function $F$ in the problem (P).
\begin{assum}\label{assum: f+g}
The objective function $F$ in the problem $\mathrm{(P)}$ satisfies:
\begin{enumerate}[leftmargin=*]
\item Function $F$ is bounded below, i.e., $F^*:=\inf_{x\in \mathbb{R}^d} F(x) > -\infty$;
\item For any $\alpha \in \mathbb{R}$, the level set $\{x: F(x) \le \alpha\}$ is compact;
\item The gradient of $f$ is $L$-Lipschitz continuous and $g$ is lower-semicontinuous and convex.
\end{enumerate}
\end{assum}
Under \Cref{assum: f+g}, we further introduce the following mapping for any $\eta>0$ and $x, u \in \mathbb{R}^d$:
\begin{align}
G_\eta(x,u) := \frac{1}{\eta} \big(x-\mathrm{prox}_{\eta g}(x-\eta u) \big). \label{eq: grad_map}
\end{align}
Such a mapping is well-defined and single-valued due to the convexity of $g$. Moreover,
the critical points of function $F$ (cf., \Cref{def:sub}) in the problem (P) can be alternatively characterized as $\mathbf{\mathop{\mathrm{crit}}}\!~F := \{ x: \mathbf{0}\in G_\eta(x,\nabla f(x)) \}$. Therefore, $G_\eta(x,\nabla f(x))$ serves as a type of `gradient' at point $x$, and we refer to such a mapping as {\em gradient mapping} in the rest of the paper. In particular, the gradient mapping reduces to the usual notion of gradient when the nonsmooth part $g\equiv 0$.
\begin{algorithm}[H]
\caption{APG-restart for nonconvex optimization}
\label{alg: Acc-PGD}
{\bf Input:} $K \in \mathbb{N}$, restart periods $q_0=0, \{q_{t}\}_{t\ge 1} \in \mathbb{N}$, stepsizes $\{\lambda_k\}_{k}, \{\beta_k\}_{k} >0.$
{\bf Define:} $Q_t:= \sum_{\ell=0}^{t}q_\ell$.
{\bf Initialize:} $x_{-1} \in \mathbb{R}^d$.
\For{$k=0, 1, \ldots, K$}
{
Denote $t$ the largest integer such that $Q_t \le k$,\\
Set: $\alpha_k = \frac{2}{k-Q_t+2}$,\\
\If{$k= Q_t~$ for some $t\in \mathbb{N}$}
{
Reset: $x_{k} = y_k = x_{k-1},$
}
$z_{k} = (1-\alpha_{k+1})y_{k} + \alpha_{k+1} x_{k}$, \\
$x_{k+1} = x_k - \lambda_{k} G_{\lambda_k}(x_k, \nabla f(z_k))$, \\
$y_{k+1} = z_{k} - \beta_{k} G_{\lambda_k}(x_k, \nabla f(z_k))$.
}
{\textbf{Output:} $x_K$.}
\end{algorithm}
To solve the nonsmooth and nonconvex problem (P), we propose the APG-restart algorithm that is presented in \Cref{alg: Acc-PGD}. APG-restart consists of new design of momentum schemes for updating the variables $x_k$ and $y_k$, the extrapolation step for updating the variable $z_k$ where $\alpha_{k+1}$ denotes the associated momentum coefficient, and the restart periods $\{q_t\}_t$. We next elaborate the two major ingredients of APG-restart: new momentum design and flexible restart scheduling with convergence guarantee.
\paragraph{New momentum design:}
We adopt new momentum steps in APG-restart for updating the variables $x_k$ and $y_k$, which are different from those of the AG method in \cite{Ghadimi2016b} and we compare our update rules with theirs as follows.
\begin{equation}
\text{(APG-restart):}
\left\{
\begin{aligned}
\!x_{k+1} \!=\! x_k \!-\! \lambda_{k} G_{\lambda_k}(x_k, \!\nabla f(z_k)), \\
\!y_{k+1} \!=\! z_{k} \!-\! \beta_{k} G_{\lambda_k}(x_k, \!\nabla f(z_k)).
\end{aligned}
\right.
\end{equation}
\begin{equation}
\text{(AG):}
\left\{
\begin{aligned}
\!x_{k+1} \!=\! \mathrm{prox}_{\lambda_k g}(x_k\!-\!\lambda_k \nabla f(z_k)) \\
\!y_{k+1} \!=\! \mathrm{prox}_{\lambda_k g}(z_k\!-\!\beta_k \nabla f(z_k))
\end{aligned}
\right.
\end{equation}
It can be seen from the above comparison that our APG-restart uses the same gradient mapping term $G_{\lambda_k}(x_k, \nabla f(z_k))$ to update both of the variables $x_k$ and $y_k$, while the AG algorithm in \cite{Ghadimi2016b} updates them using different proximal gradient terms. Consequently, our APG-restart is more computationally efficient as it requires to compute one gradient mapping per iteration while the AG algorithm needs to perform two proximal updates.
On the other hand, the update rules of the AG algorithm guarantee convergence in nonconvex optimization only for functions of $g$ with bounded domain \cite{Ghadimi2016b}. Such a restriction rules out regularization functions with unbounded domain, which are commonly used in practical applications, e.g., $\ell_1, \ell_2$ regularization, elastic net, etc. In comparison, as we show in the analysis later, the update rules of APG-restart has guaranteed convergence in nonconvex optimization and does not require the regularizer $g$ to be domain-bounded.
\paragraph{Guarantee for any restart scheduling:} APG-restart retains the convergence guarantee with any restart scheduling. In specific, by specifying an arbitrary sequence of iteration periods $\{q_{t}\}_t \in \mathbb{N}$, APG-restart calls the restart operation at the end of each period (i.e., whenever $k=Q_t$ for some $t$). Upon restart, both $x_k$ and $y_k$ are reset to be the variable $x_{k-1}$ generated at the previous iteration, and the momentum coefficient $\alpha_k$ is reset to be 1. In the subsequent iterations, the momentum coefficient is diminished inversely proportionally to the number of iterations within the restart period.
Since our APG-restart retains convergence guarantee for any restart periods $\{q_t \}_t$, it can implement any criterion that determines when to perform the parameter restart and have a convergence guarantee (see our analysis later). We list in \Cref{table: 1} some popular restart criteria from existing literature and compare their practical performance under our APG-restart framework in the experiment section later. We note that the restart criterion of the gradient mapping scheme implicitly depends on the gradient mapping, as $y_{k+1}-z_{k} \propto G_{\lambda_k}(x_k, \nabla f(z_k))$ from the update rule in \Cref{alg: Acc-PGD}.
\begin{table*}[ht]
\setlength{\tabcolsep}{4pt}
\center
{\small
\begin{tabular}{ccccc}
\toprule
\begin{tabular}{@{}c@{}} Restart \\ scheme \end{tabular}
& \begin{tabular}{@{}c@{}} Fixed restart \\ \cite{Nesterov07}\end{tabular}
& \begin{tabular}{@{}c@{}} Function value \\ \cite{Donoghue2015}\\\cite{Giselsson2014}\\\cite{Kim2018} \end{tabular}
& \begin{tabular}{@{}c@{}} Gradient mapping \\ \cite{Donoghue2015}\\\cite{Giselsson2014}\\\cite{Kim2018} \end{tabular}
& \begin{tabular}{@{}c@{}} Non-monotonic \\ \cite{Giselsson2014} \end{tabular} \\ \midrule
\begin{tabular}{@{}c@{}} Check \\ condition \end{tabular}
& \begin{tabular}{@{}c@{}} $q_t\equiv q \in \mathbb{N}$ \\ for all $t$ \end{tabular}
& \begin{tabular}{@{}c@{}} restart whenever \\ $F(x_k)>F(x_{k-1})$ \end{tabular}
& \begin{tabular}{@{}c@{}} restart whenever \\ $\inner{z_{k}\!-\!y_{k}}{y_{k+1}\!-\!z_{k}}$ \\ $\ge 0$ \end{tabular}
& \begin{tabular}{@{}c@{}} restart whenever \\ $\inner{z_{k}\!-\!y_{k}}{y_{k+1}\!-\!\frac{z_{k}\!+\!x_k}{2}}$\\ $\ge 0$ \end{tabular} \\
\bottomrule
\end{tabular}
}
\caption{Restart conditions for different parameter restart schemes.}\label{table: 1}
\end{table*}
Performing parameter restart has appealing benefits. First, synchronizing the variables $x_k$ and $y_k$ periodically can suppress the deviation between them caused by the extrapolation step. This further helps to reduce the oscillation of the generated function value sequence. Furthermore,
restarting the momentum coefficient $\alpha_k$ periodically injects more momentum into the algorithm dynamic, and therefore facilitates the practical convergence of the algorithm.
\section{Convergence Analysis of APG-restart}
In this section, we study the convergence properties of APG-restart in solving nonconvex and nonsmooth optimization problems.
We first characterize the algorithm dynamic of APG-restart.
\begin{restatable}{lemma}{LemmaDynamicPGD}[Algorithm dynamic]\label{lemma: Acc-PGD dynamic}
Let \Cref{assum: f+g} hold and apply \Cref{alg: Acc-PGD} to solve the problem (P). Set $\beta_k \equiv \frac{1}{8L}$ and $\lambda_k \in [\beta_k, (1+\alpha_{k+1})\beta_{k}]$. Then, the sequence $\{x_k \}_k$ generated by APG-restart satisfies: for all $t=1,2,...$
\begin{align}
F(x_{Q_t}) \le &F(x_{Q_{t-1}}) - \frac{L}{4}\sum_{k=Q_{t-1}}^{Q_{t}-1} \|x_{k+1} - x_k\|^2, \label{eq: 9}\\
\mathrm{dist}_{\partial F(x_{Q_t})}^2(\mathbf{0}) &\le 162L^2 {\sum_{k=Q_{t-1}}^{Q_t-1}\|x_{k+1} - x_k\|^2}. \label{eq: 10}
\end{align}
\end{restatable}
\Cref{lemma: Acc-PGD dynamic} characterizes the period-wise algorithm dynamic of APG-restart. In specific, \cref{eq: 9} shows that the function value sequence generated by APG-restart is guaranteed to decrease between two adjacent restart checkpoint (i.e., $Q_{t-1}$ and $Q_t$), and the corresponding progress $F(x_{Q_{t-1}})- F(x_{Q_t})$ is bounded below by the square length of the iteration path between the restart checkpoints, i.e., $\sum_{k=Q_{t-1}}^{Q_t-1} \|x_{k+1} - x_k\|^2$. On the other hand, \cref{eq: 10} shows that the norm of the subdifferential at the $t$-th restart checkpoint is bounded by the square length of the same iteration path. In summary, the algorithm dynamic of APG-restart is different from that of traditional gradient-based algorithms in several aspects: First, the dynamic of APG-restart is characterized at the restart checkpoints, while the dynamic of gradient descent is characterized iteration-wise \cite{Attouch2009,Attouch2013}. As we elaborate later, such a property makes the convergence analysis of APG-restart more involved; Second, APG-restart makes monotonic progress on the function value between two adjacent restart checkpoints. In other accelerated gradient algorithms, such a monotonicity property is achieved by introducing a function value check step \cite{Li:2015} or an additional proximal gradient step \cite{Li2017}.
Based on the algorithm dynamic in \Cref{lemma: Acc-PGD dynamic}, we obtain the following global convergence rate of APG-restart for nonconvex and nonsmooth optimization. Throughout the paper, we denote $f(n) = \Theta(g(n))$ if and only if for some $0<c_1<c_2$, $c_1 g(n) \le f(n) \le c_2 g(n)$ for all $n\ge n_0$.
\begin{restatable}{thm}{TheoremGlobal}[Global convergence rate]\label{thm: global}
Under the same conditions as those of \Cref{lemma: Acc-PGD dynamic}, the sequence $\{z_k \}_k$ generated by APG-restart satisfies: for all $K=1,2,...$
\begin{align*}
\min_{0\le k\le K-1}\|G_{\lambda_k}(z_k, \nabla f(z_k))\|^2 \le \Theta \Big({\frac{L\big(F(x_{0}) - F^*\big)}{K}}\Big).
\end{align*}
\end{restatable}
\Cref{thm: global} establishes the global convergence rate of APG-restart in terms of the gradient mapping, which we recall characterizes the critical point of the nonconvex objective function $F$. In particular, the order of the above global convergence rate matches that of other accelerated gradient algorithms \cite{Ghadimi2016b} for nonconvex optimization, and APG-restart further benefits from the flexible parameter restart scheme that provides extra acceleration in practice (as we demonstrate via experiments later).
\Cref{thm: global} does not fully capture the entire convergence property of APG-restart. To elaborate, convergence of the gradient mapping in \Cref{thm: global} does not necessarily guarantee the convergence of the {\em variable sequence} generated by APG-restart. On the other hand, the convergence rate estimate is based on the global Lipschitz condition of the objective function, which may not capture the local geometry of the function around critical points and therefore leads to a coarse convergence rate estimate in the asymptotic regime. To further explore stronger convergence results of APG-restart, we next exploit the ubiquitous Kurdyka-{\L}ojasiewicz (K{\L}) property (cf., \Cref{def: KL}) of nonconvex functions. We make the following assumption.
\begin{assum}\label{assum: KL}
The objective function $F$ in the problem $\mathrm{(P)}$ satisfies the K{\L}~ property.
\end{assum}
Based on the algorithm dynamic in \Cref{lemma: Acc-PGD dynamic} and further leveraging the K{\L}~ property of the objective function, we obtain the following convergence result of APG-restart in nonconvex optimization.
\begin{restatable}{thm}{TheoremVariable}[Variable convergence]\label{thm: Acc-GD variable}
Let Assumptions \ref{assum: f+g} and \ref{assum: KL} hold and apply \Cref{alg: Acc-PGD} to solve the problem (P). Set $\beta_k \equiv \frac{1}{8L}$ and $\lambda_k \in [\beta_k, (1+\alpha_{k+1})\beta_{k}]$. Define the length of iteration path of the $t$-th restart period as
$ L_t:= \sqrt{\sum_{k=Q_t}^{Q_{t+1}-1} \|x_{k+1}-x_k\|^2}.$
Then, the sequence $\{L_t \}_t$ generated by APG-restart satisfies: for all periods of iterations $t=1,2,...$
\begin{align}
\sum_{t=0}^\infty L_t < +\infty. \label{eq: finite_len}
\end{align}
Consequently, the variable sequences $\{x_k \}_k, \{y_k \}_k, \{z_k \}_k$ generated by APG-restart converge to the same critical point of the problem (P), i.e.,
\begin{align}
x_k,y_k,z_k \overset{k}{\to} x^* \in \mathbf{\mathop{\mathrm{crit}}}\!~F.
\end{align}
\end{restatable}
\Cref{thm: Acc-GD variable} establishes the formal convergence of APG-restart in nonconvex optimization. We note that such a convergence guarantee holds for any parameter restart schemes, therefore demonstrating the flexibility and generality of our algorithm. Also, unlike other accelerated gradient-type of algorithms that guarantee only convergence of function value \cite{Li:2015,Li2017}, our APG-restart is guaranteed to generate convergent variable sequences to a critical point in nonconvex optimization.
To highlight the proof technique, we first exploit the dynamic of APG-restart in \Cref{lemma: Acc-PGD dynamic} to characterize the limit points of the sequences $\{x_{Q_t} \}_t, \{F(x_{Q_t}) \}_t$ that are indexed by the restart checkpoints. Then, we further show that the entire sequences $\{x_{k} \}_k, \{F(x_{k}) \}_k$ share the same limiting properties, which in turn guarantee the sequences to enter a local parameter region of the objective function where the K{\L}~ property can be exploited. Taking advantage of the K{\L}~ property, we are able to show that the length of the optimization path is finite as iteration $k\to \infty$. Consequently, the generated variable sequences can be shown to converge to a certain critical point of the Problem (P).
Besides the variable convergence guarantee under the K{\L}~ property, we also obtain various types of convergence rate estimates of APG-restart depending on the specific parameterization of the local K{\L}~ property of the objective function. We obtain the following results.
\begin{restatable}{thm}{TheoremRates}[Convergence rate of function value]\label{thm: Acc-GD rates}
Let Assumptions \ref{assum: f+g} and \ref{assum: KL} hold and apply \Cref{alg: Acc-PGD} to solve the problem (P). Set $\beta_k \equiv \frac{1}{8L}$ and $\lambda_k \in [\beta_k, (1+\alpha_{k+1})\beta_{k}]$. Suppose the algorithm generates a sequence $\{x_k \}_k$ that converges to a certain critical point $x^*$ where the K{\L} property holds with parameter $\theta\in (0,1]$. Then, there exists a sufficiently large $t_0\in \mathbb{N}$ such that for all $t\ge t_0$,
\begin{enumerate}[leftmargin=*]
\item If $\theta = 1$, then $F(x_{Q_t}) \downarrow F(x^*)$ within finite number of periods of iterations;
\item If $\theta \in [\frac{1}{2}, 1)$, then $F(x_{Q_t}) \downarrow F(x^*)$ linearly as
$F(x_{Q_t}) - F(x^*) \le \exp \big(-\Theta(t-t_0)\big);$
\item If $\theta \in (0, \frac{1}{2})$, then $F(x_{Q_t}) \downarrow F(x^*)$ sub-linearly as
$F(x_{Q_t}) - F(x^*) \le \Theta \Big((t-t_0)^{-\frac{1}{1-2\theta}}\Big).$
\end{enumerate}
\end{restatable}
\begin{restatable}{thm}{TheoremRatesVariable}[Convergence rate of variable]\label{thm: Acc-GD var_rates}
Under the same conditions as those of \Cref{thm: Acc-GD rates}, suppose APG-restart generates a sequence $\{x_k \}_k$ that converges to a certain critical point $x^*$ where the K{\L} property holds with parameter $\theta\in (0,1]$. Then, there exists a sufficiently large $t_0\in \mathbb{N}$ such that for all $t\ge t_0$,
\begin{enumerate}[leftmargin=*]
\item If $\theta = 1$, then $x_{Q_t} \overset{t}{\to} x^*$ within finite number of periods of iterations;
\item If $\theta \in [\frac{1}{2}, 1)$, then $x_{Q_t} \overset{t}{\to} x^*$ linearly as
$\|x_{Q_t} - x^*\| \le \exp \big(-\Theta(t-t_0)\big);$
\item If $\theta \in (0, \frac{1}{2})$, then $x_{Q_t} \overset{t}{\to} x^*$ sub-linearly as $\|x_{Q_t} - x^*\| \le\Theta \Big((t-t_0)^{-\frac{\theta}{1-2\theta}}\Big)$.
\end{enumerate}
\end{restatable}
\Cref{thm: Acc-GD rates} and \Cref{thm: Acc-GD var_rates} establish the asymptotic convergence rate results for the function value sequence and variable sequence generated by APG-restart, respectively. Intuitively, after a sufficiently large number of training iterations, APG-restart enters a local neighborhood of a certain critical point. In such a case, the global convergence rate characterized in \Cref{thm: global} can be a coarse estimate because it exploits only the global Lipschitz property of the function. On the contrary, the local K{\L}~ property characterizes the function geometry in a more accurate way and leads to the above tighter convergence rate estimates. In particular, the K{\L}~ parameter $\theta$ captures the `sharpness' of the local geometry of the function, i.e., a larger $\theta$ induces a faster convergence rate.
\section{Proof of \Cref{lemma: Acc-PGD dynamic}}
\LemmaDynamicPGD*
\begin{proof}
The proof utilizes some intermediate results developed in \cite{Ghadimi2016b}, and we include the proof of these results for completeness of presentation. Throughout, we define ${\Gamma}_0 = 0, {\Gamma}_1 = 1, {\Gamma}_k = (1-{\alpha}_k) {\Gamma}_{k-1}$ for $k=2,3,...$. By the restarting nature of $\alpha_{k}$, it is easy to check that ${\Gamma}_k = 0$ whenever $k=Q_t$ for some $t$, and $\Gamma_{k}=\frac{2}{(k-Q_t)(k-Q_t+1)}$ otherwise.
Let the sequences $\{x_k\}_k, \{y_k\}_k, \{z_k\}_k$ be generated by \Cref{alg: Acc-PGD}. Let us first analyze a certain $(t-1)$-th restart period, which consists of the iterations $\{\ell: Q_{t-1}\le \ell \le Q_t-1\}$. We first bound the term $\|y_k - x_k\|$ for any iteration $k$ within this restart period. By the update rule of the momentum scheme in \Cref{alg: Acc-PGD}, we obtain that
\begin{align}
y_k - x_k &= z_{k-1} - \beta_{k-1} G_{\lambda_{k-1}}(x_{k-1}, \nabla f(z_{k-1})) - (x_{k-1} - \lambda_{k-1} G_{\lambda_{k-1}}(x_{k-1}, \nabla f(z_{k-1}))) \nonumber\\
&= (1-{\alpha}_k) (y_{k-1} - x_{k-1}) + (\lambda_{k-1} - \beta_{k-1}) G_{\lambda_{k-1}}(x_{k-1}, \nabla f(z_{k-1})).
\end{align}
Dividing both sides by ${\Gamma}_k$ and noting that $\frac{1-{\alpha}_k}{{\Gamma}_k} = \frac{1}{{\Gamma}_{k-1}}$, we further obtain that
\begin{align}
\frac{y_k - x_k}{{\Gamma}_k} &= \frac{y_{k-1} - x_{k-1}}{{\Gamma}_{k-1}} + \frac{\lambda_{k-1} - \beta_{k-1}}{{\Gamma}_k} G_{\lambda_{k-1}}(x_{k-1}, \nabla f(z_{k-1})).
\end{align}
Telescoping the above equality over the iterations $Q_{t-1},...,k$ { within this restart period}, noting that $x_{Q_{t-1}}=y_{Q_{t-1}}$ by restart and rearranging, we obtain that
\begin{align}
\|y_k - x_k \|^2 &= \|{\Gamma}_{k} \sum_{\ell=Q_{t-1}}^{k-1} \frac{\lambda_{\ell} - \beta_{\ell}}{{\Gamma}_{\ell+1}} G_{\lambda_{\ell}}(x_{\ell}, \nabla f(z_{\ell}))\|^2 \nonumber \\
&= \|{\Gamma}_{k} \sum_{\ell=Q_{t-1}}^{k-1} \frac{{\alpha}_{\ell+1}}{{\Gamma}_{\ell+1}} \frac{\lambda_{\ell} - \beta_{\ell}}{{\alpha}_{\ell+1}} G_{\lambda_{\ell}}(x_{\ell}, \nabla f(z_{\ell}))\|^2 \nonumber \\
&\overset{(i)}{\le} {\Gamma}_{k} \sum_{\ell=Q_{t-1}}^{k-1} \frac{{\alpha}_{\ell+1}}{{\Gamma}_{\ell+1}} \frac{(\lambda_{\ell} - \beta_{\ell})^2}{{\alpha}_{\ell+1}^2} \|G_{\lambda_{\ell}}(x_{\ell}, \nabla f(z_{\ell}))\|^2 \nonumber \\
&= {\Gamma}_{k} \sum_{\ell=Q_{t-1}}^{k-1} \frac{(\lambda_{\ell} - \beta_{\ell})^2}{{\Gamma}_{\ell+1}{\alpha}_{\ell+1}} \|G_{\lambda_{\ell}}(x_{\ell}, \nabla f(z_{\ell}))\|^2, \label{eq: 1}
\end{align}
where (i) uses the facts that $\{{\Gamma}_k\}_k$ is a decreasing sequence within one restart period, $\sum_{\ell=Q_{t-1}}^{k-1} \frac{{\alpha}_{\ell+1}}{{\Gamma}_{\ell+1}} = \frac{1}{{\Gamma}_k}$ for all $k\le Q_t-1$ and Jensen's inequality.
We also need the following lemma, which was established as Lemma 1 and Proposition 1 in \cite{Ghadimi2016}.
\begin{lemma}(Lemma 1 and Proposition 1, \cite{Ghadimi2016})\label{aux: 4}
Let $g$ be a proper and closed convex function. Then, for all $u, v, x\in \mathbb{R}^d$ and $\eta>0$, the following statements hold:
\begin{align*}
&\inner{u}{G_{\eta}(x,u)} \ge \|G_{\eta}(x,u)\|^2 + \frac{1}{\eta} \big(g(\mathrm{prox}_{\eta g}(x-\eta u)) - g(x) \big), \\
&\|G_{\eta}(x,u) - G_{\eta}(x,v)\| \le \|u-v\|.
\end{align*}
\end{lemma}
Next, we further bound the function value gap $F(x_k)-F(x_{k-1})$ of iteration $k$ {within this restart period}. By the Lipschitz continuity of $\nabla f$ in item 3 of \Cref{assum: f+g}, we obtain that
\begin{align}
f(x_k) &\le f(x_{k-1}) + \inner{\nabla f(x_{k-1})}{x_k - x_{k-1}} + \frac{L}{2}\|x_k - x_{k-1}\|^2 \nonumber \\
&= f(x_{k-1}) + \inner{\nabla f(x_{k-1})}{- \lambda_{k-1} G_{\lambda_{k-1}}(x_{k-1}, \nabla f(z_{k-1}))} + \frac{L\lambda_{k-1}^2}{2}\|G_{\lambda_{k-1}}(x_{k-1}, \nabla f(z_{k-1}))\|^2 \nonumber \\
&= f(x_{k-1}) - \lambda_{k-1} \inner{\nabla f(x_{k-1})-\nabla f(z_{k-1})}{G_{\lambda_{k-1}}(x_{k-1}, \nabla f(z_{k-1}))} \nonumber\\
&\quad- \lambda_{k-1} \inner{\nabla f(z_{k-1})}{G_{\lambda_{k-1}}(x_{k-1}, \nabla f(z_{k-1}))}
+ \frac{L\lambda_{k-1}^2}{2}\|G_{\lambda_{k-1}}(x_{k-1}, \nabla f(z_{k-1}))\|^2 \nonumber\\
&\overset{(i)}{\le} f(x_{k-1}) - \lambda_{k-1} \inner{\nabla f(x_{k-1})-\nabla f(z_{k-1})}{G_{\lambda_{k-1}}(x_{k-1}, \nabla f(z_{k-1}))} - \lambda_{k-1} \|G_{\lambda_{k-1}}(x_{k-1}, \nabla f(z_{k-1}))\|^2 \nonumber\\
&\quad - \big(g(\mathrm{prox}_{\lambda_{k-1} g}(x_{k-1}-\lambda_{k-1} \nabla f(z_{k-1}))) - g(x_{k-1}) \big) + \frac{L\lambda_{k-1}^2}{2}\|G_{\lambda_{k-1}}(x_{k-1}, \nabla f(z_{k-1}))\|^2 \nonumber\\
&= f(x_{k-1}) - \lambda_{k-1} \inner{\nabla f(x_{k-1})-\nabla f(z_{k-1})}{G_{\lambda_{k-1}}(x_{k-1}, \nabla f(z_{k-1}))} - \lambda_{k-1} \|G_{\lambda_{k-1}}(x_{k-1}, \nabla f(z_{k-1}))\|^2 \nonumber\\
&\quad - \big(g(x_k) - g(x_{k-1}) \big) + \frac{L\lambda_{k-1}^2}{2}\|G_{\lambda_{k-1}}(x_{k-1}, \nabla f(z_{k-1}))\|^2, \nonumber
\end{align}
where (i) follows from \Cref{aux: 4}. Rearranging the above inequality and using Cauchy-Swartz inequality yields that
\begin{align}
F(x_k) &\le F(x_{k-1}) - \lambda_{k-1}(1-\frac{L\lambda_{k-1}}{2}) \|G_{\lambda_{k-1}}(x_{k-1}, \nabla f(z_{k-1}))\|^2 \nonumber\\
&\quad + \lambda_{k-1} \|\nabla f(x_{k-1})-\nabla f(z_{k-1})\| \|G_{\lambda_{k-1}}(x_{k-1}, \nabla f(z_{k-1}))\|. \label{eq: 15}
\end{align}
Also, note that
\begin{align}
\|\nabla f(x_{k-1}) - \nabla f(z_{k-1})\| &\le L\|x_{k-1} - z_{k-1}\| \overset{(i)}{\le} L(1-{\alpha}_k) \|y_{k-1} - x_{k-1}\| \nonumber,
\end{align}
where (i) follows from the update rule of the momentum scheme. Substituting the above inequality into \cref{eq: 15} yields that
\begin{align}
F(x_k) &\le F(x_{k-1}) - \lambda_{k-1}(1-\frac{L\lambda_{k-1}}{2})\|G_{\lambda_{k-1}}(x_{k-1}, \nabla f(z_{k-1}))\|^2 \nonumber\\
&\quad+ L\lambda_{k-1}(1-{\alpha}_k) \|G_{\lambda_{k-1}}(x_{k-1}, \nabla f(z_{k-1}))\|\|y_{k-1} - x_{k-1}\| \nonumber\\
&\le F(x_{k-1}) - \lambda_{k-1}(1-\frac{L\lambda_{k-1}}{2})\|G_{\lambda_{k-1}}(x_{k-1}, \nabla f(z_{k-1}))\|^2 \nonumber\\
&\quad+ \frac{L\lambda_{k-1}^2}{2} \|G_{\lambda_{k-1}}(x_{k-1}, \nabla f(z_{k-1}))\|^2 + \frac{L(1-{\alpha}_k)^2}{2}\|y_{k-1} - x_{k-1}\|^2 \nonumber\\
&= F(x_{k-1}) - \lambda_{k-1}(\frac{1}{2}-L\lambda_{k-1}) \|G_{\lambda_{k-1}}(x_{k-1}, \nabla f(z_{k-1}))\|^2 + \frac{L(1-{\alpha}_k)^2}{2}\|y_{k-1} - x_{k-1}\|^2 \nonumber\\
&\le F(x_{k-1}) - \lambda_{k-1}(\frac{1}{2}-L\lambda_{k-1}) \|G_{\lambda_{k-1}}(x_{k-1}, \nabla f(z_{k-1}))\|^2 \nonumber\\
&\quad+ \frac{L{\Gamma}_{k-1}}{2}\sum_{\ell=Q_{t-1}}^{k-2} \frac{\lambda_{\ell} - \beta_{\ell}}{{\alpha}_{\ell+1} {\Gamma}_{\ell+1}} \|G_{\lambda_{\ell}}(x_{\ell}, \nabla f(z_{\ell}))\|^2, \label{eq: 12}
\end{align}
where the last inequality uses \cref{eq: 1} and the fact that $0<{\alpha}_k <1$. Next, telescoping the above inequality over the iterations $Q_{t-1},...,k$ { within this restart period}, we further obtain that
\begin{align}
F(x_k) &\le F(x_{Q_{t-1}}) - \sum_{j=Q_{t-1}}^{k-1} \lambda_{j}(\frac{1}{2}- L\lambda_{j}) \|G_{\lambda_{j}}(x_{j}, \nabla f(z_{j}))\|^2 \nonumber\\
&\quad + \sum_{j=Q_{t-1}}^{k-1} \frac{L{\Gamma}_{j}}{2} \sum_{\ell=Q_{t-1}}^{j-1} \frac{(\lambda_{\ell} - \beta_{\ell})^2}{{\Gamma}_{\ell+1}{\alpha}_{\ell+1}} \|G_{\lambda_{\ell}}(x_{\ell}, \nabla f(z_{\ell}))\|^2 \nonumber\\
&= F(x_{Q_{t-1}}) - \sum_{j=Q_{t-1}}^{k-1} \lambda_{j}(\frac{1}{2}- L\lambda_{j}) \|G_{\lambda_{j}}(x_{j}, \nabla f(z_{j}))\|^2 \nonumber\\
&\quad+ \frac{L}{2} \sum_{\ell=Q_{t-1}}^{k-1} \frac{(\lambda_{\ell} - \beta_{\ell})^2}{{\Gamma}_{\ell+1}{\alpha}_{\ell+1}} \|G_{\lambda_{\ell}}(x_{\ell}, \nabla f(z_{\ell}))\|^2 (\sum_{j=\ell-1}^{k-1} {\Gamma}_{j}) \nonumber\\
&\overset{(i)}{\le} F(x_{Q_{t-1}}) - \sum_{j=Q_{t-1}}^{k-1} \lambda_{j}(\frac{1}{2}- L\lambda_{j}) \|G_{\lambda_{j}}(x_{j}, \nabla f(z_{j}))\|^2 \nonumber\\
&\quad+ \frac{L}{2} \sum_{\ell=Q_{t-1}}^{k-1} \frac{2(\lambda_{\ell} - \beta_{\ell})^2}{(\ell-Q_{t-1}-1) {\Gamma}_{\ell+1}{\alpha}_{\ell+1}} \|G_{\lambda_{\ell}}(x_{\ell}, \nabla f(z_{\ell}))\|^2 \nonumber\\
&= F(x_{Q_{t-1}}) - \sum_{j=Q_{t-1}}^{k-1} \bigg[\lambda_{j}(\frac{1}{2}- L\lambda_{j}) - \frac{L(\lambda_{j} - \beta_{j})^2}{(j-Q_{t-1}-1) {\Gamma}_{j+1}{\alpha}_{j+1}} \bigg] \|G_{\lambda_{j}}(x_{j}, \nabla f(z_{j}))\|^2 \nonumber\\
&\overset{(ii)}{\le} F(x_{Q_{t-1}}) - \sum_{j=Q_{t-1}}^{k-1} \big[\frac{\lambda_{j}}{4} - \frac{L{\alpha}_{j+1} \beta_{j}^2}{(j-Q_{t-1}-1) {\Gamma}_{j+1}} \big] \|G_{\lambda_{j}}(x_{j}, \nabla f(z_{j}))\|^2 \nonumber\\
&\le F(x_{Q_{t-1}}) - \sum_{j=Q_{t-1}}^{k-1} \big[\frac{\beta_{j}}{4} - \frac{\beta_{j}}{8} \big] \|G_{\lambda_{j}}(x_{j}, \nabla f(z_{j}))\|^2 \nonumber\\
&\le F(x_{Q_{t-1}}) - \sum_{j=Q_{t-1}}^{k-1} \frac{1}{64L\lambda_{j}^2} \|x_{j+1} - x_j\|^2 \nonumber\\
&\le F(x_{Q_{t-1}}) - \sum_{j=Q_{t-1}}^{k-1} \frac{L}{4} \|x_{j+1} - x_j\|^2, \label{eq: 8}
\end{align}
where (i) follows from the fact that $\sum_{j=\ell-1}^{k-1} {\Gamma}_{j} = 2\sum_{j=\ell-1}^{k-1} \frac{1}{j-Q_{t-1}} - \frac{1}{j-Q_{t-1}+1} \le \frac{2}{\ell-Q_{t-1}-1}$, and (ii) uses the facts that $L\lambda_{j} \le 2L\beta_j \le \frac{1}{4}$, $\lambda_{j}-\beta_{j} \le {\alpha}_{j+1} \beta_{j}$. Then, setting $k$ in the above inequality to be the last iteration $Q_t-1$ within this restart period and note that $x_{Q_t} = x_{Q_t-1}$, we obtain that
\begin{align}
F(x_{Q_t}) &\le F(x_{Q_{t-1}}) - \frac{L}{4}\sum_{k=Q_{t-1}}^{Q_t-1} \|x_{k+1} - x_k\|^2. \label{eq: 7}
\end{align}
The first inequality is proved.
To prove the second inequality, by the optimality condition of the proximal gradient update for $x_{k}$, we obtain that
\begin{align}
-\nabla f(z_{k-1}) - \frac{1}{\lambda_k}(x_{k}-x_{k-1}) \in \partial g(x_{k}), \nonumber
\end{align}
which further implies that
\begin{align}
\nabla f(x_{k})-\nabla f(z_{k-1}) - \frac{1}{\lambda_{k-1}}(x_{k}-x_{k-1}) \in \partial F(x_{k}). \nonumber
\end{align}
Note that $\mathrm{dist}_{\partial F(x_{k})}(\mathbf{0}) \le u_k$ for any $u_k \in \partial F(x_{k})$. Therefore, the above inequality further implies that
\begin{align}
\mathrm{dist}_{\partial F(x_{k})}(\mathbf{0}) &\le \|\nabla f(x_{k})-\nabla f(z_{k-1}) - \frac{1}{\lambda_{k-1}}(x_{k}-x_{k-1})\| \nonumber\\
&\le \|\nabla f(x_{k})- \nabla f(x_{k-1})\| + \|\nabla f(x_{k-1}) -\nabla f(z_{k-1})\| + \frac{1}{\lambda_{k-1}} \|x_{k}-x_{k-1}\| \nonumber\\
&\overset{(i)}{\le} 9L \|x_k - x_{k-1}\| + L \sqrt{{\Gamma}_{k-1} \sum_{\ell=Q_{t-1}}^{k-2} \frac{(\lambda_{\ell} - \beta_{\ell})^2}{{\Gamma}_{\ell+1}{\alpha}_{\ell+1}} \|G_{\lambda_{\ell}}(x_{\ell}, \nabla f(z_{\ell}))\|^2}, \label{eq: 13}
\end{align}
where (i) uses the Lipschitz gradient property, the update rule of \Cref{alg: Acc-PGD} and \cref{eq: 1}. Squaring both sides of the above inequality and rearranging, we further obtain that
\begin{align}
\mathrm{dist}_{\partial F(x_{k})}^2(\mathbf{0}) &\le 162L^2 \|x_k - x_{k-1}\|^2 + 2L^2{\Gamma}_{k-1} \sum_{\ell=Q_{t-1}}^{k-2} \frac{(\lambda_{\ell} - \beta_{\ell})^2}{{\Gamma}_{\ell+1}{\alpha}_{\ell+1}} \|G_{\lambda_{\ell}}(x_{\ell}, \nabla f(z_{\ell}))\|^2 \nonumber\\
&\le 162L^2 \|x_k - x_{k-1}\|^2 + 2L^2{\Gamma}_{k-1} \sum_{\ell=Q_{t-1}}^{k-2} \frac{{\alpha}_{\ell+1} \beta_{\ell}^2}{{\Gamma}_{\ell+1} \lambda_{\ell}^2} \|x_{\ell+1} - x_{\ell}\|^2 \nonumber\\
&\le 162L^2 \|x_k - x_{k-1}\|^2 + 2L^2{\Gamma}_{k-1} \sum_{\ell=Q_{t-1}}^{k-2} \frac{{\alpha}_{\ell+1} }{{\Gamma}_{\ell+1}} \|x_{\ell+1} - x_{\ell}\|^2 \nonumber\\
&\le 162L^2 \|x_k - x_{k-1}\|^2 + 2L^2\frac{2}{(k-Q_t-1)(k-Q_t-2)} \sum_{\ell=Q_{t-1}}^{k-2} (\ell+1-Q_t) \|x_{\ell+1} - x_{\ell}\|^2 \nonumber\\
&\le 162L^2 \sum_{\ell=Q_{t-1}}^{k-1} \|x_{\ell+1} - x_{\ell}\|^2. \nonumber
\end{align}
Then, set $k$ in the above inequality to be the last iteration $Q_t-1$ within this restart period and note that $x_{Q_t} = x_{Q_t-1}$, we obtain that
\begin{align}
\mathrm{dist}_{\partial F(x_{Q_t})}^2(\mathbf{0}) \le 162L^2 \sum_{k=Q_{t-1}}^{Q_t-1} \|x_{k+1} - x_{k}\|^2. \nonumber
\end{align}
The second inequality is proved.
\end{proof}
\section{Proof of \Cref{thm: global}}
\TheoremGlobal*
\begin{proof}
Consider any iteration $K$ and the corresponding closet restart checkpoint $Q_t$ (for some $t$). In the proof of \Cref{lemma: Acc-PGD dynamic}, we have shown that (see \cref{eq: 8})
\begin{align}
F(x_{K}) &\le F(x_{Q_{t}}) - \frac{L}{4}\sum_{j=Q_{t}}^{K-1} \|x_{j+1} - x_j\|^2. \label{eq: 11}
\end{align}
On the other hand, by \Cref{lemma: Acc-PGD dynamic} we know that for all $\ell=1,..., t$,
\begin{align}
F(x_{Q_\ell}) &\le F(x_{Q_{\ell-1}}) - \frac{L}{4}\sum_{k=Q_{\ell-1}}^{Q_{\ell}-1} \|x_{k+1} - x_k\|^2.
\end{align}
Telescoping the above inequality over $\ell=1,..., t$ and combining with \cref{eq: 11}, and noting that $Q_0=q_0=0$, $x_{Q_\ell} = x_{Q_{\ell}-1}$ for all $\ell$, we obtain that
\begin{align*}
F(x_{K}) &\le F(x_{0}) - \frac{L}{4}\sum_{j=0}^{K-1} \|x_{j+1} - x_j\|^2 = F(x_{0}) - \frac{L}{4}\sum_{j=0}^{K-1} \lambda_{j}^2 \|G_{\lambda_j}(z_j, \nabla f(z_j))\|^2.
\end{align*}
Note that $\lambda_j > \beta_{j} = \frac{1}{8L}$. Then, the above inequality further implies that
\begin{align*}
\frac{1}{256L}\sum_{j=0}^{K-1} \|G_{\lambda_j}(z_j, \nabla f(z_j))\|^2 \le F(x_{0}) - F(x_{K}) \le F(x_{0}) - F^*.
\end{align*}
Ignoring the universal constants in the above inequality and taking the minimum, we obtain that
\begin{align*}
\min_{0\le k\le K-1}\|G_{\lambda_k}(z_k, \nabla f(z_k))\|^2 \le \Theta\bigg({\frac{L\big(F(x_{0}) - F^*\big)}{K}}\bigg).
\end{align*}
\end{proof}
\section{Proof of \Cref{thm: Acc-GD variable}}
\TheoremVariable*
\begin{proof}
Recall that the length of the iteration path of the $t$-th restart period is defined as
\begin{align}
L_t:= \sqrt{\sum_{k=Q_t}^{Q_{t+1}-1} \|x_{k+1}-x_k\|^2}.
\end{align}
Then, we can rewrite the results of \Cref{lemma: Acc-PGD dynamic} as
\begin{align}
F(x_{Q_t}) &\le F(x_{Q_{t-1}}) - \frac{L}{4}L_{t-1}^2, \label{eq: 2}\\
\mathrm{dist}_{\partial F(x_{Q_t})}^2(\mathbf{0}) &\le 162L^2 L_{t-1}^2.\label{eq: 3}
\end{align}
By \cref{eq: 2}, the function value sequence $\{F(x_{Q_t}) \}_t$ decreases monotonically period-wise. Since the objective function $F$ is bounded below (item 1 of \Cref{assum: f+g}), we conclude that $\{F(x_{Q_t})\}_t$ converges to a certain finite limit $F^*$. Also, since $F$ has bounded sub-level sets (item 2 of \Cref{assum: f+g}), \cref{eq: 2} further implies that the sequence $\{x_{Q_t} \}_t$ is bounded.
The above proof shows that $F(x_{Q_t}) \downarrow F^*$ and $\{x_{Q_t}\}_t$ is bounded. Next, we further show that the entire sequences $\{F(x_{k})\}_k, \{x_k\}_k$ share the same properties. Telescoping \cref{eq: 2} over $t=1,2,...T$ yields that: for any $T\in \mathbb{N}$,
\begin{align}
\sum_{t=0}^{T-1} L_{t}^2 \le L(F(x_0) - F(x_{Q_T})) \le L(F(x_0) - \inf_{x\in \mathbb{R}^d} F(x) ) < +\infty.
\end{align}
Letting $T\to \infty$ we conclude that $\sum_{t=0}^{\infty} L_{t}^2 < +\infty$ and therefore $L_t \overset{t}{\to} 0$.
Since each restart period contains a uniformly bounded number of iterations, this further implies that $\lim_{k\to \infty} \|x_{k+1} - x_k\| = 0$. Therefore, the entire sequence $\{x_k \}_k$ is bounded and we denote $\omega$ as its set of limit points ($\omega$ is a compact set). Also, by the facts that $\lim_{k\to \infty} \|x_{k+1} - x_k\| = 0$ and \cref{eq: 12}, we conclude that $\lim_{k\to \infty} F(x_{k+1}) - F(x_k) = 0$ for all $k\in \mathbb{N}$. Since $F(x_{Q_t}) \downarrow F^*$, we conclude that $F(x_k) \to F^*$. To this end, we have shown that the entire sequence $\{x_k \}_k$ has a limit point set $\omega$ and the entire sequence $\{F(x_k) \}_k$ converges to a certain finite limit $F^*$.
Now consider any limit point $x^*\in \omega$ and without loss of ambiguity we assume that $x_{k} \overset{k}{\to} x^*$ along a proper subsequence. By the proximal gradient update step of $x_k$ we obtain that
\begin{align}
g(x_{k}) + \frac{1}{2\lambda_{k-1}} \|x_k - x_{k-1}\|^2 &+ \inner{\nabla f(z_{k-1})}{x_k - x_{k-1}} \nonumber\\
&\le g(x^*) + \frac{1}{2\lambda_{k-1}} \|x^* - x_{k-1}\|^2 + \inner{\nabla f(z_{k-1})}{x^* - x_{k-1}}. \nonumber
\end{align}
Taking limsup on both sides of the above inequality and noting that $\{x_k\}_k$ is bounded, $\|x_k - x_{k-1}\| \to 0$ and $x_k \to x^*$, we conclude that $\limsup_k g(x_{k}) \le g(x^*)$. Since $g$ is lower-semicontinuous, we know that $\limsup_k g(x_{k}) \ge g(x^*)$. Combining these two inequalities yields that $\lim_k g(x_{k}) = g(x^*)$. By continuity of $f$, we further conclude that $\lim_k F(x_{k}) = F(x^*)$. Since we have shown that the entire sequence $\{F(x_{k})\}_k$ converges to a certain finite limit $F^*$, we conclude that $F(x^*)\equiv F^*$ for all $x^*\in \omega$. Also, \cref{eq: 13} and the fact that $\|x_{k+1} - x_k\| \to 0$ further imply that $\mathrm{dist}_{\partial F(x_{k})}(\mathbf{0}) \overset{k}{\to} 0$.
To this end, we have shown that for every subsequence $x_{k} \to x^* \in \omega$ we have $F(x_{k}) \to F(x^*)$ and $\mathrm{dist}_{\partial F(x_{k})}(\mathbf{0}) \to 0$. Recall the definition of limiting sub-differential, we conclude that every limit point $x^*$ of $\{x_k\}_k$ is a critical point, i.e., $\mathbf{0}\in \partial F(x^*)$.
Next, we show that the sequence $\{x_k\}_k$ has a unique limit point under the K{\L}~ property.
Consider any limit point $x^*\in \omega$. We have shown that 1) $F(x^*)\equiv F^*$ for all $x^*\in\omega$; 2)
$F(x_{Q_t}) \downarrow F^*$; and 3) $\mathrm{dist}_{\partial F(x_{k})}(\mathbf{0}) \to 0$. Collecting these facts, we are ready to apply the K{\L} property for $t$ being sufficiently large. Specifically, by the K{\L} property of the objective function, we obtain that: for all $t\ge t_1$ where $t_1$ is a sufficiently large integer,
\begin{align}
\varphi'(F(x_{Q_t}) - F^*) \ge \frac{1}{\mathrm{dist}_{\partial F(x_{Q_t})}(\mathbf{0})} \overset{(i)}{\ge} \frac{1}{15 L \cdot L_{t-1}}, \label{eq: 4}
\end{align}
where (i) follows from \cref{eq: 3}. Then, by concavity of $\varphi$ and \cref{eq: 2,eq: 4}, we further obtain that
\begin{align}
\varphi(F(x_{Q_t}) - F^*) - \varphi(f(x_{Q_{t+1}}) - F^*) &\ge \varphi'(F(x_{Q_t}) - F^*) (F(x_{Q_t}) - F(x_{Q_{t+1}})) \nonumber\\
&\ge \frac{L_t^2}{60L^2 \cdot L_{t-1}}. \label{eq: 5}
\end{align}
Rearranging the above inequality yields that
\begin{align}
L_t^2 \le 60L^2L_{t-1} \big[\varphi(F(x_{Q_t}) - F^*) - \varphi(F(x_{Q_{t+1}}) - F^*) \big]. \nonumber
\end{align}
Taking square root of both sides of the above inequality and using the fact that $\sqrt{ab} \le \frac{a+b}{2}$ for $a,b>0$, we obtain that
\begin{align}
2L_t \le L_{t-1} + 60L^2\big[\varphi(F(x_{Q_t}) - F^*) - \varphi(F(x_{Q_{t+1}}) - F^*) \big].
\end{align}
Telescoping the above inequality over $t=t_1+1,...T$ yields that
\begin{align}
2\sum_{t=t_1+1}^T L_t &\le \sum_{t=t_1+1}^T L_t + L_{t_1} + 60L^2\big[\varphi(F(x_{Q_{t_1+1}}) - F^*) - \varphi(F(x_{Q_{T+1}}) - F^*) \big] \nonumber\\
&\le \sum_{t=t_1+1}^T L_t + L_{t_1} + 60L^2\varphi(F(x_{Q_{t_1+1}}) - F^*), \nonumber
\end{align}
where the last inequality follows from the fact that $F(x_{Q_t}) \ge F^*$ for all $t\ge t_1$ and $\varphi(s)>0$ for all $s>0$. Rearranging the above inequality yields that: for all $T \ge t_1$
\begin{align}
\sum_{t=t_1+1}^T L_t &\le L_{t_1} + 60L^2\varphi(F(x_{Q_{t_1+1}}) - F^*)<+\infty. \nonumber
\end{align}
Letting $T\to \infty$ and noting that $t_1$ is a finite integer, we finally conclude that
\begin{align}
\sum_{t=0}^\infty L_t < +\infty. \nonumber
\end{align}
To further prove the convergence of the variable sequence, note that $L_t := \sqrt{\sum_{k=Q_t}^{Q_{t+1}-1} \|x_{k+1}-x_k\|^2}\ge \frac{1}{\sqrt{Q_{t+1}-Q_t}} \sum_{k=Q_t}^{Q_{t+1}-1} \|x_{k+1}-x_k\|$. Substituting into the above inequality yields that
\begin{align}
\sum_{t=0}^\infty \frac{1}{\sqrt{Q_{t+1}-Q_t}} \sum_{k=Q_t}^{Q_{t+1}-1} \|x_{k+1}-x_k\| \le \max_t \frac{1}{\sqrt{Q_{t+1}-Q_t}} \sum_{k=0}^\infty \|x_{k+1}-x_k\|< +\infty, \nonumber
\end{align}
where the last inequality uses the fact that all restart periods have uniformly bounded numbers of iterations that are uniformly bounded.
Therefore, the sequence $\{\|x_{k+1}-x_k\| \}_k$ is absolutely summable and this implies that $\{x_k \}_k$ is a convergent Cauchy sequence. Since we have shown that all the limit points of $\{x_k \}_k$ are critical points, we conclude that $\{x_k \}_k$ converges to a certain critical point of $F$. Lastly, it is clear from the previous results that $\|x_k-y_k\|\to 0, \|x_k-z_k\|\to 0$, which imply that both $\{y_k\}_k$ and $\{z_k\}_k$ converge to the same limit.
\end{proof}
\section{Proof of \Cref{thm: Acc-GD rates} and \Cref{thm: Acc-GD var_rates}}
\TheoremRates*
\TheoremRatesVariable*
\begin{proof}
Consider any $t$-th restart period and denote $r_t:= F(x_{Q_t}) - F(x^*)$ as the function value gap. Then, we can rewrite \cref{eq: 5} as: for all sufficiently large $t\ge t_0$,
\begin{align}
60L^2\big(\varphi(r_t) - \varphi(r_{t+1}) \big) \ge \frac{L_t^2}{L_{t-1}}.
\end{align}
Next, fix $\gamma \in (0,1)$ and consider any $t\ge t_0$. Suppose that $L_t\ge \gamma L_{t-1}$, then the above inequality implies that
\begin{align}
L_t \le \frac{60L^2}{\gamma^2} \big(\varphi(r_t) - \varphi(r_{t+1}) \big).
\end{align}
Otherwise, we conclude that $L_t\le \gamma L_{t-1}$. Combining these two inequalities yields that
\begin{align}
L_t \le \gamma L_{t-1} + \frac{60L^2}{\gamma^2} \big(\varphi(r_t) - \varphi(r_{t+1}) \big).
\end{align}
Summing the above inequality over $t=t_0,...,T$ yields that
\begin{align}
\sum_{t=t_0}^{T} L_t &\le \gamma \sum_{t=t_0}^{T}L_{t-1} + \frac{60L^2}{\gamma^2} \big(\varphi(r_{t_0}) - \varphi(r_{T+1}) \big) \nonumber\\
&\le \gamma \Big[\sum_{t=t_0}^{T}L_{t} + L_{t_0-1} \Big] + \frac{60L^2}{\gamma^2} \varphi(r_{t_0}). \nonumber
\end{align}
Rearranging the above inequality yields that: for all $T\ge t_0$,
\begin{align}
\sum_{t=t_0}^{T} L_t &\le \frac{\gamma}{1-\gamma} L_{t_0-1} + \frac{60L^2}{\gamma^2(1-\gamma)} \varphi(r_{t_0}). \nonumber
\end{align}
Next, define $\Delta_{t_0}:= \sum_{t=t_0}^{\infty} L_t$, which is well-defined due to \cref{eq: finite_len}. Then, letting $T\to \infty$ in the above inequality and noting that $\varphi(s)=Cs^\theta$ yields that for all sufficiently large $t$
\begin{align}
\Delta_{t} &\le \frac{\gamma}{1-\gamma} (\Delta_{t-1} - \Delta_{t}) + \frac{CL^2}{\gamma^2(1-\gamma)} r_{t}^{\theta} \nonumber \\
&\overset{(i)}{\le} \frac{\gamma}{1-\gamma} (\Delta_{t-1} - \Delta_{t}) + \frac{CL^{\frac{1}{1-\theta}}}{\gamma^2(1-\gamma)} \Big[\sum_{k=(t-1)q}^{tq-1}\|x_{k+1} - x_k\|^2\Big]^{\frac{\theta}{2(1-\theta)}} \nonumber \\
&= \frac{\gamma}{1-\gamma} (\Delta_{t-1} - \Delta_{t}) + \frac{CL^{\frac{1}{1-\theta}}}{\gamma^2(1-\gamma)} (\Delta_{t-1} - \Delta_{t})^{\frac{\theta}{1-\theta}} \nonumber \\
\end{align}
where (i) uses the K{\L}~ property and the dynamics of APG-restart in \Cref{lemma: Acc-PGD dynamic}, i.e., $r_{t} \le C\mathrm{dist}_{\partial F(x_{Q_t})}^{\frac{1}{1-\theta}}(\mathbf{0}) \le C \Big[L^2 \sum_{k=Q_{t-1}}^{Q_t-1}\|x_{k+1} - x_k\|^2\Big]^{\frac{1}{2(1-\theta)}}$. It has been shown in (Attouch \& Bolte 09) that sequence $\{\Delta_t\}_t$ satisfying the above inductive property converges to zero at different rates depending on $\theta$ as stated in the theorem. Finally, we note that Holder's inequality and triangle inequality imply that $\max_t \frac{1}{\sqrt{Q_{t+1}-Q_t}}\|x_{Q_t} - x^*\| \le \Delta_t$, and the result follows.
To prove the convergence rates of the function value gap, consider any $t$-th restart period and denote $r_t:= F(x_{Q_t}) - F(x^*)$ as the function value gap. As we have shown that $r_t \overset{t}{\to} 0$, for sufficiently large $t$ we can apply the K{\L}~ property and obtain that: for some universal constant $C>0$,
\begin{align}
r_t &\le C\mathrm{dist}_{\partial F(x_{Q_t})}^{\frac{1}{1-\theta}}(\mathbf{0}) = C\sqrt{\mathrm{dist}_{\partial F(x_{Q_t})}^{\frac{1}{1-\theta}}(\mathbf{0})} \nonumber\\
&\overset{(i)}{\le} C\sqrt{\Big(L^2\sum_{k=Q_{t-1}}^{Q_t-1} \|x_{k+1} - x_{k}\|^2 \Big)^{\frac{1}{1-\theta}}} \nonumber\\
&\overset{(ii)}{\le} CL^{\frac{3}{2(1-\theta)}} \big(r_{t-1} -r_t \big)^{\frac{1}{2(1-\theta)}}, \nonumber
\end{align}
where the constant $C$ may vary from line to line in the above derivation, (i) and (ii) follow from \Cref{lemma: Acc-PGD dynamic}. Rearranging the above inequality yields that: for all sufficiently large $t$,
\begin{align}
1\le CL^3r_t^{2(\theta -1)} (r_{t-1} - r_t). \nonumber
\end{align}
It has been shown in (Frankel et al., 2015; Li \& Lin, 2015) that sequence $\{r_t\}_t$ satisfying the above inductive property converges to zero at different rates depending on $\theta$ as stated in the theorem.
\end{proof}
\section{Restart Conditions Used in the Experiments}\label{app: exp}
\begin{enumerate}
\item For the fixed restart scheme we set the restart period to be $q=10,30,50$, respectively;
\item For the function scheme, we relax the condition to be
\begin{align*}
F(x_k) > 0.8F(x_{k-1}).
\end{align*}
\item For the gradient mapping scheme, we relax the condition to be
\begin{align*}
\inner{z_{k}-y_{k}}{y_{k+1}-z_{k}} \ge -0.2 \|z_{k}-y_{k}\|\|y_{k+1}-z_{k}\|.
\end{align*}
\item For the non-monotone scheme, we relax the condition to be
\begin{align*}
\inner{z_{k}-y_{k}}{y_{k+1}-\frac{z_{k}+x_k}{2}} \ge -0.2\|z_{k}-y_{k}\|\Big\|y_{k+1}-\frac{z_{k}+x_k}{2}\Big\|.
\end{align*}
\end{enumerate}
| {
"timestamp": "2020-04-28T02:31:44",
"yymm": "2002",
"arxiv_id": "2002.11582",
"language": "en",
"url": "https://arxiv.org/abs/2002.11582",
"abstract": "Various types of parameter restart schemes have been proposed for accelerated gradient algorithms to facilitate their practical convergence in convex optimization. However, the convergence properties of accelerated gradient algorithms under parameter restart remain obscure in nonconvex optimization. In this paper, we propose a novel accelerated proximal gradient algorithm with parameter restart (named APG-restart) for solving nonconvex and nonsmooth problems. Our APG-restart is designed to 1) allow for adopting flexible parameter restart schemes that cover many existing ones; 2) have a global sub-linear convergence rate in nonconvex and nonsmooth optimization; and 3) have guaranteed convergence to a critical point and have various types of asymptotic convergence rates depending on the parameterization of local geometry in nonconvex and nonsmooth optimization. Numerical experiments demonstrate the effectiveness of our proposed algorithm.",
"subjects": "Optimization and Control (math.OC); Machine Learning (cs.LG)",
"title": "Proximal Gradient Algorithm with Momentum and Flexible Parameter Restart for Nonconvex Optimization",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9728307692520259,
"lm_q2_score": 0.7279754371026368,
"lm_q1q2_score": 0.708196904473138
} |
https://arxiv.org/abs/physics/0701021 | Multipole structure and coordinate systems | Multipole expansions depend on the coordinate system, so that coefficients of multipole moments can be set equal to zero by an appropriate choice of coordinates. Therefore, it is meaningless to say that a physical system has a nonvanishing quadrupole moment, say, without specifying which coordinate system is used. (Except if this moment is the lowest non-vanishing one.) This result is demonstrated for the case of two equal like electric charges. Specifically, an adapted coordinate system in which the potential is given by a monopole term only is explicitly found, the coefficients of all higher multipoles vanish identically. It is suggested that this result can be generalized to other potential problems, by making equal coordinate surfaces coincide with the potential problem's equipotential surfaces. | \section{Introduction}
Multipole expansions are a standard mathematical tool in both physics research and teaching (e.g., Ref.~\cite{Morse}). For example, students of electromagnetism often calculate the electrostatic potential of a point electric charge when the coordinate system is not centered on the latter \cite{Griffiths,Panofsky}. In mathematical methods courses this problem often motivates, and serves as a means to the introduction of Legendre functions through its generating function
$$
g(t,x):=(1-2xt+t^2)^{-1/2}=\sum_{n=0}^{\infty}P_n(x)\, t^n\;\;\;\;\;\;\;,\;\;\;\;\;\;|t|<1\, .
$$
(The alternative approach to the introduction of the Legendre functions is through the angular equation obtained from the separation of variables of the Poisson equation in spherical symmetry \cite{Jackson, Pollack,Franklin,Reitz}.)
As the generating function is equal to the inverse distance between two points (in three-dimensional Euclidean space)---i.e., $1/|{\mathbf x}-\mathbf{x}'|=g(r'/r,\,\cos\theta)$, where $\mathbf{x}$,$\mathbf{x}'$ are the two points---one can readily superpose solutions to obtain the field of an arbitrary distribution of point charges. Here, $r,r'$ are their corresponding distances from the origin (so that $r'<r$) and $\theta$ is the angle between the vectors $\mathbf{x}$ and $\mathbf{x}'$ \cite{Mathews}.
The electrostatic potential then is expressed as an infinite sum over Legendre functions (see, e.g.,
\cite{Jackson}), with all multipoles present (both even and odd).
The next problem in a typical mathematical methods course may often be to calculate the potential of a pair of charges, either of opposite signs (``electric dipole") or of the same sign. In either case, half of the coefficients in the series vanish, and one ends up with a series over all odd or even terms, respectively. Specifically, in the latter case (two equal charges $q$), the potential is
\begin{equation}\label{multipoles}
\phi(r,\theta)=\frac{2q}{r_>}\left[1+P_2(\cos\theta)\left(\frac{r_<}{r_>}\right)^2+P_4(\cos\theta)\left(\frac{r_<}{r_>}\right)^4+\cdots\right]\, ,
\end{equation}
where $r_>(r_<)$ is the greater (smaller) of the evaluation point and the location of the charge from the center of the coordinate system. (We assume here, without loss of generality, that the charges are on the $z$-axis, so that the potential is azimuthal.) Notably, the potential includes a monopole term, a quadrupole term, a hexadecapole term, and so on. The different multipoles are used here describe the moments of a given charge distribution.
Specifically, the quadrupole moment equals $2qr_<^2/r_>^3$.
When $r_>\gg r_<$, the potential approximately equals the total charge divided by the distance to the evaluation point, which is the known result for a point particle at the origin of the coordinate system. Indeed, under this condition the separation of the charge from the origin is very small (compared with the distance to the evaluation point), so that the potential is indeed expected to be just such. However, when this condition is not satisfied, or indeed at finite distances, we find that the potential needs to be corrected, the correction terms being the multipole moments.
Mathematical methods texts often discuss next how to arrange the charge distribution so that one, or more, of the multipole moments vanishes (in a given coordinate system)
\cite{Arfken}. As the electrostatic problem is linear, one may superpose charges so that the lowest non-vanishing multipole moment may be as desired, e.g., one may obtain an electric dipole by having a charge $q$ at $z=a$ and a charge $-q $ at $z=-a$ thus canceling the monopole term, and the lowest non-vanishing moment is the dipole.
An interesting point that is usually not highlighted is that the multipole expansion itself is coordinate dependent, so that for a {\em given} charge distribution one can choose adapted coordinated in which certain multipole moments vanish. Specifically, we will show here in detail how, for the aforementioned example of two equal, like, static electric charges, one can find an adapted coordinate system in which {\em all the multipoles above the monopole vanish}. In such a coordinate system the potential is a function of one coordinate only, as the potential includes only a monopole term, which highlights the simplicity and elegance of the presentation. We see then that the complex multipole structure of the potential (\ref{multipoles}) is a consequence of the choice of coordinates. We can find a coordinate system in which the potential equals the total charge divided by a coordinate that plays, in some loose sense, the role of ``effective distance." It is only the lowest non-vanishing moment that cannot be nullified by a coordinate transformation.
The price to pay for this simplicity and elegance is in the form of a non-trivial coordinate transformation from, say, cartesian coordinates to the adapted ones, and in topologically non-trivial coordinates. Arguably (and only semi-seriously), there is ``conservation of complexity," which one cannot escape. However, several useful tools of mathematical physics are used in order to find this adapted coordinate system, which makes it an interesting problem, in addition to leading to deeper understanding of multipole expansions and their meaning.
\section{The Two--Distance Coordinates}
Let us first introduce a convenient coordinate system, from which we can transform to the desired adapted coordinates. Our strategy is to look for a coordinate system that is adapted to the potential in space. As the potential depends only on the distances to the two charges, a good starting point would be to base a coordinate system on the two distances from the two charges, so that equal coordinate surfaces would be spheres centered on either charge. The intersection of two such spheres spans a circle, and an azimuthal coordinate can then uniquely specify a point in space. To construct this coordinate system, position the charges along the $x$-axis, say, at $x=\pm a$. Define the distance of the evaluation point of the field from the two centers:
\begin{equation}
L:=\sqrt{(x+a)^2+y^2+z^2}\;\;\;\;\;\;R:=\sqrt{(x-a)^2+y^2+z^2}\, .
\end{equation}
These would be two coordinates, and the third one is defined by a rotation about the $x$-axis, namely,
\begin{equation}
\Phi:=\tan^{-1}\,\left(\frac{z}{y}\right)\, .
\end{equation}
The coordinate transformation from the cartesian coordinates $x,y,z$ to the coordinates $L,R,\Phi$ is a regular coordinate transformation, as is evident from the explicit expressions for the inverse transformation:
\begin{eqnarray}
x&=&\frac{L^2-R^2}{4a}\\
y&=&\sqrt{R^2-\frac{(L^2-R^2-4a^2)^2}{16a^2}}\,\sin\Phi :=Q\,\sin\Phi \\
z&=&\sqrt{R^2-\frac{(L^2-R^2-4a^2)^2}{16a^2}}\,\cos\Phi :=Q\,\cos\Phi
\end{eqnarray}
The coordinates $R,L,\Phi$ are the {\em Two--Distance Coordinates}, or 3D--TDC. We will find these coordinates to be a convenient starting point in our search for the adapted coordinate system, although they are unadapted to the potential problem themselves. The (spatial) metric in the 3D--TDC is given by
\begin{equation}\label{lr-metric}
\,d\sigma^2=\frac{L^2\,R^2}{4a^2Q^2}\,\left(\,dL^2+\,dR^2\right)-\frac{LR(L^2+R^2-4a^2)}{4a^2Q^2}\,dL\,dR+Q^2\,d\Phi^2\, ,
\end{equation}
where
\begin{equation}Q^2=R^2-\frac{(L^2-R^2-4a^2)^2}{16a^2}=\frac{[(L+R)^2-4a^2][4a^2-(L-R)^2]}{16a^2}=y^2+z^2\, .
\end{equation}
The metric (\ref{lr-metric}) is evidently non-diagonal, as it includes the ``cross term" proportional to $\,dL\,dR$. Non-diagonal metrics are of much importance in physics and mathematical physics. Nevertheless, typical mathematical method courses do not discuss them. This metric is singular at the origin (half way between the two charges)---as can be seen from the vanishing of $Q^2$ there---but as this is flat Euclidean space, this singularity is a coordinate singularity. Figure \ref{fig1} displays the 3D--TDC system.
\begin{figure}
\input epsf
\includegraphics[width=12.0cm]{TDC.eps}
\caption{The 3D--TDC: shown is the $x-y$ plane. The two centers are at $x=a,y=0$ and $x=-a,y=0$. The field evaluation point is at $p$. The coordinates $L$ and $R$ are defined by the distance of $p$ from the two centers. The third coordinate $\Phi$ is obtained by rotation about the $x$-axis. }
\label{fig1}
\end{figure}
\section{Searching for the adapted coordinates}
To search effectively for the adapted coordinate system, let us revisit regular spherical coordinates, and see why they are so effective in the description of the electric field of a single point charge. The main feature of the electric potential of a point charge is that equipotential surfaces are concentric spheres. The latter are the loci of all equidistant points from the center. These two statements imply that anywhere on an equipotential surface the value of one of the coordinates (the radial coordinate) is constant. To find the potential at some evaluation point, we only need to know on which equipotential surface it lies. For this reason, the potential is a function of only one coordinate, because only one coordiante is required to identify an equipotential surface. After the radial coordinate is chosen, one can readily find the other coordinates: the polar coordinate $\theta$ is found by requiring that it is everywhere orthogonal to constant-$r$ surfaces, and the azimuthal coordinate is obtained by rotation of the $r-\theta$ plane. Spherical coordinates are therefore adapted to the potential problem of a single charge at their center, and the potential is described in full by a monopole term only. We show this derivation in detail in Appendix A.
In searching for the adapted coordinates we follow a similar path: First, define the coordinate $\chi$ so that equal-$\chi$ surfaces are also equipotential surfaces of the static 3D Coulomb problem. Because of the problem's linearity, we may superpose solutions of two single-charge problems, for which the potentials (of unit charges) are $1/L$ and $1/R$, respectively, i.e., the total potential is $1/L+1/R$.
In fact, {\em any} monotonic function of $1/L+1/R$ is a good choice for this coordinate, i.e., $\chi=\chi(1/L+1/R)$. Specifically, we choose $\chi$ so that at great distances ($L,R\gg a$), the coordinate $\chi$ behaves like the radial coordinate $r$, because in this limit the potential is almost that of a point particle (of charge $2q$). That is, we choose
\begin{equation}
\frac{1}{\chi}:=\frac{1}{2}\,\left(\frac{1}{R}+\frac{1}{L}\right)
\end{equation}
or
\begin{equation}
\chi=2\,\frac{LR}{L+R}\, .
\end{equation}
Notice that the coordinate $\chi$ is non-trivial: it undergoes a topology change at the critical surface $\chi=a$. For $\chi <a$ equal-$\chi$ surfaces are doubly connected, and for $\chi >a$ they are singly connected. Figure \ref{fig2} shows the equal-$\chi$ surfaces.
\begin{figure}
\input epsf
\includegraphics[width=12.0cm]{roswell.eps}
\caption{The adapted coordinate system in the $x-y$ plane. Rotation about the $x$-axis yields the full 3D system. The closed contours are the equal-$\chi$ surfaces. The ``figure eight" curve is the critical surface across which the connectedness of the $\chi$ coordinate undergoes a topology change. The open curves are the equal-$\Theta$ lines.}
\label{fig2}
\end{figure}
To find the coordinate $\Theta$ we require that it is everywhere orthogonal to equal-$\chi$ surfaces (which are also the equipotential surfaces). Namely, equal-$\Theta$ lines will be the electric field lines. The
condition that two curves are orthogonal is that their gradients are. The reason why we need to require the condition of the gradients is that inner product is a map from two vectors to a scalar. Orthogonality of two vectors is defined by the requirement that their inner product vanishes. The way to create a vector from a scalar is by calculating its gradient. Specifically, $\Theta$ satisfies
\begin{equation}\label{pde}
\,\nabla\chi\cdot\,\nabla\Theta=0\, ,
\end{equation}
which in Cartesian coordinates is
\begin{equation}\label{pde-car}
\frac{\,\partial\chi}{\,\partial x}\frac{\,\partial\Theta}{\,\partial x}+\frac{\,\partial\chi}{\,\partial y}\frac{\,\partial\Theta}{\,\partial y}+\frac{\,\partial\chi}{\,\partial z}\frac{\,\partial\Theta}{\,\partial z}=0\, .
\end{equation}
The most straighforward way to try and solve the partial differential equation for $\Theta$ (recall that $\chi$ is known; the unknown in Eq.~(\ref{pde}) is $\Theta$) is to try and solve Eq.~(\ref{pde-car}) in cartesian coordinates. This turns out to be a highly non-trivial thing to do. In Appendix B we discuss this equation. The easier way to solve Eq.~(\ref{pde}) is by realizing that it is a scalar equation, so that we can express the terms on either hand side in whatever coordinates we find convenient, and the equation will retain its form. Equation (\ref{pde}) becomes particularly simple when expressed in 3D--TDC. The reason why this could be expected is that space is symmetrical under the exchange of the two centers. It is therefore reasonable to expect that geometrical expressions would be simpler when expressed in terms of the distances to the two centers (and not other points such as the origin of the cartesian grid), the only ``natural" quantities that the metric structure of space depends on.
We can express Eq.~(\ref{pde}) in 3D--TDC in two way: first, we can transform Eq.~(\ref{pde-car}) or we can use the metric (\ref{lr-metric}) to write $g^{ij}\chi_{,i}\Theta_{,j}=0$, a comma denoting partial derivative. Either way, we find that
Eq.~(\ref{pde}) transforms to
\begin{equation}
\left[ \frac{L}{2}(L^2+R^2-4a^2) +R^3 \right] L\,\frac{\,\partial\Theta}{\,\partial L}+\left[ \frac{R}{2}(L^2+R^2-4a^2) +L^3 \right] R\,\frac{\,\partial\Theta}{\,\partial R}=0
\end{equation}
whose solution is
\begin{equation}\label{theta1}
\Theta(L,R) = \Theta\left\{ \frac{R-L}{RL} \, \left[(R+L)^2-4a^2\right] \right\}\, .
\end{equation}
This solution is easy to verify, or even derive using software such as Maple or Mathematica. Notice that the solution is {\em any} function of a certain combination of $L,R$, with $a$ acting as a parameter (the only length scale in this problem). Indeed, any function of this argument is everywhere orthogonal to equal-$\chi$ surfaces. To make our solution useful, we next need to choose judiciously which function. Specifically, we make our choice by requiring that the asymptotic properties of the solution for $\Theta$ will coincide with the regular polar coordinate $\theta$. This way, our coordinate system will asymptotically approach regular spherical coordinates as it should, as at great distances the separation of the two centers becomes negligible.
At very great distances,
$$\frac{R-L}{RL} \, \left[(R+L)^2-4a^2\right]\to -8a\frac{x}{y}+O(y^{-3})\, .$$
We next require that at that limit a coordinate $\tilde \Theta$ coincides with the regular polar coordinate $\theta:=\tan^{-1}(y/x)$, which motivates us to choose
\begin{equation}
\tilde \Theta(L,R) = \tan^{-1}\left[ \frac{LR}{L-R} \, \frac{8a}{(R+L)^2-4a^2} \right]\, .
\end{equation}
This guarantees that at very great distances $\tilde\Theta$ behaves similarly to the regular polar coordinate $\theta$. However, the coordinate $\tilde \Theta$ is still not quite what we need, because it is discontinuous: Consider a semicircle at constant large distance from the center of the cartesian coordinate system (the center is half way between the two charges), starting on the positive $x$-axis and going through the upper half-plane to the negative $x$-axis. The value of $\tilde\Theta$ will vary from $\pi/4$ to $\pi/2$, jump discontinuously to $-\pi/2$ crossing the $y$-axis, and then change to $-\pi/4$. Define then
\begin{equation}\label{theta}
\Theta(L,R):=\frac{\pi}{2}-\pi\,{\rm sgn}(L-R)+2\tilde\Theta(L,R)\, .
\end{equation}
The coordinate $\Theta$ is continuous in the upper half-plane, and it is everywhere orthogonal to $\chi$. It's range is from $0$ to $\pi$. Notice that ${\rm sgn}(L-R)\equiv{\rm sgn}(x)\equiv{\rm sgn}(\pi/2-\Theta)$, and that we only need to define $\Theta$ in the upper half-plane, because we rotate the coordinates about the $x$-axis.
We now have the coordinate system $(\chi,\Theta,\Phi)$, which is the Two-Center Bi-Spherical coordinates system (TCBS). The metric in these coordinates is given by
\begin{eqnarray}\label{TCBS}
\,d\sigma^2 &=& \frac{1}{4}\,\frac{(L+R)^4}{(L+R)^4-3LR(L+R)^2-4LRa^2}\,d\chi^2 \nonumber \\
&+&\frac{1}{16\cdot 256}\, \frac{[16(L-R)^2a^4-8(L^4+R^4-10L^2R^2)a^2+(L-R)^2(L+R)^4]^2}{a^4\,Q^2\,[(L+R)^4-3LR(L+R)^2-4LRa^2]}\,d\Theta^2 \nonumber \\
&+& Q^2\,d\Phi^2\, ,
\end{eqnarray}
where $L,R$ are implicit functions of $\chi,\Theta$. The metric (\ref{TCBS}) is manifestly diagonal.
Given $\chi,\Theta$, one needs to solve the following quintic to find $R(\chi,\Theta)$ explicitly:
\begin{equation}
R^5-\chi\,R^4+4a(h^{-1}\chi-a)\,R^3-4a\chi (h^{-1}\chi-2a)\,R^2+a\chi^2(h^{-1}\chi-5a)\,R+a^2\chi^3=0\,
\end{equation}
where
$$h :=\,\tan {\tilde {\Theta}}=\,\tan \frac{1}{2}\left[ \Theta-\frac{\pi}{2} +\pi\,{\rm sgn}\left(\frac{\pi}{2}-\Theta\right) \right]\, .$$
As is well known from Abel's Impossibility Theorem \cite{Abel}, it is impossible to solve a general quintic equation in terms of radicals.
However, solutions in terms of hypergeometric functions \cite{Klein} or Jacobi Theta functions \cite{Hermite} are always possible. Numerical solutions are of course easy to find (e.g., using the Newton--Raphson method). Notice, that the Fundamental Theorem of Algebra guarantees a real solution.
It is immediately clear that the coordinate system is {\em singular} at the two centers and at the origin of the cartesian coordinate system. For example, $Q^2$ vanishes at all three singular points, so that $g_{\Theta\Theta}$ diverges, and $g_{\Phi\Phi}$ vanishes. As space is 3D flat Euclidean space, we know that this singularity is a coordinate singularity, and not a genuine geometrical one. Notably, the Jacobian of the transformation from cartesian to 3D--TCBS coordinates is regular.
Given the solution for $R(\chi,\Theta)$ we can readily find $L(\chi,\Theta)$:
\begin{equation}
L=\frac{\chi\,R}{2R-\chi}\, .
\end{equation}
\section{The potential problem in the adapted coordinates}
To express the potential problem in the newly found TCBS coordinates we first express the Laplacian in these coordinates. In any coordinate system the Laplacian of a scalar field $\Psi$ is given by
\begin{eqnarray}\label{laplacian}
\,\nabla^2\Psi &=& (\,\nabla\chi\cdot\,\nabla\chi)\; \Psi_{,\chi\chi} +
(\,\nabla\Theta\cdot\,\nabla\Theta)\; \Psi_{,\Theta\Theta}+
( \,\nabla\Phi\cdot\,\nabla\Phi)\; \Psi_{,\Phi\Phi} \nonumber \\
&+& 2(\,\nabla\chi\cdot\,\nabla\Theta)\; \Psi_{,\chi\Theta}+
2 (\,\nabla\chi\cdot\,\nabla\Phi)\; \Psi_{,\chi\Phi}+
2 (\,\nabla\Theta\cdot\,\nabla\Phi)\; \Psi_{,\Theta\Phi} \nonumber \\
&+& (\,\nabla^2\chi) \;\Psi_{,\chi}+ (\,\nabla^2\Theta) \;\Psi_{,\Theta}+ (\,\nabla^2\Phi) \;\Psi_{,\Phi}\, .
\end{eqnarray}
The coefficients in this expression can readily be calculated explicitly from the above expressions. Specifically, we find
\begin{equation}\label{lap-chi}
\,\nabla^2\chi=4\,\frac{L^4+R^4+LR(L^2+R^2-4a^2)}{LR(L+R)^3}
\end{equation}
\begin{equation}
\,\nabla^2\Theta = \frac{128\,a^3\,(L-R)\, G}{[16(L+R)^2a^4+8(L^4-10L^2R^2+R^4)a^2-(L-R)^2(L+R)^4]^2}
\end{equation}
where
\begin{eqnarray}
G &=& 64a^6-16(5L^2+2LR+5R^2)a^4+4(7L^4+28L^3R+26L^2R^2+28LR^3+7R^4)a^2\nonumber \\
&-& (L-R)^2(3L^4+20L^3R-14L^2R^2+20LR^3+3R^4)\, ,
\end{eqnarray}
\begin{equation}
\,\nabla^2\Phi=0
\end{equation}
\begin{equation}
\,\nabla\chi\cdot\,\nabla\chi=g^{\chi\chi}
\end{equation}
\begin{equation}
\,\nabla\Theta\cdot\,\nabla\Theta=g^{\Theta\Theta}
\end{equation}
\begin{equation}
\,\nabla\Phi\cdot\,\nabla\Phi=g^{\Phi\Phi}
\end{equation}
\begin{equation}
\,\nabla\chi\cdot\,\nabla\Theta=0
\end{equation}
\begin{equation}
\,\nabla\chi\cdot\,\nabla\Phi=0
\end{equation}
\begin{equation}
\,\nabla\Theta\cdot\,\nabla\Phi=0\, ,
\end{equation}
the last three relations resulting from the TCBS coordinates being orthogonal. Notice that $\Phi$ is harmonic.
We are now in a position to show that in the 3D--TCBS the solution for the potential problem of two equal like charges is given by a monopole term only. Specifically, we show that
\begin{equation}\label{lap_1chi}
\,\nabla^2 \left(\frac{1}{\chi}\right)=0\;\;\;\;\;\;\;\;\;\;\;\;(\chi\ne 0)\, .
\end{equation}
This relation can be directly verified by substitution in Eq.~(\ref{laplacian}): Clearly all the derivatives with respect to either $\Theta$ or $\Phi$ vanish, so that Eq.~(\ref{laplacian}) reduces to
\begin{eqnarray}
\,\nabla^2 \left(\frac{1}{\chi}\right) &=& g^{\chi\chi} \left(\frac{1}{\chi}\right)_{,\chi\chi}+\,\nabla^2\chi \left(\frac{1}{\chi}\right)_{,\chi}\nonumber \\
&=& \frac{2}{\chi^3}g^{\chi\chi}-\frac{\,\nabla^2\chi}{\chi^2}=0
\end{eqnarray}
after direct substitution of $g^{\chi\chi}$ from Eq.~(\ref{TCBS}) and using Eq.(\ref{lap-chi}). Equation
(\ref{lap_1chi}) is analogous to $\,\nabla^2 (1/r)=0$ (except at $r=0$) in regular spherical coordinates.
At $\chi=0$ ($x=\pm a,y=z=0$) the Laplacian of $\chi$ no longer vanishes, so that the global problem is described by Poisson's equation, $\,\nabla^2\Psi=-8\pi q\,J^{-1}\,\delta(\chi)$, where $J$ is the Jacobian determinant for the transformation from cartesian to 3D--TCBS coordinates. Comparing Poisson's equation with Eq.~(\ref{lap_1chi}), we find the solution for the potential problem to be
\begin{equation}
\Psi=\frac{2q}{\chi}\, ,
\end{equation}
which is the desired solution.
\section{Conclusions}
We showed, for the specific example of two equal like electric charges, how to find a coordinate system in which the electric potential is described by one coordinate, and for which it is given by a monopole term only. The derivation involved a large number of topics that are covered in mathematical methods courses, such as non-diagonal coordinate systems, coordinate singularities, quintic equations, multipole expansions, coordinate transformations, and potential theory. As such, it may serve as an instructive problem for such courses. In particular, it may be used to demonstrate the deep meaning of a multipole expansion, and its dependence on the choice of coordinates. Specifically, a multipole expansion can be made simple when the coordinates used are adapted to the potential problem. The reason why an infinite number of (even) multipoles are needed to describe the potential of two equal like charges using regular spherical coordinates (\ref{multipoles}) is that the equal coordinate surfaces are very different from the equipotential surfaces. By choosing the two surfaces to coincide, we are able to eliminate all the higher moltipoles from the potential, and solve the potential problem using only the monopole term. The price to pay is in a form of more involved mathematics, which is however still in the range of the usual preparation of the usual Physics education programs.
\section*{Acknowledgments}
The author wishes to thank Richard Price and Anthony Hester for discussions, and Ross Cortez for checking some of the calculations. This work was supported in part by a minigrant from the UAH Office of the Vice President for Research.
\begin{appendix}
\section{``Derivation" of spherical coordinates}
In this Appendix we apply our method to a point charge, and derive the
adapted coordinates, which are just the regular spherical coordinates. This derivation may serve as
a pedagogic illustration of the method, applied to a trivial situation for which the solution is well known.
As discussed above, for the case of a single point particle, the radial coordinate $r$ describes equipotential surfaces. We therefore choose the radial coordinate $r$ as one of the
adapted coordinates, or $\chi_{\rm s} :=r$. To find the second coordinate, we require that it is orthogonal to $r$. Therefore, it satisfies Eq. ~(11), i.e., $\,\nabla\chi_{\rm s}\cdot\,\nabla\Theta_{\rm s}=0\, ,$
or, in cartesian coordinates in the $x$--$y$ plane,
$({\,\partial r}/{\,\partial x})({\,\partial\Theta_{\rm s}}/{\,\partial x})+({\,\partial r}{\,\partial y})/({\,\partial\Theta_{\rm s}}{\,\partial y})=0$.
Recalling that $\,\partial r /\,\partial x=x/r$ and $\,\partial r /\,\partial y=y/r$, this equation becomes
$x\,({\,\partial\Theta_{\rm s}}/{\,\partial x})+y\,({\,\partial\Theta_{\rm s}}/{\,\partial y})=0$ (except for the origin which is a coordinate singularity), whose solution is $\Theta_{\rm s}=\Theta_{\rm s}(y/x)$. Next, we require that at large distances this coordinate coincides with the regular azimuthal coordinate
$\theta:=\tan^{-1}(y/x)$, so that we find that everywhere $\Theta_{\rm s}:=\theta$.
Rotating the coordinates by an angle $\phi$ about the $x$-axis yields
$x'=x=r\,\cos\theta$, $y'=y\,\cos\phi=r\,\sin\theta\,\cos\phi$, and $z'=y\,\sin\phi=r\,\sin\theta\,\sin\phi$. The regular spherical coordinates are obtained by renaming the cartesian axes $x''=y'$, $y''=z'$, and $z''=x'$:
$x=r\,\sin\theta\,\cos\phi$, $y=r\,\sin\theta\,\sin\phi$, and $z=r\,\cos\theta$, after dropping the primes for
conventionality.
\section{Trying to solve Eq.~(\ref{pde-car}) directly}
Equation (\ref{pde-car}) can be readily written explicitly. In the $x-y$ plane, the unknown is $\Theta (x,y)$, so that Eq.~(\ref{pde-car}) becomes
\begin{eqnarray*}
\left[ (x-a)\left( x^2+2ax+a^2+y^2\right)^{3/2}+ (x+a)\left(x^2-2ax+a^2+y^2\right)^{3/2}\right]\frac{\,\partial\Theta}{\,\partial x}\\
+ y\left[ \left( x^2+2ax+a^2+y^2\right)^{3/2}+\left(x^2-2ax+a^2+y^2\right)^{3/2}\right]
\frac{\,\partial\Theta}{\,\partial y}=0\, .
\end{eqnarray*}
This partial differential equation is non-trivial to solve symbolically. Standard techniques, e.g., separation of variables, prove to be ineffective. Even PDE solvers such as Maple and Mathematica, using up to 8 gigabytes RAM for a couple of weeks on a PowerMac G5 were unsuccessful in solving this equation. Indeed, having found the solution (\ref{theta1}) using the 3D--TDC it is evident why straightforward attempts to solve this equation failed. It should be noted, however, that numerical solutions for this equation can readily be found, if appropriate boundary conditions are specified.
\end{appendix}
| {
"timestamp": "2007-01-02T19:53:39",
"yymm": "0701",
"arxiv_id": "physics/0701021",
"language": "en",
"url": "https://arxiv.org/abs/physics/0701021",
"abstract": "Multipole expansions depend on the coordinate system, so that coefficients of multipole moments can be set equal to zero by an appropriate choice of coordinates. Therefore, it is meaningless to say that a physical system has a nonvanishing quadrupole moment, say, without specifying which coordinate system is used. (Except if this moment is the lowest non-vanishing one.) This result is demonstrated for the case of two equal like electric charges. Specifically, an adapted coordinate system in which the potential is given by a monopole term only is explicitly found, the coefficients of all higher multipoles vanish identically. It is suggested that this result can be generalized to other potential problems, by making equal coordinate surfaces coincide with the potential problem's equipotential surfaces.",
"subjects": "Classical Physics (physics.class-ph)",
"title": "Multipole structure and coordinate systems",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9728307676766119,
"lm_q2_score": 0.7279754371026367,
"lm_q1q2_score": 0.7081969033262752
} |
https://arxiv.org/abs/2204.01365 | Deep learning, stochastic gradient descent and diffusion maps | Stochastic gradient descent (SGD) is widely used in deep learning due to its computational efficiency, but a complete understanding of why SGD performs so well remains a major challenge. It has been observed empirically that most eigenvalues of the Hessian of the loss functions on the loss landscape of over-parametrized deep neural networks are close to zero, while only a small number of eigenvalues are large. Zero eigenvalues indicate zero diffusion along the corresponding directions. This indicates that the process of minima selection mainly happens in the relatively low-dimensional subspace corresponding to the top eigenvalues of the Hessian. Although the parameter space is very high-dimensional, these findings seems to indicate that the SGD dynamics may mainly live on a low-dimensional manifold. In this paper, we pursue a truly data driven approach to the problem of getting a potentially deeper understanding of the high-dimensional parameter surface, and in particular, of the landscape traced out by SGD by analyzing the data generated through SGD, or any other optimizer for that matter, in order to possibly discover (local) low-dimensional representations of the optimization landscape. As our vehicle for the exploration, we use diffusion maps introduced by R. Coifman and coauthors. | \section{Introduction and motivation}
The calibration of deep neural networks results in the optimization problem
\begin{align}\label{optprob}
\mathbf{x}^\star = \argmin_{\mathbf{x} \in \mathbb{R}^m} \Bigl\{ f(\mathbf{x}) := \frac1{N} \sum\nolimits_{i=1}^N f_{i}(\mathbf{x}) \Bigr\},
\end{align}
where $\mathbf{x}\in\mathbb{R}^m$ denotes the weights of the neural network and $f:\mathbb{R}^m\to\mathbb{R}$ is the loss function, which typically is non-convex as a function of $\mathbf{x}$. $f_{i}$, for $i \in \{1, \dots, N\}$, denotes the contribution to the loss function from data point $i$ and $N$ denotes the total number of data points.
A natural approach to the optimization problem in \eqref{optprob} is to use gradient descent (GD). However, when $N$ is large, it may be computationally prohibitive to compute the full gradient of the objective function $f$ and so stochastic gradient descent (SGD) provides an alternative. SGD is based on a (noisy) gradient evaluated from a single data point or a minibatch of data points, resulting in the iterative updates
\begin{align}
\mathbf{x}(t_{j+1})=\mathbf{x}_{j+1}= \mathbf{x}_{j} - \eta \nabla \widetilde{f}^{(j)}(\mathbf{x}_j)=\mathbf{x}(t_{j}) - \eta \nabla \widetilde{f}^{(j)}(\mathbf{x}(t_j)),\, t_{j+1}=t_j+\eta, \label{eqn:sgd_main}
\end{align}
where $j \in \{0, \dots, M\}$ denotes the iteration number, and $\nabla \widetilde{f}^{(j)}$ denotes the stochastic gradient at iteration $j$ defined as
\begin{align}
\label{eqn:stoch_grad}
\nabla \widetilde{f}^{(j)} (\mathbf{x}):= \frac1{n_j} \sum\nolimits_{i \in \Omega_j} \nabla f_{i}(\mathbf{x}).
\end{align}
Here, $\Omega_j \subset \{1,\dots,N\}$ is a random subset that is drawn with or without replacement at iteration $j$, and $n_j$ denotes the number of elements in $\Omega_j$. When no confusion arises, we simply write $\Omega$ and $n$. The $\eta>0$ in \eqref{eqn:sgd_main}, which can either be constant or varying with the iteration, is known as the learning rate.
Given the use of SGD, a set or sequence of points $X:=\{\mathbf x_j\}_{j=1}^M=\{\mathbf{x}(t_j)\}_{j=1}^M$ is generated, either from one sequence of runs of SGD or merged from several different runs. In particular, the set $X$ contains the information in the paths of the SGD in the high-dimensional space of parameters $\mathbb R^m$. In general, it is difficult to picture the geometry of the loss surface $$\Sigma:=\{(\mathbf{x},f(\mathbf{x})):\ \mathbf{x}\in\mathbb R^m\}$$ and insightful descriptions of this loss landscape as well as the geometry traced out by the paths of the SGD is still lacking due to the fact that while $f$ may be smooth, it is a non-linear, non-convex function in $\mathbb R^m$ with $m$ truly large. Heuristically, one way to think of the loss surface $\Sigma$, though the picture seems to be even more complex in reality, is as a landscape with peaks and valleys separated by ridges. Therefore, any optimizer including SGD could potentially get trapped in a basin and valley enclosing a local minima, finding it difficult to move from its initialized value over the ridges in the direction of the global minima.
The loss landscape or loss surface $\Sigma$ has received a lot of attention in the literature. To mention a few relevant papers, \cite{ choromanska2015loss,dauphin2014identifying} conjectured that local minima of multi-layer neural networks have similar loss function values, and proved the result in idealized settings. For linear networks, it is known \cite{kawaguchi2016deep} that all local minima are also globally optimal. Several theoretical works have explored whether a neural network has spurious valleys (non-global minima that are surrounded by other points with higher loss). \cite{freeman2016topology} showed that for a two-layer network, if it is sufficiently over-parametrized, then all the local minimizers are (approximately) connected. However, in order to guarantee a small loss along the path, they need the number of neurons to be exponential in the number of input dimensions. \cite{venturi2018spurious} proved that if the number of neurons is larger than either the number of training samples or the intrinsic dimension, then the neural network cannot have spurious valleys. \cite{liang2018understanding} proved similar results for the binary classification setting. We also refer to \cite{freeman2016topology, garipov2018loss, liang2018understanding, nguyen2019connected, nguyen2018loss, venturi2018spurious} for insightful discussions concerning the geometry of the loss landscape.
SGD is widely used in deep learning due to its computational efficiency, but understanding how SGD performs better than its full batch counterpart in terms of test accuracy remains a major challenge. While SGD seems to find zero loss solutions on the loss landscape $\Sigma$, at least in certain regimes, it appears that the algorithm finds solutions with different properties depending on how it is tuned, and a satisfactory theory explaining the success of SGD is in several ways still lacking. Empirically, it has been observed that SGD can usually find flat minima among a large number of sharp minima and local minima \cite{hochreiter1995simplifying,hochreiter1997flat}. Other papers indicate that learning flat minima is closely related to the problem of generalization \cite{dinh2017sharp, dziugaite2017computing, hardt2016train, kleinberg2018alternative, hoffer2017train, neyshabur2017exploring, wu2017towards, zhang2017understanding}. Several papers are also devoted to flatness itself, measuring flatness \cite{hochreiter1997flat, sagun2017empirical, yao2018hessian}, rescaling flatness \cite{tsuzuku2019normalized, xie2020stable}, and finding flatter minima \cite{chaudhari2017entropy, NIPS2019_8524, hoffer2017train, xie2020artificial}. Furthermore, it has been observed that most eigenvalues of the Hessian at the loss landscape of over-parameterized deep neural networks are close to zero, and in particular, only a small number of eigenvalues are large \cite{li2018visualizing, sagun2017empirical}. Zero eigenvalues indicate zero diffusion along the corresponding directions and, theoretically, one may be inclined to ignore these zero-eigenvalue directions. A small number of large eigenvalues means that the process of minima selection mainly happens in the relatively low-dimensional subspace corresponding to the top eigenvalues of the Hessian \cite{gur2018gradient}. In particular, although the parameter space is high-dimensional, SGD dynamics depends only modestly on the dimensions corresponding to small second-order directional derivatives, and SGD can heuristically be pictured as exploring the parameter space around a minimum in a much lower dimensional space. Still, a quantitative theory explaining these phenomena is lacking.
In this paper, we pursue a truly data driven approach with the ambition to contribute to the understanding of the loss landscape $\Sigma$, and in particular to the understanding of the landscape traced out by SGD, by analyzing the data generated through SGD, or any other optimizer for that matter, in order to possibly discover (local) low-dimensional representations of $\Sigma$ and the optimization landscape. Note that this discovery of low-dimensional representations of high-dimensional data, characterization of the underlying geometry, and description of the density are some of the fundamental problems in data science. In general, to achieve this, statistical tools are used on SGD paths to detect the slow variables, meta-stable states, as well as connections and transition times between these states. This is the focus of this paper as we explore a low-dimensional representation of the high-dimensional data $X$ generated by SGD. As the vehicle for our exploration, we use the insightful work of R. Coifman and collaborators on diffusion maps and geometry and the relation to Langevin dynamics and Fokker-Planck equations.
Diffusion maps and geometry \cite{Coifman2006, Lafon2006} are tools for the analysis of large datasets. A family of random walk processes on the large data set is constructed using isotropic and anisotropic diffusion kernels. Afterwhich, the eigenvalues and eigenvectors are analyzed, where the most dominant ones are known to be the principal components. These principal components contain key information regarding the geometry and statistics of the underlying space. Today, diffusion maps, based on the construction of the graph Laplacian of the data set \cite{Coifman2006a}, is an established manifold learning technique that has found application in many areas including signal processing, image processing and machine learning \cite{Coifman:2014, David2012, Farbman:2010, Gepshtein:2013, Haddad2014, Lafon2006, Mishne2013, Singer:2009, Talmon2012}.
The theme in our paper and in the works of R. Coifman and collaborators, see \cite{Coifman2006, Lafon2006, Talmon2012}, for example, is that while many dynamical systems initially may seem to require high-dimensional spaces, coarser length and time scales normally reveal an intrinsic low dimensionality. Often, this low dimensionality can be captured by only a few variables known as the reaction coordinates. Dimension reduction as well as the derivation of complex operators based on which such systems under coarser scales evolve are, therefore, central undertakings.
\subsection{Organization of the paper} The paper is organized as follows. In Section \ref{sec:diff_maps}, we introduce the necessary background concerning diffusion maps and geometry, kernels, the Mahalanobis distance and SGD. In Section \ref{emp}, the most extensive part of the paper, we analyze the high-dimensional parameter surface in the context of two different neural network architectures and two different data sets. The two data sets used are the iris flower for a classification problem and the auto miles per gallon (MPG) for a regression problem. In Section \ref{sumu}, we summarize our results, state conclusions and discuss directions for future research.
\section{Diffusion maps and geometry}\label{sec:diff_maps}
In the following, we let $X:=\{\mathbf x_i\}_{i=1}^M=\{\mathbf{x}(t_i)\}_{i=1}^M$, where $\mathbf x_i=\mathbf{x}(t_i)\in\mathbb R^m$. We stress that the integers $(N,n,M,m)$ refer to the number of samples $(N)$ used in the definition of the loss function $f$, the number of samples $(n)$ used in the calculation of the gradient in SGD, the number of samples $(M)$ in the path(s) generated by SGD, and the dimension $(m)$ of the parameter space. These as well as other notations used throughout the paper are summarized in Table \ref{table:notation}.
{\renewcommand{\arraystretch}{1.3}
\begin{table}[hbt!]
\centering
\begin{tabular}{l|p{0.55\linewidth}}
\hline
Notation & Description \\\hline
$N$ & number of samples \\
$n$ & batch size \\
$M$ & number SGD steps \\
$m$ & dimension of parameter space \\
$\mathbf{x}\in\mathbb R^m$ & model parameters, i.e., weights of the neural network\\
$f(\mathbf{x}):\mathbb{R}^m\to\mathbb{R}$ & loss function \\
$\nabla f(\mathbf{x})$ & full gradient \\
$\nabla \widetilde{f} (\mathbf{x})$ & stochastic gradient \\
$C(\mathbf{x})$ & covariance at point $\mathbf{x}$ \\
$\varepsilon$ & diffusion map parameter \\
$d$ & dimension of lower dimensional subspace \\
$\lambda_j, j=0, \ldots, M-1$ & eigenvalues obtained through diffusion mapping \\
$\Lambda_i:=\frac{\sqrt{\lambda_1^2+\lambda_2^2+\dots+\lambda_i^2}}{\sqrt{\lambda_1^2+\lambda_2^2+\dots+\lambda_N^2}}$ & energy ratio of the eigenvalues \\
\hline
\end{tabular}
\caption{Notation table.}
\label{table:notation}
\end{table}}
\subsection{The Mahalanobis distance and SGD} Recall that if two points $\mathbf z(t_1)$ and $\mathbf z(t_2)$ are drawn from an $m$-dimensional
Gaussian distribution with covariance ${C}_z$, then the Mahalanobis distance between the points, see \cite{mahalanobis1936generalized}, is defined as
\begin{equation}
\| \mathbf z(t_1) - \mathbf z(t_2) \| _{MD} = \sqrt{ (\mathbf z(t_1) - \mathbf z(t_2))^\ast {C}_z^{-1} (\mathbf z(t_1) - \mathbf z(t_2) ) },
\end{equation}
where $^\ast$ denotes the transpose. In particular, if ${C}_z^{-1} = \mathrm{diag}(\sigma_1^{-1}, \ldots, \sigma_m^{-1})$ is a constant matrix, then
\begin{equation} \label{eq:rescale_x_dist}
\| \mathbf z(t_1) - \mathbf z(t_2) \|^2_{MD} = \sum_{i=1}^m \sigma_i^{-1} \left( z_i(t_2) - z_i(t_1) \right)^2,
\end{equation}
where $z_i(\cdot)$ denotes the $i$-th coordinate of the vector $\mathbf z(\cdot)$. Note that in \eqref{eq:rescale_x_dist}, the coordinates with large volatilities or standard deviations, determined by $\sigma_i$, make negligible contributions to the Mahalanobis distance, and these coordinates or variables may be referred to as the \textit{fast variables}. In particular, the metric can be seen as implicitly insensitive, or only modestly sensitive, to changes in the fast variables. Introducing
\begin{equation} \label{eq:general_rescale}
y_i(t):= \frac 1{\sqrt{\sigma_i}} z_i(t),
\end{equation}
the metric \eqref{eq:rescale_x_dist} can be rewritten as
\begin{equation} \label{eq:norm_z}
\| {\mathbf z}(t_2) - \mathbf z(t_1) \|^2_{MD} = \|\mathbf y(t_2) - \mathbf y(t_1) \|^2_2.
\end{equation}
Using this notation, $\mathbf y(t)$ is a stochastic process, rescaled so that each variable has unit diffusivity, with the same dimensionality as $\mathbf z(t)$. By performing this rescaling, the problem is transformed from a problem of detecting the \textit{slow variables} within dynamic data, to a problem of more traditional data mining. In particular, by construction, the Mahalanobis distance takes into account information about the dynamics and relevant time scales, enabling the use of traditional data mining techniques when used with this metric to detect the slow variables in the data \cite{singer2009detecting}.
Note that the traditional Mahalanobis distance is defined for a fixed distribution,
whereas we are dealing with a distribution that possibly changes as a function of position due to
nonlinearities in the drift term of the SGD. To account for this, $\Vert\cdot\Vert_{MD}$ for us will denote a Mahalanobis distance calculated on vectors in $\mathbb R^m$ and engineered based on the (implicit) covariance structure of SGD. Indeed, given $\mathbf{x}(t_1), \mathbf{x}(t_2)\in \mathbb R^m$, we use, see \cite{Coifman2016}, the modified Mahalanobis distance
\begin{equation}
\label{eq:mahalanobis_modified}
\Vert \mathbf{x}(t_1)-\mathbf{x}(t_2)\Vert_{MD}^2 = {\frac{1}{2}}{(\mathbf{x}(t_1)-\mathbf{x}(t_2))^\ast}{\big(C^\dagger(\mathbf{x}(t_1))+C^\dagger(\mathbf{x}(t_2))\big)}{(\mathbf{x}(t_1)-\mathbf{x}(t_2))},
\end{equation}
where $C(\mathbf x(t_j))$ is the covariance at the position/point $\mathbf x(t_j)$ and $\dagger$ denotes the Moore-Penrose pseudoinverse.
The SGD covariance at $\mathbf{x}$, where $\mathbf{x}$ is the model parameters, can be expressed as, see, for example, \cite{hoffer2017train, hu2019diffusion, smith2017bayesian, wu2020noisy, xie2021covariance},
\begin{equation}
\label{eq:sgd_cov_exact}
C(\mathbf{x}) = \frac{N-n}{n(N-1)}{\Bigg[{\frac{1}{N}}{\sum_{i=1}^N \nabla_x f_i(\mathbf{x}) \nabla_x f_i(\mathbf{x})^\ast} - \nabla_x f(\mathbf{x}) \nabla_x f(\mathbf{x})^\ast \Bigg]},
\end{equation}
the proof of which is detailed in Appendix \ref{appendix:proof_cov}. When $N$ is large, it may not be feasible to compute the full gradient for every iteration. Moreover, it has been observed, see, for example, \cite{wu2020noisy, xie2021covariance}, that near critical points, the second term in \eqref{eq:sgd_cov_exact} is dominated by the first. Hence, for final stages of optimization, the covariances can be approximated by
\begin{equation}
\label{eq:sgd_cov_approx}
C(\mathbf{x}) \approx \frac{N-n}{n(N-1)}\frac{1}{N}{\sum_{i=1}^N \nabla_x f_i(\mathbf{x}) \nabla_x f_i(\mathbf{x})^\ast}.
\end{equation}
Let \begin{equation*}
\label{eq:sgd_cov_fisher}
F(\mathbf{x}):=\frac{1}{N}{\sum_{i=1}^N \nabla_x f_i(\mathbf{x}) \nabla_x f_i(\mathbf{x})^\ast}
\end{equation*}
be the Fisher information matrix. For $N \gg n$, $\frac{N-n}{N-1} \approx 1$, the approximation in \eqref{eq:sgd_cov_approx} simplifies even further and one can obtain, see \cite{zhu2018anisotropic}, that the SGD covariance is approximately proportional to the Hessian $H(\mathbf{x})$ of the loss function
\begin{equation}
\label{eq:sgd_cov_hessian}
C(\mathbf{x}) \approx \frac{1}{n}F(\mathbf{x}) \approx \frac{1}{n}{H(\mathbf{x})}.
\end{equation}
In practice, the covariance matrix can also be estimated from a short trajectory of samples in time around the sample $\mathbf{x}(t)$ by
\begin{equation}
\widehat{{C}}(\mathbf{x}(t)) = \sum \limits _{\tau = t-L}^{t+L} (\mathbf{x}(\tau) - {\mu}(t))(\mathbf{x}(\tau) - \mu(t))^\ast,
\label{eq:cov}
\end{equation}
where $\mu(t)$ is the empirical mean of the short trajectory of samples, and $2L$ is the length of the trajectory.
\subsection{Diffusion kernels and maps}\label{sec:diffusion_kernelmap} The starting point for the construction of the diffusion maps is a symmetric and non-negative kernel $k=k(\cdot,\cdot):\mathbb R^m\times\mathbb R^m\to \mathbb R$. While many kernels satisfy this property, we will, in this paper, mainly use the kernel
\begin{eqnarray}
\label{k_epsilon}
k(\mathbf{x}_i,\mathbf{x}_j):=k_\varepsilon(\mathbf{x}_i,\mathbf{x}_j):=\exp\left(-\Vert \mathbf{x}_i-\mathbf{x}_j\Vert^2/\varepsilon\right).
\end{eqnarray}
Here, $\varepsilon>0$ is a global scale parameter, a degree of freedom, and $\Vert\cdot\Vert$ could, in principle, be any relevant distance function. For us, $\Vert\cdot\Vert$ will denote the Mahalanobis distance $\Vert\cdot\Vert_{MD}$ introduced in \eqref{eq:mahalanobis_modified}, calculated on vectors in $\mathbb R^m$ and engineered based on the (implicit) covariance structure of SGD.
Given the set $X:=\{\mathbf{x}_i\}_{i=1}^M=\{\mathbf{x}(t_i)\}_{i=1}^M$ of data points, we construct a weighted graph with the data points as nodes. Given the edge connecting two nodes $\mathbf{x}_i, \mathbf{x}_j \in X$, we let the weight of the edge be equal to $k(\mathbf{x}_i,\mathbf{x}_j)=k_\varepsilon(\mathbf{x}_i,\mathbf{x}_j)$. In this context, $k(\mathbf{x}_i,\mathbf{x}_j)$ should be seen as a measure of similarity between the data points $\mathbf{x}_i, \mathbf{x}_j \in X$. Based on $X$ and $k$, we introduce a $M\times M$ dimensional matrix $K$ with entries ${K}[i,j] = {K}_\varepsilon[i,j] :=k(\mathbf{x}_i,\mathbf{x}_j)$. In practice, ${K}$ can often be computed using only the nearest neighbors of every point. In this case, ${K}[i,j]$ is defined to be zero for every $\mathbf{x}_j$ which is not among the nearest neighbors of $\mathbf{x}_i$. Naturally, a notion of nearest neighbors then has to be defined.
To construct an approximation of the Laplace-Beltrami operator on the data set, we first use a normalization of the data set; this is a natural preprocessing step and is necessary to ensure that the embeddings to be constructed do not rely on the distribution of the points~\cite{Coifman2006,Lafon2006}. Let $D$ be a $M\times M$ dimensional diagonal matrix with ${D}[i,i]:=\sum_{\mathbf{x}_j\in X}k(\mathbf{x}_i,\mathbf{x}_j)$. We then introduce a normalized matrix ${\widetilde{K}}$ with entries ${\widetilde{K}}[i,j]$,
\begin{equation*}
\label{eq:ln_norm}
{\widetilde{K}} = {D}^{-1/2}{K}{D}^{-1/2}.
\end{equation*}
Based on ${\widetilde{K}}$ we also introduce
\begin{equation}
\label{eq:random_walk}
{P} := {\widetilde{D}}^{-1}{\widetilde{K}}, \;\;\; {\widetilde{D}}[i,i]:=\sum_{j}{\widetilde{K}}[i,j].
\end{equation}
The row-stochastic matrix ${P}$ satisfies ${P}[i,j]\geq0$ and $\sum_{j}{P}[i,j]=1$ and, therefore, can be viewed as the transition matrix of a Markov chain on the data set $X$. ${P}$ has a sequence of biorthogonal left and right eigenvectors, $\phi_\ell$ and $\psi_\ell$, respectively, and a sequence of positive eigenvalues $\{\lambda_j\}_{j=0}^{M-1}$ satisfying $1 = |\lambda_0|\geq|\lambda_1|\geq ...$. Using this notation and introducing
\begin{equation}
\label{eq:eigen_decompose}
p_\tau(\mathbf{x}_i,\mathbf{x}_j):=\sum_{\ell\geq 0} \lambda^\tau_\ell\psi_\ell(\mathbf{x}_i)\phi_\ell(\mathbf{x}_j),\ \tau\geq 0,
\end{equation}
we can interpret $p_\tau(\mathbf{x}_i,\mathbf{x}_j)$ as the probability that the Markov chain, starting at $\mathbf{x}_i$ at $\tau=0$, is at $\mathbf{x}_j$ after $\tau$ steps.
We introduce a distance $d(\mathbf{x}_i,\mathbf{x}_j,\tau)$ between two points $\mathbf{x}_i,\mathbf{x}_j \in X$,
\begin{equation}
\label{eq:diffusion_distance1}
d(\mathbf{x}_i,\mathbf{x}_j,\tau ) = \sum_{\mathbf{x}_k\in X}\frac{\big(p_\tau(\mathbf{x}_i,\mathbf{x}_k)-p_\tau(\mathbf{x}_j,\mathbf{x}_k)\big)^2}{\phi_0(\mathbf{x}_k)} = \sum_{\ell\geq1} \lambda^{2\tau}_\ell(\psi_\ell(\mathbf{x}_i)-\psi_\ell(\mathbf{x}_j))^2.
\end{equation}
Here $\phi_0$ denotes the stationary probability distribution on the graph. $d(\mathbf{x}_i,\mathbf{x}_j,\tau )$ is referred to as the diffusion distance between $\mathbf{x}_i$ and $\mathbf{x}_j$ at step/time $\tau$. The distance function/metric constructed is robust to noise, as the distance between any two points is a function of all possible paths of length $\tau$ between the points. The diffusion distance can, as a consequence of the decay of the spectrum of ${P}$, be approximated using only the first, say $d$, eigenvectors. Furthermore, as a consequence of \eqref{eq:diffusion_distance1}, a mapping between the original space and the eigenvectors $\psi_\ell$ can be defined. Indeed, if one only keeps the first $d$ eigenvectors, then the data set $X$ gets embedded into the Euclidean space $\mathbb{R}^{d}$ through the map $\Psi_\tau$. In this embedding, the diffusion distance is equal to the Euclidean distance:
\begin{equation}
\label{eq:diffusion_map}
\Psi_\tau:\mathbf{x}_i\rightarrow \big( \lambda_1^\tau\psi_1(\mathbf{x}_i), \lambda_2^\tau\psi_2(\mathbf{x}_i),..., \lambda_{d}^\tau\psi_{d}(\mathbf{x}_i)\big)^\ast.
\end{equation}
As $\psi_0$ is a constant vector, $\psi_0$ is not used in \eqref{eq:diffusion_map}.
\section{Empirical Investigation}\label{emp}
To analyze the high-dimensional parameter surface in practice using the diffusion maps discussed in Section \ref{sec:diff_maps}, and based on the distance function in \eqref{eq:mahalanobis_modified}, we have conducted empirical investigations using different neural network architectures and two different data sets\footnote{Code available on GitHub.}. The two data sets used are the iris flower for a classification problem and the auto miles per gallon (MPG) for a regression problem. The iris flower data set is a collection of 150 data samples of different iris flowers. Each data sample contains four features: petal length, petal width, sepal length, and sepal width. Based on these four features, the samples are classified into three classes of iris species: setosa, versicolor, and virginica. Figure \ref{fig:iris} plots the data points based on pairwise combinations of the features.
\begin{figure}[!htbp]
\centering
\includegraphics[width=0.6\textwidth]{pic_iris.png}
\caption{Iris data set. Four features: petal length, petal width, sepal length, and sepal width. Three classes: setosa, versicolor, and virginica. Off-diagonal graphs are scatterplots of all samples based on pairwise combinations of the features shown in the x- and y-axes. Diagonal graphs are density estimates of the three classes for the particular feature shown in the x-axis.}
\label{fig:iris}
\end{figure}
The auto MPG data set is a collection of 398 data samples of different cars. Each sample contains eight attributes: number of cylinders, displacement (or engine size), horsepower, weight, acceleration, model year, origin, and fuel consumption measured in miles per gallon (mpg). The first seven attributes are then used to predict the fuel consumption. Figure \ref{fig:autompg} plots the data points based on pairwise combinations of four of the attributes.
\begin{figure}[!hbtp]
\centering
\includegraphics[width=0.6\textwidth]{pic_autompg.png}
\caption{Auto MPG data set. Eight attributes (those in bold are shown in the figure): \textbf{number of cylinders}, displacement, \textbf{horsepower}, \textbf{weight}, acceleration, model year, origin, and \textbf{fuel consumption (mpg)}. Off-diagonal graphs are scatterplots of all samples based on pairwise combinations of the attributes shown in the x- and y-axes. For example, looking at MPG vs. Horsepower, one can see that, in general, more power means higher fuel consumption. Diagonal graphs are histograms of the data samples for the attribute shown in the x-axis. For example, one can see that most of the samples have four cylinders, while very few have three and five.}
\label{fig:autompg}
\end{figure}
To maintain focus on the optimization landscape, the neural networks were designed with basic architecture. Layer activations were ReLu, except for the output layer of the regression problem, which had a softmax activation. Dropouts were not used. The optimizer was SGD; however, it should be stressed here that as our focus is on understanding the parameter space, any optimization algorithm could have also been chosen, including gradient descent. As the neural networks were trained, model parameters, i.e., weights and biases, for every iteration $i$ were extracted to create the data sets $X:=\{\mathbf{x}_i\}_{i=1}^M$ containing the points in $\mathbb R^m$ that the optimizer has visited.
\subsection{SGD covariance}
Although SGD was used for optimization, with only 150 samples for the iris flower data set and 398 samples for the auto MPG data set, it was feasible to compute the full gradients in order to calculate the exact SGD covariances. Because of this, we were able to use \eqref{eq:sgd_cov_exact} as the covariances for the Mahalanobis distance in \eqref{eq:mahalanobis_modified} rather than the approximation in \eqref{eq:sgd_cov_approx}, which would have required us to assess where the critical points are and/or would have limited our analysis to data points after convergence.
Figure \ref{fig:iris_covariance} shows (a) a section of the SGD correlation matrix for the iris flower classification problem at iteration $i = 20$, i.e., at data point $\mathbf{x}_{20}$, and (b) a histogram of the eigenvalues of the full covariance matrix at the same iteration.
\begin{figure}[htbp!]
\centering
\begin{subfigure}{.45\textwidth}\label{fig:iris_corr_matrix}
\centering
\includegraphics[trim=0cm 0cm 0.5cm 0cm, width=0.9\linewidth]{pic_corr_matrix_iris_L2.png}
\caption{}
\end{subfigure}%
\begin{subfigure}{.45\textwidth}\label{fig:iris_cov_EV}
\centering
\includegraphics[trim=0.5cm 0cm 0cm 0cm, width=0.9\linewidth]{pic_cov_EV_iris_L2.png}
\caption{}
\end{subfigure}
\caption{(a) SGD correlation matrix for classification problem (iris data set) at iteration 20 (showing only a 10x10 section of the matrix to zoom in on details). (b) Eigenvalues of the SGD covariance matrix at iteration 20. Note that y-axis is in log scale.}
\label{fig:iris_covariance}
\end{figure}
Figure \ref{fig:autompg_covariance}, on the other hand, shows the correlation and histogram of eigenvalues for the auto MPG regression.
\begin{figure}[htbp!]
\centering
\begin{subfigure}{.45\textwidth}\label{fig:autompg_cov_matrix}
\centering
\includegraphics[trim=0cm 0cm 0.5cm 0cm, width=0.9\linewidth]{pic_corr_matrix_autompg_L2.png}
\caption{}
\end{subfigure}%
\begin{subfigure}{.45\textwidth}\label{fig:autompg_cov_EV}
\centering
\includegraphics[trim=0.5cm 0cm 0cm 0cm, width=0.9\linewidth]{pic_cov_EV_autompg_L2.png}
\caption{}
\end{subfigure}
\caption{(a) SGD correlation matrix for regression problem (auto MPG data set) at iteration 5 (showing only a 10x10 section of the matrix to zoom in on details). (b) Eigenvalues of the SGD covariance matrix at iteration 5. Note that y-axis is in log scale.}
\label{fig:autompg_covariance}
\end{figure}
10x10 sections were used for the SGD correlation figures instead of the full matrices in order to zoom in on the details and more clearly illustrate the differences in values. These correlation figures, together with the histograms of the covariance eigenvalues, clearly show that the SGD correlations and covariances do not approximate the identity matrix, thus, justifying our choice of using the Mahalanobis distance. If the covariances were close to the identity, this would imply that the Euclidean distance would have been sufficient.
It is also worth noting that, as the covariance is approximately proportional to the Hessian, see \eqref{eq:sgd_cov_hessian}, the values observed for the correlation matrix and covariance eigenvalues, as well as the behavior of the eigenvalues to be concentrated around zero while only a small number of eigenvalues are large, are consistent with the findings of \cite{sagun2017empirical} where the eigenvalues of the Hessian of SGD are examined. These few eigenvalues with large magnitudes represent the principal directions, and are, therefore, the ones of interest.
\subsection{Dimension of subspace}
\label{sec:dim_subspace}
In \cite{Coifman2006}, the number of significant eigenvalues $s$ is defined through a preset accuracy $\alpha > 0$ on which $s$ depends,
\begin{equation} \label{eq:coifman_dim}
s(\alpha) := \max\{l\in \mathcal N: {|\lambda_l|} > \alpha{|\lambda_1|}\}.
\end{equation}
$s=s(\alpha)$ is then interpreted as the dimension of the underlying slow manifold. The original data set is embedded into the new $s(\alpha)$-dimensional subspace through the mapping in \eqref{eq:diffusion_map}, where the diffusion distance is equal to the Euclidean distance up to the relative accuracy $\alpha$. That is, with $\tau = 1$ in \eqref{eq:diffusion_distance1},
\begin{equation}
\label{eq:diffusion_distance_approx}
d(\mathbf{x}_i,\mathbf{x}_j) = \Bigg(\sum_{l=1}^{s(\alpha)} \lambda^{2}_\ell(\psi_\ell(\mathbf{x}_i)-\psi_\ell(\mathbf{x}_j))^2\Bigg)^{1/2}.
\end{equation}
$\alpha$ is a parameter that needs to be selected, where a smaller $\alpha$ leads to higher dimensions of the slow manifold and higher accuracy in \eqref{eq:diffusion_distance_approx}.
There is, however, not one unique way to define the intrinsic dimension. In our approach, we look at what we refer to as the energy ratio, $\Lambda_i$, defined as
\begin{align}
\Lambda_i:=\frac{\sqrt{\lambda_1^2+\lambda_2^2+\dots+\lambda_i^2}}{\sqrt{\lambda_1^2+\lambda_2^2+\dots+\lambda_N^2}}.
\end{align}
This ratio quantifies the dominance of the first $i$ eigenvalues by comparing their energy with the total energy of all eigenvalues. Figures \ref{fig:ev_iris&auto} and \ref{fig:ev_inits} in Section \ref{sec:results} show graphs of $\Lambda_i$ for different neural networks models. We define the number of significant eigenvalues, denoted by $d$, i.e., the dimension of the underlying lower dimensional subspace, to be the first $d$ eigenvalues such that $\Lambda_d>\beta$. A lower $\beta$ results in lower dimensions, and we, in the interest of employing a strict criteria, choose $\beta = 0.99$. Looking at $\Lambda_i$ and defining the significant eigenvalues as such captures the spectrum decay of the matrix $P$ (see Section \ref{sec:diffusion_kernelmap}), as does \eqref{eq:coifman_dim}, upon which the dimension of the subspace depends. In addition, we also examine the proportion of the area under the eigenvalue curve accounted for by the dominant $d$ eigenvalues. We refer to this as the AUC ratio and it can be interpreted as the explanatory capability of, or the amount of information contained in, the lower dimensional subspace in comparison to the original space.
\subsection{Diffusion map parameter $\varepsilon$}\label{sec:diff_map_epsi}
A significant parameter in the implementation of diffusion maps is the scale parameter $\varepsilon$ used in the definition of the diffusion kernel in \eqref{k_epsilon}. It represents a characteristic distance in the data and defines the local neighborhood within which we can rely on the accuracy of our metric (Mahalanobis distance in this case). Results can vary tremendously depending on its setting. Despite the importance of the parameter and the sensitivity of results, there is no agreed upon scheme as to how the appropriate range of values should be decided. Instead, the choice is dependent upon the problem and the data structure, resulting in different methods being proposed. For example, in \cite{LafonThesis}, $\varepsilon$ is set to be
\begin{equation*}
\label{eq:lafon_epsilon}
\varepsilon = \frac{1}{M}{\sum_{i=1}^M \min_{j:j \neq i}\Vert \mathbf{x}(t_i)-\mathbf{x}(t_j)\Vert^2},
\end{equation*}
which is the average of the shortest distance from each data point. Implementing this on our data using the Mahalanobis distance, however, resulted in an $\varepsilon$ that was too small compared to other values of $\Vert \mathbf{x}(t_2)-\mathbf{x}(t_1)\Vert_{MD}^2$. There were very few, if any, data points within the ball of radius $\varepsilon$, and many entries of the $K$ matrix were almost zero. In \cite{Coifman2016}, the authors looked at the error $E_{MD}(\mathbf{y}(t_1),\mathbf{y}(t_2))$ incurred by using the Mahalanobis distance on the data points $\mathbf{y}(t) = f(\mathbf{x}(t))$ in approximating the $L_2$-distance of the underlying variables. $\mathbf{x}(t)$ at times $t_1,\ldots, t_n$ are the samples of the stochastic system. The criteria used is that $\varepsilon$ should be in the order of $\Vert \mathbf{x}(t_2)-\mathbf{x}(t_1)\Vert_{MD}^2$ in the region where $|E_{MD}(\mathbf{x}(t_1),\mathbf{x}(t_2)| \ll \Vert \mathbf{x}(t_2)-\mathbf{x}(t_1)\Vert_{MD}^2$. Choosing $\varepsilon$ as such ensures that the curvatures and nonlinearities captured in the error term are negligible. This method, however, is inapplicable for our investigation as we are interested in the stochastic variable itself, not in the underlying variable, and thus we have no error term to consider. On the other hand, in \cite{bah2008diffusion, coifman2008graph, singer2009detecting}, the authors calculate the matrix $K(\varepsilon)$ for a wide range of $\varepsilon$ values and compute the sum $L(\varepsilon)$ of the entries for each matrix:
\begin{equation*}
\label{eq:singer_epsilon}
L(\varepsilon) = \sum_{i,j} K_\varepsilon[i,j].
\end{equation*}
An $\varepsilon$ that is too small compared to $\Vert \mathbf{x}(t_2)-\mathbf{x}(t_1)\Vert_{MD}^2$ will result in a lower value for $L(\varepsilon)$, since the entries for the matrix $K$ will be close to zero, indicating little to no diffusion. In contrast, an $\varepsilon$ that is too large compared to $\Vert\mathbf{x}(t_2)-\mathbf{x}(t_1)\Vert_{MD}^2$ will result in a larger $L(\varepsilon)$, as the entries of $K$ will be close to one, indicating that diffusion has already taken place. Since neither of these scenarios are interesting for diffusion maps, $\varepsilon$ should be chosen in the region between. In \cite{coifman2008graph}, assuming that the data points lie on a low-dimensional manifold $\mathcal{M}$ with finite volume, it is argued that the sum $L(\varepsilon)$ is approximated by the mean value integral. That is,
\begin{equation}
L(\varepsilon) = \sum_{i,j} K_\varepsilon[i,j] = \sum_{i,j} \exp\left(- \frac{\Vert\mathbf{x}_i-\mathbf{x}_j\Vert^2}{\varepsilon}\right) \approx \frac{N^2}{\textrm{vol}^2(\mathcal{M})} \int_\mathcal{M} \int_\mathcal{M} \exp\left(- \frac{\Vert\mathbf{x}-\mathbf{y}\Vert^2}{\varepsilon}\right)d\mathbf{x}d\mathbf{y}.
\end{equation}
Since the manifold $\mathcal{M}$ looks like its tangent space $\mathbb{R}^d$ locally,
\begin{align}
\begin{split}
\frac{N^2}{\textrm{vol}^2(\mathcal{M})} \int_\mathcal{M} \int_\mathcal{M} \exp\left(- \frac{\Vert\mathbf{x}-\mathbf{y}\Vert^2}{\varepsilon}\right)d\mathbf{x}d\mathbf{y}
&\approx \frac{N^2}{\textrm{vol}^2(\mathcal{M})} \int_\mathcal{M} \int_{\mathbb{R}^d} \exp\left(- \frac{\Vert\mathbf{x}-\mathbf{y}\Vert^2}{\varepsilon}\right)d\mathbf{x}d\mathbf{y} \\
&= \frac{N^2}{\textrm{\textrm{vol}}(\mathcal{M})}(\pi\varepsilon)^{d/2}.
\end{split}
\end{align}
Taking the logarithm,
\begin{equation}
\label{eq:linear_epsilon}
\textrm{log}L(\varepsilon) \approx \frac{d}{2}\textrm{log}\varepsilon + \textrm{log}\left(\frac{N^2\pi^{d/2}}{\textrm{vol}(\mathcal{M})}\right).
\end{equation}
Here, $\textrm{vol}(\mathcal{M})$ is the volume of the manifold. The logs of $L(\varepsilon)$ and $\varepsilon$ are, therefore, connected by an approximately straight line whose slope is $d/2$, where $d$ is the dimension of the lower-dimensional manifold. The authors suggest to choose an $\varepsilon$ within this linear region. \cite{berry2016variable} extends this approach further by setting $\varepsilon$ to be where the local slope $a_i$, given approximately by
\begin{equation}
\label{eq:dim_slope}
a_i \approx \frac{\textrm{log}(L(\varepsilon_{i+1})) - \textrm{log}(L(\varepsilon_i))}{\textrm{log}(\varepsilon_{i+1}) - \textrm{log}(\varepsilon_i)},
\end{equation}
is maximized. In this case, the slope $d/2 \approx \textrm{max}\{a_i\}$, and, hence, the dimension of the manifold is given by $d \approx 2\textrm{max}\{a_i\}$.
Figure \ref{fig:epsi_line&a} shows the results of implementing the criteria of \cite{coifman2008graph, singer2009detecting, berry2016variable} for our data set $X:=\{\mathbf{x}_i\}_{i=1}^M$ of points in $\mathbb R^m$ visited by SGD. The iris flower classification problem with a two-hidden layer neural network was used, with an original parameter space dimension of $m = 515$. Section \ref{sec:class_vs_reg} describes this neural network in more detail.
\begin{figure}[htbp!]
\centering
\includegraphics[width=1\textwidth]{pic_epsi_iris_G_a.png}
\caption{(a) As per $\varepsilon$-choosing criteria of \cite{coifman2008graph, singer2009detecting}. Log-log plot of $\varepsilon$ against $L(\varepsilon)$ showing linear region (shaded in blue) within which $\varepsilon$ should be chosen. (b) As per $\varepsilon$-choosing criteria of \cite{berry2016variable}. Graph of $\varepsilon$ and slope $a$. Green dot shows where the slope is maximized, corresponding to $a = 1.07$. Red dots in both graphs show the actual chosen $\varepsilon = 3$. From (a), it shows that $\varepsilon = 3$ is within accepted region, and from (b), although not the maximum, $\varepsilon = 3$ still gives a high value of $a$.}
\label{fig:epsi_line&a}
\end{figure}
Figure \ref{fig:epsi_line&a}(a) is the log-log plot of $\varepsilon$ against $L(\varepsilon)$, showing where $L(\varepsilon)$ grows linearly with $\varepsilon$. According to \cite{coifman2008graph} and \cite{singer2009detecting}, this is the optimal region within which $\varepsilon$ should be chosen. Figure \ref{fig:epsi_line&a}(b) shows $\varepsilon$ with corresponding slopes $a$. The green dot is where the maximum is attained, with $a = 1.07$ and $\varepsilon = 0.06$. Based on \cite{berry2016variable}'s definition, the dimension of the lower dimensional subspace of the parameter space is then $d = 2$. To err on the conservative side, this result appeared to be very optimistic. In addition, if we, instead, use our definition of dimension in Section \ref{sec:dim_subspace}, using $\varepsilon = 0.06$ in the diffusion maps resulted in $d = 868$, meaning that not even the dimension of the original parameter space was recovered.
As none of these methods showed reasonable results for our problem, we instead modified. Instead of choosing $\varepsilon$ where the slope is maximized, we instead studied the range of possible values of $\varepsilon$ in the linear region showed in Figure \ref{fig:epsi_line&a}(a). Figure \ref{fig:epsi_iris_zoomed} shows the AUC ratios and dimensions, as defined in Section \ref{sec:dim_subspace}, for these values.
\begin{figure}[htbp!]
\centering
\includegraphics[width=1\textwidth]{pic_epsi_iris_zoomed.png}
\caption{(a) Dimension of slow manifold (in blue and left y-axis) and AUC ratio (in orange and right y-axis) as functions of $\varepsilon$. (b) Scatterplot of different values of $\varepsilon$, showing resulting dimension in x-axis and AUC ratio in y-axis from application of diffusion maps. The color bar shows the range of values for the $\varepsilon$'s. $\varepsilon$'s in the upper left corner are desirable due to low-dimension and high AUC ratio. Several of these points are marked simply for reference.}
\label{fig:epsi_iris_zoomed}
\end{figure}
The results presented in this section are those of the iris flower classification problem, although the same approach was also applied for the auto MPG regression. As $\varepsilon$ increases, the dimension of the underlying subspace decreases sharply in the beginning, then "flattens out" to converge to 1, as shown in Figure \ref{fig:epsi_iris_zoomed}(a). The AUC ratio also decreases as $\varepsilon$ increases, which is expected as the dimensions decrease. However, the decrease in AUC ratio is not nearly as abrupt as that of the dimensions. The decrease is subtle for smaller values of $\varepsilon$ and gradually becomes steeper as $\varepsilon$ increases. The sudden decline in dimensions coupled with just a slight decrease in AUC ratio indicates that the decline in dimension is the result of better detection of the lower dimensional subspace due to better parameterization of the data, and that these detected lower dimensional subspaces do, in fact, account for a lot of the information in the original optimization landscape. We then choose $\varepsilon$ based on Figure \ref{fig:epsi_iris_zoomed}(b), where values on the upper left corner are desired as they result in a combination of lower dimensions and higher AUC ratios. After careful assessment, the value of the parameter was decided to be $\varepsilon = 3$ for the classification problem and $\varepsilon = 0.55$ for the regression problem. Figure \ref{fig:epsi_line&a} shows $\varepsilon = 3$ marked as a red dot in order to show that this choice does indeed fall within the linear region and, although not the maximum, does still correspond to a high slope $a$. Note that as there is not one optimal value but rather a range of accepted values, other surrounding values of the same order would have also been suitable. With these values of $\varepsilon$, the diffusion maps were applied to the data set $X$. The eigenvalues were calculated as described in Section \ref{sec:diffusion_kernelmap}, and sorted in descending order $\lambda_1\geq \lambda_2\ge \dots$.
\subsection{Results} \label{sec:results}
In our empirical investigations, we tried to ascertain whether or not the high-dimensional parameter surface does indeed have an underlying low-dimensional manifold in which the process of minima selection occurs, and, if so, determine what variables affect the dimensions of this manifold. Due to the randomness of SGD, the experiments were repeated multiple times, and the resulting dimensions as well as AUC ratios have proven to be stable throughout. For example, 30 different runs of the classification problem described in Section \ref{sec:class_vs_reg} consistently detected a subspace of dimension $d = 17$ with $AUC \approx 0.91$. The results are presented in detail in the following sections.
\subsubsection{Classification and regression} \label{sec:class_vs_reg}
To first possibly detect the lower dimensional subspace, standard neural networks were applied to the iris flower and auto MPG data sets. Using these data sets allowed us to look at both classification and regression problems, and thus, different loss functions. Categorical cross entropy (CCE) was the loss function for the classification problem
\begin{equation}
\label{cross_ent}
f(\mathbf{x})=-\frac1{N} \sum\nolimits_{i=1}^N\sum\nolimits_{j=1}^c y_{ij}\log(\hat{y}_{ij}),
\end{equation}
where $c$ is the number of classes, and $y_{ij}$ and $\hat{y}_{ij}$ are the true and predicted labels for sample $i$ and class $j$, respectively. Mean absolute error (MAE) was the loss function for the regression,
\begin{equation}
\label{mae}
f(\mathbf{x}) = \frac1{N} \sum\nolimits_{i=1}^N |y_{i}-\hat{y}_{i}|,
\end{equation}
where $y_{i}$ are the true labels, and $\hat{y}_i$ are the predicted labels. For comparability, the architectures were designed to be as similar as possible. Both had two hidden layers with neurons 24 and 14 for the iris flower, and 21 and 13 for the auto MPG. These were chosen so that the neural networks would have similar width and depth, as well as the same parameter space dimension of $m = 515$. In addition, both were trained for the same batch size of $n = 20$, with 400 epochs for the iris and 150 for the auto MPG so as to have the same number $M$ of SGD steps. Table \ref{table:ev_iris&auto} summarizes these setup together with the results.
Figure \ref{fig:ev_iris&auto} plots the eigenvalues $\lambda_i$ and energy ratios $\Lambda_i$ resulting from the application of diffusion maps. The dimensions $d$ of the subspaces as well as the AUC ratios are annotated in the graphs.
\begin{figure}[htbp!]
\centering
\includegraphics[width=1\textwidth]{pic_ev_iris_auto.png}
\caption{(a) Eigenvalues $\lambda_i$ for the iris flower classification and auto MPG regression problems. AUC ratios and dimensions of lower subspaces annotated. (b) Energy ratios $\Lambda_i$ for the classification and regression problems. For both, as $i$ increases, $\lambda_i \to 0$ and $\Lambda_i \to 1$.}
\label{fig:ev_iris&auto}
\end{figure}
For both models, only a small number of eigenvalues are actually dominant, while the others may be considered insignificant. As the eigenvalues indicate the importance of their associated direction, this result does indeed support the hypothesis that the SGD optimizer moves in a lower-dimensional subspace. For the iris flower classification, the dimension of this subspace appears to be $d=17$, while for the auto MPG regression, it is $d=18$. The classification problem has a slightly higher AUC ratio of 0.9147, with the regression having 0.9050. Both of these ratios indicate high amounts of information in these lower dimensional subspaces.
\begin{table}[htbp!]
\centering
\begin{tabular}{c|cccccccc}
\hline
Model & Hidden Layers & Neurons & $N$& $n$ & $M$ &$m$ & $d$ & AUC Ratio
\\\hline
Iris flower classification & 2 & 24x14& 150&20& 2400 & 515 & 17 & 0.9147 \\
Auto MPG regression & 2 & 21x13& 398&20& 2400 & 515 & 18 & 0.9050 \\
\hline
\end{tabular}
\caption{Summary of the number of samples $N$, batch size $n$, number of SGD data points $M$, dimension of original parameter space $m$, dimension of subspace $d$ detected by diffusion mapping, and AUC ratios.}
\label{table:ev_iris&auto}
\end{table}
It is also worth noting that the graphs in Figure \ref{fig:ev_iris&auto} are cut off at $i = 40$ in order to zoom in on the details. Beyond $i = 40$, $\lambda_i \to 0$ and $\Lambda_i \to 1$. This means that, when graphed together with all the eigenvalues, a very sharp decay in the spectrum can be observed. Figure \ref{fig:ev_iris&auto_decay} graphs the eigenvalues up until $i = 350$ to illustrate this point.
\begin{figure}[hbtp!]
\centering
\includegraphics[width=0.5\textwidth]{pic_ev_iris_auto_decay.png}
\caption{Eigenvalues resulting from diffusion maps graphed until $i = 350$ to show that they exhibit sharp decay.}
\label{fig:ev_iris&auto_decay}
\end{figure}
\subsubsection{Batch size}
With the lower dimensional manifold being detected in Section \ref{sec:class_vs_reg}, it is interesting to examine what variables influence its dimensions. The batch size is known to be an important hyperparameter to tune for neural network models. To analyze its effect on the dimension, the iris flower classification problem described in Section \ref{sec:class_vs_reg} was used. The batch size was varied from 10 to 120, where 120 corresponds to a full gradient descent\footnote{The iris flower data set has 150 samples, but only 120 was used for training as the rest was used for validation.}. Batch sizes by which the number of training data is divisible were intentionally chosen to ensure no batches with remainders are left at the end of epochs. Results, displayed in Figure \ref{fig:batch_depth}(a) and Table \ref{table:batch}, show that as the batch size is varied, the dimension of the subspace and AUC ratios remain the same. Also, the dimension of the noisy SGD is equal to the dimension of the full gradient descent, indicating the dimension's robustness to noise. This is because even though decreasing the batch size increases the noise in the SGD, the noise still have the same directions, just different magnitudes. Hence, the slow variables in the parameter surface, and, thus, the dimension of the subspace, are unchanged.
\begin{table}[hbt!]
\centering
\begin{tabular}{c|cc}
\hline
Batch size ($n$) & $d$ & AUC Ratio \\\hline
10 & 17 & 0.9140\\
20 & 17 & 0.9147 \\
30 & 17 & 0.9145 \\
40 & 17 & 0.9147 \\
60 & 17 & 0.9147 \\
120 (full batch) & 17 & 0.9147 \\
\hline
\end{tabular}
\caption{Subspace dimensions $d$ and AUC ratios for varying batch size $n$. Results show that dimension and AUC ratio remain consistent regardless of batch size.}
\label{table:batch}
\end{table}
\subsubsection{Neural network depth}
The dimension of the subspace as a function of depth was also examined. Once again, the iris flower classification problem from was used. Layers and neurons were adjusted to increase depth, while keeping the number of model parameters $m$ similar. The results are graphed in Figure \ref{fig:batch_depth}(b) and summarized in Table \ref{table:depth}.
\begin{figure}[hbt!]
\centering
\includegraphics[width=1\textwidth]{pic_batch_depth.png}
\caption{(a) Subspace dimension vs. batch size. Dimension remains consistent regardless of batch size. (b) Subspace dimension vs. number of hidden layers. Dimension fluctuates around $d = 17$.}
\label{fig:batch_depth}
\end{figure}
\begin{table}[htb!]
\centering
\begin{tabular}{c|p{0.26\linewidth}cccc}
\hline
Hidden Layer & Neurons & $m$ & $d$ & AUC Ratio & Loss \\\hline
1 & 64 & 515 & 22 & 0.9145 &0.0987\\
2 & 24, 14 & 515 & 17 & 0.9147 &0.0686\\
3 & 12, 18, 10 & 517 & 14 & 0.9044 &0.0594\\
4 & 10, 14, 12, 8 & 515 & 23 & 0.9146 &0.0641\\
5 & 9, 10, 13, 10, 6 & 515 & 12 & 0.9067 &0.0514\\
6 & 6, 8, 12, 12, 8, 5 & 517 & 21 & 0.9120 &0.0502\\
7 & 5, 8, 9, 11, 9, 8, 5 & 515 & 13 & 0.9019 &0.0513\\
8 & 5, 7, 9, 10, 9, 8, 6, 4 & 515 & 18 & 0.9108 &0.1060\\
9 & 5, 6, 7, 8, 10, 8, 7, 6, 5 & 516 & 14 & 0.9006 &0.0434\\
10 & 4, 5, 6, 8, 9, 9, 8, 6, 5, 4 & 516 & 21 & 0.9151 &0.4665\\
\hline
\end{tabular}
\caption{Dimension of subspace $d$ and AUC ratios for different number of hidden layers. Loss after 400 epochs also displayed.}
\label{table:depth}
\end{table}
One can see that the depth of the neural network does have an influence on the dimension. However, it is unclear as to what the relationship is between the two. The dimensions seem to fluctuate around $d = 17$ as the hidden layer is increased. This may indicate that the intrinsic dimensionality of the manifold in which SGD moves around depends more on the data set rather than the neural network architecture. It also appears as though the suitability of the number of hidden layers does not affect the dimension either. The loss column in \ref{table:depth} was included to illustrate this point. The neural network with ten hidden layers, for example, has a much higher loss than all other models in the table after being trained for the same number of epochs. This higher loss signifies that this neural network model is ill-designed for the problem. However, the detected dimension is still $d = 21$, the same as the six-hidden neural network that performed better in training. Again, this supports the conjecture that the dimensionality relies more on the data set, and is therefore insensitive, to some degree, to depth or suitability of network architecture.
Another interesting observation is that, apart from the one-hidden layer model, all other odd-hidden-layered models appear to have lower dimensions that those that are even. A more thorough investigation, however, needs to be conducted in order to make any conclusions.
\subsubsection{Weight Initializations}
In the preceding sections, the weight initializations were kept to be Keras' default Glorot Uniform\footnote{Uniform Distribution$[-limit,limit]$, where $limit = \sqrt{\frac{6}{fan\_in + fan\_out}}$. $fan\_in$ and $fan\_out$ are the numbers of input and output units to the weight tensor, respectively.} for consistency. However, weight initialization is also an interesting variable to study when determining possible factors of dimension. As before, the iris flower with two hidden layers described in Section \ref{sec:class_vs_reg} was used, and different weight initializations available in Keras were implemented. Figure \ref{fig:ev_inits} plots the eigenvalues and energy ratios.
\begin{figure}[hbp!]
\centering
\includegraphics[trim=0cm 0cm 0cm 0cm, width=\textwidth]{pic_ev_inits.png}
\caption{Eigenvalues $\lambda_i$ and energy ratios $\Lambda_i$ for different weight initializations. Note that in (a), the eigenvalues for constant and zeros are very close to $\lambda_i = 0$, and in (b), the energy ratios for constant and zeros are very close to $\Lambda_i = 1$.}
\label{fig:ev_inits}
\end{figure}
Notice that, for the constant\footnote{The constant initialization was set to be = 5.} and zero initializations, $d = 0$ and $AUC = 0$. These results are due to the value of the scale parameter, $\varepsilon = 3$, not being suitable for those initializations. Consequently, $\varepsilon$ had to be adjusted using the method detailed in Section \ref{sec:diff_map_epsi}. Figures \ref{fig:epsi_const} and \ref{fig:epsi_zeros} plot (a) the approximately linear regions within which $\varepsilon$ should be chosen for these two initializations, and (b) possible optimal values based on the graphs of dimension $d$ against the AUC ratio.
\begin{figure}[hbt!]
\centering
\includegraphics[width=\textwidth,center]{pic_epsi_const.png}
\caption{Re-tuning of scale parameter $\varepsilon$ for $\textrm{constant} ( = 5)$ initialized model. (a) Optimal region shaded in blue within which $\varepsilon$ should be chosen. Chosen value of $\varepsilon = 0.12$ and previously chosen value of $\varepsilon = 3$ marked as red dots for comparison. (b) Possible optimal values based on graph of dimension $d$ against AUC ratio. The colorbar shows the range of values for $\varepsilon$. Values in the upper left corner are desirable due to low-dimension and high AUC ratio. Several points marked simply for reference.}
\label{fig:epsi_const}
\end{figure}
\begin{figure}[hbt!]
\centering
\includegraphics[width=\textwidth,center]{pic_epsi_zeros.png}
\caption{Re-tuning of scale parameter $\varepsilon$ for zeros initialized model. (a) Optimal region shaded in blue within which $\varepsilon$ should be chosen. Chosen value of $\varepsilon = 0.005$ and previously chosen value of $\varepsilon = 3$ marked as red dots for comparison. (b) Possible optimal values based on graph of dimension $d$ against AUC ratio. The colorbar shows the range of values for $\varepsilon$. Values in the upper left corner are desirable due to low-dimension and high AUC ratio. Several points marked simply for reference.}
\label{fig:epsi_zeros}
\end{figure}
$\varepsilon = 3$ is marked as a red dot in the graphs to show that this value is clearly unfitting. More appropriate choices of $\varepsilon = 0.12$ for constant and $\varepsilon = 0.005$ for zeros were made. With these new values, the lower dimensional manifolds were more correctly uncovered with $d=20$ for the constant initialization and $d=178$ for the zero initialization. Table \ref{table:auc_inits} summarizes the results.
\begin{table}[hbp!]
\centering
\begin{tabular}{l|cccc}
\hline
Initialization & $m$ & $M$ & $d$ & AUC Ratio \\\hline
Constant (=5, $\varepsilon = 3$) & 515 & 2400 & 0 & 0.0000 \\
Constant (=5, $\varepsilon = 0.12$) & 515 & 2400 & 20 & 0.9150 \\
Identity & 515 & 2400 & 27 & 0.9137 \\
Orthogonal & 515 & 2400 & 18 & 0.9138 \\
Glorot Uniform & 515 & 2400 & 17 & 0.9147 \\
Random Uniform & 515 & 2400 & 23 & 0.9126 \\
Truncated Normal & 515 & 2400 & 27 & 0.9162 \\
Zeros ($\varepsilon = 3$) & 515 & 2400 & 0 & 0.0000 \\
Zeros ($\varepsilon = 0.0005$) & 515 & 2400 & 178 & 0.9098 \\
\hline
\end{tabular}
\caption{Summary of subspace dimensions $d$ and AUC ratio for different weight initializations.}
\label{table:auc_inits}
\end{table}
Significant reductions in dimension for all initializations can be observed. Only for very poorly initialized models, such as the case with zeros where $d = 178$, does SGD fail to find a much lower dimensional subspace in which to move around. This signifies that, although initial weights still should be selected carefully to ensure that SGD performs well, the dimension of the subspace does appear to be robust to initialization.
\subsubsection{Convergence and stability}
Apart from attempting to identify the variables that affect the subspace dimension, we also briefly examined how fast SGD moves into the subspace and whether it proceeds to find even lower dimensional subspaces as it continues. Rather than the two-hidden layer model for the iris flower classification that has been used previously, we looked at the model with six hidden layers (see details in Table \ref{table:depth}). The reason for this is that the optimization for this model has more interesting developments, shown in Figure \ref{fig:conv_loss&fixed}(a), where SGD first finds a potential minimum around $\textrm{loss} \approx 0.6$, then escapes to find a better minimum closer to zero before converging.
\begin{figure}[htbp!]
\centering
\begin{subfigure}{.46\textwidth}\label{fig:conv_L6_loss}
\centering
\includegraphics[trim=2cm 0.5cm 0cm 0.5cm,width=0.83\textwidth,center]{pic_conv_L6.png}
\caption{}
\end{subfigure}%
\begin{subfigure}{.46\textwidth}\label{fig:conv_iris_fixed}
\centering
\includegraphics[trim=0cm 0.5cm 0cm 0cm, width=1.1\textwidth,center]{pic_conv_iris_fixed.png}
\caption{}
\end{subfigure}
\caption{(a) Loss graph for the iris flower classification problem with six hidden layers (see details in Table \ref{table:depth}). SGD finds a potential minimum around $\textrm{loss} \approx 0.6$, then escapes to find a better minimum closer to zero before converging. (b) SGD seemingly finding lower dimensional subspaces as optimization continues. Result is erroneous, however, as it results from using a fixed $\varepsilon$ that becomes too large for later data points.}
\label{fig:conv_loss&fixed}
\end{figure}
The first 2400 points were studied to determine the dimension at the beginning of optimization, then a window of the same number of points, but incremented by 50 is rolled to evaluate how the dimension changes. Figure \ref{fig:conv_iris}(a) displays the results, showing that SGD moves into the lower dimensional subspace after just a few steps, and is stable as it stays in the same dimensional subspace even as it finds minima and converges. Figure \ref{fig:conv_iris}(b) also shows how $\varepsilon$ was readjusted.
\begin{figure}[htbp!]
\centering
\includegraphics[width=0.9\textwidth,center]{pic_conv_iris.png}
\caption{(a) Subspace dimension $d$ (in blue and left y-axis) and AUC ratio (in orange and right y-axis) as SGD progresses. (b) Readjustments made for the scale parameter $\varepsilon$. Keeping parameter fixed from the start means the value becomes too large for later SGD steps.}
\label{fig:conv_iris}
\end{figure}
The scale parameter $\varepsilon$ had to be re-tuned accordingly as SGD converges and the steps become closer together. Maintaining the same $\varepsilon$ throughout means that the parameter becomes too large for later data points, allowing too many adjacent points and noise within the ball of radius $\varepsilon$. This mislead to results that SGD converges to lower and lower dimensions as shown in Figure \ref{fig:conv_loss&fixed}(b).
\section{Summary, conclusions and future research}\label{sumu}
In this paper we pursued a truly data driven approach to the problem of getting a potentially deeper understanding of the high-dimensional parameter loss surface, and the landscape traced out by SGD, in the context of fitting (deep) neural networks to real data sets and by analyzing the data generated through SGD in order to possibly discovery (local) low-dimensional representations of the optimization landscape. As our vehicle for the exploration
we used diffusion maps introduced by R. Coifman and coauthors. Our empirical results suggest that the high-dimensional loss surface does indeed contain a lower dimensional subspace in which SGD tends to concentrate/move. The dimension of this subspace is quite significantly lower compared to the dimension of the loss surface. For example, in the case of the two-hidden layer iris flower model studied, the original parameter space has a dimension of $515$, while the subspace defined has a dimension of $d = 17$, corresponding to an approximately 97\% reduction of dimensionality. In fact, our results may lean to the conservative side, as other approaches to defining the subspace, its dimension and the criteria for choosing $\varepsilon$, lead to even lower dimensions, see Sections \ref{sec:dim_subspace} and \ref{sec:diff_map_epsi}. Our empirical results also indicate that the dimension of the subspace defined is, to some degree, robust to the noise, depth, and weight initialization of the neural network. This possibly indicates that the intrinsic dimensionality may be more dependent on the data set rather than the neural network architecture. Moreover, SGD exhibits stability by moving to the lower dimensional subspace just after a few steps, and remains in the subspace as optimization continues.
We think that our empirical results could constitute the beginning of more comprehensive studies of this interesting and relevant research problem. Finding the relationship between dimension of the subspace introduced and potential factors, in the data or in the architectures used, is complex, and further endeavors should look into different variables in order to attempt to make stronger conclusions as to what really influences the dimension of the subspace. The width of the neural network, for example, can be examined, as well as the use of more data points and different types of larger data sets, the size of which in this study was restricted due to computational resource constraints. To take full advantage of diffusion maps, subsequent investigations should also examine the actual embedding and determine characterizations of the lower dimensional subspace. Furthermore, in this paper we have focused on SGD for high dimensional problems in the context of deep learning. However, it would also be worthwhile to apply the same methods and examine the behavior of SGD in other high dimensional settings such as in latent factor models where SGD is widely adapted as a learning algorithm.
\bibliographystyle{abbrv}
| {
"timestamp": "2022-06-20T02:11:21",
"yymm": "2204",
"arxiv_id": "2204.01365",
"language": "en",
"url": "https://arxiv.org/abs/2204.01365",
"abstract": "Stochastic gradient descent (SGD) is widely used in deep learning due to its computational efficiency, but a complete understanding of why SGD performs so well remains a major challenge. It has been observed empirically that most eigenvalues of the Hessian of the loss functions on the loss landscape of over-parametrized deep neural networks are close to zero, while only a small number of eigenvalues are large. Zero eigenvalues indicate zero diffusion along the corresponding directions. This indicates that the process of minima selection mainly happens in the relatively low-dimensional subspace corresponding to the top eigenvalues of the Hessian. Although the parameter space is very high-dimensional, these findings seems to indicate that the SGD dynamics may mainly live on a low-dimensional manifold. In this paper, we pursue a truly data driven approach to the problem of getting a potentially deeper understanding of the high-dimensional parameter surface, and in particular, of the landscape traced out by SGD by analyzing the data generated through SGD, or any other optimizer for that matter, in order to possibly discover (local) low-dimensional representations of the optimization landscape. As our vehicle for the exploration, we use diffusion maps introduced by R. Coifman and coauthors.",
"subjects": "Machine Learning (stat.ML); Machine Learning (cs.LG)",
"title": "Deep learning, stochastic gradient descent and diffusion maps",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9728307668889047,
"lm_q2_score": 0.7279754371026367,
"lm_q1q2_score": 0.7081969027528436
} |
https://arxiv.org/abs/0906.0240 | A counter-intuitive correlation in a random tournament | Consider a randomly oriented graph $G=(V,E)$ and let $a$, $s$ and $b$ be three distinct vertices in $V$. We study the correlation between the events $\{a\to s\}$ and $\{s\to b\}$. We show that, when $G$ is the complete graph $K_n$, the correlation is negative for $n=3$, zero for $n=4$, and that, counter-intuitively, it is positive for $n\ge 5$. We also show that the correlation is always negative when $G$ is a cycle, $C_n$, and negative or zero when $G$ is a tree (or a forest). | \section{Introduction} \label{S:Intro}
Given a graph $G=(V,E)$ we orient each edge with equal probability for the two possible directions and independent of all other edges.
This model has been studied previously in for instance \cite{G00, SL2, CM}.
Let $a$, $s$ and $b$ be three distinct vertices in $V$.
The object of this paper is to study the correlation of the two events $\{a\to s\}$, that there exists a directed path from $a$ to $s$,
and $\{s\to b\}$. One might intuitively guess that they are always negatively correlated,
i.e. that $P(a\to s, s\to b)< P(a\to s)\cdot P(s\to b)$.
This is however not true for all graphs.
In fact, the smallest counterexample is the graph on four vertices with all edges except $\{a,b\}$ present, see Section \ref{S:CE}.
In section \ref{S:Kn} we prove that for the complete graph, $K_n$, the events are negatively correlated for $n=3$,
independent for $n=4$ and positively correlated for $n\ge 5$. The complementary events,
$A:=\{a \mathrel{\nrightarrow} s\}$, that there does not exist a directed path from $a$ to $s$, and $B:=\{s\mathrel{\nrightarrow} b\}$ have the same covariance,
and we show that their relative covariance, $(P(A\cap B)-P(A)\cdot P(B))/P(A\cap B)$ converges to $1/3$ as $n\to\infty$.
In Section \ref{S:Rec} we give exact recursions for the probabilities $P(A)$ and $P(A\cap B)$, and compute these for $n\le15$.
For completeness, in Section \ref{S:CT} we show that the events are negatively correlated when $G$ is a cycle
and negatively correlated or independent when $G$ is a tree (or a forest). We end with stating a number of conjectures and open problems.
In a coming paper, \cite{AL2}, we will study this problem when $G$ is the random graph $G(n,p)$.
\medskip
The question studied here was posed in \cite{SL2}.
There it was proved that under this model for any vertices $a,b,s,t\in V$ the events $\{s\to a\}$ and $\{s\to b\}$ are never negatively
correlated. This was shown to be true also if we first conditioned on $\{s\mathrel{\nrightarrow} t\}$, i.e.
\hbox{$P(s\to a, s\to b | s\mathrel{\nrightarrow} t )\ge P(s\to a | s\mathrel{\nrightarrow} t)\cdot P(s\to b | s\mathrel{\nrightarrow} t)$.}
As a sort of converse it was also proved that $P(s\to a, b\to t | s\mathrel{\nrightarrow} t )\le P(s\to a | s\mathrel{\nrightarrow} t)\cdot P(b\to t | s\mathrel{\nrightarrow} t)$.
The proofs in \cite{SL2} relied heavily on the results in \cite{vdBK} and \cite{vdBHK},
where similar statements were proved for edge percolation on a given graph and a result from \cite{CM} that relates the random orientation with edge percolation.
This cluster of questions on correlation of paths have been inspired by an interesting conjecture due to Kasteleyn,
named the Bunkbed conjecture by H\"aggstr\"om \cite{OH2}, see also \cite{SL1} and Remark 5 in \cite{vdBK}.
\bigskip\noindent
{\bf Acknowledgment:} We thank Svante Janson, Stanislav Volkov and Johan W\"astlund for fruitful discussions.
\section{A counter-intuitive example} \label{S:CE}
Let $G$ be a graph with four vertices and all edges except the one between $a$ and $b$ present, see Figure \ref{F:pos},
and let $C:=\{a\to s\}$ and $D:=\{s\to b\}$.
\begin{figure}[h]
\includegraphics[height=3cm]{k4-1}
\caption{A counterexample with positive correlation}
\label{F:pos}
\end{figure}
Then, $P(C)=P(D)=\frac12+\frac18+\frac1{32}=\frac{21}{32}$ and $P(C\cap D)=\frac14+\frac1{16}+\frac1{16}+\frac1{16}=\frac7{16}$,
so that
\[P(C\cap D)-P(C)\cdot P(D)=\frac7{16}-\left(\frac{21}{32}\right)^2=\frac7{1024}>0.\]
Note that if we relabel the vertices in this graph as in Figure \ref{F:neg}, we still get $P(C)=P(D)=\frac{21}{32}$, but
$P(C\cap D)=\frac14+\frac18+\frac1{32}=\frac{13}{32}$, so that
\[P(C\cap D)-P(C)\cdot P(D)=\frac{13}{32}-\left(\frac{21}{32}\right)^2=-\frac{25}{1024}<0.\]
In fact, the labeling in Figure \ref{F:pos} is the only graph with four vertices that gives a positive correlation.
\begin{figure}[h]
\includegraphics[height=30mm]{k4-2}
\caption{A slightly modified example with negative correlation}
\label{F:neg}
\end{figure}
\section{The complete graph $K_n$} \label{S:Kn}
Let $G$ be the complete graph, $K_n$, and orient the edges either way independently with probability $\frac{1}{2}$.
For three different vertices, $a$, $s$ and $b$, of $K_n$ we want to know if the event $\{a \to s\}$
and the event $\{s\to b\}$ are positively or negatively correlated.
It turns out to be easier to study the correlation of the complementary events,
i.e. $A:=\{a \mathrel{\nrightarrow} s\}$, that there does not exist a directed path from $a$ to $s$, and the event $B:=\{s\mathrel{\nrightarrow} b\}$,
which have the same correlation.
Think of the vertices of $K_n$ as $[n]:=\{1,\dots,n\}$.
\bigskip
To estimate $P(A)$, the following lemma will be useful.
\begin{lemma} \label{L:a} For all $n\ge0$,
\[ a(n):=\sum_{k=1}^{n-1} \binom{n}{k}\sum_{m=1}^{n-k} \binom{n-k}{m}\Big(\frac12\Big)^{km}\le5.6\cdot\Big(\frac74\Big)^n.
\]
\end{lemma}
\begin{proof}As $a(0)=a(1)=0$, we will assume that $n\ge2$.
We will use that $(\frac12)^{km}=(\frac14)^m\cdot(\frac12)^{(k-2)m}\le(\frac14)^m\cdot(\frac12)^{k-2}=4\cdot(\frac12)^k\cdot(\frac14)^m$,
if $m\ge1$ and $k\ge2$, and split the sum into two parts, $k=1$ and $k\ge 2$.
\begin{align*}
a(n)&=n\cdot\sum_{m=1}^{n-1} \binom{n-1}{m}\Big(\frac12\Big)^m +\sum_{k=2}^{n-1} \binom{n}{k}\sum_{m=1}^{n-k} \binom{n-k}{m}\Big(\frac12\Big)^{km}\\
&\le n\cdot\Big(\frac32\Big)^{n-1}
+4\cdot\sum_{k=2}^{n-1} \binom{n}{k}\Big(\frac12\Big)^k\sum_{m=1}^{n-k} \binom{n-k}{m}\Big(\frac14\Big)^{m}\\
&\le n\cdot\Big(\frac32\Big)^{n-1}+4\cdot\sum_{k=2}^{n-1} \binom{n}{k}\Big(\frac12\Big)^k\Big(\frac54\Big)^{n-k}\\
&\le n\cdot\Big(\frac32\Big)^{n-1}+4\cdot\Big(\frac74\Big)^n.
\end{align*}
The lemma follows by showing that $n\cdot(\frac32)^{n-1}\le1.6\cdot(\frac74)^n$ holds for all $n\ge2$.
\end{proof}
\begin{remark}
As $k=1$ contributes $n\cdot(\frac32)^{n-1}$, this gives a lower bound for $a(n)$.
This is in fact the dominating term, so that $a(n)$ asymptotically is of order $p(n)\cdot(\frac32)^n$, where $p(n)$ is a polynomial in $n$.
By splitting the sum into three parts, we can, for example, show that $a(n)\le13.6\cdot(\frac{13}8)^n$ for all $n\ge0$.
\end{remark}
\begin{theorem} \label{L:A} For all $n\ge2$,
\[ \Big(\frac12\Big)^{n-2}\left(1-\Big(\frac12\Big)^{n-1}\right)\le P(A)\le\Big(\frac12\Big)^{n-2}\left(1+3.2\cdot\Big(\frac78\Big)^{n-1}\right).
\]
\end{theorem}
\begin{proof}
A necessary condition for $A$ is that the edge between $a$ and $s$ is directed from $s$ to $a$.
Let $E$, with $P(E)=1/2$, denote this event.
Let $O_a$ and $O_s$ denote the sets of points in $[n]\setminus\{a,s\}$ that can be reached from $a$ and $s$ respectively in one step.
Similarly, let $I_a$ and $I_s$ denote the sets of points in $[n]\setminus\{a,s\}$ that can reach $a$ and $s$ respectively in one step.
For the lower bound, note that $E\cap(O_a=\emptyset)\Rightarrow A$ and $E\cap(I_s=\emptyset)\Rightarrow A$, so that
\begin{align*}
P(A)&\ge P((O_a=\emptyset)\cup(I_s=\emptyset))/2=\left(\Big(\frac12\Big)^{n-2}+\Big(\frac12\Big)^{n-2}-\Big(\frac12\Big)^{2n-4}\right)/2\\
&=\Big(\frac12\Big)^{n-2}\left(1-\Big(\frac12\Big)^{n-1}\right).
\end{align*}
For the upper bound, note that $A\Rightarrow E\cap(O_a\subset O_s)\cap F$,
where $F$ is the event that the points in $O_a$ have no directed edges to points in $I_s$.
Note that, if $k=|O_a|$ and $m=|O_s|$, then $k\le m$ is necessary, and there are $k(n-2-m)$ edges in $F$.
\begin{align*}
P(A)&\le P((O_a\subset O_s)\cap F)/2\\
&=\frac12\cdot\sum_{k=0}^{n-2}\binom{n-2}k\Big(\frac12\Big)^{n-2}\sum_{m=k}^{n-2}\binom{n-2-k}{m-k}\Big(\frac12\Big)^{n-2}\cdot\Big(\frac12\Big)^{k(n-2-m)}\\
&=\Big(\frac12\Big)^{2n-3}\sum_{k=0}^{n-2}\binom{n-2}k\sum_{m=0}^{n-2-k}\binom{n-2-k}m\Big(\frac12\Big)^{k\cdot m}\\
&=\Big(\frac12\Big)^{2n-3}\sum_{m=0}^{n-2}\binom{n-2}m+\Big(\frac12\Big)^{2n-3}\sum_{k=1}^{n-2}\binom{n-2}k\\
&\phantom{aa}+\Big(\frac12\Big)^{2n-3}\sum_{k=1}^{n-2}\binom{n-2}k\sum_{m=1}^{n-2-k}\binom{n-2-k}m\Big(\frac12\Big)^{k\cdot m}\\
&=\Big(\frac12\Big)^{2n-3}\cdot 2^{n-2}+\Big(\frac12\Big)^{2n-3}\cdot(2^{n-2}-1)\\
&\phantom{aa}+\Big(\frac12\Big)^{2n-3}\sum_{k=1}^{n-3}\binom{n-2}k\sum_{m=1}^{n-2-k}\binom{n-2-k}m\Big(\frac12\Big)^{k\cdot m}\\
&\le\Big(\frac12\Big)^{n-2}+\Big(\frac12\Big)^{2n-3}\cdot a(n-2)
\le\Big(\frac12\Big)^{n-2}+5.6\cdot\Big(\frac12\Big)^{2n-3}\Big(\frac74\Big)^{n-2}\\
&\le\Big(\frac12\Big)^{n-2}\left(1+5.6\cdot\frac47\cdot\Big(\frac78\Big)^{n-1}\right)
=\Big(\frac12\Big)^{n-2}\left(1+3.2\cdot\Big(\frac78\Big)^{n-1}\right),
\end{align*}
where the function $a$ was defined in Lemma \ref{L:a}.
\end{proof}
\begin{theorem} \label{T:A}
\[\lim_{n\to\infty} 2^{n-2}\cdot P(A)=1.\]
\end{theorem}
\begin{proof}
Follows immediately from Theorem \ref{L:A}.
\end{proof}
To estimate $P(A\cap B)$, the following lemma, in combination with Lemma \ref{L:a}, is useful.
\begin{lemma}\label{L:b}
For all $n\ge0$,
\[b(n):=\sum_{k=1}^{n-1}\binom nk\sum_{i=1}^{n-1-k}\binom{n-k}i\Big(\frac12\Big)^{ki}\sum_{m=1}^k\binom km\Big(\frac12\Big)^{m(n-k-i)}
\le4\cdot\Big(\frac74\Big)^n.\]
\end{lemma}
\begin{proof}Note first that $b(0)=b(1)=b(2)=0$, so that we may assume that $n\ge3$.
Note also that for $k,i\ge1$, $ki=(k-1)(i-1)+k+i-1\ge k+i-1$, so that $(\frac12)^{ki}\le2(\frac12)^k(\frac12)^i$. This gives
\begin{align*}
b(n)&\le4\cdot\sum_{k=1}^{n-1}\binom nk\cdot\Big(\frac12\Big)^{k}\sum_{i=1}^{n-1-k}\binom{n-k}i\cdot\Big(\frac12\Big)^{i}\cdot\Big(\frac12\Big)^{n-k-i}
\sum_{m=1}^k\binom km\cdot\Big(\frac12\Big)^{m}\\
&\le4\cdot\sum_{k=1}^{n-1}\binom nk\cdot\Big(\frac12\Big)^{k}\cdot1\cdot\Big(\frac32\Big)^{k}
=4\cdot\sum_{k=1}^{n-1}\binom nk\cdot\Big(\frac34\Big)^{k}\le4\cdot\Big(\frac74\Big)^{n}.
\end{align*}
\end{proof}
\begin{theorem}\label{L:AB}For all $n\ge3$,
\[\Big(\frac12\Big)^{2n-3}\left(3-2\Big(\frac12\Big)^{n-3}\right)\le P(A\cap B)
\le\Big(\frac12\Big)^{2n-3}\left(3+20.8\cdot\Big(\frac78\Big)^{n-3}\right).\]
\end{theorem}
\begin{proof}
A necessary condition for $A\cap B$ is that the edge between $a$ and $s$ is directed from $s$ to $a$,
that the edge between $s$ and $b$ is directed from $b$ to $s$ and that the edge between $a$ and $b$ is directed from $b$ to $a$.
Let $E$, with $P(E)=1/8$, denote this event.
Let $O_a$, $O_s$ and $O_b$ denote the sets of points in $[n]\setminus\{a,s,b\}$ that can be reached from $a$, $s$ and $b$ respectively
in one step.
Similarly, let $I_a$, $I_s$ and $I_b$ denote the sets of points in $[n]\setminus\{a,s,b\}$ that can reach $a$, $s$ and $b$ respectively
in one step.
For the lower bound, note that $E\cap(O_a=O_s=\emptyset)\Rightarrow A\cap B$, $E\cap(O_a=I_b=\emptyset)\Rightarrow A\cap B$ and
$E\cap(I_s=I_b=\emptyset)\Rightarrow A\cap B$,
so that
\begin{align*}
P(A\cap B)&\ge P((O_a=O_s=\emptyset)\cup(O_a=I_b=\emptyset)\cup(I_s=I_b=\emptyset))/8\\
&=\Big(\frac12\Big)^3\left(3\Big(\frac12\Big)^{2n-6}-2\Big(\frac12\Big)^{3n-9}\right)\\
&=\Big(\frac12\Big)^{2n-3}\left(3-2\Big(\frac12\Big)^{n-3}\right),
\end{align*}
as $(O_a=O_s=\emptyset)\cap(I_s=I_b=\emptyset)=\emptyset$.
For the upper bound, we note that $A\cap B\Rightarrow E\cap(O_a\subset O_s\subset O_b)\cap F$,
where $F$ denotes the event that the points in $O_a$ have no directed edges to points in $I_s$ and
the points in $O_s$ have no directed edges to points in $I_b$,
so that $P(A\cap B)\le P((O_a\subset O_s\subset O_b)\cap F)/8$.
Let $k=|O_s|$, $i=|I_b|$ and $m=|O_a|$. Then $0\le k\le n-3$, $0\le i\le n-3-k$ and $0\le m\le k$
and the direction of all $3(n-3)+3=3n-6$ edges connected to $a$, $s$ and $b$ are determined.
Further, the event $F$ determines the direction of $ki+m(n-3-k-i)$ edges, so that
\begin{align*}
P(A\cap B)
&\le\Big(\frac12\Big)^{3n-6}\sum_{k=0}^{n-3}\binom{n-3}k\sum_{i=0}^{n-3-k}\binom{n-3-k}i\Big(\frac12\Big)^{ki}
\sum_{m=0}^k\binom km\Big(\frac12\Big)^{m(n-3-k-i)}\\
&=\Big(\frac12\Big)^{3n-6}\cdot(S_1+S_2+S_3+S_4+S_5+S_6+S_7),
\end{align*}
where the triple sum is split into seven parts:\\
1: $k=0\Rightarrow m=0$,
2: $k=n-3\Rightarrow i=0$,
3: $i=m=0,1\le k\le n-4$,\\
4: $i=0,m\ge1,1\le k\le n-4$,
5: $m=0,i\ge1,1\le k\le n-4$,\\
6: $i=n-3-k,m\ge1,1\le k\le n-4$,\\
7: $1\le k\le n-4,1\le i\le n-4-k, 1\le m\le k$.\\
The first three cases correspond to the three cases of the lower bound,
\begin{align*}
S_1&=\sum_{i=0}^{n-3}\binom{n-3}i=2^{n-3},\\
S_2&=\sum_{m=0}^{n-3}\binom{n-3}m=2^{n-3},\\
S_3&=\sum_{k=1}^{n-4}\binom{n-3}k=2^{n-3}-2\le2^{n-3}.
\end{align*}
The next three can be expressed by the function $a$ of Lemma \ref{L:a},
\begin{align*}
S_4&=\sum_{k=1}^{n-4}\binom{n-3}k\sum_{m=1}^k\binom km\cdot\Big(\frac12\Big)^{m(n-3-k)}
=\sum_{j=1}^{n-4}\binom{n-3}j\sum_{m=1}^{n-3-j}\binom{n-3-j}m\cdot\Big(\frac12\Big)^{mj}\\
&=a(n-3)\le5.6\cdot\Big(\frac74\Big)^{n-3},\\
S_5&=\sum_{k=1}^{n-4}\binom{n-3}k\sum_{i=0}^{n-3-k}\binom{n-3-k}i\cdot\Big(\frac12\Big)^{ki}=a(n-3)\le5.6\cdot\Big(\frac74\Big)^{n-3},\\
S_6&=\sum_{k=1}^{n-4}\binom{n-3}k\cdot\Big(\frac12\Big)^{k(n-3-k)}\sum_{m=1}^k\binom km
\le\sum_{k=1}^{n-4}\binom{n-3}k\sum_{m=1}^k\binom km\cdot\Big(\frac12\Big)^{m(n-3-k)}\\
&=\sum_{i=1}^{n-4}\binom{n-3}i\sum_{m=1}^{n-3-i}\binom{n-3-i}m\cdot\Big(\frac12\Big)^{im}
=a(n-3)\le5.6\cdot\Big(\frac74\Big)^{n-3},
\end{align*}
and the last by the function $b$ of Lemma \ref{L:b},
\begin{align*}
S_7&=\sum_{k=1}^{n-4}\binom{n-3}k\sum_{i=1}^{n-4-k}\binom{n-3-k}i\cdot\Big(\frac12\Big)^{ki}\sum_{m=1}^k\binom km\cdot\Big(\frac12\Big)^{m(n-3-k-i)}\\
&=b(n-3)\le4\cdot\Big(\frac74\Big)^{n-3}.
\end{align*}
Collecting the estimates gives the lemma.
\end{proof}
\begin{theorem}\label{T:AB}
\[\lim_{n\to\infty} 2^{2n-3}\cdot P(A\cap B)=3.\]
\end{theorem}
\begin{proof}
Follows immediately from Theorem \ref{L:AB}.
\end{proof}
\begin{theorem}\label{T:relkorr}
\[\lim_{n\to\infty} \frac{P(A\cap B)-P(A)\cdot P(B)}{P(A\cap B)}=\frac13.\]
\end{theorem}
\begin{proof}
Follows immediately from Theorems \ref{T:A} and \ref{T:AB} as $P(B)=P(A)$.
\end{proof}
\begin{remark}\label{R:korr}
Note that
\[\frac{P(A\cap B)-P(A)\cdot P(B)}{P(A\cap B)}=1-\frac{P(A)\cdot P(B)}{P(A)\cdot P(B\,|\,A)}=1-\frac{P(B)}{P(B\,|\,A)},\]
so that Theorem \ref{T:relkorr} can be formulated as
\[\lim_{n\to\infty}\frac{P(B)}{P(B\,|\,A)}=\frac23,
\text{ or equivalently }
\lim_{n\to\infty}\frac{P(B\,|\,A)}{P(B)}=\frac32.\]
\end{remark}
Theorem \ref{T:relkorr} shows that the events $A=\{a\mathrel{\nrightarrow} s\}$ and $B=\{s\mathrel{\nrightarrow} b\}$ are positively correlated for sufficiently large $n$.
{}From this follows that the complementary events $C=\{a\to s\}$ and $D=\{s\to b\}$ also are positively correlated for sufficiently
large $n$. It is in fact true for all $n\ge5$ as the next theorem shows.
\begin{theorem}\label{T:korr}
The events $A=\{a\mathrel{\nrightarrow} s\}$ and $B=\{s\mathrel{\nrightarrow} b\}$ are negatively correlated for $n=3$, independent for $n=4$ and positively correlated for $n\ge5$.
\end{theorem}
\begin{proof}
{}From Lemmas \ref{L:A} and \ref{L:AB} we get
\begin{align*}
P(A\cap B)-&P(A)\cdot P(B)=P(A\cap B)-(P(A))^2\\
&\ge\Big(\frac12\Big)^{2n-3}\cdot\left(3-2\Big(\frac12\Big)^{n-3}\right)-\left\{\Big(\frac12\Big)^{n-2}\cdot\left(1+3.2\cdot\Big(\frac78\Big)^{n-1}\right)\right\}^2\\
&\ge\Big(\frac12\Big)^{2n-4}\cdot\left(6-4\cdot\Big(\frac12\Big)^{n-3}-1-6.4\cdot\Big(\frac78\Big)^{n-1}-10.24\cdot\Big(\frac{49}{64}\Big)^{n-1}\right)\\
&=\Big(\frac12\Big)^{2n-4}\cdot\left(5-\Big(\frac12\Big)^{n-5}-6.4\cdot\Big(\frac78\Big)^{n-1}-10.24\cdot\Big(\frac{49}{64}\Big)^{n-1}\right)\\
&=\Big(\frac12\Big)^{2n-4}\cdot(5-c(n)),
\end{align*}
where $c(n)$ is a decreasing function of $n$, with $c(8)<5$, so that the theorem holds for $n\ge8$.
The remaining cases, $3\le n\le7$, are proved using the recursion formulas in Lemmas \ref{L:f} and \ref{L:g} in the next section.
\end{proof}
\section{Exact recursions} \label{S:Rec}
In this section we will derive recursions for $P(A)$ and $P(A\cap B)$.
For $n\ge 2$, $s\in [n]$ and $K\subset [n]\setminus \{s\}$ let $\{K\mathrel{\nrightarrow} s\}$ denote the event $\{a\mathrel{\nrightarrow} s$ for every $a\in K\}$.
With $|K|=k$ define
\[ f(n,k):=P_n(K\mathrel{\nrightarrow} s),\]
where in particular $f(n,0)=1$. Also set $f(1,0)=1$ for convenience.
For $n\ge 3$ and $s,b\in [n], K\subset [n]\setminus \{s,b\}$, $s\neq b$ and $|K|=k$ define:
\[ g(n,k):=P_n(K\mathrel{\nrightarrow} s,s\mathrel{\nrightarrow} b),
\]
where in particular $g(n,0)=f(n,1)$. Also let $g(2,0):=f(2,1)=1/2$.
\medskip
\begin{lemma} \label{L:f} For $n\ge k+1\ge 2$ we have
\[ f(n,k)=\sum_{i=0}^{n-k-1} \binom{n-k-1}{i}\frac{(2^{k}-1)^i}{2^{k(n-k)}} f(n-k,i).
\]
\end{lemma}
\begin{proof}
We first deduce the following recursion:
\[ P_n(K\mathrel{\nrightarrow} s)=\sum_{L\subset [n]\setminus (K\cup \{s\})} \frac{1}{2^{k(n-k-|L|)}}\Big(1-\frac{1}{2^k}\Big)^{|L|} P_{n-k}(L\mathrel{\nrightarrow} s).
\]
This can be seen as follows. We think of the set $L$ as the vertices that we can reach from vertices in $K$ in one step.
Clearly, we must have $s\notin L$. Every edge from $[n]\setminus (K\cup L)$ to $K$ must be oriented in that direction,
which gives the first power of $1/2$. To reach a vertex $c\in L$ in one step it must not be the case that all edges $\{a,c\}, a\in K$
are directed away from $c$, which gives the second factor.
The edges within $K$ make no difference and we have considered all edges going between $K$ and $[n]\setminus K$.
We are left with a situation where we must make sure that there is no directed path from any vertex in $L$ to $s$ passing over
vertices in $[n]\setminus K$, which gives the last term. The recursion in the lemma is easily deduced from this,
by summing over the size, $i=|L|$, of $L$.
\end{proof}
\begin{lemma} \label{L:g} For $n\ge k+2\ge 3$ we have
\[ g(n,k)=\sum_{i=0}^{n-k-2} \binom{n-k-2}{i}\frac{(2^k-1)^i}{2^{k(n-k)}} g(n-k,i).
\]
\end{lemma}
\begin{proof}
First note that $\{K\mathrel{\nrightarrow} s\}$ and $\{s\mathrel{\nrightarrow} b\}$ implies $\{K\mathrel{\nrightarrow} b\}$.
Reasoning as in the previous proof, we obtain
\[ P_n(K\mathrel{\nrightarrow} s,s\mathrel{\nrightarrow} b)=\sum_{L\subset [n]\setminus (K\cup \{s\}\cup \{b\})} \frac{1}{2^{k(n-k-|L|)}}\Big(1-\frac{1}{2^k}\Big)^{|L|}
P_n(L\mathrel{\nrightarrow} s, s\mathrel{\nrightarrow} b).\]
\end{proof}
Using the lemmas, we can recursively compute the desired probabilities $P(A)=f(n,1)$ and $P(A\cap B)=g(n,1)$.
Exact and numerical values of $P(A)$ and $P(A\cap B)$ were computed and are given in Table \ref{Tab:corr}
together with numerical values of the
relative covariance
$(P(A\cap B)-P(A)\cdot P(B))/P(A\cap B)$ for $3\le n\le13$.
\medskip
\begin{table}[hb]
\begin{tabular}{r|r|r|r|r|r}
$n$&{\small $P(A)\cdot 2^{\binom{n}{2}}$}\hfil\hfil &{\small $P(A)$}\hfil\hfil &{\small $P(A\cap B)\cdot 2^{{\binom{n}{2}}}$}\hfil\hfil &{\small $P(A\cap B)$}\hfil\hfil &$\frac{P(A\cap B)-P(A)P(B)}{P(A\cap B)}$\\
&&&&&\\[-10pt]
\hline
&&&&&\\[-10pt]
{\small 2}&{\small 1}&{\small 0.500\,0}&{\small }&&\\
{\small 3}&{\small 3}&{\small 0.375\,0}&{\small 1}&{\small 0.125\,000\,0}&{\small $-$0.125\,000\phantom{000I}}\\
{\small 4}&{\small 16}&{\small 0.250\,0}&{\small 4}&{\small 0.062\,500\,0}&{\small 0.000\,000\phantom{000I}}\\
{\small 5}&{\small 150}&{\small 0.146\,5}&{\small 26}&{\small 0.025\,390\,6}&{\small 0.154\,898\phantom{000I}}\\
{\small 6}&{\small 2\,504}&{\small 0.076\,4}&{\small 272}&{\small 0.008\,300\,8}&{\small 0.296\,523\phantom{000I}}\\
{\small 7}&{\small 77\,472}&{\small 0.036\,9}&{\small 4\,672}&{\small 0.002\,227\,8}&{\small 0.387\,428\phantom{000I}}\\
{\small 8}&{\small 4\,677\,904}&{\small 0.017\,4}&{\small 139\,696}&{\small 0.000\,520\,4}&{\small 0.416\,449\phantom{000I}}\\
{\small 9}&{\small 571\,023\,120}&{\small 0.008\,3}&{\small 7\,928\,624}&{\small 0.000\,115\,4}&{\small 0.401\,547\phantom{000I}}\\
{\small 10}&{\small 142\,058\,571\,776}&{\small 0.004\,0}&{\small 917\,140\,928}&{\small 0.000\,026\,1}&{\small 0.374\,613\phantom{000I}}\\
{\small 11}&{\small 71\,626\,948\,215\,168}&{\small 0.002\,0}&{\small 220\,836\,999\,808}&{\small 0.000\,006\,1}&{\small 0.355\,191\phantom{000I}}\\
{\small 12}&{\small 72\,752\,562\,631\,695\,616}&{\small 0.001\,0}&{\small 109\,473\,061\,398\,784}&{\small 0.000\,001\,5}&{\small 0.344\,746\phantom{000I}}\\
{\small 13}&{\small 148\,346\,259\,329\,909\,191\,680}&{\small 0.000\,5}&{\small 110\,228\,037\,783\,934\,976}&{\small 0.000\,000\,4}&{\small 0.339\,426\phantom{000I}}\\
\end{tabular}
\medskip
\caption{Probabilities and relative covariances}\label{Tab:corr}
\end{table}
\bigskip
\section{Cycles and trees} \label{S:CT}
Let $G=C_n$, the cycle with $n$ vertices, and let $c$, $d$ and $n-c-d$ denote the distances (number of edges)
between $a$ and $s$, $s$ and $b$, and $b$ and $a$, respectively.
We assume the three vertices to be distinct, so that $c,d,n-c-d\ge1$.
Further, let $C:=\{a\to s\}$ and $D:=\{s\to b\}$.
Then,
\begin{align*}
P(C)&=\left(\frac12\right)^c+\left(\frac12\right)^{n-c}-\left(\frac12\right)^n,\\
P(D)&=\left(\frac12\right)^d+\left(\frac12\right)^{n-d}-\left(\frac12\right)^n,\\
P(C\cap D)&=\left(\frac12\right)^c\cdot\left(\frac12\right)^{d}+\left(\frac12\right)^n,
\end{align*}
where the last term corresponds to the case when there is a directed path $s\to a\to b\to s$. This gives
\begin{align*}
P(C\cap &D)-P(C)\cdot P(D)\\
&=\left(\frac12\right)^{c+d}+\left(\frac12\right)^n-\left(\Big(\frac12\Big)^c+\Big(\frac12\Big)^{n-c}-\Big(\frac12\Big)^n\right)
\cdot\left(\Big(\frac12\Big)^d+\Big(\frac12\Big)^{n-d}-\Big(\frac12\Big)^n\right)\\
&=\left(\frac12\right)^{2n}\cdot\left(2^n-2^{n-c+d}-2^{n+c-d}-2^{c+d}+2^{n-c}+2^{n-d}+2^c+2^d-1\right)\\
&=\left(\frac12\right)^{2n}\cdot\left(2^{n-c+d}\big(2\cdot2^{c-d}-1-2^{2(c-d)}\big)\right.\\
&\phantom{\left(\frac12\right)^{2n}\cdot\qquad}\left.-2^n\big(1-2^{-c}-2^{-d}\big)-\big(2^{c+d}-2^c-2^d+1\big)\right)\\
&\le-\left(\frac12\right)^{2n}\cdot\left(2^{n-c+d}\cdot\big(2^{c-d}-1)^2+\big(2^c-1\big)\cdot\big(2^d-1\big)\right)\\
&\le-\left(\frac12\right)^{2n},
\end{align*}
with equality if and only if $c=d=1$.
We have proved the following theorem.
\begin{theorem}\label{T:cycles}
When $G$ is the cyclic graph with $n$ nodes, $C_n$, the covariance between the events $\{a\to s\}$ and $\{s\to b\}$ is at most
$-\left(\frac12\right)^{2n}$, with equality if and only if the vertices $a$ and $b$ are adjacent to $s$.
\end{theorem}
\bigskip
For trees the situation is even simpler, as there are no cycles so that there is a unique path between any two vertices.
Two cases can occur.
If the path between $a$ and $b$ passes $s$, the paths between $a$ and $s$ and between $s$ and $b$ have no edges in common,
so that the events $\{a\to s\}$ and $\{s\to b\}$ are independent.
On the other hand, if the path between $a$ and $b$ does not pass $s$, then the events are disjoint,
so that $P(\{a\to s\}\cap\{s\to b\})=0$, and the covariance is strictly negative.
For a forest, if not all three vertices are in the same tree, then trivially $P(\{a\to s\}\cap\{s\to b\})=0$
and at least one of $P(a\to s)$ and $P(s\to b)$ is zero, so that the events are independent.
\begin{theorem}
When $G$ is a tree (or a forest), the events $\{a\to s\}$ and $\{s\to b\}$ are either independent or mutually exclusive.
\end{theorem}
\bigskip
\section{Open problems and conjectures} \label{S:PC}
{}From Theorem \ref{T:cycles} and the observations in Section \ref{S:CE}, we make the following conjecture.
\begin{conjecture}
For any connected graph $G=(V,E)$ and three distinct vertices $a$, $s$ and $b$ in $V$; if $s$ has degree at most two, then the events
$\{a\to s\}$ and $\{s\to b\}$ are independent or negatively correlated.
\end{conjecture}
Any connected simple graph $G=(V,E), |V|\ge 3$ belongs to (at least) one of the following classes.
\begin{itemize}
\item[I] For any three distinct vertices $a,b,s\in V(G)$, the events $\{a\to s\}$ and $\{s\to b\}$ are non-positively correlated.
\item[II] There exist three distinct vertices $a,b,s\in V(G)$, such that the events $\{a\to s\}$ and $\{s\to b\}$ are negatively
correlated and there exist three distinct vertices $a',b',s'\in V(G)$, such that the events $\{a'\to s'\}$ and $\{s'\to b'\}$ are
positively correlated.
Or there exist three distinct vertices $a,b,s\in V(G)$, such that the events $\{a\to s\}$ and $\{s\to b\}$ are independent.
\item[III] For any three distinct vertices $a,b,s\in V(G)$, the events $\{a\to s\}$ and $\{s\to b\}$ are non-negatively correlated.
\end{itemize}
We have shown that trees and cycles belong to Class I, $K_n,n\ge 5$ belongs to Class III and $K_4$ minus one edge belongs to Class II.
Note that when we have independent events there may be some overlap between the classes, in particular $K_4$ belongs to all three classes.
\begin{conjecture}
For large $n$ most graphs will belong to Class II.
\end{conjecture}
In fact we guess that for $n$ large enough, the graphs in Class I are joins (in some vague sense) of cycles and trees.
It would be interesting if it was possible to characterize the graphs in Class I. We formulate the following more specific questions.
Recall that outerplanar graphs are the graphs that do not have $K_4$ or $K_{2,3}$ as minors.
\begin{problem}
Are all graphs in Class I (with $|V(G)|\ge 5$) outerplanar?
\end{problem}
Similarly one could ask for a characterization of the graphs in Class III. The following subproblem would also be interesting if it could be solved.
\begin{problem}
For a given $n$, what is the smallest number $k$ such that there exist $k$ edges whose removal from $K_n$ gives a graph not in Class III?
\end{problem}
Finally we ask if Class I and Class III are monotone.
\begin{problem}
Is it true that if $G$ belongs to Class I, but not to Class II, then so does any connected subgraph obtained by removing one edge? Similarly, is it true that if $G$ belongs to Class III, but not to Class II, then so does $G$ plus any new edge?
\end{problem}
The results in \cite{AL2} seem to suggest that Class III is larger than Class I. Is this true?
| {
"timestamp": "2009-09-04T11:00:49",
"yymm": "0906",
"arxiv_id": "0906.0240",
"language": "en",
"url": "https://arxiv.org/abs/0906.0240",
"abstract": "Consider a randomly oriented graph $G=(V,E)$ and let $a$, $s$ and $b$ be three distinct vertices in $V$. We study the correlation between the events $\\{a\\to s\\}$ and $\\{s\\to b\\}$. We show that, when $G$ is the complete graph $K_n$, the correlation is negative for $n=3$, zero for $n=4$, and that, counter-intuitively, it is positive for $n\\ge 5$. We also show that the correlation is always negative when $G$ is a cycle, $C_n$, and negative or zero when $G$ is a tree (or a forest).",
"subjects": "Probability (math.PR); Combinatorics (math.CO)",
"title": "A counter-intuitive correlation in a random tournament",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9854964237163881,
"lm_q2_score": 0.7185944046238981,
"lm_q1q2_score": 0.7081722158594588
} |
https://arxiv.org/abs/1905.02264 | Stable multivariate generalizations of matching polynomials | The first part of this note concerns stable averages of multivariate matching polynomials. In proving the existence of infinite families of bipartite Ramanujan $d$-coverings, Hall, Puder and Sawin introduced the $d$-matching polynomial of a graph $G$, defined as the uniform average of matching polynomials over the set of $d$-sheeted covering graphs of $G$. We prove that a natural multivariate version of the $d$-matching polynomial is stable, consequently giving a short direct proof of the real-rootedness of the $d$-matching polynomial. Our theorem also includes graphs with loops, thus answering a question of said authors. Furthermore we define a weaker notion of matchings for hypergraphs and prove that a family of natural polynomials associated to such matchings are stable. In particular this provides a hypergraphic generalization of the classical Heilmann-Lieb theorem. | \section{Introduction}
The real-rootedness of the matching polynomial of a graph is a well-known result in algebraic graph theory due to Heilmann and Lieb \cite{HL}. Slightly less quoted is its stronger multivariate counterpart (see \cite{HL}) which proclaims that the multivariate matching polynomial is non-vanishing when its variables are restricted to the upper complex half-plane, a property known as \emph{stability}. Other stable polynomials occurring in combinatorics include e.g. multivariate Eulerian polynomials \cite{HV}, several bases generating polynomials of matroids (including multivariate spanning tree polynomials) \cite{COSW} and certain multivariate subgraph polynomials \cite{Wagb}.
In the present note we consider several different stable generalizations of multivariate matching polynomials.
Hall, Puder and Sawin prove in \cite{HPS} that every connected bipartite graph has a Ramanujan $d$-covering of every degree for each $d \geq 1$, generalizing seminal work of Marcus, Spielman and Srivastava \cite{MSS,MSSb} for the case $d=2$. An important object in their proof is a certain generalization of the matching polynomial of a graph $G$, called the \emph{$d$-matching polynomial}, defined by taking averages of matching polynomials over the set of $d$-sheeted covering graphs of $G$. The authors prove (via an indirect method) that the $d$-matching polynomial of a multigraph is real-rooted provided that the graph contains no loops. We prove in Theorem \ref{thm:multidmatchpoly} that the latter hypothesis is redundant by establishing a stronger result, namely that the multivariate $d$-matching polynomial is stable for any multigraph (possibly with loops). In the final section we consider a hypergraphic generalization of the Heilmann-Lieb theorem. The hypergraphic matching polynomial is not real-rooted in general (see \cite{GZM}) so it does not admit a natural stable multivariate refinement. However by relaxing the notion of matchings in hypergraphs we prove in Theorem \ref{thm:wksubgpolystab} that an associated ``relaxed'' multivariate matching polynomial is stable.
\section{Preliminaries}
\subsection{Graph coverings and group labelings}
In this subsection we outline relevant definitions from \cite{HPS}.
Let $G = (V(G), E(G))$ be a finite, connected, undirected graph on $[n]$. In particular we allow $G$ to have multiple edges between vertices and contain edges from a vertex to itself, i.e., $G$ is a multigraph with loops.
A graph homomorphism $f:H \to G$ is called a \textit{local isomorphism} if for each vertex $v$ in $H$, the restriction of $f$ to the neighbours of $v$ in $H$ is an isomorphism onto the neighbours $f(v)$ in $G$. We call $f$ a \textit{covering map} if it is a surjective local isomorphism, in which case we say that $H$ \textit{covers} $G$. If the image of $H$ under the covering map $f$ is connected, then each fiber $f^{-1}(v)$ of $v \in V(G)$ is an independent set of vertices in $H$ of the same size $d$. If so, we call $H$ a \textit{$d$-sheeted covering} (or \textit{$d$-covering} for short) of $G$.
Although $G$ is undirected we shall dually view it as an oriented graph, containing two edges with opposite orientation for each undirected edge. We denote the edges with positive (resp. negative) orientation by $E^{+}(G)$ (resp. $E^{-}(G)$) and identify $E(G)$ with the disjoint union $E^{+}(G) \sqcup E^{-}(G)$. If $e \in E^{\pm}(G)$, then we write $-e$ for the corresponding edge in $E^{\mp}(G)$ with opposite orientation. Moreover we denote by $h(e)$ and $t(e)$, the head and tail of the edge $e \in E(G)$ respectively.
A $d$-covering $H$ of a graph $G$ can be constructed via the following model, introduced in \cite{AL, Fri}. The vertices of $H$ are defined by $V(H) \coloneqq \{ v_i : v \in V(G), 1 \leq i \leq d \}$. The edges of $H$ are determined, as described below, by a labeling $\sigma:E(G) \to S_d$ (see Figure \ref{fig:grplabeling}) satisfying $\sigma(-e) = \sigma(e)^{-1}$. For notational purposes we write $\sigma(e) = \sigma_e$. For every positively oriented edge $e \in E^{+}(G)$ we introduce $d$ (undirected) edges in $H$ connecting $h(e)_i$ to $t(e)_{\sigma_e(i)}$ for $1 \leq i \leq d$, that is, we replace each undirected edge $e$ in $G$ by the perfect matching induced by $\sigma_e$, see Figure \ref{fig:covergraph}. We shall interchangeably refer to the map $\sigma$ and the covering graph $H$ which it determines, as a covering of $G$. Let $\mathcal{C}_{d,G}$ denote the probability space of all $d$-coverings of $G$ endowed with the uniform distribution.
Instead of labeling each edge in $G$ by a permutation in $S_d$ we may label the edges with elements coming from an arbitrary finite group $\Gamma$. A \textit{$\Gamma$-labeling} of a graph $G$ is a function $\gamma:E(G) \to \Gamma$ satisfying $\gamma(-e) = \gamma(e)^{-1}$. Let $\mathcal{C}_{\Gamma, G}$ denote the probability space of all $\Gamma$-labelings of $G$ endowed with the uniform distribution. Let $\pi:\Gamma \to \text{GL}_d(\mathbb{C})$ be a representation of $\Gamma$. For any $\Gamma$-labeling $\gamma$ of $G$, let $A_{\gamma,\pi}$ denote the $nd \times nd$ matrix obtained from the adjacency matrix $A_G$ of $G$ by replacing the $(i,j)$ entry in $A_G$ with the $d \times d$ block $\sum_{e \in E(G)} \pi(\gamma(e))$ (where the sum runs over all oriented edges from $i$ to $j$) and by a zero block if there are no edges between $i$ and $j$. The matrix $A_{\gamma,\pi}$ is called a \textit{$(\Gamma, \pi)$-covering} of $G$.
\begin{figure}
\begin{tikzpicture}
\coordinate (A) at (0.2,0);
\coordinate (B) at (1.25,-1);
\coordinate (C) at (2.3,0);
\coordinate (D) at (3.9,0);
\draw [very thick,red!80] (A) -- (B);
\draw [very thick,green!80] (A) -- (C);
\draw [very thick,blue!80] (B) -- (C);
\draw [very thick,orange!80] (C) -- (D);
\draw[very thick, purple!80, scale=4] (D.-10) to[in=45,out=-45,loop] (D.-10);
\draw[fill=black] (A) circle (0.08cm);
\draw[fill=black] (B) circle (0.08cm);
\draw[fill=black] (C) circle (0.08cm);
\draw[fill=black] (D) circle (0.08cm);
\node[yshift = 0.25cm] at (A) {$a$};
\node[yshift = -0.3cm] at (B) {$b$};
\node[yshift = 0.25cm] at (C) {$c$};
\node[xshift=-0.1cm ,yshift = 0.25cm] at (D) {$d$};
\node[xshift = 0.35cm,yshift = -0.7cm] at (A) {$\sigma_1$};
\node[xshift = 1.1cm,yshift = 0.2cm] at (A) {$\sigma_2$};
\node[xshift = 1.8cm,yshift = -0.7cm] at (A) {$\sigma_3$};
\node[xshift = 2.9cm,yshift = 0.2cm] at (A) {$\sigma_4$};
\node[xshift = 5.1cm,yshift = 0cm] at (A) {$\sigma_5$};
\end{tikzpicture}
\caption{A $S_4$-labeling $\gamma$ of a graph $G$ with $\gamma =(\sigma_1,\sigma_2,\sigma_3, \sigma_4,\sigma_5) = ((1 \thickspace 2),\thickspace (1 \thickspace 2)(3 \thickspace 4),\thickspace (1\thickspace 3 \thickspace 2),\thickspace (1\thickspace 2 \thickspace 3 \thickspace 4),\thickspace (1 \thickspace 2 \thickspace 3))$.}
\label{fig:grplabeling}
\end{figure}
\begin{figure}
\begin{tikzpicture}
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\coordinate (D1) at (6,0);
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\draw [very thick,red!80] (A1) -- (B2);
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\node[xshift = -0.3cm] at (A1) {$a_1$};
\node[xshift = -0.3cm] at (A2) {$a_2$};
\node[xshift = -0.3cm] at (A3) {$a_3$};
\node[xshift = -0.3cm] at (A4) {$a_4$};
\node[yshift = -0.3cm] at (B1) {$b_1$};
\node[yshift = -0.3cm] at (B2) {$b_2$};
\node[yshift = -0.3cm] at (B3) {$b_3$};
\node[yshift = -0.3cm] at (B4) {$b_4$};
\node[xshift = 0.3cm, yshift = 0.1cm] at (C1) {$c_1$};
\node[xshift = 0.3cm, yshift = 0.1cm] at (C2) {$c_2$};
\node[xshift = 0.3cm, yshift = 0.1cm] at (C3) {$c_3$};
\node[xshift = 0.3cm, yshift = -0.05cm] at (C4) {$c_4$};
\node[xshift = -0.3cm] at (D1) {$d_1$};
\node[xshift = -0.3cm, yshift = -0.1cm] at (D2) {$d_2$};
\node[xshift = -0.3cm, yshift = -0.1cm] at (D3) {$d_3$};
\node[xshift = -0.3cm, yshift = -0.1cm] at (D4) {$d_4$};
\end{tikzpicture}
\caption{The $4$-sheeted covering graph $H$ corresponding to the $S_4$-labeling $\gamma$ of $G$ in Figure \ref{fig:grplabeling}.}
\label{fig:covergraph}
\end{figure}\noindent
Consider the $d$-dimensional representation $\pi:S_d \to \text{GL}_d(\mathbb{C})$ of the symmetric group $S_d$ mapping every $\sigma \in S_d$ to its corresponding permutation matrix. The representation $\pi$ is reducible since the $1$-dimensional space $\langle \mathbf{1} \rangle \leq \mathbb{C}^d$, where $\mathbf{1} = (1,\dots, 1)$, is invariant under the action of $\pi$. The action of $\pi$ on the orthogonal complement $\langle \mathbf{1} \rangle^{\perp}$ is an irreducible $(d-1)$-dimensional representation called the \textit{standard representation}, denoted $\text{std}:S_d \to \text{GL}_{d-1}(\mathbb{C})$. As outlined in \cite{HPS}, every $d$-covering $H$ of $G$ corresponds uniquely to a $(S_d, \text{std})$-covering of $G$.
\subsection{Stable polynomials}
A polynomial $f(\mathbf{x}) \in \mathbb{C}[x_1,\dots, x_n]$ is said to be \textit{stable} if $f(x_1,\dots, x_n) \neq 0$ whenever $\text{Im}(x_j) > 0$ for all $j = 1,\dots, n$. By convention we also regard the zero polynomial to be stable. A stable polynomial with only real coefficients is said to be \textit{real stable}. Note that univariate real stable polynomials are precisely the real-rooted polynomials (i.e. real polynomials in one variable with all zeros in $\mathbb{R}$). Thus stability may be regarded as a multivariate generalization of real-rootedness. Below we collect a few facts about stable polynomials which are relevant for the forthcoming sections. For a more comprehensive background we refer to the survey by Wagner \cite{Wag} and references therein.
A common technique for proving that a polynomial $f(\mathbf{x})$ is stable is to realize $f(\mathbf{x})$ as the image of a known stable polynomial under a stability preserving linear transformation.
Stable polynomials satisfy several basic closure properties, among them are diagonalization $f \mapsto f(\mathbf{x})|_{x_i = x_j}$ for $i,j \in [n]$ and differentiation $f \mapsto \partial_i f(\mathbf{x})$ where $\partial_i \coloneqq \frac{\partial}{\partial x_i}$. The following theorem by Lieb and Sokal provides the construction for a large family of stability preserving linear transformations.
\begin{theorem}[Lieb-Sokal \cite{LS}] \label{thm:liebsokal}
If $f(x_1,\dots, x_n) \in \mathbb{C}[x_1,\dots, x_n]$ is a stable polynomial, then $f\left( \partial_1,\dots, \partial_n \right )$ is a stability preserving linear operator.
\end{theorem} \noindent
Borcea and Br\"and\'en \cite{BBa} gave a complete characterization of the linear operators preserving stability. The following is the transcendental characterization of stability preservers on infinite-dimensional complex polynomial spaces. Define the \textit{complex Laguerre-P\'olya class} to be the class of entire functions in $n$ variables that are limits, uniformly on compact sets of stable polynomials in $n$ variables.
Throughout we will use the following multi-index notation
\[
\mathbf{x}^{S} \coloneqq \prod_{i\in S}^n x_i \hspace{1cm} \mathbf{x}^{\boldsymbol{\alpha}} \coloneqq \prod_{i=1}^n x_i^{\alpha_i}, \hspace{1cm} \boldsymbol{\alpha}! \coloneqq \prod_{i=1}^n \alpha_i!,
\]
where $S\subseteq [n]$ and $\boldsymbol{\alpha} = (\alpha_i) \in \mathbb{N}^n$.
\begin{theorem}[Borcea-Br\"and\'en \cite{BBa}] \label{thm:borbra}
Let $T: \mathbb{C}[x_1,\dots, x_n] \to \mathbb{C}[x_1,\dots, x_n]$ be a linear operator. Then $T$ preserves stability if and only if either
\begin{enumerate}
\item $T$ has range of dimension at most one and is of the form
\[
T(f) = \alpha(f) P,
\]
where $\alpha$ is a linear functional on $\mathbb{C}[x_1,\dots, x_n]$ and $P$ is a stable polynomial, or
\item
\[
G_T(\mathbf{x}, \mathbf{y}) \coloneqq \sum_{\alpha \in \mathbb{N}^n} (-1)^{\alpha} T(\mathbf{x}^{\boldsymbol{\alpha}}) \frac{\mathbf{y}^{\boldsymbol{\alpha}}}{\boldsymbol{\alpha}!}
\]
belongs to the Laguerre-P\'olya class.
\end{enumerate}
\end{theorem} \noindent
A polynomial $f(x_1,\dots, x_n)$ is said to be \textit{multiaffine} if each variable $x_i$ occurs with degree at most one in $f(x_1,\dots, x_n)$, and is called \textit{symmetric} if $f(x_{\sigma(1)},\dots, x_{\sigma(n)}) = f(x_1,\dots,x_n)$ for all $\sigma \in S_n$. The Grace-Walsh-Szeg\"o coincidence theorem is a cornerstone in the theory of stable polynomials frequently used to depolarize symmetries before checking stability. One version of it is stated below, see \cite{BBa, Wag} for modern references and alternative proofs.
\begin{theorem}[Grace-Walsh-Szeg\"o \cite{Gra,Sze, Wal}] \label{thm:gws}
Let $f(x_1,\dots, x_n) \in \mathbb{C}[x_1,\dots, x_n]$ be a symmetric and multiaffine polynomial. Then $f(x_1,\dots, x_n)$ is stable if and only if $f(x,\dots, x)$ is stable.
\end{theorem} \noindent
\section{Stability of multivariate $d$-matching polynomials}
A \textit{matching} of an undirected graph $G$ is a subset $M \subseteq E(G)$ such that no two edges in $M$ share a common vertex. Let $V(M) \coloneqq \bigcup_{\{i,j\} \in M} \{i,j \}$ denote the set of vertices in the matching $M$.
For $d \in \mathbb{Z}_{\geq 1}$, the \textit{$d$-matching polynomial} of $G$ is defined by
\[
\mu_{d,G}(x) \coloneqq \mathbb{E}_{H \in \mathcal{C}_{d,G}} \mu_H(x),
\]
where
\[
\mu_G(x) \coloneqq \sum_{i=0}^{\lfloor n/2 \rfloor} (-1)^i m_i x^{n-2i} \in \mathbb{Z}[x]
\]
and $m_i$ denotes the number of matchings in $G$ of size $i$ with $m_0=1$. In particular if $d=1$, then $\mu_{d,G}(x)$ coincides with the conventional matching polynomial $\mu_G(x)$. The following results are proved in \cite{HPS}.
\begin{theorem}[Hall-Puder-Sawin \cite{HPS}] \label{thm:expdmatch}
Let $\Gamma$ be a finite group and $\pi:\Gamma \to \text{GL}_d(\mathbb{C})$ be an irreducible representation such that $\pi(\Gamma)$ is a complex reflection group, i.e., $\pi(\Gamma)$ is generated by pseudo-reflections. If $G$ is a finite connected multigraph, then
\begin{align} \label{thm:hpsavg}
\mathbb{E}_{\gamma \in \mathcal{C}_{\Gamma,G}} \det(xI - A_{\gamma, \pi}) =\mu_{d,G}(x).
\end{align}
\end{theorem} \noindent
\begin{remark}
Remarkably the expected characteristic polynomial in \eqref{thm:hpsavg} depends only on the dimension $d$ of the irreducible representation $\pi$ and not on the particular choice of group $\Gamma$, nor the specifics of the map $\pi$. Real-rooted expected characteristic polynomials have seen a surge of interest recently in light of the Kadison-Singer problem and Ramanujan coverings, see e.g. \cite{AG,HPS,MSS,MSSc, MSSb,MSSd}.
\end{remark} \noindent
\begin{example}
A classical result due to Godsil and Gutman \cite{GG} states that if $A = (a_{ij})$ is the adjacency matrix of a finite simple undirected graph $G$, then
\[
\mathbb{E}_{\mathbf{s}} \det(xI - A^{\mathbf{s}}) = \mu_G(x),
\]
where $A^{\mathbf{s}}_{ij} \coloneqq s_{e}a_{ij}$ for all $e = \{i,j\} \in E(G)$ and $\mathbf{s} = (s_{e})_{e \in E(G)} \in \{\pm 1 \}^{E(G)}$. In other words, the expected characteristic polynomial over all signings of $G$ equals the matching polynomial of $G$.
In the language of Hall, Puder and Sawin this corresponds to taking $\Gamma = \mathbb{Z}/2\mathbb{Z}$ and $\pi = \text{sgn}:\mathbb{Z}/2\mathbb{Z} \to \text{GL}_1(\mathbb{C})$ to be the sign representation in Theorem \ref{thm:hpsavg}.
\end{example}
Generalizing and extending the following theorem will be the main focus of this section.
\begin{theorem}[Hall-Puder-Sawin \cite{HPS}] \label{thm:dmatchrealroots}
If $G$ is a finite multigraph with no loops, then $\mu_{d,G}(x)$ is real-rooted.
\end{theorem} \noindent
\begin{remark}
Hall, Puder and Sawin also showed that the roots of $\mu_{d,G}(x)$ are contained inside the Ramanujan interval of $G$ (see \cite{HPS}).
\end{remark} \noindent
Define the \textit{multivariate $d$-matching polynomial} of $G$ by
\[
\mu_{d,G}(\mathbf{x}) \coloneqq \mathbb{E}_{H \in \mathcal{C}_{d,G}} \mu_H(\mathbf{x}),
\]
where
\[
\mu_G(\mathbf{x}) \coloneqq \sum_{M} (-1)^{|M|} \prod_{i \in [n]\setminus V(M)} x_i,
\]
and the sum runs over all matchings in $G$.
The real-rootedness of $\mu_{d,G}(x)$ was proved indirectly in \cite{HPS} by considering a limit of interlacing families converging to the left-hand side in Theorem \ref{thm:expdmatch}. In this section we use a more direct approach for proving the real-rootedness of $\mu_{d,G}(x)$. In fact we prove something stronger, namely that $\mu_{d,G}(\mathbf{x})$ is stable. Our proof also holds for graphs with loop edges, thus removing the redundant hypothesis in Theorem \ref{thm:dmatchrealroots}.
Coverings of graphs with loop edges have interesting properties.
In particular, consider the $|\Gamma|$-dimensional \textit{regular representation} $\text{reg}: \Gamma \to \text{GL}_{|\Gamma|}(\mathbb{C})$ sending an element $g \in \Gamma$ to the permutation matrix afforded by $g$ acting on $\Gamma$ through left translation $h \mapsto gh$. The \textit{bouquet graph} $B_r$ is the graph consisting of a single vertex with $r$ loop edges. A $(\Gamma,\text{reg})$-covering $A_{\gamma, \text{reg}}$ of $B_r$ is equivalent to the Cayley graph of $\Gamma$ with respect to the set $\gamma(E(B_r))$. In this sense $(\Gamma,\text{reg})$-coverings of finite multigraphs with loops generalize Cayley graphs of finite groups.
\begin{figure}
\begin{tikzpicture}[scale=5]
\draw[very thick, blue!80] (0,0) to[in=40+35,out=-40+35,loop] (0,0);
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\fill (0,0) circle[radius=0.6pt];
\node[xshift=-1.45cm, yshift=0.9cm] at (0,0) {$\sigma_1$};
\node[xshift=1.45cm,yshift=0.9cm] at (0,0) {$\sigma_2$};
\end{tikzpicture}
\begin{tikzpicture}[scale=1.1]
\coordinate (A) at (0+0.8, -0.1);
\coordinate (B) at (1.6+0.8, -0.3);
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\draw[very thick,blue!80] (D) to[bend left=10, bend angle=45] node {\tikz \draw[-triangle 90] (0,0) -- +(.1,0);} (A);
\draw[very thick,blue!80] (E) to[bend left=10, bend angle=45] node {\tikz \draw[-triangle 90] (0,0) -- +(.1,0);} (B);
\draw[very thick,blue!80] (F) to[bend left=10, bend angle=45] node {\tikz \draw[-triangle 90] (0,0) -- +(.1,0);} (C);
\end{scope}
\begin{scope}[very thick, red!80, every node/.style={sloped,allow upside down}]
\draw (A)-- node {\tikz \draw[-triangle 90] (0,0) -- +(.1,0);} (C);
\draw (C)-- node {\tikz \draw[-triangle 90] (0,0) -- +(.1,0);} (B);
\draw (B)-- node {\tikz \draw[-triangle 90] (0,0) -- +(.1,0);} (A);
\end{scope}
\begin{scope}[very thick, red!80, every node/.style={sloped,allow upside down}]
\draw (D)-- node {\tikz \draw[-triangle 90] (0,0) -- +(.1,0);}(E);
\draw (E)-- node {\tikz \draw[-triangle 90] (0,0) -- +(.1,0);} (F);
\draw (F)-- node {\tikz \draw[-triangle 90] (0,0) -- +(.1,0);} (D);
\end{scope}
\draw[fill=black] (A) circle (0.08cm);
\draw[fill=black] (B) circle (0.08cm);
\draw[fill=black] (C) circle (0.08cm);
\draw[fill=black] (D) circle (0.08cm);
\draw[fill=black] (E) circle (0.08cm);
\draw[fill=black] (F) circle (0.08cm);
\node[xshift=-0.2cm] at (A) {$\iota$};
\node[xshift=-0.6cm] at (C) {$(1 \thinspace 2 \thinspace 3)$};
\node[xshift=0.5cm, yshift=-0.3cm] at (B) {$(1 \thinspace 3 \thinspace 2)$};
\node[xshift=-0.4cm, yshift=0.2cm] at (D) {$(1 \thinspace 2)$};
\node[xshift=0.5cm] at (E) {$(1 \thinspace 3)$};
\node[xshift=-0.4cm, yshift = 0.2cm] at (F) {$(2 \thinspace 3)$};
\end{tikzpicture}
\caption{A $S_3$-labeling $\gamma = (\sigma_1,\sigma_2) = ((1 \thinspace 2 \thinspace 3), (1 \thinspace 2))$ of the bouquet graph $B_2$ (left) and the Cayley graph of $S_3$ with respect to $\{(1 \thinspace 2 \thinspace 3), (1 \thinspace 2) \}$ (right). }
\label{fig:cayley}
\end{figure}
\begin{example}
Let $\Gamma = S_3$, $G = B_2$ and $\pi = \text{reg}:S_3 \to \text{GL}_6(\mathbb{C})$. Consider the $S_3$-labeling $\gamma = (\sigma_1,\sigma_2) = ((1\thinspace 2 \thinspace 3), (1 \thinspace 2))$ of $B_2$ as in Figure \ref{fig:cayley} (left). Then
\[
A \coloneqq \text{reg}(\sigma_1)+ \text{reg}(\sigma_2) = \bordermatrix{
& \iota & (1 \thinspace 2) & (1 \thinspace 3) & (2 \thinspace 3) & (1 \thinspace 2 \thinspace 3) & (1 \thinspace 3 \thinspace 2) \cr
\iota & 0 & \color{blue}1 & 0 & 0 & \color{red}1 & 0 \cr
(1 \thinspace 2) & \color{blue}1 & 0 & \color{red}1 & 0 & 0 & 0 \cr
(1 \thinspace 3) & 0 & 0 & 0 & \color{red}1 & 0 & \color{blue}1 \cr
(2 \thinspace 3) & 0 & \color{red}1 & 0 & 0 & \color{blue}1 & 0 \cr
(1 \thinspace 2 \thinspace 3) & 0 & 0 & 0 & \color{blue}1 & 0 & \color{red}1 \cr
(1 \thinspace 3 \thinspace 2) & \color{red}1 & 0 & \color{blue}1 & 0 & 0 & 0
},
\]
is the adjacency matrix of the Cayley graph of $S_3$ with respect to the set $\{\sigma_1,\sigma_2 \}$, see Figure \ref{fig:cayley} (right), and the $(S_3,\text{reg})$-covering $A_{\gamma, \text{reg}}$ is given by $A + A^{T}$.
\end{example} \noindent
Choe, Oxley, Sokal and Wagner \cite{COSW} (see also \cite{BBb}) consider the \textit{multi-affine part} operator
\begin{align*}
\text{MAP}: \mathbb{C}[x_1,\dots, x_n] &\to \mathbb{C}[x_1,\dots, x_n] \\
\sum_{\boldsymbol{\alpha} \in \mathbb{N}^n} a(\boldsymbol{\alpha}) \mathbf{x}^{\boldsymbol{\alpha}} &\mapsto \sum_{\boldsymbol{\alpha} : \alpha_i \leq 1, i \in [n]} a(\boldsymbol{\alpha}) \mathbf{x}^{\boldsymbol{\alpha}}
\end{align*} \noindent
and note that it is a stability preserving linear operator. Indeed the symbol
\[
G_{\text{MAP}}(\mathbf{x}, \mathbf{y}) = \sum_{\boldsymbol{\alpha} \in \mathbb{N}^n} (-1)^{\boldsymbol{\alpha}} \text{MAP}(\mathbf{x}^{\boldsymbol{\alpha}}) \frac{\mathbf{y}^{\boldsymbol{\alpha}}}{\boldsymbol{\alpha}!} = \sum_{\substack{ \boldsymbol{\alpha} \in \mathbb{N}^n\\ \alpha_i \leq 1 }} (-1)^{\boldsymbol{\alpha}} \mathbf{x}^{\boldsymbol{\alpha}} \mathbf{y}^{\boldsymbol{\alpha}} = \prod_{i=1}^n (1-x_iy_i),
\]
is stable being a product of stable polynomials. Since the range of $\text{MAP}$ has dimension greater than one, it follows that $\text{MAP}$ preserves stability by Theorem \ref{thm:borbra}. Given the identity
\begin{align} \label{eq:subgraphgen}
P_G(\mathbf{x}) \coloneqq \sum_{E \subseteq E(G)} (-1)^{|E|} \prod_{i=1}^n x_i^{\text{deg}_{G[E]}(i)} = \prod_{\{i,j\} \in E(G)} (1- x_ix_j),
\end{align} \noindent
where $G[E]$ is the subgraph of $G$ induced by $E \subseteq E(G)$ and $\text{deg}_{G[E]}(i)$ denotes the degree of $i$ in $G[E]$, we have that
\[
\text{MAP} [P_G(\mathbf{x})] = \mu_G(\mathbf{x}),
\]
and hence that $\mu_G(\mathbf{x})$ is stable being the image of a stable polynomial under $\text{MAP}$. This result is also known as the Heilmann-Lieb theorem \cite{HL}.
By using Theorem \ref{thm:gws} and the stability preserving linear operator $\text{MAP}$ we will show below that $\mu_{d,G}(\mathbf{x})$ is stable.
\begin{theorem} \label{thm:multidmatchpoly}
If $G$ is a finite multigraph (possibly with loops), then $\mu_{d,G}(\mathbf{x})$ is stable for all $d \geq 1$.
\end{theorem}
\begin{proof}
For a $d$-covering $\sigma:E(G) \to S_d$, let
\begin{align*}
P_{\sigma,G}(\mathbf{x}) &\coloneqq \left (\prod_{e \in E^+(G) \setminus E_{\circ}^{+}(G)} \prod_{k=1}^d (1 - x_{h(e)k} x_{t(e) \sigma_e(k)}) \right ) \times \\ & \hspace{0.5cm} \left ( \prod_{e \in E_{\circ}^{+}(G)} \prod_{k : \sigma_e(k) \neq k} (1 - x_{h(e)k} x_{t(e) \sigma_e(k)}) \right ),
\end{align*} \noindent
where $E_{\circ}^{+}(G) \coloneqq \{ e \in E^{+}(G) : h(e) = t(e) \}$ denotes the set of positively oriented loops in $G$.
Since no matching may contain loops we have excluded the factors $(1-x_{ik}^2)$ from the subgraph generating polynomial $P_H(\mathbf{x})$ in (\ref{eq:subgraphgen}) where $H$ is the covering graph corresponding to $\sigma$. This explains the form of $P_{\sigma, G}(\mathbf{x})$. It follows that
\[
\text{MAP}[P_{\sigma, G}(\mathbf{x})] = \mu_H(\mathbf{x}).
\]
We have
\begin{align*}
\mathbb{E}_{\sigma \in \mathcal{C}_{d, G}} P_{\sigma, G}(\mathbf{x}) &= \sum_{\sigma \in \mathcal{C}_{d, G}} \frac{1}{|\mathcal{C}_{d, G}|} P_{\sigma, G}(\mathbf{x}) \\ &= \frac{1}{|\mathcal{C}_{d, G}|} \left (\prod_{e \in E^{+}(G) \setminus E_{\circ}^{+}(G)} \sum_{\sigma_e \in S_d} \prod_{k=1}^d (1 - x_{h(e)k} x_{t(e) \sigma_e(k)}) \right ) \times \\ & \hspace*{1.4cm}\left ( \prod_{e \in E_{\circ}^{+}(G)} \sum_{\sigma_e \in S_d} \prod_{k : \sigma_e(k) \neq k} (1 - x_{h(e)k} x_{t(e) \sigma_e(k)}) \right ).
\end{align*} \noindent
For $e\in E^{+}(G) \setminus E_{\circ}^{+}(G)$ the polynomials
\[
f_e(\mathbf{x}) = \sum_{\sigma_e \in S_d} \prod_{k=1}^d (1-x_{h(e)k}x_{t(e)\sigma_e(k)}),
\]
are symmetric and multiaffine polynomials in the two sets of variables
\[
\{ x_{h(e)k} : 1 \leq k \leq d \} \hspace{0.5cm} \text{and} \hspace{0.5cm} \{ x_{t(e)k} : 1 \leq k \leq d \},
\]
respectively. By Theorem \ref{thm:gws} we have that $f_e(\mathbf{x})$ is stable if and only if
\[
\sum_{\sigma_e \in S_d} \prod_{k=1}^d (1-xy) = d!(1-xy)^d,
\]
is stable, the latter of which is clear. Similary if $e\in E_{\circ}^{+}(G)$, then $f_e(\mathbf{x})$ is symmetric and multiaffine in the set of variables $\{x_{h(e)k} : 1\leq k\leq d \}$, so checking stability of $f_e(\mathbf{x})$ reduces by Theorem \ref{thm:gws} to checking the stability of $d!(1-x^2)^d$, which is again clear.
Hence $\mathbb{E}_{\sigma \in \mathcal{C}_{S_d, G}} P_{\sigma, G}(\mathbf{x})$ is stable being a product of stable polynomials. Finally we have
\begin{align*}
\text{MAP}\left [ \mathbb{E}_{\sigma \in \mathcal{C}_{d, G}} P_{\sigma, G}(\mathbf{x}) \right ] &= \mathbb{E}_{\sigma \in \mathcal{C}_{d, G}} \text{MAP} [P_{\sigma, G}(\mathbf{x})] \\ &= \mathbb{E}_{H \in \mathcal{C}_{d, G}} \mu_H(\mathbf{x}) \\ &= \mu_{d,G}(\mathbf{x}).
\end{align*} \noindent
Hence $\mu_{d,G}(\mathbf{x})$ is stable.
\end{proof}
\begin{corollary}
If $G$ is a finite multigraph (possibly with loops), then $\mu_{d,G}(x)$ is real-rooted for all $d \geq 1$.
\end{corollary}
\begin{proof}
Follows by putting $\mathbf{x} = (x,\dots, x)$ in Theorem \ref{thm:multidmatchpoly}
\end{proof} \noindent
\section{Stable expected matching polynomials over induced subgraphs}
In the previous section we considered stable averages of multivariate matching polynomials over the set of $d$-sheeted covering graphs of $G$. In this section we consider stable averages over (vertex-) induced subgraphs of $G$. To this end, if $S \subseteq [n]$, let $G[S]$ denote the subgraph of $G$ induced by the vertices in $S$.
Let $\mathbb{P}$ be a probability distribution on the power set $\mathcal{P}([n]) \coloneqq \{S : S \subseteq [n] \}$. The polynomial
\begin{align*}
Z_{\mathbb{P}}(\mathbf{x}) = \sum_{S \subseteq [n]} \mathbb{P}(S) \mathbf{x}^S \in \mathbb{R}[x_1,\dots, x_n],
\end{align*}
is called the \textit{partition function} of $\mathbb{P}$. The probability distribution $\mathbb{P}$ is called \textit{Rayleigh} if
\begin{align}\label{eq:rayleigh}
Z_{\mathbb{P}}(\mathbf{x}) \frac{\partial^2 Z_{\mathbb{P}}(\mathbf{x})}{\partial x_i \partial x_j} \leq \frac{\partial Z_{\mathbb{P}}(\mathbf{x})}{\partial x_i} \frac{\partial Z_{\mathbb{P}}(\mathbf{x})}{\partial x_j}
\end{align} \noindent
for all $\mathbf{x} \in \mathbb{R}_{+}^n$, $1 \leq i,j \leq n$ and is called \textit{strong Rayleigh} if \eqref{eq:rayleigh} holds for all $\mathbf{x}\in \mathbb{R}^n$, $1 \leq i,j \leq n$.
\begin{theorem}[Br\"and\'en \cite{Bra}] \label{thm:brandenrayleigh}
A probability distribution $\mathbb{P}$ is strong Rayleigh if and only if $Z_{\mathbb{P}}(\mathbf{x})$ is stable.
\end{theorem}
\begin{proposition} \label{prop:rayleighstable}
Let $G = (V(G), E(G))$ be a finite undirected graph on $[n]$ and let $\mathbb{P}$ be a probability distribution on $\mathcal{P}([n])$. If $\mathbb{P}$ is strong Rayleigh, then
$\mathbb{E}_{S \subseteq [n]}^{\mathbb{P}} \mu_{G[S]}(\mathbf{x})$ is stable.
\end{proposition}
\begin{proof}
By Theorem \ref{thm:liebsokal} the linear operator
\[
T_G \coloneqq \prod_{\{i,j\} \in E(G)} (1 - \partial_i\partial_j)
\]
preserves stability. Moreover it is easy to see that for $S \subseteq [n]$,
\[
T_G(\mathbf{x}^S) = \mu_{G[S]}(\mathbf{x}).
\]
If $\mathbb{P}$ is strong Rayleigh, then $Z_{\mathbb{P}}(\mathbf{x})$ is stable by Theorem \ref{thm:brandenrayleigh}. Hence
\begin{align*}
T_G(Z_{\mathbb{P}}(\mathbf{x})) &= \sum_{S \subseteq [n]} \mathbb{P}(S) T_G(\mathbf{x}^S) \\ &= \sum_{S \subseteq [n]} \mathbb{P}(S) \mu_{G[S]}(\mathbf{x}) \\ &= \mathbb{E}_{S \in \subseteq [n]}^{\mathbb{P}} \mu_{G[S]}(\mathbf{x}),
\end{align*} \noindent
is stable.
\end{proof}
\begin{corollary} \label{cor:avgreal}
If $\mathbb{P}$ is a strong Rayleigh probability distribution, then $\mathbb{E}_{S \subseteq [n]}^{\mathbb{P}} \mu_{G[S]}(x)$ is real-rooted.
\end{corollary}
\begin{example}
The following example demonstrates that the converse to Proposition \ref{prop:rayleighstable} is false. Consider the graph $G = \bullet\textendash \bullet$ on two vertices and one edge. If $\mathbb{P}$ is a probability distribution with $\mathbb{P}(\{1,2\}) = a$, $\mathbb{P}(\{1\}) = b$, $\mathbb{P}(\{1,2\}) = c$ and $\mathbb{P}(\emptyset) = d$, then
\begin{align*}
\mathbb{E}_{S \subseteq [2]}^{\mathbb{P}} \mu_{G[S]}(\mathbf{x}) &= a \mu_G(\mathbf{x}) + b\mu_{G[1]}(\mathbf{x}) + c\mu_{G[2]}(\mathbf{x}) + d\mu_{G[\emptyset]}(\mathbf{x}) \\ &= a(-1+x_1x_2) + bx_1 + cx_2 + d,
\end{align*} \noindent
which is stable if and only of $bc - a(-a+d) \geq 0$. On the other hand
\[
Z_{\mathbb{P}}(\mathbf{x}) = ax_1x_2 + bx_1 + cx_2+d,
\]
is stable if and only if $bc-ad\geq 0$. Hence there exists probability distributions $\mathbb{P}$ which are not strong Rayleigh for which $\mathbb{E}_{S \subseteq [n]}^{\mathbb{P}} \mu_{G[S]}(\mathbf{x})$ is stable. An interesting question would be to characterize the probability distributions for which $\mathbb{E}_{S \subseteq [n]}^{\mathbb{P}} \mu_{G[S]}(\mathbf{x})$ is stable.
\end{example}
\begin{example}
A natural probability distribution $\mathbb{P}$ on the set of induced subgraphs of $G$ is the Bernoulli distribution $\mathbb{B}$ where a vertex $i \in [n]$ is selected independently with probability $p_i$ and not selected with probability $1-p_i$. Note that $\mathbb{B}$ is a strong Rayleigh probability distribution since
\[
Z_{\mathbb{B}}(\mathbf{x}) = \sum_{S \subseteq [n]} \mathbb{B}(S) \mathbf{x}^S = \sum_{S \subseteq [n]} \prod_{i \in S}p_i \prod_{i \in [n] \setminus S} (1-p_i) \mathbf{x}^S = \prod_{i=1}^n ((1-p_i)+ p_ix_i),
\]
is stable.
Hence $\mathbb{E}_{S \subseteq [n]}^{\mathbb{B}} \mu_{G[S]}(\mathbf{x})$ is stable by Proposition \ref{prop:rayleighstable}.
\end{example} \noindent
Next we shall provide bounds for the real roots of $\mathbb{E}_{S \subseteq [n]}^{\mathbb{P}} \mu_{G[S]}(x)$.
Let $i \in V(G)$ and define a graph $U_i(G)$ with vertex set being the set of all non-backtracking walks in $G$ starting from $i$, i.e., sequences $(i_0,i_1,\dots, i_k)$ such that $i_0 = i$, $i_r$ and $i_{r+1}$ are adjacent and $i_{r+1} \neq i_{r-1}$. Two such walks are connected by an edge in $U_i(G)$ if one walk extends the other by one vertex, i.e., $(i_0,\dots, i_k, i_{k+1})$ is adjacent to $(i_0,\dots, i_k)$. The graph thus constructed is a tree that covers $G$. It is called the \textit{universal covering tree} of $G$. The universal covering tree $U_i(G)$ of $G$ is unique up to isomorphism and has the property that it covers every other covering of $G$. Thus we henceforth remove reference to the root $i$ and write $U(G)$ for the universal covering tree of $G$. The tree $U(G)$ is countably infinite, unless $G$ is a finite tree, in which case $U(G) = G$.
The \textit{spectral radius} $r(G)$ of a finite graph $G$ is the largest absolute eigenvalue of the adjacency matrix $A_G$ of $G$.
By a theorem of Mohar \cite{Moh} the spectral radius of an infinite graph $U$ can be defined as follows,
\[
r(U) \coloneqq \text{sup}\{ r(G) : \text{$G$ is a finite induced subgraph of $U$} \}.
\]
If $G$ is a finite undirected graph, then let $\rho(G) \coloneqq r(U(G))$ denote the spectral radius of its universal covering tree. Say that a probability distribution $\mathbb{P}$ on $\mathcal{P}([n])$ has \textit{constant parity} if the set $\{ |S| : S\subseteq [n], \thickspace \mathbb{P}(S) > 0 \}$ consists of numbers with the same parity (i.e. are either all odd or all even).
\begin{proposition} \label{prop:univspanbnd}
Let $G$ be a finite undirected graph with $n$ vertices and $\mathbb{P}$ a probability distribution on $\mathcal{P}([n])$. Then the real roots of $\mathbb{E}_{S \subseteq [n]}^{\mathbb{P}} \mu_{G[S]}(x)$ are bounded above by $\rho(G)$. Moreover if $\mathbb{P}$ has constant parity, then the real roots of $\mathbb{E}_{S \subseteq [n]}^{\mathbb{P}} \mu_{G[S]}(x)$ are contained in $[-\rho(G), \rho(G)]$.
\end{proposition}
\begin{proof}
Let $S \subseteq [n]$. There is a clear injective embedding of $U(G[S])$ into $U(G)$ such that any finite induced subgraph of $U(G[S])$ is an induced subgraph of $U(G)$. Therefore $\rho(G[S])\leq \rho(G)$. Heilmann and Lieb \cite{HL} showed that for any finite graph $G$, the roots of $\mu_G(x)$ are contained in $[-\rho(G), \rho(G)]$. Therefore $\mathbb{E}_{S \subseteq [n]}^{\mathbb{P}} \mu_{G[S]}(x) > 0$ for all $x \in (\rho(G), \infty)$, being a convex combination of monic polynomials with the same property. Hence the real roots of the expectation are bounded above by $\rho(G)$. If $\mathbb{P}$ also has constant parity, then $\mathbb{E}_{S \subseteq [n]}^{\mathbb{P}} \mu_{G[S]}(x)$ is a convex combination of monic polynomials with same degree parity and are therefore, by above, strictly positive or strictly negative on the interval $(-\infty, -\rho(G))$. Hence the real roots are contained in $[-\rho(G), \rho(G)]$.
\end{proof}
\begin{corollary}
Let $G$ be a finite undirected graph on $n$ vertices and $k \in [n]$. Then the uniform average of all matching polynomials over the set of induced size $k$-subgraphs of $G$ is a real-rooted polynomial with all roots contained in the interval $[-\rho(G), \rho(G)]$.
\end{corollary}
\begin{proof}
Let $\mathbb{P}$ be the probability distribution on $\mathcal{P}([n])$ with uniform support on $\binom{[n]}{k}$. Then
\[
Z_{\mathbb{P}}(\mathbf{x}) = \frac{1}{\binom{n}{k}} e_k(\mathbf{x}),
\]
where $e_k(\mathbf{x})$ denotes the elementary symmetric polynomial of degree $k$. The polynomial $e_k(\mathbf{x})$ is stable, e.g. by Theorem \ref{thm:gws}. Therefore $\mathbb{P}$ is a strong Rayleigh probability distribution by Theorem \ref{thm:brandenrayleigh}, so the statement follows by Corollary \ref{cor:avgreal} and Proposition \ref{prop:univspanbnd}.
\end{proof}
\section{Stable relaxed matching polynomials}
A hypergraph $H= (V(H), E(H))$ is a set of vertices $V(H) = [n]$ together with a family of subsets $E(H)$ of $V(H)$ called \textit{hyperedges} (or \textit{edges} for short). The \textit{degree} of a vertex $i \in V(H)$ is defined as $\text{deg}_H(i) \coloneqq |\{e \in E(H) : i \in e \}|$.
In analogy with graph matchings, a matching in a hypergraph consists of a subset of edges with empty pairwise intersection.
Although the matching polynomial $\mu_G(x)$ of a graph $G$ is real-rooted, the analogous polynomial for hypergraphs is not real-rooted in general, see e.g. \cite{GZM}. From the point of view of real-rootedness we consider a weaker notion of matchings that provide a natural generalization of the real-rootedness property of $\mu_G(x)$ to hypergraphs.
Let $H = (V(H),E(H))$ be a hypergraph.
Define a \textit{relaxed matching} in $H$ to be a collection $M = (S_e)_{e \in E}$ of edge subsets such that $E \subseteq E(H)$, $S_e \subseteq e$, $|S_e| > 1$ and $S_e \cap S_{e'} = \emptyset$ for all pairwise distinct $e,e' \in E$ (see Figure \ref{fig:relaxmatch}).
\begin{remark}
If $H$ is a graph then the concept of relaxed matching coincides with the conventional notion of graph matching. Note also that a conventional hypergraph matching is a relaxed matching $M = (S_e)_{e \in E}$ for which $S_e = e$ for all $e \in E$.
\end{remark}
\begin{remark}
The subsets $S_e$ in the relaxed matching are labeled by the edge they are chosen from in order to avoid ambiguity. However if $H$ is a \textit{linear hypergraph}, that is, the edges pairwise intersect in at most one vertex, then the subsets uniquely determine the edges they belong to and therefore no labeling is necessary. Graphs and finite projective geometries (viewed as hypergraphs) are examples of linear hypergraphs.
\end{remark}
\begin{figure}
\begin{tikzpicture}
\coordinate (v1) at (0.5,1.7);
\coordinate (v2) at (2.2,0.75);
\coordinate (v3) at (-1.5,1.5);
\coordinate (v4) at (-2.2,0);
\coordinate (v5) at (-0.5,0.5);
\coordinate (v6) at (-1.5,-1.5);
\coordinate (v7) at (-0.5,-0.65);
\coordinate (v8) at (0.8,-1.2);
\coordinate (v9) at (2.3,-0.8);
\draw (0, 0) node[inner sep=0] {\includegraphics[width=0.5\linewidth]{hyperbase.pdf}};
\draw[fill=red!80] (v1) circle (0.13cm);
\draw[fill=black] (v1) circle (0.08cm);
\draw[fill=red!80] (v2) circle (0.13cm);
\draw[fill=black] (v2) circle (0.08cm);
\draw[fill=black] (v3) circle (0.08cm);
\draw[fill=green!80] (v4) circle (0.13cm);
\draw[fill=black] (v4) circle (0.08cm);
\draw[fill=green!80] (v5) circle (0.13cm);
\draw[fill=black] (v5) circle (0.08cm);
\draw[fill=blue!80] (v6) circle (0.13cm);
\draw[fill=black] (v6) circle (0.08cm);
\draw[fill=blue!80] (v7) circle (0.13cm);
\draw[fill=black] (v7) circle (0.08cm);
\draw[fill=blue!80] (v8) circle (0.13cm);
\draw[fill=black] (v8) circle (0.08cm);
\draw[fill=black] (v9) circle (0.08cm);
\node[xshift = 0.3cm] at (v1) {$1$};
\node[xshift = 0.3cm] at (v2) {$2$};
\node[xshift = 0.3cm] at (v3) {$3$};
\node[xshift = 0.3cm] at (v4) {$4$};
\node[xshift = 0.3cm] at (v5) {$5$};
\node[xshift = 0.3cm] at (v6) {$6$};
\node[xshift = 0.3cm, yshift = 0.1cm] at (v7) {$7$};
\node[xshift = 0.3cm] at (v8) {$8$};
\node[xshift = 0.3cm] at (v9) {$9$};
\node[xshift = 1.3cm, yshift=-0.3cm] at (v1) {$e_1$};
\node[xshift = -1cm, yshift=-0.5cm] at (v3) {$e_2$};
\node[xshift = -0.1cm, yshift=-0.8cm] at (v4) {$e_3$};
\node[xshift = 1.4cm, yshift=0.1cm] at (v6) {$e_4$};
\node[xshift = 0.cm, yshift=-0.45cm] at (v9) {$e_5$};
\end{tikzpicture}
\caption{A relaxed matching $M =(S_{e_1}, S_{e_3}, S_{e_4})$ in a hypergraph $H$ with $S_{e_1} = \{1,2 \}$, $S_{e_3} = \{ 4,5\}$ and $S_{e_4} = \{ 6,7,8 \}$.}
\label{fig:relaxmatch}
\end{figure}
Let $V(M) \coloneqq \bigcup_{S_e \in M} S_e$ denote the set of vertices in the relaxed matching. Moreover let $m_k(M) \coloneqq |\{ S_e \in M : |S_e| = k \}|$ denote the number of subsets in the relaxed matching of size $k$.
Define the \textit{multivariate relaxed matching polynomial} of $H$ by
\[
\eta_{H}(\mathbf{x}) \coloneqq \sum_{M} (-1)^{|M|} W(M) \prod_{i \in [n]\setminus V(M)} x_i,
\]
where the sum runs over all relaxed matchings of $H$ and
\[
W(M) \coloneqq \prod_{k=1}^{n-1} k^{m_{k+1}(M)}.
\]
Let $\eta_H(x) \coloneqq \eta_H(x\mathbf{1})$ denote the \textit{univariate relaxed matching polynomial}.
\begin{remark}
Note that $\eta_H(x) = \mu_H(x)$ if $H$ is a graph.
\end{remark}
Our aim is to prove that $\eta_{H}(\mathbf{x})$ is a stable polynomial. In fact we shall prove the stability of a more general polynomial accommodating for arbitrary degree restrictions on each vertex.
Define a \textit{relaxed subgraph} of $H$ to be a hypergraph $K = (E(K),V(K))$ with edges $E(K) \coloneqq (S_e)_{e \in E}$ such that $E \subseteq E(H)$, $S_e \subseteq e$ and $|S_e| > 1$ for $e \in E$ with $V(K) \coloneqq \bigcup_{e \in E} S_e$. Again if $H$ is a graph, then the notion of a relaxed subgraph coincides with the conventional notion of a (edge-induced) subgraph of $H$. Let $\boldsymbol{\kappa} = (\kappa_1,\dots \kappa_n) \in \mathbb{N}^n$. Define a \textit{relaxed $\boldsymbol{\kappa}$-subgraph} of $H$ to be a relaxed subgraph $K^{\boldsymbol{\kappa}}$ of $H$ such that $\text{deg}_{K^{\boldsymbol{\kappa}}}(i) \leq \kappa_i$ for $i \in V(K^{\boldsymbol{\kappa}})$. Let $m_k(K^{\boldsymbol{\kappa}}) \coloneqq |\{ S_e \in E(K^{\boldsymbol{\kappa}}) : |S_e| = k \}|$ and let $(n)_k = n(n-1)\cdots (n-k+1)$ denote the \textit{Pochhammer symbol}.
Define the \textit{multivariate relaxed $\boldsymbol{\kappa}$-subgraph polynomial} of $H$ by
\[
\eta_{H}^{\boldsymbol{\kappa}}(\mathbf{x}) \coloneqq \sum_{K^{\boldsymbol{\kappa}}} (-1)^{|E(K^{\boldsymbol{\kappa}})|} W(K^{\boldsymbol{\kappa}}) \prod_{i \in [n] \setminus V(K^{\boldsymbol{\kappa}})} x_{i}^{\kappa_i - \text{deg}_{K^{\boldsymbol{\kappa}}}(i)},
\]
where the sum runs over all relaxed $\boldsymbol{\kappa}$-subgraphs $K^{\boldsymbol{\kappa}}$ of $H$ and
\[
W(K^{\boldsymbol{\kappa}}) \coloneqq \prod_{k=1}^{n-1} k^{m_{k+1}(K^{\boldsymbol{\kappa}})} \prod_{i \in V(K^{\boldsymbol{\kappa}})} (\kappa_i)_{\text{deg}_{K^{\boldsymbol{\kappa}}}(i)}.
\]
\begin{remark}
Note that a relaxed matching in $H$ is the same as a relaxed $(1,\dots, 1)$-subgraph of $H$ and that $\eta_{H}^{(1,\dots, 1)}(\mathbf{x}) = \eta_{H}(\mathbf{x})$.
\end{remark} \noindent
In the rest of this section we will adopt the following notation,
\begin{align*}
\boldsymbol{\partial}_S \coloneqq \sum_{i \in S} \partial_i, \hspace{0.2cm} \boldsymbol{\partial}^S \coloneqq \prod_{i \in S} \partial_i, \hspace{0.2cm} \boldsymbol{\partial}^{\boldsymbol{\alpha}} \coloneqq \prod_{i=1}^n \partial_i^{\alpha_i},
\end{align*} \noindent
where $S \subseteq [n]$ and $\boldsymbol{\alpha} = (\alpha_i) \in \mathbb{N}^n$.
With abuse of notation we shall let the multiaffine part operator $\text{MAP}$ act analogously on polynomial spaces of differential operators as follows,
\begin{align*}
\text{MAP}:\mathbb{C}[\partial_1,\dots, \partial_n] &\to \mathbb{C}[\partial_1,\dots, \partial_n] \\ \sum_{\boldsymbol{\alpha} \in \mathbb{N}^n} a(\boldsymbol{\alpha}) \boldsymbol{\partial}^{\boldsymbol{\alpha}} &\mapsto \sum_{\boldsymbol{\alpha}: \alpha_i \leq 1, i \in [n]} a(\boldsymbol{\alpha}) \boldsymbol{\partial}^{\boldsymbol{\alpha}}.
\end{align*} \noindent
The following lemma follows from Theorem \ref{thm:liebsokal}.
\begin{lemma}\label{lem:mapstab}
If $P(\boldsymbol{\partial}) \in \mathbb{C}[\partial_1,\dots, \partial_n]$ is a linear operator such that $P(\mathbf{x}) \in \mathbb{C}[x_1,\dots, x_n]$ is stable, then $\text{MAP}\left [ P(\boldsymbol{\partial}) \right ]$ preserves stability.
\end{lemma}
\begin{proof}
Write $P(\boldsymbol{\partial}) = \sum_{\boldsymbol{\alpha} \in \mathbb{N}} a(\boldsymbol{\alpha})\boldsymbol{\partial}^{\boldsymbol{\alpha}}$. Since $\text{MAP}:\mathbb{C}[x_1,\dots,x_n] \to \mathbb{C}[x_1,\dots, x_n]$ is a stability preserver we have that $\text{MAP}\left[ \sum_{\boldsymbol{\alpha} \in \mathbb{N}^n} a(\boldsymbol{\alpha}) \mathbf{x}^{\boldsymbol{\alpha}} \right ] = \sum_{\boldsymbol{\alpha}: \alpha_i \leq 1, i \in [n]} \mathbf{x}^{\boldsymbol{\alpha}}$ is stable and hence by Theorem \ref{thm:liebsokal} that $\sum_{\boldsymbol{\alpha}: \alpha_i \leq 1, i \in [n]} \boldsymbol{\partial}^{\boldsymbol{\alpha}} = \text{MAP}\left [ \sum_{\boldsymbol{\alpha} \in \mathbb{N}^n} a(\boldsymbol{\alpha}) \boldsymbol{\partial}^{\boldsymbol{\alpha}} \right ]$ is a stability preserving linear operator.
\end{proof}
\begin{theorem} \label{thm:wksubgpolystab}
Let $H = (V(H),E(H))$ be a hypergraph and $\boldsymbol{\kappa} = (\kappa_i) \in \mathbb{N}^n$. Then the multivariate relaxed $\boldsymbol{\kappa}$-subgraph polynomial $\eta_H^{\boldsymbol{\kappa}}(\mathbf{x})$ is stable with
\[
\eta_H^{\boldsymbol{\kappa}}(\mathbf{x}) = \prod_{e \in E(H)} \text{MAP}\left [ (1- \boldsymbol{\partial}_e) \prod_{i \in e} (1+\partial_i) \right ] \mathbf{x}^{\boldsymbol{\kappa}}.
\]
\end{theorem}
\begin{proof}
Let $e \in E(H)$. Then
\begin{align*}
(1-\boldsymbol{\partial}_e) \prod_{i \in e}(1+\partial_i) &= (1-\boldsymbol{\partial}_e)\left (1 + \sum_{\emptyset \neq S \subseteq e} \boldsymbol{\partial}^S \right ) \\ &= 1 + \sum_{\substack{\emptyset \neq S \subseteq e\\ |S| > 1}} \boldsymbol{\partial}^S - \sum_{\emptyset \neq S \subseteq e} \boldsymbol{\partial}_e \boldsymbol{\partial}^S \\ &= 1 + \sum_{\substack{\emptyset \neq S \subseteq e\\ |S| > 1}} \boldsymbol{\partial}^S - \sum_{i \in e} \left ( \sum_{\substack{\emptyset \neq S \subseteq e\\ i \in S}} \partial_i\boldsymbol{\partial}^S + \sum_{\substack{\emptyset \neq S \subseteq e\\ i \not \in S}} \boldsymbol{\partial}^{S \cup i} \right ) \\ &= 1 - \sum_{\substack{\emptyset \neq S \subseteq e \\ |S|>1}} (|S|-1)\boldsymbol{\partial}^S - \sum_{i \in e} \sum_{\substack{\emptyset \neq S \subseteq e\\ i\in S}} \partial_i\boldsymbol{\partial}^S.
\end{align*} \noindent
Thus since
\[
\left (1-\sum_{i\in e}x_i \right )\prod_{i \in e} (1+x_i),
\]
is a stable polynomial, being a product of stable linear factors, it follows by Lemma \ref{lem:mapstab} that
\[
\text{MAP} \left [ (1-\boldsymbol{\partial}_e)\prod_{i \in e}(1+\partial_i) \right ] = 1-\sum_{\substack{\emptyset \neq S \subseteq e \\ |S|>1}} (|S|-1)\boldsymbol{\partial}^S,
\]
is stability preserving. Hence
\begin{align*}
\prod_{e \in E(H)} \text{MAP}\left [ (1- \boldsymbol{\partial}_e) \prod_{i \in e} (1+\partial_i) \right ] \mathbf{x}^{\boldsymbol{\kappa}} &= \prod_{e \in E(H)} \left ( 1-\sum_{\substack{\emptyset \neq S \subseteq e \\ |S|>1}} (|S|-1)\boldsymbol{\partial}^S \right )\mathbf{x}^{\boldsymbol{\kappa}} \\ &= \sum_{E \subseteq E(H)} (-1)^{|E|} \sum_{\substack{(S_e)_{e \in E} \\ S_e \subseteq e \\ |S_e|>1}} \prod_{e \in E} (|S_e|-1) \boldsymbol{\partial}^{S_e} \mathbf{x}^{\boldsymbol{\kappa}} \\ &= \sum_{K^{\boldsymbol{\kappa}}} (-1)^{|E(K^{\boldsymbol{\kappa}})|} W(K^{\boldsymbol{\kappa}}) \prod_{i \in [n]\setminus V(K^{\boldsymbol{\kappa}})} x_i^{\kappa_i- \text{deg}_{K^{\boldsymbol{\kappa}}}(i)} \\ &= \eta_{H}^{\boldsymbol{\kappa}}(\mathbf{x}),
\end{align*} \noindent
is a stable polynomial.
\end{proof}
The following corollary is immediate from Theorem \ref{thm:wksubgpolystab}.
\begin{corollary}
The multivariate relaxed matching polynomial $\eta_H(\mathbf{x})$ is stable with
\[
\eta_H(\mathbf{x}) = \prod_{e \in E(H)} (1-\boldsymbol{\partial}_e) \prod_{i=1}^n (1+\partial_i)^{\text{deg}_H(i)} \mathbf{x}^{\mathbf{1}}.
\]
In particular the univariate relaxed matching polynomial
\[
\eta_H(x) = \sum_{M} (-1)^{|M|} W(M) x^{n-|V(M)|},
\]
is a real-rooted polynomial for any hypergraph $H$.
\end{corollary} \noindent
Below follows a generalization of the standard identities for the multivariate matching polynomial $\mu_G(\mathbf{x})$. Let $i \in V(H)$. Recall that the \textit{(weak) vertex-deletion} $H \setminus i$ is the hypergraph with vertex set $V(H)\setminus i$ and edges $\{ e \cap (V(H) \setminus i) : e \in E(H)\}$. Let $e \in E(H)$. The \textit{edge-deletion} $H \setminus e$ is the subhypergraph of $H$ with vertex set $V(H)$ and edges $E(H)\setminus e$.
Let $I_H(i) \coloneqq \{ e \in E(H): i \in e\}$ denote the \textit{incidence set} of $i \in V(H)$. The following identities are straightforward to verify.
\begin{proposition}
Let $H= (V(H),E(H))$ be a hypergraph, $i \in V(H)$ and $e \in E(H)$. Then $\eta_H(\mathbf{x})$ satisfies the following identities:
\begin{enumerate}
\item $\displaystyle \eta_H(\mathbf{x}) = \eta_{H\setminus e}(\mathbf{x}) - \sum_{\substack{S \subseteq e\\ |S|>1}} (|S|-1)\eta_{(H\setminus e)\setminus S}(\mathbf{x})$,
\item $\displaystyle \eta_H(\mathbf{x}) = x_i\eta_{H\setminus i}(\mathbf{x}) - \sum_{e \in I_H(i)} \sum_{\substack{S \subseteq e \\ i \in S \\ |S|> 1}} (|S|-1) \eta_{(H\setminus e) \setminus S}(\mathbf{x})$,
\item $\displaystyle \eta_{H_1 \sqcup H_2}(\mathbf{x}) = \eta_{H_1}(\mathbf{x}) \eta_{H_2}(\mathbf{x})$,
\item $\displaystyle \partial_i \eta_H(\mathbf{x}) = \eta_{H\setminus i}(\mathbf{x})$.
\end{enumerate}
\end{proposition} \noindent
It would be interesting to understand what parts of the matching theory for graphs can be extended to relaxed matchings.
\newline \newline
\textbf{Acknowledgements:} The author would like to thank Petter Br\"and\'en for helpful discussions and the anonymous referee for pointing out an issue with Theorem \ref{thm:multidmatchpoly} in an earlier version of this paper.
| {
"timestamp": "2019-05-10T02:02:24",
"yymm": "1905",
"arxiv_id": "1905.02264",
"language": "en",
"url": "https://arxiv.org/abs/1905.02264",
"abstract": "The first part of this note concerns stable averages of multivariate matching polynomials. In proving the existence of infinite families of bipartite Ramanujan $d$-coverings, Hall, Puder and Sawin introduced the $d$-matching polynomial of a graph $G$, defined as the uniform average of matching polynomials over the set of $d$-sheeted covering graphs of $G$. We prove that a natural multivariate version of the $d$-matching polynomial is stable, consequently giving a short direct proof of the real-rootedness of the $d$-matching polynomial. Our theorem also includes graphs with loops, thus answering a question of said authors. Furthermore we define a weaker notion of matchings for hypergraphs and prove that a family of natural polynomials associated to such matchings are stable. In particular this provides a hypergraphic generalization of the classical Heilmann-Lieb theorem.",
"subjects": "Combinatorics (math.CO)",
"title": "Stable multivariate generalizations of matching polynomials",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9854964228644458,
"lm_q2_score": 0.7185944046238981,
"lm_q1q2_score": 0.7081722152472578
} |
https://arxiv.org/abs/1006.3592 | Boundary quasi-orthogonality and sharp inclusion bounds for large Dirichlet eigenvalues | We study eigenfunctions and eigenvalues of the Dirichlet Laplacian on a bounded domain $\Omega\subset\RR^n$ with piecewise smooth boundary. We bound the distance between an arbitrary parameter $E > 0$ and the spectrum $\{E_j \}$ in terms of the boundary $L^2$-norm of a normalized trial solution $u$ of the Helmholtz equation $(\Delta + E)u = 0$. We also bound the $L^2$-norm of the error of this trial solution from an eigenfunction. Both of these results are sharp up to constants, hold for all $E$ greater than a small constant, and improve upon the best-known bounds of Moler--Payne by a factor of the wavenumber $\sqrt{E}$. One application is to the solution of eigenvalue problems at high frequency, via, for example, the method of particular solutions. In the case of planar, strictly star-shaped domains we give an inclusion bound where the constant is also sharp. We give explicit constants in the theorems, and show a numerical example where an eigenvalue around the 2500th is computed to 14 digits of relative accuracy. The proof makes use of a new quasi-orthogonality property of the boundary normal derivatives of the eigenmodes, of interest in its own right. | \section{Introduction and main results}
\label{s:i}
\begin{figure}
\mbox{\raisebox{0.9in}{\parbox{3.6in}{\ig{width=3.5in}{rf_tmin_E5-100.eps}\\
\ig{width=3.5in}{rf_tmin_E1e4+100.eps}}}
\hspace{-3ex}
\ig{width=1.8in}{rf_efn_10005.eps}}
\ca{Tension $t[\tilde{u}_\tbox{min}]$ versus energy $E$
for the domain shown on the right.
$\tilde{u}_\tbox{min}$ is the optimal trial Helmholtz solution lying in the
span of a numerical basis set (see Section~\ref{s:num}).
a) Low frequency, showing the minima corresponding to the lowest
20 Dirichlet eigenvalues.
b) Medium-high frequency, showing a similar interval
starting at eigenvalue number $j\approx 2552$;
note the new vertical scale.
c) Density plot of eigenfunction $\phi_j \approx \tilde{u}_\tbox{min}$
corresponding to the eigenvalue $E_j=10005.02135797\cdots$
shown by the dot in b) (black indicates large values of
$|\phi_j|^2$, white zero).
}{f:t}
\end{figure}
The computation of eigenvalues and eigenmodes of
Euclidean domains is a classical problem
(in two dimensions this is the `drum problem', reviewed in \cite{KS,tref06})
with a wealth of applications
to engineering and physics, including acoustic, electromagnetic
and optical cavity and resonator design,
micro-lasers \cite{hakan05}, and data analysis \cite{saito}.
It also has continued interest in mathematical community
in the areas of quantum chaos \cite{zencyc,que} and spectral geometry
\cite{gww}.
Let $\phi_j$ be a sequence of orthonormal eigenfunctions and $E_j$ the
respective eigenvalues ($0<E_1<E_2\le E_3\le \cdots$ counting
multiplicities) of $-\Delta$, where
$\Delta:=\sum_{m=1}^n \partial^2/\partial x_m^2$
is the Laplacian in a bounded domain $\Omega\in\mathbb{R}^n$, $n\ge 2$,
with Dirichlet boundary condition. That is, $\phi_j$ satisfies
\bea
(\Delta + E_j) \phi_j &=& 0 \qquad \mbox{in } \Omega
\label{e:evp}
\\
\phi_j&=&0 \qquad \mbox{on } {\partial\Omega}
\\
\| \phi_j \|_{L^2(\Omega)} &=& 1.
\label{e:bc}
\end{eqnarray}
We will call the spectrum $\si:=\{E_j\}_{j=1}^\infty$.
Many of the applications mentioned
demand high frequencies, that is,
mode numbers $j$ from $10^2$ to as high as $10^6$.
Efficient solution of the problem thus requires specialized numerical
approaches that scale with wavenumber
better than conventional discretization methods.
The goal of this paper is to bound the errors of approximate
eigenvalues and eigenfunctions computed using trial functions
that satisfy exactly the homogeneous Helmholtz equation in $\Omega$.
As we will review below, such computational methods have proven very powerful.
Recently one of the authors \cite{incl} improved upon the
classical eigenvalue bound of Moler--Payne
\cite{molerpayne} by a factor of the wavenumber;
however, this result has limited utility since it applies only
to Helmholtz parameters lying in neighborhoods of $\sigma$ of unknown size.
In the present paper we go well beyond this result
by giving new theorems, which
i) hold for {\em all} Helmholtz parameters (greater than an $O(1)$ constant),
ii) retain the improved high-frequency
asymptotic behavior of \cite{incl} and show that this behavior
is sharp,
and iii) improve upon the best-known eigenfunction estimates,
again by a factor of the wavenumber.
To achieve
this we make use of a new form of quasi-orthogonality of the eigenfunctions
on the boundary, Theorem~\ref{t:qow}, of independent interest.
Before presenting our results, we need to review some known inclusion bounds
and their importance for applications.
Given an energy parameter\footnote{The Helmholtz parameter $E$ may be
interpreted as energy, or as the square of frequency, depending on the application.}
$E>0$,
let $u$ be a non-trivial solution to the homogeneous Helmholtz equation
$(\Delta+E)u=0$ in $\Omega$
with no imposed boundary condition, and define
its boundary error norm (or `tension')
\begin{equation}
t[u] := \frac{\|u\|_{L^2(\pO)}}{\|u\|_{L^2(\Omega)}}
~.
\label{e:t}
\end{equation}
Clearly, $t[u]=0$ implies that $E$ is an eigenvalue.
It is reasonable to expect that if $t[u]$ is small for some Helmholtz solution $u$, then $E$ is close to an eigenvalue.
Moler--Payne \cite{molerpayne} (building upon \cite{fhm})
quantified this:
there is a constant $\cmp$ depending only on the domain, such that
\begin{equation}
\di{E} \;\le\; \cmp E\, t[u]
~,
\label{e:mp}
\end{equation}
where $\di{E}:=\min_{j}|E_j-E|$ denotes the distance of $E$ from the spectrum.
An important application is to solving \eqref{e:evp}-\eqref{e:bc}
via global approximation methods,
including the method of particular solutions (MPS) \cite{mps,incl}.
One writes a trial eigenmode
$u = \sum_{n=1}^N c_n \xi_n$
via basis functions $\xi_n$
which are closed-form Helmholtz solutions in $\Omega$ but which need not satisfy
any particular boundary condition.
By adjusting the coefficients $c_n$ (via a
generalized eigenvalue \cite{incl} or singular value problem \cite{gsvd})
one may minimize $t[u]$ at fixed $E$;
by repeating this
in a search for $E$ values where the minimum $t[u]$ is very small,
as illustrated by Fig.~\ref{f:t}a and b,
one may then locate approximate
eigenvalues whose error is bounded above by \eqref{e:mp}.
(This is sometimes called the method of {\em a priori-a posteriori}
inequalities \cite[Sec.~16]{KS}.)
Due to the work of Betcke--Trefethen \cite{mps} and others, such
methods have enjoyed a recent revival, at least in $n=2$,
due to their high (often spectral) accuracy
and their efficiency
at high frequency when
compared to direct discretization methods such as finite elements.
For example, in various domains, 14 digits may be achieved in double precision
arithmetic \cite{mps},
and with an MPS variant known as the
scaling method, tens of thousands of
eigenmodes as high as $j\sim 10^6$ have been computed \cite{que,mush}.
(There are also successful variants \cite{descloux,driscoll} by
Descloux--Tolley, Driscoll, and others,
in which subdomains are used, which we will not pursue here.)
If we instead interpret $u$ as the solution error for an interior
Helmholtz boundary-value problem (solved, for instance,
via MPS or boundary integral methods), then \eqref{e:mp}
states that the interior error is controlled by the boundary error;
this aids the numerical analysis of such problems \cite{litrefftz,mfs}.
Similar estimates (which, however, rely on impedance boundary conditions)
enable the analysis of least-squares non-polynomial
finite element methods \cite[Thm 3.1]{monkwang}.
Improving such estimates could thus be of general benefit
for the numerical solution of Helmholtz problems.
Recently one of the authors \cite{incl}
observed numerical evidence that \eqref{e:mp} is not sharp
for large $E$,
and showed that there is a constant $\cb$ depending only on $\Omega$,
such that, for each $\varepsilon>0$,
\begin{equation}
\di{E} \; \le \; \cb(1+\varepsilon) \sqrt{E}\,t[u]
\label{e:incl}
\end{equation}
holds whenever $E$ lies in some open, possibly disconnected, subset
of the real axis containing $\si$.
This is an improvement over \eqref{e:mp} by a factor of the wavenumber
$\sqrt{E}$, which
in problems of interest can be as high as $10^3$.
However, since the proof relied on analytic perturbation in the parameter $E$,
there was no knowledge about the {\em size} of this ($\varepsilon$-dependent) subset,
hence
no way to know in a given practical situation whether the error bound holds.
The point of the present work is then
to remedy this problem
by removing any restriction to an unknown subset,
and also to extend the $\sqrt{E}$ improvement to bounds on approximate
eigenfunctions.
We assume the domain $\Omega\subset\mathbb{R}^n$ has unit area (or volume for $n>2$),
and obeys the following rather weak geometric condition.
\begin{cond}
The domain $\Omega\subset\mathbb{R}^n$ is bounded, with piecewise smooth boundary
in the sense of Zelditch--Zworski \cite{zzw}.
This means that $\Omega$ is given by an intersection
$$
\Omega = \bigcap_{i=1}^N \{ \mbf{x} \mid f_i(\mbf{x}) > 0 \},
$$
where the $f_i$
are smooth functions defined on a neighborhood of $\overline{\Omega}$ such that
\begin{itemize}
\item $\nabla f_i \neq 0$ on the set $\{ f_i = 0 \}$,
\item $\{ f_i = f_j = 0 \}$ is an embedded submanifold of $\RR^n$, $1 \leq i < j \leq N$, and
\item $\Omega$ is locally Lipschitz, i.e. for any boundary point $\mbf{x}_0 \in \Omegab$, there is a Euclidean coordinate system $z_1, \dots, z_n$ and a Lipschitz function $k$ of $n-1$ variables such that in some neighborhood of $\mbf{x}_0$,
we have
\begin{equation}
\Omegab = \{ z_n = k(z_1, \dots, z_{n-1}) \}.
\label{e:Lip}\end{equation}
\end{itemize}
\label{c:a}
\end{cond}
Our main result on eigenvalue inclusion is the following.
\begin{theorem}
Let $\Omega\subset\mathbb{R}^n$ be a domain satisfying condition \ref{c:a}.
Then there are constants $C,c$ depending only on $\Omega$,
such that the following holds.
Let $E>1$ and suppose $u$ is a non-trivial solution of
$(\Delta+E)u=0$ in $C^\infty(\Omega)$,
with $t[u] := \|u\|_{L^2(\pO)}/\|u\|_{L^2(\Omega)}$.
Then,
\begin{equation}
\di{E} \; \le\; C \sqrt{E}\, t[u]
~,
\label{e:b}
\end{equation}
and for the normalized Helmholtz solution $u_\tbox{min}$ minimizing $t[u]$ at the
given $E$,
\begin{equation}
c\sqrt{E}\, t[u_\tbox{min}] \; \le\; \di{E}
\; \le \; C \sqrt{E} t[u_\tbox{min}].
\label{e:ulbnds}
\end{equation}
\label{t:b}
\end{theorem}
\begin{remark}
The estimate \eqref{e:ulbnds} states that \eqref{e:b} is sharp, i.e.,
using $t[u]$ alone
one cannot localize the spectrum any more tightly than this, apart from optimizing the constants $c$ and $C$.
\end{remark}
\begin{remark}
The existence of a minimizer for $t[u]$ follows from Lemma~\ref{compact}, in the case that $E$ is not a Dirichlet eigenvalue (and is trivial when $E$ is a Dirichlet eigenvalue).
The lower bound on the distance to the spectrum in \eqref{e:ulbnds}
is of use when
the numerical scheme is known to produce a good approximation to
$u_\tbox{min}$.
\end{remark}
We will also prove the following corresponding bound on the error of the
trial eigenfunction $u$,
which improves by a factor $\sqrt{E}$ the previous best known result
(Moler--Payne~\cite[Thm.~2]{molerpayne}).
\begin{theorem}
Let $\Omega$ be as in Theorem~\ref{t:b}. Then there is a constant
$C$ depending only on $\Omega$, such that the following holds.
Let $E>1$, let $E_j$ be the eigenvalue nearest to $E$,
and let $E_k$ the next nearest distinct eigenvalue.
Suppose $u$ is a solution of
$(\Delta+E)u=0$ in $C^\infty(\Omega)$ with $\|u\|_{L^2(\Omega)}=1$,
and let $\hat{u}_j$ be the projection of $u$ onto the $E_j$ eigenspace.
Then,
\begin{equation}
\|u - \hat{u}_j\|_{L^2(\Omega)} \; \le \; C\frac{\sqrt{E}\,t[u]}{|E-E_k|}
~.
\label{e:e}
\end{equation}
\label{t:e}
\end{theorem}
\begin{remark}
The left-hand side above is equal to
$\sin\theta$, where $\theta$ is the subspace angle
between $u$ and the $E_j$ eigenspace (this viewpoint is
elaborated in \cite[Sec.~6]{mps}).
For example, when $E_j$ is a simple eigenvalue, we may
write $\|u-\phi_j\|_{L^2(\Omega)} = 2\sin(\theta/2)$.
\end{remark}
\begin{remark}
This result is also sharp, in a certain sense: see Remark~\ref{efnopt}.
\end{remark}
\
To conclude the introduction, we present some key ingredients
of the proofs.
Define the boundary functions of the eigenmodes by
\begin{equation}
\psi_j(s) := \partial_n \phi_j(s) \qquad s\in{\partial\Omega}
\label{e:psi}
\end{equation}
where $\partial_n = \mbf{n}\cdot\nabla$ is the usual normal derivative.
Our main tools will be two theorems stating that
boundary functions $\psi_j$ lying close in eigenvalue are almost orthogonal.
The first
is the following new result which we prove in Section~\ref{s:win}.
\begin{theorem}[spectral window quasi-orthogonality]
Let $\Omega\subset\mathbb{R}^n$ be a domain satisfying Condition \ref{c:a}.
There exists a constant $\cht$ depending only on $\Omega$ such that
the operator norm bound
\begin{equation}
\Bigl\|\sum_{|E_j-E|\le E^{1/2}} \!\!
\psi_j\langle\psi_j,\cdot\rangle\Bigr\|_{L^2(\Omegab) \to L^2(\Omegab)}
\;\le\; \cht E
\end{equation}
holds for all $E \geq 1$. (Here, $\langle \cdot, \cdot \rangle$ denotes the inner product in $L^2(\Omegab)$.)
\label{t:qow}
\end{theorem}
\begin{remark}
By Weyl's Law \cite[Ch. 11]{garab} there are
$O(E^{(n-1)/2})$ terms in the above sum.
Since each term already has norm $\geq cE$ \cite{rellich, hasselltao},
the theorem expresses
essentially complete mutual orthogonality, up to a constant.
Only the scaling of the window width with $E$ is important:
the theorem also holds for a window $|E_j-E|\le c E^{1/2}$ for any fixed $c$
($\cht$ will then depend on $c$ as well as $\Omega$).
On the other hand, one could not expect it to hold over a
spectral window of width $O(E^\beta)$ for $\beta>1/2$, since
the boundary functions are approximately band-limited to spatial
wavenumber $E^{1/2}$ and thus no more than $O(E^{(n-1)/2})$
of them could be orthogonal on the boundary.
\end{remark}
The second result is a pairwise estimate on the inner product of boundary functions lying close in eigenvalue, with respect to a special inner product:
(Here, $\mbf{x}(s)$ refers to the location of boundary
point $s$ relative to a fixed origin, which may or may not be inside $\Omega$.)
\begin{theorem}[pairwise quasi-orthogonality]
Let $\Omega\subset\mathbb{R}^n$
be a bounded Lipschitz domain, and let $S:=
\frac{1}{2} \sup_{\mbf{x}\in\Omega} \|\mbf{x}\|$.
Then, for all $i, j\ge 1$,
\begin{equation}
\Big| \int_{{\partial\Omega}} (\mbf{x}(s)\cdot\mbf{n}(s))\, \psi_i(s) \psi_j(s) \, ds
- 2E_i \delta_{ij} \Big|
\;\leq\;
S^2 (E_i-E_j)^2
\label{e:qo}
\end{equation}
\label{t:qo}
\end{theorem}
\begin{remark}
This theorem was proved by the first-named author in \cite[Appendix~B]{que}.
It may be viewed as an off-diagonal generalization of a theorem
of Rellich \cite{rellich} which gives the $i=j$ case.
The boundary weight $\mbf{x}\cdot\mbf{n}$ (also known as the Morawetz
multiplier)
is the only one known that gives quadratic growth away
the diagonal yet also gives non-zero diagonal elements.
\end{remark}
Note that neither of the above quasi-orthogonality theorems implies the other.
We also note that B\"{a}cker et al.\
derived a completeness property of the boundary functions in
a (smoothed) spectral window \cite[Eq.~(53)]{backerbdry},
that is closely related to Theorem~\ref{t:qow}.
After proving Theorem~\ref{t:qow},
we combine it with a boundary operator defined in Section~\ref{s:A}
to prove the main theorems, in Section~\ref{s:main}.
In Section~\ref{s:star} we state and prove a variant of Theorem~\ref{t:b}
for strictly star-shaped planar domains, which has an optimal
constant $C$.
This builds on Theorem~\ref{t:qo} combined with the
Cotlar-Stein lemma (see Lemma~\ref{l:ao}).
In the main Theorems~\ref{t:b}, \ref{t:e} and \ref{t:bs},
the domain-dependent constants are not explicit;
we discuss their explicit values in Section~\ref{s:size}.
We present a high-accuracy
numerical example using the MPS, and sketch some of
the implementation aspects, in Section~\ref{s:num}.
Finally, we conclude in Section~\ref{s:conc}.
\section{Quasi-orthogonality in an eigenvalue window}
\label{s:win}
Here we prove Theorem~\ref{t:qow} using a ``$TT^\ast$ argument''.
We need the fact that the
upper bound $\| \psi_j \|_{L^2(\Omegab)}^2 \leq C E_j$ on eigenmode normal derivatives, proved for example in \cite{hasselltao},
generalizes to quasimodes living in an $O(E^{1/2})$ spectral window.
The proof is almost the same as in \cite{hasselltao}.
\begin{lemma}
Let $\Omega\subset\mathbb{R}^n$ satisfy Condition~\ref{c:a}.
Let $E>1$, and let
\begin{equation}
\phi:=\sum_{|E_j-E|\le E^{1/2}} \!\! c_j \phi_j
\label{e:co}
\end{equation}
with real coefficients $c_j$, and $\sum_j c_j^2=\|\phi\|^2_{L^2(\Omega)}=1$.
Then,
\begin{equation}
\|\partial_n \phi\|^2_{L^2(\pO)} \;\le\; \cht E
\label{e:qht}
\end{equation}
where the constant $\cht$ depends only on $\Omega$.
\label{l:qht}
\end{lemma}
\bp
To prove this we need the following lemma, proved in Appendix~\ref{a:outgoing},
stating that for any piecewise smooth domain (in the sense of Condition~\ref{c:a}) there is a smooth vector field that is outgoing at each boundary point.
\begin{lemma} Let $\Omega$ satisfy Condition~\ref{c:a}. Then there exists a smooth vector field $\mbf{a}$, defined on a neighborhood of $\Omegac$, such that
\begin{equation}
\mbf{a} \cdot \mbf{n} \geq 1
\label{outgoing}\end{equation}
almost everywhere on $\Omegab$.
\label{l:outgoing}\end{lemma}
The main tool for proving Lemma~\ref{l:qht} is the identity
\begin{equation}
\int_{\partial\Omega} (D\phi)\partial_n\phi =
-\int_\Omega \phi[\Delta,D]\phi + \int_\Omega (D\phi) (\Delta+E)\phi
-\int_\Omega\phi D(\Delta+E)\phi
\label{e:com}
\end{equation}
for any first order differential operator $D$, which follows from\footnote{The computation, involving a total of three derivatives, is justified for our class of domains, since Dirichlet eigenfunctions are in $H^{3/2}(\Omega)$ for any Lipschitz $\Omega$; see \cite{JK}, Theorem B, p164. Rellich-type computations are also justified on Lipschitz domains in \cite{Ancona}.} Green's 2nd identity, the definition of the commutator,
and $\phi|_{\partial\Omega}=0$.
Choosing $D:=\mbf{a}\cdot\nabla$, where $\mbf{a}$ is as in Lemma~\ref{l:outgoing},
we notice that the left-hand side of \eqref{e:com}
bounds the left-hand side of \eqref{e:qht}, since
\begin{equation}
\int_{\partial\Omega} \psi_j^2 \; \le\; \int_{\partial\Omega} (\mbf{a}\cdot\mbf{n}) \psi_j^2
\end{equation}
by Condition~\ref{c:a}.
We may now bound each of the terms on the right-hand side of \eqref{e:com}.
Defining $C_a = \sup_{\mbf{x}\in\Omega} |\mbf{a}(\mbf{x})|$, we have
\begin{equation}
\|D\phi\|_{L^2(\Omega)}^2 \le
C_a^2 \int_\Omega \|\nabla \phi\|^2 = -C_a^2 \int_\Omega \phi \Delta \phi
= C_a^2\sum_{j} |c_j|^2 E_j \le C_a^2 F
\label{e:Dp}
\end{equation}
where $F:=E+E^{1/2}$ is the upper end of the window. Similarly,
\bea
\|D(\Delta+E)\phi\|_{L^2(\Omega)}^2 &\le&
C_a^2\int_\Omega\|\nabla(\Delta+E)\phi\|^2
=
C_a^2
\sum_{ij}c_i c_j \int_\Omega (\Delta+E)\phi_i (-\Delta) (\Delta+E)\phi_j
\nonumber \\
&=&
C_a^2 \sum_j c_j^2 E_j (E-E_j)^2
\le C_a^2 EF
~.
\label{e:DHp}
\end{eqnarray}
Using Cauchy-Schwarz,
the sum of the last two terms in \eqref{e:com} is then bounded by
$2 C_a \sqrt{EF}$.
For the first term on the right of \eqref{e:com},
we use Einstein notation
$[\Delta,D] = \partial_{ii}(a_j \partial_j \cdot) - a_j \partial_{iij}$.
After several steps, using integration by parts and $\phi|_{\partial\Omega}=0$, we get
\begin{equation}
-\int_\Omega \phi[\Delta,D]\phi =
2 \int_\Omega (\partial_i a_j)(\partial_i\phi)\partial_j\phi
+\int_\Omega (\partial_{ii}a_j)\phi\partial_j\phi
\label{e:ein}
\end{equation}
The constants
$C'_a := \sup_{\mbf{x}\in\Omega} \|\mathbb{A}(\mbf{x})\|_2$ where the matrix
$\mathbb{A}\in\mathbb{R}^{n\times n}$ has entries $\partial_i a_j$,
and $C''_a := \sup_{\mbf{x}\in\Omega, j=1,\ldots,n} |\Delta a_j(\mbf{x})|$,
exist and are finite.
Then \eqref{e:ein} is bounded by $2C'_a F + C''_a F^{1/2}$.
Adding all bounds on terms in \eqref{e:com} we get
\begin{equation}
\|\partial_n \phi\|^2_{L^2(\pO)} \le 2(C_a +C'_a)F + C''_a \sqrt{F}
~,
\label{e:L2bnd}
\end{equation}
which is bounded by a constant times $E$ for $E>1$.
\ep
\emph{Proof of Theorem~\ref{t:qow}.}
Consider the coefficient vector $\mbf{c}:=\{c_j\}
\in \mathbb{R}^N$ appearing in \eqref{e:co}, where $N$ is the
number of eigenvalues (counting multiplicity) in the spectral window.
Define the linear operator $T:\mathbb{R}^N\to{L^2(\pO)}$ by
\begin{equation}
T\mbf{c} = \sum_j c_j \psi_j
\label{e:T}
\end{equation}
Lemma~\ref{l:qht} states that $\|T\|_{l^2\to {L^2(\pO)}} \le (\cht E)^{1/2}$.
Thus $\|TT^\ast\|_{L^2(\pO)} \le \cht E$.
But $TT^\ast$ is the operator in the statement of Theorem~\ref{t:qow},
which completes its proof.
\section{Relating tension to a boundary operator}
\label{s:A}
In this section, we show, following Barnett \cite{incl}, that the
tension $t[u]$ is related to the operator norm of a natural boundary
operator.
For $E$ a non-eigenvalue of $\Omega$, let
$\pois(E):{L^2(\pO)} \to {L^2(\Omega)}$ be the solution operator (Poisson kernel) for the
interior Dirichlet boundary-value problem,
\bea
(\Delta + E)u &=& 0 \qquad \mbox{ in } \Omega
\label{e:helm}
\\
u &=& f \qquad \mbox{ on } {\partial\Omega}
\label{e:data}
~,
\end{eqnarray}
that is, $u = \pois f$. (For existence and uniqueness for $L^2$ data on
a Lipschitz boundary see for example \cite[Thm.~4.25]{mclean}.)
Since the eigenbasis is complete in ${L^2(\Omega)}$,
we may write $u = \sum_{j=1}^\infty c_j \phi_j$.
We evaluate each $c_j$ by applying Green's 2nd identity,
\begin{equation}
(E-E_j)(\phi_j,u)_{L^2(\Omega)} = \int_\Omega (u\Delta \phi_j - \phi_j \Delta u)
= \int_{\partial\Omega} (f \psi_j - \phi_j \partial_n u) ds
~,
\label{e:G2I}
\end{equation}
thus $c_j = \langle \psi_j, f\rangle / (E-E_j)$.
The solution operator may therefore be written as a sum of rank-1 operators,
\begin{equation}
\pois(E) = \sum_{j=1}^\infty \frac{\phi_j \langle \psi_j, \cdot \rangle}{E-E_j}
~.
\label{e:Ksum}
\end{equation}
By the definition \eqref{e:t} we have, now for any $u$
satisfying $(\Delta+E)u=0$ in $\Omega$, that $t[u]^{-1} \le \|\pois(E)\|$.
Since $\|\pois^\ast\pois\| = \|\pois\|^2$, then by defining
the boundary operator in ${L^2(\pO)}\to{L^2(\pO)}$,
\begin{equation}
A(E) := \pois(E)^\ast\pois(E)
~,
\label{e:A}
\end{equation}
we have an estimate on the tension that will be the main tool in our
analysis,
\begin{equation}
t[u]^{-2}\;\le\; \|A(E)\|
~.
\label{e:tA}
\end{equation}
Inserting \eqref{e:Ksum} into \eqref{e:A} and using orthogonality
(or see \cite[Sec.~3.1]{incl}),
we have that $A$ also may be written as the sum of rank-1 operators,
\begin{equation}
A(E) = \sum_{j=1}^\infty \frac{\psi_j\langle\psi_j,\cdot\rangle}{(E-E_j)^2}
\label{e:Asum}
\end{equation}
This sum is conditionally convergent: the sum of the operator
norm of each term diverges.
For instance, for $n=2$, Weyl's
law \cite[Ch. 11]{garab}
states that the density of eigenvalues $E_j$ is asymptotically constant,
but since $\|\psi_j\|^2=\Omega(E_j)$
the sum of norms is logarithmically divergent;
for $n>2$ the divergence is worse.
Despite this, we have the following, which improves
upon the results of \cite{incl}.
\begin{lemma}\label{compact}
Let $\Omega\subset \mathbb{R}^n$, $n\ge 2$, satisfy Condition~\ref{c:a}, and let $E>0$.
Then
\begin{equation}
\lim_{N \to \infty} \ \ \sum_{j=1}^N \frac{\psi_j\langle\psi_j,\cdot\rangle}{(E-E_j)^2}
\label{e:Asum1}
\end{equation}
converges in the
norm operator topology. Furthermore, the limit operator $A(E)$
is compact in ${L^2(\pO)}$.
\end{lemma}
\bp This follows immediately
from \eqref{e:Im} in the proof of Lemma~\ref{l:tail} below, which
shows that the tail of the sum in \eqref{e:Asum} has vanishing operator norm.
$A$ is therefore also the norm limit of a sequence of finite-rank
operators.
\ep
\section{Proof of Theorems~\ref{t:b} and \ref{t:e}}
\label{s:main}
In the previous section we related tension to the norm of
a boundary operator which itself can be written as a sum involving
mode boundary functions.
Here we place upper bounds on $\|A(E)\|$ in order to prove Theorems~\ref{t:b}
and \ref{t:e}.
Firstly we note that when $E$ is an eigenvalue, Theorem~\ref{t:b} is
trivially satisfied, since $t[u_\tbox{min}] = 0$.
When $E$ is a non-eigenvalue, formula \eqref{e:Asum}
enables us to split up contributions from different parts
of the Dirichlet spectrum,
\begin{equation}
A(E) = A_\tbox{near}(E)+A_\tbox{far}(E)+A_\tbox{tail}(E)
\label{e:split}
\end{equation}
where
\bea
A_\tbox{near}(E) &=&
\sum_{|E_j-E|\le E^{1/2}} \frac{\psi_j\langle\psi_j,\cdot\rangle}{(E-E_j)^2}
\label{e:Anear}
\\
A_\tbox{far}(E) &=&
\sum_{E/2\le E_j\le 2E, \, |E_j-E|>E^{1/2}}
\frac{\psi_j\langle\psi_j,\cdot\rangle}{(E-E_j)^2}
\label{e:Afar}
\\
A_\tbox{tail}(E) &=&
\sum_{E_j<E/2} \frac{\psi_j\langle\psi_j,\cdot\rangle}{(E-E_j)^2}
+
\sum_{E_j>2E} \frac{\psi_j\langle\psi_j,\cdot\rangle}{(E-E_j)^2}
\label{e:Atail}
\end{eqnarray}
It is sufficient (due to the operator triangle inequality)
to bound the norms of these three terms independently.
We first tackle the ``far'' and ``tail'' terms.
\begin{lemma}
There is a constant $C$ dependent only on $\Omega$ such that
\begin{equation}
\bigl\|A_\tbox{far}(E)\bigr\| \;\le \; C \qquad \mbox{ for all } E>1
\end{equation}
\label{l:far}
\end{lemma}
\bp
For any $E>1$,
consider the spectral interval $I_m:=[E+mE^{1/2}, E+(m+1)E^{1/2}]$.
For any such interval lying in $[E/2,2E]$ we may
apply Theorem~\ref{t:qow}, with $E$ replaced by at most $2E$,
to bound $\|\sum_{E_j\in I_m} \psi_j\langle \psi_j,\rangle\|$
by $2\cht E$.
For terms in \eqref{e:Afar} associated with this interval,
the denominators are no less than $m^2E$. Thus
\begin{equation}
\Bigl\|\sum_{E_j\in I_m}
\frac{\psi_j\langle\psi_j,\cdot\rangle}{(E-E_j)^2}\Bigr\|
\;\le\;
\frac{2 \cht}{m^2}
\label{e:mwindow}\end{equation}
Covering $[E+E^{1/2},2E]$ by summing over $m=1,2,\ldots$ gives
a constant, since $\sum m^{-2} = \pi^2/6$.
The same argument applies for intervals covering $[E/2,E-E^{1/2}]$.
\ep
\begin{lemma}
There is a constant $C$ dependent only on $\Omega$ such that
\begin{equation}
\bigl\|A_\tbox{tail}(E)\bigr\| \;\le\; C E^{-1/2}\qquad\mbox{ for all } E>1
\end{equation}
\label{l:tail}
\end{lemma}
\bp
Consider a spectral interval $I_m:=[2^m E, 2^{m+1} E]$. We may cover
this with at most $2^{m/2 -1}E^{1/2} + 1$ windows of half-width
at most $2^{m/2}E^{1/2}$; for each of these windows Theorem~\ref{t:qow}
applies to bound $\|\sum_{E_j\in I_m} \psi_j\langle \psi_j,\rangle\|$
by $\cht 2^{m+1}E$.
For each $E_j\in I_m$, the denominator is no smaller than $(2^{m-1}E)^2$.
Thus
\begin{equation}
\Bigl\|\sum_{E_j\in I_m}
\frac{\psi_j\langle\psi_j,\cdot\rangle}{(E-E_j)^2}\Bigr\|
\;\le\;
(2^{m/2 -1}E^{1/2} + 1) \frac{\cht 2^{m+1} E}{(2^{m-1}E)^2}
= \cht (2^{-m/2 -2} E^{-1/2} + 2^{-m+1} E^{-1})
\label{e:Im}
\end{equation}
The infinite sum over $m=1,2,\ldots$ gives
\begin{equation}
\Bigl\|\sum_{E_j>2E} \frac{\psi_j\langle\psi_j,\cdot\rangle}{(E-E_j)^2}\Bigr\|
\le
\cht\left(\frac{E^{-1/2}}{4(\sqrt{2}-1)} + 2E^{-1}\right) \le C E^{-1/2}
\; \mbox{ for all } E>1.
\label{e:toptail}
\end{equation}
We treat the interval $(0,E/2)$ similarly, using a sequence
of intervals $J_m:=[2^{-m-1}E,2^{-m}E]$. Each such
interval may be covered by at most $2^{-(m+3)/2} E^{1/2} +1$ windows
of half-width $2^{-(m+1)/2}E^{1/2}$. For each $E_j\in J_m$,
the denominator is no smaller than $E^2/4$. In a similar manner as before, the
operator norm of the partial sum associated with $J_m$ is then
$O(2^{-m}E^{-1/2})$, thus the infinite sum over $m$ is $O(E^{-1/2})$.
Note that Theorem~\ref{t:qow} does not apply for $E<1$, but that there
are $O(1)$ such $E_j$ values and each contributes $O(E^{-1})$.
This proves the Lemma.
\ep
\emph{Proof of Theorem~\ref{t:b}.}
Examining the ``near'' term \eqref{e:Anear}, we use
Theorem~\ref{t:qow} on the sum of numerators, and get a bound by taking the
minimum denominator,
\begin{equation}
\bigl\|A_\tbox{near}(E)\bigr\|
\;\le\;
\frac{\cht E}{\di{E}^2} \qquad \mbox{ for all } E>1
\label{e:Anearbnd}
\end{equation}
Using this and the above Lemmas to sum the terms in \eqref{e:split}
gives
\begin{equation}
\bigl\|A(E)\bigr\|
\;\le\;
\frac{\cht E}{\di{E}^2} + C \qquad \mbox{ for all } E>1
\label{e:Abnd2}
\end{equation}
From Lemma~\ref{l:dist}, an upper bound on the distance to the spectrum,
we see that the second term is bounded by
at most a constant times the first, so may be absorbed into it to give
\begin{equation}
\bigl\|A(E)\bigr\|
\;\le\;
\frac{C E}{\di{E}^2} \qquad \mbox{ for all } E>1
\label{e:Abnd}
\end{equation}
Combining this with \eqref{e:tA} proves \eqref{e:b}, hence also the
second inequality in \eqref{e:ulbnds}.
The first inequality in \eqref{e:ulbnds}
simply follows from the fact that, since $A$ is a sum of positive operators,
\begin{equation}
t[u_\tbox{min}]^{-2} = \bigl\|A(E)\bigr\|
\ge
\Bigl\| \frac{\psi_j\langle\psi_j,\cdot\rangle}{(E-E_j)^2}\Bigr\|
=
\frac{\|\psi_j\|^2}{\di{E}^2}
~,
\label{e:tumin}
\end{equation}
where $E_j$ is the eigenvalue closest to $E$.
Using the lower bound $\|\psi_j\|^2\ge c E_j$ from \cite{hasselltao}
this becomes
\begin{equation}
\di{E} \ge c \sqrt{E_j} t[u_\tbox{min}]
~.
\label{e:dtumin}
\end{equation}
With a change of constant,
$E_j$ may be replaced here by
$E$ to give the first inequality in \eqref{e:ulbnds},
since Lemma~\ref{l:dist} insures that $E_j$ is relatively close to $E$.
(The lemma is not useful for $E$ less than some constant and $E_j<E$, but then
the ratio $E/E_j$ is still bounded by a constant because $E_j\ge E_1$).
\emph{Proof of Theorem~\ref{t:e}.}
We next prove the eigenfunction error bound \eqref{e:e},
first considering $E$ a non-eigenvalue.
We denote the boundary data by $U:=u|_{\partial\Omega}$.
From orthogonality,
then using the formula for the $c_i$ coefficients below \eqref{e:G2I},
we get,
\begin{equation}
\|u-\hat{u}_j\|^2_{L^2(\Omega)} = \sum_{E_i\ne E_j} |(\phi_i,u)_{L^2(\Omega)}|^2
=\sum_{E_i\ne E_j} \frac{|\langle \psi_i, U \rangle|^2}{(E-E_i)^2}
\le \Bigl\|\sum_{E_i\neq E_j}
\frac{\psi_i\langle\psi_i,\cdot\rangle}{(E-E_i)^2}\Bigr\|
\|U\|_2^2 .
\label{e:proj}
\end{equation}
The operator in the last expression is identical to \eqref{e:Asum} except with
the $E_j$-eigenspace terms omitted.
Therefore, its norm may be bounded in the same way as that of $A(E)$,
the only difference being that
the $\di{E}$ introduced in \eqref{e:Anearbnd} is replaced by
$\min_{E_i\neq E_j}|E-E_i| = |E-E_k|$.
Thus the bound analogous to \eqref{e:Abnd} is
\[
\bigl\|\sum_{E_i\neq E_j}
\frac{\psi_j\langle\psi_j,\cdot\rangle}{(E-E_j)^2}\bigr\|
\;\le\; \frac{CE}{(E-E_k)^2}
\qquad \mbox{ for all } E>1
~,
\]
and inserting this and $\|U\|_{L^2(\pO)} = t[u] \|u\|_{L^2(\Omega)} = t[u]$ into
\eqref{e:proj} gives \eqref{e:e}.
Finally, if $E$ is an eigenvalue, i.e. $E=E_j$,
the solution operator
\eqref{e:Ksum} is undefined,
since a solution to \eqref{e:helm}-\eqref{e:data} exists if and only if
$f$ is orthogonal to the normal derivative functions in the $E$-eigenspace.
This can be seen by applying Green's 2nd identity to $\phi$, any function
in the $E$-eigenspace, and $u$, giving $\langle \partial_n \phi, U \rangle=0$.
However, the solution
coefficients $c_i$ for which $E_i\neq E$ are uniquely defined
by the same formula as before.
Thus \eqref{e:proj} and the rest of the proof carries through.
\begin{remark}\label{efnopt}
Theorem~\ref{t:e} is sharp, as can be seen in the following way:
if $u$ is such that $t[u]$ is close to $t[u_{\tbox{min}}]$ (say, less than $2 t[u_{\tbox{min}}]$), then we have, by combining \eqref{e:ulbnds} and \eqref{e:e},
\begin{equation}
\|u - \hat{u}_j\|_{L^2(\Omega)} \; \le \; C\frac{|E-E_j|}{|E-E_k|}
~.
\label{e:e2}
\end{equation}
Apart from the value of the constant, one cannot expect to do better than this. For example, if $E$ is midway between $E_j$ and $E_k$, then the error $\|u - \hat{u}_j\|_{L^2(\Omega)}$ cannot be expected to be better than $1/\sqrt{2}$.
\end{remark}
\section{Star-shaped planar domains}
\label{s:star}
The purpose of this section is to say something stronger than Theorem~\ref{t:b}
in the special case of star-shaped domains
in $n=2$. We take weighted boundary functions
\begin{equation}
\psi^{(s)}_j(s) := (\mbf{x}(s)\cdot\mbf{n}(s))\partial_n \phi_j(s)
~, \qquad s\in{\partial\Omega}
\end{equation}
and our boundary inner product as
\begin{equation}
\langle f,g\rangle_s := \int_{\partial\Omega} (\mbf{x}(s)\cdot\mbf{n}(s))^{-1} f(s) g(s) ds
\end{equation}
hence norm $\|f\|_s := \sqrt{\langle f,f\rangle}$, and
$t_s[u]:= \|U\|_s/\|u\|_{L^2(\Omega)}$.
The significance of the weight $(\mbf{x}\cdot\mbf{n})$ is twofold:
it is strictly positive for strictly star-shaped domains, and
also turns the inner product in \eqref{e:qo} into $\langle \psi^{(s)}_i,
\psi^{(s)}_j\rangle_s$, enabling us to benefit from pairwise
quasi-orthogonality.
The Rellich theorem $\|\psi^{(s)}_j\|_s^2 = 2E_j$ states that,
with this special weight,
there is no fluctuation in the $L^2$-norms of the boundary functions.
As shown in \cite{incl}, the function $t_s[u_\tbox{min}]$ vs
$E$ has slope $1/\|\psi^{(s)}_j\|_s^2$ in the neighborhood of $E_j$
(this arises from dominance of a single term in \eqref{e:Assum} below).
Hence these slopes are predictable
{\em independently} of the particular form of each mode $\phi_j$.
This enables us to get the following eigenvalue inclusion result
analogous to Theorem~\ref{t:b}.
\begin{theorem}
Let $\Omega\subset\mathbb{R}^2$
be a strictly star-shaped bounded domain with piecewise
smooth boundary. Then there are constants $c_1$, $c_2$, $c_3$
depending only on $\Omega$,
such that the following holds. Let $E>1$,
and suppose $u$ is a non-trivial solution of $(\Delta+E)u=0$ in
$C^\infty(\Omega)$, with $c_2t_s[u]^2<1$. Let $F:=E+\sqrt{E}$. Then,
\begin{equation}
\di{E} \; \le \; \sqrt{2F}\, t_s[u]
\frac{1 + c_1 \sqrt{F} t_s[u]}{1-c_2 t_s[u]^2}
~.
\label{e:bs}
\end{equation}
For the Helmholtz solution $u_\tbox{min}$ minimizing $t_s[u]$ at the
given $E$,
\begin{equation}
\sqrt{2(E-c_3E^{1/2})}\,t_s[u_\tbox{min}] \; \le \;
\di{E}
\label{e:lbndss}
\end{equation}
\label{t:bs}
\end{theorem}
\begin{remark}
In the limit of high frequency $E\gg 1$ and small tension $t_s[u] \ll E^{-1/2}$,
the right-hand side of \eqref{e:bs} and the left hand side of \eqref{e:lbndss} are both $\sqrt{2E}(1 + o(1))t_s$.
This proves that both the power of $E$ and the constant $\sqrt{2}$ are sharp.
\end{remark}
\begin{remark}
Notice that this theorem is not applicable for all $E$ since there may
be large spectral gaps where $c_2t_s[u]^2<1$ cannot be satisfied.
Due to the numerator, it becomes far from optimal when $t_s[u]$
is $O(E^{-1/2})$ or larger.
In these respects it is less general than Theorem~\ref{t:b},
even though it gives better bounds in the small tension limit.
\end{remark}
The main tool used in the proof of Theorem~\ref{t:bs} is the pairwise quasi-orthogonality result, Theorem~\ref{t:qo}, together with the Cotlar-Stein lemma, which we state here for the special case of self-adjoint operators:
\begin{lemma}[Cotlar-Stein~\cite{cotlar,stein,Comech}]
Let $\{T_j\}_{j\in J}$ be a countable set of bounded self-adjoint operators,
$J \subset \mathbb{N}$. Then
\[
\Bigl\|\sum_{j\in J} T_j \Bigr\| \le
\max_{j\in J} \sum_{i\in J} \sqrt{\| T_i T_j \|}
~.
\]
\label{l:ao}
\end{lemma}
\emph{Proof of Theorem~\ref{t:bs}.}
The weighted equivalent of \eqref{e:Asum} is the operator
\begin{equation}
A^{(s)}(E) = \sum_{j=1}^\infty \frac{\psi^{(s)}_j\langle\psi^{(s)}_j,\cdot\rangle_s}{(E-E_j)^2}
\label{e:Assum}
\end{equation}
which, by analogy with \eqref{e:tA}, satisfies
\begin{equation}
t_s[u]^{-2}\;\le\; \|A^{(s)}(E)\|_s
~.
\label{e:tsAs}
\end{equation}
The lower bound \eqref{e:lbndss} follows by analogy with
\eqref{e:tumin}-\eqref{e:dtumin},
using $\|\psi^{(s)}_j\|_s^2 = 2E_j$, and $E_j\ge E-c_3 E^{1/2}$
from Lemma~\ref{l:dist}.
Using the same splitting into ``near'', ``far'', and ``tail'' parts as
in Section~\ref{s:main}, we can bound the norm of the ``near'' part
in a new way, as follows.
\begin{lemma}
There is a constant $c_1>0$ depending only on $\Omega$ such that
\[
\|A^{(s)}_\tbox{near}(E)\|_s \le \frac{2F}{\di{E}^2} +
\frac{\sqrt{2}c_1 F}{\di{E}}
\qquad \mbox{ for all } E > 1
\]
\label{l:Anear}
\end{lemma}
The first term in this bound will arise simply from the
single term in the sum \eqref{e:Assum} with $E_j$ nearest to $E$.
The second term requires more work, as we now show.
\bp
Let $J = \{j: |E_j-E|\le E^{1/2}\}$.
Using $T_j = \frac{\psi_j\langle\psi_j,\cdot\rangle}{(E-E_j)^2}$
in Lemma~\ref{l:ao} gives
\begin{equation}
\Bigl\|\sum_{j\in J} T_j\Bigr\|_s
\;\le\;
\max_{j\in J} \frac{\|\psi^{(s)}_j\|_s^{1/2}}{|E-E_j|}
\sum_{i\in J}
\frac{ \bigl(\langle\psi^{(s)}_i,\psi^{(s)}_j \rangle_s \,
\|\psi^{(s)}_i\|_s\bigr)^{1/2}}
{|E-E_i|}
\label{e:aoapply}
\end{equation}
Applying quasi-orthogonality (Theorem~\ref{t:qo}) for the inner product,
and $\|\psi^{(s)}_j\|_s^2=2E_j$, and separating diagonal ($i=j$) from off-diagonal terms, we get,
\begin{equation}
\Bigl\|\sum_{j\in J}\frac{\psi_j\langle\psi_j,\cdot\rangle}{(E-E_j)^2} \Bigr\|
\;\le\;
\max_{j\in J} \frac{2E_j}{(E-E_j)^2} +
\sqrt{2} S \max_{j\in J} \frac{E_j^{1/4}}{|E-E_j|}
\sum_{i\in J}
\frac{E_i^{1/4}|E_i-E_j|}{|E-E_i|}
\label{e:qoao}
\end{equation}
Here $S$ is as in Theorem~\ref{t:qo}.
The first term is bounded by $2F/\di{E}^2$. Using
$|E_i-E_j| \le |E_i-E| + |E-E_j|$
bounds the second term by
\bea
&&\sqrt{2} S \max_{j\in J} \frac{E_j^{1/4}}{|E-E_j|}
\sum_{i\in J}E_i^{1/4}\left(1+\frac{|E-E_j|}{|E-E_i|}\right)
\nonumber \\
&\le&
\sqrt{2F} S\,|J|\, \max_{j\in J}
\Bigl( \frac{1}{|E-E_j|} + \frac{1}{\di{E}}\Bigr)
\;\le\;
\frac{2\sqrt{2F} S|J|}{\di{E}}
\label{e:term2}
\end{eqnarray}
Recall Weyl's law for the asymptotic density of eigenvalues,
which states that, for $n=2$ and $\vol\Omega=1$,
\begin{equation}
N(E) := \#\{j:E_j<E\} = \frac{1}{4\pi}E + R(E)
~,
\label{e:weyl}
\end{equation}
where the remainder is $R(E)=O(\sqrt{E})$ (\cite{SafVas};
for the case of piecewise-smooth boundary see \cite[Eq.~(0.3)]{Seeley}).
Since the remainder is bounded for small $E$,
there is a constant $C_\tbox{W}$ such that $|R(E)| \le C_\tbox{W} \sqrt{E}$
for all $E>1$.
Thus $|J|$, the number of terms in the ``near'' window, is bounded by
\[
|J| \le \Bigl(\frac{1}{4\pi} + 2C_\tbox{W}\Bigr) \sqrt{F}
~.
\]
Inserting this into \eqref{e:term2} proves the Lemma,
and we may take $c_1 = 2S(1/4\pi + 2C_\tbox{W})$.
\ep
\emph{Completion of the proof of Theorem~\ref{t:bs}.}
The proofs of analogously weighted versions of
Lemmas~\ref{l:far} and \ref{l:tail} are unchanged.
So we may combine them with Lemma~\ref{l:Anear} and \eqref{e:tsAs}
to get, for some constant $c_2$,
\[
t_s[u]^{-2} \;\le\; \|A^{(s)}(E)\|_s \;\le\;
\frac{2F}{\di{E}^2} + \frac{\sqrt{2}c_1 F}{\di{E}} + c_2
\qquad \mbox{ for all } E > 1
\]
Multiplying through by $\di{E}^2$
we solve the quadratic inequality for $\di{E}$,
\[
\di{E} \; \le\;
\frac{c_1 F/\sqrt{2} + \sqrt{c_1^2 F^2/2 + 2(t_s[u]^{-2}-c_2)F}}
{t_s[u]^{-2}-c_2}
\]
Using the subadditivity of the square-root completes the
proof of \eqref{e:bs}.
\section{Discussion of explicit constants}
\label{s:size}
For the practical application of Theorems~\ref{t:b} and \ref{t:e},
it is important to have an explicit value for the constant $C$
(from the discussion after \eqref{e:proj} we notice that
$C$ in the two theorems is the same.)
We now compute an explicit value of this $C$ that holds for all $E>1$.
Examining \eqref{e:L2bnd} we see that a choice of constant in
Lemma~\ref{l:qht}, and hence Theorem~\ref{t:qow}, that holds for
all $E>1$ is $\cht = 4(C_a+C'_a) + \sqrt{2}C''_a$.
To compute this we need sup norms of the value, and first and second
derivative, of a vector field $\mbf{a}$ as in Lemma~\ref{l:outgoing}.
The proof of Lemma~\ref{l:outgoing} shows such a construction; the values will
depend on the size of the vectors $\mbf{a}_{x_i}$ and the choice of partition of unity used to cover $\overline{\Omega}$. The vectors $\mbf{a}_{x_i}$ will be large (order $1/\epsilon$) if $\Omega$ has corners with angles less than $\epsilon$ or greater than $2\pi - \epsilon$.
We note that a numerical procedure for this construction
could be useful.
In some special cases, a simpler prescription for the vector field can be given:
\begin{itemize}
\item
For strictly star-shaped domains in $\mathbb{R}^n$, we may choose
$\mbf{a}=\mbf{x} / \inf_{\partial\Omega}(\mbf{x}\cdot\mbf{n})$, which
gives $C_a = \sup_{\partial\Omega} (\mbf{x}\cdot\mbf{n})/ \inf_{\partial\Omega}(\mbf{x}\cdot\mbf{n})$,
$C'_a=1 / \inf_{\partial\Omega}(\mbf{x}\cdot\mbf{n})$, and $C''_a=0$.
\item
For a domain with $C^2$ boundary,
let $\delta > 0$ be the largest number such that for each $\mbf{x}_0 \in \Omegab$, a ball of radius $\delta$ can be placed within $\Omega$ so as to be tangent to $\Omegab$ at $\mbf{x}_0$.
We may then
choose $\mbf{a} = (1-r/\delta)^2\mbf{n}_r$, for $r<\delta$, $\mbf{a}=\mbf{0}$
otherwise, where the coordinate $r$ is the distance from ${\partial\Omega}$,
and $\mbf{n}_r$ is the unit vector in the local decreasing $r$ direction.
This gives constants $C_a = 1$ and $C'_a = 2/\delta$.
$C''_a$ depends on $\delta$ and an upper bound on the rate of change
of surface curvature.
(Also note that a slight modification of the proof of Theorem~\ref{t:qow}
would allow estimation purely in terms of $C_a$ and $C'_a$, but with a doubling
of the numerical constants).
\end{itemize}
Summing the terms \eqref{e:mwindow} above and below $E$ we have that the
constant in Lemma~\ref{l:far} is $2\pi^2\cht/3$.
Similarly, using \eqref{e:toptail} and its equivalent for $(0,E/2)$
gives the constant in Lemma~\ref{l:tail} as
$\cht(\frac{1}{4(\sqrt{2}-1)}+\frac{1}{4-\sqrt{2}}+6)<7\cht$.
Summing these two constants gives
a constant $C$ in \eqref{e:Abnd2} as $14\cht$.
A choice of constant in \eqref{e:Abnd} is then $\cht + 14 \cht \max[E_1^2,C_d^2]$,
where from Appendix~\ref{a:dist} we have $C_d = 2\sqrt{E_1}$, and the max
accounts for the case $1<E\le E_1$.
Finally, the constant in \eqref{e:b} is the square-root of this,
$C = \sqrt{\cht(1 + 14\max[E_1,4]E_1)}$.
Requiring that the above estimates hold for all $E>1$
caused non-optimality in the choice of constant.
It is more sensible in high frequency applications
to use a better constant which is approached for $E\gg1$, and small
tension $t\ll1$. We now give this explicitly.
In this limit, in \eqref{e:L2bnd}, $F$ tends to $E$,
and we drop lower-order terms to get $\cht = 2(C_a+C'_a)$,
which in the star-shaped case is
\begin{equation}
\cht = 2\frac{1+\sup_{\partial\Omega}(\mbf{x}\cdot\mbf{n})}{\inf_{\partial\Omega}(\mbf{x}\cdot\mbf{n})}
\qquad \mbox{ for } E\gg 1, \mbox{ $\Omega$ star-shaped .}
\label{e:chtsshf}
\end{equation}
If tension is small (i.e.\ $E$ is not in a large spectral gap), the
second term in \eqref{e:Abnd} is negligible, so
we may approximate the constant in \eqref{e:b} as
\begin{equation}
C = \sqrt{\cht} \qquad \mbox{ for } E\gg 1, \;t\ll 1
~.
\label{e:chf}
\end{equation}
\begin{remark}
The limiting constant \eqref{e:chf} does not reflect the
limiting slopes of the graph $t[u_\tbox{min}]$ vs $E$ near eigenvalues.
These slopes are known \cite{incl} to be
$1/\|\psi_j\|^2$, which is bounded by $(2C_a' E_j)^{-1}$ \cite{hasselltao}.
\end{remark}
We end by discussing the constants $c_1$ and $c_2$ in Theorem~\ref{t:bs}. Constant
$c_2$ may be estimated easily, as above, using the weighted
versions of Lemmas~\ref{l:far} and \ref{l:tail}.
In the proof of Lemma~\ref{l:Anear}, $c_1$
involves the Weyl constant $C_\tbox{W}$;
we know of no explicit estimates for $C_\tbox{W}$ in the literature
(the closest we know are estimates
of the form $|R(E)| < C \sqrt{E}\ln E$ with explicit constants
\cite{netrusov,chenhua}).
However, these constants are effectively
irrelevant for practical purposes, when $E\gg1$ and $t\ll E^{-1/2}$,
since in these limits, one may replace \eqref{e:bs}
by $\di{E}\le \sqrt{2E} t_s[u]$ and still have an error bound very close
to that given by the full expression.
\section{Numerical example}
\label{s:num}
In Fig.~\ref{f:t}c we show a planar nonconvex
domain given by the radial function
$r(\theta) = 1 + 0.3 \cos[3(\theta+ 0.2\sin\theta)]$.
The domain is star-shaped and smooth (we will not address numerical
issues raised by corners here; see
\cite{fhm,descloux,mps,driscoll,que,timodd}.)
For high-frequency eigenvalue problems,
a convenient computational basis of Helmholtz solutions are
`method of fundamental solutions' basis functions
$\xi_n(\mbf{x}) = Y_0(\sqrt{E}|\mbf{x}-\mbf{y}_n|)$,
where $Y_0$ is the irregular Bessel function of order zero,
and
$\{\mbf{y}_n\}_{n=1}^N$ are a set of `charge points'
in $\mathbb{R}^2\setminus\overline{\Omega}$. The latter were chosen
by a displacement of the boundary parametrization
$\mbf{x}(\theta)$, $0<\theta\le2\pi$, in the imaginary direction (see \cite{mfs});
specifically $\mbf{y}_n = \mbf{x}(2\pi n/N - 0.025 i)$.
We compute the data plotted in Fig.~\ref{f:t}a, b as follows.
At each $E$,
$t[\tilde{u}_\tbox{min}]$ is given by the square-root of the smallest
generalized eigenvalue of
a generalized eigenvalue problem (GEVP) involving
$N\times N$ symmetric real dense matrices $F$ and $G$
(the basis representations of the boundary and interior norms
respectively.)
Both matrices are evaluated using $M$-point
periodic trapezoidal quadrature in $\theta$, that is,
quadrature points $\mbf{x}_m = \mbf{x}(2\pi m/M)$,
$m=1,\ldots,M$, and weights $w_m = 2\pi|\mbf{x}'(2\pi m/M)|/M$.
For instance, $F = P^\ast P$, where $P\in\mathbb{R}^{M\times N}$ has elements
\begin{equation}
P_{mn} = \sqrt{w_m} \xi_n(\mbf{x}_m)
~,
\label{e:P}
\end{equation}
and $G$ is similarly found \cite[Sec.~4.1]{incl}
using $P$ and the matrices
$P^{(1)}$ and $P^{(2)}$ whose entries are the $x_1$- and $x_2$-derivatives
of those in $P$.
Since the GEVP is numerically singular,
regularization was first performed, similarly to \cite[Sec.~6]{gsvd}, by
restricting to an orthonormal basis for the numerical column space of
$[P;P^{(1)};P^{(2)}]$ given by the
left singular vectors with singular values at least $10^{-14}$ times the
largest singular value.
For low frequencies (Fig.~\ref{f:t}a), 8-digit accuracy requires
$N=100$ basis functions and $M=200$ quadrature points. For higher frequencies
corresponding to 40 wavelengths across the domain
(Fig.~\ref{f:t}b, c), it requires
$N=400$ and $M=500$,
and the above GEVP procedure takes 3 seconds per $E$ value.%
\footnote{All computation times are reported for a laptop with
2GHz Intel Core Duo processor and 2GB RAM, running MATLAB 2008a
on a linux kernel.}
Very small ($<10^{-8}$) tensions cannot be found this way,
and instead are best
approximated via the GSVD \cite{gsvd}:
the optimal tension at a given $E$ is the lowest generalized singular value
of the matrix pair $(P,Q)$, where matrix $Q$ has entries
\begin{equation}
Q_{mn} = \sqrt{\frac{\mbf{x}_m\cdot \mbf{n}_m}{2E}} \sqrt{w_m}
\,\frac{\partial \xi_n}{\partial n}(\mbf{x}_m)
~,
\label{e:Q}
\end{equation}
where $\mbf{n}_m$ is the normal at $\mbf{x}_m$, and regularization as before.
Note that $G = Q^\ast Q$ well
approximates the interior norm in the subspace with zero Dirichlet data,
due to the Rellich formula (case $i=j$ of Theorem~\ref{t:qo}).
Any single-variable function minimization algorithm may then be used to
search for a local minimum of $t[\tilde{u}_\tbox{min}]$ vs $E$;
we prefer iterated fitting of
a parabola to $t[\tilde{u}_\tbox{min}]^2$ at three nearby $E$ values,
which converges typically in 5 iterations.
Using this with the GSVD
(with $N=500$, $M=700$, i.e.\ 6 points per wavelength on
${\partial\Omega}$, and taking 8 seconds per iteration),
we find the tension
\begin{equation}
t[\tilde{u}_\tbox{min}] = 2.2\times 10^{-12}
\qquad \mbox{at } E = 10005.0213579739
~.
\label{e:dp}
\end{equation}
This is shown by the dot in Fig.~\ref{f:t}b.
The GSVD right singular vector gives the basis coefficients
of the corresponding trial function $\tilde{u}_\tbox{min}$, which is
plotted in Fig.~\ref{f:t}c (this took 34 seconds to evaluate on a square grid
of size 0.005, i.e.\ $1.3\times 10^5$ points.)
Armed with datapoint \eqref{e:dp}, what
can we deduce about Dirichlet eigenpairs of $\Omega$
using our new theorems, and how much better are they than
previous results?
The constant in the Moler--Payne bound
\eqref{e:mp} is $\cmp = q_1^{-1/2}$ where $q_1$ is the
lowest eigenvalue of a Stekloff eigenproblem on $\Omega$
\cite[(2.11)]{KSstek}.
Since $\Omega$ is star-shaped, the bound
$q_1 \ge E_1^{1/2}\inf_{\partial\Omega}(\mbf{x}\cdot\mbf{n})/2\sup_{\partial\Omega}(\mbf{x}\cdot\mbf{n})$
from \cite[Table I]{KSstek} applies,
giving $\cmp = 1.31$ as a valid choice.
Thus \eqref{e:mp} states that there is an
eigenvalue $E_j$ a distance no more than $2.9\times10^{-8}$ from the above $E$.
On the other hand, \eqref{e:chf} and \eqref{e:chtsshf} give
the constant in Theorem~\ref{t:b} as $C
= 2.9$.
Applying the theorem gives a distance from the spectrum of no more than
$6.3\times 10^{-10}$.
Taking the small-tension limit of the star-shaped planar result \eqref{e:bs},
and recomputing the weighted tension $t_s[\tilde{u}_\tbox{min}]$
from Section~\ref{s:star},
we get an even smaller distance of
$3.5\times 10^{-10}$, that is, an error of $\pm3$ in
the last digit of \eqref{e:dp}.
The latter is a 80-fold improvement over Moler--Payne.
(Also see \cite{incl}
for an example at higher frequency with 3 digits of improvement).
How good an approximation is $\tilde{u}_\tbox{min}$ to the
eigenfunction $\phi_j$?
Using the observation that the next nearest eigenvalue is $E_k
=10007.339\cdots$,
the eigenfunction bound of
Moler--Payne \cite{molerpayne} gives an $L^2$-error of $1.2\times 10^{-8}$.
With the same data, using the constant $C$ above,
Theorem~\ref{t:e} gives an $L^2$-error of $2.7\times 10^{-10}$,
a 50-fold improvement over that achievable with previously known theorems.
\section{Conclusions}
\label{s:conc}
We have improved, by a factor of the wavenumber,
the Moler--Payne bounds on Dirichlet eigenvalues and eigenfunctions
which have been the standard for the last 40 years.
This makes rigorous the conjectures
based on numerical observations in \cite{incl}.
We expect this to be useful
since high-frequency wave and eigenvalue calculations are
finding more applications in recent years.
Of independent interest is a new quasi-orthogonality result in a spectral window
(Theorem~\ref{t:qow}).
For numerical utility,
throughout we have been explicit with constants, and
have specified a lower bound on $E$ for which the estimates hold
(this being stronger than merely a `big-O' asymptotic estimate).
For star-shaped domains we strengthened the inclusion bounds
(Theorem~\ref{t:bs}), achieving a sharp power of $E$ and sharp constant,
in the limit of small tension, when tension is weighted by
a special geometric function.
This weight allowed pairwise quasi-orthogonality to be used, but
since an upper bound for the number of eigenvalues
in a $\sqrt{E}$ window is needed,
this is only useful for $n=2$ (planar domains).
We applied our theorems to a numerical example,
enabling close to 14 digits accuracy in a high-lying eigenvalue,
and 10 digits in the eigenfunction. Both are two digits beyond
what could be claimed with previously-known theorems.
Our estimate $C \sqrt{E} t[u]$ on the distance to the spectrum is
sharp (up to constants) if the tension $t[u]$ (or $t_s[u]$) is comparable to $t[u_{\tbox{min}}]$ ($t_s[u_{\tbox{min}}]$).
However, numerically, one generally has access to other properties
of $u$ (e.g. its normal derivative)
which can give more detailed information about the spectrum.
For example, the powerful `scaling method' \cite{v+s,que}
is able to locate many eigenvalues using an operator computed
at a single energy $E$. In another direction,
Still \cite[Thm.~4]{still} obtains improved inclusion bounds when the
approximate eigenvalue is equal to the Rayleigh quotient; in this case, the bound is proportional to $t[u]^2$, but scaling as $E^2$ for large energy.
\remove{Note that we show that the $\sqrt{E}$ scaling in our theorems
is sharp if the only information
used about the trial function $u$ is the tension $t[u]$ (or $t_s[u]$).
However, numerically, one generally has access to other properties
of $u$ (its normal derivative, whether it is close to $u_\tbox{min}$, etc)
which can give more detailed
statements about the spectrum than merely the distance of $E$ from it.
One example is Still \cite[Thm.~4]{still}
who achieves a bound scaling as $t[u]^2$ via the Kato-Temple inequality.
Another example is the powerful `scaling method' \cite{v+s,que},
which is able to locate many eigenvalues using an operator computed
at a single energy $E$.}
An open problem with practical benefits is the generalization of
these results to Neumann and Robin boundary conditions, and to
multiple subdomains with different trial functions on
each subdomain and least-square errors on artificial boundaries
\cite{descloux,driscoll,timodd} (these are known as Trefftz
or non-polynomial finite element methods).
\section*{Acknowledgments}
The authors are grateful for discussions with Dana Williams, Timo Betcke,
and Chen Hua.
The work of AHB was supported by NSF grant DMS-0811005,
and a Visiting Fellowship to ANU in February 2009 as part of the
{\it ANU 2009 Special Year on Spectral Theory and Operator Theory}.
The work of AH was supported by Australian Research Council Discovery Grant DP0771826.
\newpage
| {
"timestamp": "2010-06-21T02:00:35",
"yymm": "1006",
"arxiv_id": "1006.3592",
"language": "en",
"url": "https://arxiv.org/abs/1006.3592",
"abstract": "We study eigenfunctions and eigenvalues of the Dirichlet Laplacian on a bounded domain $\\Omega\\subset\\RR^n$ with piecewise smooth boundary. We bound the distance between an arbitrary parameter $E > 0$ and the spectrum $\\{E_j \\}$ in terms of the boundary $L^2$-norm of a normalized trial solution $u$ of the Helmholtz equation $(\\Delta + E)u = 0$. We also bound the $L^2$-norm of the error of this trial solution from an eigenfunction. Both of these results are sharp up to constants, hold for all $E$ greater than a small constant, and improve upon the best-known bounds of Moler--Payne by a factor of the wavenumber $\\sqrt{E}$. One application is to the solution of eigenvalue problems at high frequency, via, for example, the method of particular solutions. In the case of planar, strictly star-shaped domains we give an inclusion bound where the constant is also sharp. We give explicit constants in the theorems, and show a numerical example where an eigenvalue around the 2500th is computed to 14 digits of relative accuracy. The proof makes use of a new quasi-orthogonality property of the boundary normal derivatives of the eigenmodes, of interest in its own right.",
"subjects": "Analysis of PDEs (math.AP); Numerical Analysis (math.NA); Spectral Theory (math.SP)",
"title": "Boundary quasi-orthogonality and sharp inclusion bounds for large Dirichlet eigenvalues",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.985496423290417,
"lm_q2_score": 0.7185943985973772,
"lm_q1q2_score": 0.7081722096142434
} |
https://arxiv.org/abs/1810.12214 | Lower central series, surface braid groups, surjections and permutations | Generalising previous results on classical braid groups by Artin and Lin, we determine the values of m, n $\in$ N for which there exists a surjection between the n-and m-string braid groups of an orientable surface without boundary. This result is essentially based on specific properties of their lower central series, and the proof is completely combinatorial. We provide similar but partial results in the case of orientable surfaces with boundary components and of non-orientable surfaces without boundary. We give also several results about the classification of different representations of surface braid groups in symmetric groups. | \section{Introduction}\label{sec:intro}
In 1947, E.~Artin published two seminal papers in the Annals of Mathematics, sometimes considered as the foundation of the theory of braid groups. The paper~\cite{Art1} is devoted to a determining a presentation for the braid group $B_n$ on $n$ strings, and its interpretation in terms of automorphisms of the free group of rank $n$, while the subject of~\cite{Art2} is the study of possible homomorphisms from $B_n$ to the symmetric group $S_n$ on $n$ letters. The main result of~\cite{Art2} is the description of all \emph{transitive} homomorphisms (see \resec{other} for the definition) between $B_n$ and $S_n$. Artin considered this characterisation to be the first step in determining the group of automorphisms $\aut{B_{n}}$ of $B_{n}$, a solution of which was given in~\cite{DG}. In~\cite{L0}, Lin generalised Artin's results by characterising the homomorphisms between $B_n$ and $S_m$ and between $B_n$ and $B_m$, for all $n>m$ (see~\cite{L} for a proof of these results and a survey of this topic). Other (partial) results for homomorphisms between
$B_n$ and $B_m$ with $n < m$ were recently obtained in~\cite{BM} and in~\cite{Ca}.
The main subject of this paper is surface braid groups and the existence of surjective homomorphisms between them. These groups generalise both Artin's braid groups and fundamental groups of surfaces. As well as their geometric interpretation, they may be defined in terms of fundamental groups of configuration spaces as follows~\cite{FoN}. Let $\Sigma$ be a compact, connected surface, with or without boundary, orientable or non orientable, and let $\ensuremath{\mathbb F}_n(\Sigma)=\Sigma^n \setminus \Delta$, where $\Delta$ is the set of $n$-tuples $(x_1, \ldots, x_n)$ of elements of $\Sigma$ for which $x_i=x_j$ for some $1\leq i,j\leq n$, where $i \not= j$. The
fundamental group $\pi_1(\ensuremath{\mathbb F}_n(\Sigma))$ is called the \emph{pure braid group} on $n$ strings of $\Sigma$ and shall be denoted by $P_n(\Sigma)$. The symmetric group $S_n$ acts freely on $\ensuremath{\mathbb F}_n(\Sigma)$ by permutation of coordinates, and the fundamental group $\pi_1(\ensuremath{\mathbb F}_n(\Sigma)/S_n)$ of the resulting quotient space, denoted by $B_n(\Sigma)$, is the \emph{braid group} on $n$ strings of $\Sigma$. Further, $\ensuremath{\mathbb F}_n(\Sigma)$ is a regular $n!$-fold covering of $\ensuremath{\mathbb F}_n(\Sigma)/S_n$, from which we obtain the following short exact sequence:
\begin{equation}\label{eq:permutation}
1\to P_n(\Sigma) \to B_n(\Sigma) \to S_{n}\to 1.
\end{equation}
If $\Sigma$ is the $2$-disc $\ensuremath{\mathbb D}^{2}$, it is well known that $B_{n}(\ensuremath{\mathbb D}^2)\cong B_{n}$ and that $P_{n}(\ensuremath{\mathbb D}^2)\cong P_{n}$.
The fibration of configuration spaces of a surface $\Sigma$ without boundary defined by Fadell and Neuwirth~\cite{FN} gives rise to an exact sequence involving the pure braid groups of $\Sigma$, from which one may see that the forgetful homomorphism from $P_{n+m}(\Sigma)$ to $P_n(\Sigma)$ given geometrically by forgetting $m$ strings is well defined and is a surjection. Recently, it was stated in~\cite[Theorem~1.1]{Ch} that forgetful homomorphisms are `essentially' the only possible surjections between pure braid groups of orientable surfaces, and it was conjectured that if $\Sigma$ is a compact, orientable surface of genus $g>1$, with or without boundary, and if $m,n\in \ensuremath{\mathbb N}$, where $m\neq n$, then there is no surjective homomorphism from $B_{n}(\Sigma)$ to $B_{m}(\Sigma)$~\cite[Conjecture~1.3]{Ch}. One of the aims of this paper is to study this problem for compact, connected surfaces, with or without boundary, orientable or non orientable. We summarise our main results in this direction as follows. In what follows, $\ensuremath{\mathbb{S}^{2}}$ (resp.\ $\ensuremath{\mathbb{R}P^{2}}$) will denote the $2$-sphere (resp.\ the real projective plane), $\ensuremath{\mathbb{T}^2}$ (resp.\ $\ensuremath{\mathbb{K}^2}$) will denote the $2$-torus (resp.\ the Klein bottle), $\Sigma_{g}$ (resp.\ $\Sigma_{g,b}$) will be a compact, connected, orientable surface of genus $g\geq 0$ without boundary (resp.\ with $b\geq 1$ boundary components), and $U_{g}$ will be a compact, connected, non-orientable surface of genus $g\geq 1$ without boundary (in other words, $U_{g}$ is the connected sum of $g$ projective planes).
\begin{thm}\label{th:gensurj}
Let $m,n\in \ensuremath{\mathbb N}$ be such that $m\neq n$.
\begin{enumerate}[wide=0em,labelsep=0.2em]
\item\label{it:main1a}
\begin{enumerate}[wide=1em]
\item\label{it:main1ai} There is a surjective homomorphism from $B_{n}(\ensuremath{\mathbb{S}^{2}})$ to $B_{m}(\ensuremath{\mathbb{S}^{2}})$ if and only if $m\in \brak{1,2}$ and $n>m$.
\item\label{it:main1aii} If $g\geq 1$, there is a surjective homomorphism from $B_{n}(\Sigma_{g})$ to $B_{m}(\Sigma_{g})$ if and only if $m=g=1$.
\end{enumerate}
\item\label{it:main1b} Let $g\geq 1$, and let $\Sigma$ be either $\Sigma_{g,b}$, where $b\geq 1$, or $U_{g+1}$.
Suppose that one of the following conditions holds:
\begin{enumerate}
\item\label{it:main1bi} $n<m$ and $n\in \brak{1,2}$.
\item\label{it:main1bii} $n>m$ and $m\in \brak{1,2}$.
\item\label{it:main1biii} $n>m\geq 3$ and $n\neq 4$.
\end{enumerate}
Then there is no surjective homomorphism from $B_{n}(\Sigma)$ to $B_{m}(\Sigma)$.
\end{enumerate}
\end{thm}
Parts~\ref{it:main1a}\ref{it:main1ai} and~\ref{it:main1a}\ref{it:main1aii} of \reth{gensurj} will be proved in \resec{sphere} and \resec{suror} respectively, and
part~\ref{it:main1b} will be proved in \resec{orboundary} in the orientable case, and in \resec{nonor} in the non-orientable case. For the case of the projective plane, in \reth{rp}, using the knowledge of the torsion of its braid groups, we will obtain results that are slightly stronger than those of \reth{gensurj}\ref{it:main1b}, notably with respect to the case where $n<m$.
\reth{gensurj} proves~\cite[Conjecture~1.3]{Ch} completely in the case where the surface is orientable and without boundary, and partially in the case where the surface orientable with boundary, or non-orientable and without boundary. The cases not covered by the conditions~\ref{it:main1bi}--\ref{it:main1biii} of part~\ref{it:main1b} are likely to be difficult. Note that even in the case of the Artin braid groups, the question of whether there exists a surjective homomorphism from $B_{n}$ onto $B_{m}$ remains open in many of these cases.
As a consequence of \reth{gensurj} and some basic facts about the lower central series of surface braid groups, we give an elementary proof of~\cite[Theorem~1.2]{Ch} in \reco{pure}, and we generalise the result of this corollary to the case of orientable surfaces with boundary (\reco{pureboundary}), and to the non-orientable case (\reco{purenor}). In the cases of the sphere and real projective plane, the techniques are somewhat different to those used for other surfaces, since their braid groups have torsion~\cite{FV,VB}.
Another interesting and open problem is the study of possible surjective homomorphisms between braid groups of different surfaces. One important case occurs when the domain is a braid group of a non-orientable surface $U_{g}$, and the target is a braid group of the orientable double covering $\Sigma_{g-1}$. It is known that there exists a natural injection on the level of configuration spaces that induces an injective homomorphism between $B_{n}(U_{g})$ and $B_{2n}(\Sigma_{g-1})$~\cite{GGjlms}. In \resec{nonor}, we prove the following result concerning surjections when the number of strings is the same.
\begin{prop}\label{prop:ornor}
Let $n,g\ge1$. Then there exists a surjective homomorphism of $B_n(U_g)$ onto $B_n(\Sigma_{g-1})$ if and only if $g=1$ and $n\in \brak{1,2}$.
\end{prop}
Together with surjections between surface braid groups, another of our aims is to characterise homomorphisms between surface braid groups and symmetric groups following the approaches of~\cite{Iv,L0,L}. One of the main results of~\cite{L} is the following.
\begin{thm}[{\cite[Theorem~A]{L}}]\label{th:Lin}
Let $n>m\ge 3$ and $n\not=4$. Any homomorphism $\phi\colon\thinspace B_n \to S_m$ is cyclic, i.e.\ $\phi(B_n)$ is a cyclic group.
\end{thm}
This implies that if $n>m\ge 3$ and $n\not=4$, there is no surjective homomorphism from $B_n$ onto $S_m$. We shall show that \reth{Lin} also holds for braid groups of compact surfaces without boundary.
\begin{thm}\label{th:LinSurfcomb}
Let $n>m\ge 2$, let $g \ge 0$, and let $\Sigma$ be either $\Sigma_{g}$ or $U_{g+1}$. Then there is a surjective homomorphism from $B_n(\Sigma)$ onto $S_m$ if and only if either $m=2$, or $(n,m)=(4,3)$.
\end{thm}
If $g\geq 1$ (resp.\ $g=0$) and $\Sigma=\Sigma_{g}$, the statement of \reth{LinSurfcomb} will be proved in \resec{suror} (resp.\ in \resec{sphere}), while in the case $g\geq 0$ and $\Sigma=U_{g+1}$, the result will be proved in \resec{nonor}.
\medskip
Let $g \geq 0$,
and let $n>m \geq 1$. We recall that a representation $\rho_{n,m}\colon\thinspace B_n(\Sigma_g) \to S_m$ is said to be \emph{transitive} if the action of the image $\im{\rho_{n,m}}$ of $\rho_{n,m}$ on the set $\brak{1,\ldots, m}$ is transitive and is \emph{primitive} if the only partitions of this set that are left invariant by the action of $\im{\rho_{n,m}}$ are the set itself, or the partition consisting of singletons. By abuse of notation, we say that a subgroup of $S_m$ is primitive if its action on the set $\brak{1,\ldots, m}$ is primitive.
Notice that, if $m>2$, a primitive representation is clearly transitive.
\medskip
Inspired by Artin's characterisation of (transitive) homomorphisms between $B_n$ and $S_{n}$, Ivanov determined all of the homomorphisms between $B_n(\Sigma_{g,b})$ and $S_{n}$, but under the stronger assumption that the homomorphisms are primitive~\cite[Theorem~1]{Iv}. We prove the following theorem for homomorphisms between $B_n(\Sigma_{g}) $ and $ S_m$ when $n>m$. This result may also be compared with the classification of the homomorphisms between $B_n$ and $S_{m}$, where $n>m$, given in \reth{LinTrans}.
\begin{thm}\label{th:LinSurfaceTrans}
Let $n>m\geq 2$, and let $g\geq 1$. There exists a primitive representation
$\rho_{n,m}\colon\thinspace $ $B_n(\Sigma_{g}) \to S_m$ if and only if $m$ is prime. This being the case, one of the following statements holds:
\begin{enumerate}
\item\label{it:LinSurfaceTransa} the image $\im{\rho_{n,m}}$ of $\rho_{n,m}$ is generated by an $m$-cycle, unless $m=2$, in which case $\im{\rho_{n,2}}$ can also be equal to $\brak{\operatorname{Id}}$.
\item\label{it:LinSurfaceTransb} $n=4$ and $m=3$, and up to a suitable renumbering of the elements of the set $\brak{1, 2, 3}$,
$\rho_{4,3}(\sigma_1)=\rho_{4,3}(\sigma_3)=(1,2)$, $\rho_{4,3}(\sigma_2)=(2,3)$, and for all $1\leq i\leq g$, the permutations $\rho_{4,3}(a_i)$ and $\rho_{4,3}(b_{i})$ are trivial, where $\brak{\sigma_1, \sigma_2,\sigma_3, a_1,b_1, \ldots, a_g,b_g}$ is the generating set of $B_4(\Sigma_{g})$ given in the statement of \reth{presbng}.
\end{enumerate}
\end{thm}
More information about arbitrary (not necessarily primitive) representations of $B_n(\Sigma_{g})$ in $S_m$ is given in \repr{contraints} and the examples that follow it.
The rest of this paper is divided into four sections. Sections~\ref{sec:closed} and~\ref{sec:orboundary} deal with the braid groups of compact, orientable surfaces without boundary and with boundary respectively, \resec{nonor} is devoted to the braid groups of compact, non-orientable surfaces without boundary. In each of these sections, we give a presentation of the braid groups in question, we recall some known results about their lower central series and whether they are residually nilpotent or not, and we prove the relevant parts of Theorems~\ref{th:gensurj} and~\ref{th:LinSurfcomb}. In \resec{other} we explore representations of surface braid groups in symmetric groups. In particular, in \reth{LinSurfaceTrans}, we classify primitive representations, a result that was more or less implicitly expected in~\cite{Iv}, and in \repr{contraints} we give some constraints on general (non-primitive) homomorphisms, and we answer a question of~\cite{Iv} by providing some examples of transitive, non-primitive, non-Abelian representations.
In this paper, we do not discuss the braid groups of non-orientable surface with boundary components. This choice is motivated by two different considerations, first that these groups have rarely been studied in the literature, and secondly, that the techniques used in the case of non-orientable surfaces without boundary apply almost verbatim to the case with boundary. This is in contrast with the orientable case, where the lower central series is a stronger tool in the case without boundary than in the case with boundary.
\subsubsection*{Acknowledgements}
The second and third authors of this paper were partitially supported by the CNRS/FAPESP PRC project n\textsuperscript{o}~275209 (France) and n\textsuperscript{o}~2016/50354-1 (Brazil). The second author is also partially supported by the FAPESP Projeto Tem\'atico `Topologia Alg\'ebrica, Geom\'etrica e Diferencial' n\textsuperscript{o}~2016/24707-4 (Brazil).
\section{Orientable surfaces without boundary}\label{sec:closed}
In \resec{orlcs}, we start by recalling a presentation of the braid groups of compact, orientable surfaces without boundary, as well as some facts about their lower central series. In \resec{suror}, we generalise certain results of~\cite{GGzeit} about the minimal number of generators of these groups, and we prove Theorems~\ref{th:gensurj}\ref{it:main1a}\ref{it:main1aii} and \ref{th:LinSurfcomb} in the case where $\Sigma=\Sigma_{g}$, with $g\geq 1$. In \resec{sphere}, we prove \reth{gensurj}\ref{it:main1a}\ref{it:main1ai} and \reth{LinSurfcomb} in the case $g=0$, which is that of the sphere.
\subsection{Presentations and the lower central series of surface braid groups}\label{sec:orlcs}
In this paper, many of our techniques will be combinatorial and will make use of the lower central series of surface (pure) braid groups. Given a group $G$, recall that the \emph{lower central series} of $G$ is given by $\brak{\Gamma_i(G)}_{i\in \ensuremath{\mathbb N}}$, where $G=\Gamma_1(G)$, and $\Gamma_i(G)=[G,\Gamma_{i-1}(G)]$ for all $i\geq 2$. We thus have a filtration $\Gamma_1(G) \supseteq \Gamma_2(G) \supseteq \cdots$.
The group $G$ is said to be \emph{perfect} if $G=\Gamma_2(G)$. We shall denote the \emph{Abelianisation} $\Gamma_1(G)/\Gamma_2(G)$ of $G$ by $G_{\text{Ab}}$. If $\mathcal{P}$ is a group-theoretic property, let $\mathcal{FP}$ denote the class of groups that possess property $\mathcal{P}$. Following P.~Hall, $G$ is said to be \emph{residually $\mathcal{P}$} if for any (non-trivial) element $x\in G$, there exists a group $H$ possessing property $\mathcal{P}$ and a surjective homomorphism $\phi\colon\thinspace G \to H$ such that $\phi(x) \not=1$. It is well known that a group $G$ is residually nilpotent if and only if $\bigcap_{i \ge 1}\Gamma_i(G)=\{ 1\}$.
The lower central series of the Artin braid groups is well known.
\begin{prop}[see~\cite{BGeG,L}]\label{prop:braidlcs}
If $n\ge 3$,
$\Gamma_1(B_n)/\Gamma_2(B_n) \cong \ensuremath{\mathbb Z}$, and $\Gamma_2(B_n)=\Gamma_3(B_n)$.
\end{prop}
\repr{braidlcs} also holds trivially if $n=2$ since $B_2\cong \ensuremath{\mathbb Z}$ (and therefore $\Gamma_2(B_n)= \Gamma_3(B_n)=1$). Using~\cite{FR,Ko} and the fact that $P_2$ is isomorphic to $\ensuremath{\mathbb Z}$, we see that the group $P_n$ is residually (torsion-free) nilpotent for all $n\ge 2$.
We recall a presentation of $B_n(\Sigma_g)$ for $g\geq 1$.
\begin{thm}[{\cite[Theorem~6]{BGeG}}]\label{th:presbng}
Let $g,n\in\ensuremath{\mathbb N}$. Then $B_n(\Sigma_g)$ admits the following group presentation:
\begin{itemize}
\item \textbf{generators:} $a_1, b_1, \ldots, a_g, b_g, \sigma_1, \ldots, \sigma_{n-1}$.
\item \textbf{relations:}
\begin{gather}
\text{$\sigma_i\sigma_j=\sigma_j\sigma_i$ if $\lvert i-j \rvert \geq
2$}\label{eq:artin1}\\
\text{$\sigma_i\sigma_{i+1}\sigma_i= \sigma_{i+1}\sigma_i \sigma_{i+1}$
for all $1\leq i\leq n-2$}\label{eq:artin2}\\
\text{$c_i\sigma_j= \sigma_j c_i$ for all $j\geq 2$, $c_i=a_i$ or
$b_i$ and $i=1, \ldots, g$}\label{eq:cs}\\
\text{$c_i \sigma_1 c_i \sigma_1= \sigma_1 c_i \sigma_1 c_i$ for
$c_i=a_i$ or $b_i$ and $i=1, \ldots, g$}\label{eq:cici}\\
\text{$a_i \sigma_1 b_i = \sigma_1 b_i \sigma_1 a_i \sigma_1$ for $i=1, \ldots, g$}\label{eq:ab}\\
\text{$c_i \sigma_1^{-1} c_j \sigma_1=\sigma_1^{-1} c_j \sigma_1 c_i$
for $c_i=a_i$ or $b_i$, $c_j=a_j$ or $b_j$ and $1\le j<i\le g$}\label{eq:cicj}\\
\text{$\prod_{i=1}^g [a_i^{-1},b_i]= \sigma_1\cdots \sigma_{n-2} \sigma_{n-1}^2 \sigma_{n-2} \cdots \sigma_1$}\label{eq:tot}.
\end{gather}
\end{itemize}
\end{thm}
Throughout this paper, relations~(\ref{eq:artin1}) and~(\ref{eq:artin2}) will be referred to as the \emph{braid} or \emph{Artin relations}. Observe that if we take $g=0$ in the presentation of \reth{presbng}, we obtain the presentation of $B_{n}(\ensuremath{\mathbb{S}^{2}})$ due to Fadell and Van Buskirk~\cite{FV}, the relations being the braid relations and the `surface relation':
\begin{equation}\label{eq:totsph}
\sigma_1\cdots \sigma_{n-2} \sigma_{n-1}^2 \sigma_{n-2} \cdots \sigma_1=1,
\end{equation}
so \reth{presbng} is also valid in this case.
If $g\geq 1$, the lower central series of the braid groups of $\Sigma_{g}$ were studied in~\cite{BGeG}. The
statement of the following theorem contains some of the results of that paper, and provides some minor improvements, notably in the case $n=2$.
\begin{thm}\label{th:gam3closed}
Let $g,n\geq 1$. Then:
\begin{enumerate}
\item\label{it:gam12g} $\Gamma_1(B_n(\Sigma_g))/\Gamma_2(B_n(\Sigma_g)) \cong
\begin{cases}
\ensuremath{\mathbb Z}^{2g} & \text{if $n=1$}\\
\ensuremath{\mathbb Z}^{2g} \oplus \ensuremath{\mathbb Z}_2 & \text{if $n\geq 2$.}
\end{cases}$
\item\label{it:gam23g}\begin{enumerate}[wide=0em,labelsep=0.2em]
\item\label{it:gam23gn} $\Gamma_2(B_n(\Sigma_g))/\Gamma_3(B_n(\Sigma_g)) \cong
\begin{cases}
\ensuremath{\mathbb Z}^{g(2g-1)-1} & \text{if $n=1$}\\
\ensuremath{\mathbb Z}_{n-1+g} & \text{if $n\geq 3$.}
\end{cases}$
\item\label{it:gam23gn2} If $n=2$, $\Gamma_2(B_2(\ensuremath{\mathbb{T}^2}))/\Gamma_3(B_2(\ensuremath{\mathbb{T}^2}))\cong \ensuremath{\mathbb Z}_{2}^{3}$, and if $g>1$, $\Gamma_2(B_2(\Sigma_g))/\Gamma_3(B_2(\Sigma_g))$ is a non-trivial quotient of $\ensuremath{\mathbb Z}_2^{2g} \oplus \ensuremath{\mathbb Z}_{g+1}$.
\end{enumerate}
\item\label{it:gam3g} $\Gamma_3(B_n(\Sigma_g))=\Gamma_4(B_n(\Sigma_g))$ if and only if $n\geq 3$. Moreover $\Gamma_3(B_n(\Sigma_{g}))$ is perfect if and only if $n\ge 5$.
\item\label{it:gam4g} The group $B_n(\Sigma_g)$ is residually nilpotent if and only if $n\leq 2$.
\end{enumerate}
\end{thm}
Parts~\ref{it:gam12g} and~\ref{it:gam23g} of \reth{gam3closed} imply that the braid groups of orientable surfaces without boundary may be distinguished by their lower central series (and indeed by the first two lower central series quotients). A presentation of the group $B_n(\Sigma_{g})/\Gamma_3(B_n(\Sigma_{g}))$ was given in~\cite[eq.~(10)]{BGeG} and may be found in \rex{exo1}. Many of the statements of this theorem were proved in~\cite[Theorem~1]{BGeG} in the case $n\geq 3$, and may be deduced from~\cite{LA} in the case $n=1$. More information about the lower central series quotients of $B_1(\Sigma_{g})$ may be found in~\cite{LA}. Taking into account these papers, at the end of this section, we prove \reth{gam3closed}. We first give some preliminary results and properties regarding the remaining parts of the statement, notably in the case where $n=2$. If $g=1$, $\ensuremath{\mathbb{T}^2}$ is the $2$-torus $\ensuremath{\mathbb{T}^2}$, and we have the following result for $B_{2}(\ensuremath{\mathbb{T}^2})$.
\begin{thm}[{\cite[Theorem~3]{BGeG}}]\label{th:renil}
The group $B_2(\ensuremath{\mathbb{T}^2})$ is residually nilpotent, but is not residually torsion-free nilpotent. Further, $\Gamma_2(B_2(\ensuremath{\mathbb{T}^2}))/\Gamma_{3}(B_2(\ensuremath{\mathbb{T}^2}))\cong\ensuremath{\mathbb Z}^3_2$, and $\Gamma_3(B_2(\ensuremath{\mathbb{T}^2}))/\Gamma_{4}(B_2(\ensuremath{\mathbb{T}^2}))\cong\ensuremath{\mathbb Z}_{2}^{5}$.
\end{thm}
\begin{proof}
The first part of the statement is~\cite[Theorem~3(a) and~(c)]{BGeG}. To prove the second part, using ideas from~\cite{GG1}, it was shown in~\cite[Theorem~3(b)]{BGeG} that with the exception of the first term, the lower central series of $B_2(\ensuremath{\mathbb{T}^2})$ and the free product $\ensuremath{\mathbb Z}_2 \ast \ensuremath{\mathbb Z}_2 \ast \ensuremath{\mathbb Z}_2$ coincide. With the help of results of~\cite{Ga}, for all $i\geq 2$, it follows that lower central series quotient $\Gamma_i(B_2(\ensuremath{\mathbb{T}^2}))/\Gamma_{i+1}(B_2(\ensuremath{\mathbb{T}^2}))$ is isomorphic to the direct sum of $R_i$ copies of $\ensuremath{\mathbb Z}_2$, where $R_{i}$ is given by an explicit formula involving the M\"obius function, from which one may check that $R_{2}=3$ and $R_{3}=5$. This yields the second part of the statement.
\end{proof}
If $g>1$, $B_{2}(\Sigma_{g})$ is residually nilpotent.
\begin{prop}[{\cite[Corollary~10]{BB}}] \label{prop:n=2orientable}
If $g\ge 1$, the group $B_2(\Sigma_g)$ is residually $2$-finite. In particular, it is residually nilpotent.
\end{prop}
\begin{rem}\label{rem:notnilp}
To prove some of our results, we will need to be sure that our residually nilpotent groups are not nilpotent, in particular that all of their lower central series quotients are non trivial. We claim that this is the case for the group $B_2(\Sigma_g)$ for all $g\geq 1$. If $g=1$, the result follows from~\cite[Theorem~3]{BGeG} (note that the group $\ensuremath{\mathbb Z}_2 \ast \ensuremath{\mathbb Z}_2 \ast \ensuremath{\mathbb Z}_2$ contains a subgroup that is a free group of rank $2$).
So assume that $g>1$, and suppose on the contrary that there exists $i\in \ensuremath{\mathbb N}$ such that $\Gamma_i(B_2(\Sigma_g))=\brak{1}$. Without loss of generality, we may suppose that $i$ is minimal with respect to this property. Since $B_2(\Sigma_g)$ is non Abelian, it follows from \reth{gam3closed}\ref{it:gam12g} that $i\geq 3$. Now $\Gamma_{i}(B_2(\Sigma_g))= [\Gamma_{i-1}(B_2(\Sigma_g)), B_2(\Sigma_g)]=\brak{1}$, and hence $\Gamma_{i-1}(B_2(\Sigma_g))$ is contained in the centre of $B_2(\Sigma_g)$. This centre is trivial~\cite{GGmpcps,PR}, so $\Gamma_{i-1}(B_2(\Sigma_g))=\brak{1}$, but this contradicts the minimality of $i$, and so proves the result in this case.
\end{rem}
The computation of the lower central series quotients in the case $n=2$ and $g>1$, namely the generalisation of \reth{renil} and~\cite[Theorem~3]{BGeG} to surfaces of arbitrary genus, remains an open problem. The following result nevertheless gives some information about the quotient $\Gamma_2(B_2(\Sigma_g))/\Gamma_3(B_2(\Sigma_g))$.
\begin{prop}\label{prop:pre}
If $g\geq 1$,
the group $\Gamma_2(B_2(\Sigma_g))/\Gamma_3(B_2(\Sigma_g))$ is non-trivial, and is a quotient of $\ensuremath{\mathbb Z}_2^{2g} \oplus \ensuremath{\mathbb Z}_{g+1}$.
\end{prop}
\begin{proof}
If $g=1$ then by \reth{renil}, $\Gamma_2(B_2(\ensuremath{\mathbb{T}^2}))/\Gamma_3(B_2(\ensuremath{\mathbb{T}^2}))\cong \ensuremath{\mathbb Z}_{2}^{3}$, and the result holds. So suppose that $g>1$. In what follows we will make use freely of the Witt-Hall identities~\cite[Theorem~5.1]{MKS}. By relation~(\ref{eq:cicj}), we have $1=[c_i ,\sigma_1^{-1} c_j \sigma_1]$ for $c_i=a_i$ or $b_i$, $c_j=a_j$ or $b_j$ and for all $1\le j<i\le g$ in $B_2(\Sigma_g)$, from which it follows that $1=[c_i, c_j]$ in $B_2(\Sigma_g)/\Gamma_3(B_2(\Sigma_g))$.
Using relation~(\ref{eq:cici}), we have $1=[c_i, \sigma_1 c_i \sigma_1]$ in $B_2(\Sigma_g)$ for $c_i=a_i$ or $b_i$ and for all $i=1, \ldots, g$, which implies that $1=[c_i, \sigma_1]^2$ in $B_2(\Sigma_g)/\Gamma_3(B_2(\Sigma_g))$.
Similarly, from relation~(\ref{eq:ab}), we obtain $[b_i^{-1}, \sigma_1^{-1} a_i \sigma_1]=\sigma_1^2$ in $B_2(\Sigma_g)$ for all $i=1, \ldots, g$, and therefore $[b_i^{-1}, a_i]=[b_i, a_i]^{-1}=\sigma_1^2$ in $B_2(\Sigma_g)/\Gamma_3(B_2(\Sigma_g))$. Since $\prod_{i=1}^g\, [a_i^{-1},b_i]=\prod_{i=1}^g\, [b_i, a_i]$ in $B_2(\Sigma_g)/\Gamma_3(B_2(\Sigma_g))$ by relation~(\ref{eq:tot}), we see that $\sigma_1^{-2g}=\sigma_1^2$, and thus the order of $\sigma_1^2$ in $\Gamma_2(B_2(\Sigma_g))/\Gamma_3(B_2(\Sigma_g))$ divides $g+1$. These computations imply that $\Gamma_2(B_2(\Sigma_g))/\Gamma_3(B_2(\Sigma_g))$ is an Abelian group that is generated by the commutators
$[c_i, \sigma_1]$ for $c_i=a_i$ or $b_i$ and $i=1, \ldots, g$, which are all of order at most $2$, and
the commutators $[b_i,a_i]$, where $i=1, \ldots, g$, and which are all identified to a single element $\sigma_{1}^{2}$ of order at most $g+1$. Consequently, $\Gamma_2(B_2(\Sigma_g))/\Gamma_3(B_2(\Sigma_g))$ is a quotient of $\ensuremath{\mathbb Z}_2^{2g} \oplus \ensuremath{\mathbb Z}_{g+1}$. \rerem{notnilp} implies that this quotient is non trivial, which proves the result.
\end{proof}
\begin{prop}\label{prop:perfect}
If $g\geq 1$ and $n\in \brak{3,4}$, the group $\Gamma_3(B_n(\Sigma_{g}))$ is not perfect.
\end{prop}
\begin{proof}
Let $g\geq 1$ and $n\in \brak{3,4}$. Let $\pi_n\colon\thinspace B_n(\Sigma_{g}) \to S_n$ be the homomorphism that arises in~(\ref{eq:permutation}), and for $i\geq 2$, let $\pi_{n,i}\colon\thinspace \Gamma_i(B_n(\Sigma_{g})) \to \Gamma_i(S_n)$ denote the induced surjective homomorphism between the corresponding terms of the lower central series. A straightforward computation shows that $\Gamma_2(S_n)=\Gamma_3(S_n)=A_{n}$, where $A_{n}$ is the alternating group, and that $\Gamma_3(S_n)/[\Gamma_3(S_n),\Gamma_3(S_n)]$ is isomorphic to $\ensuremath{\mathbb Z}_3$. Now the homomorphism $\pi_{n,3}$ induces a surjection at the level of Abelianisations, and since $\Gamma_3(S_n)/[\Gamma_3(S_n),\Gamma_3(S_n)]$ is non trivial, we conclude that $\Gamma_3(B_n(\Sigma_{g}))$ cannot be perfect.
\end{proof}
\begin{proof}[Proof of Theorem \ref{th:gam3closed}]
First assume that $n=1$. We have $B_1(\Sigma_{g})= \pi_1(\Sigma_{g})$, which is residually free, and therefore residually (torsion free) nilpotent. Further, by~\cite[Main Theorem]{LA}, $\Gamma_i(B_1(\Sigma_g))/\Gamma_{i+1}(B_1(\Sigma_g))$ is isomorphic to $\ensuremath{\mathbb Z}^{2g}$ if $i=1$, to $\ensuremath{\mathbb Z}^{g(2g-1)-1}$ if $i=2$, and to $\ensuremath{\mathbb Z}^{4g(g^{2}-1)}$ if $i=3$. In particular $\Gamma_3(B_1(\Sigma_g))$ is not perfect. Therefore all statements of the theorem pertaining to the case $n=1$ hold.
Now suppose that $n=2$. It follows in a straightforward manner from \reth{presbng} that $\Gamma_1(B_2(\Sigma_g))/\Gamma_2(B_2(\Sigma_g)) \cong \ensuremath{\mathbb Z}^{2g} \oplus \ensuremath{\mathbb Z}_{2}$. Part~\ref{it:gam23g}\ref{it:gam23gn2} follows from \reth{renil} and \repr{pre}, and parts~\ref{it:gam3g} and~\ref{it:gam4g} in the case $n=2$ are a consequence of \reth{renil}, \repr{n=2orientable} and \rerem{notnilp}.
Finally, let $n\ge 3$. Parts~\ref{it:gam12g},~\ref{it:gam23g}\ref{it:gam23gn},~\ref{it:gam4g}, and the sufficiency of the condition in part~\ref{it:gam3g} were proved in~\cite[Theorem~1]{BGeG}. Part~\ref{it:gam3g} in the case $n\in \brak{3,4}$ is a consequence of \repr{perfect}. This completes the proof of the theorem.
\end{proof}
\subsection{Surjections between braid groups of orientable surfaces of non-zero genus without boundary}\label{sec:suror}
With the notation of~\cite{GGzeit}, if $\Gamma$ is a finitely-generated group, let $G(\Gamma)$ denote the minimal cardinality among all generating sets of $\Gamma$. By~\cite[Proposition~8]{GGzeit}, if $\Gamma'$ is another finitely-generated group such that there exists a surjective homomorphism from $\Gamma$ to $\Gamma'$ then:
\begin{equation}\label{eq:Abelian}
\text{$G(\Gamma)\geq G(\Gamma')$ and $G(\Gamma)\geq G(\Gamma_{\text{Ab}})$,}
\end{equation}
the second inequality following from the first by taking the homomorphism to be Abelianisation.
The following proposition generalises some of the principal results of~\cite{GGzeit} to the case of orientable surfaces of genus $g\geq 1$.
\begin{prop}\label{prop:mingen}
Let $g,m\in \ensuremath{\mathbb N}$. Then $G(B_{m}(\Sigma_{g}))=\begin{cases}
2g+m-1 & \text{if $m\in \{ 1,2\}$}\\
2g+2 & \text{if $m\geq 3$.}
\end{cases}$
\end{prop}
\begin{proof}
If $m\in \{ 1,2\}$, then $(B_{m}(\Sigma_{g}))_{\text{Ab}}\cong \ensuremath{\mathbb Z}^{2g}\oplus \ensuremath{\mathbb Z}_2^{m-1}$ using \reth{gam3closed}\ref{it:gam12g}, so $G(B_{m}(\Sigma_{g}))\geq 2g+m-1$ by~\ref{eq:Abelian}.
By taking the generating set of $B_{m}(\Sigma_{g})$ given in \reth{presbng}, we see also that $G(B_{m}(\Sigma_{g}))\leq 2g+m-1$, which proves the result in this case. So assume that $m\geq 3$. As in the proof of~\cite[Proposition~4]{GGzeit}, using \reth{gam3closed}, we see that $\{ a_{1}, b_{1},\ldots, a_{g}, b_{g}, \sigma_{1}, \sigma_{1}\cdots \sigma_{m-1} \}$ is a generating set for $B_{m}(\Sigma_{g})$, and so $G(B_{m}(\Sigma_{g}))\leq 2g+2$. Conversely, with respect to the presentation given by \reth{presbng}, let $f\colon\thinspace B_{m}(\Sigma_{g}) \to \ensuremath{\mathbb Z}^{2g}$ be the surjective homomorphism whose kernel is the normal closure of $\{ \sigma_{1},\ldots, \sigma_{m-1}\}$ in $B_{m}(\Sigma_{g})$, and let $h\colon\thinspace B_{m}(\Sigma_{g}) \to S_{m}$ be the surjective homomorphism given by equation~(\ref{eq:permutation}) whose kernel is $P_{m}(\Sigma_{g})$. Using \reth{presbng}, $h$ may also be seen to be the projection onto the quotient of $B_{m}(\Sigma_{g})$ by the normal closure of the set $\brak{a_{1}, b_{1},\ldots, a_{g},b_{g}}$.
The map $f\times h\colon\thinspace B_{m}(\Sigma_{g}) \to \ensuremath{\mathbb Z}^{2g} \times S_{m}$ is clearly a homomorphism. To see that it is surjective, note that if $(w,\tau)\in \ensuremath{\mathbb Z}^{2g} \times S_{m}$, there exist $\alpha\in P_{m}(\Sigma_{g})$ and $\beta$ belonging to the subgroup of $B_{m}(\Sigma_{g})$ generated by $\{ \sigma_{1},\ldots, \sigma_{m-1}\}$ such that $f(\alpha)=w$ and $h(\beta)= \tau$. From the description of $f$ and $h$, we have $(f\times h)(\alpha\beta)=(w,\tau)$, which proves that $f\times h$ is surjective. So by~\ref{eq:Abelian},
$G(B_{m}(\Sigma_{g})) \geq G(\ensuremath{\mathbb Z}^{2g} \times S_{m})=2g+G(S_{m})\geq 2g+2$ because $m\geq 3$. Thus $G(B_{m}(\Sigma_{g}))=2g+2$, and the statement then follows in this case.
\end{proof}
\begin{cor}\label{cor:special}
Let $m\geq 3$, $n\in \brak{1,2}$ and $g\geq 1$. Then there is no surjective homomorphism from $B_{n}(\Sigma_{g})$ to $B_{m}(\Sigma_{g})$.
\end{cor}
\begin{proof}
If $m\geq 3$, $n\in \brak{1,2}$ and $g\geq 1$, the result follows from~\ref{eq:Abelian} using the fact that $G(B_{m}(\Sigma_{g})) > G(B_{n}(\Sigma_{g}))$ by \repr{mingen}.
\end{proof}
\begin{proof}[Proof of \reth{LinSurfcomb} in the case where $\Sigma=\Sigma_{g}$ and $g\geq 1$]
Let $n>m\ge 2$, and consider the map from $B_n$ to $B_n(\Sigma_g)$ defined on the generators of $B_{n}$ by sending $\sigma_i$ to $\sigma_i$ for all $i=1, \ldots, n-1$. It is a homomorphism (note that by~\cite{PR}, it is also an embedding).
Suppose first that $n>m\ge 3$ and $n\neq 4$, and let $\Phi \colon\thinspace B_n(\Sigma_g) \to S_{m}$ be a homomorphism.
By~\reth{Lin}, the elements $\Phi(\sigma_i)$, where $i=1, \ldots, n-1$, are powers of a single element, and therefore commute pairwise. Using the braid relations, the fact that $\Phi(\sigma_{i})$ commutes with $\Phi(\sigma_{i+1})$ for all $i=1, \ldots, n-2$ implies that $\Phi(\sigma_1)= \cdots= \Phi(\sigma_{n-1})$. We denote this common element by $\sigma$. We see from relations~(\ref{eq:cs}) that $\sigma$ commutes with $\Phi(a_j)$ and $\Phi(b_j)$ for all $j=1, \ldots, g$. Suppose now that $\Phi$ is surjective. Then $\sigma$ belongs to the centre of $S_m$, which is trivial since $m\geq 3$, so $\sigma$ is trivial. Therefore the homomorphism $\Phi$ factors through the surjective homomorphism $\Phi'\colon\thinspace B_n(\Sigma_g)/\langle\!\langle \sigma_1\rangle\!\rangle \to S_m$, where $\langle\!\langle \sigma_1\rangle\!\rangle$ denotes the normal closure of $\sigma_{1}$ in $B_n(\Sigma_g)$. But using \reth{presbng} (\emph{cf.} the proof of \repr{mingen}), $B_n(\Sigma_g)/\langle\!\langle \sigma_1\rangle\!\rangle$ is isomorphic to $\ensuremath{\mathbb Z}^{2g}$, so is Abelian, while $S_{m}$ is not. This yields a contradiction, and hence $\Phi$ is not surjective.
Conversely, if $(n,m)=(4,3)$, the map from $B_4(\Sigma_g)$ to $S_3$ defined by sending the elements $a_1,b_1,\ldots, a_g, b_g$ to the identity element, $\sigma_1$ and $\sigma_3$ to $(1,2)$, and $\sigma_2$ to $(2,3)$, extends to a well-defined, surjective homomorphism by \reth{presbng}.
\end{proof}
We are now able to prove \reth{gensurj} for the braid groups of orientable surfaces without boundary of genus $g\geq 1$.
\begin{proof}[Proof of \reth{gensurj}\ref{it:main1a}\ref{it:main1aii}]
Suppose first that $m=g=1$, and that $n\geq 2$. Since $B_{1}(\ensuremath{\mathbb{T}^2})\cong \ensuremath{\mathbb Z}^{2}$, the result follows by considering the surjective homomorphism $f\colon\thinspace B_{n}(\ensuremath{\mathbb{T}^2}) \to \ensuremath{\mathbb Z}^{2}$ defined in the proof of \repr{mingen}.
To prove the converse, we will show that if $(g,m)\neq (1,1)$, there is no surjective homomorphism from $B_{n}(\Sigma_{g})$ to $B_{m}(\Sigma_{g})$. We split the proof into the following three cases.
\begin{enumerate}[label=\textit{(\arabic*)}]
\item $n<m$. If $n\in \brak{1,2}$, the result follows from \reco{special}.
So suppose that $n\geq 3$. \reth{gam3closed}\ref{it:gam23g}\ref{it:gam23gn} implies that there is no surjective homomorphism from $\Gamma_2(B_n(\Sigma_g))/\Gamma_3(B_n(\Sigma_g))$ onto $\Gamma_2(B_m(\Sigma_g))/\Gamma_3(B_m(\Sigma_g))$, and hence there is no surjective homomorphism from $B_{n}(\Sigma_{g})$ onto $B_{m}(\Sigma_{g})$.
\item\label{it:ngrm} $n>m$, where either $g>1$ and $m\in \brak{1,2}$, or $g=1$ and $m=2$.
If $n\geq 3$, by \reth{gam3closed}\ref{it:gam3g}, $\Gamma_3(B_{n}(\Sigma_{g}))/\Gamma_{4}(B_{n}(\Sigma_{g}))$ is trivial, while $\Gamma_3(B_{m}(\Sigma_{g}))/\Gamma_{4}(B_{m}(\Sigma_{g}))$ is not, and this implies that there is no surjective homomorphism from $B_{n}(\Sigma_{g})$ onto $B_{m}(\Sigma_{g})$.
If $n=2$, $m=1$ and $g>1$, $\Gamma_2(B_{2}(\Sigma_{g}))/\Gamma_{3}(B_{2}(\Sigma_{g}))$ is finite by \reth{gam3closed}\ref{it:gam23g}\ref{it:gam23gn2}, and so it cannot surject onto $\Gamma_2(\pi_1(\Sigma_g))/\Gamma_3(\pi_1(\Sigma_g))$, which is a (non-trivial) free Abelian group by \reth{gam3closed}\ref{it:gam23g}\ref{it:gam23gn}.
\item\label{it:ngrmc} $n>m \geq 3$. Assume first that $n\neq 4$. There can be no surjection homomorphism from $B_n(\Sigma_g)$ onto $B_{m}(\Sigma_{g})$, for otherwise its composition with the projection $B_{m}(\Sigma_{g})$ onto $S_{m}$ of~(\ref{eq:permutation}) would yield a surjective homomorphism from $B_n(\Sigma_g)$ onto $S_{m}$, which contradicts \reth{LinSurfcomb}.
So assume that $n=4$. Then $m=3$, and there can be no surjective homomorphism from $B_4(\Sigma_g)$ to $B_3(\Sigma_g)$ because otherwise by \reth{gam3closed}\ref{it:gam23g}\ref{it:gam23gn}, there would be a surjective homomorphism from $\Gamma_2(B_4(\Sigma_g))/\Gamma_3(B_4(\Sigma_g))$, which is isomorphic to $\ensuremath{\mathbb Z}_{3+g}$, onto $\Gamma_2(B_3(\Sigma_g))/\Gamma_3(B_3(\Sigma_g))$, which is isomorphic to $\ensuremath{\mathbb Z}_{2+g}$, but this is impossible.\qedhere
\end{enumerate}
\end{proof}
\begin{cor}\label{cor:pure}
Let $g\geq 1$, and let $n,m\in \ensuremath{\mathbb N}$. There is a surjective homomorphism of $B_n(\Sigma_g)$ onto $P_m(\Sigma_g)$ if and only if $n=m=1$ for $g \ge1$ and $m=1$ for $g=1$.
\end{cor}
\begin{proof}
Let $g\geq 1$. We first prove that the conditions are sufficient. If $n=m=1$, the result is clear since the given groups coincide with the fundamental group of the surface.
If $g=m=1$ then the result follows from \reth{gensurj}\ref{it:main1a}\ref{it:main1aii}. Conversely, suppose that there exists a surjective homomorphism $\Phi\colon\thinspace B_n(\Sigma_g)\to P_m(\Sigma_g)$. Then $\Phi$ induces a surjective homomorphism of the corresponding Abelianisations, but since $(P_m(\Sigma_g))_{\text{Ab}}\cong \ensuremath{\mathbb Z}^{2gm}$ from the presentation of $P_m(\Sigma_g)$ given in~\cite{B} for instance, it follows from \reth{gam3closed}\ref{it:gam12g}
that $m=1$. Then either $n=1$, or $n>1$, in which case $g=1$ by \reth{gensurj}\ref{it:main1a}\ref{it:main1aii}, and in both cases, the conclusion holds.
\end{proof}
\begin{rem}
With the exception of the case $g=1$, \reco{pure} was proved in~\cite[Theorem~1.2]{Ch} using different methods.
\end{rem}
\subsection{Surjections between braid groups of the sphere}\label{sec:sphere}
In this section, we complete the analysis of surjections between braid groups of orientable surfaces withour boundary by studying the case $g=0$, which is that of the sphere $\ensuremath{\mathbb{S}^{2}}$. Theorems~\ref{th:gensurj} and~\ref{th:LinSurfcomb} hold also in this case, but the arguments are somewhat different. As we mentioned just after the statement of \reth{presbng}, if $n\in \ensuremath{\mathbb N}$, the presentation of $B_{n}(\ensuremath{\mathbb{S}^{2}})$ in~\cite{FV} may be obtained from the standard presentation of $B_{n}$ by adding the relation~(\ref{eq:totsph}), so $B_{n}(\ensuremath{\mathbb{S}^{2}})$ is a quotient of $B_{n}$. It follows from this presentation that $B_{1}(\ensuremath{\mathbb{S}^{2}})$ is trivial, $B_2(\ensuremath{\mathbb{S}^{2}})=\ensuremath{\mathbb Z}_2$, $B_3(\ensuremath{\mathbb{S}^{2}})=\ensuremath{\mathbb Z}_3 \rtimes \ensuremath{\mathbb Z}_4$ (with non-trivial action), and $B_n(\ensuremath{\mathbb{S}^{2}})$ is an infinite group for all $n\ge 4$~\cite[third theorem, p.~255]{FV}.
The following result summarises some known results about the lower central series of the braid groups of the sphere.
\begin{thm}[\cite{GGsph}]\label{th:gam3sph}\mbox{}
\begin{enumerate}
\item\label{it:gam1sp} $\Gamma_1(B_n(\ensuremath{\mathbb{S}^{2}}))/\Gamma_2(B_n(\ensuremath{\mathbb{S}^{2}})) \cong \ensuremath{\mathbb Z}_{2(n-1)}$ for $n\ge 2$.
\item\label{it:gam2sp} $\Gamma_2(B_n(\ensuremath{\mathbb{S}^{2}})))=\Gamma_3(B_n(\ensuremath{\mathbb{S}^{2}}))$ for $n\ge 2$.
\item\label{it:gam3sp} $\Gamma_2(B_n(\ensuremath{\mathbb{S}^{2}})$ is perfect if and only if $n\ge 5$.
\end{enumerate}
\end{thm}
The proofs of parts~\ref{it:gam1sp},~\ref{it:gam2sp} and~\ref{it:gam3sp} of \reth{gam3sph} may be found in Proposition~2.1, and Theorems~1.3 and~1.4 respectively of \cite{GGsph}. We now prove \reth{LinSurfcomb} in the case of the sphere and \reth{gensurj}\ref{it:main1a}\ref{it:main1ai}.
\begin{proof}[Proof of \reth{LinSurfcomb} in the case where $\Sigma=\Sigma_{0}$]
Let $n>m\ge 3$ and $n\neq 4$, and suppose that there exists a surjective homomorphism $\Phi\colon\thinspace B_n(\ensuremath{\mathbb{S}^{2}}) \to S_m$. Since $B_n(\ensuremath{\mathbb{S}^{2}})$ is a quotient of $B_n$, $B_{n}$ surjects homomorphically onto $B_n(\ensuremath{\mathbb{S}^{2}})$, so its composition with $\Phi$ would gives rise to a surjective homomorphism from $B_n$ to $S_m$, which contradicts \reth{Lin}.
\end{proof}
\begin{proof}[Proof of \reth{gensurj}\ref{it:main1a}\ref{it:main1ai}]
We start by showing that the condition for the existence of a surjective homomorphism from $B_{n}(\ensuremath{\mathbb{S}^{2}})$ to $B_{m}(\ensuremath{\mathbb{S}^{2}})$ is sufficient. Since $B_{1}(\ensuremath{\mathbb{S}^{2}})$ is trivial, the result is clear if $m=1$, and if $n\geq 3$ and $m=2$, $B_n(\ensuremath{\mathbb{S}^{2}})$ surjects homomorphically onto $B_2(\ensuremath{\mathbb{S}^{2}})$ since $(B_{n}(\ensuremath{\mathbb{S}^{2}}))_{\text{Ab}}\cong \ensuremath{\mathbb Z}_{2(n-1)}$ and $B_2(\ensuremath{\mathbb{S}^{2}})\cong \ensuremath{\mathbb Z}_2$, so there exists a surjective homomorphism from $B_{n}(\ensuremath{\mathbb{S}^{2}})$ to $B_{2}(\ensuremath{\mathbb{S}^{2}})$ that factors through $(B_{n}(\ensuremath{\mathbb{S}^{2}}))_{\text{Ab}}$. To show that the condition is necessary, we consider the following two cases.
\begin{enumerate}[label=\textit{(\arabic*)}]
\item Suppose that $n<m$. Since $B_{1}(\ensuremath{\mathbb{S}^{2}})$ is trivial and $(B_{n}(\ensuremath{\mathbb{S}^{2}}))_{\text{Ab}}\cong \ensuremath{\mathbb Z}_{2(n-1)}$ for all $n\geq 2$, there does not exist a surjective homomorphism between $(B_{n}(\ensuremath{\mathbb{S}^{2}}))_{\text{Ab}}$ and $(B_{m}(\ensuremath{\mathbb{S}^{2}}))_{\text{Ab}}$, so there cannot exist a surjective homomorphism between $B_{n}(\ensuremath{\mathbb{S}^{2}})$ and $B_{m}(\ensuremath{\mathbb{S}^{2}})$.
\item Now let $n>m\ge 3$. If $n\neq 4$, the fact that there does not exist a surjective homomorphism from $B_n(\Sigma_g)$ onto $S_m$ by \reth{LinSurfcomb} implies that there does not exist a surjective homomorphism from $B_n(\ensuremath{\mathbb{S}^{2}})$ onto $B_m(\ensuremath{\mathbb{S}^{2}})$.
The remaining case, $(n,m)=(4,3)$, may be dealt with by studying the finite subgroups of the braid groups of $\ensuremath{\mathbb{S}^{2}}$ as follows. Let $\Phi\colon\thinspace B_{4}(\ensuremath{\mathbb{S}^{2}}) \to B_{3}(\ensuremath{\mathbb{S}^{2}})$ be a homomorphism, and using the notation of \reth{presbng} in $B_{4}(\ensuremath{\mathbb{S}^{2}})$, let $\alpha_{0}=\sigma_{1}\sigma_{2}\sigma_{3}$ and $\alpha_{1}= \sigma_{1}\sigma_{2} \sigma_{3}^{2}$, and let $\Delta_{4}=\sigma_{1}\sigma_{2} \sigma_{3}\sigma_{1} \sigma_{2}\sigma_{1}$ be the half-twist braid. By~\cite[Theorem~3]{GGjktr}, $B_{4}(\ensuremath{\mathbb{S}^{2}})=\langle \alpha_{0}, \alpha_{1} \rangle$. Now $\alpha_{0}$ is of order $8$, and the maximal torsion of $B_{3}(\ensuremath{\mathbb{S}^{2}})$ is equal to $6$, so the order of $\Phi(\alpha_{0})$ is a divisor of $4$~\cite{GVB,M}. But the full-twist braid $\Delta_{4}^{2}$ is the unique element of $B_{4}(\ensuremath{\mathbb{S}^{2}})$ of order $2$~\cite{GVB}. This implies that $\Delta_{4}^{2}$ belongs to the centre of $B_{4}(\ensuremath{\mathbb{S}^{2}})$, and also that $\alpha_{0}^{4}=\Delta_{4}^{2}$, from which we conclude that $\Delta_{4}^{2}$ belongs to $\ker{\Phi}$.
Let $H=\langle \alpha_{0}, \Delta_{4} \rangle$. By~\cite[Remark, p.~234]{GGblms}, $H$ is isomorphic to the generalised quaternion group $\mathcal{Q}_{16}$ of order $16$, where the relations are of the form $\alpha_{0}^{4}=\Delta_{4}^{2}$ and $\Delta_{4}\alpha_{0} \Delta_{4}^{-1}=\alpha_{0}^{-1}$. Consider the restriction $\Phi\vert_{H}\colon\thinspace H \to \im{\Phi\vert_{H}}$. Since $\Delta_{4}^{2}$ belongs to $H\cap \ker{\Phi\vert_{H}}$ and to the centre of $B_{4}(\ensuremath{\mathbb{S}^{2}})$, we see that $\Phi\vert_{H}$ factors through the quotient $H/\langle \Delta_{4}^{2}\rangle$. Using the relations of $H$ in terms of its generators, this quotient is isomorphic to the dihedral group of order $8$, and hence $\im{\Phi\vert_{H}}$ is a subgroup of $B_{3}(\ensuremath{\mathbb{S}^{2}})$ that is a quotient of this dihedral group.
On the other hand, the quotients of dihedral groups are either dihedral, the trivial group, or cyclic of order $2$. Further, $B_{3}(\ensuremath{\mathbb{S}^{2}})\cong \ensuremath{\mathbb Z}_{3}\rtimes \ensuremath{\mathbb Z}_{4}$, the action being the non-trivial one~\cite{FV}, so $B_{3}(\ensuremath{\mathbb{S}^{2}})$ has no dihedral subgroups. We conclude that $\im{\Phi\vert_{H}}\subset \langle \Delta_{3}^{2} \rangle$. Hence $\ker{\Phi\vert_{H}}$ is either equal to $H$, or is a subgroup of $H$ of index $2$. If $\ker{\Phi\vert_{H}}$ is of index $2$ in $H$, then by analysing the images of $\alpha_{0}$ and $\Delta_{4}$ by a surjective homomorphism from $H$ to $\ensuremath{\mathbb Z}_{2}$, we see that $\ker{\Phi\vert_{H}}$ is equal to $\ang{\alpha_{0}}$, to $\ang{\alpha_{0}^{2}, \Delta_{4}}$, or to $\ang{\alpha_{0}^{2}, \alpha_{0}\Delta_{4}}$. So if either $\ker{\Phi\vert_{H}}$ is equal to $H$, or is a subgroup of $H$ of index $2$, we conclude from these possibilities that $\alpha_{0}^{2} \in \ker{\Phi}$. It follows again from the fact that $\Delta_{3}^{2}$ is the unique element of $B_{3}(\ensuremath{\mathbb{S}^{2}})$ of order $2$ that $\Phi(\alpha_{0})\in \ang{\Delta_{3}^{2}}$, and so is central in $B_{3}(\ensuremath{\mathbb{S}^{2}})$. Since $B_{4}(\ensuremath{\mathbb{S}^{2}})=\langle \alpha_{0}, \alpha_{1} \rangle$, we conclude that $\im{\Phi}$ is cyclic, and hence $\Phi$ cannot be surjective.\qedhere
\end{enumerate}
\end{proof}
\begin{rem
It follows from \reth{gensurj}\ref{it:main1a}\ref{it:main1ai} that there is no surjective homomorphism from $ B_4(\ensuremath{\mathbb{S}^{2}})$ to $B_3(\ensuremath{\mathbb{S}^{2}})$. However, the maps from $B_4(\ensuremath{\mathbb{S}^{2}})$ to $S_3$ defined by sending the generators $\sigma_1$ and $\sigma_3$ to $(1,2)$ and $\sigma_2$ to $(2,3)$ and from $B_{4}$ to $B_{3}$ defined by sending the generators $\sigma_1$ and $\sigma_3$ to $\sigma_{1}$ and $\sigma_2$ to $\sigma_{2}$, extend to well-defined, surjective homomorphisms.
\end{rem}
\section{Surjections between braid groups of orientable surfaces with boundary}\label{sec:orboundary}
Let $\Sigma_{g,b}$ be a compact, connected orientable surface of genus $g$ with $b\geq 0$ boundary
components. A presentation for $B_n(\Sigma_{g,b})$ may be found in~\cite[Proposition~3.1]{BGoG}, and in the case $b=1$, a presentation for $B_n(\Sigma_{g,1})$ may be obtained from that of $B_n(\Sigma_{g})$ given in \reth{presbng} by deleting relation~(\ref{eq:tot}). The case $b=0$ was dealt with in \resec{closed}, so we shall assume henceforth that $b\geq 1$. The following two results generalise those of \reth{gam3closed} to the braid groups of $\Sigma_{g,b}$.
\begin{thm}[{\cite[Theorem~2]{BGeG}}]\label{th:gam3open}
Let $g,b\geq 1$, and let $n\geq 3$.
Then:
\begin{enumerate}
\item\label{it:g1sigma} $\Gamma_1(B_n(\Sigma_{g,b}))/ \Gamma_2(B_n(\Sigma_{g,b})) \cong\ensuremath{\mathbb Z}^{2g+b-1} \oplus \ensuremath{\mathbb Z}_2$.
\item\label{it:g2sigma} $\Gamma_2(B_n(\Sigma_{g,b}))/ \Gamma_3(B_n(\Sigma_{g,b}))\cong \ensuremath{\mathbb Z}$.
\item\label{it:g3sigma} $\Gamma_3(B_n(\Sigma_{g,b}))= \Gamma_4(B_n(\Sigma_{g,b}))$. Moreover
$\Gamma_3(B_n(\Sigma_{g,b}))$ is perfect for $n\ge 5$.
\item\label{it:resid} $B_n(\Sigma_{g,b})$ is not residually nilpotent.
\end{enumerate}
\end{thm}
The following proposition treats the case $n=2$.
\begin{prop} \label{prop:n=2orientablebound}
Let $g,b\ge 1$.
\begin{enumerate}
\item\label{it:resboundary1} The group $B_2(\Sigma_{g,b})$ is residually $2$-finite and therefore residually nilpotent, but is not nilpotent.
\item\label{it:resboundary12} $\Gamma_1(B_2(\Sigma_{g,b}))/ \Gamma_2(B_2(\Sigma_{g,b})) \cong\ensuremath{\mathbb Z}^{2g+b-1} \oplus \ensuremath{\mathbb Z}_2$.
\item\label{it:resboundary2} The group $\Gamma_2(B_2(\Sigma_{g,b}))/\Gamma_3(B_2(\Sigma_{g,b}))$ is a non-trivial quotient of $\ensuremath{\mathbb Z}_2^{2g+b-1} \oplus \ensuremath{\mathbb Z}$.
\end{enumerate}
\end{prop}
Presentations for $B_n(\Sigma_{g,b})/ \Gamma_3(B_n(\Sigma_{g,b}))$ were exhibited in~\cite[eq.~(10)]{BGeG} for $b=1$, and in~\cite[Proposition~3.13]{BGoG} for $b\ge 1$.
\begin{proof}[Proof of \repr{n=2orientablebound}]
Let $n=2$, and consider the short exact sequence~(\ref{eq:permutation}), where we take $\Sigma= \Sigma_{g,b}$.
Since $S_2\cong \ensuremath{\mathbb Z}_2$ and $P_2(\Sigma_{g,b})$ is residually torsion free nilpotent~\cite[Theorem~4]{BGeG}, and therefore $2$-finite, the hypotheses of~\cite[Lemma~1.5]{Gr} are fulfilled, so $B_2(\Sigma_{g,b})$ is residually nilpotent. To see that it is not nilpotent, suppose on the contrary that there exists $i\in \ensuremath{\mathbb N}$ such that $\Gamma_i(B_2(\Sigma_{g,b}))=\brak{1}$. Without loss of generality, we may suppose that $i$ is minimal with respect to this property. Since $B_2(\Sigma_{g,b})$ is non Abelian, it follows from \reth{gam3open}\ref{it:g1sigma} that $i\geq 3$. Now $\Gamma_{i}(B_2(\Sigma_{g,b}))= [\Gamma_{i-1}(B_2(\Sigma_{g,b})), B_2(\Sigma_{g,b})]=\brak{1}$, and hence $\Gamma_{i-1}(B_2(\Sigma_{g,b}))$ is contained in the centre of $B_2(\Sigma_{g,b})$. This centre is trivial~\cite{GGmpcps,PR}, so $\Gamma_{i-1}(B_2(\Sigma_{g,b}))=\brak{1}$, but this contradicts the minimality of $i$. Part~\ref{it:resboundary1} follows.
For part~\ref{it:resboundary12}, we just give the proof in the case $b=1$. The general case may be obtained in a similar manner using the presentation of $B_2(\Sigma_{g,b})$ given in~\cite{BGoG}. As we mentioned above, a presentation of $B_2(\Sigma_{g,1})$ may be obtained by deleting relation~(\ref{eq:tot}) from the presentation of \reth{presbng}. Thus the proof given in \repr{pre} for $\Sigma_{g}$ is also valid in the case of $\Sigma_{g,b}$, except that we can no longer conclude that $\sigma_{1}^{2}$ is of finite order, so the second factor in the direct product decomposition of $B_2(\Sigma_{g,1})/\Gamma_3(B_2(\Sigma_{g,1}))$ is $\ensuremath{\mathbb Z}$. Part~\ref{it:resboundary2} is a consequence of part~\ref{it:resboundary1}.
\end{proof}
\begin{rem}\label{rem:n1bound}
\reth{gam3open} (in the case $n\geq 3$) and \repr{n=2orientablebound} (in the case $n=2$)
generalise \reth{gam3closed}. If $n=1$, $B_1(\Sigma_{g,b})$ is a free group of rank $2g+b-1$, and
its lower central series is well known, see~\cite{LA} for instance. Note that in particular $B_1(\Sigma_{g,b})$ is residually nilpotent. It follows from \reth{gam3open}\ref{it:resid} and \repr{n=2orientablebound}\ref{it:resboundary1} that $B_n(\Sigma_{g,b})$ is residually nilpotent if and only if $n\le 2$. As in the proof of \repr{perfect}, we see that $\Gamma_3(B_n(\Sigma_{g,b}))$ is not perfect if $n\in \brak{3,4}$.
Hence using \reth{gam3open}\ref{it:g3sigma}$,\Gamma_3(B_n(\Sigma_{g,b}))$ is perfect if and only if $n\ge 5$.
\end{rem}
We now prove \reth{gensurj}\ref{it:main1b} in the orientable case and \reco{pureboundary}. We first require the following result.
\begin{lem}\label{lem:gb1}
There is no surjective homomorphism from $B_2(\Sigma_{1,1})$ onto $\pi_1(\Sigma_{1,1})$.
\end{lem}
\begin{proof}
Suppose on the contrary that there exists a surjective homomorphism $\phi\colon\thinspace B_2(\Sigma_{1,1}) \to \pi_1(\Sigma_{1,1})$. Let $\alpha= a_1 \sigma_1$, $\beta = b_1 \sigma_1$. Then $\alpha, \beta, \sigma_1$ generate $B_2(\Sigma_{1,1})$, and the defining relations of \reth{presbng} become:
\begin{align}
\alpha^2 &= \sigma_1 \alpha^2 \sigma_1^{-1} \label{eq:alpha}\\
\beta^2 &= \sigma_1 \beta^2 \sigma_1^{-1} \label{eq:beta}\\
\alpha \beta \sigma_1^{-1} &= \sigma_1 \beta\alpha.\label{eq:abs}
\end{align}
Now $\pi_1(\Sigma_{1,1})$ is a free group of rank $2$, and so if $u,v \in \pi_1(\Sigma_{1,1})$, the relation $u^2=v^2$ implies that $u=v$. Applying this to relations~(\ref{eq:alpha}) and~(\ref{eq:beta}), we deduce that $\phi(\sigma_1)$ is central in $\pi_1(\Sigma_{1,1})$, and so $\phi(\sigma_1)=1$. Since $\phi$ is surjective, it follows that $\pi_1(\Sigma_{1,1})=\ang{\phi(\alpha), \phi(\beta)}$. Relation~(\ref{eq:abs}) implies that $\phi(\alpha)$ and $\phi(\beta)$ commute. Consequently, $\ang{\phi(\alpha), \phi(\beta)}$ is cyclic, and this contradicts the assumption that $\phi$ is surjective.
\end{proof}
In contrast with the case of $\Sigma_{g}$, \reth{gam3open} implies that the lower central series does not distinguish the number of strings for braid groups of orientable surfaces with boundary if $n\geq 3$. Nevertheless, we are able to show that in certain cases, there does not exist a surjective homomorphism between $B_n(\Sigma_{g,b})$ and $B_m(\Sigma_{g,b})$.
\begin{proof}[Proof of \reth{gensurj}\ref{it:main1b} in the orientable case]\mbox{}
We consider in turn the three cases given in the statement.
\begin{enumerate}[label=\textit{(\roman*)}]
\item Let $n<m$ and $n\in \brak{1,2}$. The arguments used in \repr{mingen} apply verbatim to the case with boundary. In particular $G(B_{1}(\Sigma_{g,b}))=2g+b-1$, $G(B_{2}(\Sigma_{g,b}))= 2g+b$, and $G(B_{m}(\Sigma_{g,b}))= 2g+b+1$ for all $m\ge 3$. It follows that there does not exist a surjective homomorphism in this case.
\item Suppose that $n>m$ and $m\in \brak{1,2}$. First let $n\geq 3$. Then $B_m(\Sigma_{g,b})$ is residually nilpotent by \repr{n=2orientablebound} and \rerem{n1bound}. Since $B_n(\Sigma_{g,b})$ is not residually nilpotent by \reth{gam3open}\ref{it:resid}, it cannot surject homomorphically onto $B_m(\Sigma_{g,b})$. So suppose that $n=2$ and $m=1$. The case $(g,b) =(1,1)$ was dealt with in \relem{gb1}, so we may assume that $(g,b)\neq (1,1)$, in which case $2g+b\geq 4$. By \repr{n=2orientablebound}\ref{it:resboundary2}, the Abelian group $\Gamma_2(B_2(\Sigma_{g,b}))/\Gamma_3(B_2(\Sigma_{g,b}))$ is of rank at most $1$. On the other hand, $\Gamma_2(\pi_1(\Sigma_{g,b}))/\Gamma_3(\pi_1(\Sigma_{g,b}))$ is free Abelian of rank $(2g+b-1)(2g+b-2)/2$~\cite{LA}, and this rank is strictly greater than $1$. Thus $B_2(\Sigma_{g,b})$ cannot surject homomorphically onto $B_1(\Sigma_{g,b})$.
\item Suppose that $n>m\ge3$ and $n\neq 4$. Using the presentation of $B_m(\Sigma_{g,b})$ given in~\cite[Proposition~3.1]{BGoG}, the proof of \reth{LinSurfcomb} goes through in this case, the only difference being that $B_n(\Sigma_{g,b})/\langle\!\langle \sigma_1\rangle\!\rangle$ is isomorphic to $\ensuremath{\mathbb Z}^{2g+b-1}$. The result then follows by an argument similar to that given in case~\ref{it:ngrmc} of the proof of \reth{gensurj}\ref{it:main1a}\ref{it:main1aii} in \resec{suror}.\qedhere
\end{enumerate}
\end{proof}
The following result is the analogue of \reco{pure} in the case where the surface has boundary.
\begin{cor}\label{cor:pureboundary}
Let $g\geq 1$, and let $n,m\in \ensuremath{\mathbb N}$. Then there exists a surjective homomorphism of $B_n(\Sigma_{g,b})$ onto $P_m(\Sigma_{g,b})$ if and only if $n=m=1$.
\end{cor}
\begin{proof}
Let $g\geq 1$. The proof is similar to that of \reco{pure}. If $n=m=1$, the result is clear, so suppose that there exists a surjective homomorphism $\Phi\colon\thinspace B_n(\Sigma_{g,b})\to P_m(\Sigma_{g,b})$, where $(n,m)\neq (1,1)$. Then $\Phi$ induces a surjective homomorphism of the corresponding Abelianisations, but since $(P_m(\Sigma_{g,b}))_{\text{Ab}}$ is isomorphic to $\ensuremath{\mathbb Z}^{(2g+b-1)m}$ using a presentation of $P_m(\Sigma_{g,b})$ (see~\cite{B} for instance), it follows from \reth{gam3open}\ref{it:g1sigma}, \repr{n=2orientablebound}\ref{it:resboundary12} and the fact that $(B_{1}(\Sigma_{g,b}))_{\text{Ab}}$ is isomorphic to $\ensuremath{\mathbb Z}^{2g+b-1}$ that $m=1$. By \reth{gensurj}\ref{it:main1b}\ref{it:main1bii}, we conclude that $n=1$.
\end{proof}
\section{Surjections between braid groups of non-orientable surfaces}\label{sec:nonor}
We start this section by recalling a presentation of the braid groups of compact, non-orientable surfaces without boundary.
\begin{thm}[\cite{B}]\label{th:presnor}
Let $g\geq 1$, let $n\geq 2$, and let $U_{g}$ be a compact, connected non-orientable surface without boundary of genus $g$. Then $B_n(U_g)$ admits the following group presentation:
\begin{itemize}
\item \textbf{generators:} $\rho_1, \ldots, \rho_g, \sigma_1, \ldots, \sigma_{n-1}$.
\item \textbf{relations:}
\begin{gather}
\text{$\sigma_i\sigma_j=\sigma_j\sigma_i$ if $\lvert i-j \rvert \geq
2$}\label{eq:artin1or}\\
\text{$\sigma_i\sigma_{i+1}\sigma_i= \sigma_{i+1}\sigma_i \sigma_{i+1}$
for all $1\leq i\leq n-2$}\label{eq:artin2or}\\
\text{$\rho_i\sigma_j= \sigma_j \rho_i$ for all $j\geq 2$ and $i=1, \ldots, g$}\label{eq:rs1}\\
\text{$\rho_i \sigma_1 \rho_i \sigma_1= \sigma_1^{-1} \rho_i \sigma_1 \rho_i$ for
$i=1, \ldots, g$}\label{eq:rr}\\
\text{$\rho_r \sigma_1^{-1} \rho_s \sigma_1=\sigma_1^{-1} \rho_s \sigma_1 \rho_r$
for $1\le s<r \le g$}\label{eq:rs2}\\
\text{$\prod_{i=1}^g \rho_i^{-2}= \sigma_1\cdots \sigma_{n-2} \sigma_{n-1}^2 \sigma_{n-2} \cdots \sigma_1$}\label{eq:totnor}.
\end{gather}
\end{itemize}
\end{thm}
The presentation of $B_n(U_g)$ of~\cite[Theorem~A.3]{B} is slightly different from that given in \reth{presnor}, but one can obtain the first presentation from the second by replacing each generator $a_i$ in~\cite[Theorem~A.3]{B} by $\rho_i^{-1}$ in \reth{presnor} for all $i=1, \ldots, g$.
\begin{rem}
Notice that~\cite[Theorem~A.3]{B} was stated for $g>1$, but the presentation is also valid if $g=1$, in which case the relation~(\ref{eq:rs2}) does not exist.
This may be seen by showing that the map from $B_{n}(U_{1})$ to itself that sends the generator $\sigma_i$ (resp.\ $\rho_1$) of~\cite[Theorem~A.3]{B} to the generator $\sigma_i$ (resp.\ $\rho_1^{-1}$) of~\cite{VB} for all $1\leq i\leq n-1$ is well defined, and that it is an isomorphism. The presentation also holds if $n=1$. In particular, $B_{1}(U_{g})$ is a one-relator group, and the results of~\cite{LA} apply.
\end{rem}
The following theorem summarises some of the known results about the lower central series of braid groups of non-orientable surfaces without boundary~\cite{GG,GP,MP}, and is the analogue of Theorems~\ref{th:gam3closed} and~\ref{th:gam3open}. One may consult~\cite{BGe} for the case of pure braid groups.
\begin{thm}[\cite{GG,GP}]\label{th:gam3nor}
Let $g\geq 1$.
Then:
\begin{enumerate}
\item\label{it:g1sigmaor} $\Gamma_1(B_n(U_{g}))/ \Gamma_2(B_n(U_{g}))=\ensuremath{\mathbb Z}^{g-1} \oplus \ensuremath{\mathbb Z}_2 \oplus \ensuremath{\mathbb Z}_2$ for all $n\geq 2$.
\item\label{it:g2sigmaor} $\Gamma_2(B_n(U_{g}))= \Gamma_3(B_n(U_{g}))$ for all $n\geq 3$.
\item\label{it:g1sigmaorc} $\Gamma_2(B_n(U_{g}))$ is perfect if and only if $n\geq 5$.
\item\label{it:residor} $B_n(U_{g})$ is residually nilpotent if and only if $n\leq 2$.
\end{enumerate}
\end{thm}
\begin{proof}
If $g=1$, the four statements were proved in~\cite[Theorem~1 and Proposition~6]{GG}. So in the rest of the proof, we assume that $g\geq 2$.
\begin{enumerate}
\item The statement follows in a straightforward manner using the presentation of \reth{presnor}.
\item If $n\geq 3$, the fact that $\Gamma_2(B_n(U_{g}))= \Gamma_3(B_n(U_{g}))$ is a consequence of the proof of~\cite[Proposition~5.21]{GP} (resp.\ of~\cite[Theorem~6.1]{GP}) if $g=2$ (resp.\ if $g\geq 3$).
\item The `if' part is a consequence of~\cite[Theorem~1.4]{GP}.
The proof of the `only if' part is similar to that of \repr{perfect}, and is left to the reader.
\item This follows from~\cite[Theorem~1.4]{GP}.\qedhere
\end{enumerate}
\end{proof}
\begin{proof}[Proof of \reth{LinSurfcomb} in the case where $\Sigma=U_{g}$, and $g\geq 1$]
Let $n>m\ge 3$, where $n\neq 4$. Let $\Phi\colon\thinspace B_n(U_g) \to S_{m}$. As in the proof of the orientable case, we see that $\Phi(\sigma_1)= \cdots= \Phi(\sigma_{n-1})$. We denote this common element by $\sigma$. From relations~(\ref{eq:rs1}), $\sigma$ commutes with $\Phi(\rho_j)$ for all $j=1, \ldots, g$, so $\sigma$ belongs to the centre of $S_m$, and we conclude once more that $\sigma$ is trivial, and hence the homomorphism $\Phi$ factors through a surjective homomorphism $\Phi'\colon\thinspace B_n(U_g)/\langle\!\langle \sigma_1\rangle\!\rangle \to S_m$, where $\langle\!\langle \sigma_1\rangle\!\rangle$ denotes the normal closure of $\sigma_{1}$ in $B_n(U_g)$.
But $B_n(U_g)/\langle\!\langle \sigma_1\rangle\!\rangle$ is Abelian by relations~(\ref{eq:rs2}), which yields a contradiction because an Abelian group cannot surject homomorphically onto a non-Abelian group. So there is no surjective homomorphism from $B_{n}(U_{g})$ onto $S_{m}$.
\end{proof}
As in the case of orientable surfaces, we may obtain more information about the lower central series of $B_2(U_g)$.
\begin{prop} \label{prop:b2nor}
Let $g\ge1$. Then the group $B_2(U_g)$ is residually $2$-finite, and so is residually nilpotent.
Moreover, the group $\Gamma_2(B_2(U_g))/\Gamma_3(B_2(U_g))$ is a non-trivial quotient of $\ensuremath{\mathbb Z}_2^{g}$.
\end{prop}
\begin{proof}
The case $g=1$ is straightforward because $B_2(\ensuremath{\mathbb{R}P^{2}})$ is isomorphic to the generalised quaternion group of order $16$~\cite[Theorem, p.~94]{VB}. In particular, $\Gamma_2(B_2(\ensuremath{\mathbb{R}P^{2}}))/\Gamma_3(B_2(\ensuremath{\mathbb{R}P^{2}}))\cong \ensuremath{\mathbb Z}_{2}$.
If $g\geq 2$, the residual nilpotence of $B_2(U_g)$ follows by arguing as in the proof of \repr{n=2orientablebound}\ref{it:resboundary1}, using the fact that $P_2(U_g)$ is residually $2$-finite~\cite[Theorem~1.1]{BGe}. Note that $B_2(U_2)$ is not nilpotent, since otherwise $P_2(U_2)$ would be nilpotent, but we know from~\cite[Theorem~5.4]{GP} that this is not the case. If $g\geq 3$, the centre of the group $B_2(U_g)$ is trivial~\cite[Proposition~1.6]{PR}, and as in \rerem{notnilp}, we can prove that the group $\Gamma_2(B_2(U_g))/\Gamma_3(B_2(U_g))$ is non-trivial. To see that this group is a quotient of $\ensuremath{\mathbb Z}_2^{g}$, observe that for all $1\leq s<r\leq g$, we have $[\rho_r,\sigma_1^{-1} \rho_s \sigma_1]=1$ in $B_2(U_g)$ by relation~(\ref{eq:rs2}), so $[\rho_r,\rho_s]=1$ in $B_2(U_g)/\Gamma_3(B_2(U_g))$. Thus $\Gamma_2(B_2(U_g))/\Gamma_3(B_2(U_g))$ is generated by the commutators of the form $[\rho_i, \sigma_1]$, where $1\leq i\leq g$. Since $\sigma_1^2=1$ in $B_2(U_g)/\Gamma_3(B_2(U_g))$ by relation~(\ref{eq:rr}), these commutators are of order at most $2$.
\end{proof}
At this point, we may prove \repr{ornor} concerning the existence of a surjective homomorphism between $B_n(U_g)$ onto $B_n(\Sigma_{g-1})$.
\begin{proof}[Proof of \repr{ornor}]
First suppose that $g=1$, in which case $U_1=\ensuremath{\mathbb{R}P^{2}}$ and $\Sigma_0=\ensuremath{\mathbb{S}^{2}}$. If $n=1$, $B_1(\ensuremath{\mathbb{S}^{2}})$ is trivial, and so there is clearly a surjection of $B_1(\ensuremath{\mathbb{R}P^{2}})$ onto $B_1(\ensuremath{\mathbb{S}^{2}})$, and if $n=2$ then $B_2(\ensuremath{\mathbb{R}P^{2}})$ is isomorphic to the generalised quaternion group of order $16$~\cite[Theorem, p.~94]{VB}, and $B_2(\ensuremath{\mathbb{S}^{2}})\cong \ensuremath{\mathbb Z}_2$ \cite[third theorem, p.~255]{FV}, and $B_2(\ensuremath{\mathbb{R}P^{2}})$ surjects homomorphically onto $B_2(\ensuremath{\mathbb{S}^{2}})$. Finally, if $n\geq 3$, $(B_n(\ensuremath{\mathbb{R}P^{2}}))_{\text{Ab}}\cong \ensuremath{\mathbb Z}_2\oplus \ensuremath{\mathbb Z}_2$ by \reth{gam3nor}\ref{it:g1sigmaor}, and $(B_n(S^2))_{\text{Ab}}\cong \ensuremath{\mathbb Z}_{2(n-1)}$ by~\reth{gam3sph}\ref{it:gam1sp}, which implies that there is no surjection homomorphism from $B_n(\ensuremath{\mathbb{R}P^{2}})$ onto $B_n(\ensuremath{\mathbb{S}^{2}})$. This proves the result in the case $g=1$.
Now assume that $g\ge 2$. If $n \ge 2$, $(B_n(U_g))_{\text{Ab}}\cong \ensuremath{\mathbb Z}^{g-1} \oplus \ensuremath{\mathbb Z}_2\oplus \ensuremath{\mathbb Z}_2$ by \reth{gam3nor}\ref{it:g1sigmaor} and $(B_n(\Sigma_{g-1}))_{\text{Ab}}= \ensuremath{\mathbb Z}^{2(g-1)} \oplus \ensuremath{\mathbb Z}_2$ by \reth{gam3closed}\ref{it:gam12g} while if $n=1$, $(B_1(U_g))_{\text{Ab}}\cong \ensuremath{\mathbb Z}^{g-1} \oplus \ensuremath{\mathbb Z}_2$ and $(B_n(\Sigma_{g-1}))_{\text{Ab}}= \ensuremath{\mathbb Z}^{2(g-1)}$. Therefore it is not possible to surject $B_n(U_g)$ onto $B_n(\Sigma_{g-1}$ in this case..
\end{proof}
To prove \reth{gensurj} in the non-orientable case, we will require the following lemma for the Klein bottle.
\begin{lem}\label{lem:garrafe}
Let $\ensuremath{\mathbb{K}^2}$ be the Klein bottle. If $x$ and $y$ are elements of $\pi_1(\ensuremath{\mathbb{K}^2})$, then $xyxy=y^{-1}xyx$ if and only if $y=1$.
\end{lem}
\begin{proof}
If $y=1$ then the relation clearly holds. Conversely, suppose that there exist $x,y\in \pi_1(\ensuremath{\mathbb{K}^2})$ that satisfy the relation.
Recall that $\pi_1(\ensuremath{\mathbb{K}^2})$ is isomorphic to the semi-direct product $\ensuremath{\mathbb Z} \rtimes \ensuremath{\mathbb Z}$, where the action is given by multiplication by $-1$. With respect to this decomposition, let $x=(a,b)$ and $y=(c,d)$. Substituting these elements into the given relation, the second coordinate yields $2b+2d=2b$, so $d=0$, and computing the first coordinate, we obtain $a+(-1)^{b}c+(-1)^{b}(a+(-1)^{b}c)= -c+a+(-1)^b(c+a)$. Therefore $-c=c$, so $c=0$ and hence $y$ is the trivial element of $\pi_{1}(\ensuremath{\mathbb{K}^2})$.
\end{proof}
We now prove \reth{gensurj}\ref{it:main1b} in the non-orientable case, where $g>1$.
\begin{proof}[Proof of \reth{gensurj}\ref{it:main1b}, where $\Sigma=U_{g}$, and $g> 1$]
We study the three cases of the statement of \reth{gensurj}\ref{it:main1b} separately.
\begin{enumerate}
\item Let $n<m$ and $n\in \brak{1,2}$. Using \reth{gam3nor} and the fact that $\Gamma_1(B_1(U_{g}))/ \Gamma_2(B_1(U_{g}))=\ensuremath{\mathbb Z}^{g-1} \oplus \ensuremath{\mathbb Z}_2$, the arguments used in the proof of \repr{mingen} also apply to the non-orientable case. In particular, $G(B_{1}(U_{g}))=g$, $G(B_{2}(U_{g}))= g+1$ and $G(B_{m}(U_{g}))= g+2$ for all $m\ge 3$. It follows that there is no surjective homomorphism in this case.
\item Suppose that $n>m$ and $m\in \brak{1,2}$. If $n\geq 3$, we have $\Gamma_2(B_n(U_{g}))= \Gamma_3(B_n(U_{g}))$ by \reth{gam3nor}\ref{it:g2sigmaor}. On the other hand, $B_m(U_{g})$ is residually nilpotent if $m=1$ (and is in fact residually $2$-finite, see for instance~\cite[proof of Theorem~4.5]{BGe}), or if $m=2$ by \repr{n=2orientablebound}. So
$B_{n}(U_{g})$ cannot surject homomorphically onto $B_{m}(U_{g})$ if $n\geq 3$ and $m\in \brak{1,2}$. So assume that $n=2$ and $m=1$. If $g>2$, the result follows in a similar manner by noting that $\Gamma_2(B_{2}(U_g))/\Gamma_3(B_{2}(U_g))$ is finite by \repr{b2nor}, but that $\Gamma_2(\pi_1(U_g))/\Gamma_3(\pi_1(U_g))$ is infinite~\cite{LA}. So suppose that $g=2$, and assume that there exists a surjective homomorphism $\Phi\colon\thinspace B_2(U_2) \to \pi_1(U_{2})$. Applying $\Phi$ to relation~(\ref{eq:rr}) with $i=1$, we have that $ \Phi(\rho_1) \Phi(\sigma_1) \Phi(\rho_1) \Phi(\sigma_1)= \Phi(\sigma_1)^{-1} \Phi(\rho_1) \Phi(\sigma_1) \Phi(\rho_1) $. The relation given in the statement of \relem{garrafe} is therefore satisfied if we take $x=\Phi(\rho_1)$ and $y=\Phi(\sigma_1)$, and thus $\Phi(\sigma_1)=1$. It follows from relation~(\ref{eq:rs2}) that $\Phi(\rho_1)$ and $ \Phi(\rho_2)$ commute. We conclude that the image of $\Phi$ is an Abelian subgroup of $\pi_1(U_{2})$, and since this latter group is non Abelian, $\Phi$ cannot be surjective.
\item\label{it:proofbiii} If $n>m \geq 3$ and $n\neq 4$, it suffices to argue as in the proof of part~\ref{it:ngrmc} of \reth{gensurj}\ref{it:main1a}\ref{it:main1aii} given in \resec{suror}, and apply \reth{LinSurfcomb} in the non-orientable case.\qedhere
\end{enumerate}
\end{proof}
The following theorem gives some results in the case where $g=1$.
\begin{thm}\label{th:rp}\mbox{}
\begin{enumerate}
\item\label{it:rpa}
Suppose that one of the following conditions holds:
\begin{enumerate}
\item\label{it:rpai} $n<m$ and $n\in \brak{1,2}$.
\item\label{it:rpaii} $n>m\geq 2$.
\end{enumerate}
Then there is no surjective homomorphism from $B_{n}(\ensuremath{\mathbb{R}P^{2}})$ to $B_{m}(\ensuremath{\mathbb{R}P^{2}})$.
\item\label{it:rpb} Let $m,n\geq 2$, let $n'=2\lfloor \frac{n}{2} \rfloor$ and let $m'=2\lfloor \frac{m}{2} \rfloor$, where $\lfloor x \rfloor$ denotes the greatest integer less than or equal to $x$. Set $n'=2^{l}s$ and $m'=2^{k}r$, where $l,k\in \ensuremath{\mathbb N}$, and $s,r$ are odd integers. If $l>k$ then the image of any homomorphism $f\colon\thinspace B_{n}(\ensuremath{\mathbb{R}P^{2}}) \to B_{m}(\ensuremath{\mathbb{R}P^{2}})$ is finite cyclic. In particular, there is no surjective homomorphism from $B_{n}(\ensuremath{\mathbb{R}P^{2}})$ to $B_{m}(\ensuremath{\mathbb{R}P^{2}})$ in this case.
\end{enumerate}
\end{thm}
\begin{rems}\mbox{}\label{rems:rp2}
\begin{enumerate}
\item\label{it:rp2b} If $n\geq 2$ then $B_{n}(\ensuremath{\mathbb{R}P^{2}})_{\text{Ab}}\cong \ensuremath{\mathbb Z}_{2}\oplus \ensuremath{\mathbb Z}_{2}$ by \reth{gam3nor}\ref{it:g1sigmaor}, and since $B_{1}(\ensuremath{\mathbb{R}P^{2}})\cong \ensuremath{\mathbb Z}_{2}$, we see that $B_{n}(\ensuremath{\mathbb{R}P^{2}})$ surjects homomorphically onto $B_{1}(\ensuremath{\mathbb{R}P^{2}})$ via Abelianisation.
\item Comparing the statement of \reth{gensurj}\ref{it:main1b} with that of \reth{rp}\ref{it:rpa}, if $n>2$, then by the previous remark, there is a surjective homomorphism from $B_{n}(\Sigma)$ to $B_{2}(\Sigma)$ if $\Sigma=\ensuremath{\mathbb{R}P^{2}}$, which is not the case if $\Sigma=\Sigma_{g,b}$ or $U_{g+1}$, where $b,g\geq 1$. Further, if $n=4$ and $m=3$, we do not know whether there exists a surjective homomorphism from $B_{4}(\Sigma)$ to $B_{3}(\Sigma)$ if $\Sigma=\Sigma_{g,b}$ or $U_{g+1}$, where $b,g\geq 1$, but \reth{rp}\ref{it:rpaii} shows that there does not exist such a homomorphism if $\Sigma=\ensuremath{\mathbb{R}P^{2}}$.
\end{enumerate}
\end{rems}
\begin{proof}[Proof of \reth{rp}]\mbox{}
\begin{enumerate}
\item \begin{enumerate}
\item If $n<m$ and $n\in \brak{1,2}$, the conclusion follows from the fact that $B_{1}(\ensuremath{\mathbb{R}P^{2}})\cong \ensuremath{\mathbb Z}_{2}$, $B_{2}(\ensuremath{\mathbb{R}P^{2}})$ is isomorphic to the binary dicyclic group of order $16$, and if $m\geq 3$, $B_{m}(\ensuremath{\mathbb{R}P^{2}})$ is infinite~\cite[Theorem, p.94]{VB}.
\item Assume that $n>m\geq 2$. If $m=2$, the result is a consequence of the fact that $B_{2}(\ensuremath{\mathbb{R}P^{2}})$ is residually nilpotent, while $B_{n}(\ensuremath{\mathbb{R}P^{2}})$ is not for all $n\geq 3$ by \reth{gam3nor}. Now suppose that $m\geq 3$. If $n\neq 4$, the result follows as in the proof of part~\ref{it:proofbiii} of \reth{gensurj}\ref{it:main1b} by applying \reth{LinSurfcomb} in the non-orientable case. We defer the proof of the case $n=4$ and $m=3$ to part~\ref{it:rpb}.
\end{enumerate}
\item Let $m,n\geq 2$, let $n'=2\lfloor \frac{n}{2} \rfloor$ and let $m'=2\lfloor \frac{m}{2} \rfloor$, and let $n'=2^{l}s$ and $m'=2^{k}r$, where $l,k\in \ensuremath{\mathbb N}$, and $s,r$ are odd integers. Let $\phi\colon\thinspace B_{n}(\ensuremath{\mathbb{R}P^{2}}) \to B_{m}(\ensuremath{\mathbb{R}P^{2}})$ be a homomorphism. Consider the elements $a=\rho_{n}\sigma_{n-1} \cdots \sigma_{1}$ and $b=\rho_{n-1}\sigma_{n-2} \cdots \sigma_{1}$ of $B_{n}(\ensuremath{\mathbb{R}P^{2}})$, where we use Van Buskirk's presentation of $B_{n}(\ensuremath{\mathbb{R}P^{2}})$~\cite[p.~83]{VB}. By~\cite[Proposition~26]{GGagt}, $a$ (resp.\ $b$) is of order $4n$ (resp.\ $4(n-1)$). Let $x=a$ and $x'=b$ (resp.\ $x=b$ and $x'=a$) if $n$ is even (resp.\ is odd). Then $x$ is of order $4n'$, which in terms of the notation introduced in the statement, is equal to $2^{l+2}s$. Observe also that from the proof of~\cite[Theorem~6]{GGzeit}, $B_{n}(\ensuremath{\mathbb{R}P^{2}})=\langle x,x' \rangle$.
By~\cite[Theorem~4]{GGagt}, the (maximal) torsion of $B_{n}(\ensuremath{\mathbb{R}P^{2}})$ (resp.\ of $B_{m}(\ensuremath{\mathbb{R}P^{2}})$) is $4n$ and $4(n-1)$ (resp.\ $4m$ and $4(m-1)$), and so the maximal torsion in $B_{n}(\ensuremath{\mathbb{R}P^{2}})$ that is a power of $2$ is equal to $2^{l+2}$ in $B_{n}(\ensuremath{\mathbb{R}P^{2}})$, and is realised by $x^{s}$, and the maximal torsion in $B_{m}(\ensuremath{\mathbb{R}P^{2}})$ that is a power of $2$ is equal to $2^{k+2}$. It follows that the order of $f(x^{s})$ is a divisor of $2^{k+2}$, in particular $f(x^{2^{k+2}s})=1$ in $B_{m}(\ensuremath{\mathbb{R}P^{2}})$. Now $l\geq k+1$ by hypothesis, and so $1=(f(x^{2^{k+2}s}))^{2^{l-k-1}}=f(x^{2^{l+1}s})$. Since $x$ is of order $2^{l+2}s$, $x^{2^{l+1}s}$ is of order $2$, so is equal to the full twist braid $\Delta_{n}^{2}$ of $B_{n}(\ensuremath{\mathbb{R}P^{2}})$, using the fact that $\Delta_{n}^{2}$ is the unique element of $B_{n}(\ensuremath{\mathbb{R}P^{2}})$ of order $2$~\cite[Proposition~23]{GGagt}. We conclude that $\Delta_{n}^{2}\in \ker{f}$.
Now let $H=\ang{x,y}$, where $y=\Delta_{n}$ (resp.\ $y=\Delta_{n}a^{-1}$) if $n$ is even (resp.\ $n$ is odd). By~\cite[Proposition~15]{GGjlms}, $H$ is isomorphic to the dicyclic group $\operatorname{Dic}_{4n'}$ of order $4n'$, and the generators satisfy the relations $x^{n'}=y^{2}$ and $yxy^{-1}=x^{-1}$. Using once more the fact that $\Delta_{n}^{2}$ is the unique element of $B_{n}(\ensuremath{\mathbb{R}P^{2}})$ of order $2$, we have $\Delta_{n}^{2}\in \ker{\phi} \cap H$. Further, $\Delta_{n}^{2}$ is central in $B_{n}(\ensuremath{\mathbb{R}P^{2}})$~\cite[Proposition~6.1]{M}, and hence the restriction $f\left\lvert_{H}\right. \colon\thinspace H \to f(H)$ of $f$ to $H$ factors through the quotient $H/\langle \Delta_{n}^{2} \rangle$. But using the relations of $H$, this quotient is isomorphic to the dihedral group of order $2n'$, so $f(H)$ is a subgroup of $B_{m}(\ensuremath{\mathbb{R}P^{2}})$ that is a quotient of $H/\langle \Delta_{n}^{2} \rangle$. Now the quotients of dihedral groups are either the trivial group, cyclic of order $2$ or dihedral, and since the braid groups of $\ensuremath{\mathbb{R}P^{2}}$ do not have dihedral subgroups~\cite[Theorem~5]{GGjlms}, it follows that $f(H)$ is either trivial or cyclic of order $2$, so $\ker{f\left\lvert_{H}\right.}$ is either equal to $H$, or is a subgroup of $H$ of index $2$. If $\ker{f\left\lvert_{H}\right.}$ is of index $2$ in $H$, then by analysing the images of $x$ and $y$ by a surjective homomorphism from $H$ to $\ensuremath{\mathbb Z}_{2}$, we see that either $\ker{f\left\lvert_{H}\right.}=\ang{x}$, or if $n'$ is even, additionally $\ker{f\left\lvert_{H}\right.}=\ang{x^{2},y}$, or $\ker{f\left\lvert_{H}\right.}=\ang{x^{2},xy}$. So if either $\ker{f\left\lvert_{H}\right.}$ is equal to $H$, or is an subgroup of $H$ of index $2$, we conclude from these possibilities that $x^{2} \in \ker{f}$. It follows again from the fact that $\Delta_{m}^{2}$ is the unique element of $B_{m}(\ensuremath{\mathbb{R}P^{2}})$ of order $2$ that $f(x)\in \ang{\Delta_{m}^{2}}$. Since $f(x')$ is of finite order and $f(x)$ is central in $B_{m}(\ensuremath{\mathbb{R}P^{2}})$, using the fact mentioned above in the first paragraph that $B_{n}(\ensuremath{\mathbb{R}P^{2}})=\langle x,x' \rangle$, we see that the image of $f$ is finite cyclic as required. In particular, in the outstanding case of the proof of part~\ref{it:rpa}\ref{it:rpaii}, where $n=4$ and $m=3$, there is no surjective homomorphism from $B_{4}(\ensuremath{\mathbb{R}P^{2}})$ to $B_{3}(\ensuremath{\mathbb{R}P^{2}})$.\qedhere
\end{enumerate}
\end{proof}
\begin{cor}\label{cor:purenor}
Let $g\geq 1$, and let $m,n\in \ensuremath{\mathbb N}$.
Then there exists a surjective homomorphism of $B_n(U_g)$ onto $P_m(U_g)$ if and only if either $g=m=1$ or $n=m=1$.
\end{cor}
\begin{proof}
If $n=m=1$ and $g\geq 1$, the result is clear, and if $g=m=1$, the result follows from \rerems{rp2}\ref{it:rp2b}.
Conversely, suppose that there exists a surjective homomorphism $\Phi\colon\thinspace B_n(U_{g})\to P_m(U_{g})$. Then $\Phi$ induces a surjective homomorphism of the corresponding Abelianisations, but since $(P_m(U_{g}))_{\text{Ab}}\cong \ensuremath{\mathbb Z}^{(g-1)m} \oplus \ensuremath{\mathbb Z}_2^{m} $ using a presentation of $P_m(U_{g})$ (see~\cite[Theorem~3]{GGjpaa3} for instance), it follows from \reth{gam3nor}\ref{it:g1sigmaor} and the fact that $(B_{1}(U_{g}))_{\text{Ab}}\cong \ensuremath{\mathbb Z}^{g-1} \oplus \ensuremath{\mathbb Z}_2$ that $m=1$. So either $n=1$ or $g=1$, and thus the conclusion holds,
or else $n>1$ and $g>1$, in which case we obtain a contradiction using \reth{gensurj}\ref{it:main1b}\ref{it:main1bii}.
\end{proof}
\section{Surjections between braid groups of orientable surfaces and symmetric groups} \label{sec:other}
In this section, we start by recalling \reth{LinTrans}, due to Ivanov~\cite{Iv}, about transitive representations of $B_n$ and $S_m$, where $n>m \geq 2$ (the definitions of primitive and transitive representations were given in \resec{intro}). We then prove~\reth{LinSurfaceTrans} that generalises \reth{LinTrans} to braid groups of compact, orientable surfaces. We shall assume that the surfaces are without boundary, but the results extend easily to the case with boundary.
In~\cite{Iv}, Ivanov gave some transitive, imprimitive representations of $B_n(\Sigma_g)$ in $S_n$, where $g\geq 1$ and $n\geq 3$. These representations have the property that their images are Abelian subgroups of $S_{n}$. We shall construct some transitive, imprimitive representations of $B_n(\Sigma_g)$ in $S_m$ whose images are non Abelian, so they are different from those of Ivanov.
The following result is a variant of \reth{Lin} for transitive representations.
\begin{thm}[{\cite[Lemma~3]{Iv}}]\label{th:LinTrans}
Let $n>m \geq 2$, and let $\rho\colon\thinspace B_n \to S_m$ be a transitive representation. Then one of the following statements holds:
\begin{enumerate}
\item\label{it:LinTransa} $\rho(\sigma_1)= \cdots =\rho(\sigma_{n-1})$, and this permutation is an $m$-cycle.
\item\label{it:LinTransb} if $n=4$ and $m=3$,
up to a suitable renumbering of the elements of the set $\brak{1, 2, 3}$,
$\rho(\sigma_1)=\rho(\sigma_3)=(1,2)$ and $\rho(\sigma_2)=(2,3)$.
\end{enumerate}
\end{thm}
We can give an alternative proof of \reth{LinTrans} using \reth{Lin} due to Lin.
\begin{proof}[Proof of \reth{LinTrans}]
Suppose that $n>m \geq 2$. If $m=2$ then $\im{\rho}$ is either trivial, which contradicts the transitivity hypothesis, or is equal to $S_{2}$, and statement~\ref{it:LinTransa} holds. So suppose that $n>m \geq 3$, and assume that $n\neq 4$. Arguing as in the first part of the proof of \reth{LinSurfcomb} in \resec{suror}, it follows that $\rho(\sigma_1)= \cdots =\rho(\sigma_{n-1})$, and the fact that $\rho$ is transitive implies that the permutation $\rho(\sigma_1)$ of $S_{m}$ is an $m$-cycle, so once more statement~\ref{it:LinTransa} holds. Finally, assume that $(n,m)=(4,3)$. We claim that $\rho(\sigma_1)$ and $\rho(\sigma_2)$ have the same cycle type. To see this, first note that if one of $\rho(\sigma_1)$ or $\rho(\sigma_2)$ is equal to the identity permutation then the Artin relations imply that the other is also equal to the identity, which proves the claim in this case. So suppose that one of these two elements is a transposition and the other is a $3$-cycle. Then $\rho(\sigma_1 \sigma_2 \sigma_1)$ and $\rho(\sigma_2 \sigma_1 \sigma_2)$ have opposite signatures, which yields a contradiction using the Artin relations, and proves the claim.
It follows in a similar manner that $\rho(\sigma_2)$ and $\rho(\sigma_3)$ have the same cycle type, hence $\rho(\sigma_1)$, $\rho(\sigma_2)$ and $\rho(\sigma_3)$ all have the same cycle type. By the transitivity hypothesis, they cannot be equal to the identity permutation, and they cannot be equal to the same transposition. So we are reduced to analysing the following two cases:
\begin{enumerate}[label=\textit{(\roman*)}]
\item $\rho(\sigma_1)$, $\rho(\sigma_2)$ and $\rho(\sigma_3)$ are transpositions. Since $\sigma_{1}$ and $\sigma_{3}$ commute, it follows that $\rho(\sigma_1)=\rho(\sigma_3)$, and the fact that $\rho(\sigma_1)$, $\rho(\sigma_2)$ and $\rho(\sigma_3)$ do not coincide implies that the condition given in part~\ref{it:LinTransb} is satisfied.
\item $\rho(\sigma_1)$, $\rho(\sigma_2)$ and $\rho(\sigma_3)$ are $3$-cycles. Using the Artin relations, it follows that $(\rho(\sigma_1))^{-1}\neq \rho(\sigma_2)$, so $\rho(\sigma_1)=\rho(\sigma_2)$. In a similar fashion, $\rho(\sigma_2)=\rho(\sigma_3)$, and thus the condition given in part~\ref{it:LinTransa} is satisfied.\qedhere
\end{enumerate}
\end{proof}
We now recall the following result of~\cite{BGP} about the structure of the centraliser $C_{S_m}(u)$ of a permutation $u$ in $S_{m}$. Note that $C_{S_m}(u)$ is equal to the centraliser $C_{S_m}(\ang{u})$ of the subgroup $\ang{u}$ in $S_{m}$. If $k\in \ensuremath{\mathbb N}$, let $C_k$ denote the cyclic group of order $k$.
\begin{prop}[{\cite[Lemma~1.1]{BGP}}]\label{prop:berrickparis}
Let $u \in S_m$ be a permutation whose cycle type is equal to $(1)^{\ell_1} (2)^{\ell_2}\ldots(m)^{\ell_m}$, and let $I(u) = \setl{k \in \brak{1,2,\ldots,m}}{\ell_k > 0}$, so that $\Sigma_{k\in I(u)}\, k\ell_k =m$.
Then the centraliser $C_{S_m}(u)$ of $u$ in $S_{m}$ is isomorphic to $\prod_{k\in I(u)}\, C_k^{\ell_k}\rtimes S_{\ell_k}= \prod_{k\in I(u)}\, C_k \wr S_{\ell_k}$.
\end{prop}
In the semi-direct product $\prod_{k\in I(u)}\, C_k^{\ell_k}\rtimes S_{\ell_k}$ given in the statement of \repr{berrickparis}, the action of an element $\tau$ of $S_{\ell_k}$ is given by indexing the copies of $C_{k}$ by $\brak{1,\ldots, \ell_k}$, and by sending a given element of $C_{k}$ to the corresponding element of $C_{\tau(k)}$. Further, the partition associated with the cycle decomposition of $u$ is left invariant by the elements of $C_{S_m}(u)$.
Let $n>m \geq 1$, and let $\rho_{n,m}\colon\thinspace B_n(\Sigma_g) \to S_m$ be a representation. Considering $B_{n}$ to be a subgroup of $B_n(\Sigma_g)$ induced by the inclusion of a topological disc in $\Sigma_{g}$, by abuse of notation, we also denote the restriction of $\rho_{n,m}$ to $B_n$ by $\rho_{n,m}$. We now prove \reth{LinSurfaceTrans} that generalises \reth{LinTrans}.
\begin{proof}[Proof of \reth{LinSurfaceTrans}] Suppose that $n>m\geq 2$. If $m=2$ then $\im{\rho_{n,2}}$ is either equal to $\brak{\operatorname{\text{Id}}}$ or is isomorphic to $\ensuremath{\mathbb Z}_{2}$, and statement~\ref{it:LinSurfaceTransa} of the theorem holds. So assume that $n>m\geq3$, and suppose additionally that $n\neq 4$. Since $m\geq 3$, $\im{\rho_{n,m}} \neq \brak{\operatorname{\text{Id}}}$. By considering the composition of $\rho_{n,m}$ with the inclusion of $B_{n}$ in $B_{n}(\Sigma_{g})$, we see as in the proof of \reth{LinSurfcomb} in \resec{suror} that $\rho_{n,m}(\sigma_1)= \cdots =\rho_{n,m}(\sigma_{n-1})$.
We denote this common element of $S_{m}$ by $\sigma$. Relation~(\ref{eq:cs}) implies that $\sigma$ commutes with $\rho_{n,m}(a_i)$ and with $\rho_{n,m}(b_i)$ for all $1\leq i\leq g$. So if
$\sigma$ is an $m$-cycle then $\rho_{n,m}(a_i)$ and $\rho_{n,m}(b_i)$ are powers of $\sigma$ for all $1\leq i\leq g$, in which case $\im{\rho_{n,m}}$
is generated by $\sigma$, and statement~\ref{it:LinSurfaceTransa} of the theorem holds.
So assume that $\sigma$ is not an $m$-cycle. Then the decomposition of $\sigma$ as a product of disjoint cycles gives rise to a partition of the set $\{1, \ldots , m\}$ that is different from the set $\{1, \ldots , m\}$ itself and that is invariant under the action of $\sigma$. Since $\rho_{n,m}(a_i)$ and $\rho_{n,m}(b_i)$ commute with $\sigma$ for all $1\leq i\leq g$, they also leave this partition invariant,
and it follows from the hypothesis that $\rho_{n,m}$ is primitive that $\sigma$ is the identity permutation. Thus $\rho_{n,m}$ factors through the quotient $B_n(\Sigma_g)/\langle\!\langle \brak{\sigma_{1}, \ldots, \sigma_{n-1}}\rangle\!\rangle$ of $B_{n}(\Sigma_{g})$ by the normal closure $\langle\!\langle \brak{\sigma_{1}, \ldots, \sigma_{n-1}}\rangle\!\rangle$ of $\brak{\sigma_{1}, \ldots, \sigma_{n-1}}$ in $B_{n}(\Sigma_{g})$, and induces a homomorphism $\overline{\rho}_{n,m}\colon\thinspace B_n(\Sigma_g)/\langle\!\langle \brak{\sigma_{1}, \ldots, \sigma_{n-1}}\rangle\!\rangle \to S_m$. But from the proof of \repr{mingen}, $B_n(\Sigma_g)/\langle\!\langle \brak{\sigma_{1}, \ldots, \sigma_{n-1}}\rangle\!\rangle$ is isomorphic to $\ensuremath{\mathbb Z}^{2g}$. So $\im{\overline{\rho}_{n,m}}=\im{\rho_{n,m}}$ is non trivial and Abelian, and $\overline{\rho}_{n,m}$ is primitive.
Since $\im{\overline{\rho}_{n,m}}$ is Abelian, any non-trivial element $u\in \im{\overline{\rho}_{n,m}}$ commutes with all of the elements of $\im{\overline{\rho}_{n,m}}$, from which we see that $\im{\overline{\rho}_{n,m}}$ is contained in the centraliser of $u$ in $S_{m}$. If $u$ is an $m$-cycle then $\im{\overline{\rho}_{n,m}}$ coincides with $C_{S_{m}}(u)$, which is equal to $\ang{u}$, and thus part~\ref{it:LinSurfaceTransa} of the statement holds. So assume that $\im{\overline{\rho}_{n,m}}$ contains no $m$-cycle. Then the cycle decomposition of $u$ contains a non-trivial cycle of length strictly less than $m$, so by \repr{berrickparis}, $C_{S_{m}}(u)$ is imprimitive. But since $\im{\overline{\rho}_{n,m}}\subset C_{S_{m}}(u)$, this implies that $\overline{\rho}_{n,m}$ is also imprimitive, which yields a contradiction. This argument also implies that $m$ has to be prime, and that $u$ is an $m$-cycle.
This completes the proof of the case $n>m\geq 3$ and $n\neq 4$.
Finally, let $n=4$ and $m=3$. Suppose first that the restriction of the representation $\rho_{4,3} \colon\thinspace B_{4}(\Sigma_{g})\to S_{3}$ to $B_{4}$ is intransitive. Thus $\rho_{4,3}(B_4)$ is equal to a subgroup of $S_{3}$ of order $1$ or $2$, and in either case, it follows that $\rho_{4,3}(\sigma_1)=\rho_{4,3}(\sigma_2)=\rho_{4,3}(\sigma_3)$ using the Artin relations~\reqref{artin1} and~\reqref{artin2}. We denote this element by $\sigma$. As in the discussion of the previous paragraph of the case where $\sigma$ is not an $m$-cycle for $n>m\geq 3$ and $n\neq 4$,
we obtain a contradiction. Therefore the restriction of the representation $\rho_{4,3}$ to $B_{4}$ is transitive, and by \reth{LinTrans}, we just have to consider the following two cases.
\begin{enumerate}[label=\textit{(\roman*)}]
\item $\rho_{4,3}(B_4)$
is generated by a $3$-cycle, and $\rho_{4,3}(\sigma_1)=\rho_{4,3}(\sigma_2)=\rho_{4,3}(\sigma_3)$. Using once more relation~(\ref{eq:cs}) of \reth{presbng}, we see that $\rho_{4,3}(a_i)$ and $\rho_{4,3}(b_{i})$ commute with the $3$-cycle $\rho_{4,3}(\sigma_1)$ for all $1\leq i \leq g$, so they are powers of $\rho_{4,3}(\sigma_1)$. Thus $\im{\rho_{4,3}}=\ang{\rho_{4,3}(\sigma_1)}$, and hence part~\ref{it:LinSurfaceTransa} of the statement holds.
\item Up to a suitable renumbering of the elements of the set $\brak{1, 2, 3}$, $\rho_{4,3}(\sigma_1)=\rho_{4,3}(\sigma_3)=(1,2)$ and $\rho_{4,3}(\sigma_2)=(2,3)$. Relation~(\ref{eq:cs}) of \reth{presbng} implies once more that $\rho_{4,3}(a_i)$ and $\rho_{4,3}(b_{i})$ commute with the elements $\rho_{4,3}(\sigma_2)$ and $\rho_{4,3}(\sigma_3)$ for all $1\leq i\leq g$. Since these transpositions generate $S_{3}$, it follows that the permutations $\rho_{4,3}(a_i)$ and $\rho_{4,3}(b_{i})$ belong to the centre of $S_{3}$ for all $1\leq i\leq g$, so are trivial. Hence part~\ref{it:LinSurfaceTransb} of the statement holds.\qedhere
\end{enumerate}
\end{proof}
\begin{rem}
Using the methods of the proof of \reth{LinSurfaceTrans} and the presentation given by~\cite[Proposition~3.1]{BGoG}, the statement of \reth{LinSurfaceTrans} also holds if the surface has boundary.
\end{rem}
We may obtain some information about an arbitrary representation $\rho_{n,m}\colon\thinspace B_n(\Sigma_g)\to S_m$ in a more general setting.
\begin{prop}\label{prop:contraints}
Let $g\geq 1$, let $n>m\geq 2$, and assume that $(n,m)\neq (4,3)$.
Suppose that $\rho_{n,m}\colon\thinspace B_n(\Sigma_g)\to S_m$ is a homomorphism, and let $\rho_{n,m}(B_n)$ be the image of the subgroup $B_{n}$ of $B_n(\Sigma_g)$ under $\rho_{n,m}$.
\begin{enumerate}
\item\label{it:contraintsa} The subgroup $\rho_{n,m}(B_n)$ of $S_{m}$ is cyclic, and therefore $\rho_{n,m}(\sigma_1)=\cdots =\rho_{n,m}(\sigma_{n-1})$.
\item\label{it:contraintsb} The subgroup $\im{\rho_{n,m}}$
is contained in the centraliser $C_{S_m}(\rho_{n,m}(B_n))$ of $\rho_{n,m}(B_n)$ in $S_{m}$.
This centraliser is described by \repr{berrickparis}.
\item\label{it:contraintsc} There is an inclusion $\Gamma_3(B_n(\Sigma_g)) \subset \ker{\rho_{n,m}}$, so the homomorphism $\rho_{n,m}$ factors through the quotient $B_n(\Sigma_g)/\Gamma_3(B_n(\Sigma_g))$, and the subgroup $\im{\rho_{n,m}}$ is nilpotent of nilpotency degree at most $2$.
\end{enumerate}
\end{prop}
\begin{proof}
Parts~\ref{it:contraintsa} and~\ref{it:contraintsb} follow from the group presentation of $B_n(\Sigma_g)$ and by repeating the arguments given for instance in the proof of \reth{LinSurfaceTrans}.
If $n\geq 3$, the first statement of part~\ref{it:contraintsc} is a consequence of part~\ref{it:contraintsa} and the fact that the subgroup $\Gamma_3(B_n(\Sigma_g))$ is isomorphic to the normal closure of the element $\sigma_1 \sigma_2^{-1}$ in $B_n(\Sigma_g)$~\cite[proof of Theorem~1(c)]{BGeG}, which is contained in $\ker{\rho_{n,m}}$ using part~\ref{it:contraintsa}.
The second statement of part~\ref{it:contraintsc} then follows.
\end{proof}
In~\cite[p.~317]{Iv}, Ivanov gave some transitive, imprimitive representations of $B_n(\Sigma_g)$ in $S_n$ for $g\geq 1, n\geq 3$,
and he commented that `I do not know to what extent these examples exhaust the imprimitive representations'. All of the examples he proposed are representations whose images are Abelian. We now describe some imprimitive representations $\rho_{n,m}\colon\thinspace B_n(\Sigma_g) \to S_m$ whose images are non Abelian, so are different from those of Ivanov.
\begin{exo}\label{ex:exo1}\mbox{}
\begin{enumerate}
\item By~\cite[eq.~(10)]{BGeG}, if $g\geq 1$ and $n\geq 3$, $B_n(\Sigma_g)/\Gamma_3(B_n(\Sigma_g))$ admits the following presentation:
\begin{enumerate}
\item[\textbf{generators:}] $a_1, b_1,\ldots,a_g, b_g$ and $\sigma$.
\item[\textbf{relations:}] $\sigma^{2(n-1+g)}=1$, and the elements of $\brak{a_1, b_1,\ldots,a_g, b_g,\sigma}$ commute pairwise, except for the pairs $(a_i, b_i)_{i=1,\ldots,g}$, for which $[a_1,b_1]=\cdots=[a_g,b_g]=\sigma^2$.
\end{enumerate}
Let $n>2$ be even, and let $g=1$. From the above presentation, we have:
\begin{equation*}
B_n(\ensuremath{\mathbb{T}^2})/\Gamma_3(B_n(\ensuremath{\mathbb{T}^2}))=\setangl{a_1, b_1, \sigma}{[a_1,\sigma]=[b_1,\sigma]=1,\, [a_1, b_1]=\sigma^2,\, \sigma^{2n}=1}.
\end{equation*}
We define a map $\theta\colon\thinspace B_n(\ensuremath{\mathbb{T}^2})/\Gamma_3(B_n(\ensuremath{\mathbb{T}^2})) \to S_8$ on the generators of $B_n(\ensuremath{\mathbb{T}^2})/\Gamma_3(B_n(\ensuremath{\mathbb{T}^2}))$ by:
\begin{equation*}
\theta (a_1)=(1, 3)(2, 4),\, \theta (b_1)=(1, 5)(2, 6)(3, 7)(4, 8) \; \text{and}\; \theta (\sigma)=(1, 2, 3, 4)(5, 6, 7, 8).
\end{equation*}
It is straightforward to check that $\theta$ respects the relations of $B_n(\ensuremath{\mathbb{T}^2})/\Gamma_3(B_n(\ensuremath{\mathbb{T}^2}))$, the equality $(\theta(\sigma))^{2n}=1$ being a consequence of the fact that $2n$ is divisible by $4$, so $\theta$ is a homomorphism.
If $p\colon\thinspace B_n(\ensuremath{\mathbb{T}^2}) \to B_n(\ensuremath{\mathbb{T}^2})/\Gamma_3(B_n(\ensuremath{\mathbb{T}^2}))$ is the canonical projection, then the representation $\theta\circ p\colon\thinspace B_n(\ensuremath{\mathbb{T}^2}) \to S_{8}$ is transitive, and it is imprimitive since the non-trivial partition $\brak{\brak{1,2,3,4}, \brak{5,6,7,8}}$ is preserved by the subgroup $\im{\theta\circ p}$ of $S_{8}$.
This is perhaps the simplest example of an imprimitive representation $\rho_{n,m}\colon\thinspace B_n(\Sigma_g) \to S_m$ whose image is non Abelian. In particular, if we take $n=8$, we obtain a transitive, imprimitive representation of $B_8(\ensuremath{\mathbb{T}^2})$ in $S_8$ whose image is non Abelian, so it is not included in the examples of~\cite{Iv}.
\item
If $g+n$ is odd and $m=2^{g+2}$, \rex{exo1} may be generalised to construct a homomorphism $\theta_g \colon\thinspace B_n(\Sigma_g)/\Gamma_3(B_n(\Sigma_g))\to S_m$ such that $\im{\theta_{g}}$ is non Abelian. Composing $\theta_g$ with the projection $p_g \colon\thinspace B_n(\Sigma_g) \to B_n(\Sigma_g)/\Gamma_3(B_n(\Sigma_g))$, we thus obtain a homomorphism $\theta_g\circ p_g \colon\thinspace B_n(\Sigma_g) \to S_m$ such that $\im{\theta_{g}\circ p_{g}}$ is non Abelian. To do so, first let us denote the image by $\theta_g$ of the element $\sigma\in B_n(\Sigma_g)/\Gamma_3(B_n(\Sigma_g))$ given in \rex{exo1} by $\overline{\sigma}\in S_m$. By \repr{contraints}, $\im{\theta_g}$ is contained in the centraliser of $\overline{\sigma}$ in $S_{m}$, which is described in \repr{berrickparis}. Our strategy is to make use of the structure of this centraliser to construct imprimitive representations whose images are non Abelian.
In \rex{exo1}, the image of $\theta$ is isomorphic to $(\ensuremath{\mathbb Z}_4 \oplus \ensuremath{\mathbb Z}_4)\rtimes \ensuremath{\mathbb Z}_2$, and is the centraliser of $\overline{\sigma}=(1_4,1_4 ; 0_2)$ given by \repr{berrickparis}. In the general case, $n+g$ is odd, $m=2^{g+2}$, and the centraliser of $\overline{\sigma}$ is isomorphic to $\ensuremath{\mathbb Z}_4^{2^g}\rtimes \ensuremath{\mathbb Z}_{2^g}$. We now give two examples of this construction, one in the case where $g$ is odd, and in the other in the case where $g$ is even.
\begin{enumerate}
\item Suppose that $g=3$, so $m=32$, and $n\geq 4$ is even. Consider $\ensuremath{\mathbb Z}_4^{8}\rtimes S_{8}$, which is interpreted as a subgroup of $S_{32}$. Define the homomorphism $\theta_3 \colon\thinspace B_n(\Sigma_3)/\Gamma_3(B_n(\Sigma_3)) \to S_{32}$ by $\theta_3(a_1)=(2,0,2,0,2,0,2,0)$, $\theta_3(a_2)=(2,2,0,0,2,2,0,0)$, $\theta_3(a_3)=(2,2,2,2,0,0,0,0)$, $\theta_3(\sigma)=(1,1,1,1,1,1,1,1)$, regarded as elements of $S_{32}$, where each factor $1$ denotes the cyclic permutation of length $4$ associated to the four integers corresponding to these four positions, $2$ denotes the square of this cyclic permutation, and $0$ denotes the identity permutation associated to these four integers.
Finally, let $\theta_3(b_1)=(1,2)(3,4)(5,6)(7,8)$, $\theta_3(b_2)=(1,3) (2,4)(5,7)(6,8)$, $\theta_3(b_3)=(1,5) (2,6)(3,7)(4,8)$, all regarded as elements of $S_{8}\subset \ensuremath{\mathbb Z}_4^{8}\rtimes S_{8}$.
In terms of explicit elements of $S_{32}$, we have:
\begin{align*}
\theta_3(a_1) =& (1, 3)(2, 4)(9, 11)(10, 12)(17, 19)(18, 20)(25, 27)(26, 28)\\
\theta_3(a_2) =& (1, 3)(2, 4)(5, 7)(6, 8)(17, 19)(18, 20)(21, 23)(22, 24)\\
\theta_3(a_3) =& (1, 3)(2, 4)(5, 7)(6, 8)(9, 11)(10, 12)(13, 15)(14, 16)\\
\theta_3(b_1) =& (1, 5)(2, 6)(3, 7)(4, 8) (9, 13 )(10, 14)(11, 15)(12, 16)(17, 21 )(18, 22)(19, 23)(20, 24)\cdot\\
& (25, 29)(26, 30)(27, 31)(28, 32)\\
\theta_3(b_2) =&(1, 9 )(2, 10)(3, 11)(4, 12) (5, 13 )(6, 14)(7, 15)(8, 16)(17, 25 )(18, 26)(19, 27)(20, 28)\cdot\\
& (21, 29)(22, 30)(23, 31)(24, 32)\\
\theta_3(b_3) =&(1, 17 )(2, 18)(3, 19)(4, 20) (5, 21 )(6, 22)(7, 23)(8, 24)(9, 25 )(10, 26)(11, 27)(12, 28)\cdot\\
& (13, 29)(14, 30)(15, 31)(16, 32)\\
\theta_3(\sigma) =&(1, 2, 3, 4)(5, 6, 7, 8)(9, 10, 11, 12)(13, 14, 15, 16)(17, 18, 19,20)(21, 22, 23, 24)\cdot\\
& (25, 26, 27, 28)(29, 30, 31, 32).
\end{align*}
Using these expressions, we may check that $[\theta_3(a_i),\theta_3(a_j)]=[\theta_3(b_i),\theta_3(b_j)]= [\theta_3(a_i),\theta_3(b_j)]$ for all $1\leq i<j\leq 3$, and $[\theta_3(a_l),\theta_3(b_l)]= (\theta_3(\sigma))^2$ for all $1\leq l\leq 3$, so $\theta_3$ is a homomorphism.
\item Now suppose that $g=2$, so $m=16$, and $n\geq 3$ is odd. Consider the subgroup $\ensuremath{\mathbb Z}_4^{4}\rtimes S_{4}$, which we interpret as a subgroup of the symmetric group $S_{16}$. We define a homomorphism $\theta_{2,1}\colon\thinspace B_n(\Sigma_2)/\Gamma_3(B_n(\Sigma_2)) \to S_{16}$
as follows. Let $\theta_{2,1}(a_1)=(2,0,2,0)$ and $\theta_{2,1}(a_2)=(2,2,0,0)$ in $S_{16}$,
where as in the previous example, each factor $1$ denotes the cyclic permutation of length $4$ associated to the integers corresponding to the these four positions, $2$ is the square of this cyclic permutation, and $0$ is the identity permutation associated to these four integers. We also take $\theta_{2,1}(b_1)=(1,2) (3, 4 )\in S_4$, $\theta_{2,1}(b_2)=(1,3) (2,4)\in S_4$,
and $\theta_{2,1}(\sigma)=(1,1,1,1) \in S_{16}$.
Since $n+2$ is odd, $\theta_{2,1}$ defines a homomorphism. In $S_{16}$, the elements are given explicitly by:
\begin{align*}
\theta_{2,1}(a_1) &= (1, 3)(2, 4)(9, 11)(10, 12)\\
\theta_{2,1}(a_2) &= (1, 3)(2, 4)(5, 7)(6, 8)\\
\theta_{2,1}(b_1) &=(1, 5 )(2, 6)(3, 7)(4, 8) (9, 13 )(10, 14)(11, 15)(12, 16)\\
\theta_{2,1}(b_2) &=(1, 9 )(2, 10)(3, 11)(4, 12)(5, 13 )(6, 14)(7, 15)(8, 16)\\
\theta_{2,1}(\sigma) &=(1, 2, 3, 4)(5, 6, 7, 8)(9, 10, 11, 12)(13, 14, 15, 16).
\end{align*}
\end{enumerate}
\end{enumerate}
\end{exo}
\begin{exo}\label{ex:exo2}
Let $n>2$, and consider the group $\ensuremath{\mathbb Z}_{2n}^n\rtimes S_{n}$ seen as subgroup of $S_{2n^2}$. Let $\theta\colon\thinspace B_{n}(\ensuremath{\mathbb{T}^2})/\Gamma_3(B_{n}(\ensuremath{\mathbb{T}^2})) \to S_{2n^2}$ be the homomorphism defined by $\theta(a_1)=(a, a+2, a+4,\ldots, a-2)\in \ensuremath{\mathbb Z}_{2n}^n$ for $a$ any element of $\ensuremath{\mathbb Z}_{2n}$, $\theta(b_1)=(1, 2, \ldots, n)\in S_n$, and $\theta(\sigma)=(1_{2n},\ldots,1_{2n})\in \ensuremath{\mathbb Z}_{2n}^n$. It follows that $\theta(\sigma)$ is of order $2n$, $\theta(\sigma)$ commutes with $\theta(a_1)$ and $\theta(b_1)$, and that $\theta([a_1,b_1])=\theta(\sigma)^2$.
The image of $\theta$ is the subgroup generated by $\brak{\theta(a_1), \theta(b_1), \theta(\sigma)}$ and the image is non Abelian.
\end{exo}
We conclude this paper with the following remarks.
\begin{rems}\mbox{}
\begin{enumerate}
\item The construction of \rex{exo2} also enables us to obtain an example of a homomorphism $\theta\colon\thinspace B_n(\ensuremath{\mathbb{T}^2})/\Gamma_3(B_n(\ensuremath{\mathbb{T}^2})) \to S_m$, where $n>m$, and the order of $\theta(\sigma)$ is equal to $2n$. First note that if $l\geq 3$ and $l$ divides $n$, then it follows from the presentation given at the beginning of \rex{exo1} that the map $\tau_{l} \colon\thinspace B_n(\ensuremath{\mathbb{T}^2})/\Gamma_3(B_n(\ensuremath{\mathbb{T}^2}))\to B_l(\ensuremath{\mathbb{T}^2})/\Gamma_3(B_l(\ensuremath{\mathbb{T}^2}))$ defined by sending the generators $a_{1}$, $b_{1}$ and $\sigma$ of $B_n(\ensuremath{\mathbb{T}^2})/\Gamma_3(B_n(\ensuremath{\mathbb{T}^2}))$ to the generators $a_{1}$, $b_{1}$ and $\sigma$ respectively of $B_l(\ensuremath{\mathbb{T}^2})/\Gamma_3(B_l(\ensuremath{\mathbb{T}^2}))$ extends to a (well-defined) surjective homomorphism. Let $n=3\cdot 5 \cdot 7 \cdot 11=1155$. For $l=3,5,7,11$, let $\theta_l \colon\thinspace B_l(\ensuremath{\mathbb{T}^2})/\Gamma_3(B_l(\ensuremath{\mathbb{T}^2})) \to S_{2l^{2}}$ be the homomorphism given as in \rex{exo2}, and let $\theta \colon\thinspace B_{n}(\ensuremath{\mathbb{T}^2})/\Gamma_3(B_{n}(\ensuremath{\mathbb{T}^2})) \to S_{408}$ be defined by $\theta(x)=(\theta_{3}\circ \tau_{3}(x), \theta_{5}\circ \tau_{5}(x), \theta_{7}\circ \tau_{7}(x), \theta_{11}\circ \tau_{11}(x))\in S_{18}\times S_{50}\times S_{98} \times S_{242}$ for all $x\in B_n(\ensuremath{\mathbb{T}^2})/\Gamma_3(B_n(\ensuremath{\mathbb{T}^2}))$. Interpreting $S_{18}\times S_{50}\times S_{98} \times S_{242}$ as a subgroup of $S_{408}$, we may thus take $m=408$, and the element $\theta(\sigma)$ is of order $2310$.
\item Let $g\geq 1$, and let $G$ be the group that admits the following presentation:
\begin{enumerate}
\item[\textbf{generators:}] $a_1, b_1,\ldots,a_g, b_g$ and $\sigma$.
\item[\textbf{relations:}] $\sigma^{2(1+g)}=1$, and the elements of $\brak{a_1, b_1,\ldots,a_g, b_g,\sigma}$ commute pairwise, except for the pairs $(a_i, b_i)_{i=1,\ldots,g}$, for which $[a_1,b_1]=\cdots=[a_g,b_g]=\sigma^2$.
\end{enumerate}
Observe that this is the group obtained by taking $n=2$ in the presentation of the quotient $B_{n}(\Sigma_g)/\Gamma_{3}(B_{n}(\Sigma_g))$ given in \rex{exo1} (we suspect that $B_{2}(\Sigma_g)/\Gamma_{3}(B_{2}(\Sigma_g))$ is not isomorphic to $G$ in this case). Using the presentation of $B_{2}(\Sigma_g)$ given by \reth{presbng}, the map $\rho\colon\thinspace B_{2}(\Sigma_g) \to G$ given by sending the generators $a_i$, $b_i$ and $\sigma_{1}$ of $B_{2}(\Sigma_g)$ to the generators $a_i$, $b_i$ and $\sigma$ respectively of $G$ for all $1\leq i\leq g$ may be seen to extend to a well-defined surjective homomorphism. To check that relation~\reqref{tot} is respected by $\rho$, note that in $G$:
\begin{equation*}
\prod_{i=1}^g [a_i^{-1},b_i]=\prod_{i=1}^g a_i^{-1}[b_i,a_i]a_i=\prod_{i=1}^g a_i^{-1}\sigma^{-2}a_i=\sigma^{-2g}=\sigma^2\; \text{since}\; \sigma^{2g+2}=1.
\end{equation*}
Hence $\rho$ induces a surjective homomorphism $\overline{\rho}\colon\thinspace B_{2}(\Sigma_g)/\Gamma_{3}(B_{2}(\Sigma_g)) \to G/\Gamma_{3}(G)$. Using the presentation of $G$ and the fact that $\Gamma_{2}(G)$ is the normal closure in $G$ of the commutators of the generators of $G$, we see that $\Gamma_{2}(G)=\ang{\sigma^{2}}$, and thus $\Gamma_{3}(G)$ is trivial. Therefore $\overline{\rho}$ is a surjective homomorphism from $B_{2}(\Sigma_g)/\Gamma_{3}(B_{2}(\Sigma_g))$ to $G$. Observe that $\overline{\rho}$ is not an isomorphism if $g=1$ because $\Gamma_{2}(B_{2}(\ensuremath{\mathbb{T}^2}))/\Gamma_{3}(B_{2}(\ensuremath{\mathbb{T}^2}))\cong \ensuremath{\mathbb Z}_2^3$ by \reth{gam3closed}\ref{it:gam23gn2}, and $\Gamma_{2}(G)/\Gamma_{3}(G)=\ang{\sigma^{2}} \cong \ensuremath{\mathbb Z}_2$. The construction of \rex{exo1} may be applied to $G$ if $g$ is odd, and composing with $\overline{\rho}$, shows that it may also be extended to the case $n=2$ to yield a representation of $B_2(\Sigma_g)$ in $S_{2^{g+2}}$ whose image is non Abelian.
\end{enumerate}
\end{rems}
| {
"timestamp": "2018-10-30T01:26:18",
"yymm": "1810",
"arxiv_id": "1810.12214",
"language": "en",
"url": "https://arxiv.org/abs/1810.12214",
"abstract": "Generalising previous results on classical braid groups by Artin and Lin, we determine the values of m, n $\\in$ N for which there exists a surjection between the n-and m-string braid groups of an orientable surface without boundary. This result is essentially based on specific properties of their lower central series, and the proof is completely combinatorial. We provide similar but partial results in the case of orientable surfaces with boundary components and of non-orientable surfaces without boundary. We give also several results about the classification of different representations of surface braid groups in symmetric groups.",
"subjects": "Geometric Topology (math.GT); Group Theory (math.GR)",
"title": "Lower central series, surface braid groups, surjections and permutations",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9854964228644458,
"lm_q2_score": 0.7185943985973772,
"lm_q1q2_score": 0.708172209308143
} |
https://arxiv.org/abs/1706.00896 | Using Negative Curvature in Solving Nonlinear Programs | Minimization methods that search along a curvilinear path composed of a non-ascent nega- tive curvature direction in addition to the direction of steepest descent, dating back to the late 1970s, have been an effective approach to finding a stationary point of a function at which its Hessian is positive semidefinite. For constrained nonlinear programs arising from recent appli- cations, the primary goal is to find a stationary point that satisfies the second-order necessary optimality conditions. Motivated by this, we generalize the approach of using negative curvature directions from unconstrained optimization to nonlinear ones. We focus on equality constrained problems and prove that our proposed negative curvature method is guaranteed to converge to a stationary point satisfying second-order necessary conditions. A possible way to extend our proposed negative curvature method to general nonlinear programs is also briefly discussed. |
\section{Introduction}
This paper is concerned with solving general smooth nonlinear optimization problems. Considering the difficulty of obtaining a global optimal solution, our goal is to develop an efficient method to locate a stationary point that satisfies the second-order necessary conditions. This goal is motivated by recent applications and developments, where stationary points satisfying second-order necessary conditions are of primary interest. Specifically, in many nonlinear programs arising from dictionary learning \cite{SQW152}, tensor decomposition \cite{GHJY15}, phase retrieval \cite{SQW16}, and semidefinite programming \cite{BM03}, stationary points satisfying the second-order necessary conditions are the targets of the application and can often be proven to be global optimizers.
Even though the scope of our concern is more general than unconstrained nonlinear programs, studying such problems helps us to better understand how to achieve our goal. To find a stationary point of an unconstrained function satisfying second-order necessary conditions, apparently, using gradient information alone in a deterministic method is not sufficient as such methods can be trapped by saddle points. Therefore, second-order information should be utilized in (hopefully) some computationally efficient manner. Methods, e.g. {\em Newton's method} and {\em trust-region methods}, that involve computing the inverse, Cholesky factorization or eigen-decomposition of the Hessian or some modifications of it, might become impractical when the number of variables grow to tens of thousands. As a remedy, methods that utilize negative curvature directions were proposed in \cite{MC77, MS79, GOLD80} and later further actively developed in \cite{grandinetti1984nonlinear, martinez1990algorithm, ferris1996nonmonotone, lucidi1998curvilinear, gould2000exploiting, conforti2001curvilinear, apostolopoulou2010curvilinear} based on inexpensive computations involving the Hessian. Specifically, these methods imitate the steepest descent method but also incorporate a negative curvature direction besides the negative of the gradient when computing the step to take at each iteration. Consequently, saddle points are usually avoided and these methods can be guaranteed to converge to {\em a stationary point satisfying the second-order necessary conditions} without requiring much additional computational cost.
Inspired by the unconstrained case, we {\em generalize the negative curvature approach to equality constrained nonlinear programs}. In order to accomplish this, one natural question is how to generalize the notions of the gradient and the Hessian when constraints are imposed. For unconstrained problems, we can regard the gradient and the Hessian as quantities that characterize local optimality. Specifically, according to the first and second-order necessary optimality conditions, a local minimizer must have a zero gradient and a positive semidefinite Hessian. For constrained nonlinear programs, it seems plausible that a generalized gradient and Hessian could be developed based on the first-order and second-order optimality conditions. Essentially, this is the approach we adopt here. Our generalized gradient and Hessian are closely related to the Riemannnian gradient and Hessian in the context of optimization on Riemannian manifolds \cite{petersen2006riemannian, AMS09, absil2013extrinsic, absil2009all}. This explicit connection is discussed in the paper. Our approach of deriving generalized gradient and Hessian from the classical optimality point of view is quite well-motivated and self-contained, and moreover keeps the technical difficulty to a minimum level.
\paragraph{Organization.} The rest of the paper is organized as follows. In Section \ref{sec:prelim}, we review some basic results on equality constrained optimization. In Section \ref{sec:eq}, we present our negative curvature line search method for equality-constrained nonlinear programs in particular. The notions of gradient and Hessian are generalized based on local optimality conditions. Using these generalizations, we propose and study a negative curvature method for equality constrained problem. Last, we briefly discuss how to extend the negative curvature method proposed for the equality constrained problem to general nonlinear programming problems.
\section{Preliminaries}\label{sec:prelim}
In this section, we review several fundamental results regarding the equality-constrained problem:
\begin{flalign} \label{eqn:problem_to_solve}
\text{minimize} \quad f(\bm x) \qquad
\text{subject to} \quad c_i(\bm x) = 0, \;\;\; i\in \set{1, 2, \ldots, m},
\end{flalign}
where the objective function $f:\mathbb{R}^n \to \mathbb{R}$ and constraint functions $c_i:\mathbb{R}^n \to \mathbb{R}$ are smooth so that it is possible to characterize local optimality conditions based on their derivatives. The results and notation in this section are quite classical \cite{nocedal2006numerical}.
Regarding the feasible set $\Omega:=\set{\bm x\in \mathbb{R}^n \;\vert\; c_i(\bm x) = 0}$, we first impose a regularity condition which is known as LICQ in optimization literature.
\begin{definition}[LICQ]
We say that the linear independence constraint qualification (LICQ) holds at $\bm x \in \Omega$, if the set of constraint gradients $\set{\nabla c_1(\bm x), \nabla c_2{(\bm x)}, \ldots, \nabla c_m{(\bm x)}}$ are linearly independent, i.e. the matrix $\nabla \boldsymbol{c}({\bm x}) := [\nabla c_1(\bm x), \nabla c_2{(\bm x)}, \ldots, \nabla c_m{(\bm x)}] \in \mathbb{R}^{n \times m}$ has full column rank.
\end{definition}
From a geometric point of view, the LICQ condition guarantees ${\Omega}$ to be a differentiable manifold of dimension $n-m$. Now we are ready to state the definitions of tangent and normal subspaces, which are particularly useful geometry concepts to characterize the variational properties of $f(\cdot)$.
\begin{definition}[Tangent and Normal Subspaces]
The normal subspace $\mathcal N_{\Omega}(\bm x)$ at a point $\bm x \in \Omega$ is defined as the subspace spanned by the set of constraint gradients $\set{\nabla c_1(\bm x), \ldots, \nabla c_m{(\bm x)}}$, i.e. $\mathcal N_{\Omega}(\bm x):= \mathfrak R(\nabla\boldsymbol c({\bm x})) \subseteq \mathbb{R}^n$ the range of the matrix $\nabla\boldsymbol c({\bm x})$. We denote by $\mathcal P_{\mathcal N_{\Omega}(\bm x)}$ projection onto the normal subspace $\mathcal N_{\Omega}(\bm x)$.
The tangent subspace $\mathcal T_{\Omega}(\bm x)$ at a point $\bm x \in \Omega$ complements the normal subspace in $\mathbb{R}^n$, i.e. $\mathcal T_{\Omega}(\bm x) = \mathcal N^\perp_{\Omega}(\bm x) = \mathfrak R(\nabla \boldsymbol c({\bm x}))^\perp = \mathfrak N(\nabla \boldsymbol c({\bm x})^\top)$, is the nullspace of the matrix $\nabla \boldsymbol c({\bm x})^\top$.
We denote by $\mathcal P_{\bm x}$ the projection onto the tangent subspace $\mathcal T_{\Omega}(\bm x)$.
\end{definition}
\begin{remark}
We may drop the subscripts $\Omega$ and $\bm x$, respectively, from $\mathcal N_{\Omega}(\bm x)$ and $\mathcal T_{\Omega}(\bm x)$ if they are clear from the context.
\end{remark}
We are now fully prepared to state the first-order and second-order necessary optimality conditions for problem (\ref{eqn:problem_to_solve}) based on the Lagrangian function,
\begin{flalign}
\mathcal L(\bm x, \bm \lambda):= f(\bm x) - \sum_{i = 1}^m\; \lambda_i \bm c_i(\bm x),
\end{flalign}
where $\lambda_i$ is the Lagrange multiplier associated with the $i$-th constraint $c_i(\bm x) = 0$.
\begin{theorem}[Necessary Optimality Conditions \cite{nocedal2006numerical}]
Suppose that $\bm x^\star$ is a local minimizer of problem \eqref{eqn:problem_to_solve} and satisfies the LICQ condition.
Then $\boldsymbol{x}^\star$ satisfies the following first-order and second-order conditions
\begin{flalign}
& \mathcal{G}(\bm x^\star):=\nabla_{\bm x} \mathcal L\left(\bm x^\star, \bm \lambda^\star\left(\bm x^\star\right)\right) = \bm 0 \label{eqn:first_order_opt}\\
& \mathcal{H}(\bm x^\star):= \mathcal{P}_{\bm x^\star}^\top\nabla_{\bm x\bm x}^2 \mathcal L\left(\bm x^\star, \bm \lambda^\star\left(\bm x^\star \right)\right)\mathcal{P}_{\bm x^\star} \succeq \boldsymbol 0
\label{eqn:second_order_opt}
\end{flalign}
where $\bm \lambda^\star(\bm x^\star) \in \arg \min_{\bm \lambda}\; \norm{\nabla_{\bm x} \mathcal L(\bm x^\star, \bm \lambda)}{}.$
\end{theorem}
Feasible points satisfying conditions \eqref{eqn:first_order_opt} and \eqref{eqn:second_order_opt} are typically referred as {\em second-order critical points}. The gist of our paper is to design iterative numerical methods to locate them. We next briefly describe examples of optimization problems that arise in signal processing, machine learning and statistics where such solutions are essentially sought.
\paragraph{Symmetric Orthogonal Tensor Decomposition (SOTD).} Tensor is a multidimensional array and the symmetric orthogonal tensor decomposition (SOTD) naturally generalizes the spectral decomposition of a symmetric matrix. Here, we focus on the $p$-way $n$-dimensional symmetric orthogonal (SOD \cite{mu2015successive, wang2017tensor, mu2017}) tensor
\begin{flalign}
\mcb T = \sum_{i=1}^n \underbrace{\bm v_i\otimes \bm v_i \otimes \cdots \otimes \bm v_i}_{p\; times} \in \mathbb{R}^{\overbrace{n \times n \times \cdots \times n}^{p \; times}},
\end{flalign}
where $\boldsymbol V = [\bm v_1, \bm v_2, \ldots, \bm v_n] \in \mathbb{R}^{n \times n}$ is an orthogonal matrix and $\otimes$ denotes the usual outer product, so the $(i_1, i_2, \ldots, i_p)$-th entry of $\bm v \otimes \bm v \otimes \cdots \otimes \bm v$ is the scalar $v_{i_1} v_{i_2}\cdots v_{i_p}$. The problem addressed by SOTD, is to find (up to sign and permutation) the components $\bm v_i$'s given $\mcb T$, has many applications, including higher-order statistical estimation \cite{mccullagh1987tensor}, independent component analysis \cite{comon1994independent, comon2010handbook}, and parameter estimation for latent variable models \cite{anandkumar2014tensor}).
Ge et al. \cite{GHJY15} consider the SOTD specifically for the $p=4$ case and analyze the geometry of the following minimization problem
\begin{flalign}\label{eqn:tensor_opt}
\min_{\bm x \in \mathbb{R}^n} \quad T(\bm x, \bm x, \bm x, \bm x) = \sum_{i=1}^n\sum_{j=1}^n\sum_{k=1}^n\sum_{l=1}^n \mathcal T_{ijkl} x_i x_j x_k x_l \qquad \mbox{s.t.} \quad \norm{\bm x}{2} = 1.
\end{flalign}
They prove that for problem \eqref{eqn:tensor_opt}, the points satisfying the second-order necessary condition coincide with the component $\bm v_i$'s. Specifically, they show that
\begin{flalign}
\set{\bm x \in \mathbb{R}^n \; \vert \; G(\bm x) = 0,\; H(\bm x) \succeq \boldsymbol 0} = \set{\pm \bm v_1, \ldots, \pm \bm v_n}.
\end{flalign}
Therefore, solving problem \eqref{eqn:tensor_opt} yields one component $\bm {v}_i$, and after that, one can apply standard deflation procedures to obtain the others one by one.
Ge et al. \cite{GHJY15} further propose a larger optimization problem
\begin{flalign}\label{eqn:tensor_opt_2}
\min_{[\bm x_1, \bm x_2, \ldots, \bm x_n] \in \mathbb{R}^{n \times n}} \quad \sum_{i = 1}^n \sum_{ j \neq i} \mcb T(\bm x_i, \bm x_i, \bm x_j, \bm x_j)\qquad \mbox{s.t.} \quad \norm{\bm x_i}{2} = 1, \quad \; i = 1, \ldots, n,
\end{flalign}
to find all the components at one shot. Although problem \eqref{eqn:tensor_opt_2} is substantially more complicated than problem \eqref{eqn:tensor_opt}, similar geometrical phenomenon and the advantageous property of (\ref{eqn:tensor_opt}) are preserved. Specifically, Ge et al. \cite{GHJY15} proves that any solution of problem \eqref{eqn:tensor_opt_2} that satisfies the second-order necessary conditions in Theorem 1 is a signed and permuted version of $[\bm v_1, \bm v_2, \ldots, \bm v_n] \in \mathbb{R}^{n \times n}$.
\paragraph{Semidefinite Programming (SDP).} SDP is one of the most exciting developments in mathematical optimization and has been successfully applied to model and solve problems in traditional convex constrained optimization, control theory, and combinatorial optimization. In general, an SDP problem can be defined as
\begin{flalign}\label{eqn:sdp_opt}
\min_{\boldsymbol X \in \mathcal S_+^{n \times n}}\quad \mbox{Tr}(\boldsymbol C \boldsymbol X) \qquad \mbox{s.t.} \quad \mbox{Tr}(\boldsymbol A_i \boldsymbol X) = b_i \quad \; i = 1, \ldots, m,
\end{flalign}
where $\mathcal S_+^{n \times n}$ is the cone of symmetric positive semidefinite matrices.
Although interior-point methods can solve SDP problems in polynomial time, scalability is a problem and in practice interior-point methods run out of memory and time once $n$ is greater than $1e3$. To address this issue, Burer and Monteiro \cite{burer2003nonlinear, burer2005local}, take advantage of the low-rank structure of SDP optimal solutions characterized by Pataki \cite{pataki1998rank} and Barvinok \cite{barvinok1995problems}. Explicitly they replaced $\boldsymbol X = \boldsymbol L \boldsymbol L^\top$ where $\boldsymbol L$ has size $n\times p$, which leads to the following non-convex but low-dimensional surrogate:
\begin{flalign}\label{eqn:sdp_opt_low_rank}
\min_{\boldsymbol L \in \mathbb{R}^{n \times p}}\quad \mbox{Tr}(\boldsymbol L^\top \boldsymbol C \boldsymbol L) \qquad \mbox{s.t.} \quad \mbox{Tr}(\boldsymbol L^\top \boldsymbol A_i \boldsymbol L) = b_i \quad \; i = 1, \ldots, m.
\end{flalign}
Burer and Monteiro \cite{burer2003nonlinear, burer2005local} also applied local optimization methods to solve the above general nonlinear program \eqref{eqn:sdp_opt_low_rank} with surprisingly good performance, even though the theoretical justifications between problems \eqref{eqn:sdp_opt} and \eqref{eqn:sdp_opt_low_rank} were unclear.
Recently, Boumal et al. \cite{boumal2016non} proved that whenever $p$ satisfies $p(p+1) > 2m$ (and some other technical but mild conditions hold), problem \eqref{eqn:sdp_opt_low_rank} is equivalent to problem \eqref{eqn:sdp_opt} in the sense that for any {\em second-order critical point} of \eqref{eqn:sdp_opt_low_rank} $\boldsymbol L^\star$, the square matrix $\boldsymbol L^\star (\boldsymbol L^\star)^\top$ is optimal to problem \eqref{eqn:sdp_opt}.
One straightforward but powerful application of the above discussion is to solve the SDP problem derived as relaxation for the max-cut problem. In their celebrated work \cite{goemans1995improved}, Goemans and Williamson tackled the NP-hard max-cut problem by the following SDP
\begin{flalign}\label{eqn:sdp_opt_mc}
\min_{\boldsymbol X \in \mathcal S_+^{n \times n}}\quad \mbox{Tr}(\boldsymbol C \boldsymbol X) \qquad \mbox{s.t.} \quad \diag{\boldsymbol X} = \bm 1,
\end{flalign}
whose optimal solution (after rounding) will yield an approximate solution to max-cut within a ratio of .878. The corresponding Burer-Monteiro reformulation can be written as
\begin{flalign}\label{eqn:sdp_opt_low_rank_mc}
\min_{\boldsymbol L \in \mathbb{R}^{n \times p}}\quad \mbox{Tr}(\boldsymbol L^\top \boldsymbol C \boldsymbol L) \qquad \mbox{s.t.} \quad \norm{\bm e_i^\top \boldsymbol L}{} = 1 \quad \; i = 1, \ldots, n.
\end{flalign}
Based on the above discussion, when we choose $p = \lceil \sqrt{2n} \rceil$, even though problem \eqref{eqn:sdp_opt_low_rank_mc} has substantially smaller dimension than problem \eqref{eqn:sdp_opt_mc}, any second-order critical point $\boldsymbol L^\star$ of \eqref{eqn:sdp_opt_low_rank_mc} is able to recover a global optimal solution of \eqref{eqn:sdp_opt_mc} with $\boldsymbol X^\star = \boldsymbol L^\star (\boldsymbol L^\star)^\top$.
\hspace{5mm}
There is a large body of recent literature, especially in signal processing and learning theory, where finding second-order critical points is the fundamental target for the application at hand. Interested readers can find examples in complete dictionary recovery \cite{SQW152}, generalized phase retrieval \cite{SQW16}, matrix completion \cite{ge2016matrix}, phase synchronization and community detection \cite{boumal2016nonconvex, bandeira2016low}.
\section{Equality-Constrained Problem}\label{sec:eq}
In this section, we extend the definitions \eqref{eqn:first_order_opt} and \eqref{eqn:second_order_opt} of $\mathcal{G}(\bm{x}^\star)$ and $\mathcal{H}(\bm x^\star)$ at a local minimizer $\bm x^\star$ to a general feasible point $\bm x\in \Omega$, and verify that $\mathcal{G}(\bm x)$ and $\mathcal{H}(\bm x)$ can be regarded as natural generalizations of the conventional gradient and Hessian. Based on this, we extend the classical negative curvature algorithm from the unconstrained problem to the equality constrained one \eqref{eqn:problem_to_solve}, and prove this method converges to a second-order critical point of \eqref{eqn:problem_to_solve}.
\subsection{Generalized gradient and Hessian}\label{sub:generalized}
Based on Theorem 1, we note that $\mathcal{G}(\bm x^\star)$ and $\mathcal{H}(\bm x^\star)$ provide an elegant characterization of local optimality for the equality constrained problem in analogy with the roles of the gradient and Hessian for the unconstrained one. Inspired by this, we extrapolate the definitions of \eqref{eqn:first_order_opt} and \eqref{eqn:second_order_opt} from constrained local minimizers $\bm x^\star$ to any feasible point $\bm x \in \Omega,$ with the intention of using these generalized quantities as one uses the gradient and Hessian in the algorithmic design for unconstrained optimizations.
\begin{definition}[Generalized Gradient and Hessian]
For any feasible $\bm x \in \Omega$, let
\begin{flalign}\label{eqn:lambda_def}
\bm \lambda^\star(\bm x) \in \arg \min_{\bm \lambda}\; \norm{\nabla_{\bm x} \mathcal L(\bm x, \bm \lambda)}{}.
\end{flalign}
Then we define the generalized gradient at $\bm x$ as
\begin{flalign}\label{eqn:G}
&\mathcal{G}(\bm x):=\nabla_{\bm x} \mathcal L\left(\bm x, \bm \lambda^\star\left(\bm x\right)\right)
\end{flalign}
and the generalized Hessian at $\bm x$ as
\begin{flalign}\label{eqn:H}
\mathcal{H}(\bm x):= \mathcal{P}_{\bm x}^\top\nabla_{\bm x\bm x}^2 \mathcal L\left(\bm x, \bm \lambda^\star\left(\bm x\right)\right)\mathcal{P}_{\bm x}.
\end{flalign}
\end{definition}
In the following, we verify that $\mathcal{G}(\cdot)$ and $\mathcal{H}(\cdot)$ behave like the gradient and Hessian not only for a local optimal solution but also for any feasible point $\bm x \in \Omega$ in an approximate sense, under the mild conditions described below:
\begin{assumption}
\label{ass: key_ass}
\begin{description}
\item[(a)]\hspace{0.4cm} $\nabla f$, $\nabla^2 f$, $\nabla c_i$ and $\nabla^2 c_i$ are Lipschitz continuous over $\Omega$ with Lipschitz constants $L_{f, 1}$, $L_{f, 2}$, $L_{c_i, 1}$ and $L_{c_i, 2}$.
\item[(b)]\hspace{0.4cm} $\sup_{{\bm x} \in \Omega}\norm{\nabla f({\bm x})}{} \leq \gamma_{f, 1}$, $\sup_{{\bm x}\in \Omega}\norm{\nabla^2 f({\bm x})}{} \leq \gamma_{f, 2}$, $\sup_{{\bm x} \in \Omega}\norm{\nabla c_i({\bm x})}{} \leq \gamma_{c_i, 1}$ and $\\\sup_{{\bm x} \in \Omega}\norm{\nabla^2 c_i({\bm x})}{} \leq \gamma_{c_i, 2}$.
\item[(c)][$\sigma_0$-LICQ] \hspace{0.4cm}For every ${\bm x} \in \Omega$, $\sigma_{\text{min}}(\nabla \boldsymbol{c}({\bm x})) \ge \sigma_0$ for some $\sigma_0 > 0$, where $\sigma_{\text{min}}(\nabla \boldsymbol{c}({\bm x}))$ is the smallest singular value of the matrix $\nabla \boldsymbol{c}(\bm{x})$
\end{description}
\end{assumption}
\begin{remark}
The last assumption is slightly stronger that LICQ as LICQ only requires $\sigma_{\text{min}}(\nabla \boldsymbol{c}({\bm x})) > 0.$
\end{remark}
We next prove a key lemma regarding $\mathcal{G}(\cdot)$ and $\mathcal{H}(\cdot)$. Loosely speaking, we show that for every $\boldsymbol{x} \in \Omega$, given a small perturbation $\bm \delta \in \mathcal{T}_\Omega(\boldsymbol{x})\subset \mathbb{R}^n$, $f({\bm x})+\mathcal{G}({\bm x})^\top{\bm \delta}$ and $f({\bm x})+\mathcal{G}({\bm x})^\top{\bm \delta} +\frac{1}{2}{\bm \delta}^\top \mathcal{H}({\bm x}){\bm \delta}$, respectively approximates $f(\bm x + \bm \delta)$ up to the first order and the second order. This result is crucial for our generalization of negative curvature methods. But one issue we need to fix in advance is that $\bm x + \bm \delta$ may possibly lie outside of $\Omega$ and thus the objective $f$ might not even be defined at $\bm x + \bm \delta$. In order to resolve this infeasibility issue, we introduce the following projection operator:
\begin{definition}\label{def:proj_Omega}
For any $\bm y \in \mathbb{R}^n$, we denote by $\Pi_{\Omega}(\bm y)\in \Omega$ a point that is closest to $\bm y$, i.e. $\Pi_{\Omega}(\bm y)\in \arg\min_{\bm x \in \Omega} \; \norm{\bm x - \bm y}{}$, where $\norm{\cdot}{}$ denotes the Euclidean norm.
\end{definition}
We are now ready to state the key lemma.
\begin{lemma}\label{key_lemma}
Under Assumptions 2(b), 2(c) and 2(d),
for any ${\bm x}_0 \in \Omega$ and ${\bm \delta} \in \mathcal{T}_{\Omega}(\bm x_0)$, we have
\begin{flalign}
&\left|f(\Pi_\Omega({\bm x}_0 + {\bm \delta})) - f({\bm x}_0) -\mathcal{G}({\bm x}_0)^\top{\bm \delta} \right| \le C_0 \norm{{\bm \delta}}{}^2 ,\quad\text{and} \label{1st_order_taylor} \\
&\left|f(\Pi_\Omega({\bm x}_0 + {\bm \delta})) - f({\bm x}_0) -\mathcal{G}({\bm x}_0)^\top{\bm \delta} - \frac{1}{2}{\bm \delta}^\top \mathcal{H}({\bm x}_0){\bm \delta}\right| \le C_5 \norm{{\bm \delta}}{}^3 \label{key_taylor}
\end{flalign}
where $C_0 = \gamma_{f, 1}\{1 + \frac{\Gamma_1}{\sigma_0^2}\}\frac{4}{R^2} + 4(L_{f, 1} + \frac{\gamma_{f, 1}\sqrt{\Gamma_1\Lambda_1}}{\sigma_0^2} )$, $C_5 = 8C_1 + C_2 + C_3 + C_4, C_1 = \frac{L_{f, 2}}{2} + \frac{\gamma_{f, 1}}{2\sigma_0^2}\sqrt{\Gamma_1\Lambda_2}$, $C_2 = \{\gamma_{f, 2} + \frac{\gamma_{f, 1}}{2\sigma_0^2}\sqrt{\Gamma_1\Gamma_2}\}\frac{4}{R}$, $C_3 = \{\gamma_{f, 2} + \frac{\gamma_{f, 1}}{2\sigma_0^2}\sqrt{\Gamma_1\Gamma_2}\}\frac{2}{R}$, $C_4 = \{\gamma_{f, 1} + \frac{\gamma_{f, 1}}{\sigma_0^2}\Gamma_1\}(\frac{2\sqrt{\Gamma_1\Lambda_1}}{\sigma_0^2} + \frac{2\sqrt{\Gamma_1^3\Lambda_1}}{\sigma_0^4}) \frac{8}{R}$, $R = \sigma_0 / \sqrt{\Lambda_1}$, $\Gamma_1 = \sum_{i=1}^m \gamma_{c_i, 1}^2$, $\Gamma_2 = \sum_{i=1}^m \gamma_{c_i, 2}^2$, $\Lambda_1 = \sum_{i=1}^m L_{c_i, 1}^2$, and $\Lambda_2 = \sum_{i=1}^m L_{c_i, 2}^2$.
\end{lemma}
\begin{proof}
See the appendix.
\end{proof}
\begin{remark}
If $\Omega$ is a differentiable manifold, our $\mathcal{G}(\cdot)$ and $\mathcal{H}(\cdot)$ defined in \eqref{eqn:G} and \eqref{eqn:H} can be proven to be the Riemannian gradient and Riemannian Hessian of $f$ over $\Omega$ (see subsection \ref{sub:relations} for details). The expressions \eqref{eqn:G} and \eqref{eqn:H} concretize these abstract geometrical concepts by providing an explicit algebraic way to compute them.
\end{remark}
\subsection{Negative curvature method}
In this subsection, we will present a general framework for using negative curvature of the generalized Hessian $\mathcal{H}(\bm x)$ to solve (\ref{eqn:problem_to_solve}) and show that the sequence generated by this framework converges to a second-order critical point.
It is clear that using the simple projected gradient method
\begin{align}
\bm x_{k+1} \gets \Pi_\Omega\left({\bm x}_k - t \mathcal{G}(\bm x_k)\right) \label{project_gradient_step}
\end{align}
might be trapped by the saddle points of \eqref{eqn:problem_to_solve}.
To overcome this difficulty, it is natural to consider using second-order information, that is, information about the Hessian of $f$ and the $c_i$'s. In particular, we can use a negative curvature direction of $\mathcal{H}(\bm x)$ defined in subsection \ref{sub:generalized}. We say $\boldsymbol{d}$ is a negative curvature direction of $\mathcal{H}(\bm x)$ if it has the property that $\boldsymbol{d}^\top\mathcal{H}(\bm x)\boldsymbol{d} < 0$. Based on the second-order approximation result revealed in Lemma \ref{key_lemma}, it is intuitive that moving in a negative curvature direction will enable an algorithm to escape from a saddle point. Therefore, it is tempting to move along a direction that combines the negative gradient direction with a descent direction of negative curvature:
\begin{align}
\bm x_{k+1} \gets \Pi_\Omega\left({\bm x}_k - t_1 \mathcal{G}(\bm x_k) - t_2 \bm d \right) \label{project_curvilinear_step}
\end{align}
especially for the regions close to saddle points.
Moreover, negative-curvature directions can be obtained at a relatively small cost. For example, when a Hessian matrix $H(\bm x)$ is indefinite, the eigenvector corresponding to its algebraically smallest eigenvalue is a negative curvature direction. It can be obtained by executing the power iteration method on $H(\bm x)$ to obtain a eigenvalue, eigenvector pair $(\lambda_{\text{dom}}, \bm{d}_{\text{dom}})$ corresponding to the eigenvalue of largest magnitude, the so-called {\it dominant} eigenvalue $\lambda_{\text{dom}}$. If $\lambda_{\text{dom}} < 0$, $\bm d_{\text{dom}}$ is the direction of most negative curvature. Otherwise, we can perform the method again on $\mathcal{H} - \lambda_{\text{dom}}\mathcal{I}$. More efficient and robust methods using variants of Lanzos algorithm to compute the algebraically smallest eigenpair can be found in \cite{larsen2004propack, stathopoulos2010primme}.
In Algorithm \ref{curvlinear_search}, we provide a general framework for using negative curvature directions to solve problem \eqref{eqn:problem_to_solve}. This algorithm integrates the first-order and second-order methods. Specifically, when the iterate $\bm x_k$ is far from any saddle point, we only use the first-order information to make progress:
\[
\hat{{\bm x}} \leftarrow \Pi_\Omega({\bm x}_k - t\mcb{G}_k).
\]
When $\bm x_k$ is near saddle points, we combine the negative gradient and the negative curvature direction:
\begin{flalign}
\hat{{\bm x}} \leftarrow \Pi_\Omega({\bm x}_k - t\mcb{G}_k + t^\alpha \boldsymbol{d}_k).
\label{eqn:cur_search}
\end{flalign}
If $\alpha$ is chosen to be $2$ and $\Omega = \mathbb{R}^n$ , \eqref{eqn:cur_search} is reduced to the one leveraged in \cite{GOLD80}; if $\alpha$ is set to $1/2$ and $\Omega = \mathbb{R}^n$, \eqref{eqn:cur_search} is equivalent to the one used in \cite{MS79}.
In the rest of this section, we will confirm our intuition that Algorithm \ref{curvlinear_search} is capable of escaping saddle points by proving the following theorem:
\begin{center}
\begin{algorithm}[h]
\caption{Negative Curvature Method for Problem \eqref{eqn:problem_to_solve}}
\label{curvlinear_search}
\begin{algorithmic}[1]
\renewcommand\algorithmicrequire{\textbf{input}}
\Require parameters $0 < \sigma < 1$, $0< \rho < 1$, $\alpha > 0$, $\epsilon > 0$ and $t_0 > 0$.
\State initialize ${\bm x}_0 \in \Omega$;
\For{$k = 0, 1, \ldots, $}
\State $\mcb{G}_k \leftarrow \mathcal{G}(\boldsymbol{x_k})$
\If{$\|\mcb{G}_k\| \geq \epsilon$}
\State $t \leftarrow t_0$
\State $\hat{{\bm x}} \leftarrow \Pi_\Omega({\bm x}_k - t\mcb{G}_k)$
\While{$f(\hat{{\bm x}}) - f({\bm x}_k) > -\sigma t \|\mcb{G}_k\|^2$}
\State $t \leftarrow \rho t$
\State $\hat{{\bm x}} \leftarrow \Pi_\Omega({\bm x}_k - t\mcb{G}_k)$
\EndWhile
\Else
\State $\mcb{H}_k \leftarrow \mathcal{H}({\bm x}_k)$
\State $(\lambda_k^\text{min}, \boldsymbol{v}_k) \leftarrow$ the algebraically smallest eigenpair of $\mcb{H}_k$
\State $\lambda_k \leftarrow \min\{\lambda_k^\text{min}, 0\}$
\State $\boldsymbol{d}_k \leftarrow \vert \lambda_k \vert \text{sign}(-\boldsymbol{v}_k^\top\mcb{G}_k)\boldsymbol{v}_k$
\State $t \leftarrow t_0$
\State
$\hat{{\bm x}} \leftarrow \Pi_\Omega({\bm x}_k - t\mcb{G}_k + t^\alpha \boldsymbol{d}_k)$
\While{$f(\hat{{\bm x}}) - f({\bm x}_k) > \sigma \big( -t\|\mcb{G}_k\|^2 - \frac{1}{2}t^{2\alpha}\vert\lambda_k\vert^3\big)$}
\State $t \leftarrow \rho t$
\State $\hat{{\bm x}} \leftarrow \Pi_\Omega({\bm x}_k - t\mcb{G}_k + t^\alpha \boldsymbol{d}_k)$
\EndWhile
\EndIf
\State $t_k \leftarrow t$
\State ${\bm x}_{k+1} \leftarrow \hat{{\bm x}}$
\EndFor
\end{algorithmic}
\end{algorithm}
\end{center}
\begin{theorem}\label{thm:alg}
For the sequences $\{\mcb{G}_k\}$ and $\{\lambda_k\}$ generated by Algorithm \ref{curvlinear_search}, one has $\mcb{G}_k \rightarrow \boldsymbol{0}$ and $\lambda_k \rightarrow 0$ as $k \rightarrow \infty$.
\end{theorem}
\begin{remark}
Based on Theorem \ref{thm:alg}, any cluster point of $\set{\bm x_k}$ generated by Algorithm \ref{curvlinear_search} is a second-order critical point of problem \eqref{eqn:problem_to_solve}.
\end{remark}
To prove Theorem \ref{thm:alg}, we need several lemmas.
\begin{lemma} \label{expan_lemma}
Consider the $k$-th iteration.
\begin{description}
\item[(a)]For the parametrized curve ${\bm x}(t) = \Pi_\Omega({\bm x}_k - t\mcb{G}_k)$, we have
\begin{align}
f({\bm x}(t)) - f({\bm x}_k) \leq -\|\mcb{G}_k\|^2t + C_0\|\mcb{G}_k\|^2t^2;
\label{eqn:first_ineq}
\end{align}
\item[(b)]For the parametrized curve ${\bm x}(t) = \Pi_\Omega({\bm x}_k - t\mcb{G}_k + t^\alpha\boldsymbol{d}_k)$, we have
\begin{align}
f({\bm x}(t)) - f({\bm x}_k) \leq -\|\mcb{G}_k\|^2t + \frac{1}{2}\|\mcb{H}_k\|\|\mcb{G}_k\|^2t^2 - \frac{1}{2}\vert \lambda_k\vert^3 t^{2\alpha} + 8C_5\|\mcb{G}_k\|^3t^3+ 8C_5\vert\lambda_k\vert^3t^{3\alpha}.
\label{eqn:second_ineq}
\end{align}
\end{description}
\end{lemma}
\begin{proof}
For part (a), we can directly apply (\ref{1st_order_taylor}) in Lemma \ref{key_lemma}:
\begin{flalign}
f({\bm x}(t)) - f({\bm x}_k) \leq \innerprod{\mcb{G}_k}{-t\mcb{G}_k}+ C_0\|\mcb{G}_k\|^2t^2 = -\|\mcb{G}_k\|^2t + C_0\|\mcb{G}_k\|^2t^2.
\nonumber
\end{flalign}
Now let us focus on part (b).
For part (b), based on \eqref{key_taylor} of Lemma \ref{key_lemma}, we first have
\begin{flalign}
&f({\bm x}(t)) - f({\bm x}_k) \nonumber\\
&\leq \langle \mcb{G}_k, -t\mcb{G}_k + t^\alpha \boldsymbol{d}_k\rangle + \frac{1}{2}(-t\mcb{G}_k + t^\alpha \boldsymbol{d}_k)^\top \mcb{H}_k (-t\mcb{G}_k + t^\alpha \boldsymbol{d}_k) + C_5\|-t\mcb{G}_k + t^\alpha \boldsymbol{d}_k\|^3 \nonumber\\
&\leq -\|\mcb{G}_k\|^2 t + \langle \mcb{G}_k, \boldsymbol{d}_k \rangle t^\alpha + \frac{1}{2}\mcb{G}_k^\top \mcb{H}_k\mcb{G}_kt^2 - \mcb{G}_k^\top \mcb{H}_k \boldsymbol{d}_kt^{\alpha + 1} + \frac{1}{2}\boldsymbol{d}_k^\top\mcb{H}_k\boldsymbol{d}_kt^{2\alpha} + C_5\|-t\mcb{G}_k + t^\alpha \boldsymbol{d}_k\|^3. \label{eqn:total}
\end{flalign}
Next, we will bound each term in \eqref{eqn:total}:
\begin{flalign}
&\langle \mcb{G}_k, \boldsymbol{d}_k \rangle t^\alpha \leq |\lambda_k| \mbox{sign}(-\bm v_k^\top \mcb{G}_k)\bm v_k^\top \mcb{G}_k t^\alpha \le 0 \label{eqn:sub_1}\\
&\frac{1}{2} \mcb{G}_k^\top \mcb{H}_k \mcb{G}_k t^2 \leq \frac{1}{2} \norm{\mcb{H}_k}{} \norm{\mcb{G}_k}{}^2 t^2 \label{eqn:sub_2}\\
&-\mcb{G}_k^\top \mcb{H}_k \bm d_k t^{\alpha+1} = - \mcb{G}_k^\top |\lambda_k| \mbox{sign}(-\bm v_k^\top \mcb{G}_k) \mcb{H}_k \bm v_k t^{\alpha+1} = \lambda_k |\lambda_k| \mbox{sign}(-\bm v_k^\top \mcb{G}_k) (- \mcb{G}_k^\top\bm v_k) t^{\alpha+1} \le 0 \label{eqn:sub_3}\\
&\frac{1}{2} \bm d_k^\top \mcb{H}_k \bm d_k t^{2\alpha} = \frac{1}{2} |\lambda_k|^2 \bm v_k^\top \mcb H_k \bm v_k t^{2\alpha} = -\frac{1}{2} |\lambda_k|^3 t^{2 \alpha} \label{eqn:sub_4} \\
&\|-t\mcb{G}_k + t^\alpha \boldsymbol{d}_k\|^3 \le \left( t\norm{\mcb{G}_k}{} + t^\alpha \norm{\bm d_k}{} \right)^3 \le 8 \max(t \norm{\mcb G_k}{}, t^\alpha \norm{\bm d_k}{})^3 \le 8 t^3 \norm{\mcb G_k}{}^3 + 8 t^{3 \alpha} |\lambda_k|^3. \label{eqn:sub_5}
\end{flalign}
With \eqref{eqn:sub_1}-\eqref{eqn:sub_5} plugged in \eqref{eqn:total}, we reach \eqref{eqn:second_ineq}.
\begin{lemma}
There exists a constant $\gamma_h \ge 0$ such that $\sup_{\bm x \in \Omega} \norm{\mathcal H(\bm x)}{} \le \gamma_h.$
\end{lemma}
\begin{proof}
For any $\bm x \in \Omega$, one has
\begin{flalign}
\norm{\mathcal H(\bm x)}{} &\le \norm{\mathcal P_{\bm x}^\top \nabla_{\bm x \bm x}^2 \mathcal L(\bm x, \bm \lambda^\star (\bm x)) \mathcal P_{\bm x}}{} \nonumber \\
& \le \norm{ \nabla_{\bm x \bm x}^2 \mathcal L(\bm x, \bm \lambda^\star (\bm x))}{} \nonumber \\
& \le \norm{\nabla_{}^2 f(\bm x) - \sum_{i \in [m]} \lambda_i^\star (\bm x) \nabla^2 c_i(\bm x)}{} \nonumber \\
& \le \norm{\nabla^2 f(\bm x)}{} + \norm{\lambda^\star (\bm x)}{\infty} \sum_{i \in [m]} \norm{\nabla^2 c_i(\bm x)}{}. \label{H_bound}
\end{flalign}
Based on Assumption \ref{ass: key_ass}, we have
\begin{flalign}
\norm{\nabla^2 f(\bm x)}{} \le \gamma_{f,2} \;\; \mbox{and} \;\; \sum_{i \in [m]} \norm{\nabla^2 c_i(\bm x)}{} \le \sum_{i \in [m]} \gamma_{c_i, 2}.
\label{eqn:first_bd}
\end{flalign}
From the definition \eqref{eqn:lambda_def}. we can derive that
\begin{flalign}
\bm \lambda^\star({\bm x}) = ({\nabla \bm c}({\bm x})^\top{\nabla\bm c}({\bm x}))^{-1}{\nabla \bm c}({\bm x})^\top\nabla f({\bm x}). \label{lambda_express}
\end{flalign}
Together with Assumption \ref{ass: key_ass}, it can be obtained that
\begin{flalign}
\norm{\bm \lambda^\star({\bm x})}{\infty} & \le \norm{\bm \lambda^\star({\bm x})}{2} \nonumber\\
& = \norm{({\nabla \bm c}({\bm x})^\top{\nabla\bm c}({\bm x}))^{-1}{\nabla \bm c}({\bm x})^\top\nabla f({\bm x})}{2} \nonumber \\
& \le \norm{({\nabla \bm c}({\bm x})^\top{\nabla\bm c}({\bm x}))^{-1}}{} \norm{\nabla \bm c(\bm x)}{} \norm{\nabla f(\bm x)}{} \nonumber \\
& \le \norm{({\nabla \bm c}({\bm x})^\top{\nabla\bm c}({\bm x}))^{-1}}{} \norm{\nabla \bm c(\bm x)}{F} \norm{\nabla f(\bm x)}{} \nonumber \\
& \le \frac{1}{\sigma_0^2} \sqrt{\sum_{i \in [m]} \gamma^2_{c_i,1}} \cdot \gamma_{f,1}.
\label{eqn:lam_inf}
\end{flalign}
The lemma can be established by substituting \eqref{eqn:first_bd} and \eqref{eqn:lam_inf} into (\ref{H_bound}). Hence $\gamma_h = \sum_{i\in [m]}\gamma_{c_i, 2} + \frac{1}{\sigma_0^2} \sqrt{\sum_{i \in [m]} \gamma^2_{c_i,1}} \cdot \gamma_{f,1}\sum_{i \in [m]} \gamma_{c_i, 2}$
\end{proof}
\end{proof}
\begin{lemma}
$\{t_k\}$ is uniformly bounded from below, i.e., there exists $\beta > 0$ such that $t_k \geq \beta$ for every $k \in \mr{N}$. \label{lem:t_k}
\end{lemma}
\begin{proof}
Consider the first case $\|\mcb{G}_k\| \ge \epsilon$. For ${\bm x}(t) = \Pi_\Omega({\bm x}_k - t\mcb{G}_k)$, as shown in Lemma \ref{expan_lemma}, one has
\begin{flalign}
f({\bm x}(t)) - f({\bm x}_k) \leq -\|\mcb{G}_k\|^2t + C_0 \|\mcb{G}_k\|^2t^2.
\end{flalign}
Therefore, whenever $t \le (1-\sigma)/C_0$, we have
\begin{flalign}
f({\bm x}(t)) - f({\bm x}_k) \leq -\|\mcb{G}_k\|^2t + C_0 \|\mcb{G}_k\|^2t^2 \le -\sigma t \norm{\mcb G_k}{}^2.
\end{flalign}
Thus, $t_k \geq t_0 \rho^{\lceil \log_\rho\{(1 - \sigma) / (C_0 t_0)\}\rceil}$ for this case.
Now let us consider the other case $\|\mcb{G}_k\| < \epsilon$. For ${\bm x}(t) = \Pi_\Omega({\bm x}_k - t\mcb{G}_k + t^\alpha\boldsymbol{d}_k)$, it follows from part (b) of Lemma \ref{expan_lemma} that
\begin{align}
f({\bm x}(t)) - f({\bm x}_k) \leq -\|\mcb{G}_k\|^2t + \frac{1}{2}\|\mcb{H}_k\|\|\mcb{G}_k\|^2t^2 - \frac{1}{2}\vert \lambda_k\vert^3 t^{2\alpha} + 8C_5\|\mcb{G}_k\|^3t^3+ 8C_5\vert\lambda_k\vert^3t^{3\alpha}.
\end{align}
When $t \le \underline{t} := \min \set{\left(\frac{1 - \sigma}{16C_5}\right)^{1/\alpha}, \;\frac{2 - 2\sigma}{\gamma_h + 16C_5\epsilon}, \;1}$, it can be verified that
\begin{align}
-\frac{1}{2}\vert\lambda_k\vert^3t^{2\alpha} + 8C_5\vert\lambda_k\vert^3t^{3\alpha}&\leq -\frac{1}{2}\sigma t^{2\alpha}\vert \lambda_k \vert^3 \quad
\text{and} \label{first_ieq} \\
-\|\mcb{G}_k\|^2t + \frac{1}{2}\|\mcb{H}_k\|\|\mcb{G}_k\|^2t^2 + 8C_5\|\mcb{G}_k\|^3t^3 &\leq -\sigma t\|\mcb{G}_k\|^2\label{second_ieq}
\end{align}
Combining \eqref{first_ieq} and \eqref{second_ieq}, we have
\begin{align*}
f({\bm x}(t)) - f({\bm x}_k) \leq \sigma \big(-t\|\mcb{G}_k\|^2 - \frac{1}{2}t^{2\alpha}\vert\lambda_k\vert^3\big).
\end{align*}
Therefore,
$
t_k \geq t_0 \rho^{\lceil \log_\rho \underline t/ t_0 \rceil}
$
for this case.
Taking both cases into consideration, we have proved this lemma.
\end{proof}
Now we are ready to prove Theorem \ref{thm:alg}.
\begin{proof}[Proof of Theorem \ref{thm:alg}]
Based on Lemma \ref{lem:t_k}, we have
\begin{align*}
f({\bm x}_{k+1}) - f({\bm x}_k) \leq -\sigma \{\beta \epsilon^2, \beta\|\mcb{G}_k\|^2 + \frac{1}{2} \beta^{2\alpha}\vert\lambda_k\vert^3\}.
\end{align*}
Since $\lim_{k\rightarrow\infty} f({\bm x}_k) > -\infty$, we must have
\begin{align*}
\lim_{k\rightarrow \infty}\mcb{G}_k =\boldsymbol{0}, \quad\text{and}\quad \lim_{k\rightarrow\infty}\lambda_k = 0.
\end{align*}
\end{proof}
\begin{remark}
In Algorithm \ref{curvlinear_search}, we can relax the requirement of finding the algebraically smallest eigenpair of $\mcb{H}_k$. Instead, it is sufficient to find $(\bar \lambda, \bar{\bm v})$ satisfying
\begin{flalign}
\norm{\bar{\bm v}}{}= 1, \; \bar{\bm v}^\top \mcb{H}_k \bar{\bm v} &\le \max\{-\delta, \;\lambda_k^{\min}\}, \; \bar{\bm v}^\top \mcb{G}_k \le 0 \;\; \mbox{and} \;\; \bar{\bm v}^\top \mcb{H}_k \bar{\bm v} \le 0
\end{flalign}
where $\delta >0$ is a prescribed constant.
\end{remark}
\subsection{Examples of $\Omega$ and $\Pi_{\Omega}(\cdot)$}
For Algorithm 1 to be practical, the projection of a point onto the feasible set $\Omega$ must be affordable.
In this subsection, we enumerate a number feasible sets $\Omega$, defined by equality constraints, that are frequently encountered in optimization problems and have computationally tractable projections $\Pi_{\Omega}(\cdot)$:
\begin{table}[ht]
\caption{Examples of $\Omega$ and $\Pi_{\Omega}(\cdot)$}
\centering
\begin{tabular}{l |c | c }
\hline
Constraint Sets & $\Omega$ & $\Pi_{\Omega}(\cdot)$ \\ [1ex]
\hline
spherical &$\set{\boldsymbol X \in \mathbb{R}^{n \times m} \;\vert\; \norm{\boldsymbol X}{F} = 1 }$ & $\boldsymbol X / \norm{\boldsymbol X}{F}$ \\[1ex]
multiple spherical & $\bigcup_i \{ \boldsymbol x_i \in \mathbb{R}^{n_i} \;\vert\; \norm{\boldsymbol x_i}{2} = 1 \} $ & $ \boldsymbol x_i / \norm{\boldsymbol x_i}{2} , ~ \forall ~ i $
\\ [1ex]
orthogonality & $\set{ \boldsymbol X \in \mathbb{R}^{n \times m} \;\vert\; \boldsymbol X^T \boldsymbol X =I }$ &
$\boldsymbol U \boldsymbol V^\t, ~\mbox{with}~ \boldsymbol X = \boldsymbol U \boldsymbol \Sigma \boldsymbol V^\t \mbox{ as SVD}$
\\ [1ex]
\hline
\end{tabular}
\label{table:ex_omega}
\end{table}
There is a wide range of applications in which one is interested in solving optimization problems with constraints as listed in Table \ref{table:ex_omega}. The paper \cite{wen2013feasible} provides an extensive list of such problems and references to particular applications. These include eigenvalue and subspace tracking problems arising in signal processing; low-rank matrix optimization problems such as those that arise in SDP relaxation of combinatorial problems (e.g., the max-cut problem described in section 2); $p$-harmonic flows and other problems involving normal preserving constraints such as those that arise in $1$-bit compressive sensing, color image denoising, micromagnetics, liquid crystal theory, and directional diffusion; homogeneous polynomial optimization with spherical constraints arising in tensor eigenvalue problems signal processing, MRI, data training, approximation theory, portfolio selection and computation of the stability number of a graph; sparse principal component analysis, electronic structures computation, etc.
\subsection{Discussion: extension to general constrained problems}
We briefly discuss possibilities of generalizing the proposed algorithm to optimization problems with inequality constraints, i.e., problems of the form:
\begin{align*}
\begin{array}{ll}
\text{minimize} & f({\bm x}) \\
\mbox{subject to}& c_i({\bm x}) = 0, \; i = 1, 2, \ldots, m\\
& c_i({\bm x}) \leq 0, \; i = m+1, 2, \ldots, k\\
& {\bm x} \in \mr{R}^n.
\end{array}
\end{align*}
By adding a squared slack variable to each inequality constraint, since $c_i({\bm x}) \leq 0 \Leftrightarrow c_i({\bm x}) + z_i^2 = 0$, one can transform a problem
with inequality constraints to one that has only equality constraints. Moreover,
it can be verified that the conditions imposed in Assumption 2 on the constraints $c_i({\bm x})$, carry over to the transformed constraints. Therefore, algorithm \ref{curvlinear_search} can be applied again. But one caveat of this reformulation is that additional second-order critical points might be introduced. We leave it as future work to investigate better approaches to handling general inequality constraints.
\section{Appendix}\label{sec:append}
\subsection{Proof of Lemma \ref{key_lemma}}
We need the proposition below in the proof.
\begin{proposition}\label{the_prop}(\cite[Lemma 33]{GHJY15})
Assume that Assumption 2 holds and define $R = \sqrt{1 / (\sum_{i=1}^m L_{c_i,1}^2/\sigma_0^2)} $. For any ${\bm x}_0 \in \Omega$, any $\boldsymbol{v} \in \mr{R}^m$, let ${\bm x}_1 = {\bm x}_0 + \boldsymbol{v}$ and ${\bm x}_2 = {\bm x}_0 + \mathcal{P}_{{\bm x}_0}(\boldsymbol{v})$. Then we have
\begin{align*}
\norm{\Pi_\Omega({\bm x}_1) - {\bm x}_2} \leq \frac{4\|\boldsymbol{v}\|^2}{R}.
\end{align*}
\end{proposition}
Now we are ready to prove Lemma \ref{key_lemma}.
\begin{proof}[Proof of Lemma \ref{key_lemma}]
To prove (\ref{1st_order_taylor}), for any $\bm x \in \Omega$, let $\bm y = \Pi_\Omega(\bm x + \bm\delta)$ and consider the Lagrangian $\mathcal{L}(\bm y, \bm\lambda^*(\bm x)) = f(\bm y) - \sum_{i=1}^m \lambda^*_i(\bm{x})c_i(\bm y)$. Since $\bm x, \bm y \in \Omega$, $\bm c(\bm x) = \bm c (\bm y) = \bm 0$ and hence $\mathcal{L}(\bm y, \bm\lambda^*(\bm x)) = f(\bm y)$ and $\mathcal{L}(\bm x, \bm\lambda^*(\bm x)) = f(\bm x)$. Therefore by Taylor expansion, we have for some $s \in (0, 1)$
\begin{align}
f(\bm y) &= \mathcal{L}(\bm y, \bm\lambda^*(\bm x)) = \mathcal{L}(\bm x, \bm\lambda^*(\bm x)) + \nabla_{\bm x}\mathcal{L}(\bm x + s(\bm y - \bm x), \bm\lambda^*(\bm x))^\top (\bm y - \bm x) \nonumber\\
&= f(\bm x) + \nabla_{\bm x} \mathcal{L}(\bm x, \bm \lambda^*(\bm x))^\top (\bm y - \bm x) + \{\nabla_{\bm x}\mathcal{L}(\bm x + s(\bm y - \bm x), \bm\lambda^*(\bm x)) - \nabla_{\bm x} \mathcal{L}(\bm x, \bm \lambda^*(\bm x))\}^\top (\bm y - \bm x) \nonumber \\
&= f(\bm x) + \mathcal{G}(\bm x)^\top \bm \delta + \mathcal{G}(\bm x)^\top\{\bm y - (\bm x + \bm \delta)\} \nonumber \tag{by definition of $\mathcal{G}(\bm x)$} \\
&\quad +\{\nabla f(\bm x + s(\bm y - \bm x)) - \nabla f(\bm x)\}^\top (\bm y - \bm x) - \sum_{i=1}^m\lambda_i^*(\bm x)\{\nabla c_i(\bm x + s(\bm y - \bm x)) - \nabla c_i(\bm x)\}^\top (\bm y - \bm x). \nonumber
\end{align}
Hence using Assumption 2(b), Cauchy-Schwartz and the fact that $s < 1$, we have
\begin{align}
\vert f(\bm y) - f(\bm x) - \mathcal{G}(\bm x)^\top \bm \delta\vert &\leq \|\mathcal{G}(\bm x)\|\|\bm y - (\bm x + \bm \delta)\| + \{L_{f, 1} + \sum_{i=1}^m\lambda^*_i(\bm x)L_{c_i, 1} \} \| \bm y - \bm x \|^2 \nonumber\\
&\leq \|\mathcal{G}(\bm x)\|\|\bm y - (\bm x + \bm \delta)\| + \{L_{f, 1} + \|\bm\lambda^*(\bm x)\|\sqrt{\Lambda_1}\}\| \bm y - \bm x \|^2 . \label{first_order_taylor}
\end{align}
Both $\| \bm\lambda^*(\bm x) \|$ and $\|\mathcal{G}(\bm x) \|$ can be bounded by constants. From (\ref{lambda_express}), use of the assumptions that $\inf_{\bm x \in \Omega}\sigma_{\text{min}}(\nabla\bm{c}({\bm x})) \ge \sigma_0 $, $\sup_{\bm x \in \Omega} \|\nabla f(\bm x)\| \leq \gamma_{f, 1}$, $\sup_{\bm x \in \Omega} \|\nabla c_i(\bm x)\| \leq \gamma_{c_i, 1}$ and the definition of Frobenius norm,
\begin{align}
\|\boldsymbol \lambda^\star(\boldsymbol x)\| \leq \frac{\gamma_{f, 1}\sqrt{\Gamma_1}}{\sigma_0^2} \label{lambda_bound}
\end{align}
While
\begin{align}
\| \mathcal{G}(\bm x) \| &= \|\nabla f(\bm x) - \sum_{i=1}^m \lambda_i^*(\bm x) \nabla c_i(\bm x) \| \leq \| \nabla f(\bm x) \| + \| \bm \lambda^*(\bm x) \| \| \nabla \bm c(\bm x) \|_F \nonumber \\
&\leq \gamma_{f, 1} + \frac{\gamma_{f, 1}\Gamma_1}{\sigma_0} = \gamma_{f, 1}\{1 + \frac{\Gamma_1}{\sigma_0^2}\}. \label{G_bound}
\end{align}
To bound $\| \bm y - \bm x\|$, by definition of $\bm y$ and $\Pi_\Omega(\cdot)$ we have
\begin{align}
\| \bm y - \bm x \| = \| \Pi_\Omega(\bm x + \delta) - \bm x \| \leq \| \Pi_\Omega(\bm x + \delta) - (\bm x + \bm \delta)\| + \| \bm \delta \| \leq 2\|\bm \delta \| \label{diff_bound}
\end{align}
As above $\|\bm y - (\bm x + \bm \delta) \| \leq \| \bm\delta \|$. However, we also need to bound $\|\bm y - (\bm x + \bm \delta) \|$ in terms of $\| \bm \delta \|^2$ to facilitate our analysis. Specifically, since $\bm \delta \in \mathcal{T}_{\Omega}(\bm x)$ and $\bm x \in \Omega$, by Proposition 1, we have
\begin{align}
\|\bm y - (\bm x + \bm \delta) \| = \|\Pi_\Omega(\bm x + \bm \delta) -(\bm x + \bm \delta) \| \leq \frac{4}{R^2}\| \bm\delta\|^2 \label{tricki_bound}
\end{align}
where $R = \sigma_0 / \sqrt{\Lambda_1}$. Now plugging (\ref{tricki_bound}), (\ref{diff_bound}), (\ref{G_bound}), and (\ref{lambda_bound}) into (\ref{first_order_taylor}) we obtain the desired result (\ref{1st_order_taylor}), that is,
\begin{align*}
| f(\bm y) - f(\bm x) - \mathcal{G}(\bm x)^\top \bm \delta | \leq C_0 \| \bm \delta \|^2.
\end{align*}
where $C_0 = \gamma_{f, 1}\{1 + \frac{\Gamma_1}{\sigma_0^2}\}\frac{4}{R^2} + 4(L_{f, 1} + \frac{\gamma_{f, 1}\sqrt{\Gamma_1\Lambda_1}}{\sigma_0^2} )$.
To prove (\ref{key_taylor}), for every ${\bm x, \bm y} \in {\Omega}$, by the definition of the Lagrangian, we have $f({\bm x}) = \mathcal{L}({\bm x}, \boldsymbol\lambda^\star({\bm x}))$ and $f(\bm{y}) = \mathcal{L}(\bm{y}, \boldsymbol\lambda^\star({\bm x}))$. Let $\bm \eta = {\bm y} - {\bm x}$. Then by Taylor's theorem, there exists a $t \in (0, 1)$ such that
\begin{align}
f({\bm y}) = \mathcal{L}({\bm y}, \boldsymbol\lambda^\star({\bm x})) &= \mathcal{L}({\bm x}, \boldsymbol\lambda^\star({\bm x})) + \nabla_{\bm x}\mathcal{L}({\bm x}, \boldsymbol\lambda^\star({\bm x}))^\top{\bm \eta} + \frac{1}{2}{\bm \eta}^\top \nabla_{\mathbf{xx}}^2\mathcal{L}({\bm x} + t{\bm \eta}, \boldsymbol\lambda^\star({\bm x})){\bm \eta} \nonumber\\
&= f({\bm x}) + \mathcal{G}({\bm x})^\top{\bm \eta} + \frac{1}{2}{\bm \eta}^\top \nabla^2_{\mathbf{xx}}\mathcal{L}({\bm x}, \boldsymbol\lambda^\star({\bm x})){\bm \eta} \nonumber\\
&+ \frac{1}{2}{\bm \eta}^\top (\nabla^2_{\mathbf{xx}}\mathcal{L}({\bm x} + t{\bm \eta}, \boldsymbol\lambda^\star({{\bm x}})) - \nabla^2_{\mathbf{xx}}\mathcal{L}({\bm x}, \boldsymbol\lambda^\star({\bm x}))){\bm \eta} \label{feasible_expansion_estimate}.
\end{align}
Then the last term in $(\ref{feasible_expansion_estimate})$ can be bounded as follows
\begin{align*}
&\quad\quad \vert \frac{1}{2}{\bm \eta}^\top \{\nabla^2_{\bm x \bm x}\mathcal{L}({\bm x} + t{\bm \eta}, \boldsymbol\lambda^\star({\bm x})) -
\nabla^2_{\bm x \bm x}\mathcal{L}({\bm x}, \boldsymbol\lambda^\star({\bm x}))\}{\bm \eta} \vert \\
&\leq \frac{1}{2}\|{\bm \eta}\|^2 \|\nabla^2_{\bm x \bm x}\mathcal{L}({\bm x} + t{\bm \eta}, \boldsymbol\lambda^\star({\bm x})) - \nabla^2_{\bm x \bm x}\mathcal{L}({\bm x}, \boldsymbol\lambda^\star({\bm x}))\|\\
&=\frac{1}{2}\|{\bm \eta}\|^2 \|(\nabla^2f({\bm x} + t{\bm \eta}) - \nabla^2f({\bm x})) + \sum_{i = 1}^m\lambda^\star_i({\bm x}) (\nabla^2c_i({\bm x} + t{\bm \eta}) - \nabla^2c_i({\bm x}))\| \\
&\leq \frac{1}{2}\|{\bm \eta}\|^2 \big( \|\nabla^2f({\bm x} + t{\bm \eta}) - \nabla^2f({\bm x})\| + \sum_{i=1}^m\vert\lambda^\star_i({{\bm x}})\vert \ \|\nabla^2c_i({\bm x} + t{\bm \eta}) - \nabla^2c_i({\bm x})\| \big) \\
&\leq \frac{1}{2}\|{\bm \eta}\|^3 L_{f, 2}t + \frac{1}{2}\|{\bm \eta}\|^2\|\boldsymbol\lambda^\star({\bm x})\|\sqrt{\sum_{i=1}^m\|\nabla^2 c_i({\bm x} + t{\bm \eta}) - \nabla^2c_i({\bm x})\|^2} \tag{Cauchy-Schwartz inequailty}\\
&\leq \frac{1}{2}\|{\bm \eta}\|^3 L_{f, 2} + \frac{\gamma_{f,1} \sqrt{\Gamma_1}}{2\sigma_0^2}\sqrt{\sum_{i=1}^m L_{c_i, 2}^2t^2}\cdot \|{\bm \eta}\|^3 \tag{by (\ref{lambda_bound})}\\
&\leq \big\{\frac{L_{f, 2}}{2} + \frac{\gamma_{f, 1}}{2\sigma_0^2}\sqrt{\Gamma_1\Lambda_2}\big\}\|\bm\eta \|^3\\
&= C_1\|{\bm \eta}\|^3,
\end{align*}
where $C_1 = \big\{\frac{L_{f, 2}}{2} + \frac{\gamma_{f, 1}}{2\sigma_0^2}\sqrt{\Gamma_1\Lambda_2}\big\}$
Combining the above inequality with (\ref{feasible_expansion_estimate}), we obtain
\begin{align}\label{eqn:standard_taylor}
\big \vert f({\bm y}) - f({\bm x}) - \mathcal{G}({\bm x})^\top \bm\eta - \frac{1}{2}{\bm \eta}^\top\{ \nabla^2_{\bm x \bm x}\mathcal{L}({\bm x}, \boldsymbol\lambda^\star({\bm x}))\}{\bm \eta}\big \vert \leq C_1\|{\bm \eta}\|^3.
\end{align}
We consider two case: (i) $\bm{x}_0 + \boldsymbol\delta \in \Omega$, and (ii) $\bm{x}_0 + \boldsymbol\delta \not\in \Omega$.\\
Case (i): Substituting $\bm y = \bm x_0 + \bm \delta = \Pi_{\Omega}(\bm x_0 + \bm \delta)$ and $\bm x = \bm x_0$ into \eqref{eqn:standard_taylor}, and noting that $\boldsymbol\eta = \bm y - \bm x = \boldsymbol\delta$, we have from (\ref{eqn:standard_taylor}) and the facts that $\mathcal{H}(\bm{x}_0) = \mathcal{P}_{\bm{x}_0}^\top \nabla_{\bm x \bm x}^2 \mathcal{L}(\bm x_0, \bm \lambda^\star(\bm x_0)) \mathcal{P}_{\bm{x}_0}$ and $\bm \delta \in \mathcal{T}_\Omega(\bm x_0)$, hence that $\mathcal{P}_{\bm x_0}\bm\delta = \bm\delta$.
\begin{align*}
&\quad \big \vert f(\Pi_{\Omega}(\bm x_0 + \bm \delta)) - f(\bm x_0) - \mathcal{G}({\bm x_0})^\top \bm\delta - \frac{1}{2}{\bm \delta}^\top\mathcal{H}(\bm x_0){\bm \delta}\big \vert \leq C_1\|{\bm \delta}\|^3.
\end{align*}
Case (ii): ${\bm x}_0 + {\bm \delta} \not\in \Omega$. Letting ${\bm y_0} = \Pi_\Omega({\bm x}_0 +{\bm \delta})$, we have from (\ref{diff_bound}) and then from (\ref{eqn:standard_taylor}) that
\begin{flalign}\label{eqn:inequality_1}
&\quad\big\vert f({\bm y_0}) - f({\bm x}_0) - \mathcal{G}({\bm x}_0)^\top ({\bm y_0} - {\bm x}_0) - \frac{1}{2}({\bm y_0} - {\bm x}_0)^\top \{ \nabla^2_{\bm x \bm x}\mathcal{L}({\bm x_0}, \boldsymbol\lambda^\star({\bm x_0}))\}({\bm y_0} - {\bm x}_0) \big\vert \leq 8 C_1 \norm{\bm \delta}{}^3
\end{flalign}
Let $\bm \xi = (\bm x_0 + \bm \delta) - \bm y_0 = (\bm x_0 + \bm \delta) - \Pi_\Omega({\bm x}_0 +{\bm \delta})$. Then clearly $\bm y_0 - \bm x_0 = \bm \delta - \bm \xi$, and \eqref{eqn:inequality_1} can be rewritten as
\begin{flalign} \label{expansion}
\big\vert f({\bm y_0}) - f({\bm x}_0) - \mathcal{G}({\bm x}_0)^\top (\bm \delta - \bm \xi) - \frac{1}{2}(\bm \delta - \bm \xi)^\top \{ \nabla^2_{\bm x \bm x}\mathcal{L}({\bm x_0}, \boldsymbol\lambda^\star({\bm x_0}))\}(\bm \delta - \bm \xi) \big\vert &\le 8 C_1 \norm{\bm \delta}{}^3.
\end{flalign}
We further note that
\begin{align}
&\quad f({\bm y_0}) - f({\bm x}_0) - \mathcal{G}({\bm x}_0)^\top (\bm \delta - \bm \xi) - \frac{1}{2}(\bm \delta - \bm \xi)^\top \{ \nabla^2_{\bm x \bm x}\mathcal{L}({\bm x_0}, \boldsymbol\lambda^\star({\bm x_0}))\}(\bm \delta - \bm \xi)\\
&= f({\bm y_0}) - f({\bm x}_0) - \mathcal{G}(\bm x_0)^\top\bm \delta - \frac{1}{2}\bm \delta^\top \{ \nabla^2_{\bm x \bm x}\mathcal{L}({\bm x_0}, \boldsymbol\lambda^\star({\bm x_0}))\}\bm \delta \nonumber \\
&+ \mathcal{G}(\bm x_0)^\top\bm \xi + \bm \delta^\top \{ \nabla^2_{\bm x \bm x}\mathcal{L}({\bm x_0}, \boldsymbol\lambda^\star({\bm x_0}))\}\bm \xi - \frac{1}{2}\bm \xi^\top \{ \nabla^2_{\bm x \bm x}\mathcal{L}({\bm x_0}, \boldsymbol\lambda^\star({\bm x_0}))\}\bm \xi \label{eqn:last_three_terms}
\end{align}
Next, we will show that the last three terms in \eqref{eqn:last_three_terms} satisfy
\begin{flalign}
\mathcal{G}(\bm x_0)^\top\bm \xi + \bm \delta^\top \{ \nabla^2_{\bm x \bm x}\mathcal{L}({\bm x_0}, \boldsymbol\lambda^\star({\bm x_0}))\}\bm \xi - \frac{1}{2}\bm \xi^\top \{ \nabla^2_{\bm x \bm x}\mathcal{L}({\bm x_0}, \boldsymbol\lambda^\star({\bm x_0}))\}\bm \xi = O(\norm{\bm \delta}{}^3).
\end{flalign}
The following are helpful in establishing this. First, from the definition of $\bm \xi = (\bm x_0 + \bm \delta) - \Pi_\Omega({\bm x}_0 +{\bm \delta})$ and Proposition 1, one has $\|{\bm \xi}\| \leq 4\|{\bm \delta}\|^2/R$ as well as the bound $\|\bm\xi\| \leq \|\bm \delta\|$. Second, since $\mathcal{G}({\bm x}_0)\in \mathcal{T}_\Omega(\bm x_0)$, we have $\mathcal{G}({\bm x}_0)^\top{\bm \xi} = \mathcal{G}({\bm x}_0)^\top\mathcal{P}_{{\bm x}_0}{\bm \xi}$. Third, since $\bm y_0 = \Pi_\Omega(\bm x_0 + \bm \delta)$, $\mathcal{P}_{\bm y_0}\{(\bm x_0 + \bm\delta) - \Pi_\Omega(\bm x_0 + \bm \delta)\} = \bm 0$. Using these observations and the fact that $\|{\bm y_0} - {\bm x}_0\| \leq 2\|{\bm \delta}\|$, we are ready to bound these three terms one by one,
\begin{align}
\vert {\bm \delta}^\top \{ \nabla^2_{\bm x \bm x}\mathcal{L}({\bm x_0}, \boldsymbol\lambda^\star({\bm x_0}))\}{\bm \xi} \vert &\leq \{ \| \nabla^2f({\bm x}_0)\| + \sum_{i=1}^m \vert \lambda_i^*({\bm x}_0)\vert \ \|\nabla^2 c_i({\bm x}_0)\|\} \|{\bm \delta}\|\|{\bm \xi}\| \nonumber \\
&\leq \Big\{\|\nabla^2f({\bm x}_0)\| + \|\boldsymbol\lambda^*({\bm x}_0)\|\sqrt{\sum_{i=1}^m \|\nabla^2 c_i({\bm x}_0)\|^2}\Big\} \|{\bm \delta}\|\|{\bm \xi}\| \tag{by Cauchy-Schwartz} \nonumber\\
&\leq \{\gamma_{f, 2} + \frac{\gamma_{f, 1}}{2\sigma_0^2}\sqrt{\Gamma_1\Gamma_2}\}\frac{4}{R}\|{\bm \delta}\|^3 \tag{by (\ref{lambda_bound}) and $\norm{\bm\xi}{}\leq 4\norm{\bm\delta}{}^2/R$} \nonumber\\
&= C_2 \|{\bm \delta}\|^3, \label{part_1}
\end{align}
where $C_2 = \{\gamma_{f, 2} + \frac{\gamma_{f, 1}}{2\sigma_0^2}\sqrt{\Gamma_1\Gamma_2}\}\frac{4}{R}$.
Applying exactly the same method to bound $\frac{1}{2}{\bm \xi}^\top\{ \nabla^2_{\bm x \bm x}\mathcal{L}({\bm x_0}, \boldsymbol\lambda^\star({\bm x_0}))\}{\bm \xi}$ we have
\begin{align}
\frac{1}{2}\vert {\bm \xi}^\top\{ \nabla^2_{\bm x \bm x}\mathcal{L}({\bm x_0}, \boldsymbol\lambda^\star({\bm x_0}))\}{\bm \xi}\vert &\leq \frac{1}{2}\{ \| \nabla^2f({\bm x}_0)\| + \sum_{i=1}^m \vert \lambda_i^*({\bm x}_0)\vert \ \|\nabla^2 c_i({\bm x}_0)\|\}\|{\bm \xi}\|^2 \nonumber\\
&\leq \{\gamma_{f, 2} + \frac{\gamma_{f, 1}}{2\sigma_0^2}\sqrt{\Gamma_1\Gamma_2}\}\frac{1}{2}\|{\bm \xi}\|^2 \tag{by (\ref{lambda_bound})}\\
&\leq \{\gamma_{f, 2} + \frac{\gamma_{f, 1}}{2\sigma_0^2}\sqrt{\Gamma_1\Gamma_2}\}\frac{2}{R}\|{\bm \delta}\|^3 \tag{ by $\norm{\bm\xi}{}\leq 4\norm{\bm\delta}{}^2/R$ and $\|\bm\xi\| \leq \|\bm\delta\|$} \nonumber \\
&= C_3 \|{\bm \delta}\|^3, \label{part_2}
\end{align}
where $C_3 = \{\gamma_{f, 2} + \frac{\gamma_{f, 1}}{2\sigma_0^2}\sqrt{\Gamma_1\Gamma_2}\}\frac{2}{R}$.
To bound the last term $\mathcal{G}(\bm{x}_0)^\top\bm\xi$, we first note that $\mathcal{G}(\bm{x}_0)^\top\bm\xi = \mathcal{G}(\bm{x}_0)^\top\mathcal{P}_{\bm{x}_0}\bm\xi$ since $\mathcal{G}(\bm{x}_0) \in \mathcal{T}_{\Omega}(\bm{x}_0)$ and $\mathcal{P}_{\bm{y}_0}\bm\xi = \bm 0$ since $\bm\xi = \bm{y}_0 - \Pi_\Omega \bm{y}_0$. Therefore
\begin{align}
\mathcal{G}(\bm{x}_0)^\top \bm\xi = \mathcal{G}(\bm{x}_0)^\top \mathcal{P}_{\bm{x}_0}\bm\xi = \mathcal{G}(\bm{x}_0)^\top\{\mathcal{P}_{\bm{x}_0}\bm\xi - \mathcal{P}_{\bm{y}_0}\bm\xi\} \leq \norm{\mathcal{G}(\bm{x}_0)}{}\norm{\mathcal{P}_{\bm{x}_0}\bm\xi - \mathcal{P}_{\bm{y}_0}\bm\xi}{}. \label{tough_bound}
\end{align}
We first derive a bound for $\norm{\mathcal{P}_{\bm{x}_0}\bm\xi - \mathcal{P}_{\bm{y}_0}\bm\xi}{}$. Note that $\mathcal{P}_{\bm{x}} = \nabla\bm{c}({\bm x})\{\nabla\bm{c}({\bm x})^\top \nabla\bm{c}({\bm x})\}^{-1}\nabla\bm{c}({\bm x})^\top$; thus
\begin{align}
\norm{\mathcal{P}_{\bm{x}_0}\bm\xi - \mathcal{P}_{\bm{y}_0}\bm\xi}{} &\leq \norm{\mathcal{P}_{\bm{x}_0} - \mathcal{P}_{\bm{y}_0}}{}\norm{\bm\xi}{} \nonumber \\
&= \norm{\bm\xi}{}\norm{\nabla\bm{c}({\bm x_0})\{\nabla\bm{c}({\bm x_0})^\top \nabla\bm{c}({\bm x_0})\}^{-1}\nabla\bm{c}({\bm x_0})^\top -\nabla\bm{c}({\bm y_0})\{\nabla\bm{c}({\bm y_0})^\top \nabla\bm{c}({\bm y_0})\}^{-1}\nabla\bm{c}({\bm y_0})^\top }{}. \nonumber \\
&= \norm{\bm\xi}{} \|[\nabla\bm{c}({\bm x_0})\{\nabla\bm{c}({\bm x_0})^\top \nabla\bm{c}({\bm x_0})\}^{-1}\nabla\bm{c}({\bm x_0})^\top - \nabla\bm{c}({\bm x_0})\{\nabla\bm{c}({\bm x_0})^\top \nabla\bm{c}({\bm x_0})\}^{-1}\nabla\bm{c}({\bm y_0})^\top] \nonumber \\
&\quad + [\nabla\bm{c}({\bm x_0})\{\nabla\bm{c}({\bm x_0})^\top \nabla\bm{c}({\bm x_0})\}^{-1}\nabla\bm{c}({\bm y_0})^\top - \nabla\bm{c}({\bm y_0})\{\nabla\bm{c}({\bm y_0})^\top \nabla\bm{c}({\bm y_0})\}^{-1}\nabla\bm{c}({\bm y_0})^\top] \| \tag{adding and subtracting a term} \nonumber \\
&\leq \norm{\bm\xi}{} \Big\{ \|\nabla\bm{c}({\bm x_0})\{\nabla\bm{c}({\bm x_0})^\top \nabla\bm{c}({\bm x_0})\}^{-1}\nabla\bm{c}({\bm x_0})^\top - \nabla\bm{c}({\bm x_0})\{\nabla\bm{c}({\bm x_0})^\top \nabla\bm{c}({\bm x_0})\}^{-1}\nabla\bm{c}({\bm y_0})^\top \| \nonumber \\
&\quad +\| \nabla\bm{c}({\bm x_0})\{\nabla\bm{c}({\bm x_0})^\top \nabla\bm{c}({\bm x_0})\}^{-1}\nabla\bm{c}({\bm y_0})^\top - \nabla\bm{c}({\bm y_0})\{\nabla\bm{c}({\bm y_0})^\top \nabla\bm{c}({\bm y_0})\}^{-1}\nabla\bm{c}({\bm y_0})^\top \| \Big\} \tag{triangle inequality} \\
&\leq \norm{\bm\xi}{} \Big\{ \|\nabla\bm{c}({\bm x_0})\{\nabla\bm{c}({\bm x_0})^\top \nabla\bm{c}({\bm x_0})\}^{-1} \| \| \nabla\bm{c}({\bm x_0}) - \nabla\bm{c}({\bm y_0})\| \label{norm_bound_1}\\
&\quad + \| \nabla\bm{c}({\bm x_0})\{\nabla\bm{c}({\bm x_0})^\top \nabla\bm{c}({\bm x_0})\}^{-1} - \nabla\bm{c}({\bm y_0})\{\nabla\bm{c}({\bm y_0})^\top \nabla\bm{c}({\bm y_0})\}^{-1} \| \| \nabla\bm{c}({\bm y_0}) \| \Big\} \label{norm_bound_2}.
\end{align}
Since
\begin{align}
\norm{\nabla \bm c(\bm x_0) - \nabla \bm c(\bm y_0)}{} \leq \norm{\nabla \bm c(\bm x_0) - \nabla \bm c(\bm y_0)}{F} &= \sqrt{\sum_{i=1}^m \vert \nabla c_i(\bm x_0) - \nabla c_i(\bm y_0) \vert^2} \nonumber \\
&\leq \sqrt{\sum_{i=1}^m L_{c_i, 1}^2} \norm{\bm x_0 - \bm y_0}{} = \sqrt\Lambda_1\norm{\bm x_0 - \bm y_0}{}, \label{difference_bound}
\end{align}
and
\begin{align}
\norm{\nabla \bm c(\bm y_0)}{} \leq \norm{\nabla \bm c(\bm y_0)}{F} = \sqrt{\sum_{i=1}^m \|\nabla c_i(\bm y_0) \|^2} \leq \sqrt{\sum_{i=1}^m \gamma_{c_i, 1}^2} = \sqrt{\Gamma_1}, \label{single_bound}
\end{align}
Moreover, by
\begin{align*}
\|\nabla\bm{c}({\bm x_0})\{\nabla\bm{c}({\bm x_0})^\top \nabla\bm{c}({\bm x_0})\}^{-1} \| &\leq \| \nabla\bm{c}({\bm x_0}) \| \|\{\nabla\bm{c}({\bm x_0})^\top \nabla\bm{c}({\bm x_0})\}^{-1} \| \\
&\leq \frac{\sqrt{\Gamma_1}}{\sigma_0^2}. \tag{by (\ref{single_bound}) and $\sigma_0$-LICQ condition}
\end{align*}
Therefore $\|\nabla\bm{c}({\bm x_0})\{\nabla\bm{c}({\bm x_0})^\top \nabla\bm{c}({\bm x_0})\}^{-1} \|\|\nabla \bm c(\bm x_0) - \nabla \bm c(\bm y_0)\| $ in (\ref{norm_bound_1}) can be bounded by
\begin{align}
\|\nabla\bm{c}({\bm x_0})\{\nabla\bm{c}({\bm x_0})^\top \nabla\bm{c}({\bm x_0})\}^{-1} \|\|\nabla \bm c(\bm x_0) - \nabla \bm c(\bm y_0)\| \leq \frac{\sqrt{\Gamma_1\Lambda_1}}{\sigma_0^2}\| \bm x_0 - \bm y_0 \|. \label{result_part_1}
\end{align}
We need to further simplify $\| \nabla\bm{c}({\bm x_0})\{\nabla\bm{c}({\bm x_0})^\top \nabla\bm{c}({\bm x_0})\}^{-1} - \nabla\bm{c}({\bm y_0})\{\nabla\bm{c}({\bm y_0})^\top \nabla\bm{c}({\bm y_0})\}^{-1} \|$ in order to obtain an upper bound for (\ref{norm_bound_1}).
\begin{align}
&\quad \| \nabla\bm{c}({\bm x_0})\{\nabla\bm{c}({\bm x_0})^\top \nabla\bm{c}({\bm x_0})\}^{-1} - \nabla\bm{c}({\bm y_0})\{\nabla\bm{c}({\bm y_0})^\top \nabla\bm{c}({\bm y_0})\}^{-1} \| \nonumber \\
&= \| [\nabla\bm{c}({\bm x_0})\{\nabla\bm{c}({\bm x_0})^\top \nabla\bm{c}({\bm x_0})\}^{-1} - \nabla\bm{c}({\bm y_0})\{\nabla\bm{c}({\bm x_0})^\top \nabla\bm{c}({\bm x_0})\}^{-1}] \nonumber \\
&\quad + [\nabla\bm{c}({\bm y_0})\{\nabla\bm{c}({\bm x_0})^\top \nabla\bm{c}({\bm x_0})\}^{-1} - \nabla\bm{c}({\bm y_0})\{\nabla\bm{c}({\bm y_0})^\top \nabla\bm{c}({\bm y_0})\}^{-1}] \| \nonumber\tag{add and subtract terms} \\
&\leq \| \nabla\bm{c}({\bm x_0})\{\nabla\bm{c}({\bm x_0})^\top \nabla\bm{c}({\bm x_0})\}^{-1} - \nabla\bm{c}({\bm y_0})\{\nabla\bm{c}({\bm x_0})^\top \nabla\bm{c}({\bm x_0})\}^{-1} \| \nonumber \\
&\quad + \| \nabla\bm{c}({\bm y_0})\{\nabla\bm{c}({\bm x_0})^\top \nabla\bm{c}({\bm x_0})\}^{-1} - \nabla\bm{c}({\bm y_0})\{\nabla\bm{c}({\bm y_0})^\top \nabla\bm{c}({\bm y_0})\}^{-1}] \| \nonumber \tag{triangle inequality} \\
&\leq \| \nabla \bm c(\bm x_0) - \nabla \bm c(\bm y_0) \| \{\nabla\bm{c}({\bm x_0})^\top \nabla\bm{c}({\bm x_0})\}^{-1} \| \label{sub_bound_1}\\
&\quad + \| \nabla \bm c(\bm y_0) \| \|\{\nabla\bm{c}({\bm x_0})^\top \nabla\bm{c}({\bm x_0})\}^{-1} - \{\nabla\bm{c}({\bm y_0})^\top \nabla\bm{c}({\bm y_0})\}^{-1} \| \label{sub_bound_2}.
\end{align}
An upper bound of (\ref{sub_bound_1}) can be obtained by combining (\ref{difference_bound}) and the $\sigma_0$-LICQ condition, that is,
\begin{align}
\| \nabla \bm c(\bm x_0) - \nabla \bm c(\bm y_0) \| \{\nabla\bm{c}({\bm x_0})^\top \nabla\bm{c}({\bm x_0})\}^{-1} \| \leq \frac{\sqrt{\Lambda_1}}{\sigma^2}\|\bm x_0 - \bm y_0 \|. \label{bound_for_sub_bound_1}
\end{align}
To upper bound $ \|\{\nabla\bm{c}({\bm x_0})^\top \nabla\bm{c}({\bm x_0})\}^{-1} - \{\nabla\bm{c}({\bm y_0})^\top \nabla\bm{c}({\bm y_0})\}^{-1} \|$ in (\ref{sub_bound_2}), we need to utilize the fact that for any invertible matrices $A, B$, $\norm{A^{-1} - B^{-1}}{} \leq \norm{A^{-1}}{}\norm{A - B}{}\norm{B^{-1}}{}$ and the $\sigma_0$-LICQ condition. More specifically,
\begin{align}
&\quad \|\{\nabla\bm{c}({\bm x_0})^\top \nabla\bm{c}({\bm x_0})\}^{-1} - \{\nabla\bm{c}({\bm y_0})^\top \nabla\bm{c}({\bm y_0})\}^{-1} \| \nonumber \\
&\leq \|\{\nabla\bm{c}({\bm x_0})^\top \nabla\bm{c}({\bm x_0})\}^{-1} \| \| \{\nabla\bm{c}({\bm y_0})^\top \nabla\bm{c}({\bm y_0})\}^{-1} \| \|\nabla\bm{c}({\bm x_0})^\top \nabla\bm{c}({\bm x_0}) - \nabla\bm{c}({\bm y_0})^\top \nabla\bm{c}({\bm y_0}) \|\nonumber \\
&\leq \frac{1}{\sigma_0^4}\|\nabla\bm{c}({\bm x_0})^\top \nabla\bm{c}({\bm x_0}) - \nabla\bm{c}({\bm y_0})^\top \nabla\bm{c}({\bm y_0}) \| \tag{$\sigma_0$-LICQ condition} \nonumber \\
&= \frac{1}{\sigma_0^4} \|\nabla\bm{c}({\bm x_0})^\top \nabla\bm{c}({\bm x_0}) - \nabla\bm{c}({\bm x_0})^\top \nabla\bm{c}({\bm y_0}) + \nabla\bm{c}({\bm x_0})^\top \nabla\bm{c}({\bm y_0}) - \nabla\bm{c}({\bm y_0})^\top \nabla\bm{c}({\bm y_0}) \| \nonumber \\
&\leq \frac{1}{\sigma_0^4}\| \nabla \bm c(\bm x_0) - \nabla \bm c(\bm y_0) \|\big\{\| \nabla \bm c(\bm x_0) \| + \| \nabla \bm c(\bm{y}_0) \| \} \nonumber \\
&\leq \frac{2\sqrt{\Gamma_1\Lambda_1}}{\sigma_0^4}\|\bm x_0 - \bm y_0 \|. \label{inverse_bound}
\end{align}
Therefore we can bound $\| \nabla\bm{c}({\bm x_0})\{\nabla\bm{c}({\bm x_0})^\top \nabla\bm{c}({\bm x_0})\}^{-1} - \nabla\bm{c}({\bm y_0})\{\nabla\bm{c}({\bm y_0})^\top \nabla\bm{c}({\bm y_0})\}^{-1} \|$ in (\ref{norm_bound_2}) by
\begin{align}
&\quad \| \nabla\bm{c}({\bm x_0})\{\nabla\bm{c}({\bm x_0})^\top \nabla\bm{c}({\bm x_0})\}^{-1} - \nabla\bm{c}({\bm y_0})\{\nabla\bm{c}({\bm y_0})^\top \nabla\bm{c}({\bm y_0})\}^{-1} \| \nonumber \\
&\leq \| \nabla \bm c(\bm x_0) - \nabla \bm c(\bm y_0) \| \{\nabla\bm{c}({\bm x_0})^\top \nabla\bm{c}({\bm x_0})\}^{-1} \| \nonumber \\
&\quad + \| \nabla \bm c(\bm y_0) \| \|\{\nabla\bm{c}({\bm x_0})^\top \nabla\bm{c}({\bm x_0})\}^{-1} - \{\nabla\bm{c}({\bm y_0})^\top \nabla\bm{c}({\bm y_0})\}^{-1} \| \nonumber \\
&\leq(\frac{\sqrt{\Lambda_1}}{\sigma^2_0} + \frac{2\Gamma_1\sqrt{\Lambda_1}}{\sigma_0^4}) \|\bm x_0 - \bm y_0 \| \label{bound_for_norm_bound_2}.
\end{align}
With the upper bounds for (\ref{norm_bound_1}) and (\ref{norm_bound_2}), we bound $\norm{\mathcal{P}_{\bm x_0}\bm\xi - \mathcal{P}_{\bm y_0}\bm\xi}{}$ by
\begin{align*}
&\quad \norm{\mathcal{P}_{\bm x_0}\bm\xi - \mathcal{P}_{\bm y_0}\bm\xi}{} \\
&\leq \norm{\bm\xi}{} \Big\{ \|\nabla\bm{c}({\bm x_0})\{\nabla\bm{c}({\bm x_0})^\top \nabla\bm{c}({\bm x_0})\}^{-1} \| \| \nabla\bm{c}({\bm x_0}) - \nabla\bm{c}({\bm y_0})\| \\
&\quad + \| \nabla\bm{c}({\bm x_0})\{\nabla\bm{c}({\bm x_0})^\top \nabla\bm{c}({\bm x_0})\}^{-1} - \nabla\bm{c}({\bm y_0})\{\nabla\bm{c}({\bm y_0})^\top \nabla\bm{c}({\bm y_0})\}^{-1} \| \| \nabla\bm{c}({\bm y_0}) \| \Big\} \\
&\leq \{\frac{\sqrt{\Gamma_1\Lambda_1}}{\sigma_0^2}\| \bm x_0 - \bm y_0 \| + \sqrt{\Gamma_1}(\frac{\sqrt{\Lambda_1}}{\sigma^2_0} + \frac{2\Gamma_1\sqrt{\Lambda_1}}{\sigma_0^4}) \|\bm x_0 - \bm y_0 \|\} \|\bm\xi\| \tag{by (\ref{result_part_1}), (\ref{single_bound}), and (\ref{bound_for_norm_bound_2})} \\
&= (\frac{2\sqrt{\Gamma_1\Lambda_1}}{\sigma_0^2} + \frac{2\sqrt{\Gamma_1^3\Lambda_1}}{\sigma_0^4})\|\bm x_0 - \bm y_0 \|\|\bm\xi\| \\
&\leq (\frac{2\sqrt{\Gamma_1\Lambda_1}}{\sigma_0^2} + \frac{2\sqrt{\Gamma_1^3\Lambda_1}}{\sigma_0^4}) \frac{8\|\bm\delta\|^3}{R}. \tag{$\|\bm x_0 - \bm y_0\|\leq 2 \| \bm \delta \|$ and $\| \bm \xi \| \leq 4\|\bm \delta \|^2 / R$}
\end{align*}
Let us go back to the task of bounding $\mathcal{G}(\bm x_0)^\top \bm\xi$. By (\ref{tough_bound}) and the inequality above, we have
\begin{align}
\mathcal{G}(\bm x_0)^\top \bm\xi &\leq \norm{\mathcal{G}(\bm{x}_0)}{}\norm{\mathcal{P}_{\bm{x}_0}\bm\xi - \mathcal{P}_{\bm{y}_0}\bm\xi}{} \nonumber\\
&\leq \norm{\mathcal{G}(\bm{x}_0)}{}(\frac{2\sqrt{\Gamma_1\Lambda_1}}{\sigma_0^2} + \frac{2\sqrt{\Gamma_1^3\Lambda_1}}{\sigma_0^4}) \frac{8\|\bm\delta\|^3}{R} \nonumber \\
&\leq \Big\{\|\nabla f({\bm x}_0)\| + \|\boldsymbol\lambda^*(\bm x_0)\|\sqrt{\sum_{i=1}^m\|\nabla c_i({\bm x}_0)\|^2}\Big\}(\frac{2\sqrt{\Gamma_1\Lambda_1}}{\sigma_0^2} + \frac{2\sqrt{\Gamma_1^3\Lambda_1}}{\sigma_0^4}) \frac{8\|\bm\delta\|^3}{R} \tag{by the definition of $\mathcal{G}(\bm x_0)$ and triangle inequality} \nonumber \\
&\leq \{\gamma_{f, 1} + \frac{\gamma_{f, 1}}{\sigma_0^2}\Gamma_1\}(\frac{2\sqrt{\Gamma_1\Lambda_1}}{\sigma_0^2} + \frac{2\sqrt{\Gamma_1^3\Lambda_1}}{\sigma_0^4}) \frac{8\|\bm\delta\|^3}{R}. \tag{by (\ref{lambda_bound}) and Assumption 2} \nonumber \\
&= C_4 \|\bm\delta\|^3, \label{part_3}
\end{align}
where $C_4 = \{\gamma_{f, 1} + \frac{\gamma_{f, 1}}{\sigma_0^2}\Gamma_1\}(\frac{2\sqrt{\Gamma_1\Lambda_1}}{\sigma_0^2} + \frac{2\sqrt{\Gamma_1^3\Lambda_1}}{\sigma_0^4}) \frac{8}{R}$.
Finally, we can bound the last three terms in \eqref{eqn:last_three_terms} by combining (\ref{part_1}), (\ref{part_2}, and (\ref{part_3}), that is,
\begin{align}
\vert \mathcal{G}(\bm x_0)^\top\bm \xi + \bm \delta^\top \{ \nabla^2_{\bm x \bm x}\mathcal{L}({\bm x_0}, \boldsymbol\lambda^\star({\bm x_0}))\}\bm \xi - \frac{1}{2}\bm \xi^\top \{ \nabla^2_{\bm x \bm x}\mathcal{L}({\bm x_0}, \boldsymbol\lambda^\star({\bm x_0}))\}\bm \xi \vert \leq (C_2 + C_3 + C_4)\| \bm\delta \|^3. \label{bound_last_three_terms}
\end{align}
Plugging (\ref{bound_last_three_terms}) into (\ref{expansion}) yields the final result, we obtain,
\begin{align*}
&\quad \vert f(\bm y_0) - f(\bm x_0) - \mathcal{G}(\bm x_0)^\top\bm\delta-\frac{1}{2}\bm\delta^\top\{\nabla^2_{\bm{xx}}\mathcal{L}(\bm x_0, \bm\lambda^*(\bm x_0))\}\bm\delta \vert \\
&=\vert f(\bm y_0) - f(\bm x_0) - \mathcal{G}(\bm x_0)^\top\bm\delta-\frac{1}{2}\bm\delta^\top\mathcal{H}(\bm x_0)\bm\delta \vert \tag{by $\mathcal{P}_{\bm x_0}\bm\delta = \bm\delta$ and definition of $\mathcal{H}(\bm{x}_0)$} \\
&\leq (8C_1 + C_2 + C_3 + C_4)\| \bm\delta \|^3.
\end{align*}
\end{proof}
\subsection{Riemannian gradient and Riemannian Hessian}\label{sub:relations}
Recall that $\Omega = \{ {\bm x} \in \mr{R}^n \; \vert \; c_i({\bm x}) = 0, i= 1, \ldots, m\}$, $\nabla\bm{c}({\bm x}) = [\nabla c_1({\bm x}), \cdots, \nabla c_m({\bm x})]$ and $\sigma_{\min}({\nabla\bm c}({\bm x}))$ the minimum singular value of ${\nabla \bm c}({\bm x})$. Assume that $\inf\{\sigma_{\min}({\nabla \bm c}({\bm x})) \ \vert \ {\bm x} \in \Omega\} > \alpha$ for some $\alpha > 0$. For second-order differentiable functions $f$ and $c_i(\bm{x}), i = 1, \ldots, m$, let $\mathcal{L}({\bm x},\boldsymbol\lambda) = f({\bm x}) - \sum_{i=1}^m \lambda_i c_i({\bm x})$ and $\boldsymbol \lambda^\star({\bm x}) = \arg\!\min_{\boldsymbol\lambda}\|\nabla_{\bm x} \mathcal{L}({\bm x}, \boldsymbol\lambda)\|$. Define $\mathcal{G}({\bm x}) = \nabla_{\bm x} \mathcal{L}({\bm x}, \boldsymbol\lambda^\star)$ and $\mathcal{H}({\bm x}) = \nabla^2_{{\bm x\bm x}} \mathcal{L}({\bm x}, \boldsymbol\lambda^\star)$. Note that $\Omega$ can be considered as a $n-m$ dimensional Riemannian sub-manifold of $\mr{R}^n$ under the LICQ assumption. We will show that $\mathcal{G}(\cdot)$ and $\mathcal{H}(\cdot)$ are Riemannian gradient and Riemannian Hessian of $f$ over $\Omega$ in the following lemma. A similar argument can be found in \cite{absil2013extrinsic} and \cite{absil2009all}
\begin{lemma}
$\mathcal{G}(\cdot)$ and $\mathcal{H}(\cdot)$ defined in \ref{eqn:G} and \ref{eqn:H} are the Riemannian gradient and Riemannian Hessian of $f(\cdot)$ over $\Omega$.
\end{lemma}
\begin{proof}
Let $\text{grad} f({\bm x})$ denote the Riemannian gradient of $f$ at ${\bm x} \in \Omega$. By definition, for every $\eta \in \mathcal{T}_\Omega({{\bm x}})$, $Df({\bm x})[\eta] = \langle \text{grad} f({\bm x}), \eta \rangle_{\bm x}$ where $Df({\bm x})[\eta]$ denotes the directional derivative of $f$ at ${\bm x}$ along the direction $\eta$ and $\langle \cdot, \cdot \rangle_{\bm x}$ denotes the Riemannian metric on $\mathcal{T}_\Omega({{\bm x}})$. Since $\Omega$ is an embedded Riemannian sub-manifold of $\mr{R}^n$, $\langle \cdot, \cdot \rangle_{\bm x}$ coincides with the Euclidean inner product. Therefore $\mathcal{G}({\bm x})$ is the projection of $\nabla f({\bm x})$ onto $\mathcal{T}_{\Omega}({{\bm x}})$. Note that for every $\boldsymbol\eta \in \mathcal{T}_{\Omega}({{\bm x}})$, we have ${\nabla\bm c}({\bm x})\eta = \boldsymbol 0$. Then, we have
\begin{align*}
\langle \mathcal{G}({\bm x}), \boldsymbol\eta \rangle &= \langle \nabla f({\bm x}) - {\nabla\bm c}({\bm x})\boldsymbol\lambda^\star({\bm x}), \boldsymbol\eta \rangle = \langle \nabla f({\bm x}), \boldsymbol\eta \rangle - \boldsymbol\lambda^\star({\bm x})^\top{\nabla \bm c}({\bm x})^\top\boldsymbol\eta = \langle \nabla f({\bm x}), \boldsymbol\eta \rangle = \text{D}f({\bm x})[\boldsymbol\eta].
\end{align*}
The last equality follows from the definition of directional derivative. Therefore $\langle \text{grad} f({\bm x}), \boldsymbol\eta \rangle_{\bm x} = \langle \mathcal{G}({\bm x}), \boldsymbol\eta\rangle_{\bm x}$ for every $\boldsymbol\eta \in \mathcal{T}_\Omega({{\bm x}})$, that is, $\mathcal{G}({\bm x}) = \text{grad}f({\bm x})$. Next, we will show that $\mathcal{H}({\bm x})$ is the Riemannian Hessian of $f$.
Let $\text{Hess}f({\bm x})$ denote the Riemannian Hessian of $f$ at ${\bm x} \in \Omega$. By definition, for all ${\bm \xi}, \boldsymbol\eta \in \mathcal{T}_\Omega({\bm x})$, $\text{Hess}f({\bm x})[{\bm \xi}, \boldsymbol\eta] = \langle \overline{\nabla}_\xi\text{grad}f({\bm x}), \boldsymbol\eta \rangle_{\bm x}$ where $\overline{\nabla}$ denotes the Riemannian connection on $\Omega$. First note that under the LICQ assumption, ${\nabla \bm c}({\bm x})^\top{\nabla\bm c}({\bm x})$ is invertible for every ${\bm x} \in \Omega$ and straight forward calculation gives $\boldsymbol\lambda^\star({\bm x}) = ({\nabla \bm c}({\bm x})^\top{\nabla\bm c}({\bm x}))^{-1}{\nabla\bm c}({\bm x})^\top\nabla f({\bm x})$. By the inverse function theorem, $\boldsymbol\lambda^\star({\bm x})$ is differentiable. By Proposition 5.3.2 in \cite{AMS09} and $\mathcal{G}({\bm x}) = \text{grad} f({\bm x})$,
\begin{align*}
\overline{\nabla}_\xi\text{grad}f({\bm x}) &= \mathcal{P}_{\bm{x}}(\text{D}\ \text{grad}f({\bm x})[\bm\xi]) \\
&= \mathcal{P}_{\bm{x}}(\nabla \mathcal{G}(\boldsymbol(x)) \bm\xi) \\
&= \mathcal{P}_{\bm{x}}\{\nabla^2 f({\bm x})\bm\xi -\sum_{i=1}^m (\nabla\lambda^\star_i({\bm x})\nabla c_i({\bm x})^\top\bm\xi + \lambda^\star_i\nabla^2c_i({\bm x})\bm\xi ) \} \\
&= \mathcal{P}_{\bm{x}}\{\nabla^2 f({\bm x})\bm\xi -\sum_{i=1}^m\lambda^\star_i\nabla^2c_i({\bm x})\bm\xi \} \\
&= \mathcal{P}_{\bm{x}}\nabla_{{\bm x\bm x}}^2 \mathcal{L}({\bm x}, \boldsymbol\lambda^\star({\bm x}))\bm\xi\\
&= \mathcal{P}_{\bm{x}}\mathcal{H}({\bm x})\mathcal{P}_{\bm{x}}\bm\xi.
\end{align*}
Therefore $\text{Hess}f({\bm x})[{\bm \xi}, \boldsymbol\eta] = \mathcal{H}({\bm x})({\bm \xi}, \boldsymbol\eta)$.
\end{proof}
\section*{Acknowledgment}
This research was supported in part by NSF Grant CCF1527809.
\bibliographystyle{unsrt}
| {
"timestamp": "2017-06-06T02:03:06",
"yymm": "1706",
"arxiv_id": "1706.00896",
"language": "en",
"url": "https://arxiv.org/abs/1706.00896",
"abstract": "Minimization methods that search along a curvilinear path composed of a non-ascent nega- tive curvature direction in addition to the direction of steepest descent, dating back to the late 1970s, have been an effective approach to finding a stationary point of a function at which its Hessian is positive semidefinite. For constrained nonlinear programs arising from recent appli- cations, the primary goal is to find a stationary point that satisfies the second-order necessary optimality conditions. Motivated by this, we generalize the approach of using negative curvature directions from unconstrained optimization to nonlinear ones. We focus on equality constrained problems and prove that our proposed negative curvature method is guaranteed to converge to a stationary point satisfying second-order necessary conditions. A possible way to extend our proposed negative curvature method to general nonlinear programs is also briefly discussed.",
"subjects": "Optimization and Control (math.OC)",
"title": "Using Negative Curvature in Solving Nonlinear Programs",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.985496421586532,
"lm_q2_score": 0.7185943985973773,
"lm_q1q2_score": 0.7081722083898413
} |
https://arxiv.org/abs/1905.05397 | The scaling limit of a critical random directed graph | We consider the random directed graph $\vec{G}(n,p)$ with vertex set $\{1,2,\ldots,n\}$ in which each of the $n(n-1)$ possible directed edges is present independently with probability $p$. We are interested in the strongly connected components of this directed graph. A phase transition for the emergence of a giant strongly connected component is known to occur at $p = 1/n$, with critical window $p= 1/n + \lambda n^{-4/3}$ for $\lambda \in \mathcal{R}$. We show that, within this critical window, the strongly connected components of $\vec{G}(n,p)$, ranked in decreasing order of size and rescaled by $n^{-1/3}$, converge in distribution to a sequence $(\mathcal{C}_1,\mathcal{C}_2,\ldots)$ of finite strongly connected directed multigraphs with edge lengths which are either 3-regular or loops. The convergence occurs the sense of an $\ell^1$ sequence metric for which two directed multigraphs are close if there are compatible isomorphisms between their vertex and edge sets which roughly preserve the edge-lengths. Our proofs rely on a depth-first exploration of the graph which enables us to relate the strongly connected components to a particular spanning forest of the undirected Erdős-Rényi random graph $G(n,p)$, whose scaling limit is well understood. We show that the limiting sequence $(\mathcal{C}_1,\mathcal{C}_2,\ldots)$ contains only finitely many components which are not loops. If we ignore the edge lengths, any fixed finite sequence of 3-regular strongly connected directed multigraphs occurs with positive probability. | \section{Introduction and main result}
Many real-world networks are inherently directed in nature. Consider, for example, the World Wide Web: hyperlinks point from one webpage to another but the link in the other direction is not necessarily present. However, this structurally important feature is often ignored in modelling, and the corresponding mathematical literature is much less well-developed. In this paper, we consider the simplest possible model of a random directed graph and endeavour to understand the way in which its directed connectivity properties change as we adjust its parameters.
Let $\vec{G}(n,p)$ be a random directed graph with vertex set $[n]:= \{1,\ldots,n\}$ and random edge set where each of the $n(n-1)$ possible edges $(i,j)$, $i \neq j$, is present independently with probability $p$. We are interested in the \emph{strongly connected components} of $\vec{G}(n,p),$ that is the maximal subgraphs for which there exists a directed path from a vertex to any other.
The usual Erd\H{o}s--R\'enyi random graph, $G(n,p)$, in which each of the $n(n-1)/2$ possible \emph{undirected} edges is present independently with probability $p$, will play an important role in our results. It is well known that $G(n,p)$ undergoes a phase transition \cite{ErdosRenyi}: if $np \to c > 1$ as $n \to \infty$ then $G(n,p)$ has a unique giant component with high probability, while if $np \to c < 1$ as $n \to \infty$ then the components of $G(n,p)$ are of size $O_{\mathbb{P}}(\log n)$. In the so-called \emph{critical window}, where $p = \frac{1}{n} + \lambda n^{-4/3}$, Aldous~\cite{aldous1997} proved that the sequence of sizes of the largest components possesses a distributional limit when renormalised by $n^{-2/3}$.
Previous work by Karp~\cite{karp1990} and \L uczak~\cite{L1990} has shown that $\vec{G}(n,p)$ undergoes a similar phase transition to that of $G(n,p)$: if $np \to c>1$ as $n \to \infty$, then $\vec{G}(n,p)$ has a unique giant strongly connected component with high probability, while if $np\to c<1$ as $n \to \infty$, then all the strongly connected components are $o_{\mathbb{P}}(n)$ in size. These results were strengthened by \L uczak and Seierstad~\cite{LS}, who showed that $\vec{G}(n,p)$ has, in fact, the same critical window as $G(n,p)$.
\begin{thm}[\L uczak and Seierstad~\cite{LS}]
Let $\gamma_n = (np-1)n^{1/3}$.
\begin{itemize}
\item[(i)] If $\gamma_n \to \infty$ then the largest strongly connected component of $\vec{G}(n,p)$ has size $(4+o_{\mathbb{P}}(1)) \gamma_n^2 n^{1/3}$ and the second largest has size $O_{\mathbb{P}}(\gamma_n^{-1} n^{1/3})$.
\item[(ii)] If $\gamma_n \to -\infty$ then the largest strongly connected component of $\vec{G}(n,p)$ has size $O_{\mathbb{P}}(|\gamma_n^{-1}| n^{1/3})$.
\end{itemize}
\end{thm}
However, in contrast to $G(n,p)$, \L uczak and Seierstad also show that within the critical window, the complex strongly connected components (that is, those which do not just consist of a single directed cycle) occupy only $O_{\mathbb{P}}(n^{1/3})$ vertices in total. This shows that the critical components are very much ``thinner'' objects than in the setting of $G(n,p)$, where the complex components occupy $O_{\mathbb{P}}(n^{2/3})$ vertices.
In a recent preprint \cite{Coulson}, Coulson shows that, on rescaling by $n^{-1/3}$, the size of the largest strongly connected component of $\vec{G}(n,p)$ in the critical window is tight, with explicit upper and lower tail bounds.
\begin{figure}[h]
\centering
\includegraphics[scale=0.7]{biggraph2.pdf}
\caption{A directed graph on $[17].$ Its strongly connected components have vertex sets $\{3,6,8,9,14,16\},\{1,2,5,17\},\{7,11\},\{4\},\{10\},\{12\},\{13\}$ and $\{15\}.$}
\label{fig1}
\end{figure}
In this paper, we investigate the behaviour within the critical window in more detail, and in particular we prove a scaling limit for the strongly connected components. We do this by relating a particular subgraph of $\vec{G}(n,p)$ to a spanning forest of $G(n,p)$, and the convergence of that spanning forest (thought of as a collection of discrete metric spaces, one per component) to a collection of random ${\mathbb{R}}$-trees. Similar tools have already been used to study the components of $G(n,p)$ in the same critical window, leading to the main theorem of \cite{A-BBG12}.
\begin{thm}[Addario-Berry, Broutin and Goldschmidt~\cite{A-BBG12}] \label{thm:Gnp}
Let $p = p(n) = \frac{1}{n} + \lambda n^{-4/3}$ for fixed $\lambda \in {\mathbb{R}}$. Let $(A_1(n),A_2(n),\ldots)$ be the connected components of $G(n,p)$, each considered as a metric space by endowing the vertex-set with the graph distance. Then
\[
\left(\frac{A_i(n)}{n^{1/3}},i\in\mathbb{N} \right) \ensuremath{\overset{\mathrm{(d)}}\longrightarrow} (\mathcal{A}_i,i\in\mathbb{N}),
\]
where $\mathcal{A}=(\mathcal{A}_i,i\in\mathbb{N})$ is a random sequence of compact metric spaces, and the convergence is in distribution for the $\ell^4$ metric for sequences of compact metric spaces based on the Gromov--Hausdorff distance.
\end{thm}
Let us immediately give a description of the scaling limit $\mathcal{A}$, since it plays an important role in the sequel. Define $W^{\lambda}(t)=W(t)+\lambda t -t^2/2$ for $t \ge 0$, where $W$ is a standard Brownian motion, and let $(\sigma_i,i\in\mathbb{N})$ be the collection of excursion lengths above the running infimum of $W^{\lambda}$, ranked in decreasing order. For $\sigma>0,$ let $\tilde{\mathbf{e}}^{(\sigma)}$ be a Brownian excursion with length $\sigma$ biased by the exponential of its area, so that if $\mathbf{e}^{(\sigma)}$ is a Brownian excursion of length $\sigma$ then, for any non-negative measurable test function $g$, we have
\[
\mathbb{E} \left[g \left(\tilde{\mathbf{e}}^{(\sigma)} \right) \right] = \frac{ \mathbb{E} \left[\exp \left( \int_0^{\sigma} \mathbf{e}^{(\sigma)}(u) \mathrm{d} u \right) g \left(\mathbf{e}^{(\sigma)} \right) \right] }{\mathbb{E} \left[\exp \left( \int_0^{\sigma} \mathbf{e}^{(\sigma)}(u) \mathrm{d} u \right) \right] }.
\]
Let $\mathcal{T}_{\sigma}$ be the ${\mathbb{R}}$-tree encoded by $2\tilde{\mathbf{e}}^{(\sigma)}$ (see Section~\ref{subsec:rtrees} below for a description of how this is done). We make some additional point-identifications in this tree. Let $(t_1,\ldots,t_K)$ be the points of a Poisson random measure on $[0,\sigma]$ with intensity $\tilde{\mathbf{e}}^{(\sigma)}(t)\mathrm dt.$ The point $t_j \in [0,\sigma]$ corresponds to a point $x_j$ in $\mathcal{T}_{\sigma}$ at distance $2 \tilde{\mathbf{e}}^{(\sigma)}(t_j)$ from the root. For all $1 \le j \le K$, we identify $x_j$ with a uniformly chosen point on its path to the root. Write $\mathcal{G}_{\sigma}$ for the resulting metric space. Finally, conditionally on $(\sigma_i,i\in\mathbb{N}),$ the metric spaces $\mathcal{A}_1, \mathcal{A}_2, \ldots$ are independent and, for each $i\in\mathbb{N},$ $\mathcal{A}_i$ has the law of $\mathcal{G}_{\sigma_i}$.
\bigskip
While metric spaces provide the natural setting in which to consider scaling limits of undirected graphs, this is no longer the case in the directed setting: we need some extra structure to encode the orientations. Let us make some useful definitions.
By a \emph{directed multigraph}, we mean a triple $(V,E,r)$ where
\begin{itemize}
\item $V$ and $E$ are finite sets.
\item $r=(r_1,r_2)$ is a function from $E$ to $V\times V$, with $r_1(e)$ and $r_2(e)$ for $e\in E$ being respectively the \emph{tail} and \emph{head} of the directed edge $e$.
\end{itemize}
We will refer to the case where $V = \{v\}$, $E = \{e\}$ and $r_1(e) = r_2(e) = v$ as a \emph{loop}. $X=(V,E,r,\ell)$ is a \emph{metric directed multigraph} (henceforth MDM) if $(V,E,r)$ is a directed multigraph and $\ell$ is a function from $E$ to $(0,\infty)$ which assigns each edge a length. A special role will be played by the degenerate case of a loop whose single edge is assigned length 0, which we denote by $\mathfrak{L}$. The \emph{length} $\mathrm{len}(X)$ of $X$ is given by $\sum_{e \in E} \ell(e)$.
We now define a distance between MDMs $X=(V,E,r,\ell)$ and $X'=(V',E',r',\ell')$ in such a way that they are close if there is a graph isomorphism from $X$ to $X'$ which changes the lengths very little. Specifically, let $\mathrm{Isom}(X,X')$ be the set of graph isomorphisms from $X$ to $X',$ that is pairs of bijections $f$ from $V$ to $V'$ and $g$ from $E$ to $E'$ such that, for all $e\in E$, $r'(g(e))=(f(r_1(e)),f(r_2(e))).$ Then set
\[
d_{\vec{\mathcal{G}}}(X,X')=\inf_{(f,g)\in\mathrm{Isom}(X,X')} \ \sup_{e\in E} \ |\ell(e)-\ell'(g(e))|.
\]
Note that if $X$ and $X'$ do not have the same graph structure, then $\mathrm{Isom}(X,X')$ is empty and $d_{\vec{\mathcal{G}}}(X,X')$ is set to infinity. Let $\vec{\mathcal{G}}$ be the set of (isometry classes of) MDMs. Then $(\vec{\mathcal{G}}, d_{\vec{\mathcal{G}}})$ is a Polish space.
Let $C_i(n)$ for $i \ge 1$ be the strongly connected components of $\vec{G}(n,p)$, listed in decreasing order of size, breaking ties by increasing order of the lowest labelled vertex. We view these strongly connected components as MDMs, by assigning to each edge a length of $1$, and then removing all vertices with degree $2$ and merging their corresponding edges into paths of length greater than $1$. In the case of a strongly connected component which consists of a single directed cycle with $k\ge 2$ vertices, we think of it as a loop of length $k$. Similarly, we think of isolated vertices as loops of length $0$. Finally, since there are at most $n$ components, we complete the list with an infinite repeat of $\mathfrak{L},$ the loop of length $0$.
We can now state our main theorem.
\begin{thm}\label{thm:main} Suppose $p = p(n) = \frac{1}{n} + \lambda n^{-4/3} + o(n^{-4/3})$. There exists a sequence $\mathcal{C}=(\mathcal{C}_i,i\in\mathbb{N})$ of random strongly connected MDMs such that, for each $i \ge 1$, $\mathcal{C}_i$ is either 3-regular or a loop, and such that
\begin{equation}\label{eq:mainconvergence}
\left(\frac{C_i(n)}{n^{1/3}},i\in\mathbb{N}\right) \ensuremath{\overset{\mathrm{(d)}}\longrightarrow} (\mathcal{C}_i,i\in\mathbb{N})
\end{equation}
with respect to the distance $d$ defined by
\[
d(\mathbf{A},\mathbf{B})=\sum_{i=1}^{\infty} d_{\vec{\mathcal{G}}}(A_i,B_i),
\]
for $\mathbf{A}, \mathbf{B} \in \vec{\mathcal{G}}^{\mathbb{N}}$.
\end{thm}
In particular, the limit object $\mathcal{C}$ has finite total length. We will show later that $\mathcal{C}$ has only finitely many complex components (i.e.\ components which are not loops). So Theorem \ref{thm:main} implies the convergence in distribution of the number of complex components of $\vec{G}(n,p),$ their rescaled numbers of vertices, and their excesses (where the excess of a component is given by its number of edges minus its number of vertices). This, in particular, significantly strengthens Theorems 13 and 14 of \cite{LS}. Finally, we also show that, if we ignore the edge lengths, then any fixed finite sequence of 3-regular strongly connected directed multigraphs occurs with positive probability.
\bigskip
We defer a proper description of $\mathcal{C}$, which is rather involved, to Section~\ref{sec:scalinglimit} below. As is the case for $(\mathcal{A}_i,i\in\mathbb{N}),$ the $(\mathcal{C}_i,i\in\mathbb{N})$ are derived from the ${\mathbb{R}}$-trees encoded by the excursions of $W^{\lambda}$. However, the strongly connected components $(\mathcal{C}_i,i\in\mathbb{N})$ are much simpler objects than $(\mathcal{A}_i, i \in \mathbb{N})$ which, for example, have a rich fractal structure coming from their relationship to the Brownian continuum random tree. A closer analogy is obtained by instead looking at the scaling limit of a special subgraph of $G(n,p)$: its \emph{core}. The core of a graph is defined to be the maximal subgraph of minimum degree 2, and consists of the the vertices and edges which lie in cycles, as well as those in paths joining cycles. (It can be obtained by successively deleting leaves and their incident edges from the graph until no leaves remain.) It is possible to define an analogous notion of a core for the scaling limit $\mathcal{A}$ of the critical random undirected graph. This comprises the cycle structure created by the point-identifications we make in the ${\mathbb{R}}$-trees encoded by the excursions of $W^{\lambda}$. Indeed, for each $i \ge 1$, $\mathrm{core}(\mathcal{A}_i)$ is an undirected multigraph with edge lengths which is empty if there are no point-identifications, is a loop if there is a single point-identification, and is otherwise 3-regular. The MDMs $(\mathcal{C}_i, i \in \mathbb{N})$ are similarly obtained by making (a different collection of) point-identifications in the ${\mathbb{R}}$-trees encoded by the excursions of $W^{\lambda}$. In this context, a single ${\mathbb{R}}$-tree may give rise to one or more strongly connected components, or indeed none. The fact that we obtain an $\ell^1$ convergence in Theorem~\ref{thm:main}, comes from the property that for very small $\sigma,$ an ${\mathbb{R}}$-tree with the same distribution as $\mathcal{T}_{\sigma}$ is very unlikely to produce any strongly connected components at all.
It would be interesting to know if the distributions of the undirected version of $(\mathcal{C}_i, i \in \mathbb{N})$ and of the decreasing ordering of $(\mathrm{core}(\mathcal{A}_i), i \in \mathbb{N})$ are mutually absolutely continuous. We leave this as an open problem.
\bigskip
The rest of this paper is structured as follows. In Section~\ref{sec:graphtheory}, we introduce some standard terminology and then describe the depth-first exploration which we use in order to understand the directed graph $\vec{G}(n,p)$. A key role is played by a particular class of edges known as back edges, and we discuss back edges in both the discrete and continuum settings in Section~\ref{sec:backedges}. In Section~\ref{sec:scalinglimit}, we prove some useful properties of the scaling limit $\mathcal{C}$. Section~\ref{sec:mainproof} contains the proof of Theorem~\ref{thm:main}. In Section~\ref{sec:furtherprops}, we prove the further properties of the scaling limit which were mentioned immediately after the main theorem.
\section{Some graph theory} \label{sec:graphtheory}
\subsection{Basic terminology}
We recall here some elementary graph theoretic terminology which we will use throughout the paper.
\medskip
\noindent\textbf{Directed graphs and strongly connected components.} Let $\vec{G}$ be a directed graph. For a directed edge $(x,y)$ of $\vec{G}$, we say that $x$ is the \emph{tail} of the edge and $y$ is its \emph{head}. For two vertices $x$ and $y,$ we also say that $x$ is a \emph{parent} of $y$ (and $y$ is a \emph{child} of $x$) if there is an edge from $x$ to $y$, and that $x$ is an \emph{ancestor} of $y$ (and $y$ is a \emph{descendant} of $x$) if there is a directed path from $x$ to $y$.
A directed graph $\vec{G}$ is \emph{strongly connected} if for every pair $\{u,v\}$ of distinct vertices of $\vec{G}$ there exists a directed path from $u$ to $v$ and a directed path from $v$ to $u$. For a general directed graph $\vec{G}$, its \emph{strongly connected components} are the maximal strongly connected subgraphs. The strongly connected components partition the vertex set but note that, unlike for undirected graphs, edges of $\vec{G}$ may lead from one strongly connected component to another.
\medskip
\noindent\textbf{Trees and plane trees.} A discrete tree is a connected undirected graph $T$ with no cycles. For two vertices $x$ and $y$ in $T$, we write $[\hspace{-.10em} [ x,y ] \hspace{-.10em}]$ for the unique path between $x$ and $y.$ Our trees will often be rooted at a specified vertex $\rho$. This allows us to think of $T$ as a \emph{directed} graph, by orienting all of its edges away from $\rho.$ We write $|T|$ for the size of the vertex set of $T$ and $\|T\|$ for the \emph{height} of $T$, that is the largest distance between $\rho$ and another vertex.
A \emph{planar ordering,} also known as \emph{topological sort}, of a rooted tree $T$ is any total order $>$ on its vertex set such that every directed edge $(u,v)$ of $T$ is ``increasing", in the sense that $v > u$. A rooted plane tree is then a rooted tree endowed with a planar ordering.
\medskip
\noindent\textbf{Directed multigraphs.}
Recall the definition of a directed multigraph from the introduction. Directed multigraphs have the same notion of ancestor and descendant as directed graphs, and have strongly connected components in the same way. Note that the loop is strongly connected. The \emph{excess} of a strongly connected directed multigraph $(V,E,r)$ is defined to be $|V| - |E|$. If the excess is strictly positive then we say that the multigraph is \emph{complex}.
\subsection{The exploration process}\label{sec:exploration}
The strongly connected components of any directed graph can be found in time which is linear in the sum of the sizes of the vertex and edge sets. Several linear-time algorithms, including Tarjan's algorithm~\cite{Tarjan72} and the so-called path-based algorithms (see \cite{Gabow2000} for an example), rely on a \emph{depth-first search}, that is a procedure which consists in exploring the graph in such a way that, after we visit a vertex, we visit all of its as-yet unseen descendants before backtracking. Broadly speaking, as we traverse the graph, some information is kept in the form of a stack, which allows us to determine the strongly connected components.
For our study of $\vec{G}(n,p),$ we use a variant of these ideas to give a simple algorithm which does not directly yield the strongly connected components, but instead uses the fact that the vertex set is $[n]$ to give a specific \emph{plane} spanning forest which will be a key part of the structure of the strongly connected components. We use the now-standard ordered depth-first search exposed, for example, in \cite{A-BBG12}, but with the modification that we only allow ourselves to follow edges in the direction of their orientation. Let us give a precise definition of the construction and, along the way, remind the reader of the depth-first exploration for undirected graphs. Let $\vec{G}$ (resp.\ $G$) be any directed graph (resp.\ undirected graph) on $[n].$ Inductively on $i\in \{0,\ldots,n\}$, we define an ordered list $\mathcal{O}_i$ of open vertices (the \emph{stack}) which have been seen but not yet explored, and a set $\mathcal{A}_i$ of explored vertices:
\begin{itemize}
\item[$\bullet$] $i=0$: let $\mathcal{O}_0=(1)$ and $\mathcal{A}_0=\emptyset.$
\item[$\bullet$] Induction step: given $\mathcal{O}_i$ and $\mathcal{A}_i$, let $v_i$ be the first vertex of $\mathcal{O}_i$ and let $\mathcal{A}_{i+1}=\mathcal{A}_i \cup \{v_i\}.$ Let $\mathcal{N}_i$ be the set of out-neighbours (resp.\ neighbours) of $v_i$ which are not in $\mathcal{O}_i\cup \mathcal{A}_i.$ Construct $\mathcal{O}_{i+1}$ by removing $v_i$ from $\mathcal{O}_i$, and adding in the elements of $\mathcal{N}_i$ in increasing order, so that the smallest element of $\mathcal{N}_i$ is now at the start of $\mathcal{O}_{i+1}.$ If, however, this leads to $\mathcal{O}_{i+1}=\emptyset,$ then add to it the smallest element of $\{1,\ldots,n\} \setminus \mathcal{A}_{i+1}.$
\end{itemize}
This procedure builds a directed spanning forest $\mathcal{F}_{\vec{G}}$ of $\vec{G}$, by saying that two vertices $x$ and $y$ are linked by an edge from $x$ to $y$ if there exists $i$ for which $x=v_i$ and $y\in \mathcal{N}_i.$ This is illustrated in Figure~\ref{fig2}, for the graph given by Figure~\ref{fig1}. We call $\mathcal{F}_{\vec{G}}$ the \emph{forward depth-first forest} of $\vec{G}$.
We also obtain a total order of $[n]$, given by $(v_0,\ldots,v_{n-1}),$ which is a planar ordering of $\mathcal{F}_{\vec{G}},$ in the sense that it is a topological sort of each its trees and it also functions as a total order on the set formed by the trees. From this, the edges of $\vec{G}$ are partitioned into two categories: the \emph{forward edges}, which are increasing for this order, and the \emph{back edges}, which are decreasing. The forward edges can themselves also be separated into two sets: those which are edges of $\mathcal{F}_{\vec{G}},$ and those which are not, which we call \emph{surplus} edges. (In the case of the undirected graph $G$, we still get a forest $\mathcal{F}_G$, but all edges of $G$ are either part of the forest or are surplus edges.)
The combination of forward edges and back edges is what creates the strongly connected components of $\vec{G}.$ Notice in particular that, since there are no forward edges going between different trees of $\mathcal{F}_{\vec{G}},$ each strongly connected component lies within a single such tree. Moreover, since strongly connected components are made of cycles, any strongly connected component with at least two vertices must contain at least one forward and one back edge. As a consequence of the forthcoming Proposition~\ref{prop:starcomponents}, if such a component does not contain a surplus edge, then it must contain an \emph{ancestral} back edge -- that is one which goes from a vertex to one of its ancestors. We deduce from this a useful bound: the number of strongly connected components of $\vec{G}$ is smaller than the sum of its numbers of surplus edges and ancestral back edges.
\begin{figure}[h]
\centering
\includegraphics[scale=0.7]{exploration.pdf}
\caption{The planar embedding of exploration forest of the graph in Figure \ref{fig1}. Surplus edges and back edges are then dotted, and respectively straight and curved. (The strongly connected components have vertex sets $\{3,6,8,9,14,16\}, \{1, 2, 5, 17\}, \{7,11\}, \{4\}, \{10\}, \{12\}, \{13\}$ and $\{15\}$.)}
\label{fig2}
\end{figure}
Note that the surplus edges of $G$ are taken from the set of edges \emph{permitted} by $\mathcal{F}_{\vec{G}},$ which are the pairs $(u,v)$ such that there exists $i$ such that $u$ and $v$ are both in $\mathcal{O}_i$. In this case, $v$ is a sibling of an ancestor of $u$ which occurs later in the planar ordering.\footnote{Note also that $\mathcal{F}_{\vec{G}}$ determines the exploration process fully, so defining the permitted edges using the $\mathcal{O}_i$ is unambiguous.} In fact, given $\mathcal{F}_{\vec{G}},$ we can add or remove any permitted edge to $\vec{G},$ and this will not change $\mathcal{F}_{\vec{G}}.$ The same holds true for back edges. Thus, conditionally on $\mathcal{F}_{\vec{G}(n,p)},$ the permitted surplus edges and back edges of $\vec{G}(n,p)$ appear independently with probability $p$. This leads to the following proposition, which allows us to relate $\vec{G}(n,p)$ to $G(n,p)$ by their explorations.
\begin{prop}\label{prop:coupling} For any directed graph $\vec{G}$ on $[n]$ we call $\vec{G}_{\mathrm{fwd}}$ the undirected graph whose edges are the forward edges of $\vec{G}.$ We then have the following:
\begin{itemize}
\item[$(i)$] $\mathcal{F}_{\vec{G}(n,p)}\overset{(d)}=\mathcal{F}_{G(n,p)}$
\item[$(ii)$] $(\vec{G}(n,p))_{\mathrm{fwd}}\overset{(d)}=G(n,p)$
\item[$(iii)$] One can couple $G(n,p)$ and $\vec{G}(n,p)$ in the following way: first sample $G(n,p)$, which creates in particular a depth-first ordering on $\{1,\ldots,n\}.$ Then let $\vec{G}_{\mathrm{fwd}}(n,p)=G(n,p),$ and add to it each of the possible back edges $(v_i,v_j)$ for $j<i$ independently with probability $p$.
\end{itemize}
\end{prop}
\begin{proof}
The proof of \emph{(i)} is straightforward by induction: notice that, in the explorations of both $\vec{G}(n,p)$ and $G(n,p),$ for all $i$, given $\mathcal{O}_i$ and $\mathcal{A}_i$, the neighbourhood $\mathcal{N}_i$ contains each element of $\{1,\ldots,n\}\setminus (\mathcal{O}_i\cup \mathcal{A}_i)$ independently with probability $p$. Thus, each step of the forward exploration of $\vec{G}(n,p)$ has the same distribution as the corresponding step of the depth-first exploration of $G(n,p),$ and in particular the forests they build have the same distribution.
Part \emph{(ii)} is obtained by observing that, both for $\vec{G}(n,p)$ and $G(n,p),$ given the exploration forest, each permitted surplus edge is present independently with probability $p$. Similarly, \emph{(iii)} follows from the fact that, given $\vec{G}_{\mathrm{fwd}}(n,p)$, each back edge is present independently with probability $p$.
\end{proof}
This proposition motivates the study of a process which adds back edges to trees. The next section will formalise this, especially for the continuum trees which arise in the scaling limit of $G(n,1/n + \lambda n^{-4/3}).$
\section{Back edges on discrete and continuum trees} \label{sec:backedges}
We show that, when considering a plane tree with additional back edges, we can safely ignore a portion of the back edges and keep the same strongly connected components. We then adapt this idea to give a procedure for building a random finite set of backward identifications on a continuum tree, which is how $\mathcal{C}$ will be built.
\subsection{The discrete case}\label{sec:discretebackedges}
Let $T = (V(T), E(T))$ be a finite rooted plane tree, with root $\rho$. Recall that for two vertices $x$ and $y$ of $T$, $[\hspace{-.10em} [ x,y ] \hspace{-.10em}]$ is the path between $x$ and $y.$ We think of $T$ as a directed graph, by orienting all the edges away from $\rho.$
Consider a set $B$ of additional edges between elements of $V(T)$ which go backwards for the planar order. Such an edge is called \emph{ancestral} if it leads from a vertex $v$ to an ancestor of $v$. We sort $B$ in lexicographic order, and mark a subset of $B$ as $\big((x_i,y_i),i\leq N\big)$ inductively as follows.
\begin{itemize}
\item Let $(x_1,y_1)$ be the first ancestral back edge in $B.$
\item Assume that we are given $(x_j,y_j)$ for $j \le i.$ Now for $x\in T,$ let $T(i,x)=\bigcup_{j=1}^{i}[\hspace{-.10em} [ \rho,x_j ] \hspace{-.10em}]\cup[\hspace{-.10em} [ \rho,x ] \hspace{-.10em}],$ and let $(x_{i+1},y_{i+1})$ be the smallest element $(x,y)$ of $B\setminus\big\{(x_j,y_j),j\in \{1,\ldots,i\}\big\}$ such that $y\in T_k(x).$ If there is no such element, then we end the procedure, and set $N=i.$
\end{itemize}
\begin{prop}\label{prop:starcomponents} Let $X$ be the directed graph obtained by taking $T$ (with edges directed away from $\rho$) and adding all the edges of $B$. Let $X^*$ be the subgraph of $X$ where we remove any element of $B$ that is not of the type $(x_i,y_i)$ for some $1\leq i\leq N.$ Then $X$ and $X^*$ have the same strongly connected components.
\end{prop}
\begin{proof} Notice that, since there are no surplus edges in this setting, any cycle of $X$ must feature at least one ancestral back edge. Thus, for any edge $(x,y)$ of $B,$ it is possible to reach a back edge starting from $y$ and following edges of $X,$ and by construction this means that $(x,y)$ is an edge of $X^*.$
\end{proof}
This seemingly innocuous lemma is, in fact, a key tool for us. Indeed, if $T$ is taken to be a large tree of $\mathcal{F}_{\vec{G}(n,p)}$ (meaning it has size of order $n^{2/3}$), and $B$ is the set of back edges of $\vec{G}(n,p)$ which join elements of $T$, then $B$ has size of order $n^{1/3}$. However, as we will see later, the number of back edges in $X^*$ remains of order $1.$ This means that the reduction from $X$ to $X^*$, while not changing the strongly connected components, allows us to ignore the majority of the back edges at no cost.
\subsection{The continuum case} \label{sec:continuum}
\subsubsection{${\mathbb{R}}$-trees and notation} \label{subsec:rtrees}
We recall here some basic terminology about ${\mathbb{R}}$-trees; more information concerning their use in probability may be found in the survey paper \cite{LG2005}. An \emph{${\mathbb{R}}$-tree} is any metric space $(\mathcal{T},d)$ such that
\begin{itemize}
\item For all $x,y \in \mathcal{T}$, there exists a unique distance-preserving map $\phi_{x,y}$ from $[0,d(x,y)]$ into $\mathcal{T}$ such $\phi_{x,y}(0)=x$ and $\phi_{x,y}(d(x,y))=y.$ We write $[\hspace{-.10em} [ x,y ] \hspace{-.10em}]$ for the image of $\phi_{x,y}.$
\item For all continuous and one-to-one functions $c$: $[0,1] \to \mathcal{T} $, we have $\\ c([0,1]) = [\hspace{-.10em} [ c(0),c(1) ] \hspace{-.10em}].$
\end{itemize}
Our ${\mathbb{R}}$-trees will be typically be \emph{rooted}, which means we distinguish a point of $\mathcal{T}$ called the root, usually denoted $\rho.$ For $x,y\in\mathcal{T}$, we say that $x$ is an \emph{ancestor} of $y$, or that $y$ is a \emph{descendant} of $x,$ if $x\in [\hspace{-.10em} [ \rho,y ] \hspace{-.10em}],$ and we call the the point $x\wedge y$ such that $[\hspace{-.10em} [ \rho,x ] \hspace{-.10em}]\cap[\hspace{-.10em} [ \rho,y ] \hspace{-.10em}]=[\hspace{-.10em} [ \rho,x\wedge y ] \hspace{-.10em}]$ the \emph{most recent common ancestor of $x$ and $y$}. The \emph{degree} of a point $x \in \mathcal{T}$ is the number of connected components of $\mathcal{T} \setminus \{x\}$. If $x$ has degree 1 we call it a \emph{leaf}. The \emph{height} $\|\mathcal{T}\|$ of $\mathcal{T}$ is the largest distance from $\rho$ to another point.
The ${\mathbb{R}}$-trees we encounter will be all be encoded by functions. A function $f:[0,\sigma]\to {\mathbb{R}}_+$ is called an \emph{excursion function} if it is continuous and $f(x)=0$ if and only if $x=0$ or $\sigma.$ Let $\hat{f}:[0,\sigma]^2 \to {\mathbb{R}}_+$ be defined by $\hat{f}(x,y)=\underset{s\in[x,y]}\min f(s).$ Then $f$ encodes a pseudo-distance $d_f$ on $[0,\sigma]$, defined by $d_f(x,y)=f(x)+f(y)-2\hat{f}(x,y),$ and an ${\mathbb{R}}$-tree $\mathcal{T}_f,$ defined by
\[\mathcal{T}_f = [0,\sigma]/\{d_f=0\}.\]
The {natural projection} from $[0,\sigma]$ to $\mathcal{T}_f$ will be called $p_f$, and we let the root of $\mathcal{T}_f$ be $p_f(0)=p_f(\sigma).$ $\mathcal{T}_f$ also inherits a natural total order from $[0,1]$ which we call the \emph{planar order.} Finally, for $x,y\in[0,\sigma]^2$, we let $x\overset{f}\wedge y=\inf \{t\in[x,y]: f(t)=\hat f(x,y)\}.$
In the sequel, it will often be the case that the functions $f$ we consider have unique local minima. If this holds then the resulting ${\mathbb{R}}$-tree $\mathcal{T}_f$ is binary (meaning its points all have degree at most 3).
\subsubsection{Constructing the identifications}\label{sec:continuousbackedges}
We now describe a random process which will give us a finite number of point identifications which go backward for this planar ordering: pairs of points of the form $(x,y)$ with $x>y$ for the planar ordering and an ``arrow with zero length" pointing from $x$ to $y$. Specifically, we will define times $(s_i,i\in \{1,\ldots,N\})$ in $[0,\sigma]$, their projections $x_i=p_f(s_i)$ and points $y_i$ such that $y_i$ is in the subtree $\mathcal{T}_i=\bigcup_{j=1}^{i}[\hspace{-.10em} [ \rho,x_j ] \hspace{-.10em}].$
We start with the base case $i=1.$ Let $s_1$ be the first point of a Poisson point process on $[0,\sigma]$ with intensity $f(x)\mathrm d x$, and $x_1=p_f(s_1).$ Then let $y_1$ be a uniform random point on the segment $[\hspace{-.10em} [ \rho,x_1 ] \hspace{-.10em}].$ If the Poisson point process has no points on $[0,\sigma],$ we let $N=0.$
Now let us assume that $(s_j,x_j,y_j)$ have been built for $j\in\{1,\ldots,i\}.$ For $t\in [s_i,\sigma],$ consider the subtree $\mathcal{T}_i(t)=\mathcal{T}_i \cup [\hspace{-.10em} [ \rho,p_f(t) ] \hspace{-.10em}]$ and its length $\ell_i(t).$ Then, straightforwardly,
\begin{equation}\label{eq:rate}
\ell_i(t)=\sum_{j=1}^i \left(f(s_j)-\hat{f} (s_{j-1},s_j) \right) + f(t)-\hat{f}(t,s_i).
\end{equation}
Now let $s_{i+1}$ be the first arrival time of an independent Poisson point process on $[s_i,\sigma]$ of intensity $\ell_i(t)\mathrm d t,$ let $x_{i+1}=p_f(s_{i+1}),$ and let $y_{i+1}$ be uniform on the finite length space $\mathcal{T}_i(s_i),$ independently of everything else. If the Poisson point process has no points, we let $N=i.$
\begin{figure}[h]
\begin{center}
\includegraphics[scale=0.8]{ipetest2.pdf}
\end{center}
\caption{Some identifications on a tree. Lengths of the segments are represented by $a,\ldots,t$ for the next figures.}
\label{fig:twogen}
\end{figure}
Observe that we necessarily have $s_1 \leq s_2 \leq \ldots \leq \sigma$ (so that, in particular, the first back edge in the planar ordering is always ancestral). It is, however, in principle possible for the sequence $(s_i)_{k \ge 1}$ to accumulate, with the consequence that there are infinitely many back edges. In that case, we set $N=\infty.$ However, this in fact occurs with probability $0$. Indeed, by (\ref{eq:rate}), we have that $\ell_i(t) \le i \| f \|$ and so, for all $i\in\mathbb{N}$ and $t\geq 0,$
\[\mathbb{P}[N\geq i \text{ and } s_i-s_{i-1}\leq t \mid N\geq i-1] \leq \mathbb{P} [E_i \leq t]\]
where $(E_i,i\in\mathbb{N})$ are independent exponential variables with respective parameters $(i \|f\|,i\in\mathbb{N}),$ and $s_{0}=0$ by convention. Hence we also have
\[\mathbb{P}[N=\infty]=\mathbb{P} \left[N=\infty \text{ and } \sup_{i\in\mathbb{N}}s_i\leq \sigma \right]\leq \mathbb{P} \left[\sum_{i=1}^{\infty} E_i \leq \sigma \right]. \]
However, $\sum_{i=1}^{\infty} E_i=\infty$ a.s.\ because the harmonic series diverges, and so the above probability is $0.$
We end this section with two elementary technical points.
\begin{lem}\label{lem:explicit} The joint distribution of $N$ and the $(s_i,i\leq N)$ can be written explicitly as:
\begin{align*}
& \mathbb{P}[N=n,s_i\in\mathrm{d}t_i \,\forall i\leq n]= \\
&\qquad \qquad \prod_{k=1}^n \left(\sum_{i=1}^k \left( f(t_i) - \hat{f} (t_{i-1},t_i) \right) \right) \\
& \qquad \qquad \times \exp{\left(-\int_0^{\sigma} \Bigg(f(t)-\hat{f}(t_{I(t)},t) + \sum_{i=1}^{I(t)}\left(f(t_i)-\hat{f}(t_{i-1},t_i) \right)\Bigg)\mathrm{d}t\right)} \mathrm{d}t_1\ldots\mathrm{d}t_n,
\end{align*}
where $t_0=0$ and, for $t\in[0,\sigma]$, $I(t)=\max\{i: t_i<t\}.$
\end{lem}
\begin{rem}\label{rem:otherconstruction} It will at times be convenient to consider the pairs $(x_i,y_i)$ which are \emph{ancestral}, i.e. such that $y_i$ is an ancestor of $x_i.$ Thus we let $\big((x_i^a,y_i^a),k\leq N^a\big)$ be those pairs, and $(s_i^a,k\leq N_a)$ the corresponding times in $[0,\sigma].$ Note that these are the points of a Poisson point process with intensity $f(x)\mathrm dx$ on $[0,\sigma].$ In particular, $N^a$ is a Poisson variable with parameter $\int_0^{\sigma} f(t)\mathrm d t,$ and conditionally on $N^a=1,$ $s_1^a$ has density proportional to $f$ on $[0,\sigma].$
On the other hand, if we let $\big((x_i^b,y_i^b),k\leq N^b\big)$ be the pairs which are \emph{not} ancestral, then the corresponding times $(s_i^b,k\leq N_b)$ can, conditionally on $(s_i^a,k\leq N_a),$ also be seen as the points of a Poisson point process. We will not need this full description, but the following expression will be useful:
\[\mathbb{P}[N^b=0 \mid N^a=1,s_1^a]=\exp\left(-\int_{s_1^a}^{\sigma} \left( f(t)-\hat{f}(s_1^a,t) \right) \mathrm d t\right). \]
\end{rem}
\subsubsection{The resulting strongly connected components}
Let $\mathcal{T}_f^{\mathrm{mk}}=\bigcup_{i=0}^{\infty}\,\mathcal{T}_i$ be the subtree spanned by the root and the marked leaves, and quotient it by the equivalence relation $\sim$ which identifies $x_i$ and $y_i$ for $i\in \{1,\ldots,N\}$, to obtain a rooted metric space
\[
\mathcal{M}_f=\mathcal{T}_f^{\mathrm{mk}}/\sim.
\]
Since $\mathcal{T}_f^{\mathrm{mk}}$ has only finitely many leaves, we may also view $\mathcal{M}_f$ as a finite rooted directed multigraph $M_f$ whose edges are endowed with lengths: the vertices of $M_f$ are the images of the $(y_i)$ and of the branchpoints of $\mathcal{T}_f$, and the directions are inherited from $\mathcal{T}_f^{\mathrm{mk}}$ (which we always think of as having edges directed away from the root). We observe that, with the exception of the root (which is a leaf almost surely), the vertices of $M_f$ all have degree at least 3. Now remove all edges which do not lie in a strongly connected component of $M_f$ and delete any isolated vertices thus created. This yields a collection of strongly connected components of minimum degree 2. If there remain vertices of degree precisely two, we repeatedly apply the following merging operation. Pick an arbitrary vertex of degree 2 and merge its two incident edges as long as they are different edges, summing their lengths. This yields a collection $\mathcal{C}_f$ of strongly connected MDMs, as illustrated in Figure~\ref{fig:strongconnect}.
\begin{figure}[h]
\begin{center}
\includegraphics[scale=0.7]{ipetestgraph.pdf}
\vspace{1cm}
\includegraphics[scale=0.7]{ipetestcomponents.pdf}
\caption{A representation of $\mathcal{M}_f$ obtained from the identifications given in Figure~\ref{fig:twogen}, and the resulting strongly connected components.} \label{fig:strongconnect}
\end{center}
\end{figure}
\section{The scaling limit} \label{sec:scalinglimit}
\subsection{Excursions of Brownian motion with parabolic drift}
Let $(W(t),t\geq0)$ be a standard Brownian motion. For $\lambda\in {\mathbb{R}}$ and $t\geq0,$ let $W^{\lambda}(t)=W(t)+\lambda t -t^2/2$ and let $\underline{W}^{\lambda}(t)=\underset{0\leq s\leq t} \inf W^{\lambda}(s).$ Let $B^{\lambda}(t)=W^{\lambda}(t)-\underline{W}^{\lambda}(t)$, and let $\Gamma^{\lambda}$ be the set of excursions of $B^{\lambda}.$ For an excursion $\gamma \in \Gamma^{\lambda}$, let $|\gamma|$ denote its length.
\begin{prop} \label{prop:excursionlengths}
\begin{itemize}
\item[(i)]For $\alpha\in\{2,3\}$, we have $\mathbb{E}\left[\sum_{\gamma\in\Gamma^{\lambda}} |\gamma|^{\alpha}\right]< \infty$.
\item[(ii)] $\sum_{\gamma\in\Gamma^{\lambda}} |\gamma|^{3/2}=\infty$ a.s.
\end{itemize}
\end{prop}
The $\alpha = 2$ case of $(i)$ is Lemma 25 of Aldous~\cite{aldous1997}, which we extend here to $\alpha=3$. (Our method also works for all $\alpha>3/2$ but we omit the details for the sake of brevity.) We first need a standard result on moments of hitting times of Brownian motion with constant drift.
\begin{lem} For $\mu>0$ and $b>0$, let $T(b,\mu)=\inf\{t\geq0:\, W(t)-\mu t = -b\}.$ Then we have
\[\mathbb{E}[T(b,\mu)]=\frac{b}{\mu} \quad\text{and}\quad \mathbb{E} \left[\big(T(b,\mu)\big)^2\right]=\frac{b(1+ b \mu)}{\mu^3}.\]
\end{lem}
\begin{proof} The Laplace transform of $T(b,\mu)$ is given by
\[
\mathbb{E}\left[e^{-\theta T(b,\mu)}\right]=\exp\left( b\mu - b\sqrt{\mu^2+2\theta}\right), \:\theta>0,
\]
(see, for example, Exercise 5.10 in Chapter 3 of \cite{KaratzasShreve}) and the first two moments of $T(b,\mu)$ follow from differentiating twice.
\end{proof}
\begin{proof}[Proof of Proposition \ref{prop:excursionlengths} $(i)$.] Let $\gamma$ be an excursion of $B^{\lambda},$ and $l$ and $r$ its endpoints. We have
\[|\gamma|^{3}=3 \int_l^r(r-t)^{2} \mathrm d t.\]
If we write, for $t\geq 0$, $H_t=\min \{s>0: B^{\lambda}(t+s)=0\},$ we then have
\[\sum_{\gamma\in\Gamma^{\lambda}} |\gamma|^{3}=3\int_0^{\infty}H_t^{2}\mathrm d t\]
and so we only need to prove that $\int_0^{\infty} \mathbb{E}[H_t^2]\mathrm d t<\infty.$ To do this we split the integral into $\int_0^{\tau} \mathbb{E}[H_t^2]\mathrm d t$ and $\int_{\tau}^{\infty} \mathbb{E}[H_t^2]\mathrm d t$ where $\tau= \max (0,2\lambda+1).$
For $t >\max(0,\lambda)$ and $s\geq 0$, we have $\lambda s - (t+s)^2/2 + s^2/2 \leq (\lambda-t)s.$ Thus, conditionally on $B^{\lambda}(t)$, we get that $H_t$ is stochastically dominated by $T(B^{\lambda}(t), t - \lambda),$ which leads to
\[\mathbb{E}[H_t\mid B^{\lambda}(t)]\leq\frac{B^{\lambda}(t)}{(t-\lambda )^2} \quad\text{and}\quad \mathbb{E}[(H_t)^2\mid B^{\lambda}(t)]\leq\frac{B^{\lambda}(t)(1+(t-\lambda) B^{\lambda}(t))}{(t-\lambda)^3}.\]
In particular, we have
\[\int_{\tau}^{\infty} \mathbb{E}[H_t^{2}]\mathrm d t\leq \int_{\tau}^{\infty} \frac{E[B^{\lambda}(t)]+(t-\lambda)\mathbb{E}\left[\left(B^{\lambda}(t)\right)^2\right]}{(\lambda-t)^3}\mathrm d t.\]
However, it is also established in the proof of Lemma 25 of \cite{aldous1997} that, for $t>2\lambda,$ the random variable $B^{\lambda}(t)$ is stochastically dominated by an exponential random variable with parameter $t-2\lambda$, implying that $E[B^{\lambda}(t)]\leq 1/(t-2\lambda)$ and $E[(B^{\lambda}(t))^2]\leq (1-2\lambda+t)/(t-2\lambda)^2.$ In consequence, $\int_{\tau}^{\infty} \mathbb{E}[H_t^{2}]\mathrm d t<\infty.$
To bound $\int_0^{\tau} \mathbb{E}[H_t^2]\mathrm d t,$ notice that we have $H_t \leq \tau-t + H_{\tau}\leq \tau + H_\tau$ for $t\leq \tau$. Hence,
\begin{align*}
\mathbb{E}[H_t^2]&\leq \tau^2+2\tau\mathbb{E}[H_{\tau}]+\mathbb{E}[H_{\tau}^2]<\infty \\
&\leq \tau^2+2\tau\frac{\mathbb{E}[B^{\lambda}(\tau)]}{\tau-\lambda}+\frac{\mathbb{E}[B^{\lambda}(\tau)]}{(\tau-\lambda)^3}+\frac{\mathbb{E}[(B^{\lambda}(\tau))^2]}{(\tau-\lambda)^2},
\end{align*}
and this uniform upper bound is finite since $B^{\lambda}(\tau)$ is stochastically dominated by an exponential variable which has moments of all orders. It follows that we do indeed have $\int_0^{\tau} \mathbb{E}[H_t^2]\mathrm d t<\infty.$
\end{proof}
Part $(ii)$ also requires a preliminary lemma, which allows us to work directly with $B^{\lambda}$ instead of powers of its excursion lengths.
\begin{lem}\label{lem:sumintegral}
If $\int_0^{\infty} B^{\lambda}(t) \mathrm{d} t =\infty$ a.s.\ then $\sum_{\gamma\in\Gamma^{\lambda}} |\gamma|^{3/2}=\infty$ a.s.
\end{lem}
\begin{proof}
Let $(\sigma_1, \sigma_2, \ldots)$ be the lengths of the excursions of $B^{\lambda}$ listed in decreasing order, and let $\tilde{e}_1, \tilde{e}_2, \ldots$ be the excursions themselves, so that
\[
\int_0^{\infty} B^{\lambda}(t) \mathrm{d}t = \sum_{i \ge 1} \int_0^{\sigma_i} \tilde{e}_i(x) \mathrm{d} x.
\]
Now, we have by~\cite[Section 5]{A-BBG12} that $\tilde{e}_i \overset{d}{=} \tilde{\mathbf{e}}^{(\sigma_i)}$ for $i \ge 1,$ and the excursions $(\tilde{e}_i, i \ge 1)$ are conditionally independent given their lengths. Moreover,
\[
\mathbb{E}\left[\int_0^{s} \tilde{\mathbf{e}}^{(s)}(x) \mathrm{d} x\right]=s^{3/2}\frac{\mathbb{E}\left[\int_0^1\mathbf{e}(x)dx\exp \left(s^{3/2} \int_0^1\mathbf{e}(x)dx \right)
\right]}{\mathbb{E}\left[\exp \left( s^{3/2}\int_0^1\mathbf{e}(x)dx \right)\right]}\underset{s \to 0}\sim s^{3/2}\mathbb{E}\left[\int_0^1\mathbf{e}(x)dx\right].
\]
It follows that, almost surely
\[
\mathbb{E} \left[ \sum_{i \ge 1} \int_0^{\sigma_i} \tilde{e}_i(x) \mathrm{d} x \ \Bigg| \ (\sigma_j)_{j \ge 1} \right] = \sum_{i \ge 1} \mathbb{E} \left[ \int_0^{\sigma_i} \tilde{\mathbf{e}}^{(\sigma_i)}(x) \mathrm{d} x \Big| \sigma_i \right] < \infty
\]
if and only if
\[
\sum_{i \ge 1} \sigma_i^{3/2} \mathbb{E} \left[\int_0^{1} \mathbf{e}(x) \mathrm{d} u \right] < \infty,
\]
which itself occurs if and only if $\sum_{i \ge 1} \sigma_i^{3/2} < \infty$. But by assumption we have $\int_0^{\infty} B^{\lambda}(t) \mathrm{d} t = \infty$ a.s., and so $\sum_{\gamma \in \Gamma^{\lambda}} |\gamma|^{3/2} = \sum_{i \ge 1} \sigma_i^{3/2} = \infty$ a.s.
\end{proof}
The proof of Proposition~\ref{prop:excursionlengths} $(ii)$ is due to \'Eric Brunet.
\begin{proof}[Proof of Proposition \ref{prop:excursionlengths} $(ii)$.] Recall that
\[
W^{\lambda}(t)=W(t)+\lambda t -t^2/2,
\]
$\underline{W}^{\lambda}(t)=\underset{0\leq s\leq t} \inf W^{\lambda}(s)$ and $B^{\lambda}(t)=W^{\lambda}(t)-\underline{W}^{\lambda}(t)$.
By Lemma \ref{lem:sumintegral}, it is sufficient to show that $\int_0^{\infty} B^{\lambda}(t) \mathrm{d} t=\infty$ a.s. We will construct a lower bound for $B^{\lambda}$, built on the same probability space, whose integral we can more easily show to be infinite.
Let $(Z(t),t\geq0)$ be defined by $Z(0)=0$ and
\[
\mathrm d Z(t) = \mathrm d W(t) - \left(3t + \frac{Z(t)}{t}\right) \mathrm{d} t.
\]
Now define $\theta: {\mathbb{R}}_+ \to {\mathbb{R}}_+$ by $\theta(s) = (3s)^{1/3}$ and let
\[
Y(s)=\theta(s) Z(\theta(s)).
\]
Then straightforward stochastic calculus gives that
\[
\mathrm{d} Y(s) = \theta(s) \mathrm{d} W(\theta(s)) - 3 \mathrm{d}s.
\]
Since $\sqrt{\theta'(s)} = 1/\theta(s)$, by the Dubins--Schwarz theorem we get that
\[
\widetilde{W}(s) := \int_0^s \theta(r) \mathrm{d} W(\theta(r))
\]
is a standard Brownian motion and so $Y$ is a Brownian motion with drift $-3$:
\[
Y(s)=\widetilde{W}(s)-3s, \quad s \ge 0.
\]
As a consequence, there exists a random time $S_1\geq 0$ such that, for all $s\geq S_1,$
\[
-\frac{9}{2}s < Y(s) < 0.
\]
Letting $T_1=\theta(S_1),$ since $\theta^{-1}(t) = t^3/3$, we get
\[
-\frac{3}{2} t^2<Z(t)<0
\]
for $t \geq T_1.$ In particular, for $t\geq T_1 \vee 2 |\lambda|$, we have the following bounds on the drift term in the SDE defining $Z$:
\[
-\left(3t + \frac{Z(t)}{t} \right) \le -\frac{3}{2} t \le \lambda - t.
\]
Using the fact that $Z$ and $W^{\lambda}$ are constructed from the same Brownian motion $W$, we get that after time $T_2 := T_1\vee 2|\lambda|$, $Z$ has smaller increments than $W^{\lambda}.$
Now choose a time $T_3>T_2$ large enough such that the minima of $Z$ and $W^{\lambda}$ on $[0,T_3]$ are both attained after $T_2.$ Then, for $t\geq T_3$, the minimum of $W^{\lambda}$ on $[0,t]$ is attained at some $u \in [T_2,t]$, and so
\[
B^{\lambda}(t)=W^{\lambda}(t)- \inf_{u \in [T_2,t]} W^{\lambda}(u) = \sup_{u \in [T_2,t]} (W^{\lambda}(t) - W^{\lambda}(u)) \geq \sup_{u \in [T_2,t]} (Z(t)-Z(u)) \geq Z(t)-\underline{Z}(t),
\]
where $\underline{Z}$ is the running infimum of $Z$. Moreover, using the fact that $Y(s)<0$ for $s\geq S_1,$ we get
\[
Z(t)-\underline{Z}(t)=\frac{1}{t}Y(t^3/3) - \underset{u \in[T_2,t]}\inf\frac{1}{u}Y(u^3/3) \geq \frac{1}{t}Y(t^3/3) - \frac{1}{t}\underset{u \in[T_2,t]}\inf Y(u^3/3)=\frac{1}{t}(Y(t^3/3)-\underline{Y}(t^3/3)),
\]
where $\underline{Y}$ is the running infimum of $Y$. This yields $B^{\lambda}(t)\geq \frac{1}{t}(Y(t^3/3)-\underline{Y}(t^3/3))$ for $t\geq T_3.$ In particular,
\[
\int_0^{\infty} B^{\lambda}(t)\mathrm d t \geq \int_{T_3}^{\infty} \left(Y(t^3/3)-\underline{Y}(t^3/3) \right) \frac{\mathrm d t}{t}=\int_{(3T_3)^{1/3}}^{\infty}(Y(s) -\underline{Y}(s))\frac{\mathrm d s}{3s}.
\]
We now show that the final integral is infinite. Observe that the reflected drifting Brownian motion $Y - \underline{Y}$ is a positive recurrent Markov process, its stationary distribution being exponential with parameter $6$ (see, for example, p.94 of Harrison~\cite{Harrison}). In particular, the sets of times at which it is above $2$ and below $1$ are both unbounded, and we can define two sequences $(\tau_n)_{n \ge 0}$ and $(\eta_n)_{n \ge 1}$ of stopping times as follows. Let $\tau_0 = (3T_3)^{1/3}$ and then, for $n \ge 1$,
\begin{align*}
\eta_n & = \inf\{s > \tau_{n-1}: Y(s) - \underline{Y}(s) > 2\}, \\
\tau_n & = \inf\{s > \eta_n: Y(s) - \underline{Y}(s) < 1\}.
\end{align*}
On a downcrossing interval $[\eta_n, \tau_n]$, we trivially have $Y(s) - \underline{Y}(s) \ge 1$, so
\[\int_{(3T_3)^{1/3}}^{\infty}(Y(s) -\underline{Y}(s))\frac{\mathrm d s}{s}\geq \sum_{n=1}^{\infty}\frac{(\tau_n - \eta_n)}{\tau_{n}}.
\]
By the strong Markov property, we have that $\{\tau_n - \tau_{n-1}: n \ge 2\}$ are i.i.d.\ and so by the law of large numbers, as $n \to \infty$,
\[
\frac{\tau_n}{n} \to \mathbb{E}[\tau_2 - \tau_1] < \infty \quad \text{a.s.}
\]
We also have that $\{\tau_n - \eta_n: n \ge 2\}$ are i.i.d.\ and so $\sum_{n \ge 1} (\tau_n - \eta_n)/n$ diverges a.s. It follows that
\[
\sum_{n \ge 1} \frac{(\tau_n - \eta_n)}{\tau_n} = \infty \quad \text{a.s.} \qedhere
\]
\end{proof}
\subsection{Bounds for a single tree}
Let $\sigma>0.$ We let $f=2\tilde{\mathbf{e}}^{(\sigma)}$ be a tilted Brownian excursion with length $\sigma$, whose distribution is determined by
\[
\mathbb{E}[g(\tilde{\mathbf{e}}^{(\sigma)})]=\frac{\mathbb{E}\left[g\big(\sqrt{\sigma}\mathbf{e}(\cdot/\sigma)\big) \exp \left(\sigma^{3/2}\int_0^1\mathbf{e}(x)dx \right) \right]}{\mathbb{E}\big[\exp \left( \sigma^{3/2}\int_0^1\mathbf{e}(x)dx \right)\big]},
\]
for any non-negative measurable function $g$, where $\mathbf{e}$ is a standard Brownian excursion. We perform the construction detailed in Section~\ref{sec:continuum}, defining the ${\mathbb{R}}$-tree $\mathcal{T}_{\sigma}$ (we now replace the subscript $f$ by $\sigma$ since henceforth all of our coding functions will be of this type), performing $N_{\sigma}$ identifications, $N_{\sigma}^a$ of them being ancestral and $N_{\sigma}^b$ being non-ancestral, and thus build the MDM $\mathcal{M}_{\sigma}.$ The following proposition will enable us to control the number of strongly connected components of $\mathcal{M}_{\sigma}.$
\begin{prop}\label{prop:Poissonbounds} Let $c=\mathbb{E}[\int_0^1 \mathbf{e}(t)\mathrm dt]=F'(0)$ where $F(z)=\mathbb{E}[e^{z\int_0^1 \mathbf{e}(t)\mathrm d t}]$ is the moment generating function of the Airy distribution, which is an entire function \cite{janson2007}. We have the following asymptotics: as $\sigma \to 0$,
\begin{itemize}
\item[(i)] $\mathbb{P}[N_{\sigma}^a=0]= 1 - 2c\sigma^{3/2} + O(\sigma^3)$
\item[(ii)] $\mathbb{P}[N_{\sigma}^a=1,N_{\sigma}^b=0]= 2c\sigma^{3/2} + O(\sigma^3)$
\item[(iii)] $\mathbb{P}[N_{\sigma}^a\geq 2\text{ or } N_{\sigma}^b\geq 1]= O(\sigma^3).$
\end{itemize}
Moreover,
\begin{itemize}
\item[(iv)] $\underset{\sigma>0}\sup \; \sigma^{-3}\mathbb{E}[N_{\sigma}^a\mathbf{1}_{\{N_{\sigma}^a\geq 2\}}] <\infty.$
\end{itemize}
\end{prop}
\begin{proof}
Instead of working with $\tilde{\mathbf{e}}^{(\sigma)},$ we express the probabilities in terms of a standard Brownian excursion $\mathbf{e}$ and its area $\mathcal{A}=\int_0^1 \mathbf{e}(t)\mathrm dt.$ For $(i)$, recall that, conditionally on $\tilde{\mathbf{e}}^{(\sigma)},$ we have that $N_{\sigma}^a$ has a Poisson distribution with parameter $\int_0^{\sigma} 2\tilde{\mathbf{e}}^{(\sigma)}(x)\mathrm dx$. Therefore,
\[
\mathbb{P}[N_{\sigma}^a=0]=\frac{\mathbb{E}\left[e^{-\sigma^{3/2}\int_0^1 2 \mathbf{e}(t)\mathrm dt} e^{\sigma^{3/2}\int_0^1 \mathbf{e}(t)\mathrm dt}\right]}{\mathbb{E}\left[e^{\sigma^{3/2}\int_0^1 \mathbf{e}(t)\mathrm dt}\right]}
=\frac{F(-\sigma^{3/2})}{F(\sigma^{3/2})}=1 - 2c\sigma^{3/2} + O(\sigma^3). \]
We begin the proofs of $(ii)$ and $(iii)$ by computing
\[
\mathbb{P}[N_{\sigma}^a=1]=\frac{\mathbb{E}\left[2\sigma^{3/2}\mathcal{A}e^{-\sigma^{3/2}\mathcal{A}}\right]}{\mathbb{E}\left[e^{\sigma^{3/2}\mathcal{A}}\right]}=\frac{2\sigma^{3/2}F'(-\sigma^{3/2})}{F(\sigma^{3/2})}=\frac{2\sigma^{3/2}(c+O(\sigma^{3/2}))}{1+O(\sigma^{3/2})}=2c\sigma^{3/2} + O(\sigma^3).
\]
Next, recalling Remark \ref{rem:otherconstruction}, we write
\[
\mathbb{P}[N_{\sigma}^a=1,N_{\sigma}^b=0]=\frac{\mathbb{E}\left[e^{-2\sigma^{3/2}\mathcal{A}}\sigma^{3/2}\int_0^1 2\mathbf{e}(x) \exp \left(-\sigma^{3/2}\int_{x}^1 2(\mathbf{e}(y)-\hat{\mathbf{e}}(x,y)\mathrm dy\right) \mathrm d x \right]}{\mathbb{E}[e^{\sigma^{3/2}\mathcal{A}}]}.
\]
Using $|1-e^{-u}|\leq u$ for $u\geq 0,$ we obtain
\begin{align*}
\mathbb{P}[N_{\sigma}^a=1]-\mathbb{P}[N_{\sigma}^a=1,N_{\sigma}^b=0]&\leq 4\sigma^3\mathbb{E}\left[e^{-2\sigma^{3/2}\mathcal{A}}\int_0^1 \mathbf{e}(x)\mathrm dx\int_x^1(\mathbf{e}(x)-\hat{\mathbf{e}}(x,y))\mathrm d y \right] \\
&\leq 4\sigma^{3}\mathbb{E}\left[\int_0^1 (\mathbf{e}(x))^2 \mathrm d x\right].
\end{align*}
Now note that $\int_0^1 (\mathbf{e}(x))^2 \mathrm d x$ has finite expectation because it is smaller than $(\sup \mathbf{e})^2,$ which is indeed integrable ($\sup \mathbf{e}$ has sub-Gaussian tails, see \cite{kennedy76}). So the above quantity is $O(\sigma^3)$. This finishes the proof of $(ii)$ and $(iii)$.
Finally, since $N_{\sigma}^a$ is integer-valued, we have
\begin{align*}
\mathbb{E} \left[N_{\sigma}^a\mathbf{1}_{\{N_{\sigma}^a\geq 2\}} \right]&=\mathbb{E}[N_{\sigma}^a]-\mathbb{P}[N_{\sigma}^a=1] \\
&=\frac{\mathbb{E}[2\sigma^{3/2}\mathcal{A}e^{\sigma^{3/2}\mathcal{A}}]}{\mathbb{E}[e^{\sigma^{3/2}\mathcal{A}}]} -\mathbb{P}[N_{\sigma}^a=1] \\
&= \frac{2\sigma^{3/2}\left(F'(\sigma^{3/2})-F'(-\sigma^{3/2})\right)}{F(\sigma^{3/2})}.
\end{align*}
This proves that $\mathbb{E}[N_{\sigma}^a\mathbf{1}_{\{N_{\sigma}^a\geq 2\}}]=O(\sigma^3)$ as $\sigma \to 0$, but we also want the bound as $\sigma$ tends to infinity. To this end, we write
\[\mathbb{E}[N_{\sigma}^a\mathbf{1}_{\{N_{\sigma}^a\geq 2\}}]\leq 2\sigma^{3/2}\frac{F'(\sigma^{3/2})}{F(\sigma^{3/2})}\]
and simply aim to prove that $F'(x)=O(xF(x))$ as $x\to \infty.$ Quoting \cite[Section 7]{janson2007}, we have
\[F(x)=\sum_{n=0}^{\infty} a_nx^n,\]
where
\[a_n \underset{n\to\infty}\sim \frac{3\sqrt{2}}{(n-1)!}\left(\frac{n}{12 e}\right)^{n/2}.\]
The desired domination will follow from the fact that $\frac{(n+1)a_{n+1}}{a_{n-1}}$ (the ratio of the coefficients of $x^n$ in $F'(x)$ and $xF(x)\,$) is uniformly bounded for $n\geq 1,$ which is true, since the sequence in fact converges:
\begin{align*}
\frac{(n+1)a_{n+1}}{a_{n-1}} &\sim \frac{n+1}{n(n-1)}\frac{1}{12e}\frac{(n+1)^{(n+1)/2}}{(n-1)^{(n-1)/2}} \sim \frac{1}{12e} \frac{(n+1)^{n/2}}{(n-1)^{n/2}} \to \frac{1}{12}.
\end{align*}
This completes the proof.
\end{proof}
\subsection{Some properties of the scaling limit}
Let $(\sigma_1,\sigma_2,\ldots)$ be the lengths of the excursions of $B^{\lambda}$, listed in decreasing order. For each $i\geq 1$, let $\mathcal{D}_i$ be an independent copy of $\mathcal{M}_{\sigma_i}$ and let $\mathcal{D}=\bigcup_{i=1}^{\infty} \mathcal{D}_i$. We think of $\mathcal{D}$ as a countable MDM.
\begin{thm}
\begin{itemize}
\item[(i)] The number of complex connected components of $\mathcal{D}$ has finite expectation.
\item[(ii)] The number of loops of $\mathcal{D}$ is a.s.\ infinite.
\end{itemize}
\end{thm}
\begin{proof}
We start with part $(i).$ For each $i \ge 1$, let $K_i$ be the number of complex components in $\mathcal{D}_i.$ Each complex component contains at least one ancestral identification and so $K_i\leq N_{\sigma_i}^a$. Furthermore, if there is exactly one ancestral identification, there must also be at least one which is non-ancestral in order to obtain a complex component, so that $\mathbb{P}[K_i=1]\leq \mathbb{P}[N_{\sigma_i}^a=1,N_{\sigma_i}^b\geq 1]+\mathbb{P}[N_{\sigma_i}^a\geq 2]$. Hence, by $(ii)$ and $(iv)$ from Proposition~\ref{prop:Poissonbounds},
\begin{align*}
\mathbb{E}[K_i \, | \, \sigma_i]&= \mathbb{P}[K_i=1 \, | \, \sigma_i] + \mathbb{E}[K_i\mathbf{1}_{K_i\geq 2} \, | \, \sigma_i] \\
&\leq \mathbb{P}[N_{\sigma_i}^a=1,N_{\sigma_i}^b\geq 1 \, |\, \sigma_i]+\mathbb{P}[N_{\sigma_i}^b\geq 2 \, | \, \sigma_i] + \mathbb{E}[N_{\sigma_i}^a\mathbf{1}_{\{N_{\sigma_i}^a\geq 2\}}\mid \sigma_i] \\
& \leq C\sigma_i^3
\end{align*}
for some $C>0.$ Thus,
\[
\mathbb{E}\left[\sum_{i=1}^{\infty} K_i\right] \leq C\mathbb{E}\left[\sum_{i=1}^{\infty} \sigma_i^3\right] < \infty.
\]
For part $(ii),$ notice first that, since we now know that $\mathcal{D}$ has finitely many complex components a.s., it is sufficient to show that there are infinitely many ancestral identifications, i.e.\ that $\sum_{i=1}^{\infty} N_{\sigma_i}^a=\infty$ a.s. But since $\mathbb{P}[N_{\sigma_i}^a \geq 1 | (\sigma_j,j\in\mathbb{N})]$ is asymptotically equivalent to $2c\sigma_i^{3/2}$ by Proposition \ref{prop:Poissonbounds}, and Proposition \ref{prop:excursionlengths} gives $\sum_i \sigma_i^{3/2}=\infty$ a.s., the claim follows from an application of the Borel-Cantelli lemma.
\end{proof}
The following property of $\mathcal{D}$ is not surprising, but nonetheless requires proof.
\begin{prop}\label{prop:differentlengths}
The strongly connected components of $\mathcal{D}$ all have different lengths a.s.
\end{prop}
This follows straightforwardly from the following lemma, in which we work on a single tree.
\begin{lem} Let $\sigma>0$.
\begin{itemize}
\item[$(i)$] For all $x>0$, $\mathbb{P}[\mathcal{M}_{\sigma} \text{ has a strongly connected component of length } x]=0$.
\item[$(ii)$] $\mathbb{P}[\mathcal{M}_{\sigma} \text{ has two strongly connected components with equal lengths}]=0$.
\end{itemize}
\end{lem}
\begin{proof} Let $f = 2\tilde{\mathbf{e}}^{(\sigma)}$ be the excursion function encoding the tree $\mathcal{T}_{\sigma}$ from which $\mathcal{M}_{\sigma}$ is obtained, let the selected leaves be $(x_i,i\in \{1,\ldots,N\}),$ and let $\mathcal{C}^{(\sigma)}_1,\ldots,\mathcal{C}^{(\sigma)}_K$ be the strongly connected components of $\mathcal{M}_{\sigma}$, listed in the order of appearance of their first elements in the planar ordering of $\mathcal{T}_{\sigma}$. For each $k\in\mathbb{N},$ on the event where $k\leq K,$ let $E_k=\{i\in\{1,\ldots,N\}:\; x_i\in \mathcal{C}^{(\sigma)}_k\}$ be the set of indices of the leaves implicated in the construction of the $k$th strongly connected component, let $u_k= x_{\min E_k} \wedge x_{\max E_k}$ be the MRCA of those leaves, let $\rho_k = \sup\, \{x\in [\hspace{-.10em} [ \rho,u_k ] \hspace{-.10em}]: x\notin \mathcal{C}^{(\sigma)}_k\}$ be the root of the subtree giving rise to the $k$-th strongly connected component, and let $\mathcal{T}_k= \bigcup_{i\in E_k} [\hspace{-.10em} [ \rho_k, x_i ] \hspace{-.10em}]$ be that subtree. Finally, let $n_k= \#\{i\in E_k, z_i \in [\hspace{-.10em} [ \rho_k,u_k ] \hspace{-.10em}]\}$, be the number of heads along the line-segment separating $\rho_k$ from $u_k$. Notice then that the length of $\mathcal{C}^{(\sigma)}_k$ is exactly that of $\mathcal{T}_k$, minus the initial part between $\rho_k$ and the first $y_i$ to be encountered. However, since the $y_i$ are chosen uniformly from the length measure, this means that $[\hspace{-.10em} [ \rho_k,u_k ] \hspace{-.10em}]$ is split according to a Dirichlet distribution with $n_k+1$ components. More specifically, we have
\[\mathrm{len}(\mathcal{T}_k)-\mathrm{len}(\mathcal{C}^{(\sigma)}_k)=(f(u_k)-f(\rho_k))\Delta^k_1\]
where, conditionally on $n_k,$ $Z^k_1$ is the first component of a vector $\Delta^k=(\Delta_1^k,\ldots,\Delta_{n_k+1}^k)$ which has Dirichlet$(1,\ldots,1)$ distribution. Since Dirichlet distributions have a density, we obtain
\[\mathbb{P}[\mathrm{len}(\mathcal{C}^{(\sigma)}_k)=x \mid k\leq K, \mathrm{len}(\mathcal{T}_k),f(u_k),n_k]=0,\]
and integrating and taking the union over all $k$ gives us $(i).$ (See Figure~\ref{fig:lengths} for an illustration.)
\begin{figure}
\centering
\includegraphics[scale=0.8]{length.pdf}
\caption{For point $(i)$, focusing on the second component, $\mathcal{T}_2$ contains the leaves $x_3,x_4,x_5$, and the length of its initial segment $f(u_2)-f(\rho_2)$ is split by a Dirichlet$(1,1,1)$ into $(a,b,c).$ For $(ii),$ conditioning on $\rho_2$ being in $\mathcal{T}_1,$ then this split is still $(1,1,1).$}
\label{fig:lengths}
\end{figure}
To prove $(ii)$, consider two integers $k$ and $l$. If $k\leq K$ and $l\leq K$, let
\[A_k = \{\rho_k\not\in \mathcal{C}^{(\sigma)}_l\}\]
and
\[A_l = \{\rho_l\not\in \mathcal{C}^{(\sigma)}_k\}.\]
Observe that $\mathbb{P}[A_k\cup A_l]=1,$ since $\mathcal{C}^{(\sigma)}_k$ and $\mathcal{C}^{(\sigma)}_l$ do not intersect. Now, on the event $A_l,$ $\mathcal{T}_k$ and $\mathcal{T}_l$ intersect either at point $\rho_k$ or not at all, and we can still write
\[\mathrm{len}(\mathcal{T}_k)-\mathrm{len}(\mathcal{C}^{(\sigma)}_k)=(f(u_k)-f(\rho_k))\Delta^l_1\]
where, conditionally on $\mathcal{T}_l,$ $n_k$ and the event $A_l,$ $\Delta^l_1$ is the first component of a Dirichlet$(1,\ldots,1)$ vector. This means that the length of $\mathcal{C}^{(\sigma)}_k$ has a (conditional) density, and integrating, we get
\[\mathbb{P} \left[\mathrm{len}(\mathcal{C}^{(\sigma)}_k)=\mathrm{len}(\mathcal{C}^{(\sigma)}_l) \Big| A_l,\; k,l\leq K\right]=0.\]
Symmetrising then yields that
\[\mathbb{P}\left[\mathrm{len}(\mathcal{C}^{(\sigma)}_k)=\mathrm{len}(\mathcal{C}^{(\sigma)}_l) \Big| k,l\leq K\right]=0,\]
and taking a countable union yields $(ii)$.
\end{proof}
\begin{proof}[Proof of Proposition \ref{prop:differentlengths}] We label the strongly connected components of $\mathcal{D}$ in such a way that, for $i\in\mathbb{N},$ those which belong to $\mathcal{D}_i$ are called $\mathcal{C}_{i,1},\ldots,\mathcal{C}_{i,K_i}.$ Consider $\mathcal{C}_{i,k}$ and $\mathcal{C}_{j,l}$ for $i,j,k,l$ in $\mathbb{N}.$ We can assume $i\neq j$ as the case where $i=j$ has already been treated. Conditionally on the excursion lengths $(\sigma_i,i\in \mathbb{N})$, $\mathcal{C}_{i,k}$ and $\mathcal{C}_{j,l}$ are independent and we have $\mathbb{P}[\mathrm{len}(\mathcal{C}_{i,k})=x]=0$ for all $x>0.$ Thus we have $\mathbb{P}[\mathrm{len}(\mathcal{C}_{i,k})=\mathrm{len}(\mathcal{C}_{j,l})\mid \mathrm{len}(\mathcal{C}_{j,l})]=0,$ and integrating to remove the conditioning yields $\mathbb{P}[\mathrm{len}(\mathcal{C}_{i,k})=\mathrm{len}(\mathcal{C}_{j,l})]=0$. (Again see Figure~\ref{fig:lengths}.) This completes the proof.
\end{proof}
\section{Convergence of the strongly connected components}\label{sec:mainproof}
For $n\in\mathbb{N},$ let $p=p(n)$ be such that $p=1/n+\lambda n^{-4/3} + o(n^{-4/3})$ as $n\to\infty.$ Recall that $(C_i(n),i\in\mathbb{N})$ are the strongly connected components of $\vec{G}(n,p)$, listed in decreasing order of size (with ties broken by using the increasing order of smallest vertex-label), where we treat isolated vertices as copies of the loop of zero length, and additionally append infinitely many copies of the loop of zero length. Let $(\mathcal{C}_i,i\in\mathbb{N})$ be the strongly connected components of $\mathcal{D},$ listed in decreasing order of length.
We restate the main theorem.
\medskip
\noindent \textbf{Theorem~\ref{thm:main}.}
\[\left(\frac{C_i(n)}{n^{1/3}},i\in\mathbb{N} \right) \ensuremath{\overset{\mathrm{(d)}}\longrightarrow} (\mathcal{C}_i,i\in\mathbb{N})\]
\emph{with respect to the distance $d$ defined by
\[
d(\mathbf{A},\mathbf{B})=\sum_{i=1}^{\infty} d_{\vec{\mathcal{G}}}(A_i,B_i),
\]
for $\mathbf{A}, \mathbf{B} \in \vec{\mathcal{G}}^{\mathbb{N}}$.}
\medskip
The aim of this section is to prove this theorem. We begin by discussing some topological issues related to $d_{\vec{\mathcal{G}}}$. We then prove a series of preliminary results, before finally turning to the proof of Theorem~\ref{thm:main}.
\subsection{The relationship between $d_{\vec{\mathcal{G}}}$ and the Gromov--Hausdorff distance}
Recall from the introduction the definition of metric directed multigraphs (MDMs), and that the distance between two such objects $X=(V,E,r,\ell)$ and $X'=(V',E',r',\ell')$ is defined by
\[
d_{\vec{\mathcal{G}}}(X,X')= \inf_{(f,g)\in\mathrm{Isom}(X,X')} \ \sup_{e\in E} \ |\ell(e)-\ell'(g(e))|.
\]
Elements of $\vec{\mathcal{G}}$ can also be viewed as metric spaces, by thinking of each edge as a line segment and forgetting the orientation of the edges. This means that we can also compare them using the Gromov--Hausdorff distance $d_{\mathrm{GH}}$ (see Chapter 7 of \cite{Burago} for an introduction to the Gromov--Hausdorff distance and its use). The resulting topology is however weaker, as the following lemma shows.
\begin{lem}\label{lem:distancedomination} For $X\in\vec{\mathcal{G}}$ and $X'\in\vec{\mathcal{G}},$ we have
\[d_{\mathrm{GH}}(X,X')\leq \frac{1}{2}|E| \, d_{\vec{\mathcal{G}}}(X,X')\]
\end{lem}
\begin{proof} If $X$ and $Y$ do not have the same graph structure, then $d_{\vec{\mathcal{G}}}(X,X')=\infty$ and the statement holds trivially. If they do have the same graph structure then, up to applying an optimal isomorphism $(f,g)$, we can assume that they have the same vertex and edge sets, i.e.\ $X=(V,E,r,\ell)$ and $X'=(V,E,r,\ell'),$ where the length assignments $\ell$ and $\ell'$ are such that $\sup_{e \in E} |\ell(e)-\ell'(e)|=d_{\vec{\mathcal{G}}}(X,X')$. We let $\phi$ be the natural bijection from $X$ to $X'$ when viewed as metric spaces, which acts identically on $V$ and follows the edges ``linearly''. Viewing $\phi$ as a correspondence (again, see Chapter 7 of \cite{Burago}), its distortion can be bounded above by
\[
\sum_{e \in E} |\ell(e)-\ell'(e)| \leq |E| \, \sup_{e \in E}\, |\ell(e)-\ell'(e)|= |E| \, d_{\vec{\mathcal{G}}}(X,X'). \qedhere
\]
\end{proof}
In the case of trees, it is possible to recover a convergence for $d_{\vec{\mathcal{G}}}$ from a \emph{pointed} Gromov--Hausdorff convergence (see \cite{MiermontTessellations} for a definition).
\begin{prop}\label{prop:cvtreesG} Fix $k\in\mathbb{N}.$ For $n\in\mathbb{N}$, let $(\mathcal{T}_n,n\in\mathbb{N})$ (resp.\ $\mathcal{T}$) be ${\mathbb{R}}$-trees with roots $\rho_n$ (resp.\ $\rho$) and $k$ selected distinct leaves $(x_{i,n},1\leq i \leq k)$ (resp.\ $(x_i,i\leq k)$). Then let
\[
\overline{\mathcal{T}}_n= \bigcup_{i=1}^k [\hspace{-.10em} [ \rho_n,x_{i,n} ] \hspace{-.10em}]
\]
be the subtree spanned by the $k$ selected leaves and the root (and define $\overline{\mathcal{T}}$ similarly). View it as an element of $\vec{\mathcal{G}}$ by taking as vertices the root, the leaves, and all the branch points, orienting each edge away from $\rho_n$ (resp. $\rho$) and giving each edge the length of its corresponding metric path.
Suppose that $(\mathcal{T}_n,\rho_n, x_{1,n}, \ldots, x_{k,n})$ converges to $(\mathcal{T},\rho,x_1, \ldots x_k))$ for the $(k+1)$-pointed Gromov--Hausdorff topology, and that $\overline{\mathcal{T}}$ is binary. Then $\overline{\mathcal{T}}_n$ converges to $\overline{\mathcal{T}}$ for $d_{\vec{\mathcal{G}}}.$ Specifically, the map which sends $\rho_n$ to $\rho$ and $x_{i,n}$ to $x_i$ for each $1 \leq i \leq k$ extends uniquely to a graph isomorphism, under which the length of each edge in $\overline{\mathcal{T}}_n$ converges to that of the corresponding edge in $\overline{\mathcal{T}}.$
\end{prop}
\begin{proof}
First we prove that the reduced tree $(\overline{\mathcal{T}}_n,\rho_n, x_{1,n}, \ldots, x_{k,n})$ converges for the $(k+1)$-pointed Gromov--Hausdorff topology. Take a common embedding of the $\mathcal{\mathcal{T}}_n$ and $\mathcal{T}$ in a certain compact space $(\mathcal{Z},d)$ such that $\mathcal{T}_n\to\mathcal{T}$ in the Hausdorff sense, $\rho_n\to\rho$ and $x_{i,n}\to x_i$ for all $i$ and let us show that we also have the Hausdorff convergence of $\overline{\mathcal{T}}_n$ to $\overline{\mathcal{T}}.$ To do this, first notice that, for each $i\in\{1,\ldots,k\}$, the segment $[\hspace{-.10em} [ \rho_n,x_{i,n} ] \hspace{-.10em}]$ converges to $[\hspace{-.10em} [ \rho,x_i ] \hspace{-.10em}].$ Now noting that any point $y \in [\hspace{-.10em} [ \rho_n,x_{i,n} ] \hspace{-.10em}]$ satisfies $d(y,\rho_n)+d(y,x_{i,n})=d(\rho_n,x_{i,n}),$ we obtain by passing to the limit that any limit point $y$ of a sequence of points in $[\hspace{-.10em} [ \rho_n,x_{i,n} ] \hspace{-.10em}]$ satisfies $d(y,\rho)+d(y,x_i)=d(\rho,x_i)$ and thus lies in $[\hspace{-.10em} [ \rho,x_i ] \hspace{-.10em}].$ Conversely, any point $y$ of $[\hspace{-.10em} [ \rho,x_i ] \hspace{-.10em}]$ with $d(\rho,y)=t$ is the limit of $y_n\in [\hspace{-.10em} [ \rho_n,x_{i,n} ] \hspace{-.10em}]$ with $d(\rho_n,y_n)=t$ (if $t=d(\rho,x_i),$ then $y_n=x_{i,n}$ instead).
We also have that, for each $i,j\in\{1,\ldots,n\}$, the MRCA $x_{i,n}\wedge x_{j,n}$ converges to $x_i\wedge x_j$ in the embedding above, since any limit point $y$ must be in $[\hspace{-.10em} [ \rho, x_i ] \hspace{-.10em}]$ and passing the relation
\[
2d(x_{i,n}\wedge x_{j,n},x_{i,n})=d(x_{i,n},x_{j,n})+d(\rho,x_{i,n})-d(\rho,x_{j,n})
\]
to the limit yields $d(y,x_i)=d(x_i\wedge x_j,x_i).$
The $d_{\vec{\mathcal{G}}}$ convergence, with the specific isomorphism mentioned in the statement, can then be proved by induction. The base case $k=1$ is immediate, as the Gromov--Hausdorff convergence of a line segment implies convergence of its length. Let us then focus on the induction step: assume the proposition at rank $k\in\mathbb{N}$, and let $(\mathcal{T}_n,n\in\mathbb{N})$ be rooted trees with $k+1$ leaves $(x_{i,n},1\leq i\leq k+1)$ converging for the $(k+2)$-pointed Gromov--Hausdorff topology to $\mathcal{T}$ with root $\rho$ and leaves $(x_i,1\leq i\leq k+1).$
Consider for all $n\in \mathbb{N}$ the subtree $\mathcal{T}^k_n$ of $\mathcal{T}_n$ spanned by the root and the first $k$ leaves $x_{i,n}$ with $i\leq k$. Then $(\mathcal{T}^k_n,\rho_n,x_{1,n},\ldots,x_{k,n})$ converges to $(\mathcal{T}^k,\rho,x_1,\ldots,x_k)$ for the $(k+1)$-pointed Gromov--Hausdorff topology. By the induction hypothesis, this is also convergence for $d_{\vec{\mathcal{G}}}$ and, in particular, the graph structure of $\mathcal{T}^k_n$ is the same as that of $\mathcal{T}^k$ for $n$ large enough.
The graph structure of $\mathcal{T}^{k+1}_n,$ the tree spanned by the root and the $k+1$ leaves, is then determined by that of $\mathcal{T}^{k}_n$ together with the knowledge of which edge of $\mathcal{T}^{k}_n$ contains the projection $p_{n,k}(x_{k+1,n})$ of $x_{k+1,n}.$ However, $p_{k,n}(x_{k+1,n})$ converges to $p_k(x_{k+1})$ under the Gromov--Hausdorff convergence, since it is the maximum (in the line segment $[\hspace{-.10em} [ \rho_n,x_{k+1} ] \hspace{-.10em}]$) of $x_{i,n}\wedge x_{k+1,n}$ for $i\leq k$ and each of those terms also converges. Thus, for $n$ large enough, $p_{k,n}(x_{k+1,n})$ lies in the line segment of $\mathcal{T}^k_n$ corresponding to the one in $\mathcal{T}^k$ containing $p_k(x_{k+1}),$ and the map sending $x_{i,n}$ to $x_i$ for $i\leq k+1$ does indeed extend to a graph isomorphism.
Once we know the graph structure of $\mathcal{T}^{k+1}_n,$ each edge is either of the form $(x_{i,n}\wedge x_{j,n},x_{j,n}\wedge x_{k,n})$ where $x_{i,n}\wedge x_{j,n}$ is an ancestor of $x_{j,n}\wedge x_{k,n}$, or $(\rho,x_{i,n}\wedge x_{j,n}),$ and their lengths converge because, as noticed earlier, the branch points $x_{i,n}\wedge x_{j,n}$ can be added to the pointed Gromov--Hausdorff convergence.
\end{proof}
\begin{prop}\label{prop:graphtocc} If the connected components of an MDM $X$ all have different total lengths, and $(X_n,n\in \mathbb{N})$ is a sequence which converges to $X$ for $d_{\vec{\mathcal{G}}}$, then the strongly connected components of $X_n$, listed in decreasing order of length and seen as elements of $\vec{\mathcal{G}}$, converge to those of $X$.
\end{prop}
\begin{proof}
Writing $X=(V,E,r,\ell),$ let $(C_1,\ldots,C_k)$ be the strongly connected components of $X,$ ordered by decreasing length. For $n\in\mathbb{N}$ large enough, may assume we have $X_n=(V,E,r, \ell_n),$ where $\ell_n(C_i) \to \ell(C_i)$ as $n \to \infty$ for all $i$. In particular, for $n$ large enough, $\ell_n(C_i)$ is strictly decreasing in $i$, and so $(C_1,\ldots,C_k)$ is also the ordered sequence of the strongly connected components of $X_n,$ which completes the proof.
\end{proof}
\subsection{The components originating from a single tree} \label{sec:onetree}
The first part of the proof will consist in proving the convergence of the components originating from a single tree. For $m\in\mathbb{N}$, we take a plane tree $T_m$ which has the distribution of a tree component of $\mathcal{F}_{\vec{G}(n,p)}$ conditioned to have size $m$. We are interested in $m \sim \sigma n^{2/3}$ so that, in particular, we have $mp^{2/3}\to \sigma$ as $m\to\infty$. From \cite{A-BBG12}, up to an unimportant relabelling of the vertices, $T_m$ has the same distribution as a uniform random labelled tree on $[m]$, biased by $(1-p)^{-a(T_m)},$ where $a(T_m)$ is the number of permitted edges in $T_m$. We give this tree a planar embedding by rooting at the vertex labelled 1 and then simply using the increasing order on the labels of the children of any vertex. Let $H^m: \{0,\ldots,m-1\} \to \mathbb{Z}_+$ be the height function of $T_m$, such that $H^m(i)$ is the height of the $i$-th vertex in the planar order, starting with $H^m(0)=0.$ We recall that $\|T_m\| = \max_{0 \leq i \leq m-1} H^m(i)$ is the height of the tree $T_m$. Theorem 15 of \cite{A-BBG12} states that
\[
\big((m/\sigma)^{-1/2}H^m(\lfloor(m/\sigma)t\rfloor),0\leq t\leq \sigma\big) \ensuremath{\overset{\mathrm{(d)}}\longrightarrow} (2\tilde{\mathbf{e}}^{(\sigma)}(t),0\leq t\leq \sigma)
\]
uniformly as $m \to \infty$.
As in Proposition \ref{prop:coupling}, we include each of the $\binom{m}{2}$ possible back edges and $a(T_m)$ possible surplus edges independently with probability $p$, and let $X_m$ be the resulting directed graph. The aim of this section is to show that the rescaled strongly connected components of $X_m$ converge in distribution to those of $\mathcal{M}_{\sigma}.$ In order to do this, we will use the structure of back edges outlined in Section \ref{sec:discretebackedges}. Specifically, let $\big((x_{i,m},y_{i,m}),k\in\mathbb{N},i\leq N_m)$ be the back edges obtained with this procedure, and let $X^*_m$ be the subgraph of $X_m$ obtained by removing all back edges which are not of the above form, as well as the surplus edges. We will first show that the strongly connected components of $X^*_m$ converge in distribution to those of $\mathcal{M}_{\sigma}$, and then that $X_m$ and $X^*_m$ have the same strongly connected components with high probability. (In particular, we show that the surplus edges with high probability do not play any role in creating the strongly connected components.)
\subsubsection{Convergence of the marked points}
Our next step is to improve the convergence of the tree $T_m$ to include the marked points $(x_{i,m})$ and $(y_{i,m})$.
\begin{prop}\label{prop:cvtreewithpoints}
There exists a realisation of $T_m,\mathcal{T}_{\sigma},N_m,N,x_{i,m},y_{i,m},x_j,y_j$ for all $m\in\mathbb{N}$, $n\in\mathbb{N},$ and $i\leq N_m$ and $j\leq N$ on a single probability space such that $N_m \to N$ a.s. as $m$ tends to infinity, and
\[\left(\left(\frac{\sigma}{m}\right)^{1/2}T_m, \rho, \big((x_{i,m},y_{i,m}),i\leq N_m \big)\right) \underset{m\to\infty}{\longrightarrow}\Big(\mathcal{T}_{\sigma},\big((x_i,y_i),i\leq N \big)\Big)\]
a.s. for the $\left(1+2N\right)$-pointed Gromov--Hausdorff topology.
\end{prop}
Note that the above convergence makes sense since, for $m$ large enough, $N_m=N$.
\begin{proof}
By Skorokhod's representation theorem, we may assume that the convergence of $(\frac{\sigma}{m})^{1/2}H^m$ occurs almost surely. As a consequence, there exists a specific metric space $(\mathcal{Z},d_{\mathcal{Z}})$ and embeddings of $\mathcal{T}_{\sigma}$ and each $(\frac{\sigma}{m})^{1/2} T_m$ into $\mathcal{Z},$ such that $T_m$ converges almost surely in the Hausdorff sense to $\mathcal{T}_{\sigma}.$
As usual, we write $f = 2 \tilde{\mathbf{e}}^{(\sigma)}$, and recall that $p_f$ is the projection map $[0,\sigma] \to \mathcal{T}_{\sigma}$. Let $p_m: [0,m] \to \mathcal{Z}$ be the projection of $\{0,\ldots,m-1\}$ onto $(\frac{\sigma}{m})^{1/2}T_m$ in order of the depth-first exploration process, linearly interpolated (with $p_m(m)=p_m(0)$). The construction of $\mathcal{Z}$ can be done in such a way that $p_m(\frac{\sigma}{m}\cdot)$ converges pointwise to $p_f$ on $\mathcal{Z},$ and this convergence is in fact uniform. Indeed, $d_{\mathcal{Z}}(p_m(\frac{\sigma}{m}x),p_m(\frac{\sigma}{m}y))$ converges uniformly to $f(x)+f(y)-2\hat{f}(x,y),$ and so the family $p_m(\frac{\sigma}{m}\cdot)$ of functions is equicontinuous. So by the Arzel\`a--Ascoli theorem it converges uniformly.
Building on this, we will find a probability space on which, additionally, $N_m\to N$, $x_{i,m} \to x_i$ and $y_{i,m} \to y_i$, a.s.\ for all $i\leq N$. This is done by induction on $i\in\mathbb{N}$.
We start with $i=1,$ and first focus on $x_{1,m}.$ Let $k_{1,m}\in\{1,\ldots,m\}$ be the position of $x_{1,m}$ in the planar ordering of $T_m.$ Since the number of ancestral back edges originating at the $k$-th point has distribution $\mathrm{Bin}(H^m(k),p),$ and $pH^m(\lfloor xm \rfloor)\sim (\frac{m}{\sigma})^{-1}f(x),$ standard results on Poisson random measures imply that $\mathbb{P}[N_m\geq 1]\to \mathbb{P}[N\geq 1]$ and that $((\frac{m}{\sigma})^{-1} k_{1,m}),$ conditionally on $N_m\geq 1$, converges in distribution to the first point of a Poisson point process with intensity $f(x)\mathrm d x$, also conditioned to have at least one point (See the proof of Lemma 19 of \cite{A-BBG12} for a more detailed version of an essentially identical argument.) Since the projections converge uniformly, we can apply them to obtain that $x_{1,m} \to x_1$ as well.
Still for $i=1,$ we may now assume that the convergence of the $x_{1,m}$ occurs almost surely, and focus next on $y_{1,m}.$ This vertex belongs to the ancestral line of $x_{1,m}$, and its height is uniform on $\{0,\ldots,d(\rho,x_{1,m}^1)-1\}.$ Since the relation $d(\rho,y_{1,m})+d(y_{1,m},x_{1,m})=d(\rho,x_{1,m})$ passes to the limit, any subsequential limit in distribution of $(y_{1,m})$ must be an ancestor of $x_1$, and its height must be uniform in $[0,f(s_1)].$ Thus $y_{1,m}$ converges in distribution to a uniform ancestor of $x_1,$ which is none other than $y_1.$
We may now moreover assume that all these convergences occur almost surely.
The induction step uses the same ideas. In $T_m,$ let for $k\geq k_{i,m}$ $T_m(i,k)=\bigcup_{j=1}^i [\hspace{-.10em} [ \rho, x_{j,m} ] \hspace{-.10em}]\cup[\hspace{-.10em} [ \rho,p_m(k) ] \hspace{-.10em}]$ be the set of possible heads for a back edge originating at $p_m(k).$ We also let $l_m(i,k)$ be the number of vertices in $T_m(i,k).$ With the induction hypothesis and Proposition \ref{prop:cvtreesG}, we have that $\bigcup_{j=1}^i [\hspace{-.10em} [ \rho, x_{j,m} ] \hspace{-.10em}]$ converges a.s.\ to $\cup_{j=1}^i [\hspace{-.10em} [ \rho,x_j ] \hspace{-.10em}]$ for $d_{\vec{\mathcal{G}}}.$ Moreover, since the $(\frac{m}{\sigma})^{-1} k_{i,m}$ converge to the $s_i$, we have that $T_m(i,\lfloor\frac{m}{\sigma}t\rfloor)$ and $\mathcal{T}_i(t)$ take a uniformly close amount of length of each edge, and thus $(\frac{\sigma}{m})^{1/2} l_i(\lfloor \frac{m}{\sigma} t\rfloor)$ converges uniformly to $\ell_i(t)$. Since $x_{i+1,m}=p_m(k_{i+1,m})$ is obtained by giving to each $k$ a $\mathrm{Bin}(l_i(k),p)$ number of marks, we obtain that $(\frac{m}{\sigma})^{-1} k_{i+1,m}$ converges to $s_{i+1}.$ The argument for the head of the directed edge generalises similarly.
\end{proof}
The proof of Proposition \ref{prop:cvtreewithpoints} also implies the following corollary.
\begin{cor}\label{cor:cvcolouredtree} Let $T^{\mathrm{mk}}_m=\bigcup_{i=1}^{N_m} T_m(i)$ and $\mathcal{T}^{\mathrm{mk}}_{\sigma}=\bigcup_{i=1}^{N} \mathcal{T}_i$ be the marked subtrees of $T_m$ and $\mathcal{T}_{\sigma}$ respectively. Then
\[
\left(\left(\frac{\sigma}{m}\right)^{1/2}T_m^{\mathrm{mk}}, \rho, \big((x_{i,m},y_{i,m}),i\leq N_m \big)\right) \underset{m\to\infty}{\longrightarrow} \Big(\mathcal{T}_{\sigma}^{\mathrm{mk}},\big((x_i,y_i),k\in\mathbb{N},i\leq N \big)\Big),
\]
a.s.\ for the $\left(1+2 N\right)$-pointed Gromov--Hausdorff topology. The corresponding lengths also converge a.s.:
\[
\left(\frac{\sigma}{m}\right)^{1/2}|T^{\mathrm{mk}}_m| \underset{m\to\infty}{\longrightarrow} \mathrm{len}(\mathcal{T}^{\mathrm{mk}}_{\sigma}).
\]
\end{cor}
\subsubsection{Convergence of the marked graph}
Let $X^*_m=T^{\mathrm{mk}}_m$ along with all back edges $(x_{i,m},y_{i,m})$ for $i\leq N_m$, and recall that $\mathcal{M}_{\sigma}=\mathcal{T}_{\sigma}^{\mathrm{mk}} / \sim,$ where $\sim$ is the equivalence relation which identifies $x_i$ with $y_i$ for $i\leq N.$ We view them as elements of $\vec{\mathcal{G}},$ in a way which will fit the metric on $\vec{\mathcal{G}}.$ Specifically, we take the vertex set of $X_m^*$ to consist of $\rho$, the heads $y_{i,m}$ of the back edges for $i \le N_m$, and the branch points $x_i \wedge x_j$ for $i \neq j \le N_m$. We take the vertices of $\mathcal{M}_{\sigma}$ to be $\rho$, $y_i$ for $i \le N$ (note that post-identification we have $x_i=y_i$), and the branch points $x_i \wedge x_j$ for $i \neq j \le N$. Because the Brownian continuum random tree is almost surely binary and the law of $\mathcal{T}_{\sigma}$ is absolutely continuous with respect to that of the Brownian continuum random tree, $\mathcal{T}_{\sigma}^{\mathrm{mk}}$ is also binary almost surely. It follows that $\mathcal{M}_{\sigma}$ has $2N$ vertices and, as we will see, the same must also be true for $X_m^*$ for sufficiently large $m$.
\begin{prop}
$(\frac{\sigma}{m})^{1/2}X_m^* \ensuremath{\overset{\mathrm{(d)}}\longrightarrow} \mathcal{M}_{\sigma}$ in $\vec{\mathcal{G}}.$
\end{prop}
\begin{proof}
We keep the setting of Proposition \ref{prop:cvtreewithpoints} and work almost surely. By Corollary~\ref{cor:cvcolouredtree}, \\ $((\frac{\sigma}{m})^{1/2}T^{\mathrm{mk}}_m,\rho,x_{1,m},\ldots,x_{N_m,m})$ converges for the $(N+1)$-pointed Gromov--Hausdorff topology to $(\mathcal{T}^{\mathrm{mk}}_{\sigma},\rho,x_1,\ldots,x_N)$. Proposition~\ref{prop:cvtreesG} then makes this a convergence in $\vec{\mathcal{G}}$ (taking the root, $(x_{i,m})$ and branch points as vertices). In particular they have the same underlying graph structure for $m$ large enough.
For $m$ large enough, no $x_{i,m}$ is an ancestor of a $x_{j,m}$ or $y_{j,m},$ so the graph structure of $X_m^*$ can be obtained from that of $T_m^{\mathrm{mk}}$ by removing $x_{i,m}$ and instead connecting the edge ending in $x_{i,m}$ back into $y_{i,m}$, for each $i$. Since $y_{i,m}$ converges to $y_i$ in the Gromov--Hausdorff sense, it will in particular always be on the same edge of $T_m^{\mathrm{mk}}$ for $m$ sufficiently large. Thus the discrete structure is constant for $m$ large, and the same as that of $\mathcal{M}_{\sigma}.$
Once we know the discrete structure, the lengths of all the edges then also converge since they can be expressed in terms of the distances between the root, the $(x_{i,m})$ and the $(y_{i,m}).$
Using also Propositions~\ref{prop:graphtocc} and~\ref{prop:differentlengths}, we then obtain that the connected components of $\left(\frac{\sigma}{m}\right)^{1/2}X_m^*,$ listed in decreasing order of size, converge in the sense of $d_{\vec{\mathcal{G}}}$ to those of $\mathcal{M}_{\sigma},$ listed in decreasing order of length.
\end{proof}
\subsubsection{Surplus edges do not contribute}
As mentioned before, we now want to prove that the surplus edges contribute to the strongly connected components of $X_m$ with vanishingly small probability. Specifically, we aim to prove the following proposition.
\begin{prop}\label{prop:surplus} $\mathbb{P}\left[X_m\text{ and } X^*_m \text{ have different strongly connected components}\right] \to 0$ as $m \to \infty$.
\end{prop}
Let $R(m)$ be the number of surplus edges in $X_m$. For $1 \le i \le R(m)$, let $\alpha_{i,m}$ and $\beta_{i,m}$ be the tail and head respectively of the $i$-th surplus edge in increasing planar order of their tails. Let $W_i(m)$ be the number of vertices descending from $\beta_{i,m}$ in $T_m$. Proposition \ref{prop:surplus} will follow if we can establish that the family $\left(\sum_{i=1}^{R(m)}W_i(m),m\in\mathbb{N} \right)$ is tight, namely if
\begin{equation}\label{eq:tightness}
\lim_{K \to \infty} \limsup_{m \to \infty} \; \mathbb{P}\left[\sum_{i=1}^{R(m)} W_i(m) > K \right]=0.
\end{equation}
Indeed, for a strongly connected component of $X_m$ to feature a surplus edge, we need at least one back edge to originate from a descendant of some $\beta_{i,m}$ (since any surplus edge in a strongly connected component is part of a cycle and must thus lead to a back edge). Conditionally on $\sum_{i=1}^{R(m)} W_i(m) \leq K$, the probability of this event is smaller than the probability that a $\mathrm{Bin}(mK,p)$ variable is non-zero. Assuming (\ref{eq:tightness}) and fixing $\varepsilon>0$, we may find a $K$ sufficiently large that
\[
\limsup_{m \to \infty} \;\mathbb{P}\left[\sum_{i=1}^{R(m)} W_i(m) > K\right]\leq \varepsilon/3, \] and $m$ large enough such that $\mathbb{P}\left[\sum_{i=1}^{R(m)} W_i(m) \geq K\right]\leq \varepsilon/2$ and $1-(1-p)^{mK}\leq \varepsilon/2$ (recall that $p\sim \sigma^{3/2}m^{-3/2}$ ). Then
\begin{align*}
& \mathbb{P}\left[X_m\text{ and } X^*_m \text{ have different strongly connected components}\right] \\
& \qquad \leq\mathbb{P}\left[\sum_{i=1}^{R(m)} W_i(m) \geq K\right] + 1-(1-p)^{mK} \leq\frac{\varepsilon}{2}+\frac{\varepsilon}{2} = \varepsilon.
\end{align*}
As we have already mentioned, it is shown in \cite{A-BBG12} that $T_m$ is a biased version of the uniform labelled tree $\mathsf{T_m}$ on $[m]$ (with a canonical planar embedding): for non-negative measurable test functions $f$,
\begin{equation}\label{eq:defbias}
\mathbb{E}[f(T_m)]=\frac{\mathbb{E}[(1-p)^{-a(\mathsf{T}_m)}f(\mathsf{T}_m)]}{\mathbb{E}[(1-p)^{-a(\mathsf{T}_m)}]}.
\end{equation}
We recall that $a(T)$ denotes the the number of surplus edges permitted by the planar structure of a tree $T$, called its \emph{area} in \cite{A-BBG12}. We know from Theorem 12 and Lemma 14 of \cite{A-BBG12} that
\[
(1-p)^{-a(\mathsf{T}_m)} \ensuremath{\overset{\mathrm{(d)}}\longrightarrow} e^{\int_0^{\sigma}\mathbf{e}^{(\sigma)}(t)\mathrm dt},
\]
and that the sequence on the left-hand side is bounded in $L^2$. We will prove (\ref{eq:tightness}) by first showing the analogous statement for $\mathsf{T}_m$ (this is Lemma~\ref{lem:notilttightness} below) and then using the measure change. We need the following lemma, which makes use of Kesten's tree, that is the tree $\widehat{\mathsf{T}}$ consisting of a copy of $\mathbb{Z}_{+}$ (the \emph{spine}), at each point of which we graft an independent Galton--Watson tree with Poisson(1) offspring distribution. We root the resulting infinite tree at 0. (This is the local weak limit of $\mathsf{T}_m$ \cite{Grimmett}.)
\begin{lem} \label{lem:gwsubtree}
\begin{itemize}
\item[(i)] Let $\mathsf{Y}(m)$ be the number of vertices of $\mathsf{T}_m$ which lie outside the largest subtree descending from a child of the root. Then
\[\mathsf{Y}(m) \underset{m\to\infty}{\ensuremath{\overset{\mathrm{(d)}}\longrightarrow} } \mathsf{Y},\]
where $\mathsf{Y}$ is the number of vertices of $\widehat{\mathsf{T}}$ which have no ancestors on the spine apart from the root.
\item[(ii)] Write $(v_i,i\in\{1,\ldots,m\})$ for the vertices of $\mathsf{T}_m$ in planar order. For $v\in\mathsf{T}_m,$ let $\mathsf{Z}_v(m)$ be the number of vertices in the subtree rooted at $v$. Let $\mathsf{Y}_v(m)$ be the number of such vertices which lie outside the largest of the subtrees rooted at a child of $v.$ Then the $(\mathsf{Y}_{v_i}(n),n\in\mathbb{N},i\leq n)$ are tight:
\[\underset{M \to \infty}\lim \;\underset{n\in\mathbb{N}} \limsup \underset{i\in\{1,\ldots,n\}}\sup \mathbb{P}[\mathsf{Y}_i(n)>M]=0.\]
\end{itemize}
\end{lem}
\begin{proof} It is well-known that $\mathsf{T}_m$ is a Galton-Watson tree with Poisson(1) offspring distribution, conditioned to have $m$ vertices and assigned a uniformly random labelling from $[m]$. Knowing this, a proof of $(i)$ can be found within the proof of Proposition 5.2 in \cite{pagnard2017}, so we will just give an informal argument. Let $\mathsf{T}_1(n),\ldots,\mathsf{T}_D(n)$ be the subtrees of $\mathsf{T}_n$ rooted at its first generation, with $D$ being the degree of the root, listed in decreasing order of size. For any non-increasing finite sequence $\lambda$ of positive integers, one can show that $\mathbb{P}[(\#\mathsf{T}_2(n),\ldots\#\mathsf{T}_D(n))=\lambda]$ converges to $\mathbb{P}[(\#\widehat{\mathsf{T}}_2,\ldots,\#\widehat{\mathsf{T}}_D)=\lambda],$ where the $\widehat{\mathsf{T}}_i$ are defined similarly, and are well-known to be finite since they are off the spine of $\widehat{\mathsf{T}}$. Thus, this limit is a probability distribution. Hence, the sequence $(\#\mathsf{T}_2(n),\ldots\#\mathsf{T}_D(n))$ converges in distribution, and so does its sum.
Part $(ii)$ follows from the fact that, for all $i$ and $n$, the conditional distribution of $\mathsf{Y}_{v_i}(n)$ given $\mathsf{Z}_{v_i}(n)$ is the same as that of $\mathsf{Y}(\mathsf{Z}_{v_i}(n)).$ (This is part of the \emph{Markov branching property} of conditioned Galton-Watson trees, see \cite{HM12}.) Hence, the distributions of all the $\mathsf{Y}_{v_i}(n)$ are mixtures of the distributions of the $(\mathsf{Y}(k),k\in\mathbb{N}),$ which form a tight sequence, and thus are also tight.
\end{proof}
Now add to the tree $\mathsf{T}_m$ each of the $a(\mathsf{T}_m)$ permitted surplus edges independently with probability $p$. Conditionally on $a(\mathsf{T}_m)$ this yields a $\mathrm{Bin}(a(\mathsf{T}_m),p)$ number of surplus edges, for which we write $\mathsf{R}(m)$. Write the tails and heads of these surplus edges as $\mathsf{a}_{i,m}$ and $\mathsf{b}_{i,m}$ respectively, listed in increasing planar order of $\mathsf{a}_{i,m}$, for $i\leq \mathsf{R}(m)$. We also write $\mathsf{b}_{i,m}^-$ for the parent of $\mathsf{b}_{i,m}$ in $\mathsf{T}_m$. Let $\mathsf{W}_i(m)$ be the number of descendants of $\mathsf{b}_{i,m}$. The following lemma is a version of (\ref{eq:tightness}) for $\mathsf{T}_m$.
\begin{lem}\label{lem:notilttightness}
\[
\lim_{K \to \infty} \limsup_{m \to \infty} \; \mathbb{P}\left[\sum_{i=1}^{\mathsf{R}(m)} \mathsf{W}_i(m) > K\right]=0.\]
\end{lem}
\begin{proof} Fix $\varepsilon > 0$. In Lemma 19 of \cite{A-BBG12}, it is proved that $R(m)$ converges in distribution as $m \to \infty$. An identical argument shows that $\mathsf{R}(m)$ converges in distribution as $m \to \infty$ and, in particular, is tight. Therefore, there exists $I>0$ such that $\mathbb{P}[\mathsf{R}(m)>I]<\frac{\varepsilon}{2}$ for all $m.$ Moreover,
\begin{align*}
\mathbb{P} \left[\sum_{i=1}^{\mathsf{R}(m)} \mathsf{W}_i(m)>K \right] &\leq \mathbb{P} \left[\mathsf{R}(m)>I \right] + \mathbb{P}\left[\mathsf{R}(m)\leq I , \sum_{i=1}^{\mathsf{R}(m)} \mathsf{W}_i(m)>K \right] \\
&\leq \frac{\varepsilon}{2} + \mathbb{P} \left[\sum_{i=1}^{\mathsf{R}(m) \wedge I} \mathsf{W}_i(m)>K \right].
\end{align*}
We then split the event where $\sum_{i=1}^{\mathsf{R}(m) \wedge I} \mathsf{W}_i(m)>K$ in two: either, for all $i\leq \mathsf{R}(m) \wedge I$, the vertex $\mathsf{a}_{i,m}$ lies in the largest of the subtrees rooted at the children of $\mathsf{b}_{i,m}^-$, in which case we also have $\sum_{i=1}^{\mathsf{R}(m) \wedge I} \mathsf{Y}_{\mathsf{b}^-_{i,m}}(m)>K$, or there exists $i$ for which $\mathsf{a}_{i,m}$ is \emph{not} in this largest subtree, which then implies, in particular, that $\mathsf{Y}_{\mathsf{b}^-_{i,m}}(m)\geq d(\mathsf{a}_{i,m},\mathsf{b}_{i,m}).$ This leads to
\begin{align*}
& \mathbb{P}\left[\sum_{i=1}^{\mathsf{R}(m)} \mathsf{W}_i(m)> K\right] \\
& \qquad \leq \frac{\varepsilon}{2} + \mathbb{P}\left[\sum_{i=1}^{\mathsf{R}(m) \wedge I} \mathsf{Y}_{\mathsf{b}^-_{i,m}}(m)>K
\right] + \mathbb{P}\left[\exists i \le \mathsf{R}(m) \wedge I: \mathsf{Y}_{\mathsf{b}^-_{i,m}}(m)\geq d(\mathsf{a}_{i,m},\mathsf{b}_{i,m})\right].
\end{align*}
By Lemma~\ref{lem:gwsubtree}, the $(Y_{\mathsf{b}_i^{-}(m)}(m),i\leq \mathsf{R}(m))$ are tight as $m \to \infty$, and thus so is the sum of at most $I$ of them:
\[
\mathbb{P}\left[\sum_{i=1}^{\mathsf{R}(m) \wedge I} \mathsf{Y}_{\mathsf{b}^-_{i,m}}(m)>K\right] \le \frac{\varepsilon}{4}
\]
for all $m$, for $K$ large enough. For the final term, we may again adapt the argument from Lemma 19 of \cite{A-BBG12} to see that for each $i$, $m^{-1/2}d(\mathsf{a}_{i,m},\mathsf{b}_{i,m})$ converges in distribution, where $d$ denotes the graph distance in $\mathsf{T}_m.$ In particular there exists $\eta>0$ such that $\mathbb{P}[d(\mathsf{a}_{i,m},\mathsf{b}_{i,m}) \leq m^{1/2}\eta]\leq \frac{\varepsilon}{8I}.$ We then have
\[\mathbb{P} \left[i\leq \mathsf{R}(m),\mathsf{Y}_{\mathsf{b}^-_{i,m}}(m)\geq d(\mathsf{a}_{i,m},\mathsf{b}_{i,m}) \right]\leq \frac{\varepsilon}{8I} + \mathbb{P} \left[i\leq \mathsf{R}(m),\mathsf{Y}_{\mathsf{b}^-_{i,m}}(m)\geq m^{1/2}\eta \right],\]
and by Lemma \ref{lem:gwsubtree} again, $\mathbb{P}[i\leq \mathsf{R}(m),\mathsf{Y}_{\mathsf{b}^-_{i,m}}(m)\geq m^{1/2}\eta] < \varepsilon/8I$ for all $m$ sufficiently large, so that for such $m$,
\[
\mathbb{P} \left[\exists i\leq \mathsf{R}(m) \wedge I :\; \mathsf{Y}_{\mathsf{b}^-_{i,m}}(m) \geq d(\mathsf{a}_{i,m},\mathsf{b}_{i,m}) \right] \leq \frac{\varepsilon}{4}.
\]
Combining all the terms yields
\[
\limsup_{m \to \infty} \mathbb{P}\left[\sum_{i=1}^{\mathsf{R}(m)} W_i(m)>K\right]\leq \varepsilon. \qedhere
\]
\end{proof}
\begin{proof}[Proof of Proposition~\ref{prop:surplus}]
It remains to show that (\ref{eq:tightness}) holds. We use the change of measure to pass from $\mathsf{T}_m$ to $T_m$. Call $A(m,K)$ the event where $\sum_{i=1}^{R(m)} W_i(m)>K$ and $\mathsf{A}(m,K)$ the event where $\sum_{i=1}^{\mathsf{R}(m)} \mathsf{W}_i(m) >K$. Then we have
\[\mathbb{P}[A(m,K)]= \frac{\mathbb{E}[(1-p)^{-a(\mathsf{T}_m)}\mathbf{1}_{\mathsf{A}(m,K)}]}{\mathbb{E}[(1-p)^{-a(\mathsf{T}_m)}]}
\leq \frac{\sqrt{\mathbb{E}[(1-p)^{-2a(\mathsf{T}_m)}]}}{\mathbb{E}[(1-p)^{-a(\mathsf{T}_m)}]}\sqrt{\mathbb{P}[\mathsf{A}(m,K)]}.\]
We know that $\mathbb{E}[(1-p)^{-2a(\mathsf{T}_m)}]$ is bounded and that $\mathbb{E}[(1-p)^{-a(\mathsf{T}_m)}]$ converges to a positive limit. So by Lemma \ref{lem:notilttightness}, we obtain
\[
\lim_{K \to \infty} \limsup_{m \to \infty} \mathbb{P}[A(m,K)] = 0,
\]
as required.
\end{proof}
\subsection{Proof of Theorem \ref{thm:main}}
We first prove that the convergence in Theorem~\ref{thm:main} occurs in the weaker product topology, namely that for any $k \in \mathbb{N}$,
\[
n^{-1/3} (C_1(n), C_2(n), \ldots, C_k(n)) \ensuremath{\overset{\mathrm{(d)}}\longrightarrow} (\mathcal{C}_1, \mathcal{C}_2, \ldots, \mathcal{C}_k)
\]
with respect to $d_{\vec{\mathcal{G}}}^k$. We will later improve this to a convergence with respect to $d$.
\subsubsection{Convergence in the product topology}
Let $(T_1^n,T_2^n,\ldots)$ be the forward exploration trees of $\vec{G}(n,p).$ We list them in decreasing order of their sizes $(Z_1^n,Z_2^n,\ldots)$, and recall that we write $(\| T_1^n\|, \|T_2^n\|,\ldots)$ for their heights. We also let $(X_1^n,X_2^n,\ldots)$ be the subgraphs of $\vec{G}(n,p)$ induced by the vertex-sets of these trees (which include both surplus and back edges).
By \cite{aldous1997}, we have the following convergence for the $\ell^2$ topology on sequences:
\begin{equation}\label{eq:cvaldous}
n^{-2/3}(Z_i^n,i \in \mathbb{N}) \ensuremath{\overset{\mathrm{(d)}}\longrightarrow} (\sigma_i,i\in\mathbb{N}).
\end{equation}
Again, using Skorokhod's theorem, we may work on a probability space for which this convergence occurs almost surely. Moreover, conditionally on $(Z_1^n, Z_2^n, \ldots)$, the $(X_i^n,i\in\mathbb{N})$ are independent, each having the distribution of $X_{Z_n^i}$ as in Section~\ref{sec:onetree}. Since $Z_n^i p^{2/3} \to \sigma_i$, we have that the rescaled strongly connected components of $X_i^n$ converge in distribution to those of $\mathcal{M}_{\sigma_i}$, and this holds jointly for any finite set of indices $i$. Taking into account Proposition \ref{prop:graphtocc}, the following proposition will give the convergence in Theorem \ref{thm:main} for the product topology.
\begin{prop}\label{prop:aa} For all $k\in\mathbb{N},$ we have
\[\underset{N\to\infty}\lim \, \mathbb{P}[\forall i\leq k,\exists j\leq N:\;\mathcal{C}_i\subseteq \mathcal{D}_j]=1\]
and, for all $n\in\mathbb{N},$
\[\underset{N\to\infty}\lim\, \underset{n\to\infty}\liminf \, \mathbb{P}[\forall i\leq k,\exists j\leq N:\; C_i(n) \subseteq X_j^n]=1.\]
\end{prop}
Informally, Proposition~\ref{prop:aa} states that, with high probability, large strongly connected components of $\vec{G}(n,p)$ and $\mathcal{D}$ will only be found in large trees of the forward depth-first forest, making the ordering of both trees and strongly connected components by their lengths compatible. Its proof relies on two lemmas.
\begin{lem}\label{lem:smallcontinuous}
As $\sigma\to 0,$ we have
\begin{equation}\label{eq:smallcontinuous1}
\mathbb{P}[\mathcal{M}_{\sigma}\text{ has a complex component}]= O(\sigma^3).\end{equation}
For all $\varepsilon>0,$ we have as $\sigma\to 0$
\begin{equation}\label{eq:smallcontinuous2}
\mathbb{P}[\|\mathcal{T}_{\sigma}\|\ge \varepsilon] =O (\sigma^3).\end{equation}
Consequently, for all $\varepsilon>0,$
\begin{equation}\label{eq:smallcontinuous3}
\mathbb{P}\Big[\mathcal{M}_{\sigma}\text{ has a component with length greater than }\varepsilon \Big]= O(\sigma^3).\end{equation}
\end{lem}
\begin{lem}\label{lem:smalldiscrete}
There exists $C>0$ such that, for all $n$ large enough and $1\le m\le n^{2/3},$
\begin{equation}\label{eq:smalldiscrete1}\mathbb{P}\Big[X_m\text{ has a complex component}\Big]\leq C\frac{m^3}{n^{2}}\end{equation}
and
\begin{equation}\label{eq:smalldiscrete15}
\mathbb{P}\Big[X_m\text{ has a component which contains a surplus edge}\Big]\leq C\frac{m^3}{n^{2}}.\end{equation}
Moreover, for all $\varepsilon>0,$ there exists $C>0$ such that, for $n$ large enough, and $1\leq m\leq n^{2/3},$
\begin{equation}\label{eq:smalldiscrete2}
\mathbb{P}[\|T_m\| \geq n^{1/3}\varepsilon]\leq C\frac{m^2}{n^{4/3}}.\end{equation}
Consequently, for all $\varepsilon>0,$ there exists $C>0$ such that, for $n$ large enough,
\begin{equation}\label{eq:smalldiscrete3}\mathbb{P}\Big[X_{m}\text{ has a component with length greater than }n^{1/3}\varepsilon \Big]\leq C\frac{m^2}{n^{4/3}}.\end{equation}
\end{lem}
Note that for both of these lemmas, the final statement is a consequence of the previous ones by noticing that any component consisting of a single ancestral cycle has length smaller than the height of the tree (plus one in the discrete case).
\begin{proof}[Proof of Lemma~\ref{lem:smallcontinuous}]
By Proposition \ref{prop:Poissonbounds},
\[
\mathbb{P}\Big[\mathcal{M}_{\sigma}\text { has a complex component }\varepsilon\Big] \leq \mathbb{P}\big[N_{\sigma}^a\geq 2 \text{ or } N_{\sigma}^b\geq 1] =O(\sigma^3).
\]
Recalling that the height $\|\mathcal{T}_{\sigma}\|$ has the same distribution as $\sup 2\tilde{\mathbf{e}}^{(\sigma)}$ and that it has exponential moments \cite{kennedy76}, we have, for $\sigma<1,$
\begin{align*}
\mathbb{P}[\|\mathcal{T}_{\sigma}\| > \varepsilon]&= \frac{\mathbb{E}\left[\mathbf{1}_{\{\sup \mathbf{e}\geq \varepsilon/2\sqrt{\sigma}\}} \exp \left(\sigma^{3/2}\int_0^1\mathbf{e}(x)dx \right) \right]}{\mathbb{E}\big[\exp \left( \sigma^{3/2}\int_0^1\mathbf{e}(x)dx \right)\big]}\\
&\le \mathbb{E}[e^{\sup \mathbf{e}-\varepsilon/2\sqrt{\sigma}}\,e^{\sup\mathbf{e}}]\leq \mathbb{E}[e^{2\sup \mathbf{e}}]e^{-\varepsilon/2\sqrt{\sigma}}=O(\sigma^3).
\end{align*}
This proves~(\ref{eq:smallcontinuous1}) and~(\ref{eq:smallcontinuous2}); (\ref{eq:smallcontinuous3}) then follows.
\end{proof}
For Lemma \ref{lem:smalldiscrete}, we require some preliminary bounds on the height and area of $T_m$.
\begin{lem}\label{lem:boundsheightarea} There exists a constant $M>0$ such that, for all $n$ large enough such that $1/(2n)<p<2/n$ and all
$1\leq m \leq n^{2/3},$
\begin{align}
\label{eq:heightbound} \mathbb{E}[\|T_m\|^4] & \leq M m^2 \\
\intertext{and}
\label{eq:areabound} \mathbb{E}[(a(T_m))^2] & \leq M m^3.
\end{align}
\end{lem}
\begin{proof} Lemma 25 from \cite{A-BBG12} gives $\mathbb{E}[\|T_m\|^4]\leq M \cdot \max (m^6n^{-4},1) \cdot m^2$ for all $n$ large enough and $m\leq n,$ and restricting ourselves to $m\leq n^{2/3}$ yields (\ref{eq:heightbound}).
For (\ref{eq:areabound}), we follow the beginning of the proof of Lemma 25 from \cite{A-BBG12}. Let $q=\max(m^{-3/2},p)$. Then (\ref{eq:defbias}) and Markov's inequality together yield
\begin{align*}
\mathbb{P}[a(T_m)> xm^{3/2}] &\leq \frac{\mathbb{E}[(1-q)^{-a(T_m)}]}{(1-q)^{-xm^{3/2}}} \leq \frac{\mathbb{E}[((1-p)(1-q))^{-a(\mathsf{T}_m)}]}{(1-q)^{-xm^{3/2}}} \leq \frac{\mathbb{E}[(1-q)^{-2a(\mathsf{T}_m)}]}{(1-q)^{-xm^{3/2}}}.
\end{align*}
From Lemma 14 in \cite{A-BBG12}, we obtain that $\mathbb{E}[(1-q)^{-2a(\mathsf{T}_m)}]\leq K \exp{4\kappa\delta^2}$ where $\delta=\max(2m^{3/2}/n,1).$ Since $qm^{3/2}\geq \delta/4$, we get
\[\mathbb{P}[a(T_m)> xm^{3/2}]\leq Ke^{4\kappa\delta^2-x\delta/4},\]
and for $1\leq m\leq n^{3/2},$ we have $1\leq \delta \leq 2,$ so that
\[\mathbb{P}[a(T_m)> xm^{3/2}]\leq Ke^{64\kappa-x/4}.\]
It follows that
\[
\mathbb{E}\left[\frac{(a(T_m))^2}{m^3} \right]\leq Ke^{64\kappa} \int_0^{\infty} e^{-\sqrt{x}/4}\mathrm d x = 32Ke^{64\kappa},
\]
which completes the proof.
\end{proof}
\begin{proof}[Proof of Lemma \ref{lem:smalldiscrete}]
We take $n$ large enough for (\ref{eq:heightbound}) and (\ref{eq:areabound}) to hold, and $m\leq n^{2/3}.$
Notice first that~(\ref{eq:smalldiscrete2}) follows from (\ref{eq:heightbound}) and Markov's inequality:
\[\mathbb{P}[\|T_m\|\geq n^{1/3}\varepsilon]\leq \frac{\mathbb{E}[\|T_m\|^4]}{\varepsilon^4n^{4/3}}\leq \frac{M m^2}{\varepsilon^4n^{4/3}}.\]
We now want to show that the probability that $X_m$ contains a strongly connected component which is complex or features surplus edges is also bounded by $m^3n^{-2}$. Such a component can only arise if one of the following four events occurs:
\begin{align*}
A_m & = \{X_m \text{ has at least two ancestral back edges.}\} \\
B_m & =\{X_m \text{ has one ancestral back edge, and at least one other back edge which points} \\
& \qquad \text{ inside the created cycle.}\} \\
C_m & =\{X_m \text{ has at least two surplus edges.}\} \\
D_m & =\{X_m \text{ has one surplus edge $(a,b)$ and at least one back edge pointing to} \\
& \qquad \text{ an ancestor of }a.\}
\end{align*}
We will bound the probabilities of each of these events separately.
Conditionally on the tree $T_m$, the number of ancestral back edges in $X_m$ has distribution $\mathrm{Bin}(S_m,p),$ where is $S_m$ the sum of the heights of all vertices in $T_m.$ By using the well-known stochastic domination of $\mathrm{Bin}(k,p)$ by $\mathrm{Poi}(-k\log(1-p))$ and the fact that $\mathbb{P}[\mathrm{Poi}(\mu)\geq 2]\leq \mu^2$, we have
\[\mathbb{P}[A_m\mid T_m]\leq (-S_m \log(1-p))^2 \leq M(S_m p)^2. \]
From now on, the constant $M$ can vary from line to line, but never depends on $n$ or $m.$
Since $S_m\leq m \|T_m\|,$ by using (\ref{eq:heightbound}) again, for $n$ large enough we end up with
\[
\mathbb{P}[A_m ] \leq M\frac{m^2}{n^2}\mathbb{E}[\|T_m\|^2] \leq M\frac{m^3}{n^2}.
\]
Given that there is exactly one ancestral back edge in $X_m$, the number of back edges which point back into the cycle created is stochastically dominated by $\mathrm{Bin}(m\|T_m\|,p).$ Hence we have
\begin{align*}
\mathbb{P}[B_m \mid T_m] &\leq p S_m (1-p)^{S_m-1}(1-(1-p)^{m\|T_m\|}) \\
&\leq M p S_m (m \|T_m\| \log(1-p)) \\
&=M n^{-2}(m \|T_m\|)^2.
\end{align*}
This is the same bound as above, thus leading to
\[
\mathbb{P}[B_m]\leq M\frac{m^3}{n^2}
\]
Since the number of surplus edges has distribution $\mathrm{Bin}(a(T_m),p),$ we get $\mathbb{P}[C_m \mid T_m] \leq M p^2 a(T_m)^2$ and
\[\mathbb{P}[C_m]\leq Mn^{-2}\mathbb{E}[a(T_m)^2].\] A similar argument as for $B_m$ also yields
\[\mathbb{P}[D_m]\leq M n^{-2}m \mathbb{E}[\|T_m\| a(T_m)\mid m]\leq Mn^{-2}m \sqrt{\mathbb{E}[\|T_m\|^2]}\sqrt{\mathbb{E}[a(T_m)^2]},\]
and an application of (\ref{eq:areabound}) concludes the proof.
\end{proof}
We can now prove the proposition.
\begin{proof}[Proof of Proposition~\ref{prop:aa}]
Fix $k\in\mathbb{N}$ and $\eta>0$, and let $\varepsilon>0$ be small enough that
\[
\mathbb{P}[\mathrm{len}(\mathcal{C}_k) >\varepsilon] > 1-\eta.
\]
By Lemmas \ref{lem:smallcontinuous} and \ref{lem:smalldiscrete}, there exists $C>0$ such that
\[\mathbb{P}\Big[\exists i> N: \mathcal{D}_{i}\text{ contains a component with length greater than }\varepsilon \Big]\leq \mathbb{P}[\sigma_{N+1}>1]+C\mathbb{E}\left[\sum_{i> N}\sigma_i^3\right]\]
and
\begin{align*}
& \mathbb{P}\Big[\exists i> N: X_i^n\text{ contains a component with length greater than }n^{1/3}\varepsilon \Big] \\
& \qquad \leq \mathbb{P}[Z_{N+1}^n > n^{2/3}] +C\mathbb{E}\left[\sum_{i> N}\frac{(Z_i^n)^2}{n^{4/3}}\right].
\end{align*}
By Proposition \ref{prop:excursionlengths} and (\ref{eq:cvaldous}), there exists $N$ sufficiently large that both of these are smaller than $\eta.$ Then
\[\mathbb{P}[\mathcal{C}_1, \ldots, \mathcal{C}_k \text{ are in }\mathcal{D}_1,\ldots,\mathcal{D}_N, \ \mathrm{len}(\mathcal{C}_k) > \varepsilon]\geq 1-2\eta.\]
From the fact that $(n^{-1/3}X_1^n,\ldots,n^{-1/3}X_N^n) \ensuremath{\overset{\mathrm{(d)}}\longrightarrow} (\mathcal{D}_1,\ldots,\mathcal{D}_N)$, we deduce that, for $n$ greater than some $n_0\in\mathbb{N}$,
\[\mathbb{P}[C_1(n), \ldots, C_k(n) \text{ are in }X_1^n,\ldots,X_N^n, \ \mathrm{len}(C_k(n)) > \varepsilon n^{1/3}]\geq 1-3\eta\]
and hence
\[\mathbb{P}[C_1(n),\ldots,C_k(n) \text{ are in } X_1^n,\ldots, X_N^n]\geq 1-4\eta. \qedhere
\]
\end{proof}
\subsubsection{Controlling the tail}
The proof of Theorem \ref{thm:main} will be completed if we can show that, for all $\varepsilon>0,$
\[\underset{k\to\infty}\lim \mathbb{P}\left[\sum_{i=k+1}^{\infty} d_{\vec{\mathcal{G}}}(\mathcal{C}_i,\mathfrak{L}) >\varepsilon\right]=0\]
and
\[\underset{k\to\infty}\lim\, \underset{n\to\infty}\limsup\, \mathbb{P}\left[\sum_{i=k+1}^{\infty} d_{\vec{\mathcal{G}}}(C_i(n),\mathfrak{L}) >n^{1/3}\varepsilon\right]=0.\]
Fix $\eta>0.$ For $k\in \mathbb{N},$ let $Q(k)$ be the largest integer such that
\[
\mathbb{P} \left[\text{all components in $\mathcal{D}_1,\ldots,\mathcal{D}_{Q(k)}$ have lengths exceeding $\mathrm{len}(\mathcal{C}_k)$} \right] > 1-\eta.
\]
Then by Proposition~\ref{prop:aa} and the convergence of $n^{-1/3} C_k(n)$ to $\mathcal{C}_k$, it also holds that, for $n$ large enough, all the components of $X_1^n,\ldots,X^n_{Q(k)}$ have lengths exceeding that of $C_k(n)$ with probability at least $1-2\eta.$ Thus we have
\[\mathbb{P}\left[\sum_{i=k+1}^{\infty} d_{\vec{\mathcal{G}}}(\mathcal{C}_i,\mathfrak{L}) >\varepsilon\right]\leq \eta + \mathbb{P}\left[\left(\sum_{i=Q(k)+1}^{\infty}\sum_{j: \, \mathcal{C}_j\subset \mathcal{D}_i}d_{\vec{\mathcal{G}}}(\mathcal{C}_j,\mathfrak{L})\right) >\varepsilon\right]\]
and similarly
\begin{align*}
& \mathbb{P}\left[\sum_{i=k+1}^{\infty} d_{\vec{\mathcal{G}}}(C_i(n),\mathfrak{L}) >n^{1/3}\varepsilon\right] \\
& \qquad \leq 2\eta + \mathbb{P}\left[\left(\sum_{i=Q(k)+1}^{\infty}\sum_{j: \, C_j(n)\subset X_i^n}d_{\vec{\mathcal{G}}}(C_j(n),\mathfrak{L})\right) >n^{1/3}\varepsilon\right].
\end{align*}
Note that $Q(k) \to \infty$ as $k \to \infty$: indeed, it is non-decreasing, and so if did possess a finite limit $Q$, then the probability of $\mathcal{D}_1,\ldots,\mathcal{D}_{Q+1}$ containing a smallest component of $\mathcal{C}$ would be at least $\eta,$ a contradiction since there is no smallest component. It is therefore enough to prove that
\[\underset{N\to\infty}\lim \mathbb{P}\left[\sum_{i=N+1}^{\infty}\sum_{j: \, \mathcal{C}_j\subset \mathcal{D}_i} d_{\vec{\mathcal{G}}}(\mathcal{C}_j,\mathfrak{L}) >\varepsilon\right]=0\]
and
\[\underset{N\to\infty}\lim\, \underset{n\to\infty}\limsup\, \mathbb{P}\left[\sum_{i=N+1}^{\infty} \sum_{j: \, C_j(n)\subset X_i^n} d_{\vec{\mathcal{G}}}(C_j(n),\mathfrak{L}) >n^{1/3}\varepsilon\right]=0.\]
However, by~(\ref{eq:smallcontinuous1}),~(\ref{eq:smalldiscrete1}) and~ (\ref{eq:smalldiscrete15}), for $N$ large enough, all the components contained in $(\mathcal{D}_i,i\ge N+1)$ are single ancestral cycles with probability at least $1-\eta,$ and for $n$ large enough, this also holds for those contained in $(X_i^n,i\ge N+1)$. Noting that such components have length at most the height of the underlying tree (plus one in the discrete case), and that their number is at most the number of ancestral back edges, we are reduced to proving the following statements:
\begin{equation}\label{eq:smalldiamcont}
\underset{K\to\infty}\lim \mathbb{P}\left[\sum_{i=K+1}^{\infty} N_{\sigma_i}^a\|\mathcal{T}_{\sigma_i}\| >\varepsilon\right]=0
\end{equation}
and
\begin{equation}\label{eq:smalldiamdisc}
\underset{K\to\infty}\lim\, \underset{n\to\infty}\limsup\, \mathbb{P}\left[\sum_{i=K+1}^{\infty}A_i^n(1) \|T_i^n\| >n^{1/3}\varepsilon\right]=0,
\end{equation}
where $A_i^n$ is the number of ancestral back edges in $X_i^n.$ These may be obtained using the following lemma.
\begin{lem}
\begin{itemize}
\item[$(i)$] There exists $C>0$ such that, for $\sigma<1,$
\[\mathbb{E}[N_{\sigma}^a \|\mathcal{T}_{\sigma}\|\,]\leq C \sigma^2.\]
\item[$(ii)$] There exists $C>0$ such that, for $n$ large enough, and $1\leq m\leq n^{2/3},$
\[\mathbb{E}\left[A_m \|T_m\|\,\right]\leq C \frac{m^2}{n},\]
where $A_m$ is the number of ancestral back edges in $X_m.$
\end{itemize}
\end{lem}
\begin{proof} Part $(i)$ is straightforward: assuming $\|\mathcal{T}_{\sigma}\|$ and $N_{\sigma}^a$ are built from a tilted excursion $\tilde{\mathbf{e}}^{(\sigma)},$ and remembering that $\mathbb{E}[N_{\sigma}^a\mid \tilde{\mathbf{e}}^{(\sigma)}] = \int_0^\sigma 2\tilde{\mathbf{e}}^{(\sigma)}(t)\mathrm dt$, we have
\begin{align*}
\mathbb{E}[N_{\sigma}^a \|\mathcal{T}_{\sigma}\|\,]&=\mathbb{E} \left[\sup 2\tilde{\mathbf{e}}^{(\sigma)}\int_0^{\sigma}2\tilde{\mathbf{e}}^{(\sigma)}(t)\mathrm dt \right] \\
&\leq 4\sigma\mathbb{E}[\sup (\tilde{\mathbf{e}}^{(\sigma)})^2] \\
&\leq 4\sigma\frac{\mathbb{E}\left[\sup (\sqrt{\sigma}\mathbf{e})^2 \exp \left(\sigma^{3/2}\int_0^1\mathbf{e}(t)dt \right) \right]}{\mathbb{E}\big[\exp \left( \sigma^{3/2}\int_0^1\mathbf{e}(t)dt \right)\big]} \\
&\leq 4\sigma^2 \mathbb{E} \left[e^{\int_0^{1}\mathbf{e}(t)\mathrm dt}\sup (\mathbf{e})^2 \right],
\end{align*}
the latter expectation being finite (see the proof of Lemma~\ref{lem:smallcontinuous}).
For part $(ii),$ recall that, conditionally on $T_m$ the distribution of $A_m$ is stochastically dominated by $\mathrm{Bin}(m \|T_m\|,p).$ Thus, we have
\[\mathbb{E}[A_m \|T_m\|]\leq p\mathbb{E}[m \|T_m\|^2],\]
and applying Lemma \ref{lem:boundsheightarea} concludes the proof.
\end{proof}
We leave the straightforward adaptation of the arguments used for Proposition \ref{prop:aa} to prove (\ref{eq:smalldiamcont}) and (\ref{eq:smalldiamdisc}) to the reader. This completes the proof of Theorem \ref{thm:main}.
\section{Further properties of the scaling limit} \label{sec:furtherprops}
We write $\mathcal{C}$ for the list of strongly connected components of $\mathcal{D},$ and $\mathcal{C}_{\sigma}$ for that of $\mathcal{M}_{\sigma},$ in decreasing order of length. Let also $\mathcal{C}_{\mathrm{cplx}}$ be the list of \emph{complex} components of $\mathcal{C},$ i.e.\ those that are not cycles, also in decreasing order of length. We have not yet been able to find the exact distribution of $\mathcal{C}$ and $\mathcal{C}_{\sigma}$ for $\sigma>0$: this will be the subject of future research. However, we show here that $\mathcal{C}_{\sigma}$ and $\mathcal{C}_{\mathrm{cplx}}$ have a positive probability of being equal to any appropriate fixed family of directed multigraphs.
For sequences $(G_1,\ldots,G_k)$ and $(H_1,\ldots,H_j)$ of directed multigraphs, we write $(G_1,\ldots,G_k)\equiv (H_1,\ldots,H_j)$ if $j=k$ and $G_i$ is isomorphic to $H_i$ for each $i\leq j.$ We extend this notation naturally to the case where one or both of the sequences has edge lengths by simply ignoring the edge lengths.
\begin{prop}\label{prop:fullsupport} Let $G_1,\ldots,G_k$ be a finite sequence consisting of $3$-regular strongly connected directed multigraphs or loops. We have
\[\mathbb{P}[\mathcal{C}_{\sigma}\equiv(G_1,\ldots,G_k)>0].\]
Assuming that $G_1,\ldots,G_k$ are all complex, we also have
\[\mathbb{P}[\mathcal{C}_{\mathrm{cplx}}\equiv (G_1,\ldots,G_k)>0].\]
Let $(e_i,1\leq i\leq K)$ be an arbitrary ordering of the edges of $(G_1,\ldots,G_k).$ Then, conditionally on $\mathcal{C}_{\sigma}\equiv(G_1,\ldots,G_k)$ (resp. $\mathcal{C}_{\mathrm{cplx}}\equiv (G_1,\ldots,G_k)$), $\mathcal{C}_{\sigma}$ (resp. $\mathcal{C}_{\mathrm{cplx}})$ gives lengths $(\ell(e_i),1\leq i\leq K)$ to these edges, and their joint distribution has full support in
\[\left\{\mathbf{x}=(x_1,\ldots,x_K)\in{\mathbb{R}}_+^K: \forall 1\leq i\leq k-1, \sum_{j:e_j\in E(G_i)}x_j \geq \sum_{j:e_j\in E(G_{i+1})}x_j\right\}.\]
\end{prop}
\noindent\textbf{Constructing $3$-regular directed multigraphs from trees and back edges.} First, we want to show that any of the graphs in which we are interested can be constructed by a procedure which adds back edges to a plane tree. We set this up in a discrete framework. Let $\mathsf{t}$ be a discrete plane tree whose vertices have outdegrees in $\{0,1,2\}.$ We think of this as a directed graph, with edges pointing away from the root. We assume that $\mathsf{t}$ has as many leaves as internal vertices of outdegree one, which we call $x_1,\ldots,x_n$ and $y_1,\ldots,y_n$ respectively, in the planar order. We assume, moreover, that for each $i \ge 1$, the internal vertex $y_i$ is visited before the leaf $x_i$ in the depth-first exploration. By identifying $x_i$ and $y_i$ for all $i,$ we obtain a directed graph, whose strongly connected components we then extract. Each strongly connected component will have exactly one vertex of degree $2$, which we erase, merging its two incident edges. The result is a set of $3$-regular strongly connected directed multigraphs. The next lemma asserts that any appropriate collection of such multigraphs can be obtained by this procedure, and Figure \ref{fig6} provides an example.
\begin{figure}[h]
\includegraphics[scale=1]{anygraph.pdf}
\caption{Obtaining a $3$-regular connected directed multigraph from a tree with backward identifications. The tree was built using the method presented in the proof of Lemma \ref{lem:existstree}.}
\label{fig6}
\end{figure}
\begin{lem}\label{lem:existstree} For any $(G_1,\ldots,G_k),$ there exist a discrete plane tree $\mathsf{t}$ and pairings $(x_i,y_i)$ such that the above construction results in $(G_1,\ldots,G_k).$
\end{lem}
\begin{proof}
Notice first that we can focus on the case where $k=1$. Once this case is treated, the general case can be solved by taking a tree $\mathsf{t}$ which contains distinct subtrees corresponding to each $G_i.$
So let $G$ be a fixed strongly connected 3-regular directed multigraph. Noticing that it cannot have vertices with outdegree $0$ or $3$ and that the sum of the outdegrees of all the vertices is equal to that of all the indegrees, we deduce that there exists $n\in\mathbb{N}$ such that $G$ has $n$ vertices with indegree $1$ and outdegree $2$, and $n$ vertices with indegree $2$ and outdegree $1$. Let $a_1,\ldots,a_n$ be the former and $b_1,\ldots,b_n$ the latter, for any ordering such that the edge $(b_1,a_1)$ exists.
We will give a method to construct the necessary plane tree as well as the backward links between leaves and edges. At each step $\mathsf{t}$ will contain a certain number of vertices of $G,$ as well as some ``open" edges, which have their tails at points in $\mathsf{t}$ but are missing their heads.
Start with $\mathsf{t}$ initially containing three vertices: a root with outdegree 1, its child (which we arbitrarily call $\rho_0$) which has outdegree $1$ as well, and its next neighbour $a_1,$ from which originate two open edges. At each step of the algorithm, let $z$ be the leftmost of the deepest vertices of $\mathsf{t}$ which have open edges, choose any edge of $G$ starting at $z$ which is not yet featured in $\mathsf{t}$, call $u$ the head of that edge, and do the following:
\begin{itemize}
\item If $u$ is not already in $\mathsf{t},$ add it at the end of the leftmost open edge, and add one or two open edges at $u$ corresponding to its outdegree in $G$. The edge $(z,u)$ is then a tree edge in $\mathsf{t}.$
\item If $u$ is already in $\mathsf{t}$ but $u\neq a_1,$ add a leaf at the end of the leftmost open edge, label that leaf $x_j$ for the smallest available $j$ and let also $u=y_j.$ The edge $(z,u)$ is then featured in $\mathsf{t}$ as the tree edge $(z,x_j),$ identifying $x_j$ with $u.$
\item If $u=a_1,$ put a leaf at the end of the leftmost open edge, label that leaf $x_j$ for the smallest available $j$, and let $y_j=\rho_0.$ The edge $(z,u)$ is then featured in $\mathsf{t}$ as the merging of the tree edges $(z,x_j)$ and $(\rho_0,a_1),$ identifying $x_j$ with $\rho_0.$
\end{itemize}
Note that this algorithm terminates, and that identifying the pairs $(x_i,y_i)$ in $\mathsf{t}$ and removing the root (which is not in its strongly connected component) and $\rho_0$ (which has degree $2$ in the strongly connected component) gives us $G.$
Moreover, by construction, the successive vertices appearing as $z$ follow the planar ordering of $\mathsf{t}$. This means that at any step, any other vertex of $\mathsf{t}$ can be found earlier than $z$ in the contour process, and thus in every pair $(x_i,y_i),$ the vertex $y_i$ is seen earlier than $x_i$ in the exploration process, and the identifications indeed go backwards. This completes the proof.
\end{proof}
\noindent\textbf{The marked tree has full support.} If $T$ is a discrete plane tree and $\mathcal{T}$ is a discrete plane tree with edge lengths (equivalently an ${\mathbb{R}}$-tree with finitely many leaves which are ordered), we write $\mathcal{T}\equiv T$ if the discrete plane structure underlying $\mathcal{T}$ is $T$. If $\mathcal{T}\equiv T$ then the lengths of the edges of $\mathcal{T}$, in planar order, form a vector in ${\mathbb{R}}_+^k$ where $k$ is the number of edges of $T$.
Let $T$ be a fixed binary rooted discrete plane tree with $n\in\mathbb{N}$ leaves. For an excursion function $f: [0,\sigma]\to {\mathbb{R}}_+,$ we let $D_T(f)$ be the set of increasing sequences $\mathbf{t}=(t_1,\ldots,t_n)\in[0,\sigma]^n$ such that the $\mathcal{T}_f(t_1,\ldots,t_n)\equiv T.$ This is an open subset of $[0,\sigma]^n$ which can be written explicitly as
\begin{align*}
D_T(f)= & \left\{\mathbf{t}\in[0,\sigma]^n: \; t_1<t_2\ldots <t_n\text{ and } \forall k\in\{3,\ldots,n\},\, \right. \\
& \left. \qquad \qquad \qquad \hat{f}(t_{i(k)},t_{k-1})< \hat{f}(t_{k-1},t_k) < \hat{f}(t_{j(k)},t_{k-1}) \right\}.
\end{align*}
Here the indices $i(k)$ and $j(k)$ are defined as follows, and illustrated by Figure \ref{fig7}. Let $L_1,\ldots,L_n$ be the leaves of $T$ in planar order (we add $L_{0}=\rho$ for the sake of convenience). For $k\in \{3,\ldots,n\}$, we then take $i(k)<j(k)$ to be any two integers in $\{0,1,2,\ldots,k-1\}$ such that, on the path $[\hspace{-.10em} [ \rho, L_{k-1} ] \hspace{-.10em}],$ the two points $L_{i(k)} \wedge L_{k-1}$ and $L_{j(k)} \wedge L_{k-1}$ are respectively maximal and minimal such that $L_{i(k)} \wedge L_{k-1}\leq L_{k-1} \wedge L_k \leq L_{j(k)}\wedge L_{k-1}$ for the genealogical/planar order.
\begin{figure}[h]
\centering
\includegraphics[scale=1]{set.pdf}
\caption{For this tree, $i(3)=1$, $j(3)=2$, $i(4)=0$, and $j(4)=1.$ Given an excursion function $f,$ a sequence $t_1<t_2<t_3<t_4$ will then be in $D_T(f)$ iff $\hat{f}(t_1,t_2)<\hat{f}(t_2,t_3)<f(t_2)$ and $0<\hat{f}(t_3,t_4)<\hat{f}(t_1,t_4).$}
\label{fig7}
\end{figure}
\begin{lem}\label{lem:chargetree} We have
\begin{align*}
\mathbb{P}[\mathcal{T}_f^{\mathrm{mk}}=T]= \int_{\mathbf{t}\in D_T(f)}&\mathrm{d}\mathbf{t} \prod_{k=1}^n \left(\sum_{i=1}^k f(t_i) - \hat{f} (t_{i-1},t_i)\right) \\
&\exp{\left(-\int_0^{\sigma} \Bigg(f(t)-\hat{f}(t_{I(t)},t) + \sum_{i=1}^{I(t)}f(t_i)-\hat{f}(t_{i-1},t_i)\Bigg)\mathrm{d}t\right)},
\end{align*}
where $t_0=0$ and, for $t\in[0,\sigma]$, $I(t)=\max\{i: t_i<t\}.$
Moreover, if we take $f=2\tilde{\mathbf{e}}^{(\sigma)}$ for $\sigma>0$, then
\[\mathbb{P}[\mathcal{T}_{\sigma}^{\mathrm{mk}}\equiv T]>0, \]
and conditionally on $\mathcal{T}_{\sigma}^{\mathrm{mk}}\equiv T$, the joint distribution of the edge lengths of $\mathcal{T}_{\sigma}^{\mathrm{mk}}$ has full support in ${\mathbb{R}}_+^{2n-1}.$
\end{lem}
\begin{proof}
The first statement comes from Lemma \ref{lem:explicit}. For the second statement, we use a comparison with the scaling limit of the undirected random graph. Specifically, Lemma 10 of \cite{A-BBG10} gives the joint distribution of the tree shape and the edge lengths in the subtree of $\mathcal{T}_{\sigma}$ spanned by the root and a random collection of leaves obtained as the projection of a Poisson point process on $[0,\sigma]$ with intensity $\tilde{\mathbf{e}}^{(\sigma)}(\cdot)$ onto the tree.\footnote{The sampled leaves in the undirected graph setting come from a Poisson point process with intensity $\tilde{\mathbf{e}}^{(\sigma)}(\cdot)$ rather than the intensity $2\tilde{\mathbf{e}}^{(\sigma)}(\cdot)$ we have in our construction for the directed graph. This is because (as seen in~\cite{A-BBG12}) in the setting of the undirected graph the identifications arise as the limit of the surplus edges: the number of potential surplus edges originating at a single vertex is given not by the height of the vertex but rather by the number of vertices sitting on the stack in the depth-first exploration (the so-called \emph{depth-first walk}). The depth-first walk is asymptotically half the size of the height, and so has scaling limit $\tilde{\mathbf{e}}^{(\sigma)}$ rather than $2\tilde{\mathbf{e}}^{(\sigma)}$.} In particular, the probability that this procedure gives the tree shape $T$ and that the lengths of the edges (in planar order) lie in an open set $A\subset {\mathbb{R}}_+^{2n-1}$ is positive, that is
\[\mathbb{E}\left[\int_{\mathbf{t}\in D_T(2\tilde{\mathbf{e}}^{(\sigma)})}\mathrm{d}\mathbf{t} \mathbf{1}_{\{(2\tilde{\mathbf{e}}^{(\sigma)}(\mathbf{t}))\in A'\}}\prod_{k=1}^n \tilde{\mathbf{e}}^{(\sigma)}(t_k) \exp{\left(-\int_0^{\sigma} \tilde{\mathbf{e}}^{(\sigma)}(t)\mathrm{d}t\right)}\right]>0,\]
where $A'\in {\mathbb{R}}_+^n$ is the open set such that the heights the leaves of $T$ are in $A'$ iff its edge lengths are in $A$.
This implies that $\mathbb{E}[G]>0$ where
\[
G=\int_{\mathbf{t}\in D_T(2\tilde{\mathbf{e}}^{(\sigma)})}\mathrm d \mathbf t\mathbf{1}_{\{(2\tilde{\mathbf{e}}^{(\sigma)}(\mathbf{t}))\in A'\}}\prod_{k=1}^n \left(\sum_{i=1}^k 2 \tilde{\mathbf{e}}^{(\sigma)}(t_i) \right),
\]
(since $G$ is larger than the random variable in the expectation above). We then have
\begin{align*}
\mathbb{P}[\mathcal{T}_{\sigma}^{\mathrm{mk}}\equiv T,\text{lengths in }A] &\geq \mathbb{E}\left[G\exp{(-\sigma(n+1)\sup \tilde{\mathbf{e}}^{(\sigma)})}\right],
\end{align*}
and this is positive since $\sup \tilde{\mathbf{e}}^{(\sigma)}$ is a.s.\ finite.
\end{proof}
\begin{proof}[Proof of Proposition \ref{prop:fullsupport}]
We first show the result for $\mathcal{C}_{\sigma}$. Let $\mathsf{t}$ and $((x_i,y_i),i\in\{1,\ldots,n\})$ be the discrete tree and pairing of leaves and outdegree-$1$ vertices given by Lemma \ref{lem:existstree}. Moreover, let $T$ be obtained from $\mathsf{t}$ by erasing the vertices of degree $2$, and merging their adjacent edges. Let
\begin{itemize}
\item $(e_1,\ldots,e_K)$ be the edges of $(G_1,\ldots,G_k),$ in any order;
\item $(e_1,\ldots,e_K,{\mathrm e}_{K+1},\ldots,e_N)$ be those of $\mathsf{t},$ in any order completing the previous one;
\item $(f_1,\ldots,f_M)$ be those of $T,$ in planar order.\footnote{Note that in fact we have $M=2n-1$, $N=3n-1$ and $K=3(n-k)+k',$ where $k'$ is the number of unicycles amongst $(G_1,\ldots,G_k)$; however, this fact is not useful here.}
\end{itemize}
By construction, each edge of $(G_1,\ldots,G_k),$ is an edge of $\mathsf{t}$, justifying the notation for the edges of $\mathsf{t}.$ Moreover, each edge of $T$ is obtained by merging edges of $\mathsf{t}$, so there exists a partition of $\{1,\ldots,N\}$ with blocks $(S(i), 1 \le i \le M)$ such that for each $i\in\{1,\ldots,M\}$, $f_i$ is obtained by merging $e_j$ for $j\in S(i).$ For $i\in\{1,\ldots,n\}$, let $e_T(y_i)$ be the edge of $T$ containing $y_i$.
Given this information, we call a collection of positive lengths $\ell(e_i)$, and $\ell(f_i)$
such that $\ell(f_i)=\sum_{j\in S(i)} \ell(e_j)$ an \emph{admissible length assignment}.
Recall that, from the construction given in Section \ref{sec:continuousbackedges}, conditionally on $\mathcal{T}_{\sigma}^{\mathrm{mk}}$ with leaves $L_1,\ldots,L_p,$ the marked internal points $z_1,\ldots,z_p$ are independent and, for each $j$, $z_j$ is uniform on $\cup_{k=1}^j [\hspace{-.10em} [ \rho,L_k ] \hspace{-.10em}].$ If $\mathcal{T}_{\sigma}^{\mathrm{mk}}\equiv T$ then this gives rise to a length assignment $\ell$ on $T,$ and we have
\[\mathbb{P}\left[z_j \in e_T(y_j),\, \forall j\in\{1,\ldots,n\}\mid \mathcal{T}_{\sigma}^{\mathrm{mk}}\equiv T\right]\geq
\prod_{j=1}^n\frac{\ell(g(y_j))}{\mathrm{len}(\mathcal{T}_{\sigma}^{\mathrm{mk}})}.\]
Moreover, conditionally on the event $\{z_j \in e_T(y_j),\, \forall j\in\{1,\ldots,n\}, \mathcal{T}_{\sigma}^{\mathrm{mk}}\equiv T\}$, for any edge $f_i$ of $T$, the probability that $z_j$, for $j$ such that $y_j\in f_i$, are in the right order on $f_i$ is $\frac{1}{|S(i)|!}.$ If this occurs, then it gives rise to a length assignment $\ell$ on $\mathsf{t}$ as well, making the whole length assignment admissible. We then have $(\ell(e_j),j\in S(i))=(D_1(i)\ell(f_i),\ldots,D_{|S(i)|}(i)\ell(f_i))$ where $D(i)=(D_1(i),\ldots,D_{|S(i)|}(i))\in \Delta_{|S(i)|}$ has the Dirichlet$(1,\ldots,1)$ distribution on the $(|S(i)|-1)$-dimensional simplex $\Delta_{|S(i)|}$. These events occur independently for different $i\in\{1,\ldots,M\}.$
Let $A$ be an open set in ${\mathbb{R}}_+^K.$ Take open sets $B\subset {\mathbb{R}}_+^M$ and $C_i\in \Delta_{|S(i)|}$ for $i\in \{1,\ldots,M\}$ such that, for any admissible length assignment, if $(\ell(f_i),i\in\{1,\ldots,M\})\in B$ and, for all $i$, $\left(\frac{\ell(e_j)}{\ell(f_i)},j\in S(i)\right)\in C_i$, then we have $\ell(e_i),i\in\{1,\ldots,K\})\in A.$
Then
\begin{align*}
\mathbb{P}\Big[C_{\sigma}\equiv (G_1,\ldots,G_k), \ & (\ell(e_i),i\in\{1,\ldots,K\})\in A\Big] \\
&\geq\mathbb{E}\left[\mathbf{1}_{\{\mathcal{T}_{\sigma}^{\mathrm{mk}}\equiv T,(\ell(f_i),i\in\{1,\ldots,M\})\in B\}}\prod_{j=1}^n\frac{\ell(f(y_j))}{\mathrm{len}(\mathcal{T}_{\sigma}^{\mathrm{mk}})}\prod_{i=1}^M \, \frac{1}{|S(i)|!}\mathbf{1}_{\{D(i)\in C_i\}}\right].
\end{align*}
By Lemma \ref{lem:chargetree}, the event $\{\mathcal{T}_{\sigma}^{\mathrm{mk}}\equiv T,(\ell(f_i),i\in\{1,\ldots,M\})\in B\}$ occurs with positive probability, and since Dirichlet distributions charge the full simplex, we do indeed have that
\[\mathbb{P}\Big[C_{\sigma}\equiv (G_1,\ldots,G_k),(\ell(e_i),i\in\{1,\ldots,K\})\in A\Big]>0.\]
We finally turn to the result for $\mathcal{C}_{\mathrm{cplx}}$. Recall that $(\sigma_i,i\geq 1)$ are the ranked excursion lengths of a Brownian motion with parabolic drift and that, conditionally on the lengths, $\mathcal{C}_i, i\geq 1$ are independent copies of $\mathcal{C}_{\sigma_i}.$ Notice that
\begin{align*}
& \mathbb{P}[\mathcal{C}_{\mathrm{cplx}}\equiv(G_1,\ldots,G_k), \text{ lengths in }A] \\
& \qquad \geq \mathbb{P}[\mathcal{C}_{1}\equiv(G_1,\ldots,G_k),\text{lengths in }A ,\, \mathcal{C}_i \text{ has no complex components } \forall i\geq 2].
\end{align*} From Propositions \ref{prop:excursionlengths} and \ref{prop:Poissonbounds}, we deduce that $(\mathcal{C}_i, i\geq 2)$ has no complex components with positive probability. An application of the first part of the proposition then completes the proof.
\end{proof}
\section*{Acknowledgements}
This research was supported by EPSRC Fellowship EP/N004833/1. We would like to thank Nicolas Broutin and Julien Berestycki for helpful discussions. We are very grateful to \'Eric Brunet for the proof of Proposition~\ref{prop:excursionlengths} $(ii)$.
\bibliographystyle{abbrv}
| {
"timestamp": "2019-10-31T01:03:28",
"yymm": "1905",
"arxiv_id": "1905.05397",
"language": "en",
"url": "https://arxiv.org/abs/1905.05397",
"abstract": "We consider the random directed graph $\\vec{G}(n,p)$ with vertex set $\\{1,2,\\ldots,n\\}$ in which each of the $n(n-1)$ possible directed edges is present independently with probability $p$. We are interested in the strongly connected components of this directed graph. A phase transition for the emergence of a giant strongly connected component is known to occur at $p = 1/n$, with critical window $p= 1/n + \\lambda n^{-4/3}$ for $\\lambda \\in \\mathcal{R}$. We show that, within this critical window, the strongly connected components of $\\vec{G}(n,p)$, ranked in decreasing order of size and rescaled by $n^{-1/3}$, converge in distribution to a sequence $(\\mathcal{C}_1,\\mathcal{C}_2,\\ldots)$ of finite strongly connected directed multigraphs with edge lengths which are either 3-regular or loops. The convergence occurs the sense of an $\\ell^1$ sequence metric for which two directed multigraphs are close if there are compatible isomorphisms between their vertex and edge sets which roughly preserve the edge-lengths. Our proofs rely on a depth-first exploration of the graph which enables us to relate the strongly connected components to a particular spanning forest of the undirected Erdős-Rényi random graph $G(n,p)$, whose scaling limit is well understood. We show that the limiting sequence $(\\mathcal{C}_1,\\mathcal{C}_2,\\ldots)$ contains only finitely many components which are not loops. If we ignore the edge lengths, any fixed finite sequence of 3-regular strongly connected directed multigraphs occurs with positive probability.",
"subjects": "Probability (math.PR); Combinatorics (math.CO)",
"title": "The scaling limit of a critical random directed graph",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.985496421586532,
"lm_q2_score": 0.7185943985973772,
"lm_q1q2_score": 0.7081722083898412
} |
https://arxiv.org/abs/1711.11369 | On the normalized $p$-parabolic equation in arbitrary domains | The boundary regularity for the normalized $p$-parabolic equation $u_t =\frac{1}{p}|Du|^{2-p}\Delta_pu$ is studied. Perron's method is used to construct solutions in arbitrary domains. We classify the regular boundary points in terms of barrier functions, and prove an Exterior Sphere result. A fundamental solution is identified. A Petrovsky criterion is established, and we examine the convergence of solutions as $p \to \infty$. |
\section{Introduction}
\noindent We investigate Perron solutions of the \emph{normalised} $p$-parabolic equation
\begin{equation}
\label{eq:evoplap}
\begin{split}
u_t &=\frac{1}{p}|Du|^{2-p}\Delta_pu \\
&=\frac{1}{p}\text{tr}(D^2u) +\frac{p-2}{p}\left\langle D^2u\frac{Du}{|Du|}, \frac{Du}{|Du|} \right\rangle,
\end{split}
\end{equation}
where $1<p <\infty$, in general domains $\Omega \subset \mathbb{R}^n\times (-\infty, \infty)$. Here $\Delta_pu$ denotes the $p$-Laplace operator of $u$,
\[
\Delta_pu=\text{div}(|Du|^{p-2}Du),
\]
and we let
\[
\mathcal{A}_pu:=\frac{1}{p}|Du|^{2-p}\Delta_pu.
\]
The operator $\mathcal{A}_p$ is called normalised since it is homogeneous of degree 1, that is $\mathcal{A}_p(\alpha u)= \alpha \mathcal{A}_pu$. In contrast, the $p$-Laplace operator is homogeneous of degree $p-1$.
Since the equation cannot be written on divergence form, the distributional weak solutions are not available to us. The correct notion is the viscosity solutions, introduced in \cite{crandall1983viscosity}.
Sternberg \cite{sternberg1929gleichung} observed that Perron's method for solving the Dirichlet boundary value problem for Laplace's equation in \cite{perron1923neue} could be extended to the heat equation. We adapt Perron's method to the non-linear equation \eqref{eq:evoplap} in general domains, not necessarily space-time cylinders.
Equation \eqref{eq:evoplap} was studied by Does in connection with image processing, see \cite{does2009evolution}. The existence of viscosity solutions on cylinders $Q_T=Q \times (0, T )$ was established by Perron's method.
Integral to the potential theory is the regularity of boundary points. A point $\zeta_0\in \partial \Omega$ is called regular if, for \emph{every} continuous function $f: \partial \Omega \to \mathbb{R}$, we have
\[
\lim_{\eta \to \zeta_0}u(\eta)=f(\zeta_0), \ \eta \in \Omega.
\]
The boundary values are prescribed as for an elliptic problem. For example, the points on $Q \times \{t=T\}$ are not regular boundary points of the cylinder $Q_T$.
We characterize the regularity of boundary points in terms of \emph{barriers}. We say that $w$ is a barrier at $\zeta$ if $w$ is a positive supersolution of \eqref{eq:evoplap} defined on the entire domain, such that $w(\zeta)=0$ and $w(\eta)>0$ for $\zeta \neq \eta\in \partial \Omega$.
To keep the presentation within reasonable limits, we investigate only the case where the boundary data $f$ is bounded. This is not a serious restriction.
The related equation
\begin{equation}
\label{eq:banerjeeeq}
u_t=|Du|^{2-p}\Delta_pu,
\end{equation}
without the factor $1/p$ present, was investigated in \cite{banerjee2015dirichlet}.
For $p \geq 2$, it was proved that a boundary point $\zeta$ is regular if, and only if, there exists a barrier at $\zeta$. The authors also showed that in the case of space - time cylinder $Q_T$, $(x, t) \in \partial Q\times (0, T]$, is a regular boundary point if and only if $x \in \partial Q$
is a a regular boundary point for the elliptic $p$-Laplacian.
The regularity of a point is a very delicate issue. Using the Petrovsky criterion, one can construct a domain where the origin is regular for the equation $u_t =\Delta u$, while it is \emph{irregular} for $u_t =\frac{1}{2}\Delta u$, cf \cite{watson2012introduction}. Therefore it is quite remarkable that as $p \to \infty$, the domain in our Petrovsky criterion converges precisely to the domain in the Petrovsky criterion for the normalised $\infty$-parabolic equation \eqref{eq:norminftylaplace} derived in \cite{ubostad1}, namely that the origin is an irregular point for the domain enclosed by the hypersurfaces
\begin{equation}
\label{eq:infinitypetrovsky}
\begin{split}
&\{(x, t)\in \mathbb{R}^n\times (0, \infty)\ : \ |x|^2 =-4t\log|\log|t||\} \\ &\text{and} \ \{t=-c\},
\end{split}
\end{equation}
for $0<c<1$.
We consider \eqref{eq:evoplap} instead of \eqref{eq:banerjeeeq} because of the convergence properties as $p \to \infty$. Results similar to ours could easily be established for \eqref{eq:banerjeeeq} by the same methods.
Indeed, this was recently proven in \cite{bjorn2017tusk} .
The regular, non-normalised $p$-parabolic equation
\begin{equation}
\label{eq:pparabolic}
u_t = \Delta_pu,
\end{equation}
$1 < p <\infty$, has an interesting history. The initial value problem was first studied by Barenblatt in connection with the propagation of heat after thermonuclear detonations in the atmosphere, cf. \cite{barenblatt}. The equation has several applications, for example
in image processing with variable $p=p(x)$ in \cite{imageenhancement}. The $p$-parabolic equation, together with its stationary counterpart the $p$-Laplace equation, also have interesting applications related to game theory and "Tug-of-War'' games, see \cite{manfredi2010asymptotic} and \cite{peres2008tug}. For the regularity theory regarding equations of this type, we mention \cite{dibenedetto1995}.
For $p=1$, the equation is connected to motion by mean curvature, investigated by Evans and Spruck in \cite{evans1991motion}. The Dirichlet problem in general domains for \eqref{eq:pparabolic} was studied by Kilpeläinen and Lindqvist in \cite{kilpelainen1996dirichlet}. See also \cite{Juutinen2006, crandall2003another}.
Also worth mentioning is the case $p=\infty$, when we get the $\infty$-parabolic equation
\begin{equation}
\label{eq:inftylaplace}
u_t=\Delta_{\infty}u,
\end{equation}
and the related \emph{normalised} $\infty$-parabolic equation
\begin{equation}
\label{eq:norminftylaplace}
u_t=|Du|^{-2}\Delta_{\infty}u := \Delta_{\infty}^Nu,
\end{equation}
where the $\infty$-Laplace operator is given by
\[
\Delta_{\infty} u =\left\langle D^2u \ Du, Du \right\rangle.
\]
Both of these have an interesting theory in their own right, see \cite{Juutinen2006} regarding the normalised case and \cite{crandall2003another} for \eqref{eq:inftylaplace}.
As $p \to \infty$, \eqref{eq:evoplap} converges to \eqref{eq:norminftylaplace} and \emph{not} to \eqref{eq:inftylaplace}.
Our first result is a characterization of regular boundary points via exterior spheres:
\begin{theorem}[Exterior Sphere]
\label{thm:exterior sphere}
Let $\zeta_0 =(t_0, x_0) \in \partial \Omega$, and suppose that there exists a closed ball $\{(x, t) \ : \ |x-x'|^2+(t-t')^2 \leq R_0^2\}$ intersecting $\overline{\Omega}$ precisely at $\zeta_0$. Then $\zeta_0$ is regular, if the intersection point is \underline{not} the south pole, that is $(x_0, t_0) \neq (x', t'-R_0)$. If the point of intersection is the north pole, we must restrict the radius of the sphere.
\end{theorem}
We use a barrier function to prove the following Petrovsky criterion:
\begin{theorem}
\label{thm:petrovsky}
The origin $(0, 0)$ is a regular point for the domain enclosed by the hypersurfaces
\begin{equation*}
\{(x, t)\in \mathbb{R}^n\times (0, \infty)\ : \ |x|^2 =-\beta t\log|\log|t||\} \ \text{and} \ \{t=-c\},
\end{equation*}
for $0<c<1$, where
\[
\beta = 4 \ \frac{p-1}{p}.
\]
\end{theorem}
We also have the following irregularity result, showing that Theorem \ref{thm:petrovsky} is in some sense sharp:
\begin{theorem}
\label{thm:not regular}
The origin is \emph{not} a regular point of the domain $\Omega$ defined by
\[
|x|^2 =-\beta(1+\epsilon)t\log|{\log{|t|}}|, \ \ t=-c
\]
for \emph{any} $\epsilon >0$, and $\beta$ as in Theorem \ref{thm:petrovsky}.
\end{theorem}
These results are similar to the classical Petrovsky criterion for the heat equation, derived in \cite{petrovsky1935ersten}.
The article is structured as follows. In Section \ref{sec:severalsolutions} we investigate several explicit solutions of \eqref{eq:evoplap}. We also transform it into a heat equation with variable coefficient. Basic facts regarding viscosity solutions, Perron solutions, a comparison principle and the barrier characterization are displayed in Section \ref{sec:comparison}. The exterior sphere condition in Theorem \ref{thm:exterior sphere} is derived in Section \ref{sec:exterior sphere}. As a demonstration of the necessity of eliminating the south pole, we show that the latest moment of the $p$-parabolic ball is not regular. Section \ref{sec:petrovsky} is dedicated to the Petrovsky criterion, that is the proof of Theorem \ref{thm:petrovsky} and Theorem \ref{thm:not regular}.
\subsection{Notation}
In what follows $\Omega$ is an \emph{arbitrary} domain in $\mathbb{R}^n \times (-\infty, \infty)$. $Q_T$ is a space-time cylinder: $Q_T = Q\times (0, T)$, $\partial \Omega$ is the Euclidean boundary of $\Omega$ and $\partial_pQ_T$ is the \emph{parabolic} boundary of $Q_T$, i.e. $(\overline{Q}\times \{0\}) \cup (\partial Q\times (0, T])$. (the "bottom" and the sides of the cylinder. The top is excluded.) $\zeta, \eta \in \mathbb{R}^n \times \mathbb{R}$ are points in space-time, that is $\zeta =(x, t)$.
We denote by $Du$ the gradient of $u(x, t)$ taken with respect to the spatial coordinates $x$, and $D^2u$ is the spatial Hessian matrix of $u$. $\langle a, b \rangle$ is the Euclidean inner product of the vectors $a, b \in \mathbb{R}^n$.
and $x \otimes y$ denotes the tensor product of the vectors $x, y$, that is $(x\otimes y)_{i, j}=x_iy_j$.
The space of lower semi-continuous functions from $\Omega$ to $\mathbb{R}\cup \{\infty\}$ is denoted by $\operatorname{LSC}(\Omega)$, while $\operatorname{USC}(\Omega)$ contains the upper semicontinuous ones.
\section{Several solutions}
\label{sec:severalsolutions}
\noindent We derive several explicit solutions to \eqref{eq:evoplap}, and identify the fundamental solution.
\subsection{Uniform propagation}
Assume $u(x, t) = w(\langle a, x \rangle-bt)$, $a\in \mathbb{R}^n$, $b\in \mathbb{R}$. We then get
\begin{align*}
& u_t =-bw' ,\\
& u_{x_i} =a_iw', \\
& u_{x_ix_j}=a_ia_jw'',
\end{align*}
and hence
\[
\Delta u =\Delta_{\infty}^N =|a|^2w''.
\]
Inserting this into \eqref{eq:evoplap}, we get that $w$ must satisfy
\[
w'+w''|a|^2\frac{p-1}{bp} =0,
\]
with solution
\[
w(\zeta) = A+B\operatorname{e}^{-\frac{\zeta}{m}},
\]
or
\[
u(x, t) = A+B\operatorname{e}^{-\frac{1}{m}(\langle a, x \rangle-bt)},
\]
with $m =|a|^2\frac{p-1}{bp}$.
and $\zeta =\langle a ,x \rangle-bt$.
\subsection{Separable solution}
Assume $u(x, t)=f(r)+g(t)$, $r=|x|$. We get
\begin{align*}
u_t =g', \\
\Delta u = f''+\frac{n-1}{r}f', \\
\Delta_{\infty}^Nu = f''.
\end{align*}
Thus
\[
g'(t) =c,
\]
and
\[
f''(r)+\frac{n-1}{(p-1)r}f'(r) =c
\]
So
\[
f(r)= \frac{1}{2}\frac{cp}{n+p}r^2 + c_1\frac{p-1}{p-n}r^{\frac{p-n}{p-1}}+c_2,
\]
Setting $c_1=c_2=0$, we get the solution
\[
u(x, t) =c\left(\frac{p}{n+p}|x|^2+2\frac{p-1}{p-n}|x|^{\frac{p-n}{p-1}}+2t\right).
\]
\subsection{Heat Equation transformation}
We search for solutions on the form
$u(x, t) =v(r^{\nu}, t)$, where $r=|x|$ and $\nu$ is a critical exponent to be determined. This gives
\begin{align*}
&\Delta u = \frac{\partial^2}{\partial r^2}v(r^{\nu}, t) + \frac{n-1}{r}\frac{\partial}{\partial r}v(r^{\nu}, t), \\
&\Delta_{\infty}^Nu = \frac{\partial^2}{\partial r^2}v(r^{\nu}, t).
\end{align*}
Calculating, we get
\begin{align*}
&\frac{\partial}{\partial r}v(r^{\nu}, t) = \nu r^{\nu-1}v', \\
&\frac{\partial^2}{\partial r^2}v(r^{\nu}, t) = \nu(\nu-1)r^{\nu-2}v'+\nu^2r^{2\nu-2}v'',
\end{align*}
where
\[
v' =\frac{\partial v}{\partial \rho}, \ \rho =r^{\nu}
\]
Hence we get
\begin{align*}
&\Delta u = \nu(\nu-1)r^{\nu-2}v'+\nu^2r^{2\nu-2}v''+\nu r^{\nu-2}v', \\
&\Delta_{\infty}^Nu = \nu(\nu-1)r^{\nu-2}v'+\nu^2r^{2\nu-2}v''.
\end{align*}
Inserting this into \eqref{eq:evoplap} and collecting terms we get
\begin{equation}
\label{eq:almostheat}
v_t= \nu^2\frac{p-1}{p}r^{2\nu-2}v''+\frac{\nu}{p}r^{\nu-2}[(p-2)(\nu-1)+n+\nu]v'.
\end{equation}
We want to eliminate the first order terms in \eqref{eq:almostheat}, so we demand
\[
(p-2)(\nu-1)+n+\nu = 0,
\]
or
\[
\nu = \frac{p-n}{p-1}.
\]
Then \eqref{eq:evoplap} reads
\[
v_t =\frac{(p-n)^2}{p(p-1)}\rho^{\frac{2-n}{p-n}}v_{\rho \rho},
\]
where
\[
\rho = |x|^{\frac{p-n}{p-1}}.
\]
\subsection{Similarity I}
We make the ansatz
\[
u(x, t) =F(\zeta), \ \ \zeta =\frac{|x|^2}{t},
\]
We calculate
\begin{align*}
& u_t =-\frac{F'(\zeta)\zeta}{t}, \\
& Du =\frac{2F'(\zeta)x}{t}, \\
& D^2u =\frac{2F'(\zeta)}{t}I+\frac{4F''(\zeta)}{t^2}(x \otimes x).
\end{align*}
Hence
\begin{align*}
\text{tr}(D^2u) =\frac{2F'(\zeta)}{t}n+\frac{4F''(\zeta)}{t^2}|x|^2, \\
\Delta_{\infty}^Nu =\frac{2F'(\zeta)}{t}+\frac{4F''(\zeta)}{t^2}|x|^2.
\end{align*}
So, if $u$ is a solution then
\[
-\frac{F'(\zeta)\zeta}{t}-\alpha \frac{F'(\zeta)}{t}- \beta\frac{F''(\zeta)\zeta}{t} =0,
\]
with $\alpha, \beta$ as in \eqref{eq:bestsolution}. If $t \neq 0$;
\[
\frac{F''(\zeta)}{F'(\zeta)}=\frac{d}{d\zeta}\log F'(\zeta) = -\frac{\alpha}{\beta \zeta} -\frac{1}{\beta}.
\]
Integrating the above gives
\[
\log F'(\zeta) =-\frac{\zeta}{\beta}-\frac{\alpha}{\beta}\log\zeta,
\]
or
\[
F(\zeta) = C\int_0^{\zeta}s^{-\frac{\alpha}{\beta}}\operatorname{e}^{-\frac{s}{\beta}} \ ds.
\]
This leads to the solution
\begin{equation}
\label{eq:tardsoln}
u(x, t) = C\int_{0}^{|x|^2/t}s^{-\frac{\alpha}{\beta}}\operatorname{e}^{-\frac{s}{\beta}} \ ds.
\end{equation}
This solution is not differentiable where $x=0$. However, \eqref{eq:tardsoln} is a solution outside the line $\{0\}\times(0, \infty)\subset \mathbb{R}^n\times (0, \infty)$, and a subsolution or supersolution depending on the sign of $C$ in all of $\mathbb{R}^n\times (0, \infty)$.
\subsection{Similarity II}
\label{sec:similarity1}
We note that if $u(x, t)$ is a solution of \eqref{eq:evoplap}, then so is $v(x, t) =u(Ax, A^2t)$. We search for solutions on the form
\[
u(x, t) =g(t)f(\zeta), \ \ \zeta =\frac{|x|^2}{t}.
\]
Inserting this into \eqref{eq:evoplap}, we get
\[
u_t =g'(t)f(\zeta) -\frac{g(t)f'(\zeta)\zeta}{t},
\]
\[
Du = \frac{2g(t)f'(\zeta)}{t}x,
\]
and
\[
D^2u = \frac{2g(t)f'(\zeta)}{t} I +\frac{4g(t)f''(\zeta)}{t^2}(x\otimes x).
\]
Hence we see
\[
\text{tr}(D^2u) = \frac{2g(t)f'(\zeta)}{t} n+\frac{4g(t)f''(\zeta)}{t^2}|x|^2,
\]
and
\[
\Delta_{\infty}^Nu = \frac{2g(t)f'(\zeta)}{t} +\frac{4g(t)f''(\zeta)}{t^2}|x|^2.
\]
Therefore, $u$ is a solution to \eqref{eq:evoplap} if
\[
g'(t)f(\zeta)-\frac{g(t)f'(\zeta)\zeta}{t} =2\frac{p+n-2}{p}\frac{g(t)f'(\zeta)}{t}+4\frac{p-1}{p}\frac{g(t)f''(\zeta)\zeta}{t}.
\]
with $\alpha = 2\ \frac{p+n-2}{p}$ and $\beta =4\ \frac{p-1}{p}$,
\[
tg'(t)f(\zeta)-\alpha g(t)f'(\zeta) =g(t)\zeta(f'(\zeta)+\beta f''(\zeta)).
\]
for $t>0$. The right hand side of this is zero if $f(\zeta)=\operatorname{e}^{-\frac{\zeta}{\beta}}$. Inserting this back in, we see that
\[
f(\zeta) \left(tg'(t)+\frac{\alpha}{\beta}g(t)\right) =0,
\]
with solution
\[
g(t) =t^{-\frac{\alpha}{\beta}}.
\]
Together, this gives
\begin{equation}
\label{eq:bestsolution}
\begin{split}
& u(x, t) =t^{-\frac{\alpha}{\beta}}\operatorname{e}^{-\frac{|x|^2}{\beta t}}, \\
&\alpha = 2\ \frac{p+n-2}{p}, \ \beta =4\ \frac{p-1}{p}.
\end{split}
\end{equation}
This is a solution for $t>0$, and if we replace $t$ by $-t$ we get a solution for negative $t$, as well.\footnote{It has recently come to the author's attention that this solution also was found in \cite{banerjee2013gradient}.}
\begin{remark}
As $p\to \infty$, $ \alpha \to 2 $ and $\beta \to 4$. This gives that \eqref{eq:bestsolution} converges to the fundamental solution
\[
W(x, t) =\frac{1}{\sqrt{t}}\operatorname{e}^{-\frac{|x|^2}{4t}}.
\]
of the normalised $\infty$-parabolic equation found in \cite{Juutinen2006}. Compare also \eqref{eq:bestsolution} with the fundamental solution to the heat equation,
\[
H(x, t) = \frac{1}{(4\pi t)^{n/2}}\mathrm{e}^{-\frac{|x|^2}{4t}}.
\]
\end{remark}
\section{Comparison and Perron solutions}
\label{sec:comparison}
\noindent In this Section we present several basic facts regarding the existence of solutions to \eqref{eq:evoplap}, and present Perron's method. We start with the definition of viscosity solutions. If $Du=0$, we replace the operator $\mathcal{A}_p$ with its lower or upper semicontinuous envelope:
\begin{definition}
\label{def:viscsuper}
A lower semicontinuous function $u \in L^{\infty}(\Omega)$ is a \emph{viscosity supersolution} of \eqref{eq:evoplap} provided that, if $ u - \phi$ has a minimum at $\zeta_0 \in \Omega$ for $\phi \in C^2 (\Omega)$, then
\[
\begin{cases}
\phi_t(\zeta_0)-\mathcal{A}_p\phi(\zeta_0) \geq 0, \ &\text{if} \ D\phi(\zeta_0)\neq 0, \\
\phi_t(\zeta_0) -\frac{1}{p}\text{tr}(D^2\phi(\zeta_0))-\frac{p-2}{p}\lambda(D^2\phi(\zeta_0))\geq 0, \ &\text{if} \ D\phi(\zeta_0)=0.
\end{cases}
\]
An upper semicontinuous function $u \in L^{\infty}(\Omega)$ is a \emph{viscosity subsolution} of \eqref{eq:evoplap} provided that, if $ u - \phi$ has a maximum at $\zeta_0 \in \Omega$ for $\phi \in C^2 (\Omega)$, then
\[
\begin{cases}
\phi_t(\zeta_0)-\mathcal{A}_p\phi(\zeta_0) \leq 0, \ &\text{if} \ D\phi(\zeta_0)\neq 0, \\
\phi_t(\zeta_0) -\frac{1}{p}\text{tr}(D^2\phi(\zeta_0))-\frac{p-2}{p}\Lambda(D^2\phi(\zeta_0))\leq 0, \ &\text{if} \ D\phi(\zeta_0)=0.
\end{cases}
\]
A function that is both a viscosity sub- and supersolution is called a \emph{viscosity solution}.
Here $\lambda(D^2\phi(\zeta_0))$, $\lambda(D^2\phi(\zeta_0))$ denotes the smallest and largest eigenvalues of the Hessian matrix $D^2\phi(\zeta_0)$, and $\text{tr}(D^2\phi(\zeta_0))$ is its trace.
\end{definition}
It turns out that the second condition in Definition \ref{def:viscsuper} can be relaxed. This is the Lemma 2 in \cite{manfredi2010asymptotic}.
\begin{lemma}
\label{def:p-parabolic1}
An upper semicontinuous function $u \in L^{\infty}(\Omega)$, is a viscosity subsolution of \eqref{eq:evoplap} provided that, if $ u - \phi$ has a maximum at $\zeta_0 \in \Omega$ for $\phi \in C^2 (\Omega)$, then either
\[
\phi_t(\zeta_0)-\mathcal{A}_p\phi(\zeta_0) \leq 0, \ \text{if} \ D\phi(\zeta_0)\neq 0,
\]
or
\[
\phi_t(\zeta_0) \leq 0, \ \text{if} \ D\phi(\zeta_0)=0, \ D^2\phi(\zeta_0) =0.
\]
A similar result holds for viscosity supersolutions.
\end{lemma}
We define $p$-parabolic functions in $\Omega$ as follows:
\begin{definition}
\label{def:p-parabolic2}
A function $u\in LSC(\Omega)\cap L^{\infty}(\Omega)$ is a \emph{supersolution} to \eqref{eq:evoplap} if
it satisfies the following comparison principle: \\
On each set of the form $Q_{t_1, t_2} =Q \times (t_1, t_2 )$ with closure in $\Omega$, and for each solution $h$ to \eqref{eq:evoplap} continuous up to the closure of $Q_{t_1, t_2}$, and $h \leq u $ on $\partial_p Q_{t_1, t_2}$, then
\[
h \leq u \ \text{in} \ Q_{t_1, t_2}.
\]
\end{definition}
In Banerjee–Garofalo \cite{banerjee2015dirichlet}, they use the name \emph{generalized super/subsolution} instead of super/subparabolic function. They prove that these are the same as the viscosity super/subsolutions in a given domain. Hence we can use the term parabolic interchangeably with viscosity solution.
The assumption that supersolutions are bounded is not needed. See Theorem 2.6 in \cite{bjorn2017tusk}.
We shall improve this to include comparison on general domains $\Sigma$ compactly contained in $\Omega$, see Lemma \ref{thm:bettercomparison}.
Using the classical comparison principle for cylindrical domains $Q_T=Q\times (0, T)$ and a covering argument, we can prove Theorem 3.10 from \cite{banerjee2015dirichlet}. This is the comparison principle, essential for Perron's method to work.
\begin{theorem}
\label{thm:comparison}
Suppose $u$ is a supersolution bounded from above and $v$ is a subsolution bounded from below of \eqref{eq:evoplap} in a bounded
open set $\Omega \subset \mathbb{R}^{n+1}$. If at each point $\zeta_0 \in \partial \Omega$ we have
\[
\limsup_{\zeta \to \zeta_0} v(\zeta) \leq \liminf_{\zeta \to \zeta_0} u(\zeta),
\]
then $v \leq u$ in $\Omega$.
\end{theorem}
\subsection{The Perron Method}
\label{subsec:perron}
We start with a definition. Let $f : \partial \Omega \to \mathbb{R}$ be a continuous function.
\begin{definition}
A function $u$ belongs to the \emph{upper class} $\mathcal{U}_f$ if $u$ is a viscosity supersolution in $\Omega$ and
\begin{equation*}
\liminf_{\eta \to \zeta}u(\eta) \geq f(\zeta)
\end{equation*}
for $\zeta \in \partial \Omega$.
Likewise, a function $v$ belongs to the \emph{lower class} $\mathcal{L}_f$ if $v$ is a viscosity subsolution in $\Omega$ and
\begin{equation*}
\limsup_{\eta \to \zeta}v(\eta) \leq f(\zeta)
\end{equation*}
for all $\zeta \in \partial \Omega$.
We define the \emph{upper solution}
\[
\overline{H}_f(\zeta)=\inf\{u(\zeta) : u \in \mathcal{U}_f \},
\]
and the \emph{lower solution}
\[
\underline{H}_f(\zeta)=\sup\{v(\zeta) : v \in \mathcal{L}_f \}.
\]
Note that at each point the $\inf$ and $\sup$ are taken over the \emph{functions}.
\end{definition}
\begin{remark}
The Comparison Principle, Theorem \ref{thm:comparison}, gives immediately that $v \leq u$ in $\Omega$, for $v \in\mathcal{L}_f$ and $u \in \mathcal{U}_f$, and hence
\[\underline{H}_f\leq \overline{H}_f.\]
Whether $\underline{H}_f= \overline{H}_f$ holds in general, is a more subtle question.
\end{remark}
We need that the Upper and Lower Perron solutions, indeed, are viscosity solutions to \eqref{eq:evoplap}.
\begin{theorem*}[Theorem 3.12 in \cite{banerjee2015dirichlet}]
The upper Perron solution $\overline{H}_f$ and the lower Perron solution $\underline{H}_f$
are solutions to \eqref{eq:evoplap} in $\Omega$.
\end{theorem*}
An integral part of the theory of Perron solutions are the boundary regularity and the barrier functions.
\begin{definition}
We say that $\zeta_0 \in \partial \Omega $ is a \emph{regular} boundary point if
\[
\lim_{\zeta \to \zeta_0}\underline{H}_f(\zeta) =f(\zeta_0).
\]
for every continuous function $f: \partial \Omega \to \mathbb{R}$.
\end{definition}
Note that we instead could have used $\overline{H}_f$ in the above, since $\underline{H}_f =-\overline{H}_{-f}$.
\begin{remark}
The Petrovsky condition in Theorem \ref{thm:not regular} shows that a point can be regular for $u_t=\mathcal{A}_pu$ but \emph{not} for $u_t=\mathcal{A}_qu$, $p<q$. Hence it would be more accurate to use the term $p$-\emph{regular}, but we use regular where no confusion will arise.
\end{remark}
\begin{definition}
\label{def:barrier}
A function $w$ is a barrier at $\zeta_0 \in \partial \Omega$ if
\begin{enumerate}
\item $w >0$ and $w$ is $p$-superparabolic in $\Omega$,
\item $\liminf_{\zeta \to \eta}w(\zeta) >0$ for $\zeta_0 \neq \eta \in \partial \Omega$,
\item $\lim_{\zeta \to \zeta_0}w(\zeta) =0.$
\end{enumerate}
\end{definition}
Using barrier functions, we can prove the following classical result, which is
Theorem 4.2 in \cite{banerjee2015dirichlet}:
\begin{theorem}
\label{thm:barrier}
A boundary point $\zeta_0$ is regular if and only if there exists a barrier at $\zeta_0.$
\end{theorem}
The existence of a barrier is a local property in the following sense: Let $\tilde{\Omega}$ be another domain such that
\[
\overline{B}\cap \tilde{\Omega} =\overline{B}\cap \Omega
\]
for an open ball $B$ centered at $\zeta_0$. Suppose there is a barrier $w$ at $\zeta_0$, and let
\[
m= \inf\{w(\zeta)\ : \ \zeta \in \partial B \cup \tilde{\Omega}\}.
\]
It now follows that the function
\[
v=
\begin{cases}
\min(w, m) \ &\text{in} \ B\cup \tilde{\Omega}, \\
m \ &\text{in} \ \Omega \setminus B
\end{cases}
\]
is a barrier in $\Omega$. Indeed, since $w$ is assumed to be a barrier in $\tilde{\Omega}$, $\left.w \right|_{\tilde{\Omega}}> 0$ by the definition, and therefore $m>0$ and $v>0$. Since $\tilde{\Omega}\cap \overline{B}=\Omega \cap \overline{B}$ we see that $\liminf_{\eta \to \zeta}v(\eta)=\liminf_{\eta \to \zeta}w(\eta)>0$ on $\partial \tilde{\Omega} \cap \overline{B}$ and $\liminf_{\eta \to \zeta}v(\eta)=m>0$ elsewhere. At last, $\lim_{\zeta \to \zeta_0}v(\zeta)=\lim_{\zeta \to \zeta_0}w(\zeta)=0$.
From this we get the following useful corollary.
\begin{corollary}
\label{cor:irregular}
Let $\tilde{\Omega} \subset \Omega$, and let $\zeta_0$ be a common boundary point. If $\zeta_0$ is \emph{not} a regular point for $\tilde{\Omega}$, then it is not a regular point for $\Omega$.
\end{corollary}
\begin{proof}
Let $\tilde{\Omega} \subset \Omega$, and let $\zeta_0$ be an irregular boundary point for $\tilde{\Omega}$. Assume that $\zeta_0$ is regular for $\Omega$. Then Theorem \ref{thm:barrier} gives that there exists a barrier, $w$, in $\Omega$. The above implies the existence of a barrier in $\tilde\Omega$, contradicting the irregularity of $\zeta_0$.
\end{proof}
A classical application of the theory of viscosity solutions is the following convergence lemma.
\begin{lemma}
\label{thm:convergenceofeq}
Assume that $\{u_p\}_p$ is a sequence of viscosity solutions of
\[
u_t-\mathcal{A}_pu=0.
\]
Assume further that $\{u_p\}_p$ contains a subsequence $\{u_{p_j}\}_j$ that converges uniformly to a function $u_{\infty}$ in $\Omega$. Then, as $j \to \infty$, the $u_{p_j}$
converge to $u$, the viscosity solution of the normalised $\infty$-parabolic equation \eqref{eq:norminftylaplace}, that is
\[
u_t-\Delta^N_{\infty}u=0.
\]
\end{lemma}
\begin{proof}
We show that viscosity subsolutions of \eqref{eq:evoplap} converge to viscosity subsolutions of \eqref{eq:inftylaplace}. The proof for supersolutions is similar.
We say that $u \in \operatorname{USC}(\Omega)$ is a \emph{viscosity subsolution} to \eqref{eq:norminftylaplace} if, for every function $\phi \in C^2(\Omega)$ such that $u-\psi$ has a maximum at $\zeta_0$, we have
\begin{equation*}
\begin{cases}
\phi_t(\zeta_{\infty})-\Delta_{\infty}^N \psi(\zeta_{\infty}) \leq 0, \ \text{for} \ D\phi(\zeta_{\infty}) \neq 0 \\
\phi_t(\zeta_{\infty}) -\Lambda(D^2\psi(\zeta_{\infty})) \leq 0, \ \text{for} \ D\phi(\zeta_{\infty}) =0.
\end{cases}
\end{equation*}
Assume that $u_{\infty}-\phi$ has a maximum at $\zeta_{\infty}$ for $\phi \in C^2(Q_T)$.
{\bf 1.} Assume first that $D\phi(\zeta_{p_j})\neq0$ for $j$ greater than some number $N$. By definition of viscosity subsolution, we then have
\begin{equation}
\label{eq:converges}
\phi_t(\zeta_{p_j})- \frac{1}{{p}_j}\Delta \phi(\zeta_{p_j}) -\frac{p_j-2}{p_j}\Delta^N_{\infty}\phi(\zeta_{p_j}) \leq 0.
\end{equation}
Since $u_{p_j}\to u_{\infty}$ uniformly, standard arguments, cf \cite{lindqvist2016notes} gives that the maximum points $\zeta_{p_j}$ converge to a maximum point $\zeta_{\infty}$ of $u_{\infty}-\phi$. Hence, letting $j\to \infty$ in \eqref{eq:converges} we see that
\[
\phi_t(\zeta_{\infty})-\Delta_{\infty}^N\phi(\zeta_{\infty}) \leq 0.
\]
{\bf 2. } If $D\phi(\zeta_{p_j})=0$ for $j>N$, we have
\[
\phi_t(\zeta_{p_j}) -\frac{1}{p}\text{tr}(D^2\phi(\zeta_{p_j}))-\frac{p-2}{p}\Lambda(D^2\phi(\zeta_{p_j}))\leq 0
\]
and arguing as in the first case, we get as $p_j \to \infty$;
\[
\phi_t(\zeta_{\infty})-\Lambda(D^2\phi(\zeta_{\infty}))\leq 0.
\]
This shows that $u_{\infty}$ is indeed a viscosity subsolution of the normalised $\infty$-parabolic equation.
\end{proof}
\begin{remark}
The existence of a uniformly convergent subsequence of $u_p$ is not known to exist in general. Does finds such an example for the initial-boundary value problem with smooth boundary data in \cite{does2009evolution}.
\end{remark}
\section{Exterior Sphere Condition}
\label{sec:exterior sphere}
\noindent We use the barrier characterization to prove Theorem \ref{thm:exterior sphere}. We repeat the result here for completeness.
\begin{theorem*}[Exterior sphere]
Let $\zeta_0 =(t_0, x_0) \in \partial \Omega$, and suppose that there exists a closed ball $\{(x, t) \ : \ |x-x'|^2+(t-t')^2 \leq R_0^2\}$ intersecting $\overline{\Omega}$ precisely at $\zeta_0$. Then $\zeta_0$ is regular, if the intersection point is \underline{not} the south pole, that is $(x_0, t_0) \neq (x', t'-R_0)$.
\end{theorem*}
\begin{proof}
We use the exterior sphere to construct a suitable barrier function at $\zeta_0$. Define
\[
w(x, t) = \operatorname{e}^{-aR^2_0}-\operatorname{e}^{-aR^2}, \ R^2= |x-x'|^2+(t-t')^2,
\]
for a constant $a>0$ to be determined. Clearly $w(x_0, t_0)=0$, and close to $(x_0, t_0)$ we have
\begin{equation}
\label{eq:ballestimates}
\delta <|x-x'|, \ \ -2R_0 <t-t'.
\end{equation}
We prove that $w$ is a viscosity supersolution. Calculating the derivatives, we get
\begin{equation}
\begin{split}
\label{eq:w-derivatives}
&Dw(x, t) = 2a\mathrm{e}^{-aR^2}(x-x'),
\\ &w_t(x, t)=2a\mathrm{e}^{-aR^2}(t-t'), \\
&D^2w(x, t)=2a\mathrm{e}^{-aR^2}(\mathbb{Id}_n-2a(x-x')\otimes(x-x')).
\end{split}
\end{equation}
This shows that $Dw=0$ precisely when $x=x'$. According to Definition \ref{def:viscsuper}, we need to check the cases $x=x'$ and $x\neq x'$ separately.
{\bf 1.} Assume that $x\neq x'$. Then the point of contact is not the north pole. It suffices to show that $w$ is a classical supersolution.
Inserting the derivatives into \eqref{eq:evoplap} we get
\begin{align*}
& w_t-\frac{1}{p}\Delta w +\frac{p-2}{p}\Delta_{\infty}^Nw \\
&=2a(t-t')\operatorname{e}^{-aR^2}-\frac{1}{p}\operatorname{e}^{-aR^2}\left[(p-1)(2a-4a^2r^2)+2ar\frac{n-1}{r}\right] \\
&= 2a\operatorname{e}^{-aR^2}\left[(t-t')+2a\frac{p-1}{p}|x-x'|^2-\frac{p+n-2}{p}\right].
\end{align*}
In light of \eqref{eq:ballestimates}, we have
\begin{equation*}
\label{eq:mustbepos}
w_t-\mathcal{A}_pw> 2a\operatorname{e}^{-aR^2}\left[-2R_0+2a\frac{p-1}{p}\delta^2-\frac{p+n-2}{p}\right]
\end{equation*}
For the right hand side of this to be positive, we must have
\[
-R_0+a\frac{p-1}{p}\delta^2>\frac{p+n-2}{2p},
\]
and choosing $a$ big enough to ensure this, shows that $w$ is superparabolic.
{\bf 2.} If the point of intersection is the north pole, i.e $(x_0, t_0) = (x', t'+R_0)$, we can find points arbitrarily close to the line $x=x'$ such that
\[
w_t-\mathcal{A}_pw =2a\operatorname{e}^{-aR^2}\left[R_0-\frac{p+n-2}{p}\right] +\epsilon,
\]
for any $\epsilon >0$.
We see that we must demand that the radius $R_0$ satisfies
\[
R_0 \geq \alpha/2, \ \alpha = 2\frac{p+n-2}{p}
\]
for $w$ to be a barrier in this case.
Assume now that $x=x'$. We need to verify that for every $\phi\in C^2(\Omega)$ touching $w$ from below at $(x', t)$ we have
\begin{equation}
\label{eq:w-visc-super}
\phi_t(x', t) \geq\frac{1}{p}\text{tr}(D^2\phi(x', t))+ \frac{p-2}{p} \lambda(D^2\phi(x', t)).
\end{equation}
Assume to the contrary that there is a $\phi$ such that $w-\phi$ has a minimum at $(x', t)$, but that
\[
\phi_t(x', t) <\frac{1}{p}\text{tr}(D^2\phi(x', t))+ \frac{p-2}{p} \lambda(D^2\phi(x', t)).
\]
Since $w-\phi$ has a minimum, we must have
\[
\phi_t(x', t)=u_t(x', t), \ D\phi(x', t)=Du(x', t), \ D^2u(x', t) \geq D^2\phi(x', t).
\]
This implies, for any $z\in \mathbb{R}^n$
\[
\langle D^2w\ z, z\rangle \geq \langle D^2\phi \ z, z\rangle
\]
and, since $D^2w$ is a scalar multiple of the identity matrix,
\[
\text{tr}(D^2w)|z|^2\geq \text{tr}(D^2\phi)|z|^2
\]
at $(x', t)$. Hence
\begin{align*}
\frac{1}{p}\text{tr}(D^2w)|z|^2 +\frac{p-2}{p}\langle D^2w\ z, z\rangle \\
\geq \frac{1}{p}\text{tr}(D^2\phi)|z|^2 +\frac{p-2}{p}\langle D^2\phi\ z, z\rangle \\
\geq \frac{1}{p}\text{tr}(D^2\phi)|z|^2 +\frac{p-2}{p}\lambda(D^2\phi)|z|^2 \\
>\phi_t|z|^2=w_t|z|^2.
\end{align*}
Inserting $x=x'$ in \eqref{eq:w-derivatives} and dividing by $|z|^2$ this is
\[
\frac{1}{p}2an\operatorname{e}^{-aR^2}+\frac{p-2}{p}2a\operatorname{e}^{-aR^2} > 2a\operatorname{e}^{-aR^2}(t-t'),
\]
or
\[
\frac{n+p-2}{p}>(t-t').
\]
This is a contradiction because of our restriction on the radius, and hence \eqref{eq:w-visc-super} must hold, and $w$ is a supersolution even in this case.
The condition $R_0 \geq \alpha/2$ restricts the set of exterior spheres usable in a positive way. The author does not know if this restriction can be circumvented.
The exclusion of the south pole $(x_0, t_0)=(x', t'-R_0)$ in the above is strictly necessary, since then for $(x, t)$ close to $(x_0, t_0)$ we could have
\[
(t-t')<0 \ \text{and} \ |x-x'|=|x'-x'|=0,
\]
and so
\begin{align*}
w_t-\mathcal{A}_pw =2a\operatorname{e}^{-aR^2}\left[(t-t')-\frac{p+n-2}{p}\right] <0
\end{align*}
for any positive $a$, since $p+n\geq 2$.
\end{proof}
Another way to see that it is necessary to exclude the south pole is to consider the Dirichlet problem on the cylinder $Q_T =Q\times (0, T)$.
\begin{example}
Suppose that $f : \partial Q_T\to \mathbb{R}$ is continuous. Theorem \ref{thm:comparison} and Theorem 2.6 in \cite{banerjee2015dirichlet} gives the existence of a unique viscosity solution $h$ in $Q_T$.
Now construct the upper and lower Perron solutions $\overline{H}_f$ and $\underline{H}_f$. Since both are $p$-parabolic in $Q_T$, uniqueness gives that $\underline{H}_f = \overline{H}_f=h$, regardless of what values we choose at that part of the boundary where $t=T$. Indeed, $h$ itself need not be in either the upper or lower class, because we may not have that either $h>f$ or $h<f$ on the plane $t=T$. However, if we define
\begin{equation*}
\tilde{h} =h(x, t)+\frac{\epsilon}{T-t},
\end{equation*}
we see that
\begin{equation*}
\tilde{h}_t-\mathcal{A}_p\tilde{h} =0+\frac{\epsilon}{(T-t)^2},
\end{equation*}
so $\tilde{h}$ is in $\mathcal{U}_f$ for $\epsilon >0$, and in $\mathcal{L}_f$ for $\epsilon <0$. Therefore, it is possible for every point on the top of the cylinder to be irregular. we can say that $f$ is \emph{resolutive} in this case.
\end{example}
We provide another example of an irregular boundary point.
\begin{example}[Latest moment on heat balls]
Recall the self-similar solution derived in Section \ref{sec:similarity1}. We define the \emph{fundamental} solution to \eqref{eq:evoplap} as
\begin{equation*}
H_p(x, t) = t^{-\frac{\alpha}{\beta}}\operatorname{e}^{-\frac{|x|^2}{\beta t}}.
\end{equation*}
Analogous to the heat equation and the $p$-parabolic equation, we define the \emph{normalised} $p$-\emph{parabolic} \emph{balls} by the level sets
\begin{equation}
\label{eq:level sets}
H_p(x_0-x, t_0-t) >c
\end{equation}
We want to prove that the latest moment, or "centre" $(x_0, t_0)$ of \eqref{eq:level sets} is \emph{not} a regular point. Fix $c>0$. We can assume that $(x_0, t_0)=(0,0)$, so that \eqref{eq:level sets} reads
\[
(-t)^{-\frac{\alpha}{\beta}}\operatorname{e}^{-\frac{|x|^2}{\beta (-t)}}>c
\]
for $t<0$.
But this is equivalent to
\[
|x|^2<t(\frac{\log{c}}{\beta}+\alpha\log|t|),
\]
and this inequality defines a domain containing the one in the Petrovsky criterion \ref{thm:petrovsky}. Hence the origin must be irregular.
\end{example}
We prove that it suffices to consider arbitrary domains in the equivalent definition of $p$-parabolic functions. The proof follows the same idea as in \cite{kilpelainen1996dirichlet}.
\begin{lemma}
\label{thm:bettercomparison}
A function $u\in LSC(\Omega)\cap L^{\infty}(\Omega)$ is $p$-superparabolic if and only if
for each domain $\Sigma$ with compact closure in $\Omega$, and for each solution $h \in C(\overline{\Sigma})$ to \eqref{eq:evoplap}, the condition $h \leq u $ on $\partial \Sigma$ implies $h \leq u$ in $\Sigma$.
\end{lemma}
\begin{proof}
Assume first that Definition \ref{def:p-parabolic2} holds. If $\Sigma$ is a box or finite union of boxes, the result is clearly true. The case where $\Sigma$ is arbitrary follows by covering the set $\{h \geq u+\epsilon\}$ with finitely many boxes.
For the other direction, let $Q_{t_1, t_2}$ be a box with closure in $\Omega$ and let $h \in C(\overline{Q}_{t_1, t_2})$ be $p$-parabolic, and so that $h \leq u$ on $\partial_pQ_{t_1, t_2}$. Assume that
\[
Q =(a_1, b_1) \times \cdots \times (a_n, b_n).
\]
Let $\delta>0$ be so that $\delta <t_2-t_1$, and choose a hyperplane $P_{\delta}$ such that the points $(x, t_2-\delta)$ with $x_1=a_1$ and $(y, t_2)$ with $y_1=b_1$ belong to $P_{\delta}$. let $\Sigma$ be the subset of $Q_{t_1, t_2}$ that contains all the points below the hyperplane, that is all $(x, t)$ with $t<s$ and $(x, s) \in P_{\delta}$.
The Exterior sphere condition Theorem \ref{thm:exterior sphere} immediately gives that every point on $\partial \Sigma$ is regular. Fix $\epsilon>0$, and choose $\delta$ so small that
\[
u(x, t) \geq h(x, t)-\frac{\epsilon}{t_2+\frac{\delta}{2}-t}
\]
for $(x, t) \in P_{\delta}\cap \Sigma$. Let $\overline{H}_{\theta}$ be the upper Perron solution in $\Sigma$ with
\[
\theta = h-\frac{\epsilon}{t_2+\frac{\delta}{2}-t}
\]
as boundary function.
Then $\overline{H}_{\theta}$ is continuous up to $\partial \Sigma$, and we have
\[
u\geq \overline{H}_{\theta}
\]
in all of $\Sigma$ since the inequality holds on $\partial \Sigma$. Hence
\[
u(x, t) \geq h(x, t)-\frac{\epsilon}{t_2+\frac{\delta}{2}-t}
\]
in $\Sigma$, and letting $\epsilon, \delta \to 0$, we get
\[
u \geq h
\]
in the box $Q_{t_1, t_2}$.
\end{proof}
\section{The Petrovsky Criterion}
\label{sec:petrovsky}
\noindent We provide the proof of the Petrovsky Criterion, repeated here for completeness.
\begin{theorem*}
\label{thm:petrowski}
The origin $(x, t)=(0, 0)$ is a regular point for \eqref{eq:evoplap}
in the domain $\Omega$ enclosed by the hypersurfaces
\begin{equation}
\label{eq:thedomain}
\{(x, t)\in \mathbb{R}^n\times (-\infty, 0)\ : \ |x|^2 =-\beta t\log|\log|t||\} \ \text{and} \ \{t=-c\},
\end{equation}
for a small constant $0<c<1$. Recall that
\[
\beta =4\frac{p-1}{p}.
\]
\end{theorem*}
According to Theorem \ref{thm:barrier}, it suffices to find a barrier function $w$ so that
\begin{enumerate}
\item $w$ is a supersolution in $\Omega$,
\item $w(x, t) >0$ for $(x, t)\in \Omega$,
\item $\liminf_{(y, s) \to (x, t)}w(y, s) >0$ for $(x, t) \neq (0, 0) \in \partial \Omega$,
\item $\lim_{(x, t) \to (0, 0)}w(x, t) =0.$
\end{enumerate}
Our barrier will be on the form
\begin{equation*}
\label{eq:barrierform}
w(x, t)=f(t)e^{-\frac{|x|^2}{\beta t}}+ g(t),
\end{equation*}
for smooth functions $f$ and $g$. Differentiating formally, we get
\begin{equation}
\label{eq:timederivative}
w_t(x, t)=e^{-\frac{|x|^2}{\beta t}}\left(f'(t)+{\frac{|x|^2}{\beta t^2}}f(t)\right)+g'(t),
\end{equation}
\begin{equation}
\label{eq:gradient}
Dw(x, t)=-x\frac{2f(t)}{\beta t}e^{-\frac{|x|^2}{\beta t}},
\end{equation}
and
\begin{equation}
\label{eq:hessian}
\begin{split}
D^2w(x, t)&=
-\frac{2f(t)}{\beta t}e^{-\frac{|x|^2}{\beta t}}\mathbb{Id}_n+\frac{4f(t)}{t^2\beta^2}e^{-\frac{|x|^2}{\beta t}}x\otimes x \\
&=\frac{2f(t)}{\beta t}e^{-\frac{|x|^2}{\beta t}}\left(-\mathbb{Id}_n+\frac{2}{\beta t}x \otimes x\right).
\end{split}
\end{equation}
From \eqref{eq:hessian} we see that
\begin{equation}
\label{eq:trace}
\text{tr}(D^2w(x, t))=\frac{2f(t)}{\beta t}e^{-\frac{|x|^2}{\beta t}}\left(-n+\frac{2|x|^2}{\beta t}\right).
\end{equation}
From \eqref{eq:gradient} and \eqref{eq:hessian}, (or observing that $w(x, t)=G(r, t)$, and so $\Delta_{\infty}^Nw =G_{rr}$), we get
\begin{equation}
\label{eq:inftylaplace_p}
\left\langle D^2w\frac{Dw}{|Dw|}, \frac{Dw}{|Dw|} \right\rangle =\frac{2f(t)}{\beta t}e^{-\frac{|x|^2}{\beta t}}\left(-1+\frac{2|x|^2}{\beta t}\right).
\end{equation}
From \eqref{eq:inftylaplace_p} and \eqref{eq:trace}, we calculate
\begin{align*}
\mathcal{A}_pw =&\frac{1}{p}\text{tr}(D^2w) +\frac{p-2}{p}\left\langle D^2w\frac{Dw}{|Dw|}, \frac{Dw}{|Dw|} \right\rangle, \\
&=\frac{1}{p}\cdot\frac{2f(t)}{\beta t}e^{-\frac{|x|^2}{\beta t}}\left(-n+\frac{2|x|^2}{\beta t}\right) +\frac{p-2}{p}\cdot\frac{2f(t)}{\beta t}e^{-\frac{|x|^2}{\beta t}}\left(-1+\frac{2|x|^2}{\beta t}\right) \\
&=\frac{2f(t)}{\beta t}e^{-\frac{|x|^2}{\beta t}}\left(-\frac{n}{p}-\frac{p-2}{p}+\frac{2|x|^2}{\beta t}\left(\frac{1}{p}+\frac{p-2}{p}\right)\right)\\
&=\frac{f(t)}{\beta t}e^{-\frac{|x|^2}{\beta t}}\left(-2\left(\frac{n+p-2}{p}\right)+\frac{|x|^2}{\beta t}\left(4\frac{p-1}{p}\right)\right) \\
&=\frac{f(t)}{\beta t}e^{-\frac{|x|^2}{\beta t}}\left(-\alpha+\frac{|x|^2}{t}\right),
\end{align*}
where
\[
\alpha =2\frac{n+p-2}{p}.
\]
This, together with \eqref{eq:timederivative}, gives
\begin{equation}
\label{eq:p-parabolicw}
\begin{split}
w_t-\mathcal{A}_pw \\
=&e^{-\frac{|x|^2}{\beta t}}\left(f'(t)+{\frac{|x|^2}{\beta t^2}}f(t)\right)+g'(t)-\frac{f(t)}{\beta t}e^{-\frac{|x|^2}{\beta t}}\left(-\alpha+\frac{|x|^2}{t}\right) \\
=&e^{-\frac{|x|^2}{\beta t}}\left(f'(t)+\frac{|x|^2f(t)}{\beta t^2}+\frac{\alpha f(t)}{\beta t}-\frac{|x|^2f(t)}{\beta t^2}\right) +g'(t) \\
=&e^{-\frac{|x|^2}{\beta t}}\left(f'(t)+\frac{\alpha f(t)}{\beta t} +g'(t)e^{\frac{|x|^2}{\beta t}}\right).
\end{split}
\end{equation}
Choose
\[
f(t)=-c\frac{1}{|\log|t||^{\delta+1}}, \ g(t)=\frac{1}{|\log|t||^{\delta}},
\]
for constants $0<c<1$, $\delta$ to be determined. We are now in position to prove the following theorem:
\begin{theorem}
The smooth function $w:\Omega \to \mathbb{R}$ given by
\begin{equation}
\label{eq:the barrier}
w(x, t)=-c\frac{1}{|\log|t||^{\delta+1}}e^{-\frac{|x|^2}{\beta t}} +\frac{1}{|\log|t||^{\delta}}
\end{equation}
is a barrier at $(0, 0)$.
\end{theorem}
\begin{proof}
We check the requirements listed in Definition \ref{def:barrier}.
{\bf 1.} We must check that $w$ is a viscosity supersolution in $\Omega$. Equation \eqref{eq:gradient} shows that $Dw=0$ precisely when $x=0$, so assume first that $x\neq 0$. It suffices to show that $w$ is a classical solution in this case. We first differentiate $f$ and $g$:
\[
f'(t)=-c(\delta+1)\frac{1}{t|\log|t||^{\delta+2}}, \ g'(t)=\delta\frac{1}{t|\log|t||^{\delta+1}}.
\]
Inserting the derivatives into \eqref{eq:p-parabolicw} gives
\begin{align*}
&w_t-\mathcal{A}_pw \\
=&e^{-\frac{|x|^2}{\beta t}}\left(-c(\delta+1)\frac{1}{t|\log|t||^{\delta+2}}-c\frac{\alpha}{\beta}\frac{1}{t|\log|t||^{\delta+1}}+\delta\frac{1}{t|\log|t||^{\delta+1}}e^{\frac{|x|^2}{\beta t}}\right) \\
=&\frac{1}{t|\log|t||^{\delta+1}}e^{-\frac{|x|^2}{\beta t}}\left(\frac{-c(\delta+1)}{|\log|t||}-c\frac{\alpha}{\beta}+\delta e^{\frac{|x|^2}{\beta t}}\right).
\end{align*}
$t$ is negative, so $e^{\frac{|x|^2}{\beta t}}<1$, hence
\begin{align*}
w_t-\mathcal{A}_pw >\frac{1}{t|\log|t||^{\delta+1}}e^{\frac{|x|^2}{\beta t}}\left(\frac{-c(\delta+1)}{|\log|t||}-c\frac{\alpha}{\beta}+\delta\right)
\end{align*}
For this to be positive, the expression inside the parentheses must be negative. Choosing
\begin{equation}
\label{eq:choosing_delta}
\delta =c\frac{\alpha}{\beta}
\end{equation}
ensures this, and with this choice $w$ is superparabolic in this case.
Assume that $x=0$ so that $Dw=0$. From \eqref{eq:hessian} and \eqref{eq:trace} we deduce
\[
\text{tr}(D^2w(0, t))=c\frac{2n}{\beta t|\log|t||^{\delta+1}},
\]
and
\[
\lambda(D^2w(0, t))=c\frac{2}{\beta t|\log|t||^{\delta+1}}.
\]
Since
\[
w_t(0, t)=f'(t)+g'(t) =\frac{1}{t|\log|t||^{\delta+1}}\left(\frac{-c(\delta+1)}{|\log|t||}+\delta\right),
\]
Definition \ref{def:viscsuper} demands that we verify
\begin{align*}
&\frac{1}{t|\log|t||^{\delta+1}}\left(\frac{-c(\delta+1)}{|\log|t||}+\delta\right) \\
\geq &\frac{1}{p}\cdot c\frac{2n}{\beta t|\log|t||^{\delta+1}} +\frac{p-2}{p}\cdot c\frac{2}{\beta t|\log|t||^{\delta+1}}
\end{align*}
for $t<0$.
This is equivalent to
\[
\frac{-c(\delta+1)}{|\log|t||}+\delta\leq \frac{2cn}{p\beta} +\frac{2c(p-2)}{p\beta} \\
=2c\frac{p+n-2}{p\beta}=c\frac{\alpha}{\beta}.
\]
Because of our choice of $\delta$ in \eqref{eq:choosing_delta}, the above inequality is satisfied for all $t$, and \eqref{eq:the barrier} satisfies the second condition in Definition \eqref{def:viscsuper}.
It remains to show that $w$ is a viscosity supersolution. Let $\phi\in C^2(\Omega)$ touch $w$ from below at $(0, t)$. Since $w-\phi$ has a minimum at $(0, t)$, we have
\[
w_t=\phi_t, \ Dw=D\phi, \ D^2w>D^2\phi
\]
at this point. Since $D^2w(0, t)=\frac{2f(t)}{\beta t}\mathbb{Id}_n$, a scalar multiple of the identity matrix, this implies
\[
\lambda(D^2w((0, t))>\Lambda(D^2\phi(0, t))>\lambda(D^2\phi(0, t)),
\]
where $\lambda$ is the smallest eigenvalue and $\Lambda$ is the greatest. Since
\[
\text{tr}(D^2\phi(0, t)) =\sum_{i=1}^{n}\lambda_i(D^2\phi(0, t)),
\]
we get
\begin{align*}
\phi_t(0, t)&=w_t(0, t) \\
&\geq\frac{1}{p}\text{tr}(D^2w(0, t))-\frac{p-2}{p}\lambda(D^2w(0, t)) \\
&\geq\frac{1}{p}\text{tr}(D^2\phi(0, t))-\frac{p-2}{p}\lambda(D^2\phi(0, t)),
\end{align*}
which implies that $w$ is indeed a viscosity supersolution.
{\bf 2.} Since \eqref{eq:thedomain} implies $-\frac{|x|^2}{\beta t}<\log|\log|t||$, we see
\[
w(x, t)>-c\frac{1}{|\log|t||^{\delta+1}}e^{\log|\log|t||} +\frac{1}{|\log|t||^{\delta}} =\frac{1-c}{|\log|t||^{\delta}}>0,
\]
for $0<c<1$, as desired, and (2) in the Definition holds.
{\bf 3.} $w$ is continuous in $\overline{\Omega}$, so we only need to check that the restriction of $w$ to $\partial \Omega$ is positive. We see
\[
\left.w(x, t)\right|_{\partial \Omega}=-c\frac{1}{|\log|t||^{\delta+1}}e^{\frac{-\beta t\log|\log|t||}{\beta t}} +\frac{1}{|\log|t||^{\delta}} =\frac{1-c}{|\log|t||^{\delta}}>0.
\]
{\bf 4.} We see that
\[
\lim_{t \to 0^-}f(t)=\lim_{t \to 0^-}g(t)=0.
\]
Since $|x|^2<-\beta t\log|\log|t||\to 0$, we see
\[
e^{-\frac{|x|^2}{\beta t}} =\mathcal{O}(|\log|t||)
\]
as $t \to 0^-$. Therefore
\[
\lim_{(x, t)\to (0, 0^-)}w(x, t)=0,
\]
and (4) in the Definition is satisfied.
Together, these points show that \eqref{eq:the barrier} is indeed a barrier at $(0, 0)$, and hence the origin is a regular point for the domain \eqref{eq:thedomain}.
\end{proof}
\begin{remark}
Since $\beta \to 4$ as $p\to \infty$, we see that \eqref{eq:thedomain} converges to the Petrovsky criterion for the $\infty$-parabolic equation \eqref{eq:infinitypetrovsky}. Note also that the result is completely independent of the number of spatial variables $n$.
\end{remark}
We now turn to the proof that Theorem \ref{thm:petrovsky} is sharp; any constant greater than $\beta$ in \eqref{eq:thedomain} will produce domain containting $\Omega$ where the origin is irregular.
\begin{theorem*}
The origin is \emph{not} a regular point for the domain $\Omega$ enclosed by the hypersurfaces
\begin{equation}
\label{eq:irregulardomain}
\begin{split}
&\{(x, t)\in \mathbb{R}^n\times(-\infty, 0) \ : \ |x|^2 =-\beta(1+\epsilon)t\log|{\log{|t|}}|\} \\ &\text{and}\{t=-c \},
\end{split}
\end{equation}
for any $\epsilon >0$.
\end{theorem*}
\begin{proof}
The proof proceeds by constructing a domain $\tilde{\Omega}$ contained in $\Omega$, with the origin as common boundary point. We then show that $(0, 0)$ is irregular for $\tilde{\Omega}$, and Lemma \ref{cor:irregular} then implies that $(0, 0)$ regarded as a boundary point of $\Omega$ is irregular, too.
We shall construct a smooth function $w$ so that
\begin{enumerate}
\item $w$ is subparabolic in $\tilde\Omega$,
\item $w$ is continuous on $\overline{\tilde\Omega}\setminus\{(0, 0)\}$,
\item The upper limit of $w$ at interior points converging to $(0, 0)$ is greater than its upper limit for the points converging to $(0,0)$ along the boundary.
\end{enumerate}
To see why the existence of such a $w$ implies that the origin is irregular, consider the boundary data $f: \partial \tilde\Omega \to \mathbb{R}$ defined as follows. Let $f=w$ near $(0, 0)$, and set
\[
f(0, 0)=\lim_{\partial \tilde\Omega\ni(x, t)\to(0, 0)}v(x, t).
\]
As we shall see, this limit exists.
For the rest of the boundary, continuously extend $f$ to a large constant $b$.
If $b$ is large enough, the comparison principle implies that every function $\overline{u}\in \mathcal{U}_f$ which satisfies $\overline{u}\geq f$ on $\partial \tilde\Omega$ also satisfies $\overline{u}\geq w$ in $\tilde\Omega$ since $w$ is a subsolution by (1). Taking the infimum over all such $\overline{u}$, we see $\overline{H}_f\geq w$ in $\tilde\Omega$, and hence by point (3) in the definition of $w$;
\begin{align*}
\limsup_{\tilde\Omega\ni(x, t) \to (0, 0)}\overline{H}_f(x, t) \geq &\limsup_{\tilde\Omega\ni(x, t) \to (0, 0)} w(x, t) \\
>&\limsup_{\partial \tilde\Omega\ni(y, s) \to (0, 0)}w(y, s) =f(0, 0),
\end{align*}
and so $(0, 0)$ is not a regular point for $\tilde \Omega$.
Our function $w$ will be on the form
\begin{equation}
\label{eq:irregularitatsbarriare}
w(x, t)=f(t)\mathrm{e}^{\frac{-|x|^2}{\beta t}k} +g(t),
\end{equation}
for suitable functions $f$ and $g$. Here $k\in (\frac{1}{2}, 1)$ will be chosen later, and $-1<t<0$. Indeed, we shall choose $t$ to be very close to 0. Calculating, we get
\[
w_t(x, t ) = f'(t)\mathrm{e}^{\frac{-|x|^2}{\beta t}k}+\frac{f(t)|x|^2k}{\beta t^2}\mathrm{e}^{\frac{-|x|^2}{\beta t}k}+g'(t)
\]
and
\begin{align*}
& w_r = -f(t)\frac{2rk}{\beta t}\mathrm{e}^{\frac{-|x|^2}{\beta t}k}, \\
& w_{rr} = f(t)\mathrm{e}^{\frac{-|x|^2}{\beta t}k}\left(\frac{4r^2k^2}{\beta^2 t^2}-\frac{2k}{\beta t}\right)
\end{align*}
Inserting this into \eqref{eq:evoplap}, we get
\begin{equation}
\label{eq:choosefg}
w_t-\mathcal{A}_pw = \mathrm{e}^{\frac{-|x|^2}{\beta t}k}\left[f'(t)+f(t)\frac{|x|^2(k-k^2)}{\beta t^2}+f(t)\frac{\alpha k}{\beta t}\right]+g'(t),
\end{equation}
with $\alpha$ as in \eqref{eq:bestsolution}.
Choose
\[
f(t)=\frac{-1}{{|\log{|t|}|}^{1+\epsilon_1}} \ \text{and} \ g(t)= \frac{1}{\log|\log{|t|}|},
\]
where $\epsilon_1$ is a positive constant.
{\bf The case $\bf{x\neq 0}$.} We see that $Dw=0$ precisely when $x=0$. We show that \eqref{eq:irregularitatsbarriare} is a \emph{classical} subsolution when $x \neq 0$.
Inserting derivatives into \eqref{eq:choosefg}, we get
\begin{equation}
\label{eq:bigstuff}
\begin{split}
w_t-\mathcal{A}_pw = \mathrm{e}^{\frac{-|x|^2}{\beta t}k}\left[\frac{-(1+\epsilon_1)}{t|\log{|t|}|^{2+\epsilon_1}}-\frac{|x|^2(k-k^2)}{\beta t^2|\log{|t|}|^{1+\epsilon_1}} - \right. \\\left. \frac{\alpha k}{\beta t|\log{|t|}|^{1+\epsilon_1}}+\mathrm{e}^{\frac{|x|^2}{\beta t}k}\frac{1}{t\cdot \log^2{|\log{|t|}|}\cdot|\log{|t|}|}\right].
\end{split}
\end{equation}
Multiplying by $t\cdot |\log{|t|}|^{1+\epsilon_1}<0$, we see that the sign of \eqref{eq:bigstuff} coincides with the sign of
\begin{align*}
-\frac{1+\epsilon_1}{\log{|t|}}+\frac{|x|^2}{\beta t}(k-k^2)+\frac{\alpha k}{\beta}-\mathrm{e}^{\frac{|x|^2}{\beta t}k}\frac{|\log{|t|}|^{\epsilon_1}}{\log^2{|\log{|t|}|}}.
\end{align*}
We can choose $|t|$ small enough that
\[
\left|\frac{1+\epsilon_1}{\log{|t|}}\right|<\frac{\alpha k}{\beta},
\]
and then
\begin{equation}
\label{eq:negative}
\frac{|x|^2}{\beta t}(k-k^2)+\frac{2\alpha k}{\beta}-\mathrm{e}^{\frac{|x|^2}{\beta t}k}\frac{|\log{|t|}|^{\epsilon_1}}{\log^2{|\log{|t|}|}} <0.
\end{equation}
This inequality is satisfied if $|x|$ is so small that
\begin{equation}
\label{eq:x small}
\frac{2\alpha k}{\beta } <\mathrm{e}^{\frac{|x|^2}{\beta t}k}\frac{|\log{|t|}|^{\epsilon_1}}{\log^2{|\log{|t|}|}}.
\end{equation}
or if $|x|$ so large that
\begin{equation}
\label{eq:x large}
\frac{|x|^2(k-k^2)}{\beta |t|}>\frac{2\alpha k}{\beta}.
\end{equation}
We argue that at least one of these inequalities must hold. Indeed, fix $|t|$ so that
\begin{equation}
\label{eq:t- condition}
\frac{\epsilon_1}{2}\log|\log{|t|}|>4\frac{\alpha}{\beta}.
\end{equation}
{\bf 1. } In the case \eqref{eq:x small}, we take logarithms to get
\[
\log{\frac{2\alpha k}{\beta}} < \frac{|x|^2}{\beta t}k +\epsilon_1\log|\log{|t|}|-2\log\log|\log{|t|}|,
\]
or
\begin{align*}
\frac{|x|^2}{\beta|t|}k &<\epsilon_1\log|\log{|t|}|-2\log\log|\log{|t|}|-\log{\frac{2 \alpha k}{\beta}} \\
& <\epsilon_1\log|\log{|t|}|-\log{k} \\
&< \epsilon_1\log|\log{|t|}|-\frac{\epsilon_1}{2}\log|\log{|t|}|\\
& =\frac{\epsilon_1}{2}\log|\log{|t|}|>2
\end{align*}
for $|t|$ small enough. Hence \eqref{eq:negative} is satisfied.
{\bf 2.} On the other hand, if \eqref{eq:x large} holds, we calculate
\[
\frac{|x|^2}{\beta |t|}>\frac{2\alpha}{\beta(1-k)}>4\frac{\alpha}{\beta}.
\]
Since we chose $|t|$ according to \eqref{eq:t- condition}, we have that at least one of the inequalities \eqref{eq:x small} or \eqref{eq:x large} is satisfied for any $x\neq 0$, and $w$ is a subsolution.
{\bf The case $\bf{x= 0}$.} Then $Dw=0$, and according to Definition \ref{def:viscsuper} we need to show that for every $\phi \in C^2(\Omega)$ so that $w-\phi$ has a maximum at $(0, t)$, we have
\begin{equation}
\label{eq:subparabolic}
\phi_t(0, t)\leq \frac{1}{p}\text{tr}(D^2\phi(0, t))+\frac{p-2}{p}\Lambda(D^2\phi(0, t)).
\end{equation}
We show that $w$ itself satisfies this condition. An argument similar to the one in the proof of Theorem \ref{thm:petrovsky} then shows that $w$ is a viscosity subsolution.
Inserting the derivatives at $(0, t)$, we see that \eqref{eq:subparabolic} reads
\[
f'(t)+g'(t) \leq -\frac{\alpha f(t)}{\beta t}k,
\]
or
\[
-\frac{1+\epsilon_1}{t\cdot|\log{|t|}|^{2+\epsilon_1}} +\frac{1}{\log^2|\log{|t|}|\cdot|\log{|t|}|\cdot t} \leq \frac{\alpha k}{\beta t\cdot|\log{|t|}|^{1+\epsilon_1}}.
\]
This is the same as
\[
\frac{1+\epsilon_1}{|\log{|t|}|}-\frac{|\log{|t|}|^{\epsilon_1}}{\log^2|\log{|t|}|}+\frac{\alpha k}{\beta}\leq 0,
\]
but this inequality is the same as the one in \eqref{eq:negative}, and because of our choices of $|t|$ and $k$. This shows that the condition \eqref{eq:subparabolic} holds, and $w$ is a subsolution even in this case.
Now we consider the level set $w(x, t)=m$, $m<0$, and calculate
\begin{align*}
w(x, t)&=\frac{-1}{|\log{|t|}|^{\epsilon_1+1}}\mathrm{e}^{-\frac{|x|^2}{\beta t}k}+\frac{1}{\log|\log{|t|}|} =m \\
&\iff \frac{-1}{|\log{|t|}|^{\epsilon_1+1}}\mathrm{e}^{-\frac{|x|^2}{\beta t}k} =m-\frac{1}{\log|\log{|t|}|} \\
&\iff \mathrm{e}^{-\frac{|x|^2}{ \beta t}k} =|\log{|t|}|^{\epsilon_1+1}\left(\frac{1}{\log|\log{|t|}|}-m\right) \\
&\iff -\frac{|x|^2}{\beta t}k = (\epsilon_1+1)\log|\log{|t|}|+\log\left(\frac{1}{\log|\log{|t|}|}-m\right),
\end{align*}
or simply
\begin{equation}
\label{eq:smalldomain}
x^2 =-\beta t\left(\frac{\epsilon_1+1}{k}\log|\log{|t|}|+\frac{1}{k}\log\left(\frac{1}{\log|\log{|t|}|}-m\right)\right).
\end{equation}
Letting $\tilde{\Omega}$ denote the domain enclosed by \eqref{eq:smalldomain} and the hyperplane $t=c<0$, we have that for $m<0$, the function $v$ \eqref{eq:irregularitatsbarriare} is negative in $\tilde{\Omega}$, and $w(x, 0)=0$. This shows that the origin is an irregular boundary point for $\tilde{\Omega}$.
The inclusion $\tilde \Omega \subset \Omega$ requires that
\[
\frac{\epsilon_1+1}{k}\log|\log{|t|}|+\frac{1}{k}\log\left(\frac{1}{\log|\log{|t|}|}-m\right) <(1+\epsilon)\log|\log|t||
\]
for small $|t|$. Fix $k$ close to 1 and $\alpha$ close to 0 so that
\[
\frac{\epsilon_1+1}{k}<1+\frac{\epsilon}{2}.
\]
Thus we have to verify that
\[
\left(\frac{1}{\log|\log{|t|}|}+|m|\right)^{\frac{1}{k}} \leq |\log|t||^{\frac{\epsilon}{2}},
\]
but this obviously holds for small $|t|$ since the left-hand side is bounded.
Hence $\tilde{\Omega} \subset \Omega$ for $\epsilon_1$ and $c$ close to 0, and $k$ close to 1, $(0,0)$ is an irregular boundary point for $\Omega$ as well.
\end{proof}
\section*{Acknowledgement}
\noindent The author would like to thank Jana Björn and Vesa Julin for discovering a flaw in the original proof of the Petrovsky criterion.
\bibliographystyle{alpha}
| {
"timestamp": "2018-09-19T02:08:20",
"yymm": "1711",
"arxiv_id": "1711.11369",
"language": "en",
"url": "https://arxiv.org/abs/1711.11369",
"abstract": "The boundary regularity for the normalized $p$-parabolic equation $u_t =\\frac{1}{p}|Du|^{2-p}\\Delta_pu$ is studied. Perron's method is used to construct solutions in arbitrary domains. We classify the regular boundary points in terms of barrier functions, and prove an Exterior Sphere result. A fundamental solution is identified. A Petrovsky criterion is established, and we examine the convergence of solutions as $p \\to \\infty$.",
"subjects": "Analysis of PDEs (math.AP)",
"title": "On the normalized $p$-parabolic equation in arbitrary domains",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9854964211605607,
"lm_q2_score": 0.7185943985973772,
"lm_q1q2_score": 0.7081722080837406
} |
https://arxiv.org/abs/2207.03238 | A variational principle for the metric mean dimension of level sets | We prove a variational principle for the upper and lower metric mean dimension of level sets \[ \left\{x\in X: \lim_{n\to\infty}\frac{1}{n}\sum_{j=0}^{n-1}\varphi(f^{j}(x))=\alpha\right\} \] associated to continuous potentials $\varphi:X\to \mathbb R$ and continuous dynamics $f:X\to X$ defined on compact metric spaces and exhibiting the specification property. This result relates the upper and lower metric mean dimension of the above mentioned sets with growth rates of measure-theoretic entropy of partitions decreasing in diameter associated to some special measures. Moreover, we present several examples to which our result may be applied to. Similar results were previously known for the topological entropy and for the topological pressure. | \section{Introduction}
One of the most important notions in Dynamical Systems is that of \emph{topological entropy}. It is a topological invariant and, roughly speaking, measures how chaotic a system is. In particular, it is an effective tool to decide whether two systems are conjugated or not. Nevertheless, there are plenty of systems with infinite topological entropy (for instance, they form a $C^0$-generic set in the space of homeomorphisms of a compact manifold \cite{Yano} with dimension greater than one) and thus, in this context, the entropy is not useful anymore. Therefore, in order to study these types of systems, new dynamical quantities are required and an example of such a quantity is the \emph{metric mean dimension}.
The notion of metric mean dimension was introduced by Lindenstrauss and Weiss in \cite{LW} as metric-dependent analog of the \emph{mean dimension}, a topological invariant associated to a dynamical system which was introduced by Gromov \cite{Gromov}. This last notion has several applications, like in the study of embedding problems \cite{GT}, and the metric mean dimension presents an upper bound to it. But more than that, the metric mean dimension turned out to be useful in several contexts like in the study of compression \cite{GS2, GS3}.
In the present paper we give a modest contribution to the study of ergodic theoretical aspects of the metric mean dimension by presenting a variational principle. Previous connections between ergodic theory and metric mean dimension were presented, for instance, by Lindenstrauss and Tsukamoto \cite{LT}, Velozo and Velozo \cite{VV}, Tsukamoto \cite{TSU2020}, Shi \cite{Shi}, Gutman and \'Spiewak \cite{GS} and Yang, Chen and Zhou \cite{YCZ}. For more on these works, see Section \ref{sec: related results}. The main novelty of our work with respect to the previously mentioned ones is that our variational principle holds for special subsets and not only for the whole phase space. More precisely, we consider level sets
\begin{displaymath}
\left\{x\in X: \lim_{n\to\infty}\frac{1}{n}\sum_{j=0}^{n-1}\varphi(f^{j}(x))=\alpha\right\}
\end{displaymath}
associated to continuous potentials $\varphi:X\to \mathbb R$ and continuous dynamics $f:X\to X$ defined on compact metric spaces exhibiting the specification property and present a relation between the \emph{upper metric mean dimension} of the above mentioned sets and growth rates of measure-theoretic entropy of partitions decreasing in diameter associated to some special measures. This is the content of Theorem \ref{thm1}. In Section \ref{sec: examples} we present several examples to which our result is applicable. We start introducing some definitions and notations.
\section{Definitions and Statements}
Let $(X,d)$ be a compact metric space and $f \colon X \to X$ be a continuous map. Given $n\in \mathbb{N}$, we define the dynamical metric $d_n \colon X \times X \, \to \,[0,\infty)$ by
\begin{displaymath}
d_n(x,z)=\max\,\Big\{d(x,z),\,d(f(x),f(z)),\,\dots,\,d(f^{n-1}(x),f^{n-1}(z))\Big\}.
\end{displaymath}
It is easy to see that $d_n$ is indeed a metric and, moreover, generates the same topology as $d$. Furthermore, given $\varepsilon > 0$, $n \in \mathbb{N}$ and a point $x \in X$, we define the open $(n, \varepsilon)$-ball around $x$ by
\begin{displaymath}
\mathit{B}_{n}(x, \varepsilon) = \{ y \in X ; d_n( x, y) < \varepsilon \}.
\end{displaymath}
We sometimes call these $(n, \varepsilon)$-balls \emph{dynamical balls} of radius $\varepsilon$ and length $n$. We say that a set $E \subset X$ is \emph{$(n,\varepsilon)$--separated} by $f$ if $d_n(x,z) > \varepsilon$ for every $x,z \in E$.
\subsection{The metric mean dimension}\label{sse.mmd}
Given $n\in \mathbb{N}$ and $\varepsilon>0$, let us denote by $s(f,n,\varepsilon)$ the maximal cardinality of all $(n,\varepsilon)$--separated subsets of $X$ by $f$ which, due to the compactness of $X$, is finite.
The \emph{upper metric mean dimension} of $f$ with respect to $d$
is given by
$$\mathrm{\overline{mdim}_M}\,\Big(X,f,d\Big) = \limsup_{\varepsilon\,\to\, 0} \,\frac{h(f,\varepsilon)}{|\log \varepsilon|}$$
where
$$h(f,\varepsilon) = \limsup_{n\, \to\, \infty}\,\frac{1}{n}\,\log s(f,n,\varepsilon).$$
Recall that the \emph{topological entropy} of the map $f$ is given by
\begin{displaymath}
h_{\text{top}}(f) = \lim_{\varepsilon \, \to \, 0}\,h(f,\varepsilon).
\end{displaymath}
Consequently, $ \mathrm{\overline{mdim}_M}\,\Big(X,f,d\Big) = 0$ whenever the topological entropy of $f$ is finite. In particular, the metric mean dimension is a suitable quantity to study systems with infinite topological entropy. For more on this quantity see \cite{LW,LT,TSU2020} and references therein.
\subsection{The metric mean dimension for non-compact subset} We now present the notion of metric mean dimension on non-compact sets introduced in \cite{CHeng}.
Given a set $Z\subset X$, let us consider
$$
m(Z,s, N,\varepsilon)=\inf_{\Gamma}\left\{\sum_{i\in I}\exp{\left(-sn_i\right)}\right\},
$$
where the infimum is taken over all covers $\Gamma=\{B_{n_i}(x_i,\varepsilon)\}_{i\in I}$ of $Z$ with $n_i\geq N$. We also consider
$$
m(Z,s,\varepsilon)=\lim_{N\to\infty}m(Z,s, N,\varepsilon).
$$
One can show (see for instance \cite{Pesin}) that there exists a certain number $s_0\in [0,+\infty)$ such that $m(Z,s,\varepsilon)=0$ for every $s>s_0$ and $m(Z,s,\varepsilon)=+\infty$ for every $s<s_0$. In particular, we may consider
$$
h\,\Big(Z,f,\varepsilon\Big)=\inf\{s:m(Z,s,\varepsilon)=0\}=\sup\{s:m(Z,s,\varepsilon)=+\infty\}.
$$
The \textit{upper metric mean dimension of f on $Z$} is then defined as the following limit
\begin{equation}
\overline{\text{mdim}}_M\,\Big(Z,f,d\Big)=\limsup_{\varepsilon\to 0}\frac{h\,\Big(Z,f,\varepsilon\Big)}{|\log \varepsilon|}.
\end{equation}
In the case when $Z=X$ one can check that the two definitions of metric mean dimension given above actually coincide.
\subsection{Level sets of a continuous map} Let $C(X,\mathbb{R})$ denote the set of all continuous maps $\varphi:X\to\mathbb R$ and take $\varphi \in C(X,\mathbb{R})$. For $\alpha\in\mathbb R$, let
\begin{align}\label{def-target-set}
K_\alpha=\left\{x\in X: \lim_{n\to\infty}\frac{1}{n}\sum_{j=0}^{n-1}\varphi(f^{j}(x))=\alpha\right\}.
\end{align}
We also consider the set
\begin{displaymath}
\mathcal L_\varphi=\{\alpha\in\mathbb R: K_\alpha\not=\emptyset\}.
\end{displaymath}
It is easy to see that $\mathcal L_\varphi $ is a bounded and non-empty set \cite[Lemma 2.1]{TV}. Moreover, if $f$ satisfies the so called specification property (see Section \ref{sec: specification}) then $\mathcal L_\varphi$ is a closed interval of $\mathbb R$ and, moreover, $\mathcal L_\varphi=\{\int \varphi d\mu ; \mu \in \mathcal{M}_f(X)\}$ where $\mathcal{M}_f(X)$ stands for the set of all invariant measures (see \cite[Lemma 2.5]{Tho}.
\subsection{The auxiliary quantity $\Lambda_\varphi\mathrm{\overline{mdim}_M}\,(f,\alpha,d)$}
Fix $\alpha\in\mathbb R$ and $\varphi\in C(X,\mathbb{R})$. For $\delta>0$ and $n\in\mathbb N$ define the set
\begin{align*}
P(\alpha,\delta,n)=\left\{x\in X: \left|\frac{1}{n}\sum_{j=0}^{n-1}\varphi(f^{j}(x))-\alpha\right|<\delta\right\}.
\end{align*}
Let $N(\alpha,\delta,n,\varepsilon)$ denote the minimal number of $(n,\varepsilon)$-balls needed to cover $P(\alpha,\delta,n)$. Define
\begin{align*}
\Lambda_{\varphi}(\alpha,\varepsilon)=\lim_{\delta\to0}\liminf_{n\to\infty}\frac{1}{n}\log N(\alpha,\delta,n,\varepsilon)
\end{align*}
and
\begin{align}\label{def_Lamb}
\Lambda_\varphi\mathrm{\overline{mdim}_M}\,(f,\alpha,d)=\limsup_{\varepsilon\to0}\frac{\Lambda_{\varphi}(\alpha,\varepsilon)}{|\log\varepsilon|}.
\end{align}
\begin{remark} \label{remark: sep x span}
Observe that, if $M(\alpha,\delta,n,\varepsilon)$ denotes the maximal cardinality of a $(n,\varepsilon)$-separated set contained in $P(\alpha,\delta,n)$, then we have have that
$$
N(\alpha,\delta,n,\varepsilon)\leq M(\alpha,\delta,n,\varepsilon)\leq N(\alpha,\delta,n,\varepsilon/2).
$$
In particular,
\begin{align}\label{eq:ineq-sep-cov}
\Lambda_{\varphi}(\alpha,\varepsilon)=\lim_{\delta\to0}\liminf_{n\to\infty}\frac{1}{n}\log M(\alpha,\delta,n,\varepsilon).
\end{align}
\end{remark}
\subsection{The main quantity $\mathrm{H_\varphi\overline{mdim}_M}(f,\alpha,d)$} \label{sec: def H varphi}
Given $\varphi\in C(X,\mathbb R)$ and $\alpha \in\mathbb R$, let us consider
$$\mathcal M_{f}(X,\varphi,\alpha)=\left\{\mu\text{ is $f$-invariant and }\int \varphi\;d\mu=\alpha\right\}.$$
A simple observation is that $\mathcal M_{f}(X,\varphi,\alpha)\neq \emptyset$ for every $\alpha\in \mathcal{L}_\varphi$ (see \cite[Lemma 4.1]{TV}).
Let $\mu\in\mathcal M_f(X)$. We say that $\xi=\{C_1,\ldots,C_k\}$ is a measurable partition of $X$ if every $C_i$ is a measurable set, $\mu\left(X\setminus\cup_{i=1}^kC_i\right)=0$ and $\mu\left(C_i\cap C_j\right)=0$ for every $i\neq j$. The \emph{entropy} of $\xi$ with respect to $\mu$ is given by
\[
H_\mu(\xi)=-\sum_{i=1}^{k}\mu(C_i)\log(\mu(C_i)).
\]
Given a measurable partition $\xi$, we consider $\xi^n=\bigvee_{j=0}^{n-1}f^{-j}\mathcal \xi$. Then, the \emph{metric entropy of $(f,\mu)$ with respect to $\xi$} is given by
\[
h_\mu(f,\xi)=\lim_{n\to +\infty}\frac{1}{n} H_\mu(\xi^n).
\]
Using this quantity we define
\begin{align}\label{def_Hmdim}
\mathrm{H_\varphi\overline{mdim}_M}\,(f,\alpha,d)=\limsup_{\varepsilon\to0}\frac{1}{|\log\varepsilon|}\sup_{\mu\in \mathcal M_{f}(X,\varphi,\alpha)}\inf_{|\xi|<\varepsilon}h_\mu(f,\xi)
\end{align}
where $|\xi|$ denotes the diameter of the partition $\xi$ and the infimum is taken over all finite measurable partitions of $X$ satisfying $|\xi|<\varepsilon$.
We also recall that the \emph{metric entropy of $(f,\mu)$} is given by
\[
h_\mu(f)=\sup_\xi h_\mu(f,\xi)
\]
where the supremum is taken over all finite measurable partitions $\xi$ of $X$.
\subsection{Specification property} \label{sec: specification}
We say that $f$ satisfies the \emph{specification property} if for every $\epsilon > 0$, there exists an integer $m = m(\epsilon )$ such that for any collection of finite intervals $ I_j = [a_j, b_j ] \subset \mathbb{N}$, $j = 1, \ldots, k $, satisfying $a_{j+1} - b_j \geq m(\epsilon )$ for every $j = 1, \ldots, k-1 $ and any $x_1, \ldots, x_k$ in $X$, there exists a point $x \in X$ such that
\begin{equation*}
d(f^{p + a_j}x, f^p x_j) < \epsilon \mbox{ for all } p = 0, \ldots, b_j - a_j \mbox{ and every } j = 1, \ldots, k.
\end{equation*}
The specification property is present in many interesting examples. For instance, every topologically mixing locally maximal hyperbolic set has the specification property and factors of systems with specification have specification (see for instance \cite{KH}). Other examples of systems satisfying this property which are more adapted to our purposes will appear in Section \ref{sec: examples}.
\subsection{Main result} Our main result may be seen as an extension of \cite[Theorem 5.1]{TV} to the infinite entropy setting. We assume that all the quantities $\mathrm{\overline{mdim}_M}\,\Big(K_\alpha,f, d\Big)$, $\Lambda_\varphi\mathrm{\overline{mdim}_M}\,(f,\alpha,d)$ and $\mathrm H_\varphi\mathrm{\overline{mdim}_M}\,(f,\alpha,d)$ are finite (see Remark \ref{remark: finiteness} about this hypothesis).
\begin{maintheorem}\label{thm1}
Suppose $f:X \to X$ is a continuous transformation with the specification
property. Let $\varphi\in C(X,\mathbb R)$ and $\alpha\in\mathbb R$ be such that $K_\alpha\not=\emptyset$.
Then
$$
\mathrm{\overline{mdim}_M}\,\Big(K_\alpha,f, d\Big)=\Lambda_\varphi\mathrm{\overline{mdim}_M}\,(f,\alpha,d)= \mathrm H_\varphi\mathrm{\overline{mdim}_M}\,(f,\alpha,d).
$$
\end{maintheorem}
We consider the equality between $\mathrm{\overline{mdim}_M}\,\Big(K_\alpha,f, d\Big)$ and $\mathrm H_\varphi\mathrm{\overline{mdim}_M}\,(f,\alpha,d)$ to be the most important part of the result because it relates a topological quantity with one that as an ergodic-theoretical flavor. An interesting question is whether we can change the order between the limit and the supremum in the definition of $\mathrm H_\varphi\mathrm{\overline{mdim}_M}\,(f,\alpha,d)$. This would allow, for instance, to talk about the existence of ``maximizing measures": measures that realize the supremum. Such a measure would capture the complexity of the system over all scales $\varepsilon >0$.
It was observed in \cite[Section VIII]{LT} that a similar question involving different ergodic quantities is, in general, false. Nevertheless, under the additional assumption that $f$ has the marker property, one can do such a change (in the setting of \cite{LT}) as observed by Yang, Chen and Zhou \cite{YCZ}.
\subsection{Related results}\label{sec: related results}
As already mentioned, for the topological entropy a result similar to Theorem \ref{thm1} was obtained in \cite{TV}. In fact, our result was inspired by that one. Moreover, \cite{TV} was extended to the framework of topological pressure in \cite{Tho}.
As for variational results involving the upper metric mean dimension, there are several works dealing with this problem. For instance, \cite{LT} presented a variational principle relating the metric mean dimension with the supremum of certain rate distortion functions over invariant measures of the system. This was further explored in \cite{VV}. More recently, \cite{Shi} obtained variational principles for the metric mean dimension in terms of Brin-Katok local entropy and Shapira's
entropy of an open cover. One result that is more connected to ours is the one obtained in \cite{GS} which says that
\begin{align}\label{vari_Hmdim}
\mathrm{\overline{mdim}_M}\,(X,f,d)=\limsup_{\varepsilon\to0}\frac{1}{|\log\varepsilon|}\sup_{\mu\in \mathcal M_{f}(X)}\inf_{|\xi|<\varepsilon}h_\mu(f,\xi).
\end{align}
This is a variational result for the upper metric mean dimension of the entire space $X$ while Theorem \ref{thm1} applies also to level sets of continuous maps $\varphi$. Observe that in the case when $\varphi$ is a constant map equal to $\alpha$, the $\alpha$-level set of it coincides with $X$. In particular, whenever $f$ has the specification property, \eqref{vari_Hmdim} may be seen as a particular case of our result. We stress however that the results in \cite{GS} do not assume such property.
\begin{remark}\label{remark: finiteness}
We observe that, since $K_\alpha \subset X$, $\mathrm{\overline{mdim}_M}\,\Big(K_\alpha,f, d\Big)\leq \mathrm{\overline{mdim}_M}\,\Big(X,f, d\Big)$. Similarly, applying \eqref{vari_Hmdim},
$$\mathrm H_\varphi\mathrm{\overline{mdim}_M}\,(f,\alpha,d)\leq \limsup_{\varepsilon\to0}\frac{1}{|\log\varepsilon|}\sup_{\mu\in \mathcal M_{f}(X)}\inf_{|\xi|<\varepsilon}h_\mu(f,\xi)=\mathrm{\overline{mdim}_M}\,(X,f,d).$$
Moreover, using that $M(\alpha,\delta,n,\varepsilon)\leq s(f,n,\varepsilon)$, it follows by Remark \ref{remark: sep x span} that $\Lambda_\varphi\mathrm{\overline{mdim}_M}\,(f,\alpha,d)\leq \mathrm{\overline{mdim}_M}\,\Big(X,f, d\Big)$.
Therefore, combining all these observations it follows that the three quantities $\mathrm{\overline{mdim}_M}\,\Big(K_\alpha,f, d\Big)$, $\Lambda_\varphi\mathrm{\overline{mdim}_M}\,(f,\alpha,d)$ and $\mathrm H_\varphi\mathrm{\overline{mdim}_M}\,(f,\alpha,d)$ are finite whenever $\mathrm{\overline{mdim}_M}\,\Big(X,f, d\Big)$ is finite. In particular, Theorem \ref{thm1} may be applied whenever $f$ has the specification property and $\mathrm{\overline{mdim}_M}\,\Big(X,f, d\Big)<+\infty$.
\end{remark}
\section{Proof of Theorem \ref{thm1}}
In order to prove Theorem \ref{thm1}, we split its statement into three main propositions. We emphasize that this proof is an adaptation of the proof of Theorem 5.1 of \cite{TV} to our setting. Fix $\varphi\in C(X,\mathbb R)$ and $\alpha\in\mathbb R$ such that $K_\alpha\not=\emptyset$.
\begin{proposition}\label{lemma:20}
Under the hypotheses of Theorem \ref{thm1} we have that
\[
\mathrm{\overline{mdim}_M}\,(K_\alpha,f,d)\leq \Lambda_\varphi\mathrm{\overline{mdim}_M}\,(f,\alpha,d).
\]
\end{proposition}
\begin{proof}
Let $\{\varepsilon_j\}_{j\in\mathbb N}$ be a sequence of positive numbers converging to zero such that
\[
\mathrm{\overline{mdim}_M}\,(K_\alpha,f,d)=\lim_{j\to\infty}\frac{h(K_\alpha,f,\varepsilon_j)}{|\log \varepsilon_j|}.
\]
In particular we have that
$$
\limsup_{j\to\infty}\frac{\Lambda_{\varphi}(\alpha,\varepsilon_j)}{|\log\varepsilon_j|}\leq\limsup_{\varepsilon\to0}\frac{\Lambda_{\varphi}(\alpha,\varepsilon)}{|\log\varepsilon|}= \Lambda_\varphi\mathrm{\overline{mdim}_M}\,(f,\alpha,d).
$$
Given $\delta>0$ and $k\in\mathbb N$, let us consider the set
\begin{align*}
G(\alpha,\delta,k)&=\bigcap_{n=k}^\infty P(\alpha,\delta, n)\\
&=\bigcap_{n=k}^\infty\left\{x\in X:\left|\frac{1}{n}\sum_{j=0}^{n-1}\varphi(f^{j}(x))-\alpha\right|<\delta \right\}.
\end{align*}
As a consequence of the definition we have that $K_\alpha\subset\bigcup_{k\in\mathbb N} G(\alpha,\delta,k)$.
Now, given $k\in \mathbb{N}$, since $G(\alpha,\delta,k)\subset P(\alpha,\delta,n)$ for $n\geq k$, it follows that $G(\alpha,\delta,k)$ may be covered by $N(\alpha,\delta,n,\varepsilon_j)$ dynamical balls of radius $\varepsilon_j$ and length $n$. Thus, for every $s\geq0$ and $n\geq k$ we have
\[
m(G(\alpha,\delta,k),s,\varepsilon_j)\leq N(\alpha,\delta,n,\varepsilon_j)\exp(-ns).
\]
Let $s=s(\varepsilon_j)>\Lambda_\varphi(\alpha,\varepsilon_j)$ and $\gamma(\varepsilon_j)=(s-\Lambda_\varphi(\alpha,\varepsilon_j))\slash2$. Then, if $\delta_j>0$ is small enough, there exists an increasing sequence
$\{n_\ell\}_{\ell\in\mathbb N}\subset \mathbb N$ such that
\[
N(\alpha,\delta_j,n_\ell, \varepsilon_j)\leq \exp(n_\ell(\Lambda_\varphi(\alpha,\varepsilon_j)+\gamma(\varepsilon_j))).
\]
Thus, assuming without lost of generality that $n_1\geq k$ and combining the previous observations we conclude that
\[
m(G(\alpha,k,\delta_j),s(\varepsilon
_j),\varepsilon_j)\leq \exp(-n_\ell \gamma(\varepsilon_j)).
\]
In particular, as $\gamma(\varepsilon_j)>0$, letting $n_\ell\to\infty $ we obtain $m(G(\alpha,k,\delta_j),s(\varepsilon_j),\varepsilon_j)=0$. Consequently,
\[
h(G(\alpha,k,\delta_j),f,\varepsilon_j)\leq s(\varepsilon_j)
\]
which implies that
\[
h(K_\alpha,f,\varepsilon_j)\leq \sup_k h(G(\alpha,k,\delta_j),f,\varepsilon_j)\leq s(\varepsilon_j).
\]
Hence,
\begin{align*}
\mathrm{\overline{mdim}_M}\,(K_\alpha,f,d)&=\limsup_{j\to\infty}\frac{h(K_\alpha,f,\varepsilon_j)}{|\log \varepsilon_j|}\\
&\leq \limsup_{j\to\infty}\frac{s(\varepsilon_j)}{|\log \varepsilon_j|}\\
&=\limsup_{j\to\infty}\frac{2\gamma(\varepsilon_j)}{|\log \varepsilon_j|}+\limsup_{j\to\infty}\frac{\Lambda_\varphi(\alpha,\varepsilon_j)}{|\log \varepsilon_j|}\\
&\leq \limsup_{j\to\infty}\frac{2\gamma(\varepsilon_j)}{|\log \varepsilon_j|}+ \Lambda_\varphi\mathrm{\overline{mdim}_M}\,(f,\alpha,d).
\end{align*}
Therefore, as we can choose $s(\varepsilon_j)$ arbitrarily close to $\Lambda_\varphi(\alpha,\varepsilon_j)$, the limsup in the last step is zero for an adequate choice of $s(\varepsilon_j)$. Then,
$ \mathrm{\overline{mdim}_M}\,(K_\alpha,f,d)\leq \Lambda_\varphi\mathrm{\overline{mdim}_M}\,(f,\alpha,d)$ completing the proof of the proposition.
\end{proof}
\begin{proposition}\label{lemma:21}
Under the hypotheses of Theorem \ref{thm1} we have that
\[
\mathrm {H_\varphi\overline{mdim}_M}\,(f,\alpha,d)\leq \mathrm{\overline{mdim}_M}\,(K_\alpha,f,d).
\]
\end{proposition}
The strategy of the proof consists in constructing a fractal set $F$ contained in $K_\alpha$ and a special probability measure $\eta $ supported on $F$ that satisfies the hypothesis of the so called Entropy Distribution Principle (see Lemma \ref{lemma:entropy-dist-principles}). This will be enough to get the desired inequality. As a step towards the definition of $F$, we introduce a family of finite sets $\mathcal{S}_k$ which play a major role in the construction.
In order to prove Proposition \ref{lemma:21} we will need the following auxiliary quantity. For $\mu\in\mathcal M_f(X)$, $\delta >0$ and $n\in\mathbb N$, let us denote by $N_\mu(\delta,\varepsilon,n)$ the minimal number $(n,\varepsilon)$-balls needed to cover a set of $\mu$-measure bigger than $1-\delta$. Then, we define
\begin{equation}\label{eq: h mu f eps delta}
h_\mu(f,\varepsilon,\delta)=\limsup_{n\to\infty}\frac{1}{n}\log N_\mu(\delta,\varepsilon,n).
\end{equation}
\begin{proof}[Proof of Proposition \ref{lemma:21}]
Fix $\gamma>0$ and let $\{\delta_k\}_{k\in\mathbb{N}}$ be a decreasing sequence converging to $0$. Take $\varepsilon
=\varepsilon(\gamma)>0$ and $\mu\in \mathcal M_{f}(X,\varphi,\alpha)$ so that
\begin{align*}
\frac{\inf_{|\xi|<5\varepsilon}h_{\mu}(f,\xi)}{|\log5\varepsilon|} \geq\mathrm {H_\varphi\overline{mdim}_M}\,(f,\alpha,d)-\frac\gamma2
\end{align*}
and
\begin{equation}\label{eq:choice varepsilon for h}
\frac{h(K_\alpha, f,\varepsilon/2)}{|\log \varepsilon/2|}\leq \mathrm{\overline{mdim}_M}\,(K_\alpha,f,d) +\gamma.
\end{equation}
Let $\mathcal U$ be a finite open cover of $X$ with diameter $\mathrm{diam}(\mathcal U)\leq 5\varepsilon$ and Lebesgue number $\mathrm{Leb}(\mathcal U)\geq \frac{5\varepsilon}{4}$. We now construct an auxiliary measure which is a finite combination of ergodic measures and ``approximates" $\mu$. To prove this lemma we follow the idea from \cite[p. 535]{You}. In what follows, $\partial \xi$ will denote the boundary of the partition $\xi$ which is just the union of the boundaries of all the elements of the partition and $\xi\succ\mathcal U$ means that $\xi$ refines $\mathcal{U}$, that is, each element of $\xi$ is contained in an element of $\mathcal U$.
\begin{lemma}\label{lem: approx}
For each $k\in\mathbb N$, there exists a measure $\nu_k \in \mathcal M_{f}(X)$ satisfying
\begin{itemize}
\item[(a)] $\nu_k=\displaystyle\sum_{i=1}^{j(k)} \lambda_i\nu^k_i$, where $\lambda_i>0$, $\displaystyle\sum_{i=1}^{j(k)} \lambda_i=1$ and $\nu^k_i\in \mathcal M_f^{\text{erg}}(X) $;
\item[(b)] $ \displaystyle \inf_{\xi\succ\mathcal U} h_\mu\left(f,\xi\right)\leq \inf_{\xi\succ\mathcal U}\displaystyle h_{\nu_k}\left(f,\xi\right)+\delta_k/2$;
\item[(c)] $\left|\displaystyle\int_X \varphi\; d\nu_k-\displaystyle\int_X\varphi\; d\mu\right|<\delta_k$.
\end{itemize}
\end{lemma}
\begin{proof}[Proof of Lemma \ref{lem: approx}] Given $k\in \mathbb N$, let $\beta_k>0$ be such that for every $\tau_1,\tau_2 \in \mathcal{M}_{f}(X)$,
\begin{displaymath}
d_{\mathcal{M}_{f}(X)}(\tau_1,\tau_2)<\beta_k \implies \left| \int \varphi d\tau_1-\int \varphi d\tau_1\right|<\delta_k
\end{displaymath}
where $d_{\mathcal{M}_{f}(X)}$ is a metric in $\mathcal{M}_{f}(X)$. Let $\mathcal{P}=\{P_1,\ldots,P_{j(k)}\}$ be a partition of $\mathcal{M}_{f}(X)$ whose diameter with respect to $d_{\mathcal{M}_{f}(X)}$ is smaller than $\beta_k$. By the Ergodic Decomposition Theorem there exists a measure $\hat{\mu}$ on $\mathcal{M}_{f}(X)$ satisfying $\hat{\mu}(\mathcal{M}_{f}^{\text{erg}}(X))=1$ such that
$$\int \psi (x) d\mu(x)=\int_{\mathcal{M}_{f}(X)} \left(\int_X \psi(x) d\tau(x)\right)d\hat{\mu}(\tau) \text{ for every } \psi \in C(X,\mathbb R).$$
Let us consider now $\lambda_i=\hat{\mu}(P_i)$ and take $\nu^k_i\in P_i\cap \mathcal{M}_{f}^{\text{erg}}(X)$ such that $\inf_{\xi\succ\mathcal U} h_{\nu^k_i}\left(f,\xi\right)\geq \inf_{\xi\succ\mathcal U} h_\tau\left(f,\xi\right)-\delta_k/2$ for $\hat{\mu}$-almost every $\tau \in P_i\cap \mathcal{M}_{f}^{\text{erg}}(X) $. Observe that such a measure $\nu_i^k$ exists because $\sup_{\tau \in \mathcal{M}_f^{\text{erg}}(X)} \inf_{\xi\succ\mathcal U} h_\tau\left(f,\xi\right)<+\infty$. This latter fact follows from Lemma 3 and Theorem 5 of \cite{Shi} and the fact that the upper metric mean dimension is finite. Finally, define $\nu_k=\sum_{i=1}^{j(k)} \lambda_i\nu^k_i$. It is easy to see that $\nu_k$ satisfies properties a) and c) from the statement. Let us now check that it also satisfies b). By \cite[Proposition 5]{HMRY} we know that
$$\inf_{\xi\succ\mathcal U} h_{\mu}\left(f,\xi\right)=\int_{M_{f}(X)} \inf_{\xi\succ\mathcal U} h_{\tau}\left(f,\xi\right)d\hat{\mu}(\tau).$$
Thus, by our choice of the measures $\nu^k_i$ it follows that
\begin{displaymath}
\begin{split}
\inf_{\xi\succ\mathcal U} h_{\mu}\left(f,\xi\right)&=\int_{M_{f}(X)} \inf_{\xi\succ\mathcal U} h_{\tau}\left(f,\xi\right)d\hat{\mu}(\tau)\\
&\leq \sum_{i=1}^{j(k)} \lambda_i \inf_{\xi\succ\mathcal U} h_{\nu_i^k}\left(f,\xi\right)+\delta_k/2\\
&\leq \inf_{\xi\succ\mathcal U} h_{\nu_k}\left(f,\xi\right) +\delta_k/2
\end{split}
\end{displaymath}
completing the proof of the lemma.
\end{proof}
Let $\nu_k$ be as in the previous lemma. Using the fact that each measure $\nu^k_i$ is ergodic, by the proof of \cite[Theorem 9]{Shi} there exists a finite Borel measurable partition $\xi_k$ which refines $\mathcal U$ so that
\begin{align}\label{eq: relation h part x sep}
h_{\nu^k_i}(f,5\varepsilon,\gamma)\leq h_{\nu_i^k}(f,\xi_k)\leq h_{\nu^k_i}(f,5\varepsilon/4,\gamma)+\delta_k.
\end{align}
Now, take a finite Borel partition $\xi$ refining $\mathcal{U}$ with $\mu(\partial \xi)=0$ such that
\begin{displaymath}
h_\mu\left(f,\xi\right)-\delta_k\leq \inf_{\zeta \succ\mathcal U}\displaystyle h_{\nu_k}\left(f,\zeta \right).
\end{displaymath}
In particular, since $\xi_k \succ \mathcal{U}$,
\begin{equation}\label{eq: hmu X hnu}
h_\mu\left(f,\xi\right)-\delta_k\leq h_{\nu_k}\left(f,\xi_k\right).
\end{equation}
Moreover, since $\xi\succ\mathcal U$ it follows that $|\xi|<5\varepsilon$ and thus
\begin{align}\label{eq:escolha de xi}
\frac{h_{\mu}(f,\xi)}{|\log5\varepsilon|} \geq\mathrm {H_\varphi\overline{mdim}_M}\,(f,\alpha,d)-\gamma.
\end{align}
Again, since each $\nu^k_i$ is ergodic, there exists $\ell_k\in \mathbb N$ large enough for which the set
\[
Y_{i}(k)=\left\{x\in X: \left|\frac{1}{n}\sum_{j=0}^{n-1}\varphi(f^j(x))-\int_X\varphi\; d\nu^k_i\right|<\delta_k\;\; \forall \; n\geq \ell_k\right\}
\]
has $\nu^k_i$-measure bigger than $1-\gamma$ for every $k\in\mathbb N$ and $i\in\{1,\dots,j(k)\}$.
By \cite[Lemma 3.6]{Tho}, there exists $\hat n_k\to\infty$ with $[\lambda_i\hat n_k]\geq \ell_k$ so that the maximal cardinality of an $([\lambda_i\hat n_k],5\varepsilon/4)$-separated set in $Y_i(k)$, denoted by $M_{k,i}$, satisfies \begin{equation}\label{eq: Mki inequality}
M_{k,i}\geq \exp \left( [\lambda_i\hat n_k]\left(h_{\nu^k_i}(f,5\varepsilon/4,\gamma)-\frac{4\gamma}{j(k)}\right)\right) .
\end{equation}
Furthermore, the sequence $\hat n_k$ can be chosen such that $\hat n_k\geq 2^{ m_k}$ where $m_k=m(\varepsilon/2^{k+2})$ is as in the definition of the specification property. Let $n_k:=m_k({j(k)}-1)+\sum_{i}[\lambda_i\hat n_k]$. Observe that $n_k/\hat n_k\to 1$.
Denote by $E_{i,k}([\lambda_i\hat n_k],5\varepsilon/4)$ a maximal $([\lambda_i\hat n_k],5\varepsilon/4)$-separated set in $Y_i(k)$. By the specification property, for each
$$x_1\in E_{1,k}(n_1,5\varepsilon/4),\; x_2\in E_{2,k}(n_2,5\varepsilon/4), \dots,\; x_{j(k)}\in E_{j(k),k}(n_{j(k)},5\varepsilon/4),$$
there exists $y=y(x_1,\dots,x_{j(k)})\in X$ so that the pieces of orbits
\begin{displaymath}
\{x_i,f(x_i),\dots,f^{[\lambda_i\hat n_k]-1}(x_i); \; i= 1,\ldots,j(k)\}
\end{displaymath}
are $\varepsilon/2^k$-shadowed by $y$ with gap $m_k$. We claim that if $(x_1,\dots,x_{j(k)})\not=(x'_1,\dots,x'_{j(k)})$ then $y(x_1,\dots,x_{j(k)})\not=y'(x'_1,\dots,x'_{j(k)})$. Indeed, if $x_i\not=x'_i$,
\begin{align*}
\frac{5 \varepsilon}{4} &<d_{[\lambda_i\hat n_k]}(x_i,x'_i)\\
&\leq d_{[\lambda_i\hat n_k]}(x_i, f^{[\lambda_1\hat n_k]+\dots+[\lambda_{i-1}\hat n_k]+(i -1)m_k}(y))\\
&+d_{[\lambda_i\hat n_k]}(x'_i, f^{[\lambda_1\hat n_k]+\dots+[\lambda_{i-1}\hat n_k]+(i -1)m_k}(y'))\\
&+d_{[\lambda_i\hat n_k]}( f^{[\lambda_1\hat n_k]+\dots+[\lambda_{i-1}\hat n_k]+(i -1)m_k}(y), f^{[\lambda_1\hat n_k]+\dots+[\lambda_{i-1}\hat n_k]+(i -1)m_k}(y'))\\
&<2\frac\varepsilon{2^{k+2}}+d_{[\lambda_i\hat n_k]}( f^{[\lambda_1\hat n_k]+\dots+[\lambda_{i-1}\hat n_k]+(i -1)m_k}(y), f^{[\lambda_1\hat n_k]+\dots+[\lambda_{i-1}\hat n_k]+(i -1)m_k}(y')),
\end{align*}
which implies that $d_{n_k}(y,y')>9\varepsilon/8$ proving our claim. Moreover, as a by-product of this observation we get that
\begin{align*}
\mathcal S_k=\{y(x_1,\dots,x_{j(k)}): x_i\in E_{i,k}([\lambda_i\hat n_k],5\varepsilon/4) \text{ for } i=1,\ldots,j(k)\}
\end{align*}
is a $(n_k, 9\varepsilon/8)$-separated set with cardinality $M_k:=\prod_{i=i}^{j(k)}M_{k,i}$. Combining \eqref{eq: relation h part x sep}, \eqref{eq: hmu X hnu} and \eqref{eq: Mki inequality} with the the choices of $\varepsilon$, $\gamma$ and $n_k$ and recalling that $n_k/\hat n_k\to 1$ we get that for $k$ sufficiently large
\begin{align}\label{eq:200}
M_k=\prod_{i=i}^{j(k)}\sharp E_{i,k}([\lambda_i\hat n_k],9\varepsilon/8)
&\geq \exp{\left(\sum_{i=1}^{j(k)}[\lambda_i\hat n_k]\left(h_{\nu^k_i}(f,5\varepsilon/4,\gamma)-\frac{4\gamma}{j(k)}\right)\right)}\\ \nonumber
&\geq \exp{\left(\hat n_k\sum_{i=1}^{j(k)}\lambda_ih_{\nu^k_i}(f,5\varepsilon/4,\gamma)-4\hat n_k\gamma\right)}\\ \nonumber
&\geq \exp{\left(\hat n_k\sum_{i=1}^{j(k)}\lambda_ih_{\nu^k_i}(f,\xi_k)-4\hat n_k\gamma-\hat n_k\delta_k\right)}\\ \nonumber
&\geq \exp{\left(\hat n_k(h_{\nu_k}(f,\xi_k)-4\gamma-\delta_k)\right)}\\ \nonumber
&\geq \exp{\left(R_k n_k(h_{\nu_k}(f,\xi_k,\gamma)-4\gamma-\delta_k)\right)}\\ \nonumber
&\geq \exp{\left(R_k n_k(h_{\mu}(f,\xi)-4\gamma - 2\delta_k)\right)}\\ \nonumber
&\geq \exp{\left(R_k n_k(h_{\mu}(f,\xi)-5\gamma)\right)}
\end{align}
for some $R_k\in(0,1)$.
Let $y=y(x_1,\dots,x_k) \in \mathcal S_k$. Then,
\begin{align*}
\left|S_{n_k}\varphi(y)-n_k\alpha\right|
&\leq \left|S_{n_k}\varphi(y)-n_k\left(\int\varphi d\nu_k-\delta_k\right)\right|\\
&\leq\sum_{i=1}^{j(k)-1}\left|S_{[\lambda_i\hat n_k]}\varphi(f^{\sum_{t=1}^{i-1}[\lambda_t\hat n_k]+(i-1)m_k}(y))-n_k\lambda_i\int\varphi d\nu^k_i\right|\\
&+n_k\delta_k+m_k(j(k)-1)\|\varphi\|\\
&\leq\sum_{i=1}^{j(k)-1}\left|S_{[\lambda_i\hat n_k]}\varphi(x_i)-[\lambda_in_k]\int\varphi d\nu^k_i\right|\\
&+n_k\delta_k+m_kj(k)\|\varphi\|+n_k\mathrm{Var}(\varphi,\varepsilon/2^k)\\
&<\delta_k\sum_{i=1}^{j(k)-1}[\lambda_i\hat n_k]+m_kj(k)\|\varphi\|+n_k\delta_k+n_k\mathrm{Var}(\varphi,\varepsilon/2^k).
\end{align*}
Thus, for sufficiently large $k$,
\begin{align}\label{eq:201}
\left|\frac{1}{n_k}S_{n_k}\varphi(y)-\alpha\right|\leq \delta_k+\mathrm{Var}(\varphi,\varepsilon/2^k)+\frac{1}{k}.
\end{align}
We now choose a sequence $\{N_k\}_{k\in\mathbb N}$ of positive integers such that $N_1=1$ and
\begin{itemize}
\item[(1)] $[R_kN_k]\geq 2^{n_{k+1}+m_{k+1}}$, for $k\geq 2$;
\item[(2)] $[R_{k+1}N_{k+1}]\geq 2^{[R_1N_1n_1]+\dots +[R_kN_k(n_k+m_k)]}$, for $k\geq 1$.
\end{itemize}
Observe that this sequence $\{N_k\}_{k\in\mathbb N}$ grows very fast and
\begin{align}\label{eq:0}
\lim_{k\to \infty}\frac{n_{k+1}+m_{k+1}}{R_kN_k}=0 \text{ and }\lim_{k\to \infty}\frac{R_1N_1n_1+\dots +R_kN_k(n_k+m_k)}{R_{k+1}N_{k+1}}=0.
\end{align}
Moreover, we enumerate the points in $\mathcal{S}_k$ as
\begin{displaymath}
\mathcal S_k=\{x_i^k:\; i=1,\ldots,M_k\}.
\end{displaymath}
For any $(i_1,\dots,i_{N_k})\in \{1,2,\dots,M_{k}\}^{[R_kN_k]}$, let $y(i_1,\dots,i_{[R_kN_k]})\in X$ be given by the specification property so that its orbit
$\varepsilon\slash 2^k$-shadows, with gap $m_k$, the pieces of orbits $\{x_{i_j}^k,f(x_{i_j}^k),\dots,f^{n_k-1}(x_{i_j}^k)\}$, $j=1,2,\ldots, [R_kN_k]$. Then, define
$$
\mathcal C_k=\{y(i_1,\dots,i_{[R_kN_k]})\in X:(i_1,\dots,i_{[R_kN_k]})\in \{1,2,\dots,M_k\}^{[R_kN_k]}\}.
$$
Moreover, consider
\begin{equation*}
c_k=[R_kN_k]n_k+([R_{k}N_k]-1)m_k.
\end{equation*}
We now observe that different sequences in $\{1,2,\dots,M_k\}^{[R_kN_k]}$ give rise to different points in $\mathcal C_k$ and that such points are uniformly separated with respect to $d_{c_k}$.
\begin{lemma}[Lemma 5.1 of \cite{TV}] \label{lemma:10}
If $(i_1,\dots,i_{[R_kN_k]})\not=(j_1,\dots,j_{[R_kN_k]})$, then
$$
d_{c_k}(y(i_1,\dots,i_{[R_kN_k]}),y(j_1,\dots,j_{[R_kN_k]}))>\varepsilon.
$$
In particular $\sharp\mathcal C_k=M_k^{[R_kN_k]}$.
\end{lemma}
Our next step is to construct inductively an auxiliary sequence of finite sets $\mathcal{T}_k$.
Let $\mathcal T_1=\mathcal C_1$ and $t_1=c_1$. Now, suppose that we have already constructed the set $\mathcal T_k$ and we will describe how to construct $\mathcal T_{k+1}$. Consider
\begin{align}\label{eq:1}
t_{k+1}&=t_k+m_{k+1}+c_{k+1} \nonumber \\
&\phantom{=} =[R_1N_1]n_1+[R_2N_2](n_2+m_2)+\dots+[R_{k+1}N_{k+1}](n_{k+1}+m_{k+1}).
\end{align}
For $x\in \mathcal T_k$ and $y\in \mathcal C_{k+1}$, let $z=z(x,y)$ be some point such that
\begin{align}\label{eq:4}
d_{t_k}(x,z)<\frac{\varepsilon}{2^{k+1}} \text{ and } d_{c_{k+1}}(y,f^{t_k+m_{k+1}}(z))<\frac{\varepsilon}{2^{k+1}}.
\end{align}
Observe that the existence of such a point is guaranteed by the specification property of $f$. Then, let us consider
$$
\mathcal T_{k+1}=\{z(x,y):x\in \mathcal T_k, \; y\in \mathcal C_{k+1}\}.
$$
By proceeding as in the proof of the Lemma \ref{lemma:10} we can see that different pairs $(x,y)$, $x\in \mathcal T_k$, $y\in \mathcal C_{k+1}$, produce different points $z=z(x,y)$. In particular, $\sharp \mathcal T_{k+1}=\sharp \mathcal T_k \cdot \sharp\mathcal C_{k+1}$. Therefore, proceeding inductively,
$$
\sharp \mathcal T_k=\sharp \mathcal C_1\dots \sharp\mathcal C_k= M_1^{[R_1N_1]}\dots M_k^{[R_kN_k]}.
$$
In particular, by Lemma \ref{lemma:10} and \eqref{eq:4} we have that for every $x\in \mathcal T_k$ and $y,y'\in \mathcal C_{k+1}$ with $y\not=y'$,
\begin{align}\label{eq:500}
d_{t_k}(z(x,y),z(x,y'))<\frac{\varepsilon}{2^{k+2}} \text{ and } d_{t_{k+1}}(z(x,y),z(x,y'))>\frac{3\varepsilon}{4}.
\end{align}
For every $k\in\mathbb N$ let us consider
$$
F_k:=\bigcup_{x\in \mathcal T_k}\overline B_{t_k}(x, \varepsilon/2^{k+1}),
$$
where $\overline B_{ t_k}(x, \varepsilon/2^{k+1})$ denotes the closure of the open ball $ B_{t_k}(x, \varepsilon/2^{k+1})$. As a simple consequence of \eqref{eq:500} we have the following observation.
\begin{lemma}[Lemma 5.2 of \cite{TV}] \label{lemma:11}
For every $k$ the following is satisfied:\\
\noindent(1) for any $x, x'\in \mathcal T_k$, $x \not= x'$, the sets $\overline B_{t_k}(x, \varepsilon/2^{k+1})$ and
$\overline B_{t_k}(x', \varepsilon/2^{k+1})$ are disjoint;\\
\noindent(2) if $z\in \mathcal T_{k+1}$ is such that $z=z(x,y)$ for some $x \in\mathcal T_k$ and $y\in \mathcal{C}_{k+1}$, then
$$
\overline B_{t_{k+1}}\left(z, \frac{\varepsilon}{2^{k+2}}\right)\subset \overline B_{t_k}\left(x, \frac{\varepsilon}{2^{k+1}}\right).
$$
Hence, $F_{k+1}\subset F_k$.
\end{lemma}
Consider
$$F:=\bigcap_{k\in\mathbb N}F_k.$$
Observe that, since each $F_k$ is a closed and non-empty set and, moreover, $F_{k+1}\subset F_k$, the set $F$ is a non-empty and closed set too. Furthermore, using \eqref{eq:201} we may prove that
\begin{lemma}[Lemma 5.3 of \cite{TV}] \label{lemma: F_subset_K_alpha}
Under the above conditions,
$$F\subset K_\alpha.$$
\end{lemma}
Now, for every $k\geq 1$, let us consider the probability measure $\eta_k$ given by
\begin{align*}
\eta_k=\frac{1}{\sharp \mathcal T_k}\sum_{z\in \mathcal T_k}\delta_z.
\end{align*}
Observe that, as $\mathcal{T}_k\subset F_k$, $\eta_k(F_k)=1$. Moreover,
\begin{lemma}[Lemma 5.4 of \cite{TV}] \label{lemma:12}
The sequence of probability measures $(\eta_k)_{k\in \mathbb N}$ converges in the weak$^{\ast}$-topology to some probability measure $\eta$. Furthermore, the limiting measure $\eta$ satisfies $\eta(F)=1$.
\end{lemma}
An important feature of the measure $\eta$ that can be obtained by exploring its definition and \eqref{eq:200} is that the $\eta$-measure of some appropriate dynamical balls decay exponentially fast. More precisely,
\begin{lemma}[Lemma 5.5 of \cite{TV}] \label{lemma: exp decay measure}
For every $n$ sufficiently large and $q\in X$ so that $B_n\left(q,\frac{\varepsilon}{2}\right)\cap F\not=\emptyset$ one has
$$\eta\left(B_n\left(q,\frac{\varepsilon}{2}\right)\right)\leq \exp{\left(-n(h_\mu(f,\xi)-8\gamma)\right)}.$$
\end{lemma}
In order to conclude our proof we need a simple yet interesting fact whose proof we include for the sake of completeness. This is a version of the \emph{Entropy Distribution Principle} of \cite{TV} (see \cite[Theorem 3.6]{TV}) Observe that for this result, the measure involved does not need to be invariant, as it is the case of the measure $\eta$ obtained in the previous lemmas.
\begin{lemma} \label{lemma:entropy-dist-principles}
Let $f : X \to X$ be a continuous transformation and $\varepsilon >0$. Given a set $Z \subset X$ and a constant $s \geq0$, suppose there exist a constant $C>0$ and a Borel probability measure $\eta$ satisfying:
\begin{itemize}
\item[(i)] $\eta(Z)>0$;
\item[(ii)] $\eta(B_n(x,\varepsilon))\leq C e^{-ns}$ for every ball $B_n(x,\varepsilon)$ such that $B_n(x,\varepsilon)\cap Z\not=\emptyset$.
\end{itemize}
Then $h(Z,f,\varepsilon)\geq s$.
\end{lemma}
\begin{proof}[Proof of Lemma \ref{lemma:entropy-dist-principles}]
Let $\Gamma=\{B_{n_i}(x_i,\varepsilon)\}_i$ be some cover of $Z$. Without loss of generality we may assume that $B_{n_i}(x_i,\varepsilon)\cap Z\not=\emptyset$ for every $i$. In such case we have that
\begin{align*}
\sum_{i}\exp(-sn_i)&\geq C^{-1}\sum_{i}\eta(B_{n_i}(x,\varepsilon))\geq C^{-1}\eta\left(\bigcup_iB_{n_i}(x,\varepsilon)\right)\\
&\geq C^{-1}\eta(Z)>0.
\end{align*}
Therefore, $m(Z,s,\varepsilon)>0$ and hence $h(Z,f,\varepsilon)\geq s$.
\end{proof}
By Lemma \ref{lemma: F_subset_K_alpha} we have that $h(K_\alpha,f,\varepsilon/2)\geq h(F,f,\varepsilon/2)$. Lemmas \ref{lemma: exp decay measure} and \ref{lemma:entropy-dist-principles} gives us that $h(F,f,\varepsilon/2)\geq h_{\mu}(f,\xi)-8\gamma$. Consequently,
\begin{align*}
h(K_\alpha,f,\varepsilon/2)\geq
h_{\mu}(f,\xi)-8\gamma.\\
\end{align*}
Thus, combining this observation with \eqref{eq:choice varepsilon for h} and \eqref{eq:escolha de xi} we get that
\begin{align*}
\mathrm H_\varphi\mathrm{\overline{mdim}_M}\,(f,\alpha,d)-9\gamma
&\leq \frac{h_{\mu}(f,\xi)-8\gamma}{|\log5\varepsilon |}\\
&\leq \frac{h(K_\alpha,f,\varepsilon/2)}{|\log\varepsilon/2|+\log 10}\\
&\leq \mathrm{\overline{mdim}_M}\,(K_\alpha,f,d) +\gamma.
\end{align*}
Thus, since $\gamma>0$ is arbitrary, the proof of the proposition is complete.
\end{proof}
\begin{proposition}\label{lemma:22}
Under the hypotheses of Theorem \ref{thm1} we have that
$$
\mathrm {H_\varphi\overline{mdim}_M}\,(f,\alpha,d)\geq \Lambda_\varphi\mathrm{\overline{mdim}_M}\,(f,\alpha,d).
$$
\end{proposition}
\begin{proof}
Fix $\gamma>0$. Let $\{\varepsilon_j\}_{j\in\mathbb N}$ be a sequence of positive numbers which converges to zero and satisfies
$$
\Lambda_\varphi\mathrm{\overline{mdim}_M}\,(f,\alpha,d)=\lim_{j\to\infty}\frac{\Lambda_\varphi(\alpha,\varepsilon_j)}{|\log\varepsilon_j|}.
$$
Then, there exists $\varepsilon_0>0$ so that for all $\varepsilon_j\in(0,\varepsilon_0]$ we have
$$
\frac{\Lambda_\varphi(\alpha,\varepsilon_j)}{|\log\varepsilon_j|}>\Lambda_\varphi\mathrm{\overline{mdim}_M}\,(f,\alpha,d)-\frac{1}{3}\gamma.
$$
In particular, for every $\varepsilon_j\in(0,\varepsilon_0]$,
$$
\Lambda_\varphi(\alpha,\varepsilon_j)>\left(\Lambda_\varphi\mathrm{\overline{mdim}_M}\,(f,\alpha,d)-\frac{1}{3}\gamma\right)\cdot |\log\varepsilon_j|.
$$
Fix $j\in\mathbb N$ such that $\varepsilon_j\in(0,\varepsilon_0]$. By the alternative expression of $\Lambda_\varphi(\alpha,\varepsilon_j)$ given in \eqref{eq:ineq-sep-cov} it follows that there exists a sequence of positive numbers $(\delta_{j,k})_{k\in \mathbb N}$ converging to zero and such that for every $k\in \mathbb{N}$,
\begin{align*}
\liminf_{n\to\infty}\frac{1}{n}\log M(\alpha,\delta_{j,k},n,\varepsilon_{j})
&>\Lambda_\varphi(\alpha,\varepsilon_{j})-\frac{2}{3}\gamma\\
&>\left(\Lambda_\varphi\mathrm{\overline{mdim}_M}\,(f,\alpha,d)-\frac{1}{3}\gamma\right)\cdot|\log\varepsilon_{j}|-\frac{2}{3}\gamma.
\end{align*}
Similarly, there exists a sequence $(n_{j,k})_{k\in \mathbb{N}}$ in $\mathbb N$ satisfying $\lim_{k\to \infty}n_{j,k}=+\infty $ and
\begin{align}\label{eq: def Njk}
M_{j,k}&:=M(\alpha,\delta_{j,k},n_{j,k},\varepsilon_{j})\\
&\phantom{=} >\exp\left(n_{j,k}\left(\left(\Lambda_\varphi\mathrm{\overline{mdim}_M}\,(f,\alpha,d)-\frac{1}{3}\gamma\right)\cdot|\log\varepsilon_{j}|-\gamma\right)\right). \nonumber
\end{align}
Consider a maximal $(n_{j,k},\varepsilon_j)$-separated set $C_{j,k}$ of $P(\alpha,\delta_{j,k},n_{j,k})$. For each $j,k\in\mathbb N$ consider
$$
\sigma_k^{(j)}=\frac{1}{M_{j,k}}\sum_{x\in C_{j,k}}\delta_x,
$$
and
$$
\mu_k^{(j)}=\frac{1}{n_{j,k}}\sum_{i=0}^{n_{j,k}-1}(f^i)_\ast(\sigma^{(j)}_k)=\frac{1}{M_{j,k}}\sum_{x\in C_{j,k}}\frac{1}{n_{j,k}}\sum_{i=0}^{n_{j,k}-1}\delta_{f^i(x)}.
$$
It is not difficult to see that any accumulation point of $\{\mu_k^{(j)}\}_{k\in\mathbb N}$, say $\mu^{(j)}$, is $f$-invariant (see \cite[Theorem 6.9]{Walters}). Moreover, $\displaystyle\int_X\varphi \;d\mu^{(j)}=\alpha$ for every $j\in \mathbb{N}$. Indeed,
we may assume without loss of generality that $\displaystyle\lim_{k\to\infty}\mu_k^{(j)}=\mu^{(j)}$. Then, for every $j$ and $k$ in $\mathbb{N}$ we have
$$
\left|\int_X \varphi\;d\mu^{(j)}_k-\alpha\right|\leq \frac{1}{M_{j,k}}\sum_{x\in C_{j,k}}\left|\frac{1}{n_{j,k}}\sum_{i=0}^{n_{j,k}-1}\varphi (f^i(x))-\alpha\right| \leq \delta_{j,k}.
$$
Thus,
\begin{align*}
\left|\int_X \varphi\;d\mu^{(j)}-\alpha\right| &\leq \left|\int_X \varphi\;d\mu^{(j)}-\int_X \varphi\;d\mu^{(j)}_k\right| +\left|\int_X \varphi\;d\mu^{(j)}_k-\alpha\right|\\
&\leq\left|\int_X \varphi\;d\mu^{(j)}-\int_X \varphi\;d\mu^{(j)}_k\right|+\delta_{j,k}.
\end{align*}
Consequently, making $k\to +\infty$ we conclude that $\displaystyle\int_X\varphi \;d\mu^{(j)}=\alpha$ for every $j\in \mathbb N$ as claimed.
For every $j\in \mathbb N$ choose a Borel partition $\xi(j)=\{A_1,\dots,A_\ell\}$ of $X$ so that $\mathrm{diam}(\xi(j))<\varepsilon_{j}$ and $\mu^{(j)}(\partial A_i)=0$ for $0\leq i\leq \ell$ (see \cite[Lemma 8.5(ii)]{Walters}). Then,
$$H_{\sigma^{(j)}_{k}}\left(\bigvee_{i=0}^{n_{j,k}-1}f^{-i}\xi(j)\right)=\log M(\alpha,\delta_{j,k},n_{j,k},\varepsilon_j).$$
Indeed, observe that if $x$ and $y$ belong to the same element of $\bigvee_{i=0}^{n_k-1}f^{-i}\xi(j)$ then $d_{n_{j,k}}(x,y)<\varepsilon_j$. In particular, no element of $\bigvee_{i=0}^{n_k-1}f^{-i}\xi(j)$ can contain more than one point of a maximal $(n_{j,k},\varepsilon_{j})$-separated set. Thus, exactly $M(\alpha,\delta_{j,k},n_{j,k},\varepsilon_j)$ elements of $\bigvee_{i=0}^{n_{j,k}-1}f^{-i}\xi(j)$ have $\sigma^{(j)}_k$-measure equal to $\frac{1}{M(\alpha,\delta_{j,k},n_{j,k},\varepsilon_j)}$. All others have zero $\sigma^{(j)}_k$-measure.
Fix natural numbers $q$ and $n_{j,k}$ with $1<q<n_{j,k}$ and define, for $0\leq s\leq q-1$, $a(s)=[(n_{j,k}-s)/q]$ where $[p]$ denotes the integer part of $p$. Fix $0\leq s\leq q-1 $. Then, by \cite[Remark 2(ii), p. 188]{Walters} we have that
$$
\bigvee_{i=0}^{n_{j,k}-1}f^{-i}\xi(j)=\bigvee_{r=0}^{a(s)-1}f^{-(rq+s)}\left(\bigvee_{i=0}^{q-1}f^{-i}\xi(j)\right)\vee \bigvee_{t\in L}f^{-t}\xi(j)
$$
where $L$ is a set with cardinality at most $2q$. Therefore, using \cite[Theorem 4.3(viii)]{Walters} and \cite[Corollary 4.2.1]{Walters},
\begin{align*}
\log M(\alpha,\delta_{j,k},n_{j,k},\varepsilon_{j})&=H_{\sigma^{(j)}_{k}}\left(\bigvee_{i=0}^{n_{j,k}-1}f^{-i}\xi(j)\right)\\
&\leq \sum_{i=0}^{a(s)-1}H_{\sigma^{(j)}_{k}}f^{-(rq+s)}\left(\bigvee_{i=0}^{q-1}f^{-i}\xi(j)\right)+ \sum_{t\in L}H_{\sigma^{(j)}_{k}}(f^{-t}\xi(j) )\\
&\leq \sum_{i=0}^{a(s)-1}H_{\sigma^{(j)}_{k}\circ f^{-(rq+s)}}\left(\bigvee_{i=0}^{q-1}f^{-i}\xi(j)\right)+2q\log(\ell).
\end{align*}
Summing the previous inequality over $s$ from $0$ to $q-1$ and using \cite[Remark 2(iii), p. 188]{Walters}, we get that that
$$
q \log M(\alpha,\delta_{j,k},n_{j,k},\varepsilon_{j})\leq \sum_{p=0}^{n_{j,k}-1}H_{\sigma^{(j)}_{k}\circ f^{-p}}\left(\bigvee_{i=0}^{q-1}f^{-i}\xi(j)\right)+2q^2\log(\ell).
$$
Thus, dividing everything by $n_{j,k}$ in the above inequality and using \eqref{eq: def Njk} we obtain
\begin{align}\label{eq:02}
\nonumber &q \left(\left(\Lambda_\varphi\mathrm{\overline{mdim}_M}\,(f,\alpha,d)-\frac{1}{3}\gamma\right)\cdot|\log\varepsilon_{j}|-\gamma\right)\\ \nonumber
&<\frac{q}{n_{j,k}} \log M(\alpha,\delta_{j,k},n_{j,k},\varepsilon_{j})\\
&\leq H_{\mu^{(j)}_{k}}\left(\bigvee_{i=0}^{q-1}f^{-i}\xi(j)\right)+\frac{2q^2\log(\ell)}{n_{j,k}}.
\end{align}
Now, since the elements of $\bigvee_{i=0}^{q-1}f^{-i}\xi(j)$ have boundaries of $\mu^{(j)}$-measure zero, it follows from the weak convergence of the measures $\mu^{(j)}_{k}$ to $\mu^{(j)}$ that $\displaystyle\lim_{k\to\infty}\mu^{(j)}_{k}(B)=\mu^{(j)}(B)$
for each element $B$ of $\bigvee_{i=0}^{q-1}f^{-i}\xi(j)$ and, therefore,
$$
\displaystyle\lim_{k\to\infty}H_{\mu^{(j)}_k}\left(\bigvee_{i=0}^{q-1}f^{-i}\xi(j)\right)=H_{\mu^{(j)}}\left(\bigvee_{i=0}^{q-1}f^{-i}\xi(j)\right).
$$
Thus, by \eqref{eq:02} we have that
\begin{align*}
\nonumber q \left(\left(\Lambda_\varphi\mathrm{\overline{mdim}_M}\,(f,\alpha,d)-\frac{1}{3}\gamma\right)\cdot|\log\varepsilon_{j}|-\frac{2}{3}\gamma\right) &\leq H_{\mu^{(j)}}\left(\bigvee_{i=0}^{q-1}f^{-i}\xi(j)\right).
\end{align*}
Dividing both sides of the previous inequality by $q$ and letting $q$ go to $+\infty$ we obtain
$$
\left(\Lambda_\varphi\mathrm{\overline{mdim}_M}\,(f,\alpha,d)-\frac{1}{3}\gamma\right)\cdot|\log\varepsilon_{j}|- \frac{2\gamma}{3}\leq h_{\mu^{(j)}}(f,\xi(j)), \text{ for all }j\in\mathbb N,
$$
which implies that
$$
\Lambda_\varphi\mathrm{\overline{mdim}_M}\,(f,\alpha,d)-\frac{1}{3}\gamma\leq \frac{h_{\mu^{(j)}}(f,\xi(j))+\frac{2}{3}\gamma}{|\log\varepsilon_{j}|}, \text{ for all }j\in\mathbb N.
$$
Therefore,
\begin{align*}
\Lambda_\varphi\mathrm{\overline{mdim}_M}\,(f,\alpha,d)-\frac{1}{3}\gamma
\leq\frac{\inf_{|\xi|<\varepsilon_j} h_{\mu^{(j)}}(f,\xi)+\gamma}{|\log\varepsilon_{j}|} \text{ for all }j\in\mathbb N
\end{align*}
and consequently,
\begin{align*}
\Lambda_\varphi\mathrm{\overline{mdim}_M}\,(f,\alpha,d)-\frac{1}{3}\gamma
&\leq\limsup_{j\to\infty}\frac{\sup_{\nu\in\mathcal M_f(X,\alpha,d)}\inf_{|\xi|<\varepsilon_j} h_\nu(f, \xi)}{|\log\varepsilon_{j}|}\\
&\leq \limsup_{\varepsilon\to0}\frac{\sup_{\nu\in\mathcal M_f(X,\alpha,d)}\inf_{|\xi|<\varepsilon} h_\nu(f,\xi)}{|\log\varepsilon|}\\
&=\mathrm H_\varphi\mathrm{\overline{mdim}_M}\,(f,\alpha,d)
\end{align*}
completing the proof of the proposition.
\end{proof}
Finally, Theorem \ref{thm1} follows directly by combining Propositions \ref{lemma:20}, \ref{lemma:21} and \ref{lemma:22}.
\section{Examples}\label{sec: examples}
In this section we present some examples where our results may be applied (recall Remark \ref{remark: finiteness}).
\begin{example}
Let $(Z,D)$ be a compact metric space with upper box-counting dimension $\mathrm{\overline{dim}_B}\;Z<\infty$. Let us consider $X=Z^\mathbb N$ endowed with the metric
$$
d((x_n)_{n\in\mathbb N},(y_n)_{n\in\mathbb N})=\sum_{n=1}^\infty\frac{1}{2^n}D(x_n,y_n)
$$
and let $\sigma: X\to X$ be the shift map. It is well known that $\sigma$ has the specification property and
$\mathrm{\overline{mdim}_M}\,(X,\sigma,d)=\mathrm{\overline{dim}_B}\;Z$ (see for instance \cite{Acevedo-Rodrigues}). In particular, we may apply Theorem \ref{thm1} to it getting that for any $\varphi\in C^0(X,\mathbb R)$ and $\alpha\in\mathbb R$,
$$
\mathrm{\overline{mdim}_M}\,\Big(K_\alpha,\sigma, d\Big)=\Lambda_\varphi\mathrm{\overline{mdim}_M}\,(\sigma,\alpha,d)= \mathrm{H_\varphi\overline{mdim}_M}\,(\sigma,\alpha,d).
$$
\end{example}\label{ex1}
\begin{example}
Let $X=[0,1]^{\mathbb N}$ be endowed with the metric induced by the Euclidian distance in $[0,1]$ as in the previous example and consider the set
$$
E=\left\{\{x^{(i,j)}\}_{i,j\in\mathbb N}\in X:x^{(i,j)}_n=\frac{1}{2^j} \text{ if }i=n\text{ and }x^{(i,j)}_n=0 \text{ if }i\not=n\right\}\cup\{e\},
$$
where $e=(0,0,\dots)$, which is closed and shift invariant. If $2^E$ denotes the space of subsets of $X$ endowed with the Hausdorff distance $d_H$, by \cite[Proposition 3.6]{HW} we have that
$$
\mathrm{\overline{mdim}_{M}}\,\Big(E,\sigma,d\Big)=0 \text{ and } \mathrm{\overline{mdim}_{M}}\,\Big(2^E,\sigma_\sharp,d_H\Big)=1,
$$
where $\sigma_\sharp$ is the induced map by $\sigma$ on the hyperspace $2^E$. By \cite[Proposition 4]{BAS} we have that $\sigma_\sharp$ has the specification property and then Theorem \ref{thm1} may be applied.
\end{example}
\begin{example}
It was proved in \cite{CRV} that for $C^0$-generic homeomorphisms acting on a compact smooth boundaryless manifold $X$ with dimension greater than one, the upper metric mean dimension with respect to the smooth metric coincides with the dimension of the manifold. Now, in order to be able to apply Theorem \ref{thm1} to elements of this generic set, we need to guarantee that they have the specification property. For this purpose we restrict ourselves to the set of conservative homeomorphisms, where the specification property holds $C^0$-generically.
We fix a good Borel probability measure $\mu\in\mathcal M(X)$, i.e., a probability measure that satisfies the following conditions:
\medskip
\noindent $(C_1)\quad$ [Non-atomic] For every $x \in X$ one has $\mu(\{x\})=0$;
\smallskip
\noindent $(C_2) \quad$ [Full support] For every nonempty open set $U \subset X$ one has $\mu(U)>0$;
\smallskip
\noindent $(C_3) \quad$ [Boundary with zero measure] $\mu(\partial X)=0$.
\medskip\\
In a forthcoming paper by S. Roma\~na and G. Lacerda it is proved that there exists a Baire generic subset of $\mathrm{Homeo}_\mu(X,d)$ (the set of conservative homeomorphisms on $X$) with metric mean dimension equal to the dimension of $X$. Consequently, since according to \cite{GL} the specification property is a Baire generic property in $\mathrm{Homeo}_\mu(X,d)$, there exists a $C^0$-open and dense subset of $\mathrm{Homeo}_\mu(X,d)$ where Theorem \ref{thm1} may be applied.
\end{example}
In the next two examples we consider the specification property for linear operators acting on Banach spaces and we start by recalling the appropriate definition for this setting. Let $B$ be a Banach space over $\mathbb K$ ($= \mathbb R$ or $\mathbb{C}$) and $T:B\to B$ be a linear operator. We say that $T$ has the \emph{operator specification property} if there exists a sequence of $T$-invariant sets $\{K_m\}_{m\in\mathbb N}$ with $B=\overline{\cup_{m\in \mathbb N}K_m}$ for which $T|_{K_m}:K_m\to K_m$ satisfies the specification property. We emphasize that the sets $K_m$ do not need be compact, although in the all known examples we have compactness for such sets.
\begin{example}
Fix $\nu=(\nu_n)_{n\in\mathbb N}\in \mathbb R^{\mathbb N}$ so that $\nu_n>0$ for all $n\in\mathbb N$ and $\displaystyle\sum_{n=1}^\infty\nu_n<\infty$. Consider
$$
\ell^p(\nu)=\left\{(x_n)_{n\in\mathbb N}\in \mathbb K^{\mathbb N}: \|(x_n)_{n\in\mathbb N}\|_{\ell_p(\nu)}:=\left(\sum_{n=1}^\infty|x_n|^p\nu_n\right)^{\frac{1}{p}}<\infty\right\},
$$
which is a Banach space, and the shift map $\sigma:\ell^p(\nu)\to \ell^p(\nu)$. By \cite[Theorem 2.1]{BMP} we have that $\sigma:\ell^p(\nu)\to\ell^p(\nu)$
has the operator specification property with $K_m=mK$, $m\in \mathbb{N}$, where $K$ is the compact set $K=\{(x_n)_{n\in\mathbb N}\in \ell^p(\nu): |x_n|\leq 1 \text{ for all }n\in\mathbb N\}$. We now observe that $T|_{K}:K\to K$ has positive upper metric mean dimension. More precisely,
\begin{lemma}\label{lem: metric mean Linear op}
$$
\mathrm{\overline{mdim}_M}\,\Big(K,\sigma, \|\cdot\|_{\ell_p(\nu)}\Big)=1.
$$
\end{lemma}
\begin{proof}
Given $\varepsilon >0$ and $n\in\mathbb N$, we observe that
$$
\left\{x\in K: x_i\in\left \{0,\frac{\varepsilon}{\sqrt[p]{\nu_1}},\frac{2\varepsilon}{\sqrt[p]{\nu_1}},\dots,\left\lfloor1/\frac{\varepsilon}{\sqrt[p]{\nu_1}}\right\rfloor\frac{\varepsilon}{\sqrt[p]{\nu_1}}\right\} \text{ for all }1\leq i \leq n\right\}
$$
is a $(n,\varepsilon)$-separated set in $K$. In particular,
\begin{displaymath}
\begin{split}
\mathrm{\overline{mdim}_M}\,\Big(K,\sigma, \|\cdot\|_{\ell_p(\nu)}\Big)&=\limsup_{\varepsilon\to0}\frac{h(\sigma,\varepsilon)}{|\log\varepsilon|}\\
&\geq \limsup_{\varepsilon\to0}\frac{\limsup_{n\to\infty}\frac1n\left|\log \left(\left\lfloor1/\frac{\varepsilon}{\sqrt[p]{\nu_1}}\right\rfloor\right)^n\right|}{|\log\varepsilon|}
=1.
\end{split}
\end{displaymath}
In order to get the reverse inequality, let $\ell\in\mathbb N$ be so that $\displaystyle\sum_{n\geq \ell}\nu_n<\frac{\varepsilon}{2}$ and define $M=\left(\displaystyle\sum_{k\in \mathbb N}\nu_k\right)^{1/p}>0$. We consider an open cover of $[-1,1]$ by
$$
I_k=\left(\frac{(k-1)\varepsilon}{12M},\frac{(k+1)\varepsilon}{12M}\right), \text{ for }-\lfloor12M/\varepsilon\rfloor\leq k \leq \lfloor12M/\varepsilon\rfloor.
$$
Note that each $I_k$ has length $\displaystyle\frac{\varepsilon}{6M}$. Given $n\geq 1$, let us consider the following
open cover of $K^\mathbb{N}$:
$$
\{x:x_1\in I_{k_1},x_2\in I_{k_2},\dots,x_{n+\ell}\in I_{k_{n+\ell}}\},
$$
where $-\lfloor12M/\varepsilon\rfloor\leq k_1,\dots,k_{n+\ell} \leq \lfloor12M/\varepsilon\rfloor$.
Observe that each element of this open cover has diameter less than $\varepsilon$ with respect to the metric $d_n$ (induced by $\|\cdot\|_{\ell_p(\nu)}$).
So,
\begin{align*}
\limsup_{\varepsilon\to0}\frac{h(\sigma,\varepsilon)}{|\log\varepsilon|}
\leq \limsup_{\varepsilon\to0}\frac{\limsup_{n\to\infty}\frac1n\log \left(2\left\lfloor12M/\varepsilon\right\rfloor\right)^{n+\ell+1}}{|\log\varepsilon|}=1.
\end{align*}
Hence,
$$
\mathrm{\overline{mdim}_M}\,\Big(K,\sigma, \|\cdot\|_{\ell_p(\nu)}\Big)=1
$$
as claimed.
\end{proof}
As a consequence of the previous proof we also get that
$$
\mathrm{\overline{mdim}_M}\,\Big(K_m,\sigma, \|\cdot\|_{\ell_p(\nu)}\Big)=1
$$
for all $m\in\mathbb N$. In particular, we may apply Theorem \ref{thm1} to $\sigma|K_m:K_m\to K_m$ for every $m\in \mathbb N$.
\end{example}
\begin{example}
Another class of examples is given by the weighted shifts. Let $\nu=(\nu_n)_{n\in\mathbb N}=(1)_{n\in\mathbb N}$ and consider $\ell^p(\nu)$ as in the previous example. Observe that in this case $\ell^p(\nu)=\ell^p$. Let $w=(w_n)_{n\in\mathbb N}$ be a weight sequence and define the \emph{weighted shift} on $\ell^p$ as
$B_w((x_n)_{n\in\mathbb N})=(w_{n+1}x_{n+1})_{n\in\mathbb N}$. It was observed in \cite[p. 602]{BMP} that if one considers $a=(a_n)_{n\in\mathbb N}$ given by
$$
a_1=1 \text{ and }a_n:=w_2\dots w_n, \text{ for all }n>1,
$$
and $\bar \nu = (\bar \nu_n)_{n\in \mathbb N}$ given by
$$
\bar\nu_n=\frac{1}{\prod_{j=2}^n|w_j|^p}, \text{ for all }n\in\mathbb N,
$$
then
$$
\phi_a:(x_{n})_{n\in\mathbb N}\in \ell^p\mapsto \phi_a((x_{n})_{n\in\mathbb N})=(a_1x_1,a_2x_2,\dots)\in \ell^p(\bar\nu)
$$
defines a topological conjugacy between the weighted shift and the backward shift given in the previous example. Moreover, they observed that this topological conjungacy is also an isometry, which implies that $\mathrm{\overline{mdim}_M}\,\Big(\phi_a^{-1}(K_m),B_w, \|\cdot\|_{\ell_p}\Big)=1$, for all $m\in\mathbb N$. In particular, if $\sum_{n=1}^\infty\bar\nu_n<\infty$ we have that
$B_w$ has the operator specification property (see \cite[Theorem 2.3]{BMP}) and then we are in the context of Theorem \ref{thm1}.
\end{example}
\medskip{\bf Acknowledgements.} L.B. was partially supported by a CNPq-Brazil PQ fellowship under Grant No. 307633/2021-7.
| {
"timestamp": "2022-07-08T02:14:28",
"yymm": "2207",
"arxiv_id": "2207.03238",
"language": "en",
"url": "https://arxiv.org/abs/2207.03238",
"abstract": "We prove a variational principle for the upper and lower metric mean dimension of level sets \\[ \\left\\{x\\in X: \\lim_{n\\to\\infty}\\frac{1}{n}\\sum_{j=0}^{n-1}\\varphi(f^{j}(x))=\\alpha\\right\\} \\] associated to continuous potentials $\\varphi:X\\to \\mathbb R$ and continuous dynamics $f:X\\to X$ defined on compact metric spaces and exhibiting the specification property. This result relates the upper and lower metric mean dimension of the above mentioned sets with growth rates of measure-theoretic entropy of partitions decreasing in diameter associated to some special measures. Moreover, we present several examples to which our result may be applied to. Similar results were previously known for the topological entropy and for the topological pressure.",
"subjects": "Dynamical Systems (math.DS)",
"title": "A variational principle for the metric mean dimension of level sets",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9854964207345893,
"lm_q2_score": 0.7185943985973772,
"lm_q1q2_score": 0.70817220777764
} |
https://arxiv.org/abs/2206.00514 | The volume of random simplices from elliptical distributions in high dimension | Random simplices and more general random convex bodies of dimension $p$ in $\mathbb{R}^n$ with $p\leq n$ are considered, which are generated by random vectors having an elliptical distribution. In the high-dimensional regime, that is, if $p\to\infty$ and $n\to\infty$ in such a way that $p/n\to\gamma\in(0,1)$, a central and a stable limit theorem for the logarithmic volume of random simplices and random convex bodies is shown. The result follows from a related central limit theorem for the log-determinant of $p\times n$ random matrices whose rows are copies of a random vector with an elliptical distribution, which is established as well. | \section{Introduction}\setcounter{equation}{0}
The probabilistic analysis of convex hulls in $\mathbb{R}^n$ generated by $p$ random points is one of the central themes of geometric probability and stochastic geometry. A variety of different models and results are known in the literature, mainly in the asymptotic regime $p\to\infty$, while the dimension parameter $n$ is kept fixed. Examples include the expectation asymptotics for the number of faces or the intrinsic volumes and their tight relation to affine surface areas, related upper and lower variance bounds as well as results on the asymptotic normality or concentration properties for these combinatorial and geometric parameters. We refer the reader to the survey articles of B\'ar\'any \cite{Bar07}, Hug \cite{Hug13} and Reitzner \cite{ReitznerSurvey} for motivation, background material and references. Recent interest in random polytopes has triggered an analysis in which the number $p$ of generating points and the space dimension $n$ tend to infinity in a suitably coupled way. In this context, the case where $p\leq n$ and $n\to\infty$ is of particular interest and has been considered in \cite{ABGK21,EK17,GKT17,Paindaveine}, building on earlier works \cite{Mae80,Math82,Rub77} in which $p$ stays constant. In this situation the random polytopes are just simplices of dimension $p$. In particular, these papers prove asymptotic normality of the volume of the random simplices in high dimensions under special distributional assumptions on the generating random points. More precisely, the papers \cite{EK17,GKT17} deal with the case in which the random points are distributed according to a beta or beta-prime distribution in $\mathbb{R}^n$, which includes the uniform distribution in the $n$-dimensional unit ball and, as a limiting case, the uniform distribution on the $(n-1)$-dimensional unit sphere or the standard Gaussian distribution. The article \cite{Paindaveine} treats the Gaussian case together with applications to multivariate medians in statistics. In \cite{ABGK21} the distribution of the points arises from general product measures or a class of $q$-radial distributions on the $\ell_q$-ball in $\mathbb{R}^n$. In the latter setting, the particularly interesting case of high-dimensional pinned random simplices has been considered, where one of the generating points is fixed (pinned) at the origin. In this case, $p!$ times the volume of the random simplex is the same as the volume of the parallelotope spanned by the random points.
The aim of the present paper is to provide central limit theorems for the logarithmic volume of $p$-dimensional pinned random simplices whose generating points follow a general elliptical distribution in $\mathbb{R}^n$. In particular, we will be interested in the high-dimensional regime, where $p=p_n$ is a function of the space dimension $n$ satisfying $p\to\infty$, as $n\to\infty$, and is such that $p/n\to\gamma\in(0,1)$. For convenience, let us recall that an $n$-dimensional random vector ${\mathbf x}$ follows an elliptical distribution if it takes the form
$$
{\mathbf x}=R{\mathbf A}{\mathbf u},
$$
where $R\geq 0$ is an arbitrary real-valued random variable, ${\mathbf A}$ is a fixed $n\times n$ matrix of full rank and ${\mathbf u}$ is a uniform random direction, that is, a uniform random point on the unit sphere in $\mathbb{R}^n$, which is independent from $R$. Suppose now that we have $p$ independent random copies ${\mathbf x}_1,\ldots,{\mathbf x}_p$ of the vector ${\mathbf x}$ that form the matrix ${\mathbf X}:=({\mathbf x}_1,\ldots,{\mathbf x}_p)^{\top}\in\mathbb{R}^{p\times n}$.
The (pinned) random simplex with vertex set $\{\mathbf{0},{\mathbf x}_1,\ldots,{\mathbf x}_p \}$ can now be defined as
\begin{align}\label{def:simplex}
\Delta {\mathbf X} := \bigg\{ \sum_{i=1}^ps_i {\mathbf x}_i \,:\, s_i \geq 0 \quad\text{and}\quad \sum_{i = 1}^p s_i \leq 1 \bigg\},
\end{align}
and its $p$-volume admits the following representation (see \cite[Section 8.7]{shilov:2012}) in terms of the matrix~${\mathbf X}$:
\begin{align} \label{eq:simplexVol}
\operatorname{Vol}_p \left( \Delta{\mathbf X} \right)
= \frac{1}{p!} \sqrt{\det({\mathbf X}\X^{\top})} \,,
\end{align}
where $\det(M)$ denotes the determinant of a square matrix $M$ and $M^{\top}$ its transpose. Note that since $p\le n$ the vectors ${\mathbf x}_1,\ldots,{\mathbf x}_p$ are almost surely linearly independent and $\Delta {\mathbf X}$ is a $p$-dimensional random convex polytope in $\mathbb{R}^n$ with non-zero $p$-volume. The representation \eqref{eq:simplexVol} immediately motivates a probabilistic analysis of random determinants of the form $\det({\mathbf X}\X^\top)$, which are in the focus of the present paper as well. We remark that the study of random determinants has a long history starting with the works in \cite{GirkoCLT79,GirkoBook}, which have later been extended by many authors, see \cite{bao:lin:pan:zhou:2015,NguyenVuDet,TaoVuCLTWigner,wang:han:pan:2018, dette:doernemann:2020, heiny:johnston:prochno:2022, dornemann2022likelihood}, for example, as well as the references cited therein. Our main results can be summarized as follows:
\begin{itemize}
\item[(i)] Assuming upper and lower bounds on the eigenvalues of the matrix ${\mathbf A}\A^\top$ we derive in Theorem \ref{thm:main} asymptotic normality in high dimensions, that is, as $p\to\infty$ and $n\to\infty$ such that $p/n\to\gamma\in(0,1)$, for the logarithmic $p$-volume of the pinned random simplices $\Delta{\mathbf X}$ generated by random vectors following an elliptical distribution with $R=1$. We also extend this result by allowing arbitrary distributions for the radius $R\ge 0$ which might yield non-normal limiting laws for the logarithmic volume if the distribution of $\log R$ has infinite variance.
\item[(ii)]
Then we take a more general point of view by considering for a fixed convex body $\Sigma\subset\mathbb{R}^p$ the random convex set
$$
\Upsilon_{p,n}(\Sigma,{\mathbf X}) := \left\{\sum_{i=1}^ps_i {\mathbf x}_i:(s_1,\ldots,s_p)\in\Sigma \right\},
$$
where ${\mathbf x}_1,\ldots,{\mathbf x}_p$ are the column vectors of ${\mathbf X}^\top$. In the special case $p=n$ this random set model has first been considered in \cite{PP13} in the context of small-ball probabilities and later in \cite{ABGK21} under the angle of high-dimensional central limit theorems. Note that by choosing for $\Sigma$ the $p$-dimensional standard simplex, $\Upsilon_{p,n}(\Sigma,{\mathbf X})$ reduces to the pinned random simplex $\Delta{\mathbf X}$ discussed above. We establish in Theorem \ref{thm:RandomConvexBody} a central limit theorem for the logarithmic $p$-volume of $\Upsilon_{p,n}(\Sigma,{\mathbf X})$, as $p\to\infty$ and $n\to\infty$ such that $p/n\to\gamma\in(0,1)$, for matrices ${\mathbf X}$ generated by random vectors satisfying the conditions of Theorem \ref{thm:main}.
\end{itemize}
The remaining parts of this text are structured as follows. In Section \ref{sec:Results}, we formally introduce our set-up together with the necessary notation. Our main results, Theorem \ref{thm:main} for the log-determinant and its geometric counterpart Theorem \ref{thm:RandomConvexBody}, are the contents of Section \ref{subsec:LogDet} and Section \ref{subsec:RCB}, respectively. Section \ref{sec:proofmain} is devoted to the proof of Theorem \ref{thm:main}, while the Appendix contains some auxiliary lemmas.
\section{Limit theorems for the logarithmic volume of elliptical simplices}\label{sec:Results}\setcounter{equation}{0}
We consider independent and identically distributed (i.i.d.)~$n$-dimensional random vectors ${\mathbf x}_1,\ldots,{\mathbf x}_p$ following an elliptical distribution, that is,
\begin{equation}\label{eq:dataell}
{\mathbf x}_i=R_i{\mathbf A}{\mathbf u}_i\,, \qquad i=1,\ldots,p\,,
\end{equation}
where ${\mathbf A}\in \mathbb{R}^{n\times n}$ is a deterministic matrix of full rank,
$R_i\ge 0$ is a scalar random variable representing the radius of ${\mathbf x}_i$, and ${\mathbf u}_i$ is the random direction, which is uniformly distributed on the unit sphere $\mathbb{S}^{n-1}$. Moreover, we suppose that the random variables $R_i, {\mathbf u}_i, i=1,\ldots,p$ are independent.
The elliptically distributed random vectors are collected in the $p\times n$ matrix
\begin{equation}\label{eq:datam}
{\mathbf X}:={\mathbf X}_n:=({\mathbf x}_1,\ldots,{\mathbf x}_p)^{\top}\,.
\end{equation}
{\em Throughout this paper, ${\mathbf X}, {\mathbf x}_i, R_i, {\mathbf u}_i, {\mathbf A}$ depend on the dimension $n$, i.e.\ ${\mathbf X}={\mathbf X}_n, {\mathbf x}_i={\mathbf x}_{i,n}, R_i=R_{i,n}, {\mathbf u}_i={\mathbf u}_{i,n}, {\mathbf A}={\mathbf A}_n$. For simplicity we will typically suppress the dependence on $n$ in the notation.}
Let us consider the random simplex $\Delta{\mathbf X}$, defined in \eqref{def:simplex}, induced by the matrix ${\mathbf X}$. From \eqref{eq:simplexVol} we get the identity
\begin{equation}\label{eq:081220A}
\log \operatorname{Vol}_p \left( \Delta{\mathbf X} \right)=-\log(p!) +\frac{1}{2} \log \det({\mathbf X}\X^{\top})\,.
\end{equation}
Thus, in order to establish a central limit theorem for $\log \operatorname{Vol}_p \left( \Delta{\mathbf X} \right)$ it is enough to establish such a result for $\log \det({\mathbf X}\X^{\top})$. This problem is treated in Section \ref{subsec:LogDet}, whereas its geometric implications will be discussed in Section \ref{subsec:RCB}.
\begin{comment}
Introduce the matrix ${\mathbf Y}:={\mathbf Y}_n:=\{\operatorname{diag}({\mathbf X}\X^{\top})\}^{-1/2} {\mathbf X}$. Then we have
\begin{equation}\label{}
\det({\mathbf X}\X^{\top})= \det\left(\{\operatorname{diag}({\mathbf X}\X^{\top})\}^{1/2}{\mathbf Y}\Y^{\top} \{\operatorname{diag}({\mathbf X}\X^{\top})\}^{1/2}\right)
=\det({\mathbf Y}\Y^{\top}) \det(\operatorname{diag}({\mathbf X}\X^{\top}))\,.
\end{equation}
Taking logarithm this yields
\begin{equation*}
\begin{split}
\log \det({\mathbf X}\X^{\top})&= \log \det(\operatorname{diag}({\mathbf X}\X^{\top})) + \log \det({\mathbf Y}\Y^{\top})\\
&= \log \det(\operatorname{diag}({\mathbf X}\X^{\top})) + \log \det({\mathbf Y}\Y^{\top})
\end{split}
\end{equation*}
\end{comment}
\subsection{Central limit theorem for the log-determinant}\label{subsec:LogDet}
As anticipated above, in this section we deal with a central limit theorem for the log-determinant of ${\mathbf X}\X^\top$. Introducing the diagonal matrix ${\mathbf R}$ with diagonal entries $R_1,\ldots, R_p$, and the matrix
$$
{\mathbf Y}:={\mathbf Y}_n:=({\mathbf A} {\mathbf u}_1,\ldots,{\mathbf A} {\mathbf u}_p)^{\top},
$$
we have
\begin{equation*}
\det({\mathbf X}\X^{\top})= \det({\mathbf R} {\mathbf Y}\Y^{\top} {\mathbf R})
=\det({\mathbf Y}\Y^{\top}) (\det{\mathbf R})^2\,
\end{equation*}
and taking the logarithm yields
\begin{equation}\label{eq:081220B}
\begin{split}
\log \det({\mathbf X}\X^{\top})&= 2 \log \det {\mathbf R} + \log \det({\mathbf Y}\Y^{\top})\\
&= 2 \sum_{i=1}^p \log R_i + \log \det({\mathbf Y}\Y^{\top})\,,
\end{split}
\end{equation}
where the the last two terms are independent.
The following CLT for $\log \det({\mathbf Y}\Y^{\top})$ is crucial for us. We start by spelling out our assumptions under which we are able to derive the central limit theorem:
\begin{itemize}
\item[(A)] The parameters $p=p_n$ and $n$ are of the same order, that is, there exists a constant $\gamma\in(0,1)$ such that $p/n\to \gamma$ as $n \to \infty$.
\item[(B)] There exists a constant $C>0$ not depending on $n$ such that the ordered eigenvalues
\begin{equation}\label{eq:B1}
\lambda_{max}({\mathbf A}\A^{\top})=\lambda_1({\mathbf A}\A^{\top})\ge \lambda_2({\mathbf A}\A^{\top})\ge\cdots \ge \lambda_n({\mathbf A}\A^{\top})=\lambda_{min}({\mathbf A}\A^{\top})
\end{equation}
of ${\mathbf A}\A^{\top} ={\mathbf A}_n {\mathbf A}_n^{\top}$ satisfy
$C^{-1}\le \lambda_{min}({\mathbf A}\A^{\top})\le \lambda_{max}({\mathbf A}\A^{\top})\le C$. In addition, we assume that
\begin{equation}\label{eq:B2}
\lim_{n \to \infty} \operatorname{tr}\Big( {\mathbf A}_n{\mathbf A}_n^{\top} - \frac{\operatorname{tr}({\mathbf A}_n{\mathbf A}_n^{\top})}{n} {\mathbf I}_n\Big)^2 =0\,,
\end{equation}
where ${\mathbf I}_n$ is the $n\times n$ identity matrix.
\end{itemize}
Next, we introduce some quantities that will play an important role in our results. For $1\le i\le p-1$ and $1\le k\le n$, we set
\begin{equation}\label{def:tik}
t_{i,k}({\mathbf A})= \mathbb{E}\left[\frac{1}{1+ \lambda_k({\mathbf A}\A^{\top}) w_{ik}^{\top} \big(\sum_{\ell=1; \ell \neq k}^n \lambda_{\ell}({\mathbf A}\A^{\top}) w_{i\ell} w_{i\ell}^{\top}\big)^{-1}w_{ik}}\right]\,,
\end{equation}
where $w_{i1}, \ldots, w_{in}$ are i.i.d.\ $i$-dimensional random column vectors whose components are independent standard normal random variables.
Recalling that ${\mathbf Y}={\mathbf Y}_n$ and ${\mathbf A}={\mathbf A}_n$, we are now prepared to present our central limit theorem for the log-determinant of ${\mathbf Y}\Y^{\top}$, whose proof is presented in Section \ref{sec:proofmain}.
\begin{theorem}\label{thm:main}
Let $({\mathbf Y})_{n \ge 1}$ be a sequence of random $p\times n$ matrices defined as follows: ${\mathbf Y}=({\mathbf A} {\mathbf u}_1,\ldots,{\mathbf A} {\mathbf u}_p)^{\top}$, where $({\mathbf A})_{n\ge 1}$ is a sequence of deterministic $n\times n$ matrices satisfying assumption (B), and ${\mathbf u}_1,\ldots, {\mathbf u}_p$ are independent $n$-dimensional random vectors, distributed uniformly on the unit sphere $\mathbb{S}^{n-1}$.
Under assumption (A), as $n \to \infty$, it holds that
\begin{equation}\label{eq:main}
\frac{\log \det({\mathbf Y}\Y^{\top}) -\mu_n}{\sigma_n} \stackrel{d}{\longrightarrow} N(0,1)\,,
\end{equation}
where the centering and normalizing sequences $(\mu_n)_{n\geq 1}$ and $(\sigma_n)_{n\geq 1}$ are given by
\begin{align}\label{eq:meanvar}
\begin{split}
\mu_n&= \log \operatorname{tr}({\mathbf A}\A^{\top}) -p \log n -\frac{\sigma_n^2}{2}+ \sum_{i=1}^{p-1} \log \bigg(\sum_{k=1}^n \lambda_k({\mathbf A}\A^{\top}) t_{i,k}({\mathbf A})\bigg)\,,\\
\sigma_n^2&=-2\frac{p}{n} + 2 \sum_{i=1}^{p-1} \frac{\sum_{k=1}^n \lambda_k^2({\mathbf A}\A^{\top}) t_{i,k}({\mathbf A})}{(\sum_{k=1}^n \lambda_k ({\mathbf A}\A^{\top}) t_{i,k}({\mathbf A}))^2}\,.
\end{split}
\end{align}
\end{theorem}
\begin{remark}\label{rem:3.2}\rm
\begin{enumerate}
\item[(1)] The centering and normalizing sequences only depend on ${\mathbf A}$ through the eigenvalues $\lambda_1({\mathbf A}\A^{\top}), \ldots, \lambda_n({\mathbf A}\A^{\top})$. This is due to the rotational invariance of the vectors $u_i$ that was used in \eqref{eq:repU} of the proof.
\item[(2)] Assumption (A) and \eqref{eq:B1} in Assumption (B) guarantee that $\sigma_n^2$ is of constant order (see Lemma~\ref{lem:sigma}), which is convenient in view of the existing results on log-determinants of random matrices \cite{BaoPanZhou2015,GirkoCLT79,GirkoBook,NguyenVuDet,TaoVuCLTWigner}.
\item[(3)] The quantities $t_{i,1}({\mathbf A}), \ldots, t_{i,n}({\mathbf A})$ satisfy $t_{i,1}({\mathbf A})+ \cdots +t_{i,n}({\mathbf A})=n-i$ for any matrix ${\mathbf A}\in \mathbb{R}^{n\times n}$ and $1\le i\le p-1$. For details we refer to Remark~\ref{rem:tik}.
\item[(4)] If ${\mathbf A}={\mathbf I}_n$ (the $n$-dimensional identity matrix), one sees from \eqref{def:tik} that $t_{i,k}({\mathbf I}_n)=t_{i,\ell}({\mathbf I}_n)$ for any $k\neq \ell$, which in conjunction with the previous point implies $t_{i,k}({\mathbf I}_n)=(n-i)/n$. Plugging this into \eqref{eq:meanvar}, we recover \cite[Theorem~1]{jiang:yang:2013} and a particular case of \cite[Theorem~2.1]{parolya:heiny:kurowicka:2021} as a special case of our Theorem~\ref{thm:main}.
\item[(5)] Instead of \eqref{eq:B1} in Assumption (B), it is possible to require
$$\frac{\lambda_{max}({\mathbf A}\A^{\top})}{\lambda_{min}({\mathbf A}\A^{\top})} \le C^2$$
for some constant $C>0$ not depending on $n$, which would, for example, allow that the largest and smallest eigenvalues of ${\mathbf A}\A^{\top}$ tend to zero or infinity at the same speed. Up to a rescaling of ${\mathbf A}$, this assumption is, however, equivalent to \eqref{eq:B1}.
\item[(6)] Equation \eqref{eq:B2} in Assumption (B) is a technical condition that can likely be relaxed. It means that ${\mathbf A}\A^{\top}$ is close in Frobenius norm to a diagonal matrix. Thus, \eqref{eq:B1} is a condition on the eigenvalues of ${\mathbf A}\A^{\top}$.
\end{enumerate}
\end{remark}
\subsection{Central limit theorem for the volume of random simplices and convex bodies}\label{subsec:RCB}
After having discussed the central limit theorem for random determinants in the previous section, we return to our geometric application to random simplices and convex bodies. We start by recalling from \eqref{eq:081220A} and \eqref{eq:081220B} that
$$
\log\operatorname{Vol}_p(\Delta{\mathbf X}) = -\log(p!) + \sum_{i=1}^p\log R_i + \frac{1}{2}\log\det({\mathbf Y}\Y^\top),
$$
where we use the same notation as in the previous section. Especially we recall that $p=p_n$ and that ${\mathbf X}$ is a matrix generated by $p$ independent random elliptically distributed random vectors with radial parts $R_1,\ldots,R_p$. Since the random variables $R_1,\ldots,R_p$ are independent and identically distributed for every $n$, the generalized central limit theorem for sums of i.i.d.\ random variables (see, e.g., Theorem~\ref{thm:petrovnormal} and Theorem~\ref{thm:petrovstable}) implies that there are sequences $(m_n)_{n\geq 1}$ and $(s_n)_{n\geq 1}$ such that
\begin{equation}\label{eq:CLTRi}
\frac{\sum_{i=1}^p\log R_i- m_n}{s_n} \overset{d}{\longrightarrow} S_\alpha,\qquad n\to\infty\,,
\end{equation}
where $S_\alpha$ stands for the $\alpha$-stable distribution with stability index $\alpha\in(0,2]$. For example, if $\alpha=1$, $S_\alpha$ is the Cauchy distribution or if $\alpha=2$ then $S_\alpha$ corresponds to a centered Gaussian distribution. Sufficient conditions on $\log R_1$ such that \eqref{eq:CLTRi} holds are provided in Theorem~\ref{thm:petrovnormal} for the case $\alpha=2$ and in Theorem~\ref{thm:petrovstable} for the case $\alpha\in (0,2)$. Writing now
\begin{align*}
V_n:&={\log\operatorname{Vol}_p(\Delta{\mathbf X})-{1\over 2}\mu_n-m_n+ \log(p!) \over\max\{\sigma_n/2,s_n\}}\\
&={\sum_{i=1}^p\log R_i- m_n\over\max\{\sigma_n/2,s_n\}} + {{1\over 2}\log\det({\mathbf Y}\Y^\top)-{1\over 2}\mu_n\over\max\{\sigma_n/2,s_n\}}
\end{align*}
and noting that both summands on the right-hand side are independent, we conclude the following corollary from Theorem \ref{thm:main}, part (2) of Remark~\ref{rem:3.2} and \eqref{eq:CLTRi}. We note that this result generalizes Theorems E-F in \cite{heiny:johnston:prochno:2022} for $p/n\to \gamma$, where only the case ${\mathbf A}={\mathbf I}_n$ was analyzed.
\begin{corollary}\label{cor:SimplexWithRadialPart}
Consider the random simplex $\Delta{\mathbf X}$ defined in \eqref{def:simplex}, where ${\mathbf X}=({\mathbf x}_1,\ldots,{\mathbf x}_p)^{\top}$ is as in \eqref{eq:datam} with i.i.d.\ random vectors ${\mathbf x}_i=R_i {\mathbf A} {\mathbf u}_i$ following an $n$-dimensional elliptical distribution as in \eqref{eq:dataell}. Under assumptions (A) and (B), and assuming that $\log R_1$ satisfies \eqref{eq:CLTRi} for some $\alpha \in (0,2]$, the following statements hold.
\begin{enumerate}
\item[(i)] If $s_n\to 0$, then $V_n\stackrel{d}{\longrightarrow} N(0,1)$, as $n\to\infty$.
\item[(ii)] If $s_n\to \infty$, then $V_n \stackrel{d}{\longrightarrow} S_\alpha$, as $n\to\infty$.
\item[(iii)] If $s_n/(\sigma_n/2)\to \tau\in(0,\infty)$, then $V_n \stackrel{d}{\longrightarrow} \max\big\{1,{1\over\tau}\big\}S_\alpha+\min\{1,\tau\}N(0,1)$, as $n\to\infty$.
\end{enumerate}
\end{corollary}
In a next step, we turn to a generalization of this result. We begin by noting that for any $p\times p$ matrix $M$, one has
\begin{equation*}
\det(M {\mathbf Y}\Y^{\top} M^{\top})= (\det(M))^2 \det({\mathbf Y}\Y^{\top})\,.
\end{equation*}
Assuming that $M$ has full rank, this yields the following representation of the log-volume:
\begin{align}
\log \operatorname{Vol}_p \left( \Delta (M{\mathbf Y}) \right)
&= -\log(p!)+ \frac{1}{2} \log \det(M {\mathbf Y}\Y^{\top}M^{\top}) \nonumber\\
&=-\log(p!)+ \frac{1}{2} \log \det({\mathbf Y}\Y^{\top}) +\log |\det M|\,. \label{eq:simplexnew}
\end{align}
Thus the fluctuations of $\log \operatorname{Vol}_p \left( \Delta (M{\mathbf Y}) \right)$ can be directly derived from our theorems above.
Theorem \ref{thm:main} has an immediate geometrical interpretation, which we will present next. For a compact convex subset $\Sigma \subset \mathbb{R}^p$ with non-empty interior and random $n$-dimensional vectors $Y_1,\ldots, Y_p$ we define the random convex body
\begin{equation}\label{}
\Upsilon_{p,n}(\Sigma,{\mathbf Y}) :=\bigg\{ \sum_{i=1}^p s_i Y_i \,:\, (s_1,\dots,s_p)\in \Sigma \bigg\}\,,
\end{equation}
where $p \le n$ and the $n\times p$ matrix ${\mathbf Y}^{\top}=(Y_1,\ldots,Y_p)$ is treated as a linear operator, applied to the body $\Sigma$. It should be noted here, that for any $n,p\in\mathbb{N}$, $p\le n$ and for any convex body $\Sigma$, $\Upsilon_{p,n}(\Sigma,{\mathbf Y})$ is a random $p$-dimensional closed convex set in $\mathbb{R}^n$ in the usual sense of stochastic geometry (see \cite[Chapter 2]{SW08}). The $p$-dimensional volume of $\Upsilon_{p,n}(\Sigma,{\mathbf Y}_n)$ is hence an ordinary random variable. Depending on the choice of the convex body $\Sigma$ the construction above includes a number of common geometrical objects, which are of particular interest, see \cite{PP12,PP13, ABGK21}.
\begin{itemize}
\item [(a)] If $\Sigma$ is the standard simplex $T^p$,
$$
T^p:=\bigg\{(x_1,\ldots,x_p)\in\mathbb{R}^p\colon x_i\ge 0, \sum\limits_{i=1}^p x_i\leq 1\bigg\},
$$
then $\Upsilon_{p,n}(T^p,{\mathbf Y})$ coincides with the pinned simplex $\Delta {\mathbf Y}$, which is the convex hull of points $\mathbf{0},Y_1,\ldots, Y_p$ (see \eqref{def:simplex}).
\item [(b)] If $\Sigma$ is the unit cube $C^p=[0,1]^p$, then $\Upsilon_{p,n}(C^p,{\mathbf Y})$ is the parallelotope spanned by vectors $Y_1,\ldots, Y_p$.
\item [(c)] If $\Sigma$ is the symmetric cube $B_{\infty}^p=[-1,1]^p$, then $\Upsilon_{p,n}(B^p_{\infty},{\mathbf Y})$ is a zonotope, generated by the random segments $[-Y_i,Y_i]$, $1\leq i\leq p$.
\item [(d)] If $\Sigma$ is the cross-polytope $B^p_1$,
$$
B_1^p:=\{(x_1,\ldots,x_p)\in\mathbb{R}^p\colon \sum_{i=1}^p|x_i|\leq 1\},
$$
then $\Upsilon_{p,n}(B^p_1,{\mathbf Y})$ is the convex hull of the $2p$ random points $\pm Y_1,\ldots,\pm Y_p$.
\item[(e)]If $\Sigma$ is the unit ball $B^p_2$, then $\Upsilon_{p,n}(B^p_2,{\mathbf Y})$ is an ellipsiod, which is an intersection of the $n$-dimensional ellipsoid
$$
\mathcal{E}({\mathbf Y}):=\Big\{x\in\mathbb{R}^n\colon x^{\top}({\mathbf Y}^{\top}{\mathbf Y})^{-1}x\leq 1\Big\}
$$
with the $p$-dimensional linear subspace, spanned by the vectors $Y_1,\ldots, Y_p$.
\end{itemize}
The following theorem provides asymptotic normality for the logarithmic volume of the random convex bodies $\Upsilon_{p,n}(\Sigma,{\mathbf Y})$, when $p$ and $n$ tend to infinity simultaneously and the generating vectors have an elliptic distribution with all radial parts equal to $1$. We remark that a limit theorem for the logarithmic volume $\log\operatorname{Vol}_p(\Upsilon_{p,n}(\Sigma,{\mathbf X}))$ with ${\mathbf X}$ having a general elliptic distribution with random radial parts $R_1,\ldots,R_p$ can be derived as above and leads to a result very similar to that of Corollary~\ref{cor:SimplexWithRadialPart}. For simplicity, we decided to restrict our attention to random convex bodies generated by ${\mathbf Y}$ only.
\begin{theorem}\label{thm:RandomConvexBody}
Assume the conditions of Theorem \ref{thm:main} and let $Y_1,\ldots, Y_p$ be the columns of the matrix ${\mathbf Y}^{\top}$ defined in Theorem \ref{thm:main}. Let $(\Sigma_p)_{p\in\mathbb{N}}$ be a sequence of convex bodies, s.t. $\Sigma_p\subset\mathbb{R}^p$. Then, as $n\to\infty$, we have
$$
{\log\operatorname{Vol}_p(\Upsilon_{p,n}(\Sigma_p,{\mathbf Y}))-{1\over 2}\mu_n-\log\operatorname{Vol}_p(\Sigma_p)
\over {1\over 2}\sigma_n}\stackrel{d}{\longrightarrow} N(0,1),
$$
where $\mu_n$ and $\sigma_n$ are defined in \eqref{eq:meanvar}.
\end{theorem}
\begin{proof}
In the first step, we will show the equality
\begin{equation}\label{eq_19.11.20}
\operatorname{Vol}_p(\Upsilon_{p,n}(\Sigma_p, {\mathbf Y}))=\sqrt{\det({\mathbf Y}\Y^{\top})}\operatorname{Vol}_p(\Sigma_p).
\end{equation}
We note that for any integer $m$, any $m$-dimensional convex body $\tilde\Sigma$ and any $m$-dimensional vectors $X_1,\ldots,X_m$ with ${\mathbf X}_{(m)}=(X_1,\ldots,X_m)^{\top}$ we have
\begin{equation}\label{eq_19.11.20_2}
\operatorname{Vol}_m(\Upsilon_{m,m}(\tilde\Sigma,{\mathbf X}_{(m)}))=|\det( {\mathbf X}_{(m)})|\operatorname{Vol}_m(\tilde\Sigma)=m!\operatorname{Vol}_m(\Delta {\mathbf X}_{(m)})\operatorname{Vol}_m(\tilde\Sigma),
\end{equation}
see, for example, \cite[Proposition 2.1]{PP13}.
Denote by $L:=\operatorname{lin}(Y_1,\ldots,Y_p)$ the linear hull of the vectors $Y_1,\ldots,Y_p$. Then for any $s_1,\ldots,s_p\in\mathbb{R}$ we have
$$
\sum_{i=1}^ps_iY_i\in L,
$$
and, hence, $\Upsilon_{p,n}(\Sigma_p,{\mathbf Y})\subset L$. If $\operatorname{rank}({\mathbf Y})<p$, then $\det({\mathbf Y}\Y^{\top})=0$ and $\dim\Upsilon_{p,n}(\Sigma_p,{\mathbf Y})\leq\dim L<p$, leading to $\operatorname{Vol}_p(\Upsilon_{p,n}(\Sigma_p,{\mathbf Y}))=0$. Thus, \eqref{eq_19.11.20} trivially holds.
Let $\dim L=p$, $e_1,\ldots,e_n$ be the orthonormal basis of $\mathbb{R}^n$ and let $O_L$ be a rotation operator, such that $O_L(L)=\operatorname{lin}(e_1,\ldots,e_p)$. The linear hull $\operatorname{lin}(e_1,\ldots,e_p)$ can be identified with $\mathbb{R}^p$. Writing $\widetilde{Y}_i:=O_LY_i\in\mathbb{R}^p$, using that the volume is invariant with respect to rotations and applying \eqref{eq_19.11.20_2} we conclude
\begin{align*}
\operatorname{Vol}_p(\Upsilon_{p,n}(\Sigma_p, {\mathbf Y}))&=\operatorname{Vol}_p(\Upsilon_{p,n}(\Sigma_p, O_L({\mathbf Y}))\\
&=\operatorname{Vol}_p(\Upsilon_{p,p}(\Sigma_p, \widetilde{{\mathbf Y}}))\\
&=p!\operatorname{Vol}_p(\widetilde{{\mathbf Y}})\operatorname{Vol}_p(\Sigma_p)\\
&=p!\operatorname{Vol}_p(\Delta{\mathbf Y})\operatorname{Vol}_p(\Sigma_p),
\end{align*}
which is equivalent to \eqref{eq_19.11.20}. The desired conclusion now follows directly from \eqref{eq_19.11.20} and Theorem~\ref{thm:main}.
\end{proof}
\section{Proof of Theorem \ref{thm:main}}\setcounter{equation}{0}\label{sec:proofmain}
\subsection{Some notation}
In preparation for the proof of Theorem~\ref{thm:main}, we need to introduce some useful notation.
We start by noting that the entries of the vector ${\mathbf u}_1=(U_{11},\ldots,U_{1n})^{\top}$ are symmetric and exchangeable. We have the stochastic representation
\begin{equation}\label{def:R}
(U_{11},\ldots,U_{1n}) \overset{d}{=} \frac{1}{\sqrt{N_{11}^2+\cdots+N_{1n}^2}} (N_{11},\ldots,N_{1n})\,,
\end{equation}
where $(N_{ij})_{i,j\ge 1}$ is a field of independent standard normal random variables.
By \cite[Example 2.1]{heiny:mikosch:2017:corr}, we have for positive integers $m_1,\ldots,m_r$
\begin{equation}\label{lem:moment24}
\mathbb{E}[U_{11}^{2m_1} U_{12}^{2m_2}\cdots U_{1r}^{2m_r}] = \frac{\Gamma(n/2) \prod_{j=1}^r (2m_j-1)!!}{2^{m_1+\cdots+m_r} \Gamma(n/2+m_1+\cdots+m_r)} =\frac{\prod_{j=1}^r (2m_j-1)!!}{\prod_{j=0}^{m_1+\cdots+ m_r-1} (n+2j)}\,.
\end{equation}
In particular, we have $\mathbb{E}[U_{11}^4] = 3/(n(n+2))$ and we note that $\mathbb{E}[U_{11}^{2m_1} U_{12}^{2m_2}\cdots U_{1r}^{2m_r}]$ is of order $n^{-(m_1+\cdots+m_r)}$. By symmetry of the normal distribution, $\mathbb{E}[U_{11}^{m_1} U_{12}^{m_2}\cdots U_{1r}^{m_r}]$ is zero if one of the $m_i$'s is odd.
Throughout this section, we will use the notation
$$\beta_{2m_1,\ldots, {2m_r}}:=\mathbb{E} [ U_{11}^{2m_1} \cdots U_{1r}^{2m_r} ]\,.$$
Since $\beta_{2m_1,\ldots, {2m_r}}=\beta_{2m_{\pi(1)},\ldots, 2m_{\pi(r)}}$ for any permutation $\pi$ on $\{1,\ldots,r\}$ we will typically write the indices in decreasing order. For example, instead of $\beta_{2,4}$ we prefer writing $\beta_{4,2}$.
\par
For any symmetric matrix ${\mathbf M}\in \mathbb{R}^{d\times d}$, we will write
$$\lambda_{max}({\mathbf M})=\lambda_{1}({\mathbf M})\ge \lambda_{2}({\mathbf M}) \ge \cdots \ge \lambda_{d}({\mathbf M})=\lambda_{min}({\mathbf M})$$
for its ordered eigenvalues, $\norm{{\mathbf M}}$ for its spectral norm, that is, $\norm{{\mathbf M}}=\sqrt{\lambda_1({\mathbf M} {\mathbf M}^{\top})}$,
and denote its spectral decomposition by
\begin{equation}\label{eq:SpectralDecomposition}
{\mathbf M}=\mathcal{O}_{{\mathbf M}} \Lambda_{{\mathbf M}} \mathcal{O}_{{\mathbf M}}^{\top}\,,
\end{equation}
where $ \Lambda_{{\mathbf M}}$ is the diagonal matrix whose $i$-th diagonal element is $\lambda_i({\mathbf M})$, and $\mathcal{O}_{{\mathbf M}}$ is an orthogonal matrix.
\subsection{Opening: Proof of Theorem~\ref{thm:main}}
We set ${\mathbf U}=({\mathbf u}_1,\ldots,{\mathbf u}_p)^{\top}$ and ${\mathbf U}_{(i)}=({\mathbf u}_1,\ldots,{\mathbf u}_i)^{\top}$ for $1\le i \le p$. Due to the rotational invariance of the rows of ${\mathbf U}$, we have ${\mathbf U}\stackrel{d}{=} {\mathbf U}\mathcal{O}$ for any orthogonal matrix $\mathcal{O}$ with the appropriate dimension.
In view of \eqref{def:R}, we will use the representation
\begin{equation}\label{eq:repU}
{\mathbf U} \overset{d}{=} (\operatorname{diag}({\mathbf N} {\mathbf N}^{\top}))^{-1/2} {\mathbf N}\,,
\end{equation}
where ${\mathbf N}$ is a $p\times n$ matrix with independent standard normal entries $(N_{ij})$. Recalling that ${\mathbf Y}={\mathbf U} {\mathbf A}^{\top}$ and using the spectral decomposition ${\mathbf A}^{\top} {\mathbf A}=\mathcal{O}_{{\mathbf A}^{\top} {\mathbf A}} \Lambda_{{\mathbf A}^{\top} {\mathbf A}} \mathcal{O}_{{\mathbf A}^{\top} {\mathbf A}}^{\top}\,,$ and the rotational invariance of the rows of ${\mathbf U}$, we see that
\begin{equation}\label{eq:dkdkdd}
{\mathbf Y} {\mathbf Y}^{\top}={\mathbf U} {\mathbf A}^{\top} {\mathbf A} {\mathbf U}^{\top} \stackrel{d}{=} {\mathbf U} \Lambda_{{\mathbf A}^{\top} {\mathbf A}} {\mathbf U}^{\top}\,.
\end{equation}
Therefore, we may assume without loss of generality that ${\mathbf A}=\Lambda_{{\mathbf A}^{\top} {\mathbf A}}^{1/2}$ and, hence, that ${\mathbf A}$ is a diagonal matrix.
Next, we see that
\begin{equation}\label{eq:dkdkdd1}
\log \det \left({\mathbf U} {\mathbf A}^{\top} {\mathbf A} {\mathbf U}^{\top}\right) =p \log \frac{\operatorname{tr}({\mathbf A}^{\top} {\mathbf A})}{n} +\log \det \left( \frac{n}{\operatorname{tr}({\mathbf A}^{\top} {\mathbf A})} {\mathbf U} {\mathbf A}^{\top} {\mathbf A} {\mathbf U}^{\top} \right)\,.
\end{equation}
{Thus, in the remainder of this section, we will assume that ${\mathbf A}$ is a diagonal matrix with positive diagonal elements $A_{jj}=\sqrt{\lambda_j({\mathbf A}^{\top} {\mathbf A})}$, $j=1,\ldots,n$ and $\operatorname{tr}({\mathbf A}^{\top} {\mathbf A})=n$. By Assumption (B), there exists a constant $C>0$ not depending on $n$ such that $C^{-1}\le A_{nn}^2 \le A_{11}^2\le C$.}
\medskip
With the same matrix algebra as in Wang et al.~\cite[p.~85-86]{wang:han:pan:2018} who derived an expression for the log determinant of the sample covariance matrix, we get
\begin{equation}\label{eq:fedfse}
\log \det ({\mathbf Y}\Y^{\top})= -p \log n+ \sum_{i=0}^{p-1} \log( Z_{i+1})\,,
\end{equation}
where
\begin{equation*}
Z_{i+1}=n\, b_{i+1}^{\top} P_i b_{i+1} \quad \text{ and } \quad P_i={\mathbf I}_n-B_{(i)}^{\top} (B_{(i)} B_{(i)}^\top)^{-1} B_{(i)}\,.
\end{equation*}
Here $P_0={\mathbf I}_n$, $B_{(i)}=( b_1,\ldots, b_i)^{\top}$ and $b_i=(Y_{i1}, \ldots,Y_{in})^{\top}$ denotes the $i$-th row of the matrix ${\mathbf Y}$, that is $b_i={\mathbf A} {\mathbf u}_i$, and $P_i=(p_{i,kl})$ is a projection matrix. It is easy to check that $P_i=P_i^2$ and $\operatorname{tr}(P_i)=n-i$.
Using \eqref{eq:repU}, we can write $P_i$ as
\begin{equation}\label{eq:repPi}
P_i={\mathbf I}_n-{\mathbf A} {\mathbf N}_{(i)}^{\top} \big({\mathbf N}_{(i)}{\mathbf A}^2 {\mathbf N}_{(i)}^{\top}\big)^{-1} {\mathbf N}_{(i)} {\mathbf A}\,,
\end{equation}
where ${\mathbf N}_{(i)}$ is the $i\times n$ matrix comprised of the first $i$ rows of ${\mathbf N}$.
It holds $\lambda_{min}({\mathbf N}_{(i)}{\mathbf A}^2 {\mathbf N}_{(i)}^{\top}) \ge \lambda_{min}({\mathbf A}^2) \lambda_{min}({\mathbf N}_{(i)} {\mathbf N}_{(i)}^{\top})\ge C^{-1} \lambda_{min}({\mathbf N}_{(i)} {\mathbf N}_{(i)}^{\top})$.
Moreover, due to \cite[Proposition 2.1]{wang:han:pan:2018} all matrices ${\mathbf N}_{(i)} {\mathbf N}_{(i)}^{\top}$ are invertible with overwhelming probability, so that in combination with the previous statement all ${\mathbf N}_{(i)}{\mathbf A}^2 {\mathbf N}_{(i)}^{\top}$ are invertible.
By \cite[Lemma 2.1]{mohammadi:2016} and \cite[Lemma 3.1]{mohammadi:2016}, we have for $0\le i\le p-1$ and $1\le k,l\le n$,
\begin{equation}\label{eq:boundelements}
0\le p_{i,kk} \le 1 \quad \text{and} \quad -\frac{1}{2} \le p_{i,kl}\le \frac{1}{2}\,.
\end{equation}
We proceed by rewriting \eqref{eq:fedfse}. To this end, we define, for $0\le i\le p-1$,
\begin{equation}\label{eq:Q}
T_i:= \operatorname{tr}( {\mathbf A}^2\mathbb{E}[P_i] )= \sum_{k=1}^n \mathbb{E}[ p_{i,kk}] A_{kk} ^2 \quad \text{ and } \quad
Q_i:=\frac{{\mathbf A} P_i {\mathbf A}}{T_i}=(q_{i,kl})\,,
\end{equation}
and set
\begin{equation*}
\wt Z_{i+1}:=\frac{n\,{\mathbf u}_{i+1}^{\top} {\mathbf A} P_i {\mathbf A} {\mathbf u}_{i+1} -T_i}{T_i}=n\,{\mathbf u}_{i+1}^{\top} Q_i {\mathbf u}_{i+1}-1\,.
\end{equation*}
Using this notation, we have
\begin{equation}\label{eq:segtse1}
\log \det ({\mathbf Y}\Y^{\top})= \underbrace{-p \log n+\sum_{i=0}^{p-1} \log T_i}_{=:c_n}+ \sum_{i=0}^{p-1} \log(1+\wt Z_{i+1})\,.
\end{equation}
\subsection{Properties of $\wt Z_{i+1}$} Before we continue analyzing \eqref{eq:segtse1}, we collect some essential results about the random variables $\wt Z_{i+1}$. Let $\mathcal{F}_k=\mathcal{F}_k^{(n)}$ be the sigma algebra generated by the first $k$ rows of ${\mathbf N}$. Since $Q_i$ and ${\mathbf u}_{i+1}$ are independent, we see that, for $0\le i\le p-1$,
\begin{equation}\label{eq:supercool}
{\mathbf u}_{i+1}^{\top} Q_i {\mathbf u}_{i+1} \stackrel{d}{=} {\mathbf u}_{i+1}^{\top} \Lambda_{Q_i} {\mathbf u}_{i+1}\,.
\end{equation}
Taking expectation, we deduce
\begin{equation*}
\begin{split}
n\,\mathbb{E}\left[{\mathbf u}_{i+1}^{\top} Q_i {\mathbf u}_{i+1}\right]&= n\,\mathbb{E}\left[\mathbb{E}[{\mathbf u}_{i+1}^{\top} Q_i {\mathbf u}_{i+1}\,|\,\mathcal{F}_i]\right]= n\,\underbrace{\mathbb{E}[U_{11}^2]}_{=1/n}
\mathbb{E}\left[\operatorname{tr}(Q_i) \right]=1\,,
\end{split}
\end{equation*}
from which we conclude that $\wt Z_{i+1}$ is centered.
We compute the variance of $\wt Z_{i+1} $ in the next lemma.
\begin{lemma}\label{lem:secondmoment}
For $0\le i\le p-1$, one has
\begin{equation}\label{eq:s2}
\mathbb{E}[\wt Z_{i+1}^2]=(n^2\beta_4-1) \Big(\mathbb{E}[\operatorname{tr}(\Lambda_{Q_i}^2)] -\frac{1}{n-1}\Big) +{\mathbb{E}[\operatorname{tr}(\Lambda_{Q_i}^2)](n^2 \beta_4-1)\over n-1} +n^2\beta_{2,2} \operatorname{Var}(\operatorname{tr}(Q_i))\,.
\end{equation}
\end{lemma}
\begin{proof}
In view of \eqref{eq:supercool}, we have
\begin{equation*}
\operatorname{Var}(\wt Z_{i+1}^2)= n^2 \mathbb{E}\left[({\mathbf u}_{i+1}^{\top} \Lambda_{Q_i} {\mathbf u}_{i+1})^2 \right]-1\,.
\end{equation*}
From Lemma~\ref{lem:quf}, we get by conditioning on $\mathcal{F}_i$ that
\begin{equation*}
n^2 \mathbb{E}\left[({\mathbf u}_{i+1}^{\top} \Lambda_{Q_i} {\mathbf u}_{i+1})^2 \right]=n^2 \beta_4 \mathbb{E}[\operatorname{tr}(\Lambda_{Q_i}^2)] +n^2\beta_{2,2}(1- \mathbb{E}[\operatorname{tr}(\Lambda_{Q_i}^2)]) +n^2\beta_{2,2} \operatorname{Var}(\operatorname{tr}(Q_i))
\end{equation*}
since $\operatorname{Var}(\operatorname{tr}(Q_i))=\operatorname{Var}(\operatorname{tr}(\Lambda_{Q_i}))=\mathbb{E}[(\operatorname{tr}(\Lambda_{Q_i}))^2]-1$.
In conjunction with $n\beta_4+n(n-1) \beta_{2,2}=1$, which follows from taking expectation of the identity $(U_{11}^2+\cdots+U_{1n})^2=1$, the above equalities establish \eqref{eq:s2}.
\end{proof}
By our moment formula \eqref{lem:moment24}, it holds $\beta_4=3/(n(n+2))$. Now we study traces of powers of the matrices $\Lambda_{Q_i}$.
\begin{lemma}\label{lem:orderSj}
For $0\le i\le p-1$, we have $\norm{\Lambda_{Q_i}}\le T_i^{-1}C$,
\begin{equation*}
\operatorname{tr}(\Lambda_{Q_i}^2)=\frac{1}{T_i^2} \operatorname{tr}({\mathbf A}^4 P_i)\, \quad \text{ and } \quad \frac{C^{-j-1}}{(n-i)^{j-1}}\le \operatorname{tr}(\Lambda_{Q_i}^j) \le \frac{C^{j+1}}{(n-i)^{j-1}}\,, \quad j\ge 1\,.
\end{equation*}
\end{lemma}
\begin{proof} Let $0\le i\le p-1$ and recall that $Q_i=T_i^{-1} {\mathbf A} P_i {\mathbf A}$. Hence, $\norm{\Lambda_{Q_i}}=T_i^{-1} \lambda_1({\mathbf A} P_i {\mathbf A})\le T_i^{-1}C\lambda_1(P_i)=T_i^{-1}C$ is immediate. Since $P_i=P_i^2$, we deduce that
\begin{align}\label{eq:sdsdfd1}
Q_i {\mathbf A}^{-2} Q_i &= \frac{1}{T_i^2} {\mathbf A} P_i {\mathbf A} {\mathbf A}^{-2} {\mathbf A} P_i {\mathbf A}= \frac{Q_i}{T_i}\,.
\end{align}
Using that ${\mathbf A}$ is a diagonal matrix, $Q_i=Q_i^{\top}$ and the fact that $Q_i$ and ${\mathbf A} Q_i {\mathbf A}^{-1}$ have the same eigenvalues, it follows
\begin{align*}
\operatorname{tr} (Q_i^2)&= \sum_{j=1}^n \lambda_j^2(Q_i) =\sum_{j=1}^n \lambda_j^2({\mathbf A} Q_i {\mathbf A}^{-1})\\
&= \operatorname{tr}\left( ({\mathbf A} Q_i {\mathbf A}^{-1}) ({\mathbf A} Q_i {\mathbf A}^{-1})^{\top} \right) =
\operatorname{tr}({\mathbf A}^2 Q_i {\mathbf A}^{-2} Q_i)\,.
\end{align*}
In combination with \eqref{eq:sdsdfd1} and since $\operatorname{tr}(\Lambda_{Q_i}^2)=\operatorname{tr}(Q_i^2)$, we obtain
\begin{equation*}
\operatorname{tr}(\Lambda_{Q_i}^2)=\operatorname{tr}(Q_i^2)=\frac{1}{T_i} \operatorname{tr}({\mathbf A}^2 Q_i)=\frac{1}{T_i^2} \operatorname{tr}({\mathbf A}^4 P_i)\,.
\end{equation*}
Now let $j\ge 1$. Since $\operatorname{tr}(\Lambda_{Q_i}^j)=\sum_{k=1}^n \lambda_k^j(Q_i)$ and
$$ C^{-1} \lambda_k(P_i)\le \lambda_n({\mathbf A}^2) \lambda_k(P_i)\le \lambda_k({\mathbf A} P_i {\mathbf A})\le \lambda_1({\mathbf A}^2) \lambda_k(P_i)\le C\lambda_k(P_i)$$
one gets the upper bound
\begin{equation*}
\operatorname{tr}(\Lambda_{Q_i}^j)\le C T_i^{-j} \sum_{k=1}^n \lambda_k^j(P_i)= \frac{C (n-i)}{T_i^{j}} \le \frac{C^{j+1}}{(n-i)^{j-1}}\,,
\end{equation*}
where \eqref{eq:orderTi} was used for the last inequality.
The derivation of the lower bound is analogous.
\end{proof}
The following result will be useful.
\begin{proposition}\label{prop:useful}
There exists a constant $c_0$ such that
\begin{equation*}
\mathbb{E}[\wt Z_{i+1}^4] \le c_0 n^{-2}\,, \qquad 0\le i\le p-1\,.
\end{equation*}
\end{proposition}
\begin{proof}
From the definition of $\wt Z_{i+1}$ and \eqref{eq:supercool} we get for $0\le i\le p-1$,
\begin{align}\label{eq:fgdgdfs}
&\mathbb{E}[(\wt Z_{i+1}-\operatorname{tr} Q_i +1)^4]=n^4 \mathbb{E}\E\Big[\Big({\mathbf u}_{i+1}^{\top} \Lambda_{Q_i} {\mathbf u}_{i+1}-n^{-1} \operatorname{tr} \Lambda_{Q_i} \Big)^4 \, \Big| \mathcal{F}_i \Big]\,.
\end{align}
By Lemma \ref{lem:johnstonelemma6} and since all moments of the normal distribution are finite, we have
\begin{equation*}
\mathbb{E}\Big[\Big({\mathbf u}_{i+1}^{\top} \Lambda_{Q_i} {\mathbf u}_{i+1}-\frac{\operatorname{tr} \Lambda_{Q_i}}{n} \Big)^4 \, \Big| \mathcal{F}_i \Big]
\le C_4 \left[n^{-4} \big( \operatorname{tr} (\Lambda_{Q_i}^4) +( \operatorname{tr} (\Lambda_{Q_i}^2))^2 \big)+ \norm{\Lambda_{Q_i}}^4 n^{-2} \right]
\end{equation*}
where $C_4$ is a constant not depending on $n$ or $i$.
Due to $p/n\to \gamma\in (0,1)$ it holds $(1-\gamma)n \sim n-p \le n-i\le n$, so that $n-i$ is of order $n$ for all $0\le i\le p-1$, where we write $a_n\sim b_n$ for two sequences $(a_n)$ and $(b_n)$ whenever $a_n/b_n\to 1$ as $n\to\infty$. A combination of this fact with Lemma~\ref{lem:orderSj} yields that for sufficiently large $n$ there exists a constant $c_1$ such that $|\operatorname{tr} (\Lambda_{Q_i}^j)|\le c_1 n^{1-j}$ and $\norm{\Lambda_{Q_i}}\le c_1/n$ for $j\in\{1,2,3,4\}$ and $0\le i\le p-1$.
Therefore, it follows that
\begin{align*}
\mathbb{E}\Big[\Big({\mathbf u}_{i+1}^{\top} \Lambda_{Q_i} {\mathbf u}_{i+1}-\frac{\operatorname{tr} \Lambda_{Q_i}}{n} \Big)^4 \, \Big| \mathcal{F}_i \Big]
\le C_4 \left[n^{-4} \big( c_1 n^{-3} +c_1^2 n^{-2} \big)+ c_1^4 n^{-4} n^{-2} \right]\,
\end{align*}
and by \eqref{eq:fgdgdfs}, this establishes that there exists a constant $c_0$ such that
$\mathbb{E}[(\wt Z_{i+1}-\operatorname{tr} Q_i +1)^4] \le c_0 n^{-2}$. Now we note that
\begin{equation*}
\mathbb{E}[\wt Z_{i+1}^4] \le 16 \big(\mathbb{E}[(\wt Z_{i+1}-\operatorname{tr} Q_i +1)^4] +\mathbb{E}[(\operatorname{tr} Q_i -1)^4]\big)\,.
\end{equation*}
Since $|\operatorname{tr} Q_i -1|= |\operatorname{tr}\Lambda_{Q_i} -1|\le c_1+1$ we get $\mathbb{E}[(\operatorname{tr} Q_i -1)^4]\le (c_1+1)^2 \operatorname{Var}(\operatorname{tr} Q_i) \le c_2 n^{-2}$, where the variance bound above \eqref{eq:dfhdfdfd} was used and $c_2$ is some absolute constant independent of $i$. The desired claim of the proposition follows.
\end{proof}
\begin{lemma}\label{lem:4.2}
Under the conditions of Theorem~\ref{thm:main}, we have
\begin{equation}\label{eq:maxZ}
\max_{i=0,\ldots,p-1} |\wt Z_{i+1}| \stackrel{\P}{\rightarrow} 0\,, \qquad n \to \infty\,.
\end{equation}
\end{lemma}
\begin{proof}
Using the union bound, Markov's inequality and Proposition \ref{prop:useful}, we get
\begin{equation}\label{eq:sumz4}
\P\Big(\max_{i=0,\ldots,p-1} |\wt Z_{i+1}|>\varepsilon\Big) \le \sum_{i=0}^{p-1} \P(|\wt Z_{i+1}|>\varepsilon)\le \sum_{i=0}^{p-1}\frac{\mathbb{E}[\wt Z_{i+1}^4]}{\varepsilon^4}\to 0 \,, \qquad n\to\infty,
\end{equation}
for any $\varepsilon>0$.
\end{proof}
\subsection{Middlegame: Proof of Theorem~\ref{thm:main}}
Now we decompose the last term in \eqref{eq:segtse1}. By Taylor's theorem, we get
\begin{equation}\label{eq:taylornew}
\sum_{i=0}^{p-1} \log(1+\wt Z_{i+1})= \sum_{i=0}^{p-1} (\wt Z_{i+1}-\frac{\wt Z_{i+1}^2}{2}) +\sum_{i=0}^{p-1} \Omega_{i+1}\,,
\end{equation}
where the remainder in Lagrange form is given by
\begin{equation}\label{eq:Ri+1}
\Omega_{i+1}=\frac{1}{3} \Big(\frac{\wt Z_{i+1}}{1+\theta \wt Z_{i+1}} \Big)^3 \quad \text{ for some } \theta=\theta(\wt Z_{i+1})\in (0,1)\,.
\end{equation}
The Taylor expansion is justified by Lemma~\ref{lem:4.2}.
We have
\begin{equation}\label{eq:decomposecor}
\begin{split}
\sum_{i=0}^{p-1} (\wt Z_{i+1}-\frac{\wt Z_{i+1}^2}{2})&=\sum_{i=0}^{p-1} \wt Z_{i+1}-\sum_{i=0}^{p-1} \underbrace{\tfrac{1}{2} (\wt Z_{i+1}^2-\mathbb{E}[\wt Z_{i+1}^2 | \mathcal{F}_i])}_{=:\wt Y_{i+1}} - \sum_{i=0}^{p-1}\tfrac{1}{2}\mathbb{E}[\wt Z_{i+1}^2 | \mathcal{F}_i]\,.
\end{split}
\end{equation}
Our next goal is to show
\begin{equation}\label{eq:mainintermediate}
\frac{\log \det({\mathbf Y}\Y^{\top}) -\wt\mu_n}{\wt\sigma_n} \stackrel{d}{\longrightarrow} N(0,1)\,, \qquad n \to \infty\,,
\end{equation}
where, for $n\ge 1$, $\wt \sigma_n^2:= \sum_{i=0}^{p-1}\mathbb{E}[\wt Z_{i+1}^2]$ and the centering sequence $\wt \mu_n$ is given by
\begin{equation}\label{eq:mean1}
\wt \mu_n:=p \log \frac{\operatorname{tr}({\mathbf A}^{\top} {\mathbf A})}{n} -\frac{\wt\sigma_n^2}{2}+ \sum_{i=0}^{p-1} \log T_i -p\log n=-\frac{\wt\sigma_n^2}{2}+ \sum_{i=0}^{p-1} \log T_i -p\log n\,.
\end{equation}
In view of \eqref{eq:segtse1} and \eqref{eq:taylornew}, one gets
\begin{equation}\label{eq:dddd}
\log \det ({\mathbf Y}\Y^{\top}) -\wt\mu_n =
\sum_{i=0}^{p-1} \wt Z_{i+1}-\sum_{i=0}^{p-1} \wt Y_{i+1}+\sum_{i=0}^{p-1} \Omega_{i+1}
- \sum_{i=0}^{p-1}\tfrac{1}{2}\mathbb{E}[\wt Z_{i+1}^2 | \mathcal{F}_i]+c_n-\wt\mu_n\,,
\end{equation}
By virtue of \eqref{eq:dddd}, distributional convergence \eqref{eq:mainintermediate} follows from the next four limit relations by an application of the Slutsky lemma,
\begin{align}
\frac{1}{\wt\sigma_n} \sum_{i=0}^{p-1} \wt Z_{i+1} &\stackrel{d}{\longrightarrow} N(0,1)\,, \label{sumZ}\\
\sum_{i=0}^{p-1} \wt Y_{i+1}&\stackrel{\P}{\rightarrow} 0\,, \label{sumY_i}\\
\sum_{i=0}^{p-1} \Omega_{i+1}&\stackrel{\P}{\rightarrow} 0\,, \label{sumR_i}\\
\sum_{i=0}^{p-1} \tfrac{1}{2}\mathbb{E}[\wt Z_{i+1}^2 | \mathcal{F}_i]-c_n+\wt\mu_n&\stackrel{\P}{\rightarrow} 0\,, \label{sumconst}
\end{align}
as $n \to \infty$. Note that, by Lemma~\ref{lem:sigma}, $\wt\sigma_n^2$ is of constant order. For this reason we omitted $\wt\sigma_n^{-1}$ in \eqref{sumY_i}, \eqref{sumR_i} and \eqref{sumconst}. Equations \eqref{sumZ}, \eqref{sumY_i}, \eqref{sumR_i}, \eqref{sumconst} are proved in Sections \ref{sec:sumZ}, \ref{sec:sumY_i}, \ref{sec:sumR_i} and \ref{sec:sumconst}, respectively. This establishes \eqref{eq:mainintermediate}.
\subsection{Endgame: Fine-tuning the norming sequences}
In this subsection we complete the proof of Theorem~\ref{thm:main} by providing simpler formulas for the mean and variance. We start by showing that the sequences $\wt\mu_n$ and $\wt\sigma_n^2$ in \eqref{eq:mainintermediate} can be replaced by
\begin{equation}\label{defmu}
\mu_n:=p \log \frac{\operatorname{tr}({\mathbf A}^{\top} {\mathbf A})}{n} -\frac{\sigma_n^2}{2}+ \sum_{i=1}^{p-1} \log \frac{\operatorname{tr}({\mathbf A}^2 \mathbb{E}[P_i])}{n}
\end{equation}
and
\begin{equation}\label{defsigma}
\sigma_n^2:=-2\frac{p}{n} + 2 \sum_{i=1}^{p-1} \frac{\operatorname{tr}({\mathbf A}^4 \mathbb{E}[P_i])}{(\operatorname{tr}({\mathbf A}^2 \mathbb{E}[P_i]))^2}\,,
\end{equation}
respectively. Here, the summations start at $i=1$ since $P_0={\mathbf I}$ and $T_0=n$. The dependence on ${\mathbf A}$ of these new sequences is more explicit.
\begin{lemma}\label{lem:sigma}
It holds $\sigma_n^2 \sim \wt\sigma_n^2$, as $n \to \infty$, and
\begin{equation*}
-2\frac{p}{n} + 2 \sum_{i=0}^{p-1} \frac{1}{n-i}\le \sigma_n^2 \le -2\frac{p}{n} + 2C^4 \sum_{i=0}^{p-1} \frac{1}{n-i} \,
\end{equation*}
\begin{equation}\label{eq:var1}
-2\frac{p}{n} + 2 \sum_{i=0}^{p-1} \frac{1}{n-i} \sim -2\frac{p}{n} -2\log\Big(1-\frac{p}{n}\Big)\to -2 \gamma -2\log(1-\gamma)>0\,,\qquad n \to \infty\,.
\end{equation}
\end{lemma}
\begin{proof}
From Lemma~\ref{lem:secondmoment}, Lemma~\ref{lem:orderSj}, \eqref{eq:dfhdfdfd} and the facts that $n^2 \beta_4 \to 3$ and $n^2 \beta_{2,2} \to 1$, we get
\begin{align*}
\wt \sigma_n^2&= \sum_{i=0}^{p-1} \Bigg( (n^2\beta_4-1) \Big(\mathbb{E}[\operatorname{tr}(\Lambda_{Q_i}^2)] -\frac{1}{n-1}\Big) +{\mathbb{E}[\operatorname{tr}(\Lambda_{Q_i}^2)](n^2 \beta_4-1)\over n-1}
+n^2\beta_{2,2} \operatorname{Var}(\operatorname{tr}(Q_i))\Bigg)\\
&= -2 \frac{p}{n} +2 \sum_{i=0}^{p-1} \frac{\operatorname{tr}({\mathbf A}^4 \mathbb{E}[P_i])}{(\operatorname{tr}({\mathbf A}^2 \mathbb{E}[P_i]))^2} +o(1)\,, \qquad n \to \infty\,.
\end{align*}
The fact that $\sum_{i=0}^{p-1} \frac{1}{n-i} \sim -\log(1-p/n)$ follows from asymptotic properties of the harmonic series, more precisely, by using that $\sum_{k=1}^n 1/k -\log n$ tends to the Euler--Mascheroni constant.
The last limit in \eqref{eq:var1} follows from $p/n\to \gamma\in (0,1)$. To complete the proof of the lemma, it suffices to show
$$\frac{1}{n-i} \le \frac{\operatorname{tr}({\mathbf A}^4 \mathbb{E}[P_i])}{(\operatorname{tr}({\mathbf A}^2 \mathbb{E}[P_i]))^2}\le \frac{C^4}{n-i} .$$
For $\ell\in\{2,4\}$ and conditionally on $\mathcal{F}_i$, define the random variables $\eta_{\ell}$ through
$$\P(\eta_{\ell}=A_{kk})=\frac{p_{i,kk}}{n-i}\,, \qquad k=1,\ldots,n\,,$$
where we recall that $\operatorname{tr} P_i=n-i$. By Jensen's inequality, we deduce
$$ (n-i) \operatorname{tr}({\mathbf A}^4 \mathbb{E}[P_i]) = (n-i)^2 \mathbb{E}[\eta_{\ell}^4] \ge (n-i)^2 \big(\mathbb{E}[\eta_{\ell}^2]\big)^2 = (\operatorname{tr}({\mathbf A}^2 \mathbb{E}[P_i]))^2\,.$$
Together with $C^{-1}\le A_{nn}^2 \le A_{11}^2\le C$ this yields the desired claim.
\end{proof}
An immediate consequence of Lemma~\ref{lem:sigma} is $\wt\mu_n-\mu_n\to 0$. Next, we observe that only the expectations of the diagonal elements of $P_i$ are needed in \eqref{defmu} and \eqref{defsigma}. Indeed, we have
$$\operatorname{tr}({\mathbf A}^2 \mathbb{E}[P_i])=\sum_{k=1}^n A_{kk}^2 \mathbb{E}[p_{i,kk}] \quad \text{ and } \quad \operatorname{tr}({\mathbf A}^4 \mathbb{E}[P_i])=\sum_{k=1}^n A_{kk}^4 \mathbb{E}[p_{i,kk}]\,.$$
Thus our final goal is to find $t_{i,k}({\mathbf A}):= \mathbb{E}[p_{i,kk}]$ for $1\le i\le p-1$ and $1\le k\le n$.
\begin{lemma}\label{lem:tik}
For $1\le i\le p-1$ and $1\le k\le n$, let $w_{i1}, \ldots, w_{in}$ be i.i.d.\ $i$-dimensional random vectors whose components are independent standard normal random variables. Then it holds
\begin{equation}\label{eq:tikdag}
t_{i,k}({\mathbf A})= \mathbb{E}\left[\frac{1}{1+ A_{kk}^2 w_{ik}^{\top} \big(\sum_{\ell=1; \ell \neq k}^n A_{\ell\ell}^2 w_{i\ell} w_{i\ell}^{\top}\big)^{-1}w_{ik}}\right]\,.
\end{equation}
\end{lemma}
\begin{proof}
Denote the $k$-th column of ${\mathbf N}_{(i)}$ by $w_{ik}$ and write ${\mathbf N}_{(i,k)}$ for the matrix obtained from ${\mathbf N}_{(i)}$ by removing its $k$-th column. Let ${\mathbf A}_{(k)}$ be the matrix ${\mathbf A}$ without its $k$-th row and column.
Using \eqref{eq:repPi} and the Sherman-Morrison formula, we get
\begin{equation}\label{eq:pikk}
\begin{split}
p_{i,kk}
&= 1- A_{kk}^2 w_{ik}^{\top} \big({\mathbf N}_{(i)}{\mathbf A}^2 {\mathbf N}_{(i)}^{\top} \big)^{-1}w_{ik}\\
&= 1- A_{kk}^2 w_{ik}^{\top} \big({\mathbf N}_{(i,k)}{\mathbf A}_{(k)}^2 {\mathbf N}_{(i,k)}^{\top} +A_{kk}^2 w_{ik} w_{ik}^{\top}\big)^{-1}w_{ik}\\
&= \frac{1}{1+ A_{kk}^2 w_{ik}^{\top} \big({\mathbf N}_{(i,k)}{\mathbf A}_{(k)}^2 {\mathbf N}_{(i,k)}^{\top} \big)^{-1}w_{ik}}\,.
\end{split}
\end{equation}
Noting that the entries of ${\mathbf N}_{(i)}$ are independent standard normal random variables finishes the proof.
\end{proof}
\begin{remark}\label{rem:tik}{\em
If ${\mathbf A}={\mathbf I}$, one has $t_{i,k}({\mathbf I})=(n-i)/n$ since $\operatorname{tr} P_i=n-i$ and $p_{i,11},\ldots, p_{i,nn}$ are identically distributed. We stress that the formula for $t_{i,k}({\mathbf A})$ in \eqref{eq:tikdag} is valid for diagonal matrices ${\mathbf A}$ and observe
\begin{equation}\label{eq:tricks}
t_{i,k}({\mathbf A})= t_{i,k}(c \,{\mathbf A})\,, \qquad c\in \mathbb{R}\backslash\{0\}\,.
\end{equation}
}\end{remark}
For $s\in\{2,4\}$ we get
$\operatorname{tr}({\mathbf A}^s \mathbb{E}[P_i])=\sum_{k=1}^n A_{kk}^s t_{i,k}({\mathbf A})$
and therefore, we may write \eqref{defmu} and \eqref{defsigma} as follows:
\begin{align*}
\mu_n&=p \log \frac{\operatorname{tr}({\mathbf A}^{\top} {\mathbf A})}{n} -\frac{\sigma_n^2}{2}+ \sum_{i=1}^{p-1} \log \frac{\sum_{k=1}^n A_{kk}^2 t_{i,k}({\mathbf A})}{n}\,,\\
\sigma_n^2&=-2\frac{p}{n} + 2 \sum_{i=1}^{p-1} \frac{\sum_{k=1}^n A_{kk}^4 t_{i,k}({\mathbf A})}{(\sum_{k=1}^n A_{kk}^2 t_{i,k}({\mathbf A}))^2}\,.
\end{align*}
So far we have assumed that ${\mathbf A}$ is a diagonal matrix with positive diagonal elements $A_{jj}=\sqrt{\lambda_j({\mathbf A}^{\top} {\mathbf A})}$, $j=1,\ldots,n$ and $\operatorname{tr}({\mathbf A}^{\top} {\mathbf A})=n$. For the general case we need to replace $t_{i,k}({\mathbf A})$ with $t_{i,k}(\wt{\mathbf A})$, where
$$\wt{\mathbf A}:= \sqrt{\frac{n}{\operatorname{tr}({\mathbf A}^{\top}{\mathbf A})}} \, \Lambda_{{\mathbf A}^{\top}{\mathbf A}}^{1/2}\,,$$
with $k$-th diagonal element of $\wt{\mathbf A}^2$ given by
$\frac{n}{\operatorname{tr}({\mathbf A}^{\top}{\mathbf A})} \lambda_k({\mathbf A}^{\top}{\mathbf A})$. In view of \eqref{eq:tricks}, we have $t_{i,k}(\wt{\mathbf A})=t_{i,k}\big( \Lambda_{{\mathbf A}^{\top}{\mathbf A}}^{1/2} \big)$. To unify notation, we define for (not necessarily diagonal) matrices ${\mathbf A}$,
\begin{equation}\label{eq:tikdag1}
t_{i,k}({\mathbf A})= \mathbb{E}\left[\frac{1}{1+ \lambda_k({\mathbf A}^{\top}{\mathbf A}) w_{ik}^{\top} \big(\sum_{\ell=1; \ell \neq k}^n \lambda_{\ell}({\mathbf A}^{\top}{\mathbf A}) w_{i\ell} w_{i\ell}^{\top}\big)^{-1}w_{ik}}\right]\,,
\end{equation}
which coincides with \eqref{eq:tikdag} in case ${\mathbf A}$ is a diagonal matrix.
The above considerations establish that the sequences $\wt\mu_n$ and $\wt\sigma_n^2$ in \eqref{eq:mainintermediate} can be replaced by
\begin{align*}
\mu_n&=p \log \frac{\operatorname{tr}({\mathbf A}^{\top} {\mathbf A})}{n} -\frac{\sigma_n^2}{2}+ \sum_{i=1}^{p-1} \log \frac{\sum_{k=1}^n \frac{n}{\operatorname{tr}({\mathbf A}^{\top}{\mathbf A})}\lambda_k({\mathbf A}^{\top}{\mathbf A}) t_{i,k}({\mathbf A})}{n}\\
&= \log \operatorname{tr}({\mathbf A}^{\top}{\mathbf A}) -p \log n -\frac{\sigma_n^2}{2}+ \sum_{i=1}^{p-1} \log \bigg(\sum_{k=1}^n \lambda_k({\mathbf A}^{\top}{\mathbf A}) t_{i,k}({\mathbf A})\bigg)\,,\\
\sigma_n^2&=-2\frac{p}{n} + 2 \sum_{i=1}^{p-1} \frac{\sum_{k=1}^n \lambda_k^2({\mathbf A}^{\top}{\mathbf A}) t_{i,k}({\mathbf A})}{(\sum_{k=1}^n \lambda_k ({\mathbf A}^{\top}{\mathbf A}) t_{i,k}({\mathbf A}))^2}\,,
\end{align*}
respectively, which completes the proof of Theorem~\ref{thm:main}.
\subsection{Proof of \eqref{sumZ}} \label{sec:sumZ}
Set $\bar{Z}_{i+1} := \wt Z_{i+1}-\mathbb{E}[\wt Z_{i+1} | \mathcal{F}_i]=\wt Z_{i+1}- (\operatorname{tr} Q_i -1)$. By Lemma \ref{lem:yaskov}, we have $\wt\sigma_n^{-1} \sum_{i=0}^{p-1}(\operatorname{tr} Q_i -1)=o_{\P}(1)$, as $n \to \infty$, and thus it suffices to show
\begin{align}\label{sumZ11}
\frac{1}{\wt\sigma_n} \sum_{i=0}^{p-1} \bar{Z}_{i+1} &\stackrel{d}{\longrightarrow} N(0,1)\,.
\end{align}
To this end, we will use the following CLT for martingale differences.
\begin{lemma}[e.g. Hall and Heyde \cite{hall:heyde:1980}]\label{lem:martingaleclt}
Let $\{S_{ni},\mathcal{F}_{ni}, 1\le i\le k_n, n\ge 1\}$ be a zero-mean, square integrable martingale array with differences $Z_{ni}$. Suppose that $\mathbb{E}[\max_i Z_{ni}^2]$ is bounded in $n$ and that
\begin{equation*}
\max_i |Z_{ni}|\stackrel{\P}{\rightarrow} 0 \quad \text{ and } \quad \sum_i Z_{ni}^2 \stackrel{\P}{\rightarrow} 1\,.
\end{equation*}
Then we have $S_{nk_n}\stackrel{d}{\longrightarrow} N(0,1)$ as $n\to\infty$.
\end{lemma}
In view of $\mathbb{E}[\bar{Z}_{i+1} | \mathcal{F}_i]=0$, we observe that $(\bar{Z}_{i+1})_i$ is a martingale difference sequence with respect to the filtration $(\mathcal{F}_i)$.
We apply Lemma~\ref{lem:martingaleclt} to the martingale differences $\wt\sigma_n^{-1} \bar{Z}_{i+1}$.
From \eqref{eq:maxZ} and \eqref{eq:fdfdd1}, we have
$$\max_{i=0,\ldots,p-1} | \wt\sigma_n^{-1} \,\bar{Z}_{i+1}| \le \wt\sigma_n^{-1} \Big( \max_{i=0,\ldots,p-1} | \wt Z_{i+1}|+ \max_{i=0,\ldots,p-1} | \operatorname{tr} Q_i -1|\Big)\stackrel{\P}{\rightarrow} 0\,, \qquad n \to \infty\,.$$
Next, we see that
\begin{equation*
\wt\sigma_n^{-2} \mathbb{E}\Big[\max_{i=0,\ldots,p-1} \bar{Z}_{i+1}^2\Big]\le 2\,\wt\sigma_n^{-2} \sum_{i=0}^{p-1} \Big(\mathbb{E}[\wt Z_{i+1}^2] +\operatorname{Var}\big(\operatorname{tr}(Q_i)\big) \Big)=2+o(1)\,,
\end{equation*}
where the last equality follows from the definition of $\wt\sigma_n^2$ and \eqref{eq:dfhdfdfd}.
Using Markov's inequality and \eqref{eq:dfhdfdfd}, it can be checked that $\sum_{i=0}^{p-1}\big( \bar{Z}_{i+1}^2-\wt Z_{i+1}^2\big) \stackrel{\P}{\rightarrow} 0$.
Due to $\wt\sigma_n^{-2} \sum_{i=0}^{p-1}\mathbb{E}[\wt Z_{i+1}^2]=1$, the condition $\wt\sigma_n^{-2} \sum_{i=0}^{p-1}\bar{Z}_{i+1}^2\stackrel{\P}{\rightarrow} 1$ is therefore implied by
\begin{equation}\label{eq:1212}
\sum_{i=0}^{p-1}(\wt Z_{i+1}^2-\mathbb{E}[\wt Z_{i+1}^2\,| \mathcal{F}_i])\stackrel{\P}{\rightarrow} 0 \,,\qquad n \to \infty\,,
\end{equation}
and
\begin{equation}\label{eq:reess}
\sum_{i=0}^{p-1}(\mathbb{E}[\wt Z_{i+1}^2\,| \mathcal{F}_i]-\mathbb{E}[\wt Z_{i+1}^2]) \stackrel{\P}{\rightarrow} 0 \,,\qquad n \to \infty\,.
\end{equation}
Observe that \eqref{eq:1212} is equivalent to \eqref{sumY_i}. Hence, it remains to show \eqref{eq:reess}. To this end, recall that in Lemma \ref{lem:secondmoment} and its proof we obtained
\begin{align*}
\sum_{i=0}^{p-1}\big(\mathbb{E}[\wt Z_{i+1}^2\,| \mathcal{F}_i]-\mathbb{E}[\wt Z_{i+1}^2]\big) &=
\Big(n^2\beta_4-1+{(n^2 \beta_4-1)\over n-1}\Big) \sum_{i=0}^{p-1} \Big(\operatorname{tr}(\Lambda_{Q_i}^2)-\mathbb{E}[\operatorname{tr}(\Lambda_{Q_i}^2)]\Big) \\
&\quad +n^2\beta_{2,2} \sum_{i=0}^{p-1}\Big( (\operatorname{tr}(Q_i))^2-\mathbb{E}\big[ (\operatorname{tr}(Q_i))^2\big]\Big)=:S^{(1)}+S^{(2)}\,.
\end{align*}
Using $n^2 \beta_4\to 3$, $n^2 \beta_{2,2}\to 1$ and Lemma~\ref{lem:yaskov}, we get
\begin{align*}
S^{(1)}&\sim 2 \sum_{i=0}^{p-1} \Big(\operatorname{tr}(\Lambda_{Q_i}^2)-\mathbb{E}[\operatorname{tr}(\Lambda_{Q_i}^2)]\Big)\stackrel{\P}{\rightarrow} 0 \quad \text{ and } \quad S^{(2)}\stackrel{\P}{\rightarrow} 0\,, \qquad n \to \infty\,.
\end{align*}
Thus, we have verified the conditions of Lemma~\ref{lem:martingaleclt} which now yields \eqref{sumZ} and finishes the proof.
\subsection{Proof of \eqref{sumY_i}}\label{sec:sumY_i}
By Markov's inequality, one has for $\varepsilon>0$,
\begin{equation}\label{eq:boundY}
\P\Big(\Big|\sum_{i=0}^{p-1} \wt Y_{i+1}\Big|>\varepsilon\Big)\le \varepsilon^{-2} \mathbb{E}\Big[\Big(\sum_{i=0}^{p-1} \wt Y_{i+1}\Big)^2 \Big]\,.
\end{equation}
If $j\neq i$ one can show by conditioning on $\mathcal{F}_{\max(i,j)}$ that
$\mathbb{E}[\wt Y_{i+1} \wt Y_{j+1}]=0$. Therefore, one gets
\begin{equation*}
\begin{split}
\mathbb{E}\Big[\Big(\sum_{i=0}^{p-1} \wt Y_{i+1}\Big)^2 \Big]&= \sum_{i=0}^{p-1} \mathbb{E}[\wt Y_{i+1}^2]= \frac14 \sum_{i=0}^{p-1} \mathbb{E}\Big[(\wt Z_{i+1}^2-\mathbb{E}[\wt Z_{i+1}^2 | \mathcal{F}_i])^2\Big]\\
&\le \frac12 \sum_{i=0}^{p-1} \mathbb{E}[\wt Z_{i+1}^4]
+ \frac12 \sum_{i=0}^{p-1} \mathbb{E}[(\mathbb{E}[\wt Z_{i+1}^2 | \mathcal{F}_i])^2] \,,
\end{split}
\end{equation*}
where the first term in the last line is $o(1)$ by Proposition \ref{prop:useful}.
Using Lemma~\ref{lem:secondmoment}, Lemma~\ref{lem:orderSj} and \eqref{lem:moment24}, we see that, uniformly in $i$,
\begin{align*}
\mathbb{E}[\wt Z_{i+1}^2 | \mathcal{F}_i] -n^2\beta_{2,2} (\operatorname{tr}(Q_i)-1)^2&=
(n^2\beta_4-1) \Big(\operatorname{tr}(\Lambda_{Q_i}^2) -\frac{1}{n-1}\Big) +{\operatorname{tr}(\Lambda_{Q_i}^2)(n^2 \beta_4-1)\over n-1} =O(n^{-1})\,.
\end{align*}
therefore we have by \eqref{eq:dfhdfdfd} and since $\operatorname{tr}(Q_i)+1$ is uniformly bounded by some constant $c$,
\begin{align*}
\sum_{i=0}^{p-1}\mathbb{E}[(\mathbb{E}[\wt Z_{i+1}^2 | \mathcal{F}_i])^2]&\le O(n^{-1})+2 (n^2\beta_{2,2})^2 \sum_{i=0}^{p-1} \mathbb{E}\big[\big((\operatorname{tr}(Q_i))^2-1\big)^2\big]\\
&\le O(n^{-1})+2 \sum_{i=0}^{p-1}\mathbb{E}\big[(\operatorname{tr}(Q_i)-1)^2(\operatorname{tr}(Q_i)+1)^2\big]\\
&\le O(n^{-1})+2 c^2 \sum_{i=0}^{p-1}\mathbb{E}\big[(\operatorname{tr}(Q_i)-1)^2\big]=o(1)\,.
\end{align*}
We conclude
$$\lim_{n \to \infty} \mathbb{E}\Big[\Big(\sum_{i=0}^{p-1} \wt Y_{i+1}\Big)^2 \Big] =0\,.$$
In view of \eqref{eq:boundY}, we have proved \eqref{sumY_i}.
\subsection{Proof of \eqref{sumR_i}}\label{sec:sumR_i}
Set $\delta=1/2$ and define the event $E_n(\delta)=\{\max_{i=0,\ldots,p-1} |\wt Z_{i+1}|\le \delta \}$. For any $\varepsilon>0$, it follows that
\begin{equation*}
\P\left( \Big|\sum_{i=0}^{p-1} \Omega_{i+1}\Big| >\varepsilon\right)\le \P\left( \Big|\sum_{i=0}^{p-1} \Omega_{i+1}\Big| \mathds{1}_{E_n(\delta)}>\varepsilon\right)+\P\left(\max_{i=0,\ldots,p-1} |\wt Z_{i+1}|>\delta\right)\,.
\end{equation*}
The second term on the right-hand side~tends to zero by virtue of \eqref{eq:maxZ}.
From \eqref{eq:Ri+1}, recall that
\begin{equation*}
\Omega_{i+1}=\frac{1}{3} \Big(\frac{\wt Z_{i+1}}{1+\theta \wt Z_{i+1}} \Big)^3 \quad \text{ for some } \theta=\theta(\wt Z_{i+1})\in (0,1)\,.
\end{equation*}
On the event $E_n(\delta)$ we have $1+\theta \wt Z_{i+1}\in (1-\delta,1+\delta)$ and therefore
\begin{equation*}
\begin{split}
\P\left( \Big|\sum_{i=0}^{p-1} \Omega_{i+1}\Big| \mathds{1}_{E_n(\delta)}>\varepsilon\right)
&\le \frac{1}{3\varepsilon} \sum_{i=0}^{p-1} \mathbb{E}\Big[\Big|\frac{\wt Z_{i+1}}{1+\theta \wt Z_{i+1}}\Big|^3 \mathds{1}_{E_n(\delta)}\Big]\\
&\le \frac{1}{3\varepsilon} \sum_{i=0}^{p-1} (1-\delta)^{-3} \mathbb{E}\Big[|\wt Z_{i+1}|^3 \mathds{1}_{E_n(\delta)}\Big]\\
&\le \frac{1}{3\varepsilon (1-\delta)^{3}} \sum_{i=0}^{p-1} \Big(\mathbb{E}\Big[|\wt Z_{i+1}|^4\Big]\Big)^{3/4}\\
&\le \frac{1}{3\varepsilon (1-\delta)^{3}} p \big(c_0n^{-2}\big)^{3/4}\to 0\,,\qquad n \to \infty\,,
\end{split}
\end{equation*}
where Hölder's inequality was used for the third inequality and Proposition~\ref{prop:useful} for the last.
\subsection{Proof of \eqref{sumconst}}\label{sec:sumconst}
In view of \eqref{eq:reess}, equation \eqref{sumconst} follows from
\begin{equation*
\lim_{n \to \infty}\Big[ \sum_{i=0}^{p-1} \tfrac{1}{2}\mathbb{E}[\wt Z_{i+1}^2] -c_n+\wt\mu_n\Big]=\lim_{n \to \infty}\big[ \wt\sigma_n^2/2-c_n+\wt\mu_n\big]=0\,.
\end{equation*}
| {
"timestamp": "2022-06-02T02:21:36",
"yymm": "2206",
"arxiv_id": "2206.00514",
"language": "en",
"url": "https://arxiv.org/abs/2206.00514",
"abstract": "Random simplices and more general random convex bodies of dimension $p$ in $\\mathbb{R}^n$ with $p\\leq n$ are considered, which are generated by random vectors having an elliptical distribution. In the high-dimensional regime, that is, if $p\\to\\infty$ and $n\\to\\infty$ in such a way that $p/n\\to\\gamma\\in(0,1)$, a central and a stable limit theorem for the logarithmic volume of random simplices and random convex bodies is shown. The result follows from a related central limit theorem for the log-determinant of $p\\times n$ random matrices whose rows are copies of a random vector with an elliptical distribution, which is established as well.",
"subjects": "Probability (math.PR)",
"title": "The volume of random simplices from elliptical distributions in high dimension",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9854964194566753,
"lm_q2_score": 0.7185943985973772,
"lm_q1q2_score": 0.7081722068593381
} |
https://arxiv.org/abs/0810.3164 | Linear Dynamical Systems over Finite Rings | The problem of linking the structure of a finite linear dynamical system with its dynamics is well understood when the phase space is a vector space over a finite field. The cycle structure of such a system can be described by the elementary divisors of the linear function, and the problem of determining whether the system is a fixed point system can be answered by computing and factoring the system's characteristic polynomial and minimal polynomial. It has become clear recently that the study of finite linear dynamical systems must be extended to embrace finite rings. The difficulty of dealing with an arbitrary finite commutative ring is that it lacks of unique factorization. In this paper, an efficient algorithm is provided for analyzing the cycle structure of a linear dynamical system over a finite commutative ring. In particular, for a given commutative ring $R$ such that $|R|=q$, where $q$ is a positive integer, the algorithm determines whether a given linear system over $R^n$ is a fixed point system or not in time $O(n^3\log(n\log(q)))$. | \section{Introduction}
A finite dynamical system is a function $f: X\longrightarrow X$, where $X$ is a finite set. The dynamics of the system is obtained by iterating the function $f$. Such dynamical systems have a variety of applications, such as in engineering, computer science, and computational biology [1,3,4].
\par It is a well-known fact in finite field theory that a function $f:\mathbb{F}_{q}^{n}\longrightarrow \mathbb{F}_{q}$, where $\mathbb{F}_q$ is a finite field of $q$ elements, can be represented by a polynomial function. Thus any function $f:\mathbb{F}_{q}^{n}\longrightarrow \mathbb{F}_{q}^{n}$ can be represented by $f=(f_1,\ldots, f_n)$, where $f_i\in \mathbb{F}_q{[x_1,\ldots, x_n]}$. When $f$ is a linear system, the dynamics of $f$ can be described using its characteristic polynomial and minimal polynomial, and the computation can be done in polynomial time [1,6]. For general polynomial systems, there have been only limited successes in determining the dynamics of such systems, except for monomial dynamical systems, where all the coordinate functions $f_i$ are monomials.
\par
In [3], monomial dynamical systems over $\mathbb{Z}_2$, i.e. Boolean monomial systems, were studied. In [4], the problem of determining whether a monomial dynamical system over a finite field $\mathbb{F}_q$ is a fixed point system was reduced to the same question of an associated Boolean monomial system and a linear system over a ring of the form $\mathbb{Z}/(q-1)$. In [1], the study of fixed point systems was further developed. In particular, linear systems were defined for modules over a ring, and a necessary and sufficient condition for a linear system to be a fixed point system was derived using Fitting's lemma.
\par
Though the result in [1] does not lead to an efficient algorithm for determining whether a linear system over a general finite commutative ring is a fixed point system, the computational problem, which is ultimately needed in applications, was discussed in some detail in the special case where the ring is a finite field, and a computational method via the factorization of the characteristic polynomial and the minimal polynomial of the linear function was described. As pointed out in [4], the approach via characteristic polynomial and minimal polynomial for a linear dynamical system over a finite commutative ring faces considerable difficulties due to the lack of unique factorization (see also the comment after Example 4 in [1]). The following example illustrates this point.
\begin{example} Let $f:\mathbb{Z}_{8}^{2}\longrightarrow \mathbb{Z}_{8}^{2}$ be defined by the $2\times 2$ matrix
\[
A=\left(\begin{array}{cc} 2 & 6\\ 1 & 0\end{array}\right).
\]
Then $A^k\ne 0$ for $1\le k <6$ and $A^6=0$. Thus $f$ is a fixed point system with the only fixed point $0$. The characteristic polynomial of $A$ is $ch_{A}(\lambda)=\lambda^2+6\lambda+2$, which has no root in $\mathbb{Z}_8$, though $A$ clearly has an eigenvector $(0,4)^T$ corresponds to the eigenvalue $0$ (the eigenvalues and eigenvectors of a matrix over a commutative ring are defined as usual, see [2]). Note that
\[
\lambda^6=(\lambda^2+6\lambda+2)(\lambda^4+2\lambda^3+2\lambda^2+4)\;\mbox{$\pmod{8}$}.
\]
\end{example}
\par
In this paper, we consider a different approach. Our approach is based on the fact that there are efficient algorithms for the computation of the powers of a matrix: the multiplication of two $n\times n$ matrices takes at most $n^3$ operations (the state of the art algorithms use close to $n^2$ operations). If $A$ is an $n\times n$ matrix, then to compute $A^m$, where $m$ is a positive integer, it will take about $n^3\log_2m$ operations. Therefore, one can just work with the matrix of a linear dynamical system directly to avoid the difficulties of dealing with the factoring problems over an arbitrary commutative ring.
\par
In order for this approach to work, one must have a reasonable upper bound on the exponents of the powers of the matrix, that is, a reasonable upper bound on the number of iterations, that one must compute in order to determine the dynamics of a given system.
\par
Our first observation is, although Fitting's lemma tells us that a linear system will be stabilized after a certain number of iterations (see [1]), the lemma itself is a fairly general statement: it applies to any group $G$ that satisfies both ACC and DCC conditions on normal subgroups and any normal endomorphism $f$ of $G$ (see [7, p. 84]). While for the systems that we are interested in, the groups involved are finite abelian groups, and therefore, we should be able to derive more precise information on how many iterations it will need in order for a given system to reach a certain type of stabilization status.
\par
Our second observation is, the upper bound on the number of iterations also depends on the size of space. This can be seen from Example 1.1, where it takes $6$ iterations for the system to be stabilized. This can also be seen by just considering the simplest type of linear systems on $\mathbb{Z}_{q}$, where $q$ is a positive integer, namely the ones defined by a scalar multiplication. For such a system, the matrix size is 1, but the dynamics of the system depend on $q$. If the system is defined by the multiplication of an element $1<a<q$, then one either needs to know the prime factorizations of $a$ and $q$ or needs to compute the powers of $a$ to derive the dynamics of the system. Therefore, certain assumption on the size of $q$ must be made. Here we assume that the size of $q$ is comparable to the size of any integer that we maybe able to factor in the foreseeable future. We believe it is reasonable to make this assumption. Under this assumption, the numbers $\log_2q$ and $\log_2(\log_2q)$ are relatively small: the RSA keys are typically $1024-2048$ bits long and $\log_2(\log_22^{2048})=11$.
\par
This paper is organized as follows. In Section 2, we develop the basic theory that lays the foundation for an efficient algorithm. In section 3, we describe an algorithm for determining whether a linear dynamical system over a finite ring is a fixed point system or not and give two examples of linear fixed point systems over finite rings which are not fields. In Section 4, we conclude with some discussions and an example.
\par\medskip
\section{Main results}
\par
Let $R$ be a finite commutative ring with $q>1$ elements. Let the prime factorization of $q$ be
\begin{eqnarray*}
q=\prod_{i=1}^{t}{p_{i}^{t_i}}.
\end{eqnarray*}
\par
We shall view the elements of $R^n$, where $n$ is positive integer, as column vectors, and denote by $e_i$, $1\le i\le n$, the canonical basis (if $R$ has $1$). For a function $f$ from a set to the same set, we write
\begin{eqnarray*}
f^m=\underbrace{f\circ f\circ\cdots\circ f}_{\mbox{$m$ copies}}
\end{eqnarray*}
if $m$ is a positive integer. If $m$ is a positive number, not necessary an integer, then by writing $f^m$ we mean $f^{\lceil m\rceil}$, where $\lceil m\rceil$ is the smallest integer greater than or equal to $m$. Our first theorem upper bounds the number of iterations needed for a linear system to reach a certain stable status.
\par
\begin{theorem} Let $n$ be a positive integer, and let $f:R^n\longrightarrow R^n$ be a linear function. Then for any nonnegative integer $k$, we have
\begin{eqnarray*}
f^{n\log_{2}(q)+k}(R^n)=f^{n\log_{2}(q)}(R^n).
\end{eqnarray*}
If $R$ is a field, then the factor $\log_{2}(q)$ is not needed, that is
\begin{eqnarray*}
f^{n+k}(R^n)=f^{n}(R^n).
\end{eqnarray*}
\end{theorem}
\begin{proof} We first consider the general case when $R$ is a commutative ring. View $R^n$ as an $f$-module, set $M_0=R^n$, and consider a sequence of $f$-submodules of $M_0$ defined by
\begin{eqnarray}
M_0\supseteq M_1=f(M_0)\supseteq\cdots\supseteq M_r=f^r(M_0)\supseteq\cdots.
\end{eqnarray}
Since each $M_r$ ($r\ge 0$) is a finite abelian group and
\begin{eqnarray*}
|M_0|=q^n=\prod_{i=1}^{t}{p_{i}^{nt_i}},
\end{eqnarray*}
by Lagrange's theorem, we have
\begin{eqnarray*}
|M_r|=\prod_{i=1}^{t}{p_{i}^{r_i}},
\end{eqnarray*}
where $0\le r_i\le nt_i$. Thus, if $M_r\ne M_{r+1}$, then
\begin{eqnarray*}
|M_{r+1}|\le |M_r|/p_i
\end{eqnarray*}
for some $1\le i\le t$. Therefore either there is an
\begin{eqnarray}
r<\sum_{i=1}^{t}{nt_i}=n\sum_{i=1}^{t}{t_i}:=s,
\end{eqnarray}
such that $M_r=M_{r+1}$, or we must have $|M_s|=1$. In any case, $f(M_s)=M_s$. Since $s\le n\log_{2}(q)$, the first statement follows.
\par
If $R$ is a field, then the modules $M_i$ are vector spaces over $R$. So if $M_i\supsetneq M_{i+1}$, then $\dim M_{i+1}\le\dim M_{i}-1$. Since $\dim M_0=n$, the desired result follows.
\end{proof}
\par
Next, we give a general lemma about fixed point systems on a finite set. We remark that one can almost read out the proof of the lemma from the proof of Theorem 2 in [1]. Here we give a proof which sheds some light from a different view.
\begin{lemma} If $X$ is a finite set and $f:X\longrightarrow X$ is a function such that $f(X)=X$, then $f$ is a fixed point system if and only if $f$ is the identity function.
\end{lemma}
\begin{proof} Since $X$ is a finite set, $f(X)=X$ implies that $f$ is also injective. Thus $f$ is a permutation of the set $X$. Writing $f$ as a disjoint product of cycles, we see immediately that $f$ is a fixed point system if and only if all the cycles have length one, that is, $f$ is the identity function.
\end{proof}
\par
Recall that an element $u$ in a commutative ring $R$ with $1$ is called a unit if it is invertible. The following is an immediate consequence of Lemma 2.1.
\begin{corollary}
Let $R$ be a finite commutative ring with $1$. Let $A:R^n\longrightarrow R^n$, where $A$ is an $n\times n$ matrix over $R$, be a linear dynamical system. If $A\ne I$ and $\det A$ is a unit in $R$, then $A$ is not a fixed point system.
\end{corollary}
Now we give a criterion for a linear dynamical system over a finite ring to be a fixed point system.
\begin{theorem} Let $R$ be a finite commutative ring with $q$ elements, let $n$ be a positive integer, let $f:R^n\longrightarrow R^n$ be a linear system, and let $A$ be the matrix of $f$ with respect to the canonical basis (if $R$ has $1$). Then $f$ is a fixed point system if and only if $f^{n\log_{2}(q)+1}=f^{n\log_{2}(q)}$, or equivalently $A^{n\log_{2}(q)+1}=A^{n\log_{2}(q)}$. If $R$ is a field, then the condition simplifies to $f^{n+1}=f^{n}$ or $A^{n+1}=A^{n}$.
\end{theorem}
\begin{proof} By Theorem 2.1,
\begin{eqnarray*}
f(f^{n\log_{2}(q)}(R^n))=f^{n\log_{2}(q)}(R^n).
\end{eqnarray*}
Thus, by Lemma 2.1, $f$ is a fixed point system if and only if
\begin{eqnarray*}
f|_{f^{n\log_{2}(q)}(R^n)}=id|_{f^{n\log_{2}(q)}(R^n)},
\end{eqnarray*}
which is equivalent to
\begin{eqnarray*}
f(f^{n\log_{2}(q)}(x))=f^{n\log_{2}(q)}(x),\quad \forall x\in R^n.
\end{eqnarray*}
That is $f^{n\log_{2}(q)+1}=f^{n\log_{2}(q)}$.
\end{proof}
\par
Theorem 2.2 provides an efficient algorithm to determine whether a linear dynamical system over a finite commutative ring is a fixed point system, which will be discussed in the next section. The results in this section also reduce the study of a general linear dynamical system over a finite commutative ring to an invertible non-fixed point system.
\medskip
\section{Algorithms and Examples}
In this section, we first describe an algorithm based on Theorem 2.2 for determining
whether a linear system $A:\mathbb{Z}_{q}^{n}\longrightarrow\mathbb{Z}_{q}^{n}$ is a fixed point system or not, where $q>1$ is an integer and $A$ is taken to be the form of an $n\times n$ matrix. We choose $\mathbb{Z}_{q}$ as the base ring for the simplicity of the statements, the same algorithm works for any ring of the type
\begin{eqnarray*}
\mathbb{Z}_{q_1}\times \mathbb{Z}_{q_2}\times\cdots\times\mathbb{Z}_{q_k},
\end{eqnarray*}
as well as for any finite commutative ring with $1$ as long as the operations of the ring are implemented.
\par
The algorithm is called an {\sl LFPS (Linear Fixed Point System) test}.
\begin{algorithm} {\bf LFPS test}.\\
{\bf Input:} Two positive integers $n$ and $q>1$, an $n\times n$ matrix $A$ over $\mathbb{Z}_{q}$, and $b_{t-1}2^{t-1}+b_{t-2}2^{t-2}\cdots +b_12+b_0$, the binary representation of $\lceil n\log_2 q\rceil$.\\
{\bf Output:} {\bf true} or {\bf false}. \\
\begin{enumerate}
\item $X\gets I$
\item {\bf for} $i$ from $t-1$ down to $0$ {\bf do}\\
\mbox{}\hspace{5mm} $X\gets XX$\\
\mbox{}\hspace{5mm} {\bf if} $b_i = 1$ {\bf then}\\
\mbox{}\hspace{10mm} $X\gets AX$\\
\item {\bf if} $X=XA$ {\bf then}\\
\mbox{}\hspace{5mm} {\bf return true}\\
{\bf else}\\
\mbox{}\hspace{5mm} {\bf return false}\\
\end{enumerate}
\end{algorithm}
Let us explain this algorithm in more detail. In step (1) the
(matrix) variable $X$ is initialized by the identity matrix $I$. The
main computation of $A^{\lceil n\log_2 q\rceil}$ is performed in
step (2) using the ``square and multiply'' method.
Since $b_{t-1} = 1$ (the leading bit of $\lceil n\log_2
q\rceil$), at the beginning (i.e., $i=t-1$), $X$ first becomes
$II=I$, then becomes $X = AI=A$. After this, for each $i$ with
$t-2\ge i\ge 0$, the value in $X$ becomes the square of the value
previously stored in $X$. If $b_i=1$, then the value of $X$ is
further updated to be the product of $A$ and the previous
value. At the end of step (2), the value in $X$ is $A^{\lceil
n\log_2 q\rceil}$. For example, if $A$ is a $6\times 6$ matrix over
$\mathbb{Z}_{3\cdot 7}$, then $\lceil 6\log_2 21\rceil = 27$ and by the
``square and multiply'' method:
\[
A^{27} = A^{1\cdot 2^4+1\cdot 2^3+0\cdot 2^2+1\cdot 2+1} =
\bigg(\bigg(\big((A)^2A\big)^2 \bigg)^2A\bigg)^2A.
\]
In step (3), the result of Theorem 2.2 is applied. Since the value
of $X$ is now $A^{\lceil n\log_2 q\rceil}$, the system is a
fixed point system if $X=XA$, and the program returns {\bf true};
otherwise, the system is not a fixed point system and the program
returns {\bf false}.
\par
\medskip
Suppose two matrices over $\mathbb{Z}_{q}$ can be multiplied
with $O(n^{\omega})$ operations, by using Strassen's algorithm,
$\omega \le \log_27$. This number can be further reduced, see
[5]. The cost of running {\sl LFPS test} is
$O(n^{\omega}(\log_2n +\log_2\log_2 q))$. Under our assumption that
$\log_2\log_2 q$ is small, determine whether a linear system over a finite ring is a fixed point system or not can be done with $O(n^3)$ operations. If $R$ is a field, then the number of operations required is $O(n^{\omega}\log_2n)$.
As long as the problem of determining whether a linear system is a fixed point system is concerned, a comparison of the computational cost analysis given in [1] with the analysis given above shows, in addition to its simplicity, that our algorithm is at least as efficient as the approach via the characteristic polynomial and the minimal polynomial even for the case of finite fields.
Next we give two examples of fixed point linear systems over
finite rings. The first example is over the ring $\mathbb{Z}_{2^4}$.
\begin{example}
The system $A:\mathbb{Z}_{2^4}^4\longrightarrow \mathbb{Z}_{2^4}^4$ defined by
\[
A=\begin{pmatrix}
15 & 7 & 7 & 1\\
0 & 7 & 11 & 7\\
7 &7 &7 &11\\
14&8 &15 &6
\end{pmatrix},
\]
is a fixed point system. This can be verified by using Algorithm 3.1 to compute $A^{4\log_2 2^4}=A^{16}$ ($4$ iterations) and verify that
$A^{16}=A^{17}$. The ``stabilized'' matrix is
\[
A^{16} =\begin{pmatrix}
12 & 1 & 2 & 11\\
0 & 4 & 8 & 12\\
4 & 3 & 6 & 1\\
12& 1 & 2 & 11
\end{pmatrix}.
\]
\end{example}
The second example describes a fixed point system over the ring
$\mathbb{Z}_{3^2\cdot 5}$.
\begin{example}
The system $A:\mathbb{Z}_{3^2\cdot 5}^4\longrightarrow\mathbb{Z}_{3^2\cdot 5}^4$ defined by
\[
A=\begin{pmatrix}
36 &23 &32 &9\\
27 &32 & 30& 25\\
32& 25& 13 &28\\
32 &8& 41& 40
\end{pmatrix}.
\]
is a fixed point system. The ``stabilized'' matrix is
\[
A^{4\lceil\log_2(3^2\cdot 5)\rceil}=A^{24}=\begin{pmatrix}
0 &9& 9& 27\\
10 &27 &12 &26\\
35 &18 &33 &19\\
5 &27 &42 &31
\end{pmatrix}
\]
\end{example}
We remark that the number $r$ such that $A^r=A^{r+1}$ can be smaller than our theoretical bound $n\log_2q$ in some cases. In the second example above, $r=6$, i.e., we have $A^6=A^7$. Our algorithm can be refined so it terminates before the iteration process reaches the theoretical bound if $r$ is small enough, say $r<\sqrt{n\log_2q}$. But we believe that the gain is not significant by doing so.
\section{Conclusions}
We have provided an efficient algorithm to determine whether a linear dynamical system over a finite commutative ring is a fixed point system. As an application, our result together with the results in [3] and [4] should settle the problem of determining whether a monomial dynamical system over a finite field is a fixed point system.
\par
When the system is not a fixed point system, a natural problem is finding the cycles of the system. If $R$ is a field, then under the assumption that the elementary divisors of a linear system and their orders (the order of a polynomial $g$ is the least positive integer $k$ such that $g$ divides $x^k-1$) can be computed efficiently, the cycles can be computed by a theorem due to Elspas (see [6]). Obviously, the implementation via such approach is quite involved, in particular the computation of the orders of the elementary divisors. The orders of the elementary divisors are the lengths of the cycles. If the lengths of the cycles can be found, then the cycles can be obtained. For example, suppose that $f$ is linear dynamical system over a finite commutative ring $R$ with $1$, and suppose that the lengths of its cycles, say
\begin{eqnarray*}
0=k_0<k_1<\ldots<k_m,
\end{eqnarray*}
are known. Then the cycles can be computed by solving the linear systems:
\begin{eqnarray*}
(A^{k_i}-I)X=0,\quad 0\le i\le m.
\end{eqnarray*}
\begin{example} Consider the system $A:\mathbb{Z}_{105}^4\longrightarrow\mathbb{Z}_{105}^4$ defined by
\[
A=\begin{pmatrix}
70 &27 &5 &26\\
35 &98 & 104& 99\\
81& 85& 78 &102\\
27 &97& 13& 69
\end{pmatrix}.
\]
Since $\det A=2 \pmod{105}$ is a unit in $\mathbb{Z}_{105}$, Corollary 2.1 implies that $A$ is not a fixed point system. Since $A^{24}=I$ and $A^k\ne I$ for $0<k<24$, we see that the cycles lengths are the factors of $24$. With some computation, one can find the cycles lengths, they are $1, 2, 24$. The only cycle of length 1 is $0$, there are $5512$ cycles of length 2, and $5064150$ cycles of length $24$.
\end{example}
However, the search for the cycle lengths seems to be exponential.
\par\medskip
Computations of linear systems over finite commutative rings are basic, since one typically handles the other computational problems by reducing them to the ones about linear systems, and for systems over finite fields, the reduction can result in linear systems over commutative rings which are not necessary fields. Developing efficient algorithms over commutative rings deserves further attention (see also [1]).
\subsection*{Acknowledgment}
The first author gratefully acknowledges partial support from
the National 973 Project of China (No. 2007CB807900).
| {
"timestamp": "2008-10-17T16:35:02",
"yymm": "0810",
"arxiv_id": "0810.3164",
"language": "en",
"url": "https://arxiv.org/abs/0810.3164",
"abstract": "The problem of linking the structure of a finite linear dynamical system with its dynamics is well understood when the phase space is a vector space over a finite field. The cycle structure of such a system can be described by the elementary divisors of the linear function, and the problem of determining whether the system is a fixed point system can be answered by computing and factoring the system's characteristic polynomial and minimal polynomial. It has become clear recently that the study of finite linear dynamical systems must be extended to embrace finite rings. The difficulty of dealing with an arbitrary finite commutative ring is that it lacks of unique factorization. In this paper, an efficient algorithm is provided for analyzing the cycle structure of a linear dynamical system over a finite commutative ring. In particular, for a given commutative ring $R$ such that $|R|=q$, where $q$ is a positive integer, the algorithm determines whether a given linear system over $R^n$ is a fixed point system or not in time $O(n^3\\log(n\\log(q)))$.",
"subjects": "Dynamical Systems (math.DS); Commutative Algebra (math.AC)",
"title": "Linear Dynamical Systems over Finite Rings",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9854964186047326,
"lm_q2_score": 0.7185943985973772,
"lm_q1q2_score": 0.7081722062471368
} |
https://arxiv.org/abs/0904.4027 | More supplements to a class of logarithmically completely monotonic functions associated with the gamma function | In this article, a necessary and sufficient condition and a necessary condition are established for a function involving the gamma function to be logarithmically completely monotonic on $(0,\infty)$. As applications of the necessary and sufficient condition, some inequalities for bounding the psi and polygamma functions and the ratio of two gamma functions are derived. | \section{Introduction}
Recall \cite{Atanassov, minus-one-rgmia} that a positive function $f$ is said to be logarithmically completely monotonic on an interval $I$ if $f$ has derivatives of all orders on $I$ and
\begin{equation}
(-1)^n[\ln f(x)]^{(n)}\ge0.
\end{equation}
This kind of functions has very closer relationships with the Laplace transforms, Stieltjes transforms and infinitely divisible completely monotonic functions. For more detailed information, please refer to \cite{CBerg, grin-ismail, Guo-Qi-Srivastava2007.tex, Guo-Qi-Srivastava2007-02.tex, sandor-gamma-2-ITSF.tex, compmon2, auscm} and related references therein.
\par
It is well-known that the classical Euler gamma function is defined for $x>0$ by
\begin{equation}\label{egamma}
\Gamma(z)=\int^\infty_0t^{x-1} e^{-t}\td t.
\end{equation}
The logarithmic derivative of $\Gamma(z)$, denoted by $\psi(z)=\frac{\Gamma'(z)}{\Gamma(z)}$, is called the psi function, and $\psi^{(k)}$ for $k\in\mathbb{N}$ are called the polygamma functions.
\par
For $\alpha\in\mathbb{R}$ and $\beta\ge0$, define
\begin{equation}\label{ffabrx}
f_{\alpha,\beta,\pm1}(x)=\biggl[\frac{e^x\Gamma(x+\beta)}{x^{x+\beta-\alpha}}\biggr]^{\pm1},\quad
x\in(0,\infty).
\end{equation}
The investigation of the function $f_{\alpha,\beta,\pm1}(x)$ has a long history. In what follows, we would like to give a short survey in the literature.
\par
In \cite[Theorem~1]{keckic}, for showing
\begin{equation}\label{kec}
\frac{b^{b-1}}{a^{a-1}} e^{a-b}<\frac{\Gamma(b)}{\Gamma(a)}
<\frac{b^{b-1/2}}{a^{a-1/2}} e^{a-b}
\end{equation}
for $b>a>1$, monotonic properties of the functions $\ln f_{\alpha,0,+1}(x)$ and $\ln f_{\alpha,0,+1}(x)$ on $(1,\infty)$ were obtained.
\par
In \cite[Theorem~2.1]{Muldoon-78}, the function $f_{\alpha,0,+1}(x)$ for $\alpha\le1$ was proved to be logarithmically completely monotonic on $(0,\infty)$. In~\cite[Theorem~2.1]{Ismail-Lorch-Muldoon}, the functions $f_{\alpha,0,+1}(x)$ and $f_{\alpha,0,-1}(x)$ were proved to be logarithmically completely monotonic on $(0,\infty)$ if and only if $\alpha\le\frac12$ and $\alpha\ge1$ respectively. These results were mentioned in \cite[Theorem~2.1]{Ismail-Muldoon-119} later. However, we do not think the proof in \cite{Ismail-Lorch-Muldoon} for the necessity is convincible.
\par
In~\cite[Theorem~3.2]{Anderson}, it was recovered that the function $f_{1/2,0,+1}(x)$ is decreasing and logarithmically convex from $(0,\infty)$ onto $\big(\sqrt{2\pi}\,,\infty\big)$ and that the function $f_{1,0,+1}(x)$ is increasing and logarithmically concave from $(0,\infty)$ onto $(1,\infty)$.
\par
In \cite[p.~376, Theorem~2]{psi-alzer}, it was presented that the function $f_{\alpha,0,+1}(x)$ is decreasing on $(c,\infty)$ for $c\ge0$ if and only if $\alpha\le\frac12$ and increasing on $(c,\infty)$ if and only if
\begin{equation}
\alpha\ge
\begin{cases}
c[\ln c-\psi(c)]&\text{if $c>0$},\\1&\text{if $c=0$}.
\end{cases}
\end{equation}
\par
The necessary and sufficient conditions for the functions $f_{\alpha,0,+1}(x)$ and $f_{\alpha,0,-1}(x)$ to be logarithmically completely monotonic on $(0,\infty)$, stated in \cite[Theorem~2.1]{Ismail-Lorch-Muldoon} and \cite[Theorem~2.1]{Ismail-Muldoon-119} and mentioned above, were recovered in~\cite[Theorem~2]{chen-qi-log-jmaa}. Moreover, the function $f_{\alpha,\beta,+1}(x)$ was proved in~\cite[Theorem~1]{chen-qi-log-jmaa} to be logarithmically completely monotonic on $(0,\infty)$ if $2\alpha \leq 1 \leq \beta$. Using monotonic properties of the functions $f_{1/2,0,+1}(x)$ and $f_{1,0,-1}(x)$, the inequality~\eqref{kec} was extended in~\cite[Remark~1]{chen-qi-log-jmaa} from $b>a>1$ to $b>a>0$.
\par
In \cite{chen-ASCM-07}, after proving once again the logarithmically completely monotonic property of the functions $f_{1/2,0,+1}(x)$ and $f_{1,0,-1}(x)$, in virtue of Jensen's inequality for convex functions, the upper and lower bounds were established: For positive numbers $x$ and $y$, the inequality
\begin{equation}\label{chen-gurland-ineq}
\frac{x^{x-1/2}y^{y-1/2}}{[(x+y)/2]^{x+y-1}} \le \frac{\Gamma(x)\Gamma(y)}{[\Gamma((x+y)/2)]^2} \le\frac{x^{x-1}y^{y-1}}{[(x+y)/2]^{x+y-2}}
\end{equation}
holds, where the middle term in \eqref{chen-gurland-ineq} is called Gurland's ratio \cite{Merkle-Gurland}. The left-hand side inequality in \eqref{chen-gurland-ineq} is same as the corresponding one in \cite[Theorem~1]{Merkle-Gurland}, but their upper bounds do not include each other.
\par
Recently, some new conclusions on logarithmically completely monotonic properties of the function $f_{\alpha,\beta,+1}(x)$ were procured in \cite[Theorem~1]{Guo-Qi-Srivastava2007-02.tex}:
\begin{enumerate}
\item
If $\beta\in(0,\infty)$ and $\alpha\le0$, then $f_{\alpha,\beta,+1}(x)$ is logarithmically completely monotonic on $(0,\infty)$;
\item
If $\beta\in(0,\infty)$ and $f_{\alpha,\beta,+1}(x)$ is a logarithmically completely monotonic function on $(0,\infty)$, then $\alpha\le\min\bigl\{\beta,\frac12\bigr\}$;
\item
If $\beta \ge 1$, then $f_{\alpha,\beta,+1}(x)$ is logarithmically completely monotonic on $(0,\infty)$ if and only if $\alpha\le\frac12$.
\end{enumerate}
As direct consequences of these monotonic properties, it is deduced immediately that if $x$ and $y$ are positive numbers with $x\ne y$, then
\begin{enumerate}
\item
the inequality
\begin{equation}\label{guo-neces-suff-ineq}
\frac{\Gamma(x+\beta)}{\Gamma(y+\beta)} <\frac{x^{x+\beta-\alpha}}{y^{y+\beta-\alpha}}e^{1/(y-x)}
\end{equation}
for $\beta\ge1$ and $x>y>0$ holds true if and only if $\alpha\le\frac12$;
\item
the inequality \eqref{guo-neces-suff-ineq} for $\beta\in(0,\infty)$ holds true also if $\alpha\le0$.
\end{enumerate}
In this paper, for continuing the work in the paper~\cite{Guo-Qi-Srivastava2007-02.tex}, we consider logarithmically completely monotonic properties of the function $f_{\alpha,\beta,-1}(x)$ on $(0,\infty)$.
\par
The main results of this paper are as follows.
\begin{thm}\label{fth1}
If the function $f_{\alpha,\beta,-1}(x)$ is logarithmically completely monotonic on $(0,\infty)$, then either $\beta>0$ and $\alpha\ge \max\bigl\{\beta,\frac12\bigr\}$ or $\beta=0$ and $\alpha\ge 1$.
\end{thm}
\begin{thm}\label{fth3}
If $\beta\ge\frac12$, the necessary and sufficient condition for the function $f_{\alpha,\beta,-1}(x)$ to be logarithmically completely monotonic on $(0,\infty)$ is $\alpha\ge\beta$.
\end{thm}
As the first application of Theorem~\ref{fth3}, the following inequalities are derived by using logarithmically completely monotonic properties of the function $f_{\alpha,\beta,\pm1}(x)$ on $(0,\infty)$.
\begin{thm}\label{kevic-type-ineq}
Let $\beta$ be a positive number.
\begin{enumerate}
\item
For $k\in\mathbb{N}$, double inequalities
\begin{equation}\label{qi-psi-ineq-1}
\ln x-\frac1x\le\psi(x)\le\ln x-\frac1{2x}
\end{equation}
and
\begin{equation}\label{qi-psi-ineq}
\frac{(k-1)!}{x^k}+\frac{k!}{2x^{k+1}}\le(-1)^{k+1}\psi^{(k)}(x)
\le\frac{(k-1)!}{x^k}+\frac{k!}{x^{k+1}}
\end{equation}
hold in $(0,\infty)$.
\item
When $\beta>0$, inequalities
\begin{equation}\label{beta>0-1}
\psi(x+\beta)\le \ln x+\frac{\beta}x
\end{equation}
and
\begin{equation}\label{beta>0-2}
(-1)^{k}\psi^{(k-1)}(x+\beta)\ge \frac{(k-2)!}{x^{k-1}}-\frac{\beta(k-1)!}{x^{k}}
\end{equation}
hold on $(0,\infty)$ for $k\ge2$.
\item
When $\beta\ge\frac12$, inequalities
\begin{equation}\label{beta>1/2-dou-ineq}
\psi(x+\beta)\ge \ln x\quad \text{and}\quad (-1)^{k}\psi^{(k-1)}(x+\beta)\le \frac{(k-2)!}{x^{k-1}}
\end{equation}
hold on $(0,\infty)$ for $k\ge2$.
\item
When $\beta\ge1$, inequalities
\begin{equation}\label{beta>1/2-dou-ineq=1}
\psi(x+\beta)\le \ln x+\frac{\beta-1/2}x
\end{equation}
and
\begin{equation}\label{beta>1/2-dou-ineq=2}
(-1)^{k}\psi^{(k-1)}(x+\beta)\ge \frac{(k-2)!}{x^{k-1}}-\frac{(\beta-1/2)(k-1)!}{x^{k}}
\end{equation}
holds on $(0,\infty)$ for $k\ge2$.
\end{enumerate}
\end{thm}
As the second application of Theorem~\ref{fth3}, the following inequalities are derived by using logarithmically convex properties of the function $f_{\alpha,\beta,\pm1}(x)$ on $(0,\infty)$.
\begin{thm}\label{gurland-deriv-thm}
Let $n\in\mathbb{N}$, $x_k>0$ for $1\le k\le n$, $p_k\ge0$ satisfying $\sum_{k=1}^np_k=1$. If either $\beta>0$ and $\alpha\le0$ or $\beta\ge1$ and $\alpha\le\frac12$, then
\begin{equation}\label{n-gurland-ineq}
\frac{\prod_{k=1}^n[\Gamma(x_k+\beta)]^{p_k}}{\Gamma\bigl(\sum_{k=1}^np_kx_k+\beta\bigr)} \ge\frac{\prod_{k=1}^nx_k^{p_k(x_k+\beta-\alpha)}} {\bigl(\sum_{k=1}^np_kx_k\bigr)^{\sum_{k=1}^np_kx_k+\beta-\alpha}}.
\end{equation}
If $\alpha\ge\beta\ge\frac12$, then the inequality \eqref{n-gurland-ineq} reverses.
\end{thm}
As the final application of Theorem~\ref{fth3}, the following inequality may be derived by using the decreasingly monotonic property of the function $f_{\alpha,\beta,-1}(x)$ on $(0,\infty)$.
\begin{thm}\label{gurland-deriv-thm-mon}
If $\alpha\ge\beta\ge\frac12$, then
\begin{equation}\label{guo-neces-suff-ineq}
I(x,y)<\biggl[\biggl(\frac{x}y\biggr)^{\alpha-\beta} \frac{\Gamma(x+\beta)}{\Gamma(y+\beta)}\biggr]^{1/(x-y)}
\end{equation}
holds true for $x,y\in(0,\infty)$ with $x\ne y$, where
\begin{equation}
I(a,b)=\frac 1e\biggl(\frac{b^b}{a^a}\biggr)^{1/(b-a)}
\end{equation}
for $a>0$ and $b>0$ with $a\ne b$ is the identric or exponential mean.
\end{thm}
\begin{rem}
For $\beta\in\mathbb{R}$, let
\begin{equation}\label{guo-funct-g}
h_{\beta,\pm1}(x)=\biggl[\frac{e^x\Gamma(x+1)}{(x+\beta)^{x+\beta}}\biggr]^{\pm1}
\end{equation}
on $(\max\{0,-\beta\},\infty)$. In \cite{s-guo-ijpam, Guo-Qi-Srivastava2007.tex}, it was showed that the functions $h_{\beta,+1}(x)$ and $h_{\beta,-1}(x)$ are logarithmically completely monotonic if and only if $\beta\ge1$ and $\beta\le\frac12$ respectively. As consequences of monotonic results of the function $h_{\beta,\pm1}(x)$, the following two-sided inequality was derived in \cite{Guo-Qi-Srivastava2007.tex}:
\begin{equation}\label{guosl-ineq}
\frac{(x+1)^{x+1}}{(y+1)^{y+1}}\;e^{y-x}<\frac{\Gamma(x+1)}{\Gamma(y+1)} <\frac{(x+1/2)^{x+1/2}}{(y+1/2)^{y+1/2}}\;e^{y-x}
\end{equation}
for $y>x>0$, where the constants $1$ and $\frac12$ in the very left and the very right sides of the two-sided inequality \eqref{guosl-ineq} cannot be replaced, respectively, by smaller and larger numbers.
\end{rem}
\begin{rem}
In \cite{note-on-li-chen.tex}, it was showed that the function
\begin{equation}
h(x)=\frac{e^x\Gamma(x)}{x^{x[1-\ln x+\psi(x)]}}
\end{equation}
on $(0,\infty)$ has a unique maximum $e$ at $x=1$, with the following two limits
\begin{equation}\label{2limits}
\begin{aligned}
\lim_{x\to0^+}h(x)&=1 & \text{and} &&\lim_{x\to\infty}h(x)&=\sqrt{2\pi}\,.
\end{aligned}
\end{equation}
As consequences of the monotonicity of $h(x)$, it was concluded in \cite{note-on-li-chen.tex} that the following inequality
\begin{equation}\label{note-li-chen-ineq}
\frac{x^{x[\ln x-\psi(x)-1]}}{y^{y[\ln y-\psi(y)-1]}} e^{y-x} <\frac{\Gamma(y)}{\Gamma(x)}
\end{equation}
holds true for $y>x\ge1$. If $0<x<y\le1$, the inequality \eqref{note-li-chen-ineq} is reversed.
\end{rem}
\begin{rem}
It is worthwhile to point out that \cite[Thorem~1.3]{sandor-gamma-2-ITSF.tex} is equivalent to necessary and sufficient conditions in \cite[Theorem~2]{chen-qi-log-jmaa} and \cite[Theorem~2.1]{Ismail-Lorch-Muldoon} for the functions $f_{1/2,0,+1}(x)$ and $f_{1,0,-1}(x)$ to be logarithmically completely monotonic on $(0,\infty)$. See also \cite{Guo-Qi-Srivastava2007.tex, Guo-Qi-Srivastava2007-02.tex} and related references therein.
\end{rem}
\section{Proofs of main results}
Now we are in a position to prove our theorems.
\begin{proof}[Proof of Theorem \ref{fth1}]
Suppose that $f_{\alpha,\beta,-1}(x)$ is logarithmically completely monotonic on $(0,\infty)$. Then
\begin{equation}\label{f500}
[\ln f_{\alpha,\beta,-1}(x)(x)]'=\ln x-\psi(x+\beta)+\frac{\beta-\alpha}x \le0,
\end{equation}
from which we have
\begin{equation}\label{f505}
\beta-\alpha \le x[\psi(x+\beta)-\ln x], \quad x\in(0,\infty).
\end{equation}
If $\beta>0$, then
\begin{equation*}\label{f510}
\beta-\alpha \le \lim_{x\to 0^+}[x\psi(x+\beta)-x\ln x]=0.
\end{equation*}
That is
\begin{equation}\label{f515}
\alpha \ge \beta.
\end{equation}
\par
Using the asymptotic formula
\begin{equation}\label{flm22}
\psi(x)=\ln x-\frac1{2x}+O\bigg(\frac1{x^2}\bigg), \quad x\to \infty,
\end{equation}
see \cite[p.~47]{er}, in \eqref{f505} for $\beta>0$, we obtain
\begin{align*}
\beta-\alpha&\le\lim_{x\to\infty}x\biggl[\ln(x+\beta)-\frac{1}{2(x+\beta)} +O\biggl(\frac{1}{x^2}\biggr)-\ln x\biggr]\\
\quad &=\lim_{x\to \infty}\biggl[x\ln\biggl(1+\frac{\beta}x\biggr)\biggr]-\frac{1}{2}\\
\quad &=\beta\lim_{x\to \infty}\biggl[\frac{x}{\beta} \ln\biggl(1+\frac{\beta}x\biggr)\biggr]-\frac{1}{2}\\
\quad &=\beta -\frac{1}{2},
\end{align*}
from which we get
\begin{equation}\label{f530}
\alpha \ge \frac{1}{2}.
\end{equation}
Combining \eqref{f515} and \eqref{f530} yields
\begin{equation}\label{f550}
\alpha \ge \max\biggl\{\beta,\frac12\biggr\} \quad \text{if} \quad \beta>0.
\end{equation}
\par
If $\beta=0$, considering $f_{\alpha,0,-1}(x)=f_{\alpha,1,-1}(x)$ and \eqref{f550} yields $\alpha\ge\max\bigl\{1,\frac12\bigr\}=1$. The proof is complete.
\end{proof}
\begin{proof}[Proof of Theorem \ref{fth3}]
By Theorem \ref{fth1}, the necessary condition is obtained readily.
\par
Differentiating \eqref{f500} and making use of
\begin{equation*}
\psi ^{(n)}(x)=(-1)^{n+1}\int_{0}^{\infty}\frac{t^{n}}
{1-e^{-t}}e^{-xt}\td t,\quad x\in (0,\infty)
\end{equation*}
and
\begin{equation*}
\frac1{x^n}=\frac1{\Gamma(n)}\int_0^\infty t^{n-1}e^{-xt}\td t,\quad x\in (0,\infty),
\end{equation*}
see \cite[p.~884]{grads}, gives
\begin{gather}
(-1)^{n}[\ln f_{\alpha,\beta,-1}(x)]^{(n)} =\frac{(n-2)!}{x^{n-1}}-(-1)^{n}\psi^{(n-1)}(x+\beta)-\frac{(\beta-\alpha)(n-1)!}{x^{n}}\notag\\
\begin{aligned}\label{f595}
&=\int_{0}^{\infty}t^{n-2}e^{-xt}\td t-\int_{0}^{\infty}\frac{t^{n-1}}{1-e^{-t}}e^{-(x+\beta)t}\td t
-(\beta-\alpha)\int_{0}^{\infty}t^{n-1}e^{-xt}\td t\\
&=\int_{0}^{\infty} \bigg[\alpha-\beta-\frac1t\bigg(e^{(1-\beta)t}\frac{t}{e^t-1}-1\bigg)\bigg] {t^{n-1}e^{-xt}}\td t
\end{aligned}
\end{gather}
for $n\ge2$. The inequality
\begin{equation}\label{f800}
\frac{t}{e^t-1}< \frac{1}{e^{t/2}}, \quad t\in(0,\infty)
\end{equation}
was ever used in \cite{mathieu-rostock, mathieu-rostock-rgmia, mathieu-ijmms}. Substituting it into \eqref{f595} leads to
\begin{equation*}\label{f801}
(-1)^{n}[\ln f_{\alpha,\beta,-1}(x)]^{(n)}\ge \int_{0}^{\infty}
\bigg[\alpha-\beta-\frac{e^{(1/2-\beta)
t}-1}{t}\bigg]t^{n-1}e^{-xt}\td t,\quad n\ge2.
\end{equation*}
Since the function
\begin{equation}
\frac{e^{(1/2-\beta)t}-1}t,\quad t\in(0,\infty)
\end{equation}
is increasing, if $\alpha \ge \beta \ge \frac12$, then
\begin{equation}\label{f806}
(-1)^{n}[\ln f_{\alpha,\beta,-1}(x)]^{(n)}\ge (\alpha-\beta)
\int_{0}^{\infty} t^{n-1}e^{-xt}\td t \ge 0,\quad n\ge2.
\end{equation}
\par
By using \eqref{flm22}, it follows that
\begin{equation*}
[\ln f_{\alpha,\beta,-1}(x)(x)]'=\ln\biggl(1+\frac{\beta}{x}\biggr) -\frac{1}{2(x+\beta)}+\frac{\alpha-\beta}{x}+O\biggl(\frac{1}{x^2}\biggr)
\end{equation*}
as $x\to\infty$, thus for all $\alpha$ and $\beta$,
\begin{equation}\label{f718}
\lim_{x\to\infty}[\ln f_{\alpha,\beta,-1}(x)(x)]'=0.
\end{equation}
From \eqref{f718} and \eqref{f806}, we have
\begin{equation*}
[\ln f_{\alpha,\beta,-1}(x)]' \le 0 \quad \text{if $\alpha\ge\beta\ge\frac12$}.
\end{equation*}
Thus $(-1)^{n}[\ln f_{\alpha,\beta,-1}(x)]^{(n)}\ge 0$ are valid for all $n\in\mathbb{N}$. The proof is complete.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{kevic-type-ineq}]
If $f_{\alpha,\beta,-1}(x)$ is logarithmically completely monotonic on $(0,\infty)$, then
$$
(-1)^k[\ln f_{\alpha,\beta,-1}(x)]^{(k)}\ge0
$$
on $(0,\infty)$ for $k\in\mathbb{N}$, which is equivalent to
\begin{equation}\label{reason-ineq-1}
\psi(x+\beta)\ge \ln x+\frac{\beta-\alpha}x
\end{equation}
and, for $k\ge2$,
\begin{equation}\label{reason-ineq-2}
(-1)^{k}\psi^{(k-1)}(x+\beta)\le \frac{(k-2)!}{x^{k-1}}-\frac{(\beta-\alpha)(k-1)!}{x^{k}}.
\end{equation}
Hence, Theorem~\ref{fth3} implies inequalities in \eqref{beta>1/2-dou-ineq}.
\par
If $\beta \ge 1$, Theorem~1 in \cite{Guo-Qi-Srivastava2007-02.tex} said that the function $f_{\alpha,\beta,+1}(x)$ is logarithmically completely monotonic on $(0,\infty)$ if and only if $\alpha\le\frac12$, this means that inequalities in \eqref{reason-ineq-1} and \eqref{reason-ineq-2} are reversed, and so inequalities in \eqref{beta>1/2-dou-ineq=1} and \eqref{beta>1/2-dou-ineq=2} are valid.
\par
If $\beta>0$ and $\alpha\le0$, Theorem~1 in \cite{Guo-Qi-Srivastava2007-02.tex} also said that the function $f_{\alpha,\beta,+1}(x)$ is logarithmically completely monotonic on $(0,\infty)$, this means that inequalities in \eqref{reason-ineq-1} and \eqref{reason-ineq-2} are also reversed, and so inequalities \eqref{beta>0-1} and \eqref{beta>0-2} are valid.
\par
When $\beta=0$, several mathematicians have proved that the functions $f_{\alpha,0,+1}(x)$ and $f_{\alpha,0,-1}(x)$ are logarithmically completely monotonic on $(0,\infty)$ if and only if $\alpha\le\frac12$ and $\alpha\ge1$ respectively, which implies by reasoning as above the double inequalities \eqref{qi-psi-ineq-1} and \eqref{qi-psi-ineq}. The proof of Theorem~\ref{kevic-type-ineq} is complete.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{gurland-deriv-thm}]
The first conclusion in \cite[Theorem~1]{Guo-Qi-Srivastava2007-02.tex} implies that the function $f_{\alpha,\beta,+1}(x)$ is logarithmically convex for $\beta>0$ and $\alpha\le0$ on $(0,\infty)$. Combining this with Jensen's inequality for convex functions yields
\begin{equation}\label{jensen-discreate}
\ln\frac{\exp\bigl(\sum_{k=1}^np_kx_k\bigr)\Gamma\bigl(\sum_{k=1}^np_kx_k+\beta\bigr)} {\bigl(\sum_{k=1}^np_kx_k\bigr)^{\sum_{k=1}^np_kx_k+\beta-\alpha}} \le\sum_{k=1}^np_k\ln\frac{\exp(x_k)\Gamma(x_k+\beta)}{x_k^{x_k+\beta-\alpha}},
\end{equation}
where $n\in\mathbb{N}$, $x_k>0$ for $1\le k\le n$, $p_k\ge0$ satisfying $\sum_{k=1}^np_k=1$, $\beta>0$ and $\alpha\le0$. Rearranging it leads to the inequality \eqref{n-gurland-ineq}.
\par
The third conclusion in \cite[Theorem~1]{Guo-Qi-Srivastava2007-02.tex} implies that the function $f_{\alpha,\beta,+1}(x)$ is also logarithmically convex for $\beta\ge1$ and $\alpha\le\frac12$ on $(0,\infty)$, hence, the inequality~\eqref{jensen-discreate} is also valid for $\beta\ge1$ and $\alpha\le\frac12$.
\par
Theorem~\ref{fth3} above implies that the function $f_{\alpha,\beta,+1}(x)$ is logarithmically concave for $\alpha\ge\beta\ge\frac12$ on $(0,\infty)$, therefore, the inequality~\eqref{jensen-discreate} is reversed. Theorem~\ref{gurland-deriv-thm} is proved.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{gurland-deriv-thm-mon}]
Theorem~\ref{fth3} implies that the function $f_{\alpha,\beta,-1}(x)$ is decreasing on $(0,\infty)$ if $\alpha\ge\beta\ge\frac12$, this is,
\begin{equation*}
\frac{e^y\Gamma(y+\beta)}{y^{y+\beta-\alpha}}>\frac{e^x\Gamma(x+\beta)}{x^{x+\beta-\alpha}}
\end{equation*}
for $y>x>0$, which can be rearranged as
\begin{gather*}
\frac{\Gamma(y+\beta)}{\Gamma(x+\beta)}>e^{x-y}\frac{y^{y+\beta-\alpha}}{x^{x+\beta-\alpha}},\\
\biggl[\biggl(\frac{y}{x}\biggr)^{\alpha-\beta}\frac{\Gamma(y+\beta)}{\Gamma(x+\beta)}\biggr]^{1/(y-x)} >\frac1e\biggl(\frac{y^{y}}{x^{x}}\biggr)^{1/(y-x)}.
\end{gather*}
The proof of Theorem~\ref{gurland-deriv-thm-mon} is complete.
\end{proof}
| {
"timestamp": "2009-04-26T16:04:11",
"yymm": "0904",
"arxiv_id": "0904.4027",
"language": "en",
"url": "https://arxiv.org/abs/0904.4027",
"abstract": "In this article, a necessary and sufficient condition and a necessary condition are established for a function involving the gamma function to be logarithmically completely monotonic on $(0,\\infty)$. As applications of the necessary and sufficient condition, some inequalities for bounding the psi and polygamma functions and the ratio of two gamma functions are derived.",
"subjects": "Classical Analysis and ODEs (math.CA)",
"title": "More supplements to a class of logarithmically completely monotonic functions associated with the gamma function",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9854964173268185,
"lm_q2_score": 0.7185943985973772,
"lm_q1q2_score": 0.708172205328835
} |
https://arxiv.org/abs/2107.13282 | Dense Graph Partitioning on sparse and dense graphs | We consider the problem of partitioning a graph into a non-fixed number of non-overlapping subgraphs of maximum density. The density of a partition is the sum of the densities of the subgraphs, where the density of a subgraph is its average degree, that is, the ratio of its number of edges and its number of vertices. This problem, called Dense Graph Partition, is known to be NP-hard on general graphs and polynomial-time solvable on trees, and polynomial-time 2-approximable. In this paper we study the restriction of Dense Graph Partition to particular sparse and dense graph classes. In particular, we prove that it is NP-hard on dense bipartite graphs as well as on cubic graphs. On dense graphs on $n$ vertices, it is polynomial-time solvable on graphs with minimum degree $n-3$ and NP-hard on $(n-4)$-regular graphs. We prove that it is polynomial-time $4/3$-approximable on cubic graphs and admits an efficient polynomial-time approximation scheme on graphs of minimum degree $n-t$ for any constant $t\geq 4$. | \section{Introduction}
The research around communities in social networks can be seen as a contribution to the well establish
research of clustering and graph partitioning. Graph partitioning problems have
been intensively studied with various measures in order to evaluate clustering quality, see e.g.~\cite{New2004,Sch2007,For2010,BulucMSS016} for an overview. In the context of social networks, a ‘community’ is a collection of individuals who are relatively well connected compared to other parts of the social network graph and a ‘community structure’ corresponds to a partition of the whole social network into communities.
We consider a classical definition of the density of a subgraph induced by a subset~$S$ of vertices (see, for example, \cite{bib:density:darlay2012DENSE,bib:density:goldberg1984finding}) given by the ratio between the number of edges and the number of vertices in $S$. The density of a partition is the sum of the densities of all its parts.
For this definition of density, there are several papers on finding the densest subgraph. This problem was shown solvable in polynomial time by Goldberg~\cite{bib:density:goldberg1984finding} but if the size of the subgraph is a part on the input, the problem called \textsc{$k$-Densest Subgraph} becomes NP-hard even restricted to bipartite or chordal graphs~\cite{bib:density:corneil1984clustering}. The approximability of \textsc{$k$-Densest Subgraph} was also studied, see~\cite{Khot06,FeigePK01,BhaskaraCCFV10}.
In this paper, we study the problem \MDGP{} of finding a partition $\mathcal{P} = \{ V_1, \dots, V_k\}$, $k \geq 1$, of a given undirected graph $G$, such that the density of the partition, denoted by $d(\mathcal{P})$, is maximised.
Note that the general concept of a community structure does not put any restriction on the
number of communities. We therefore address the problem \MDGP{} of finding a partition of maximum density, without fixing the number of classes of
the partition.
Indeed, when the number of classes is given, the problem is a generalization of a partition into $k$ cliques.
Darley et al.~\cite{bib:density:darlay2012DENSE} studied \MDGP, and its complement {\sc Min Sparse Graph Partition}. They defined the sparsity of a partition $\mathcal{P}$ as $F(\mathcal{P})= \frac{|\mathcal{P}|}{2} + d(\mathcal{P})$ and the problem {\sc Min Sparse Graph Partition} as finding a partition of a given undirected graph $G$ such that the sparsity of the partition is minimised. Observe that these two problems \MDGP{} and {\sc Min Sparse Graph Partition} are duals in the sense that solving the first one on a graph $G$ is the same as solving the second one on the complement of $G$. In~\cite{bib:density:darlay2012DENSE} it is shown that both problems are NP-complete, and that there is no constant factor approximation for {\sc Min Sparse Graph Partition} unless $P=NP$. Moreover, a polynomial time algorithm for \MDGP{} on trees is given.
We point out that their proof of NP-completeness is a polynomial time reduction from \textsc{$k$-Coloring}. By construction, the same reduction when starting from \textsc{3-Coloring} on graphs of degree at most 4 (proved NP-complete in~\cite{GJS1976}) yields as instance of \MDGP{} a graph on $n$ vertices and of minimum degree greater than~$n-4n^{4/5}$. Thus it follows that \MDGP{} is NP-complete restricted to graphs of minimum degree~$n-4n^{4/5}$.
Aziz et al.~\cite{bib:density:aziz2015welfare} studied the problem \textsc{Fractional Hedonic Game}, and more particularly the \textsc{Max Utilitarian Welfare} problem as the simple symmetric version of the game defined as follows. Let $N$ be a set of agents, the utility of $i \in N$ in a coalition $S \subseteq N$ is $u_i(S) = \tfrac 1{|S|}{\sum_{j \in S}u_i(j)}$ where $u_i(j)$ is such that $u_i(j) \in \{0,1\}$ for a simple game and $u_i(j) = u_j(i)$ for a symmetric one. For \textsc{Max Utilitarian Welfare} one tries to find a partition~$C$ of~$N$ into coalitions that maximizes $\sum_{S \in C}\sum_{i \in S}u_i(S)$. This game can be seen as a graph~$G$ where agents are vertices and there is an edge between two agents $i$ and $j$ if and only if $u_i(j)=1$.
In this context, $u_i(S) = \tfrac 1{|S|}{\sum_{j \in S}u_i(j)} = \tfrac 1{|S|}deg_{G[S]}(u_i)$. We deduce that $\sum_{S \in C}\sum_{i \in S}u_i(S) =\tfrac 1{|S|} \sum_{S \in C}\sum_{i \in S} deg_{G[S]}(u_i) = \tfrac 1{|S|}\sum_{S \in C} {2|E(S)|} = 2 \cdot d(C)$. Hence, the problems \textsc{Max Utilitarian Welfare} and \MDGP{} are equivalent to within a constant, which means that the 2-approximation for the former given in~\cite{bib:density:aziz2015welfare} directly translates to the latter.
\medskip
\noindent
\emph{Our contributions.} The following overview summarises the results achieved in this paper concerning \MDGP{} (MDGP).
\begin{itemize}
\item MDGP is trivially solvable on graphs of maximum degree 2, we prove its NP-hardness for 3-regular (cubic) graphs.
\item We establish that on bipartite complete graphs an optimal partition consists of one part, that is the whole graph. Moreover if the size of the two independent sets are relatively prime numbers then this optimal solution is unique. We use this result to show that MDGP is $W[2]$-hard with respect to (an upper bound on) the number of clusters in an optimal solution on dense bipartite graphs. Our reduction is polynomial and hence in particular implies the NP-hardness of MDGP on dense bipartite graphs.
\item MDGP is trivial on complete graphs since the optimal solution is the whole graph as one part of the partition. Moreover, as we previously explained, it is NP-hard on graphs of minimum degree $n-4n^{4/5}$. We show that for graphs of minimum degree $\geq n-3$, the problem is solvable in polynomial time and any optimal solution has two parts. Moreover on $(n-4)$-regular graphs, the problem becomes NP-hard.
\item We show that MDGP admits an approximation with ratio $4/3$ on cubic graphs, and an eptas (i.e.~a $(1+\varepsilon)$-approxiation for any $\varepsilon >0$) on $(n-4)$-regular graphs, improving on the 2-approximation on general graphs~\cite{bib:density:aziz2015welfare}
\end{itemize}
Our paper is organized as follows. Notations and formal definitions are given in
Section~\ref{sec2}. The study of (dense) bipartite graphs is established in Section~\ref{sec3}. Section~\ref{sec4} presents the results on cubic graphs. In Section~\ref{sec5} we study
dense graphs.
Some conclusions are given at the end of the paper.
\section{Preliminaries}\label{sec2}
In this paper we assume that all graphs are undirected, without loops or multiple edges, and not necessary connected. We use $G=(V,E)$ to denote an undirected graph with a set~$V$ of vertices and a set $E$ of edges. We use $|V|$ to denote the number of vertices in $G$, i.e., the order of $G$, and we use $|E|$ to denote the number of edges in $G$, i.e., the size of $G$. We denote by $deg_G(v)$ the degree of $v\in V$ in $G$ that is the number of edges incident to $v$.
The maximum degree of $G$, denoted by $\Delta(G)$, is the degree of the vertex with the greatest number of edges incident to it. The minimum degree of $G$, denoted by $\delta(G)$, is the degree of the vertex with the least number of edges incident to it. For any vertex $v\in V$, $N_G(v)$ is the set of neighbors of~$v$ in $G$ and $N_G[v]= N_G(v)\cup\{v\}$. Moreover, $N_G(S)=\bigcup_{v \in S} N_G(v)$. For a graph $G=(V,E)$ and a subset $S\subseteq V$ we denote by $E(S)$ the set of the edges of $G$ with both endpoints in~$S$. For a given partition $\{A,B\}$ of $V$, we denote by $E(A,B) = \{uv \in E:~u \in A,~ v \in B\}$. Further, $G[S]$ denotes the graph induced by $S$, defined as $G[S]=(S, E(S))$.
A triangle graph is the cycle graph $C_{3}$ or the complete graph $K_{3}$.
A diamond graph has 4 vertices and 5 edges, it consists of a complete graph $K_{4}$ minus one edge. A graph is called cubic if all its vertices are of degree three. A graph is bipartite if its vertices can be partitioned into two sets $A$ and $B$ such that every edge connects a vertex in $A$ to one in $B$. A complete bipartite graph is a special kind of bipartite graph where every vertex of $A$ is connected to every vertex in $B$. A graph on $n$ vertices is $\delta$-dense if its minimum degree is at least $\delta n$. A set of instances is called dense if there is a constant $\delta >0$ such that all instances in this set are $\delta$-dense (this notion was introduced in \cite{AroraKK95} and called everywhere-dense).
The density $d(G)$ of a graph $G=(V,E)$ is the ratio between the number of edges and the number of vertices in $G$, that is, $d(G)=\tfrac{|E|}{|V|}$. Moreover, for $S\subseteq V$, $d(S)=d(G[S])=\tfrac{|E(S)|}{|S|}$. We use $\mathcal{P}$ to denote a partition of the set $V$
of vertices of $G$, that is, $\mathcal{P} = \{ V_1, \dots, V_k\}$, where $\cup_{i=1}^k V_i = V$, and $V_i \cap V_j = \emptyset$ for each $i, j \in \{1, \dots, k\}$. Then the density of the partition $\mathcal{P}$ of $G$ is defined as $d(\mathcal{P})=\sum_{i=1}^k d(G[V_i])$, where $G[V_i]$ is the subgraph of $G$ induced by the subset $V_i$ of vertices, that is, $G[V_i]= (V_i, E_i)$, $E_i = \{\{u,v\} : \{u,v\} \in E \land u,v\in V_i\}$.
We study the problem of finding a partition $\mathcal{P} = \{ V_1, \dots, V_k\}$ of a given graph~$G$, such that $k \geq 1$
and that, among all such partitions, $d(\mathcal{P})$ is maximised. We refer to this problem as {\sc Max Dense Graph Partition} and we define its decision version as follows.
\begin{center}
\fbox{\begin{minipage}{.95\textwidth}
\noindent{\sc Dense Graph Partition}\\\nopagebreak
{\bf Input:} An undirected graph $G=(V,E)$, a positive rational number $r$. \\\nopagebreak
{\bf Question:} Is there a partition $\mathcal{P}$ such that $d(\mathcal{P}) \geq r$ ?
\end{minipage}}
\end{center}
We use some concepts from parameterized complexity and refer to~\cite{CFK+15,DowFel2013} for details of this terminology. A parameterized problem is a decision problem given together with a parameter, that is, an integer $k$ depending on the instance.
Such a parameterized problem is fixed-parameter tractable (fpt for short) if it can be solved in time $f(k)\cdot|I|^c$ for an instance $I$ of size $|I|$ with parameter $k$, where $f$ is a computable function and $c$ is a constant. If a parameterized problem is hard for the complexity class W[2], it is unlikely (under certain complexity theoretic assumptions) to be fixed-parameter tractable. An fpt-reduction between two parameterized problems~$P$ and $Q$ maps instances $(I,k)$ of $P$ to instances $(I',k')$ of $Q$ in time $f(k)|I|^{O(1)}$ for some computable functions $f$ and $g$, such that $k'\leq g(k)$ and $(I,k)$ is a yes-instance of~$P$ if and only if $(I',k')$
is a yes-instance of~$Q$. If problem $P$ is W[2]-hard, then such an fpt-reduction shows that $Q$ is W[2]-hard as well.
Given an optimization problem in NPO and an instance $I$ of this problem, we denote by $|I|$ the size of $I$, by $opt(I)$
the optimum value of $I$, and by $val(I, S)$ the value of a feasible solution $S$ of instance $I$. The performance ratio of $S$ (or
approximation factor) is $r(I, S) = \max \{\frac{val(I,S)}{opt(I)},\frac{opt(I)}{val(I,S)}\}\geq 1$.
For a function $f$, an algorithm is an $f(|I|)$-approximation, if for every instance $I$ of the problem, it returns a solution $S$
such that $r(I, S) \leq f (|I|)$. Moreover if the algorithm runs in polynomial time in $|I|$, then this algorithm gives a polynomial time $f(|I|)$-approximation.
We consider in this paper only polynomial time algorithms. When $f$ is a constant $\alpha$, the problem is polynomial-time $\alpha$-approximable. When $f=1+\varepsilon$, for any $\varepsilon >0$, the problem admits a polynomial-time approximation scheme. When the running time of an approximation scheme is of the form $O(g(1/\varepsilon)poly(|I|)$ the problem has an efficient polynomial-time approximation scheme (eptas).
Before we start studying specific graph classes, we observe the following helpful structural properties that hold for {\sc Dense Graph Partition} on general graphs.
\begin{remark*}\label{remarkconnexe}
We can assume that for any optimal partition $\mathcal{P}$ and for any part $P_i \in \mathcal{P}$, $G[P_i]$ is connected, since otherwise turning each connected component into its own part does not decrease the density.
\end{remark*}
\begin{lemma}\label{lemmacomplete}
Among all partitions of $G$ into $t\geq 2$ parts, those where the parts correspond to complete graphs, if there exists such, have the largest density.
\end{lemma}
\begin{proof}
Consider a partition of $G$ into $t$ parts $\{V_1,\ldots,V_t\}$ of size $n_1,\ldots,n_t$. If $G[V_i]$ has $o_i$ missing edges for any $1\leq i\leq t$, then the density of this partition is $\frac{n-t}{2}- \frac{o_1}{n_1}-\ldots -\frac{o_t}{n_t}$.
Consider a partition of $G$ into $t$ parts of size $n_1',\ldots,n_t'$ such that each part induces a complete graph for any $1\leq i\leq t$. Then the density of this partition is $\frac{n-t}{2}$ and thus it is larger than the density of any partition in $t$ parts where at least one edge is missing inside $G[V_i]$ for some $1\leq i\leq t$.
\end{proof}
A direct consequence of this is the following.
\begin{lemma}
\label{lem:densMax}
Let $G = (V,E)$ be a graph and $\mathcal{P}$ be any partition of $V$. Then $d(\mathcal{P}) \leq \frac{|V|}{2} - \frac{|\mathcal{P}|}{2}$.
\end{lemma}
\section{Dense Bipartite Graphs}\label{sec3}
\label{sec:bipartiDense}
In this section we show that \MDGP{} has a trivial solution on complete bipartite graphs. Moreover, using this result we show that the problem is NP-hard on dense bipartite graphs and even $W[2]$-hard with respect to the number of clusters in an optimal solution as parameter.
In the first part, we consider a complete bipartite graph $G_{n,m}$ with the two subsets that are independent sets of size $n$ and $m$ and we first prove the following result.
\begin{lemma} \label{complete_bipartite_lem}
The density $d(G_{n,m})$ of a complete bipartite graph $G_{n,m}$ is greater than or equal to the density $d(\mathcal{P})$ of any partition $\mathcal{P}$ of $G_{n,m}$.
\end{lemma}
\begin{proof}
The density of the complete bipartite graph $G_{n,m}=(A,B,E)$, with $|A|=n, |B|=m$ is given by $d(G_{n,m}) = \frac{nm}{n+m}$
It suffices to show that $d(G_{n,m})$ is greater than or equal to the density of any partition $\mathcal{P} = \{ V_1, V_2\}$ that splits the set of vertices into exactly $2$ nonempty subsets. Indeed, if this holds and we have a partition $\mathcal{P} = \{ V_1, \dots, V_k\}$ where $k \geq 3$, we can show recursively that $d(G_{n,m}) \geq d(G[V_1])+ d(G[V_2 \cup \dots \cup V_k]) \geq \dots \geq d(G[V_1])+ \dots + d(G[V_k])$.
We first consider a partition $\mathcal{P}_1 = \{ V_1, V_2\}$ where $A\subseteq V_1$. Without loss of generality we may assume that $V_2=B\setminus V_1$ contains $m_2$ vertices from $B$. Then
\begin{equation*}
d(\mathcal{P}_1) = \frac{n(m-m_2)}{n+m-m_2} + 0 \leq \frac{nm}{n+m}
\label{density_of_P_1}
\end{equation*}
Now, consider a partition $\mathcal{P}_1 = \{ V_1, V_2\}$ such that each of the graphs $G[V_i]$ contains at least one edge, so let $G[V_1]=G_{n_1,m_1}$ with $0<n_1<n$ and $0<m_1<m$. Then $G[V_2]=G_{n-n_1,m-m_1}$ and
\begin{equation*}
d(\mathcal{P}_1) = \frac{n_1m_1}{n_1+m_1} + \frac{(n-n_1)(m-m_1)}{n+m-n_1-m_1} = \frac{nm(n_1+m_1)-mn_1^2-nm_1^2}{(n+m-n_1-m_1)(n_1+m_1)}\,,
\end{equation*}
which yields
\begin{equation*}
d(G_{n,m})-d(\mathcal{P}_1) =\frac{(nm_1-mn_1)^2}{(n+m-n_1-m_1)(n_1+m_1)(n+m)}\geq 0
\end{equation*}
\end{proof}
It follows that an optimal solution of any complete bipartite graph is the whole graph. From the calculations in the previous proof, we can inductively deduce the following result.
\begin{corollary}\label{bipar_cond}
For any complete bipartite graph $G=(A,B,E)$ with $|A|=n$ and $|B|=m$, a partition $\mathcal{P}=\{V_1,\dots,V_k\}$ of $A\cup B$ satisfies $d(\mathcal{P}) = \frac{nm}{n+m}$ if and only if $G[V_i]=G_{n_i,m_i}$ with $n_i\not=0$ and $m_i\not=0$ and $\frac{n_i}{m_i}=\frac{n}{m}$ for all $i\in\{1,\dots,k\}$.
\end{corollary}
Consequently, for any complete bipartite graph $G_{n,m}$, if $n$ and $m$ are relatively prime the only
optimal solution of $G_{n,m}$ is the whole graph. Otherwise, several optimal solutions exist and are characterized exactly by Corollary~\ref{bipar_cond}.
\medskip
In the second part of this section, we study the role of the number of sets in an optimum solution for {\sc Dense Graph Partition}. We specifically consider parameterization in the following sense. To formally give a parameter as input, we consider instances of the form $((G,r),k)$ for the decision problem: Is there a partition $\mathcal P$ of the vertices of $G$ into at most $k$ sets with $d(\mathcal P)\geq r$? Formally, the algorithm can hence also answer no, if the bound $k$ given in the input is not large enough to build a partition of density $r$.
Note that although the task of partitioning into \emph{exactly} $k$ sets is NP-hard even for $k=3$ (covering with $k$ cliques), the complexity with an upper bound on the number of sets is open; while there exists a partition into exactly $k$ sets of density $(n-k)/2$ if and only if the input graph can be partitioned into $k$ cliques (see Lemma~\ref{lemmacomplete}), there can be a partition into less than $k$ sets with a density even higher than $(n-k)/2$ even if the input cannot be partitioned into $k$ cliques. As an example, consider a complete graph of an even number $n$ of vertices and turn four of the vertices into an independent set by removing all edges among them. The resulting graph cannot be partitioned into 3 cliques (at least one set contains two of the four independent vertices), but it has a partition into two sets with density $(n-2)/2 -4/n$.
\begin{figure}
\begin{center}
\begin{minipage}[c]{0.2\linewidth}
\begin{tikzpicture}[thick,scale=1]
\tikzset{node/.style 2 args={draw, circle,draw=black,scale=0.7, label=#1:{\large #2}}};
\node[node={135}{\small $v_1$} ] (0) at (0,0) {};
\node[node={0}{\small $v_2$} ] (1) at (1,0) {};
\node[node={225}{\small $v_3$} ] (2) at (0,-1) {};
\node[node={315}{\small $v_4$} ] (3) at (1,-1) {};
\node[node={90}{\small $v_5$} ] (4) at (1,1) {};
\draw (0) -- (1);
\draw (0) -- (2);
\draw (2) -- (3);
\draw (3) -- (1);
\draw (1) -- (4);
\end{tikzpicture}
\end{minipage}
\begin{minipage}[c]{0.05\linewidth}
$\Rightarrow$
\end{minipage}
\begin{minipage}[c]{0.7\linewidth}
\begin{tikzpicture}[thick,scale=0.42]
\tikzset{node/.style 2 args={draw, circle,draw=black,scale=0.7, label=#1:{\large #2}}};
\node[node={90}{\small $v_1$} ] (0) at (0,0) {};
\node[node={90}{\small $v_2$} ] (1) at (1,0) {};
\node[node={90}{\small $v_3$} ] (2) at (2,0) {};
\node[node={90}{\small $v_4$} ] (3) at (3,0) {};
\node[node={90}{\small $v_5$} ] (4) at (4,0) {};
\node[node={90}{\small $w_1^1$} ] (5) at (6.5,0) {};
\node[node={90}{\small $w^1_2$} ] (6) at (7.5,0) {};
\node[node={90}{\small $w^1_3$} ] (7) at (8.5,0) {};
\node[node={90}{\small $w^2_1$} ] (8) at (13.5,0) {};
\node[node={90}{\small $w^2_2$} ] (9) at (14.5,0) {};
\node[node={90}{\small $w^2_3$} ] (10) at (15.5,0) {};
\node[node={90}{\small $z$} ] (11) at (19,0) {};
\node[node={270}{\small $v'_1$} ] (12) at (0,-8) {};
\node[node={270}{\small $v'_2$} ] (13) at (1,-8) {};
\node[node={270}{\small $v'_3$} ] (14) at (2,-8) {};
\node[node={270}{\small $v'_4$} ] (15) at (3,-8) {};
\node[node={270}{\small $v'_5$} ] (16) at (4,-8) {};
\foreach \i in {1,...,7}{
\node[node={270}{\small $x^{1}_{\i}$} ] (a\i) at (4 + \i ,-8) {};}
\foreach \i in {1,...,7}{
\node[node={270}{\small $x^{2}_{\i}$} ] (b\i) at (4 + \i + 7,-8) {};}
\node[node={270}{\small $z_1$} ] (31) at (19,-8) {};
\node[node={270}{\small $z_2$} ] (32) at (20,-8) {};
\foreach \i in {0,...,4}{
\foreach \j in {1,...,7}{
\draw (\i) edge[gray!60] (a\j);
}
}
\foreach \i in {0,...,4}{
\foreach \j in {1,...,7}{
\draw (\i) edge[gray!60] (b\j);
}
}
\foreach \j in {0,...,4}{
\draw (\j) edge[gray!60] (31);
\draw (\j) edge[gray!60] (32);
}
\foreach \i in {12,...,16}{
\foreach \j in {5,...,10}{
\draw (\i) edge[gray!60] (\j);
}
}
\foreach \i in {5,6,7}{
\foreach \j in {1,...,6}{
\draw (\i) edge[red] (a\j);
}
}
\foreach \i in {8,9,10}{
\foreach \j in {1,...,6}{
\draw (\i) edge[red] (b\j);
}
}
\draw (0) -- (12);
\draw (0) -- (13);
\draw (0) -- (14);
\draw (1) -- (12);
\draw (1) -- (13);
\draw (1) -- (15);
\draw (1) -- (16);
\draw (2) -- (12);
\draw (2) -- (14);
\draw (2) -- (15);
\draw (3) -- (13);
\draw (3) -- (14);
\draw (3) -- (15);
\draw (4) -- (13);
\draw (4) -- (16);
\foreach \j in {2,...,7}{
\draw (11) edge[cyan] (a\j);
}
\foreach \j in {2,...,7}{
\draw (11) edge[cyan] (b\j);
}
\draw (31) edge[cyan] (11);
\draw (32) edge[cyan] (11);
\end{tikzpicture}
\end{minipage}
\caption{A graph $G$, instance of \textsc{Dominating Set} and the bipartite graph $G'$ obtained from $G$, for $k=2$ and $n=5$.}
\end{center}
\end{figure}\label{bipartitefigure}
\begin{theorem} \label{bipartite}
\DGP~parameterized by (an upper bound on) the number of clusters in an optimal solution is W[2]-hard on dense bipartite graphs.
\end{theorem}
\begin{proof}
We give a reduction from {\sc Dominating Set}. Given a graph $G=(V,E)$ with $V=\{v_1,\dots,v_n\}$ and an integer $k$ as instance of {\sc Dominating Set}, we construct a bipartite graph $G'=(V_1,V_2,E')$ as input for \DGP~as follows:
\begin{itemize}
\item $V_1= V\cup \{w_i^j\colon 1\leq i\leq n-k, \ 1\leq j\leq k\}\cup\{z\}$
\item $V_2=\{v_1',\dots,v_n'\}\cup\{x_r^j\colon 1\leq r\leq N, \ 1\leq j\leq k\}\cup\{z_i\colon 1\leq i\leq N-n\}$ where $N\in \mathbb{N}$ is chosen as follows.
Let $c\in \mathbb N$ be the smallest integer such that $c(n-k+1)-1 > n$ (note that $1\leq c\leq n$) and define $N=c(n-k+1)-1$. For this choice of $N$ it follows that the greatest common divisor of $N$ and $n-k+1$ is 1, and $n<N\leq 2n$.
\item $E'=E_d\cup E_{wx}\cup E_c \cup E_z$ with
$E_d=\{\{v_i,v_j'\}\colon \{v_i,v_j\}\in E\}\cup \{\{v_i,v_i'\}\colon 1\leq i\leq n \}$,\\
$E_{wx}=\{\{w_i^j,x_r^j\}\colon 1\leq i\leq n-k, \ 1\leq r\leq N-1, 1\leq j\leq k \}$, \\
$E_c=\{\{w_i^j,v_s'\}\colon 1\leq i\leq n-k, 1\leq j\leq k,\ 1\leq s\leq n\}\cup\{\{v_s,x_r^j\}\colon 1\leq s\leq n, \ 1\leq r\leq N, \ 1\leq j\leq k\}$ and \\ $E_z=\{\{z,z_j\}\colon 1\leq j\leq N-n\} \cup \{\{z,x^j_r\} : 2\leq r\leq N, 1\leq j\leq k \} \cup\{\{v_i,z_j\}\colon 1\leq i\leq n, \ 1\leq j\leq N-n \}$
\end{itemize}
Notice that $G'$ is a bipartite graph with $|V_1|=n+1+ k(n-k)$ and $|V_2|= (k+1)N$.
We show that there exits a dominating set of cardinality at most $k$ in $G$ if and only if there exists a partition $\mathcal{P}$ of $G'$ with $d(\mathcal{P})=(k+1)d(G_{n-k+1,N})$.
Suppose there exists a dominating set $D$ in $G$ with $|D|=k$.
Let $D=\{v_{i_1},\dots,v_{i_k}\}$ and $N'(v_{i_j})=N_G[v_{i_j}]\setminus (D\cup N_G(\{v_{i_1},\dots, v_{i_{j-1}}\})$. Define the partition $\mathcal{P}=\{P_1,\dots,P_{k+1}\}$ by:\\ $P_j=\{v_{i_j}\}\cup \{v_r'\colon v_r \in N'(v_{i_j})\} \cup \{w_r^j\colon 1\leq r \leq n-k\} \cup\{x_r^j\colon 1\leq r\leq N-|N'(v_{i_j})| \}$ for $1\leq j\leq k $ and $P_{k+1}=V_1\cup V_2\setminus (\cup_{j=1}^k P_j)$.
It is not hard to see that the vertices in $P_j$ induce a complete bipartite graph $G_{n-k+1,N}$ for each $j$. Thus $d(\mathcal{P})=(k+1)d(G_{n-k+1,N})$.
Conversely, let $\mathcal{P}$ be a partition of $G'$ of density $(k+1)d(G_{n-k+1,N})$.
Thus, Corollary~\ref{bipar_cond} implies that the vertices for each set $P\in \mathcal{P}$ induce a complete bipartite graph $G_{r,s}$ such that $\frac rs=\frac{|V_1|}{|V_2|}=\frac{k(n-k)+n+1}{(k+1)N}=\frac{n-k+1}{N}$. Since the greatest common divisor of $n-k+1$ and $N$ is one, this yields $r\geq n-k+1$ and $s\geq N$ and especially $\mathcal{P}$ can contain at most $k+1$ sets.
For all $w_i^j$ and $w_{\ell}^{t}$, if $j \neq t$, $w_i^j$ and $w_{\ell}^{t}$ have $n$ common neighbours, and $n < N$ then there is no part $P\in \mathcal{P}$ such that $w_i^j, w_{\ell}^{t} \in P$. Moreover, for all $i,j$, $w_i^j$ and $z$ have $N-1$ common neighbours so they cannot be in the same $P\in \mathcal{P}$. Hence, there are exactly $k+1$ parts in $\mathcal{P}$ that are complete bipartite graphs $G_{n-k+1,N}$.
For all $1 \leq j \leq k$, denote by $P_j$ the set containing the vertices $w^j_i$ for all $1 \leq i \leq n-k$ and $P_z$ the part containing $z$. To reach the cardinality exactly $n-k+1$, $P_j\cap V_1$ has to contain exactly one vertex from $V$ for each $1\leq j\leq k$.
Further, since for any $i$, $v_i'$ is not adjacent to $z$, $V' \subseteq \cup_{j=1}^k P_j$. Moreover, each $P_j$ contains exactly one vertex of $V$.
As each $P \in \mathcal{P}$ induces a complete bipartite graph in $G'$, $D= V \cap \cup_{j=1}^k P_j$ is a set of size $k$, such that each vertex in $V'$ is adjacent to at least one vertex in $D$, so we deduce that $D$ is dominating set of size $k$ in $G$.
At last, in case of a yes-instance of \textsc{Dominating Set}, there exists an optimum solution with $k+1$ sets for \textsc{Dense Graph Partition} on $G'$. With parameter $k'=k+1$, the instance $((G',(k+1)d(G_{n-k+1,N})),k')$ describes an fpt-reduction from \textsc{Dominating Set} parameterized by solution size to \textsc{Max Dense Graph Partition} parameterized by an upper bound on the number of sets, which shows the claimed W[2]-hardness.
\medskip
We extend the construction of the proof to create from $G'$ a dense bipartite graph $G''=(V'',E'')$ by adding four sets of vertices $V_1^u, V_1^d, V_2^u, V_2^d$ with $|V_1^u|=|V_1^d|=kn|V_1|=kn(k(n-k)+n+1)$ and $|V_2^u|=|V_2^d|=kn|V_2|=knN(k+1)$. Further, we add edges to turn the pairs $(V_1^u, V_2^u)$, $(V_1^d, V_2^d)$, $(V_1^u, V_1)$, and $(V_2^d,V_2)$ each into complete bipartite graphs. Observe that with this construction $G''$ has $|V''|=(2kn+1)(k(n-k)+n+1)+(2kn+1)N(k+1)< 10k^2n^2$ vertices and that all vertices have degree at least $kn|V_1|\geq \frac 12 k^2 n^2\in \Theta(|V''|)$ (Note that the fpt-reduction can solve the instance of {\sc Dominating Set} exactly in the trivial case of $k\geq \frac n2$, so we can assume that $k\leq \frac n2$.).
We claim that there exists a partition $\mathcal{P'}$ of $G''$ with $d(\mathcal{P'})=(k+1)d(G_{n-k+1,N})+2kn(k+1)d(G_{n-k+1,N})$ if and only if there exists a dominating set of size $k$ for $G$. Corollary~\ref{bipar_cond} again implies that this density for $G''$ can only be achieved by a partition into complete bipartite graphs $G_{r,s}$ with $\frac rs=\frac{(2kn+1)(k(n-k)+n+1)}{(2kn+1)N(k+1)}=\frac{n-k+1}{N}$. The vertices in $V_1^d$ are only adjacent to vertices in $V_2^d$, and the vertices in $V_2^u$ are only adjacent to vertices in $V_1^u$. Clustering these in a ratio $\frac rs$ results in clusters containing exactly all newly added vertices, and this can be done with just two sets in total.
What remains is to cluster the graph $G'$ into complete bipartite graphs $G_{r,s}$ such that $\frac rs=\frac{|V_1|}{|V_2|}=\frac{k(n-k)+n+1}{(k+1)N}=\frac{n-k+1}{N}$ as before.
At last, in case of a yes-instance of \textsc{Dominating Set}, there exists an optimum solution with $k+3$ sets for \textsc{Dense Graph Partition} on $G''$. With parameter $k'=k+3$, the instance $((G'',(2kn+1)(k+1)d(G_{n-k+1,N})),k')$ describes an fpt-reduction from \textsc{Dominating Set} parameterized by solution size to \textsc{Max Dense Graph Partition} parameterized by an upper bound on the number of sets, which shows the claimed W[2]-hardness
\end{proof}
\section{Cubic Graphs }\label{sec4}
\label{sec:cubicDense}
We show that the problem \DGP~is NP-complete even for cubic graphs by giving a reduction from \textsc{Exact Cover By 3-Sets} where each element appears in exactly 3 sets, denoted \textsc{Restricted Exact Cover By 3-Sets}, known to be NP-hard by~\cite{Gonzalez85}.
\defprob{Restricted Exact Cover By 3-Sets (RX3C)}{A set $X$ of elements with $|X| = 3q$ and a collection $C$ of 3-element subsets of $X$ where each element appears in exactly 3 sets.}{Does $C$ contain an exact cover for $X$, i.e. a subcollection $C' \subseteq C$ such that every element occurs in exactly one member of $C'$ ?}
\noindent
Before describing the reduction, we give useful notions for utility.
\begin{definition}
For $S \subseteq V$, the utility of a vertex $v \in S$ is defined by $u_S(v) = \frac{d(S)}{|S|}$, and the utility of $S$ is defined by $u(S)=u_S(v)$ for any $v \in S$. For a partition $\mathcal{P}= \{ V_1, \ldots, V_k\}$, the utility of a vertex $v$ in $\mathcal{P}$ is defined by $u_{\mathcal{P}}(v)=u_{V_i}(v)$ with $i$ such that $v\in V_i$.
\end{definition}
Considering these definitions, we can remark that:
\begin{itemize}
\item For any subset $S \subseteq V$, and $v,w \in S $, $u_S(v)=u_S(w)$.
\item If $S=\{v\}$ then $u_S(v)=0$.
\item For any partition $\mathcal{P}$ of $G$, $\sum\limits_{V_i \in \mathcal{P}}d(V_i) = \sum\limits_{v \in V}u_{\mathcal{P}}(v)$.
\end{itemize}
\noindent
The following definition gives the construction to reduce \textsc{RX3C} to \DGP.
\begin{figure}
\begin{minipage}[c]{0.49\linewidth}
\centering
\begin{tikzpicture}[thick,scale=0.35]
\tikzset{node/.style 2 args={draw, circle,draw=black,scale=0.7, label=#1:{\large #2}}};
\node[node={90}{$v_x$},fill ] (1) at (0,0) {};
\node[node={90}{$v_{xy_1z_1}^x$} ] (2) at (3,2) {} edge (1);
\node[node={270}{$v_{xy_2z_2}^x$} ] (3) at (0,-3) {} edge (1);
\node[node={90}{$v_{xy_3z_3}^x$} ] (4) at (-3,2) {} edge (1) ;
\end{tikzpicture}
\caption{Subgraph containing one vertex of type 1, $v_x$, and its neighbors in $G$}
\label{t1}
\end{minipage}
\begin{minipage}[c]{0.49\linewidth}
\centering
\begin{tikzpicture}[thick,scale=0.35]
\tikzset{node/.style 2 args={draw, circle,draw=black,scale=0.7, label=#1:{\large #2}}};
\node[node={90}{$v_{xyz}^x$} ] (1) at (0,0) {};
\node[node={90}{$v_{xyz}^y$} ] (2) at (3,2) {} edge (1);
\node[node={270}{$v_x$},fill ] (3) at (0,-3) {} edge (1);
\node[node={90}{$v_{xyz}^z$} ] (4) at (-3,2) {} edge (1) edge(2) ;
\end{tikzpicture}
\caption{Subgraph containing one vertex of type 1, $v_x$, and three of type 2}
\label{t2}
\end{minipage}
\end{figure}
\begin{definition}\label{construction}
Let $I=(X,C)$ be an instance of $RX3C$. We define the construction $\sigma$ transforming the instance $I$ into the graph $G:= \sigma(I)$ where $G=(V,E)$ as follows:
\begin{itemize}
\item for each element $x \in X$, we add the vertex $v_x$ in $V$ (called vertices of type 1 or black vertices).
\item for each subset of the collection $\{x,y,z\} \in C$, we add the vertices $v_{xyz}^x$, $v_{xyz}^y$, $v_{xyz}^z$ in $V$ (called vertices of type 2 or white vertices).
\item we add the edges $\{v_{xyz}^x,v_{xyz}^y\}$, $\{v_{xyz}^x,v_{xyz}^z\}$ and $\{v_{xyz}^y,v_{xyz}^z\}$ to $E$
\item we add the edges $\{v_{xyz}^x,v_x\}$, $\{v_{xyz}^y,v_y\}$ and $\{v_{xyz}^z,v_z\}$ to $E$
\end{itemize}
\noindent
\normalfont{Notice that $G$ is a cubic graph on $|X|$ vertices of type 1, $3|X|$ vertices of type 2.
}
\end{definition}
\noindent
Case distinction on the subgraphs in $\sigma(I)$ shows:
\begin{lemma}\label{casescubic}
For any subset $S\subseteq V$ of the vertices of the graph $\sigma(I)$, the only subgraphs $G[S]$ with $u(S) \geq \frac{1}{4}$ are:
\begin{itemize}
\item a triangle where all the vertices are of type 2 and then $u(S) = \frac{1}{3}$.
\item a matching between two type 2 vertices or between two vertices of different types and then $u(S) = \frac{1}{4}$.
\item the subgraph described in Figure \ref{t2} and then $u(S) = \frac{1}{4}$.
\end{itemize}
\label{lem:utiliteG}
\end{lemma}
\begin{proof}
Let $S\subseteq V$ such that $u(S) \geq \frac{1}{4}$. We show in the following that there are exactly three possible subgraphs $G[S]$ such that $u(S) \geq \frac{1}{4}$.
First, observe that by its construction $G$ does not contain $C_4$ as subgraph, since there are no two vertices $u,v\in V$ that have more than one common neighbor. Note that this also implies that $G$ is diamond-free.
As $G$ is cubic, $|E(G[S])| \leq \frac{3}{2}|S|$ and so $d(S) \leq \frac{3}{2}|S|\cdot \frac{1}{|S|} = \frac{3}{2}$. Since $\frac{1}{4} \leq u(S) \leq \frac{3}{2|S|}$ then $|S| \leq 6$. We study the five following cases :
\begin{itemize}
\item Case $|S| = 6$ : Since $u(S)= \frac{|E(S)|}{6^2} \geq \frac{1}{4}$, we have $|E(S)| \geq 9$. Since $G[S]$ cannot be cubic ($G$ is connected and $|V| > 6$), a subgraph with $|S|=6$ and $|E(S)| \geq 9$ does not exist.
\item Case $|S| = 5$ : Since $u(S)= \frac{|E(S)|}{5^2} \geq \frac{1}{4}$, we have $|E(S)| \geq 7$. Assuming such a subgraph with $|S| = 5$ and $|E(S)| \geq 7$, $G[S]$ must contain a diamond or a clique of size 4. Since $G=\sigma(I)$ is diamond-free and $G$ is cubic, such a subgraph does not exist.
\item Case $|S| = 4$ : Since $u(S)= \frac{|E(S)|}{4^2} \geq \frac{1}{4}$, we have $|E(S)| \geq 4$. Since $G$ is diamond-free, $G$ is not a $K_4$ or a $C_4$ the only possibility for $G[S]$ is the subgraph described in Figure \ref{t2}.
\item Case $|S| = 3$ : Since $u(S) = \frac{|E(S)|}{3^2} \geq \frac{1}{4}$, we have $|E(S)| \geq 3$ and thus $S$ is a triangle where all the vertices are of type 2 and $u(S) = \frac{1}{3}$.
\item Case $|S| = 2$ : Since $u(S) = \frac{|E(S)|}{2^2} \geq \frac{1}{4}$, we have $|E(S)| \geq 1$ and thus $S$ is a matching between two type 2 vertices or between two vertices of different types and $u(S) = \frac{1}{4}$.
\end{itemize}
\end{proof}
\begin{remark}
For any subset $S\subseteq V$ of the vertices of the graph $\sigma(I)$, if $v$ is of type 2 then $u_S(v) \leq \frac{1}{3}$, otherwise $u_S(v) \leq \frac{1}{4}$.
\end{remark}
With these observations about the construction of $\sigma(I)$, it can be shown that $I=(X,C)$ is a yes-instance of RX3C if and only if $I'=(\sigma(I),d)$ is a yes-instance of \DGP{} which yields the following.
\begin{theorem}\label{cubichardness}
\DGP~is NP-complete on cubic graphs.
\end{theorem}
\begin{proof}
Let $I=(X,C)$ be an instance of RX3C and consider the following instance $I'$ of \DGP{} on the graph $G=\sigma(I)$ and $d=\frac{7|X|}{6}$.
We claim that $I=(X,C)$ is a yes-instance of RX3C if and only if $I'=(G,d)$ is a yes-instance of \DGP.
\medskip
Let $C' \subseteq C$ be an exact cover for $X$ of size $\frac{|X|}{3}$. Consider the following partition $\mathcal{P}$ with $\frac{5|X|}{3}$ parts : for any $c \in C'$, $c=\{x,y,z\}$, we define three parts of size 2 $\{v_x, v_{xyz}^x\}$, $\{v_y, v_{xyz}^y\}$, $\{v_z, v_{xyz}^z\}$ and for any $c \notin C'$, $c=\{x,y,z\}$, we define the following part of size 3 $\{v_{xyz}^x, v_{xyz}^y, v_{xyz}^z\}$. Since $C'$ is an exact cover, $\mathcal{P}$ is a partition and its density is $\frac{3}{2}\cdot\frac{|X|}{3} + \frac{2}{3}|X| = \frac{7}{6}|X|$.
\medskip
Let $\mathcal{P'}$ be a partition of $G$ of density $d(\mathcal{P'})=\frac{7}{6}|X|$. Firstly, we show that $\mathcal{P'}$ has necessarily the following shape: $\frac{2|X|}{3}$ parts of size 3 containing only vertices of type 2 forming a triangle in $G$ and $|X|$ parts of size 2 containing one vertex of type 1 and one of type 2 adjacent in $G$ (see Figures \ref{t1} and \ref{t2}). From Remark~\ref{remarkconnexe}, we can consider that for every part $P_i \in \mathcal{P'}$, $G[P_i]$ is connected.
We prove in the following that since $d(\mathcal{P'})=\frac{7|X|}{6}$ then there are at least $\frac{2|X|}{3}$ parts in $\mathcal{P'}$ corresponding to triangles in $G$.
Assume by contradiction that $\mathcal{P'}$ has $\frac{2|X|}{3} - \ell $ triangles, with $\ell > 0$. Since $G$ has $4|X|$ vertices, there are $2|X|+ 3 \ell$ vertices that do not belong to a part in $\mathcal{P'}$ that corresponds to a triangle in $G$. By Lemma \ref{lem:utiliteG} the utility of these last vertices is smaller than or equal to $\frac{1}{4}$. Then the density of $\mathcal{P'}$ is
\[
d(\mathcal{P}') \leq \frac{2|X|}{3} - \ell + (2|X|+ 3 \ell) \cdot\frac{1}{4} \leq \frac{7|X|}{6} - \frac{\ell}{4} < \frac{7|X|}{6}
\]
This contradicts the choice of $\mathcal{P'}$ such that $d(\mathcal{P'})=\frac{7|X|}{6}$, hence there are at least $\frac{2|X|}{3}$ triangles in $\mathcal{P'}$.
Now, we will prove that there are at most $\frac{2|X|}{3}$ parts in $\mathcal{P'}$ corresponding to triangles in $G$. Assume by contradiction that $\mathcal{P'}$ has $\frac{2|X|}{3} + \ell$ triangles, with $\ell > 0$. Since there are $3|X|$ vertices of type 2 and among these vertices $3\cdot(\frac{2|X|}{3} + \ell)$ belong to a triangle then $|X|-3 \ell$ vertices of type 2 do not belong to a triangle. But each neighbour of a vertex $v_x$ of type 1 is of type 2, so if the utility of $v_x$ is positive, then there exists a vertex of type 2, $v^x_{xyz}$, neighbour of $v_x$, that is in the same part as $v_x$ and $v^x_{xyz}$ does not belong to a triangle. Moreover, as all type 1 vertices have no common neighbours, for each type 1 vertex with positive utility, there is a type 2 vertex that is not in a triangle. Since there are at most $|X|-3 \ell$ type 2 vertices that do not belong to a triangle, there are at most $|X|-3 \ell$ type 1 vertices with positive utility. Then the density of $\mathcal{P'}$ is at most
\[
d(\mathcal{P'}) \leq \frac{2|X|}{3}+ \ell + \frac{|X| - 3 \ell}{4} + \frac{|X| - 3 \ell}{4} \leq \frac{7|X|}{6} - \frac{\ell}{2} < \frac{7|X|}{6}
\]
This contradicts the choice of $\mathcal{P'}$ such that $d(\mathcal{P'})=\frac{7|X|}{6}$, and then there are exactly $\frac{2|X|}{3}$ triangles in $\mathcal{P'}$.
We will show now that $d(\mathcal{P'}) = \frac{7|X|}{6}$ implies that all type 1 vertices are in a part that is a matching with a type 2 vertex. There are $|X|$ type 1 vertices and $|X|$ type 2 vertices that are not in some triangle in $\mathcal{P'}$. Since there are exactly $\frac{2|X|}{3}$ parts in $\mathcal{P'}$ forming a triangle and the utility of each other vertex is smaller than or equal to $\frac{1}{4}$, to reach a density of $\frac{7|X|}{6}$ it is necessary that each of the $2|X|$ vertices outside the parts that are triangles has a utility of exactly $\frac{1}{4}$. To reach this utility, by Lemma \ref{lem:utiliteG} there are two possibilities, the graph described in Figure \ref{t2} and an edge. Since there are exactly $|X|$ vertices of type 1 and $|X|$ vertices of type 2 outside the triangles in $\mathcal{P'}$, and vertices of type 1 only have neighbors of type 2, the only possibility for all these vertices to have utility $\frac{1}{4}$ is if each type 1 vertex is matched with one type 2 vertex.
Consider now the following subcollection $C''\subseteq C$: for each triple $v^x_{xyz}$,$v^y_{xyz}$,$v^z_{xyz}$ that does not belong to a triangle, we add the set $\{x,y,z\}$ to $C''$. The subcollection $C''$ is a cover since each type 1 vertex is a neighbour of one of these vertices and it is an exact cover since there are exactly $\frac{|X|}{3}$ 3-element subsets that do not belong to a triangle.
\end{proof}
In the following we give a polynomial time $\frac{4}{3}$-approximation for \MDGP{} on cubic graphs. We start with some preliminary results.
\begin{lemma}
\label{lem:utilPasTriDia}
Let $G=(V,E)$ be a cubic graph. Let $\mathcal{P}$ any partition of $V$.
If a part $P$ of $\mathcal P$ is not a triangle or a diamond, then $u_\mathcal{P}(v) \leq \frac{1}{4}$ for any $v\in P$.
\end{lemma}
\begin{proof}
Since the graph is cubic, $d(P) \leq \frac{3|P|}{2|P|} = \frac{3}{2}$. Then $u_{\mathcal{P}}(v) \leq \frac{3}{2|P|}$. If $|P| \geq 6$, $u_{\mathcal{P}}(v) \leq \frac{3}{2 \cdot 6} = \frac{1}{4}$. If $|P| < 6$ and since $P$ is not a triangle or a diamond, by exhaustive search, $u_{\mathcal{P}}(v)$ is maximised when $P$ is a matching and its value is $\frac{1}{4}$.
\end{proof}
\begin{lemma}
\label{lem:UtilMatchCubique}
Let $G$ be a cubic graph. For every vertex $v$ of $G$ that does not belong to a triangle of $G$, $u_{\mathcal{P}}(v) \leq \frac{1}{4}$ for any partition $\mathcal{P}$ of $V$.
\end{lemma}
\begin{proof}
The lemma is a direct consequence of \autoref{lem:utilPasTriDia}.
\end{proof}
\begin{lemma}
\label{lem:UtilDiamCubique}
Let $G$ be a cubic graph with at least one diamond. Let $v_1,v_2,v_3,v_4$ be a diamond in $G$ where $v_1$ and $v_3$ are the induced degree two vertices. Then $u_{\mathcal{P}}(v_1) + u_{\mathcal{P}}(v_2) + u_{\mathcal{P}}(v_3) + u_{\mathcal{P}}(v_4) \leq \frac{5}{4}$ for any partition $\mathcal{P}$.
\end{lemma}
\begin{figure}[ht]
\centering
\begin{minipage}[c]{0.3\textwidth}
\centering
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\begin{minipage}[c]{0.3\textwidth}
\centering
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\begin{minipage}[c]{0.3\textwidth}
\centering
\begin{tikzpicture}[thick, label distance=0.15cm,scale=0.9]
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\draw[gray] \convexpath{2,4}{0.3 cm};
\node (n1) at (0,-1.3) {Case 3};
\end{tikzpicture}
\end{minipage}
}
\caption{Different cases of \autoref{lem:UtilDiamCubique}}\label{fig:utilDiamants}
\end{figure}
\begin{proof}
Let $\mathcal{P}$ be any partition of $V$. Let $P_1 \in \mathcal{P}$ (resp. $P_2$, $P_3$ and $P_4$) be the part that contains $v_1$ (resp. $v_2$, $v_3$ and $v_4$). We distinguish several cases.
\noindent
\textbf{Case 1:} The four vertices $v_i$ are in the same part and this part is a diamond. Then $d(P_1) = \frac{5}{4}$ and thus $u_{\mathcal{P}}(v_1) + u_{\mathcal{P}}(v_2) + u_{\mathcal{P}}(v_3) + u_{\mathcal{P}}(v_4) = \frac{5}{4}$.
\noindent
\textbf{Case 2:} Three among the four vertices of the diamond forming a triangle are in the same part. The other one cannot belong to a part that is a triangle or an diamond. Then, by \autoref{lem:utilPasTriDia}, $u_{\mathcal{P}}(v_1) + u_{\mathcal{P}}(v_2) + u_{\mathcal{P}}(v_3) + u_{\mathcal{P}}(v_4) \leq 1 + \frac{1}{4} = \frac{5}{4}$.
\noindent
\textbf{Case 3:} $P_1$, $P_2$, $P_3$ and $P_4$ are not triangles or diamonds then, by \autoref{lem:utilPasTriDia}, for all $v \in \cup_{i \leq 4} P_i$, $u_{\mathcal{P}}(v) \leq \frac{1}{4}$. We conclude that $u_{\mathcal{P}}(v_1) + u_{\mathcal{P}}(v_2) + u_{\mathcal{P}}(v_3) + u_{\mathcal{P}}(v_4) \leq 1$.
\end{proof}
\begin{lemma}
\label{lem:UtilTriangleCubique}
Let $G$ be a cubic graph and $v$ a vertex of $G$. Then $u_{\mathcal{P}}(v) \leq \frac{1}{3}$ in any partition $\mathcal{P}$ of $V$.
\end{lemma}
\begin{proof}
Let $P$ be the part of $\mathcal{P}$ that contains $v$. Using the same reasoning as in the proof of \autoref{lem:utilPasTriDia}, we deduce that if $|P| \geq 5$ then $u_{\mathcal{P}}(v) < \frac{1}{3}$. If $|P| < 5$, by exhaustive search, $u_{\mathcal{P}}(v)$ is maximised when $P$ is a triangle and in this case $u_{\mathcal{P}}(v) = \frac{1}{3}$.
\end{proof}
\begin{lemma}
\label{lem:ApproxBorneCubique}
Let $G$ be a cubic graph on $n$ vertices and let $D$ be the set of diamonds in $G$ and $T$ the set of triangles in $G$ that do not belong to a diamond. For any partition $\mathcal{P}$, $d(\mathcal{P}) \leq \frac{5}{4}|D| + |T| + \frac{1}{4}(n - 3|T| - 4|D|)$.
\end{lemma}
\begin{proof}
By \autoref{lem:UtilDiamCubique}, we know that the sum of the utilities of the vertices constituting a diamond is at most $\frac{5}{4}$. By \autoref{lem:UtilTriangleCubique}, we deduce that the sum of the utilities of the vertices constituting a triangle is at most $3 \cdot \frac{1}{3} = 1$. The other vertices that do not belong to a triangle or a diamond have a utility of at most $\frac{1}{4}$ (\autoref{lem:UtilMatchCubique}). We deduce that $d(G) \leq \frac{5}{4}|D| + |T| + \frac{1}{4}(n - 3|T| - 4|D|)$.
\end{proof}
\begin{theorem}
\MDGP{} is polynomial time $\frac{4}{3}$-approximable on cubic graphs.
\end{theorem}
\begin{proof}
Let $I=G$ be a cubic graph, instance of \MDGP{}. Let~$D$ be the set of all diamonds in $G$, and $T$ the set of all triangles that do not belong to a diamond.
Diamonds (resp. triangles) can be found in polynomial time simply by enumerating all 4-tuples (resp. 3-tuples) of vertices and checking if they induce a diamond (resp. triangle) as subgraph.
Let $G'$ be the graph obtained from $G$ after removing the vertices of $D$ and $T$. Let $M$ be the set of edges that constitute a maximal matching of $G'$. Let $G''$ be the graph obtained from $G'$ after removing the vertices of $M$. Since $M$ is a maximal matching, $G''$ is an independent set. Since $G$ is cubic, the maximum size of an independent set is $\frac{|V|}{4}$, thus $|V(G'')| \leq \frac{|V|}{4}$.
Consider the partition $\mathcal{P} = D \cup T \cup M \cup V(G'')$ in the sense that $\mathcal{P}$ contains a set for each diamond in $D$, one set for each triangle in $T$, one set for each edge in the matching $M$ and one set for each vertex in $V(G'')$.
Then $d(\mathcal{P}) = \frac{5}{4}|D| + |T| + \frac{1}{4} |M| \geq \frac{5}{4}|D| + |T| + \frac{1}{4}(n - 3|T| - 4|D| - \frac{n}{4})$. By \autoref{lem:ApproxBorneCubique} we know that $opt(I) \leq \frac{5}{4}|D| + |T| + \frac{1}{4}(n - 3|T| - 4|D|)$. Then $\frac{opt(I)}{d(\mathcal{P})} \leq \frac{\frac{5}{4}|D| + |T| + \frac{1}{4}(n - 3|T| - 4|D|)}{\frac{5}{4}|D| + |T| + \frac{1}{4}(n - 3|T| - 4|D| - \frac{n}{4})} = \frac{\frac{1}{4}|D| + \frac{1}{4}|T| + \frac{n}{4}}{\frac{1}{4}|D| + \frac{1}{4}|T| + \frac{3n}{16}} = 1+\frac{{n}}{4|D|+4|T|+{3n}}$. This function is maximized when $|D| = |T| = 0$. Then $\frac{opt(I)}{d(\mathcal{P})} \leq \frac{n}{\frac{3n}{4}} = \frac{4}{3}$.
\end{proof}
\section{Dense Graphs}\label{sec5}
\label{sec:densGrapheDensite}
In this section we consider graphs $G=(V,E)$ on $n$ vertices such that $G$ can be viewed as $G=K_n-H$ where $H$ is a graph of small maximum degree. The edges of $H$ are called \emph{missing edges} in $G$. We first consider graphs $G=(V,E)$ on $n$ vertices such that $\delta(G) \geq n-3$, that is $G=K_n-H$ where $H$ has $\Delta(H)=2$ and has $q\leq n$ edges and show that \MDGP{} is solvable in polynomial time on these graphs.
\begin{lemma}\label{threesets}
For any graph $G$ on $n$ vertices such that $\delta(G) \geq n-3$, its density $d(G)$ is greater than or equal to the density of any partition $\mathcal{P}$ of $G$ into $t\geq 3$ parts.
\end{lemma}
\begin{proof}
The density of $G$ is given by $d(G)= \frac{\frac{n(n-1)}{2}-q}{n}=\frac{n-1}{2} - \frac{q}{n}$. From Lemma~\ref{lemmacomplete}, among all partitions of $G$ into $t\geq 3$ parts, those where the parts correspond to complete graphs have the largest density. The density of such a partition in $t$ parts of size $n_1,\ldots,n_t$ is $\frac{n-t}{2}$. Thus, the density of $G$ is at least as large as the density of this last partition since $t\geq 3$ and $q\leq n$ (note here that a graph with minimum degree $n-3$ has at most $n$ non-edges).
\end{proof}
Remark that in the proof of the previous lemma when $q=n$ and $t=3$, the density of a partition in 3 parts corresponding to complete subgraphs and the density of the entire graph are the same. This previous lemma implies that for any graph $G$ such that $\delta(G) \geq n-3$, there exists a partition into one or two parts of maximum density.
\begin{lemma}\label{odd}
For any graph $G$ on $n$ vertices such that $\delta(G) \geq n-3$, in any partition into two parts of $G$, the number of missing edges inside the two parts is at least $o$, where $o$ is the number of odd cycles defined by the missing edges of $G$.
\end{lemma}
\begin{proof}
Let $C$ be and odd cycle of missing edges in $G$. Since $C$ is not bipartite, there is no partition $\{V_1,V_2\}$ of $V$ such that all the edges of $C$ have one endpoint in $V_1$ and one endpoint in $V_2$. Hence, for any partition $\{V_1,V_2\}$ at least one of the missing edges from $C$ is in inside $G[V_1]\cup G[V_2]$.
\end{proof}
\begin{lemma}\label{part1}
Among all partitions into 2 parts of fixed size containing $x$ missing edges, the one containing all missing edges in the larger part has the best density.
\end{lemma}
\begin{proof}
Consider two partitions $\{V_1,V_2\}$ and $\{V_1',V_2'\}$ such that $|V_1|=|V_1'|=n_1$ and $|V_2|=|V_2'|=n_2$ with $n_1\leq n_2$ and $G[V_1]$ (resp. $G[V_2]$) containing $x_1$ (resp. $x_2$) missing edges and $G[V_1']$ (resp. $G[V_2']$) containing $0$ (resp. $x=x_1+x_2$) missing edges.
$d(\{V_1,V_2\})= \frac{n-2}{2}- \frac{x_1}{n_1} - \frac{x_2}{n_2}$
$d(\{V_1',V_2'\})= \frac{n-2}{2}- \frac{x}{n_2}$
Since $x=x_1+x_2$ and $n_1\leq n_2$, we have $d(\{V_1,V_2\}) \leq d(\{V_1',V_2'\})$.
\end{proof}
\begin{lemma}\label{part2}
Among all partitions into 2 parts containing 0 (resp. $x$) missing edges in the smaller (resp. larger) part, the one with a maximum number of vertices in the larger part has the best density. \
\end{lemma}
\begin{proof}
Consider two partitions $\{V_1,V_2\}$ and $\{V_1',V_2'\}$ such that $|V_1|=n_1$, $|V_2|=n_2$ with $n_1\leq n_2$ and $|V_1'|=n_1'$, $|V_2'|=n_2'$ with $n_1'\leq n_2'$ and $G[V_1]$ (resp. $G[V_2]$) containing $0$ (resp. $x$) missing edges and $G[V_1']$ (resp. $G[V_2']$) containing $0$ (resp. $x$) missing edges. Moreover suppose $n_2\leq n_2'$.
$d(\{V_1,V_2\})= \frac{n-2}{2}- \frac{x}{n_2}$
$d(\{V_1',V_2'\})= \frac{n-2}{2}- \frac{x}{n_2'}$
Since $n_2\leq n_2'$, we have $d(\{V_1,V_2\}) \leq d(\{V_1',V_2'\})$.
\end{proof}
\begin{theorem}\label{polydense}
\MDGP{} is solvable in polynomial time on graphs $G$ with $n$ vertices and $\delta(G)\geq n-3$.
\end{theorem}
\begin{figure}
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\caption{Construction of $V_1$ and $V_2$ in Theorem~\ref{polydense}}
\label{fig:polyDens}
\end{figure}
\begin{proof} We define a partition $\{V_1,V_2\}$ where $V_1$ (resp. $V_2$) contains vertices of color 1 (resp. 2). An example is given in Figure \ref{fig:polyDens}.
Each vertex of degree $n-1$ has color 2. The graph $H$ of missing edges contains paths or cycles. The vertices on paths or cycles with an even number of vertices are colored alternating by 1 and 2. The vertices on paths or cycles with an odd number of vertices are colored alternating by 1 and 2 but starting with color 2. Thus cycles of odd size have two adjacent vertices of color 2. The partition $\{V_1,V_2\}$ defined above is such that it contains $o$ missing edges in $V_2$ and $|V_2|$ is maximized among all such partitions.
Its density is equal to $\frac{n-2}{2}- \frac{o}{n_2}$, where $n_2=|V_2|$. Denote by $d_{n-1}$ the number of vertices of~$G$ of degree $n-1$ and by $p_o$ the number of paths with an odd number of vertices (even length) among the missing edges. Thus $n_2=\frac12({n+d_{n-1}+p_o+o})$. We claim that there is no partition into two parts that has a higher density.
By Lemma~\ref{odd}, any partition into two sets contains at least $o$ missing edges inside the two parts. By construction we have maximized the number of vertices in the part with the missing edges among all partitions with the minimum number $o$ of missing edges, i.e., there is no partition into two parts $\{V'_1,V'_2\}$ with $o$ missing edges all contained in $V'_2$ and $|V'_2|>|V_2|$. Hence, by Lemma~\ref{part1} and~\ref{part2}, it remains to show that any partition $\{V'_1,V'_2\}$ with $n'_2=|V_2|$ such that $n'_2=n_2+y$ with $o+x>o$ missing edges has a smaller density than $\{V_1,V_2\}$.
By definition of the partition $\{V_1,V_2\}$, it follows that $|E(H)|=2n_1-r+o$, where $r$ is the number of paths of odd length in $H$. For the partition $\{V'_1,V'_2\}$, it follows that $|E(H)|\leq 2(n_1-y)-r_1+(o+x)$, for some $r_1\geq r-x$ (number of vertices in $V'_1$ adjacent to only one edge in $H$). Observe that all non-edges have to either be among the $o+x$ missing edges in the partition or in the cut between $V'_1$ and $V'_2$. In the cut between $V'_1$ and $V'_2$, each vertex in $V'_1$ is adjacent to at most two such edges, and further every path of odd length either results in a vertex in $V'_1$ adjacent to only one edge in $E(H)$ ($r_1$) or in a missing edge in $V'_2$, hence $r_1\geq r-x$. These inequalities imply that $y\leq x$, and hence the density of $\{V'_1,V'_2\}$ is at most $\frac{n-2}{2}- \frac{o+x}{n_2+y} \leq \frac{n-2}{2}- \frac{o+y}{n_2+y}\leq \frac{n-2}{2}- \frac{o}{n_2}$. Note that the last inequality follows from $o\leq {n_2}$, which simply holds since $H$ is of degree at most 2.
\end{proof}
In the rest of the section we consider graphs $G=(V,E)$ on $n$ vertices, $(n-4)$-regular, that is $G=K_n-H$ where $H$ is a cubic graph.
We show that \DGP~is NP-hard on $(n-4)$-regular graphs, by showing a reduction from \UC{} on cubic graphs, that is the complement of \textsc{Max Cut}. This last problem on cubic graphs was proved NP-hard and even not polynomial-time 1.003-approximable, unless P=NP \cite{BK99}.
\defprob{Min UnCut}{A graph $G=(V,E)$, an integer $k$.}{Does $G$ contain a partition of V into two parts $A, B$ such that the number of edges with both endpoints in the same part is at most $k$?}
Since we did not find a reference for the following result in the literature.
\begin{lemma}\label{perhapsknown}
\label{lem:borneUnCut}
Let $G=(V,E)$ be a cubic graph. There exists a partition $\{A,B\}$ of $G$ with a cut of size at least $|V|$ and it can be found in polynomial time.
\end{lemma}
\begin{proof}
Let $\mathcal{P} = \{A,B\}$ be a partition of $V$. Consider the following operation: if there is a vertex $v \in A$ (resp. $B$) with at least two neighbours in $A$ (resp. $B$) then $A = A \setminus \{v\} $ (resp. $B = B \setminus \{v\} $) and $B = B \cup \{v\} $ (resp. $A = A \cup \{v\} $). Since the graph is cubic, this operation increases the number of edges between $A$ and $B$ by at least one. Since the number of edges is finite, we can repeat this operation until we obtain a partition $P'=\{A',B'\}$ with no vertex $v \in A'$ (resp. $B'$) with at least two neighbours in $A'$ (resp. $B'$). Since the graph is cubic, if every vertex in $A'$ (resp. $B'$) has at most one neighbour in $A'$, then it has at least two neighbours in $B'$ (resp. $A'$). Consequently $P'$ has a cut of size at least $ \frac{2(|A'| + |B'|)}{2} = |V|$.
\end{proof}
\begin{figure}
\begin{center}
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\caption{The construction of $G'$ in Definition~\ref{denseNPdef}}
\label{fig:hardDens}
\end{center}
\end{figure}
\begin{definition}\label{denseNPdef}
Let $I = (G,k)$ be an instance of \UC~ where $G=(V,E)$ is a cubic graph. We define the construction $\sigma$ transforming the graph $G$ into the graph $G':= (V',E') =\sigma(G)$ (see Figure \ref{fig:hardDens}) as follows:
\begin{itemize}
\item let $G_0=(V_0,E_0)$ be the union of $\frac{n^2 - n}{6}$ copies of $K_{3,3}$ (see remark below). Thus $G_0$ is a cubic bipartite graph with $n^2 - n$ vertices and $V_0$ is the union of two independent sets $L, R$ such that $|L|=|R|$.
\item let $G_1 = (V \cup V_0, E \cup E_0)$.
\item let $G' = \overline{G_1}$.
\end{itemize}
\end{definition}
\begin{remark}
Note that we can assume that the number of vertices of a cubic graph $G$ is a multiple of $6$. Since $G$ is cubic, $n$ is a multiple of 2.
If $n$ is not a multiple of 3, we consider the instance $I_{triple}$ defined as follows: $G_{triple}$ is the union of 3 copies of $G$ and $k_{triple}=3k$, and thus in the new instance $I_{triple}$ the graph has $3n$ vertices.
\end{remark}
Let $n=|V|$, $m=|E|$, $n'=|V'|$ and $m'=|E'|$. Remark that $n' = n^2$, and $G'$ is a $(n'-4)$-regular graph.
Since we reduce from {Min UnCut} on cubic graphs, we use the following straight forward observation on any partition in such graphs.
\begin{lemma} For any cubic graph $G$ and any $\{A,B\}$ partition of $V$, we have
$ |A| + \frac{2}{3}\cdot |E(B)| = |B| + \frac{2}{3} \cdot |E(A)|$,
where $E(A)$, resp.~$E(B)$, are the set of edges with both endpoints in $A$, resp~$B$.
\label{lem:interneexterne}
\end{lemma}
\begin{proof}
Since the graph $G$ is cubic,
$E(A,B) = 3\cdot |A| - 2 \cdot|E_A| = 3\cdot |B| - 2 \cdot |E_B|$.
We can deduce that $|A| + \frac{2}{3} \cdot |E_B| = |B| + \frac{2}{3} \cdot |E_A|$
\end{proof}
\begin{theorem}\label{denseNP}
\DGP{} is NP-complete on $(n-4)$-regular graphs with $n$ vertices.
\end{theorem}
\begin{proof}
Let $I=(G=(V,E),k)$ be an instance of \UC, where $G$ is a cubic graph. Consider the following instance $I'$ of \DGP{} on the graph $G' = \sigma(G)$ and $d=\frac{n^2}{2} - 1 - \frac{2k}{n^2}$. We claim that $I= (G,k)$ is a yes-instance of \UC~ if and only if $I' = (G',d)$ is a yes-instance of \DGP.
\medskip
Let $\{A,B\}$ be a partition of $V$ whose uncut value is at most $k$. Since $V_0=L\cup R$, where $L,R$ are independent sets in $G_0$ such that $|L|=|R|$, the sets $L,R$ form two cliques of the same size in $G'$. Let $A' = A \cup L$ and $B' = B \cup R$ and $\mathcal{P}=\{A',B'\}$ be a partition of $G'$.
Let ${M}_{A'}$ and ${M}_{B'}$, be the set of missing edges in $G'[A']$ and $G'[B']$, respectively.
Due to the construction of $G'$, there is no missing edge between $A$ and $L$ and between $B$ and $R$. Thus all missing edges are inside $G'[A\cup B]$, then $|{M}_{A'}| + |{M}_{B'}| = k$. Thus, the density of the partition $\mathcal{P}$ is:
$$ d(\mathcal{P}) = \frac{|A'|-1}{2} - \frac{|{M}_{A'}|}{|A'|} + \frac{|B'|-1}{2} - \frac{|{M}_{B'}|}{|B'|}
= \frac{n^2-2}{2} - \frac{|{M}_{A'}|}{|A'|} - \frac{|{M}_{B'}|}{|B'|}
$$
We will prove in the following that $d(\mathcal{P})
\geq d=\frac{n^2}{2} - 1 - \frac{2k}{n^2}$ that is equivalent to proving that $\frac{|{M}_{A'}|}{|A'|} + \frac{|{M}_{B'}|}{|B'|} \leq \frac{2(|M_{A'}|+|M_{B'}|)}{|A'|+|B'|}$.
Thus the difference
$$ \frac{2(|M_{A'}|+|M_{B'}|)}{|A'|+|B'|} - \left(\frac{|M_{A'}|}{|A'|} + \frac{|M_{B'}|}{|B'|}\right)=$$
$$=\frac{1}{|A'|+|B'|} \left( 2|M_{A'}|+2|M_{B'}| - \frac{|A'|+|B'|}{|A'|} |M_{A'}|-\frac{|A'|+|B'|}{|B'|} |M_{B'}|\right)=$$
$$=\frac{1}{|A'|+|B'|} \frac{1}{|A'|} \frac{1}{|B'|} (|A'||B'| |M_{A'}|+ |A'||B'| |M_{B'}|-|B'|^2 |M_{A'}|- |A'|^2 |M_{B'}|)=
$$
$$ =\frac{1}{|A'|+|B'|} \frac{1}{|A'|} \frac{1}{|B'|} (|A'|-|B'|)(|B'||M_{A'}|-|A'||M_{B'}|)
$$
Wlog we can consider that $|A'|\geq |B'|$, that implies $|B'|\leq \frac{n^2}{2}$.
From Lemma~\ref{lem:interneexterne} for the cubic graph $G_1$ and partition $\{A', B'\}$, we have
$|A'|+ \frac{2}{3}\cdot |M_{B'}| = |B'| + \frac{2}{3} \cdot |M_{A'}|$. Using that $|A'|=n^2-|B'|$ and $|M_{A'}|=k-|M_{B'}|$, we have $n^2-|B'|+ \frac{2}{3}\cdot |M_{B'}| = |B'| + \frac{2}{3} \cdot (k-|M_{B'}|)$ and thus $ |M_{B'}|= \frac{3}{4} (2 |B'|+ \frac{2}{3} k-n^2)$.
Thus,
$$|B'||M_{A'}|-|A'||M_{B'}|= |B'| (k-|M_{B'}|) - (n^2-|B'|)|M_{B'}|=|B'|k-n^2 |M_{B'}|=$$
$$=|B'|k-n^2\frac{3}{4} \left(2 |B'|+ \frac{2}{3} k-n^2\right)=\left(|B'|-\frac{n^2}{2}\right)\left(k-\frac{3n^2}{2}\right)$$
Since $|B'|\leq \frac{n^2}{2}$ and $k \leq \frac{n}{2}\leq \frac{3n^2}{2}$ we can conclude that
$$ \frac{2(|M_{A'}|+|M_{B'}|)}{|A'|+|B'|} - \left(\frac{|M_{A'}|}{|A'|} + \frac{|M_{B'}|}{|B'|}\right) \geq 0$$
Thus, the partition $\mathcal{P}=\{A',B'\}$ has the density $d(\mathcal{P})
\geq d=\frac{n^2}{2} - 1 - \frac{2k}{n^2}$.
\bigskip
Let $\mathcal{P'}$ be a partition of $G'$ of density $d(\mathcal{P'}) \geq d = \frac{n^2-2}{2} - \frac{2k}{n^2}$. We will prove that $\mathcal{P'}$ has exactly two parts $A'$ and $B'$ such that $A = A' \cap V$ and $B = B' \cap V$ is a partition of $G$ whose uncut value is at most $k$.
Suppose that $|\mathcal{P'}| \geq 3$. Then, using Lemma~\ref{lem:densMax}, we have $d(\mathcal{P'}) \leq \frac{n^2 - |\mathcal{P'}|}{2} \leq \frac{n^2-3}{2} = \frac{n^2-2}{2} - \frac{1}{2}$. Since $k \leq \frac{n}{2}$ and $n \geq 6$ then $\frac{2k}{n^2} < \frac{1}{2}$. Then $d(\mathcal{P'}) < \frac{n^{2}-2}{2} - \frac{2k}{n^2} = d$ which is a contradiction. Then $|\mathcal{P'}| < 3$.
Suppose that $|\mathcal{P'}| = 1$. Since $G'$ is $(n^2-4)$-regular, its density is $d(\mathcal{P'}) = \frac{n^2-1}{2} - \frac{3}{2} = \frac{n^2-2}{2} - 1 < \frac{n^2-2}{2} - \frac{2k}{n^2} = d$ which is a contradiction. Then $|\mathcal{P'}| > 1$. We conclude that $|\mathcal{P}| = 2$.
Let $A'$ and $B'$ be the two parts of $\mathcal{P}$. Let ${M}_{A'}$, resp. ${M}_{B'}$, be the set of missing edges in $G'[A']$, resp. $G'[B']$. Remark that if $|M_{A'}| + |M_{B'}| \leq k$ then $|M_{A}| + |M_{B}| \leq k$ and then there is a cut of size at least $k$ between $A$ and $B$ in $G$. What it remains to prove is that $|M_{A'}| + |M_{B'}| \leq k$.
As a first step we will show the following inequality we need later $\frac{|{M}_{A'}| + |{M}_{B'}|}{\frac{n^2}{2} + \frac{|{M}_{A'}| + |{M}_{B'}|}{3}} \leq \frac{|{M}_{A'}|}{|A'|} + \frac{|{M}_{B'}|}{|B'|}$. In order to prove this, we consider the following difference
$$
\frac{|{M}_{A'}|}{|A'|} + \frac{|{M}_{B'}|}{|B'|} - \frac{|{M}_{A'}|+|{M}_{B'}|}{\frac{|A'|+|B'|}{2} + \frac{|{M}_{A'}| + |{M}_{B'}|}{3}}
$$
By removing the denominator we get
$$
|{M}_{A'}||B'|\left(\frac{|A'| + |B'|}{2} + \frac{|{M}_{A'}|+|{M}_{B'}|}{3}\right) + |{M}_{B'}||A'|\left(\frac{|A'| + |B'|}{2} + \frac{|{M}_{A'}|+|{M}_{B'}|}{3}\right)
$$
$$
- (|{M}_{A'}| + |{M}_{B'}|)|A'||B'| =
$$
$$
= |{M}_{A'}||B'| \left(\frac{|B'|}{2} + \frac{|{M}_{A'}|}{3} + \frac{|{M}_{B'}|}{3} - \frac{|A'|}{2}\right) + |{M}_{B'}||A'|\left(\frac{|A'|}{2} + \frac{|{M}_{B'}|}{3} + \frac{|{M}_{A'}|}{3} - \frac{|B'|}{2}\right) =
$$
From Lemma~\ref{lem:interneexterne} for the cubic graph $G_1$ and partition $\{A', B'\}$, we have $|A'| + \frac{2}{3}|{M}_{B'}| = |B'| + \frac{2}{3}|{M}_{A'}|$, which implies that $\frac{|A'|}{2} = \frac{|B'|}{2} + \frac{|{M}_{A'}|}{3} - \frac{|{M}_{B'}|}{3}$ and $\frac{|B'|}{2} = \frac{|A'|}{2} + \frac{|{M}_{B'}|}{3} - \frac{|{M}_{A'}|}{3}$ and then we get
$$
= |{M}_{A'}||B'| \left(\frac{|B'|}{2} + \frac{|{M}_{A'}|}{3} + \frac{|{M}_{B'}|}{3} - \left(\frac{|B'|}{2} + \frac{|{M}_{A'}|}{3} - \frac{|{M}_{B'}|}{3}\right)\right)
$$
$$
+ |{M}_{B'}||A'|\left(\frac{|A'|}{2} + \frac{|{M}_{B'}|}{3} + \frac{|{M}_{A'}|}{3} - \left(\frac{|A'|}{2} + \frac{|{M}_{B'}|}{3} - \frac{|{M}_{A'}|}{3}\right)\right) =
$$
$$
= |{M}_{A'}||B'|\frac{2|{M}_{B'}|}{3} + |{M}_{B'}||A'|\frac{2|{M}_{A'}|}{3}
$$
Since $|{M}_{A'}|$, $|{M}_{B'}|$, $|A'|$ and $|B'|$ are positive integers then $\frac{|{M}_{A'}|}{|A'|} + \frac{|{M}_{B'}|}{|B'|} - \frac{|{M}_{A'}| + |{M}_{B'}|}{\frac{n^2}{2} + \frac{|{M}_{A'}| + |{M}_{B'}|}{3}} \geq 0$. We conclude that $\frac{|{M}_{A'}| + |{M}_{B'}|}{\frac{n^2}{2} + \frac{|{M}_{A'}| + |{M}_{B'}|}{3}} \leq \frac{|{M}_{A'}|}{|A'|} + \frac{|{M}_{B'}|}{|B'|}$.
\medskip
Finally, we show that $|{M}_{A'}| + |{M}_{B'}| \leq k$ using the previous inequality. Let $x= |{M}_{A'}| + |{M}_{B'}|$. In order to finalize the proof, we suppose that $x> k$ and we will arrive at a contradiction, that is $d(\mathcal{P'}) < d$. Consider the following difference
$$
d - d(\mathcal{P'}) = \frac{n^2 - 2}{2} - \frac{2k}{n^2} - \left(\frac{n^2 - 2}{2} - \frac{|{M}_{A'}|}{|A'|} - \frac{|{M}_{B'}|}{|B'|}\right) = \frac{|{M}_{A'}|}{|A'|} + \frac{|{M}_{B'}|}{|B'|} - \frac{2k}{n^2}
$$
Since $\frac{x}{\frac{n^2}{2} + \frac{x}{3}} \leq \frac{|{M}_{A'}|}{|A'|} + \frac{|{M}_{B'}|}{|B'|}$ then
$$
d - d(\mathcal{P'}) \geq \frac{x}{\frac{n^2}{2}+\frac{x}{3}} - \frac{2k}{n^2} = \frac{x\cdot n^2 - k \cdot n^2 -\frac{2x\cdot k}{3}}{(\frac{n^2}{2}+\frac{x}{3}) \cdot n^2}
$$
Since $x$ and $k$ are integers, then $x\geq k+1$, and by removing the denominator, we get
$$
\geq (k+1) \cdot (n^2 - \frac{2}{3}\cdot k) - k\cdot n^2 = n^2 - \frac{2}{3}\cdot k^2 - \frac{2}{3}\cdot k
$$
Since $k \leq \frac{n}{2}$ it follows that $n^2 - \frac{2}{3}\cdot k^2 - \frac{2}{3}\cdot k > 0$. This finally gives $d(\mathcal{P'}) < d$, a contradiction to the choice of $\mathcal P'$ as partition with density at least $d$, and we hence conclude that $|{M}_{A'}| + |{M}_{B'}| \leq k$.
Overall, it follows that if $d(\mathcal{P'}) \geq \frac{n^{'}-2}{2} - \frac{2k}{n^2}$ then there is a partition $\{A,B\}$ with an uncut of size at most $k$.
\end{proof}
At the end of this section we show that a partition into three cliques provides a good approximation of the problem.
\begin{lemma}
\label{lem:borneDensNReg} Let $G=(V,E)$ be a $(n-4)$-regular graph and $\mathcal{P}$ any partition of $V$. Then $d(\mathcal{P}) \leq \frac{n}{2} - 1$.
\end{lemma}
\begin{proof}
If $|\mathcal{P}| = 1$ then $d(\mathcal{P}) = \frac{n-4}{2}$. Suppose that $|\mathcal{P}| \geq 2$, the density is maximized when for every $P \in \mathcal{P}$, $G[P]$ is a clique. Then $d(\mathcal{P}) = \sum\limits_{P \in \mathcal{P}}$ $\frac{|P| - 1}{2} \leq \frac{n}{2} - 1$.
\end{proof}
\begin{theorem}
There is an efficient polynomial time approximation scheme for \MDGP{} on $(n-4)$-regular graphs.
\end{theorem}
\begin{proof}
Let $I=G$ be a graph on $n$ vertices and $(n-4)$-regular, instance of \MDGP. Let $\overline{G}$ be the complementary graph of $G$. By Brooks' theorem, we know that there is a 3-coloration of $\overline{G}$ that can be found in polynomial time \cite{Karloff89}.
We establish in the following an eptas. Given $\varepsilon>0$, consider two cases. If $n\geq 3+\frac{1}{\varepsilon}$, then
let $\mathcal{P}$ be a partition, that corresponds to a 3-coloration of $\overline{G}$, that is each part is a clique in $G$. Then $d(\mathcal{P}) = \frac{n}{2} - \frac{3}{2}$. By Lemma~\ref{lem:borneDensNReg} we know that $opt(I) \leq \frac{n}{2} - 1$. Thus $d(\mathcal{P}) \geq \frac{n/2-1}{1+\varepsilon}\geq \frac{opt(I)}{1+\varepsilon}$.
Otherwise, that is
$n < 3+\frac{1}{\varepsilon}$, enumerate all the partitions of $G$ and consider the best one. Since the number of partitions of $G$ is the Bell number of order $|V|=n$, $B_n$, and that $B_n\leq n^n$, we get an optimal solution in time $O((1/\varepsilon)^ {1/\varepsilon})$. \end{proof}
\section{Conclusion}
In order to have a better understanding of the complexity of \MDGP{} it would be nice to study it on other graph classes. It was proved to be polynomial-time solvable on trees, but the complexity on graphs of bounded treewidth remains open. Moreover no result exists on split graphs.
Concerning the approximation, no lower bound was established, it would be nice to improve the 2-approximation algorithm or to show that no polynomial-time approximation scheme exist on general instances.
\bibstyle{plainurl}
| {
"timestamp": "2021-08-24T02:39:17",
"yymm": "2107",
"arxiv_id": "2107.13282",
"language": "en",
"url": "https://arxiv.org/abs/2107.13282",
"abstract": "We consider the problem of partitioning a graph into a non-fixed number of non-overlapping subgraphs of maximum density. The density of a partition is the sum of the densities of the subgraphs, where the density of a subgraph is its average degree, that is, the ratio of its number of edges and its number of vertices. This problem, called Dense Graph Partition, is known to be NP-hard on general graphs and polynomial-time solvable on trees, and polynomial-time 2-approximable. In this paper we study the restriction of Dense Graph Partition to particular sparse and dense graph classes. In particular, we prove that it is NP-hard on dense bipartite graphs as well as on cubic graphs. On dense graphs on $n$ vertices, it is polynomial-time solvable on graphs with minimum degree $n-3$ and NP-hard on $(n-4)$-regular graphs. We prove that it is polynomial-time $4/3$-approximable on cubic graphs and admits an efficient polynomial-time approximation scheme on graphs of minimum degree $n-t$ for any constant $t\\geq 4$.",
"subjects": "Computational Complexity (cs.CC)",
"title": "Dense Graph Partitioning on sparse and dense graphs",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9854964237163882,
"lm_q2_score": 0.7185943925708562,
"lm_q1q2_score": 0.7081722039812292
} |
https://arxiv.org/abs/2001.06994 | On the directions determined by a Cartesian product in an affine Galois plane | We prove that the number of directions contained in a set of the form $A \times B \subset AG(2,p)$, where $p$ is prime, is at least $|A||B| - \min\{|A|,|B|\} + 2$. Here $A$ and $B$ are subsets of $GF(p)$ each with at least two elements and $|A||B| <p$. This bound is tight for an infinite class of examples. Our main tool is the use of the Rédei polynomial with Szőnyi's extension. As an application of our main result, we obtain an upper bound on the clique number of a Paley graph, matching the current best bound obtained recently by Hanson and Petridis. | \section{Introduction}
Let $U$ be a subset of the Desargusian affine plane $AG(2,p)$, where $p$ is a prime number. A direction is \emph{determined} by $U$ if two points of $U$ lie on a line in that direction. We can coordinatize $AG(2,p)$ so that $U = \{(a_i,b_i) \colon 1 \leq i \leq |U|\}$, where $a_i,b_i \in GF(p)$ for all $1 \leq i \leq |U|$, and then the set of directions determined by $U$ is given by
\[ D = \left\{ \frac{b_i-b_j}{a_i-a_j} \colon 1 \leq i<j \leq n \right\}.\]
Note that $D$ is a subset of $GF(p)\cup \{\infty\}$.
The possible values of $|D|$ have been studied by various authors. For a survey of results on this topic see \cite{ts2} and \cite{BB}. A key tool in this area are the properties of \emph{lacunary polynomials}, which are polynomials with several consecutive coefficients equal to zero. R\'edei's monograph \cite{r}, as well as Ball and Blokhuis's chapter \cite{BB} contain many results on lacunary polynomials and their applications, one of which is a sharp lower bound of $(p+3)/2$ on the size of $D$ for sets of size exactly $p$, excepting lines. The size of $|D|$ has also been considered in the setting $AG(2,q)$, $q$ a prime power, see for example \cite{bbbss}.
R\'edei's method was later extended by Sz\H{o}nyi \cite{ts} to sets of size smaller than $p$. Our result uses Sz\H{o}nyi's extension of R\'edei's method and relies on the fact that in the case when the set is a Cartesian product, the relevant polynomials have a very special structure that can be exploited. We prove the following theorem, improving Sz\H{o}nyi's bound by a factor of two (up to lower order terms) for Cartesian product sets.
\begin{theorem}\label{t1}
Let $A,B\subset GF(p)$ be sets each of size at least two such that $|A||B| < p$. Then the set of points $A\times B\subset AG(2,p)$ determines at least $|A||B| - \min\{|A|,|B|\} + 2$ directions.
\end{theorem}
Let $d>1$ be a divisor of $p-1$ and let $Z_d$ be a multiplicative subgroup of size $d$ inside $GF(p)$. If a set $A$ satisfies $A-A\subset Z_d\cup\{0\}$, then all of its directions are elements of $Z_d\cup\{0,\infty\}$. Thus, as a corollary of Theorem 1, we obtain the following result, which was recently proved by Hanson and Petridis \cite{hp} using Stepanov's method.
\begin{cor}
Let $A\subset GF(p)$ be a set such that $A-A\subset Z_d\cup\{0\}$. Then
\[ |A|(|A|-1)\leq d.\]
\end{cor}
In particular, if $p$ is congruent to 1 modulo 4 and $d=(p-1)/2$ this gives an upper bound of $(\sqrt{2p-1}+1)/2$ on the clique number of the Paley graph $G_p$. Recall that the vertices of $G_p$ are the elements of $GF(p)$ with an edge between elements whose difference is a square in $GF(p)$.
Estimating the size of sets of the form $(A-A)/(A-A)$ played a crucial role in early sum-product estimates over finite fields, \cite{g,ks}, and it is still an important tool in proving sum-product type bounds (see e.g. in \cite{mprs}). For $A \subset GF(p)$ with $|A|^2 < p$, Theorem~\ref{t1} gives that
\[ \#\left\{ \frac{a-b}{c-d} \colon (a,b,c,d) \in A^4, a \neq b, c\neq d \right\} \geq |A|^2-|A|. \]
Hence there is a nonzero $x \in (A-A)/(A-A)$ such that the number of representations $x = (a-b)/(c-d)$ with $(a,b,c,d) \in A^4$ is at most $|A|^2-|A|$. Consider the set $A-x A = \{\alpha - x \beta \colon (\alpha,\beta) \in A^2 \}$. The number of representations of a $y \in \alpha - x \beta$ cannot be too high on average, otherwise $x$ would have many representations in $(A-A)/(A-A)$. This idea can be made rigorous using the Cauchy-Schwarz inequality. The corresponding result is recorded in the following corollary.
\begin{cor} Let $A$ be a subset of $GF(p)$ such that $|A|^2 <p$. There exist $a,b,c,d \in A$ such that $|(a-b)A + (c-d)A| \geq |A|^3/(2|A|-1)$.
\end{cor}
Our arguments for Cartesian products can be extended to a set in $GF(p)$ consisting of a union of two Cartesian products. A corollary of this is as follows.
\begin{cor}\label{cu} Let $A,B \subset GF(p)$ be disjoint sets each of size at least two such that each of the difference sets $A-A$, $A-B$, $B-B$ contain either only squares, or only non-squares, in addition to $0$. Then
\[ \min\{|A|^2-2|A|,|B|^2-2|B|\} + |A||B| + 2 \leq \frac{p+3}{2}.\]
\end{cor}
For a subset $A \subseteq GF(p)$, the directions determined by $A \times \{0,1\}$ is the set $(A-A)\cup \{\infty\}$. Hence by Theorem~\ref{t1} we recover an instance of the well known Cauchy-Davenport Theorem~\cite{D}.
\begin{cor} Let $A \subseteq GF(p)$ be nonempty, then $|A-A| \geq \min\{p,2|A|-1\}$.
\end{cor}
\section{R\'edei polynomials}
Let $U= \{(a_i,b_i) \colon 1 \leq i \leq |U|\}$ be a subset of the affine plane $AG(2,p)$, and $D$ be the set of directions determined by $U$. Suppose that $AG(2,p)$ is coordinatized so that $\infty \in D$. Put $n = |U|$. The R\'edei polynomial of $U$ is
\[ H(x,y) = \prod_{i=1}^n (x+a_iy-b_i) .\]
Consider $H_y(x) = H(x,y)$ as a polynomial with indeterminate $x$ and coefficients in $GF(p)[y]$. Define the set $A_y = \{-a_iy+b_i\}_{i=1}^n$. Observe that $H_y(x)$ divides $x^p-x$ if and only if the elements of $A_y$ are all distinct, and this is equivalent to $y \not\in D$. In the case $y \not\in D$, we see that $(x^p-x)/H_y(x)$ has a root at every element of $GF(p) \setminus A_y$, i.e. the coefficients of $(x^p-x)/H_y(x)$ are symmetric polynomials of the form $\sigma_k(GF(p)\setminus A_y)$, $k = 1,2,\ldots,p-n$. We can determine the symmetric polynomials $\sigma_k(GF(p)\setminus A_y)$ in terms of the symmetric polynomials $\{\sigma_i( A_y)\}_{i=1}^k$ recursively as follows.
For $1 \leq k< p-1$ we have $\sigma_k(GF(p)) = 0$ and so
\[ \sum_{i=0}^k \sigma_i(A_y)\sigma_{k-i}(GF(p)\setminus A_y) = 0.\]
This gives, for example
\[ \sigma_1(GF(p)\setminus A_y) = -\sigma_1(A_y), \quad \text{and} \quad \sigma_2(GF(p)\setminus A_y) = \sigma_1^2(A_y)-\sigma_2(A_y).\]
Continuing recursively we see that $\sigma_k(GF(p)\setminus A_y)$ is a polynomial in $GF(p)[y]$ of degree at most $k$ and can be defined even when the elements of $A_y$ are not all distinct. Put $m=p-n$ and define
\begin{equation}\label{extpoly} f(x,y) = x^m - \sigma_1(GF(p) \setminus A_y)x^{m-1} + \sigma_2(GF(p) \setminus A_y)x^{m-2} + \cdots +(-1)^m \sigma_m(GF(p) \setminus A_y). \end{equation}
Note that $f$ is a degree $m$ polynomial in $GF(p)[x,y]$ and crucially we have
\[ H(x,y)f(x,y) = x^p-x\]
for all $y \not\in D$.
For more on the construction and properties of $H$ and $f$ see \cite{ts,ts2}. Let
\[ H(x,y)f(x,y) = x^p + h_1(y)x^{p-1} + h_2(y)x^{p-2} + \cdots + h_p(y),\]
and note that deg$(h_i) \leq i$. Since $H(x,y)f(x,y) = x^p-x$, for every $y \not\in D$ we see that if $i \neq p-1$ then $h_i(y) = 0$ for all $y \not\in D$. Recall that there are $p+1$ directions in $AG(2,p$). Since $\infty \in D$, there are $p+1-|D|$ directions not in $D$, and all such directions are in $GF(p)$. This implies that $h_i \equiv 0$ if $i< p+1-|D|$. Equivalently, if $h_i \not\equiv 0$, then $|D| \geq p+1-i$. Therefore showing that there is a high degree term in this polynomial with a nonzero coefficient results in a lower bound on $|D|$. This is how we will proceed.
\medskip
\medskip
\section{Directions in Cartesian products}
Let $U$ be a Cartesian product set in $AG(2,p)$, i.e. there exists a coordinatization such that $U = A \times B$, where $A,B \subset GF(p)$. Assume that the elements of $A$ and $B$ are all distinct, and put $|A|=m$, $|B|=n$. Let $A = \{ a_i \colon 1 \leq i \leq m\}$ and $B = \{b_j \colon 1 \leq j \leq n\}$.
If $m=1$ or $n=1$ then $U$ is contained in a line and spans only one direction. Notice also that any subset of $AG(2,p)$ with at least $p+1$ elements determines all directions. This is because there are only $p$ parallel lines in each direction, and so for each direction there must be a line in that direction containing at least two points from the set. Consequently, we will assume that $m,n \geq 2$ and $mn<p$. Translating preserves the number of directions, and so we will assume $0 \in B$.
The R\'edei polynomial of $A \times B$ is
\[ H(x,y) = \prod_{i,j } (x+a_iy-b_j) .\]
Let $A_y = \{-a_iy+b_j \colon 1\le i\le m; 1\le j\le n \}$. Put $k = p-mn$ and define
\[ f(x,y) = x^k - \sigma_1(GF(p) \setminus A_y)x^{k-1} + \sigma_2(GF(p) \setminus A_y)x^{k-2} + \cdots + (-1)^k\sigma_k(GF(p) \setminus A_y).\]
We will consider the R\'edei polynomial in the horizontal direction, $y = 0$. Let
\begin{equation}\label{red0} H(x,0)f(x,0) = f(x,0)\prod_j (x-b_j)^m = x^p + c_1x^{p-1} + c_2x^{p-2} + \cdots + c_p,
\end{equation}
for some coefficients $c_1,\ldots,c_p \in GF(p)$. We will exploit the product structure of the polynomial above to obtain our result. We begin with a lemma.
\begin{lemma}\label{lem} Let $R,S \in GF(p)[x]$ be polynomials each with constant term equal to $1$ and $\deg{R} \geq 1$. Suppose that $R$ and $R'$ are relatively prime and that $R$ does not divide $S$. Then $x^{\text{deg}(R)+\text{deg}(S)+1}$ does not divide $R^m(x)S(x)-1$ for any positive integer $m$ such that $p$ does not divide $m$.
\end{lemma}
\begin{proof} Suppose for a contradiction that there exist $R$, $S$, and $m$ satisfying the conditions of the Lemma. Then there exists a polynomial $P(x) \in GF(p)[x]$ such that
\begin{equation}\label{lemeq} R^m(x) S(x) = 1 + x^{\text{deg}(R)+\text{deg}(S)+1}P(x) .
\end{equation}
Let $k = \text{deg}(R)$ and $n = \text{deg}(S)$. By differentiating (\ref{lemeq}) we obtain
\[ R^{m-1}(x) ( mR'(x)S(x) + R(x) S'(x) ) = x^{k+n} ((k+n+1)P(x) + xP'(x)).\]
Since the constant term in $R^{m-1}(x)$ is 1, we see that $x^{k+n}$ divides $mR'(x)g(x) + R(x) S'(x)$. But the degree of $mR'(x)S(x) + R(x) S'(x)$ is at most $k+n-1$ and so $mR'(x)S(x) + R(x) S'(x) = 0$. Since $R$ and $R'$ are relatively prime, it must be the case that $R$ divides $mS$. Since $m \neq 0$ in $GF(p)$ we see $R$ divides $S$, a contradiction.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{t1}]
Recall that if $c_i \neq 0$, then there are at least $p - i+1$ directions in $A \times B$. Suppose for a contradiction that $c_1 = c_2 = \ldots = c_{k+n-1} = 0$. Put $R(y) = \prod_{j=1}^n (1-b_jy)$, and $S(y) = y^kf(y^{-1},0)$. We see that $R(y),S(y) \in GF(p)[y]$, deg$(R) = n-1$ and deg$(S) \leq k$. Substitute $x = y^{-1}$ in (\ref{red0}) and multiply by $y^p$ to obtain
\[ R^m(y)S(y) = 1 + c_1y + c_2y^2 + \cdots + c_py^p = 1 + y^{k+n} Q(y),\]
for some polynomial $Q(y) \in GF(p)[y]$. Since the elements of $B$ are distinct, all roots of $R$ have multiplicity 1, and so $R$ is relatively prime to $R'$. Let $q$ be the highest power of $R$ dividing $S$. From the above we have
\begin{equation*}\label{aplem} R^{m+q} \left( \frac{S}{R^q} \right) = 1+y^{k-q(n-1)+n}[y^{q(n-1)}Q(y)].\end{equation*}
We have the following relations on the above variables.
\[ mn+k = p, \quad \text{and} \quad k-q(n-1) \geq 0.\]
It is easy to obtain the relation $m+q \leq p - m/(n-1)<p$ from the above. Therefore by Lemma~\ref{lem} we conclude that $\deg{R} = 0$, i.e. $R(y) = 1$. This gives $B = \{0\}$, which is a contradiction since we assumed $|B| \geq 2$. It follows that at least one of $c_1,\ldots,c_{k+n-1}$ is nonzero, and so there are at least $p-k-n+2 = mn-n+2$ directions in $A \times B$. By rotating the affine plane $90^{\circ}$ and repeating the argument we obtain the result
\begin{equation*}
\# \{ \text{Directions in } A \times B \} \geq |A||B| - \min\{|A|,|B|\} + 2 .
\end{equation*}
\end{proof}
We remark that in the proof of Theorem~\ref{t1}, Lemma 6 could be substituted by a similar and simple different argument. We conclude with a proof of Corollary~\ref{cu}.
\begin{proof}[Proof of Corollary~\ref{cu}] Let $A,B \subset GF(p)$ be as in the statement of the corollary. Let $|A| = m$, $|B|=n$, and put $A = \{a_i\}_{i=1}^m$, $B = \{b_j\}_{j=1}^n$, and $U =A \times A \cup B \times B$. Our strategy will be to bound the number of directions in $U$. Consequently, we can assume $0 \in A$ by translating $U$. Without loss of generality, we'll assume $m \geq n$. The R\'edei polynomial $H(x,y)$ of $U$ evaluated at $y=0$ is
\[ H(x,0) = \prod_{i=1}^m (x-a_i)^m \prod_{j=1}^n (x-b_j)^n .\]
Let $f(x,y)$ be the polynomial defined in (\ref{extpoly}) corresponding to $U$. Define $k = \deg(f) = p-m^2-n^2$. Put $H(x,0)f(x,0) = x^p + G(x)$ for some $G[x] \in GF(p)[x]$. Define $R(y) = \prod_{i=1}^m (1-a_iy) \prod_{j=1}^n (1-b_jy)$, and $S(y) = y^kf(y^{-1},0) \left( \prod_{i=1}^m (1-a_iy) \right)^{m-n}$. Note that $R(y),S(y) \in GF(p)[y]$, $\deg(R) =m+n-1$, and $\deg{S} \leq k + (m-1)(m-n)$. We make a similar substitution to that in the proof above as follows.
\[ y^p H(y^{-1},0)f(y^{-1},0) = R^n(y)S(y) = 1 + y^p G(y^{-1}) .\]
Let $q$ be the highest power of $R$ dividing $S$. The above gives
\begin{equation}\label{coreq} R^{m+q} \left( \frac{S}{R^q} \right) =1 + y^p G(y^{-1}) .\end{equation}
The following relations hold.
\[ m^2+n^2+k = p, \quad \text{and} \quad k+(m-1)(m-n) - q(m+n-1) \geq 0.\]
It is easy to check that the above implies $m+q <p$. The lowest degree term in $y^{p}G(y^{-1})$ is $p - \deg(G)$. Applying Lemma~\ref{lem} to (\ref{coreq}) gives
\[ p - \deg{G} \leq k + (m-1)(m-n) + m+n-1.\]
Recall that the number of directions in $U$ is at least $\deg(G) +1$. Therefore $U$ determines at least $n^2+mn-2n+2$ directions. Every direction in $U$ is a quotient of two squares, or two non-squares in $GF(p)$. Hence all directions are squares, or zero, or infinity. This amounts to no more than $\frac{p+3}{2}$ directions, thereby giving the required result.
\end{proof}
\section{Concluding remarks}
The bound of Theorem~\ref{t1} is sometimes tight. For example if $p = 41$ and $A = \{0,1,5,9,10\}$, i.e. a maximal Paley clique, then the directions determined by $A \times A$ are the quadratic residues and $0$ and $\infty$. This totals $22$ directions, matching the lower bound $5^2 - 5 +2=22$ given by Theorem~\ref{t1}. Interestingly, this is the largest square grid we have found in which Theorem~\ref{t1} is tight. An infinite class of examples achieving exactly the lower bound are the long rectangles $A=\{0,1\}$, $B = \{0,1,\ldots,n-1\}$, and $p > 2n$ or $A = \{0,1,2\}$, $B = \{0,1,\ldots,n-1\}$, $n$ odd, and $p>3n$.
It is worth noting that in the proof of Theorem~\ref{t1}, to show that the number of directions determined by $A \times B$ is at least $mn-n+2$, we used that the R\'edei polynomial at $y=0$ was of the form $H(x,0)=\prod_j (x-b_j)^m$. A set of points in $AG(2,p)$ has a R\'edei polynomial of this form if $n$ horizontal lines each contain exactly $m$ points, and so does not necessarily need to be a Cartesian product. For example, let $A = \{a_i\}_{i=1}^n$, $B = \{b_i\}_{i=1}^n$, be subsets of $GF(p)$ such that $n^2<p$ and $0 \not\in B$. Consider the following sets in $AG(2,p)$
\begin{enumerate}[i.]
\item $\{(a_i+a_j^2 , a_j) \colon 1 \leq i,j \leq n\}$,
\item $\{(b_i+b_j^{-1} , b_j) \colon 1 \leq i,j \leq n\}$.
\end{enumerate}
Note that the vertical direction is not necessarily determined by either of the above sets, and so the number of directions determined is only at least $n^2-n+1$. The below sets describe the reciprocal directions of the sets above, but exclude $\infty$ (formerly the direction $0$). Therefore the sets below each have size at least $n^2-n$.
\begin{enumerate}[I.]
\item $\{(x-y)(z-w)^{-1} + (z+w) \colon w,x,y,z \in A, z \neq w \}$,
\item $\{(x-y)(z-w)^{-1} - (zw)^{-1} \colon w,x,y,z \in B, z \neq w \}$.
\end{enumerate}
\section{Acknowledgements} The research of the first author was supported in part by a Four Year Doctoral Fellowship from the University of British Columbia. The research of the second author was supported in part by an NSERC Discovery grant and OTKA K 119528 grant. The work of the second author was also supported by the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No. 741420, 617747, 648017). The research of the third author was supported in part by Killam and NSERC doctoral scholarships. We also thank Sammy Luo for helpful comments.
| {
"timestamp": "2020-06-25T02:02:25",
"yymm": "2001",
"arxiv_id": "2001.06994",
"language": "en",
"url": "https://arxiv.org/abs/2001.06994",
"abstract": "We prove that the number of directions contained in a set of the form $A \\times B \\subset AG(2,p)$, where $p$ is prime, is at least $|A||B| - \\min\\{|A|,|B|\\} + 2$. Here $A$ and $B$ are subsets of $GF(p)$ each with at least two elements and $|A||B| <p$. This bound is tight for an infinite class of examples. Our main tool is the use of the Rédei polynomial with Szőnyi's extension. As an application of our main result, we obtain an upper bound on the clique number of a Paley graph, matching the current best bound obtained recently by Hanson and Petridis.",
"subjects": "Combinatorics (math.CO); Number Theory (math.NT)",
"title": "On the directions determined by a Cartesian product in an affine Galois plane",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
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"lm_q1q2_score": 0.7081722036751286
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https://arxiv.org/abs/2202.02637 | Proof of two conjectures on Askey-Wilson polynomials | We give positive answer to two conjectures posed by M. E. H Ismail in his monograph [Classical and quantum orthogonal polynomials in one variable, Cambridge University Press, 2005]. | \section{Introduction and main result}
The Askey-Wilson divided difference operator is defined by
\begin{align}
(\mathcal{D}_q f)(x)=\frac{\breve{f}\big(q^{1/2} z\big)
-\breve{f}\big(q^{-1/2} z\big)}{\breve{e}\big(q^{1/2}z\big)-\breve{e}\big(q^{-1/2} z\big)},\quad
z=e^{i\theta}, \label{0.3}
\end{align}
where $\breve{f}(z)=f\big((z+1/z)/2\big)=f(\cos \theta)$ for each polynomial $f$ and $e(x)=x$.
Here $0<q<1$ and $\theta$ is not necessarily a real number (see \cite[p. 300]{I05}). Hereafter, we denote $x(s)=(q^{s}+q^{-s})/2$ with $0<q<1$. Taking $e^{i\theta}=q^s$ in \eqref{0.3}, $\mathcal{D}_q$ reads
\begin{align*
\mathcal{D}_q f(x(s))= \frac{f\big(x(s+\frac{1}{2})\big)-f\big(x(s-\frac{1}{2})\big)}{x(s+\frac{1}{2})-x(s-\frac{1}{2})}.
\end{align*}
Set $\mathcal{D}^0_q\, f=f$ and $\mathcal{D}^1_q=\mathcal{D}_q$, and define $\mathcal{D}^k_q=\mathcal{D}_q(\mathcal{D}^{k-1}_q)$ for each $k=1,2, \dots$. The following two conjectures, which generalize the Sonin-Hahn problem, were posed by M. E. H Ismail in his monograph on Orthogonal Polynomials and Special Functions published in 2005 (see \cite[Conjecture 24.7.10 and Conjecture 24.7.11]{I05}) and revised in 2009:
\begin{conj}\label{C1}
If $(p_n)_{n\geq 0}$ and $(\mathcal{D}_q p_{n})_{n\geq 0}$, or the latter with a limiting case of $\mathcal{D}_q$, are two sequences of orthogonal polynomials, then $(p_n)_{n\geq 0}$ are multiples of the Askey-Wilson polynomials, or special or limiting cases of them.
\end{conj}
\begin{conj}\label{C2}
If $(p_n)_{n\geq 0}$ and $(\mathcal{D}^k_q p_{n+k})_{n\geq 0}$, or the latter with a limiting case of $\mathcal{D}_q$, are two sequences of orthogonal polynomials for some $k$, $k=1,2,\dots$, then $(p_n)_{n\geq 0}$ are multiples of the Askey-Wilson polynomials, or special or limiting cases of them.
\end{conj}
Define the average operator $\mathcal{S}_q$ by
\begin{align*
\mathcal{S}_q f(x(s))=\frac{f\big(x(s+\frac{1}{2})\big)+f\big(x(s-\frac{1}{2})\big)}{2}.
\end{align*}
for every polynomial $f$. In 2003, Ismail proved the following result (see \cite[Theorem 20.1.3]{I05}):
\begin{theorem}\label{T}
A second order operator equation of the form
\begin{align}\label{ismail}
f(x)\mathcal{D}^2_q\, y+g(x) \mathcal{S}_q \mathcal{D}_q \, y+h(x)\, y=\lambda_n\, y
\end{align}
has a polynomial solution $y_n(x)$ of exact degree $n$ for each $n=0,1,\dots$, if and only if $y_n(x)$ is a multiple of the Askey-Wilson polynomials, or special or limiting cases of them. In all these cases $f$, $g$, $h$, and $\lambda_n$ reduce to
\begin{align*}
f(x)&=-q^{-1/2}(2(1+\sigma_4)x^2-(\sigma_1+\sigma_3)x-1+\sigma_2-\sigma_4),\\[7pt]
g(x)&=\frac{2}{1-q} (2(\sigma_4-1)x+\sigma_1-\sigma_3), \quad h(x)=0,\\[7pt]
\lambda_n&=\frac{4 q(1-q^{-n})(1-\sigma_4 q^{n-1})}{(1-q)^2},
\end{align*}
or a special or limiting case of it, $\sigma_j$ being the jth elementary symmetric function of the Askey-Wilson parameters.
\end{theorem}
Virtually the above conjectures are summed up in one if we are able to prove Conjecture \ref{C2}. To do this, we prove that the sequences of polynomials appearing in Conjecture \ref{C2} satisfy, for each $k$, a second order operator equation of the form \eqref{ismail}. The important point to note here is that this argument would not lead to a satisfactory conclusion if we were not looking for the whole space of ``Askey-Wilson polynomials, or special or limiting cases of them".
\begin{theorem}\label{L}
If $(p_n)_{n\geq 0}$ and $(\mathcal{D}^k_q p_{n+k})_{n\geq 0}$, or the latter with a limiting case of $\mathcal{D}_q$, are two sequences of orthogonal polynomials for some $k$, $k=1,2,\dots$, then, for each $k$, $(\mathcal{D}^{k-1}_q p_{n+k-1})_{n\geq 0}$ are multiples of the Askey-Wilson polynomials, or special or limiting cases of them.
\end{theorem}
Fix $k$, $k=1,2,\dots$. It is easily seen that $(p_n)_{n\geq 0}$ is a sequence of orthogonal polynomials satisfying
\begin{align}\label{estructura}
\pi(x) \mathcal{D}^k_q p_{n}(x)=\sum_{j=-m}^m c_{n,j}\, p_{n+j}(x),\quad c_{n,-m}\not=0,
\end{align}
for a polynomial $\pi$ which does not depend on $n$, if and only if $(p_n)_{n\geq 0}$ and $(\mathcal{D}^k_q p_{n+k})_{n\geq 0}$ are sequences of orthogonal polynomials. Now, we can apply Theorem \ref{L} to conclude that for each $k$, $k=1,2,\dots$, $(\mathcal{D}^{k-1}_q p_{n+k-1})_{n\geq 0}$ are multiples of the Askey-Wilson polynomials, or special or limiting cases of them. In particular, taking $k=2$ and $m=2$ in \eqref{estructura}, we have the main result proved in \cite{KJ19}: $(p_n)_{n\geq 0}$ are multiples of the Askey-Wilson polynomials, or special or limiting cases of them. Neither in this work nor in \cite{KJ19} was possible to exclude the `limiting cases' in the last statement. If so, we would have positive answer to a particular case of another conjecture posed by Ismail (see \cite[Conjecture 24.7.9]{I05}).
\section{Proof of Theorem \ref{L}}
The following properties are well known:
\begin{align*}
\mathcal{D}_q (fg)&= (\mathcal{D}_q f)(\mathcal{S}_q g)+(\mathcal{S}_q f)(\mathcal{D}_q g),\\[7pt]
\mathcal{S}_q ( fg)&=\mathrm{U}_2 (\mathcal{D}_q f) (\mathcal{D}_q g) +(\mathcal{S}_q f) (\mathcal{S}_q g),\\[7pt]
\mathcal{S}^2_q f&=\alpha \mathrm{U}_2 \mathcal{D}^2_q f+\mathrm{U}_1 \mathcal{S}_q \mathcal{D}_q f+f,\\[7pt]
\mathcal{D}_q \mathcal{S}_q f&=\alpha \mathcal{S}_q \mathcal{D}_q f+\mathrm{U}_1\mathcal{D}^2_q f,
\end{align*}
for polynomials $f$ and $g$, where $\mathrm{U}_1(x)=(\alpha^2-1)x$, $\mathrm{U}_2(x)=(\alpha^2-1)(x^2-1)$, and $2 \alpha=q^{1/2}+q^{-1/2}$.
We leave it to the reader to verify by induction that
\begin{align*
\mathcal{D}^k_q(X f(x))=\gamma_k \mathcal{S}_q \mathcal{D}^{k-1}_q f(x)+\frac{q^{k/2}+q^{-k/2}}{2} X \mathcal{D}^k_q f(x).
\end{align*}
where
\begin{align*}
\gamma_k=\frac{q^{k/2}-q^{-k/2}}{q^{1/2}-q^{-1/2}}.
\end{align*}
Set $P_n^{[k]}=\gamma_n!/\gamma_{n+k}!\,\mathcal{D}^k_q P_{n+k}$, and so $P_n^{[k]}=\gamma_{n+1}^{-1}\mathcal{D}_q P^{[k-1]}_{n+1}$. Since $(P_n)_{n\geq 0}$ and $(P^{[k]}_n)_{n\geq 0}$, for a certain fixed $k$, are sequences of (monic) orthogonal polynomials, any three consecutive elements of these sequences satisfy
\begin{align}
\label{rec1}x P_{n+k}(x)&=P_{n+k+1}(x)+B_{n+k}P_{n+k}(x)+C_{n+k} P_{n+k-1}(x),\\[7pt]
\label{rec2}x P^{[k]}_{n-1}(x)&=P^{[k]}_{n}(x)+B^{[k]}_{n-1}P^{[k]}_{n-1}(x)+C^{[k]}_{n-1} P^{[k]}_{n-2}(x),
\end{align}
with $C_{n+k}\not=0$ and $C^{[k]}_{n-1}\not=0$. We apply $\mathcal{D}^{k}_q$ to \eqref{rec1} to get
\begin{align}
\label{1a}&\gamma_{k}\mathcal{S}_q P^{[k-1]}_{n}(x)+\frac{q^{k/2}+q^{-k/2}}{2} x \mathcal{D}_q P^{[k-1]}_{n}(x)\\[7pt]
\nonumber &\quad =\frac{\gamma_{n+k}}{\gamma_{n+1}}\mathcal{D}_q P^{[k-1]}_{n+1}(x)+B_{n+k-1}\mathcal{D}_q P^{[k-1]}_{n}(x)+\frac{\gamma_{n}}{\gamma_{n+k-1}}C_{n+k-1}\mathcal{D}_q P^{[k-1]}_{n-1}(x).
\end{align}
From \eqref{rec2} we have
\begin{align}
\label{1b}&\frac{1}{\gamma_n}x\mathcal{D}_q P^{[k-1]}_{n}(x)\\[7pt]
\nonumber &\quad =\frac{1}{\gamma_{n+1}}\mathcal{D}_q P^{[k-1]}_{n+1}(x)+\frac{1}{\gamma_n}B^{[k]}_{n-1} \mathcal{D}_q P^{[k-1]}_{n}(x)+\frac{1}{\gamma_{n-1}}C^{[k]}_{n-1} \mathcal{D}_q P^{[k-1]}_{n-1}(x).
\end{align}
We now apply $\mathcal{S}_q$ to \eqref{1a} and \eqref{1b} and, by combining the resulting equations, we can eliminate $\mathcal{S}_q \mathcal{D}_q P^{[k-1]}_{n-1}(x)$, and obtain the equation
\begin{align}
&\label{3} D_n(x) \mathcal{S}_q \mathcal{D}_q P^{[k-1]}_{n}(x)+E_n \mathrm{U}_2(x) \mathcal{D}^2_qP^{[k-1]}_{n}(x)+\frac{\gamma_k}{\gamma_{n-1}} C_{n-1}^{[k]}P^{[k-1]}_{n}(x)\\[7pt]
\nonumber &\quad =F_n\mathcal{S}_q \mathcal{D}_q P^{[k-1]}_{n+1}(x),
\end{align}
where
\begin{align*}
D_n(x)&=\left(\frac{q^{(k+1)/2}+q^{-(k+1)/2}}{2}\frac{1}{\gamma_{n-1}}C_{n-1}^{[k]}-\frac{\alpha}{\gamma_{n+k-1}}C_{n+k-1} \right)x-\frac{1}{\gamma_{n-1}}B_{n+k-1}C^{[k]}_{n-1}\\[7pt]
&\quad +\frac{1}{\gamma_{n+k-1}}B^{[k]}_{n-1}C_{n+k-1},\\[7pt]
E_n&=\frac{\gamma_{k+1}}{\gamma_{n-1}} C_{n-1}^{[k]}-\frac{1}{\gamma_{n+k-1}}C_{n+k-1},\\[7pt]
F_n&=\frac{\gamma_{n+k}}{\gamma_{n+1}\gamma_{n-1}} C^{[k]}_{n-1}-\frac{\gamma_{n}}{\gamma_{n+1}\gamma_{n+k-1}} C_{n+k-1}.
\end{align*}
Similarly, we can eliminate $\mathcal{S}_q \mathcal{D}_q P^{[k-1]}_{n+1}(x)$, and shift $n$ to $n+1$ to obtain the equation
\begin{align}
\label{4}&\widetilde{D}_n(x) \mathcal{S}_q \mathcal{D}_q P^{[k-1]}_{n+1}(x)-\widetilde{E}_n \mathrm{U}_2(x) \mathcal{D}^2_qP^{[k-1]}_{n+1}(x)-\frac{\gamma_k}{\gamma_{n+2}} P^{[k-1]}_{n+1}(x)\\[7pt]
\nonumber &\quad=F_{n+1}\mathcal{S}_q \mathcal{D}_q P^{[k-1]}_{n}(x),
\end{align}
where
\begin{align*}
\gamma_{n+2} \widetilde{D}_n(x)&=\frac{q^{k/2}+q^{-k/2}}{2} \frac{\gamma_{n}}{\gamma_{n+1}} x+B_{n+k}-\frac{\gamma_{n+k+1}}{\gamma_{n+1}} B_n^{[k]},\\[7pt]
\widetilde{E}_n&=\frac{\gamma_n \gamma_k}{\gamma_{n+1}\gamma_{n+2}}.
\end{align*}
We now apply $\mathcal{D}_q$ to \eqref{1a} and \eqref{1b} and, by combining the resulting equations, we can eliminate $\mathcal{D}^2_q P^{[k-1]}_{n-1}(x)$, and obtain the equation
\begin{align}\label{3b}
D_n(x)\mathcal{D}^2_q P^{[k-1]}_{n}(x)+E_n \mathcal{S}_q \mathcal{D}_q P^{[k-1]}_{n}(x)=F_{n} \mathcal{D}^2_q P^{[k-1]}_{n+1}(x).
\end{align}
Similarly, we can eliminate $\mathcal{D}^2_q P^{[k-1]}_{n+1}(x)$, and shift $n$ to $n+1$ to obtain the equation
\begin{align}\label{3a}
\widetilde{D}_n(x)\mathcal{D}^2_q P^{[k-1]}_{n+1}(x)-\widetilde{E}_{n} \mathcal{S}_q \mathcal{D}_q P^{[k-1]}_{n+1}(x)=F_{n+1}\mathcal{D}^2_q P^{[k-1]}_{n}(x).
\end{align}
Note that if $\mathcal{S}_q P^{[k]}_{n}(x(s_1))=0$ we have $P^{[k]}_{n}(x(s_1+1/2))=-P^{[k]}_{n}(x(s_1-1/2))$. Suppose that $\mathcal{D}_q P^{[k]}_{n}(x(s_1))=0$. So
$$
P^{[k]}_{n}(x(s_1+1/2))=P^{[k]}_{n}(x(s_1-1/2))=0,
$$
which is impossible. Thus $\mathcal{D}_q P^{[k]}_{n}(x)$ and $\mathcal{S}_q P^{[k]}_{n}(x)$ have no common zeros. After shifting $n$ to $n-1$ in \eqref{3a}, to obtain a contradiction, suppose that $F_n=0$, i.e.
$$
\widetilde{D}_{n-1}(x)\mathcal{D}_q P^{[k]}_{n}(x)=\widetilde{E}_{n-1} \mathcal{S}_q P^{[k]}_{n}(x).
$$
Since $\mathcal{D}_q P^{[k]}_{n}(x)$ and $\mathcal{S}_q P^{[k]}_{n}(x)$ have no common zeros, we have $\widetilde{E}_{n-1}=0$, which is impossible. (We can also conclude that $F_{n+1}\not=0$.) Multiplying \eqref{3a} by $F_n$ and using \eqref{3b} and \eqref{3}, we get
\begin{align}\label{6}
f_n(x)\mathcal{D}^2_q P^{[k-1]}_{n}(x)+g_n(x) \mathcal{S}_q \mathcal{D}_q P^{[k-1]}_{n}(x)+\frac{\gamma_k}{\gamma_{n-1}}\widetilde{E}_n C^{[k]}_{n-1}P^{[k-1]}_{n}(x)=0,
\end{align}
where
\begin{align*}
f_n(x)&= E_n \widetilde{E}_n\mathrm{U}_2-D_n(x) \widetilde{D}_n(x)+F_n F_{n+1},\\[7pt]
g_n(x)&=\widetilde{E}_n D_n(x)-E_n \widetilde{D}_n(x).
\end{align*}
We next claim that there exist nonzero numbers $r_n$ and two polynomials $f(x)$ and $g(x)$ of degree at most two and one, respectively, not simultaneously zero, such that
$$
f_n(x)=r_n\, f(x), \quad g_n(x)=r_n\, g(x).
$$
Indeed, multiplying \eqref{3b} by $F_{n+1}$ and \eqref{3a} by $D_n(x)$, we can eliminate $\mathcal{D}^2_q P^{[k-1]}_{n}(x)$, and obtain, using \eqref{4} and shifting $n$ to $n-1$, the equation
\begin{align}\label{7}
f_{n-1}(x)\mathcal{D}^2_q P^{[k-1]}_{n}(x)+g_{n-1}(x) \mathcal{S}_q \mathcal{D}_q P^{[k-1]}_{n}(x)+\frac{\gamma_k}{\gamma_{n+1}}E_{n-1}P^{[k-1]}_{n}(x)=0.
\end{align}
(If $D_n(x)=0$, we combine directly \eqref{3b} and \eqref{4} to obtain \eqref{7}.)
Suppose that $E_{n-1}=0$, i.e.
$$
f_{n-1}(x)\mathcal{D}_q P^{[k]}_{n-1}(x)=-g_{n-1}(x) \mathcal{S}_q P^{[k]}_{n-1}(x).
$$
Since $\mathcal{D}_q P^{[k]}_{n-1}(x)$ and $\mathcal{S}_q P^{[k]}_{n-1}(x)$ have no common zeros, we have $f_{n-1}(x)=g_{n-1}(x)=0$, which is imposible according to \eqref{6} after shifting $n$ to $n-1$. Thus, by combining \eqref{6} and \eqref{7}, we can eliminate $P^{[k-1]}_{n}(x)$, and obtain the equation
\begin{align*}
&\left(f_n(x)-\frac{\gamma_{n+1}}{\gamma_{n-1}}\frac{\widetilde{E}_n}{E_{n-1}} C^{[k]}_{n-1} f_{n-1}(x)\right) \mathcal{D}_q P^{[k]}_{n}(x)\\[7pt]
&\quad =-\left(g_n(x)-\frac{\gamma_{n+1}}{\gamma_{n-1}}\frac{\widetilde{E}_n}{E_{n-1}} C^{[k]}_{n-1} g_{n-1}(x)\right) \mathcal{S}_q P^{[k]}_{n}(x).
\end{align*}
Again, since $\mathcal{D}_q P^{[k]}_{n}(x)$ and $\mathcal{S}_q P^{[k]}_{n}(x)$ have no common zeros, the desired conclusion follows. This allows us to rewrite \eqref{6} as
\begin{align}\label{final}
f(x)\mathcal{D}^2_q P^{[k-1]}_{n}(x)+g(x) \mathcal{S}_q \mathcal{D}_q P^{[k-1]}_{n}(x)+\lambda_n\,P^{[k-1]}_{n}(x)=0,
\end{align}
where $f(x)$ and $g(x)$ are polynomials of degree at most two and one, respectively, not simultaneously zero, and $\lambda_n=\gamma_k/(r_n \gamma_{n-1})\widetilde{E}_n C^{[k]}_{n-1}\not=0$. Thus, by Theorem \ref{T}, $(P^{[k-1]}_{n})_{n\geq 0}$ are multiples of the Askey-Wilson polynomials, or special or limiting cases of them. Since now $(P_n)_{n\geq 0}$ and $(P^{[k-1]}_{n})_{n\geq 0}$ are two sequences of orthogonal polynomials, repeating the previous argument we conclude, for each $j=k-2, \dots, 0$, that $(P^{[j]}_{n})_{n\geq 0}$ are multiples of the Askey-Wilson polynomials, or special or limiting cases of them. The rest of the proof is trivial.
\section*{Acknowledgements}
The authors thank to Professor T. H. Koornwinder for helpful discussions and comments. This work was supported by the Centre for Mathematics of the University of Coimbra-UIDB/00324/2020, funded by the Portuguese Government through FCT/ MCTES. The second author thanks the support of the ERDF and Consejería de Economía, Conocimiento, Empresas y Universidad de la Junta de Andalucía (grant UAL18-FQM-B025-A).
| {
"timestamp": "2022-03-29T02:48:15",
"yymm": "2202",
"arxiv_id": "2202.02637",
"language": "en",
"url": "https://arxiv.org/abs/2202.02637",
"abstract": "We give positive answer to two conjectures posed by M. E. H Ismail in his monograph [Classical and quantum orthogonal polynomials in one variable, Cambridge University Press, 2005].",
"subjects": "Classical Analysis and ODEs (math.CA)",
"title": "Proof of two conjectures on Askey-Wilson polynomials",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9854964224384746,
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https://arxiv.org/abs/2102.01986 | A unified half-integral Erdős-Pósa theorem for cycles in graphs labelled by multiple abelian groups | Erdős and Pósa proved in 1965 that there is a duality between the maximum size of a packing of cycles and the minimum size of a vertex set hitting all cycles. Such a duality does not hold if we restrict to odd cycles. However, in 1999, Reed proved an analogue for odd cycles by relaxing packing to half-integral packing. We prove a far-reaching generalisation of the theorem of Reed; if the edges of a graph are labelled by finitely many abelian groups, then there is a duality between the maximum size of a half-integral packing of cycles whose values avoid a fixed finite set for each abelian group and the minimum size of a vertex set hitting all such cycles.A multitude of natural properties of cycles can be encoded in this setting, for example cycles of length at least $\ell$, cycles of length $p$ modulo $q$, cycles intersecting a prescribed set of vertices at least $t$ times, and cycles contained in given $\mathbb{Z}_2$-homology classes in a graph embedded on a fixed surface. Our main result allows us to prove a duality theorem for cycles satisfying a fixed set of finitely many such properties. | \section{Introduction}
A classical theorem of Erd\H{o}s and P\'{o}sa~\cite{ErdosP1965} states that every graph contains either~$k$ vertex-disjoint cycles
or a vertex set of size at most~${\mathcal{O}(k \log k)}$ that hits all cycles of~$G$.
Such a theorem does not hold if we restrict to odd cycles; Lov\'{a}sz and Schrijver~(see~\cite{Thomassen1988})
found a class of graphs having no two vertex-disjoint odd cycles and no small vertex set hitting all odd cycles.
In the setting of odd cycles, Reed~\cite{Reed1999} obtained an analogue of the theorem of Erd\H{o}s and P\'{o}sa by relaxing the ``vertex-disjoint'' condition.
A \emph{half-integral packing} is a set of subgraphs such that no vertex is contained in more than two of them.
Reed~\cite{Reed1999} proved that there is a function~$f$ such that
every graph has a half-integral packing of at least~$k$ odd cycles
or has a vertex set of size at most~${f(k)}$ hitting all odd cycles.
As an easy corollary of Reed's result, given a graph whose edges are labelled with~$\mathbb{Z}_2$, there is a half-integral packing of at least~$k$ cycles, each of non-zero total weight,
or a vertex set of size at most~${f(k)}$ hitting all such cycles.
Very recently, Thomas and Yoo~\cite{YooR2020} extended this result to arbitrary abelian groups:
they showed that there is a function~$f$ such that given a graph whose edges are labelled by an abelian group, there is a half-integral packing of at least~$k$ cycles each of non-zero total weight,
or a vertex set of size at most~${f(k)}$ hitting all such cycles.
Kakimura and Kawarabayashi~\cite{KakimuraK2013} proved a different kind of strengthening of the theorem of Reed.
They showed that there is a function~$f$ such that every graph
contains a half-integral packing of~$k$ odd cycles each of which intersects a prescribed set~$S$ of vertices
or a vertex set of size at most~${f(k)}$ hitting all such cycles.
When~$S$ is the entire vertex set of the graph, this is equivalent to the theorem of Reed.
This result can be encoded in the setting of graphs labelled with two abelian groups~$\mathbb{Z}$ and~$\mathbb{Z}_2$:
in~$\mathbb{Z}$ we label each edge incident with a vertex in~$S$ with~$1$, and all other edges with~$0$,
and in~$\mathbb{Z}_2$ we label each edge with~$1$.
The cycles which are of non-zero total weight with respect to both of these group labellings are precisely the cycles described by Kakimura and Kawarabayashi.
Note that, since two groups are required for the encoding, this result is not covered by the previously mentioned result of Thomas and Yoo.
Our main theorem generalises all of these results to the setting of cycles in graphs labelled with a bounded number of abelian groups, whose values avoid a bounded number of elements of each group.
For an abelian group~${\Gamma}$ and a graph~$G$, a function~${\gamma \colon E(G) \to \Gamma}$ is called a \emph{$\Gamma$-labelling} of~$G$.
The \emph{$\gamma$-value} of a subgraph~$H$ of~$G$ is the sum of~${\gamma(e)}$ over all edges~$e$ in~$H$.
For an integer~$m$, we write~${[m]}$ for the set of positive integers at most~$m$.
We call a set of vertices which hits all subgraphs in a set~$\mathcal{H}$ a \emph{hitting set for~$\mathcal{H}$}.
\begin{restatable}{theorem}{mainthm}
\label{thm:main}
For every pair of positive integers~$m$ and~$\omega$, there is a function~${f_{m,\omega} \colon \mathbb{N} \to \mathbb{N}}$ satisfying the following property.
For each~${i \in [m]}$, let~$\Gamma_i$ be an abelian group, and let~$\Omega_i$ be a subset of~$\Gamma_i$.
Let~$G$ be a graph and for each~${i \in [m]}$, let~${\gamma_i}$ be a~${\Gamma_i}$-labelling of~$G$, and let~${\mathcal{O}}$ be the set of all cycles of~$G$ whose $\gamma_i$-value is in~${\Gamma_i \setminus \Omega_i}$ for all~${i \in [m]}$.
If~${\abs{\Omega_i} \leq \omega}$ for all~${i \in [m]}$,
then for all~${k \in \mathbb{N}}$ there exists
either a half-integral packing of~$k$ cycles in~$\mathcal{O}$,
or a hitting set for~$\mathcal{O}$ of size at most~${f_{m,\omega}(k)}$.
\end{restatable}
We point out that the function~${f_{m, \omega}}$ does not depend on the choice of groups~${\Gamma_i}$ or the subsets~$\Omega_i$.
For a graph labelled with a single abelian group~$\Gamma$,
Wollan~\cite{Wollan2011} showed that if~$\Gamma$ has no element of order~$2$, then there are arbitrarily many vertex-disjoint cycles of non-zero $\gamma$-value, or a hitting set of bounded size for the $\gamma$-non-zero cycles.
However, if~$\Gamma$ has an element of order~$2$, then as in the case of odd cycles, this does not hold~\cite{Thomassen1988, Reed1999}.
Hence, even when~${m = \omega = 1}$, the half-integral condition cannot be removed in Theorem~\ref{thm:main}.
If the number~$m$ of given groups~${\Gamma_i}$ is unbounded or the size of~${\Omega_i}$ is unbounded in Theorem~\ref{thm:main}, then such a function~$f_{m, \omega}$ does not exist, and moreover, for every integer $n\ge 2$, a ${(1/n)}$-integral analogue of the Erd\H{o}s-P\'{o}sa theorem does not hold.
We discuss this in Section~\ref{sec:overview}.
\medskip
We now present some corollaries which help illustrate the power of this theorem. In the following corollaries, we use the function $f_{m,\omega}$ in Theorem~\ref{thm:main}.
As already mentioned, the cycles in a graph~$G$ which intersect a prescribed set of vertices can be encoded as precisely the non-zero cycles with respect to the $\mathbb{Z}$-labelling which assigns value~$1$ to edges incident with vertices in~$S$ and~$0$ to all other edges.
If instead each edge of~$G$ is assigned the integer that is the number of its endvertices which lie in~$S$, then the cycles of total weight at least~${2t}$ are precisely the cycles which intersect the set~$S$ at least~$t$ times.
Using multiple labellings with~$\mathbb{Z}$, we can encode the set of cycles which intersect each of a bounded number of sets at least~$t$ times each.
Thus we obtain our first corollary.
\begin{corollary}
\label{cor:S_i-intersecting}
Let~$m$ and~$t$ be positive integers.
For each~${i \in [m]}$, let~${S_i}$ be a subset of the vertices of a graph~$G$ and~${t_i \in [t]}$, and let~$\mathcal{O}$ be the set of all cycles of~$G$ containing at least~$t_i$ vertices of~$S_i$ for all~${i \in [m]}$.
Then for all~${k \in \mathbb{N}}$ there exists
either a half-integral packing of~$k$ cycles in~$\mathcal{O}$,
or a hitting set for~$\mathcal{O}$ of size at most~${f_{m,t}(k)}$.
\end{corollary}
In fact, the construction above can be generalised, allowing us to convert a group labelling of the vertices of a graph to a group labelling of its edges.
For an abelian group~$\Gamma$ and a graph~$G$, a \emph{$\Gamma$-vertex-labelling of~$G$} is a function~${\gamma \colon V(G) \to \Gamma}$, and the \emph{$\gamma$-value} of a subgraph~$H$ of~$G$ is the sum of~${\gamma(v)}$ over all vertices in~$H$.
In Section~\ref{sec:overview}, we will discuss in detail how such a conversion works in general, and thus obtain the following corollary.
\begin{restatable}{corollary}{vxlabel}
\label{cor:vxlabelling}
Let~$m$ and~$\omega$ be positive integers.
For each~${i \in [m]}$, let~$\Gamma_i$ be an abelian group and let~$\Omega_i$ be a subset of~$\Gamma_i$.
Let~$G$ be a graph, and for each~${i \in [m]}$, let~${\gamma_i}$ be a~${\Gamma_i}$-vertex-labelling of~$G$, and let~${\mathcal{O}}$ be the set of all cycles of~$G$ whose $\gamma_i$-value is in~${\Gamma_i \setminus \Omega_i}$ for all~${i \in [m]}$.
If~${\abs{\Omega_i} \leq \omega}$ for all~${i \in [m]}$,
then for all~${k \in \mathbb{N}}$ there exists
either a half-integral packing of~$k$ cycles in~$\mathcal{O}$,
or a hitting set for~$\mathcal{O}$ of size at most~${f_{m,\omega}(k)}$.
\end{restatable}
Our next corollary relates to graphs labelled with a fixed finite abelian group, where we obtain a similar result for the set of cycles of any specified value.
In particular, this shows that for every pair of positive integers~${p}$ and~${q}$, cycles of length~${p}$ modulo~${q}$ satisfy a half-integral analogue of the Erd\H{o}s-P\'{o}sa theorem.
Dejter and Neumann-Lara~\cite{DejterN1988} showed that without the half-integral relaxation, the analogous Erd\H{o}s-P\'{o}sa type result fails for cycles of length~$p$ modulo~$q$ whenever the least common multiple~${\lcm(p,q)}$ of~$p$ and~$q$ is divisible by~$2p$ (see also~\cite{Wollan2011}).
When this condition is not met, Gollin, Hendrey, Kwon, Oum, and Yoo~\cite{GollinHKOY21} show that the half-integral relaxation is required for an Erd\H{o}s-P\'{o}sa type result in this setting if and only if~${\lcm(p,q)/p}$ is divisible by three distinct primes.
\begin{corollary}
\label{cor:specific-value-hiEP}
Let~${\Gamma}$ be a finite abelian group and let~${g \in \Gamma}$.
Let~$G$ be a graph, and~$\gamma$ be a~${\Gamma}$-labelling of~$G$,
and let~$\mathcal{O}$ be the set of all cycles of~$\gamma$-value~$g$.
Then for all~${k \in \mathbb{N}}$ there exists
either a half-integral packing of~$k$ cycles in~$\mathcal{O}$,
or a hitting set for~$\mathcal{O}$ of size at most~${f_{1, \abs{\Gamma}-1}(k)}$.
\end{corollary}
Our next corollary relates to graphs embedded on a fixed compact surface, where we obtain a similar result for the set of cycles contained in any given set of (first) $\mathbb{Z}_2$-homology classes.
Huynh, Joos, and Wollan~\cite{HuynhJW2017} proved that for graphs embedded on a fixed surface, cycles not homologous to zero in the $\mathbb{Z}$-homology group satisfy a half-integral analogue of the Erd\H{o}s-P\'{o}sa theorem.
They used a different type of graph labelling, called a directed $\Gamma$-labelling.
With (undirected) $\Gamma$-labellings, we can do the same thing with respect to $\mathbb{Z}_2$-homology classes. Since a compact surface has a finite abelian group as its $\mathbb{Z}_2$-homology group, we can obtain a half-integral analogue of the Erd\H{o}s-P\'{o}sa theorem for cycles contained in any given set of $\mathbb{Z}_2$-homology classes.
We discuss this further in Section~\ref{sec:overview}.
\begin{corollary}
\label{cor:surface}
Let~$\Sigma$ be a compact surface with $\mathbb{Z}_2$-homology group $\Gamma$ and let~${\mathcal{C}}$ be a set of $\mathbb{Z}_2$-homology classes of~$\Sigma$.
Let~$G$ be a graph embedded on~$\Sigma$, and let~$\mathcal{O}$ be the set of all cycles of~$G$
whose $\mathbb{Z}_2$-homology classes are contained in~$\mathcal{C}$.
Then for all~${k \in \mathbb{N}}$ there exists
either a half-integral packing of~$k$ cycles in~$\mathcal{O}$,
or a hitting set for~$\mathcal{O}$ of size at most~${f_{1, \abs{\Gamma}-\abs{\mathcal{C}}}(k)}$.
\end{corollary}
One nice feature of our main theorem is that it allows us to combine various properties of cycles together to obtain new results,
as long as we take a bounded number of properties and can encode each of them with a bounded number of group labellings.
Thus, we could combine any subset of these corollaries together and obtain a result of the same form.
\medskip
Huynh, Joos, and Wollan~\cite{HuynhJW2017} obtained a result similar to our main theorem for graphs with two directed group labellings, where the value of an edge is inverted if it is traversed in the reverse direction.
They showed that a half-integral analogue of the Erd\H{o}s-P\'{o}sa theorem holds for cycles whose values are non-zero in each coordinate.
They conjectured that their result can be extended to graphs with more than two directed labellings.
Because the~${\Gamma}$-labellings in this paper are equivalent to directed ${\Gamma}$-labellings when all elements of~${\Gamma}$ have order~$2$,
Theorem~\ref{thm:main} implies that the conjecture of Huynh, Joos, and Wollan hold for graphs with a fixed number of directed labellings with such groups.
The conjecture otherwise remains open, although there is a large overlap between the motivation of the conjecture and the consequences of our main theorem.
We discuss directed group labellings in more detail in Section~\ref{sec:conclusion}.
\medskip
The structure of this paper is as follows.
In Section~\ref{sec:prelim}, we introduce preliminary concepts, and we give a high-level overview of the proof of our main theorem and present proofs of corollaries in Section~\ref{sec:overview}.
In Section~\ref{sec:pack}, we define a packing function and provide its application.
In Section~\ref{sec:cleanwalls}, we define a concept of a clean wall which is well-behaved for each of the labellings~$\gamma_i$.
In Section~\ref{sec:handles}, we prove a key lemma to find many vertex-disjoint paths attached to a wall.
In Section~\ref{sec:abelian}, we prove useful lemmas on a product of abelian groups that will be used in the last step of the main theorem.
Section~\ref{sec:handles2cycles} discusses how to obtain a desired cycle from a wall together with attached disjoint paths.
We prove our main result in Section~\ref{sec:main},
and we discuss some open problems in Section~\ref{sec:conclusion}.
\section{Preliminaries}
\label{sec:prelim}
In this paper, all graphs are undirected simple graphs having no loops and multiple edges. For every abelian group, we regard its operation as an additive operation and denote its zero by~$0$.
Even though we work on simple graphs, all the results are extended to multigraphs; given a multigraph instance, we can take a subdivision to produce an equivalent simple graph with group value~$0$ on new edges.
For an integer~$m$, we write~${[m]}$ for the set of positive integers at most~$m$.
Let~$G$ be a graph.
We denote by~${V(G)}$ and~${E(G)}$ the vertex set and the edge set of~$G$, respectively.
For a vertex set~$A$ of~$G$, we denote by~${G - A}$ the graph obtained from~$G$ by deleting all the vertices in~$A$ and all edges incident with vertices in~$A$,
and denote by~${G[A]}$ the subgraph of~$G$ induced by~$A$, which is~${G-(V(G)\setminus A)}$.
If~${A = \{v\}}$, then we write~${G - v}$ for~${G - A}$.
For an edge~$e$ of~$G$, we denote by~${G - e}$ the graph obtained by deleting~$e$.
For two graphs~$G$ and~$H$, let
\[
{G \cup H := (V(G) \cup V(H), E(G) \cup E(H))}
\ \textnormal{ and } \
{G \cap H := (V(G) \cap V(H), E(G) \cap E(H))}.
\]
For a set~$\mathcal{G}$ of graphs, we denote by~${\bigcup \mathcal{G}}$ the union of the graphs in~$\mathcal{G}$.
For an integer~${t}$, a graph~$G$ is \emph{$t$-connected} if it has more than~$t$ vertices and~${G-S}$ is connected for all vertex sets~$S$ with~${\abs{S}<t}$.
\emph{Subdividing} an edge~$uv$ in a graph~$G$ is an operation that yields a graph containing one new vertex~$w$, and with an edge set replacing~$uv$ by two new edges,~$uw$ and~$wv$.
A graph~$H$ is a \emph{subdivision} of a graph~$G$ if~$H$ can be obtained from~$G$ by subdividing edges repeatedly.
Let~$A$ and~$B$ be vertex sets of~$G$.
An \emph{${(A, B)}$-path} is a path from a vertex in~$A$ to a vertex in~$B$ such that all internal vertices are not contained in~${A \cup B}$.
We also denote an~${(A,A)}$-path as an \emph{$A$-path}.
For a subgraph~$H$ of~$G$, we shortly write as an $H$-path for a ${V(H)}$-path.
A path is \emph{$A$-intersecting} if it contains a vertex of~$A$.
For a graph~$G$, we denote by~$\ensuremath{V_{\neq 2}}(G)$ the set of all vertices of~$G$ whose degrees are not equal to~$2$.
A \emph{corridor} of a graph~$G$ is a $\ensuremath{V_{\neq 2}}(G)$-path of length at least~$1$.
\begin{remark}
Every graph in which no block is a cycle is the edge-disjoint union of its corridors.
\end{remark}
For a family~${\mathcal{F} = ( x_i \colon i \in I )}$ we write $\abs{\mathcal{F}}=\abs{I}$, called the \emph{size} of~$\mathcal{F}$.
\subsection{Walls}
Let~${c,r \geq 3}$ be integers.
The \emph{elementary $(c,r)$-wall $W_{c,r}$} is the graph obtained from the graph on the vertex set~${[2c] \times [r]}$ whose edge set is
\begin{align*}
\left\{ (i,j) (i+1,j) \, \colon \, i \in [2c-1],\, j \in [r] \right\}
\cup
\left\{ (i,j) (i,j+1) \, \colon \, i \in [2c],\, j \in [r-1],\, i+j \textnormal{ is odd} \right\}
\end{align*}
by deleting both degree~$1$ vertices.
\begin{figure}[h]
\centering
\begin{tikzpicture}[scale=0.5, decoration={markings, mark=at position 0.5 with {\arrow{>}}}]
\tikzstyle{w}=[circle,draw,fill=black!50,inner sep=0pt,minimum width=3pt]
\draw[gray] (1,0)--(9,0);
\draw[gray] (0,6)--(8,6);
\foreach \y in {1,2,3,4,5}{
\draw[gray] (0,\y)--(9,\y);
}
\foreach \x in {0, 2, 4, 6, 8}{
\foreach \y in {1,3,5}{
\draw[gray] (\x, \y+1)-- (\x, \y);
\draw[gray] (\x+1, \y)-- (\x+1, \y-1);
}
}
\foreach \x in {2, 4, 6}{
\foreach \y in {3,5}{
\draw[very thick] (\x, \y+1)-- (\x, \y);
\draw[very thick] (\x+1, \y)-- (\x+1, \y-1);
\draw[very thick] (\x, \y)-- (\x+1, \y);
\draw[very thick] (\x+1, \y-1)-- (\x, \y-1);
}
}
\foreach \x in {2,4,6}{
\foreach \y in {1}{
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\draw[very thick] (\x+1, \y)-- (\x+1, \y-1);
\draw[very thick] (\x, \y)-- (\x+1, \y);
\draw (\x+1, \y-1)-- (\x, \y-1);
}
}
\foreach \x in {2,4}{
\draw[very thick] (\x+1, 7-1)-- (\x, 7-1);
}
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\draw[very thick] (3, \y)--(5,\y);
\draw[very thick] (5, \y)--(7,\y);
}
\draw[very thick] (3, 6)--(6,6);
\node at (2, .5) {};
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\node at (1, 1.5) {};
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}
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}
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\node at (\x,0) [w] {};
}
\end{tikzpicture}
\qquad
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\draw [gray](v\x\y)--(v\nextx\y);
}{}
\pgfmathtruncatemacro{\nexty}{\y+1}
\pgfmathtruncatemacro{\sum}{\x+\y}
\ifthenelse{\isodd{\sum}\and \y<6}{
\draw [gray](v\x\y)--(v\x\nexty);
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\ifthenelse{\x>1}{
\pgfmathtruncatemacro{\start}{1};
}{
\pgfmathtruncatemacro{\start}{2};
}
\ifthenelse{\x<8}{
\pgfmathtruncatemacro{\t}{5};
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\pgfmathtruncatemacro{\t}{4};
}
\foreach \y in {\start,...,\t} {
\pgfmathtruncatemacro{\nextx}{\x+1}
\ifthenelse{\x<7}{
\draw [very thick](v\x\y)--(v\nextx\y);
}{}
\ifthenelse{\x=7 \and \y<5}{
\draw [very thick](v\x\y)--(v\nextx\y);
}{}
\pgfmathtruncatemacro{\nexty}{\y+1}
\pgfmathtruncatemacro{\sum}{\x+\y}
\ifthenelse{\isodd{\sum}\and \y<5}{
\draw [very thick](v\x\y)--(v\x\nexty);
}{}
}
}
\end{tikzpicture}
\caption{An elementary ${(5, 7)}$-wall~$W$ depicted on the left with a $3$-column-slice, and depicted on the right with a ${(4,5)}$-subwall that is ${\ensuremath{V_{\neq 2}}(W)}$-anchored.}
\label{fig:wall}
\end{figure}
For ${j \in [r]}$, the \emph{$j$-th row}~$R_j$ of~$W_{c,r}$ is the path~${W_{c,r} \big[ \big\{ (i, j)\in V(W_{c,r}) \, \colon \, i \in [2c] \big\} \big]}$.
For ${i \in [c]}$, the \emph{$i$-th column}~$C_i$ of~$W_{c,r}$ is the path~${W_{c,r} \big[ \big\{ (i',j)\in V(W_{c,r}) \, \colon \, i' \in \{ 2i - 1, 2i \}, \, j \in [r] \big\} \big]}$.
A \emph{$(c,r)$-wall} is a subdivision~$W$ of the elementary $(c,r)$-wall.
If~$W$ is a~$(c,r)$-wall for some suitable integers~$c$ and~$r$, then we say~$W$ is a \emph{wall of order~$\min\{c,r\}$}.
We call a branch vertex corresponding to the vertex~${(i,j)}$ of the elementary wall a \emph{nail} of~$W$, and denote by~$N^W$ the set of nails of~$W$.
\begin{remark}
\label{rmk:wall3connected}
Any wall is a subdivision of a $3$-connected planar graph.
\end{remark}
For a subgraph~$H$ of the elementary wall, we denote by~$H^W$ the subgraph of~$W$ corresponding to a subdivision of~$H$.
We call~$R_j^W$ or $C_i^W$ the \emph{$j$-th row} or \emph{$i$-th column} of~$W$, respectively.
A subgraph~$W'$ of a wall~$W$ that is itself a wall is called a \emph{subwall of~$W$}.
For a set~$S$ of vertices, we say a wall~$W$ is \emph{$S$-anchored} if~${N^W \subseteq S}$.
We observe the following.
\begin{remark}
\label{rmk:nicesubwall}
Let~$W$ be a~$(c,r)$-wall for integers~${c, r \geq 5}$.
Then~$W$ contains a~$(c-1,r-2)$-subwall~$W'$ which is~$\ensuremath{V_{\neq 2}}(W)$-anchored.
See Figure~\ref{fig:wall}.
\qed
\end{remark}
For an integer~${c \geq 3}$, we call a subwall~$W'$ of a wall~$W$ a \emph{$c$-column-slice of~$W$} if the set of nails of~$W'$ is exactly~${N^W \cap V(W')}$,
there is a column of~$W'$ which is a column of~$W$,
and~$W'$ has exactly~$c$ columns, see Figure~\ref{fig:wall} for an example.
Similarly, for an integer~${r \geq 3}$, we call a subwall~$W'$ of a wall~$W$ an \emph{$r$-row-slice of~$W$} if the set of nails of~$W'$ is exactly~${N^W \cap V(W')}$,
there is a row of~$W'$ which is a row of~$W$,
and~$W'$ has exactly~$r$ rows.
Note that in an $r$-row-slice~$W'$ of~$W$, depending on the location, the first column of~$W'$ may be in the last column of~$W$ by the definition of a wall.
See Figure~\ref{fig:rowslice} for an illustration.
\begin{figure}[h]
\centering
\begin{tikzpicture}[scale=0.5, decoration={markings, mark=at position 0.5 with {\arrow{>}}}]
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\end{tikzpicture}
\quad
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\end{tikzpicture}
\quad
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\tikzstyle{w}=[circle,draw,fill=black!50,inner sep=0pt,minimum width=3pt]
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\ifthenelse{\x<9}{
\pgfmathtruncatemacro{\start}{1};
}{
\pgfmathtruncatemacro{\start}{2};
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\ifthenelse{\x>0}{
\pgfmathtruncatemacro{\t}{5};
}{
\pgfmathtruncatemacro{\t}{4};
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\pgfmathtruncatemacro{\nextx}{\x+1}
\ifthenelse{\x<8}{
\draw [very thick](v\x\y)--(v\nextx\y);
}{}
\ifthenelse{\x=8 \and \y>1}{
\draw [very thick](v\x\y)--(v\nextx\y);
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\pgfmathtruncatemacro{\nexty}{\y+1}
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\ifthenelse{\isodd{\sum}\and \y<5}{
\draw [very thick](v\x\y)--(v\x\nexty);
}{}
}
}
\end{tikzpicture}
\caption{A $3$-row-slice $W_3$, a $4$-row-slice $W_4$, and a $5$-row-slice $W_5$ of a $(5,6)$-wall~$W$. Notice that the first column of $W_3$ is in the first column of $W$ but the first column of $W_4$ or $W_5$ is in the last column of $W$.}
\label{fig:rowslice}
\end{figure}
Given a wall~$W$, a \emph{$W$-handle} is a non-trivial $W$-path whose endvertices are degree-$2$ nails of~$W$ contained in the union of the first and last column of~$W$.
Let~$W$ be a ${(c,r)}$-wall, let~$W'$ be a $c'$-column-slice of~$W$ for some~${3 \leq c' \leq c}$,
and let~$P$ be a $W$-handle.
We define the \emph{row-extension of~$P$ to~$W'$ in~$W$} as the unique non-trivial $W'$-path in~${P \cup \bigcup \{ R_i^W \colon i \in [r]\}}$.
Note that such a~$P$ is a $W'$-handle.
For a set~$\mathcal{P}$ of vertex-disjoint $W$-handles, we define the \emph{row-extension of~$\mathcal{P}$ to~$W'$ in~$W$} to be the set of row-extensions of the paths in~$\mathcal{P}$ to~$W'$ in~$W$.
Note that these $W'$-handles are vertex-disjoint.
\subsection{Linkages, separations, and tangles}
Let~$G$ be a graph.
For vertex sets~$A$ and~$B$ in~$G$,
a set~$\mathcal{P}$ of vertex-disjoint ${(A, B)}$-paths of~$G$ is called a \emph{linkage} from~$A$ to~$B$,
and its \emph{order} is defined to be~$\abs{\mathcal{P}}$.
A \emph{separation} of~$G$ is a pair~${(A, B)}$ of subsets of~$V(G)$ such that~${G[A] \cup G[B] = G}$.
Its \emph{order} is defined to be~$\abs{A \cap B}$.
We will use Menger's theorem.
\begin{theorem}[Menger~\cite{Menger27}]
\label{thm:menger}
Let~$A$ and~$B$ be vertex sets in a graph~$G$, and~$k$ be a positive integer.
Then~$G$ contains either a linkage of order~$k$ from~$A$ to~$B$,
or a separation~${(A', B')}$ of order less than~$k$ such that~${A \subseteq A'}$ and~${B \subseteq B'}$.
\end{theorem}
We need a concept of a large wall dominated by a tangle, see~\cite[(2.3)]{RST1994}.
For a positive integer~$t$,
a set~$\cT$ of separations of order less than~$t$ is a \emph{tangle of order~$t$} in~$G$ if it satisfies the following.
\begin{enumerate}
[label=(\arabic*)]
\item If ${(A, B)}$ is a separation of~$G$ of order less than~$t$, then~$\cT$ contains exactly one of~${(A, B)}$ and~${(B, A)}$.
\item If~${(A_1, B_1), (A_2, B_2), (A_3, B_3) \in \cT}$, then~${G[A_1] \cup G[A_2] \cup G[A_3] \neq G}$.
\end{enumerate}
Let~$W$ be a wall of order~$g$ with~${g \geq 3}$ in a graph~$G$.
Let~$\cT_W$ be the set of all separations~${(A, B)}$ of~$G$ of order less than~$g$ such that~${G[B]}$ contains a row of~$W$.
By the following simple lemma, we may replace the row with the column.
The proof in~\cite{RS1991} is for the grid but one can easily modify it for the wall.
\begin{lemma}[{Robertson and Seymour~\cite[(7.1)]{RS1991}}]
\label{lem:rowcol}
Let~$W$ be a wall of order~$g$ in a graph~$G$.
Let~${(A,B)}$ be a separation of order less than~$g$.
Then~${G[B]}$ contains a row of~$W$
if and only if it contains a column of~$W$.
\end{lemma}
Kleitman and Saks (see~\cite[(7.3)]{RS1991}) showed that~$\cT_W$ is a tangle of order~$g$.
A tangle~$\cT$ in~$G$ \emph{dominates} the wall~$W$ if~${\cT_W \subseteq \cT}$.
\begin{theorem}[Robertson, Seymour, and Thomas~\cite{RST1994}]
\label{thm:wall}
There exists a function~${f_{\ref*{thm:wall}} \colon \mathbb{N} \to \mathbb{N}}$
such that
if~${g \geq 3}$ is an integer and~$\cT$ is a tangle in a graph~$G$ of order at least~${f_{\ref*{thm:wall}}(g)}$,
then~$\cT$ dominates a ${(g, g)}$-wall~${W}$ in~${G}$.
\end{theorem}
We will show that if a tangle dominates a wall~$W$, then it also dominates every large subwall of~$W$.
We first prove the following lemma.
\begin{lemma}
\label{lem:wallseparations}
Let~$W$ be a wall in a graph~$G$ and let~${S}$ be a subset of~${V(G)}$ of size exactly~$t$.
For each column~$C^W_x$ and row~$R^W_y$ of~$W$,
there are no more than~$t^2$ nails of $W$ which belong to components of~${W-S}$ that do not intersect~${C^W_x \cup R^W_y}$.
\end{lemma}
\begin{proof}
We proceed by induction on~$t$.
The statement is trivial if~${t = 0}$.
We may assume that~${G = W}$.
Let~${S =: \{ s_i \colon i \in [t] \}}$ and
let~${T := V(C^W_x\cup R^W_y)}$.
Suppose there is a vertex~$v$ in~${S \setminus N^W}$, and let~$P$ be the $N^W$-path in~$W$ containing~$v$.
If both or neither of the endvertices of~$P$ are in components of~${W-S}$ that intersect~$T$,
then we may apply the inductive hypothesis to~${S \setminus \{v\}}$.
Otherwise, replacing~$v$ in~$S$ with the unique endvertex of~$P$ which is in a component of~${W-S}$ that intersects~$T$ does not decrease the number of nails in components of~${W-S}$ that do not intersect~$T$.
Hence, we may assume that~${S \subseteq N^W}$.
For each~${i \in [t]}$, let~${c(i)}$ be the integer such that~$s_i$ is in~$C^W_{c(i)}$ and let~${r(i)}$ be the integer such that~$s_i$ is in~$R^W_{r(i)}$.
For each~${i \in [t]}$, let~$S^c_i$ be the set of nails~$v$ of~$W$ in~${C^W_{c(i)}-S}$ such that the ${(v,R^W_y)}$-subpath of~$C^W_{c(i)}$ contains the vertex~$s_i$,
and let~$S^r_i$ be the set of nails~$v$ of~$W$ in~${R^W_{r(i)}-S}$ such that the ${(v,C^W_x)}$-subpath of~$R^W_{r(i)}$ contains the vertex~$s_i$.
Note that for every nail~$v$ of~$W$, if $v$ is in a component of~${W-S}$ not intersecting~$T$, then there exist~${i,j \in [t]}$ such that~${v \in S^c_i \cap S^r_j}$.
Also note that~${\abs{S^c_i \cap S^r_j} \leq 2}$, and that if~${\abs{S^c_i \cap S^r_j} > 0}$ and~${i \neq j}$, then~${\abs{S^c_j \cap S^r_i} = 0}$.
Furthermore, for~${i \in [t]}$, the nails in~${S^c_i \cap S^r_i}$ are in~${(C^W_{c(i)} \cap R^W_{r(i)}) - s_i}$, so~${\abs{S^c_i \cap S^r_i} \leq 1}$.
It follows that the number of nails of~$W$ which are in components of~${W-S}$ that do not intersect~${C^W_x \cup R^W_y}$ is at most~${2{\binom{t}{2}} + t = t^2}$.
\end{proof}
\begin{lemma}
\label{lem:dominatedsubwall}
Let~${w \geq t \geq 3}$ be integers, let~$W$ be a wall of order~$w$ in a graph~$G$, and let~$\mathcal{T}$ be a tangle dominating~$W$.
If~$W'$ is a subwall of~$W$ of order~$t$ and $\abs{N^{W'}\cap N^W}> (2t-1)(t-1)$,
then~$\mathcal{T}$ dominates~$W'$.
In particular, if $W'$ is $N^W$-anchored, then~$\mathcal{T}$ dominates~$W'$.
\end{lemma}
\begin{proof}
Suppose for a contradiction that~$\mathcal{T}$ does not dominate~$W'$.
Then~$G$ has a separation~${(A,B)}$ of order less than~${t}$ such that~${G[B]}$ contains some row~$R'$ of~$W'$
and~${(A,B) \notin \mathcal{T}}$.
The order of~$\mathcal{T}$ is at least~$w$, so~$\mathcal{T}$ contains ${(B,A)}$ and therefore~${G[A]}$ contains some row~$R$ of~$W$.
Let~${S := A \cap B}$.
Since~$W$ has more than~$\abs{S}$ columns and~$R$ intersects each of them, ${G[A]}$ contains some column~$C$ of~$W$, and similarly~${G[B]}$ contains some column~$C'$ of~$W'$.
By Lemma~\ref{lem:wallseparations} applied to~$W$, there are at most~${(t-1)^2}$ nails of~$W$ in components of~${G-S}$ which do not intersect~${R \cup C}$.
Similarly, there are at most~${(t-1)^2}$ nails of~$W'$ in components of~${G-S}$ which do not intersect~${R'\cup C'}$.
Since~${\abs{N^{W'} \cap N^W} > (2t-1)(t-1) \geq 2(t-1)^2 + \abs{S}}$,
there is a vertex in~${(N^{W'} \cap N^W) \setminus S}$
that is in a component of~${G-S}$ intersecting~${R \cup C}$
and also in a component of~${G-S}$ intersecting~${R' \cup C'}$.
However, every component of~${G-S}$ intersecting~${R \cup C}$ is in~${G[A \setminus B]}$
and every component of~${G-S}$ intersecting~${R' \cup C'}$ is in~${G[B \setminus A]}$, a contradiction.
\end{proof}
\subsection{Groups}
Let $\Gamma_i$ be a group for each $i\in [m]$.
We refer to the \emph{direct product} of these groups by~${\prod_{i \in [m]} \Gamma_i}$ and denote by~$\pi_j$ the projection map from~${\prod_{i \in [m]} \Gamma_i}$ to~$\Gamma_j$ for~${j \in [m]}$.
For an element~$g$ of~$\prod_{i \in [m]} \Gamma_i$, we refer to the image~$\pi_i(g)$ as the \emph{$i$-th coordinate of~$g$}.
When we say~${\Gamma = \prod_{i \in [m]} \Gamma_i}$ is a product of groups, we implicitly use this notation.
For a non-empty set of elements~${S = \{a_i \colon i \in [t] \}}$ in a group~$\Gamma$, we denote by~${\gen{S}}$ or~${\gen{ a_i \colon i \in [t] }}$ the \emph{subgroup generated by~$S$}, which is the intersection of all subgroups of~$\Gamma$ containing~$S$.
\subsection{Group-labelled graphs}
Let~$\Gamma$ be an abelian group.
A \emph{$\Gamma$-labelled graph} is a pair of a graph~$G$ and a function~${\gamma \colon E(G) \to \Gamma}$.
We say that~$\gamma$ is a \emph{$\Gamma$-labelling} of~$G$.
A \emph{subgraph} of a $\Gamma$-labelled graph~${(G,\gamma)}$ is a $\Gamma$-labelled graph~$(H,\gamma')$ such that~$H$ is a subgraph of~$G$ and~$\gamma'$ is the restriction of~$\gamma$ to~$E(H)$.
By a slight abuse of notation, we may refer to this $\Gamma$-labelled graph by~$(H,\gamma)$.
For a $\Gamma$-labelled graph~${(G,\gamma)}$ and a subgraph~${H \subseteq G}$, we define~$\gamma(H)$ as~${\sum_{e \in E(H)} \gamma(e)}$,
which we call the \emph{$\gamma$-value of~$H$}.
Note that this definition implies that the $\gamma$-value of the empty subgraph is~$0$.
We say that a subgraph~$H$ is \emph{$\gamma$-non-zero} if~${\gamma(H) \neq 0}$, and otherwise, we call it \emph{$\gamma$-zero}.
We will often consider the special case where~$\Gamma$ is the product~${\prod_{i \in [m]} \Gamma_i}$ of~$m$ abelian groups for a positive integer~$m$.
In this case, we denote by~$\gamma_i$ the composition of~$\gamma$ with the projection to~$\Gamma_i$.
A $\Gamma$-labelled graph~${(G,\gamma)}$ is \emph{$\gamma$-bipartite} if every cycle of~$G$ is $\gamma$-zero.
We frequently take a subgroup~$\Lambda$ of~$\Gamma$ and consider a new labelling using the quotient group~${\Gamma/\Lambda}$.
For a $\Gamma$-labelled graph~${(G, \gamma)}$ and a subgroup~$\Lambda$ of~$\Gamma$,
the~${\Gamma/\Lambda}$-labelling~$\lambda$ defined by~${\lambda(e) := \gamma(e) + \Lambda}$ for all edges~${e \in E(G)}$ is the \emph{induced $(\Gamma/\Lambda)$-labelling} of~$(G,\gamma)$.
We will use the following duality theorem between packing and covering of $\gamma$-non-zero $A$-paths.
\begin{theorem}[Wollan~\cite{Wollan2010}]
\label{thm:tpath}
Let~$k$ be a positive integer, let~$\Gamma$ be an abelian group, let~$(G,\gamma)$ be a $\Gamma$-labelled graph, and let~${A \subseteq V(G)}$.
Then~$G$ contains~$k$ vertex-disjoint $\gamma$-non-zero $A$-paths or
there exists a set~${X \subseteq V(G)}$ of size at most~${f_{\ref*{thm:tpath}}(k) := 50k^4}$ such that~${G-X}$ has no $\gamma$-non-zero $A$-paths.
\end{theorem}
Let~$x$ be a vertex of~$G$ and let~${\delta \in \Gamma}$ be an element of order~$2$.
For each edge~$e$ of~$G$, let
\[
\gamma'(e)=
\begin{cases}
\gamma(e) + \delta & \text{if $e$ is incident with $x$,}\\
\gamma(e) &\text{otherwise.}
\end{cases}
\]
We say that~$\gamma'$ is obtained from~$\gamma$ by \emph{shifting by~$\delta$ at~$x$}.
Observe that this shift does not change the weight sum of a cycle because~${\delta + \delta = 0}$.
We say two $\Gamma$-labellings~$\gamma_1$ and~$\gamma_2$ of~$G$ are \emph{shifting-equivalent}
if~$\gamma_1$ can be obtained from~$\gamma_2$ by a sequence of shifting operations.
The following lemma asserts that for $\gamma$-bipartite graphs we can find a shifting-equivalent $\Gamma$-labelling~$\gamma'$ in which every corridor is $\gamma'$-zero.
Similar ideas appear in Geelen and Gerards~\cite{GeelenG2009}.
\begin{lemma}
\label{lem:shifting}
Let~$\Gamma$ be an abelian group, let~${(G,\gamma)}$ be a~$\Gamma$-labelled graph and let~${H \subseteq G}$ be a subdivision of a $3$-connected
graph~${\hat{H}}$.
If~$H$ is $\gamma$-bipartite, then~$\gamma$ is shifting-equivalent to a $\Gamma$-labelling~$\gamma'$ such that every corridor of~$H$ is $\gamma'$-zero.
\end{lemma}
\begin{proof}
Let~$T$ be a spanning tree of~${\hat{H}}$ rooted at some~${r \in V({\hat{H}})}$.
It is enough to find a $\Gamma$-labelling~$\gamma'$ of~$G$ which is shifting-equivalent to~$\gamma$, such that all corridors of~$H$ corresponding to edges in~$T$ are $\gamma'$-zero,
because~$H$ is $\gamma'$-bipartite.
Choose a $\Gamma$-labelling~$\gamma'$ shifting-equivalent to~$\gamma$ and a subtree~$T'$ of~$T$ containing~$r$ such that all corridors of~$H$ corresponding to edges in~$T'$ are $\gamma'$-zero, and subject to these conditions,~$\abs{V(T')}$ is maximised.
Suppose that~${T' \neq T}$.
Then there is an edge~$vw$ of~$T$ such that~${v \in V(T')}$ and~${w \notin V(T')}$.
Let~$P$ be the corridor of~$H$ corresponding to the edge~$vw$.
Since~$\hat{H}$ is $3$-connected,
there is a cycle~$O$ in~${H-E(P)}$ containing~$v$ and~$w$.
Let~$O_1$ and~$O_2$ denote the distinct cycles in~${O \cup P}$ containing~$P$.
Since $H$ is $\gamma'$-bipartite, we have that ${\gamma'(P) + \gamma'(P) = \gamma'(O_1) + \gamma'(O_2) - \gamma'(O) = 0}$,
and hence~$\gamma'(P)$ is an element of order at most~$2$.
Let~$\gamma''$ be a $\Gamma$-labelling of~$G$ obtained from~$\gamma'$
by shifting by~$\gamma'(P)$ at~$w$.
Let~${T'' = T[V(T') \cup \{w\}]}$.
Then all corridors of~$H$ corresponding to edges of~$T''$ are $\gamma''$-zero, contradicting the choice of~$\gamma'$ and~$T'$.
\end{proof}
\section{Discussion}
\label{sec:overview}
\subsection{Proof Sketch}
\label{subsec:sketch}
We now sketch the proof of Theorem~\ref{thm:main}, which will proceed by induction on~$k$.
We consider the group~${\Gamma := \prod_{i \in [m]} \Gamma_i}$ and a single $\Gamma$-labelling, which simplifies the arguments we present and in particular allows us to consider quotient groups.
The goal will be to show that if the smallest hitting set~$T$ for the cycles in~$\mathcal{O}$ is sufficiently larger than~${f_m(k-1)}$, then there is a half-integral packing of~$k$ cycles in~$\mathcal{O}$.
To construct this packing, in Section~\ref{sec:pack}, we first find a tangle whose order is correlated with~${\abs{T}/f_m(k-1)}$, which allows us to construct a large wall as a subgraph in~$G$.
In particular, this wall will have the property that no cycle in~$\mathcal{O}$ can be separated from the nails of the wall by deleting a small set of vertices.
Our strategy will be to find disjoint sets of $\gamma$-non-zero paths and use the structure of the wall to connect them up to form cycles.
But before we begin to do this, in Section~\ref{sec:cleanwalls} we find a subwall~$W$ of the original wall such that the $N^{W}$-paths in~$W$ have some nice homogeneity properties with respect to the group labelling.
It would be simplest if we could guarantee that all $N^{W}$-paths in~$W$ were $\gamma$-zero, but this is not feasible with arbitrary abelian groups.
Instead, we deal separately with the factors~$\Gamma_i$ of~$\Gamma$ for which we can guarantee that all $N^{W}$-paths in~$W$ are $\gamma_i$-zero, and the factors of~$\Gamma$ for which we cannot find any large subwall of~$W$ with this property.
Applying Theorem~\ref{thm:tpath}, we can find for every factor~$\Gamma_i$ of~$\Gamma$ a large set of disjoint~$N^{W}$-paths which are $\gamma_i$-non-zero.
The difficulty here lies in combining these paths together such that for all~${i \in [m]}$ the total $\gamma_i$-value is not in~$\Omega_i$.
To achieve this, in Section~\ref{sec:handles} we show how to iteratively find sets of disjoint paths which are non-zero with respect to a quotient group defined in terms of the previously constructed paths, and link them up to the boundary of a subwall of~$W$.
In Section~\ref{sec:abelian}, we analyse conditions under which we can find a set of elements from a product of abelian groups whose sum avoids a finite set~$\Omega_i$ in each coordinate.
In Section~\ref{sec:handles2cycles}, we discuss how to combine the disjoint sets of paths into cycles using the wall.
This allows us to find a half-integral packing of~$k$ cycles whose $\gamma_i$-values are in~${\Gamma_i \setminus \Omega_i}$ for every factor~$\Gamma_i$ of~$\Gamma$ for which the $N^{W}$-paths in~$W$ are $\gamma_i$-zero.
To deal with each remaining factor~$\Gamma_i$, we observe that since no large subwall of~$W$ is $\gamma_i$-bipartite, every large subwall of~$W$ contains a $\gamma_i$-non-zero cycle.
In fact, we iteratively find disjoint cycles in~$W$ which are non-zero with respect to quotient groups defined in terms of the previously constructed cycles.
We link these cycles up to the half-integral packing of $k$-cycles we have constructed, and by rerouting through them, transform each cycle in our half-integral packing into a cycle in~$\mathcal{O}$.
\subsection{%
\texorpdfstring{Obstructions for integral and half-integral Erd\H{o}s-P\'{o}sa type results}%
{Obstructions for integral and half-integral Erdos-Posa type results}}
\label{subsec:obstructionsEP}
One fundamental obstruction to Erd\H{o}s-P\'{o}sa type results, known as the Escher wall, is due to Lov\'{a}sz and Schrijver~(see~\cite{Thomassen1988}).
An \emph{Escher wall of height~$n$} is obtained from an ${(n, n)}$-wall~$W$ by adding a family~${(P_i \colon i \in [n])}$ of vertex-disjoint $W$-paths, such that for each~${i \in [n]}$, one endvertex of~$P_i$ is in~${R^W_1 \cap C^W_i}$ and the other is in~${R^W_n \cap C^W_{n+1-i}}$. See Figure~\ref{fig:escher} for an illustration.
They observed that there are no two disjoint cycles in such an Escher wall which each contain an odd number of paths in~${(P_i \colon i \in [n])}$,
but for any set~$S$ of at most~${n-1}$ vertices, there is a cycle of the Escher Wall containing exactly one path in~${(P_i \colon i \in [n])}$.
Using this construction, Thomassen~\cite{Thomassen1988} argued that an analogue of the Erd\H{o}s-P\'{o}sa theorem does not hold for odd cycles.
It can be further used to show that the same holds for cycles of length $\ell$ modulo $m$ whenever $m$ is an even positive integer and $\ell$ is an odd integer with $0<\ell<m$, see Wollan~\cite{Wollan2011}.
\begin{figure}
\centering
\begin{tikzpicture}[scale=.6]
\tikzstyle{w}=[circle,draw,fill=black!50,inner sep=0pt,minimum width=3pt]
\useasboundingbox (0,-1) rectangle (7.5,4);
\foreach \x in {0,...,13}{
\ifthenelse{\x>0}{
\pgfmathtruncatemacro{\start}{0};
}{
\pgfmathtruncatemacro{\start}{1};
}
\ifthenelse{\x<13}{
\pgfmathtruncatemacro{\t}{6};
}{
\pgfmathtruncatemacro{\t}{5};
}
\foreach \y in {\start,...,\t} {
\node at (\x/2,\y/2) [w] (v\x\y) {};
}
}
\foreach \x in {0,1,...,13}{
\ifthenelse{\x>0}{
\pgfmathtruncatemacro{\start}{0};
}{
\pgfmathtruncatemacro{\start}{1};
}
\ifthenelse{\x<13}{
\pgfmathtruncatemacro{\t}{6};
}{
\pgfmathtruncatemacro{\t}{5};
}
\foreach \y in {\start,...,\t} {
\pgfmathtruncatemacro{\nextx}{\x+1}
\ifthenelse{\x<12}{
\draw [gray](v\x\y)--(v\nextx\y);
}{}
\ifthenelse{\x=12 \and \y<6}{
\draw [gray](v\x\y)--(v\nextx\y);
}{}
\pgfmathtruncatemacro{\nexty}{\y+1}
\pgfmathtruncatemacro{\sum}{\x+\y}
\ifthenelse{\isodd{\sum}\and \y<6}{
\draw [gray](v\x\y)--(v\x\nexty);
}{}
}
}
\draw [out=20](v06) .. controls (11,6) and (11,-3) .. (v120);
\draw [out=20](v26) .. controls (11,6) and (11,-3) .. (v100);
\draw [out=20](v56).. controls (11,6) and (11,-3) ..(v80);
\draw [out=20](v66) .. controls (11,6) and (11,-3) .. (v70);
\draw [out=20](v96) .. controls (11,6) and (11,-3) .. (v40);
\draw [out=20](v116).. controls (11,6) and (11,-3) .. (v20);
\draw [out=20](v126) .. controls (11,6) and (11,-3) .. (v10);
\end{tikzpicture}
\caption{An Escher wall of height~$7$.}
\label{fig:escher}
\end{figure}
Many previous half-integral Erd\H{o}s-P\'{o}sa type results have relied on characterising Escher walls as the fundamental obstructions to finding integral packings of certain classes of cycles.
Half-integral packings are then obtained using the structure of the Escher Wall.
In our setting, the Escher Wall is not the only possible obstruction which can arise.
In fact, unlike Escher walls, the following type of obstruction can occur even in planar graphs.
For this reason, some of the structural results which have been used to obtain other half-integral Erd\H{o}s-P\'{o}sa type results are unlikely to be useful in our setting.
For example, we do not use the flat wall theorem of Robertson and Seymour~\cite{RobertsonS95, RobertsonS1986} in this paper, and there is no obvious way to significantly simplify our proofs by doing so.
\begin{proposition}
\label{prop:obstruction2}
Let~$G$ be the ${n \times n}$-grid, and let~$\mathcal{O}$ be the set of cycles of~$G$
which contain at least one edge of the top row~$R_t$ of~$G$,
at least one edge of the bottom row~$R_b$ of~$G$,
and at least one edge from the leftmost column~$C_{\ell}$ of~$G$,
and let~$\mathcal{O}'$ be the set of cycles of~$G$
which contain exactly one edge of the top row~$R_t$ of~$G$,
exactly one edge of the bottom row~$R_b$ of~$G$,
and exactly one edge from the leftmost column~$C_{\ell}$ of~$G$.
Then every pair of cycles in~$\mathcal{O}$ intersect, but there is no hitting set for~$\mathcal{O}'$ of size less than~${(n-1)/2}$.
\end{proposition}
\begin{proof}
We may assume that~${n \geq 3}$.
Note that the graph~$G'$ obtained from~$G$ by adding a vertex~$z$ adjacent to every vertex of~${R_t \cup R_b \cup C_{\ell}}$ is planar.
Suppose for contradiction that there are disjoint cycles~$O_1$ and~$O_2$ in~$\mathcal{O}$,
and let~$H$ be the component of~${G-O_1}$ containing~$O_2$.
Since~$C_{\ell}$ is connected~$O_1$ intersects~${C_{\ell} - (R_t \cup R_b)}$, there is a vertex~$v_1$ in~${(O_1 \cap C_{\ell}) - (R_t \cup R_b)}$ adjacent to a vertex of~$H$,
and similarly a vertex~$v_2$ in~${O_1 \cap R_t}$ adjacent to a vertex of~$H$ and a vertex~$v_2$ in~${O_1 \cap R_t}$ adjacent to a vertex of~$H$.
Contracting~$C_1$ to a triangle on~${\{v_1,v_2,v_3\}}$ and~$H$ to a single vertex, we find a~$K_5$-minor in~$G'$, contradicting Wagner's Theorem.
Now consider~${S \subseteq V(G)}$ of size less than~${(n-1)/2}$.
Note that there are two adjacent columns~$C_i$ and~$C_{i+1}$ which do not intersect~$S$,
and likewise two adjacent rows~$R_j$ and~$R_{j+1}$ which do not intersect~$S$.
It is easy to see that the subgraph of~$G$ induced on the vertices in~${C_i \cup C_{i+1} \cup R_j \cup R_{j+1}}$ contains a cycle in~$\mathcal{O}'$.
\end{proof}
As an example, consider an~${n \times n}$-grid in which all edges on the top row are subdivided exactly~$14$ times,
all edges of the bottom row are subdivided exactly~$69$ times,
all edges of the leftmost column are subdivided exactly~$20$ times,
and all other edges are subdivided exactly~$104$ times.
By Proposition~\ref{prop:obstruction2}, there are no two vertex-disjoint cycles of length~${1 \mod 105}$,
and no hitting set for these cycles of size less than~${(n-1)/2}$.
Another consequence of Proposition~\ref{prop:obstruction2} is that an analogue of the Erd\H{o}s-P\'osa theorem does not hold for cycles intersecting three prescribed vertex sets.
To see this, consider an~${n \times n}$-grid,
let~$S_1$ be the set of all vertices on the top row,
let~$S_2$ be the set of all vertices of the bottom row,
and let~$S_3$ be the set of all vertices of the leftmost column.
By Proposition~\ref{prop:obstruction2}, there are no two vertex-disjoint cycles each containing at least one vertex from each of~$S_1$,~$S_2$,~$S_3$, and no hitting set for these cycles of size less than~${(n-1)/2}$.
\medskip
Our main theorem demonstrates that it is not easy to find non-trivial obstructions for half-integral Erd\H{o}s-P\'{o}sa type results.
However, the following proposition does allow us to describe settings in which even a half-integral Erd\H{o}s-P\'{o}sa type result is not possible.
\begin{proposition}
\label{prop:counterexample-for-half-integral}
Let~$t$ and~$c$ be positive integers, let~$\Gamma$ be an abelian group and let~$\Omega$ be a subset of~$\Gamma$ such that there is an element~${g \in \Gamma}$ and an integer~${d > (c-1)t}$ such that~$d$ is the minimum integer greater than~$2$ for which~${dg \notin \Omega}$.
Then there is a graph $G$ with $\Gamma$-labelling $\gamma$ such that, for the set~$\mathcal{O}$ of all cycles of~$G$ whose $\gamma$-values are not in~${\Omega}$, every~$c$ cycles in~$\mathcal{O}$ share a common vertex, but there is no hitting set for~$\mathcal{O}$ of size less than~$t$.
\end{proposition}
\begin{proof}
The~${c = 1}$ case is trivial, so we may assume~${c \geq 2}$.
Let~${n := \lceil cd/(c-1) \rceil - 1}$,
let~${G := K_n}$,
and let~$\gamma$ be the $\Gamma$-labelling assigning~$g$ to every edge of~$G$.
By construction, every cycle in~$\mathcal{O}$ has length greater than~${(c-1)n/c}$, and so every~$c$ cycles in~$\mathcal{O}$ share a common vertex.
However, every cycle of length~$d$ in~$G$ is in~$\mathcal{O}$, so the smallest hitting set for~$\mathcal{O}$ has size~${n-(d-1) > t}$.
\end{proof}
As an example, consider an infinite abelian group which contains arbitrarily large finite cyclic subgroups.
One consequence of Proposition~\ref{prop:counterexample-for-half-integral} is that there is no half-integral Erd\H{o}s-P\'{o}sa type result for cycles of weight zero in graphs labelled with such a group.
\medskip
As another consequence, we obtain a lower bound on the functions mentioned in Theorem~\ref{thm:main} which depends on both~$m$ and~$\omega$.
\begin{corollary}
For any function~${f_{m, \omega}}$ as in Theorem~\ref{thm:main}, we have~${f_{m,\omega}(3) > (2+ m \omega )/ 2}$.
\end{corollary}
\begin{proof}
Consider for each~${i \in [m]}$ the group~${\Gamma_i := \mathbb{Z}}$ and the subset~${\Omega_i := [2+\omega i] \setminus [2+\omega (i-1)]}$.
The result follows
with~${c = 3}$,~${g = ( 1 \colon i \in [m])}$,
and~${\Omega = \bigcup_{i\in [m]} \{ g' \in \Gamma \colon \pi_i(g') \in \Omega_i \}}$ from Proposition~\ref{prop:counterexample-for-half-integral}.
\end{proof}
\subsection{Relating vertex-labellings to edge-labellings}
\label{subsec:vxlabel}
We now demonstrate how to convert a group labelling of the vertices of a graph to a group labelling of its edges.
Corollary~\ref{cor:vxlabelling} follows easily from Theorem~\ref{thm:main} after applying the following lemma to each of the vertex-labellings it mentions.
\begin{lemma}
Let~$\Gamma$ be an abelian group, let~$\Omega \subseteq \Gamma$ be a finite subset, let~$G$ be a graph, and let~$\gamma$ be a $\Gamma$-vertex-labelling of~$G$.
Then there is a group~$\Gamma'$, a subset~$\Omega' \subseteq \Gamma'$ with~${\abs{\Omega'} = \abs{\Omega}}$, and a $\Gamma'$-labelling~$\gamma'$ of~$G$ such that for every cycle~$O$ of~$G$, we have
${\gamma(O) \in \Omega}$ if and only if~${\gamma'(O) \in \Omega'}$.
\end{lemma}
\begin{proof}
Let~${\Gamma'' = \gen{ \Omega \cup \{\gamma(v) \colon v \in V(G)\}}}$.
By the fundamental theorem of finitely generated abelian groups, there exist an integer~$m$ and an isomorphism~$\varphi$ from~$\Gamma''$ to a product~${\prod_{i \in [m]} \Gamma_i}$, where each~$\Gamma_i$ is either~$\mathbb{Z}$, or a cyclic group~$\mathbb{Z}_{n_i}$ of order~$n_i$.
For~${i \in [m]}$, let~${\Gamma_i' := \mathbb{Z}_{2n_i}}$ if the order of~$\Gamma_i$ is finite, and~${\Gamma_i' := \Gamma_i}$ otherwise.
Let~${\Gamma' := \prod_{i \in [m]} \Gamma_i'}$.
For~${i \in [m]}$, let~$e_i$ denote the element of~$\Gamma''$ such that~${\pi_j(\varphi(e_i)) = 1}$ if~${i = j}$, and~${\pi_j(\varphi(e_i)) = 0}$ if~${i \neq j}$.
Then~$\Gamma'' = \gen{ e_i \colon i \in [r']}$.
For~${i \in [m]}$, let~$e_i'$ denote the element of~$\Gamma'$ such that~${\pi_j(e_i') = 1}$ if~${i = j}$, and~${\pi_j(e_i') = 0}$ if~${i \neq j}$.
For each~${j \in [2]}$, we define a homomorphism~$\psi_j$ from~$\Gamma''$ to~$\Gamma'$ by setting~${\psi_j(e_i) := j e_i'}$ on the generators.
Note that~$\psi_2$ is injective since the kernel of~$\psi_2$ is trivial,
that the image of~$\psi_2$ is~${2 \Gamma'}$,
and that~${\psi_1(2g) = \psi_2(g)}$ for all~${g \in \Gamma''}$.
We define~${\Omega' := \psi_2(\Omega)}$ and a $\Gamma'$-labelling~$\gamma'$ of~$G$ for an edge~$e = vw$ of~$G$ by setting~${\gamma'(e) = \psi_1\big(\gamma(v) + \gamma(w)\big)}$.
Note that for a cycle~$O$ of~$G$, we have that
\[
\gamma'(O)
= \sum_{e \in E(O)} \gamma'(e)
= \sum_{vw \in E(O)} \psi_1( \gamma(v) + \gamma(w) )
= \psi_1 \left( 2 \sum_{v \in V(O)} \gamma(v) \right)
= \psi_2 ( \gamma(O) ).
\]
Hence, the result follows from the injectivity of~$\psi_2$.
\end{proof}
\subsection{Graphs embedded on a surface}
\label{subsec:surface}
We now discuss how our result applies to graphs embedded on a surface, where we consider the first homology group with coefficients in~$\mathbb{Z}_2$.
Huynh, Joos, and Wollan~\cite[Proposition 5]{HuynhJW2017} demonstrated that given a graph~$G$ embedded on a surface whose $\mathbb{Z}$-homology group is~$\Gamma$, there is a directed $\Gamma$-labelling of~$G$ so that the set of cycles in~$G$ that are homologous to zero is exactly the set of cycles having group value~$0$ in the labelling.
This allowed them to obtain for graphs embedded on a surface a half-integral Erd\H{o}s-P\'{o}sa result for the non-null-homologous cycles of the embedding.
Our result works in essentially the same way.
A graph~$H$ is called \emph{even} if every vertex of~$H$ has even degree.
For a graph~$G$, let~$\mathcal{C}(G)$ denote the \emph{cycle space} of~$G$ over~$\mathbb{Z}_2$, that is the vector space of all even subgraphs~$H$ of~$G$ with the symmetric difference as the operation.
\begin{proposition}
\label{prop:cyclespace-hom-to-gamma}
Let~$G$ be a graph, let~$\Gamma$ be an abelian group, and let~${\phi \colon \mathcal{C}(G) \to \Gamma}$ be a group homomorphism.
Then there is a $\Gamma$-labelling~$\gamma$ of~$G$ such that~${\gamma(H) = \phi(H)}$ for every even subgraph~$H$ of~$G$.
\end{proposition}
\begin{proof}
Without loss of generality, we may assume that~$G$ is connected.
Let~$T$ be a spanning tree of~$G$.
For each edge~${e \in E(G) \setminus E(T)}$, let~$C_{e,T}$ denote the unique cycle in~${T + e}$.
We define~${\gamma(e) := 0}$ for each~${e \in E(T)}$ and~${\gamma(e) := \phi(C_{e,T})}$ for each~${e \in E(G) \setminus E(T)}$.
The statement now trivially follows, because the set~$\{C_{e,T} \colon e \in E(G) \setminus E(T)\}$ forms a basis of the cycle space (see~\cite[Theorem~1.9.5]{Diestel5}).
\end{proof}
Now for a graph~$G$ embedded in a surface~$\Sigma$, the map assigning each even subgraph its $\mathbb{Z}_2$-homology class is a group homomorphism from~$\mathcal{C}(G)$ to the $\mathbb{Z}_2$-homology group of~$\Sigma$.
Hence, Corollary~\ref{cor:surface} follows with Proposition~\ref{prop:cyclespace-hom-to-gamma} from Theorem~\ref{thm:main}.
Note that for a closed orientable surface, the set of simple closed curves homologous to zero for the $\mathbb{Z}_2$-homology is exactly the same as for the $\mathbb{Z}$-homology.
This follows the universal coefficient theorem (see \cite{Hatcher02}), which allows us to relate the $\mathbb{Z}$-homology with the $\mathbb{Z}_2$-homology by taking all coefficients modulo~${2}$.
We then apply a classical result which states that no simple closed curve has $\mathbb{Z}$-homology class~${kh}$ for any integer~${k \geq 2}$ and any non-zero element~$h$ of the $\mathbb{Z}$-homology (see for example \cite{Schafer1976}).
Hence, in the case of graphs embedded on closed orientable surfaces, we recover the result of Huynh, Joos and Wollan for non-null-homologous cycles.
\section{Packing functions and hitting sets}
\label{sec:pack}
In this section, we introduce the concept of packing functions as a tool to generalise the ideas of both integral and half-integral packings of subgraphs,
which enables us to discuss these and similar ideas in a unified way.
For a function~$\nu$ from the set of subgraphs of a graph~$G$ to the set of non-negative integers, we say that
\begin{itemize}
\item $\nu$ is \emph{monotone} if~$\nu(H) \leq \nu(H')$ whenever~$H$ is a subgraph of~$H'$,
\item $\nu$ is \emph{additive} if~$\nu(H \cup H') = \nu(H) + \nu(H')$ whenever~$H$ and~$H'$ are vertex-disjoint subgraphs of $G$, and
\item $\nu$ is a \emph{packing function for~$G$} if it is monotone and additive.
\end{itemize}
Now let~${\nu}$ be a packing function for a graph~$G$.
For a subgraph~${H \subseteq G}$, we say a set~${T \subseteq V(H)}$ is an \emph{$\nu$-hitting set for~$H$} if~${\nu(H - T) = 0}$.
We define~$\tau_\nu(H)$ as the size of a smallest $\nu$-hitting set of~$H$.
Note that in the traditional sense of the word, a $\nu$-hitting set of~$G$ is a hitting set for the minimal subgraphs~${H \subseteq G}$ for which~${\nu(H) \geq 1}$.
For example, a function mapping a subgraph~$H$ of~$G$ to the maximum number of vertex-disjoint cycles in~$H$ is a packing function of~$G$.
\medskip
The following lemma argues that if~${\nu(G)}$ is small but~$G$ has no small $\nu$-hitting set, then every minimum $\nu$-hitting set
induces a tangle of large order.
Similar arguments for specific packing functions appear many times in the literature,
see~\cite{HuynhJW2017} and~\cite{ReedRST1996} for instance.
\begin{lemma}
\label{lem:welllinked}
Let~$\nu$ be a packing function for a graph~$G$
and let~${T \subseteq V(G)}$ be a minimum $\nu$-hitting set for~$G$ of size~${t}$.
Let~$\mathcal{T}_T$ be the set of all separations~${(A,B)}$ of~$G$
of order less than~${t/6}$ such that~${\abs{B \cap T} > 5t/6}$.
If~${\tau_\nu(H) \leq t/12}$ whenever~$H$ is a subgraph of~$G$ with~${\nu(H) < \nu(G)}$,
then~$\mathcal{T}_T$ is a tangle of order~${\lceil t/6 \rceil}$.
\end{lemma}
\begin{proof}
First, we show the following claim.
\begin{claim*}
Let~${X, Y \subseteq T}$ be disjoint sets with~${\abs{X} = \abs{Y} \geq t/6}$.
Then there is a linkage in~$G$ from~$X$ to~$Y$ of order~$\abs{X}$ containing no vertex in~${Z := T \setminus (X \cup Y)}$.
\end{claim*}
\begin{proofofclaim}
Suppose for a contradiction that there is no such linkage.
By Menger's theorem applied to~${G - Z}$,
there is a separation~${(A, B)}$ of~$G$ of order strictly less than~${\abs{X} + \abs{Z}}$ with~${Z \subseteq A \cap B}$, ${X \subseteq A}$, and ${Y \subseteq B}$.
Let~${S := A \cap B}$.
Now observe that
\[
{\nu(G - (A \cup T))+\nu(G - (B \cup T)) = \nu(G - (S \cup T)) \leq \nu(G - T) = 0},
\]
and so~${\nu(G - (A \cup T)) = \nu(G - (B \cup T)) = 0}$.
Hence
\[
{\nu(G - B) = \nu(G - B) + \nu(G - (A \cup T)) = \nu(G - B) + \nu(G - (A \cup Y)) = \nu(G - (S \cup Y))},
\]
and so by the minimality of~$T$ and the fact that~${\abs{S \cup Y} \leq \abs{S} + \abs{Y} < \abs{X} + \abs{Z} + \abs{Y} = \abs{T}}$ we have that~${\nu(G - B) \geq 1}$.
By symmetry~${\nu(G - A) \geq 1}$,
and since~${\nu(G - A) + \nu(G - B) \leq \nu(G)}$, both~${\nu(G - A)}$ and~${\nu(G - B)}$ are strictly less than~${\nu(G)}$.
By the assumption, ${\tau_\nu(G - A), \tau_\nu(G - B) \leq t/12}$.
Let~$T_{A}$ and~$T_{B}$ be $\nu$-hitting sets of minimum size for~${G-A}$ and~${G-B}$, respectively.
Then
\[
{\nu(G - (T_A \cup T_B \cup S)) = \nu(G - (T_{A} \cup A)) + \nu(G - (T_{A} \cup B)) = 0},
\]
but
${\abs{T_A} + \abs{T_B} + \abs{S}
\leq (t/6) + \abs{S}
\leq \abs{Y} + \abs{S}
< \abs{T}}$, contradicting the assumption that~$T$ is a minimum $\nu$-hitting set.
\end{proofofclaim}
Let~${(A,B)}$ be a separation of order less than~${t/6}$ with~${\abs{B \cap T} \geq \abs{A \cap T}}$, and let~${S := A \cap B}$.
Clearly,~${(B,A) \notin \mathcal{T}_T}$.
Suppose for a contradiction that~${(A,B) \notin \mathcal{T}_T}$ and hence~${A \setminus B}$ contains at least~${t/6}$ vertices of~$T$.
Since~${B \setminus A}$ contains at least as many vertices of~$T$ as~${A \setminus B}$ does, by the claim there is a linkage of size~${\lceil t/6 \rceil}$ in~$G$ from~${A \setminus B}$ to~${B \setminus A}$, contradicting the assumption on the order of~${(A,B)}$.
Hence, ${(A,B) \in \mathcal{T}_T}$.
Note that~${\abs{T \cap A} < t/3}$ for each~${(A,B) \in \mathcal{T}_T}$.
Hence for~${(A_1,B_2), (A_2,B_2), (A_3,B_3) \in \mathcal{T}_T}$ we have that~${\abs{T \cap (A_1 \cup A_2 \cup A_3)} < t}$, and hence~${G[A_1] \cup G[A_2] \cup G[A_3] \neq G}$.
Thus we conclude that~$\mathcal{T}_T$ is a tangle of order~${\lceil t/6\rceil}$.
\end{proof}
Let us now turn our attention to packing functions~$\nu$ for a $\Gamma$-labelled graph~${(G,\gamma)}$ for an abelian group~$\Gamma$.
The following lemma is useful for converting between $\gamma$-non-zero cycles and $\gamma$-non-zero paths.
We will appeal to it in the final lemma of this section, and again in Lemma~\ref{lem:breaking}.
\begin{lemma}
\label{lem:non-zero-path-in-cycle}
Let~$\Gamma$ be an abelian group,
let~${(G, \gamma)}$ be a $\Gamma$-labelled graph,
let~$O$ be a $\gamma$-non-zero cycle in~$G$, and
let~${T \subseteq V(G)}$.
If~$G$ contains three vertex-disjoint ${(V(O),T)}$-paths~$P_1$, $P_2$, $P_3$,
then~${H := O \cup P_1\cup P_2\cup P_3}$ contains a $\gamma$-non-zero $T$-path.
\end{lemma}
\begin{proof}
We may assume that~${\abs{V(P_i) \cap V(O)} = \abs{V(P_i) \cap T} = 1}$ for all~${i \in [3]}$ by taking a subpath if necessary.
For~${i \in [3]}$,
let~$Q_i$ and~$Q'_i$ be the two paths in~$H$ each having~${\bigcup_{j \in [3] \setminus \{i\}} V(P_j) \cap T}$ as its set of endvertices,
where~$Q'_i$ is the path that is disjoint from~$P_i$.
Now,
\[
\sum \limits_{i \in [3]} \left( \gamma(Q_{i}) - \gamma(Q'_{i}) \right)
= 2 \cdot \sum \limits_{i \in [3]} \left (\gamma(P_i) - \gamma(P_i) \right) + 2 \cdot \gamma(O) - \gamma(O)
= \gamma(O)
\neq 0.
\]
Hence, for some~${i \in [3]}$, one of the paths~$Q_{i}$ or~$Q'_{i}$ is~$\gamma$-non-zero.
And since this path is the edge-disjoint union of $T$-paths, it contains a~$\gamma$-non-zero $T$-path, as desired.
\end{proof}
Given an abelian group~$\Gamma$ and a $\Gamma$-labelled graph ${(G,\gamma)}$, we are interested in the packing function~$\nu$ for~$G$ which maps a subgraph~$H$ of~$G$ to the size of the largest half-integral packing
of the type of cycles of~$H$ we are considering.
To prove our main result, we will need the following tool for finding disjoint sets of paths in~$G$ which are non-zero with respect to the induced labelling of certain quotient groups, which we will later construct.
\begin{lemma}
\label{lem:cover}
Let~$u$, $k$ be positive integers such that~${f_{\ref*{thm:tpath}}(k) < u - 2}$.
Let~$\Gamma$ be an abelian group,
let~${(G, \gamma)}$ be a $\Gamma$-labelled graph, and let~$\nu$ be a packing function for~$G$ such that
\begin{itemize}
\item every minimal subgraph~${H}$ of~$G$ with~${\nu(H) \geq 1}$ is a $\gamma$-non-zero cycle,
\item ${\tau_\nu(H) \leq 3 u}$ for every subgraph~${H}$ of $G$ with~${\nu(H) < \nu(G)}$, and
\item ${\tau_\nu(G) \geq u}$.
\end{itemize}
Let~${T \subseteq V(G)}$ be a minimum $\nu$-hitting set for~$G$ and let~${N \subseteq V(G)}$ such that for every~${S \subseteq V(G)}$ of size less than~${u}$, there is a component of~${G-S}$ containing a vertex of~$N$ and at least~$4u$ vertices of~$T$.
Then~$G$ contains~$k$ vertex-disjoint $\gamma$-non-zero $N$-paths.
\end{lemma}
\begin{proof}
Suppose that $G$ does not contain~$k$ vertex-disjoint $\gamma$-non-zero $N$-paths.
By Theorem~\ref{thm:tpath}, there exists~${S \subseteq V(G)}$ of size less than~${u-2}$ hitting all $\gamma$-non-zero $N$-paths.
Since~${\abs{S} < u \leq \tau_\nu(G)}$, we have that~${\nu(G - S) \geq 1}$, so $G-S$ has a $\gamma$-non-zero cycle~$O$ with~${\nu(O) \geq 1}$.
By Lemma~\ref{lem:non-zero-path-in-cycle}, $G-S$ does not have three vertex-disjoint ${(V(O),N)}$-paths.
By Menger's theorem applied to~${G-S}$, there exists~${S'\subseteq V(G)}$ of size at most~${\abs{S}+2}$ separating~$O$ from~$N$.
Since~${\abs{S'} < u}$, by the given assumption on~$N$, the graph $G-S'$ has a component~$H$ containing a vertex of~$N$ and at least~$4u$ vertices of~$T$.
Now~${\nu(H) \leq \nu(H\cup O) - \nu(O) < \nu(H\cup O) \leq \nu(G)}$, so there is a $\nu$-hitting set~$T_H$ for~$H$ of size at most~$3u$.
Let~${T' := T_H \cup S' \cup (T \setminus V(H))}$, and observe that~${\abs{T'} \leq \abs{T_H} + \abs{S'} + \abs{T} - 4u < \abs{T}}$.
But~${\nu(G-T') \leq \nu(G - (S' \cup V(H) \cup T)) + \nu(H-T_H) = 0}$,
contradicting the assumption that~$T$ is a minimum $\nu$-hitting set.
\end{proof}
\section{Clean walls}
\label{sec:cleanwalls}
In the proof of our main theorem in Section~\ref{sec:main}, we will apply Theorem~\ref{thm:wall} and Lemma~\ref{lem:welllinked} to construct a wall~$W$ in a group-labelled graph.
However, it will be useful to move to a large subwall of~$W$ which has some nice homogeneity properties.
For this purpose, we introduce the following notion of cleanness.
Let ${\Gamma = \prod_{i \in [m]} \Gamma_i}$ be a product of~$m$ abelian groups
and let~$(G,\gamma)$ be a $\Gamma$-labelled graph.
Given a subset~${Z \subseteq [m]}$ and an integer~$\ell$,
we say that a wall~$W$ in~$G$ is \emph{$(\gamma,Z,\ell)$-clean} if
\begin{enumerate}
[label=(\arabic*)]
\item\label{item:clean1} every $N^W$-path in~$W$ is $\gamma_i$-zero for all~${i \in Z}$, and
\item\label{item:clean2} $W$ has no ${(\ell,\ell)}$-subwall which is $\gamma_i$-bipartite for all~${i \in [m] \setminus Z}$.
\end{enumerate}
\begin{lemma}
\label{lem:cleansubwall}
Let~${\Gamma = \prod_{i \in [m]} \Gamma_i}$ be a product of~$m$ abelian groups,
let~${(G,\gamma)}$ be a $\Gamma$-labelled graph,
let~${\psi \colon \{0\} \cup [m+1] \to \mathbb{N}_{\geq 3}}$ be a function,
and let~$W$ be a wall of order $\psi(0)+2$ in~$G$.
Then there exist a $\Gamma$-labelling~$\gamma'$ of~$G$ shifting-equivalent to~$\gamma$,
a subset~$Z$ of~${[m]}$,
and a ${(\gamma',Z,\psi(\abs{Z}+1)+2)}$-clean $\ensuremath{V_{\neq 2}}(W)$-anchored ${(\psi(\abs{Z}),\psi(\abs{Z}))}$-subwall of~$W$.
\end{lemma}
\begin{proof}
Let~$Z$ be a maximal subset of~${[m]}$ such that there is a $(\psi(\abs{Z})+2,\psi(\abs{Z})+2)$-subwall~$W'$ of~$W$ which is $\gamma_i$-bipartite for all~${i \in Z}$.
Such a set~$Z$ exists because~${Z := \emptyset}$ satisfies the requirement.
Since~$Z$ is maximal,
there is no~${j \in [m] \setminus Z}$ such that~$W'$ has a ${(\psi(\abs{Z}+1)+2,\psi(\abs{Z}+1)+2)}$-subwall which is $\gamma_j$-bipartite.
Among all $\Gamma$-labellings~$\gamma'$ of~$G$ shifting-equivalent to~$\gamma$,
we choose~$\gamma'$ maximising the number of elements~${i \in Z}$ such that all corridors of~$W'$ are $\gamma'_i$-zero.
If there is~${i \in Z}$ such that some corridor of~$W'$ is not $\gamma'_i$-zero, then
Lemma~\ref{lem:shifting} applied to~$\Gamma_i$ yields the $\Gamma$-labelling~$\gamma''$ for which every corridor of~$W'$ is~$\gamma''_i$-zero,
thus contradicting the choice of~$\gamma'$.
Thus all corridors of~$W'$ are $\gamma_i'$-zero for all~${i \in Z}$.
By Remark~\ref{rmk:nicesubwall}, $W'$ has a $\ensuremath{V_{\neq 2}}(W')$-anchored ${(\psi(\abs{Z}),\psi(\abs{Z}))}$-subwall~$W''$.
Then~$W''$ is $\ensuremath{V_{\neq 2}}(W)$-anchored since~${\ensuremath{V_{\neq 2}}(W') \subseteq \ensuremath{V_{\neq 2}}(W)}$.
Now the property~\ref{item:clean1} holds since every~$N^{W''}$-path in~$W''$ is a corridor of~$W'$.
\end{proof}
In a sense, the notion of cleanness helps us to generalise the ideas of Thomassen~\cite{Thomassen1988} who proved the following result.
\begin{proposition}[Thomassen~\cite{Thomassen1988}]
\label{prop:thomassen}
There exists a function~${w_{\ref*{prop:thomassen}} \colon \mathbb{N}^2 \to \mathbb{N}}$ satisfying the following.
Let~$t$ and~${w \geq 3}$ be integers, let~$\Gamma$ be an abelian group generated by an element of order at most~$t$,
and let~${(W, \gamma)}$ be a $\Gamma$-labelled wall of order~${w_{\ref*{prop:thomassen}}(t,w)}$.
Then~$W$ contains a ${(w,w)}$-subwall~$W'$ such that~${\gamma(P) = 0}$ for all corridors~$P$ of~$W'$.
\end{proposition}
We extend Proposition~\ref{prop:thomassen} to a group generated by a fixed number of generators.
\begin{lemma}
\label{lem:smallorder}
There exists a function~$w_{\ref*{lem:smallorder}} \colon \mathbb{N}^3 \to \mathbb{N}$ satisfying the following.
Let~$q$, $t$ and~${w \geq 3}$ be integers, let~$\Gamma$ be an abelian group generated by~$q$ elements
each of order at most~$t$,
and let~${(W, \gamma)}$ be a $\Gamma$-labelled wall of order $w_{\ref*{lem:smallorder}}(q, t, w)$.
Then~$W$ contains a ${(w,w)}$-subwall~$W'$ such that~${\gamma(P) = 0}$ for all corridors~$P$ of~$W'$.
\end{lemma}
\begin{proof}
We define
\begin{itemize}
\item ${w_{\ref*{lem:smallorder}}(1,t,w) := w_{\ref*{prop:thomassen}}(t,w)}$, and
\item ${w_{\ref*{lem:smallorder}}(q,t,w) := w_{\ref*{prop:thomassen}}(t,f_{\ref*{lem:smallorder}}(q-1,t,w))}$ for all integers~${q \geq 2}$.
\end{itemize}
We prove the lemma by induction on~$q$ with Proposition~\ref{prop:thomassen} as the base case.
So let~${q \geq 2}$, and
let~${\Gamma = \gen{ \{ x_i \colon i \in [q] \}}}$ for a suitable set of~$q$ generators each of order at most~$t$.
Let~${\Gamma_1 := \gen{x_q}}$ and~${\Gamma_2 := \gen{\{x_i \colon i \in [q-1]\}}}$, and let~$\gamma'$ be a $\Gamma_1$-labelling of~$W$ such that for every edge~$e$ of~$W$, we have~${\gamma(e) + \gamma'(e) \in \Gamma_2}$.
By Proposition~\ref{prop:thomassen}, $W$ has a ${(w_{\ref*{lem:smallorder}}(q-1,t,w),w_{\ref*{lem:smallorder}}(q-1,t,w))}$-subwall~$W'$ such that~${\gamma(P) \in \Gamma_2}$ for all corridors~$P$ of~$W'$.
By the induction hypothesis, $W'$ has a ${(w,w)}$-subwall~$W''$ such that~${\gamma(P) = 0}$ for all corridors~$P$ of~$W''$.
Note that as~$W'$ is a subwall of~$W$, all corridors of~$W''$ in~$W'$ are corridors of~$W''$ in~$W$.
\end{proof}
The following variation allows us to take advantage of our notion of cleanness and will be needed for Lemma~\ref{lem:omega-avoiding-cycle}.
\begin{corollary}
\label{cor:smallorder}
There exists a function~${w_{\ref*{cor:smallorder}} \colon \mathbb{N}^3 \to \mathbb{N}}$ satisfying the following.
Let~$q$, $t$ and~${w \geq 3}$ be integers, let~$\Gamma$ be an abelian group, and let~$\Lambda$ be a subgroup of~$\Gamma$ generated by~$q$ elements each of order at most~$t$,
and let~${(W, \gamma)}$ be a $\Gamma$-labelled wall of order $w_{\ref*{cor:smallorder}}(q, t, w)$.
If~${\gamma(O) \in \Lambda}$ for all cycles~$O$ of~$W$,
then~$W$ contains a $\gamma$-bipartite ${(w,w)}$-subwall.
\end{corollary}
\begin{proof}
Let~${w_{\ref*{cor:smallorder}}(q,t,w) := w_{\ref*{lem:smallorder}}(t^q,2t^q,w)}$.
For each~$g$ in~${\Lambda \cap 2 \Gamma}$, let $\sigma(g)$ be an element of~$\Gamma$ such that~${2 \sigma(g) = g}$.
Let~$\hat{\Lambda}$ be the subgroup of~$\Gamma$ generated by~${S := (\Lambda \setminus 2\Gamma) \cup \{ \sigma(g) \colon g \in \Lambda \cap 2\Gamma \}}$.
Note that~${\Lambda \subseteq \hat{\Lambda}}$,
that~${\abs{S} \leq \abs{\Lambda} \leq t^q}$,
and that each element in~$S$ has order at most~${2 \abs{\Lambda} \leq 2t^q}$.
We will show that there is a $\Gamma$-labelling~$\gamma'$
shifting-equivalent to~$\gamma$ such that~${\gamma'(P) \in \hat{\Lambda}}$ for every corridor~$P$ of~$W$.
The wall~$W$ is a subdivision of some $3$-connected planar graph~${\hat{H}}$.
Let~$T$ be a spanning tree of~${\hat{H}}$, rooted at an arbitrary vertex~$r$.
Choose a $\Gamma$-labelling~$\gamma'$ shifting-equivalent to~$\gamma$ and a subtree~$T'$ of~$T$ containing~$r$ such that~${\gamma'(P) \in \hat{\Lambda}}$ for all corridors~$P$ of~$W$ corresponding to edges in~$T'$, and subject to these conditions,~$\abs{V(T')}$ is maximised.
Suppose that~${T' \neq T}$.
Then there is an edge~$vw$ of~$T$ such that~${v \in V(T')}$ and~${w \notin V(T')}$.
Let~$Q$ be the corridor of~$W$ corresponding to the edge~$vw$.
Since~$\hat{H}$ is $3$-connected, there is a cycle~$O$ in~${W-E(Q)}$ containing~$v$ and~$w$.
Let~$O_1$ and~$O_2$ denote the distinct cycles in~${O \cup Q}$ containing~$Q$.
Since~$\gamma'$ is shifting-equivalent to~$\gamma$,
from the assumption on $W$ we deduce that ${\gamma'(O), \gamma'(O_1), \gamma'(O_2) \in \Lambda}$.
Hence, ${2 \gamma'(Q) = \gamma'(O_1) + \gamma'(O_2) - \gamma'(O)\in \Lambda}$ and so $\sigma(2\gamma'(Q))$ is well defined.
Observe that
\[
2\left(\sigma(2\gamma'(Q))-\gamma'(Q)\right)
= 2\gamma'(Q) -2\gamma'(Q)
= 0.
\]
Let~$\gamma''$ be the $\Gamma$-labelling of~$G$ obtained from~$\gamma'$ by shifting by~${\sigma(2\gamma'(Q))-\gamma'(Q)}$ at~$w$.
Then $\gamma''(Q)=\gamma'(Q)+\sigma(2\gamma'(Q))-\gamma'(Q)=\sigma(2\gamma'(Q))\in \hat\Lambda$.
Let~${T'' = T[V(T')\cup \{w\}]}$.
Then~${\gamma''(P) \in \hat{\Lambda}}$ for all corridors~$P$ of~$W$ corresponding to edges of~$T''$, contradicting our choice of~$\gamma'$ and~$T'$. Therefore $T'=T$.
Now, observe that~${\gamma'(P) \in \hat{\Lambda}}$ for every corridor~$P$ of~$W$,
because~${\Lambda \subseteq \hat{\Lambda}}$ and for every cycle~$O$ of~$W$, we have~${\gamma'(O) = \gamma(O) \in \Lambda}$.
Let~$\gamma''$ be the $\hat{\Lambda}$-labelling of~$W$ which assigns an arbitrary edge~$e_P$ of each corridor~$P$ the value~${\gamma'(P)}$, and all other edges the value~$0$.
By Lemma~\ref{lem:smallorder}, there is a ${(w,w)}$-subwall~$W'$ of~$W$ which in particular is $\gamma''$-bipartite, and hence $\gamma'$-bipartite.
Since~$\gamma'$ and~$\gamma$ are shifting-equivalent, $W'$ is $\gamma$-bipartite.
\end{proof}
\section{Handling handles}
\label{sec:handles}
This section is dedicated to proving the following key lemma, which allows us to iteratively find sets of vertex-disjoint handles.
\begin{lemma}
\label{lem:addlinkage}
There exist functions~${w_{\ref*{lem:addlinkage}}\colon \mathbb{N}^2 \to \mathbb{N}}$ and~$f_{\ref*{lem:addlinkage}} \colon \mathbb{N} \to \mathbb{N}$ satisfying the following.
Let~${k, t}$ and~$c$ be positive integers with~${c \geq 3}$, let~${\Gamma}$ be an abelian group, and let~${(G, \gamma)}$ be a $\Gamma$-labelled graph.
Let~$W$ be a wall in~$G$ of order at least~${w_{\ref*{lem:addlinkage}}(k,c)}$ such that all corridors of~$W$ are $\gamma$-zero.
For each~${i \in [t-1]}$, let~$\mathcal{P}_i$ be a set of~$4k$ $W$-handles in~$G$ such that the paths in~${\bigcup_{i \in [t-1]} \mathcal{P}_i}$
are vertex-disjoint.
If~$G$ contains at least~${f_{\ref*{lem:addlinkage}}(k)}$ vertex-disjoint $\gamma$-non-zero ${\ensuremath{V_{\neq 2}}(W)}$-paths,
then there exist a $c$-column-slice~$W'$ of~$W$ and a set~$\mathcal{Q}_i$ of~$k$ vertex-disjoint $W'$-handles for each~${i \in [t]}$ such that
\begin{enumerate}
[label=(\roman*)]
\item for each~${i \in [t-1]}$, the set~$\mathcal{Q}_i$ is a subset of the row-extension of~$\mathcal{P}_i$ to~$W'$ in~$W$,
\item the paths in~${\bigcup_{i \in [t]} \mathcal{Q}_i}$ are vertex-disjoint,
\item the paths in~$\mathcal{Q}_t$ are $\gamma$-non-zero.
\end{enumerate}
\end{lemma}
Before we can prove this lemma, we need to establish a variety of other lemmas.
At the heart of the proof, we have the following natural result, which we will iteratively apply to decouple the sets of vertex-disjoint handles that we will construct.
Huynh, Joos, and Wollan~\cite[Lemma~27]{HuynhJW2017} proved a somewhat similar result for oriented group-labelled graphs.
\begin{lemma}
\label{lem:separating}
Let~${k, t}$ be positive integers,
let~${\Gamma}$ be an abelian group,
let~${(G,\gamma)}$ be a $\Gamma$-labelled graph,
and let~$T$ be a subset of~${V(G)}$.
For each ${i \in [t-1]}$, let~$\mathcal{P}_i$ be a set of $T$-paths of size~${4k}$
such that the paths in~${\bigcup_{i \in [t-1]} \mathcal{P}_i}$ are vertex-disjoint.
If~$G$ contains~$k$ vertex-disjoint ${\gamma}$-non-zero $T$-paths,
then there exist a set~$\mathcal{Q}_t$ of $k$ vertex-disjoint ${\gamma}$-non-zero $T$-paths and a subset~${\mathcal{Q}_i \subseteq \mathcal{P}_i}$ of size~$k$ for each~${i \in [t-1]}$
so that the paths in~${\bigcup_{i \in [t]} \mathcal{Q}_i}$ are vertex-disjoint.
\end{lemma}
\begin{proof}
Let~$\mathcal{Q}_t$ be a set of~$k$ vertex-disjoint $\gamma$-non-zero $T$-paths such that
the number of edges of paths in~${\mathcal{Q}_t}$ that are not contained in any path in~${\mathcal{P} := \bigcup_{i \in [t-1]} \mathcal{P}_i}$ is as small as possible.
For each~${j \in [t-1]}$, let~$\mathcal{P}_j^\ast$ be the set of paths in~$\mathcal{P}_j$ that do not contain an endvertex of a path in~$\mathcal{Q}_t$.
We have~${\abs{\mathcal{P}_j^\ast} \geq \abs{\mathcal{P}_j}-2\abs{\mathcal{Q}_t} = 2k}$.
Let~$\mathcal{P}_j^{**}$ be the set of all paths in~$\mathcal{P}_j^\ast$
intersecting a path in~$\mathcal{Q}_t$.
Assume that for some~${j \in [t-1]}$ we have~${\abs{ \mathcal{P}_j^{**} } \geq k+1}$.
Then there are two paths~${P_1, P_2 \in \cP_j^{**}}$ such that when traversing from an endvertex~$p_i$ of~$P_i$ for each~${i \in [2]}$, $P_1$ and~$P_2$ first meet the same path~${Q \in \mathcal{Q}_t}$.
For each~${i \in [2]}$, let~$q_i$ be the first intersection of~$P_i$ and~$Q$ when traversing~$P_i$ from~$p_i$.
\begin{figure}
\begin{tikzpicture}
\tikzstyle{w}=[circle,draw,fill=black!50,inner sep=0pt,minimum width=3pt]
\node at (0,2) [w,label=left:$p_1$] (vp1) {};
\node at (1,1) [w,label=left:$p_2$] (vp2) {};
\node at (1,4.5) [w,label=below:$v$] (v){};
\node at (4,1) [w,label=left:$w$] (w){};
\draw [name path=p1] (vp1) .. controls (3,3) and (2,4) .. (4,4.5)
node[pos=0.1,label=$P_1'$]{}
node [pos=1,label=right:$P_1$]{};
\draw [name path=q] (v) .. controls (3,3) and (4,3) .. (w)
node [pos=0,label=left:$Q$]{}
node [pos=0.55,label=$Q_2$]{}
node [pos=0.2,label=$Q_1$]{}
node [pos=0.9,label=right:$Q_3$]{} ;
\draw [name path=p2] (vp2) .. controls (2,2) and (3,2) .. (5,2)
node [pos=0.2,label=$P_2'$]{}
node [pos=1,label=right:$P_2$]{};
\node [name intersections={of=p1 and q}] at (intersection-1) [w,label=left:$q_1$](vq1) {};
\node [name intersections={of=p2 and q}] at (intersection-1) [w,label=above left:$q_2$](vq2) {};
\end{tikzpicture}
\caption{Segments of the paths~$P_1$,~$P_2$ and~$Q$ mentioned in Lemma~\ref{lem:separating}.}
\label{fig:separating}
\end{figure}
Let~$v$ and~$w$ be the endvertices of~$Q$ such that the distance between~$v$ and~$q_1$ in~$Q$ is smaller than the distance between~$v$ and~$q_2$ in~$Q$.
Let~${Q_1, Q_2, Q_3}$ be the subpaths of~$Q$ from~$v$ to~$q_1$, from~$q_1$ to~$q_2$, and
from~$q_2$ to~$w$, respectively.
Also, for each~${i \in [2]}$, let~$P_i'$ be the subpath of~$P_i$ from~$p_i$ to~$q_i$,
see Figure~\ref{fig:separating}.
Since the paths of~${\bigcup_{i \in [t-1]} \mathcal{P}_i}$ are vertex-disjoint and~${v, w \notin V(P_1 \cup P_2)}$, both~$Q_1$ and~$Q_3$ contain edges not in a path of~${\bigcup_{i \in [t-1]} \mathcal{P}_i}$; for instance, edges incident with~$q_1$ or~$q_2$.
By assumption, ${\gamma(Q) = \gamma(Q_1) + \gamma(Q_2) + \gamma(Q_3)}$ is non-zero.
If there is a~${\{v, w, p_1, p_2\}}$-path~$R$ in~${Q \cup P_1' \cup P_2'}$ such that~${R \neq Q}$ and~${\gamma(R) \neq 0}$,
then by replacing~$Q$ with~$R$, the number of edges of paths in~$\mathcal{Q}_t$ that are not contained in any path in~${\bigcup_{i \in [t-1]} \mathcal{P}_i}$ decreases.
Therefore, by the assumption on~$\mathcal{Q}_t$, we have~${\gamma(R) = 0}$ for every such path~$R$.
It implies that
\begin{enumerate}
[label=(\arabic*)]
\item \label{item:separating1} ${\gamma(Q_1)+\gamma(Q_2)+\gamma(P_2') = 0}$,
\item \label{item:separating2} ${\gamma(P_1')+\gamma(Q_2)+\gamma(P_2') = 0}$,
\item \label{item:separating3} ${\gamma(P_1')+\gamma(Q_2)+\gamma(Q_3) = 0}$.
\end{enumerate}
The equations~\ref{item:separating1} and~\ref{item:separating2} imply that~${\gamma(Q_1) = \gamma(P_1')}$
and, similarly, the equations~\ref{item:separating2} and~\ref{item:separating3} imply that~${\gamma(Q_3) = \gamma(P_2')}$.
But these imply that
\[
0 = \gamma(P_1')+\gamma(Q_2)+\gamma(P_2') = \gamma(Q_1) + \gamma(Q_2) + \gamma(Q_3) \neq 0,
\]
which is a contradiction.
We conclude for all~${j \in [t-1]}$ that~${\abs{ \mathcal{P}_j^{**} } \leq k}$,
and thus~${\abs{\mathcal{P}_j^\ast\setminus \mathcal{P}_j^{**}} \geq k}$.
For each~${j \in [t-1]}$, let~$\mathcal{Q}_j$ be a set of~$k$ paths in~${\mathcal{P}_j^\ast\setminus \mathcal{P}_j^{**}}$.
Then for each~${i \in [t-1]}$, we have that~${\mathcal{Q}_i \subseteq \mathcal{P}_i}$, and
the paths in~${\bigcup_{i \in [t]} \mathcal{Q}_i}$
are vertex-disjoint, as required.
\end{proof}
Lemma~\ref{lem:addlinkage} mentions a set of vertex-disjoint $\ensuremath{V_{\neq 2}}(W)$-paths in~$G$, but note that these may arbitrarily intersect the internal vertices of corridors of~$W$.
The following technical lemma allows us to take subpaths of these paths which intersect the corridors of~$W$ in a more controlled manner.
\begin{lemma}
\label{lem:breaking}
Let~${\Gamma}$ be an abelian group,
let~${(G, \gamma)}$ be a $\Gamma$-labelled graph
and let~${H \subseteq G}$ be a subdivision of a $3$-connected graph
such that every corridor of~$H$ is $\gamma$-zero.
If~$G$ contains a $\gamma$-non-zero $\ensuremath{V_{\neq 2}}(H)$-path~$P$, then
there exist a subpath~$U$ of~$P$
and a set~$\mathcal{X}$ of at most~$12$ corridors of~$H$ satisfying the following properties:
\begin{enumerate}[label=(\roman*)]
\item\label{item:b1} ${H \cap U}$ is a subgraph of~${\bigcup \mathcal{X}}$.
\item\label{item:b2} For any subgraph~${H' \subseteq H}$ which is a subdivision of a $3$-connected graph with~${\bigcup \mathcal{X} \subseteq H'}$,
and any subset~${T \subseteq \ensuremath{V_{\neq 2}}(H')}$
with~${\abs{T} \geq 3}$,
there is a $\gamma$-non-zero $T$-path in~${H'\cup U}$.
\end{enumerate}
\end{lemma}
\begin{proof}
For each vertex~$z$ of~$H$, we define~${x_{z,1}, x_{z,2} \in \Gamma}$
and a path~$X_z$ as follows.
\begin{itemize}
\item If~$z$ has degree~$2$ in~$H$, then
let~$X_z$ be the corridor of~$H$ containing~$z$,
let~${x_{z,1} := \gamma(X_{z,1})}$,
and~${x_{z,2} := \gamma(X_{z,2})}$
where~$X_{z,1}$ and~$X_{z,2}$ are the two distinct subpaths of~$X_z$ from~$z$ to the endvertices of~$X_z$.
\item Otherwise let~$X_z$ be a path of length~$0$ containing~$z$, let~${x_{z,1} := 0}$, and~${x_{z,2} := 0}$.
\end{itemize}
A path~$Q$ from~${a \in V(H)}$ to~${b \in V(H)}$ in~$G$
with~${x_{a,1} = x_{a,2}}$ and~${x_{b,1} = x_{b,2}}$
is \emph{$\gamma$-preserving} if~${x_{a,1} + \gamma(Q) + x_{b,1} = 0}$,
and is \emph{$\gamma$-breaking} otherwise.
We first prove the following claim.
\begin{claim*}
${P \cup H}$ contains a $\gamma$-breaking path~$U$ such that
\begin{enumerate}
[label=(\alph*)]
\item\label{item:breaking1}
both endvertices of~$U$ are in~$H$,
\item\label{item:breaking3}
at most two corridors of~$H$ intersect the set of internal vertices of~$U$, and
\item\label{item:breaking4}
for each endvertex~$z$ of~$U$, either~${z \in \ensuremath{V_{\neq 2}}(H)}$
or~$X_z$ contains no internal vertex of~$U$.
\end{enumerate}
\end{claim*}
\begin{proofofclaim}
Suppose that this claim does not hold.
We first show that
\begin{enumerate}[label=($\ast$)]
\item \label{item:breakingast} if~$P$ contains a $V(H)$-path~$Q$ from~$a$ to~$b$ where~${X_a \neq X_b}$, then~$Q$ is a $\gamma$-preserving path.
\end{enumerate}
Because there are no two distinct corridors of~$H$ with the same set of endvertices, $X_a$ intersects at most one of~$X_{b,1}$ and~$X_{b,2}$.
If~${\gamma(X_{a,1}) \neq \gamma(X_{a,2})}$ and $X_{b,i}$ does not intersect~$X_a$ for some~${i \in \{1,2\}}$, then ${X_{a,1} \cup Q \cup X_{b,i}}$ or~${X_{a,2} \cup Q \cup X_{b,i}}$ is $\gamma$-breaking.
It is not difficult to verify that such a $\gamma$-breaking path satisfies the required properties, contradicting the assumption.
Thus, ${\gamma(X_{a,1}) = \gamma(X_{a,2})}$, and by symmetry,
${\gamma(X_{b,1})= \gamma(X_{b,2})}$.
Now, by the assumption, $Q$ is $\gamma$-preserving.
This shows~\ref{item:breakingast}.
\smallskip
Let~$M_1$ be the set of $V(H)$-paths in~$P$ whose endvertices are internal vertices of distinct corridors of~$H$.
Let~$M_2$ be the set of maximal subpaths of~${P - \bigcup_{Q \in M_1} E(Q)}$ of length at least~$1$.
Note that for each~${R \in M_2}$,
at most one corridor of~$H$ intersects the set of internal vertices of~$R$.
Let~$v$ and~$w$ be the endvertices of~$P$, and let~${t := \abs{M_1}+\abs{M_2}}$.
Note that $M_1\cup M_2$ is a partition of~$P$ into $t$ edge-disjoint subpaths, each having length at least~$1$.
Let~$P_1$ be the path in~${M_1 \cup M_2}$ containing~$v$, and for each~${i \in [t-1]}$ let~$P_{i+1}$ be the unique path in~${(M_1 \cup M_2) \setminus \{P_j \colon j \in [i]\}}$ sharing an endvertex, say~$v_i$, with~$P_i$.
By~\ref{item:breakingast}, we have~${\gamma(X_{v_i,1}) = \gamma(X_{v_i,2})}$ for all~${i \in [t-1]}$.
Note that~${\gamma(X_{v,2}) = \gamma(X_{w,1}) = 0}$.
We claim that there is a $\gamma$-breaking path in~$M_2$.
Suppose for a contradiction that all paths in~$M_2$ are $\gamma$-preserving.
By~\ref{item:breakingast}, all paths in~$M_1$ are $\gamma$-preserving and therefore
\[
\sum_{i=0}^{t-1} \big( \gamma(X_{v_i,2}) + \gamma(P_{i+1}) + \gamma(X_{v_{i+1},1}) \big) = 0.
\]
As every corridor of~$H$ is $\gamma$-zero,
we know that
\[
\sum_{i=0}^{t-1} \big( \gamma(X_{v_i,2}) + \gamma(X_{v_{i+1},1}) \big) = 2 \sum_{i =1}^{t-1} \gamma(X_{v_i}) = 0.
\]
This implies that~${\sum_{i =0}^{t-1} \gamma(P_{i+1}) = \gamma(P) = 0}$, which contradicts the fact that~$P$ is $\gamma$-non-zero.
So, we conclude that there exists~${j \in [t]}$ such that~${P_j \in M_2}$ and~$P_j$ is $\gamma$-breaking.
We obtain that the path~$P'$ defined by
\[
P' :=
\begin{cases}
P_1 \cup P_2 & \textnormal{ if } j = 1,\\
P_{j-1} \cup P_j\cup P_{j+1} & \textnormal{ if } j \in [t-1] \setminus \{1\},\\
P_{t-1}\cup P_t & \textnormal{ if } j = t,
\end{cases}
\]
has the desired properties.
\end{proofofclaim}
Let~$U$ be a path obtained by the previous claim.
Let~$\mathcal{X}_1$ be the set of corridors of~$H$ intersecting the set of internal vertices of~$U$.
By the previous claim, we have~${\abs{\mathcal{X}_1} \leq 2}$.
Let~$a$,~$b$ be the endvertices of~$U$.
For~${x \in \{a,b\}}$, let~$Y_x$ be a corridor of~$H$ containing~$x$.
Thus if $x$ has degree~$2$ in~$H$, then~${Y_x = X_x}$
and otherwise~$Y_x$ is an arbitrary corridor of~$H$ ending at~$x$.
Let~$\mathcal{X}_2$ be a minimal set of corridors of~$H$ such that~${Y_a, Y_b \in \mathcal{X}_2}$ and each endvertex of~$Y_a$ and~$Y_b$ is contained in at least three corridors in~$\mathcal{X}_2$.
Then~${\abs{\mathcal{X}_2} \leq 10}$.
Let~${\mathcal{X} := \mathcal{X}_1 \cup \mathcal{X}_2}$.
Then~${\abs{\mathcal{X}} \leq 12}$ and~\ref{item:b1} holds.
It remains to show~\ref{item:b2}.
Let~$H'$ be a subgraph of~$H$ which is a subdivision of a $3$-connected graph~$\hat{H}'$
such that~${\bigcup \mathcal{X}}$ is a subgraph of~$H'$
and let~$T$ be subset of~${\ensuremath{V_{\neq 2}}(H')}$ of size at least~$3$.
Note that~${\ensuremath{V_{\neq 2}}(H') \subseteq \ensuremath{V_{\neq 2}}(H)}$ and
therefore every corridor of~$H'$ is $\gamma$-zero.
By the construction of~$\mathcal{X}_2$,
each endvertex of~$U$ is contained in some corridor of~$H$ which is also a corridor of~$H'$ and every corridor of~$H$ intersecting the set of internal vertices of~$U$ is also a corridor of~$H'$.
Hence from the claim, we deduce that
\begin{enumerate}
[label=(\alph*$'$)]
\item\label{item:breaking1'}
both endvertices of~$U$ are in~$H'$,
\item\label{item:breaking3'}
at most two corridors of~$H'$ intersect the set of internal vertices of~$U$, and
\item\label{item:breaking4'}
for each endvertex~$z$ of~$U$, either~${z \in \ensuremath{V_{\neq 2}}(H')}$
or the corridor of $H'$ containing $z$ contains no internal vertex of~$U$.
\end{enumerate}
Since~$\hat{H}'$ is $3$-connected, there are two disjoint paths~$Q_1$, $Q_2$ in~$H'$ between the endvertices of~$U$ and the set~$T$.
If~${Q_1 \cup Q_2}$ does not contain an internal vertex of~$U$, then~${Q_1 \cup U \cup Q_2}$ is as desired, since~$U$ is $\gamma$-breaking and all corridors of~$H$ are $\gamma$-zero.
If~${Q_1 \cup Q_2}$ contains an internal vertex of~$U$, then~${Q_1 \cup Q_2}$ contains a corridor~$R$ of~$H'$ intersecting the set of internal vertices of~$U$.
Choose~$x$ among two endvertices of~$R$ that is closer to~$T$ in~${Q_1 \cup Q_2}$.
Then~$x$ is not an endvertex of~$U$.
Since~$x$ is in at least~$3$ corridors of~$H'$, by property~\ref{item:breaking3'},~${x \notin V(U)}$.
Since~$\hat{H}'$ is $3$-connected, by properties~\ref{item:breaking3'} and~\ref{item:breaking4'}, ${H'-E(\bigcup \mathcal{X}_1)}$ is connected.
Thus,~$H'$ has a path~$Q$ connecting the endvertices of~$U$ which contains no internal vertex of~$U$.
The cycle~${O := Q \cup U}$ is $\gamma$-non-zero since~$U$ is $\gamma$-breaking.
Since~${\hat{H}'}$ is~$3$-connected, there are three vertex-disjoint paths from~$T$ to~${\{x\} \cup \ensuremath{V_{\neq 2}}(U)}$ in~$H'$.
By extending one of the paths ending at~$x$ to an internal vertex of~$U$ through~$R$ if~${x \notin V(O)}$,
we obtain three vertex-disjoint~${(V(O),T)}$-paths in~${H' \cup U}$.
Hence Lemma~\ref{lem:non-zero-path-in-cycle} yields the desired result.
\end{proof}
In the next lemma, we extend subpaths from the previous lemma to handles of some suitable column-slice.
\begin{lemma}
\label{lem:addlinkage1}
There exist functions~${w_{\ref*{lem:addlinkage1}}\colon \mathbb{N}^2 \to \mathbb{N}}$ and~$f_{\ref*{lem:addlinkage1}} \colon \mathbb{N} \to \mathbb{N}$ satisfying the following.
Let~$k$ and~$c$ be positive integers with~${c \geq 3}$, let~${\Gamma}$ be an abelian group, and let~${(G, \gamma)}$ be a $\Gamma$-labelled graph.
Let~$W$ be a wall in~$G$ of order at least~${w_{\ref*{lem:addlinkage1}}(k,c)}$ such that all corridors of~$W$ are $\gamma$-zero.
If~$G$ contains ${f_{\ref*{lem:addlinkage1}}(k)}$ vertex-disjoint
$\gamma$-non-zero
$\ensuremath{V_{\neq 2}}(W)$-paths,
then there exist a $c$-column-slice~$W'$ of~$W$ and~$k$ vertex-disjoint $\gamma$-non-zero $W'$-handles in~$G$.
\end{lemma}
\begin{proof}
Let~${h(k) := 3 f_{\ref*{thm:tpath}}(k) + 1}$
and~${f_{\ref*{lem:addlinkage1}}(k) := 2 \cdot 240^2 h(k)^2 }$.
Let~$\mathcal{P}$ be a set of~${f_{\ref*{lem:addlinkage1}}(k)}$ vertex-disjoint $\gamma$-non-zero $\ensuremath{V_{\neq 2}}(W)$-paths.
Let~${w_{\ref*{lem:addlinkage1}}(k,c) := (48 h(k) + 1)(c - 1) + 144h(k) + 1 }$.
\begin{claim*}
For all~${i \in [h(k)]}$ there exist a set~$\mathcal{C}_i$ of $3$-column-slices of~$W$, a set~$\mathcal{R}_i$ of $3$-row-slices of~$W$ and a subpath~$U_i$ of a path in~$\mathcal{P}$ such that, with ${H_i := \bigcup(\mathcal{C}_i \cup \mathcal{R}_i)}$, we have
\begin{enumerate}
[label=(\alph*)]
\item\label{item:slicenumbers} ${1 \leq \abs{\mathcal{C}_i} \leq 48}$ and~${1 \leq \abs{\mathcal{R}_i} \leq 48}$,
\item\label{item:disjointcolumns} every~${C \in \mathcal{C}_i}$ is vertex-disjoint from every~${C' \in \mathcal{C}_j}$ for all~${j \in [i-1]}$,
\item\label{item:disjointrows} every~${R \in \mathcal{R}_i}$ is vertex-disjoint from every~${R' \in \mathcal{R}_j}$ for all~${j \in [i-1]}$,
\item\label{item:vertexdisjU} $U_i$ and~$U_j$ are vertex-disjoint for all~${j \in [i-1]}$,
\item\label{item:pathintersections-col} every $3$-column-slice of~$W$ that intersects~$U_i$ also intersects some $3$-column-slice in~${\bigcup_{j \in[i]} \mathcal{C}_{j}}$,
\item\label{item:gammabreakingTpath} for any column-slice~$W'$ of~$W$ which is disjoint from~$U_i$, there is a $\gamma$-non-zero $W'$-handle in~${H_i \cup U_i}$.
\end{enumerate}
\end{claim*}
\begin{proofofclaim}[Proof of Claim]
We proceed by induction on~$i$.
For~${i \in [h(k)]}$, assume that the claim holds for all~${j \in [i-1]}$.
We define
\begin{itemize}
\item $\widetilde{\mathcal{C}}_i$ to be the set of all $3$-column-slices of~$W$ which intersect no $3$-column-slices in~${\bigcup_{j \in [i-1]} \mathcal{C}_{j}}$,
\item $\widetilde{\mathcal{R}}_i$ to be the set of all $3$-row-slices of~$W$ which intersect no $3$-row-slices in~${\bigcup_{j \in [i-1]} \mathcal{R}_{j}}$, and
\item ${\widetilde{H}_i := \bigcup \left( \widetilde{\mathcal{C}}_i \cup \widetilde{\mathcal{R}}_i \right)}$.
\end{itemize}
We will first show that the number of vertices in~${\ensuremath{V_{\neq 2}}(W) \setminus V(\widetilde{H}_i)}$ is small.
Let~$I$ be the set of all column indices~$a$ such that
the $a$-th column~$C^W_a$ intersects no $3$-column-slice in~${\bigcup_{j \in [i-1]} \mathcal{C}_j}$.
Then~$I$ admits a partition into intervals consisting of consecutive integers such that the number of intervals is bounded by~${\sum_{j \in [i-1]} \abs{\mathcal{C}_j} + 1 \leq 48(i-1)+1}$.
Observe that at least one interval of~$I$ has size at least~$3$ because~${w_{\ref*{lem:addlinkage1}}(k,c) > 3 \cdot 48 (i-1) + 2 \cdot (48 (i-1) + 1)}$.
This implies that~$\widetilde{C}_i$ is nonempty.
Suppose that a vertex~$v$ in~${\ensuremath{V_{\neq 2}}(W)}$ is not in~${\widetilde{H}_i}$.
Let us say that~${v \in V(C^W_x) \cap V(R^W_y)}$ for some~$x$ and~$y$.
Since~$v$ is not in~$\widetilde{H}_i$, either~$C^W_x$ intersects some $3$-column-slice in~$\mathcal{C}_j$ for some~${j < i}$
or~$x$ belongs to an interval of~$I$ of size at most~$2$.
Since at least one interval of~$I$ has size more than~$2$,
the number of possible values of~$x$ is at most
\[
3 \cdot \sum_{j \in [i-1]} \abs{\mathcal{C}_j} +
2 \cdot \sum_{j \in [i-1]} \abs{\mathcal{C}_j}
\leq 240(i-1).
\]
By the same argument,
we deduce that~$\widetilde{R}_i$ is nonempty
and the number of possible values of~$y$ is at most~${240(i-1)}$.
Thus, the number of vertices in~${\ensuremath{V_{\neq 2}}(W)}$ not in~$\widetilde{H}_i$ is at most~${2(240(i-1))^2}$,
because there are at most two vertices of~${\ensuremath{V_{\neq 2}}(W)}$ in~${V(C^W_x) \cap V(R^W_y)}$ for each~$x$ and~$y$.
Since~${\abs{\mathcal{P}} \geq 2(240i)^2 > 2(240(i-1))^2+(i-1)}$,
there is a path~$P_i$ in~$\mathcal{P}$ both of whose endvertices are in~$\widetilde{H}_i$ such that~${U_j}$ is not a subpath of~${P_i}$ for all~${j < i}$.
Let~$\mathcal{X}_i$ be the set of at most~$12$ corridors of~$\widetilde{H}_i$ and let~$U_i$ be a subpath of~${P_i}$ guaranteed by Lemma~\ref{lem:breaking}.
Let~$\mathcal{C}_i$ be a minimal non-empty subset of~$\widetilde{C}_i$ containing all $3$-column-slices in~$\widetilde{\mathcal{C}}_i$
which intersect some corridor in~$\mathcal{X}_i$.
Since each corridor of~$\widetilde{H}_i$ intersects at most four $3$-column-slices,~${\abs{\mathcal{C}_i} \leq 4 \cdot 12}$.
Similarly, let~$\mathcal{R}_i$ be a minimal non-empty subset of $\widetilde{R}_i$ containing all $3$-row-slices in~$\widetilde{\mathcal{R}}_i$ which intersect some corridor in~$\mathcal{X}_i$.
Then~${\abs{\mathcal{R}_i} \leq 48}$.
Now~\ref{item:slicenumbers}--\ref{item:pathintersections-col} are true by construction.
To see~\ref{item:gammabreakingTpath},
let~$W'$ be a column-slice disjoint from~$U_i$.
Note that~${H_i \cup W'}$ is a subdivision of a $3$-connected graph.
Applying Lemma~\ref{lem:breaking}\ref{item:b2}
with~${H' := H_i \cup W'}$ and~${T := \ensuremath{V_{\neq 2}}(H') \cap V(W')}$,
there is a $\gamma$-non-zero $T$-path~$P$ in~${H' \cup U_i}$.
Then~$P$ must use at least one edge of~$U_i$ because ${\gamma(P') = 0}$ for every $T$-path~$P'$ in~$H'$.
This implies that ${P \subseteq H_i \cup U_i}$, because~$T$ separates~$U_i$ from~$W'$ in~$H'$.
It follows that~$P$ is a $W'$-handle.
\end{proofofclaim}
Let~$I$ be the set of all column indices~$a$ such that
$C^W_a$ intersects no $3$-column-slice in~${\bigcup_{j \in [h(k)]} \mathcal{C}_j}$.
By~\ref{item:slicenumbers},~$I$ admits a partition into at most~${48h(k)+1}$ disjoint intervals, each consisting of consecutive integers.
Since~${w_{\ref*{lem:addlinkage1}}(k,c) - 3 \cdot 48 h(k) > (48 h(k) + 1)(c-1)}$, there exist~$c$ consecutive columns of~$W$ that do not intersect any $3$-column-slice in~${\bigcup_{i \in [h(k)]} \mathcal{C}_i}$.
Thus they form a $c$-column-slice~$W'$ of~$W$ which is disjoint from~${\bigcup_{i \in [h(k)]} U_i}$ by~\ref{item:pathintersections-col}.
By~\ref{item:gammabreakingTpath}, for each~${i \in[h(k)]}$, there is a $\gamma$-non-zero $W'$-handle~$X_i$ in~${H_i \cup U_i}$.
By construction, each vertex of~${W - V(W')}$ is contained in at most two graphs in~${\{ H_i \colon i \in [h(k)] \}}$
and in at most one path in~${\{ U_i \colon i \in [h(k)] \}}$.
Hence, every vertex of~$W$ is contained in at most three paths in~${\{ X_i \colon i \in [h(k)] \}}$.
Since~${h(k) = 3 f_{\ref*{thm:tpath}}(k) + 1}$, any vertex set of size at most~${f_{\ref*{thm:tpath}}(k)}$ cannot hit all these paths.
By Theorem~\ref{thm:tpath}, there exist~$k$ vertex-disjoint $\gamma$-non-zero $W'$-handles, as desired.
\end{proof}
Now we obtain Lemma~\ref{lem:addlinkage} as a corollary of Lemmas~\ref{lem:separating} and~\ref{lem:addlinkage1}.
\begin{proof}[Proof of Lemma~\ref{lem:addlinkage}]
Let~$w_{\ref*{lem:addlinkage}}$ and~$f_{\ref*{lem:addlinkage}}$ be the functions~$w_{\ref*{lem:addlinkage1}}$ and~$f_{\ref*{lem:addlinkage1}}$ respectively, as given by Lemma~\ref{lem:addlinkage1}.
By Lemma~\ref{lem:addlinkage1},
there exists a $c$-column-slice~$W'$ of~$W$ and a set~$\mathcal{P}'$ of~$k$ vertex-disjoint $\gamma$-non-zero $W'$-handles.
For each~${i \in [t-1]}$ and each~${P \in \mathcal{P}_i}$, let~${R_{P,1}}$ and~${R_{P,2}}$ be the rows of~$W$ containing the endvertices of~$P$,
and let~$Q_P$ be the row-extension of~$P$ to $W'$, that is the unique $V(W')$-path such that~${P \subseteq Q_P \subseteq P \cup R_{P,1} \cup R_{P,2}}$.
We remark that it is possible that~${R_{P,1} = R_{P,2}}$.
Now applying Lemma~\ref{lem:separating} to the sets~${\{ Q_P \colon P \in \mathcal{P}_i \}}$ for~${i \in [t-1]}$ together with~$\mathcal{P}'$ yields the desired result.
\end{proof}
\section{Basic lemmas for products of abelian groups}
\label{sec:abelian}
In this section, we prove some basic lemmas on products of abelian groups that will be useful throughout Sections~\ref{sec:handles2cycles} and~\ref{sec:main}.
\medskip
An \emph{arithmetic progression} is a set of integers~$A$ such that there are integers~$a$ and~${b \neq 0}$ for which~${A = \{ a + bn \colon n \in \mathbb{Z} \}}$.
For a set~${\mathcal{A} = \{A_i \colon i \in [k]\}}$ of arithmetic progressions, we say~$\mathcal{A}$ \emph{covers} a set~$S$ if~${S \subseteq \bigcup_{i \in[k]} A_i}$.
We will use the following fact about arithmetic progressions, conjectured by Erd\H{o}s in 1962 and proven by Crittenden and Vanden Eynden~\cite{CrittendenVE1970} in 1969.
We cite an equivalent version of Balister et~al.~\cite{BalisterBMST2020}, who presented a simple proof.
\begin{theorem}[\cite{CrittendenVE1970,BalisterBMST2020}]
\label{thm:arithmetic}
Let~${\mathcal{A} = \{ A_i \colon i \in [k] \}}$ be a set of~$k$ arithmetic progressions.
If~$\mathcal{A}$ covers a set of~$2^k$ consecutive integers, then~$\mathcal{A}$ covers~$\mathbb{Z}$.
\end{theorem}
\begin{corollary}
\label{cor:omega-avoiding}
Let~$m$, $t$, and~$\omega$ be positive integers, let~${\Gamma = \prod_{j \in [m]} \Gamma_j}$ be a product of~$m$ abelian groups, and for all~${j \in [m]}$ let~$\Omega_j$ be a subset of~$\Gamma_j$ of size at most~$\omega$.
For all~${i \in [t]}$ and~${j \in [m]}$, let~${g_{i,j}}$ be an element of~$\Gamma_j$.
If for each~${i \in [t]}$ there exists an integer~$c_i$
such that~${\sum_{i=1}^t c_i g_{i,j} \notin \Omega_j}$
for all~${j \in [m]}$,
then for each~${i \in [t]}$ there exists an integer~${d_i \in [2^{m\omega}]}$
such that~${\sum_{i=1}^t d_i g_{i,j} \notin \Omega_j}$ for all~${j \in [m]}$.
\end{corollary}
\begin{proof}
Pick an integer~$d_i$ for each~${i \in [t]}$ such that~${\sum_{i=1}^t d_i g_{i,j} \notin \Omega_j}$ for all~${j \in [m]}$, and subject to this ${\abs{\{i \in [t] \colon d_i \in [2^{m\omega}]\}}}$ is maximised.
Suppose for a contradiction that for some~${x \in [t]}$, we have that ${d_x \notin [2^{m\omega}]}$.
Without loss of generality, we may assume~${x = t}$.
For all~${j \in [m]}$ and~${g \in \Omega_j}$, let~$A_{j,g}$ be the set of integers~$d$ such that~${d g_{t,j} + \sum_{i=1}^{t-1} d_i g_{i,j} = g}$.
Note that~$A_{j,g}$ is an arithmetic progression or contains at most one integer.
Let $A_{j,g}'$ be an arithmetic progression such that~${A_{j,g} \subseteq A_{j,g}'}$ and~${d_t \notin A_{j,g}'}$.
Such an~$A_{j,g}'$ exists;
if~$A_{j,g}$ is an arithmetic progression then let~${A'_{j,g} := A_{j,g}}$, and if~$A_{j,g}$ contains a unique integer~$a_j$,
then let~${A'_{j,g}}$ be the arithmetic progression~${\{ a_j + 2 (d_t - a_j) k \colon k \in \mathbb{Z}\}}$.
Now~${\mathcal{A} := \{ A'_{j,g} \colon j \in [m], g \in \Omega_j \}}$ is a set of~${m\omega}$ arithmetic progressions not covering~$d_t$.
By Theorem~\ref{thm:arithmetic}, there exists~${d'_t \in [2^{m\omega}]}$ such that~${d'_t g_{t,j} + \sum_{i=1}^{t-1} d_i g_{i,j} \notin \Omega_j}$ for all~${j \in [m]}$, contradicting our choice of~$d_t$.
\end{proof}
\medskip
For a sequence~${\mathfrak{a} = ( a_i \colon i \in [t] )}$ over an abelian group~$\Gamma$,
we let~${\Sigma(\mathfrak{a})}$ denote the set of all sums of subsequences of~$\mathfrak{a}$.
We write~${\abs{\mathfrak{a}} := t}$, the length of~$\mathfrak{a}$.
We say~${a \in \Gamma}$ is \emph{repeated} in~$\mathfrak{a}$ if~${a = a_i = a_j}$ for some~${1 \leq i < j \leq t}$.
We say that~$\mathfrak{a}$ is \emph{good} if
$\abs{\Sigma(\mathfrak{a})} \geq \abs{\mathfrak{a}}$.
Obviously, a sequence of pairwise distinct elements of~$\Gamma$ is good.
Also, observe that for an element~$g$ of order at least~${t}$, if~${a_i := g}$ for all~${i \in [t]}$, then the sequence~${\mathfrak{a} = (a_i \colon i \in [t])}$ is good,
because ${\abs{\Sigma(\mathfrak{a})} \geq \abs{ \{kg \colon k \in [t]\}} \geq t}$.
\begin{lemma}\label{lem:smallgoodset}
Let~$\Gamma$ be an abelian group and let~${\mathfrak{a} = (a_i \colon i \in [t])}$ be a sequence of length~$t$ over~$\Gamma$.
If all repeated elements in~$\mathfrak{a}$ have order at least~$t$, then~$\mathfrak{a}$ is good.
\end{lemma}
\begin{proof}
We proceed by induction on~$t$.
We may assume that~${t \geq 2}$.
If~$\mathfrak{a}$ has no repeated elements, then~${\{ a_i \colon i \in [t]\} \subseteq \Sigma(\mathfrak{a})}$ and therefore~$\mathfrak{a}$ is good.
Thus,
without loss of generality, we may assume that~$a_t$ is a repeated element.
Let~${\mathfrak{a}' := (a_i \colon i \in [t-1])}$
and~${S := \Sigma(\mathfrak{a}')}$.
By induction~${\abs{S} \geq t-1}$.
We may assume that~${\abs{S} = t-1}$.
Let~${T := \{ x + a_t \colon x \in S \} \subseteq \Sigma(\mathfrak{a})}$.
If~${S = T}$, then~${\sum_{x \in S} x = \sum_{x \in S} (x+a_t)}$
and therefore~${\abs{S}a_t = 0}$, contradicting the assumption on the order of~$a_t$.
Thus~${S \neq T}$ and therefore~${\abs{\Sigma(\mathfrak{a})} \geq \abs{S \cup T} \geq t}$.
\end{proof}
\medskip
The following lemma and its corollary are useful to find a
cycle whose $\gamma_i$-value is not in~$\Omega_i$ for all~${i \in [m]}$.
\begin{lemma}
\label{lem:vectorsum}
Let~$m$,~$t$, and~$\omega$ be positive integers, let~${\Gamma = \prod_{j \in [m]} \Gamma_j}$ be a product of~$m$ abelian groups and for all~${j \in [m]}$ let~$\Omega_j$ be a subset of~$\Gamma_j$ of size at most~$\omega$.
For all~${i \in [t]}$ let~$S_i$ be a subset of~$\Gamma$
such that for each~${j \in [m]}$ there exists some~${i \in [t]}$ such that~${\pi_j(g) \neq \pi_j(g')}$
for all distinct~${g, g'}$ in~$S_i$.
If~${\abs{S_i} > m\omega }$ for all~${i \in [t]}$,
then for every~${h \in \Gamma}$ there is a sequence~${(g_i \colon i \in [t])}$ of elements of~$\Gamma$ such that
\begin{enumerate}
[label=(\roman*)]
\item ${g_i \in S_i}$ for each ${i \in [t]}$, and
\item ${\pi_j\left(h + \sum_{i \in [t]} g_i \right) \notin \Omega_j}$ for all~${j \in [m]}$.
\end{enumerate}
\end{lemma}
\begin{proof}
Uniformly at random, select~${g_i \in S_i}$ independently for each~${i \in [t]}$, and consider the sum ${g := h + \sum_{i \in [t]} g_i}$.
For each~${j \in [m]}$, there exists~${i \in [t]}$ such that~$g$ and every group element obtained by replacing~$g_i$ in the sum with a different element of~$S_i$ have distinct $j$-th coordinates.
Hence, the probability that~${\pi_j(g) \in \Omega_j}$ is at most~${\omega/(m\omega+1)}$.
It follows that there is a positive probability that~${\pi_j(g) \notin \Omega_j}$ for all~${j \in [m]}$.
\end{proof}
For two sequences $\mathfrak{a} = (a_i \colon i \in[t])$, $\mathfrak{b} = (b_i \colon i \in [t])$ of length~$t$ over a product~${\Gamma = \prod_{j \in [m]} \Gamma_j}$ of~$m$ abelian groups,
we write~${\mathfrak{a} - \mathfrak{b}}$ to denote the sequence~${(a_i - b_i \colon i \in [t])}$
and for~${j \in [m]}$, we write~${\pi_j(\mathfrak{a})}$ to denote the sequence~${(\pi_j(a_i) \colon i \in [t])}$ over~$\Gamma_j$.
\begin{corollary}
\label{cor:vectorsum}
Let~$m$ and~$\omega$ be positive integers,
let~${\Gamma = \prod_{i \in [m]} \Gamma_i}$ be a product of~$m$ abelian groups,
and for each~${i \in [m]}$ let~$\Omega_i$ be a subset of~$\Gamma_i$ of size at most~$\omega$ and let~${\mathfrak{a}_i := (a_{i,j} \colon j \in [m\omega+1])}$ and ${\mathfrak{b}_i := (b_{i,j} \colon j \in [m\omega+1])}$ be two sequences over~${\Gamma}$
such that~${ \pi_i(\mathfrak{a}_{i}-\mathfrak{b}_{i})}$ is good.
Then for all~${h \in \Gamma}$, ${i \in [m]}$, and~${j \in [m\omega+1]}$,
there exists~${c_{i,j} \in \{a_{i,j},b_{i,j}\}}$
such that for all~${x \in [m]}$, we have~${\pi_x\left(h + \sum_{i \in [m]} \sum_{j \in [m\omega+1]} c_{i,j}\right)\notin \Omega_x}$.
\end{corollary}
\begin{proof}
By definition of a good sequence, for each~${i \in [m]}$, we have
${\abs{\Sigma(\pi_i(\mathfrak{a}_{i} - \mathfrak{b}_{i}))} \geq m\omega+1}$.
Let~$S_i$ be a subset of~${\Sigma(\mathfrak{a}_{i} - \mathfrak{b}_{i})}$
such that~$\pi_i$ restricted to~$S_i$ is a bijection from~$S_i$ to~${\Sigma(\pi_i(\mathfrak{a}_{i} - \mathfrak{b}_{i}))}$.
Let~${h' := h+\sum_{i \in [m]} \sum_{j \in [m\omega+1]} b_{i,j}}$.
We apply Lemma~\ref{lem:vectorsum} with~$h'$ to find a sequence~${(g_i \colon i \in [m])}$ such that~${g_i \in S_i}$ for each~${i \in [m]}$
and~${\pi_j\left(h' + \sum_{i \in [m]} g_i\right)\notin \Omega_j}$ for all $j\in [m]$.
Now, for each~${i \in [m]}$, we have that ${g_i \in \Sigma(\mathfrak{a}_i - \mathfrak{b}_i)}$, and hence for each~${j \in [m\omega+1]}$ there exists~${c_{i,j} \in \{a_{i,j},b_{i,j}\}}$ such that~${g_i+\sum_{j \in [m\omega+1]} b_{i,j} = \sum_{j \in [m\omega+1]} c_{i,j}}$.
This completes the proof.
\end{proof}
\medskip
Given positive integers~$n$ and~$k$, we write~${R(n;k)}$ for the minimum integer~$N$ such that in every $k$-colouring of the edges of~$K_N$ there is a monochromatic copy of~$K_n$.
A classical result of Ramsey~\cite{Ramsey1929} shows that~${R(n;k)}$ exists.
\begin{lemma}
\label{lem:ramsey}
There exists a function~$f_{\ref*{lem:ramsey}} \colon \mathbb{N}^2 \to \mathbb{N}$ satisfying the following.
Let~$m$,~$t$ and~$N$ be positive integers
with~${N \geq f_{\ref*{lem:ramsey}}(t,m)}$
and let~${\Gamma = \prod_{i \in [m]} \Gamma_i}$ be a product of~$m$ abelian groups.
Then for every sequence~${(g_i \colon i \in [N])}$ over~$\Gamma$,
there exists a subset~$I$ of~$[N]$ with~${\abs{I} = t}$ such that for each~${i \in [m]}$, either
\begin{itemize}
\item ${\pi_i(g_{j}) = \pi_i(g_{k})}$ for all ${j,k \in I}$, or
\item ${\pi_i(g_{j}) \neq \pi_i(g_{k})}$ for all distinct~${j,k \in I}$.
\end{itemize}
Furthermore, if~$Z$ is a subset of~${[m]}$ such that for all distinct~$i$ and~$j$ in~${[N]}$ there exists ${x \in Z}$ such that~${\pi_x(g_i) \neq \pi_x(g_j)}$, then the second condition holds for some~${i \in Z}$.
\end{lemma}
\begin{proof}
Let~${f_{\ref*{lem:ramsey}}(t,m) := R(t; 2^m)}$.
We define a $2^m$-colouring of the edges of~$K_N$ by colouring each edge~${xy}$ of~$K_N$ by the set~${\{ i \in [m] \colon \pi_i(g_x) \neq \pi_i(g_y)\}}$.
The result follows from the definition of~${R(t;2^m)}$.
Note that if~$Z$ is subset of~${[m]}$ such that for all distinct~$i$ and~$j$ in~${[N]}$ there exists an~${x \in Z}$ such that~${\pi_x(g_i) \neq \pi_x(g_j)}$, then every set used in the colouring intersects~$Z$.
\end{proof}
\section{From handles to cycles}
\label{sec:handles2cycles}
The focus of this section is proving the following key lemma, which will be the final ingredient needed in the proof of Theorem~\ref{thm:main} for constructing
the cycles from the clean subwall from Section~\ref{sec:cleanwalls} and the sets of handles from Section~\ref{sec:handles}.
\begin{lemma}
\label{lem:omega-avoiding-cycle}
There exist functions ${c_{\ref*{lem:omega-avoiding-cycle}}, r_{\ref*{lem:omega-avoiding-cycle}} \colon \mathbb{N}^4 \to \mathbb{N}}$ satisfying the following.
Let~$t$,~$\ell$,~$m$ and~$\omega$ be positive integers with~${\ell \geq 3}$,
let~${\Gamma = \prod_{i \in [m]} \Gamma_i}$ be a product of~$m$ abelian groups, for each~${i \in [m]}$ let~$\Omega_i$ be a subset of~$\Gamma_i$ of size at most~$\omega$,
and let~${(G,\gamma)}$ be a $\Gamma$-labelled graph.
Let~$Z$ be a subset of~${[m]}$,
let~$W$ be a ${(\gamma,Z,\ell)}$-clean $(c,r)$-wall
with~${c \geq c_{\ref*{lem:omega-avoiding-cycle}}(t, \ell, m, \omega)}$
and~${r \geq r_{\ref*{lem:omega-avoiding-cycle}}(t, \ell, m, \omega)}$.
For every set~$\mathcal{P}$ of at most~$t$ vertex-disjoint $W$-handles such that~${\gamma_i\left( \bigcup \mathcal{P} \right) \notin \Omega_i}$ for all~${i \in Z}$,
there is a cycle~$O$ in~${W \cup \bigcup \mathcal{P}}$ such that~${\gamma_i(O) \notin \Omega_i}$ for all~${i \in [m]}$.
\end{lemma}
We begin by linking up a set of handles of a wall into a cycle.
\begin{lemma}
\label{lem:pairlink}
Let~$t$ be a positive integer and
let~$W$ be a ${(c,r)}$-wall in a graph~$G$ with~${r \geq 3}$ and~${c \geq \max\{3,t+1\}}$.
For every set~$\mathcal{P}$ of at most~$t$ vertex-disjoint $W$-handles in~$G$,
there is a cycle~$O$ in~${W \cup \bigcup \mathcal{P}}$ that contains~${\bigcup \mathcal{P}}$ as a subgraph.
\end{lemma}
\begin{proof}
Let~$T$ be the set of endvertices of all paths in~$\mathcal{P}$.
We proceed by induction on~$t$.
If~${\abs{\mathcal{P}} \leq 2}$, then
as~${W \cup \bigcup \mathcal{P}}$ is $2$-connected, ${W \cup \bigcup \mathcal{P}}$ has a cycle~$O$ containing at least one edge from every path in~$\mathcal{P}$.
Therefore, we may assume that~${\abs{\mathcal{P}} = t > 2}$.
By symmetry, we may assume that the first column of~$W$ meets at least two paths in~$\cP$.
In the first column of~$W$, choose two degree-$2$ nails~$v_1$,~$v_2$ that are endvertices of distinct paths~$P_1$,~$P_2$ of~$\cP$ respectively
such that the distance between~$v_1$ and~$v_2$ in the first column of~$W$ is minimised.
Let~$Q$ be the path from~$v_1$ to~$v_2$ in the first column of~$W$.
Let~$P^\ast$ be the path~${P_1 \cup Q \cup P_2}$.
Let~$W'$ be the ${(c-1)}$-column-slice of~$W$ obtained by removing the first column.
Let~$\mathcal{P}'$ be the row-extension of~${(\cP \setminus \{P_1, P_2\}) \cup \{ P^\ast \}}$ to~$W'$.
Since~$\mathcal{P}'$ is a set of~${t-1}$ vertex-disjoint $W'$-handles,
by the induction hypothesis, there is a cycle~$O$ in~${W' \cup \bigcup \mathcal{P}'}$ such that~${ \bigcup \mathcal{P}' \subseteq O}$.
This completes the proof because~${\bigcup \mathcal{P} \subseteq \bigcup \mathcal{P}'}$ and~${W' \cup \bigcup \mathcal{P}' \subseteq W \cup \bigcup \mathcal{P}}$.
\end{proof}
In order to obtain a
cycle whose $\gamma_i$-value is not in~$\Omega_i$ for every~${i \in [m]}$,
it will be useful to have access to a sequence of subwalls to reroute segments of the cycle constructed by the previous lemma.
The following straightforward corollary provides this.
\begin{corollary}
\label{cor:cycle+subwalls}
There exist functions~${c_{\ref*{cor:cycle+subwalls}}, r_{\ref*{cor:cycle+subwalls}} \colon \mathbb{N}^3 \rightarrow \mathbb{N}}$ satisfying the following.
Let~$t$,~$k$ and~$w$ be positive integers with~${w \geq 4}$.
Let~$G$ be a graph containing a $(c,r)$-wall~$W$ with~${c \geq c_{\ref*{cor:cycle+subwalls}}(t,k,w)}$ and~${r \geq r_{\ref*{cor:cycle+subwalls}}(t,k,w)}$.
For every set~$\mathcal{P}$ of at most~$t$ vertex-disjoint $W$-handles in~$G$,
there exist a cycle~$O$ in~${W \cup \bigcup \mathcal{P}}$ and a set~${\{ W_i \colon i \in [k] \}}$ of~$k$ vertex-disjoint $N^W$-anchored ${(w,w)}$-subwalls of~$W$ such that~${\bigcup \mathcal{P} \subseteq O}$
and~${W_i \cap O = R_1^{W_i}}$ for all~${i \in [k]}$.
\end{corollary}
\begin{proof}
Define
\[
{c_{\ref*{cor:cycle+subwalls}}(t,k,w) := kw+t+1, } \
\textnormal{ and } \
{r_{\ref*{cor:cycle+subwalls}}(t,k,w) := (2t+1)(w-2) + 1}.
\]
The case~${\mathcal{P} = \emptyset}$ is easy to verify, so we may assume~${\abs{\mathcal{P}} = t > 0}$.
Without loss of generality, we may assume that the last column of~$W$ intersects~$\bigcup \mathcal{P}$.
Let~$W''$ be a ${kw}$-column-slice of~$W$ containing the last column of~$W$
and let~$W'$ be a ${(c-kw)}$-column-slice of~$W$ disjoint with~$W''$.
Let~$\mathcal{P}'$ denote the row-extension of~$\mathcal{P}$ to~$W'$ in~$W$.
By the pigeonhole principle, there is a ${(w-1)}$-row-slice of~$W''$ which is disjoint from~${\bigcup \mathcal{P}'}$.
Hence there is a $w$-row-slice~$S$ of~$W''$ such that~${S \cap \bigcup \mathcal{P}' = R_1^{S}}$.
We can pack~$k$ vertex-disjoint $N^W$-anchored ${(w,w)}$-subwalls~${\{ W_i \colon i \in [k]\}}$ in~$S$ so that~${W_i \cap \bigcup \mathcal{P}' = R_1^{W_i}}$.
Applying Lemma~\ref{lem:pairlink} to~$W'$ and~$\mathcal{P}'$ yields the desired cycle.
\end{proof}
Moreover, we need the following variation of Lemma~\ref{lem:non-zero-path-in-cycle}.
\begin{lemma}
\label{lem:reroutingpath}
Let~$\Gamma$ be an abelian group, and let~${(G,\gamma)}$ be a $\Gamma$-labelled graph,
let~$O$ be a $\gamma$-non-zero cycle in~$G$,
and let~$P$ be a path disjoint from~$O$.
If~$G$ contains three vertex-disjoint ${(V(P),V(O))}$-paths~$P_1$, $P_2$, $P_3$,
then there is a path~$P'$ in~${P \cup O \cup P_1 \cup P_2 \cup P_3}$ with the same endvertices as~$P$ such that~${\gamma(P') \neq \gamma(P)}$.
\end{lemma}
\begin{proof}
Let~${T := V(O) \cap (V(P_1)\cup V(P_2)\cup V(P_3))}$.
Since~${\abs{T} = 3}$, the cycle~$O$ contains three distinct $T$-paths~$Q_1$, $Q_2$, $Q_3$ such that~${V(Q_i) \cap V(P_i) = \emptyset}$ for each~${i \in [3]}$.
Since~${\gamma(O) \neq 0}$, we have that~${\gamma(O) \neq 2\gamma(O)}$, which implies
\[
\gamma(Q_1) + \gamma(Q_2) + \gamma(Q_3) \neq (\gamma(Q_2) + \gamma(Q_3)) + (\gamma(Q_3) + \gamma(Q_1)) + (\gamma(Q_1) + \gamma(Q_2)).
\]
Without loss of generality, we may assume~${\gamma(Q_3) \neq \gamma(Q_1) + \gamma(Q_2)}$.
Observe that there are paths~$P'$ and~$P''$ in~${P \cup O \cup P_1\cup P_2}$ with the same endvertices as~$P$ such that~${E(P') \setminus E(P'') = E(Q_3)}$ and~${E(P'') \setminus E(P') = E(Q_1) \cup E(Q_2)}$.
Hence, $P'$ or~$P''$ is the desired path.
\end{proof}
Finally, we can prove Lemma~\ref{lem:omega-avoiding-cycle}.
\begin{proof}[Proof of Lemma~\ref{lem:omega-avoiding-cycle}]
For convenience, let~${w_0 := w_{\ref*{cor:smallorder}}(m\omega,m\omega,\ell)}$.
We define
\begin{align*}
c_{\ref*{lem:omega-avoiding-cycle}}(t,\ell,m,\omega) &:=
\max\{ 3, t+1, c_{\ref*{cor:cycle+subwalls}}(t, \, m(m\omega+1), \, w_0 + 1)\},\\
r_{\ref*{lem:omega-avoiding-cycle}}(t,\ell,m,\omega) &:=
\max\{3, r_{\ref*{cor:cycle+subwalls}}(t, \, m(m\omega+1), \, w_0 + 1)\}.
\end{align*}
If~${Z = [m]}$, then the result follows from Lemma~\ref{lem:pairlink}.
Hence, without loss of generality, we may assume that~${Z = [m] \setminus [y]}$ for some~${y \in [m]}$.
By Corollary~\ref{cor:cycle+subwalls},
there exist a cycle~$O$ in~${W \cup \bigcup \mathcal{P}}$
and a set~${\{ W_{i,j} \colon i\in [y],\, j \in [y\omega+1] \}}$ of~$y(y\omega+1)$ vertex-disjoint $N^W$-anchored $(w_0+1,w_0+1)$-subwalls of~$W$
such that~${\bigcup \mathcal{P} \subseteq O}$
and~${W_{i,j} \cap O = R_1^{W_{i,j}} =: P_{i,j}}$ for
all~${i \in [y]}$ and~${j \in [y\omega+1]}$.
For~${i \in [y]}$ and~${j \in [y\omega+1]}$,
let~$W'_{i,j}$ be a ${w_0}$-row-slice of~$W_{i,j}$ disjoint from~$O$,
and let~$H$ be the graph obtained from~$O$ by deleting the internal vertices of the paths~$P_{i,j}$ for
all~${i \in [y]}$ and~${j \in [y\omega+1]}$.
For each~${i \in [y]}$, we now recursively define a family~${\big(Q_{i,j} \colon j \in [y\omega+1]\big)}$ of paths
and a family
${(S_{i,j} \colon j \in [y\omega+1])}$ of subsets of~$\Gamma_i$,
such that for all~${j \in [y\omega+1]}$ and~${g \in S_{i,j}}$,
\begin{itemize}
\item ${\abs{S_{i,j}} \leq j-1}$, and
\item the order of~$g$ is at most~${m \omega}$.
\end{itemize}
We first set~${S_{i,1} := \emptyset}$.
Now, for~${j \in [y\omega+1]}$, let~$\lambda_j$ be the induced~${\Gamma_i/\gen{S_{i,j}}}$-labelling of~$G$.
Note that since~${i \notin Z}$ and~$W$ is ${(\gamma,Z,\ell)}$-clean, $W$ has no $\gamma_i$-bipartite ${(\ell,\ell)}$-subwall, and in particular, each~${W_{i,j}'}$ does not have such a subwall.
As ${\abs{S_{i,j}} \leq m\omega}$ and each element of~$S_{i,j}$ has order at most~${m \omega}$,
Corollary~\ref{cor:smallorder} implies that there is a $\lambda_j$-non-zero cycle~$O_{i,j}$ in~$W'_{i,j}$.
By Lemma~\ref{lem:reroutingpath},
there is a path~$Q_{i,j}$ in~${W_{i,j}}$ with the same endvertices as~$P_{i,j}$ such that~${\lambda_j(P_{i,j}) \neq \lambda_j(Q_{i,j})}$, since there are three vertex-disjoint ${(V(P_{i,j}),V(O_{i,j}))}$-paths in~$W_{i,j}$.
We set
\[
S_{i,j+1} :=
\begin{cases}
S_{i,j} & \textnormal{ if } {\gamma_i(Q_{i,j}) - \gamma_i(P_{i,j})} \textnormal{ has order at least } {m\omega+1}, \textnormal{ and } \\
S_{i,j} \cup \big\{\gamma_i(Q_{i,j}) - \gamma_i(P_{i,j})\big\} & \textnormal{ otherwise.}
\end{cases}
\]
For~${i \in [y]}$, let~${D_i := \big(\gamma(Q_{i,j}) - \gamma(P_{i,j}) \colon j \in [y\omega+1]\big)}$.
By construction of the sets~${S_{i,j}}$, the projection of~${D_i}$ to~${\Gamma_i}$ contains no repeated elements of order at most~${m \omega}$ in~${\Gamma_i}$, and thus
this sequence is good by Lemma~\ref{lem:smallgoodset}.
Hence, by Corollary~\ref{cor:vectorsum} with~${h := \gamma(H)}$, there exists~${X_{i,j} \in \{ P_{i,j}, Q_{i,j} \}}$ for all~${i \in [y]}$ and~${j \in [y\omega+1]}$
such that for the cycle~${O' := H \cup \bigcup\big\{ X_{i,j} \colon i \in [y],\, j \in [y\omega+1] \big\}}$, we have that~${\gamma_i(O') \notin \Omega_i}$ for all~${i \in [y]}$.
By construction, ${\gamma_j(O') = \gamma_j(O) \notin \Omega_j}$ for all~${j \in Z}$ because~$W$ is ${(\gamma,Z,\ell)}$-clean.
\end{proof}
\section{Proof of the main theorem}
\label{sec:main}
We now complete the proof of Theorem~\ref{thm:main}, which we are restating for the convenience of the reader.
\mainthm*
\begin{proof}
Let~${\Gamma := \prod_{i \in [m]} \Gamma_i}$ denote the product of the~$m$ given abelian groups,
and let~$\gamma \colon E(G) \to \Gamma$ denote the $\Gamma$-labelling for which~${\gamma_i(e) = \pi_i (\gamma(e))}$ for all~${e \in E(G)}$.
We proceed by induction on~$k$.
For~${k \leq 2}$, we may trivially set~${f_{m,\omega}(k) = 1}$.
Now suppose that~${k > 2}$ and that there is some integer~${f_{m,\omega}(k-1)}$ as per the theorem.
For every subgraph~$H$ of~$G$, let~${\nu(H)}$ denote the maximum size of a set of cycles~$O$ in~$H$ with~${\gamma_i(O) \notin \Omega_i}$ for all~${i \in [m]}$ such that no three cycles in the set share a common vertex.
Observe that~$\nu$ is a packing function for~$G$.
We will show that if~$\tau_\nu(G)$ is sufficiently large relative to~$k$, then~${\nu(G) \geq k}$.
This will complete the proof of the theorem.
For non-negative integers~$p$ and~$z_0$ with~${z_0 \leq m}$, let~${\alpha(p,z_0)}$ and~${\rho(z_0)}$ be recursively defined as follows.
For every non-negative integer~$p$, we define
\begin{align*}
\alpha(p,0) &:= k2^{m\omega}+m\omega+1, &
\rho(0) &:= m + f_{\ref*{lem:ramsey}}(\alpha(1,0),m),
\intertext{and for~${z_0 > 0}$ we recursively define}
\alpha(p,z_0) &:= 4^{\max\{\rho(z_0-1)-p,0\}} \alpha(1,z_0-1),&
\rho(z_0) &:= m + f_{\ref*{lem:ramsey}}(\alpha(1,z_0),m).
\end{align*}
Let~${\hat{p} := \rho(m-1)}$.
Note that we may assume that~$f_{\ref*{lem:ramsey}}$ is increasing in its first argument, and hence~${\rho(z_0) \leq \hat{p}}$ for~${z_0 \leq m-1}$.
Let
\[
u := \max\{ \lceil f_{m,\omega}(k-1)/3\rceil,
f_{\ref*{thm:tpath}} ( f_{\ref*{lem:addlinkage}} ( f_{\ref*{lem:ramsey}}(\alpha(1,m),m) ) ) + 3 \}.
\]
We recursively define~${\beta(p,z_0,z)}$ for non-negative integers~$p$, $z_0$, and~$z$ with~${z_0 \leq z \leq m}$ and~${p \leq \hat{p}}$,
as well as~${\psi(z)}$ for a non-negative integer~$z$ with~${z \leq m+1}$ as follows.
We define
\begin{align*}
\psi(m+1) &:= 3,
\intertext{and for~${z \leq m}$ we define}
\beta(p,z_0,z) &:=
\begin{cases}
\max \big\{
u,\,
k c_{\ref*{lem:omega-avoiding-cycle}}(2^{m\omega}\hat{p}, \psi(z+1)+2, m, \omega) \big\}
& \text{if } z_0 = 0,\\
\beta(1,z_0-1,z)
& \text{if } z_0 > 0 \text{ and } p = \hat{p},\\
\max\big\{
\beta(p+1,z_0,z),\, w_{\ref*{lem:addlinkage}}(\alpha(p+1,z_0),
\beta(p+1,z_0,z)) \big\}
& \text{if } z_0 > 0 \text{ and } p < \hat{p};
\end{cases}\\
\psi(z) &:= \max \big\{
\psi(z+1),\,
\beta(0,z,z),\,
r_{\ref*{lem:omega-avoiding-cycle}}(2^{m\omega}\hat{p},\,
\psi(z+1)+2, m, \omega) \big\}.
\end{align*}
Observe that~${\beta(p,z_0,z) \geq u}$.
Lastly, we define~${f_{m,\omega}(k) := \max\big\{
6f_{\ref*{thm:wall}}(\psi(0)+2),\,
6u, \,
12f_{m,\omega}(k-1)
\big\}}$.
\medskip
Let~$T$ be a minimum $\nu$-hitting set of size~${t := \tau_\nu(G)}$,
and assume that~${t > f_{m,\omega}(k)}$.
By the induction hypothesis,~$G$ has a half-integral packing of~${k-1}$ cycles in~$\mathcal{O}$
and therefore we may assume
for a contradiction
that~${\nu(G) = k-1}$.
For each subgraph~$H$ of~$G$, if~${\nu(H) < \nu(G)}$, then
by the induction hypothesis,~${\tau_\nu(H) \leq f_{m,\omega}(k-1) \leq f_{m,\omega}(k)/12 < t/12}$.
Lemma~\ref{lem:welllinked} yields that the set~$\mathcal{T}_T$ of all separations~${(A,B)}$ of~$G$ of order less than~${t/6}$ with~${\abs{B \cap T} > 5t/6}$ is a tangle of order~${\lceil t/6 \rceil > f_{\ref*{thm:wall}}(\psi(0)+2)}$.
By Theorem~\ref{thm:wall}, there is a ${(\psi(0)+2,\psi(0)+2)}$-wall in~$G$ dominated by~$\mathcal{T}_T$.
By Lemma~\ref{lem:cleansubwall},
this wall has a ${(\psi(\abs{Z}),\psi(\abs{Z}))}$-subwall~$W$ which is ${(\gamma',Z,\psi(\abs{Z}+1)+2)}$-clean
for some subset~${Z \subseteq [m]}$ and some $\Gamma$-labelling~$\gamma'$ of~$G$ shifting-equivalent to~$\gamma$.
By Lemma~\ref{lem:dominatedsubwall}, the wall~$W$ is
dominated by~$\mathcal{T}_T$.
Since~${\gamma(O) = \gamma'(O)}$ for every cycle~$O$ in~$G$, we may assume without loss of generality that~${\gamma = \gamma'}$.
\begin{claim}
\label{clm:manyhandles}
There exist integers~$c$ and~$p$ with~${c \geq \beta(1,0,z)}$ and~${0 \leq p \leq \hat{p}}$,
a $c$-column-slice~$W'$ of~$W$,
a family~${\mathfrak{P} = ( \mathcal{P}_i \colon i \in [p])}$ of non-empty sets of $W'$-handles,
and a family~${( Z_i \colon i \in \{0\} \cup [p] )}$ of disjoint subsets of~$Z$ such that
\begin{enumerate}
[label=\textnormal{(\alph*)}, series=manyhandles]
\item \label{item:manyhandles1}
if~${P \in \mathcal{P}_i}$ and~${Q \in \mathcal{P}_j}$ are not vertex-disjoint for some~${i,j \in [p]}$, then~${i = j}$ and~${P = Q}$,
\item \label{item:manyhandles2}
${\bigcup_{i\in \{0\}\cup [p]}Z_i = Z}$,
\item \label{item:manyhandles2b}
${\abs{\mathcal{P}_i} \geq \alpha(p,\abs{Z_0})}$ for all~${i \in [p]}$,
\item \label{item:manyhandles3}
${\abs{\gamma_j(\mathcal{P}_{i})} = \abs{\mathcal{P}_i}}$ for all~${i \in [p]}$ and~${j \in Z_i}$,
\item \label{item:manyhandles4}
${\abs{\gamma_j(\mathcal{P}_{i})} = 1}$ for all~${i \in [p]}$ and ${j \in Z_{0}}$,
\item \label{item:manyhandles5}
there is some~${g \in \gen{\bigcup_{i \in [p]} \gamma(\mathcal{P}_{i})}}$ such that~${\pi_j(g) \notin \Omega_j}$ for all~${j \in Z_0}$.
\end{enumerate}
\end{claim}
\begin{proofofclaim}
For non-negative integers~$c$,~$q$, and~$p$,
we say that a triple ${(W',\mathfrak{P}, \mathcal{Z})}$ consisting of a wall~$W'$, a family $\mathfrak{P} := (\mathcal{P}_i \colon i \in [p])$ of non-empty sets of $W'$-handles, and a family $\mathcal{Z} := ( Z_i \colon i \in \{0\} \cup [p])$ of disjoint subsets of~$Z$
is a \emph{${(c,q,p)}$-McGuffin} if~$W'$ is a $c$-column-slice of~$W$,
and~$W'$,~$\mathfrak{P}$, and~$\mathcal{Z}$ satisfy~\ref{item:manyhandles1}--\ref{item:manyhandles4}, as well as
the following two conditions:
\begin{enumerate}
[resume*=manyhandles]
\item \label{item:manyhandles7}
${Z_i\neq\emptyset}$ for~${i \in [q]}$ and~${Z_i=\emptyset}$ for~${i \in [p] \setminus[q]}$,
\item \label{item:manyhandles8}
for all distinct~${i, i' \in [p] \setminus [q]}$ there is ${j \in Z_{0}}$ such that~${\gamma_j(\mathcal{P}_{i}) \cap \gamma_j(\mathcal{P}_{i'}) = \emptyset}$.
\end{enumerate}
Note that~${(W,\emptyset,(Z))}$ is a ${(\psi(\abs{Z}),0,0)}$-McGuffin.
Let~${(q,p)}$ be a lexicographically maximal pair of non-negative integers
with~${q \leq \abs{Z}}$ and~${q \leq p \leq \hat{p}}$
for which there is a ${(c,q,p)}$-McGuffin~${(W',\mathfrak{P},\mathcal{Z})}$ for some~${c \geq \beta(p,\abs{Z_0},\abs{Z})}$.
Now suppose for a contradiction that~${(W',\mathfrak{P},\mathcal{Z})}$ does not satisfy~\ref{item:manyhandles5}.
Then~$Z_0$ is nonempty.
We distinguish two cases.
If~${p = \hat{p}}$, then ${p - q \geq \hat{p} - m \geq \rho(z_0) - m \geq f_{\ref*{lem:ramsey}}(\alpha(q+1,\abs{Z_0}-1),m)}$ since~${q \leq m}$ by~\ref{item:manyhandles7} and~$\alpha$ is decreasing in its first argument.
Let~$\mathcal{P}''$ be a set of~${p-q}$ vertex-disjoint $W'$-handles containing exactly one element of~$\mathcal{P}_i$ for each~${i \in [p] \setminus [q]}$.
Such a set~$\mathcal{P}''$ exists by~\ref{item:manyhandles1}.
For~${i \in [q]}$, let~${\mathcal{P}'_i := \mathcal{P}_i}$ and~${Z'_i := Z_i}$.
Observe that by~\ref{item:manyhandles8},
for all distinct paths~$P$ and~$Q$ in~$\mathcal{P}''$, there exists ${j \in Z_0}$ such that~${\gamma_j(P) \neq \gamma_j(Q)}$.
Thus, by Lemma~\ref{lem:ramsey}, there is a subset $\mathcal{P}'_{q+1}$ of $\mathcal{P}''$
with~${\abs{\mathcal{P}'_{q+1}} = \alpha(q+1,\abs{Z_0}-1)}$
such that for each~${i \in [m]}$, either
\begin{itemize}
\item ${\gamma_i(P) = \gamma_i(Q)}$ for all~${P,Q \in \mathcal{P}'_{q+1}}$, or
\item ${\gamma_i(P) \neq \gamma_i(Q)}$ for all distinct~${P,Q \in \mathcal{P}'_{q+1}}$,
\end{itemize}
and the second condition holds for some~${i \in Z_0}$.
Let
\[
Z'_{q+1} :=
\{ i \in Z_0 \colon \gamma_i(P) \neq \gamma_i(Q) \text{ for all distinct } P, Q \in \mathcal{P}'_{q+1} \}
\text{ and }
Z'_0 := Z_0 \setminus Z'_{q+1}.
\]
Let~${\mathfrak{P}' := (\mathcal{P}_i' \colon i \in [q+1])}$ and~${\mathcal{Z}' := (Z_i' \colon i \in \{0\} \cup [q+1])}$.
Then~${(W',\mathfrak{P}', \mathcal{Z}')}$ is a ${(c,q+1,q+1)}$-McGuffin, since~\ref{item:manyhandles2b} follows from the fact that~${\abs{\mathcal{P}'_{q+1}} \geq \alpha(q+1,\abs{Z_0}-1) \geq \alpha(q+1,\abs{Z_0'})}$, and the remaining conditions (\ref{item:manyhandles1}, \ref{item:manyhandles2}, \ref{item:manyhandles3}, \ref{item:manyhandles4}, \ref{item:manyhandles7}, and~\ref{item:manyhandles8}) are easy to check.
This contradicts the maximality of~${(q,p)}$.
So we may assume that~${p < \hat{p}}$.
Let~$\Lambda$ be the subgroup of~$\Gamma$ consisting of all~${g \in \Gamma}$
for which there is ${g' \in \gen{\bigcup_{i \in [p]} \gamma(\mathcal{P}_i)}}$ such that for all~${j \in Z_{0}}$ we have~${\pi_j(g) = \pi_j(g')}$.
Let~$\lambda$ be the induced ${\Gamma/\Lambda}$-labelling of~$G$.
Note that by the negation of~\ref{item:manyhandles5},
neither ${\gen{\bigcup_{i \in [p]} \gamma(\mathcal{P}_i)}}$ nor~$\Lambda$ contains an element~$g$ such that~${\pi_j(g) \notin \Omega_j}$ for all~${j \in Z_0}$.
Therefore,
\begin{enumerate}
[label=(\arabic*)]
\item\label{eq:key} every cycle~$O$ of~$G$ for which~${\gamma_i(O) \notin \Omega_i}$ for all~${i \in [m]}$ is $\lambda$-non-zero.
\end{enumerate}
Note that~$W'$ is a subwall of~$W$ of order~${c \geq u}$.
For any~${S \subseteq V(G)}$ of size at most~${u-1}$, there is a component~$X$ of~${G-S}$ containing a row of~$W'$.
By Lemma~\ref{lem:dominatedsubwall},~$\mathcal{T}_T$ dominates~$W'$, so the separation~${(V(G) \setminus V(X), S \cup V(X))}$ is in~$\mathcal{T}_T$, so~$X$ contains a vertex of~$\ensuremath{V_{\neq 2}}(W')$ and at least~${5t/6 - (u-1) > 5f_{m,\omega}(k)/6-(u-1) > 4u}$ vertices of~$T$.
By~\ref{eq:key}, every minimal subgraph~$H$ with~${\nu(H) \geq 1}$ is a $\lambda$-non-zero cycle.
Moreover, if~$H$ is a subgraph of~$G$ with~${\nu(H) < \nu(G) = k-1}$, then
by the induction hypothesis,~${\tau_\nu(H) \leq f_{m,w}(k-1) \leq 3u}$.
Hence, by Lemma~\ref{lem:cover}, we have that~$G$ contains a set of~$f_{\ref*{lem:addlinkage}} ( f_{\ref*{lem:ramsey}}(\alpha(1,m),m) )$ disjoint $\lambda$-non-zero~$\ensuremath{V_{\neq 2}}(W')$-paths.
We may assume that function~$w_{\ref*{lem:addlinkage}}$ is increasing in both of its arguments.
As~${\abs{Z_0} > 0}$ and~${p < \hat{p}}$, we have
\begin{align*}
c \geq \beta(p, \abs{Z_0}, \abs{Z})
&\geq w_{\ref*{lem:addlinkage}}(\alpha(p+1,\abs{Z_0}),\beta(p+1,\abs{Z_0},\abs{Z})).
\end{align*}
Thus, by Lemma~\ref{lem:addlinkage} applied to~$W'$, there exist a $c'$-column-slice~$W''$ of~$W'$
for some
\[
c' \geq \beta(p+1,\abs{Z_0},\abs{Z}) \geq \beta(q+1,\abs{Z_0}-1,\abs{Z})
\]
and a set~$\mathcal{P}'_i$ of~${f_{\ref*{lem:ramsey}}(\alpha(p+1,\abs{Z_0}),m)}$ vertex-disjoint $W''$-handles for each~${i \in [p+1]}$ such that
\begin{itemize}
\item for each~${i \in [p]}$, the set~$\mathcal{P}'_i$ is a subset of the row-extension of~$\mathcal{P}_i$ to~$W''$ in~$W'$,
\item the paths in~${\bigcup_{i \in [p+1]} \mathcal{P}'_i}$ are vertex-disjoint,
\item the paths in~$\mathcal{P}'_{p+1}$ are $\lambda$-non-zero.
\end{itemize}
By Lemma~\ref{lem:ramsey}, there exist a subset~${\mathcal{R}}$ of~${\mathcal{P}'_{p+1}}$ and a subset~$Z'$ of~$Z_0$ such that
\begin{itemize}
\item ${\abs{\gamma_j(\mathcal{R})} = \abs{\mathcal{R}}}$ for all~${j \in Z'}$,
\item ${\abs{\gamma_j(\mathcal{R})} = 1}$ for all~${j \in Z_{0} \setminus Z'}$, and
\item ${\abs{\mathcal{R}} \geq \alpha(p+1,\abs{Z_0}) \geq \alpha(q+1,\abs{Z_0}-1)}$.
\end{itemize}
Let~${p'' := p+1}$ if~$Z'$ is empty and let~${p'' := q+1}$ if~$Z'$ is non-empty, and
For~${i \in \{0\} \cup [p'']}$, let
\[
Z''_i :=
\begin{cases}
Z_0 \setminus Z' & \text{if } i = 0,\\
Z_i & \text{if } i \in [p''-1],\\
Z' & \text{if } i = p''.\\
\end{cases}
\]
For~${i \in [p''-1]}$, let~${\mathcal{P}_i'' := \mathcal{P}_i'}$
and let~${\mathcal{P}_{p''} := \mathcal{R}}$.
We now show that~${\big( W'', (\mathcal{P}''_i \colon i \in [p'']), (Z''_i \colon i \in \{0\} \cup [p'']) \big)}$ is either a ${(c',q,p+1)}$-McGuffin or a ${(c', q+1, q+1)}$-McGuffin, contradicting the maximality of~${(q,p)}$.
First, observe that
since~$W$ is ${(\gamma',Z,\psi(\abs{Z}+1)+2)}$-clean,
every $N^W$-path is $\gamma_i$-zero for all~${i \in Z}$ and therefore
if~$P'$ is the row-extension of~$P$ to~$W''$ in~$W'$,
then~${\gamma_i(P') = \gamma_i(P)}$ for all~${i \in Z}$,
implying~\ref{item:manyhandles3} and~\ref{item:manyhandles4} for~${i < p''}$.
By the definition of~$Z'$, properties~\ref{item:manyhandles3} and~\ref{item:manyhandles4} hold for~${i = p''}$.
It remains to check~\ref{item:manyhandles8} when~$Z'$ is empty, ${q < i \leq p}$, and~${i' = p'' = p+1}$.
This is implied by the property that
the paths in~$\mathcal{P}'_{p+1}$ are $\lambda$-non-zero.
\end{proofofclaim}
\begin{claim}
There is a family~${(\mathcal{Q}_i \colon i \in [k])}$ of~$k$ disjoint subsets of~$\bigcup_{i \in [p]}\mathcal{P}_i$,
each of size at most $2^{\abs{Z_0}\omega}p$,
such that for each~${i \in [k]}$ and each~${j \in Z}$,
we have~${\gamma_j \big( \bigcup \mathcal{Q}_i \big) \notin \Omega_j}$.
\end{claim}
\begin{proofofclaim}
Recursively, for each~${i \in [k]}$ we define~$\mathcal{Q}_i$
containing at most~$2^{\abs{Z_0}\omega}$ elements of~$\mathcal{P}_j$ for all~${j \in [p]}$.
For each~${i \in [k]}$ and~${j \in [p]}$, let~${\mathcal{X}_{i,j} := \mathcal{P}_j \cap \bigcup_{i' \in [i-1]} \mathcal{Q}_{i'}}$, and note that we have~${\mathcal{X}_{1,j} = \emptyset}$ and~${\abs{\mathcal{X}_{i,j}} \leq (i-1) 2^{\abs{Z_0}w} \leq (k-1) 2^{mw}}$.
For each~${j \in [p]}$, select~${g_j \in \gamma(\mathcal{P}_j)}$ arbitrarily.
By Claim~\ref{clm:manyhandles}\ref{item:manyhandles4} and~\ref{item:manyhandles5},
for each~${j \in [p]}$ there exists an integer~$c_j$ such that~${\pi_x \big( \sum_{j \in [p]} c_jg_j \big) \notin \Omega_x}$ for all~${x \in Z_0}$.
Hence, by Corollary~\ref{cor:omega-avoiding}, for each~${j \in [p]}$ there exists an integer~${d_j \in \big[ 2^{\abs{Z_0}\omega} \big]}$
such that for all~${x \in Z_0}$,
we have~${\pi_x \big( \sum_{j \in [p]} d_jg_j \big) \notin \Omega_x}$.
Let~$I$ be the set of indices~${j \in [p]}$ such that~${Z_j \neq \emptyset}$.
Now~${\abs{\mathcal{P}_j} \geq \alpha(p,\abs{Z_0}) \geq \alpha(p,0) \geq k2^{mw} \geq \abs{\mathcal{X}_{i,j}} + d_j}$ for all~${j \in [p]}$ by Claim~\ref{clm:manyhandles}\ref{item:manyhandles2b}.
Hence, for each~${j \in [p]}$, we can select a set~$\mathcal{Y}_j$ of distinct $W'$-handles in~${\mathcal{P}_j \setminus \mathcal{X}_{i,j}}$ of size~$d_j -1$ if~${j \in I}$ and of size~$d_j$ otherwise.
By the definition of~$\mathcal{X}_{i,j}$ and Claim~\ref{clm:manyhandles}\ref{item:manyhandles2b},
there are at least~${\alpha(p,\abs{Z_0}) - k2^{\abs{Z_0}\omega} \geq m\omega+1}$ distinct $W'$-handles in~${\mathcal{P}_j \setminus (\mathcal{X}_i \cup \mathcal{Y}_j)}$ for every~${j \in [p]}$.
Define
\[
h := \sum_{j \in [p]} \sum_{P \in \mathcal{Y}_j} \gamma(P),
\]
and for~${j \in I}$ define~${S_j := \{ \gamma(P) \colon P \in \mathcal{P}_j \setminus (\mathcal{X}_{i,j} \cup \mathcal{Y}_j) \}}$.
By Claim~\ref{clm:manyhandles}\ref{item:manyhandles3}, for all~${j \in I}$, for all distinct~$g$,~$g'$ in~$S_j$, and for all~${x \in Z_j}$,
we have~${\pi_x(g) \neq \pi_x(g')}$
and so~${\abs{S_j} > mw}$.
Now by Lemma~\ref{lem:vectorsum},
there is a family~${(Q_j \colon j \in I)}$ of paths
such that~${Q_j \in \mathcal{P}_j \setminus (\mathcal{X}_{i,j} \cup \mathcal{Y}_j)}$
for all~${j \in I}$,
and~${\pi_x \big( h + \sum_{j \in I} \gamma(Q_j) \big) \notin \Omega_x}$ for all~${x \in Z}$.
Hence, let~${\mathcal{Q}_i := \{Q_j \colon j \in I \} \cup \bigcup_{j \in [p]} \mathcal{Y}_j}$,
and note that~${\abs{\mathcal{Q}_i} \leq \sum_{j \in [p]} d_j \leq 2^{\abs{Z_0}w}p}$.
\end{proofofclaim}
We now complete the proof of the theorem.
Since~$W'$ has at least~${\beta(1,0,\abs{Z})}$ columns,
there is a set~${\{W_i \colon i \in [k]\}}$ of~$k$ vertex-disjoint
${c_{\ref*{lem:omega-avoiding-cycle}}(2^{m\omega}\hat{p},\psi(\abs{Z}+1)+2, m, \omega)}$-column-slices of~$W'$.
Note that the number of rows of~$W'$ is at least~${\psi(\abs{Z}) \geq r_{\ref*{lem:omega-avoiding-cycle}}(2^{m\omega}\hat{p},\psi(\abs{Z}+1)+2, m, \omega)}$.
For each~${i \in [k]}$, let~$\mathcal{Q}^\ast_i$ be the row-extension of~$\mathcal{Q}_i$ to~$W_i$.
By Lemma~\ref{lem:omega-avoiding-cycle},
for each~${i \in [k]}$ there is a cycle~$O_i$ in~${W_i \cup \bigcup \mathcal{Q}^\ast_i}$ with~${\gamma_j(O_i) \notin \Omega_j}$ for all~$j\in [m]$.
Observe that for every vertex~${v \in V(G)}$, there are at most two indices~${i \in [k]}$ such that~${v \in V(W_i \cup \bigcup \mathcal{Q}^\ast_i)}$.
Hence,~${\nu(G) \geq k}$, a contradiction.
\end{proof}
\section{Conclusion}
\label{sec:conclusion}
In this work, we proved that a half-integral analogue of the Erd\H{o}s-P\'{o}sa theorem holds for
cycles in graphs labelled with a bounded number of abelian groups, whose values avoid a bounded number of elements of each group.
We conclude with some open problems.
In the proof of our theorem, the theorem of Wollan~\cite{Wollan2010} about $\gamma$-non-zero $A$-paths was important.
This theorem implies that an analogue of the Erd\H{o}s-P\'{o}sa theorem holds for the odd $A$-paths, and for $A$-paths intersecting a prescribed set of vertices.
Bruhn, Heinlein, and Joos~\cite{BruhnMJ2018} further showed that an analogue of the Erd\H{o}s-P\'{o}sa theorem holds for $A$-paths of length at least~$\ell$, and for $A$-paths of even length,
but interestingly, they also showed that for every composite integer~${m > 4}$ and every~${d \in \{0\} \cup [m-1]}$,
no such analogue holds for $A$-paths of length~$d$ modulo~$m$.
Later, Thomas and Yoo~\cite{YooT20202} characterised the abelian groups~$\Gamma$ and elements~${\ell \in \Gamma}$ where an analogue of the Erd\H{o}s-P\'{o}sa theorem holds for $A$-paths of $\gamma$-value~$\ell$.
We would like to ask whether a statement similar to Theorem~\ref{thm:main} holds for $A$-paths.
\begin{question}
\label{ques1}
For every pair of positive integers~$m$ and~$\omega$, does there exist a function~${g_{m,\omega} \colon \mathbb{N} \to \mathbb{N}}$ satisfying the following property?
\begin{itemize}
\item For each~${i \in [m]}$, let~$\Gamma_i$ be an abelian group and let~$\Omega_i$ be a subset of~$\Gamma_i$.
Let~$G$ be a graph, let~$A$ be a set of vertices in~$G$, and for each~${i \in [m]}$,
let~${\gamma_i}$ be a~${\Gamma_i}$-labelling of~$G$,
and let~$\mathcal{P}$ be the set of all $A$-paths of~$G$ whose $\gamma_i$-value is in~${\Gamma_i \setminus \Omega_i}$ for all~${i \in [m]}$.
If ${\abs{\Omega_i} \leq \omega}$ for all~${i \in [m]}$, then there exists
either a half-integral packing of~$k$ paths in~$\mathcal{P}$,
or a hitting set for~$\mathcal{P}$ of size at most~${g_{m,\omega}(k)}$.
\end{itemize}
\end{question}
Our next question relates to directed labellings of graphs.
Let~$\Gamma$ be a group (not necessarily abelian).
A \emph{directed $\Gamma$-labelling} of a graph~$G$ is a function~${\gamma}$ from the set of oriented edges ${\vec{E}(G) := \{ (e, w) \colon e = uv \in E(G), w \in \{u,v\} \}}$ to~$\Gamma$ such that~${\gamma(e,u) = -\gamma(e,v)}$ for each edge~${e = uv}$.
Given a walk~${W := v_t e_1 v_1 e_2 \cdots e_t v_t}$, we define~${\gamma(W) := \sum_{j = 1}^t \gamma(e_j, v_{j})}$, and say that~$W$ \emph{corresponds} to a cycle~$O$ if~${E(O) = E(W)}$ and~$v_t$ is the only repeated vertex of~$W$.
It is straightforward to check that if any walk corresponding to a cycle~$O$ has value~$0$, then all walks corresponding to~$O$ do, so it makes sense to consider non-zero cycles with respect to a directed labelling.
Note that if~$\Gamma$ is abelian and~$W_1$ and~$W_2$ are walks corresponding to the same cycle~$O$, then~${\gamma(O_1) = \pm \gamma(O_2)}$.
If~$\Gamma$ is not abelian, then the choice of start vertex for the corresponding walk does matter as well.
Hence, in this case, we are really considering cycles together with a specified start vertex and direction.
Huynh, Joos, and Wollan~\cite{HuynhJW2017} conjectured that a half-integral analogue of the Erd\H{o}s-P\'{o}sa theorem holds for cycles which are non-zero with respect to a fixed number of directed labellings.
We ask whether a statement similar to Theorem~\ref{thm:main} holds for directed labellings.
If it does, then it would imply the conjecture of Huynh, Joos, and Wollan.
\begin{question}
\label{ques2}
For every pair of positive integers~$m$ and~$\omega$, does there exist a function~${g_{m,\omega} \colon \mathbb{N} \to \mathbb{N}}$ satisfying the following property?
\begin{itemize}
\item For each~${i \in [m]}$, let $\Gamma_i$ be a group and let~$\Omega_i$ be a subset of~$\Gamma_i$.
Let~$G$ be a graph and for each~${i \in [m]}$, let~${\gamma_i}$ be a directed~${\Gamma_i}$-labelling of~$G$, and let~${\mathcal{O}}$ be the set of all cycles of~$G$ which have a corresponding walk $W$ such that $\gamma_i(W)$ is in~${\Gamma_i \setminus \Omega_i}$ for all~${i \in [m]}$.
If~${\abs{\Omega_i} \leq \omega}$ for all~${i \in [m]}$, then there exists
either a half-integral packing of~$k$ cycles in~$\mathcal{O}$,
or a hitting set for~$\mathcal{O}$ of size at most~${g_{m,\omega}(k)}$.
\end{itemize}
\end{question}
As discussed in Section~\ref{subsec:obstructionsEP}, an analogue of the Erd\H{o}s-P\'{o}sa theorem does not hold for the cycles described in Theorem~\ref{thm:main}.
It was shown that in graphs of sufficiently high connectivity~\cite{Thomassen2001, RautenbackR2001, KawarabayashiR2009, Joos2017},
an analogue of the Erd\H{o}s-P\'{o}sa theorem holds for odd cycles.
We ask whether a similar phenomenon happens for the cycles described in Theorem~\ref{thm:main}.
\begin{question}
\label{ques3}
For every pair of positive integers~$m$ and~$\omega$, do there exist functions~${g_{m,\omega} \colon \mathbb{N} \to \mathbb{N}}$ and~${g'_{m,\omega} \colon \mathbb{N} \to \mathbb{N}}$ satisfying the following property?
\begin{itemize}
\item For each~${i \in [m]}$, let~$\Gamma_i$ be an abelian group and let~$\Omega_i$ be a subset of~$\Gamma_i$.
Let~$G$ be a graph, and for each~${i \in [m]}$, let~${\gamma_i}$ be a~${\Gamma_i}$-labelling of~$G$,
and let~${\mathcal{O}}$ be the set of all cycles of~$G$ whose $\gamma_i$-value is in~${\Gamma_i \setminus \Omega_i}$ for all~${i \in [m]}$.
If~$G$ is ${g'_{m, \omega}(k)}$-connected and~${\abs{\Omega_i} \leq \omega}$ for all~${i \in [m]}$, then there exists
either a set of~$k$ vertex-disjoint cycles in~$\mathcal{O}$,
or a hitting set for~$\mathcal{O}$ of size at most~${g_{m,\omega}(k)}$.
\end{itemize}
\end{question}
Similar to Question~\ref{ques3}, we ask whether an analogue of the Erd\H{o}s-P\'{o}sa theorem holds for the $A$-paths described in Question~\ref{ques1},
and for the cycles described in Question~\ref{ques2},
in the case of highly connected graphs.
\bibliographystyle{abbrv}
| {
"timestamp": "2021-02-04T02:12:46",
"yymm": "2102",
"arxiv_id": "2102.01986",
"language": "en",
"url": "https://arxiv.org/abs/2102.01986",
"abstract": "Erdős and Pósa proved in 1965 that there is a duality between the maximum size of a packing of cycles and the minimum size of a vertex set hitting all cycles. Such a duality does not hold if we restrict to odd cycles. However, in 1999, Reed proved an analogue for odd cycles by relaxing packing to half-integral packing. We prove a far-reaching generalisation of the theorem of Reed; if the edges of a graph are labelled by finitely many abelian groups, then there is a duality between the maximum size of a half-integral packing of cycles whose values avoid a fixed finite set for each abelian group and the minimum size of a vertex set hitting all such cycles.A multitude of natural properties of cycles can be encoded in this setting, for example cycles of length at least $\\ell$, cycles of length $p$ modulo $q$, cycles intersecting a prescribed set of vertices at least $t$ times, and cycles contained in given $\\mathbb{Z}_2$-homology classes in a graph embedded on a fixed surface. Our main result allows us to prove a duality theorem for cycles satisfying a fixed set of finitely many such properties.",
"subjects": "Combinatorics (math.CO)",
"title": "A unified half-integral Erdős-Pósa theorem for cycles in graphs labelled by multiple abelian groups",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9854964220125033,
"lm_q2_score": 0.7185943925708562,
"lm_q1q2_score": 0.708172202756827
} |
https://arxiv.org/abs/2110.01849 | A penalty scheme to solve constrained non-convex optimization problems in $BV(Ω)$ | We investigate non-convex optimization problems in $BV(\Omega)$ with two-sided pointwise inequality constraints. We propose a regularization and penalization method to numerically solve the problem. Under certain conditions, weak limit points of iterates are stationary for the original problem. In addition, we prove optimality conditions for the original problem that contain Lagrange multipliers to the inequality constraints. Numerical experiments confirm the theoretical findings. | \section{Introduction}
Let $\Omega\in\mathbb{R}^n$ be an open bounded set
with Lipschitz boundary.
We consider the possibly non-convex optimization problem of the form
\begin{equation} \label{P_BV}
\min\limits_{u\in U_{ad} \cap BV(\Omega)}f(u)+ \BVSN{u}.
\end{equation}
Mostly, we will work with
\begin{equation} \label{eq:def_Uad}
U_{ad}=\{u\in BV(\Omega):\:u_a \le u(x)\le u_b \:\text{ f.a.a. } x\in \Omega\},
\end{equation}
where $u_a,u_b \in \mathbb{R}$.
The function space setting is $BV(\Omega)$, i.e., the space of functions of bounded variation that consists of $L^1(\Omega)$-functions with weak derivative in the Banach space $\mathcal{M}(\Omega)$ of real Borel measures on $\Omega$.
The term $\BVSN{u}$ denotes the $BV(\Omega)$-seminorm, which is equal to the total variation of the measure $\nabla u$, i.e., $\BVSN{u}=|\nabla u|(\Omega)$.
The functional $f:L^2( \Omega)\to\mathbb{R}$ is assumed to be smooth and can be non-convex. In particular, we have in mind to choose $f(u):=f(y(u))$ as the reduced smooth part of an optimal control problem, incorporating the state equation.
We will give more details on the assumptions on the optimal control problem in Section \ref{sec:2}.
Problem \eqref{P_BV} is solvable, and existence of solutions to \eqref{P_BV} can be shown by the direct method of the calculus of variations, see Theorem \ref{existece_sol0}.
The purpose of the paper is two-fold:
\begin{enumerate}[label=(\Alph{enumi})]
\item \label{purpose_A}
We prove optimality conditions for \eqref{P_BV}--\eqref{eq:def_Uad} that contain Lagrange multipliers to the inequality constraints $u_a \le u$ and $u\le u_b$.
Moreover, these multipliers belong to $L^2(\Omega)$.
\item We investigate an algorithmic scheme to solve \eqref{P_BV}--\eqref{eq:def_Uad}, where weak limit points of iterates satisfy the optimality conditions from \ref{purpose_A}.
\end{enumerate}
Both of these goals rely on the same approximation method.
The algorithmic scheme consists of the following two parts.
First, we approximate the non-differentiable total variation term by a smooth approximation and apply a continuation strategy.
Second, we address the box constraints with a classical penalty method.
Of course, solutions to \eqref{P_BV} and appearing subproblems are not unique due to the lack of convexity, which makes the analysis challenging.
In general, only stationary points of these non-convex problems can be computed.
Under suitable assumptions,
limit points of the generated sequences of stationary points of the subproblems and of the associated multipliers satisfy a certain necessary optimality condition for the original problem.
In addition, we apply this regularization and penalization approach to local minima of the original problem.
This allows us to prove optimality conditions that contains Lagrange multipliers to the inequality constraints, see Theorem \ref{thm:final_local}.
Such a result is not available in the literature.
Admittedly, we have to make the assumption that the constraints $u_a$ and $u_b$ are constant functions.
Regularization by total variation is nowadays a standard tool in image analysis.
Following the seminal contribution \cite{TV_rudin_1992}, much research was devoted to study
such kind of optimization problems. We refer to \cite{chambolle_TV,overview_TV_2006}
for a general introduction and an overview on total variation in image analysis.
Optimal control problems in $BV$-spaces were studied for instance in \cite{CasasKunisch99_BV,CasasKunisch_ocsl_BV,Casas_TV_papablic17}.
These control problems are subject to semilinear equations, which results in non-convex control problems.
Finite element discretization and convergence analysis for optimization problems in $BV(\Omega)$ were investigated for instance in \cite{CasasKunisch99_BV,Bartels_TV_12,ClasonKunisch11}.
An extensive comparisons of algorithms to solve \eqref{P_BV} with the choice $f(u):=\frac12\|u-g\|_{L^2(\Omega)}$ can be found in \cite{Milicevic17}, see also \cite{MilicevicDiss}.
In \cite{SchevenSchmidt2018}, the one-sided obstacle problem in $BV(\Omega)$ is analyzed under low regularity requirements on the obstacle.
Interestingly, we could not find any results, where the existence of Lagrange multipliers to the inequality constraints in $BV(\Omega)$ is addressed.
One natural idea to regularize \eqref{P_BV} is to replace the non-differentiable $BV$-seminorm by a smooth approximation.
This was introduced in the image processing setting in \cite{Acar94_BV_approx} with the functional
\[
u\mapsto\int_\Omega\sqrt{\epsilon+|\nabla u|^2}\,\mathrm{d}x,
\]
which is widely used in the literature. Our regularization method is similar, with the exception that our functional guarantees existence of solutions in $H^1(\Omega)$.
A similar scheme was employed in the recent work \cite{hafemeyer2020}, where a path-following inexact Newton method for a convex PDE-constrained optimal control problem in $BV(\Omega)$ is studied.
Let us comment on the structure of this work.
In Section \ref{sec:2} we give a brief introduction to the function space $BV(\Omega)$ and recall some useful facts.
Furthermore, we prove existence of solutions and a necessary first-order optimality condition for the optimization problem \eqref{P_BV}.
In Section \ref{sec3}, we introduce the regularization scheme for \eqref{P_BV} and show that limit points of the suggested smoothing and penalty method
satisfy a stationary condition for the original problem, see Theorem \ref{thm:final}.
In Section \ref{sec4},
we apply the regularization scheme to derive an optimality condition for locally optimal solutions of \eqref{P_BV}, see Theorem \ref{thm:final_local}.
These conditions are stronger than
the conditions proven in Section \ref{sec:2}, since they
contain Lagrange multipliers to the inequality constraints.
Finally, we provide numerical results and details regarding the implementation of the method in Section \ref{sec5}.
\section{Preliminaries and Background} \label{sec:2}
In this section we want to provide some definitions and results regarding the mathematical background of the paper. For details we refer also to, e.g.,
\cite{Attouch2006,ClasonKunisch11, CasasKunisch_ocsl_BV,CasasKunisch99_BV}.
First, let us recall that $\mathcal{M}(\Omega)$ is the dual space of $C_0(\Omega)$. The noram of a measure $\mu\in \mathcal{M}(\Omega)$ is given by
\[
\|\mu\|_{\mathcal{M}(\Omega)}=\sup\left\{\int_\Omega zd\mu\: :\: z\in C_0(\Omega),\ |z(x)|\le1\:\ \forall x\in\Omega\right\}.
\]
The space of functions of bounded variation $BV(\Omega)$ is a non-reflexive Banach space when endowed with the norm
\[
\|u\|_{BV(\Omega)}=\|u\|_{L^1(\Omega)}+\BVSN{u},
\]
where we define the total variation of $\nabla u$ by
\begin{equation}\label{def:total_var}
\BVSN{u}:=\sup\left\{\int_\Omega u\Div\varphi\,\mathrm{d}x:\:\varphi\in C^\infty_0(\Omega)^n,\:\ |\varphi(x)|\leq1\ \forall x\in\Omega \right\} =\|\nabla u\|_{\mathcal M(\Omega)^n}.
\end{equation}
Here, $|\cdot|$ denotes the Euclidean norm on $\mathbb{R}^n$.
In the definition \eqref{def:total_var}, $\nabla:BV(\Omega)\to\mathcal{M}(\Omega)^n$ is a linear and continuous map.
Functions in $BV(\Omega)$ are not necessarily continuous, as an example we mention the characteristic functions of a set with sufficient regularity.
If $u\in W^{1,1}(\Omega)$, then $\|u\|_{BV(\Omega)}=\|u\|_{W^{1,1}(\Omega)}$ and $\BVSN{u}=\|\nabla u\|_{L^1(\Omega)}.$
The Banach space $BV(\Omega)$ and $\BVSN{\cdot}$ have some useful properties, which are recalled in the following.
\begin{proposition}\label{prop:BV}
Let $\Omega\subset \mathbb{R}^n$ be an open bounded set with Lipschitz boundary.
\begin{enumerate}
\item
The space $BV(\Omega)$ is continuously embedded in $L^r(\Omega)$ for $1\le r\le\frac{n}{n-1}$, while the embedding
is compact for $1\le r<\frac{n}{n-1}$.
\item Let $(u_k)\subset BV(\Omega)$
be bounded in $BV(\Omega)$
with $u_k\to u$ in $L^1(\Omega)$. Then
\[
\BVSN{u}\leq \liminf\limits_{k\to\infty}\BVSN{u_k}
\]
holds.
\item
For $u\in BV(\Omega)\cap L^p(\Omega)$: $p\in[1,\infty)$, there is a sequence $(u_k)\subset C^\infty(\bar \Omega)$ such that
\begin{equation} \label{eq_intermed_conv}
u_k\to u \text{ in } L^p(\Omega) \text{ and } \BVSN{u_k}\to\BVSN{u},
\end{equation}
that is, $C^\infty(\bar \Omega)$ is dense in $BV(\Omega)\cap L^p(\Omega)$ with respect to the intermediate convergence \eqref{eq_intermed_conv}.
\end{enumerate}
\end{proposition}
\begin{proof}
(1) is \cite[Thm. 10.1.3, 10.1.4]{Attouch2006}.
(2) is \cite[Prop. 10.1.1(1)]{Attouch2006}.
(3) Can be proven analogously to \cite[Theorem 10.1.2]{Attouch2006}, which contains the case $p=1$.
A proof for $p\in (1,\infty)$ can be obtained by replacing $L^1$-norms by $L^p$-norms in the proof by \cite{Attouch2006},
which is by a standard mollification procedure.
\end{proof}
\paragraph{Notation.}
Frequently, we will use the following standard notations from convex analysis.
The indicator function of a convex set $C$ is denoted by $\delta_C$.
The normal cone of a convex set $C$ at a point $x$ is denoted by
$N_C(x)$, and $\partial h$ denotes the convex subdifferential of a convex function $h$.
It is well known that $\partial\delta_C(x)=N_C(x)$ holds for convex sets $C$.
Moreover, we introduce the notation
\[
J(u):= f(u)+\BVSN{u},
\]
which will be used thoughout the paper.
In addition, we will denote the positive and negative part of $x\in \mathbb{R}$ by $(x)_+:=\max(x,0)$ and $(x)_- := \min(x,0)$.
\subsection{Standing assumption}
In order to prove existence of solutions of \eqref{P_BV} and to analyze the regularization scheme later on, we need some assumptions on the ingredients of the optimal control problem \eqref{P_BV}.
Let us start with collecting those in the following paragraph.
\begin{assumption} \label{ass:A}
\phantom{bla}
\begin{enumerate}[label=(A\arabic{enumi})]
\item \label{ass:A1} $f:L^2(\Omega)\to \mathbb{R}$ is bounded from below and weakly lower semicontinuous.
\item \label{ass:A2} $f + \delta_{U_{ad}}$ is weakly coercive in $L^2(\Omega)$, i.e.,
for all sequences $(u_k)$ with $u_k\in U_{ad}$ and $\|u_k\|_{L^2(\Omega)} \to \infty$ it follows $f(u_k)\to +\infty$.
\item \label{ass:A3}
$U_{ad}$ is a convex and closed subset of $L^2(\Omega)$ with $U_{ad} \cap BV(\Omega) \ne \emptyset$.
\item \label{ass:A4} $f:L^2(\Omega)\to \mathbb{R}$ is continuously Fréchet differentiable.
\item \label{ass:A5}
$U_{ad}:=\{u\in L^2(\Omega):\:u_a\le u(x)\le u_b\:\text{ f.a.a. } x\in \Omega\}$ with $u_a,u_b\in \mathbb{R}$ and $u_a<u_b$.
\end{enumerate}
\end{assumption}
Here, assumptions \ref{ass:A1}--\ref{ass:A3} will be used to prove existence of solutions of \eqref{P_BV}.
Condition \ref{ass:A4} is necessary to derive necessary optimality conditions.
The assumption \ref{ass:A5} will be used in Section \ref{sec3} to prove boundedness of Lagrange multipliers
associated to the inequality constraints in $U_{ad}$.
\begin{example}
We consider
\[
f(u):= \int_\Omega L(x, y_u(x))\,\mathrm{d}x,
\]
where $y_u\in H^1_0(\Omega)$ is defined as the unique weak solution to the elliptic partial differential equation
\[
(Ay)(x)+d(x,y(x)) = u(x)\quad \text{ a.e. in } \Omega.
\]
Let us assume that $A$ is an uniformly elliptic operator with bounded coefficients and $L,d$ are Carath{\'e}odory functions, continuously differentiable with respect to $y$ such that derivatives are bounded on bounded sets and that $d$ is monotonically increasing with respect to $y$. Then it is well known that $f$ is covered by Assumption \ref{ass:A}, see for instance \cite{CasasL12012}.
\end{example}
\begin{example}
Another example is given by the functional
\[
f(u):=\int_\Omega\int_I L(x,t,y_u(x,t))\,\mathrm{d}x\,\mathrm{d}t.
\]
with $y_u\in L^2(I,H^1_0(\Omega)),\: I:=(0,T),\:T>0,\: \Omega\subset\mathbb{R}^n,$
as the solution of the parabolic equation
\[
\partial_ty(x,t) +(Ay)(x,t) +d(x,t,y(x,t)) = u(x,t) \quad\text{a.e. in }\Omega\times I.
\]
Assuming again an uniformly elliptic operator $A$ and measurable functions $L,d$ of class $C^2$ w.r.t. $y$ with bounded derivatives, such that $d$ is monotonically increasing, the functional $f$ satisfies Assumption \ref{ass:A}.
We refer to \cite[Chapter 5]{BOOK_troeltzsch}, \cite{Casas_TV_papablic17}.
\end{example}
\subsection{Existence of solutions of \eqref{P_BV}}
Next, we show that under suitable assumptions on the function $f$, Problem \eqref{P_BV} has at least one solution in $L^2(\Omega)\cap BV(\Omega)$.
\begin{theorem}\label{existece_sol0}
Let Assumptions \ref{ass:A1}--\ref{ass:A3} be satisfied. Then \eqref{P_BV} has a solution $u\in L^2(\Omega)\cap BV(\Omega)$.
\end{theorem}
\begin{proof}
The proof is standard. We recall it by following the lines of the proof of \cite[Theorem 2.1]{CasasKunisch99_BV}.
Consider a minimizing sequence $(u_k)\subset L^2(\Omega)\cap BV(\Omega)$.
Since $f$ is bounded from below by \ref{ass:A1}, $\left(\BVSN{u_k}\right)$ is bounded.
By \ref{ass:A2}, $(u_k)$ is bounded in $L^2(\Omega)$, and hence $(u_k)$ is bounded in $BV(\Omega)$.
By Proposition \ref{prop:BV}, there is a subsequence $(u_{k_n})$ and $\bar u\in L^2( \Omega)\cap BV(\Omega)$ with
$u_{k_n}\rightharpoonup \bar u$ in $L^2(\Omega)$ and $u_{k_n}\to\bar u$ in $L^1(\Omega)$.
Due to \ref{ass:A3}, $U_{ad}$ is weakly closed in $L^2(\Omega)$, and $\bar u\in U_{ad}$ follows.
By weak lower semicontinuity \ref{ass:A1} and Proposition \ref{prop:BV}, we obtain
\[
J(\bar u)\le\liminf\limits_{k_n\to\infty}J(u_{k_n}) = \inf\limits_{u\in L^2(\Omega)\cap BV(\Omega)}J(u),
\]
therefore, $\bar u$ is a solution.
\end{proof}
\subsection{Necessary optimality conditions}
Next, we provide a first-order necessary optimality condition for \eqref{P_BV}. A similar result with proof can be found in \cite[Theorem 2.3]{CasasKunisch99_BV}.
\begin{theorem}\label{thm_FONC_BV}
Let Assumptions \ref{ass:A3}--\ref{ass:A4} be satisfied.
Let $\bar u\in BV(\Omega)\cap U_{ad}$
be locally optimal for \eqref{P_BV} with respect to $BV(\Omega) {\cap L^2(\Omega)}$, i.e., there is $r>0$ such that $J(\bar u) \le J(u)$ for all $u\in U_{ad}$
with $\|u-\bar u\|_{BV(\Omega)}+\|u-\bar u\|_{L^2(\Omega)} <r$.
Then there is $\lambda\in \partial\|\cdot\|_{\mathcal{M}{(\Omega)}}(\nabla \bar u) \subset (\mathcal{M}(\Omega)^n)^*$ such that
\begin{equation}\label{nec_opt_0}
-\nabla f(\bar u)\in -\Div\lambda + N_{U_{ad}}(\bar u) \text{ in } (BV(\Omega)\cap L^2(\Omega))^*,
\end{equation}
where $-\Div:(\mathcal{M}(\Omega)^n)^*\to (BV(\Omega)\cap L^2(\Omega))^*$ is the adjoint operator of $\nabla:BV(\Omega)\cap L^2(\Omega)\to\mathcal{M}(\Omega)^n$,
and the normal cone $N_{U_{ad}}$ of $U_{ad}$ is determined with respect to $BV(\Omega)\cap L^2(\Omega)$:
\[
N_{U_{ad}} = \{ \mu \in (BV(\Omega)\cap L^2(\Omega))^*: \ \langle \mu, u-\bar u \rangle \le 0 \quad \forall u \in U_{ad} \cap BV(\Omega)\}.
\]
\end{theorem}
\begin{proof}
Let us define $X:=BV(\Omega)\cap L^2(\Omega)$.
By standard arguments, we find
\[
-\nabla f(\bar u)\in\partial(\BVSN{\cdot}+\delta_{U_{ad}})(\bar u) \subset X^*.
\]
Recall, $\BVSN{u}=\|\nabla u\|_{\mathcal{M}(\Omega)} = (\|\cdot\|_{\mathcal{M}(\Omega)}\circ\nabla)(u)$.
Following \cite[Theorem 2.3]{CasasKunisch99_BV}, let $-\Div:(\mathcal{M}(\Omega)^n)^*\to X^*$ denote the adjoint operator of $\nabla:X\to\mathcal{M}(\Omega)^n$, i.e.,
\[
\langle\Div \lambda,u\rangle=-\langle\lambda,\nabla u\rangle\quad \forall u\in X,\ \lambda \in (\mathcal{M}(\Omega)^n)^*.
\]
By the sum and chain rules for the convex subdifferential, \eqref{nec_opt_0} can be rewritten as
\[
-\nabla f(\bar u)\in -\Div \left( \partial\|\cdot\|_{\mathcal{M}{(\Omega)}}(\nabla \bar u)\right) + N_{U_{ad}}(\bar u),
\]
which is the claim.
\end{proof}
\section{Regularization scheme} \label{sec3}
In this section, we introduce the regularization scheme for \eqref{P_BV}.
We will use a smoothing of the $BV$-norm as well as a penalization of the constraints $u\in U_{ad}$.
In order to approximate the $BV$-norm by smooth functions,
we introduce the following family $(\psi_\epsilon)_{\epsilon>0}$ of smooth functions with $\psi_\epsilon(t) \approx |t|$.
For $\epsilon>0$ we define $\psi_{\epsilon}:\mathbb{R}^n\to\mathbb{R}^n$ by
\begin{equation} \label{eq:def_psi}
\psi_\epsilon(t):=\sqrt{\epsilon+|t|^2} +\epsilon |t|^2,
\end{equation}
where $|\cdot|$ denotes the Euclidian norm in $\mathbb{R}^{n}$.
As a first direct consequence of the above definition we have:
\begin{lemma}\label{ass:B}
For $\epsilon>0$,
$(\psi_\epsilon)_{\epsilon>0}$ is a family of twice
continuously differentiable functions from $\mathbb{R}^n$ to $\mathbb{R}^n$
with the following properties:
\begin{enumerate}[label=(\arabic*), ref=Lemma \ref{ass:B}(\theenumi)]
\item $t\mapsto\psi_\epsilon(t)$ is convex. \label{ass:B1}
\item $\psi_\epsilon(t)\ge |t|+\epsilon |t|^2$ and $\psi'_\epsilon(t)t\ge0$ for all $t\in \mathbb{R}^n$. \label{ass:B2}
\item For all $t\in\mathbb{R}^n$, $\epsilon\mapsto\psi_\epsilon(t)$ is monotonically increasing and $\psi_\epsilon(t)\to|t|$ as $\epsilon\searrow0.$\label{ass:B3}
\item $t\mapsto\psi'_\epsilon(t)t$ is coercive, i.e., $\psi'_\epsilon(t)t\to\infty$ as $|t|\to \infty$.\label{ass:B4}
\item $\psi'_\epsilon(t)t \ge |t| - \sqrt\epsilon$ for all $t\in \mathbb{R}^n$.\label{ass:B5}
\end{enumerate}
\end{lemma}
\begin{proof}
Properties (1)--(4) are immediate consequences of the definition.
(5) can be proven as follows:
\[
\psi'_\epsilon(t)t -|t| \ge \frac{ |t|^2 - |t| \sqrt{\epsilon + |t|^2} }{\sqrt{\epsilon + |t|^2}}
= \frac{ -\epsilon |t|^2 }{ \sqrt{\epsilon + |t|^2} (|t|^2 + |t| \sqrt{\epsilon + |t|^2})}
\ge \frac{ -\epsilon |t|^2 }{ \sqrt{\epsilon + |t|^2} |t|^2} \ge -\sqrt\epsilon.
\]
\end{proof}
The $BV$-minimization problem \eqref{P_BV} is then approximated by
\begin{equation}
\label{P_eps}\tag{$P_\epsilon$}
\min\limits_{u\in H^1(\Omega)}f(u)+\int_\Omega\psi_\epsilon(\nabla u)\,\mathrm{d}x
\quad\text{ s.t. }u\in U_{ad}.
\end{equation}
The choice \eqref{eq:def_psi} of $\psi_\epsilon$ guarantees the existence of solutions of this problem in $H^1(\Omega)$.
Note that the standard approximation $|t| \approx \sqrt{ \epsilon + |t|^2}$ does \ref{ass:B2},
which ensures that $u\mapsto \int_\Omega\psi_\epsilon(\nabla u)\,\mathrm{d}x$ is weakly coercive in $H^1(\Omega)$.
The existence of solutions $u\in H^1(\Omega)$ (and the fact that $\nabla u$ is a measurable function) is important for the subsequent analysis.
Due to the presence of the inequality constraints, Problem \eqref{P_eps} is difficult to solve.
Following existing approaches in the literature, see, e.g., \cite{Kunisch_Wa_2012,schiela_wachsmuth}, we will use a smooth penalization of these constraints.
We define the smooth function $\max_\rho$ by
\begin{equation}\label{eq:max_gamma}
{\max}_\rho(x):=\begin{cases}
\max(0,x)\quad&\text{ if } |x|\ge \frac{1}{2\rho},\\
\frac{\rho}{2}(x+\frac{1}{2\rho})^2&\text{ if }|x|\le\frac{1}{2\rho},
\end{cases}
\end{equation}
where $\rho>0$. Due to the inequalities
\begin{equation} \label{eq:est_max_rho}
0\le\max(0,x)\le{\max}_\rho(x)\le \max(x,0)+\frac{1}{2\rho}\quad \forall x\in\mathbb{R},
\end{equation}
it can be considered as an approximation of $x\mapsto \max(x,0) = (x)_+$.
In addition, one verifies $\max(0,x)\le t^{-1}{\max}_\rho(tx)$ for all $x\in \mathbb{R}$ and $t>0$.
Let us introduce
\begin{equation}\label{def_Mrho}
M_\rho(x):=\int_{-\infty}^x {\max}_\rho(t)\,\mathrm{d}t.
\end{equation}
Using this function, we define the penalized problem by
\begin{equation}\label{Peps_penal}\tag{$P_{\epsilon,\rho}$}
\min_{u\in H^1(\Omega)} f(u)+\int_\Omega\psi_\epsilon(\nabla u)\,\mathrm{d}x+
\int_\Omega\frac 1{\rho}\left(M_\rho(\rho(u_a-u))+M_\rho(\rho( u-u_b))\right)\,\mathrm{d}x.
\end{equation}
If $u\in H^1(\Omega)$ is a local solution of \eqref{Peps_penal} then it satisfies
\begin{equation}\label{OC_eps,c}
\int_\Omega\nabla f(u)v+\psi_\epsilon'(\nabla u)\nabla v
-\lambda^a_{\rho}(u)v
+\lambda_{\rho}^b(u)v \,\mathrm{d}x=0 \quad \forall\: v\in H^1(\Omega),
\end{equation}
where we used the abbreviations
\begin{equation}\label{eq_def_lambda}
\lambda^a_{\rho}(u):={\max}_\rho(\rho(u_a-u)),\:\lambda^b_{\rho}(u):={\max}_\rho(\rho(u-u_b)).
\end{equation}
Existence of solutions of \eqref{Peps_penal} and necessity of \eqref{OC_eps,c} for local optimality can be proven by standard arguments.
\begin{corollary}\label{cor_OC_Eps_exist_sol}
Let assumptions \ref{ass:A1}--\ref{ass:A5} be satisfied. Then the equation \eqref{OC_eps,c}
admits a solution $u\in H^1(\Omega)$.
\end{corollary}
We will investigate the behavior of a penalty and smoothing method to solve \eqref{P_BV}.
Since \eqref{Peps_penal} is a non-convex problem, it is unrealistic to assume that one can compute global solutions.
Instead, the iterates will be chosen as stationary points of \eqref{Peps_penal}.
Hence, we are interested in the behavior of stationary points $u_{\epsilon,\rho}$ and corresponding multipliers $\lambda_{\rho}^a(u_{\epsilon,\rho})$, $\lambda^b_{\rho}(u_{\epsilon,\rho})$ as $\rho\to\infty$
and $\epsilon\searrow0$.
The resulting method then reads as follows.
\begin{algorithm} \label{alg1}
Choose $\epsilon_0\in(0,1),\rho_0>0$ and $u_0\in H^1(\Omega)$.
\begin{enumerate}
\item
Compute $u_k$ as solution to
\begin{equation}\label{alg1:eq_k}
\int_\Omega\nabla f(u)v+\psi_{\epsilon_k}'(\nabla u)\nabla v
-\lambda^a_{\rho_k}(u)v
+\lambda_{\rho_k}^b(u)v \,\mathrm{d}x=0 \quad \forall\: v\in H^1(\Omega),
\end{equation}
where $\lambda^a_{\rho}(u),\lambda_{\rho}^b(u)$ are defined in \eqref{eq_def_lambda}.
\item Choose $\rho_{k+1}>\rho_k,\: \epsilon_{k+1}<\epsilon_k.$
\item If a suitable stopping criterion is satisfied: Stop. Else set $k:=k+1$ and go to Step 1.
\end{enumerate}
\end{algorithm}
In view of Corollary \ref{cor_OC_Eps_exist_sol}, the algorithm is well-defined.
In the following, we assume that the algorithm generates an infinite sequence of iterates $(u_k, \lambda^a_{\rho_k}(u_k),\lambda^b_{\rho_k}(u_k))$.
Here, we are interested to prove that weak limit points are stationary, i.e., they satisfy the optimality condition \eqref{nec_opt_0} for \eqref{P_BV}.
Throughout the subsequent analysis, we assume that assumptions \ref{ass:A1}--\ref{ass:A5} are satisfied
\subsection{A-priori bounds} \label{sec:a-priori}
In order to investigate the sequences of iterates $(u_k)$ and its (weak) limit points, it is reasonable to derive bounds of iterates $u_k$, i.e., solutions of \eqref{OC_eps,c}, first.
To this end, we will study solutions $u\in H^1(\Omega)$ of the nonlinear variational equation
\begin{equation}\label{eq_u_g}
\int_\Omega \psi_\epsilon'(\nabla u)\nabla v
-\lambda^a_{\rho}(u)v
+\lambda_{\rho}^b(u)v \,\mathrm{d}x=
\int_\Omega gv\,\mathrm{d}x
\quad \forall\: v\in H^1(\Omega)
\end{equation}
for $\epsilon>0$, $\rho>0$, and $g\in L^2(\Omega)$.
The functions $\lambda^a_\rho(u)$, $\lambda^b_\rho(u)$ are defined in \eqref{eq_def_lambda} as
\[
\lambda^a_{\rho}(u):={\max}_\rho(\rho(u_a-u)),\:\lambda^b_{\rho}(u):={\max}_\rho(\rho(u-u_b)).
\]
Let us start with the following lemma. It shows that the supports of the multipliers $\lambda^a_\rho(u)$, $\lambda^b_\rho(u)$
do not overlap if the penalty parameter is large enough.
\begin{lemma}\label{lem_lalb_disjoint}
Let $u\in H^1(\Omega)$ be a solution of \eqref{eq_u_g} to $g\in L^2(\Omega)$.
Suppose $\rho^2\ge\frac1{u_b-u_a}$. Then it holds
\[
\lambda^a_\rho(u) \cdot \lambda^b_\rho(u) =0 \text{ a.e.\ on } \Omega.
\]
\end{lemma}
\begin{proof}
Let $x\in \Omega$ such that $\lambda^a_\rho(u)(x) \cdot \lambda^b_\rho(u)(x) \ne 0$. Then it holds
$u(x) < u_a(x) + \frac1{2\rho^2}$ and $u(x) > u_b(x) - \frac1{2\rho^2}$.
Consequently, $\rho^2< \frac1{u_b-u_a}$ follows.
\end{proof}
Under the assumptions that the bounds $u_a$ and $u_b$ are constant, we
can prove the following series of helpful results regarding the boundedness of iterates.
Let us start with the boundedness of the multiplier sequences.
This result is inspired by related results for the $H^1$-obstacle problem,
see, e.g., \cite[Lemma 5.1]{KinderlehrerStampacchia1980}, see also \cite[Lemma 2.3]{schiela_wachsmuth}.
The results for $H^1$-obstacle problems require the assumption $\Delta u_a, \Delta u_b\in L^2(\Omega)$.
It is not clear to us, how the following proof can be generalized to non-constant obstacles $u_a,u_b$.
\begin{lemma}\label{lem_lambda_bounded}
Let $u\in H^1(\Omega)$ be a solution of \eqref{eq_u_g} to $g\in L^2(\Omega)$.
Suppose $\rho^2\ge\frac1{u_b-u_a}$. Then it holds
\[
\|\lambda^a_\rho(u)\|_{L^2(\Omega)} + \|\lambda^b_\rho(u)\|_{L^2(\Omega)} \le 2\|g\|_{L^2(\Omega)}.
\]
\end{lemma}
\begin{proof}
To show boundedness of the multipliers $\lambda^a_\rho(u),\lambda^b_\rho(u)$ in $L^2(\Omega)$, we test the optimality condition \eqref{eq_u_g} with $\lambda^a_\rho(u)$ and $\lambda^b_\rho(u)$, respectively.
We get for $\lambda^a_\rho(u)={\max}_{\rho}(\rho(u_a-u))$
\[
\|\lambda^a_\rho(u)\|^2_{L^2(\Omega)} = \int_\Omega \psi_{\epsilon}'(\nabla u) \nabla({\max}_{ \rho}(\rho(u_a-u)))\,\mathrm{d}x
- \int_\Omega g\lambda^a_\rho(u)\,\mathrm{d}x
+ \int_\Omega\lambda^b_\rho(u)\lambda^a_\rho(u)\,\mathrm{d}x.
\]
Due to Lemma \ref{lem_lalb_disjoint}, the last term is zero.
It remains to analyze the first term.
Here, we find
\[
\int_\Omega \psi_{\epsilon}'(\nabla u) \nabla {\max}_{ \rho}(\rho(u_a-u)) \,\mathrm{d}x
=\int_\Omega \psi_{\epsilon}'(\nabla u) \rho {\max}_{ \rho}'(\rho(u_a-u)) \nabla (-u) \,\mathrm{d}x \le0,
\]
where we used \ref{ass:B2} and ${\max}_{ \rho}' \ge0$.
This proves $\|\lambda^a_\rho(u)\|_{L^2(\Omega)} \le \|g\|_{L^2(\Omega)}$.
Similarly, $\|\lambda^b_\rho(u)\|_{L^2(\Omega)} \le \|g\|_{L^2(\Omega)}$ can be proven.
\end{proof}
\begin{corollary} \label{cor_constr_vio}
Let $u\in H^1(\Omega)$ be a solution of \eqref{eq_u_g} to $g\in L^2(\Omega)$.
Suppose $\rho^2\ge\frac1{u_b-u_a}$.
Then it holds
\[
\|(u-u_b)_+\|_{L^2(\Omega)}+\|(u_a-u)_+\|_{L^2(\Omega)} \le 2\rho^{-1}\|g\|_{L^2(\Omega)}.
\]
\end{corollary}
\begin{proof}
Due to the definition of $\max_\rho$, we have $\max(x,0) \le \rho^{-1}\max_\rho(\rho x)$ for all $x\in \mathbb{R}$.
This implies
\[
\|(u-u_b)_+\|_{L^2(\Omega)} \le \rho^{-1} \|{\max}_{\rho}(\rho(u-u_b))\|_{L^2(\Omega)} = \rho^{-1}\|\lambda^b_\rho(u)\|_{L^2(\Omega)},
\]
and the claim follows by Lemma \ref{lem_lambda_bounded} above.
\end{proof}
\begin{corollary}\label{cor_bound_L2}
Let $u\in H^1(\Omega)$ be a solution of \eqref{eq_u_g} to $g\in L^2(\Omega)$.
Suppose $\rho^2\ge\frac1{u_b-u_a}$.
Then it holds
\[
\|u\|_{L^2(\Omega)} \le 2\rho^{-1}\|g\|_{L^2(\Omega)} +
\| \max(|u_a|,|u_b|)\|_{L^2(\Omega)}.
\]
\end{corollary}
\begin{proof}
The claim is a consequence of Corollary \ref{cor_constr_vio} and the identity
\begin{equation}\label{eq35}
u= ( u-u_b)_+-(u_a- u)_++\proj_{[u_a,u_b]}( u).
\end{equation}
\end{proof}
\begin{lemma}\label{lem_bound_W11}
Let $u\in H^1(\Omega)$ be a solution of \eqref{eq_u_g} to $g\in L^2(\Omega)$.
Suppose $\rho^2\ge\frac1{u_b-u_a}$.
Then it holds
\[
\|\nabla u\|_{L^1(\Omega)} \le 3 \|g\|_{L^2(\Omega)} \|u\|_{L^2(\Omega)}+\sqrt{\epsilon} |\Omega|.
\]
\end{lemma}
\begin{proof}
We test the optimality condition \eqref{OC_eps,c} with $u$ and use \ref{ass:B5} to get the estimate
\[
\begin{split}
\int_\Omega |\nabla u| -\sqrt{\epsilon} \,\mathrm{d}x\le
\int_\Omega \psi'_{\epsilon}(\nabla u)\nabla u\,\mathrm{d}x
& = \left|\int_\Omega gu- \lambda^a_\rho(u) u + \lambda^b_\rho(u) u\,\mathrm{d}x\right| \\
& \le (\|g\|_{L^2(\Omega)} + \|\lambda^a_\rho(u)\|_{L^2(\Omega)}+ \|\lambda^b_\rho(u)\|_{L^2(\Omega)} ) \|u\|_{L^2(\Omega)}.
\end{split}
\]
The claim follows with the estimate of Lemma \ref{lem_lambda_bounded}.
\end{proof}
\subsection{Preliminary convergence results} \label{sec_prelim}
As next step, we derive convergence properties of solutions $u_k\in H^1(\Omega)$ of
\begin{equation}\label{eq_uk_gk}
\int_\Omega \psi_{\epsilon_k}'(\nabla u_k)\nabla v
-\lambda^a_{\rho_k}(u_k)v
+\lambda_{\rho_k}^b(u_k)v \,\mathrm{d}x=
\int_\Omega g_kv\,\mathrm{d}x
\quad \forall\: v\in H^1(\Omega)
\end{equation}
where
\begin{equation}\label{eq_ass_conv}
\epsilon_k \searrow0, \ \rho_k \to +\infty, \ g_k \rightharpoonup g \text{ in } L^2(\Omega).
\end{equation}
From the results of the previous section, we immediately obtain that $(u_k)$ is bounded in $BV(\Omega)$, and $(\lambda^a_{\rho_k}(u_k))$ and $(\lambda^b_{\rho_k}(u_k))$
are bounded in $L^2(\Omega)$.
Moreover, strong limit points of $(u_k)$ satisfy the inequality constraints in \eqref{eq:def_Uad} due to Corollary \ref{cor_constr_vio}.
In a first result, we need to lift the strong convergence of (a subsequence of) $(u_k)$ in $L^1(\Omega)$,
which is a consequence of the compact embedding $BV(\Omega) \hookrightarrow L^r(\Omega)$, $r<\frac n{n-1}$,
to strong convergence in $L^2(\Omega)$.
\begin{lemma}\label{lem_strong_conv_L2}
Assume \eqref{eq_ass_conv}.
Let $u_k$, $k\in \mathbb{N}$, be a solution of \eqref{eq_uk_gk}.
Suppose $u_k \to u$ in $L^1(\Omega)$.
Then $u_k \to u$ in $L^2(\Omega)$.
\end{lemma}
\begin{proof}
We use again the identity \eqref{eq35}:
\[
u_k= ( u_k-u_b)_+-(u_a- u_k)_++\proj_{[u_a,u_b]}( u_k).
\]
By Corollary \ref{cor_constr_vio}, the first two terms converge to zero in $L^2(\Omega)$,
which implies $u_a \le u\le u_b$ almost everywhere.
The sequence $(\proj_{[u_a,u_b]}( u_k))$ converges to $\proj_{[u_a,u_b]}(u)=u$ in $L^1(\Omega)$
and is bounded in $L^\infty(\Omega)$. By H\"older inequality it converges in $L^2(\Omega)$.
\end{proof}
We are now in the position to prove existence of suitably converging subsequence under assumption \eqref{eq_ass_conv}.
\begin{theorem}\label{thm310}
Assume \eqref{eq_ass_conv}.
Let $(u_k)$, $k\in \mathbb{N}$, be a family of solutions of \eqref{eq_uk_gk}.
Then there is a subsequence such that
$u_{k_n} \to u^*$, $\lambda^a_{\rho_{k_n}}(u_{k_n})\rightharpoonup \lambda^a$, and $\lambda^b_{\rho_{k_n}}(u_{k_n}) \rightharpoonup \lambda^b$ in $L^2(\Omega)$.
\end{theorem}
\begin{proof}
By Lemmas \ref{lem_lambda_bounded}, Corollary \ref{cor_bound_L2}, and Lemma \ref{lem_bound_W11}, $(u_k)$ is bounded in $BV(\Omega){\cap L^2(\Omega)}$, and $(\lambda^a_{\rho_k}(u_k))$ and $(\lambda^b_{\rho_k}(u_k))$
are bounded in $L^2(\Omega)$.
Then we can choose $(u_{k})$ as a subsequence that converges strongly in $L^1(\Omega)$ by Proposition \ref{prop:BV}.
Due to Lemma \ref{lem_strong_conv_L2} this convergence is strong in $L^2(\Omega)$.
Now, extracting additional weakly converging subsequences from $(\lambda^a_{\rho_k}(u_k))$ and $(\lambda^b_{\rho_k}(u_k))$ finishes the proof.
\end{proof}
The next result shows that limit points of $(u_k,\lambda^a_{\rho_k}(u_k),\lambda^b_{\rho_k}(u_k))$
satisfy the usual complementarity conditions.
\begin{lemma} \label{lem:complemtarity_cond_prelim}
Assume \eqref{eq_ass_conv}.
Let $u_k$, $k\in \mathbb{N}$, be a solution of \eqref{eq_uk_gk}.
Let $u_k \to u^*$, $\lambda^a_{\rho_k}(u_k)\rightharpoonup \lambda^a$, and $\lambda^b_{\rho_k}(u_k) \rightharpoonup \lambda^b$ in $L^2(\Omega)$. Then it follows that
\[
(\lambda^b,\,u^*-u_b) = 0 \text{ and } (\lambda^a,\,u_a-u^*) = 0.
\]
\end{lemma}
\begin{proof}
Due to \eqref{eq_ass_conv}, $(g_k)$ is bounded in $L^2(\Omega)$.
We deduce using \eqref{eq:est_max_rho}
\begin{align*}
|(\lambda^b_{\rho_k}(u_k),\,u_k-u_b)| &\le \int_\Omega {\max}_{\rho_k}(\rho_k(u_k-u_b))|u_k-u_b|\,\mathrm{d}x \\
&\le \int_\Omega\left(\rho_k(u_k-u_b)_++\frac1{2\rho_k}\right)|u_k-u_b|\,\mathrm{d}x\\
& = \rho_k\int_\Omega(u_k-u_b)_+(u_k-u_b)\,\mathrm{d}x +\int_{\Omega}\frac{1}{2\rho_k}|u_k-u_b|\,\mathrm{d}x \\
& = \rho_k \|(u_k-u_b)_+\|_{L^2(\Omega)}^2 +\frac{1}{2\rho_k}\|u_k-u_b\|_{L^1(\Omega)}.
\end{align*}
Due to Corollary \ref{cor_constr_vio} and the boundedness of $(u_k)$ in $L^2(\Omega)$, both expressions tend to zero for $k\to\infty$.
This implies
\[
(\lambda^b,\,u^*-u_b) =\lim_{k\to\infty} (\lambda^b_{\rho_k}(u_k),u_k-u_b) =0.
\]
The same argumentation yields the claim for $(\lambda^a,\,u_a-u^*)$.
\end{proof}
We will now show that weak limit points of solutions to \eqref{eq_uk_gk} satisfy a stationary condition similar to the one for the original problem \eqref{P_BV}.
We will utilize this result twice: first we apply it to iterates of Algorithm \ref{alg1}, second we will use it to prove a optimality condition for \eqref{P_BV} that
has a different structure than that of Theorem \ref{thm_FONC_BV}.
\begin{theorem} \label{thm:final_prelim}
Assume \ref{ass:A1}--\ref{ass:A5} and \eqref{eq_ass_conv}.
Let $(u_k)$, $k\in \mathbb{N}$, be a family of solutions of \eqref{eq_uk_gk}.
Let $u_k \to u^*$, $\lambda^a_{\rho_k}(u_k)\rightharpoonup \lambda^a$, and $\lambda^b_{\rho_k}(u_k) \rightharpoonup \lambda^b$ in $L^2(\Omega)$.
Then it holds
\begin{equation}\label{eq_osys_comp}
\begin{aligned}
u^* & \in U_{ad} \\
\lambda^a & \ge 0, & \lambda^b &\ge0, \\
(\lambda^a,\,u_a-u^*)&= 0, & (\lambda^b,\,u^*-u_b)&= 0.
\end{aligned}
\end{equation}
In addition, there is $\mu^*\in L^\infty(\Omega)^n$ with $\Div \mu^*\in L^2(\Omega)$ such that
\[
-\Div \mu^* -\lambda^a+\lambda^b =g
\]
and
\[
-\Div \mu^* \in \partial(\BVSN{\cdot})(u^*).
\]
Moreover, there is $\lambda^* \in \partial\|\cdot\|_{\mathcal{M}{(\Omega)}}(\nabla u^*) \subset (\mathcal{M}(\Omega)^n)^*$ with $\Div \lambda^*\in L^2(\Omega)$ such that
\[
-\Div \lambda^* -\lambda^a+\lambda^b =g.
\]
\end{theorem}
\begin{proof}
The system \eqref{eq_osys_comp} is a consequence of Corollary \ref{cor_constr_vio}, Lemma \ref{lem:complemtarity_cond_prelim},
and the non-negativity of $\max_\rho$.
In order to pass to the limit in \eqref{eq_uk_gk},
we need to analyze the term involving $\psi_{\epsilon_k}'(\nabla u_k)$.
Here, we argue similar as in the proof of \cite[Theorem 10]{CasasKunisch_ocsl_BV}.
Let $v\in C_c^\infty(\Omega)$.
Then, we find
\[
\int_\Omega \psi_{\epsilon_k}'(\nabla u_k)\nabla v \,\mathrm{d}x
=\int_\Omega \frac{\nabla u_k}{\sqrt{\epsilon_k+|\nabla u_k|^2}} \nabla v + 2\epsilon \nabla u_k \nabla v \,\mathrm{d}x
= \int_\Omega \frac{\nabla u_k}{\sqrt{\epsilon_k+|\nabla u_k|^2}} \nabla v - 2\epsilon u_k \Delta v \,\mathrm{d}x.
\]
Let us define $ \mu_k\in L^\infty(\Omega)$ by
\[
\mu_k:=\frac{\nabla u_k}{\sqrt{\epsilon_k+|\nabla u_k|^2}}.
\]
Clearly, the sequence $(\mu_k)$ is bounded in $L^\infty(\Omega)^n$,
and there exists a subsequence converging weak-star in $L^\infty(\Omega)^n$.
W.l.o.g.\@ we can assume $\mu_k\rightharpoonup^* \mu^* $ in $L^\infty(\Omega)^n$.
Since $(u_k)$ is bounded in $L^2(\Omega)$, we obtain
\[
\lim_{k\to\infty} \int_\Omega \psi_{\epsilon_k}'(\nabla u_k)\nabla v \,\mathrm{d}x
= \lim_{k\to\infty} \int_\Omega \mu_k \nabla v - 2\epsilon u_k \Delta v \,\mathrm{d}x
= \int_\Omega\mu^*\nabla v\,\mathrm{d}x.
\]
Then we can pass to the limit in \eqref{eq_uk_gk} to find
\[
\int_\Omega\mu^*\nabla v -\lambda^a v + \lambda^b v \,\mathrm{d}x= \int_\Omega gv \,\mathrm{d}x
\]
which is satisfied for all $v\in C_c^\infty(\Omega)$.
This implies
\[
-\Div \mu^* = g + \lambda^a - \lambda^b \in L^2(\Omega).
\]
Let now $v\in C^\infty(\bar \Omega)$. By convexity of $\psi_\epsilon$, we have
\begin{equation}\label{eq_subgrad_ineq_psik}
\int_\Omega\psi_{\epsilon_k}(\nabla u_k) + \psi'_{\epsilon_k}(\nabla u_k)(\nabla v-\nabla u_k)\,\mathrm{d}x \le \int_\Omega\psi_{\epsilon_k}(\nabla v)\,\mathrm{d}x.
\end{equation}
Here, we find $\int_\Omega\psi_{\epsilon_k}(\nabla v)\,\mathrm{d}x \to \|\nabla v\|_{L^1(\Omega)}$ and
\[
\liminf_{k\to\infty} \int_\Omega\psi_{\epsilon_k}(\nabla u_k) \,\mathrm{d}x \ge \liminf_{k\to\infty} \|\nabla u_k\|_{L^1(\Omega)} \ge \BVSN{u^*},
\]
cf.\@, Proposition \ref{prop:BV}.
Here, we used that $(u_k)$ is bounded in $BV(\Omega)$ due to Lemma \ref{lem_bound_W11} and \eqref{eq_ass_conv}.
Using the equation \eqref{eq_uk_gk}, we find
\[\begin{split}
\int_\Omega \psi'_{\epsilon_k}(\nabla u_k)(\nabla v-\nabla u_k)\,\mathrm{d}x
& = \int_\Omega (g_k +\lambda^a_{\rho_k}(u_k) -\lambda_{\rho_k}^b(u_k))(v-u_k)\,\mathrm{d}x\\
&\to \int_\Omega (g+\lambda^a -\lambda^b)(v-u^*) \,\mathrm{d}x
= \int_\Omega -\Div \mu^* (v-u^*) \,\mathrm{d}x.
\end{split}\]
Then we can pass to the limit in \eqref{eq_subgrad_ineq_psik} to obtain
\[
\BVSN{u^*} + \int_\Omega -\Div \mu^* (v-u^*) \,\mathrm{d}x \le \BVSN{v}
\]
for all $v\in C^\infty(\bar \Omega)$.
Due to the density result of Proposition \ref{prop:BV} with respect to intermediate convergence \eqref{eq_intermed_conv},
the inequality holds for all $v\in BV(\Omega) \cap L^2(\Omega)$.
Consequently, $-\Div \mu^* \in \partial(\BVSN{\cdot})(u^*) \subset (BV(\Omega)\cap L^2(\Omega))^*$.
Using the chain rule as in Theorem \ref{thm_FONC_BV}, we find
\[
-\Div \mu^* \in -\Div \left( \partial\|\cdot\|_{\mathcal{M}{(\Omega)}}(\nabla u^*)\right),
\]
which proves the existence of $\lambda^*$ with the claimed properties.
\end{proof}
\subsection{Convergence of iterates}
We are now going to apply the results of the previous two sections to the iterates of Algorithm \ref{alg1}.
In terms of \eqref{eq_uk_gk}, we have to set $g_k:=\nabla f(u_k)$.
As can be seen from, e.g., Theorem \ref{thm:final_prelim}, the boundedness of $(\nabla f(u_k))$ in $L^2(\Omega)$ will be crucial for any convergence analysis.
Unfortunately, this boundedness can only be guaranteed in exceptional cases.
Here, we prove it under the assumption that $\nabla f$ is globally Lipschitz continuous.
In Section \ref{sec:global} we show that convexity of $f$ or global optimality of $u_k$ is sufficient.
\begin{lemma}
Let $\nabla f: L^2(\Omega) \to L^2(\Omega)$ be globally Lipschitz continuous with modulus $L_f$.
Assume $\rho_k\to\infty$.
Then $(u_k)$ and $(\nabla f(u_k))$ are bounded in $L^2(\Omega)$.
\end{lemma}
\begin{proof}
Due to the Lipschitz continuity of $\nabla f$, we have
\[
\|\nabla f(u_k)\|_{L^2(\Omega)} \le L_f \|u_k\|_{L^2(\Omega)} + \|\nabla f(0)\|_{L^2(\Omega)}.
\]
By Corollary \ref{cor_bound_L2}, we find for $k$ sufficiently large
\[
\begin{split}
\|u_k\|_{L^2(\Omega)} & \le 2\rho_k^{-1}\|\nabla f(u_k)\|_{L^2(\Omega)} + \| \max(|u_a|,|u_b|)\|_{L^2(\Omega)}\\
&\le 2\rho_k^{-1}L_f \|u_k\|_{L^2(\Omega)} + 2\rho_k^{-1}\|\nabla f(0)\|_{L^2(\Omega)}+ \| \max(|u_a|,|u_b|)\|_{L^2(\Omega)}.
\end{split}
\]
If $k$ is such that $2\rho_k^{-1}L_f < \frac12$, then
$
\|u_k\|_{L^2(\Omega)} \le 4\rho_k^{-1}\|\nabla f(0)\|_{L^2(\Omega)}+ 2\| \max(|u_a|,|u_b|)\|_{L^2(\Omega)},
$
which proves the claim.
\end{proof}
The next observation is a simple consequence of previous results and shows the close relation between boundedness of $(u_k)$ in $L^2(\Omega)$ and $BV(\Omega)$ and the boundedness of $(\nabla f(u_k))$ in $L^2(\Omega)$.
\begin{lemma} \label{lem:L2conv}
Assume $\rho_k\to\infty$.
Then the following statements are equivalent:
\begin{enumerate}[label=(\arabic*)]
\item\label{lem314_1} $(\nabla f(u_k))$ is bounded in $L^2(\Omega)$,
\item\label{lem314_2} $(u_k)$ is bounded in $L^2(\Omega)$ and $BV(\Omega)$,
\item\label{lem314_3} $\{u_k: \ k\in \mathbb N\}$ is pre-compact in $L^2(\Omega)$.
\end{enumerate}
\end{lemma}
\begin{proof}
\ref{lem314_1} $\Rightarrow$ \ref{lem314_2}:
The boundedness of $(u_k)$
is a direct consequence of Corollary \ref{cor_bound_L2} and Lemma \ref{lem_bound_W11}.
\ref{lem314_2} $\Rightarrow$ \ref{lem314_3}
follows from Proposition \ref{prop:BV} and Lemma \ref{lem_strong_conv_L2}.
\ref{lem314_3} $\Rightarrow$ \ref{lem314_1}:
Since $u\mapsto \nabla f(u)$ is continuous from $L^2(\Omega)$ to $L^2(\Omega)$ by \ref{ass:A4}, the set $\{\nabla f(u_k): \ k\in \mathbb N\}$ is pre-compact in $L^2(\Omega)$
and thus bounded.
\end{proof}
Similarly to Theorem \ref{thm310}, we have the following result on the existence of converging subsequences.
\begin{theorem}
Suppose $\epsilon_k\searrow0$ and $\rho_k\to\infty$.
Let $(u_k)$ solve \eqref{OC_eps,c}.
Assume that $(\nabla f(u_k))$ is bounded in $L^2(\Omega)$.
Then there is a subsequence such that
$u_{k_n} \to u^*$, $\lambda^a_{\rho_{k_n}}(u_{k_n})\rightharpoonup \lambda^a$, and $\lambda^b_{\rho_{k_n}}(u_{k_n}) \rightharpoonup \lambda^b$ in $L^2(\Omega)$.
\end{theorem}
\begin{proof}
This result can be proven with similar arguments as Theorem \ref{thm310}.
\end{proof}
We finally arrive at the following convergence result for iterates of Algorithm \ref{alg1} which is a consequence of Theorem \ref{thm:final_prelim}.
\begin{theorem} \label{thm:final}
Assume \ref{ass:A1}--\ref{ass:A5}.
Suppose $\epsilon_k\searrow0$ and $\rho_k\to\infty$.
Let $(u_k)$ solve \eqref{OC_eps,c}.
Assume that there is a subsequence with $u_{k_n}\to u^*$, $\lambda_{k_n}^a \rightharpoonup\lambda^a$, and $\lambda_{k_n}^b \rightharpoonup\lambda^b$ in $L^2(\Omega)$.
Then it holds
\
\begin{aligned}
u^* & \in U_{ad} \\
\lambda^a & \ge 0, & \lambda^b &\ge0, \\
(\lambda^a,\,u_a-u^*)&= 0, & (\lambda^b,\,u^*-u_b)&= 0.
\end{aligned}
\
In addition, there is $\mu^*\in L^\infty(\Omega)^n$ with $\Div \mu^*\in L^2(\Omega)$ such that
\[
-\Div \mu^* -\lambda^a+\lambda^b = -\nabla f(u^*)
\]
and
\[
-\Div \mu^* \in \partial(\BVSN{\cdot})(u^*).
\]
Moreover, there is $\lambda^* \in \partial\|\cdot\|_{\mathcal{M}{(\Omega)}}(\nabla u^*) \subset (\mathcal{M}(\Omega)^n)^*$ with $\Div \lambda^*\in L^2(\Omega)$ such that
\[
-\Div \lambda^* -\lambda^a+\lambda^b =g.
\]
\end{theorem}
\begin{proof}
By assumption, we have $\nabla f(u_{k_n}) \to \nabla f(u^*)$.
The proof is now a direct consequence of Theorem \ref{thm:final_prelim}.
\end{proof}
\subsection{Global solutions}\label{sec:global}
The next theorem shows that global optimality is sufficient to obtain boundedness of iterates.
We note that if $f$ is convex, solutions of \eqref{OC_eps,c} are global solutions to the penalized problem
\eqref{Peps_penal}.
\begin{theorem} \label{lem:conv_case}
Assume \ref{ass:A1}--\ref{ass:A5}.
Suppose $\epsilon_k\searrow0$ and $\rho_k\to\infty$.
Suppose $(u_{k})$ is the corresponding sequence of global solutions to the penalized problems \eqref{Peps_penal}.
Then $(u_k)$ is bounded in $BV(\Omega)\cap L^2(\Omega)$
\end{theorem}
\begin{proof}
We introduce the notation
\[
j_{\epsilon,\rho}(u):= f(u)+\int_\Omega\psi_\epsilon(\nabla u)\,\mathrm{d}x+
\int_\Omega\frac 1{\rho}\left(M_\rho(\rho(u_a-u))+M_\rho(\rho( u-u_b))\right)\,\mathrm{d}x.
\
\]
Set $\tilde u:= \frac12(u_a+u_b) \in H^1(\Omega)$. Then $j_{\epsilon_{k},\rho_k}(\tilde u)=f(\tilde u)$ for $\rho_k$ large enough.
Let $u_k$ be a global minimizer of $j_{\epsilon_{k},\rho_k}$.
This implies
\
f(u_k)+\int_\Omega\psi_{\epsilon_{k}}(\nabla u_k)\,\mathrm{d}x
+ \int_\Omega\frac 1{{\rho_k}}\left(M_{\rho_k}({\rho_k}(u_a-u))+M_{\rho_k}({\rho_k}( u-u_b))\right)\,\mathrm{d}x
\le f(\tilde u).
\
Since $f$ is bounded from below, there is $K>0$ such that
\[
\int_\Omega\psi_{\epsilon_{k}}(\nabla u_k)\,\mathrm{d}x
+ \int_\Omega\frac 1{{\rho_k}}\left(M_{\rho_k}({\rho_k}(u_a-u))+M_{\rho_k}({\rho_k}( u-u_b))\right)\,\mathrm{d}x
\le K.
\]
This proves that $(\nabla u_k)$ is bounded in $L^1(\Omega)$ by \ref{ass:B2}.
By construction, we have
\[
M_\rho(x) = \int_{-\infty}^x {\max}_\rho(t)\,\mathrm{d}t \ge \int_{-\infty}^x \max(t,0) \,\mathrm{d}t = \frac12 \max(0,x)^2.
\]
This implies
\[
\frac {\rho_k}2 \left( \|(u_k-u_b)_+\|_{L^2(\Omega)}^2+\|(u_a-u)_+\|_{L^2(\Omega)} ^2 \right) \le K,
\]
and the boundedness of $(u_k)$ in $L^2(\Omega)$ is now a consequence of identity \eqref{eq35}.
\end{proof}
\section{Optimality condition by regularization} \label{sec4}
Let us assume $\bar u\in BV(\Omega) \cap L^2(\Omega)$ is locally optimal to \eqref{P_BV}.
In this section, we want to show that
there is a sequence of solutions $(u_{\rho,\epsilon})$ of certain regularized problems converging to $\bar u$.
This will allow us to prove optimality conditions for $\bar u$ that are similar to the systems obtained in Theorems \ref{thm:final_prelim} and \ref{thm:final}.
Again, we work under the assumptions \ref{ass:A1}--\ref{ass:A5}.
The solution $\bar u$ satisfies the necessary optimality condition
\begin{equation}\label{eq:OC_lok}
-\nabla f(\bar u)\in \partial\left(\BVSN{\cdot}\right)(\bar u) +N_{U_{ad}}(\bar u) \text{ in } (BV(\Omega)\cap L^2(\Omega))^*,
\end{equation}
see also Theorem \ref{thm_FONC_BV}.
It is easy to see that \eqref{eq:OC_lok} implies that $\bar u$ is the unique solution to the linearized, strictly convex problem
\begin{equation}\label{eq:P_lin}
\min\limits_{u\in BV(\Omega)\cap L^2(\Omega)} \nabla f(\bar u) \cdot u+\BVSN{u}+\frac12\|u-\bar u\|_{L^2(\Omega)}^2 + \delta_{U_{ad}}(u).
\end{equation}
In fact, let $u^*\in BV(\Omega)\cap L^2(\Omega)$ be the solution of \eqref{eq:P_lin}. Then we have the following optimality condition
\[
-\nabla f(\bar u)\in \partial\left(\BVSN{\cdot}\right)(u^*) +(u^*-\bar u) + N_{U_{ad}}(u^*) \text{ in } (BV(\Omega)\cap L^2(\Omega))^*,
\]
which is satisfied by $u^*:=\bar u$.
Let us approximate \eqref{eq:P_lin} by the family of unconstrained convex problems
\begin{equation}\label{eq:P_lin_pen}
\min\limits_{u\in H^1(\Omega)} \nabla f(\bar u)\cdot u+\int_\Omega\psi_\epsilon(\nabla u)\,\mathrm{d}x+\frac12\|u-\bar u\|_{L^2(\Omega)}^2 +\frac1\rho\int_\Omega M_\rho(\rho(u_a-u))+M_\rho(\rho(u-u_b))\,\mathrm{d}x.
\end{equation}
The optimality condition for the unique solution $u_{\epsilon,\rho}$ to \eqref{eq:P_lin_pen} is given by
\begin{equation} \label{eq:P_lin_pen_OC}
\int_\Omega \nabla f(\bar u)v +\psi'_\epsilon(\nabla u_{\epsilon,\rho})\nabla v+(u_{\epsilon,\rho}-\bar u)v
-\lambda^a_{\rho}(u_{\epsilon,\rho})v
+\lambda_{\rho}^b(u_{\epsilon,\rho})v \,\mathrm{d}x=0.
\end{equation}
for all $v\in H^1(\Omega)$.
\begin{corollary}\label{cor41}
Suppose $\epsilon_k\searrow0$ and $\rho_k\to\infty$.
Suppose $(u_{k})$ is the corresponding sequence of global solutions to the penalized problems \eqref{eq:P_lin_pen}.
Then $(u_k)$ is bounded in $BV(\Omega)\cap L^2(\Omega)$.
\end{corollary}
\begin{proof}
The claim follows by a similar argumentation as in the proof of Theorem \ref{lem:conv_case}.
\end{proof}
\begin{lemma}\label{lem42}
Suppose $\epsilon_k\searrow0$ and $\rho_k\to\infty$. Let $(u_{k})$ be the corresponding sequence of global solutions to the penalized problems \eqref{eq:P_lin_pen}.
Then $u_k\to\bar u $ in $L^2(\Omega)$,
and the sequences $(\lambda^a_{\rho_k}(u_k))$ and $(\lambda^b_{\rho_k}(u_k))$
are bounded in $L^2(\Omega)$.
\end{lemma}
\begin{proof}
Due to Corollary \ref{cor41}, $(u_k)$ is bounded in $BV(\Omega)\cap L^2(\Omega)$.
Suppose for the moment $u_k \to u^*$ in $L^1(\Omega)$ and $u_k \rightharpoonup u^*$ in $L^2(\Omega)$.
By Lemma \ref{lem_strong_conv_L2} applied to $g_k:=-\nabla f(\bar u) - (u_k -\bar u)$ and $g:=-\nabla f(\bar u) - (u^* -\bar u)$, we obtain $u_k \to u^*$ in $L^2(\Omega)$.
By Theorem \ref{thm310}, the corresponding sequences $(\lambda^a_{\rho_k}(u_k))$ and $(\lambda^b_{\rho_k}(u_k))$
are bounded in $L^2(\Omega)$.
Suppose $\lambda^a_{\rho_k}(u_k) \rightharpoonup \lambda^a$ and $\lambda^b_{\rho_k}(u_k)\rightharpoonup\lambda^b$ in $L^2(\Omega)$.
By Theorem \ref{thm:final_prelim}, we have
\[
-\nabla f(\bar u) - (u^* -\bar u) + \lambda^a - \lambda^b\in \partial(\BVSN{\cdot})(u^*).
\]
Due to the complementarity conditions \eqref{eq_osys_comp} of Theorem \ref{thm:final_prelim},
we get
\[
-\nabla f(\bar u) - (u^* -\bar u) \in \partial\left(\BVSN{\cdot}+\delta_{U_{ad}}\right)(u^*).
\]
Since $-\nabla f(\bar u) \in \partial\left(\BVSN{\cdot}+\delta_{U_{ad}}\right)(\bar u)$, we have by the monotonicity of the subdifferential
\[
(-\nabla f(\bar u) - (u^* -\bar u) - (-\nabla f(\bar u)), \ u^*-\bar u)\ge0,
\]
which implies $u^*=\bar u$.
With similar arguments, we can show that every subsequence of $(u_k)$ contains another subsequence that converges in $L^2(\Omega)$ to $\bar u$.
Hence, the convergence of the whole sequence follows.
\end{proof}
This convergence result enables us to prove that $\bar u$ satisfies an optimality condition
similar to those of Theorems \ref{thm:final_prelim} and \ref{thm:final}.
\begin{theorem} \label{thm:final_local}
Assume \ref{ass:A1}--\ref{ass:A5}.
Let $\bar u$ be locally optimal for \eqref{P_BV}.
Then there is
\[
\lambda^* \in \partial\|\cdot\|_{\mathcal{M}{(\Omega)}}(\nabla \bar u) \subset (\mathcal{M}(\Omega)^n)^*
\]
with $\Div \lambda^*\in L^2(\Omega)$
such that
\[
-\Div \lambda^* -\lambda^a+\lambda^b = \nabla f(\bar u)
\]
and
\
\begin{aligned}
\lambda^a & \ge 0, & \lambda^b &\ge0, \\
(\lambda^a,\,u_a-u^*)&= 0, & (\lambda^b,\,u^*-u_b)&= 0.
\end{aligned}
\
\end{theorem}
\begin{proof}
We define $(u_k)$ as global solutions to the penalized problems \eqref{eq:P_lin_pen} to parameter sequences
$\epsilon_k\searrow0$ and $\rho_k\to\infty$. Due to Lemma \ref{lem42}, we have $u_k \to\bar u$ in $L^2(\Omega)$.
Define $g_k:=-\nabla f(\bar u) - (u_k-\bar u)$ and $g:=-\nabla f(\bar u)$.
Now, the claim follows by Theorem \ref{thm:final_prelim}.
\end{proof}
Clearly, the optimality conditions of Theorem \ref{thm:final_local} are stronger than those of Theorem \ref{thm_FONC_BV}.
However, the proofs above only work on the strong assumptions that the bounds $u_a$ and $u_b$ are constant functions.
Here, it is not clear to us, under which assumptions the above techniques carry over to non-constant $u_a$ and $u_b$.
\section{Numerical tests}\label{sec5}
In this section, the suggested algorithm is tested with selected examples. To this end,
we implemented Algorithm \ref{alg1} in python using FEnicCS, \cite {fenicstutorial:1}.
Our examples are carried out in the optimal control setting. In particular, $f$ is given by the reduced tracking type functional
$$f(u):=\frac12\|S(u)-y_d\|_{L^2(\Omega)}^2,$$ where $S$ is the weak solution operator of some elliptic partial differential equation (PDE) specified below.
To solve the partial differential equation, the domain is divided into a regular triangular mesh, and the PDE as well as the control are discretized with piecewise linear finite elements. If not mentioned otherwise, the computations are done on a 128 by 128 grid, which results in a mesh size of $h = 0.022$.
Let us define
$j_{\epsilon,\rho}: H^{1}(\Omega)\to\mathbb{R}$
by
\begin{equation*}
j_{\epsilon,\rho}(u) := f(u)+ \int_\Omega\psi_\epsilon(\nabla u)\,\mathrm{d}x+
\int_\Omega\frac 1{\rho}\left(M_\rho(\rho(u_a-u))+M_\rho(\rho( u-u_b))\right)\,\mathrm{d}x
\end{equation*}
with $M_\rho$ as defined in \eqref{def_Mrho}. It is given in our tests by the specific choice
\[
M_\rho(x):=\begin{cases}
\frac12x^2+\frac{1}{24\rho^2}\quad&\text{ if }x>\frac{1}{2\rho},\\
\frac\rho6(x+\frac1{2\rho})^3&\text{ if }|x|<\frac{1}{2\rho},\\
0&\text{otherwise.}
\end{cases}
\]
let us recall that we use the following function to approximate the $BV$-seminorm:
\[
\psi_{\epsilon}= \sqrt{\epsilon+t^2}+\epsilon t^2.
\]
Concerning the continuation strategy for the parameters $\epsilon$ and $\rho$ in Algorithm \ref{alg1}, we set $\epsilon_0:=0.5$ in the initialization and decrease $\epsilon$ by factor $0.5$ after each iteration.
The penalty parameter is increased by factor $2$ after every iteration and is initialized with $\rho_0 := 2$.
Algorithm \ref{alg1} is stopped if the following termination criterion is satisfied:
\begin{equation}\label{alg: termination}
R^\rho_k\le10^{-4}\text{ and } R^\epsilon_{k}\le 10^{-3},
\end{equation}
where the residuals $R^\rho_k,\: R^\epsilon_k$ are given by
\[
R_k^\rho:=\|(u_a-u_k)_+\|_{L^2(\Omega)}+\|(u_k-u_b)_+\|_{L^2(\Omega)}+(\lambda_k^a,u_a-u_k)+(\lambda_k^b,u_k-u_b).
\]
and
\[
R^\epsilon_k := \|\nabla u_k\|_{L^1(\Omega)} - \langle \mu_k,\nabla u_k\rangle
\]
with
$
\mu_k:=\frac{\nabla u_k}{\sqrt{\epsilon_k+|\nabla u_k|^2}}
$
as in the proof of Theorem \ref{thm:final_prelim}.
Here, the residuum $R^\rho_k$ measures the violation of the box-constraints and of the complementarity condition in Theorem \ref{thm:final}.
Let us discuss the choice of the residuum $R^\epsilon$.
It can be interpreted as a residual in the subgradient inequality.
Since $\|\mu_k\|_{L^\infty(\Omega)^n} \le1$, we have $\int_\Omega \mu_k \cdot \nabla v\,\mathrm{d}x \le \|\nabla v\|_{L^1(\Omega)}$ for $v\in W^{1,1}(\Omega)$.
Hence, $R^\epsilon_k\ge0$.
This implies
\[
\langle \mu_k, \ \nabla v - \nabla u_k\rangle \le \|\nabla v\|_{L^1(\Omega)} - \|\nabla u_k\|_{L^1(\Omega)} + R^\epsilon_k \quad \forall v\in W^{1,1}(\Omega).
\]
Hence, $\mu_k$ can be interpreted is an element of the $\varepsilon$-subdifferential to the error level $\varepsilon:=R^\epsilon_k$.
\subsection{Globalized Newton Method for the subproblems}
To solve the variational subproblems of form \eqref{Peps_penal}, i.e.,
\[
\min_{u\in H^1(\Omega)} j_{\epsilon,\rho},
\]
we use a globalized Newton method.
Let us recall the notation
\begin{align*}
&\lambda^a(u):={\max}_{\rho}(\rho(u_a-u)),\\
&\lambda^b(u):={\max}_{\rho}(\rho(u-u_b)).
\end{align*}
and introduce
\[
\Lambda^a(u):=-{\max}_{\rho}'(\rho(u_a-u))=\begin{cases}
-1\quad &\text{if } \rho(u_a-u)>\frac{1}{2\rho},\\
-\rho\left(\rho(u_a-u)+\frac{1}{2\rho}\right)&\text{if } |\rho(u_a-u)|<\frac{1}{2\rho},
\\ 0&\text{otherwise},
\end{cases}
\]
and
$$\Lambda^b(u):={\max}_{\rho}'(\rho(u-u_b))=\begin{cases}
1\quad &\text{if } \rho(u-u_b)>\frac{1}{2\rho},\\
\rho\left(\rho(u-u_b)+\frac{1}{2\rho}\right)&\text{if } |\rho(u-u_b)|<\frac{1}{2\rho}, \\
0&\text{otherwise}.
\end{cases}$$
The Newton method with a line search strategy is given as follows:
\begin{algorithm} [Global Newton method]\label{alg:semi_smoothN}
Set $k=0$, $\rho>0,\:\epsilon\in(0,1)$, $u_a,u_b\in\mathbb{R}$, $\eta >0,\: p>2$, $\phi\in(0,1),\tau\in(0,\frac12)$.
Choose $u_0\in H^1(\Omega)$.
\begin{enumerate}
\item Compute the search direction $w_k$ by solving
\begin{equation} \label{alg_eq_GS}
j''(u_{k})w =-\nabla j(u_k)(u),
\end{equation}
where
\[
\nabla j(u):= -\Div(\psi_{\epsilon}(\nabla u))+\nabla f(u)-{\max}_{\rho}(\rho(u_a-u))+{\max}_{\rho}(\rho(u-u_b))
\]
and
\[
j''(u)d = -\Div\left(\psi_\epsilon''(\nabla u)\nabla d\right)+f''(u)d -\rho\Lambda^a(u) d+\rho\Lambda^b(u) d.
\]
\\
If $\nabla j(u_k)\cdot w_k \le -\eta \|w_k\|^p$: set $w_k:=-\nabla j(u_k)$.
\item (line search) Find $\sigma_k:=\max\{\phi^l:l=0,1,2,...\}$ such that
\[
j(u_k+\sigma_k w_k)-j_k(u_k)\le \tau\sigma_k \nabla J(u_k)\cdot w_{k}.
\]
\item Set $u_{k+1}:=u_k+\sigma w_k$.
\item If a suitable stopping criteria is satisfied: Stop.
\item Set $k:=k+1$ and go to step 1.
\end{enumerate}
\end{algorithm}
Let us provide details regarding the implementation of Algorithm \eqref{alg:semi_smoothN}.
In the initialization of Algorithm \ref{alg:semi_smoothN}, we set $\phi = 0.5$, $\tau = 10^{-4}$, $\eta = 10^{-8}$ and $p=2.1$.
In addition, we employed the following
termination criterion:
\[
\text{If } \|u_{k+1}-u_k\|_{L^2(\Omega)}+\|y_{k+1}-y_k\|_{L^2(\Omega)}+\|p_{k+1}-p_k\|_{L^2(\Omega)} < 10^{-10}: \quad \text{ Stop.}
\]
That is, if there is no sufficient change between consecutive iterates, we assume that the method resulted in a stationary (minimal) point of the subproblem.
\subsection{Example 1: linear elliptic PDE}
First, we consider the optimal control problem
\begin{equation}\label{ex1:cnvprob}
\min\limits_{u\in BV(\Omega)} \frac12\|y-y_d\|^2_{L^2(\Omega)}+\beta\BVSN{u}
\end{equation}
subject to
\[
-\Delta y = u \text{ on } \Omega,\: y=0 \text{ on } \partial\Omega
\]
and the box constraints
\[ u_a\le u(x)\le u_b \quad\text{ f.a.a. } x\in\Omega.
\]
Note that \eqref{ex1:cnvprob} as well as the subproblem
\begin{align}\label{ex1:subprob}
\min\limits_{u\in BV(\Omega)} J_{\epsilon,\rho}(y,u) := \frac12\|y-y_d\|^2_{L^2(\Omega)}&+\beta\int_\Omega\psi_\epsilon(\nabla u)\,\mathrm{d}x\notag\\
&+\int_\Omega\frac 1{\rho}\left(M_\rho(\rho(u_a-u))+M_\rho(\rho( u-u_b))\right)\,\mathrm{d}x\\
\text{s.t. } -\Delta y = u \text{ on } \Omega,&\: y=0 \text{ on } \partial\Omega,\notag
\end{align}
are convex and uniquely solvable.
Let us introduce the adjoint state $p \in H^1_0(\Omega)$ as the solution of the partial differential equation
\[
-\Delta p =y-y_d \text{ on } \Omega,\: y=0 \text{ on } \partial\Omega.
\]
Applying Algorithm \ref{alg:semi_smoothN} to the reduced functional of problem \eqref{ex1:subprob}
results in the following system of equations that has to be solved in each Newton step:
\[
G(y,p,u)(\delta y,\delta p,\delta u) = F(y,p,u),
\]
where $F$ is given by
\[
F(y,p,u) := \begin{pmatrix}
-\Delta y-u \\ -\Delta p -(y-y_d) \\
p-\beta \Div(\psi'_\epsilon(\nabla u))-{\max}_{\rho}(\rho(u_a-u))+{\max}_{\rho}(\rho(u-u_b))
\end{pmatrix}.
\]
The equation $F(y,p,u)=0$
is the optimality system to problem \eqref{ex1:subprob}.
The derivative of $F$ in direction $(\delta y,\delta p,\delta u)\in H^1_0(\Omega)\times H^1_0(\Omega)\times H^1(\Omega)$ is given by
\[
G(y,p,u)(\delta y,\delta p,\delta u) = \begin{pmatrix}
-\Delta\delta y -\delta u \\ -\Delta \delta p -\delta y\\
\delta p-\beta \Div(\psi''_\epsilon(\nabla u))\nabla \delta u -\rho\Lambda^a \delta u+\rho\Lambda^b \delta u.
\end{pmatrix}.
\]
The solution of the Newton step \eqref{alg_eq_GS} is then given by $w:=\delta u$.
We adapt the example problem data from \cite{ClasonKunisch11}.
Here, $\Omega = [-1,1]^2$ and
$$y_d:= \begin{cases}
1\quad&\text{on }(-0.5,0.5)^2\\
0&\text{otherwise}
\end{cases}.$$ In the computations we set $-u_a = u_b = 10$ and $\beta = 0.0001$.
This example (without additional box constraints) was also used in \cite{hafemeyer2020}.
In Table \ref{tab:conv_rate_Ex1}, we see the convergence behavior of iterates.
Here, the errors
$$E_{u}:= \|u_{k}-u_{ref}\|_{L^2(\Omega)}, \quad E_{J}:= |J_k-J_{ref}| $$
are presented, where $u_{ref}$ is the final iterate after the algorithm terminated at step $k=19$
and $J_{ref}:=J(u_{ref})$.
Furthermore, we
observe $R^\epsilon = O(\sqrt{\epsilon})$
and $R^\rho = O(\frac1\rho)$ as $\epsilon \sim 2^{-k}$ and $\rho_k \sim 2^k$.
\begin{table}[htb]
\centering
\begin{tabular}{ccccc}
\hline $k$ & $E_u$ &$E_J$ &$R^\epsilon_k$ & $R^\rho_k$\\
\hline
12 & 1.11 & $6.0\cdot 10^{-4}$ &$ 8.0 \cdot 10^{-3} $ & $1.3\cdot 10^{-9}$ \\
13 & 0.80 & $3.5\cdot 10^{-4}$ & $ 5.9 \cdot 10^{-3} $ & $6.7\cdot 10^{-10}$\\
14 & 0.56 & $1.9\cdot 10^{-4}$ & $ 4.2 \cdot 10^{-3}$& $3.4\cdot 10^{-10}$ \\
15& 0.34 & $1.0\cdot 10^{-4}$ & $ 3.0 \cdot 10^{-3} $ & $1.7\cdot 10^{-10}$ \\
16 & 0.17 & $5.5\cdot 10^{-5}$ & $2.1 \cdot 10^{-3} $ & $8.3\cdot 10^{-11}$\\
17 & 0.07 & $2.4\cdot 10^{-5}$ & $ 1.5 \cdot 10^{-3} $ & $4.2\cdot 10^{-11}$ \\
18 & 0.02 & $8.2\cdot 10^{-6}$ & $ 1.1\cdot 10^{-3} $ & $2.1\cdot 10^{-11}$\\
19 & --- & --- & $ 7.6 \cdot 10^{-4} $ &$1.1\cdot 10^{-11}$ \\
\hline
\end{tabular}
\caption{Computed errors during the final iterations.}
\label{tab:conv_rate_Ex1}
\end{table}
Figure \ref{fig:u_lin} shows the optimal control. The result is in agreement with the results obtained in \cite{hafemeyer2020}.
In Figure \ref{fig:lin_ref} the computed optimal controls are depicted for the unconstrained case (left), i.e., constraints are inactive during the computation process.
The right plot shows the optimal control $u$, when lower and upper bound are set to $u_a =-5$ and $u_b= 18$.
\begin{figure}
\centering
\begin{subfigure}[b]{0.4\textwidth}
\centering
\includegraphics[width=\textwidth]{NT_lin256_2d}
\end{subfigure}
\hfill
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{NT_lin256_3d}
\end{subfigure}
\caption{Optimal control $u$.}%
\label{fig:u_lin}
\end{figure}
\begin{figure}
\centering
\begin{subfigure}[b]{0.4\textwidth}
\centering
\includegraphics[width=\textwidth]{BV_global_minus518}
\end{subfigure}
\hfill
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=\textwidth]{BV_global_uncon}
\end{subfigure}
\caption{Optimal control $u$ for different choices of $u_a,u_b$.}%
\label{fig:lin_ref}
\end{figure}
\subsection{Example 2: Semilinear elliptic optimal control problem}
Let us now consider the following problem with semilinear state equation.
That is, we study the minimization problem
\[
\min_{u\in U_{ad}}f_{sl}(u)+\beta\BVSN{u},
\]
where $f_{sl}$ is given by the standard tracking type functional
$u\mapsto\|y_u-y_d\|^2_{L^2(\Omega)}$,
and $y_u$ is the weak solution of the semilinear elliptic state equation
\[
-\Delta y+ y^3=u \quad\text{in }\Omega,\quad y = 0\quad\text{on }\partial\Omega.
\]
The adjoint state $p\in H^1_0$ is given now as solution to the equation
\[
-\Delta p+3y^2p= y-y_d \quad\text{in }\Omega,\quad p = 0\quad\text{on }\partial\Omega.
\]
For this example, system of equations \eqref{alg_eq_GS} for the state $y\in H^1_0(\Omega)$, the adjoint state $p\in H^1_0(\Omega)$ and the control variable $u\in H^1(\Omega)$ is given by
\[
G(y,p,u)(\delta y,\delta p,\delta u) = F(y,p,u)
\]
with
\[
F(y,p,u) := \begin{pmatrix}
-\Delta y+y^3-u \\
-\Delta p +3y^2p-(y-y_d) \\
p-\beta \Div(\psi'_\epsilon(\nabla u))-{\max}_{\rho}(\rho(u_a-u))+{\max}_{\rho}(\rho(u-u_b))
\end{pmatrix}.
\]
The derivative in direction $(\delta y,\delta p,\delta u)\in H^1_0(\Omega)\times H^1_0(\Omega)\times H^1(\Omega)$ is given with
\[
G(y,p,u)(\delta y,\delta p,\delta u) = \begin{pmatrix}
-\Delta\delta y +3y^2\delta y-\delta u \\
-\Delta \delta p +3y^2\delta p-\delta y+6yp\delta y\\
\delta p -\beta \Div(\psi''_\epsilon(\nabla u))\nabla \delta u -\rho\Lambda^a \delta u+\rho\Lambda^b \delta u.
\end{pmatrix}.
\]
The data is given as in Example 1.
The optimal control is depicted in Figure \ref{fig:u_semlin}. It is close to the solution of Example 1.
Let us consider the performance of the algorithm on different levels of discretization for this example.
Table \ref{table:Ex2_iterations} shows the number of outer iterations ($\sharp$it), as well as the total number of newton iterations ($\sharp$newt) needed until the stopping criterion \eqref{alg: termination} holds for increasing meshsizes.
The last column shows the final objective value
$ J_{\epsilon,\rho}$.
The residuals $R^\epsilon$ and $R^\rho $ behaved as in Example 1.
\begin{table}[htbp]
\centering
\begin{tabular}{cccccc}
\hline
$h$ & $\sharp$it & $\sharp$newt &$\epsilon_{final }$ & $\rho_{final}$ & $J_{\epsilon,\rho}$ \\
\hline
\rule{0pt}{1\normalbaselineskip}
0.088 & 16 & 182 & $ 2^{-16}$& $2^{16}$ & 0.0596 \\
0.044 &19& 201 & $2^{-19} $& $2^{19}$ & 0.0685 \\
0.022 & 19& 314 & $2^{-19} $ &$2^{19}$ & 0.0737 \\
0.011 &19& 486 & $2^{-19} $ & $2^{19}$ & 0.0767\\
\hline
\end{tabular}
\caption{Number of iterations and newton steps for different mesh-sizes.}
\label{table:Ex2_iterations}
\end{table}
\begin{figure}
\centering
\includegraphics[height=6cm]{solution_SL}
\caption{Optimal control $u$ for the semilinear problem.}%
\label{fig:u_semlin}
\end{figure}
\subsection{Experiments with non-constant constraints}
So far our analysis and numerical experiments are restricted to the case where $u_a,u_b$ are constant functions.
This assumption was needed to show the boundedness of multipliers $\lambda_k^a(u),\lambda_k^b(u)$ in $L^2(\Omega)$ in Lemma \ref{lem_lambda_bounded},
which is crucial for the final result Theorem \ref{thm:final}.
For this section we tested Algorithm \ref{alg1} also for non-constant functions $u_a,u_b\in L^\infty(\Omega)$.
Here, we consider again the linear optimal control problem and data from Example 1 with different choices for $u_a,u_b$:
\begin{align}\label{ex1:ub_sin}
(i)\quad &u_a:= -100, \: u_b(x_1,x_2):=8\sin(\pi x_1)\sin(\pi x_2),\\
(ii)\quad \label{ex1:ub_cont2}
&u_a:= -100, \: u_b(x_1,x_2):= -4(x_1-0.5)^2-4x_2^2+10,
\end{align}
In Figure \ref{fig_L2_discretization} the behavior of the quantity
$\|\lambda_k^a(u)\|^2_{L^2(\Omega)}+\|\lambda_k^b(u)\|^2_{L^2(\Omega)}$ is plotted along the iterations, i.e., for increasing $\rho_k$, for different discretization levels.
In Figure \ref{fig:u_nc}, the respective solution plots are shown.
While the multipliers seem to be bounded for one example, their norm grows with $\rho$ (and thus with $\epsilon$) for the other example.
Clearly, more research has to be done to develop necessary and sufficient conditions for the boundedness of the multipliers.
\begin{figure}
\centering
\begin{subfigure}[b]{0.45\textwidth}
\centering
\includegraphics[width=\textwidth]{lambda_nl1_log}
\end{subfigure}
\hfill
\begin{subfigure}[b]{0.45\textwidth}
\centering
\includegraphics[width=\textwidth]{lambda_nl2_log}
\end{subfigure}
\caption{The $L^2$-norm of multipliers $\lambda^a_k(u)$, $\lambda_k^b(u)$ for Example \eqref{ex1:ub_sin} (left) and \eqref{ex1:ub_cont2} (right).}%
\label{fig_L2_discretization}
\end{figure}
\begin{figure}
\centering
\begin{subfigure}[b]{0.45\textwidth}
\centering
\includegraphics[height = 5cm]{sol_TV_sin}
\end{subfigure}
\hfill
\begin{subfigure}[b]{0.45\textwidth}
\centering
\includegraphics[height= 5cm]{sol_TV_cont2}
\end{subfigure}
\caption{Optimal control $u$ for Example \eqref{ex1:ub_sin} (left) and \eqref{ex1:ub_cont2} (right).}%
\label{fig:u_nc}
\end{figure}
\section*{Acknowledgement}
The authors are grateful to Gerd Wachsmuth for an inspiring discussion that led to an improvement of Theorem \ref{thm:final_prelim} and subsequent results.
| {
"timestamp": "2021-10-06T02:14:29",
"yymm": "2110",
"arxiv_id": "2110.01849",
"language": "en",
"url": "https://arxiv.org/abs/2110.01849",
"abstract": "We investigate non-convex optimization problems in $BV(\\Omega)$ with two-sided pointwise inequality constraints. We propose a regularization and penalization method to numerically solve the problem. Under certain conditions, weak limit points of iterates are stationary for the original problem. In addition, we prove optimality conditions for the original problem that contain Lagrange multipliers to the inequality constraints. Numerical experiments confirm the theoretical findings.",
"subjects": "Optimization and Control (math.OC)",
"title": "A penalty scheme to solve constrained non-convex optimization problems in $BV(Ω)$",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9854964217995176,
"lm_q2_score": 0.7185943925708561,
"lm_q1q2_score": 0.7081722026037766
} |
https://arxiv.org/abs/2012.14924 | Cutoff profile of ASEP on a segment | This paper studies the mixing behavior of the Asymmetric Simple Exclusion Process (ASEP) on a segment of length $N$. Our main result is that for particle densities in $(0,1),$ the total-variation cutoff window of ASEP is $N^{1/3}$ and the cutoff profile is $1-F_{\mathrm{GUE}},$ where $F_{\mathrm{GUE}}$ is the Tracy-Widom distribution function. This also gives a new proof of the cutoff itself, shown earlier by Labbé and Lacoin. Our proof combines coupling arguments, the result of Tracy-Widom about fluctuations of ASEP started from the step initial condition, and exact algebraic identities coming from interpreting the multi-species ASEP as a random walk on a Hecke algebra. | \section{Introduction}
We consider ASEP on the segment $[1;N]:=\{1,\ldots,N\}$ with $k\leq N$ particles. This is a continuous time Markov chain with state space
\begin{equation*}
\Omega^{N,k} :=\left\{\xi\in \{0,1\}^{N}:\sum_{i=1}^{N}\xi(i)=k\right\}.
\end{equation*}
We think of the $1's$ as particles, and of the $0's$ as holes. The dynamics of ASEP can be described as follows: Each particle waits an exponential time with parameter $1$, after which with probability $p>1/2$ it attempts to make a unit step to the right, and with probability $q=1-p<1/2$ it attempts to make a unit step to the left. The attempt is succesfull if the target site lies in $[1;N]$ and is occupied by a hole, the hole and the particle exchanging their positions when the particle moves a unit step. If the attempt is not successful, nothing happens.
For $\xi\in \Omega^{N,k},$ we denote by $\xi_{t}$ the state at time $t$ of the ASEP started from $\xi,$ and we denote by $P_{t}^{\xi}$ the law of $\xi_{t}$.
The ASEP dynamics on $[1;N]$ with $k$ particles has a unique stationary measure which we denote by $\pi_{N,k}$.
Recall that the total-variation distance of two probability measures $\mu,\mu^{\prime}$ on $\Omega^{N,k}$ is given by
\begin{equation*}
||\mu-\mu^{\prime}||_{\mathrm{TV}} :=\max_{A \subset \Omega^{N,k}}|\mu(A)-\mu^{\prime}(A)|.
\end{equation*}
We define the maximal total-variation distance between the distribution at given time and the stationary distribution as
\begin{equation*}
d^{N,k}(t) :=\max_{\xi \in \Omega^{N,k}}||P_{t}^{\xi}-\pi_{N,k}||_{\mathrm{TV}},
\end{equation*}
and for $c\in \mathbb{R}, $ we define the time point
\begin{equation}
g(k,c):=\frac{(\sqrt{k}+\sqrt{N-k})^{2}+cN^{1/3}}{p-q}.
\end{equation}
The main result of this paper is the following:
\begin{tthm}\label{main} Assume that $k=k_{N}$ satisfies $\lim_{N\to\infty}k_{N}/N = \alpha$, and $\alpha \in (0,1)$. For any $c\in \mathbb{R}$ we have
\begin{equation*}
\lim_{N\to \infty}d^{N, k_{N}}\left(g(k_{N},c)\right) =1-F_{\mathrm{GUE}}(cf(\alpha)),
\end{equation*}
where $f(\alpha)=\frac{(\alpha(1-\alpha))^{1/6}}{(\sqrt{\alpha}+\sqrt{1-\alpha})^{4/3}}$, and $F_\mathrm{GUE}$ is the $\mathrm{GUE}$ Tracy-Widom distribution defined in \eqref{FGUE2}.
\end{tthm}
\begin{proof}
This is an immediate consequence of Theorem \ref{upper} (which gives an upper bound for the limit on the lefthand side), proven in Section \ref{uppersec}, and Theorem \ref{lower} (which gives a lower bound), proven in Section \ref{lowersec}.
\end{proof}
Theorem \ref{main} gives the cutoff window and the cutoff profile (or shape) of ASEP, we refer to Chapter 18 of the textbook \cite{LPW17} by Levin-Peres (with contributions by Wilmer) for definitions and examples in the general context of Markov chains. A fortiori, Theorem \ref{main} also gives an independent proof of the cutoff itself, which was previously shown by Labb\'{e}-Lacoin in \cite[Theorem 2]{LL19}. On \cite[page 1556]{LL19} the authors mention that the cutoff window for the process is expected to be $N^{1/3}$ and that the cutoff profile is expected to be a function of the $\mathrm{Airy}_{2}$ process. Our Theorem \ref{main} confirms (and gives a precise meaning to) this conjecture.
\subsection{Historic overview}
Detailed information about the relaxation of ergodic Markov chains to equilibrium has been the goal of a vast literature, see e.g. classical works by Aldous \cite{A81}, Diaconis-Shahshahani\cite{DS81}, the review article by Diaconis \cite{D95}, the aforementioned textbook \cite{LPW17}, and references therein. Of particular interest is the so-called \textit{cutoff} phenomenon; a sequence of Markov chains exhibits this phenomenon if the distance between its distribution and the stationary measure abruptly falls from 1 to 0 on a certain time scale. There are different metrics for this distance, leading to different notions of cutoff, see the paper \cite{HLP16} by Hermon-Lacoin-Peres for their differences and similarities. The most commonly used are separation and total-variation cutoff, we study the latter in this paper. Once the cutoff phenomenon is established, it is natural to ask for a more refined information: What happens at the critical time point on a finer scale? The answer to this question is given by the \textit{cutoff window}, which is a finer time scale at which the total-variation distance goes from 1 to 0 \textit{not} abruptly, and a \textit{cutoff profile}, which gives the exact limiting function for the total-variation distance in such a critical scaling. We refer to the article \cite{Tes20} by Teyssier, and the work of Nestoridi-Thomas \cite{NT20} for recent interesting results about cutoff profiles.
Cutoff-type questions were previously investigated for ASEP as well. The first result in this direction was obtained by Diaconis-Ram in \cite{DR04}, where the so called \textit{pre-cutoff} for a discrete time variation of ASEP was proved with the use of representations of Hecke algebra. The pre-cutoff is a claim that there is a unique time scale in which the total-variation changes from 1 to 0, but not necessarily abruptly (again, we refer to \cite{LPW17} for formal definitions). In the work
\cite{BBHM} by Benjamini-Berger-Hoffman-Mossel, the pre-cutoff was shown for the ASEP on a segment, the ansatz of \cite{BBHM} to bound $d^{N,k}$ by studying hitting times is also used in the present work. Whether cutoff holds was an open question for over a decade, until it was proven in \cite{LL19} with the use of hydrodynamics of ASEP on $\mathbb{Z}$ and a careful probabilistic analysis of the system.
It is important to mention that the $p=1,q=0$ case of Theorem \ref{main} can be obtained in a quite simple way from the result of Johansson \cite[Theorem 1.6]{Jo00b}, see \cite[Theorem 1.6]{AHR09} by Angel-Holroyd-Romik. However, it was not clear how one can generalise such type of results to $q \ne 0$. Our proof of Theorem \ref{main} can be viewed as such a generalisation.
Let us also mention that Theorem \ref{main} seems to be the first example when the fluctuation term $N^{1/3},$ which governs the Kardar-Parisi-Zhang universality class (see Corwin's review \cite{Cor11}), appears as a cutoff window in the study of mixing times, and the Tracy-Widom distribution appears as a cutoff profile.
\subsection{Our tools and further questions}
Our proof combines several ingredients. The first one is the result \cite[Theorem 3]{TW08b} by Tracy-Widom about fluctuations of ASEP started from the step initial condition, see Theorem \ref{ASEPthm} below. Our argument is based on a comparison of the ASEP dynamics on a segment with the ASEP dynamics on all integers, and \cite[Theorem 3]{TW08b} is (not surprisingly) the original source of the function $F_\mathrm{GUE}$ in Theorem \ref{main}.
The second ingredient is the use of the multi-species ASEP and its close connection to random walks on Hecke algebra. Using it, we are able to use certain symmetries of Hecke algebra in order to relate the ASEP started from the step initial condition and ASEP started from initial conditions that we are interested in. Somewhat similar ideas were used for TASEP ($q=0$ case) by Borodin-Bufetov in \cite{BB19} and Bufetov-Ferrari in \cite{BF20} for the study of shocks. An important novelty of this paper is the extension of these ideas to ASEP case, which requires the use of \textit{Mallows elements} in Hecke algebra, see Section \ref{sec6} below.
The third ingredient is a variety of probabilistic coupling techniques that are needed throughout the paper for all steps of the argument.
As already mentioned earlier, our approach provides an independent (and rather short) proof of the cutoff for ASEP. Let us mention some further questions where our approach can be of use.
ASEP is arguably the most well-known representative of a fairly large class of integrable stochastic systems in the KPZ universality class. One can study other systems from this class instead. For example, one can study mixing times of the so called q-TASEP on the interval $[1;N]$. It was shown by Bufetov in \cite{Bu20} that a variety of integrable systems can be interpreted as random walks on Hecke algebras. It is possible that our technique can be used to study the mixing times for them as well.
In a different direction, it is important to note that the aforementioned papers \cite{DR04}, \cite{BBHM}, \cite{LL19}, studied the mixing times of the multi-species ASEP (this process can be also referred to as a random Metropolis scan or a biased card shuffling) on a segment as well. As shown in \cite[Section 4.4]{LL19}, on the level of cutoff this question can be reduced to a question about the single-species ASEP. The situation is significantly more delicate for the cutoff profile of the multi-species ASEP. Even for TASEP this question was resolved only very recently by Bufetov-Gorin-Romik in \cite{BGR20}; the obtained cutoff profile is the GOE Tracy-Widom distribution function (this is \textit{not} the $F_\mathrm{GUE}$ function from Theorem \ref{main}). Based on this, we conjecture that the cutoff profile for the multi-species ASEP is the GOE Tracy-Widom distribution function as well (and the cutoff window is $N^{1/3}$). Note that the result of \cite{BGR20} was based on highly nontrivial recent developments (see Borodin-Gorin-Wheeler \cite{BGW19}, Bisi-Cunden-Gibbons-Romik \cite{BCGR20}, Galashin \cite{G20}, Dauvergne \cite{D20} and Bufetov-Korotkikh \cite{BK20}). Parts of the approach from the current paper and from \cite{BGR20} definitely can be of use for this question; nevertheless, this remains an interesting open problem.
We also mention in Remark \ref{rem:further-questions} below further possible directions.
\subsection*{Acknowledgments} We are grateful to anonymous referees for their helpful comments. The work of both authors was partially supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany's Excellence Strategy -- EXC 2047 ``Hausdorff Center for Mathematics''. P. Nejjar is supported by the DFG by the CRC 1060
(Projektnummer 211504053). Data sharing not applicable to this article as no datasets were generated or analysed during the current study.
\section{Preliminaries}
\subsection{Basic Coupling and graphical construction}\label{coupling}
While our main result deals with an ASEP on $[1;N]$ which has only particles and holes, it will be important for us to consider
ASEP on $\mathbb{Z}$ with countably many colors of particles; instead of colors we may speak interchangeably of types, classes, or species of particles.
The state space of this multi-species ASEP is the set of all bijections $\mathbb{Z}\to \mathbb{Z}$ which we denote by $\mathfrak{S},$ for $w\in \mathfrak{S},$ having $w(i)=j$ is interpreted as position $i$ being occupied by the particle with color $j$.
Our convention is that particles with lower color have priority over particles with higher color
Here we briefly give the graphical construction of this multi-species ASEP which goes back to Harris \cite{Har78}. Let $(\mathcal{P}(z),z\in\mathbb{Z})$ be a collection of independent, rate $p$ Poisson processes constructed on some probability space $(\hat{\Omega},\mathcal{A}, \mathbb{P} ).$ By $\mathcal{P}_{t}(z)$ we denote the value of the Poisson process at time $t$.
For fixed $t$, the independence of the Poisson processes implies that for almost every $\omega\in \hat{\Omega}$ there is a sequence $(i_{n},n\in \mathbb{Z})$ of integers such that
\begin{equation*}
\cdots <i_{n-2}< i_{n-1}<i_{n}<i_{n+1}<i_{n+2}<\cdots,
\end{equation*}
and $\mathcal{P}_{t}(i_{n})=0,n\in \mathbb{Z}.$
Given $w \in \mathfrak{S},$ and $z\in \mathbb{Z}$, we denote the swapped configuration
\begin{equation*}
\sigma_{z,z+1}(w)(i)=
\begin{cases}
w(i+1) &\mathrm{for} \, i=z\\
w(i-1) &\mathrm{for} \, i=z+1\\
w(i) &\mathrm{else}.
\end{cases}
\end{equation*}
The dynamics of ASEP is now as follows: We fix a (possibly random) initial configuration $w_{0}\in \mathfrak{S}$ and define the parameter
\begin{equation*}
Q:=\frac{q}{p}\in [0,1).
\end{equation*} When at time $\tau$ the Poisson processes $\mathcal{P}(z)$ has a jump, we update the process as follows: If $w_{\tau^{-}}(z)<w_{\tau^{-}}(z+1)$, then we update $w_{\tau}=\sigma_{z,z+1}(w_{\tau^{-}})$, whereas if
$w_{\tau^{-}}(z)>w_{\tau^{-}}(z+1), $ we toss an independent coin such that with probability $1-Q$, $w_{\tau}=w_{\tau^{-}},$ and with probability $Q$ we have $w_{\tau}=\sigma_{z,z+1}(w_{\tau^{-}})$. Note that to construct the process up to time $t,$ it suffices to apply these update rules inside each of the finite boxes $[i_{n}+1;i_{n+1}],$ and inside each box there are a.s. only finitely many jumps of the Poisson processes (in particular, there is a.s. a well-defined first jump) during the interval $[0,t]$, and no two jumps happen at the same time, hence the graphical construction is well-defined. This construction also allows to obtain the multi-species ASEP on a finite segment $[a;b]$ using finitely many Poisson processes $\mathcal{P}(a),\ldots, \mathcal{P}(b-1).$ This is a process on the set of permutations of $[a;b]$; we denote this set of permutations by $S_{a;b}$.
To recover the ASEP which has only particles and holes, it suffices to fix an integer $k$ and identify all particles whose color lies in $(-\infty,k]$ as particles, and all particles with color in $(k, +\infty)$ as holes, i.e. we map $w(\cdot)\mapsto 1_{(-\infty,k]}(w(\cdot)).$ This defines a map $X^{k}: \mathfrak{S}\to \{0,1\}^{\mathbb{Z}}$ (resp. a map $X_{[a;b]}^{k}: S_{a;b}\to \{0,1\}^{[a;b]}$ for the finite ASEP), the image of which is the ASEP on $\mathbb{Z}$ (resp. $[a;b]$) with particles and holes. In particular for $k\in [1;N]$ we recover the ASEP in $\Omega^{N,k}$ described in the introduction.
Later, we will also consider ASEPs with second class particles; they can be obtained by fixing $k_1, k_2\in \mathbb{Z},k_{1}<k_{2},$ and identifying all particles whose color lies in $(-\infty,k_1]$ as first class particles, all particles whose color lies in $(k_1,k_2]$ as second class particles, and all particles with color in $(k_2,\infty)$ as holes.
The graphical construction allows us to couple different ASEPs together: We use the same collection of Poisson processes to construct ASEPs which start from different initial configurations and/or at different time points. We call this coupling the \textit{basic coupling}, which also allows us to couple ASEPs on a segment $[a;b]$ with ASEPs on $\mathbb{Z}$.
\subsection{Invariant measure for ASEP }\label{secinv}
Define the \textit{Mallows measure} on $S_{a;b}$ as
$$
\mathcal{Q}_{a;b}(w) := Q^{(b-a+1)(b-a)/2 - l \left( w \right)} Z_{a;b},
$$
where $l(w)$ is the number of inversions of $w$ and
\begin{equation}\label{Zab}
Z_{a;b} := \left( \sum_{w \in S_{a;b}} Q^{(b-a+1)(b-a)/2 - l \left( w \right)} \right)^{-1} = \left( \sum_{w \in S_{a;b}} Q^{ l \left( w \right)} \right)^{-1} = \prod_{i=1}^{b-a+1} \frac{1-Q}{1-Q^i}.
\end{equation}
It is immediate that the Mallows measure is invariant for the multi-species ASEP on $[a;b]$ as described in Section \ref{coupling}. Furthermore, under the map $X_{[a;b]}^{k}: S_{a;b}\to \{0,1\}^{[a;b]}$ (with $k\in [a;b]$) from Section \ref{coupling}, the Mallows measure becomes the stationary measure for the ASEP on $[a;b]$ with $k-a+1$ particles and $b-k$ holes which we denote by $\pi_{[a;b],k}$. For the case $[a;b] = [1;N]$ we will use a shortened notation $\pi_{[1;N],k}=:\pi_{N,k}.$
\subsection{Single-species ASEPs}
Given $\xi\in \Omega^{N,k}$, we attach a label (an integer) to each particle from right to left:
Let
\begin{equation*}
x_{k}^{\xi}(0)<\cdots < x_{1}^{\xi}(0)
\end{equation*}
be the initial positions of the $k$ particles of $\xi$. We denote by $x_{i}^{\xi}(t)$ the position at time $t$ of the particle that started in $x_{i}^{\xi}(0)$.
We will later often compare finite ASEPs with positive recurrent ASEPs on $\mathbb{Z}.$ The latter are supported on $\bigcup_{Z\in \mathbb{Z}}\Omega_{Z},$ where for $Z\in \mathbb{Z}$
\begin{equation}\label{omega}
\Omega_{Z}=\left\{\zeta\in \{0,1\}^{\mathbb{Z}}:\sum_{j<Z}\zeta(j)=\sum_{j\geq Z}1-\zeta(j)<\infty\right\}.
\end{equation}
We note that
there is a partial order on $\Omega_{Z}$: For $\zeta^{\prime}, \zeta^{\prime\prime} \in \Omega_{Z}$, we define
\begin{equation}\label{order}
\zeta^{\prime}\preceq\zeta^{\prime\prime}\iff \sum_{j=r}^{\infty}1-\zeta^{\prime\prime }(j)\leq \sum_{j=r}^{\infty}1-\zeta^{\prime}(j)\quad \mathrm{for \, all\,}r\in \mathbb{Z}.
\end{equation}
It is easy to see that under the basic coupling, this order is preserved, i.e. if $\zeta^{\prime}\preceq\zeta^{\prime\prime},$ then also $\zeta^{\prime}_{t}\preceq\zeta^{\prime\prime}_{t},t\geq 0.$
Likewise, we use the same symbol to denote the analogous partial order on $\{0,1\}^{[a;b]}$:
For $\xi^{\prime}, \xi^{\prime\prime}\in \{0,1\}^{[a;b]}$, we define
\begin{equation}\label{order2}
\xi^{\prime}\preceq\xi^{\prime\prime}\iff \sum_{j=r}^{b}1-\xi^{\prime\prime }(j)\leq \sum_{j=r}^{b}1-\xi^{\prime}(j)\quad \mathrm{for \, all\,}r\in [a;b].
\end{equation}
We denote the minimal and maximal element in $ \Omega^{N,k_{N}}$ w.r.t. this order
as $\xi^{0},\xi^1$: \begin{equation}\label{xi01}
\xi^{0}=\mathbf{1}_{[1; k_{N}]}\in \Omega^{N,k_{N}},\quad \xi^{1}=\mathbf{1}_{[N-k_{N}+1; N]}\in \Omega^{N,k_{N}}.
\end{equation}
We define the corresponding elements in $\Omega_{N+1-k_{N}}$ as
\begin{equation}
\label{zeta01}\zeta^{0}=\mathbf{1}_{[1;k_{N}]}+\mathbf{1}_{\mathbb{Z}_{> N}}\in \Omega_{N+1-k_{N}},\quad \zeta^{1}=\mathbf{1}_{\mathbb{Z}_{>(N-k_{N})}}\in \Omega_{N+1-k_{N}}.
\end{equation}
Analogous to the finite case, we label the particles of $\zeta^{0}$ from right to left. We set
$\{i\in \mathbb{Z}: \zeta^{0}(i)=1\}=\{x_{i}^{\zeta^{0}}(0),i\leq k_{N}\}$ with $x_{k_{N}}^{\zeta^{0}}(0)< x_{k_{N}-1}^{\zeta^{0}}(0)<\cdots$ and denote by $x_{i}^{\zeta^{0}}(t)$ the position of the particle of $\zeta^{0}$ that started in $x_{i}^{\zeta^{0}}(0)$.
We also label the holes of $\zeta^{0}, $ but from right to left, i.e. we write
$\{i\in \mathbb{Z}: \zeta^{0}(i)=0\}=\{H_{i}^{\zeta^{0}},i\leq k_{N}\}$ with $H_{k_{N}}(0)>H_{k_{N}-1}(0)>\cdots$, and $H_{i}^{\zeta^{0}}(t)$ is the position at time $t$ of the hole that started in $H_{i}^{\zeta^{0}}(0).$ We label the particles and holes of $\zeta^{1}$ in the same way.
Given a particle configuration $\zeta\in \{0,1\}^{\mathbb{Z}}$
we define the (possibly infinite) position of the leftmost particle and rightmost hole of
$\zeta$ as
\begin{equation}
\begin{aligned}
\label{parthole0}
\mathcal{L}(\zeta)=\inf\{i\in \mathbb{Z}:\zeta(i)=1\}\quad \mathcal{R}(\zeta)=\sup\{i\in \mathbb{Z}:\zeta(i)=0\}.
\end{aligned}
\end{equation}
We define analogously $\mathcal{L}(\xi),\mathcal{R}(\xi)$ for $\xi \in \{0,1\}^{[a;b]}$.
With this notation, we have in particular
\begin{equation}
\begin{aligned}
\label{parthole}
x_{k_{N}}^{\zeta^{0}}(t)=\mathcal{L}(\zeta_{t}^{0})\quad H_{k_{N}}^{\zeta^{0}}(t)=\mathcal{R}(\zeta_{t}^{0}).\end{aligned}
\end{equation}
Finally, throughout the paper, we will often omit writing integer brackets.
\subsection{ASEP with step initial data}\label{stepsec}
Let us introduce the Tracy-Widom $\mathrm{GUE}$ distribution. This probability distribution originates in the theory of random matrices \cite{TW94}, namely it is the limit law of the rescaled largest eigenvalue of
a matrix drawn from the Gaussian Unitary Ensemble (GUE). Its
cumulative distribution function is given by
\begin{equation}\label{FGUE2}
F_{\mathrm{GUE}}(s)=\sum_{n=0}^{\infty} \frac{(-1)^{n}}{n!} \int_{s}^{\infty}\mathrm{d} x_{1}\ldots \int_{s}^{\infty}\mathrm{d} x_{n}\det(K_{2}(x_{i},x_{j})_{1\leq i,j\leq n}), \end{equation}
where $K_{2}(x,y)$ is the Airy kernel $K_{2}(x,y)=\frac{Ai(x)Ai^{\prime}(y)-Ai(y)Ai^{\prime}(x)}{x-y},x\neq y, $ defined for $x=y$ by continuity and $Ai$ is the Airy function.
The ASEP with step initial data is the ASEP on $\mathbb{Z}$ which starts from the initial configuration $x_{m}^{\mathrm{step}}(0)=-m,m> 0$. The following fluctuation result plays an important role in our proof.
\begin{tthm}[ \cite{TW08b}, Theorem 3 ] \label{ASEPthm}
Consider ASEP with step initial data and $p>q$. Let $\gamma=p-q,m>0, \sigma=m/t, c_1=1-2\sqrt{\sigma},c_2 =\sigma^{-1/6}(1-\sqrt{\sigma})^{2/3}$.
Then, uniformly for $\sigma $ in a compact subset of $(0,1),$ we have
\begin{equation}
\lim_{t \to \infty}\mathbb{P}\left(\frac{x_{m}^{\mathrm{step}}(t/\gamma)-c_1 t}{-c_{2}t^{1/3}}\leq s\right)=F_{\mathrm{GUE}}(s).
\end{equation}
\end{tthm}
We will need Theorem \ref{ASEPthm} in the form of the following corollary. Note that
the $c'N^{\kappa},c'' N^{\kappa'}$ terms in the following are irrelevant in the $N\to\infty$ limit as they get absorbed by the $N^{1/3}$ fluctuations.
\begin{cor}\label{cor}
We have for $k_{N}$ with $k_N /N \to \alpha \in (0,1)$ and arbitrary $\kappa,\kappa'\in [0,1/3)$ and $c',c''\in \mathbb{R}$ that
\begin{equation}
\lim_{N \to \infty}\mathbb{P}\left(x_{k_{N}+c'N^{\kappa}}^{\mathrm{step}}(g(k_{N},c))\leq N-2k_{N} +c'' N^{\kappa'}\right)=1-F_{\mathrm{GUE}}(cf(\alpha)),
\end{equation}
where $f(\alpha)=\frac{(\alpha(1-\alpha))^{1/6}}{(\sqrt{\alpha}+\sqrt{1-\alpha})^{4/3}}.$
\end{cor}
\begin{proof}
We set $a=a(N)=k_{N}/N \to \alpha$ and $D=(\sqrt{a}+\sqrt{1-a})^{2}.$ Next we define $\tilde{N}=DN+cN^{1/3}=g(k_{N},c)(p-q) $. Writing everything in terms of $\tilde{N},$ we get (with $\sigma$ as in Theorem \ref{ASEPthm}) that \begin{equation*}\begin{aligned}
&k_{N}+c'N^{\kappa}=\frac{a}{D}\tilde{N}-\frac{ac}{D^{4/3}}\tilde{N}^{1/3}+o(\tilde{N}^{1/3})
\\&\sigma= (k_{N}+c'N^{\kappa})/\tilde{N}=\frac{a}{D}-\frac{ac}{D^{4/3}}\tilde{N}^{-2/3}+o(\tilde{N}^{-2/3})
\\&N-2k_{N}+c''N^{\kappa'}=(1-2\sqrt{\sigma})\tilde{N}-c\tilde{N}^{1/3}\left(\frac{(1-2a)}{D^{4/3}}+\frac{\sqrt{a}}{D^{5/6}}\right)+o(\tilde{N}^{1/3}).
\end{aligned}\end{equation*}
We may thus rewrite
\begin{equation*}
\begin{aligned}
&\mathbb{P}\left(x_{k_{N}+c'N^{\kappa}}^{\mathrm{step}}(g(k_{N},c))\leq N-2k_{N} +c'' N^{\kappa'}\right)
\\&=1-\mathbb{P}\left(x_{k_{N}+c'N^{\kappa}}^{\mathrm{step}}(g(k_{N},c))>N-2k_{N} +c'' N^{\kappa'}\right)
\\&=1-\mathbb{P}\left(\frac{x_{\frac{a}{D}\tilde{N}-\frac{ac}{D^{4/3}}\tilde{N}^{1/3}+o(\tilde{N}^{1/3})}(\tilde{N}/(p-q))-(1-2\sqrt{\sigma})\tilde{N}}{-\tilde{N}^{1/3}}< c\left(\frac{(1-2a)}{D^{4/3}}+\frac{\sqrt{a}}{D^{5/6}}\right)+o(1)\right).
\end{aligned}
\end{equation*}
In order to apply Theorem \ref{ASEPthm} with $\tilde{N}=t,$ we still have to divide by $\sigma^{-1/6}(1-\sqrt{\sigma})^{2/3}$.
An elementary computation reveals
\begin{equation*}
\frac{\left(\frac{(1-2a)}{D^{4/3}}+\frac{\sqrt{a}}{D^{5/6}}\right)}{\sigma^{-1/6}(1-\sqrt{\sigma})^{2/3}}=\frac{(a(1-a))^{1/6}}{D^{2/3}}+o(1)\to_{N\to\infty}\frac{(\alpha(1-\alpha))^{1/6}}{(\sqrt{\alpha}+\sqrt{1-\alpha})^{4/3}}= f(\alpha).
\end{equation*}
Using Theorem \ref{ASEPthm} thus yields
\begin{align*}
&\lim_{\tilde{N}\to \infty}\mathbb{P}\left(\frac{x_{\frac{a}{D}\tilde{N}-\frac{ac}{D^{4/3}}\tilde{N}^{1/3}+o(\tilde{N}^{1/3})}(\tilde{N}/(p-q))-(1-2\sqrt{\sigma})\tilde{N}}{-\sigma^{-1/6}(1-\sqrt{\sigma})^{2/3}\tilde{N}^{1/3}}< c\frac{\left(\frac{(1-2a)}{D^{4/3}}+\frac{\sqrt{a}}{D^{5/6}}\right)}{\sigma^{-1/6}(1-\sqrt{\sigma})^{2/3}}+o(1)\right)
\\&=F_{\mathrm{GUE}}(cf(\alpha)),
\end{align*}
finishing the proof.
\end{proof}
\section{Upper bound}\label{uppersec}
The aim of this section is to show that $1-F_{\mathrm{GUE}}(cf(\alpha))$ is an upper bound for the cutoff profile. More precisely, we will show the following Theorem, see Figure \ref{Graph} for an illustration of the proof idea.
\begin{tthm}\label{upper}Let $k=k_{N}$ with $k_N /N \to \alpha \in (0,1)$. Then
we have for $c\in \mathbb{R}$
\begin{equation*}
\limsup_{N\to \infty}d^{N,k_{N}}\left(g(k_{N},c)\right) \leq 1-F_{\mathrm{GUE}}(cf(\alpha)).
\end{equation*}
\end{tthm}
The starting point for showing Theorem \ref{upper} is the following Theorem \ref{alexey}.
The proof of Theorem \ref{alexey} exploits the link between multi-species ASEP and Hecke algebras, and is postponed to Section \ref{sec6}.
\begin{tthm}\label{alexey}
Let $\mathcal{L}(\zeta^{0}_{t}),\mathcal{R}(\zeta^{0}_{t})$ be as in \eqref{parthole}.
Define for $c\in \mathbb{R}$
\begin{equation*}
\begin{aligned}
B_{N}(c)=&\{ \mathcal{L}(\zeta^{0}_{g(k_{N},c)})> N-k_N -N^{1/10}\}
\\&\cap \{ \mathcal{R}(\zeta^{0}_{g(k_{N},c)})\leq N-k_N+N^{1/10} \}.
\end{aligned}
\end{equation*}
Then
\begin{equation*}
\lim_{N\to\infty}\mathbb{P}(B_{N}(c))=F_{\mathrm{GUE}}(c f(\alpha)).
\end{equation*}
\end{tthm}
We note that the $N^{1/10}$ term in the definition of $B_{N}(c)$ is there merely for concreteness, any term that goes to $+\infty$ with $N$ and is $o(N^{1/3})$ would do.
To get an upper bound for $d^{N,k_N }(t),$ we consider the hitting time
\begin{equation}\label{HHH}
\mathfrak{H}=\inf\{t\geq 0: \zeta^{0}_{t}=\zeta^{1}\}.
\end{equation}
\begin{figure}\begin{center}
\begin{tikzpicture}[scale=2]
\draw[thick, ->] (-1,0) -- (5.7,0)
\draw (2.5,0) node[below] {$N-k_N$};
\filldraw (1.1,0.2) circle (0.1);
\draw (1.4,0) node[below] {$\mathcal{L}(\zeta^{0}_{g(k_{N},c)})$};
\draw (3.3,0.2) circle (0.1);
\draw (3.6,0) node[below] {$\mathcal{R}(\zeta^{0}_{g(k_{N},c)})$};
\draw (4.9,0) node[below] {$N-k_N+N^{\frac{1}{10}}$};
\draw (0.1,0) node[below] {$N-k_N-N^{\frac{1}{10}}$};
\foreach \x in {2.5,3.3,1.1,4.5,0.3}
\draw[thick] (\x,0.075)--(\x,-0.075);
\foreach \x in {3.6,3.85,4.1,4.35,4.6,4.85,5.1,5.35,5.6}
\filldraw (\x,0.2) circle (0.1);
\foreach \x in {0.85,0.6,0.35,0.1,-0.15,-0.4,-0.65,-0.9}
\draw (\x,0.2) circle (0.1);
\end{tikzpicture}\end{center}
\caption{The particle configuration $\zeta_{g(k_{N},c)}^{0}$ on the event $B_{N}(c)$ from Theorem \ref{alexey}: Black balls indicate the presence of a particle, white balls of a hole, blank space may be occupied by holes or particles. On $B_{N}(c),$ the leftmost particle $\mathcal{L}(\zeta^{0}_{g(k_{N},c)})$ and the rightmost hole $\mathcal{R}(\zeta^{0}_{g(k_{N},c)})$ of $\zeta^{0}_{g(k_{N},c)}$ lie in $[N-k_{N}-N^{1/10};N-k_{N}+N^{1/10}].$ If $B_{N}(c)$ happens, the hitting time $\mathfrak{H}$ from \eqref{HHH} cannot be much larger than $g(k_{N},c)$, see Proposition \ref{0}. As $\mathfrak{H}$
induces an upper bound for $d^{N,k_N}(g(k_{N},c))$ via Proposition \ref{hineq2}, this implies that $d^{N,k_N}(g(k_{N},c))$ is asymptotically bounded from above by $1-\mathbb{P}(B_{N}(c))$, which equals $1-F_{\mathrm{GUE}}(cf(\alpha))$ asymptotically.
}
\label{Graph}
\end{figure}
The link between this hitting time and the maximal total-variation distance is as follows.
\begin{prop}\label{hineq2}
We have
\begin{equation}\label{ineqcoupl}
d^{N,k_N }(t)\leq \mathbb{P}(\mathfrak{H}> t).
\end{equation}
\end{prop}
\begin{proof}
Let $\xi, \xi^{\prime}\in \Omega^{N,k}$ and let the ASEPs $(\xi_{t},\xi^{\prime}_{t},t\geq 0)$ be coupled via the basic coupling. We define the coalescence time
\begin{equation}
\tau^{\xi,\xi^{\prime}}=\inf\{t: \xi_{t}=\xi^{\prime}_{t}\}.
\end{equation}
Then we have the general inequality (see \cite[Corollary 5.5]{LPW17})
\begin{equation}\label{5.5}
d^{N,k_{N}}(t)\leq \max_{\xi,\xi^{\prime}\in \Omega^{N,k_{N}}}\mathbb{P}(\tau^{\xi,\xi^{\prime}}>t).
\end{equation}
Define furthermore the hitting time
\begin{equation}
\mathfrak{h}=\inf\{t:\xi^{0}_{t}=\xi^{1}\}.
\end{equation}
Assuming all appearing ASEPs are coupled via the basic coupling, we prove the inequality
\begin{equation}\label{hineq}
\max_{\xi,\xi^{\prime}\in \Omega^{N,k_{N}}}\tau^{\xi,\xi^{\prime}}\leq \mathfrak{h}.
\end{equation}
For this, recall the partial order \eqref{order2}. We then have
\begin{equation}
\xi^{1}=\xi^{0}_{\mathfrak{h}}\preceq \xi_{\mathfrak{h}}\preceq \xi^{1}
\end{equation}
i.e. $\xi_{\mathfrak{h}}=\xi^{1}$ and likewise $\xi_{\mathfrak{h}}^{\prime}=\xi^{1}$ so that $\xi_{\mathfrak{h}}=\xi_{\mathfrak{h}}^{\prime}$ and $\tau^{\xi,\xi^{\prime}}\leq \mathfrak{h}$.
To proceed, we use the basic coupling of ASEPs on $[1;N]$ with ASEPs on $\mathbb{Z}$ described at the end of Section \ref{coupling}.
We will show that
\begin{equation}\label{hH}
\mathfrak{h}\leq \mathfrak{H}.
\end{equation}
To see this, we show the following random time is infinite: Let
\begin{equation*}
\mathcal{T}=\inf\{t: \mathrm{\, there\, is\, an \,}i^{*}\in\{1,\ldots,k_{N}\}\mathrm{\,such \,that\,} x^{\zeta^{0}}_{i^*}(t)>x^{\xi^{0}}_{i^*}(t)\}.
\end{equation*}
Note that $x^{\zeta^{0}}_{i}(0)=x^{\xi^{0}}_{i}(0),i=1,\ldots,k_{N}.$ Thus, to have an $i^{*}$ with $x^{\zeta^{0}}_{i^*}(t)>x^{\xi^{0}}_{i^*}(t),$ one of the Poisson processes $(\mathcal{P}(z),z\in[1;N])$ of the graphical construction must have made a jump during $[0,t].$ In particular, $\mathcal{T}>0$ almost surely, we can thus consider the left limit $\mathcal{T}^{-}.$ At time $\mathcal{T}$, there is exactly one $i^{*}$ with
$x^{\zeta^{0}}_{i^*}(\mathcal{T})>x^{\xi^{0}}_{i^*}(\mathcal{T}),$ having more than one $i^{*}$ would require two jumps to happen at the same time. Furthermore, we always have
\begin{equation}
x^{\xi^{0}}_{i^*}(\mathcal{T}^{-})=x^{\zeta^{0}}_{i^*}(\mathcal{T}^{-}), \quad x^{\xi^{0}}_{i}(\mathcal{T}^{-})\geq x^{\zeta^{0}}_{i}(\mathcal{T}^{-}), i=1,\ldots,k_{N}
\end{equation}
We now distinguish two possibilities : The first possibility to have $x^{\zeta^{0}}_{i^*}(\mathcal{T})>x^{\xi^{0}}_{i^*}(\mathcal{T})$ is that at time $\mathcal{T},$ $x^{\zeta^{0}}_{i^*}$ makes a jump to the right that $x^{\xi^{0}}_{i^*}$ does not make. One way for $x^{\xi^{0}}_{i^*}$ not to make a jump to the right is if $x^{\xi^{0}}_{i^*}(\mathcal{T}^{-})=N.$ But then also $x^{\zeta^{0}}_{i^*}(\mathcal{T}^{-})=N,$ and $x^{\zeta^{0}}_{i^*}$ cannot jump either. The other way for
$x^{\xi^{0}}_{i^*}$ not to make a jump to the right is if it is blocked by the particle $x^{\xi^{0}}_{i^* -1}$, i.e. if
\begin{equation*}
x^{\xi^{0}}_{i^* -1}(\mathcal{T}^{-})=x^{\xi^{0}}_{i^* }(\mathcal{T}^{-})+1.
\end{equation*}
However, since \begin{equation*}x^{\xi^{0}}_{i^* -1}(\mathcal{T}^{-})\geq x^{\zeta^{0}}_{i^* -1}(\mathcal{T}^{-})> x^{\zeta^{0}}_{i^* }(\mathcal{T}^{-})=x^{\xi^{0}}_{i^* }(\mathcal{T}^{-}),\end{equation*}
this implies that
\begin{equation*}
x^{\zeta^{0}}_{i^* -1}(\mathcal{T}^{-})=x^{\zeta^{0}}_{i^* }(\mathcal{T}^{-})+1,
\end{equation*}
showing that $x^{\zeta^{0}}_{i^*}$ cannot jump to the right at time $\mathcal{T}$ either.
The second possibility to have $x^{\zeta^{0}}_{i^*}(\mathcal{T})>x^{\xi^{0}}_{i^*}(\mathcal{T})$ is that at time $\mathcal{T},$ $x^{\xi^{0}}_{i^*}$ makes a jump to the left that $x^{\zeta^{0}}_{i^*}$ does not make. The only way for this to happen however is that $x^{\zeta^{0}}_{i^* +1}$ blocks the left jump of $x^{\zeta^{0}}_{i^*}$, i.e.
\begin{equation}
x^{\zeta^{0}}_{i^* +1}(\mathcal{T}^{-})=x^{\zeta^{0}}_{i^* }(\mathcal{T}^{-})-1.\end{equation}
But since $x^{\xi^{0}}_{i^* +1}(\mathcal{T}^{-})\geq x^{\zeta^{0}}_{i^* +1}(\mathcal{T}^{-}),$ this implies that
\begin{equation}x^{\xi^{0}}_{i^* +1}(\mathcal{T}^{-})=x^{\xi^{0}}_{i^* }(\mathcal{T}^{-})-1,
\end{equation}
and therefore $x^{\xi^{0}}_{i^*}$ cannot jump to the left at time $\mathcal{T}$ either.
In total, we thus have $\mathbb{P}(\mathcal{T}=\infty)=1$. Since clearly
\begin{equation}
x_{i}^{\zeta^{0}}(\mathfrak{H})=N+1-i,\quad i=1,\ldots,k_{N},
\end{equation}
we can thus conclude
\begin{equation}
x_{i}^{\xi^{0}}(\mathfrak{H})=N+1-i,\quad i=1,\ldots,k_{N},
\end{equation}
and therefore \eqref{hH} holds. Combining \eqref{5.5}, \eqref{hineq} and \eqref{hH} finishes the proof.
\end{proof}
The key observation now is that if the event $B_{N}(c)$ from Theorem \ref{alexey} happens, the hitting time $\mathfrak{H}$ cannot be much larger than $g(k_{N},c)$.
In order to show this, we will use the following result from \cite{BBHM}, which in fact is much stronger than what we will be needing.
\begin{tthm}[Theorem 1.9 of \cite{BBHM}]\label{BBHMthm}For $M\in \mathbb{Z}_{\geq 1}$, let $I^M=\mathbf{1}_{[-M;-1]}+\mathbf{1}_{\mathbb{Z}_{\geq M}},$ and $(I^{M}_{t},t\geq 0)$ be the ASEP started from $I^{M}$. Define the hitting time $\mathfrak{H}_{M}=\inf\{t\geq 0: I^{M}_{t}=\mathbf{1}_{\mathbb{Z}_{\geq 0}}\}$. Then for every $\delta>0$ there is a constant $D=D(p,\delta)$ such that
\begin{equation}\label{26}
\mathbb{P}( \mathfrak{H}_{M} \geq DM)<\frac{\delta}{M}.
\end{equation}
\end{tthm}
Now we can show that on $B_{N}(c)$, we have good control over $\mathfrak{H}$.
\begin{prop}\label{0} We have
\begin{equation}
\lim_{N\to\infty}\mathbb{P}(\{\mathfrak{H}\geq g(k_{N},c)+N^{1/5}\}\cap B_{N}(c))=0.
\end{equation}
\end{prop}
\begin{proof}We define with $\widetilde{N}:=N-k_N+N^{1/10}+1$ the configuration
\begin{equation}
\eta (j)= \mathbf{1}_{[N-k_N -N^{1/10}+1; N-k_N]}+\mathbf{1}_{\mathbb{Z}_{\geq \widetilde{N}}}.
\end{equation}
We start now \textit{at time} $g(k_{N},c)$ an ASEP from $\eta$ and couple this ASEP to all other appearing ASEPs via the basic coupling.
To make this clear in our notation, we write
\begin{equation}
\hat{\eta}_{g(k_{N},c)}:=\eta
\end{equation}
and denote $(\hat{\eta}_{\ell},\ell\geq g(k_{N},c))$ the ASEP which starts at time
$g(k_{N},c)$ from $\eta$, so that $\hat{\eta}_{t}$ for $t\geq g(k_{N},c)$ has the same law as $\eta_{t- g(k_{N},c)}.$ We define the corresponding hitting time
\begin{equation}
\mathfrak{H}^{\eta}=\inf\{t\geq g(k_{N},c): \hat{\eta}_{t}=\zeta^{1}\}.
\end{equation}
With $I^{N^{\frac{1}{10}}},\mathfrak{H}_{N^{\frac{1}{10}}}$ defined in Theorem \ref{BBHMthm}, we have $\eta(j+N-k_N+1)=I^{N^{\frac{1}{10}}}(j),j\in \mathbb{Z},$ and thus in particular,
$\mathfrak{H}^{\eta}$ has the same law as $\mathfrak{H}_{N^{\frac{1}{10}}}+g(k_{N},c)$.
It is thus an immediate corollary of \eqref{26} that for every $\varepsilon>0$ we have
\begin{equation}\label{22}
\lim_{N\to \infty}\mathbb{P}(\mathfrak{H}^{\eta}\geq g(k_{N},c)+N^{\frac{1}{10}+\varepsilon})=0.
\end{equation}
It is easy to see that we have the inclusion
\begin{equation}
B_{N}(c)\subseteq \{\zeta^{0}_{g(k_N,c)}\succeq\hat{\eta}_{g(k_{N},c)}\}.
\end{equation}
Since the partial order $\succeq$ is preserved under the basic coupling, we thus have
\begin{equation}
B_{N}(c)\subseteq \{\mathfrak{H}^{\eta}\geq \mathfrak{H}\}.
\end{equation}
Taking $\varepsilon >0$ such that $1/10+\varepsilon<1/5$, we can thus conclude from \eqref{22} that
\begin{equation}
\begin{aligned}
& \lim_{N\to\infty}\mathbb{P}(\{\mathfrak{H}\geq g(k_{N},c)+N^{1/5}\}\cap B_{N}(c))
\\&\leq \lim_{N\to\infty}\mathbb{P}(\{\mathfrak{H}^{\eta}\geq g(k_{N},c)+N^{1/5}\})=0.
\end{aligned}
\end{equation}
\end{proof}
We can now prove Theorem \ref{upper}.
\begin{proof}[Proof of Theorem \ref{upper}]
Let $\varepsilon>0$ and let $N$ be sufficiently large so that \begin{equation*}g(k_N,c-\varepsilon)+N^{1/5}<g(k_{N},c).\end{equation*} Then, using \eqref{ineqcoupl}, we get
\begin{align*}
d^{N,k_N}\left(g(k_{N},c)\right) &\leq\mathbb{P}(\{\mathfrak{H}\geq g(k_N,c-\varepsilon)+N^{1/5}\}\cap B_{N}(c-\varepsilon))\\&+1-\mathbb{P}(B_{N}(c-\varepsilon)).\end{align*}
Combining Theorem \ref{alexey} with Proposition \ref{0} we get
\begin{align*}
\limsup_{N\to \infty}d^{N,k_N}\left(g(k_{N},c)\right) \leq \lim_{N\to \infty}1-\mathbb{P}(B_{N}(c-\varepsilon))=1-F_{\mathrm{GUE}}((c-\varepsilon)f(\alpha)).\end{align*}
Since $\varepsilon $ is arbitrary, Theorem \ref{upper} follows.
\end{proof}
\section{Lower bound}\label{lowersec}
In this section we prove that the upper bound obtained in the previous section is also a lower bound. This is the content of the following Theorem.
\begin{tthm}\label{lower}Let $k=k_{N}$ with $k_N /N \to \alpha \in (0,1)$. Then
we have for $c\in \mathbb{R}$
\begin{equation}
\liminf_{N\to \infty}d^{N,k_{N}}\left(g(k_{N},c)\right) \geq 1-F_{\mathrm{GUE}}(cf(\alpha)),
\end{equation}
where $f(\alpha)=\frac{(\alpha(1-\alpha))^{1/6}}{(\sqrt{\alpha}+\sqrt{1-\alpha})^{4/3}}.$
\end{tthm}
The main tool to prove Theorem \ref{lower} is to compare the finite ASEP $(\xi_{t}^{0},t\geq 0)$ (defined in \eqref{xi01}) with the infinite ASEP started from the step initial data from Section \ref{stepsec}.
We will consider the shifted step initial data on $\mathbb{Z}$ given by
\begin{equation}\label{shift}
x_{n}^{\mathrm{step}(k_{N})}(0)=k_{N}+1-n, n\geq 1.
\end{equation}
We start by noting that under the basic coupling, the leftmost particle of $\xi^{0}_{t}$ is never to the right of $ x_{k_{N}}^{\mathrm{step}(k_{N})}(t)$.
\begin{prop}\label{3.1}
Consider the basic coupling of the finite ASEP $(\xi^{0}_{t},t\geq 0)$ and the infinite
ASEP started from the shifted step initial data \eqref{shift}. Then we have
\begin{equation}
x_{k_{N}}^{\xi^{0}}(t)\leq x_{k_{N}}^{\mathrm{step}(k_{N})}(t), \quad t \geq 0.
\end{equation}
\end{prop}
\begin{proof}
This is quite similar to the proof of the inequality \eqref{hH}, we will thus not repeat all the details. We show $\mathbb{P}(\widehat{\mathcal{T}}=\infty)=1,$ where $\widehat{\mathcal{T}}$ is the random time
\begin{equation*}
\widehat{\mathcal{T}}=\inf\{t: \mathrm{\, there\, is\, an \,}i^{*}\in\{1,\ldots,k_{N}\}\mathrm{\,such \,that\,} x^{\mathrm{step}(k_{N})}_{i^*}(t)<x^{\xi^{0}}_{i^*}(t)\}.
\end{equation*}
We again distinguish two possibilities : The first possibility to have $x^{\mathrm{step}(k_{N})}_{i^*}(\widehat{\mathcal{T}})<x^{\xi^{0}}_{i^*}(\widehat{\mathcal{T}})$ is that at time $\widehat{\mathcal{T}},$ $x^{\xi^{0}}_{i^*}$ makes a jump to the right that $x^{\mathrm{step}(k_{N})}_{i^*}$ does not make. This implies that
\begin{equation}
x^{\mathrm{step}(k_{N})}_{i^* -1}(\widehat{\mathcal{T}}^{-})=x^{\mathrm{step}(k_{N})}_{i^* }(\widehat{\mathcal{T}}^{-})+1,
\end{equation}
which however implies that
\begin{equation}
x^{\xi^{0}}_{i^* -1}(\widehat{\mathcal{T}}^{-})=x^{\xi^{0}}_{i^* }(\widehat{\mathcal{T}}^{-})+1
\end{equation}
also, meaning $x^{\xi^{0}}_{i^* }$ cannot jump to the right at time
$\widehat{\mathcal{T}}$ either.
The other possibility is that at time $\widehat{\mathcal{T}}$, $x^{\mathrm{step}(k_{N})}_{i^*}$ makes a jump to the left that $x^{\xi^{0}}_{i^*}$ does not make. For $x^{\xi^{0}}_{i^*}$ not to make a jump to the left, we must have either
$x^{\xi^{0}}_{i^*}(\widehat{\mathcal{T}}^{-})=1,$ or we have
\begin{equation}\label{40}
x^{\xi^{0}}_{i^* +1}(\widehat{\mathcal{T}}^{-})=x^{\xi^{0}}_{i^* }(\widehat{\mathcal{T}}^{-})-1.
\end{equation}
If $x^{\xi^{0}}_{i^*}(\widehat{\mathcal{T}}^{-})=1,$ we have $i^{*}=k_{N},$ and $x^{\mathrm{step}(k_{N})}_{k_{N}}(\widehat{\mathcal{T}}^{-})=1,$ so that $x^{\mathrm{step}(k_{N})}_{k_{N}}$ also cannot jump to the left at time $\widehat{\mathcal{T}},$ since $x^{\mathrm{step}(k_{N})}_{k_{N}}$ can never jump from $1$ to $0.$
If instead \eqref{40} holds, we also have
\begin{equation}\label{41}
x^{\mathrm{step}(k_{N})}_{i^* +1}(\widehat{\mathcal{T}}^{-})=x^{\mathrm{step}(k_{N})}_{i^* }(\widehat{\mathcal{T}}^{-})-1,
\end{equation}
implying that $x^{\mathrm{step}(k_{N})}_{i^* }$ cannot jump to the left at time $\widehat{\mathcal{T}}$ either.
\end{proof}
Define for $l\in [1;N-k-1]$ the event
\begin{equation}
A_{N}(l)=\left\{\xi\in \Omega^{N,k}:\sum_{i=N-k-l}^{N}\xi(i)\leq k-1\right\},
\end{equation}
so that
\begin{equation*}
\xi_{t} \in A_{N}(l) \iff \mathcal{L}(\xi_t) <N-k-l.
\end{equation*}
Recall that $\pi_{N,k}$ is the stationary measure of ASEP in $\Omega^{N,k}$. The next proposition shows in particular that $\pi_{N,k}$ gives vanishing mass to the event $A_{N}(l) $ when $l=l(N)$ goes to $+\infty$ with $N$.
\begin{prop}\label{3.2}
There are constants $C_1,C_2>0$ which depend on $p$ but not on $N,k_N$ such that we have $\pi_{N,k_{N}}(A_{N}(l))\leq C_{1}e^{-C_{2}l}.$
\end{prop}
\begin{proof}
We consider the basic coupling and define the random time
\begin{equation*}
\widetilde{\mathcal{T}}=\inf\{t: \mathrm{\, there\, is\, an \,}i^{*}\in\{1,\ldots,k_{N}\}\mathrm{\,such \,that\,} x^{\xi^{1}}_{i^*}(t)< x^{\zeta^{1}}_{i^*}(t)\}.
\end{equation*}
With a proof that is very similar to the proof of \eqref{hH} and Proposition \ref{3.1}, we show that $\mathbb{P}(\widetilde{\mathcal{T}}=\infty)=1.$
Thus in particular $\mathcal{L}(\xi^{1}_t)\geq \mathcal{L}(\zeta^{1}_t)$ holds for all $t\geq 0,$ and hence
\begin{align*}
\pi_{N,k_{N}}(A_{N}(l))&=\lim_{t\to\infty}\mathbb{P}(\mathcal{L}(\xi^{1}_t)<N-k_{N}-l)\\&\leq \lim_{t\to\infty}\mathbb{P}( \mathcal{L}(\zeta^{1}_t)<N-k_{N}-l).
\end{align*}
Finally, we apply \cite[Proposition 3.1]{N20CMP} which shows that
\begin{align*}
\lim_{t\to\infty}\mathbb{P}( \mathcal{L}(\zeta^{1}_t)<N-k_{N}-l)\leq C_{1}e^{-C_{2}l},
\end{align*}
finishing the proof.
\end{proof}
Now we can prove Theorem \ref{lower}.
\begin{proof}[Proof of Theorem \ref{lower}]
Recall $P_{t}^{\xi}$ is the law of the ASEP started from $\xi$ at time $t$. We can by definition bound
\begin{equation}
d^{N,k_{N}}(g(k_{N},c))\geq P_{g(k_{N},c)}^{\xi^{0}}(A_{N}(N^{1/4}))-\pi_{N,k_{N}}(A_N (N^{1/4})).
\end{equation}
By Proposition \ref{3.2}, $\lim_{N\to \infty}\pi_{N,k_{N}}(A_N (N^{1/4}))=0.$
Combining Proposition \ref{3.1} with Corollary \ref{cor} yields
\begin{equation}
\begin{aligned}
\liminf_{N\to\infty} P_{g(k_{N},c)}^{\xi^{0}}(A_{N}(N^{1/4}))&=\liminf_{N\to\infty}\mathbb{P}( x_{k_{N}}^{\xi^{0}}(g(k_{N},c))<N-k_{N}-N^{1/4})
\\&\geq\lim_{N\to\infty}\mathbb{P}( x_{k_{N}}^{\mathrm{step}(k_{N})}(g(k_{N},c))<N-k_{N}-N^{1/4})
\\&=1-F_{\mathrm{GUE}}(cf(\alpha)),
\end{aligned}
\end{equation}
finishing the proof.
\end{proof}
\section{Proof of Theorem \ref{alexey}}\label{sec6}
In this section we prove Theorem \ref{alexey} via a certain distribution identity coming from viewing the multi-species ASEP as a random walk on a Hecke algebra. We start by introducing the Hecke algebra and other necessary notions in Sections \ref{subsec:rand-walkHA} -- \ref{Qeq}. Then in Section \ref{plan} we explain the idea of the proof. In the remaining sections we give a formal proof.
\subsection{Random walk on Hecke algebra}
\label{subsec:rand-walkHA}
Let $S_n$ be the symmetric group of permutations of $n$ elements. For each permutation $w \in S_n,$ recall that we denote by $l(w)$ the number of inversions in it. Let $\mathfrak{S}_n$ be the set of all nearest neighbor transpositions from $S_n$.
We will fix the parameter (introduced earlier in Section \ref{coupling})
\begin{equation*}
Q:=\frac{q}{p}\in [0,1).
\end{equation*}
A \textit{Hecke algebra} $\mathcal H (S_n)$ is the algebra with a linear basis $\{ T_w \}_{w \in S_n}$ and the multiplication which satisfies the following rules for any $s \in \mathfrak{S}_n$, $w \in S_n$:
\begin{equation}
\label{eq:HeckeRules}
\begin{cases}
T_s T_w = T_{sw}, \qquad & \mbox{if $l(sw)=l(w)+1$} \\
T_s T_w = (1-Q) T_w + Q T_{sw}, \qquad & \mbox{if $l(sw)=l(w)-1$}.
\end{cases}
\end{equation}
It is clear that such rules can be used for a computation of the product $T_{w_1} T_{w_2}$ for any $w_1, w_2 \in S_n$; a non-trivial (but very well-known) part is that the rules are consistent and indeed define an (associative) multiplication.
Let $\mathfrak i: \mathcal H (S_n) \to \mathcal H (S_n)$ be a linear map such that $\mathfrak i \left( T_w \right) = T_{w^{-1}}$. The following proposition is well-known (and can be straightforwardly proved by induction in $l(w)$ with the use of \eqref{eq:HeckeRules}).
\begin{prop}
\label{prop:CPsymmetry}
The map $\mathfrak i$ is an involutive anti-homomorphism. In more detail, for any $T_1, \dots T_r \in \mathcal H (S_n)$ we have
\begin{equation*}
\mathfrak i \left( T_r T_{r-1} \dots T_2 T_1 \right) = \mathfrak i \left( T_1 \right) \mathfrak i \left( T_2 \right) \dots \mathfrak i \left( T_{r-1} \right) \mathfrak i \left( T_r \right),
\end{equation*}
and also, trivially, $\mathfrak i^2 \left( T_1 \right) = T_1$.
\end{prop}
For $a,b \in \mathbb{Z}$, $a<b$, recall that $S_{a;b}$ is the group of permutations of the set $[a;b]$. We have the natural embedding $S_{a_2;b_2} \subset S_{a_1;b_1}$ for any $a_1 \le a_2 \le b_2 \le b_1$. Denote by $\mathcal H_{a;b}$ the Hecke algebra corresponding to $S_{a;b}$. These Hecke algebras satisfy an analogous embedding relation.
Consider the following \textit{random walk on the Hecke algebra} $\mathcal H_{a;b}$. Let $\mathfrak{S}_{a;b}$ be the set of all nearest neighbor transpositions from $S_{a;b}$. We attach to every element $(z,z+1)$ of $\mathfrak{S}_{a;b}$ a Poisson process $\mathcal{P}(z)$ on $\mathbb{R}_{\ge 0}$ of rate $p$. All these Poisson processes are jointly independent. Next, we define the stochastic process $W_{a;b} (t)$ which takes values in $\mathcal H_{a;b}$. Its initial value is $W_{a;b} (0) = T_{id}$ (the basis vector corresponding to the identity permutation). When at a certain time $\tau \in \mathbb{R}_{\ge 0}$ one of the Poisson processes has a point, then we set $W_{a;b} (\tau) = T_s W_{a;b} (\tau^{-})$, where $s$ is the nearest neighbor transposition corresponding to the Poisson process with point at $\tau$. Since the points from all Poisson processes are almost surely distinct and can be linearly ordered, this rule defines a stochastic process $W_{a;b} (t)$ in continuous time.
We will need the \textit{Mallows element}
$$
\mathcal{M}_{a;b} := \sum_{w \in S_{a;b}} Q^{(b-a+1)(b-a)/2 - l \left( w \right)} Z_{a;b} T_w, \qquad \mathcal{M}_{a;b} \in \mathcal H_{a;b},
$$
with $Z_{a;b}$ as in \eqref{Zab} and furthermore we define
\begin{equation}
\mathcal{H}_{\mathrm{prob}}(S_{a;b})=\left\{h \in \mathcal H_{a;b}: h=\sum_{w\in S_{a;b}} \kappa_w T_w, \kappa_w \ge 0, \sum_{w\in S_{a;b}} \kappa_w = 1\right\}.
\end{equation}
The main property of the element $\mathcal{M}_{a;b}$ is\footnote{It is sufficient to check this property for $h_{a;b}=T_s$, for any $s \in \mathfrak{S}_{a;b}$. For such a choice it follows from a detailed balance type equation.}
\begin{equation}\label{Melement}
h_{a;b} \mathcal{M}_{a;b} = \mathcal{M}_{a;b} h_{a;b} = \mathcal{M}_{a;b}, \qquad \mbox{for any $h_{a;b} \in \mathcal{H}_{\mathrm{prob}}(S_{a;b})$}.
\end{equation}
Finally note that due to the multiplication rule \eqref{eq:HeckeRules} the elements $\mathcal{M}_{a;b}$, $W_{a;b} (t)$, as well as their products, are elements of $\mathcal{H}_{\mathrm{prob}}(S_{a;b})$.
For any element of $\mathcal{H}_{\mathrm{prob}}(S_{a;b})$ one can define the random permutation \textit{generated} by this element of Hecke algebra by assigning to a permutation $w$ the probability $\kappa_w$.
\subsection{Multi-species ASEP as a random walk on Hecke algebra}
Let us make the link here between the random walk $W_{a;b}(t)$ and the multi-species ASEP on $S_{a;b}$ as constructed in Section \ref{coupling}. Note that
the process $(W_{a;b}(t),t\geq0),$ which takes values in the Hecke algebra $\mathcal H_{a;b},$ immediately induces a stochastic process $(w_{t},t\geq0)$ on $S_{a;b}$. The definition of $W_{a;b}(t)$ and the multiplication rules \eqref{eq:HeckeRules} imply that this process is exactly the multi-species ASEP as introduced in Section \ref{coupling}: The first rule of \eqref{eq:HeckeRules} says that (with $s=(z,z+1)$) if $w(z)<w(z+1),$ and the Poisson process $\mathcal{P}(z)$ has a jump, $w$ gets updated as $\sigma_{z,z+1}(w),$ whereas if $w(z)>w(z+1),$ and the Poisson process $\mathcal{P}(z)$ has a jump, $w$ stays the same with probability $1-Q,$ and gets updated as $\sigma_{z,z+1}(w)$ with probability $Q$. Note that by definition $W_{a;b}(0)=T_{id},$ so that also $w_{0}=id,$ however if we wish to start with another configuration, e.g. from a deterministic configuration $w \in S_{a;b}$, we just need to consider the permutation generated by $W_{a;b} (t) T_w$.
\subsection{Bringing into $Q-$equilibrium}\label{Qeq}
Let $h \in\mathcal{H}_{\mathrm{prob}}(S_{a;b})$ and $[a_1;b_1]\subseteq [a;b] $. Then \textit{bringing
the segment $[a_1;b_1]$ into $Q-$equilibrium} means to multiply $h$ with the Mallows element $\mathcal{M}_{a_1;b_1}$ from the left. By \eqref{Melement}, this has the effect of distributing the colors of particles present in $[a_1;b_1]$ according to the stationary measure which is essentially the Mallows measure: It will formally coincide with the Mallows measure on $S_{a_1;b_1}$ if we relabel the colors of particles at positions inside the segment $[a_1;b_1]$ by integers from $[a_1;b_1]$ in a monotonous way.
Seeing $h$ as a random element of $S_{[a;b]}$, and projecting down the colors
$[a;b]$ to particles and holes, let $k'$ be the number of particles present in $[a_1;b_1].$ Then bringing into $Q-$equilibrium $[a_1;b_1]$ means that
inside $[a_1;b_1]$, the particles and holes are distributed according to $\pi_{[a_1; b_1],k'}.$
\subsection{Idea of the proof}\label{plan}
We have recalled all the necessary notions, and will give a proof of Theorem \ref{alexey} in the following. Before this, let us briefly describe the idea of the proof.
Theorem \ref{alexey} asks us to analyze the continuous time ASEP which starts with a nontrivial initial configuration in a very precise manner. There are currently no general tools which seem to be applicable to such sorts of questions (see, however, \cite{QS20}); ASEP with step initial data being an exception where a detailed result in the form of Theorem \ref{ASEPthm} is available. Fortunately, the algebraic framework of the Hecke algebra allows to get an \textit{exact distribution identity} which relates ASEP started with our initial configuration and the step-initial condition. It appears in the following way.
In principle, any initial configuration $w$ is generated by the element $T_w$ of the Hecke algebra. So if we are interested in the continuous time ASEP, we might want to study $W_{a;b} (t) T_w$. How to do this? The first crucial idea is that we can study
$$
\mathfrak{i} \left( T_{w^{-1}} \mathfrak{i} \left( W_{a;b} (t) \right) \right) = \mathfrak{i} \left( T_{w^{-1}} W_{a;b} (t) \right)
$$
instead due to Proposition \ref{prop:CPsymmetry}. Probabilistically, it is very important (and highly non-intuitive) that we first run continuous time ASEP from the identity permutation, which can be projected to the step initial condition, and only after this we apply the multiplication by $T_{w^{-1}}$. Since we have a precise information about the step initial condition ASEP, this gives hope to compute something about the configuration that we are interested in. Yet it is arguably impossible for general $w$. And here comes the second crucial idea --- we utilize Mallows elements which can be used to create an initial configuration which is sufficiently close to $\zeta^{0}$ (defined in \eqref{zeta01}), the particle configuration we are interested in. The use of Mallows elements requires some estimates to relate it to the deterministic initial configuration $\zeta^{0}$ , but all of them in our situation hold with a large margin.
\subsection{Distribution identity}
Let us fix three positive integers $S,M,R$ and a positive real $t$. We will consider random walks on the Hecke algebra $\mathcal{H}_{-S-R;S+M}$. Recall that the stochastic process $W_{-S-R;S+M} (t)$ and the Mallows elements were defined in Section \ref{subsec:rand-walkHA}.
\begin{prop}
\label{prop:DistrIdent1}
With $\,{\buildrel d \over =}$ denoting equality in distribution, we have
\begin{equation*}
W_{-S-R;S+M} (t) \mathcal{M}_{-S-R;0} \mathcal{M}_{-S;S+M} \,{\buildrel d \over =}\, \mathfrak i \left( \mathcal{M}_{-S;S+M} \mathcal{M}_{-S-R;0} W_{-S-R;S+M} (t) \right).
\end{equation*}
\end{prop}
\begin{proof}
Note that $\mathcal{M}_{-S-R;0}$, $\mathcal{M}_{-S;S+M}$, are invariant under the action of $\mathfrak{i}$.
As for $W_{a;b}(t), $ note that for arbitrary $k\geq 1$ and $s_{1},\ldots,s_k \in \mathfrak{S}_{a;b}$ we have $\mathfrak{i}(W_{a;b}(t))=T_{s_{k}}\cdots T_{s_{1}}$ iff $W_{a;b}(t)=T_{s_{1}}\cdots T_{s_{k}},$ so that we show $\mathfrak{i}(W_{a;b}(t))\,{\buildrel d \over =}W_{a;b}(t)$ by computing
\begin{align*}
\mathbb{P}(\mathfrak{i}(W_{a;b}(t))=T_{s_{k}}T_{s_{k-1}}\cdots T_{s_{1}})&=\mathbb{P}(W_{a;b}(t)=T_{s_{1}}T_{s_{2}}\cdots T_{s_{k}})\\&=\mathbb{P}(W_{a;b}(t)=T_{s_{k}}T_{s_{k-1}}\cdots T_{s_{1}}).
\end{align*}
Setting $a=-S-R, b=S+M$ and applying Proposition \ref{prop:CPsymmetry} yields the result.
\end{proof}
We will need one particular corollary of this distribution identity. Denote by $\pi_{S,R,M;t}$ the random permutation generated by $$W_{-S-R;S+M} (t) \mathcal{M}_{-S-R;0} \mathcal{M}_{-S;S+M},$$ and denote by $\hat \pi_{S,R,M;t}$ the random permutation generated by
$$\mathcal{M}_{-S;S+M} \mathcal{M}_{-S-R;0} W_{-S-R;S+M} (t).$$
\begin{cor}
\label{cor:Req}
For any $x \le y \in \mathbb{R}$, we have
\begin{multline}
\label{eq:DistIdent2}
\mathbb{P} \left( \min_{-S-R \le i \le 0} \pi_{S,R,M;t}^{-1} (i) > x, \max_{0 < j \le S+M} \pi_{S,R,M;t}^{-1} (j) \le y \right)
\\ = \mathbb{P} \left( \min_{-S-R \le i \le 0} \hat \pi_{S,R,M;t} (i) > x, \max_{0 < j \le S+M} \hat \pi_{S,R,M;t} (j) \le y \right).
\end{multline}
\end{cor}
\begin{proof}
Immediate from Proposition \ref{prop:DistrIdent1}.
\end{proof}
The event on the left-hand side of \eqref{eq:DistIdent2} distinguishes whether a particle is of color $\le 0$ and $>0$. Therefore, the probability of this event can be expressed as a probability of the corresponding event in a single-species ASEP. Similarly, the right-hand side of \eqref{eq:DistIdent2} distinguishes whether a particle is of color $\le x$, inside $(x;y]$, or $>y$. Therefore, the probability of this event can be expressed as a probability of the corresponding event in a two-species ASEP, i.e. ASEP which contains first-class particles, second-class particles, and holes. Let us present these simpler processes and events more formally.
\subsection{Auxiliary processes}
In this section we will introduce several auxiliary ASEP processes. All of them are continuous time processes which start from distinct initial configurations. These processes may depend on $M,R,S,x,y$, but we omit this in notations. See Figure \ref{Graph2} for an illustration.
\begin{figure}\begin{center}
\begin{tikzpicture}[scale=1.6]
\draw[thick] (-0.5,0) -- (5.2,0)
\draw (-1.8,0.5) node[below] {\large{$\mathfrak{C}_{0}:$}};
\draw (3.05,0.2) circle (0.1);
\draw (-0.5,-0.1) node[below] {$-S-R$};
\draw (5.2,-0.1) node[below] {$S+M$};
\draw (3.05,-0.1) node[below] {$M$};
\draw (2,-0.1) node[below] {$0$};
\draw (0.8,-0.1) node[below] {$-R$};
\foreach \x in {3.05,2,0.8,5.2,-0.5}
\draw[very thick] (\x,0.075)--(\x,-0.075);
\foreach \x in {-0.1,-0.4, 4.25,1.4,2.25,2.5,0.2,0.5}
\draw (\x,0.2) circle(0.1);
\filldraw (1.1,0.2) circle (0.1);
\foreach \x in {2.75, 3.35,3.65,3.95,4.55,4.85,5.15}
\filldraw (\x,0.2) circle(0.1);
\foreach \x in {2.75, 3.05, 3.35,3.65,3.95,4.55,4.85,5.15,1.7,2,0.8}
\filldraw (\x,0.2) circle(0.1);
\begin{scope}[yshift=-1.2cm]
\draw[thick] (-0.5,0) -- (5.2,0)
\draw (-1.8,0.5) node[below] {\large{$\mathfrak{D}_{0}^{(1)}:$}};
\draw (-0.5,-0.1) node[below] {$-S-R$};
\draw (5.2,-0.1) node[below] {$S+M$};
\draw (3.65,-0.1) node[below] {$y$};
\draw (2.25,-0.1) node[below] {$x$};
\foreach \x in {5.2,-0.5,3.65,2.25}
\draw[very thick] (\x,0.075)--(\x,-0.075);
\foreach \x in {2.25,2,1.7, 1.4,1.1,0.8,0.5, -0.1,-0.4,0.2}
\filldraw (\x,0.2) circle(0.1);
\foreach \x in {2.5,2.8,3.1,3.375,3.65}
\filldraw[gray] (\x,0.2) circle(0.1);
\foreach \x in {3.95,4.25,4.55,4.85,5.15 }
\draw (\x,0.2) circle(0.1);
\end{scope}
\begin{scope}[yshift=-2.4cm]
\draw (-1.8,0.5) node[below] {\large{$\mathfrak{D}_{0}^{(2)}:$}};
\draw[thick, ->] (-1,0) -- (5.75,0) node[below] {$\mathbb Z$};
\draw (3.65,-0.1) node[below] {$y$};
\draw (2.25,-0.1) node[below] {$x$};
\foreach \x in {3.65,2.25}
\draw[very thick] (\x,0.075)--(\x,-0.075);
\foreach \x in {2.25,2,1.7, 1.4,1.1,0.8,0.5, -0.1,-0.4,0.2,-0.7,-1}
\filldraw (\x,0.2) circle(0.1);
\foreach \x in {2.5,2.8,3.1,3.375,3.65}
\filldraw[gray] (\x,0.2) circle(0.1);
\foreach \x in {3.95,4.25,4.55,4.85,5.15,5.45,5.75 }
\draw (\x,0.2) circle(0.1); \end{scope}
\end{tikzpicture}\end{center}
\caption{From top to bottom: The three particle configurations $\mathfrak{C}_0, \mathfrak{D}^{(1)}_0, \mathfrak{D}^{(2)}_0. $ Black/gray/white balls represent first class particles/second class particles/holes.
$\mathfrak{C}_0$ is random except for $p=1$ in which case
$\mathfrak{C}_0=\mathbf{1}_{[-R+1;0]}+\mathbf{1}_{[M;S+M]}$, for $p<1,$ the particles remain within $\mathcal{O}(1)$ distance from $[-R+1;0],[M;S+M].$
For $x<y,$ the configuration $\mathfrak{D}^{(1)}_0$ is on $[-S-R;S+M]$ and has first class particles in $[-S-R;x], $ second class particles on $(x;y], $ and holes in $(y;S+M], $ and $\mathfrak{D}^{(2)}_0$ is the extension of $\mathfrak{D}^{(1)}_0$ to $\mathbb{Z}$.
}
\label{Graph2}
\end{figure}
$\mathfrak{C}_0$ is a (random) configuration of particles and holes on $[-S-R;S+M]$ obtained in the following way. We start with particles at $[-S-R;0]$ and holes in $[1;S+M]$. First, we bring into $Q$-equilibrium the segment $[-S;S+M]$. Second, we bring into $Q$-equilibrium the segment $[-S-R;0]$.
$\mathfrak{D}^{(1)}_0$ is a (deterministic) configuration of first class particles, second-class particles and holes on $[-S-R;S+M]$ positioned in the following way: At positions $\le x$ we have first class particles, at positions inside $(x;y]$ we have second class particles, and at positions $>y$ there are holes.
$\mathfrak{D}^{(2)}_0$ is a (deterministic) configuration of first class particles, second-class particles and holes on $\mathbb{Z}$ positioned in essentially the same way: At positions $\le x$ we have first class particles, at positions inside $(x;y]$ we have second class particles, and at positions $>y$ there are holes.
$\mathfrak{C}_t$, $\mathfrak{D}^{(1)}_t$, $\mathfrak{D}^{(2)}_t$ are the notations for configurations of these processes after time $t$.
$\mathfrak{\tilde D}^{(1)}_t$ is the (random) configuration of particles obtained from the configuration $\mathfrak{D}^{(1)}_t$ by bringing into $Q$-equilibrium the segment $[-S-R;0]$ in it. $\mathfrak{\hat D}^{(1)}_t$ is obtained from $\mathfrak{\tilde D}^{(1)}_t$ by bringing into $Q$-equilibrium the segment $[-S;S+M]$ in it. See also Figure \ref{Graph3}.
Recall that for a (possibly random) configuration $\mathfrak{A}$ we denote by $\mathcal{L} \left( \mathfrak{A} \right)$ the position of the leftmost particle in $\mathfrak{A}$, and we denote by $\mathcal{R} \left( \mathfrak{A} \right)$ the position of the rightmost hole in $\mathfrak{A}$. We are in a position to reformulate Corollary \ref{cor:Req} in the language of these processes.
\begin{prop}
\label{prop:DistId3}
We have
\begin{multline}
\label{eq:DistIdent3}
\mathbb{P} \left( \mathcal{L} \left( \mathfrak{C}_t \right) > x, \mathcal{R} \left( \mathfrak{C}_t \right) \le y \right)
\\ = \mathbb{P} \left( \mbox{all first class particles in $\mathfrak{\hat D}^{(1)}_t$ are at positions $>0$}, \right. \\ \left. \mbox{all holes in $\mathfrak{\hat D}^{(1)}_t$ are at positions $\le 0$} \right).
\end{multline}
\end{prop}
\begin{proof}
Note that $\pi^{-1}_{S,R,M;t}$ maps types of particles into positions. If we map all colors $\le 0$ into particles, and colors $> 0$ into holes, we will obtain the coupling of $\pi^{-1}_{S,R,M;t}$ and $\mathfrak{C}_t$, which will give the expression in the left-hand side of \eqref{eq:DistIdent3}. Analogously, to obtain the right-hand side, in $\hat \pi_{S,R,M;t}$ we need to map all types $> y$ into holes, all types inside $(x;y]$ into second class particles, and all types $\le x$ into first class particles.
\end{proof}
\subsection{Limit transition }
In the remainder of the section we will consider sequences of parameters $M=M(N) = N-k_N+1$, $R=R(N) = k_N$, $x=x(N)=N- 2 k_N - N^{1/10}$, $y=y(N)=N- 2 k_N + N^{1/10} $, $t = t(N)= g(k_N,c)$. We also assume from now on that $S=S(N)$ is an arbitrary sequence of numbers such that $S \ge N^N$. We will study the $N \to \infty$ limit. The first key result is the following.
\begin{prop}
\label{prop:baseMult}
We have
\begin{multline}
\label{prop:brr}
\liminf_{N\to\infty} \mathbb{P} \left( \mbox{all first class particles in $\mathfrak{\hat D}^{(1)}_t$ are at positions $>0$}, \right. \\ \left. \mbox{ all holes in $\mathfrak{\hat D}^{(1)}_t$ are at positions $\le 0$} \right) \ge F_{\mathrm{GUE}} \left( c f(\alpha) \right) .
\end{multline}
\end{prop}
\begin{proof}
Throughout the proof, $C_1 , C_2>0$ will be some constants independent of $N$ whose values are immaterial and may change from line to line.
Let $\mathcal{N}_1$ be the number of holes to the left of 0 in the configuration $\mathfrak{D}^{(1)}_t$ and let $\mathcal{N}_2$ be the number of holes to the left of 0 in the configuration $\mathfrak{D}^{(2)}_t$. Clearly, if $S$ is incomparably larger than $M$, $N$, and $t$, then these two quantities have almost the same distribution. More formally, let us couple the processes $\mathfrak{D}^{(1)}_t$ and $\mathfrak{D}^{(2)}_t$ via the basic coupling. Note that if there exists at least one position inside $[-S-R;x]$ and at least one position inside $[y;S+M]$ such that no signals from Poisson processes were received at these points, then the configurations of $\mathfrak{D}^{(1)}_t$ and $\mathfrak{D}^{(2)}_t$ inside the interval $[-S-R;S+M]$ must coincide. The probability that a given position has not received a signal can be bounded by $C_1 \exp(-C_2 t)$, and the probability that there is at least one such position in the interval $[-S; x]$ is bounded by $1 - (1 - C_1 \exp(-C_2 t))^{x+S}$. For our values of parameters, this can be estimated as $1 - C_1 \exp(-C_2 N) $. The same argument also works for the interval $[y; S+M]$. Therefore, we have $\lim_{N\to \infty}\mathbb{P}(\mathcal{N}_1 = \mathcal{N}_2)=1.$
Next, we note that
\begin{equation}\label{TWcor}
\lim_{N \to \infty} \mathbb{P} \left( \mathcal{N}_2 \ge k_N + N^{1/10} \right) = F_{\mathrm{GUE}} \left( c f(\alpha) \right).
\end{equation}
Indeed, to compute $\mathcal{N}_2$ we need not distinguish between first and second class particles. By the particle hole-duality, we then get
$$\mathbb{P} \left( \mathcal{N}_2 \ge k_N + N^{1/10} \right)=\mathbb{P}(x_{k_{N}+N^{1/10}}^{\mathrm{step}}>N-2k_{N}+N^{1/10}),
$$
so that \eqref{TWcor} follows from Corollary \ref{cor}. Since $\lim_{N\to \infty}\mathbb{P}(\mathcal{N}_1 = \mathcal{N}_2)=1,$ we obtain
\begin{equation}
\label{eq:applyTW}
\lim_{N \to \infty}\mathbb{P} \left( \mathcal{N}_1 \ge k_N + N^{1/10} \right) = F_{\mathrm{GUE}} \left( c f(\alpha) \right). \end{equation}
\begin{figure}\begin{center}
\begin{tikzpicture}[scale=1.6]
\draw[thick] (-0.5,0) -- (5.2,0)
\draw (-1.8,0.5) node[below] {\large{$\mathfrak{D}^{(1)}_t:$}};
\draw (-0.5,-0.1) node[below] {$-S-R$};
\draw (5.2,-0.1) node[below] {$S+M$};
\draw (2,-0.1) node[below] {$0$};
\foreach \x in {2,5.2,-0.5}
\draw[very thick] (\x,0.075)--(\x,-0.075);
\foreach \x in {-0.1,-0.4, 1.4,0.8, 0.2,0.5}
\draw (\x,0.2) circle(0.1);
\foreach \x in {2.75,3.05,2.5, 3.95, 2.25,3.35,4.25,4.85,1.7,-0.4}
\filldraw (\x,0.2) circle(0.1);
\foreach \x in {2,1.1,3.65,4.55,5.15}
\filldraw[gray] (\x,0.2) circle(0.1);
\begin{scope}[yshift=-1.2cm]
\draw[thick] (-0.5,0) -- (5.2,0)
\draw (-1.8,0.5) node[below] {\large{$\tilde{\mathfrak{D}}^{(1)}_t:$}};
\draw (2,-0.1) node[below] {$0$};
\draw (0.5,-0.1) node[below] {$-S$};
\draw (-0.5,-0.1) node[below] {$-S-R$};
\draw (5.2,-0.1) node[below] {$S+M$};
\foreach \x in {5.2,-0.5,0.5,2}
\draw[very thick] (\x,0.075)--(\x,-0.075);
\foreach \x in {2.25,3.05,2,1.7,3.35,3.95,4.85,4.25,2.75,2.5 }
\filldraw (\x,0.2) circle(0.1);
\foreach \x in {1.1,1.4,5.15,4.55,3.65}
\filldraw[gray] (\x,0.2) circle(0.1);
\foreach \x in {0.8,0.5, -0.1,-0.4,0.2 }
\draw (\x,0.2) circle(0.1);
\end{scope}
\begin{scope}[yshift=-2.4cm]
\draw[thick] (-0.5,0) -- (5.2,0)
\draw (-1.8,0.5) node[below] {\large{$\hat{\mathfrak{D}}^{(1)}_t:$}};
\draw (2,-0.1) node[below] {$0$};
\draw (0.5,-0.1) node[below] {$-S$};
\draw (-0.5,-0.1) node[below] {$-S-R$};
\draw (5.2,-0.1) node[below] {$S+M$};
\foreach \x in {5.2,-0.5,0.5,2}
\draw[very thick] (\x,0.075)--(\x,-0.075);
\foreach \x in {-0.4,-0.1,0.2,0.5,0.8,1.1,1.4}
\draw (\x,0.2) circle(0.1);
\foreach \x in {1.4,1.1,1.7,2,2.25}
\filldraw[gray] (\x,0.2) circle(0.1);
\foreach \x in {2.5,2.75,3.05,3.35,3.65,3.95,4.25,4.55,4.85,5.15}
\filldraw (\x,0.2) circle(0.1);
\end{scope}
\end{tikzpicture}\end{center}
\caption{From top to bottom: The particle configurations $ \mathfrak{D}^{(1)}_t, \tilde{\mathfrak{D}}^{(1)}_t, \hat{\mathfrak{D}}^{(1)}_t. $ If $ \mathcal{N}_1 \ge R + N^{1/10},$
then $\tilde{\mathfrak{D}}^{(1)}_t $ will have the segment $[-S-R;-S)$ filled only by holes with very high probability. Consequently, with very high probability $\hat{\mathfrak{D}}^{(1)}_t $ will have all its holes in $[-S-R;0],$ the second class particles will be inside or close to the segment $[-N^{\frac{1}{10}}; N^{\frac{1}{10}}],$ and all first class particles will be in $[ 0; S+R].$
}
\label{Graph3}
\end{figure}
Let us now analyze how the configuration $\mathfrak{\hat D}^{(1)}_t$ looks like conditioned on the event $ \mathcal{N}_1 \ge k_N + N^{1/10}$. See Figure \ref{Graph3} for an illustration. According to the definition, we need to first bring into $Q$-equilibrium the segment $[-S-R;0]$ in the configuration $\mathfrak{ D}^{(1)}_t$. Since we have at least $k_N + N^{1/10} = R + N^{1/10} $ holes in the segment $[-S-R;0]$, Proposition \ref{3.2} allows to conclude that the positions in the interval $[-S-R;-S)$ will \textit{all} be filled by holes with probability at least $1 - C_1\exp \left( - C_2 N^{1/10} \right)$ after this step. Let us further restrict ourselves on the event that indeed $[-S-R;-S)$ is filled by holes. Then in the remaining segment $[-S;S+M]$ we have exactly
$$
S+M - y -R = S +(N-k_N+1) - (N-2 k_N + N^{1/10} ) - k_N = S - N^{1/10}+1
$$
holes. Our second (and last) step is to bring into $Q$-equilibrium the segment $[-S;S+M]$. Again, due to Proposition \ref{3.2} the conditional probability that all these $S - N^{1/10}+1 $ holes are inside $[-S;0]$ can be bounded from below as $1 - C_1 \exp \left( - C_2 N^{1/10} \right)$. Also, the segment $[-S;S+M]$ contains all $S+N-k_N- N^{1/10} +1$ first class particles of the system, and the conditional probability that all of them will be inside the segment $[0;S+M]$ is also at least $1 - C_1 \exp \left( - C_2 N^{1/10} \right)$ again by Proposition \ref{3.2}. It remains to note that the conditional probability that all the mentioned three events happen can also be estimated as $1 - C_1 \exp \left( - C_2 N^{1/10} \right)$, and that if these three events happen, than the event in the lefthand side of \eqref{prop:brr} also happens: The holes are all to the left of 0, and all first-class particles are to the right of 0. Combining this with \eqref{eq:applyTW} implies the claim.
\end{proof}
We have now collected all necessary ingredients to prove Theorem \ref{alexey}.
\begin{proof}[Proof of Theorem \ref{alexey}]
We define the configuration
$$
\widehat{\zeta}^0:= \mathbf{1}_{[ -k_N+1; 0] } + \mathbf{1}_{\mathbb{Z}_{>(N-k_N)}}.
$$
Note that $\widehat{\zeta}^0$ is simply a shift by $(-k_N)$ of the configuration $\zeta^0$ from \eqref{zeta01}. We denote further the restriction of $\widehat{\zeta}^0$ to $[-S-R;S+M]$ by $\widehat{\zeta}^{0,S},$ i.e.
$$
\{0,1\}^{[-S-R;S+M]}\ni \widehat{\zeta}^{0,S}:= \mathbf{1}_{ [-k_N+1; 0] } + \mathbf{1}_{[N-k_{N}+1; S+ N-k_{N}+1]}.
$$
Let us prove that
\begin{equation}\label{60}
\lim_{N\to \infty}\mathbb{P}(\mathcal{L}(\widehat{\zeta}^{0,S}_{t})=\mathcal{L}(\widehat{\zeta}^{0}_{t}), \mathcal{R}(\widehat{\zeta}^{0,S}_{t})=\mathcal{R}(\widehat{\zeta}^{0}_{t}))=1.
\end{equation}
To see this, note that for the event $\{\mathcal{L}(\widehat{\zeta}^{0,S}_{t})=\mathcal{L}(\widehat{\zeta}^{0}_{t}), \mathcal{R}(\widehat{\zeta}^{0,S}_{t})=\mathcal{R}(\widehat{\zeta}^{0}_{t})\}$ to happen, it suffices that one of the Poisson processes in $[-S-R;-S/2]$ and one of the Poisson processes in $[S/2;S+M]$ make no jump during $[0,t].$ Since $S\geq N^{N},$ this will happen with probability going to $1$ as $N\to \infty$ and \eqref{60} follows.
Recall further the partial order \eqref{order2} and note that we have
\begin{equation} \label{smaller}
\mathfrak{C}_{0}\preceq \widehat{\zeta}^{0,S}.
\end{equation}
Now, one can easily compute that
under the basic coupling, \eqref{smaller} implies that
\begin{equation}\label{shorter}
\mathcal{L}(\mathfrak{C}_{t})\leq \mathcal{L}(\widehat{\zeta}^{0,S}_{t}) \mathrm{\,\,and\,\, } \mathcal{R}(\mathfrak{C}_{t})\geq \mathcal{R}(\widehat{\zeta}^{0,S}_{t})
\end{equation}
(see e.g. \cite[Lemma 3.2]{N20AAP} and \cite[Lemma 3.2]{N20CMP} for a proof for the partial order \eqref{order} which carries over to our case).
We can now conclude
\begin{align*}
\liminf_{N\to\infty} \mathbb{P}(B_{N}(c))&= \liminf_{N\to\infty}\mathbb{P}(\mathcal{L}(\widehat{\zeta}^{0,S}_{t})> x, \mathcal{R}(\widehat{\zeta}^{0,S}_{t})\leq y )
\\&\geq \liminf_{N\to\infty} \mathbb{P} \left( \mathcal{L} \left( \mathfrak{C}_t \right) > x, \mathcal{R} \left( \mathfrak{C}_t \right) \le y \right)
\\& \geq F_{\mathrm{GUE}} \left( c f(\alpha) \right),
\end{align*}
where the first inequality follows from \eqref{shorter}, the second follows from Propositions \ref{prop:DistId3} and \ref{prop:baseMult}.
Finally, using Corollary \ref{cor} we obtain
\begin{align*}
\limsup_{N\to\infty} \mathbb{P}(B_{N}(c))&\leq \lim_{N\to\infty}\mathbb{P}(\mathcal{L}(\widehat{\zeta}^{0}_{t})\geq x)\\&\leq \lim_{N \to \infty}\mathbb{P}\left(x_{k_{N}}^{\mathrm{step}}(g(k_{N},c))\geq N-2k_{N} -N^{1/10}\right)\\&=F_{\mathrm{GUE}} \left( c f(\alpha) \right),
\end{align*}
so that Theorem \ref{alexey} follows.
\end{proof}
\begin{remark} \label{rem:further-questions}
It is plausible that an extension of Theorem \ref{main} to the case of slowly varying parameters $p-q \to 0$, $N^{1/3} (p-q) \to \infty$ can be obtained by an upgrade of our method. This will require certain additional estimates, in particular, regarding the asymptotics of Mallows measure. On the other hand, if $(p-q)$ becomes to decay to 0 faster, new effects should appear, first on the level of the cutoff window, and then on the level of the cutoff itself (cf. \cite{LP16}).
Another interesting direction would be to understand whether our method can be adjusted to the study of mixing times in ASEP with open boundaries, cf. \cite{GNS20}.
\end{remark}
| {
"timestamp": "2021-11-15T02:02:29",
"yymm": "2012",
"arxiv_id": "2012.14924",
"language": "en",
"url": "https://arxiv.org/abs/2012.14924",
"abstract": "This paper studies the mixing behavior of the Asymmetric Simple Exclusion Process (ASEP) on a segment of length $N$. Our main result is that for particle densities in $(0,1),$ the total-variation cutoff window of ASEP is $N^{1/3}$ and the cutoff profile is $1-F_{\\mathrm{GUE}},$ where $F_{\\mathrm{GUE}}$ is the Tracy-Widom distribution function. This also gives a new proof of the cutoff itself, shown earlier by Labbé and Lacoin. Our proof combines coupling arguments, the result of Tracy-Widom about fluctuations of ASEP started from the step initial condition, and exact algebraic identities coming from interpreting the multi-species ASEP as a random walk on a Hecke algebra.",
"subjects": "Probability (math.PR); Discrete Mathematics (cs.DM); Mathematical Physics (math-ph)",
"title": "Cutoff profile of ASEP on a segment",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.985496421586532,
"lm_q2_score": 0.7185943925708562,
"lm_q1q2_score": 0.7081722024507263
} |
https://arxiv.org/abs/1408.4867 | Furstenberg theorem for frequently hypercyclic operators | In this paper, we show that if the direct sum $T\oplus T$ of frequently hypercyclic operators is frequently hypercyclic, then every higher direct sum $T\oplus\cdots\oplus T$ is also frequently hypercyclic. | \section{Introduction}
In this paper, we study the dynamics of linear operators on a separable $F$-space $X$. A bounded linear operator $T$ on $X$ is said to be hypercyclic if there is a vector $x\in X$ such that the orbit $O(x,T)=\{T^nx\mid n\in \mathbb{N}\}$ is dense in $X$. The operator $T$ is said to be topologically transitive if, for every pair of non-empty open subsets $U$ and $V$, there is an integer $n$ such that $T^nU\cap V\ne \emptyset$. By the Baire category theorem, topological transitivity of $T$ is equivalent to the hypercyclicity of $T$. See \cite{book-GEandPeris} and \cite{book-BayartandMatheron} for details and references. If $T\oplus T$ is hypercyclic, then the operator $T$ is called weakly mixing. It is shown in \cite{BesandPeris99} that the weakly mixing property is equivalent to the Hypercyclicity Criterion. On the other hand, as shown in \cite{RosaandRead} and \cite{BayartandMatheron2},
hypercyclic operators may not be weakly mixing, see also \cite{MatheronandBayart}. An interesting fact is so-called the Furstenberg theorem, which is given as follows: if $T$ is weakly mixing, then the $n$-fold product is $T\times\cdots\times T$ is weakly mixing for $n\ge 2$. The proof is given in \cite{book-GEandPeris} by using the 4-set trick. In the linear setting we have
\begin{thm}
Let $X$ be a separable $F$-space. If $T\oplus T$ is hypercyclic, then the higher sum $T\oplus\cdots \oplus T$ is also hypercyclic. \hfill$\qed$
\end{thm}
The $T$-orbit of a hypercyclic operator visits each non-empty open subsets of $X$. Then it is natural to ask how often the orbit visits each non-empty open sets in $X$ and it leads to the notion of the frequently hypercyclic operators which has been introduced by
Bayart and Grivaux, see \cite{BayartandGrivaux} and \cite{BonillaandGrossErdmann}. In \cite{GrosseandPeris05}, it is shown that every frequently hypercyclic operator is weakly mixing. Based on ideas given in \cite{GrosseandPeris05}, we prove the Furstenberg theorem for the frequently hypercyclic operators.
\section{Frequently Hypercyclic Operators}
Let $X$ be a separable $F$-space and let $\mathcal{L}(X)$ be the space of continuous linear operators on $X$. By definition, an operator
$T\in \mathcal{L}(X)$ is hypercyclic if there is a vector $x\in X$ such that the orbit \[O(x,T)=\{T^nx\mid n\in\mathbb{N}\}\] is dense in $X$. In other words, the $T$-orbit $O(x,T)$ intersects with each non-empty open set $U$ in $X$. For a non-empty open subset $U$ of $X$, define
\[\mathbf{N}(x,U)=\{n\in\mathbb{N}\mid T^nx\in U\}.\]
If an operator $T$ on $X$ is hypercyclic, then there is a vector $x\in X$ such that for each non-empty open set $U$ in $X$, the set $\mathbf{N}(x,U)$ are all non-empty. For any non-empty open sets $U$ and $V$, let us define the {\it return set} as follows:
\[\mathbf{N}(U,V)=\{n\in \mathbb{N}\mid T^nU\cap V\ne\emptyset\}.\]
By the topological transitivit, if $T$ is hypercyclic, then each set $\mathbf{N}(U,V)$ is non-empty.
If $T$ is weakly mixing, then there is a natural number $n$ such that for each open subsets $U_1, U_2$ and $V_1, V_2$ of $X$ such that
\[T^nU_1\cap V_1\ne\emptyset \text{ \ \ and \ \ } T^nU_2\cap V_2\ne\emptyset. \]
Then the $T\in \mathcal{L}(X)$ is weakly mixing if and only if
\begin{align}\label{weakmixing}
\mathbf{N}(U_1,V_1)\cap \mathbf{N}(U_2,V_2)\ne\emptyset.
\end{align}
See \cite{GrosseandPeris05}, \cite{book-GEandPeris} and \cite{book-BayartandMatheron} for other formulas which are equivalent to (\ref{weakmixing}).
The frequently hypercyclicity corresponds to the largeness of each sets $\mathbf{N}(x,U)$, in other words, how frequently the $T$-orbit intersects with each open set $U$. Let us first recall that the lower density of a subset $A$ in $\mathbb{N}$ which is given by
\[\underline{\text{dens}}(A)=\liminf_{N\to\infty}\frac{|A\cap [1,N]|}{N}\]
where $|A\cap [1,N]|$ denotes the cardinality of the set $A\cap [1,N]$.
\begin{dfn}{\rm Let $X$ be a topological vector space and let $T\in\mathcal{L}(X)$. The operator $T$ is called {\it frequently hypercyclic} if there is a vector $x\in X$ such that for every non-empty open set $U$, $\mathbf{N}(x,U)$ has positive lower density. Such a vector $x$ is called frequently hypercyclic for $T$ and the set of all frequently hypercyclic vectors for $T$ is denoted by $FHC(T)$.
}
\end{dfn}
If we enumerate an infinite set $A\subset\mathbb{N}$ as an increasing sequence $(n_k)_{k\in \mathbb{N}}$, then it is easy to see that $A$ has positive lower density if and only if there is a constant $C$ such that
\[n_k\le Ck \text{ \ \ \ \ for all } k\ge 1\]
Thus, a vector $x\in X$ is frequently hypercyclic for $T$ if and only if for each non-empty open subset $U$ of $X$, there is a strictly increasing sequence $(n_k)$ and some constant $C$ such that
\[T^{n_k}x\in U \text{ \ \ \ and \ \ \ } n_k\le Ck\]
for all $k\in \mathbb{N}$. We now prove the Furstenberg theorem for frequently hypercyclic operators.
\begin{thm}
Let $X$ be a separable $F$-space and let $T\in \mathcal{L}(X)$. If $T\oplus T$ is frequently hypercyclic, then $3$-fold sum $T\oplus T \oplus T$ is also frequently hypercyclic.
\end{thm}
{\it Proof. } We will show that there is a vector $x_1\oplus x_2\oplus x_3\in X\oplus X\oplus X$ satisfying the following property: for each non-empty open subsets $U_1$, $U_2$ and $U_3$ of $X$, there is a strictly increasing sequence $(n_k)_{k\in \mathbb{N}}$ of natural numbers and a constant $C$ such that for $i=1,2,3$,
\[T^{n_k}x_i\in U_i \text{ \ \ and \ \ } n_k\le Ck \text{ \ \ \ for all }k\in \mathbb{N}\]
First, we note that if $T\oplus T$ is frequently hypercyclic, then $T\oplus T$ is hypercyclic. By the Furstenberg theorem $T\oplus T \oplus T$ is also hypercyclic. Thus there is a hypercyclic vector $x_1\oplus x_2\oplus x_3\in X\oplus X\oplus X$ such that for each non-empty open subsets $U_1$, $U_2$ and $U_3$ of $X$,
\[\mathbf{N}(x_1,U_1)\cap \mathbf{N}(x_2,U_2)\cap \mathbf{N}(x_3,U_3)\ne\emptyset\]
Suppose that $x_1\oplus x_2\in FHC(T\oplus T)$. Then for non-empty open sets $U_1$ and $U_2$ there is a strictly increasing sequence $(m_k)_{k\in\mathbb{N}}$ and a constant $C_1$ such that
\[T^{m_k}x_1\in U_1, \ \ T^{m_k}x_2\in U_2 \text{ \ \ \ and \ \ } m_k\le C_1k\]
Since $T$ is hypercyclic, the set $\mathbf{N}(U_1,U_2):=\{l\in \mathbb{N}\mid T^lU_1\cap U_2\ne\emptyset\}$ is non-empty and the $T$-orbit $O(x_1,T)$ is dense in $X$, there is an increasing sequence $(b_j)_{j\in \mathbb{N}}$ such that
\begin{align}\label{bn}
x_2=\lim_{j\to\infty}T^{b_j}x_1
\end{align}
Since $T$ is continuous,
\begin{align}
T^{m_k}x_2=\lim_{j\to\infty}T^{b_j}T^{m_k}x_1\in U_2
\end{align}
Thus there is an integer $N$ such that for all $j\ge 1$
\[T^{b_{j+N}}T^{m_k}x_1\in U_2\]
In particular, for all $k\in \mathbb{N}$
\[T^{b_{k+N}}T^{m_k}\in U_2 \text{ \ \ and \ \ } T^{b_{k+N}}U_1\cap U_2\ne\emptyset \]
In other words, the sequence $(b_{k+N})_{k\in\mathbb{N}}$ is in $\mathbf{N}(U_1,U_2)$ and since $x_1$ and $x_2$ are frequently hypercyclic, the sequence $(b_{k+N})_{k\in\mathbb{N}}$ satisfies $b_{k+N}=O(k)$ ({\it cf.} \cite{MatheronandBayart}). Let
\[U_{1k}=U_1\cap T^{-b_{k+N}}U_2\]
Then $\mathbf{N}(x_1,U_{1k})\subset \mathbf{N}(x_1,U_1)$. If $l\in\mathbf{N}(x_1,U_{1k})$ then for all $k\in \mathbb{N}$,
\[T^lx_1\in U_1 \text{ \ \ and \ \ } T^{b_{k+N}}T^l x_1\in U_2\]
By (\ref{bn}) and $T^{b_{k+N}}T^l x_1=T^lT^{b_{k+N}}x_1$, we get
$l\in \mathbf{N}(x_1, U_{1k})$. Thus
\begin{align}\label{con1}
\mathbf{N}(x_1,U_{1k})\subset \mathbf{N}(x_1,U_{1})\cap \mathbf{N}(x_2,U_{2})
\end{align}
Applying the same argument for $\mathbf{N}(x_2,U_2)\cap \mathbf{N}(x_3,U_3)$ we
may obtain
\begin{align}\label{con2}
\mathbf{N}(x_2,U_{2j})\subset \mathbf{N}(x_2,U_{2})\cap \mathbf{N}(x_3,U_{3})
\end{align}
Now by (\ref{con1}) and (\ref{con2})
\begin{align}
\mathbf{N}(x_1,U_{1k})\cap \mathbf{N}(x_2,U_{2j})\subset \mathbf{N}(x_1,U_1)\cap \mathbf{N}(x_2,U_2)\cap \mathbf{N}(x_3,U_3)
\end{align}
Since $x_1\oplus x_2\in FHC(T\oplus T)$, there is a strictly increasing sequence $(n_k)_{k\in \mathbb{N}}$, which may be an enumeration of thet set $\mathbf{N}(x_1,U_{1k})\cap \mathbf{N}(x_2,U_{2j})$, such that for some constant $C$, and for $i=1,2,3$,
\[T^{n_k}x_i\in U_i \text{ \ \ and \ \ } n_k\le Ck \text{ \ \ \ for all }k\in \mathbb{N}\]
as desired. \hfill\qed
By proceeding induction, we have the main result
\begin{thm}
Let $X$ be a separable $F$-space and let $T\in \mathcal{L}(X)$. If $T\oplus T$ is frequently hypercyclic, then the higher product $T\oplus \cdots\oplus T$ is also frequently hypercyclic.\hfill\qed
\end{thm}
\vspace{1mm}
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"abstract": "In this paper, we show that if the direct sum $T\\oplus T$ of frequently hypercyclic operators is frequently hypercyclic, then every higher direct sum $T\\oplus\\cdots\\oplus T$ is also frequently hypercyclic.",
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"title": "Furstenberg theorem for frequently hypercyclic operators",
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https://arxiv.org/abs/2112.06859 | Choice-free Stone duality | The standard topological representation of a Boolean algebra via the clopen sets of a Stone space requires a nonconstructive choice principle, equivalent to the Boolean Prime Ideal Theorem. In this paper, we describe a choice-free topological representation of Boolean algebras. This representation uses a subclass of the spectral spaces that Stone used in his representation of distributive lattices via compact open sets. It also takes advantage of Tarski's observation that the regular open sets of any topological space form a Boolean algebra. We prove without choice principles that any Boolean algebra arises from a special spectral space X via the compact regular open sets of X; these sets may also be described as those that are both compact open in X and regular open in the upset topology of the specialization order of X, allowing one to apply to an arbitrary Boolean algebra simple reasoning about regular opens of a separative poset. Our representation is therefore a mix of Stone and Tarski, with the two connected by Vietoris: the relevant spectral spaces also arise as the hyperspace of nonempty closed sets of a Stone space endowed with the upper Vietoris topology. In addition to representation, we establish a choice-free dual equivalence between the category of Boolean algebras with Boolean homomorphisms and a subcategory of the category of spectral spaces with spectral maps. We show how this duality can be used to prove some basic facts about Boolean algebras. | \section{Introduction}\label{Intro}
Stone \cite{Stone1936} proved that any Boolean algebra (BA) $\mathbb{A}$ is isomorphic to the field of clopen sets of a Stone space (zero-dimensional compact Hausdorff space), namely the Stone dual of $\mathbb{A}$. As the Stone dual of $\mathbb{A}$ is the set of ultrafilters of $\mathbb{A}$ with the topology generated by $\{\widehat{a}\mid a\in \mathbb{A}\}$, where $\widehat{a}$ is the set of ultrafilters containing $a$, Stone's representation requires a nonconstructive choice principle---equivalent to the Boolean Prime Ideal Theorem---asserting the existence of sufficiently many ultrafilters.
In this paper, we describe a choice-free topological representation of BAs. This representation uses a subclass of the spectral spaces that Stone \cite{Stone1938} used in his representation of distributive lattices via compact open sets. It also takes advantage of Tarski's \cite{Tarski1937,Tarski1938} observation that the regular open sets of any topological space form a Boolean algebra. We prove without choice principles that any Boolean algebra arises from a special spectral space $X$ via the compact regular open sets of $X$; these sets may also be described as those that are both compact open in $X$ and regular open in the upset topology of the specialization order of $X$, allowing one to apply to an arbitrary BA simple reasoning about regular opens of a separative poset.\footnote{The consideration of two topologies is clearly related to Priestley's \cite{Priestley1970} alternative representation for distributive lattices using certain ordered Stone spaces: any distributive lattice arises from a Priestley space via the sets that are both clopen in the Stone topology of the space and open in the upset topology arising from the additional order. We consider a Priestley-like version of our representation of BAs in Section \ref{ChoiceSection}.} Our representation is therefore a mix of Stone and Tarski, with the two connected by Vietoris~\cite{Vietoris1922}: the relevant spectral spaces also arise as the hyperspace of nonempty closed sets of a Stone space endowed with the upper Vietoris topology. We characterize these spectral spaces, which we call UV-spaces, with several axioms including a special separation axiom, reminiscent of the Priestley separation axiom \cite{Priestley1970}. The connection with the Vietoris hyperspace construction makes clear the relation between our point-set topological approach to choice-free Stone duality, which may be called the hyperspace approach, and a point-free approach to choice-free Stone duality using Stone locales \cite{Johnstone1982,Vickers1989}.
Unlike Stone's representation of BAs via Stone spaces, the choice-free topological representation of BAs via UV-spaces does not show that every BA can be represented as a field of sets, with complement as set-theoretic complement and join as union. Such a representation implies the Boolean Prime Ideal Theorem.\footnote{If a BA is isomorphic to a field $\mathcal{F}$ of sets over a set $X$, then picking any point $x\in X$ gives us an ultrafilter $\{S\in\mathcal{F}\mid x\in S\}$. The statement that every BA contains an ultrafilter then implies that for any disjoint filter-ideal pair in a BA, the filter can be extended to an ultrafilter disjoint from the ideal. The equivalent dual statement for ideals is the Boolean Prime Ideal Theorem.} However, like Stone's representation, ours provides the benefit of a topological perspective on BAs, only now without choice.
In addition to representation, we establish a choice-free dual equivalence between the category of BAs with Boolean homomorphisms and the category of UV-spaces with special spectral maps. We show how this duality can be applied by using it to prove some basic theorems about BAs.
The axiom of choice and its variants have traditionally been of general interest to logicians. Interest in choice also arises specifically in connection with topology and Stone duality as in \cite{Johnstone1981,Johnstone1982,Johnstone1983}. In this paper, we assume the motivations summarized in \cite{Herrlich2006} for investigating mathematics without the axiom of choice---in particular, mathematics based on ZF set theory instead of only ZFC. Only starting in our applications section (Section 8) will we go beyond ZF by using the axiom of dependent choice (DC), which is widely considered to be constructively acceptable (see \cite[\S~14.76]{Schechter1996}). There we work in the style of what is called \textit{quasiconstructive mathematics} in \cite{Schechter1996}, defined as ``mathematics that permits conventional rules of reasoning plus ZF + DC, but no stronger forms of Choice'' (p.~404).
The paper is organized as follows. Sections \ref{PossSection} and \ref{RepresentationSection} present requisite background and the representation to be used in the following sections, which is redescribed in Section \ref{ROsection}. Section \ref{VietorisSection} characterizes the resulting duals of BAs as UV-spaces; Section \ref{DualitySection} establishes the dual equivalence result; and Section \ref{L&H} contrasts our hyperspace approach with a localic approach. Section \ref{DictionarySection} contains a ``duality dictionary'' for translating between BA notions and UV notions, and Section \ref{ApplicationSection} contains sample applications of the duality. Although our focus is on choice-free duality, Section \ref{ChoiceSection} considers three perspectives on UV-spaces assuming choice. Section \ref{Conclusion} concludes with a brief recap and look ahead.
\section{Background}\label{PossSection}
The choice-free topological representation of BAs that we will describe results from ``topologizing'' the choice-free representation of BAs in \cite{Holliday2015,Holliday2018}.\footnote{The focus of \cite{Holliday2015,Holliday2018} is on modal algebras, but here we present only the Boolean side of the story.} A \textit{possibility frame} from \cite{Holliday2015} is a triple $(S,\leqslant, P)$ where $(S,\leqslant)$ is a poset and $P$ is a collection of \textit{regular open} sets in the upset topology $\mathsf{Up}(S,\leqslant)$ of the poset, such that $P$ contains $S$ and is closed under intersection and the operation $\neg$ defined by
\begin{equation}
\neg U=\{x\in S\mid \forall x'\geqslant x\;\; x'\not\in U\}.\label{NegationEQ}
\end{equation}
Recall that an open set $U$ in a space is regular open iff $U=\mathsf{int}(\mathsf{cl}(U))$. Since the closure and interior operations in $\mathsf{Up}(S,\leqslant)$ are calculated by
\begin{eqnarray}
&&\mathsf{cl}_\leqslant (U)=\{x\in S\mid \exists y\geqslant x : y\in U\}, \label{IntDef}\\
&&\mathsf{int}_\leqslant (U)=S\setminus \mathsf{cl}_\leqslant (S\setminus U)= \{x\in S\mid \forall y\geqslant x \;\; y\in U\},\label{ClDef}
\end{eqnarray}
an open set $U$ in $\mathsf{Up}(S,\leqslant)$ is regular open iff
\begin{equation}U=\mathsf{int}_\leqslant(\mathsf{cl}_\leqslant(U))=\{x\in S \mid \forall x'\geqslant x\,\exists x''\geqslant x': x''\in U\}. \label{ROeq}\end{equation}
Also note that $\neg U=\mathsf{int}_\leqslant (X\setminus U)$, so $U$ is regular open iff $U=\neg\neg U$.
As Tarski \cite{Tarski1937,Tarski1938,Tarski1956} observed, the regular open sets of any topological space form a (complete) Boolean algebra with binary meet as intersection and complement as interior of set-theoretic complement, so any subalgebra thereof is also a Boolean algebra. Thus, for any possibility frame $(S,\leqslant,P)$, the set $P$ gives us a Boolean algebra.\footnote{\label{LocaleNote}From the perspective of locale theory (see Section \ref{L&H}), the collection $\mathsf{Up}(S,\leqslant)$ of upsets forms a locale with meet as intersection and join as union. Equivalently, $\mathsf{Up}(S,\leqslant)$ may be viewed as a complete Heyting algebra. Then $\neg U$ is the \textit{pseudocomplement} of $U$ in $\mathsf{Up}(S,\leqslant)$, i.e., the largest upset whose meet with $U$ is $\varnothing$, and $P$ is a subalgebra of the Boolean algebra of all \textit{regular elements} (i.e., those $U$ such that $U=\neg\neg U$) of $\mathsf{Up}(S,\leqslant)$.}
Conversely, given any Boolean algebra $\mathbb{A}$, we construct a possibility frame $(\mathrm{PropFilt}(\mathbb{A}),\subseteq,\{\widehat{a}\mid a\in \mathbb{A}\})$ where $\mathrm{PropFilt}(\mathbb{A})$ is the set of proper filters of $\mathbb{A}$, ordered by inclusion, and $\widehat{a}=\{F\in \mathrm{PropFilt}(\mathbb{A})\mid a\in F\}$; then $\{\widehat{a}\mid a\in \mathbb{A}\}$ is a collection of regular open sets from $\mathsf{Up}(\mathrm{PropFilt}(\mathbb{A}),\subseteq)$ that satisfies the required closure conditions, and under the operations $\cap$ and $\neg$ it becomes a Boolean algebra isomorphic to $\mathbb{A}$.\footnote{It can then be proved choice-free that the complete BA of all regular opens from $\mathsf{Up}(\mathrm{PropFilt}(\mathbb{A}),\subseteq)$ is a \textit{canonical extension} of $\mathbb{A}$ in the sense of \cite{Gehrke2001} (see \cite[\S~5.6]{Holliday2018} and Theorem \ref{CanonicalExt} below).} The possibility frames that arise (isomorphically) in this way, called \textit{filter-descriptive} in \cite{Holliday2015}, are exactly those satisfying the separation property that if $x\not\leqslant y$, then there is a $U\in P$ such that $x\in U$ and $y\not\in U$, and the ``filter realization'' property that if $F$ is a proper filter in $P$, then $F=\{U\in P\mid x\in U\}$ for some $x\in S$. In \cite{Holliday2015,Holliday2018} it is proved without choice principles that the category of filter-descriptive frames with appropriate morphisms (see Section \ref{DualitySection}) is dually equivalent to the category of BAs with Boolean homomorphisms.
In Section \ref{RepresentationSection}, we will show that the duality just sketched can be understood topologically as a choice-free duality between BAs and special \textit{spectral spaces}. In particular, the dual possibility frame $(S,\leqslant, P)$ of a BA gives rise to a spectral space $X$ by using $P$ as a basis for a topology on $S$. This makes $\leqslant$ the specialization order of $X$. We can then conveniently pick out among all regular opens in the upset topology of $\leqslant$ just those that give us back our original BA via $P$: those that are also \textit{compact open} in $X$. It turns out we may equivalently think of these compact sets as regular open in $X$, though thinking of them as regular open in the upset topology of $\leqslant$ has the advantage of simplifying reasoning. The story above is our starting point, but we go much further: we develop a full topological duality, including a duality dictionary for many algebraic concepts, along with sample applications via topological proofs of basic facts about BAs.
There are several precedents for the strategy of working with all proper filters of a lattice. In the context of logic, since the early 1980s logicians have studied alternative semantics for classical first-order logic and classical modal logics in which one builds a canonical model using all consistent and deductively closed sets of formulas, rather than only maximally consistent sets of formulas \cite{Roper1980,Humberstone1981,Benthem1981,Benthem1986,Benthem1988,Holliday2015,Benthem2016b}. Although not presented as such, these constructions are essentially applications of the fact indicated above that any BA $\mathbb{A}$ embeds into the BA of regular open upsets in the poset of proper filters of $\mathbb{A}$. If $\mathbb{A}$ is the Lindenbaum-Tarski algebra of a logic, then its poset of proper filters is isomorphic to the poset of consistent and deductively closed sets of formulas. The subsets of this canonical model that are definable by a formula then correspond to the sets $\widehat{a}$ above.
The idea of topologizing the set of proper filters also appears in Goldblatt's \cite{Goldblatt1975} representation of ortholattices, discussed in Section \ref{GoldblattSection}. However, Goldblatt uses a different topology on the set of proper filters with the consequence that his representation is not choice free.
After completing the following work, we learned that Moshier and Jipsen \cite{Moshier2014} propose a choice-free duality for arbitrary lattices using the space of all filters endowed with the analogous $\widehat{a}$ topology. Though we work with proper filters (since otherwise there would be only two regular open sets with respect to $\leqslant$, namely $\varnothing$ and the whole space), the more important difference is that we study what happens in the special case of BAs.
Our approach to choice-free Stone duality for BAs is also closely related to a point-free approach. The collection $\mathrm{Filt}(\mathbb{A})$ of all filters of a BA $\mathbb{A}$ ordered by inclusion is an example of what we will call a \textit{Stone locale}: a zero-dimensional compact locale (see Section \ref{L&H} for definitions). The category of Stone locales with localic maps\footnote{For the definition of localic maps, see, e.g., \cite[\S~II.2]{Picado2012}.} is dually equivalent to the category of BAs with Boolean homomorphisms. However, our aim is to provide a choice-free duality using spaces instead of locales. We do so by taking the non-zero elements of the Stone locale $\mathrm{Filt}(\mathbb{A})$ as the points of a new space with an appropriate topology, namely the upper Vietoris topology (see Section \ref{RepresentationSection}). Thus, we call our approach to choice-free Stone duality the \textit{hyperspace approach}, in contrast to the \textit{localic approach} using Stone locales.
The hyperspace approach allows us to retain the intuitiveness of reasoning with a set of points, without paying the price of choice principles. But there is a cost, or at least a currency exchange: whereas standard Stone duality represents each BA as a subalgebra of the powerset of a set, the choice-free dualities in \cite{Holliday2015} and in this paper represent each BA as a subalgebra of the regular open algebra of a separative poset.
\begin{definition} Let $(S,\leqslant)$ be a poset, and for $x\in S$, let $\mathord{\Uparrow}x=\{x'\in S\mid x\leqslant x'\}$. Then $(S,\leqslant)$ is \textit{separative} iff for any $x,y\in S$, $x\not\leqslant y$ implies that there is a $z\in \mathord{\Uparrow}y$ such that $\mathord{\Uparrow}z\cap\mathord{\Uparrow}x=\varnothing$. Equivalently, $(S,\leqslant)$ is separative iff every principal upset $\mathord{\Uparrow}x$ is regular open in $\mathsf{Up}(S,\leqslant)$.
\end{definition}
\noindent It is easy to see that the separation property mentioned for possibility frames above implies separativity of the underlying partial order.
Thus, with the choice-free duality for BAs that we will pursue, instead of reasoning about sets with intersection and set-theoretic complement, we reason about separative posets (given by the specialization orders of our spaces) with intersection and the operation $\neg$ defined in (\ref{NegationEQ}). A major difference is that for $U\subseteq S$, while $U\cup (S\setminus U)=S$, we often have ${U\cup \neg U\subsetneq S}$.\footnote{From the perspective of Footnote \ref{LocaleNote}, the observation that we often have ${U\cup \neg U\subsetneq S}$ reflects the fact that $\mathsf{Up}(S,\leqslant)$ is a Heyting algebra that is typically not Boolean.} This makes reasoning with $\neg$ more subtle, but one can quickly get used to reasoning patterns with $\neg$ of the kind shown in the following lemmas.
\begin{lemma} Let $(S,\leqslant)$ be a poset and $U$ regular open in $\mathsf{Up}(S,\leqslant)$. If $x\not\in U$, then there is an $x'\geqslant x$ such that $x'\in \neg U$.
\end{lemma}
\begin{proof} If $x\not\in U$, then since $U$ is regular open, it follows by (\ref{ROeq}) that there is an $x'\geqslant x$ such that for all $x''\geqslant x'$, $x''\not\in U$, which means $x'\in \neg U$.
\end{proof}
\begin{lemma}\label{EitherInfinite} Let $(S,\leqslant)$ be an infinite separative poset and $U$ regular open in $\mathsf{Up}(S,\leqslant)$. Then either $U$ or $\neg U$ is infinite.
\end{lemma}
\begin{proof} Let $x\sim y$ iff $\mathord{\Uparrow}x\cap U=\mathord{\Uparrow}y\cap U$. If $U$ is finite, then $\sim$ partitions the infinite set $S$ into finitely many cells, one of which must be infinite. Call it $I$, and define $f\colon I\to \wp(\neg U)$ by $f(x)=\mathord{\Uparrow}x\cap \neg U$. We claim that $f$ is injective. For if $x,y\in I$ and $x\not\leqslant y$, then by separativity, there is a $z\in \mathord{\Uparrow}y$ such that $\mathord{\Uparrow}z\cap\mathord{\Uparrow}x=\varnothing$. It follows, since $\mathord{\Uparrow}x\cap U=\mathord{\Uparrow}y\cap U$, that $\mathord{\Uparrow}z\cap U=\varnothing$, so $z\in \neg U$. Thus, $z\in f(y)$ but $z\not\in f(x)$, so $f$ is injective. Then since $I$ is infinite, it follows that $\wp(\neg U)$ is infinite and hence $\neg U$ is infinite.\end{proof}
\section{Representation of BAs using spectral spaces}\label{RepresentationSection} Before reviewing spectral spaces, let us fix some notational conventions.
We will conflate a BA $\mathbb{A}$ and its underlying set, and we will conflate a topological space $X$ and its underlying set, so that we will write, e.g., `$a\in\mathbb{A}$', `$x\in X$', etc. The top and bottom elements of a bounded lattice such as a BA are denoted `$1$' and `$0$', respectively, possibly with subscripts to indicate the relevant algebra. We will often consider filters in a BA, as well as principal upsets in the specialization order of a space. To avoid any confusion about which side a principal filter/upset is on---the algebra side or the space side---we make the following notational distinction.
\begin{notation} \textnormal{Let $\mathbb{A}$ be a BA whose underlying order is $\leq$ and $X$ a space whose specialization order is $\leqslant$. For $a\in\mathbb{A}$ and $x\in X$:
\begin{enumerate}
\item $\mathord{\uparrow} a = \{b\in\mathbb{A}\mid a\leq b \}$ and $\mathord{\downarrow} a = \{b\in\mathbb{A}\mid b\leq a \}$;
\item $\mathord{\Uparrow}x = \{y\in X\mid x\leqslant y\}$ and $\mathord{\Downarrow}x = \{y\in X\mid y\leqslant x\}$.
\end{enumerate}}
\end{notation}
It will also help to distinguish between the built-in complement operation of a BA $\mathbb{A}$ and the operation $\neg$ defined in (\ref{NegationEQ}) of Section~\ref{PossSection}.
\begin{notation} Given a BA $\mathbb{A}$ and a space $X$ whose specialization order is $\leqslant$:
\begin{enumerate}
\item let $-$ be the complement operation in $\mathbb{A}$;
\item let $\neg$ be the operation defined for $U\subseteq X$ by $\neg U=\mathsf{int}_\leqslant (X\setminus U)$.
\end{enumerate}
\end{notation}
It is important to remember that we are distinguishing two interior (resp.~closure) operations associated with a given space $X$.
\begin{notation} \textnormal{For a space $X$ whose specialization order is $\leqslant$:
\begin{enumerate}
\item $\mathsf{int}$ and $\mathsf{cl}$ are the interior and closure operations for $X$;
\item $\mathsf{int}_\leqslant$ and $\mathsf{cl}_\leqslant$ are the interior and closure operations for the upset topology with respect to $\leqslant$, as in (\ref{IntDef})--(\ref{ClDef}) of Section~\ref{PossSection}.
\end{enumerate}}
\end{notation}
As is well known, the operations $\mathsf{int}_\leqslant$ and $\mathsf{cl}_\leqslant$ coincide with $\mathsf{int}$ and $\mathsf{cl}$, respectively, if and only if $X$ is an Alexandroff space.
The following notation will be used throughout.
\begin{notation}\label{COROnotation} Let $X$ be a space. We define the following collections of subsets of $X$:
\begin{enumerate}
\item $\mathsf{O}(X)$ is the collection of sets that are open in $X$;
\item $\mathsf{C}(X)$ is the collection of sets that are compact in $X$;
\item $\mathsf{CO}(X)=\mathsf{C}(X)\cap\mathsf{O}(X)$;
\item $\mathsf{RO}(X)$ is the collection of sets that are regular open in $X$;
\item $\mathsf{CRO}(X)=\mathsf{C}(X)\cap \mathsf{RO}(X)$;
\item $\mathcal{RO}(X)$ is the collection of sets that are regular open in ${\mathsf{Up}(X,\leqslant)}$, where $\leqslant$ is the specialization order of $X$;
\item $\mathsf{O}\mathcal{RO}(X)=\mathsf{O}(X)\cap \mathcal{RO}(X)$;
\item $\mathsf{CO}\mathcal{RO}(X)=\mathsf{CO}(X)\cap \mathcal{RO}(X)$;
\item $\mathsf{Clop}(X)$ is the collection of sets that are clopen in $X$.
\end{enumerate}
\end{notation}
Let us now recall the notion of a spectral space and two theorems illustrating its importance.
\begin{definition}\label{SpectralDef} A topological space $X$ is a \textit{spectral space} if $X$ is compact, $T_0$, coherent ($\mathsf{CO}(X)$ is closed under intersection and forms a base for the topology of $X$), and sober (every completely prime filter in $\mathsf{O}(X)$ is $\mathsf{O}(x)=\{U\in \mathsf{O}(X)\mid x\in U\}$ for some $x\in X$).
\end{definition}
\begin{theorem}[Stone \cite{Stone1938}] $L$ is a distributive lattice iff $L$ is isomorphic to the lattice of compact open sets of a spectral space.
\end{theorem}
\begin{theorem}[Hochster \cite{Hochster1969}] $X$ is a spectral space iff $X$ is homeomorphic to the spectrum of a commutative ring.
\end{theorem}
We will show that every BA $\mathbb{A}$ can be represented as $\mathsf{CO}\mathcal{RO}(X)$ (or equivalently $\mathsf{CRO}(X)$, as shown in Section \ref{ROsection}) for some spectral space $X$. Using the nonconstructive Boolean Prime Ideal Theorem, one could prove this by taking $X$ to be the Stone space of $\mathbb{A}$: since the specialization order $\leqslant$ in a Stone space is the discrete order, all subsets are regular open in $\mathsf{Up}(X,\leqslant)$, and it can be proved that the compact open sets of $X$ are exactly the clopen sets used in the standard Stone representation. However, it is also possible to provide a choice-free representation, as shown below.
We first recall the \textit{upper Vietoris topology} \cite{Vietoris1922} on the hyperspace of nonempty closed sets of a Stone space. Where $\mathsf{F}(X)$ is the collection of nonempty closed subsets of $X$ and $U\in\mathsf{Clop}(X)$, let \[\Box U =\{F\in \mathsf{F}(X)\mid F\subseteq U\}.\]
Observe that $\Box U\cap \Box V = \Box (U\cap V)$, so $\{\Box U\mid U\in\mathsf{Clop}(X)\}$ is closed under binary intersection.
\begin{definition}\label{UVStoneDef} Given a Stone space $X$, define $\mathscr{UV}(X)$ to be the space of nonempty closed sets of $X$ with the topology generated by the family $\{\Box U\mid U\in\mathsf{Clop}(X)\}$.
\end{definition}
The same idea can be applied to the space of proper filters of a BA. For $a\in \mathbb{A}$, let \[\widehat{a} = \{F\in \mathrm{PropFilt}(\mathbb{A})\mid a\in F\}.\] Observe that $\widehat{a}\cap\widehat{b}=\widehat{a\wedge b}$, so $\{\widehat{a}\mid a\in \mathbb{A}\}$ is closed under binary intersection.
\begin{definition}\label{UVofBA} Given a BA $\mathbb{A}$, define $UV(\mathbb{A})$ to be the space of proper filters of $\mathbb{A}$ with the topology generated by $\{\widehat{a}\mid a\in \mathbb{A}\}$.
\end{definition}
\begin{prop}\label{UVStone} For any Stone space $X$, $\mathscr{UV}(X)$ is homeomorphic to $UV(\mathsf{Clop}(X))$, regarding $\mathsf{Clop}(X)$ as the BA of clopen subsets of $X$.
\end{prop}
\begin{proof} Let $f: C\mapsto \{U\in\mathsf{Clop}(X)\mid C\subseteq U\}$. Since $X$ is nonempty, $f(C)$ is clearly a proper filter in $\mathsf{Clop}(X)$, so $f(C)\in UV(\mathsf{Clop}(X))$. For injectivity, if $C\neq C'$, then without loss of generality suppose $x\in C\setminus C'$. Since $X$ is compact Hausdorff, it follows that there is a $U\in\mathsf{Clop}(X)$ such that $C'\subseteq U$ but $x\not\in U$, so $C\not\subseteq U$. Hence $U\in f(C')$ but $U\not\in f(C)$. For surjectivity, if $F$ is a proper filter in $\mathsf{Clop}(X)$, then $F$ has the finite intersection property, so by the compactness of $X$, we have that $\bigcap F$ is nonempty, and since $\bigcap F$ is the intersection of closed sets, it is closed. We claim that $f(\bigcap F)=F$. That $f(\bigcap F)=\{U\in\mathsf{Clop}(X)\mid \bigcap F\subseteq U\}\supseteq F$ is immediate. To see that $f(\bigcap F)\subseteq F$, if $U\in\mathsf{Clop}(X)$ and $\bigcap F\subseteq U$, so $X\setminus U\subseteq \bigcup \{X\setminus V\mid V\in F\}$, then by compactness there is a finite $F_0\subseteq F$ such that $X\setminus U\subseteq \{X\setminus V\mid V\in F_0\}$ and hence $\bigcap F_0\subseteq U$. Then since $F_0$ is finite, it follows that $U\in F$. For continuity of $f$, if $\widehat{U}$ is a basic open in $UV(\mathsf{Clop}(X))$, so $U\in\mathsf{Clop}(X)$, then we have:
\begin{eqnarray*}
f^{-1}[\widehat{U}]&=&\{C\in UV(X)\mid f(C)\in \widehat{U}\}\\
&=&\{C\in UV(X)\mid U\in f(C)\}\\
&=&\{C\in UV(X)\mid C\subseteq U\}\\
&=& \Box U.
\end{eqnarray*}
For openness of $f$, if $\Box U$ is a basic open in $\mathscr{UV}(X)$, so $U\in\mathsf{Clop}(X)$, then we have:
\begin{eqnarray*}
f[\Box U]&=&\{f(C)\mid C\in \Box U\} \\
&=& \{f(C)\mid C\subseteq U\} \\
&=& \{f(C)\mid U\in f(C)\} \\
&=& \widehat{U}.
\end{eqnarray*}
For the last equality, the left-to-right inclusion follows from the fact that $f(C)$ is a proper filter, since $C\neq \varnothing$, and the right-to-left inclusion follows from the surjectivity of $f$.
\end{proof}
\begin{remark} Assuming the Boolean Prime Ideal Theorem, one can also prove that for any BA $\mathbb{A}$, $UV(\mathbb{A})$ is homeomorphic to $\mathscr{UV}(\mathrm{Stone}(\mathbb{A}))$, where $\mathrm{Stone}(\mathbb{A})$ is the Stone dual of $\mathbb{A}$ (see Section~\ref{UVStoneSection}).
\end{remark}
\begin{prop}\label{IsSpectral} For any BA $\mathbb{A}$:
\begin{enumerate}
\item\label{Spectral} $UV(\mathbb{A})$ is a spectral space;
\item\label{Specialization} the specialization order in $UV(\mathbb{A})$ is the inclusion order.
\end{enumerate}
\end{prop}
\begin{proof} We first show that each $\widehat{a}$ is compact open in $UV(\mathbb{A})$. Since the sets $\widehat{b}$ form a basis, it suffices to show that if $\widehat{a}\subseteq \underset{i\in I}{\bigcup}\widehat{b_i}$, then there is a finite subcover. If $\widehat{a}\subseteq \underset{i\in I}{\bigcup}\widehat{b_i}$, then every proper filter that contains $a$ also contains one of the $b_i$. In particular, the principal filter $\mathord{\uparrow}a$ contains one of the $b_i$, which implies $a\leq b_i$ and hence $\widehat{a}\subseteq \widehat{b_i}$, so $\widehat{b_i}$ alone is the finite subcover. It follows that $UV(\mathbb{A})$ is compact, since $X=\widehat{1}$. It also follows by the definition of $UV(\mathbb{A})$ that the compact open sets form a basis.
To see that the compact opens are closed under binary intersection, suppose $U$ and $V$ are compact open, so $U=\underset{i\in I}{\bigcup}\widehat{a_i}$ and $V=\underset{j\in J}{\bigcup}\widehat{b_j}$ for finite $I$ and $J$. Then
\[U\cap V= \underset{i\in I,\,j\in J}\bigcup (\widehat{a_i}\cap\widehat{b_j})=\underset{i\in I,\,j\in J}\bigcup \widehat{a_i\wedge b_j},\]
which is a finite union of compact opens. Hence $U\cap V$ is compact open.
For $T_0$, if $F\neq F'$, without loss of generality suppose $a\in F\setminus F'$. Then $F\in \widehat{a}$ but $F'\not \in \widehat{a}$, and $\widehat{a}$ is open, so we are done.
For sobriety, we show that every completely prime filter $\mathcal{F}$ in $\mathsf{O}(UV(\mathbb{A}))$ is of the form $\mathsf{O}(F)=\{U\in \mathsf{O}(UV(\mathbb{A}))\mid F\in U\}$ for some $F\in UV(\mathbb{A})$. Let $F$ be the filter generated by $\{a\in\mathbb{A}\mid \widehat{a}\in\mathcal{F}\}$. Then since $\mathcal{F}$ is a proper filter in $\mathsf{O}(UV(\mathbb{A}))$, it follows that $F$ is a proper filter in $\mathbb{A}$. To see that $\mathcal{F}=\mathsf{O}(F)$, the right-to-left direction is immediate from the definition of $F$. For the left-to-right direction, suppose $U=\underset{i\in I}{\bigcup}\widehat{a_i}\in\mathcal{F}$. Then since $\mathcal{F}$ is completely prime, there is an $a_i$ such that $\widehat{a_i}\in\mathcal{F}$, which implies $a_i\in F$, so $F\in\widehat{a_i}$. Thus, $\widehat{a_i}\in \mathsf{O}(F)$ and hence $U\in\mathsf{O}(F)$.
For part \ref{Specialization}, we already saw above for $T_0$ that if $F\not\subseteq F'$, then $F\not\leqslant F'$. Conversely, if $F\subseteq F'$, then for any basic open $\widehat{a}$, if $F\in\widehat{a}$ and hence $a\in F$, then $a\in F'$ and hence $F'\in\widehat{a}$, so $F\leqslant F'$.
\end{proof}
We now provide the promised choice-free representation.
\begin{theorem}\label{MainRep}$\,$
\begin{enumerate}
\item\label{MainRep1} For each BA $\mathbb{A}$, the map $\widehat{\cdot}:\mathbb{A}\to \mathsf{CO}\mathcal{RO}(UV(\mathbb{A}))$ is an isomorphism from $\mathbb{A}$ to $\mathsf{CO}\mathcal{RO}(UV(\mathbb{A}))$ ordered by inclusion.
\item\label{MainRep2} $\mathsf{CO}\mathcal{RO}(UV(\mathbb{A}))$ is a BA with operations given by:
\begin{equation} U\wedge V=U\cap V\quad \mathord{-}U=\mathsf{int}_\leqslant(UV(\mathbb{A})\setminus U)\quad U\vee V=\mathsf{int}_\leqslant(\mathsf{cl}_\leqslant(U\cup V)).\label{MainRepEQ}\end{equation}
\end{enumerate}
\end{theorem}
\begin{proof} For part \ref{MainRep1}, we will show that
\begin{equation}\mathsf{CO}\mathcal{RO}(UV(\mathbb{A}))=\{\widehat{a}\mid a\in\mathbb{A}\},\label{MainEq}\end{equation}
for then the map $a\mapsto \widehat{a}$ is the isomorphism from $\mathbb{A}$ to $\mathsf{CO}\mathcal{RO}(UV(\mathbb{A}))$, since clearly $a\leq b$ iff $\widehat{a}\subseteq \widehat{b}$.
For the right-to-left inclusion of (\ref{MainEq}), we showed in the proof of Proposition \ref{IsSpectral}.\ref{Spectral} that each $\widehat{a}$ is compact open in $UV(\mathbb{A})$. Now we show that $\widehat{a}$ is regular open in $\mathsf{Up}(UV(\mathbb{A}),\leqslant)$, using the fact from Proposition \ref{IsSpectral}.\ref{Specialization} that the specialization order $\leqslant$ is the inclusion order $\subseteq$. First, $\widehat{a}$ is an $\leqslant$-upset, for if $F\in \widehat{a}$ and $F\leqslant F'$, so $a\in F$ and $F\subseteq F'$, then $a\in F'$ and hence $F'\in \widehat{a}$. Then to see that $\widehat{a}$ is regular open, by (\ref{ROeq}) it suffices to show that if $F\not\in \widehat{a}$, then there is a proper filter $F'\supseteq F$ such that for all proper filters $F''\supseteq F'$, we have $F''\not\in \widehat{a}$. Indeed, if $F\not\in \widehat{a}$, so $a\not\in F$, then the filter $F'$ generated by $F\cup \{-a\}$ is a proper filter with $F'\supseteq F$, and for all proper filters $F''\supseteq F'$, we have $a\not\in F''$ and hence $F''\not\in \widehat{a}$.
For the left-to-right inclusion of (\ref{MainEq}), suppose $S$ is compact open, so $S=\widehat{a_1}\cup\dots\cup \widehat{a_n}$ for some $a_1,\dots,a_n\in\mathbb{A}$. Now if in addition $\widehat{a_1}\cup\dots \cup \widehat{a_n}$ is regular open in $\mathsf{Up}(UV(\mathbb{A}),\leqslant)$, then we claim
\begin{equation}\widehat{a_1}\cup\dots\cup \widehat{a_n}=\reallywidehat{a_1\vee\dots\vee a_n}.\label{FiniteJoinEq}\end{equation}
First, we show
\begin{equation}\reallywidehat{a_1\vee\dots\vee a_n}=\mathsf{int}_\leqslant(\mathsf{cl}_\leqslant(\widehat{a_1}\cup\dots\cup \widehat{a_n})).\label{JoinEQ}\end{equation}
For the left-to-right inclusion, if $F\in \reallywidehat{a_1\vee\dots\vee a_n}$, so $a_1\vee\dots \vee a_n\in F$, then for any proper filter $F'\supseteq F$, there is some $a_i$ such that $-a_i\not\in F'$. Thus, the filter $F''$ generated by $F'\cup \{a_i\}$ is proper, and $a_i\in F''$ implies $F''\in \widehat{a_i}$ and hence $F''\in\widehat{a_1}\cup\dots\cup \widehat{a_n}$. Thus, by (\ref{ROeq}), $F\in \mathsf{int}_\leqslant(\mathsf{cl}_\leqslant(\widehat{a_1}\cup\dots\cup \widehat{a_n}))$. Conversely, if $F\not\in \reallywidehat{a_1\vee\dots\vee a_n}$, so $a_1\vee\dots\vee a_n\not\in F$, then the filter $F'$ generated by $F\cup\{-a_1\wedge\dots\wedge -a_n\}$ is a proper filter, and for every proper filter $F''\supseteq F'$, each $a_i$ is not in $F''$, so $F''\not\in \widehat{a_1}\cup\dots\cup \widehat{a_n}$. Thus, by (\ref{ROeq}), $F\not\in \mathsf{int}_\leqslant(\mathsf{cl}_\leqslant(\widehat{a_1}\cup\dots\cup \widehat{a_n}))$. Finally, if $\widehat{a_1}\cup\dots \cup \widehat{a_n}$ is regular open in $\mathsf{Up}(UV(\mathbb{A}),\leqslant)$, then $\widehat{a_1}\cup\dots \cup\widehat{a_n}=\mathsf{int}_\leqslant(\mathsf{cl}_\leqslant(\widehat{a_1}\cup\dots \cup\widehat{a_n}))$, which with (\ref{JoinEQ}) implies (\ref{FiniteJoinEq}). Thus, $S\in \{\widehat{a}\mid a\in\mathbb{A}\}$.
For part \ref{MainRep2}, since $a\mapsto\widehat{a}$ is an isomorphism, we have:
\begin{equation}\widehat{a}\wedge \widehat{b}=\widehat{a\wedge b}\qquad \mathord{-}\widehat{a}=\widehat{- a}\qquad \widehat{a}\vee\widehat{b}=\widehat{a\vee b}.\label{From1Eq}\end{equation}
We have already observed the first and third of the following equalities:
\begin{equation}\widehat{a\wedge b}=\widehat{a}\cap\widehat{b}\qquad \widehat{- a}=\mathsf{int}_\leqslant(UV(\mathbb{A})\setminus \widehat{a})\qquad \widehat{a\vee b}=\mathsf{int}_\leqslant(\mathsf{cl}_\leqslant(\widehat{a}\cup\widehat{b})).\label{AlreadyObserved}\end{equation}
For the second equality, if $F\in \widehat{-a}$, so $-a\in F$, then for every proper filter $F'\supseteq F$, we have $-a\in F'$, so $a\not\in F'$ and hence $F'\not\in\widehat{a}$. Thus, $F\in \mathsf{int}_\leqslant(UV(\mathbb{A})\setminus \widehat{a})$. If $-a\not\in F$, then the filter $F'$ generated by $F\cup \{a\}$ is a proper filter such that $F\subseteq F'\in \widehat{a}$, so $F\not\in \mathsf{int}_\leqslant(UV(\mathbb{A})\setminus \widehat{a})$.
Combining (\ref{From1Eq}) and (\ref{AlreadyObserved}), we have:
\begin{equation}\widehat{a}\wedge \widehat{b}=\widehat{a}\cap\widehat{b}\qquad \mathord{-}\widehat{a}=\mathsf{int}_\leqslant (UV(\mathbb{A})\setminus \widehat{a})\qquad \widehat{a}\vee\widehat{b}=\mathsf{int}_\leqslant (\mathsf{cl}_\leqslant (\widehat{a}\cup\widehat{b})),\end{equation}
which with (\ref{MainEq}) shows that the BA operations of $\mathsf{CO}\mathcal{RO}(UV(\mathbb{A}))$ satisfy the equations in (\ref{MainRepEQ}).
\end{proof}
\begin{cor} For each Stone space $X$, $\mathsf{Clop}(X)$ is isomorphic to $\mathsf{CO}\mathcal{RO}(\mathscr{UV}(X))$ via the map $U\mapsto \Box U$.
\end{cor}
\begin{proof} By Theorem \ref{MainRep}, we have an isomorphism between $\mathsf{Clop}(X)$ and $\mathsf{CO}\mathcal{RO}(UV(\mathsf{Clop}(X)))$ via the map that sends $U\in\mathsf{Clop}(X)$ to $\widehat{U}\in \mathsf{CO}\mathcal{RO}(UV(\mathsf{Clop}(X)))$. By the proof of Proposition \ref{UVStone}, $\mathscr{UV}(X)$ is homeomorphic to $UV(\mathsf{Clop}(X))$ via the map $f$, which satisfies $f^{-1}[\widehat{U}]=\Box U$. Thus, $\mathsf{Clop}(X)$ is isomorphic to $\mathsf{CO}\mathcal{RO}(\mathscr{UV}(X))$ via the map $U\mapsto \Box U$.
\end{proof}
\section{Regular opens in the Alexandroff and spectral topologies}\label{ROsection} In response to the representation in the previous section, Tom\'{a}\v{s} Jakl (p.~c.) observed that in the special case of compact open sets, being regular open in the Alexandroff space $\mathsf{Up}(UV(\mathbb{A}))$ is equivalent to being regular open in the spectral space $UV(\mathbb{A})$, i.e., $\mathsf{CO}\mathcal{RO}(UV(\mathbb{A}))=\mathsf{CRO}(UV(\mathbb{A}))$. We have $U\in\mathsf{RO}(UV(\mathbb{A}))$ iff $U$ is an open set such that $U=\mathsf{int}(\mathsf{cl}(U))$, where $\mathsf{int}$ and $\mathsf{cl}$ are the interior and closure operations of $UV(\mathbb{A})$. This is equivalent to $U=U^{**}$, where $^*$ is the pseudocomplement operation on $\mathsf{O}(UV(\mathbb{A}))$:
\[U^*=\mathsf{int}(UV(\mathbb{A})\setminus U).\]
It is then easy to see that
\[U^*=\bigcup \{V\in \mathsf{O}(UV(\mathbb{A}))\mid U\cap V=\varnothing\}=\bigcup\{\widehat{c}\mid U\cap\widehat{c}=\varnothing\}.\]
Thus, we can derive $\mathsf{CO}\mathcal{RO}(UV(\mathbb{A}))=\mathsf{CRO}(UV(\mathbb{A}))$ from the following more basic facts.
\begin{prop}\label{StarNeg} Let $\mathbb{A}$ be a BA.
\begin{enumerate}
\item\label{StarNeg1} If $U\in\mathsf{O}(UV(\mathbb{A}))$, then $U^*\subseteq \neg U$;
\item\label{StarNeg2} If $U\in\mathsf{CO}(UV(\mathbb{A}))$, then $\neg U\subseteq U^*$.
\end{enumerate}
\end{prop}
\begin{proof} For part (\ref{StarNeg1}), suppose $F\in U^*$, so there is some $c$ such that $F\in\widehat{c}$, i.e., $c\in F$, and $U\cap\widehat{c}=\varnothing$, i.e., no proper filter containing $c$ belongs to $U$. Thus, no proper filter extending $F$ belongs to $U$, whence $F \in \neg U$.
For part (\ref{StarNeg2}), suppose $U\in\mathsf{CO}(X)$, so $U=\widehat{a_1}\cup\dots\cup\widehat{a_n}$ for some $a_1,\dots,a_n\in\mathbb{A}$. Then assuming $F\in\neg U$, we have $\neg a_1,\dots,\neg a_n\in F$ and hence $c:=\neg a_1\wedge\dots\wedge\neg a_n \in F$. Thus, $F\in\widehat{c}$, and clearly $U\cap\widehat{c}=\varnothing$. Therefore, $F\in U^*$.
\end{proof}
As an immediate corollary of Proposition \ref{StarNeg}, we have the following.
\begin{cor}\label{Jakl} For any BA $\mathbb{A}$, $\mathsf{CO}\mathcal{RO}(UV(\mathbb{A}))=\mathsf{CRO}(UV(\mathbb{A}))$.
\end{cor}
\noindent Thus, by Theorem \ref{MainRep}, $\mathbb{A}$ is isomorphic to $\mathsf{CRO}(UV(\mathbb{A}))$. It is also easy to check that $-\widehat{a}=\mathsf{int}(UV(\mathbb{A})\setminus \widehat{a})$ and $\widehat{a}\vee\widehat{b}=\mathsf{int}(\mathsf{cl}(\widehat{a}\cup \widehat{b}))$.
If we do not restrict to compact open sets, then the operations $\neg$ and $^*$ may behave differently; however, the extent of this difference depends on one's set-theoretic assumptions. It is a theorem of $\mathrm{ZF}+\mathrm{BPI}$ that every infinite BA contains a non-principal ultrafilter (see, e.g., \cite[p.~174]{Givant2009}), in which case $\neg$ and $^*$ can be distinguished with an open set as in Proposition \ref{Nonprinc}.\ref{Nonprinc1} below. On the other hand, it is consistent with ZF that there is an infinite BA in which every filter is principal \cite{Plotkin1976} (for an overview, see \cite[p.~165]{Howard1998}), and in such a BA $\neg$ and $^*$ cannot be distinguished with open sets in light of Proposition \ref{Nonprinc}.\ref{Nonprinc2} (plus Proposition \ref{StarNeg}.\ref{StarNeg1}).
\begin{prop}\label{Nonprinc} Let $\mathbb{A}$ be a BA.
\begin{enumerate}
\item\label{Nonprinc0} $\mathsf{RO}(UV(\mathbb{A}))\subseteq \mathsf{O}\mathcal{RO}(UV(\mathbb{A}))$.
\item\label{Nonprinc1} If $F$ is a non-principal ultrafilter in $\mathbb{A}$ and $U=\bigcup\{\widehat{-a}\mid a\in F\}$, then:
\begin{enumerate}
\item\label{Nonprinc1a} $F\in \neg U\setminus U^*$;
\item\label{Nonprinc1b} $U=\neg\neg U$;
\item\label{Nonprinc1c} $U\subsetneq U^{**}$;
\item\label{Nonprinc1d} $\mathsf{O}\mathcal{RO}(UV(\mathbb{A}))\not\subseteq\mathsf{RO}(UV(\mathbb{A}))$.
\end{enumerate}
\item\label{Nonprinc2} Let $F$ be a principal filter in $\mathbb{A}$ and $U\in\mathsf{O}(UV(\mathbb{A}))$. If $F\in \neg U$, then $F\in U^*$.
\end{enumerate}
\end{prop}
\begin{proof} For part \ref{Nonprinc0}, suppose $U\in \mathsf{RO}(UV(\mathbb{A}))$, so $U=U^{**}$. Since $U^*\in\mathsf{O}(UV(\mathbb{A}))$, we have $U^*=\bigcup\{\widehat{b}\mid b\in B\}$ for some $B\subseteq\mathbb{A}$. Thus,
\begin{eqnarray*}U^{**}&=&\bigcup\{\widehat{c}\mid \bigcup\{\widehat{b}\mid b\in B \} \cap \widehat{c}=\varnothing\}\\
&=&\bigcup\{\widehat{c}\mid \forall b\in B\;\, \widehat{b} \cap \widehat{c}=\varnothing\}\\
&=&\bigcup\{\widehat{c}\mid \forall b\in B\;\, b\wedge c=0\}.
\end{eqnarray*}
Let $I:= \{c\in\mathbb{A}\mid \forall b\in B\;\, b\wedge c=0\}$, and observe that $I$ is an ideal in $\mathbb{A}$. To see that $U^{**}\in \mathcal{RO}(UV(\mathbb{A}))$, suppose $F$ is a proper filter in $\mathbb{A}$ such that $F\not\in U^{**}$. It follows that $F\cap I=\varnothing$. Let $F'$ be the filter generated by $\{a\wedge -c\mid a\in F, c\in I\}$. We claim that $F'$ is a proper filter. If not, then there are $a_1,\dots,a_n\in F$ and $c_1,\dots,c_n\in I$ such that $a_1\wedge -c_1\wedge\dots \wedge a_n\wedge -c_n=0$, so $a_1\wedge\dots\wedge a_n\leq c_1\vee\dots\vee c_n$. Then since $F$ is a filter containing $a_1,\dots,a_n$, we have $c_1\vee\dots\vee c_n\in F$, and since $I$ is an ideal containing $c_1,\dots,c_n$, we have $c_1\vee\dots\vee c_n\in I$, contradicting $F\cap I=\varnothing$. Hence $F'$ is a proper filter, and clearly every proper filter $F'' \supseteq F'$ is disjoint from $I$, so $F''\not\in U^{**}$. It follows that $F'\in\neg (U^{**})$, which with $F\subseteq F'$ implies $F\not\in \neg\neg (U^{**})$. Thus, $\neg\neg (U^{**})\subseteq U^{**}$, so we have $U^{**}=U\in \mathcal{RO}(UV(\mathbb{A}))$.
For part (\ref{Nonprinc1a}), clearly $F\in\neg U$. Suppose for contradiction that $F\in U^*$, so there is a $c$ such that $F\in\widehat{c}$ and $U\cap\widehat{c}=\varnothing$. Since $F\in\widehat{c}$, we have $c\in F$. We claim that $F$ is the principal filter generated by $c$, i.e., $c\leq a$ for all $a\in F$. For if there is an $a\in F$ such that $c\not\leq a$, then $c\wedge - a\neq 0$, so there is a proper filter $G$ containing $c\wedge - a$. Hence $c,- a\in G$, so $G\in\widehat{c}$ and $G\in\widehat{- a}$. Since $a\in F$, $G\in\widehat{- a}$ implies $G\in U$. Then since $G\in\widehat{c}$, we have $G\in U\cap\widehat{c}$, contradicting $U\cap\widehat{c}=\varnothing$ above. Thus, $F\not\in U^*$.
For part (\ref{Nonprinc1b}), $U\subseteq\neg\neg U$ always holds. To see $\neg\neg U\subseteq U$, suppose $G\not\in U$. It follows by definition of $U$ that for all $a\in F$, $G\not\in \widehat{-a}$ and hence $-a\not\in G$. We claim that $G\subseteq F$. Suppose $b\not\in F$, so $-b\in F$ since $F$ is an ultrafilter. Then by what we derived above, $\mathnormal{--}b\not\in G$, i.e., $b\not \in G$. Thus, $G\subseteq F$. Then since $F\in\neg U$, we have $G\not\in \neg\neg U$.
For part (\ref{Nonprinc1c}), again $U\subseteq U^{**}$ always holds. Recall $U^*=\bigcup\{\widehat{c}\mid U\cap\widehat{c}=\varnothing\}$. Given the definition of $U$, the condition that $U\cap\widehat{c}=\varnothing$ is equivalent to: for all $a\in F$, $\widehat{-a}\cap\widehat{c}=\varnothing$. This is in turn equivalent to: for all $a\in F$, $-a\wedge c=0$, i.e., $c\leq a$. Since $F$ is a non-principal ultrafilter, the only $c$ such that $c\leq a$ for all $a\in F$ is given by $c:=0$. Thus, $U^*=\bigcup \{\widehat{0}\}=\bigcup\{\varnothing\}=\varnothing$. It follows that $U^{**}=UV(\mathbb{A})$. Then since $F\not\in U$, we have $U\subsetneq U^{**}$.
Part (\ref{Nonprinc1d}) is immediate from parts (\ref{Nonprinc1b})--(\ref{Nonprinc1c}).
For part (\ref{Nonprinc2}), since $U$ is open, $U=\bigcup\{\widehat{a_i}\mid i\in I\}$ for some $I$. Assuming $F\in\neg U$, we have $\neg a_i\in F$ for each $i\in I$. If $F$ is a principal filter generated by some $c$, then $c\leq \neg a_i$ for each $i\in I$, so $U\cap\widehat{c}=\varnothing$. Hence $F\in U^*$.
\end{proof}
\begin{remark} The inclusions \[\mathsf{CO}\mathcal{RO}(UV(\mathbb{A}))= \mathsf{CRO}(UV(\mathbb{A}))\subseteq\mathsf{RO}(UV(\mathbb{A}))\subseteq \mathsf{O}\mathcal{RO}(UV(\mathbb{A}))\] can be understood in terms of the dual correspondence between these types of regular open sets and ideals in the BA $\mathbb{A}$, as we will show in Section \ref{DictionarySection}:
\begin{eqnarray*}
\mathsf{O}\mathcal{RO}(UV(\mathbb{A})) & \quad\mbox{corresponds to}\quad & \mbox{ideals of }\mathbb{A}\\
\mathsf{RO}(UV(\mathbb{A})) & \quad\mbox{corresponds to}\quad & \mbox{normal ideals of }\mathbb{A}\\
\mathsf{CO}\mathcal{RO}(UV(\mathbb{A})) & \quad\mbox{corresponds to}\quad & \mbox{principal ideals of }\mathbb{A}. \\
= \mathsf{CRO}(UV(\mathbb{A}))
\end{eqnarray*}
\end{remark}
Given Theorem \ref{MainRep} and the fact that $\mathsf{CO}\mathcal{RO}(UV(\mathbb{A}))=\mathsf{CRO}(UV(\mathbb{A}))$, we can reason about elements of a BA as compact open sets in $UV(\mathbb{A})$ that are regular open in either the Alexandroff space $\mathsf{Up}(UV(\mathbb{A}))$ or in the spectral space $UV(\mathbb{A})$. Since the definition of a regular open set in the Alexandroff space is especially simple, given by the first-order condition (\ref{ROeq}) involving the specialization order $\leqslant$, we will continue to use this definition of regular open for the purposes of our calculations.
\section{Characterization of choice-free duals of BAs}\label{VietorisSection}
We now wish to characterize the spectral spaces $X$ that are homeomorphic to $UV(\mathbb{A})$ for some Boolean algebra $\mathbb{A}$. For the following definition, given $x\in X$, let $\mathsf{CO}\mathcal{RO}(x)=\{U\in \mathsf{CO}\mathcal{RO}(X)\mid x\in U\}$.
\begin{definition}\label{VOspace} A \textit{UV-space} is a $T_0$ space $X$ such that:
\begin{enumerate}
\item\label{CloseProp} $\mathsf{CO}\mathcal{RO}(X)$ is closed under $\cap$ and $\mathsf{int}_\leqslant (X\setminus \cdot)$ and is a basis for $X$;
\item\label{PossCompact} every proper filter in $\mathsf{CO}\mathcal{RO}(X)$ is $\mathsf{CO}\mathcal{RO}(x)$ for some $x\in X$.
\end{enumerate}
\end{definition}
\begin{remark} An equivalent definition of a $UV$-space (in light of Section \ref{ROsection} and the proof of Theorem \ref{SecondThm} below) substitutes $\mathsf{CRO}$ for $\mathsf{CO}\mathcal{RO}$ and $\mathsf{int}$ for $\mathsf{int}_\leqslant$ in Definition \ref{VOspace}.
\end{remark}
The conditions in Definition \ref{VOspace} are reminiscent of conditions mentioned earlier: compare part 1 with the statement of coherence in Definition \ref{SpectralDef} and part 2 with the statement of sobriety in Definition \ref{SpectralDef}. Note that the basis condition implies an analogue of the Priestley separation axiom \cite{Priestley1970}: if $x\not\leqslant y$, then there is a $U\in\mathsf{CO}\mathcal{RO}(X)$ such that $x\in U$ and $y\not\in U$.
\begin{prop}\label{COROBA} For any UV-space $X$, $\mathsf{CO}\mathcal{RO}(X)$ ordered by inclusion is a BA with the following operations:
\[U\wedge V=U\cap V\qquad\neg U=\mathsf{int}_\leqslant(X\setminus U)\qquad U\vee V=\mathsf{int}_\leqslant(\mathsf{cl}_\leqslant(U\cup V)).\]
\end{prop}
\begin{proof} As noted in Section \ref{PossSection}, it is a well-known result of Tarski that the collection of all regular open sets of a space forms a BA with the operations $\wedge$, $\neg$, and $\vee$ defined above (see, e.g., \cite[\S~4]{Halmos1963}). By Definition \ref{VOspace}.\ref{CloseProp}, in a UV-space $X$, $\mathsf{CO}\mathcal{RO}(X)$ with the operations $\wedge$ and $\neg$ is a subalgebra of the full regular open algebra and therefore a BA.\end{proof}
We now prove that Definition \ref{VOspace} provides our desired characterization.
\begin{theorem}\label{SecondThm} For any BA $\mathbb{A}$ and space $X$:
\begin{enumerate}
\item\label{SecondThmA} $UV(\mathbb{A})$ is a UV-space;
\item\label{SecondThmB} $X$ is homeomorphic to $UV(\mathsf{CO}\mathcal{RO}(X))$ iff $X$ is a UV-space.
\end{enumerate}
\end{theorem}
\begin{proof} For part \ref{SecondThmA}, to see that property \ref{CloseProp} of Definition \ref{VOspace} holds, if $U,V\in\mathsf{CO}\mathcal{RO}(UV(\mathbb{A}))$, then by the proof of Theorem \ref{MainRep} we have that $U=\widehat{a}$ and $V=\widehat{b}$ for some $a,b\in \mathbb{A}$. We also saw in the proof of Theorem \ref{MainRep} that $\widehat{a}\cap \widehat{b}=\widehat{a\wedge b}\in \mathsf{CO}\mathcal{RO}(UV(\mathbb{A}))$ and $\mathsf{int}_\leqslant (UV(\mathbb{A})\setminus \widehat{a})=\widehat{-a}\in \mathsf{CO}\mathcal{RO}(UV(\mathbb{A}))$. For property \ref{PossCompact}, if $\mathcal{F}$ is a proper filter in $\mathsf{CO}\mathcal{RO}(UV(\mathbb{A}))$, then by the proof of Theorem \ref{MainRep}, $G=\{a\in\mathbb{A}\mid\widehat{a} \in \mathcal{F}\}$ is a proper filter in $\mathbb{A}$. Then $G$ is an element of $UV(\mathbb{A})$ and $\mathsf{CO}\mathcal{RO}(G)=\mathcal{F}$.
For part \ref{SecondThmB}, the left-to-right direction follows from part \ref{SecondThmA}. For the right-to-left direction, we will show that the map $\epsilon :x\mapsto \mathsf{CO}\mathcal{RO}(x)$ is the desired homeomorphism from $X$ to $UV(\mathsf{CO}\mathcal{RO}(X))$. To see that $\epsilon$ is injective, if ${x\neq y}$, then by $T_0$, either $x\not\leqslant y$ or $y\not\leqslant x$, which by Definition \ref{VOspace}.\ref{CloseProp} implies $\mathsf{CO}\mathcal{RO}(x)\neq \mathsf{CO}\mathcal{RO}(y)$. That $\epsilon$ is surjective follows from Definition \ref{VOspace}.\ref{PossCompact}. To see that $\epsilon$ is continuous, it suffices to show that the inverse image of each basic open is open. A basic open of $UV(\mathsf{CO}\mathcal{RO}(X))$ is $\widehat{U}$ for some $U\in \mathsf{CO}\mathcal{RO}(X)$. Then we have:
\begin{eqnarray*}
\epsilon^{-1}[\widehat{U}]&=&\{x\in X\mid \mathsf{CO}\mathcal{RO}(x)\in \widehat{U}\} \\
&=&\{x\in X\mid U\in \mathsf{CO}\mathcal{RO}(x)\} \\
&=& \{x\in X\mid x\in U\}\\
&=& U.
\end{eqnarray*}
Finally, to see that $\epsilon^{-1}$ is continuous, we have
\begin{eqnarray*}
\epsilon[U]&=&\{\mathsf{CO}\mathcal{RO}(x)\mid x\in U\}\\
&=&\{\mathsf{CO}\mathcal{RO}(x)\mid U\in \mathsf{CO}\mathcal{RO}(x)\}\\
&=&\widehat{U}.
\end{eqnarray*}
For the last equality, the left-to-right inclusion uses that $\mathsf{CO}\mathcal{RO}(x)$ is a proper filter, while the right-to-left follows from the surjectivity of $\epsilon$.
\end{proof}
For the following, recall that for a space $X$, its specialization order is $\leqslant$.
\begin{cor}\label{UVspectral} Let $X$ be a UV-space. Then:
\begin{enumerate}
\item\label{UVspectral1} $X$ is a spectral space;
\item\label{UVspectral3} every set in $ \mathsf{CO}(X)$ is a finite union of sets from $\mathsf{CO}\mathcal{RO}(X)$;
\item\label{UVspectral4} $(X,\leqslant)$ may be obtained from a complete Heyting algebra\footnote{In Section \ref{L&H}, we strengthen `complete Heyting algebra' to `Stone locale', but we will wait to introduce this notion.} by deleting the top element, and each $U\in\mathsf{CO}\mathcal{RO}(X)$ is a filter in $(X,\leqslant)$;
\item\label{UVspectral4.5} if $X$ is finite, then $(X,\leqslant)$ may be obtained from a Boolean algebra by deleting the top element;
\item\label{UVspectral5} if $U\in\mathsf{CO}\mathcal{RO}(X)$ and $z\in X$, then there is a unique $x\in U$ and $y\in\neg U$ such that $z=x\sqcap y$ where $\sqcap$ is the meet operation in $(X,\leqslant)$.
\item\label{UVspectral6} if $U,V\in\mathsf{CO}\mathcal{RO}(X)$, then \[U\vee V=U\cup V\cup \{x\sqcap y\mid x\in U,\, y\in V \}.\]
\end{enumerate}
\end{cor}
\begin{proof} For part \ref{UVspectral1}, by Theorem \ref{SecondThm}.\ref{SecondThmB}, each UV-space $X$ is homeomorphic to the space $UV(\mathsf{CO}\mathcal{RO}(X))$, which is spectral by Proposition \ref{IsSpectral}.\ref{Spectral}. For part \ref{UVspectral3}, if $U\in \mathsf{CO}(X)$, then it is a finite union of basic open sets, so by Definition \ref{VOspace}.\ref{CloseProp}, it is a finite union of sets from $\mathsf{CO}\mathcal{RO}(X)$.
For part \ref{UVspectral4}, as $X$ is homeomorphic to the $T_0$ space $UV(\mathsf{CO}\mathcal{RO}(X))$ of proper filters of $\mathsf{CO}\mathcal{RO}(X)$, it follows that $(X,\leqslant)$ is order-isomorphic to the poset $(UV(\mathsf{CO}\mathcal{RO}(X)),\subseteq)$ of proper filters of $\mathsf{CO}\mathcal{RO}(X)$ ordered by inclusion. As observed by Tarski \cite{Tarski1937b}, the filters of any BA (indeed, any distributive lattice) ordered by inclusion form a complete Heyting algebra, so the proper filters ordered by inclusion form a complete Heyting algebra minus the top element. Finally, suppose $U=\widehat{a}$ for $a\in \mathbb{A}$, and $F,G\in UV(\mathbb{A})$ are such that $F,G\in \widehat{a}$. Then $a\in F\cap G=F\sqcap G$, so $F\sqcap G\in \widehat{a}=U$. It follows, given that $U$ is an upset, that $U$ is a filter in $(UV(\mathbb{A}),\subseteq)$.
For part \ref{UVspectral4.5}, if $X$ is finite, then the BA $\mathsf{CO}\mathcal{RO}(X)$ is finite. As in part \ref{UVspectral4}, ${(X,\leqslant)}$ is order-isomorphic to the poset of proper filters of $\mathsf{CO}\mathcal{RO}(X)$ ordered by inclusion. Since any filter in a finite BA is principal, we obtain that $(X,\leqslant)$ is order-isomorphic to the poset of proper \textit{principal} filters of $\mathsf{CO}\mathcal{RO}(X)$ ordered by inclusion, which is obviously isomorphic to $\mathsf{CO}\mathcal{RO}(X)$ minus its top element.
For part \ref{UVspectral5}, let $X=UV(\mathbb{A})$. If $U\in \mathsf{CO}\mathcal{RO}(UV(\mathbb{A}))$, then by Theorem \ref{SecondThm}.\ref{SecondThmB} and the proof of Theorem \ref{MainRep}, we have $U=\widehat{a}$ and $\neg U=\widehat{-a}$ for some $a\in\mathbb{A}$, which implies $\mathord{\uparrow}a\in U$ and $\mathord{\uparrow}\mathord{-a}\in \neg U$. Let $\sqcap$ and $\sqcup$ be the meet and join operations in the Heyting algebra arising from $(UV(\mathbb{A}),\subseteq)$, i.e., $F\sqcap G=F\cap G$ and $F\sqcup G$ is the filter generated by $F\cup G$. Let $\top$ be the top element of the Heyting algebra, which we may identify with the improper filter in $\mathbb{A}$. Thus, $\mathord{\uparrow}a\sqcup\mathord{\uparrow}\mathord{-}a=\top$. Now for any $F\in UV(\mathbb{A})$, we have $F=(F\sqcup \mathord{\uparrow}a)\sqcap(F\sqcup\mathord{\uparrow}\mathord{-}a)$. Suppose $G\in U$ and $H\in \neg U$, which implies $\mathord{\uparrow}a\subseteq G$ and $\mathord{\uparrow}\mathord{-a}\subseteq H$, and $F=G\sqcap H$. Then we have
\[ F\sqcup \mathord{\uparrow}a =(G\sqcap H)\sqcup \mathord{\uparrow}a= (G\sqcup \mathord{\uparrow}a)\sqcap (H\sqcup \mathord{\uparrow}a)=G\sqcap \top = G,\]
and similarly $F\sqcup\mathord{\uparrow}\mathord{-}a=H$. This completes the proof of part \ref{UVspectral5}.
For part \ref{UVspectral6}, we show that $\widehat{a}\vee \widehat{b}= \widehat{a}\cup \widehat{b}\cup \{F\sqcap G\mid F\in \widehat{a},\, G\in \widehat{b} \}$. By the proof of Theorem \ref{MainRep}, $\widehat{a}\vee \widehat{b}=\widehat{a\vee b}$. To see that $\widehat{a\vee b}\supseteq \widehat{a}\cup \widehat{b}\cup \{F\sqcap G\mid F\in \widehat{a},\, G\in \widehat{b} \}$, obviously $\widehat{a\vee b}\supseteq \widehat{a}\cup \widehat{b}$. If $F\in \widehat{a}$ and $G\in \widehat{b}$, so $a\in F$ and $b\in G$, then $a\vee b\in F\cap G=F\sqcap G$, so $F\sqcap G\in \widehat{a\vee b}$. To see that $\widehat{a\vee b}\subseteq \widehat{a}\cup \widehat{b}\cup \{F\sqcap G\mid F\in \widehat{a},\, G\in \widehat{b} \}$, if $H\in \widehat{a\vee b}$, so $a\vee b\in H$, and $H\not\in \widehat{a}\cup \widehat{b}$, so $a\not\in H$ and $b\not\in H$, then we claim that $H= (H\sqcup \mathord{\uparrow}a)\sqcap (H\sqcup \mathord{\uparrow}b)$. For if $c$ is in the right-hand side, then there are $a_0\in H$ and $b_0\in H$ such that $a_0\wedge a\leq c$ and $b_0\wedge b\leq c$, which implies $a_0\wedge b_0\wedge (a\vee b)\leq c$. Then since $a_0,b_0,a\vee b\in H$, we have $c\in H$. Finally, both $H\sqcup \mathord{\uparrow}a$ and $H\sqcup \mathord{\uparrow}b$ are proper filters. For if $H\sqcup \mathord{\uparrow}a$ is improper, then $-a\in H$, which with $a\vee b\in H$ implies $b\in H$, which contradicts what we derived above. Similarly, that $H\sqcup \mathord{\uparrow}b$ is improper leads to a contradiction.\end{proof}
\begin{cor}\label{StoneCor} For any Stone space $X$, $\mathscr{UV}(X)$ is a UV-space.
\end{cor}
\begin{proof} By Proposition \ref{UVStone}, for any Stone space $X$, $\mathscr{UV}(X)$ is homeomorphic to $UV(\mathsf{Clop}(X))$, which is a UV-space by Theorem \ref{SecondThm}.\ref{SecondThmA}.
\end{proof}
\section{Morphisms and choice-free duality for BAs}\label{DualitySection}
To go beyond representation to categorical duality, we introduce appropriate morphisms. A \textit{spectral map} \cite{Hochster1969} between spectral spaces $X$ and $X'$ is a map $f\colon X\to X'$ such that $f^{-1}[U]\in\mathsf{CO}(X)$ for each $U\in\mathsf{CO}(X')$, which implies that $f$ is continuous. We combine this definition with the standard notion (in modal logic) of a {p-morphism} between ordered sets (see, e.g., \cite[p.~30]{Chagrov1997}).
\begin{definition}\label{UVmapDef} A \textit{UV-map} between UV-spaces $X$ and $X'$ is a spectral map $f\colon X\to X'$ that also satisfies the \textit{p-morphism condition}:
\[\mbox{if $f(x) \leqslant' y'$, then $\exists y: x\leqslant y$ and $f(y)=y'$}.\]
\end{definition}
\begin{figure}[h]
\begin{center}
\begin{tikzpicture}[->,>=stealth',shorten >=1pt,shorten <=1pt, auto,node
distance=2cm,thick,every loop/.style={<-,shorten <=1pt}] \tikzstyle{every state}=[fill=gray!20,draw=none,text=black]
\node[circle,draw=black!100,fill=black!100, label=below:$x$,inner sep=0pt,minimum size=.175cm] (x) at (0,0) {{}};
\draw[rotate=90] (1,0) ellipse (2cm and .7cm);
\draw[rotate=90] (1,-3) ellipse (2cm and .7cm);
\draw[rotate=90] (1,-6) ellipse (2cm and .7cm);
\draw[rotate=90] (1,-9) ellipse (2cm and .7cm);
\node[circle,draw=black!100,fill=black!100, label=below:$\;f(x)$,inner sep=0pt,minimum size=.175cm] (x') at (3,0) {{}};
\node[circle,draw=black!100,fill=black!100, label=above:$y'$,inner sep=0pt,minimum size=.175cm] (y') at (3,2) {{}};
\path (x') edge[->] node {{}} (y');
\path (x) edge[bend left,dotted,->] node {{}} (x');
\node[circle,draw=black!100,fill=black!100, label=below:$x$,inner sep=0pt,minimum size=.175cm] (x2) at (6,0) {{}};
\node[circle,draw=black!100,fill=black!100, label=above:$\exists y$,inner sep=0pt,minimum size=.175cm] (y2) at (6,2) {{}};
\node[circle,draw=black!100,fill=black!100, label=below:$\;f(x)$,inner sep=0pt,minimum size=.175cm] (x'2) at (9,0) {{}};
\node[circle,draw=black!100,fill=black!100, label=above:$y'$,inner sep=0pt,minimum size=.175cm] (y'2) at (9,2) {{}};
\path (x2) edge[->] node {{}} (y2);
\path (x'2) edge[->] node {{}} (y'2);
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\path (y2) edge[bend left,dotted,->] node {{}} (y'2);
\end{tikzpicture}
\end{center}
\caption{The p-morphism condition of UV-maps.}\label{p-morphismFig}
\end{figure}
\begin{remark} A UV-map, like any continuous map, preserves the specialization order: if $x\leqslant y$, then $f(x)\leqslant f(y)$.
\end{remark}
\begin{fact}\label{PosetMaps} Let $P$ and $P'$ be partial orders, and let $f\colon P\to P'$ be an order-preserving map satisfying the p-morphism condition. If $U\in\mathcal{RO}(P')$, then $f^{-1}[U]\in\mathcal{RO}(P)$ \textnormal{(}where we regard $P,P'$ as spaces given by their upset topologies\textnormal{)}.
\end{fact}
\begin{proof} To see that $f^{-1}[U]\in\mathsf{Up}(P)$, suppose $x\in f^{-1}[U]$ and $x\leqslant y$. Then $f(x)\in U$, and since $f$ is order-preserving, $f(x)\leqslant' f(y)$, so $U\in \mathsf{Up}(P')$ implies $f(y)\in U$ and hence $y\in f^{-1}[U]$. Now to see that $f^{-1}[U]\in\mathcal{RO}(P)$, suppose $x\not\in f^{-1}[U]$, so $f(x)\not\in U$. Then since $U\in\mathcal{RO}(P')$, there is a $y'\geqslant' f(x)$ such that for all $z'\geqslant' y'$, we have $z'\not\in U$. It follows by the p-morphism condition that there is a $y$ such that $x\leqslant y$ and $f(y)=y'$. Then for any $z$ such that $y\leqslant z$, we have $f(y)\leqslant' f(z)$ and hence $y'\leqslant' f(z)$, which implies $f(z)\not\in U$ by our reasoning above, so $z\not\in f^{-1}[U]$. Thus, we have shown that if $x\not\in f^{-1}[U]$, then there is a $y\geqslant x$ such that for all $z\geqslant y$, $z\not\in f^{-1}[U]$. By (\ref{ROeq}), this completes the proof that $f^{-1}[U]\in\mathcal{RO}(P)$.
\end{proof}
From Fact \ref{PosetMaps} and the definition of UV-maps as special spectral spaces, we have the following.
\begin{cor}\label{InverseCor} Let $X$ and $X'$ be UV-spaces and $f\colon X\to X'$ a UV-map. Then $f^{-1}[U]\in\mathsf{CO}\mathcal{RO}(X)$ for every $U\in\mathsf{CO}\mathcal{RO}(X')$.\footnote{Cf.~the notion of an \textit{R-map} in \cite{Carnahan1973}, which is a map between spaces such that the inverse image of each regular open set is regular open.}
\end{cor}
Conversely, the condition that the inverse image of a $\mathsf{CO}\mathcal{RO}$ set is also $\mathsf{CO}\mathcal{RO}$ (or simply $\mathsf{CO}$) implies that $f$ is a spectral map.
\begin{fact}\label{IsSpectralMap} Let $X$ and $X'$ be UV-spaces. If $f\colon X\to X'$ is such that $f^{-1}[U]\in\mathsf{CO}(X)$ for every $U\in\mathsf{CO}\mathcal{RO}(X')$, then $f$ is a spectral map.
\end{fact}
\begin{proof} Suppose $f\colon X\to X'$ satisfies the assumption, and $U\in\mathsf{CO}(X')$. By Proposition \ref{UVspectral}.\ref{UVspectral3}, $U$ is a finite union $\underset{i\in I}{\bigcup} U_i$ of sets $U_i\in\mathsf{CO}\mathcal{RO}(X')$. Then $f^{-1}[U]=f^{-1}[\underset{i\in I}{\bigcup} U_i]=\underset{i\in I}{\bigcup} f^{-1}[U_i]$. By the assumption, $f^{-1}[U_i]\in\mathsf{CO}(X)$, so $f^{-1}[U]$ is a finite union of compact opens and is therefore compact open. Thus, $f$ is a spectral map.
\end{proof}
The following simple lemma is also useful.
\begin{lemma}\label{SubbasisLem} Let $X$ and $Y$ be spectral spaces and $f: X\to Y$. If for each set $U$ in some subbasis for $Y$, we have $f^{-1}[U]\in\mathsf{CO}(X)$, then $f$ is a spectral map.
\end{lemma}
\begin{proof} By definition, every open set is a union of finite intersections of subbasic sets. Thus, every compact open set $V$ is a finite union $V_1\cup\dots\cup V_n$ of finite intersections of subbasic sets. Then since
\[f^{-1}[V]=f^{-1}[V_1\cup\dots\cup V_n]=f^{-1}[V_1]\cup \dots\cup f^{-1}[V_n],\]
we have that $f^{-1}[V]$ is compact open if each $f^{-1}[V_i]$ is compact open. Now each $V_i$ is $U_1\cap\dots\cap U_n$ for some subbasic sets $U_1,\dots,U_n$. Then since
\[f^{-1}[V_i]=f^{-1}[U_1\cap\dots\cap U_n]=f^{-1}[U_1]\cap\dots\cap f^{-1}[U_n],\]
we have that $f^{-1}[V_i]$ is compact open if each $f^{-1}[U_j]$ is compact open. By assumption, each $f^{-1}[U_j]$ is compact open, so we are done.\end{proof}
One can easily check that UV-spaces with UV-maps form a category. We now prove the promised categorical duality.
\begin{theorem}\label{DualityThm} The category of UV-spaces with UV-maps is dually equivalent to the category of Boolean algebras with Boolean homomorphisms.
\end{theorem}
\begin{proof} Suppose $h\colon \mathbb{A}\to\mathbb{B}$ is a BA homomorphism. Given $F\in UV(\mathbb{B})$, let $h_+(F)=h^{-1}[F]$. Then since $h$ is a homomorphism, and $F$ is a proper filter in $\mathbb{B}$, it follows that $h_+(F)$ is a proper filter in $\mathbb{A}$. Thus, \[h_+:UV(\mathbb{B})\to UV(\mathbb{A}).\] We claim that $h_+$ is a UV-map. First, to see that $h_+$ is a spectral map, it suffices by Lemma \ref{SubbasisLem} to show that for each basic open $\widehat{a}$ of $UV(\mathbb{A})$, we have $h_+^{-1}[\widehat{a}]\in\mathsf{CO}(UV(\mathbb{B}))$. Indeed,
\begin{eqnarray*}
h_+^{-1}[\widehat{a}]&=&\{F\in UV(\mathbb{B})\mid h_+(F)\in \widehat{a}\} \\
&=&\{F\in UV(\mathbb{B})\mid h^{-1}[F]\in \widehat{a}\} \\
&=& \{F\in UV(\mathbb{B})\mid a\in h^{-1}[F]\} \\
&=& \{F\in UV(\mathbb{B})\mid h(a)\in F\}\\
&=&\widehat{h(a)},
\end{eqnarray*}
and $\widehat{h(a)}$ is compact open by the proof of Proposition \ref{IsSpectral}.\ref{Spectral}.
Next, we show that $h_+$ satisfies the p-morphism condition:
\[\mbox{if $h_+(F) \leqslant' G'$, then $\exists G: F\leqslant G$ and $h_+(G)=G'$}.\]
If $G'\in UV(\mathbb{A})$ and $h_+(F)\subseteq G'$, we claim that the filter $G$ generated by $h[G']\cup F$ is a proper filter. If not, then there are some $c_1,\dots,c_n\in h[G']$ such that $-(c_1\wedge\dots\wedge c_n)\in F$. Since $c_1,\dots,c_n\in h[G']$, there are some $c'_1,\dots,c'_n\in G'$ such that $h(c'_i)=c_i$, so $-(h(c'_1)\wedge\dots\wedge h(c'_n))\in F$. Then since $h$ is a homomorphism, we have $h(-(c_1\wedge\dots\wedge c_n))\in F$, so that $-(c_1\wedge\dots\wedge c_n)\in h^{-1}[F]=h_+(F)$, which with $h_+(F)\subseteq G'$ implies $-(c_1\wedge\dots\wedge c_n)\in G'$, which contradicts the fact that $c'_1,\dots,c'_n\in G'$ and $G'$ is a proper filter. Thus, $G$ is indeed a proper filter, and we have both $F\subseteq G$ and $G'\subseteq h^{-1}[G]=h_+(G)$. Finally, we claim that $h_+(G)\subseteq G'$.\footnote{Thanks to David Gabelaia and Mamuka Jibladze for pointing out this strengthening of the original proof.} For if $c'\in h_+(G)$, so $h(c')\in G$, then by definition of $G$ there is a $b'\in G'$ and $a\in F$ such that $h(b')\wedge a\leq h(c')$, which implies $a\leq -h(b')\vee h(c')$ and hence $a\leq h(-b'\vee c')$. Then since $a\in F$, we have $h(-b'\vee c')\in F$, so $-b'\vee c'\in h^{-1}[F]=h_+(F)$. Since $h_+( F )\subseteq G'$, it follows that $-b'\vee c'\in G'$, which with $b'\in G'$ implies $c'\in G'$, which completes the proof that $h_+(G)\subseteq G'$. Thus, $h_+(G)= G'$, so $h_+$ satisfies the p-morphism condition.
Finally, it is easy to see that $(\cdot)_+$ preserves the identity and composition. Thus, together $UV(\cdot)$ and $(\cdot)_+$ give us a contravariant functor from the category of BAs with BA homomorphisms to the category of UV-spaces with UV-maps.
In the other direction, suppose $f: X\to Y$ is a UV-map. Given $U\in\mathsf{CO}\mathcal{RO}(Y)$, let $f^+(Y)=f^{-1}[Y]$. Then by Corollary \ref{InverseCor}, \[f^+: \mathsf{CO}\mathcal{RO}(Y)\to \mathsf{CO}\mathcal{RO}(X).\] We claim that $f^+$ is a BA homomorphism. First, $f^+(U\wedge V)=f^{-1}[U\cap V]=f^{-1}[U]\cap f^{-1}[V]=f^+(U)\wedge f^+(V)$. Second, since $f$ is a UV-map, we have that for all $x\in X$ and $U \in \mathsf{CO}\mathcal{RO}(Y)$, $\mathord{\Uparrow}f(x)\cap U=\varnothing$ iff ${\mathord{\Uparrow}x\cap f^{-1}[U]=\varnothing}$. It follows that $f^{-1}[\mathsf{int}_\leqslant (Y\setminus U)]=\mathsf{int}_\leqslant (X\setminus f^{-1}[U])$ and hence $f^+(\neg U)=\neg f^+(U)$. It is also easy to see that $(\cdot)^+$ preserves the identity and composition. Thus, together $\mathsf{CO}\mathcal{RO}(\cdot)$ and $(\cdot)^+$ give us a contravariant functor from the category of UV-spaces with UV-maps to the category of BAs with BA homomorphisms.
In Theorems \ref{MainRep} and \ref{SecondThm}.\ref{SecondThmB} we showed that each BA $\mathbb{A}$ is isomorphic to $\mathsf{CO}\mathcal{RO}(UV(\mathbb{A}))$ and each UV-space $X$ is homeomorphic to $UV(\mathsf{CO}\mathcal{RO}(X))$.
Finally, it is not difficult to check that the following diagrams commute for any BA homomorphism $h:\mathbb{A}\to\mathbb{B}$ and UV-map $f:X\to Y$:
\begin{center}
\begin{tikzpicture}[->,>=stealth',shorten >=1pt,shorten <=1pt, auto,node
distance=2cm,thick,every loop/.style={<-,shorten <=1pt}]
\tikzstyle{every state}=[fill=gray!20,draw=none,text=black]
\node (A) at (0,2) {{$\mathbb{A}$}};
\node (A') at (0,0) {{$\mathsf{CO}\mathcal{RO}(UV(\mathbb{A}))$}};
\node (B) at (5,2) {{$\mathbb{B}$}};
\node (B') at (5,0) {{$\mathsf{CO}\mathcal{RO}(UV(\mathbb{B}))$}};
\path (A) edge[->] node {{$h$}} (B);
\path (A') edge[<-] node {{}} (A);
\path (B) edge[->] node {{}} (B');
\path (B') edge[<-] node {{$(h_+)^+$}} (A');
\end{tikzpicture}
\end{center}
\begin{center}
\begin{tikzpicture}[->,>=stealth',shorten >=1pt,shorten <=1pt, auto,node
distance=2cm,thick,every loop/.style={<-,shorten <=1pt}]
\tikzstyle{every state}=[fill=gray!20,draw=none,text=black]
\node (A) at (0,2) {{$X$}};
\node (A') at (0,0) {{$UV(\mathsf{CO}\mathcal{RO}(X))$}};
\node (B) at (5,2) {{$Y$}};
\node (B') at (5,0) {{$UV(\mathsf{CO}\mathcal{RO}(Y))$}};
\path (A) edge[->] node {{$f$}} (B);
\path (A') edge[<-] node {{}} (A);
\path (B) edge[->] node {{}} (B');
\path (B') edge[<-] node {{$(f^+)_+$}} (A');
\end{tikzpicture}
\end{center}
This completes the proof.
\end{proof}
\section{The hyperspace approach and the localic approach}\label{L&H} In this section, we relate the hyperspace approach to choice-free duality using UV-spaces to the localic approach using Stone locales.
Recall that a \textit{locale} is a complete lattice $L$ satisfying the join-infinite distributive law for each $a\in L$ and $Y\subseteq L$:
\[a\wedge \bigvee Y= \bigvee \{a\wedge y\mid y\in Y\}.\]
The collection of open sets of any space ordered by $\subseteq$ is a locale. In point-free topology, it is locales rather than spaces that are the basic objects. If we ignore choices of signature, then a lattice is a locale iff it is a complete Heyting algebra. For more information on locales, see, e.g., \cite{Johnstone1982,Picado2012}.
A locale is \textit{compact} if $\bigvee Y=1$ implies $\bigvee Y_0=1$ for some finite $Y_0\subseteq Y$. A locale is \textit{zero-dimensional} if each element of the locale is a join of complemented elements, where an element $a$ is complemented if there exists an element $b$ such that $a\wedge b=0$ and $a\vee b=1$.
\begin{definition} A \textit{Stone locale} is a compact zero-dimensional locale.
\end{definition}
The name `Stone locale' is justified by the fact that the locale of any Stone space is a Stone locale, and assuming the Boolean Prime Ideal Theorem, every Stone locale $L$ is the locale of opens of a Stone space, namely the Stone dual of the BA of complemented elements of $L$.
As mentioned in Section \ref{PossSection}, Stone locales provide another kind of choice-free Stone duality for BAs. A proof of the following may be found in \cite{Bezhanishvili2015}.
\begin{theorem}\label{StoneLocaleThm} The category of Stone locales with localic maps\footnote{For the definition of localic maps, see, e.g., \cite[\S~II.2]{Picado2012}.} is dually equivalent to the category of BAs with Boolean homomorphisms.
\end{theorem}
The key to Theorem \ref{StoneLocaleThm} is the following correspondence.
\begin{lemma}\label{ZLem} $\,$
\begin{enumerate}
\item\label{ZLem1} For any BA $\mathbb{A}$, $(\mathrm{Filt}(\mathbb{A}),\subseteq)$ is a Stone locale.
\item\label{ZLem2} $L$ is a Stone locale iff $L$ is isomorphic to $(\mathrm{Filt}(Z(L)),\subseteq)$ where $Z(L)$ is the Boolean algebra of complemented elements of $L$.
\end{enumerate}
\end{lemma}
\begin{proof}(sketch) Part \ref{ZLem1} is straightforward to check. For part \ref{ZLem2}, the right-to-left direction follows from part \ref{ZLem1}. For the left-to-right direction, the isomorphism sends $a\in L$ to $\mathord{\uparrow}a\cap Z(L)$ (see \cite{Bezhanishvili2015}).
\end{proof}
We can characterize UV-spaces as the result of putting an appropriate topology on the (non-maximum) elements of a Stone locale. Given a Stone locale $L$, just as Johnstone \cite[\S~4.1]{Johnstone1982} defines the \textit{Vietoris space of $L$}, we may define the \textit{upper Vietoris space of $L$}. The starting observation is that in defining the upper Vietoris space of a Stone space $X$, instead of taking the points of the new space to be the nonempty closed sets of $X$, we can take the points to be the complements of such sets, i.e., the open sets of $X$ not equal to $X$. Then for $U\in\Omega(X)$, instead of defining
\[\Box U =\{F\in \mathsf{F}(X)\mid F\subseteq U\},\]
we define
\begin{eqnarray*}
\blacksquare U &=&\{V\in \Omega(X)\setminus\{X\}\mid V^c\subseteq U\} \\
& =&\{V\in \Omega(X)\setminus\{X\}\mid U\cup V=X\}
\end{eqnarray*}
and let the topology be generated by $\{\blacksquare U\mid U\in\Omega(X)\}$. With this change of perspective, we can define the Vietoris space entirely in terms of the locale $\Omega(X)$, motivating the following definition.
\begin{definition}\label{UVofLocale} The \textit{upper Vietoris space} of a Stone locale $L$ is the space whose set of points is $L^-=\{x\in L\mid x\neq 1\}$ and whose topology is generated by the sets \[\blacksquare x= \{y\in L^-\mid x\vee y=1\},\; x\in L.\]
\end{definition}
Now suppose $L$ is the Stone locale $(\mathrm{Filt}(\mathbb{A}),\subseteq)$ for a BA $\mathbb{A}$. The join $F\vee G$ of two filters $F,G\in L$ is the filter generated by $F\cup G$, and the top element $1$ of $L$ is the improper filter. Our $UV(\mathbb{A})$ is exactly the topological space based on $L^-$ with the topology generated by the sets
\[\widehat{a}=\{F\in \mathrm{PropFilt}(\mathbb{A})\mid a\in F\},\;a\in\mathbb{A}.\]
We can now see that $UV(\mathbb{A})$ is exactly the upper Vietoris space of the Stone locale $(\mathrm{Filt}(\mathbb{A}),\subseteq)$.
\begin{prop}\label{SameTopologies} Let $L$ be the Stone locale of filters of a BA $\mathbb{A}$. Then the topology on $L^-$ generated by $\{\blacksquare x\mid x\in L\}$ is equal to the topology on $L^-$ generated by $\{\widehat{a}\mid a\in\mathbb{A}\}$.\end{prop}
\begin{proof} Given $a\in\mathbb{A}$, we have: \[\widehat{a}=\blacksquare\mathord{\uparrow}\mathord{-}a.\] For $\widehat{a}\subseteq\blacksquare\mathord{\uparrow}\mathord{-}a$, if $F\in\widehat{a}$, so $F$ is a proper filter with $a\in F$, then clearly $F\vee \mathord{\uparrow}\mathord{-}a$, i.e., the filter generated by $F\cup \mathord{\uparrow}\mathord{-}a$, is the improper filter, so $F\in \blacksquare\mathord{\uparrow}\mathord{-}a$. For $\widehat{a}\supseteq\blacksquare\mathord{\uparrow}\mathord{-}a$, if $F\in \blacksquare\mathord{\uparrow}\mathord{-}a$, so $F$ is a proper filter such that the filter generated by $F\cup \mathord{\uparrow}\mathord{-}a$ is improper, then $a\in F$, so $F\in\widehat{a}$.
Given $F\in L$, we have:
\[\blacksquare F=\bigcup\{\widehat{-a}\mid a\in F\}.\]
For the left-to-right inclusion, suppose $G\in \blacksquare F$, so $G$ is a proper filter such that $F\vee G$ is the improper filter. Hence there is some $a\in F$ such that $-a\in G$, so that $G\in\widehat{-a}$ and hence $G\in \bigcup\{\widehat{-a}\mid a\in F\}$. From right to left, suppose $G\in \bigcup\{\widehat{-a}\mid a\in F\}$, so for some $a\in F$, $G\in\widehat{-a}$, which means $-a\in G$. Then clearly $F\vee G$ is the improper filter, so $G\in \blacksquare F$.
\end{proof}
Combining Proposition \ref{SameTopologies} with Definitions \ref{UVofBA} and \ref{UVofLocale}, we have the following as an immediate corollary.
\begin{cor}\label{SameTopCor} For any BA $\mathbb{A}$, $UV(\mathbb{A})$ is the upper Vietoris space of the Stone locale $(\mathrm{Filt}(\mathbb{A}),\subseteq)$.
\end{cor}
We can now justify our choice of the terminology `UV-space' with the following choice-free characterization.
\begin{theorem}\label{UVJustification} $X$ is a UV-space iff $X$ is homeomorphic to the upper Vietoris space of a Stone locale.
\end{theorem}
\begin{proof} Suppose $X$ is a UV-space. Then by Theorem \ref{SecondThm}.\ref{SecondThmB}, $X$ is homeomorphic to $UV(\mathsf{CO}\mathcal{RO}(X))$. By Corollary \ref{SameTopCor}, $UV(\mathsf{CO}\mathcal{RO}(X))$ is the upper Vietoris space of the Stone locale $(\mathrm{Filt}(\mathsf{CO}\mathcal{RO}(X)),\subseteq)$. Thus, $X$ is homeomorphic to the upper Vietoris space of a Stone locale.
Conversely, suppose $X$ is homeomorphic to the upper Vietoris space of a Stone locale $L$. By Lemma \ref{ZLem}.\ref{ZLem2}, $L$ is isomorphic to $(\mathrm{Filt}(Z(L)),\subseteq)$. Thus, $X$ is homeomorphic to the upper Vietoris space of $(\mathrm{Filt}(Z(L)),\subseteq)$, which is equal to $UV(Z(L))$ by Corollary \ref{SameTopCor}, which is a UV-space by Theorem \ref{SecondThm}.\ref{SecondThmA}. Thus, $X$ is a UV-space.
\end{proof}
Theorem \ref{UVJustification} is a choice-free point-free analogue of the statement that $X$ is a UV-space iff $X$ is homeomorphic to $\mathscr{UV}(Y)$ for a Stone space $Y$. The left-to-right direction of that statement assumes the Boolean Prime Ideal Theorem (see Section \ref{UVStoneSection}). But by switching from Stone spaces to Stone locales, one obtains Theorem \ref{UVJustification} without choice.
\begin{remark}For a Stone locale $L$, in addition to defining the Vietoris space of $L$, Johnstone \cite[\S~4.1]{Johnstone1982} defines the \textit{Vietoris locale of $L$}, also known as the \textit{Vietoris powerlocale of $L$}.\footnote{Johnstone studies these constructions for any compact regular locale $L$, but here we need only consider Stone locales.} This is a purely localic construction, and the terminology is justified by the fact that the space of points of the Vietoris locale of $L$ is homeomorphic to the Vietoris space of $L$. Similarly, one can give a purely localic construction of the \textit{upper Vietoris locale of $L$}, also known as the \textit{upper powerlocale of $L$} \cite{Vickers1997,Vickers2009}, such that its space of points is homeomorphic to the upper Vietoris space of $L$.\end{remark}
Figure \ref{Locales&Spaces} below relates the different constructions we have discussed, viewed as ways of constructing the dual UV-space of a given BA.
\begin{figure}[h]
\begin{center}
\begin{tikzpicture}[->,>=stealth',shorten >=1pt,shorten <=1pt, auto,node
distance=2cm,thick,every loop/.style={<-,shorten <=1pt}] \tikzstyle{every state}=[fill=gray!20,draw=none,text=black]
\node [circle,draw=black!100,fill=black!100, label=below:$$,inner sep=0pt,minimum size=.175cm] (a) at (0,0) {{}};
\node [circle,draw=black!100,fill=black!100, label=above:$$,inner sep=0pt,minimum size=.175cm] (b) at (3,2) {{}};
\node [circle,draw=black!100,fill=black!100, label=above:$$,inner sep=0pt,minimum size=.175cm] (b') at (9,2) {{}};
\node [circle,draw=black!100,fill=black!100, label=above:$$,inner sep=0pt,minimum size=.175cm] (c) at (3,-2) {{}};
\node [circle,draw=black!100,fill=black!100, label=above:$$,inner sep=0pt,minimum size=.175cm] (c') at (9,-2) {{}};
\path (a) edge[->] node {{}} (b);
\node at (1.3,1.3) {{$\rotatebox{34}{locale $L$ of filters}$}};
\path (b) edge[->, bend left] node {{}} (a);
\node at (2,0.4) {{$\rotatebox{34}{$Z$}$}};
\path (a) edge[->] node {{}} (c);
\node at (1.3,-1.3) {{$\rotatebox{-34}{Stone space $X$}$}};
\path (c) edge[->, bend right] node {{}} (a);
\node at (2,-0.4) {{$\rotatebox{-34}{$\mathsf{Clop}$}$}};
\path (b) edge[->] node {{upper Vietoris locale of $L$}} (b');
\path (c') edge[<-] node {{upper Vietoris space of $X$}} (c);
\path (b) edge[->] node {{pt}} (c);
\path (b') edge[->] node {{pt}} (c');
\path (b) edge[->] node {{}} (c');
\path (b') edge[->,bend right=80] node {{}} (a);
\node at (3.75,3.35) {{$\rotatebox{14}{{\footnotesize compact regular}}$}};
\node at (3.75,3.05) {{$\rotatebox{14}{{\footnotesize \, elements}}$}};
\path (c') edge[->,bend left=80] node {{}} (a);
\node at (3.75,-3.35) {{$\rotatebox{-14}{$\mathsf{CO}\mathcal{RO}=\mathsf{CRO}$}$}};
\node at (6.5,0) {{$\rotatebox{-34}{upper Vietoris space of $L$}$}};
\node at (-.75,0) {{BA $\mathbb{A}$}};
\node at (9.5,-2.4) {{$UV(\mathbb{A})$}};
\node at (-.7,2.05) {{locales:}};
\node at (-.7,-2.05) {{spaces:}};
\end{tikzpicture}
\caption{Routes to the dual UV-space of a BA and back.}\label{Locales&Spaces}
\end{center}
\end{figure}
\section{Duality dictionary}\label{DictionarySection} In this section, we explain the dictionary in Table \ref{Dictionary} for translating between BA notions and UV notions.
\begin{table}[h]
\begin{center}
\scriptsize
\begin{tabular}{l|l|l}
\hline
{\bf BA} & {\bf UV} & {\bf Stone } \\
\hline \hline
BA & UV-space & Stone space \\
\hline
homomorphism & UV-map & continuous map \\
\hline
filter & $\mathord{\Uparrow}x$, $x\in X$ & closed set \\
\hline
ideal & $U\in \mathsf{O}\mathcal{RO}(X)$ & open set\\
\hline
principal filter & $U\in \mathsf{CO}\mathcal{RO}(X)$ & clopen set \\
\hline
principal ideal & $U\in \mathsf{CO}\mathcal{RO}(X)$ & clopen set \\
\hline
maximal filter & $\{x\}$, $x\in\mathsf{Max}_\leqslant(X)$ & $\{x\}$, $x\in X$ \\
\hline
maximal ideal & $X\setminus\mathord{\Downarrow}x$, $x\in\mathsf{Max}_\leqslant(X)$ & $X\setminus \{x\}$, $x\in X$ \\
\hline
normal ideal & $U\in\mathsf{RO}(X)$ & $U\in\mathsf{RO}(X)$ \\
\hline
relativization & subspace $U\in\mathsf{CO}\mathcal{RO}(X)$ & subspace $U\in\mathrm{Clop}(X)$ \\
\hline
complete algebra & complete UV-space & ED Stone space \\
\hline
atom & isolated point & isolated point \\
\hline
atomless algebra & $X_\mathrm{iso}=\varnothing$ & $X_\mathrm{iso}=\varnothing$ \\
\hline
atomic algebra & $\mathsf{cl}(X_\mathrm{iso})=X$ & $\mathsf{cl}(X_\mathrm{iso})=X$ \\
\hline
homomorphic image & subspace induced by $\mathord{\Uparrow}x$, $x\in X$ & subspace induced by closed set \\
\hline
subalgebra & image under UV-map & image under continuous map \\
\hline
direct product & UV-sum & disjoint union \\
\hline
canonical extension & $\mathcal{RO}(X)$ & $\wp(X)$ \\
\hline
MacNeille completion & $\mathsf{RO}(X)$ & $\mathsf{RO}(X)$ \\
\hline
\end{tabular}
\end{center}
\vspace{5mm} \caption{Dictionary for \textbf{BA}, \textbf{UV}, and \textbf{Stone}.}\label{Dictionary}
\end{table}
\subsection{Filters and ideals}
For a filter $F$ and ideal $I$ in a BA $\mathbb{A}$, we define:
\begin{eqnarray*}
\eta (F)&=& \bigcap\{\widehat{a}\mid a\in F\};\\
\zeta(I)&=& \bigcup\{\widehat{a}\mid a\in I\}.
\end{eqnarray*}
\begin{fact}\label{EtaFact} Let $\mathbb{A}$ be a BA and $X$ its dual UV-space. The map $\eta$ is a dual isomorphism between the poset of proper filters of $\mathbb{A}$ \textnormal{(}ordered by inclusion\textnormal{)} and the poset of principal upsets in the specialization order of $X$ \textnormal{(}ordered by inclusion\textnormal{)}.
\end{fact}
\begin{proof}
Given a filter $F$ in $\mathbb{A}$, we have
\begin{eqnarray*}
\eta (F) &=& \bigcap\{\widehat{a}\mid a\in F\} \\
&=&\{F'\in UV(\mathbb{A})\mid \forall a\in F : F'\in\widehat{a}\} \\
&=&\{F'\in UV(\mathbb{A})\mid \forall a\in F : a\in F' \} \\
&=&\{F'\in UV(\mathbb{A})\mid F\subseteq F' \} \\
&=& \mathord{\Uparrow}F,
\end{eqnarray*}
where we recall that $\mathord{\Uparrow}F=\{G\in X\mid F\leqslant G\}$. By the same argument, any principal upset $\mathord{\Uparrow}F$ in the specialization order of the UV-space is equal to $\eta(F)$. Finally, it is clear that $F\subseteq F'$ iff $\eta(F)\supseteq \eta(F')$.
\end{proof}
\begin{fact}\label{IdealIso} Let $\mathbb{A}$ be a BA and $X$ its dual UV-space. The map $\zeta$ is an isomorphism between the poset of ideals of $\mathbb{A}$ \textnormal{(}ordered by inclusion\textnormal{)} and $(\mathsf{O}\mathcal{RO}(X),\subseteq)$.
\end{fact}
\begin{proof} First, we show that for any ideal $I$ in $\mathbb{A}$, we have $\zeta(I)\in \mathsf{O}\mathcal{RO}(X)$. The set $\zeta(I)$ is a union of basic opens and hence is open. We claim that $\zeta(I)$ is also an $\mathcal{RO}$ set. To see that it is an $\leqslant$-upset, if $F\in \zeta(I)$, so for some $a\in I$, we have $F\in\widehat{a}$ and hence $a\in F$, then for any $F'\supseteq F$, we have $a\in F'$ and hence $F'\in\widehat{a}$, so $F'\in \zeta(I)$. Then to see that $\zeta(I)$ is an $\mathcal{RO}$ set, suppose $F\not\in \zeta(I)$, so for all $a\in I$, $F\not\in\widehat{a}$ and hence $a\not\in F$. Let $F'$ be the filter generated by $F\cup \{-a\mid a\in I\}$. We claim that $F'$ is proper. If not, then there are $b\in F$ and $a_1,\dots,a_n\in I$ such that $b\wedge -a_1\wedge\dots\wedge - a_n=0$, so $b\leq a_1\vee\dots\vee a_n$. Then since $F$ is a filter, $b\in F$ implies $a_1\vee\dots \vee a_n\in F$. But since $I$ is an ideal, $a_1,\dots,a_n\in I$ implies $a_1\vee\dots\vee a_n\in I$ and hence $a_1\vee\dots\vee a_n\not\in F$ by our choice of $F$. From this contradiction, we conclude that $F'$ is proper. Then since $-a\in F'$ for each $a\in I$, it follows that for any proper filter $F''\supseteq F'$, we have $F''\not\in \zeta(I)$. This shows that $\zeta(I)$ is an $\mathcal{RO}$ set by (\ref{ROeq}). This in turn completes the proof that $\zeta(I)\in \mathsf{O}\mathcal{RO}(X)$, and it is easy to see that $I\subseteq I'$ iff $\zeta(I)\subseteq \zeta(I')$.
Finally, to see that $f$ is surjective, given any $\mathsf{O}\mathcal{RO}$ subset $U$ of $UV(\mathbb{A})$, by the proof of Theorem \ref{MainRep} we have $U=\bigcup\{\widehat{a}\mid \widehat{a}\subseteq U\}$. We claim that the set $I=\{a\mid \widehat{a}\subseteq U\}$ is an ideal in $\mathbb{A}$. If $a\in I$, so $\widehat{a}\subseteq U$, then for any $b\leq a$, we have $\widehat{b}\subseteq\widehat{a}$ and hence $\widehat{b}\subseteq U$, so $b\in I$. Finally, if $a,b\in I$, so $\widehat{a},\widehat{b}\subseteq U$ and hence $\widehat{a}\cup\widehat{b}\subseteq U$, then we have $\mathsf{int}_\leqslant(\mathsf{cl}_\leqslant(\widehat{a}\cup\widehat{b}))\subseteq \mathsf{int}_\leqslant(\mathsf{cl}_\leqslant(U))$. Since $U$ is an $\mathcal{RO}$ set, we have $\mathsf{int}_\leqslant(\mathsf{cl}_\leqslant(U))=U$, and then since $\mathsf{int}_\leqslant(\mathsf{cl}_\leqslant(\widehat{a}\cup\widehat{b}))=\widehat{a\vee b}$, it follows that $\widehat{a\vee b}\subseteq U$ and hence $a\vee b\in I$. Thus, $I$ is an ideal, and clearly $\zeta(I)=U$.
\end{proof}
\begin{fact} Let $\mathbb{A}$ be a BA and $X$ its dual UV-space. The restriction of $\eta$ to principal filters is a dual isomorphism between the poset of principal filters of $\mathbb{A}$ \textnormal{(}ordered by inclusion\textnormal{)} and $(\mathsf{CO}\mathcal{RO}(X),\subseteq)$.
\end{fact}
\begin{proof} The map $\mathord{\uparrow}a\mapsto \widehat{a}$ is the dual isomorphism, using the fact from Theorem \ref{MainRep} that $\widehat{a}\in\mathsf{CO}\mathcal{RO}(UV(\mathbb{A}))$. \end{proof}
\begin{fact} Let $\mathbb{A}$ be a BA and $X$ its dual UV-space. The restriction of $\zeta$ to principal ideals is a dual isomorphism between the poset of principal ideals of $\mathbb{A}$ \textnormal{(}ordered by inclusion\textnormal{)} and $(\mathsf{CO}\mathcal{RO}(X),\subseteq)$.
\end{fact}
\begin{proof} The map $\mathord{\downarrow}a\mapsto \widehat{a}$ is the dual isomorphism. \end{proof}
\begin{fact} Let $\mathbb{A}$ be a BA and $X$ its dual UV-space. The restriction of $\eta$ to maximal filters is a bijection between the collection of maximal filters of $\mathbb{A}$ and the collection of singleton sets $\{x\}$ for $x\in\mathsf{Max}_\leqslant(X)$.
\end{fact}
\begin{proof} Since the specialization order $\leqslant$ of $X$ is the inclusion order $\subseteq$ on proper filters of $\mathbb{A}$, the elements of $\mathsf{Max}_\leqslant(X)$ are exactly the maximal filters of $\mathbb{A}$. By Fact \ref{EtaFact}, for any filter $F$, $\eta(F)=\mathord{\Uparrow}F$, so if $F$ is a maximal filter, then $\eta(F)=\mathord{\Uparrow}F=\{F\}$.\end{proof}
\begin{fact} Let $\mathbb{A}$ be a BA and $X$ its dual UV-space. The restriction of $\zeta$ to maximal ideals is a bijection between the collection of maximal ideals of $\mathbb{A}$ and the collection of sets $X\setminus \mathord{\Downarrow}x$ for $x\in\mathsf{Max}_\leqslant(X)$.
\end{fact}
\begin{proof} If $I$ is a maximal ideal in $\mathbb{A}$, then the complement $F$ of $I$ is a maximal filter in $\mathbb{A}$ and hence an element of $\mathsf{Max}_\leqslant(X)$. We claim that $\zeta(I)= \bigcup\{\widehat{a}\mid a\in I\}=X\setminus \mathord{\Downarrow}F$. For the $\subseteq$ inclusion, if $G\in \zeta(I)$, then for some $a\in I$, we have $G\in\widehat{a}$ and hence $a\in G$, which implies $G\not\subseteq F$. Conversely, if $G\in X\setminus \mathord{\Downarrow}F$, then $G\not\subseteq F$, so there is an $a\in G$ such that $a\not\in F$. Thus, we have an $a\in I$ such that $G\in \widehat{a}$ and hence $G\in \zeta(I)$.\end{proof}
In Section \ref{CompletionSection} we will prove a correspondence between the \textit{normal} ideals of $\mathbb{A}$ and sets in $\mathsf{RO}(UV(\mathbb{A}))$.
\subsection{Relativization} As one would expect by analogy with standard Stone duality, the operation on a UV-space dual to relativizing a BA to an element is the operation of taking a $\mathsf{CO}\mathcal{RO}$ subspace of a UV-space.
\begin{prop}\label{SubSpaceProp} Let $X$ be a UV-space. If $U\in\mathsf{CO}\mathcal{RO}(X)$, then $U$ with the subspace topology is a UV-space.
\end{prop}
\begin{proof} It is well known that every compact open subspace of a spectral space is again spectral.\footnote{This fact does not use any choice. To see that an open subspace of $X$ is sober, suppose $U$ is such a subspace. To prove that $U$ is sober, it suffices to show (see \cite[p.~2]{Picado2012}) that any open $V\subsetneq U$ is meet-irreducible iff it is the complement of the closure of a point. Let $V$ be an open proper subset of $U$, and suppose $V$ is meet-irreducible, so for all open $A,B\subseteq U$, if $A\cap B\subseteq V$, then $A\subseteq V$ or $B\subseteq V$. But then note that $V$ is also a meet-irreducible proper open subset of $X$. So by the sobriety of $X$, we have $V = X\setminus \mathsf{cl}\{x\}$ for some $x\in X$. Now if $x\in X\setminus U$ and hence $\mathsf{cl}\{x\}\subseteq X\setminus U$, then together $V\subsetneq U$ and $V = X\setminus \mathsf{cl}\{x\}$ imply $U = V$, a contradiction. Thus, $x\in U$ and $V = U \setminus \mathsf{cl}^U\{x\}$.}
Thus, since $X$ is a spectral space, so is the subspace induced by $U$. We denote the interior and closure operations given by the restriction of $\leqslant$ to $U$ by $\mathsf{int}_\leqslant^U$ and $\mathsf{cl}_\leqslant^U$, respectively.
It is easy to check that $\mathsf{CO}(U) = \{V\cap U \mid V\in \mathsf{CO}(X)\}$. We will now show that $\mathsf{CO}\mathcal{RO}(U) = \{V'\cap U\mid V'\in \mathsf{CO}\mathcal{RO}(X)\}$.
Let $V\subseteq U$. We first prove that
\begin{equation}
V\in \mathsf{CO}\mathcal{RO}(U)\mbox{ iff }V\in \mathsf{CO}\mathcal{RO}(X).\label{Viff}
\end{equation}
Let $V\in \mathsf{CO}\mathcal{RO}(U)$. Then clearly $V\in\mathsf{CO}(X)$. We will show that $V\in\mathcal{RO}(X)$. Since $V$ is open in $U$, it is open in $X$. So $V$ is an $\leqslant$-upset, and $V\subseteq \mathsf{int}_\leqslant\mathsf{cl}_\leqslant (V)$.
Now suppose $x\in \mathsf{int}_\leqslant\mathsf{cl}_\leqslant (V)$. Then for each $y\in X$ with $x\leqslant y$, there is $z\in X$ with $y\leqslant z$ and $z\in V$, which with $V\subseteq U$ implies $z\in U$. Thus, $x\in \mathsf{int}_\leqslant\mathsf{cl}_\leqslant (U)$, which implies $x\in U$ since $U\in\mathcal{RO}(X)$. Together $x\in\mathsf{int}_\leqslant\mathsf{cl}_\leqslant (V)$ and $x\in U$ imply $x\in \mathsf{int}^U_\leqslant\mathsf{cl}^U_\leqslant (V)$, which implies $x\in V$ since $V\in\mathcal{RO}(U)$. Thus, $\mathsf{int}_\leqslant\mathsf{cl}_\leqslant (V)\subseteq V$, so $V\in \mathcal{RO}(X)$.
Conversely, suppose $V\in \mathsf{CO}\mathcal{RO}(X)$. Then clearly $V\in \mathsf{CO}(U)$. To show that $V\in \mathcal{RO}(U)$, suppose $x\in U$ but $x\not\in V$. Then since $V\in \mathcal{RO}(X)$, there is a $y\in X$ such that (a) $x\leqslant y$ and (b) for all $z\in X$ with $y\leqslant z$, we have $z\not\in V$. Since $U$ is an $\leqslant$-upset with $x\in U$, (a) implies $y\in U$. In addition, (b) implies that for all $z\in U$ with $y\leqslant z$, we have $z\not\in V$. Thus, we have shown that if $x\not\in V$, then there is a $y\in U$ such that $x\leqslant y$ and for all $z\in U$ with $y\leqslant z$, we have $z\not\in V$. Hence $V\in\mathcal{RO}(U)$.
The left-to-right direction of (\ref{Viff}) yields $\mathsf{CO}\mathcal{RO}(U)\subseteq \mathsf{CO}\mathcal{RO}(X)$. Now let $V'\in \mathsf{CO}\mathcal{RO}(X)$. Then $V'\cap U\in \mathsf{CO}\mathcal{RO}(X)$ and $V'\cap U\subseteq U$, so $V'\cap U\in \mathsf{CO}\mathcal{RO}(U)$ by the right-to-left direction of (\ref{Viff}). Therefore we have proved that $\mathsf{CO}\mathcal{RO}(U) = \{V'\cap U\mid V'\in \mathsf{CO}\mathcal{RO}(X)\}$.
Next, we show that if $V\in \mathsf{CO}(U)$, then $\mathsf{int}^U_\leqslant (U\setminus V)\in \mathsf{CO}(U)$. Note that for each $W\subseteq U$, we have $\mathsf{int}^U_\leqslant (W) = U\cap \mathsf{int}_\leqslant ((X\setminus U)\cup W)$. So $\mathsf{int}^U_\leqslant (U\setminus V) = U\cap \mathsf{int}_\leqslant ((X\setminus U)\cup (U\setminus V)) = U\cap \mathsf{int}_\leqslant (X\setminus V)$. Since $X$ is a UV-space, $\mathsf{int}_\leqslant (X\setminus V)\in \mathsf{CO}\mathcal{RO}(X)$. Then as $U\in\mathsf{CO}(X)$ and $\mathsf{CO}(X)$ is closed under finite intersections, $\mathsf{int}^U_\leqslant (U\setminus V)\in \mathsf{CO}(X)$. So $\mathsf{int}^U_\leqslant (U\setminus V)\in \mathsf{CO}(U)$.
Finally, let $F$ be a filter in $\mathsf{CO}\mathcal{RO}(U)$. Let $F'$ be the filter in $\mathsf{CO}\mathcal{RO}(X)$ generated by $F$. Then $F' = \mathsf{CO}\mathcal{RO}(x)$ for some $x\in X$. But then $x\in V$ for each $V\in F$.
So $x\in U$ and $F = \mathsf{CO}\mathcal{RO}(x)$. Thus, $U$ is a UV-space. \end{proof}
\begin{prop}\label{RelaProp} Let $X$ be a UV-space. For any $U\in\mathsf{CO}\mathcal{RO}(X)$, the relativization of the BA $\mathsf{CO}\mathcal{RO}(X)$ to $U$ is the dual of the subspace of $X$ induced by $U$.
\end{prop}
\begin{proof} The proposition follows from two facts. First, by Proposition \ref{SubSpaceProp}, the subspace of $X$ induced by $U$ is a UV-space, so by Theorem \ref{SecondThm}.\ref{SecondThmB}, $U$ is homeomorphic to $UV(\mathsf{CO}\mathcal{RO}(U))$. Second, $\mathsf{CO}\mathcal{RO}(U) = \{V'\cap U\mid V'\in \mathsf{CO}\mathcal{RO}(X)\}$ is the relativization of the BA $\mathsf{CO}\mathcal{RO}(X)$ to $U$.\end{proof}
\subsection{Completeness} We now characterize the UV-duals of complete BAs.
\begin{definition} A UV-space $X$ is \textit{complete} iff $\mathsf{int}( \mathsf{cl}(U))\in\mathsf{CO}\mathcal{RO}(X)$ for every open $U$.
\end{definition}
\begin{prop}\label{Comp} Let $\mathbb{A}$ be a BA and $X$ its dual UV-space.
\begin{enumerate}
\item\label{Comp1} If $\{U_i\}_{i\in I}\subseteq \mathsf{CO}\mathcal{RO}(X)$, then $\{U_i\}_{i\in I}$ has a meet in $\mathsf{CO}\mathcal{RO}(X)$ iff
\[\mathsf{int}\underset{i\in I}{\bigcap}U_i\in\mathsf{CO}\mathcal{RO}(X),\]
in which case \[\underset{i\in I}{\bigwedge}U_i=\mathsf{int}\underset{i\in I}{\bigcap}U_i.\]
\item\label{Comp2} If $\{U_i\}_{i\in I}\subseteq \mathsf{CO}\mathcal{RO}(X)$, then $\{U_i\}_{i\in I}$ has a join in $\mathsf{CO}\mathcal{RO}(X)$ iff
\[\mathsf{int} (\mathsf{cl} \underset{i\in I}{\bigcup}U_i)\in\mathsf{CO}\mathcal{RO}(X),\]
in which case \[\underset{i\in I}\bigvee U_i = \mathsf{int}( \mathsf{cl} \underset{i\in I}{\bigcup}U_i).\]
\item\label{Comp3} $\mathbb{A}$ is complete iff $X$ is complete.
\end{enumerate}
\end{prop}
\begin{proof} For part \ref{Comp1}, if $\mathsf{int}\underset{i\in I}{\bigcap}U_i\in\mathsf{CO}\mathcal{RO}(X)$, then clearly $\mathsf{int}\underset{i\in I}{\bigcap}U_i$ is the greatest lower bound in $\mathsf{CO}\mathcal{RO}(X)$ of $\{U_i\}_{i\in I}$. Conversely, if $\underset{i\in I}{\bigwedge}U_i$ exists in $\mathsf{CO}\mathcal{RO}(X)$, then we claim that $\underset{i\in I}{\bigwedge}U_i=\mathsf{int}\underset{i\in I}{\bigcap}U_i$. By the proof of Theorem \ref{MainRep}, for each $i\in I$, we have $U_i=\widehat{a_i}$ for some $a_i\in\mathbb{A}$, so $\underset{i\in I}{\bigwedge}U_i=\underset{i\in I}{\bigwedge}\widehat{a}_i$. Since $a\mapsto\widehat{a}$ is an isomorphism from $\mathbb{A}$ to $\mathsf{CO}\mathcal{RO}(X)$, we have $ \underset{i\in I}{\bigwedge}\widehat{a_i}= \widehat{\underset{i\in I}{\bigwedge}a_i}$. Thus, it suffices to show that $ \widehat{\underset{i\in I}{\bigwedge}a_i} = \mathsf{int} \underset{i\in I}{\bigcap} \widehat{a_i}$. Suppose $F\in \widehat{\underset{i\in I}{\bigwedge}a_i}$. Then since $\widehat{\underset{i\in I}{\bigwedge}a_i}\subseteq \underset{i\in I}{\bigcap} \widehat{a_i}$ and $\widehat{\underset{i\in I}{\bigwedge}a_i}$ is open, we have $F\in\mathsf{int} \underset{i\in I}{\bigcap} \widehat{a_i}$. For the reverse inclusion, suppose $F\in \mathsf{int}\underset{i\in I}{\bigcap} \widehat{a_i}$, so there is a $U\in \mathsf{CO}\mathcal{RO}(X)$ such that $F\in U\subseteq \underset{i\in I}{\bigcap} \widehat{a_i}$. Then $U=\widehat{b}$ for some $b\in\mathbb{A}$, and $\widehat{b}\subseteq \underset{i\in I}{\bigcap} \widehat{a_i}$ implies that $b$ is a lower bound of $\{a_i\}_{i\in I}$ in $\mathbb{A}$, so $b\leq \underset{i\in I}{\bigwedge}a_i$. Then we have the following chain of implications: \[F\in \widehat{b} \Rightarrow b\in F \Rightarrow \underset{i\in I}{\bigwedge}a_i\in F \Rightarrow F\in \widehat{\underset{i\in I}{\bigwedge}a_i} .\]
For part \ref{Comp2}, if $\underset{i\in I}{\bigvee}U_i$ exists in $\mathsf{CO}\mathcal{RO}(X)$, then we claim that $\underset{i\in I}{\bigvee}U_i=\mathsf{int}(\mathsf{cl}\underset{i\in I}{\bigcup}U_i)$. By the proof of Theorem \ref{MainRep}, for each $i\in I$, we have $U_i=\widehat{a_i}$ for some $a_i\in\mathbb{A}$, so $\underset{i\in I}{\bigvee}U_i=\underset{i\in I}{\bigvee}\widehat{a}_i$. Since $a\mapsto\widehat{a}$ is an isomorphism from $\mathbb{A}$ to $\mathsf{CO}\mathcal{RO}(X)$, we have $ \underset{i\in I}{\bigvee}\widehat{a_i}= \widehat{\underset{i\in I}{\bigvee}a_i}$. Thus, it suffices to show that $ \widehat{\underset{i\in I}{\bigvee}a_i} = \mathsf{int}(\mathsf{cl} \underset{i\in I}{\bigcup} \widehat{a_i})$. For the right-to-left inclusion, since $\underset{i\in I}{\bigcup} \widehat{a_i}\subseteq \widehat{\underset{i\in I}{\bigvee}a_i}$ and $\widehat{\underset{i\in I}{\bigvee}a_i}\in\mathsf{CO}\mathcal{RO}(X)=\mathsf{CRO}(X)$ (by Corollary \ref{Jakl}), we have $\mathsf{int}(\mathsf{cl}\underset{i\in I}{\bigcup} \widehat{a_i})\subseteq \mathsf{int}(\mathsf{cl}\widehat{\underset{i\in I}{\bigvee}a_i})=\widehat{\underset{i\in I}{\bigvee}a_i}$. For the left-to-right inclusion, since $\widehat{\underset{i\in I}{\bigvee}a_i}$ is open, it suffices to show $\widehat{\underset{i\in I}{\bigvee}a_i}\subseteq \mathsf{cl} \underset{i\in I}{\bigcup} \widehat{a_i}$. Consider any $F\in \widehat{\underset{i\in I}{\bigvee}a_i}$ and basic open neighborhood $U$ of $F$, so $U=\widehat{b}$ for some $b\in\mathbb{A}$. Then since $F\in \widehat{b}$ and $F\in \widehat{\underset{i\in I}{\bigvee}a_i}$, we have $b\in F$ and $\underset{i\in I}{\bigvee}a_i\in F$, so $b\wedge \underset{i\in I}{\bigvee}a_i=\underset{i\in I}{\bigvee} (b\wedge a_i) \in F$.\footnote{Here we use the join-infinite distributive law for BAs, which says that if $\underset{i\in I}{\bigvee}a_i$ exists, then $\underset{i\in I}{\bigvee} (b\wedge a_i)$ exists and $b\wedge \underset{i\in I}{\bigvee}a_i=\underset{i\in I}{\bigvee} (b\wedge a_i)$ \cite[p.~47, Lem.~3]{Givant2009}.} Since $F$ is a proper filter, it follows that for some $i\in I$, $b\wedge a_i\neq 0$ and hence $\widehat{b}\cap\widehat{a_i}\neq\varnothing$. Thus, $\widehat{b}\cap \underset{i\in I}{\bigcup} \widehat{a_i}\neq\varnothing$. This shows that $F\in \mathsf{cl} \underset{i\in I}{\bigcup} \widehat{a_i}$.
For part \ref{Comp3}, suppose $X$ is complete. For any $\{a_i\}_{i\in I}\subseteq\mathbb{A}$, the set $\underset{i\in I}{\bigcup}\widehat{a_i}$ is open, so by the completeness of $X$, we have $\mathsf{int}( \mathsf{cl} \underset{i\in I}{\bigcup}\widehat{a_i})\in\mathsf{CO}\mathcal{RO}(X)$, in which case $\underset{i\in I}{\bigvee}a_i$ exists by part \ref{Comp2}. Conversely, suppose $\mathbb{A}$ is complete and $U$ is an open set in $X$. Then by Definition \ref{VOspace}.\ref{CloseProp}, we have that $U=\bigcup\{V\in\mathsf{CO}\mathcal{RO}(X)\mid V\subseteq U\}$. Since $\mathbb{A}$ is complete, so is the isomorphic $\mathsf{CO}\mathcal{RO}(X)$, so $\bigvee \{V\in\mathsf{CO}\mathcal{RO}(X)\mid V\subseteq U\}$ exists. Then by part \ref{Comp2}, $\bigvee \{V\in\mathsf{CO}\mathcal{RO}(X)\mid V\subseteq U\}=\mathsf{int}( \mathsf{cl} \bigcup \{V\in\mathsf{CO}\mathcal{RO}(X)\mid V\subseteq U\})$, so $\mathsf{int}( \mathsf{cl} \bigcup \{V\in\mathsf{CO}\mathcal{RO}(X)\mid V\subseteq U\})\in\mathsf{CO}\mathcal{RO}(X)$, i.e., $\mathsf{int}( \mathsf{cl} U)\in\mathsf{CO}\mathcal{RO}(X)$. Hence $X$ is complete.
\end{proof}
\begin{remark} In contrast to the equality in Proposition \ref{Comp}.\ref{Comp2} for arbitrary joins, we observed in Proposition \ref{COROBA} that for finite joins, we have $U_1\vee\dots\vee U_n =\mathsf{int}_\leqslant (\mathsf{cl}_\leqslant(U_1\cup\dots\cup U_n))$. However, we cannot assert this equality for arbitrary joins, as it is refutable in ZF + Boolean Prime Ideal Theorem. To see this, suppose $F$ is a non-principal ultrafilter. Then $\bigwedge F=0$. For if $b$ is a lower bound of $F$, then since $F$ is non-principal, $b\not\in F$, and then since $F$ is an ultrafilter, $- b\in F$. But then $b\leq - b$, so $b=0$. Now since $\bigwedge F=0$, we have $\bigvee \{- a\mid a\in F\}=1$, so $\bigvee \{- a\mid a\in F\}\in F$. Thus, we have $F\in \reallywidehat{\bigvee \{- a\mid a\in F\}}=\bigvee \{\widehat{- a}\mid a\in F\}$, yet clearly $F\not\in \mathsf{int}_\leqslant (\mathsf{cl}_\leqslant \bigcup \{\widehat{- a}\mid a\in F\})$; since $F$ is an ultrafilter, it is maximal in $\leqslant$, so $F\in \mathsf{int}_\leqslant (\mathsf{cl}_\leqslant \bigcup \{\widehat{- a}\mid a\in F\})$ implies $F\in \bigcup \{\widehat{- a}\mid a\in F\}$, contradicting the fact that $F$ is a proper filter. \end{remark}
\begin{lemma}\label{SubspaceComplete} If $X$ is a complete UV-space and $U\in\mathsf{CO}\mathcal{RO}(X)$, then the subspace induced by $U$ is a complete UV-space.
\end{lemma}
\begin{proof} By Proposition~\ref{SubSpaceProp}, $U$ with the subspace topology is a UV-space. To show that $\mathsf{CO}\mathcal{RO}(U)$ is complete, it suffices to show that all meets exist. Thus, by Proposition~\ref{Comp}.\ref{Comp1}, it suffices to show that for any $\{U_i\}_{i\in I}\subseteq \mathsf{CO}\mathcal{RO}(U)$,
we have $\mathsf{int}^U\bigcap_{i\in I} U_i\in \mathsf{CO}\mathcal{RO}(U)$. We show that $\mathsf{int}^U\bigcap_{i\in I} U_i = U\cap \mathsf{int}\bigcap_{i\in I} U_i $. Suppose $x\in \mathsf{int}^U \bigcap_{i\in I} U_i$. Then there is an open set $U_x\subseteq U$ such that $x\in U_x$ and
$U_x\subseteq U_i$ for each $i\in I$. But then $x\in U\cap \mathsf{int} \bigcap_{i\in I} U_i $. Conversely, if $x\in U\cap \mathsf{int}\bigcap_{i\in I} U_i$, then $x\in U$ and there is an open set $V_x$ such that $x\in V_x$ and $V_x\subseteq U_i$ for each $i\in I$. But then
$V_x\subseteq U$ and so $x\in \mathsf{int}^U \bigcap_{i\in I} U_i$.
Therefore, by Proposition~\ref{Comp}.\ref{Comp1}, $\mathsf{int}^U(\bigcap_{i\in I} U_i)$ is the intersection of two
$\mathsf{CO}\mathcal{RO}(U)$ sets and thus, by Proposition \ref{SubSpaceProp}, $\mathsf{int}^U \bigcap_{i\in I} U_i \in \mathsf{CO}\mathcal{RO}(U)$.
\end{proof}
\subsection{Atoms}
Recall that an \textit{isolated point} of a space $X$ is an $x\in X$ such that $\{x\}$ is open.
\begin{prop}\label{Isolated} The map $a\mapsto \mathord{\uparrow}a$ is a bijection from the atoms of a BA to the isolated points of its dual UV-space.
\end{prop}
\begin{proof} If $a$ is an atom of the BA $\mathbb{A}$, then clearly $\widehat{a}=\{\mathord{\uparrow}a\}$, and $\widehat{a}$ is open in $UV(\mathbb{A})$, so $\mathord{\uparrow}a$ is an isolated point. If $a\neq b$, then $\mathord{\uparrow} a\neq\mathord{\uparrow}b$, so the map is injective. Finally, to see that the map is surjective, if $F$ is an isolated point, then $\{F\}$ is open and hence $\{F\}\in \mathsf{CO}\mathcal{RO}(UV(\mathbb{A}))$ by Definition \ref{VOspace}.\ref{CloseProp}. Thus, by the proof of Theorem \ref{MainRep}, there is some $a\in\mathbb{A}$ such that $\widehat{a}=\{F\}$, which implies that $a$ is an atom. For if $a$ is not an atom, then there is a $b< a$ with $b\neq 0$, in which case the proper filters $\mathord{\uparrow}b$ and $\mathord{\uparrow}a$ are distinct and belong to $\widehat{a}$. Since $a$ is an atom, $\widehat{a}=\{\mathord{\uparrow}a\}$, so $F=\mathord{\uparrow}a$. \end{proof}
\begin{cor} A BA is atomless iff the set of isolated points of its dual UV-space is empty.
\end{cor}
Let $X_{\mathrm{iso}}$ be the set of all isolated points of the space $X$ and $At(\mathbb{A})$ the set of all atoms of the BA $\mathbb{A}$.
\begin{prop}\label{AtomicProp} Let $\mathbb{A}$ be a BA and $X$ its dual space. The following are equivalent:
\begin{enumerate}
\item $\mathbb{A}$ is atomic;
\item $\mathsf{int}( \mathsf{cl} X_\mathrm{iso})=X$;
\item the set of isolated points is dense in $X$, i.e., $\mathsf{cl} X_\mathrm{iso}=X$.
\end{enumerate}
\end{prop}
\begin{proof} 1 $\Rightarrow$ 2. If $\mathbb{A}$ is atomic, then $1 = \bigvee \{a\in \mathbb{A}\mid a\in At(\mathbb{A})\}$. Then $X = \widehat{1} = \reallywidehat{\bigvee \{a\in \mathbb{A}\mid a\in At(\mathbb{A})\}} = \bigvee \{\widehat{a}\in \mathbb{A}\mid a\in At(\mathbb{A})\}= \mathsf{int}( \mathsf{cl} \bigcup \{\widehat{a}\mid a\in At(\mathbb{A})\})= \mathsf{int}( \mathsf{cl} X_{\mathrm{iso}})$ by Propositions \ref{Comp}.\ref{Comp2} and \ref{Isolated}.
2 $\Rightarrow$ 3. Since $\mathsf{int}( \mathsf{cl} X_\mathrm{iso})\subseteq \mathsf{cl} X_\mathrm{iso}$, $\mathsf{int}( \mathsf{cl} X_\mathrm{iso})=X$ implies $\mathsf{cl} X_\mathrm{iso}=X$.
3 $\Rightarrow$ 1. We need to show that $1 = \bigvee \{a\in A\mid a\in At(A)\}$. In dual terms this means that
$X = \widehat{1} = \reallywidehat{\bigvee \{a\in A\mid a\in At(A)\}}$. By Propositions \ref{Comp}.\ref{Comp2} and \ref{Isolated}, we have $\reallywidehat{\bigvee \{a\in A\mid a\in At(A)\}} = \mathsf{int}( \mathsf{cl} X_{\mathrm{iso}})$. As $\mathsf{cl}(X_{\mathrm{iso}}) = X$, we have $\mathsf{int}(\mathsf{cl}X_{\mathrm{iso}}) = \mathsf{int}X=X$.\end{proof}
\subsection{Subalgebras and homomorphic images}
We now characterize subalgebras and homomorphic images of BAs in terms of UV-spaces.
\begin{definition} Let $X$ and $Y$ be UV-spaces. An injective UV map $f:X\to Y$ is a \textit{UV-embedding} if for every $U\in\mathsf{CO}\mathcal{RO}(X)$ there is a $V\in \mathsf{CO}\mathcal{RO}(Y)$ such that $f[U]=f[X]\cap V$.
\end{definition}
\begin{fact}\label{thm: surj-inj} Let $\mathbb{A}$ and $\mathbb{B}$ be BAs and $h: \mathbb{A}\to \mathbb{B}$ a homomorphism. Let $h_+: UV(\mathbb{B})\to UV(\mathbb{A})$ be the UV-map dual to $h$. Then:
\begin{enumerate}
\item\label{surj1} if $h$ is injective, then $h_+$ is surjective;
\item\label{inj1} if $h$ is surjective, then $h_+$ is a UV-embedding.
\end{enumerate}
\end{fact}
\begin{proof} For part \ref{surj1}, consider a proper filter $F\in UV(\mathbb{A})$, and let $G = \{b\in \mathbb{A}\mid \exists a\in h[F]: a\leq b\}$. We show that $G$ is a proper filter such that $h^{-1}[G] = F$. Suppose $0_\mathbb{B}\in G$. Then $0_\mathbb{B}\in h[F]$, so there is an $a\in F$ such that $h(a) = 0_\mathbb{B}$. As $F$ is proper, $a\neq 0_\mathbb{A}$, which is a contradiction as $h$ is injective and $h(0_\mathbb{A}) = 0_\mathbb{B}$. Now if $c, d\in G$, then there are $a,b\in F$ such that $h(a)\leq c$ and $h(b)\leq d$. Since $F$ is a filter, $a,b\in F$ implies $a\wedge b\in F$, so $h(a\wedge b)\in h[F]$. Then since $h(a\wedge b) = h(a)\wedge h(b) \leq c\wedge d$, we have $c\wedge d\in G$. It is also obvious that $G$ is an upset. Thus, $G$ is a proper filter.
We now show that $h^{-1}[G] = F$. Clearly $F\subseteq h^{-1}[G]$. Suppose $a\in h^{-1}[G]$. Then $h(a)\in G$, so there is a $b\in F$ such that $h(b)\leq h(a)$. If $a\notin F$, then $b\nleq a$ and so $a\wedge b\neq b$. On the other hand, $h(a\wedge b) = h(a)\wedge h(b) = h(b)$, which is a contradiction as $h$ is injective. Therefore, $h^{-1}[G] = F$. As $h_+(G) = h^{-1}[G]$, we obtain that $h_+$ is a surjective {UV-map}.
For part \ref{inj1}, let $F$ and $G$ be proper filters in $\mathbb{B}$ such that $F\neq G$. Then without loss of generality there is a $b\in F$ such that $b\notin G$. As $h$ is surjective there is an $a\in \mathbb{A}$ such that $h(a) = b$. Obviously, $a\in h^{-1}[F]$ and $a\notin h^{-1}[G]$. So $ h^{-1}[F]\neq h^{-1}[G]$, implying that $h_+$ is injective. Finally, we check the UV-embedding condition. Each $U\in \mathsf{CO}\mathcal{RO}(UV(\mathbb{B}))$ is of the form $\widehat{b}$ for some $b\in\mathbb{B}$. Since $h$ is surjective, there is an $a\in\mathbb{A}$ such that $h(a)=b$, so $h_+[\widehat{b}]=h_+[\widehat{h(a)}]$. Now it suffices to show \[h_+[\widehat{h(a)}]=h_+[UV(\mathbb{B})]\cap \widehat{a}.\] From left to right, suppose $F\in h_+[\widehat{h(a)}]$, so there is a $G\in \widehat{h(a)}$ such that $h_+(G)=F$. Since $G\in\widehat{h(a)}$, we have $h(a)\in G$. Since $h_+(G)=F$, we have $h^{-1}[G]=F$. From $h(a)\in G$ and $h^{-1}[G]=F$, we have $a\in F$, so $F\in\widehat{a}$. From right to left, suppose $F\in h_+[UV(\mathbb{B})]\cap \widehat{a}$. Since $F\in h_+[UV(\mathbb{B})]$, there is a $G\in UV(\mathbb{B})$ such that $h_+(G)=F$ and hence $h^{-1}[G]=F$. Since $F\in \widehat{a}$, we have $a\in F$ and hence $h(a)\in G$. Thus, $G\in\widehat{h(a)}$, which with $h_+(G)=F$ implies $F\in h_+[\widehat{h(a)}]$. This completes the proof.\end{proof}
\begin{fact}\label{thm: surj-inj2} Let $X$ and $Y$ be UV-spaces and $f: X\to Y$ a UV-map. Let $f^+: \mathsf{CO}\mathcal{RO}(Y)\to\mathsf{CO}\mathcal{RO}(X)$ be the homomorphism dual to $h$. Then:
\begin{enumerate}
\item\label{inj2} if $f$ is surjective, then $f^+$ is injective;
\item\label{surj2} if $f$ is UV-embedding, then $f^+$ is surjective.
\end{enumerate}
\end{fact}
\begin{proof} For part \ref{inj2}, suppose for $U,V\in\mathsf{CO}\mathcal{RO}(Y)$ that $U\neq V$. Suppose $y\in U\setminus V$. Since $f$ is surjective, there is an $x\in X$ such that $f(x)=y$, so $x\in f^{-1}[U]=f^+[U]$ but $x\not\in f^{-1}[V]=f^+(V)$. Hence $f^+$ is injective.
For part \ref{surj2}, suppose $U\in\mathsf{CO}\mathcal{RO}(X)$. Then since $f$ is a UV-embedding, there is a $V\in\mathsf{CO}\mathcal{RO}(Y)$ such that $f[U]=f[X]\cap V$. Then $f^{-1}[f[U]]=f^{-1}[f[X]\cap V]=f^{-1}[f[X]]\cap f^{-1}[V]=X\cap f^{-1}[V]=f^{-1}[V]$. Since $f$ is injective, $f^{-1}[f[U]]=U$. Hence $U=f^{-1}[V]=f^+[V]$.\end{proof}
\begin{cor} $\,$
\begin{enumerate}
\item\label{subimage} There is a one-to-one correspondence between subalgebras of a BA $\mathbb{A}$ and images via onto UV-maps of its dual UV-space $X_\mathbb{A}$.
\item\label{homprinc} There is a one-to-one correspondence between homomorphic images of a BA $\mathbb{A}$ and subspaces induced by principal upsets in the specialization order of the dual UV-space $X_\mathbb{A}$.
\end{enumerate}
\end{cor}
\begin{proof} Part \ref{subimage} follows from Facts~\ref{thm: surj-inj}.\ref{surj1} and \ref{thm: surj-inj2}.\ref{inj2} and Theorem \ref{DualityThm}.
Since there is a one-to-one correspondence between homomorphic images of $\mathbb{A}$ and filters of $\mathbb{A}$, part \ref{homprinc} follows directly from Fact~\ref{EtaFact}. However, we also sketch a more direct argument. By Facts~\ref{thm: surj-inj}.\ref{inj1} and \ref{thm: surj-inj2}.\ref{surj2} and Theorem \ref{DualityThm}, there is a one-to-one correspondence between homomorphic images of $\mathbb{A}$ and UV-embeddings into its dual $X_\mathbb{A}$. Let $\mathbb{B}$ be a homomorphic image of $\mathbb{A}$ via $h$. Then $h_+:X_\mathbb{B}\to X_\mathbb{A}$ is a UV-embedding. First, since $h_+$ is an injective p-morphism, $h_+[X_\mathbb{B}]$ is a principal upset in the specialization order of $X_\mathbb{A}$. Second, if $Y$ is the subspace of $X_\mathbb{A}$ induced by $h_+[X_\mathbb{B}]$, we claim that $X_\mathbb{B}$ and $Y$ are homeomorphic via the bijection $h_+: X_\mathbb{B}\to Y$. Since $h_+$ is a continuous map from $X_\mathbb{B}$ to $X_\mathbb{A}$, it follows that $h_+$ is a continuous map from $X_\mathbb{B}$ to $Y$, and since $h_+$ is a UV-embedding from $X_\mathbb{B}$ to $X_\mathbb{A}$, it follows that $h_+$ is an open map from $X_\mathbb{B}$ to $Y$. \end{proof}
\subsection{Products} The operation on UV-spaces dual to taking direct products of BAs is the following.
\begin{definition}\label{UVunionDef} The \textit{UV-sum} of disjoint UV-spaces $X$ and $Y$ is the space $X\bigcirc Y$ whose underlying set is $X\cup Y\cup (X\times Y)$ and whose topology is generated by the collection of sets
\[U\cup V \cup (U\times V)\]
for $U\in\mathsf{CO}\mathcal{RO}(X)$ and $V\in \mathsf{CO}\mathcal{RO}(Y)$.
\end{definition}
The following lemma is helpful for visualizing UV-sums.
\begin{lemma}\label{SumLem1} Given UV-spaces $X$ and $Y$ with specialization orders $\leqslant_X$ and $\leqslant_Y$, respectively, the specialization order in $X\bigcirc Y$ is given by:
\begin{eqnarray}
&& \leqslant_X\cup\leqslant_Y\cup \nonumber \\
&& \{\langle\langle x,y\rangle, x'\rangle\mid x\leqslant_X x'\}\cup \{\langle\langle x,y\rangle, y'\rangle\mid y\leqslant_Y y'\} \cup \nonumber \\
&& \{\langle\langle x,y\rangle,\langle x',y'\rangle\rangle\mid x\leqslant_X x',y\leqslant_Yy'\}.\label{SpecialDef}
\end{eqnarray}
\end{lemma}
\begin{proof} Suppose $\langle z,z'\rangle $ belongs to the set in (\ref{SpecialDef}), and $z$ belongs to an open set $U\cup V\cup (U\times V)$ of $X\bigcirc Y$. We must show that $z'$ also belongs to the set. There are five cases. If $z\leqslant_X z'$ (resp.~$z\leqslant_Y z'$), then $z\in U$ (resp.~$z\in V$), which with $U\in\mathsf{CO}\mathcal{RO}(X)$ (resp.~$V\in \mathsf{CO}\mathcal{RO}(Y)$) implies $z'\in U$ (resp.~$z'\in V$) and hence $z'\in U\cup V\cup (U\times V)$. On the other hand, if $z=\langle x,y\rangle $, then $\langle x,y\rangle\in U\times V$, so $x\in U$ and $y\in V$. Therefore, if $x\leqslant_X z'$ (resp.~$y\leqslant_Y z'$), then $z'\in U$ (resp.~$z'\in V$) and hence $z'\in U\cup V\cup (U\times V)$. Similarly, if $z'=\langle x',y'\rangle$, and $x\leqslant_X x'$ and $y\leqslant_Y y'$, then $x'\in U$ and $y'\in V$, so $z'\in U\times V$. This completes the proof that $z\leqslant_{X\bigcirc Y}z'$.
Conversely, suppose $\langle z,z'\rangle$ does not belong to the set in (\ref{SpecialDef}). Again there are five cases. For example, if $z,z'\in X$, it follows that $z\not\leqslant_X z'$, so there is an open set $U$ of $X$ such that $z\in U$ but $z'\not\in U$, and $U$ is open in $X\bigcirc Y$, so $z\not\leqslant_{X\bigcirc Y}z'$. Similarly, if $z=\langle x,y\rangle$ and $z'\in X$, it follows that $x\not\leqslant_X z'$, so there is an open set $U$ of $X$ such that $x\in U$ but $z'\not\in U$. Thus, $\langle x,y\rangle \in U\cap Y\cup ( U\times Y)$ but $z'\not\in U\cup Y\cup (U\times Y)$, so $\langle x,y\rangle\not\leqslant_{X\bigcirc Y}z'$. The other cases are analogous.
\end{proof}
\begin{example} For finite UV-spaces, the UV-sum is easily drawn. Figure \ref{Figure} shows the UV-sum of the UV-duals of the four-element and two-element BAs, \textbf{4} and \textbf{2} (recall Corollary \ref{UVspectral}.\ref{UVspectral4.5}). Solid lines indicate the specialization order $\leqslant$ in $UV(\mathbf{4})$, so $x\leqslant y_1$ and $x\leqslant y_2$. Dashed lines indicate the new part of the relation defined in Lemma \ref{SumLem1}. Note that $UV(\mathbf{4})\bigcirc UV(\mathbf{2})=UV(\mathbf{4}\times\mathbf{2})$ in line with Proposition \ref{UnionProduct}.
\end{example}
\begin{figure}[h]
\begin{center}
\begin{tikzpicture}[->,>=stealth',shorten >=1pt,shorten <=1pt, auto, node distance=2in,thick,every loop/.style={<-,shorten <=1pt}] \tikzstyle{every state}=[fill=gray!20,draw=none,text=black]
\node at (-6,.5) {{$UV(\mathbf{4})$}};
\node (x) at (-6,1.25) {{$x$}};
\node (y1) at (-7,2.5) {{$y_1$}};
\node (y2) at (-5,2.5) {{$y_2$}};
\path (x) edge[-] node {{}} (y1);
\path (x) edge[-] node {{}} (y2);
\node at (-3.5,.5) {{$UV(\mathbf{2})$}};
\node (z) at (-3.5,2.5) {{$z$}};
\node at (0,-.75) {{$UV(\mathbf{4})\bigcirc UV(\mathbf{2})$}};
\node at (0,-1.25) {{$UV(\mathbf{4}\times\mathbf{2})$}};
\node (bot) at (0,0) {{$\langle x,z\rangle$}};
\node (A) at (-2,1.25) {{$x$}};
\node (B) at (0,1.25) {{$\langle y_1,z\rangle$}};
\node (C) at (2,1.25) {{$\langle y_2,z\rangle$}};
\node (A') at (-2,2.5) {{$y_1$}};
\node (B') at (0,2.5) {{$y_2$}};
\node (C') at (2,2.5) {{$z$}};
\path (A) edge[-] node {{}} (A');
\path (A) edge[-] node {{}} (B');
\path (B) edge[-,dashed] node {{}} (A');
\path (B) edge[-,dashed] node {{}} (C');
\path (C) edge[-,dashed] node {{}} (C');
\path (C) edge[-,dashed] node {{}} (B');
\path (A) edge[-,dashed] node {{}} (bot);
\path (B) edge[-,dashed] node {{}} (bot);
\path (C) edge[-,dashed] node {{}} (bot);
\end{tikzpicture}
\end{center}
\caption{UV-sum of the UV-duals of the BAs \textbf{4} and \textbf{2}.}\label{Figure}
\end{figure}
\begin{prop}\label{UnionProduct} For any BAs $\mathbb{A}$ and $\mathbb{B}$, $UV(\mathbb{A})\bigcirc UV(\mathbb{B})$ is homeomorphic to $UV(\mathbb{A}\times\mathbb{B})$.
\end{prop}
\begin{proof} Given $F\in UV(\mathbb{A}\times\mathbb{B})$, so that $F$ is a proper filter in $\mathbb{A}\times \mathbb{B}$, we have that $F_\mathbb{A}=\{a\mid \exists b :\langle a,b\rangle\in F \}$ and $F_\mathbb{B}=\{b\mid \exists a :\langle a,b\rangle\in F \}$ are filters in $\mathbb{A}$ and $\mathbb{B}$, respectively, at least one of which is a proper filter. We define a function $h$ from $UV(\mathbb{A}\times\mathbb{B})$ to $UV(\mathbb{A})\bigcirc UV(\mathbb{B})$ as follows:
\[h(F)=\begin{cases}F_\mathbb{A} &\mbox{if }F_\mathbb{B}\mbox{ is improper} \\
F_\mathbb{B} &\mbox{if }F_\mathbb{A}\mbox{ is improper} \\
\langle F_\mathbb{A},F_\mathbb{B}\rangle & \mbox{otherwise}.
\end{cases}\]
We claim that $h$ is a homeomorphism from $UV(\mathbb{A}\times\mathbb{B})$ to $UV(\mathbb{A})\bigcirc UV(\mathbb{B})$. Clearly $h$ is injective. For surjectivity, suppose $G\in UV(\mathbb{A})\bigcirc UV(\mathbb{B})$. If $G\in UV(\mathbb{A})$, then for $F=\{\langle a,b\rangle\mid a\in G,b\in\mathbb{B}\}\in UV(\mathbb{A}\times\mathbb{B})$, we have that $G=F_\mathbb{A}$ and $F_\mathbb{B}$ is improper, so $G=h(F)$. Similarly, if $G\in UV(\mathbb{B})$, then for $F=\{\langle a,b\rangle\mid a\in \mathbb{A},b\in G\}\in UV(\mathbb{A}\times\mathbb{B})$, we have that $G=F_\mathbb{B}$ and $F_\mathbb{A}$ is improper, so $G=h(F)$. Finally, if $G=\langle G^\mathbb{A},G^\mathbb{B}\rangle$ for $G^\mathbb{A}\in UV(\mathbb{A})$ and $G^\mathbb{B}\in UV(\mathbb{B})$, then $G^\mathbb{A}\times G^\mathbb{B}\in UV(\mathbb{A}\times\mathbb{B})$ and $G=h(G^\mathbb{A}\times G^\mathbb{B})$, since $(G^\mathbb{A}\times G^\mathbb{B})_\mathbb{A}=G^\mathbb{A}$ and $(G^\mathbb{A}\times G^\mathbb{B})_\mathbb{B}=G^\mathbb{B}$. Thus, $h$ is surjective.
To show that $h$ is continuous, it suffices to show that the inverse image of each basic open is open. By Definition \ref{UVunionDef}, each basic open in $UV(\mathbb{A})\bigcirc UV(\mathbb{B})$ is of the form $U\cup V\cup (U\times V)$ for $U\in \mathsf{CO}\mathcal{RO}(UV(\mathbb{A}))$ and $V\in \mathsf{CO}\mathcal{RO}(UV(\mathbb{B}))$. By the proof of Theorem \ref{MainRep}, $U=\widehat{a}$ and $V=\widehat{b}$ for some $a\in\mathbb{A}$ and $b\in\mathbb{B}$, so our basic open in $UV(\mathbb{A})\bigcirc UV(\mathbb{B})$ is $\widehat{a}\cup\widehat{b}\cup (\widehat{a}\times\widehat{b})$. Then we have:
\begin{eqnarray*}
h^{-1}[\widehat{a}\cup\widehat{b}\cup (\widehat{a}\times\widehat{b})] &=& h^{-1}[\widehat{a}]\cup h^{-1}[\widehat{b}]\cup h^{-1}[\widehat{a}\times\widehat{b}] \\
&=& \widehat{\langle a,0\rangle} \cup \widehat{\langle 0,b\rangle} \cup \widehat{\langle a,b\rangle},
\end{eqnarray*}
so $h^{-1}[\widehat{a}\cup\widehat{b}\cup (\widehat{a}\times\widehat{b})] $ is a union of basic opens in $UV(\mathbb{A}\times\mathbb{B})$.
Finally, to see that $h^{-1}$ is continuous, for any basic open set $\widehat{\langle a,b\rangle}$ of $UV(\mathbb{A}\times\mathbb{B})$, we have:
\begin{eqnarray*}
\widehat{\langle a,b\rangle} &=& \{F\in\mathrm{PropFilt}(\mathbb{A}\times\mathbb{B})\mid \langle a,b\rangle\in F\mbox{ and } F_\mathbb{B}\mbox{ improper} \} \cup \\
&& \{F\in\mathrm{PropFilt}(\mathbb{A}\times\mathbb{B})\mid \langle a,b\rangle\in F \mbox{ and }F_\mathbb{A}\mbox{ improper} \} \cup \\
&& \{F\in\mathrm{PropFilt}(\mathbb{A}\times\mathbb{B})\mid \langle a,b\rangle\in F\mbox{ and } F_\mathbb{A}, F_\mathbb{B}\mbox{ proper} \},
\end{eqnarray*}
which implies
\begin{eqnarray*}
h[\widehat{\langle a,b\rangle}]&=& \widehat{a}\cup \widehat{b}\cup (\widehat{a}\times\widehat{b}),
\end{eqnarray*}
so that $h[\widehat{\langle a,b\rangle}]$ is basic open in $UV(\mathbb{A})\bigcirc UV(\mathbb{B})$.
\end{proof}
\begin{cor} For any UV-spaces $X$ and $Y$, $X\bigcirc Y$ is a UV-space.
\end{cor}
\begin{proof} By Theorem \ref{SecondThm}.\ref{SecondThmB}, $X$ and $Y$ are respectively homeomorphic to $UV(\mathsf{CO}\mathcal{RO}(X))$ and $UV(\mathsf{CO}\mathcal{RO}(Y))$, which implies that $X\bigcirc Y$ is homeomorphic to $UV(\mathsf{CO}\mathcal{RO}(X))\bigcirc UV(\mathsf{CO}\mathcal{RO}(Y))$. By Proposition \ref{UnionProduct}, $UV(\mathsf{CO}\mathcal{RO}(X)) \bigcirc UV(\mathsf{CO}\mathcal{RO}(Y))$ is homeomorphic to $UV(\mathsf{CO}\mathcal{RO}(X)\times\mathsf{CO}\mathcal{RO}(Y))$, which is a UV-space by Theorem \ref{SecondThm}.\ref{SecondThmA}. Thus, by the two homeomorphisms, $X\bigcirc Y$ is a UV-space.
\end{proof}
\begin{cor} For any UV-spaces $X$ and $Y$, $\mathsf{CO}\mathcal{RO}(X\bigcirc Y)$ is isomorphic to $\mathsf{CO}\mathcal{RO}(X)\times \mathsf{CO}\mathcal{RO}(Y)$.
\end{cor}
\begin{proof} Apply Proposition \ref{UnionProduct} and duality (Theorem \ref{DualityThm}).
\end{proof}
\begin{remark}
Another natural question is how one can characterize products in the category of UV-spaces with UV-maps, which will be the duals of coproducts in the category of BAs with BA homomorphisms. We cannot characterize the product of UV-spaces $X$ and $Y$ as a topological space based on the Cartesian product of the underlying sets of $X$ and $Y$. E.g., if we take the Cartesian product of two copies of the three-element set underlying $UV(\mathbf{4})$ (Figure \ref{Figure}), then we obtain a set with nine elements; this cannot be the underlying set of any poset obtained from a BA by deleting its top element, so by Corollary \ref{UVspectral}.\ref{UVspectral4.5} it cannot be the underlying set of a UV-space. We leave for future work the problem of characterizing products in the category of UV-spaces, which is reminiscent of the open problem of characterizing products in the category of Esakia spaces \cite{Esakia2018}.\end{remark}
\subsection{Completions}\label{CompletionSection} The \textit{canonical extension} of a BA $\mathbb{A}$, as defined in \cite{Gehrke2001}, is the unique (up to isomorphism) complete BA $\mathbb{B}$ for which there is a Boolean embedding $e$ of $\mathbb{A}$ into $\mathbb{B}$ such that every element of $\mathbb{B}$ is a join of meets of $e$-images of elements of $\mathbb{A}$, and for any sets $S$ and $T$ of elements of $\mathbb{A}$, if $\bigwedge e[S]\leq \bigvee e[T]$, then there are finite sets $S'\subseteq S$ and $T'\subseteq T$ such that $\bigwedge S'\leq \bigvee T'$. It is shown in \cite[\S~5.6]{Holliday2018} that the canonical extension of a BA $\mathbb{A}$ can be constructed without choice as the BA of all regular open upsets in the poset of proper filters of $\mathbb{A}$ ordered by inclusion. Putting this in terms of UV-spaces, we have the following.
\begin{theorem}\label{CanonicalExt} Let $\mathbb{A}$ be a BA and $X$ its dual UV-space. Then $\mathcal{RO}(X)$ is \textnormal{(}up to isomorphism\textnormal{)} the canonical extension of $\mathbb{A}$.
\end{theorem}
The \textit{MacNeille completion} of a BA $\mathbb{A}$ is the unique (up to isomorphism) complete BA $\mathbb{B}$ for which there is a Boolean embedding $e$ of $\mathbb{A}$ into $\mathbb{B}$ such that every non-minimum element of $\mathbb{B}$ is above the $e$-image of some non-minimum element of $\mathbb{A}$ (see, e.g., \cite[Ch.~25]{Givant2009}). The MacNeille completion of $\mathbb{B}$ may be constructed as the lattice of normal ideals of $\mathbb{B}$ ordered by inclusion; an ideal $I$ of $\mathbb{B}$ is normal iff $I=I^{u\ell}$, where for any $A\subseteq \mathbb{B}$, $A^u$ is the set of upper bounds of $A$, and $A^\ell$ is the set of lower bounds of $A$.\footnote{The MacNeille completion of a BA $\mathbb{A}$ can also be constructed as the BA of all regular open upsets in the poset that results from deleting the bottom element of $\mathbb{A}$ and reversing the restricted order \cite[\S~5.6]{Holliday2018}.}
\begin{theorem} Let $\mathbb{A}$ be a BA and $X$ its dual space. Then $\mathsf{RO}(X)$ is \textnormal{(}up to isomorphism\textnormal{)} the MacNeille completion of $\mathbb{A}$.\end{theorem}
\begin{proof} We show an order isomorphism between $\mathsf{RO}(X)$ and the set of normal ideals of $\mathbb{A}$ ordered by inclusion. It suffices to define an inclusion-preserving map $r$ from normal ideals to $\mathsf{RO}(X)$ and an inclusion-preserving map $i$ from $\mathsf{RO}(X)$ to normal ideals such that $i(r(I))=I$ and $r(i(U))=U$.
Suppose $I$ is a normal ideal, so $I=I^{u\ell}$. Let $r(I):=\bigcup \{\widehat{c}\mid c\in I\}$. To see that $r(I)\in\mathsf{RO}(X)$, let
$U:=\bigcup \{\widehat{-a}\mid a\in I^u\}$. Then as in the proof of Proposition 4.3, we have
\begin{eqnarray*}
U^*&=&\bigcup\{\widehat{c}\mid \forall a\in I^u\;\, \mathord{-}a\wedge c=0\}\\
&=&\bigcup\{\widehat{c}\mid \forall a\in I^u\;\, c\leq a\}\\
&=&\bigcup \{\widehat{c}\mid c\in I^{u\ell}\}\\
&=&\bigcup \{\widehat{c}\mid c\in I\} = r(I).
\end{eqnarray*}
Thus, $r(I)\in\mathsf{RO}(X)$. Clearly $I\subseteq J$ implies $r(I)\subseteq r(J)$.
In the other direction, suppose $V\in \mathsf{RO}(X)$. Let $i(V)=\{-b\mid \widehat{b}\subseteq V^*\}^\ell$. It is easy to see that for any $S\subseteq\mathbb{A}$, $S^\ell$ is a normal ideal, so $i(V)$ is a normal ideal. Also observe that $i$ is inclusion-preserving:
\begin{eqnarray*}
&& V\subseteq U \\
&\Rightarrow& U^*\subseteq V^*\\
&\Rightarrow &\{-b\mid \widehat{b}\subseteq U^*\}\subseteq \{-b\mid \widehat{b}\subseteq V^*\} \\
&\Rightarrow & \{-b\mid \widehat{b}\subseteq V^*\}^\ell \subseteq \{-b\mid \widehat{b}\subseteq U^*\}^\ell \\
&\Rightarrow & i(V)\subseteq i(U).
\end{eqnarray*}
Next, observe:
\begin{eqnarray*}
i(r(I)) &=&i\big(\bigcup \{\widehat{c}\mid c\in I\}\big) \\
&=& \{-b\mid \widehat{b}\subseteq\big(\bigcup \{\widehat{c}\mid c\in I\}\big)^*\}^\ell \\
&=& \{-b\mid \widehat{b}\subseteq\bigcup \{\widehat{d}\mid \forall c\in I\;\, c\wedge d=0\}\}^\ell \\
&=& \{-b\mid \widehat{b}\subseteq\bigcup \{\widehat{d}\mid \forall c\in I\;\, c\leq -d\}\}^\ell \\
&=& \{-b\mid \mathord{\uparrow}b\in \bigcup\{\widehat{d}\mid \forall c\in I\;\, c\leq -d\}^\ell \\
&=& \{-b\mid \exists d: b\leq d\mbox{ and } \forall c\in I\;\, c\leq -d\}^\ell \\
&=& \{-b\mid \forall c\in I\;\, c\leq -b\}^\ell \\
&=& I^{u\ell}=I.
\end{eqnarray*}
Finally, observe:
\begin{eqnarray*}
r(i(U)) &=&r(\{-b\mid \widehat{b}\subseteq U^*\}^\ell)\\
&=&\bigcup \{\widehat{c}\mid c\in \{-b\mid \widehat{b}\subseteq U^*\}^\ell\}. \\
&=&\bigcup \{\widehat{c}\mid \forall b\,(\widehat{b}\subseteq U^*\Rightarrow c\leq -b)\}\\
&=&\bigcup \{\widehat{c}\mid \forall b\,(\widehat{b}\subseteq U^*\Rightarrow b\wedge c=0)\} \\
&=&U^{**} =U.
\end{eqnarray*}
This completes the proof.\end{proof}
\section{Example applications}\label{ApplicationSection} In this section, we apply our duality to prove some basic theorems about BAs in Propositions \ref{ChainsAnti}, \ref{CompleteBAProp}, and \ref{SubalgebrasProp}.
\subsection{Chains and antichains in BAs} By an \textit{antichain} in a BA, we mean a collection $C$ of elements such that for all $x,y\in C$ with $x\neq y$, we have $x\wedge y=0$.
\begin{prop}\label{ChainsAnti} Every infinite BA contains infinite chains and infinite antichains.
\end{prop}
\begin{proof} By duality, it suffices to show that in any infinite UV-space $X$, there is an infinite descending chain $U_0\supsetneq U_1\supsetneq\dots$ of sets from $\mathsf{CO}\mathcal{RO}(X)$, as well as an infinite family of pairwise disjoint sets from $\mathsf{CO}\mathcal{RO}(X)$. For this it suffices to show that for every infinite $U\in \mathsf{CO}\mathcal{RO}(X)$ (note that $X$ is such a $U$), there is an infinite $U'\in \mathsf{CO}\mathcal{RO}(X)$ with $U\supsetneq U'$ and $U\cap\neg U'\neq\varnothing$. For then by DC, there is an infinite descending chain $U_0\supsetneq U_1\supsetneq \dots$ of sets from $\mathsf{CO}\mathcal{RO}(X)$ with $U_i\cap \neg U_{i+1}\neq\varnothing$ for each $i\in\mathbb{N}$, in which case $\{U_0\cap \neg U_1,U_1\cap \neg U_2,\dots\}$ is our antichain.
Assume $U\in \mathsf{CO}\mathcal{RO}(X)$ is infinite. Since $X$ is $T_0$, there are $x,y\in U$ such that $x\not\leqslant y$. Then by the separation property of UV-spaces, there is a $V\in\mathsf{CO}\mathcal{RO}(X)$ such that $x\in V$ and $y\not\in V$, which with $y\in U$ and $U,V\in\mathcal{RO}(X)$ implies that there is a $z\geqslant y$ such that $z\in U\cap \neg V$. Since $U,V\in \mathsf{CO}\mathcal{RO}(X)$, we have $U\cap V,U\cap \neg V\in \mathsf{CO}\mathcal{RO}(X)$ by Definition \ref{VOspace}.\ref{CloseProp}; and since $z\in U\cap \neg V$ and $x\in U\cap V$, we have $z\in U\cap \neg (U\cap V)\neq \varnothing$ and $x\in U\cap \neg (U\cap \neg V)\neq \varnothing$. Thus, if $U\cap V$ is infinite, then we can set $U':=U\cap V$, and otherwise we claim that $U\cap\neg V$ is infinite, in which case we can set $U':=U\cap\neg V$. Since $U\in \mathcal{RO}(X)$, we may regard $U$ as a separative partial order. Given $V\in\mathcal{RO}(X)$, we have $U\cap V,U\cap \neg V\in \mathcal{RO}(U)$ and $U\cap \neg V=\neg_U(U\cap V)$, where $\neg_U$ is the complement operation in $\mathcal{RO}(U)$. Then since $U$ is infinite, by Lemma \ref{EitherInfinite} either $U\cap V$ or $\neg_U(U\cap V)$ is infinite, as desired.\end{proof}
\subsection{Products of BAs} Before our second example application in Proposition \ref{CompleteBAProp}, we prove a preliminary lemma. Recall from Proposition \ref{SubSpaceProp} that a subspace of a UV-space induced by a $\mathsf{CO}\mathcal{RO}$ set is also a UV-space.
\begin{lemma}\label{SumLem} If $X$ is a UV-space and $U\in\mathsf{CO}\mathcal{RO}(X)$, then $X$ is homeomorphic to the UV-sum of the subspaces induced by $U$ and $\neg U$, respectively.
\end{lemma}
\begin{proof} By Corollary \ref{UVspectral}.\ref{UVspectral4}, $(X,\leqslant)$ has a meet $x\sqcap y$ for any two elements $x,y\in X$. We define $f\colon U\bigcirc \neg U\to X$ as follows: if $z\in U\cup \neg U$, then $f(z)=z$; otherwise $z=\langle x,y\rangle$ for $x\in U$ and $y\in\neg U$, so we define $f(\langle x,y\rangle)=x\sqcap y$. That $f$ is a bijection follows from Corollary \ref{UVspectral}.\ref{UVspectral5}. To see that $f$ is continuous, we show that the inverse image of each basic open is open. Given $V\in\mathsf{CO}\mathcal{RO}(X)$, we have:
\begin{eqnarray*}
f^{-1}[V] &=& (U\cap V)\cup (\neg U\cap V)\cup \{\langle x,y\rangle\mid x\in U,y\in \neg U,x\sqcap y\in V\} \\
&=& (U\cap V)\cup (\neg U\cap V)\cup ((U\cap V) \times (\neg U\cap V)),
\end{eqnarray*}
where we have used the fact that $V$ is a filter with respect to $\sqcap$ (Corollary \ref{UVspectral}.\ref{UVspectral4}). Since $U\cap V\in \mathsf{CO}\mathcal{RO}(U)$ and $\neg U\cap V\in\mathsf{CO}\mathcal{RO}(\neg U)$, it follows from the above equation and Definition \ref{UVunionDef} that $f^{-1}[V]$ is open in $U\bigcirc\neg U$.
Finally, to see that $f^{-1}$ is continuous, each basic open of $U\bigcirc\neg U$ is of the form $V\cup V'\cup (V\times V')$ for $V\in\mathsf{CO}\mathcal{RO}(U)$ and $V'\in\mathsf{CO}\mathcal{RO}(\neg U)$ by Definition \ref{UVunionDef}. Then $V,V'\in\mathsf{CO}\mathcal{RO}(X)$ and
\begin{eqnarray*}
f[V\cup V'\cup (V\times V')] &=& f[V]\cup f[V']\cup f[V\times V'] \\
&=& V\cup V' \cup \{x\sqcap y\mid x\in V,\, y\in V'\}\\
&=& V\vee V' \in\mathsf{CO}\mathcal{RO}(X),
\end{eqnarray*}
where the last equality uses Corollary \ref{UVspectral}.\ref{UVspectral6}.
\end{proof}
\begin{prop}\label{CompleteBAProp} Any complete BA is isomorphic to the product of a complete and atomless BA and a complete and atomic BA.
\end{prop}
\begin{proof} By duality, it suffices to show that any complete UV-space $X$ is the UV-sum of a complete UV-space with no isolated points and a complete UV-space in which the isolated points form a dense subset. Since $X$ is complete, $U:=\mathsf{int} (\mathsf{cl} X_\mathrm{iso})\in\mathsf{CO}\mathcal{RO}(X)$. Form the subspaces induced by $U$ and $\neg U$. By Lemma \ref{SumLem}, $X$ is homeomorphic to the UV-sum of these subspaces. By Lemma \ref{SubspaceComplete}, both subspaces are complete UV-spaces. Clearly $(\neg U)_\mathrm{iso}\subseteq X_\mathrm{iso}$, and $X_\mathrm{iso}=U_\mathrm{iso}$, which with $\neg U\cap U=\varnothing$ implies $(\neg U)_\mathrm{iso}=\varnothing$. Thus, the subspace induced by $\neg U$ has no isolated points. Finally, in the subspace induced by $U$, we have
\[\mathsf{int}^U\mathsf{cl}^U U_\mathrm{iso}=U\cap\mathsf{int}(\mathsf{cl} U_\mathrm{iso})=U\cap U=U,\] which implies $\mathsf{cl}^U U_\mathrm{iso}=U$ by Proposition \ref{AtomicProp}.\end{proof}
\subsection{Subalgebras of BAs} Let $\mathbb{B}_n$ be the finite Boolean algebra with $n$ atoms. As our final example, we will prove using our duality that every infinite BA contains subalgebras isomorphic to $\mathbb{B}_n$ for each positive integer $n$. First, we prove some preliminary results about UV-spaces.
\begin{definition} Let $X$ be a UV-space and $\{U_0,\dots,U_n\}$ a family of $\mathsf{CO}\mathcal{RO}(X)$ sets. We say that $\{U_0,\dots,U_n\}$ is a \textit{regular partition} of $X$ iff $U_0,\dots,U_n$ are pairwise disjoint and $X=U_0\vee\dots \vee U_n$.
\end{definition}
\begin{prop}\label{prop: partition}
Let $X$ be an infinite UV-space. For each $n\in \omega$, there is a family $\{V_0,\dots, V_n\}$ of $\mathsf{CO}\mathcal{RO}$ sets that is a regular partition of $X$.
\end{prop}
\begin{proof} Consider the antichain $\{U_0\cap \neg U_1$, $U_1\cap\neg U_2$, \dots , $U_{n-1}\cap\neg U_n\}$ constructed in the proof of Proposition \ref{ChainsAnti}. Let $U_{n+1}=\varnothing$. We claim that the antichain $\{U_0\cap \neg U_1,\dots, U_{n-1}\cap\neg U_n, U_n\cap \neg U_{n+1}\}$ is a regular partition, i.e., its join is $X$. Using the equation for join in terms of $\mathsf{int}_\leqslant \mathsf{cl}_\leqslant$ and union (Proposition \ref{COROBA}), it suffices to show that for every $x\in X$, there is a $y\geqslant x$ such that $y\in U_i\cap U_{i+1}$ for some $i\in\{0,\dots, n\}$. If $x\in \neg U_1$, then since $U_0=X$, we have $x\in U_0\cap\neg U_1$, so we take $y=x$ and $i=0$. If $x\not\in U_1$, then there is an $x_1\geqslant x$ such that $x_1\in U_1$. Now if $x_1\in \neg U_2$, then $x_1\in U_1\cap\neg U_2$, so we take $y=x_1$ and $i=1$. If $x_1\not\in \neg U_2$, then there is an $x_2\geqslant x_1$ such that $x_2\in U_2$. By transitivity, $x_2\geqslant x$. If $x_2\in\neg U_3$, then $x_2\in U_2\cap \neg U_3$, so we take $y=x_2$ and $i=2$. If $x_2\not\in U_3$, then there is an $x_3\geqslant x_2$ such that $x_3\in U_3$, etc. If we do not find our $y$ and $i$ in this way by $n-1$, then we reason as follows: given $x_{n-1}\not\in \neg U_n$, there is an $x_n\geqslant x_{n-1}$ such that $x_n\in U_n$. Then since $U_{n+1}=\varnothing$, we have $x_n\in U_n\cap\neg U_{n+1}$, so we set $y=x_n$ and $i=n$.\end{proof}
To obtain Corollary \ref{cor: partition2} below from Proposition \ref{prop: partition}, we use the following topological fact.
\begin{fact}\label{TopFact} For any space $X$, $V\subseteq X$, and open $U\subseteq X$, if $U\cap V=\varnothing$, then $U\cap \mathsf{int}(\mathsf{cl}(V))=\varnothing$.
\end{fact}
\begin{cor}\label{cor: partition2}
Let $X$ be a UV-space. For each positive integer $m$, there is a family $\{U_1,\dots, U_m\}$ of pairwise disjoint $\mathsf{CO}\mathcal{RO}$ sets such that:
\begin{enumerate}
\item\label{partition2a} for every $x\in X$, there is a unique $K\subseteq \{1,\dots, m\}$ such that $x\in {\bigvee}_{k\in K}U_k$ and $x\not\in {\bigvee}_{j\in J} U_j$ for each $J\subsetneq K$;
\item\label{partition2b} for every $K\subseteq \{1,\dots, m\}$ such that $K\neq\varnothing$, there is an $x\in X$ such that $x\in \underset{k\in K}{\bigvee} U_k$ and $x\not\in \underset{j\in J}{\bigvee} U_j$ for each $J\subsetneq K$;
\item\label{partition2c} for every $K\subseteq \{1,\dots, m\}$, if $x\not\in \neg \bigvee_{k\in K} U_k \vee \bigvee_{j\in J} U_j $ for each $J\subsetneq K$, then there is a $y\geqslant x$ such that $y\in {\bigvee}_{k\in K} U_k$ and $y\not\in {\bigvee}_{j\in J} U_j$ for each $J\subsetneq K$.
\end{enumerate}
\end{cor}
\begin{proof} By Proposition \ref{prop: partition}, there is a family $\{U_1,\dots, U_m\}$ of pairwise disjoint $\mathsf{CO}\mathcal{RO}(X)$ sets such that $X= U_1\vee\dots\vee U_m$. It follows that for each $x\in X$, there is a $K\subseteq \{1,\dots, m\}$ such that $x\in \bigvee_{k\in K} U_k$ and such that $x\not\in \bigvee_{j\in J} U_j$ for each $J\subsetneq K$. It remains to show that this $K$ is unique. Suppose not, so there is a $K'\subseteq \{1,\dots, m\}$ such that $K'\neq K$, $x\in \bigvee_{k\in K'} U_k$, and $x\not\in \bigvee_{j\in J'} U_j$ for each $J'\subsetneq K'$. Since $K'\not\subseteq K$, pick $k'\in K'\setminus K$. Since $x\in \bigvee_{k\in K'} U_k$ but $x\not\in \bigvee_{j\in K'\setminus \{k'\}} U_j$, it follows that there is some $y\geqslant x$ such that $y\in U_{k'}$. Since $U_{k'}$ is disjoint from $\bigcup_{k\in K} U_k$, we have $y\not\in \bigvee_{k\in K} U_k$ by Fact \ref{TopFact} and the equation for join in terms of $\mathsf{int}_\leqslant \mathsf{cl}_\leqslant$ and union (Proposition \ref{COROBA}). But then since $y\geqslant x$, we have $x\not\in \bigvee_{k\in K} U_k$, contradicting our assumption. Thus, $K$ is unique.
For part 2, let $K\subseteq \{1,\dots, m\}$ and $K\neq\varnothing$. Since $\{V\in \mathsf{CO}\mathcal{RO}(X)\mid \bigvee_{k\in K}U_k\subseteq V\}$ is a proper filter in $\mathsf{CO}\mathcal{RO}(X)$, it follows by the definition of a UV-space (Definition \ref{VOspace}.\ref{PossCompact}) that there is some $x\in X$ such that $\mathsf{CO}\mathcal{RO}(x)=\{V\in \mathsf{CO}\mathcal{RO}(X)\mid \bigvee_{k\in K}U_k\subseteq V\}$. Now suppose $J\subsetneq K$ and consider some $i\in K\setminus J$. Hence by Fact \ref{TopFact}, $U_i$ is disjoint from $\bigvee_{j\in J}U_j$. Thus, $\bigvee_{k\in K}U_k\not\subseteq \bigvee_{j\in J}U_j$, which with $\mathsf{CO}\mathcal{RO}(x)=\{V\in \mathsf{CO}\mathcal{RO}(X)\mid \bigvee_{k\in K}U_k\subseteq V\}$ implies that $\bigvee_{j\in J}U_j\not\in \mathsf{CO}\mathcal{RO}(x)$, i.e., $x\not\in \bigvee_{j\in J}U_j$.
For part 3, let $K\subseteq \{1,\dots, m\}$, and suppose $x\not\in \neg \bigvee_{k\in K} U_k \vee \bigvee_{j\in J} U_j $ for each $J\subsetneq K$. It follows that the filter $F$ in $\mathsf{CO}\mathcal{RO}(X)$ generated by $\mathsf{CO}\mathcal{RO}(x)\cup \{ \bigvee_{k\in K}U_k\}$ is a proper filter such that $\bigvee_{j\in J} U_j\not\in F$ for each $J\subsetneq K$. Then by the definition of a UV-space (Definition \ref{VOspace}.\ref{PossCompact}), there is some $y\in X$ such that $\mathsf{CO}\mathcal{RO}(y)=F$. Hence $\mathsf{CO}\mathcal{RO}(x)\subseteq\mathsf{CO}\mathcal{RO}(y)$, which implies $x\leqslant y$ by the definition of a UV-space (Definition \ref{VOspace}), and $y\in \bigvee_{k\in K}U_k$. Finally, for $J\subsetneq K$, from $\bigvee_{j\in J} U_j\not\in F$ and $F=\mathsf{CO}\mathcal{RO}(y)$, we have $y\not\in \bigvee_{j\in J}U_j$.\end{proof}
\begin{theorem}\label{SubalgebrasProp} Every infinite BA $\mathbb{B}$ contains subalgebras isomorphic to $\mathbb{B}_n$ for each positive integer $n$.
\end{theorem}
\begin{proof}
Let $X$ be the infinite UV-space dual to $\mathbb{B}$ and $X_n$ the finite UV-space dual to $\mathbb{B}_n$. By duality, it suffices to show there is a surjective UV-map $f$ from $X$ onto $X_n$. Let $x_1,\dots, x_m$ be the maximal elements of $X_n$. By Corollary \ref{UVspectral}.\ref{UVspectral4.5}, $x_1,\dots, x_m$ are the co-atoms of a Boolean algebra obtained by adding a top node to $X_n$. Therefore, we have that (a) for every $y\in X_n$, there is a unique $K\subseteq \{1,\dots,m\}$ such that $y=\bigwedge_{k\in K} x_k$.
Take a family $\{U_1,\dots, U_m\}$ of $\mathsf{CO}\mathcal{RO}(X)$ sets as in Corollary \ref{cor: partition2}. By Corollary \ref{cor: partition2}.\ref{partition2a}, for each $x\in X$, there is a unique $K\subseteq\{1,\dots,m\}$ such that $x\in \bigvee_{k\in K} U_k$ and $x\not\in \bigvee_{j\in J} U_j$ for each $J\subsetneq K$. Let $f(x)=\bigwedge_{k\in K} x_k$. Then by (a) and Corollary \ref{cor: partition2}.\ref{partition2b}, $f$ is surjective.
Now we show that $f$ is a UV-map. First note that the compact opens of $X_n$ are exactly the upsets of $X_n$ with respect to $\leqslant$. Now let $y\in X_n$ be such that $y=\bigwedge_{i\in I} x_i$ for some $I\subseteq \{1,\dots,m\}$. Then it follows from the definition of
$f$ that $f^{-1}[{\Uparrow}y] = \bigvee_{i\in I} U_i \in \mathsf{CO}\mathcal{RO}(X)$. Now let $U\subseteq X_n$. Then $U = \bigcup_{y\in U} {\Uparrow}y$ and $f^{-1}[U] = f^{-1} [ \bigcup_{y\in U} {\Uparrow}y] = \bigcup_{y\in U} f^{-1}[{\Uparrow}y]$.
Since the collection of compact open sets is closed under finite unions, we obtain that $f^{-1}[U] $ is compact open in $X$. Therefore, $f$ is a spectral map.
Finally, suppose $f(x)\leqslant y'$. Then there are $I, K\subseteq \{1,\dots,m\}$ such that $I\subseteq K$, $y' = \bigwedge_{i\in I} x_i$, $f(x) = \bigwedge_{k\in K} x_k$, and $K$ is the unique subset of $\{1,\dots,m\}$ such that $x\in \bigvee_{k\in K}U_k$ and (b) $x\not\in \bigvee_{j\in J}U_j$ for each $J\subsetneq K$. We claim there is a $y\geqslant x$ such that (c) $y\in \bigvee_{i\in I}U_i$ and $y\not\in \bigvee_{\ell\in L}U_\ell$ for each $L\subsetneq I$. By Corollary \ref{cor: partition2}.\ref{partition2c}, it suffices to show that $x\not\in \neg \bigvee_{i\in I}U_i\vee \bigvee_{\ell\in L}U_\ell$ for each $L\subsetneq I$. For contradiction, suppose $x\in \neg \bigvee_{i\in I}U_i\vee \bigvee_{\ell\in L}U_\ell$ for some $L\subsetneq I$. Then since $I\subseteq K$ and $x\in \bigvee_{k\in K}U_k$, it follows that $x\in \bigvee_{k\in K\setminus (I\setminus L)}U_k$, which contradicts (b). By (c) and the definition of $f$, we have $f(y)=\bigwedge_{i\in I} x_i=y'$. Thus, $f$ is a UV-map.\end{proof}
\section{Perspectives on UV-spaces assuming choice}\label{ChoiceSection} In this penultimate section, we briefly discuss some results about UV-spaces that can be proved under the assumption of the Boolean Prime Ideal Theorem (BPI).
\subsection{UV-spaces as upper Vietoris spaces of Stone spaces}\label{UVStoneSection} Recall from Definition \ref{UVStoneDef} that for a Stone space $X$, $\mathscr{UV}(X)$ is the hyperspace of nonempty closed subsets of $X$ endowed with the upper Vietoris topology. We already observed (Corollary \ref{StoneCor}) that $\mathscr{UV}(X)$ is a UV-space. Assuming the BPI, every UV-space arises homeomorphically in this way.
\begin{prop} Assuming the BPI, every UV-space is homeomorphic to $\mathscr{UV}(X)$ for some Stone space $X$.
\end{prop}
\begin{proof} Let $Y$ be a UV-space. Let $X$ be the Stone dual of $\mathsf{CO}\mathcal{RO}(Y)$, so $\mathsf{Clop}(X)$ is isomorphic to $\mathsf{CO}\mathcal{RO}(Y)$. By Proposition \ref{UVStone}, $\mathscr{UV}(X)$ is homeomorphic to $UV(\mathsf{Clop}(X))$. Combining the previous two facts, we have that $\mathscr{UV}(X)$ is homeomorphic to $UV(\mathsf{CO}\mathcal{RO}(Y))$, which is homeomorphic to $Y$ by Theorem \ref{SecondThm}.\ref{SecondThmB}. Thus, $Y$ is homeomorphic to $\mathscr{UV}(X)$.\end{proof}
\subsection{Equivalent Priestley spaces assuming choice}\label{Priestley} In this subsection, we relate UV-spaces to Priestley spaces \cite{Priestley1970}. For convenience, we now keep track of the topology $\tau$ of a space explicitly.
For a spectral space $(X,\tau)$, its corresponding Priestley space $(X, \leq, \tau^+)$ is defined as follows: $\tau^+$ is the patch topology of $\tau$, i.e., the topology generated by $\tau \cup\{X\setminus U \mid U\in \tau\}$ as a subbasis, and $\leq$ is the specialization order of $\tau$. Note that the BPI, in its equivalent form as the Alexander Subbasis Theorem, is used already in showing that the patch topology of a spectral topology is
compact. Conversely, if $(X, \leq, \tau)$ is a Priestley space, then $X$ together with the topology given
by open upsets of $(X, \leq, \tau)$ is a spectral space. It is well known that $U\subseteq X$ is compact open in a spectral space iff $U$ is a clopen upset in the associated Priestley space, with the right-to-left direction using the~BPI.
Since UV-spaces are spectral, given a UV-space $(X, \tau)$ we can consider the corresponding Priestley space $(X, \leq, \tau^+)$. It is easy to see that $U\subseteq X$ is
$\mathsf{CO}\mathcal{RO}$ in $(X, \tau)$ iff $U$ is a clopen $\mathcal{RO}$ subset of $(X, \leq, \tau^+)$, where $\mathcal{RO}$ is now taken with respect to $\leq$.
Let $\mathsf{Clop}\mathcal{RO}(X)$ be the set of clopen $\mathcal{RO}$ subsets of $(X, \leq, \tau)$.
Then the definition of a UV-space easily translates into
the following definition in terms of Priestley spaces.
\begin{definition}
A Priestley space $(X, \leq, \tau)$ is a \emph{UV-Priestley space} iff:
\begin{enumerate}
\item $\mathsf{Clop}\mathcal{RO}(X)$ is closed under $\mathsf{int}_\leq (X\setminus \cdot)$;
\item if $x\nleq y$, then there is a $U\in \mathsf{Clop}\mathcal{RO}(X)$ such that $x\in U$ and $y\notin U$.\footnote{Note that if we had only required that $U$ be a clopen upset, then part 2 would be exactly the Priestley separation axiom.}
\item every proper filter in $\mathsf{Clop}\mathcal{RO}(X)$ is $\mathsf{Clop}\mathcal{RO}(x)$ for some $x \in X$, where $\mathsf{Clop}\mathcal{RO}(x) = \{U\in \mathsf{Clop}\mathcal{RO}(X) \mid x\in U\}$.
\end{enumerate}
\end{definition}
It is easy to verify that if $(X, \tau)$ is a UV-space, then $(X, \leq,\tau^+)$ is a UV-Priestley space, and if
$(X, \leq,\tau)$ is a UV-Priestley space, then $X$ together with the topology given by open upsets of $(X, \leq,\tau)$ is a UV-space.
Moreover, given a UV-Priestley space $(X, \leq,\tau)$, it is easy to see that $\mathsf{Clop}\mathcal{RO}(X)$ is a BA with meet as intersection and $\neg U = \mathsf{int}_\leqslant (X\setminus U)$. Conversely, given a BA $\mathbb{A}$, we obtain a dual UV-Priestley space $X$ based on the set of proper filters in $\mathbb{A}$ by defining $\leq$ as $\subseteq$ and generating a topology by declaring $\{\widehat{a}, \mathrm{PropFilt}(\mathbb{A})\setminus \widehat{a}\mid a\in \mathbb{A}\}$ as a subbasis. It is easy to see that this is the same as taking the UV-space dual
to $\mathbb{A}$ and considering its corresponding UV-Priestley space. Then $\mathbb{A}$ is isomorphic to the BA $\mathsf{Clop}\mathcal{RO}(X)$, and each UV-Priestley space $Y$ is order-homeomorphic to the dual of $\mathsf{Clop}\mathcal{RO}(Y)$.
Next we discuss morphisms, which are the obvious adaptation of the UV-maps of Definition \ref{UVmapDef} to the Priestley setting.
\begin{definition}
A map $f:X\to X'$ between UV-Priestley spaces is called a \emph{UV-Priestley morphism} iff it is a Priestley morphism (i.e., continuous and order-preserving) satisfying the p-morphism condition:
\begin{center} if $f(x) \leq' y'$, then $\exists y$: $x \leq y$ and $f(y)=y'$.
\end{center}
\end{definition}
Assuming the BPI, it is easy to show that the category of UV-spaces and UV-maps is isomorphic to the category of UV-Priestley spaces and UV-Priestley morphisms, which is therefore dually equivalent to the category of BAs and BA homomorphisms by Theorem \ref{DualityThm}.
One can also develop a duality dictionary for this duality similar to the one discussed in Section 6. But we will not do so here, as our primary goal is to study the setting of choice-free dualities for BAs.
Just as one can move freely between Priestley spaces and the \textit{pairwise Stone spaces} of \cite{BBGK10}, one can also move freely between UV-Priestley spaces and analogous pairwise UV-spaces. We omit the details, as they are straightforward to reconstruct based on the information above and in \cite{BBGK10}.
\subsection{Goldblatt's representation of ortholattices}\label{GoldblattSection} Our choice-free duality for BAs is related to Goldblatt's \cite{Goldblatt1975} representation of ortholattices. An ortholattice is a bounded lattice equipped with an additional unary operation $'$ such that $a\wedge a'=0$, $a\vee a'=1$, $a''=a$, and $a\leq b$ only if $b'\leq a'$. Goldblatt showed that ortholattices can be represented using a Stone space $X$ equipped with a symmetric and irreflexive relation $\bot$. A subset $U\subseteq X$ is \textit{$\bot$-regular} iff $U=U^{\bot\bot}$ where $V^\bot=\{x\in X\mid x\bot y \mbox{ for all }y\in V\}$. The collection of all $\bot$-regular subsets ordered by inclusion forms a complete ortholattice with $'$ as $^\bot$. Conversely, every complete ortholattice is isomorphic to the collection of $\bot$-regular subsets with $^\bot$ coming from a set with a symmetric and irreflexive relation $\bot$. To represent an arbitrary ortholattice $L$, Goldblatt defined a space $X$ with a binary relation $\bot$ as follows:
\begin{enumerate}
\item the underlying set of $X$ is the set of all proper filters of $L$;
\item for $F,G\in X$, let $F \bot G$ iff there is some $a\in F$ such that $a'\in G$;
\item the topology of $X$ is generated by the collection of sets $\widehat{a}$ and $X\setminus \widehat{a}$ as a subbasis, i.e., the patch topology associated with $\tau=\{\widehat{a}\mid a\in L\}$.
\end{enumerate}
Assuming the BPI, Goldblatt proved that $X$ is a Stone space and $L$ is isomorphic to the collection of \textit{clopen $\bot$-regular} sets ordered by inclusion with the operation $^\bot$. Since every BA is an ortholattice with $'$ as Boolean complement, this representation applies to BAs. Like our representation of BAs, it uses the proper filters of $L$. Indeed, Goldblatt's representation applied to BAs is essentially the UV-Priestley representation discussed in Section \ref{Priestley} but using the incompatibility relation $\bot$ between proper filters instead of the inclusion order on proper filters, which is the specialization order of $\tau$. It is easy to see that for a BA, the $\bot$-regular sets are exactly the regular open sets with respect to the inclusion order.
There are two important differences between Goldblatt's representation applied to BAs and ours. First, because we work with the spectral topology $\tau$ instead of the patch topology, we do not need the extra datum of the relation $\bot$; the regular sets can be defined simply in terms of the specialization order of the space. Thus, we can work with spaces instead of spaces plus a binary relation. Second, because we work with the spectral topology $\tau$ instead of the patch topology, we do not require the nonconstructive BPI.
\section{Conclusion}\label{Conclusion} We have developed a full choice-free duality for BAs in terms of UV-spaces. We showed how to translate, via this duality, the main algebraic concepts and constructions into topological terms. We also gave several sample applications of this duality in the form of choice-free proofs, using spatial intuition essentially, of some basic facts about BAs.
The distinguishing features of the duality for BAs in this paper are that (a) the duals of BAs are topological spaces and (b) the duality is choice-free. Standard Stone duality satisfies (a) but not (b). The pointfree duality using Stone locales satisfies (b) but not (a). To draw a contrast with the localic approach, we characterized our approach to choice-free Stone duality as the \textit{hyperspace approach}. The choice-freeness is achieved by not working with Stone spaces, but rather with UV-spaces, examples of which are given by the upper Vietoris hyperspace of a Stone space. Assuming choice, all UV-spaces arise homeomorphically in this way; but we do not need this assumption to carry out our duality for BAs.
Though we have concentrated on BAs, we believe that choice-free
duality does not end here. In future work, we aim to generalize the strategy of
this paper to obtain choice-free spatial dualities for other classes of algebras (connecting with work in \cite{Massas2016}), giving rise to choice-free completeness proofs for non-classical logics. We hope that this can be the beginning of a new area of choice-free
duality in non-classical logic and beyond.\\
\subsection*{Acknowledgement}
For helpful feedback, we wish to thank Johan van Benthem, Benno van den Berg, Guram Bezhanishvili, Yifeng Ding, David Gabelaia, Tom\'{a}\v{s} Jakl, Mamuka Jibladze, Frederik Lauridsen, Vincenzo Marra, Shezad Mohamed, Floris Sluijter, Joran van Weel, and the anonymous referee for \textit{The~Journal~of~Symbolic~Logic}. We are also grateful for comments from audiences at ALCOP 2017 in Glasgow, the 2017 Algebra and Coalgebra Seminar at the ILLC, University of Amsterdam, ToLo 2018 in Tbilisi, and BLAST 2018 in Denver.
\bibliographystyle{asl}
| {
"timestamp": "2021-12-14T02:44:08",
"yymm": "2112",
"arxiv_id": "2112.06859",
"language": "en",
"url": "https://arxiv.org/abs/2112.06859",
"abstract": "The standard topological representation of a Boolean algebra via the clopen sets of a Stone space requires a nonconstructive choice principle, equivalent to the Boolean Prime Ideal Theorem. In this paper, we describe a choice-free topological representation of Boolean algebras. This representation uses a subclass of the spectral spaces that Stone used in his representation of distributive lattices via compact open sets. It also takes advantage of Tarski's observation that the regular open sets of any topological space form a Boolean algebra. We prove without choice principles that any Boolean algebra arises from a special spectral space X via the compact regular open sets of X; these sets may also be described as those that are both compact open in X and regular open in the upset topology of the specialization order of X, allowing one to apply to an arbitrary Boolean algebra simple reasoning about regular opens of a separative poset. Our representation is therefore a mix of Stone and Tarski, with the two connected by Vietoris: the relevant spectral spaces also arise as the hyperspace of nonempty closed sets of a Stone space endowed with the upper Vietoris topology. In addition to representation, we establish a choice-free dual equivalence between the category of Boolean algebras with Boolean homomorphisms and a subcategory of the category of spectral spaces with spectral maps. We show how this duality can be used to prove some basic facts about Boolean algebras.",
"subjects": "Logic (math.LO); Category Theory (math.CT)",
"title": "Choice-free Stone duality",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9854964198826467,
"lm_q2_score": 0.7185943925708562,
"lm_q1q2_score": 0.708172201226324
} |
https://arxiv.org/abs/math/9902097 | Subsequences of frames | Every frame in Hilbert space contains a subsequence equivalent to an orthogonal basis. If a frame is n-dimensional then this subsequence has length (1 - \epsilon) n. On the other hand, there is a frame which does not contain bases with brackets. | \section{Introduction}
The notion of frame goes back to R.Duffin and A.Schaeffer \cite{D-S} and was studied
extensively since then with relation to nonharmonic Fourier analysis, see \cite{He}.
From a geometrical point of view, a frame in a Hilbert space $H$ is the image
of an orthonormal basis in a larger Hilbert space under an orthogonal projection
onto $H$, up to equivalence \cite{Ho}
(the equivalence constant is called the frame constant).
Since frames have nice representation properties (see \cite{D-S}, \cite{A}),
much attention was paid to their subsequences that inherit these properties.
The most interesting questions arise about subsequences equivalent to an orthogonal
basis \cite{Ho}, \cite{S}, \cite{C1}, \cite{C-C1}.
P.Casazza \cite {C2} proved that, given an $\varepsilon > 0$,
any $n$-dimensional frame whose norms are well bounded below
contains a subsequence of length $(1 - \varepsilon) n$ equivalent to an orthogonal basis
(the constant of equivalence does not depend of $n$).
In the present paper this is proved for all frames,
without restrictions on norms of the elements.
If a frame is $n$-dimensional then it contains a subsequence
of length $(1 - \varepsilon) n$ which is $C$-equivalent to an orthogonal basis.
Here $C$ depends only on the frame constant and $\varepsilon$.
To put the result in other words, orthogonal projections in Hilbert space
preserve orthogonal structure in almost whole range.
Namely, any orthogonal projection $H$ of an orthogonal basis
contains a subset of cardinality $(1 - \varepsilon) {\rm rank} (P)$
which is $C(\varepsilon)$-equivalent to an orthogonal system.
This is proved in Section \ref{secfinitedim}.
An infinite dimensional version of this result is considered in Section \ref{basicsubs}.
Every infinite dimensional frame has an infinite subsequence equivalent to an
orthogonal basis.
However, for some frames this subsequence can not be complete,
as was shown by K.Seip \cite{S} and P.Casazza and O.Christensen \cite{C-C2}.
This result is generalized in Section \ref{secwithoutbrackets}
by constructing a frame which does not contain bases with brackets.
So our frame $(x_j)$ is "asymptotically indecomposable" in the following sense.
If $(y_j)$ is any complete subsequence of $(x_j)$, then the distance
from ${\rm span}(y_j)_{j \le n}$ to ${\rm span}(y_j)_{j > n}$ tends to zero as $n \rightarrow \infty$.
In the rest of this section we recall standard definitions and simple
known facts about frames.
In what follows, $H$ will denote a separable Hilbert space,
finite or infinite dimensional.
Absolute constants will be denoted by $c_1, c_2, \ldots$.
A sequence $(x_j)$ in $H$ is called a {\em frame}
if there exist positive numbers $A$ and $B$ such that
$$
A \|x\|^2 \le \sum_j | \langle x, x_j \rangle |^2 \le B \|x\|^2
\ \ \ \ \mbox{for $x \in H$.}
$$
The number $(B/A)^{1/2}$ is called a {\em constant} of the frame.
We call $(x_j)$ a {\em tight frame} if $A = B = 1$.
Two sequences $(x_j)$ and $(y_j)$ in possibly different Banach spaces
are called {\em equivalent} if there is an isomorphism
$T : [x_j] \rightarrow [y_j]$ such that $Tx_j = y_j$ for all $j$.
Here $[x_j]$ denotes the closed linear span of $(x_j)$.
Let $c = \|T\| \|T^{-1}\|$ then the sequences $(x_j)$ and $(y_j)$
are called {\em $c$-equivalent}.
\medskip
The next observation (see \cite{Ho}) allows to look at frames
as at projections of the canonical vector basis $(e_j)$ in $l_2$.
\begin{proposition}
\label{view}
Let $(x_n)_{n=1}^m$ be a frame in $H$ with constant $c$,
where $m$ can be equal to infinity.
Then there is an orthogonal projection $P$ in $l_2^m$ such that
$(x_n)$ is $c$-equivalent to $(P e_n)$.
Conversely, if $P$ is an orthogonal projection in $l_2^m$ onto a
subspace $H$,
then $(P e_n)_{n=1}^m$ is a tight frame in $H$.
\end{proposition}
\begin{corollary} \label{frameistight}
Let $(x_n)$ be a frame with constant $c$.
Then $(x_n)$ is $c$-equivalent to a tight frame.
\end{corollary}
Now we present another view at frames.
We can regard them as the columns of a row-orthogonal matrix (either
finite or infinite).
\begin{lemma} \label{columnrow}
Let $n, m \in {\bf N} \cup \infty$ and $A$ be an $n \times m$ matrix
whose rows are orthonormal.
Then the columns of $A$ form a tight frame in $l_2^n$.
Conversely, let $(x_j)_{j=1}^m$ be a frame in $H$.
Then there exists an $n \times m$ matrix $A$ with $n = \dim H$
whose rows are orthonormal and such that the columns
form a tight frame equivalent to $(x_j)$.
\end{lemma}
\noindent {\bf Proof.}\ \
If $A$ is as above then $A^*$ acts as an isometric embedding
of $l_2^n$ into $l_2^m$.
Then $A$ acts as a quotient map in a Hilbert space, and we can regard
it
as an orthogonal projection. On the other side, the columns of $A$
are equal to $Ae_j$. Proposition \ref{view} finishes the proof of the
first statement.
The converse can also be proved by this argument.
{\mbox{}\nolinebreak\hfill\rule{2mm}{2mm}\par\medbreak}
\begin{lemma} \label{square}
Let $(x_j)$ be a tight frame in $H$.
Then $\sum_j \|x_j\|^2 = \dim H$
(which is possibly equal to infinity).
\end{lemma}
\noindent {\bf Proof.}\ \
By Proposition \ref{view} we may assume that $H$ is a subspace of $l_2$
and $x_j = P e_j$, where $P$ is the orthogonal projection in $l_2$
onto $H$.
Then the Hilbert-Schmidt norm
$\|P\|_{\rm HS} = ( \sum_j \|x_j\|^2 )^{1/2}$.
On the other hand, $\|P\|_{\rm HS} = (\dim H)^{1/2}$.
{\mbox{}\nolinebreak\hfill\rule{2mm}{2mm}\par\medbreak}
\section{Finite dimensional frames} \label{secfinitedim}
In this section we prove
\begin{theorem} \label{quantitative}
There is a function $h : {\bf R}_+ \rightarrow {\bf R}_+$ such that the following holds.
Suppose $(x_j)$ is an $n$-dimensional frame with constant $c$.
Then for every $\varepsilon > 0$ there is a set of indices $\sigma$
with $|\sigma| > (1-\varepsilon) n$ such that
the system $(x_j)_{j \in \sigma}$
is $C$-equivalent to an orthogonal basis,
where $C = h(\varepsilon) c$.
\end{theorem}
We will need a result of A.Lunin on norms of restriction of operators
onto coordinate subspaces \cite{L} (for improvements see \cite{K-Tz}).
\begin{theorem} (A.Lunin). \label{Lunin}
Let $T : l_2^m \rightarrow l_2^n$ be a linear operator.
Then there is a set $\sigma \subset \{1, \ldots, m\}$ with $|\sigma| = n$
such that
$$
\| T |_{{\bf R}^\sigma} \| \le c_1 {\sqrt \frac{n}{m}} \|T\|.
$$
\end{theorem}
Given an $h>0$, a system of vectors $(x_j)$ in a Hilbert space is
called
{\bf $h$-Hilbertian} if
$$
\Big\| \sum_j a_j x_j \Big\| \le h \Big( \sum_j |a_j|^2 \Big)^{1/2}
$$
for all sequences of scalars $(a_j)$.
Then Theorem \ref{Lunin} can be reformulated as follows.
Suppose $(x_j)_{1 \le j \le m}$ is a $1$-Hilbertian system
in $l_2^n$. Then there is a set $\sigma \subset \{1, \ldots, m\}$
with $|\sigma| = n$ such that
$({\sqrt \frac{m}{n}} x_j)_{j \in \sigma}$ is $c_1$-Hilbertian.
Next, we will use a result of J.Bourgain and L.Tzafriri
on invertibility of large submatrices \cite{B-Tz} Theorem 1.2:
\begin{theorem} (J.Bourgain, L.Tzafriri).
\label{BourgainTzafriri}
Let $T : l_2^n \rightarrow l_2^n$ be a linear operator
such that $\|T e_j\| = 1$ for all $j$.
Then there is a set $\sigma \subset \{1, \ldots, n\}$
with $|\sigma| \ge c_2 n / \|T\|^2$ such that
$$
\|Tx\| \ge c_2 \|x\| \ \ \ \ \mbox{for every $x \in {\bf R}^\sigma$}.
$$
\end{theorem}
Given a $b>0$, a system of vectors $(x_j)$ in a Hilbert space is called
{\bf $b$-Besselian} if
$$
b \Big\| \sum_j a_j x_j \Big\| \ge \Big( \sum_j |a_j|^2 \Big)^{1/2}
$$
for all sequences of scalars $(a_j)$.
Then Theorem \ref{BourgainTzafriri} can be reformulated as follows.
Suppose $(x_j)_{1 \le j \le n}$ is an $h$-Hilbertian system
in $l_2^n$ and $\|x_j\| \ge \alpha$ for all $1 \le j \le n$.
Then there is a set $\sigma \subset \{1, \ldots, n\}$
with $|\sigma| \ge c_2 (\alpha / h)^2 n$ such that
the system $(\alpha^{-1} x_j)_{j \in \sigma}$ is $c_3$-Besselian.
Clearly, every tight frame is $1$-Hilbertian.
\begin{lemma}
\label{sizeoftau}
Let $(y_j)_{1 \le j \le m}$ be a tight frame in $l_2^n$
with $\|y_j\| = {\sqrt \frac{n}{m}}$ for all $j$.
Let $P$ be a $k$-dimensional orthogonal projection in $l_2^n$.
Then for $\delta > 0$
$$
\Big| \Big\{ j : \| (I - P) y_j \| \ge \delta {\sqrt \frac{n}{m}} \Big\} \Big|
\ge \Big( 1 - \delta^2 - \frac{k}{n} \Big) m.
$$
\end{lemma}
\noindent {\bf Proof.}\ \
Let $\tau = \Big\{ j : \| (I - P) y_j \| \ge \delta {\sqrt \frac{n}{m}} \Big\}$.
Since $((I - P)y_j)_{1 \le j \le m}$ is a tight frame in an
$(n-k)$-dimensional space $(I - P) l_2^n$, Lemma \ref{square} yields
\begin{eqnarray*}
n - k &=& \sum_{j=1}^m \|(I - P) y_j\|^2
\le \sum_{j \in \tau} \|y_j\|^2 + \sum_{j \in \tau^c} \|(I -
P)y_j\|^2 \\
&\le& |\tau| \cdot (n/m) + m \cdot \delta^2 (n/m)
= (|\tau| / m + \delta^2) n.
\end{eqnarray*}
The required estimate follows.
{\mbox{}\nolinebreak\hfill\rule{2mm}{2mm}\par\medbreak}
\vspace{0.5cm}
Now we proceed to the proof of Theorem \ref{quantitative}.
As in P.Casazza's proof \cite{C2}, the set $\sigma$ will be constructed by
an iteration procedure. Our proof consists of several parts.
{\bf I. Splitting.} \ \
By Corollary \ref{frameistight}, we may assume that
the frame $(x_j) \subset l_2^n$ is tight and all of its terms are nonzero.
First we will split $(x_j)$ to get almost equal norms of the terms.
Note that if we substitute any member $x_j$ of the frame by $k$ elements
$x_j / \sqrt{k}, \ldots, x_j / \sqrt{k}$, we will still get a tight
frame.
Fix a $\nu > 0$.
Splitting each element $x_j$ as above, we can obtain a new tight frame
$(y_j)_{1 \le j \le m}$ such that
(i) elements of $(y_j)$ are multiples of the ones from $(x_j)$;
(ii) there is a $\lambda > 0$ such that
$\lambda \le \|y_j\| \le (1 + \nu) \lambda$ for all $j = 1, \ldots, m$.
\noindent The constant $\lambda$ be evaluated using Lemma \ref{square}:
$$
(1 + \nu)^{-1} {\sqrt \frac{n}{m}} \le \|y_j\| \le (1 + \nu) {\sqrt \frac{n}{m}}
\ \ \ \ \mbox{for $j = 1, \ldots, m$}.
$$
Clearly, it is enough to prove the theorem for $(y_j)$ instead of
$(x_j)$.
We can choose the parameter $\nu =\nu(\varepsilon) > 0$ arbitrarily small.
To make the proof more readable, we simply assume that $\nu = 0$
which is a slight abuse of rules. The reader will easily adjust the
arguments
to the general case. So we have
$$
\|y_j\| = {\sqrt \frac{n}{m}}, \ \ \ \ j=1, \ldots, m.
$$
We can also assume that $(\varepsilon / 2) m \ge n$.
{\bf II. Iterative construction.} \ \
Let $\delta = \sqrt{\varepsilon / 2}$.
{\em Step 1.} \
Set $\tau_0 = \{1, \ldots, m\}$.
The system $(y_j)_{j \in \tau_0}$ is $1$-Hilbertian.
Lunin's theorem yields the existence of a set $\sigma'_1 \subset \tau_0$
with $|\sigma'_1| = n$ such that
$$
\mbox{the system $({\sqrt \frac{m}{n}} y_j)_{j \in \sigma'_1}$ is $c_1$-Hilbertian.}
$$
Note that $\| {\sqrt \frac{m}{n}} y_j \| = 1$ for $j \in \sigma'_1$.
Then Bourgain-Tzafriri's theorem gives us a set
$\sigma_1 \subset \sigma'_1$ with $|\sigma_1| \ge (c_2 / c_1^2) n$ such that
$$
\mbox{the system $({\sqrt \frac{m}{n}} y_j)_{j \in \sigma_1}$ is $c_3$-Besselian.}
$$
So we have already found a subsequence $(y_j)_{j \in \sigma_1}$
of length proportional to $n$ which is well equivalent
to an orthogonal basis. If $|\sigma_1| \ge (1 - \varepsilon) n$,
then we are done and stop here. Otherwise proceed to the next step.
{\em Step 2.} \
Let $P_1$ be the orthogonal projection in $l_2^n$
onto $[y_j]_{j \in \sigma_1}$.
Let
$$
\tau_1 = \Big\{ j : \| (I - P_1) y_j \| \ge \delta {\sqrt \frac{n}{m}} \Big\}.
$$
Clearly, $\tau_1 \subset \sigma_1^c$.
By Lemma \ref{sizeoftau}
$$
|\tau_1| \ge \Big( 1 - \delta^2 - \frac{|\sigma_1|}{n} \Big) m.
$$
As $|\sigma_1| < (1 - \varepsilon) n$,
$$
|\tau_1| > \Big( 1 - \delta^2 - (1 - \varepsilon) \Big) m
= (\varepsilon / 2) m.
$$
The system $(y_j)_{j \in \tau_1}$ is $1$-Hilbertian
and $|\tau_1| \ge n$ by the choise of $m$.
Lunin's theorem yields the existence of a set $\sigma'_2 \subset \tau_1$
with $|\sigma'_2| = n$ such that
$$
\mbox{the system
$( \sqrt{\frac{|\tau_1|}{n}} y_j )_{j \in \sigma'_2}$
is $c_1$-Hilbertian.}
$$
Then the system $(\sqrt{\frac{|\tau_1|}{n}} (I - P_1) y_j)_{j \in \sigma'_2}$
is also $c_1$-Hilbertian.
By the definition of $\tau_1$, it has not too small norms:
$$
\Big\| \sqrt{\frac{|\tau_1|}{n}} (I - P_1) y_j \Big\|
\ge \delta \sqrt{\frac{|\tau_1|}{m}}, \ \ \ \ j \in \sigma'_2.
$$
Then Bourgain-Tzafriri's theorem gives us a set $\sigma_2 \subset
\sigma'_2$
with
$$
|\sigma_2| \ge c_2 \Big( \delta^2 \frac{|\tau_1|}{m} / c_1^2 \Big) n
\ge (c_2 / c_1^2) \delta^2 \Big( (1 - \delta^2) n - |\sigma_1| \Big)
$$
such that
$$
\mbox{the system
$( {\sqrt \frac{m}{n}} (I - P_1) y_j )_{j \in \sigma_2}$
is $(c_3 \delta^{-1})$-Besselian.}
$$
If $|\sigma_1| + |\sigma_2| \ge (1 - \varepsilon) n$, then we stop here.
Otherwise proceed to the next step.
{\em Step $k+1$.} \ We assume that the sets $\sigma_1, \ldots, \sigma_k$
are already constructed and
\begin{equation} \label{notyet}
\sum_{i=1}^k |\sigma_i| < (1 - \varepsilon) n.
\end{equation}
Let $P_k$ be the orthogonal projection in $l_2^n$
onto $[y_j]_{j \in \sigma_1 \cup \ldots \cup \sigma_k}$.
Let
$$
\tau_k = \Big\{ j : \| (I - P_k) y_j \| \ge \delta {\sqrt \frac{n}{m}} \Big\}.
$$
Clearly, $\tau_k \subset (\sigma_1 \cup \ldots \cup \sigma_k)^c$.
By Lemma \ref{sizeoftau}
$$
|\tau_k| \ge \Big( 1 - \delta^2 - \frac{ \sum_{i=1}^k |\sigma_i| }{n} \Big) m.
$$
By (\ref{notyet})
$$
|\tau_k| > \Big( 1 - \delta^2 - (1 - \varepsilon) \Big) m
= (\varepsilon / 2) m.
$$
The system $(y_j)_{j \in \tau_k}$ is $1$-Hilbertian
and $|\tau_k| \ge n$ by the choise of $m$.
Lunin's theorem yields the existence of a set $\sigma'_{k+1} \subset \tau_k$
with $|\sigma'_{k+1}| = n$ such that
$$
\mbox{the system
$( \sqrt{\frac{|\tau_k|}{n}} y_j )_{j \in \sigma'_{k+1}}$
is $c_1$-Hilbertian.}
$$
Then the system $(\sqrt{\frac{|\tau_k|}{n}} (I - P_k) y_j)_{j \in \sigma'_{k+1}}$
is also $c_1$-Hilbertian.
By the definition of $\tau_k$, it has not too small norms:
$$
\Big\| \sqrt{\frac{|\tau_k|}{n}} (I - P_k) y_j \Big\|
\ge \delta \sqrt{\frac{|\tau_k|}{m}}, \ \ \ \ j \in \sigma'_{k+1}.
$$
Then Bourgain-Tzafriri's theorem gives us a set $\sigma_{k+1} \subset \sigma'_{k+1}$
with
\begin{equation}
\label{sizeofsigma}
|\sigma_{k+1}| \ge c_2\Big( \delta^2 \frac{|\tau_k|}{m} / c_1^2 \Big) n
\ge (c_2 / c_1^2) \delta^2 \Big( (1 - \delta^2) n - \sum_{i=1}^k|\sigma_i| \Big)
\end{equation}
such that
$$
\mbox{the system
$( {\sqrt \frac{m}{n}} (I - P_k) y_j )_{j \in \sigma_{k+1}}$
is $(c_3 \delta^{-1})$-Besselian.}
$$
If $\sum_{i=1}^{k+1} |\sigma_i| \ge (1 - \varepsilon) n$, then we stop here.
Otherwise proceed to the next step.
{\bf III. When we stop.} \ \
Let $k_0$ be the number of the last step, that is the smallest integer
such that
$$
\sum_{i=1}^{k_0} |\sigma_i| \ge (1 - \varepsilon) n.
$$
We claim that such $k_0$ exists
and there is a function $K(\varepsilon)$ such that $k_0 \le K(\varepsilon)$.
Indeed, let $K(\varepsilon) = [4 c_1^2 c_2^{-1} \varepsilon^{-2}] + 2$.
If the claim were not true, then
$$
\sum_{i=1}^k |\sigma_i| < (1 - \varepsilon) n \ \ \ \ \mbox{for $k = 1, \ldots, K(\varepsilon)$}.
$$
Then by (\ref{sizeofsigma}) for all $k = 2, \ldots, K(\varepsilon)$
\begin{eqnarray*}
|\sigma_k| &\ge& (c_2 / c_1^2) \delta^2 \Big( (1 - \delta^2) - (1 - \varepsilon) \Big) n \\
& = & (c_2 / c_1^2) (\varepsilon^2 / 4) n.
\end{eqnarray*}
Thus
$$
\sum_{i=1}^{K(\varepsilon)} |\sigma_i|
\ge (K(\varepsilon) - 1) \cdot (c_2 / c_1^2) (\varepsilon^2 / 4) n \ge n.
$$
This contradiction proves the claim.
Now set $\sigma = \sigma_1 \cup \ldots \cup \sigma_{k_0}$,
then $|\sigma| > (1 - \varepsilon) n$.
To complete the proof of the theorem, it remains to check
that the system $({\sqrt \frac{m}{n}} y_j)_{j \in \sigma}$ is well equivalent
to an orthonormal basis.
{\bf IV. Equivalence to the orthogonal basis within blocks $\sigma_k$.} \ \
Recall that for every $k < k_0$ the size of $\tau_k$ is comparable with $m$,
namely $|\tau_k| \ge (\varepsilon / 2) m$.
Then we conclude from the construction the existence of functions
$c_1(\varepsilon)$ and $c_2(\varepsilon)$ such that for every $k = 1, \ldots, k_0$
\begin{equation}
\label{hilbertian}
\mbox{the system
$({\sqrt \frac{m}{n}} y_j )_{j \in \sigma_k}$
is $c_1(\varepsilon)$-Hilbertian,}
\end{equation}
\begin{equation}
\label{besselian}
\mbox{the system
$({\sqrt \frac{m}{n}} (I - P_{k-1}) y_j )_{j \in \sigma_k}$
is $c_2(\varepsilon)$-Besselian.}
\end{equation}
{\bf V. The system $({\sqrt \frac{m}{n}} y_j)_{j \in \sigma}$ is $h$-Hilbertian
for some function $h = h(\varepsilon)$.} \ \
Indeed, fix scalars $(a_j)_{j \in \sigma}$ such that $\sum_{j \in \sigma}
|a_i|^2 = 1$.
Then
\begin{eqnarray*}
\Big\| \sum_{j \in \sigma} a_j \Big( {\sqrt \frac{m}{n}} y_j \Big) \|
&\le& \sum_{k=1}^{k_0} \Big\| \sum_{j \in \sigma_k} a_j \Big( {\sqrt \frac{m}{n}} y_j
\Big) \| \\
&\le& \sqrt{k_0} \left(
\sum_{k=1}^{k_0} \Big\| \sum_{j \in \sigma_k} a_j \Big( {\sqrt \frac{m}{n}} y_j
\Big) \Big\|^2
\right)^{1/2} \\
&\le& \sqrt{k_0} \; c_1(\varepsilon) \left(
\sum_{k=1}^{k_0} \sum_{j \in \sigma_k} |a_j|^2
\right)^{1/2} \ \ \ \ \mbox{by (\ref{hilbertian})} \\
& = & \sqrt{K(\varepsilon)} \; c_1(\varepsilon).
\end{eqnarray*}
{\bf VI. The system $({\sqrt \frac{m}{n}} y_j)_{j \in \sigma}$ is $b$-Besselian
for some function $b = b(\varepsilon)$.} \ \
We follow P.Casazza \cite{C2}. Choose $r = r(\varepsilon) > 2$ large enough
(to be specified later). Let $a = a(\varepsilon) > 0$ be such that
$r^{k_0+1} a < 1$.
Fix scalars $(a_j)_{j \in \sigma}$ such that $\sum_{j \in \sigma} |a_j|^2 = 1$.
Suppose
\begin{equation}
\label{iolargest}
\mbox{$1 \le k' \le k_0$ is the largest so that} \ \
\Big( \sum_{j \in \sigma_{k'}} |a_j|^2 \Big)^{1/2}
\ge r^{k_0 - k'} a.
\end{equation}
Such $k'$ must exist, otherwise
\begin{eqnarray*}
\Big( \sum_{j \in \sigma} |a_j|^2 \Big)^{1/2}
&\le& \sum_{k=1}^{k_0} \Big( \sum_{j \in \sigma_k} |a_j|^2 \Big)^{1/2}
\\
&\le& \sum_{k=1}^{k_0} r^k a
\le r^{k_0 + 1} a < 1,
\end{eqnarray*}
contradicting the choise of $a$.
We have
\begin{eqnarray*}
\Big\| \sum_{j \in \sigma} a_j \Big( {\sqrt \frac{m}{n}} y_j \Big) \Big\|
&\ge& \Big\| \sum_{k=1}^{k'} \sum_{j \in \sigma_k} a_j \Big( {\sqrt \frac{m}{n}} y_j \Big) \Big\|
- \sum_{k=k' + 1}^{k_0} \Big\|\sum_{j \in \sigma_k} a_j \Big( {\sqrt \frac{m}{n}} y_j \Big) \Big\| \\
&\ge& \Big\| (I - P_{k'-1}) \sum_{k=1}^{k'} \sum_{j \in \sigma_k} a_j \Big( {\sqrt \frac{m}{n}} y_j \Big) \Big\| - \\
& & \ \ \ \ \ \ \ - c_1(\varepsilon) \sum_{k=k' + 1}^{k_0} \Big( \sum_{j \in \sigma_k} |a_j|^2 \Big)^{1/2}
\ \ \ \ \mbox{by (\ref{hilbertian})} \\
&\ge& \Big\| \sum_{j \in \sigma_{k'}} a_j \Big( {\sqrt \frac{m}{n}} (I - P_{k' - 1}) y_j \Big) \Big\|
- c_1(\varepsilon) \sum_{k=k' + 1}^{k_0} r^{k_0 - k} a
\ \ \ \ \mbox{by (\ref{iolargest})} \\
&\ge& c_2(\varepsilon)^{-1} \Big( \sum_{j \in \sigma_{k'}} |a_j|^2 \Big)^{1/2}
- c_1(\varepsilon) \frac{r^{k_0 - k'}}{r-1} a
\ \ \ \ \mbox{by (\ref{besselian})} \\
&\ge& \Big( c_2(\varepsilon)^{-1} - c_1(\varepsilon) (r - 1)^{-1} \Big) r^{k_0 - k'} a
\ \ \ \ \mbox{by (\ref{iolargest})} \\
&\ge& \Big( c_2(\varepsilon)^{-1} - c_1(\varepsilon) (r - 1)^{-1} \Big) a.
\end{eqnarray*}
If $r$ was chosen so that
$c_2(\varepsilon)^{-1} - c_1(\varepsilon) (r - 1)^{-1} > c_2(\varepsilon)^{-1} / 2$, we are done.
The proof is complete.
{\mbox{}\nolinebreak\hfill\rule{2mm}{2mm}\par\medbreak}
\noindent {\bf Remark 1. \ }
$C$ tends to $1$ as $\varepsilon \rightarrow 1$.
This is a consequence of a restriction theorem \cite{K-Tz} which we use
in the following special case (see aslo \cite{B-Tz} Theorem 1.6).
\begin{theorem} (B.Kashin, L.Tzafriri). \label{zeroes}
Let $T$ be a linear operator in $l_2^n$ with $0$'s on the diagonal and $\|T\| = 1$.
Let $1/n \le \delta < 1$.
Then there exists a set $\sigma \subset \{1, \ldots, n\}$ with $|\sigma| \ge \delta n / 4$
for which
$$
\| R_\sigma T R_\sigma \| \le c_5 \delta^{1/2}.
$$
\end{theorem}
First, Theorem \ref{quantitative} gives us a set of indices $\sigma_1$ with $|\sigma_1| \ge n/2$
such that the system $(x_j / \|x_j\|)_{j \in \sigma_1}$ is $c_6 c$-equivalent to the
canonical vector basis of $l_2^{\sigma_1}$.
Let $\delta = 1 - \varepsilon$ and $z_j = x_j / \|x_j\|$ for $j \in \sigma_1$.
Consider the linear operator $T$ in $l_2^{\sigma_1}$ which sends $e_j$ to $z_j$ for $j \in \sigma_1$.
Then the operator $T^* T - I$ has $0$'s on the diagonal and is of norm at most $2 c_6^2 c^2$.
Applying Theorem \ref{zeroes} we get a set $\sigma \subset \sigma_1$ with $\sigma \ge \delta |\sigma_1| / 4$
such that the following holds.
For any sequence of scalars $(a_j)$
$$
\Big\| \Big\langle
(T^* T - I) \sum_{j \in \sigma} a_j e_j , \sum_{j \in \sigma} a_j e_j
\Big\rangle \Big\|
\le (2 c_6^2 c^2) c_5 \delta^{1/2} = c_7 c^2 \delta^{1/2}.
$$
Thus
$$
\Big| \Big\langle
\sum_{j \in \sigma} a_j z_j , \sum_{j \in \sigma} a_j z_j
\Big\rangle
- \sum_{j \in \sigma} |a_j|^2 \Big|
\le c_7 c^2 \delta^{1/2}.
$$
Therefore the sequence $(z_j)_{j \in \sigma}$ is $g(\delta)$-equivalent to $(e_j)_{j \in \sigma}$
for a function $g(\delta)$ which tends to $1$ as $\delta \rightarrow 0$.
This proves Remark 1.
\vspace{0.5cm}
\noindent {\bf Remark 2. \ }
$h(\varepsilon)$ tends to infinity as $\varepsilon \rightarrow \ 0$.
This is verified for the following tight frame $(x_j)_{1 \le j \le n+1}$, $n \ge 2$,
considered by P.Casazza and O.Christensen in \cite{C-C2}:
\begin{eqnarray*}
x_j &=& e_j - n^{-1} \sum_{j=1}^n e_j \ \ \ \ \mbox{for $j = 1, \ldots, n;$} \\
x_{n+1} &=& n^{-1/2} \sum_{j=1}^n e_j.
\end{eqnarray*}
Indeed, let $\sigma \subset \{1, \ldots, n\}$ be such that $|\sigma| > (1 - \varepsilon) n$
and the system $(x_j)_{j \in \sigma}$ is $M$-equivalent to an orthogonal basis.
By change of coordinates, the system $(x_j)_{1 \le j \le |\sigma|-1}$ must be $M$-equivalent
to an orthogonal basis as well. However,
$$
\Big\| \sum_{j=1}^{|\sigma| - 1} x_j \Big\|^2 \le 2 (\varepsilon n + 1)
$$
while $\|x_j\| \ge 1/2$ for all $j$.
Therefore $M$ can not be bounded independently of $n$ as $\varepsilon \rightarrow 0$.
This proves Remark 2.
\section{Almost orthogonal subsequences of frames} \label{basicsubs}
In this section we prove an infinite dimensional version of Theorem \ref{quantitative}.
\begin{theorem} \label{main}
Given an $\varepsilon > 0$,
every infinite dimensional frame has a subsequence
$(1 - \varepsilon)$-equivalent to an orthogonal basis of $l_2$.
\end{theorem}
Given two sets $A$ and $B$ in $H$, we put by definition
$$
\theta (A, B) = \sup_{a \in A} \ {\rm dist} (a, B)
= \sup_{a \in A} \ \inf \{ \|a - b\| : b \in B \}.
$$
\begin{lemma} \label{distone}
Let $(x_j)$ be a frame in an infinite-dimensional $H$.
Let $A = \{ x_j / \|x_j\| \}$.
Then for any finite-dimensional subspace $E \subset H$
$$
\theta (A, E) = 1.
$$
\end{lemma}
\noindent {\bf Proof.}\ \
Let $z_j = x_j / \|x_j\|$ for all $j$.
Assume for the contrary that there is a $\delta < 1$ such that
$$
{\rm dist} (z_j, E) < \delta \ \ \ \ \mbox{for all $j$}.
$$
Let $P$ be the orthogonal projection in $H$ onto $E$. Then
$$
\|P z_j \| > \sqrt{1 - \delta^2} \ \ \ \ \mbox{for all $j$},
$$
so that
\begin{equation} \label{pxj}
\|P x_j \| \ge \sqrt{1 - \delta^2} \cdot \|x_j\| \ \ \ \ \mbox{for all $j$}.
\end{equation}
Since $P$ is finite-dimensional,
Lemma \ref{square} yields that the sequence $\|P x_j \|$ is square
summable.
Then, by (\ref{pxj}), $\| x_j \|$ must be square summable, too.
Thus $(x_j)$ is finite-dimensional. This contradiction completes the
proof.
{\mbox{}\nolinebreak\hfill\rule{2mm}{2mm}\par\medbreak}
\begin{lemma} \label{stability}
Let $\varepsilon_j$ be a sequence of quickly decreasing positive numbers
($2^{-j-1}$ will do).
Let $(z_j)$ be a normalized sequence in $H$ such that
$$
\langle z_i, z_j \rangle < \varepsilon_j \ \ \ \ \mbox{whenever $i < j$}.
$$
Then $(z_j)$ is equivalent to an orthonormal basis.
\end{lemma}
The proof is simple.
\vspace{0.5cm}
{\bf Proof of Theorem \ref{main}}. \ \
First note that, given an $\varepsilon > 0$, every subsequence equivalent to the
canonical vector basis of $l_2$ is weakly null, therefore has a subsequence
which is $(1 - \varepsilon)$-equivalent to the canonical vector basis of $l_2$.
Hence by Corollary \ref{frameistight} we may assume that our given
frame $(x_j)$ is tight.
Let $z_j = x_j / \|x_j\|$ for all $j$.
We will find a subsequence $(z_{j_k})$ equivalent to an orthogonal
basis by induction.
Put $j_1 = 1$. Let $j_1, \dots, j_{k-1}$ be defined
and let $E = {\rm span}( z_{j_1}, \dots, z_{j_{k-1}} )$.
Choose $j_k$ from Lemma \ref{distone} so that
$$
{\rm dist} ( z_{j_k}, E ) > 1 - 2^{-2k}.
$$
Then it is easy to check that the constructed subsequence $(z_{j_k})$
satisfies the assumpiton of Lemma \ref{stability}. This finishes the
proof.
{\mbox{}\nolinebreak\hfill\rule{2mm}{2mm}\par\medbreak}
\section{A frame not containing bases with brackets}
\label{secwithoutbrackets}
\begin{definition}
\label{basiswithbrackets}
A sequence $(x_n)_{n=1}^\infty$ in a Banach space $X$
is called a {\em basis with brackets} if there are numbers
$1 < n_1 < n_2 < \ldots$ such that every vector $x \in X$
admits a unique representation of the form
$$
x = \lim_j \sum_{n=1}^{n_j} a_n x_n, \ \ \ \ a_n \in {\bf R}.
$$
\end{definition}
Clearly, every basis is a basis with brackets.
The difference between bases and bases with brackets is that
the latter require the convergence only of {\em some} partial sums
in the representation.
The following lemma is known \cite{L-T}.
\begin{lemma}
\label{triangle}
Let $(x_n)_{n=1}^\infty$ be a basis with brackets,
and numbers $1 < n_1 < n_2 < \ldots$ be as in
Definition \ref{basiswithbrackets}.
Consider the projection $P_j$ onto $[x_n : n \le n_j]$
parallel to $[x_n : n > n_j]$.
Then $\sup_j \|P_j\| < \infty$.
\end{lemma}
Clearly, the converse also holds: if $\sup_j \|P_j\| < \infty$,
for some sequence $1 < n_1 < n_2 < \ldots$,
then $(x_n)$ is a basis with brackets.
\vspace{0.5cm}
In this section we prove
\begin{theorem} \label{withoutbrackets}
There exists a frame not containing bases with brackets.
\end{theorem}
\noindent Moreover, this frame is tight and have norms bounded from below.
\begin{lemma} \label{eitheror}
There is an orthonormal basis $(z_j)$ in $l_2^n$ such that,
given any set $J \subset \{1, \ldots, n\}$, $|J| \ge n-2$, one has
\begin{eqnarray*}
{\rm dist}(e_1, [z_j : j \in J, j \ge j_0]) \le 4 / \sqrt{n}
\ \ \ \ & & \mbox{for $1 \le j_0 < n/2$}, \\
{\rm dist}(e_n, [z_j : j \in J, j < j_0]) \le 4 / \sqrt{n}
\ \ \ \ & & \mbox{for $n/2 \le j_0 \le n$}.
\end{eqnarray*}
\end{lemma}
\noindent {\bf Proof.}\ \
By rotation, it is enough to find normalized vectors $v_1, v_2$ in
$l_2^n$
such that $\langle v_1, v_2 \rangle = 0$
and, given a set $J$ as in the hypothesis,
\begin{eqnarray*}
{\rm dist}(v_1, [e_j : j \in J, j \ge j_0]) \le 4 / \sqrt{n}
\ \ \ \ & & \mbox{for $1 \le j_0 < n/2$}, \\
{\rm dist}(v_2, [e_j : j \in J, j < j_0]) \le 4 / \sqrt{n}
\ \ \ \ & & \mbox{for $n/2 \le j_0 \le n$}.
\end{eqnarray*}
Clearly, one may take
$$
v_1 = {\lceil n/2 \rceil}^{-1/2} \cdot
( \underbrace{1, \ldots, 1}_{\lceil n/2 \rceil}, 0, \ldots, 0)
\ \ \mbox{and} \ \
v_2 = {\lceil n/2 \rceil}^{-1/2} \cdot
( 0, \ldots, 0, \underbrace{1, \ldots, 1}_{\lceil n/2 \rceil}).
$$
This completes the proof.
{\mbox{}\nolinebreak\hfill\rule{2mm}{2mm}\par\medbreak}
We will construct our frame $(x_j)$ by blocks $(x_j : j \in J(n))$,
where
$$
J(1) = \{1\}, \ \
J(2) = \{2, 3\}, \ \
J(3) = \{4, 5, 6\}, \ \
J(4) = \{7, 8, 9, 10\}, \ldots
$$
The supports of $x_j$'s from block $J(n)$ will lie in an interval
$I(n)$, where
$$
I(1) = \{1\}, \ \
I(2) = \{1, 2\}, \ \
I(3) = \{2, 3, 4\}, \ \
I(4) = \{4, 5, 6, 7\}, \ldots
$$
Let $i(n)$ be the first element in $I(n)$.
$$
\begin{array}{ccccccccccccc}
* & * & * & & & & &\mbox{\large $0$}& & & \\
& * & * & * & * & * & & & & & \\
& & & * & * & * & & & & & \\
& & & * & * & * & * & * & * & * & \\
& & & & & & * & * & * & * & \\
& &\mbox{\large $0$}& & & & * & * & * & * & \\
& & & & & & * & * & * & * & \cdots
\end{array}
$$
The columns of this infinite matrix form the frame elements $x_j$,
the asterisks marking their support.
Consider the shift operator $T_n : l_2^n \rightarrow l_2$
which sends $(e_i)_{i=1}^n$ to $(e_i : i \in I(n))$.
Choose an orthonormal basis $(z_j : j \in J(n))$ in $l_2^n$
satisfying the conclusion of Lemma \ref{eitheror},
and define
$$
x_j = T_n z_j \ \ \ \ \mbox{for $j \in J(n)$}.
$$
\begin{lemma}
$(x_j)$ is a frame.
\end{lemma}
\noindent {\bf Proof.}\ \
Indeed, look at the rows in the picture, that is the vectors
$y_i = (x_1(i), x_2(i), \ldots)$.
Since the vectors $x_j$, $j \in J(n)$ are orthonormal for a fixed $n$,
the vectors $y_i$ are orthogonal. Moreover, their norm is either equal
to $2$ (if $i = i(n)$ for some $n$) or to $1$ (otherwise).
Now we pass again from the rows $y_i$ to the columns $x_j$.
Lemma \ref{columnrow} yields that $(x_j)$ is a frame.
{\mbox{}\nolinebreak\hfill\rule{2mm}{2mm}\par\medbreak}
Let $J$ be a set of positive integers such that
the sequence $(x_j)_{j \in J}$ is complete in $l_2$.
We shall prove that it is not a basis with brackets.
\begin{lemma}
$|J(n) \cap J| \ge n - 2$ for every $n$.
\end{lemma}
\noindent {\bf Proof.}\ \
Let $P$ be the orthogonal projection onto those $n-2$ coordinates in
$I(n)$
which don't belong to the other blocks $I(n_1)$, i.e. onto
$[e_i : i \in I(n) \setminus \{i(n), i(n+1)\} ]$.
Thus $P$ sends to zero all $x_j$ with $j \not\in J(n)$.
Hence ${\rm Im}(P) = P( [x_j : j \in J(n) \cap J] )$.
Since ${\rm Im}(P)$ is an $(n-2)$-dimensional space, the lemma follows.
{\mbox{}\nolinebreak\hfill\rule{2mm}{2mm}\par\medbreak}
In the sequel we consider large blocks $J(n)$, i.e. with $n \rightarrow \infty$.
Given a vector $v$ and a subspace $L$ in $l_2$ (both possibly
dependent on $n$),
we say that {\em $v$ is close to $L$} if ${\rm dist}(x, L) \le c / \sqrt{n}$.
Here $c$ is some absoulte constant, whose value may be different in
different occurences.
\begin{lemma}
\label{onetwothree}
1) $e_{i(n)}$ is close to $[x_j : j \in J(n-1) \cap J]$.
2) $e_{i(n+1)}$ is close to $[x_j : j \in J(n+1) \cap J]$.
3) Given a $j_0 \in J(n)$,
either $e_{i(n)}$ is close to $[x_j : j \in J(n) \cap J, j \ge
j_0]$,
or $e_{i(n+1)}$ is close to $[x_j : j \in J(n) \cap J, j <
j_0]$.
\end{lemma}
\noindent {\bf Proof.}\ \
Note that $T_n$ sends $e_1$ to $e_{i(n)}$ and $e_n$ to $e_{i(n+1)}$.
Then all three statements of the lemma follow from Lemma \ref{eitheror}.
{\mbox{}\nolinebreak\hfill\rule{2mm}{2mm}\par\medbreak}
The next and the last lemma, in tandem with Lemma \ref{triangle},
completes the proof of Theorem \ref{withoutbrackets}.
\begin{lemma}
For every $j_0 \in J(n)$ there is a normalized vector $x$ in $l_2$
which is close to both subspaces
$E = [x_j : j \in J, j \ge j_0]$ and
$F = [x_j : j \in J, j < j_0]$.
\end{lemma}
\noindent {\bf Proof.}\ \
We make use of Lemma \ref{onetwothree}.
By 3), we take either
$x = e_{i(n)}$ to have $x$ close to $E$, or
$x = e_{i(n+1)}$ to have $x$ close to $F$.
In the first case $x$ is also close to $F$ by 2),
and in the second case $x$ is close to $E$ by 1).
The proof is complete.
{\mbox{}\nolinebreak\hfill\rule{2mm}{2mm}\par\medbreak}
\vspace{0.5cm}
A part of this work was accomplished when the author was visiting
Friedrich-Schiller-Universit\"{a}t Jena.
The author is grateful to M.Rudelson and P.Wojtaszczyk for helpful discussions,
and to V.Kadets for his constant encouragement.
{\small
| {
"timestamp": "1999-02-15T19:16:24",
"yymm": "9902",
"arxiv_id": "math/9902097",
"language": "en",
"url": "https://arxiv.org/abs/math/9902097",
"abstract": "Every frame in Hilbert space contains a subsequence equivalent to an orthogonal basis. If a frame is n-dimensional then this subsequence has length (1 - \\epsilon) n. On the other hand, there is a frame which does not contain bases with brackets.",
"subjects": "Functional Analysis (math.FA)",
"title": "Subsequences of frames",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9854964198826467,
"lm_q2_score": 0.7185943925708561,
"lm_q1q2_score": 0.7081722012263238
} |
https://arxiv.org/abs/1710.04627 | On the connectedness of the set of Riemann surfaces with real moduli | The moduli space ${\mathcal{M}}_{g}$, of genus $g\geq2$ closed Riemann surfaces, is a complex orbifold of dimension $3(g-1)$ which carries a natural real structure i.e. it admits an anti-holomorphic involution $\sigma$. The involution $\sigma$ maps each point corresponding to a Riemann surface $S$ to its complex conjugate $\overline{S}$. The fixed point set of $\sigma$ consists of the isomorphism classes of closed Riemann surfaces admitting an anticonformal automorphism. Inside $\mathrm{Fix}(\sigma)$ is the locus ${\mathcal{M}}_{g}(\mathbb{R})$, the set of real Riemann surfaces, which is known to be connected by results due to P. Buser, M. Seppälä and R. Silhol. The complement $\mathrm{Fix}(\sigma)-{\mathcal{M}}_{g}(\mathbb{R})$ consists of the so called pseudo-real Riemann surfaces, which is known to be non-connected. In this short note we provide a simple argument to observe that $\mathrm{Fix}(\sigma)$ is connected. | \section{Introduction}
The moduli space ${\mathcal{M}}_{g}$, of genus $g\geq2$ closed Riemann
surfaces, is a complex orbifold of dimension $3(g-1)$. The study of this
moduli space was already started by F. Klein. This space carries a natural
real structure given by an involution $\sigma$ which sends each Riemann
surface to its complex conjugate. The fixed point set of $\sigma$ consists of
the isomorphism classes of closed Riemann surfaces admitting an anticonformal
automorphism and those surfaces are said to have real moduli. The quotient
space ${\mathcal{M}}_{g}/\left\langle \sigma\right\rangle $ is the moduli
space of Riemann surfaces of genus $g$ considered as Klein surfaces, i. e. two
surfaces are equivalent if they are holomorphic or anti-holomorphically
equivalent. Inside $\mathrm{Fix}(\sigma)$ is the locus ${\mathcal{M}}%
_{g}(\mathbb{R})$, consisting of those admitting an anticonformal involution
(Riemann surfaces corresponding to real algebraic curves), which is known to
be connected by results due to P. Buser, M. Sepp\"{a}l\"{a} and R. Silhol
\cite{BSS,Seppala} (see also a proof in \cite{CI}). The locus ${\mathcal{M}%
}_{g}(\mathbb{R})$ has been studied by Klein in \cite{K} and the fact that in
general $\mathrm{Fix}(\sigma)\neq{\mathcal{M}}_{g}(\mathbb{R})$ was noted by
C. Earle in \cite{E}. The complement ${\mathcal{P}}_{g}=\mathrm{Fix}%
(\sigma)-{\mathcal{M}}_{g}(\mathbb{R})$ consists of the so called pseudo-real
Riemann surfaces. In \cite{BCC} it was observed that ${\mathcal{P}}_{g}$ is
non-empty for every $g\geq2$ (then $\mathrm{Fix}(\sigma)\neq{\mathcal{M}}%
_{g}(\mathbb{R})$ for all $g\geq2$). In \cite{BC} it was observed that
${\mathcal{P}}_{2}$ and ${\mathcal{P}}_{3}$ are connected and ${\mathcal{P}%
}_{4}$ is non-connected (it has three connected components). There are
infinite many integers $n_{i}$ such that ${\mathcal{P}}_{n_{i}}$ is not
connected (this fact follows from Theorems 3.4, 5.4, 5.5 and Section 6 of
\cite{BCC} to construct families of pseudo-real surfaces and from the
Corollary of Theorem 2 of \cite{Si} to prove that these families are in
different connected components).
In this short note we provide a simple argument to observe that $\mathrm{Fix}%
(\sigma)$ is connected.
\begin{theo}
\label{conexo} The set $\mathrm{Fix}(\sigma)$ is connected.
\end{theo}
This connectedness fact may not be a surprise, but it seems it has not been
noted in the existent literature.
\textbf{Acknowledgements}. We wish to thank the referee for corrections and suggestions.
\section{The set of Riemann surfaces with real moduli is connected}
In this section, we proceed to prove Theorem \ref{conexo}. Let $S$ be a closed
Riemann surface of genus $g\geq2$, admitting an anticonformal automorphism
$\tau$ of order $2n$, where $n\geq2$ is even. The quotient orbifold
${\mathcal{O}}=S/\langle\tau\rangle$ is homeomorphic to the connected sum of
some $\gamma$ real projective planes and has exactly $r$ cone points, say of
orders $n_{1},\ldots,n_{r}\in\{2,\ldots,n\}$, where each $n_{j}$ is a divisor
of $n$. This means that there is an NEC group $\Delta$, acting on the unit
disc ${\mathbb{D}}$, with signature $(\gamma;-;[n_{1},\ldots,n_{r}])$ and
there is a surjective homomorphism $\rho:\Delta\rightarrow C_{2n}=\langle
\tau\rangle$, whose kernel $\Gamma$ is a Fuchsian group uniformizing $S$ and
$\langle\tau\rangle$ is induced by $\Delta$, that is, ${\mathbb{D}}%
/\Gamma=S\rightarrow{\mathbb{D}}/\Delta=S/\langle\tau\rangle$.
The locus ${\mathcal{O}}(S,\tau)$ in moduli space ${\mathcal{M}}_{g}$
consisting of those (classes of) closed Riemann surfaces $\widehat{S}$
admitting an anticonformal automorphism $\widehat{\tau}$ of order $2n$ so
there is an orientation preserving homeomorphism $\phi:S\rightarrow
\widehat{S}$ conjugating $\tau$ to $\widehat{\tau}$ is connected \cite{BC}.
Now, since the locus ${\mathcal{M}}_{g}(\mathbb{R})$ is connected, in order to
check the connectivity of $\mathrm{Fix}(\sigma)$, we only need to find a point
$[\widehat{S}]\in{\mathcal{O}}(S,\tau)$ so that $\widehat{S}$ admits also an
anticonformal involution. In the NEC group setting, this is equivalent to
finding an NEC group $K$ containing reflections (i.e. the group $K$
uniformizes a bordered Klein surface) and a subgroup $\widehat{\Delta}$ of $K$
so that there is an isomorphism $\iota:\Delta\rightarrow\widehat{\Delta}$ with
$\iota(\Gamma)$ being a normal subgroup of $K$. Note that $\iota(\Gamma)$ is a
Fuchsian group with $\widehat{S}={\mathbb{D}}/\iota(\Gamma)$ a closed Riemann
surface which has an automorphism $\widehat{\tau}$ topologically equivalent to
$\tau$ ($\widehat{\tau}$ is an automorphism group of the cyclic covering
${\mathbb{D}}/\iota(\Gamma)=\widehat{S}\rightarrow{\mathbb{D}}/\widehat{\Delta
}$) and $\widehat{S}$ has anticonformal involutions too, produced by the
lifting of the reflections of $K$ to $\widehat{S}$ (note that $\widehat{S}$
has empty boundary but ${\mathbb{D}}/K$ is bordered). In this way we have
${\mathcal{O}}(S,\tau)\cap{\mathcal{M}}_{g}(\mathbb{R})\neq\varnothing$ for
every pseudo-real surface $S $; this implies $\mathrm{Fix}(\sigma)$ is connected.
\subsection{The construction of $K$}
Let $K$ be an NEC group uniformizing the closed disc, with $\gamma$ interior
cone points of order $2$ and $r$ cone points in its border of orders
$n_{1},\ldots,n_{r}$, that is, an NEC\ group of signature
$(0;+;[2,\overset{\gamma}{...},2],\{(n_{1},...,n_{r})\})$. A canonical
presentation for $K$ is as follows
\[
K=\langle x_{1},\ldots,x_{\gamma},e,\tau_{1},\ldots,\tau_{r+1}:x_{1}%
^{2}=\cdots=x_{\gamma}^{2}=\tau_{1}^{2}=\cdots=\tau_{r+1}^{2}=1,
\]%
\[
e^{-1}\tau_{r+1}e=\tau_{1},x_{\gamma}\cdots x_{2}x_{1}e=1,(\tau_{1}\tau
_{2})^{n_{1}}=\cdots=(\tau_{r}\tau_{r+1})^{n_{r}}=1\rangle,
\]
where the elements $x_{j}$ are elliptic transformations of order two, the
elements $\tau_{j}$ are reflections and $e$ is an hyperbolic or elliptic
element (see \cite{BEGG} and Figure \ref{fig:FD1} for a fundamental domain of
$K$).
\begin{figure}[ptb]
\centering
\includegraphics[width=3.0cm]{FD1.eps}\caption{A fundamental domain for $K$}%
\label{fig:FD1}%
\end{figure}
\subsection{A subgroup $\protect\widehat{\Delta}$ of $K$}
Let us consider the surjective homomorphism
\[
\theta:K\rightarrow C_{2}=\langle a:a^{2}=1\rangle
\]%
\[
x_{1},\ldots,x_{\gamma},\tau_{1},\ldots,\tau_{r+1}\mapsto a,\;e\mapsto1.
\]
The kernel $\widehat{\Delta}$ of $\theta$ has no reflections and contains
orientation reversing elements (for instance $c_{1}x_{1}$); so its signature
must be of the form $(h;-;[m_{1},...,m_{s}])$, that is, ${\mathbb{D}%
}/\widehat{\Delta}$ is the connected sum of $h$ real projective planes and
contains exactly $s$ cone points, these having orders $m_{1},\ldots,m_{s}$.
Using the Riemann-Hurwitz formula and the usual methods to compute the
signature of an NEC\ subgroup (see \cite{BEGG}), we have that $h=\gamma$,
$s=r$ and $n_{j}=m_{j}$. So, $\Delta$ is isomorphic to $\widehat{\Delta}$; let
$\iota:\Delta\rightarrow\widehat{\Delta}$ be such an isomorphism. A
fundamental domain for $\widehat{\Delta}$ is shown in Figure \ref{fig:FD2},
this given as the union of the previous fundamental domain for $K$ with its
image under the reflection $\tau_{1}$. By the Poincar\'{e} polygon theorem (or
using the Schreier-Reidemeister method) a presentation of $\widehat{\Delta}$,
in terms of the generators of $K$, may be obtained. We have as generators
\[
\delta_{1}=\tau_{1}x_{1},\ldots,\delta_{\gamma}=\tau_{1}x_{\gamma},c_{1}%
=\tau_{1}\tau_{2},\ldots,c_{r}=\tau_{1}\tau_{r+1},e_{1}=e,e_{2}=\tau_{1}%
e\tau_{1}%
\]
satisfying the following relations
\[
c_{1}^{n_{1}}=1,
\]%
\[
(c_{1}^{-1}c_{2})^{n_{2}}=(c_{2}^{-1}c_{3})^{n_{3}}\cdots=(c_{r-1}^{-1}%
c_{r})^{n_{r}}=1,
\]%
\[
e_{1}e_{2}^{-1}c_{r}=1,
\]%
\[
\left.
\begin{array}
[c]{lll}%
\delta_{1}^{-1}\delta_{2}\delta_{3}^{-1}\cdots\delta_{\gamma-1}^{-1}%
\delta_{\gamma} & = & e_{1}\\
\delta_{1}\delta_{2}^{-1}\delta_{3}\cdots\delta_{\gamma-1}\delta_{\gamma}^{-1}
& = & e_{2}%
\end{array}
\right\} \;(\mbox{if $\gamma$ is even})
\]%
\[
\left.
\begin{array}
[c]{lll}%
\delta_{1}\delta_{2}^{-1}\delta_{3}\cdots\delta_{\gamma-1}^{-1}\delta_{\gamma}
& = & e_{1}\\
\delta_{1}^{-1}\delta_{2}\delta_{3}^{-1}\cdots\delta_{\gamma-1}\delta_{\gamma
}^{-1} & = & e_{2}%
\end{array}
\right\} \;(\mbox{if $\gamma$ is odd})
\]
\begin{figure}[ptb]
\centering
\includegraphics[width=6.0cm]{FD2.eps}\caption{A fundamental domain for
$\protect\widehat{\Delta}$}%
\label{fig:FD2}%
\end{figure}
\subsection{The final step}
Let us consider the surjective homomorphism $\eta=\rho\circ\iota
^{-1}:\widehat{\Delta} \to C_{2n}$, whose kernel is $\widehat{\Gamma}%
=\iota(\Gamma) $; a torsion free Fuchsian group that uniformizes a closed
Riemann surface ${\widehat{S}}$. In order to finish our proof, we only need to
check that $\widehat{\Gamma}$ is a normal subgroup of $K$. This is what the
following general lemma asserts.
\begin{lemm}
\label{lema1} Let $A$ be an abelian group and let $\zeta:\widehat{\Delta} \to
A$ be a homomorphism. Then $e_{1}e_{2} \in\ker(\zeta)$ and $\ker(\zeta) \lhd K
$.
\end{lemm}
\begin{proof}
We assume $\gamma$ even; the odd case is similar. The relations
\[
\delta_{1}^{-1}\delta_{2}\delta_{3}^{-1}\cdots\delta_{\gamma-1}^{-1}%
\delta_{\gamma}=e_{1},\;\mbox{ and }\;\delta_{1}\delta_{2}^{-1}\delta
_{3}\cdots\delta_{\gamma-1}\delta_{\gamma}^{-1}=e_{2},
\]
assert that $\zeta(e_{1})=\zeta(e_{2})^{-1}$, so, $\zeta(e_{1}e_{2})=1$. Next,
since
\[
\tau_{1}\delta_{j}\tau_{1}=\delta_{j}^{-1},j=1,\ldots,\gamma,
\]%
\[
\tau_{1}c_{k}\tau_{1}=c_{k}^{-1},k=1,\ldots,r,
\]%
\[
\zeta(e_{1})=\zeta(e_{2})^{-1},
\]%
\[
\widehat{\Delta}^{\prime}\lhd\ker(\zeta)\;\mbox{(as $A$ is an abelian
group)},
\]
we may see that $\tau_{1}$ induces the inverse automorphism of $A$, i.e.,
\[
a\in A\mapsto a^{-1}\in A.
\]
In particular, $\tau_{1}\ker(\zeta)\tau_{1}=\ker(\zeta)$. Since,
$K=\langle\widehat{\Delta},\tau_{1}\rangle$, we obtain that $\ker(\zeta)\lhd
K$.
\end{proof}
\begin{coro}
If $S$ is a pseudo-real Riemann surface admitting an anticonformal
automorphism $\tau$ of order $2n$, then there is a real Riemann surface
$\widehat{S}$ admitting an anticonformal automorphism $\widehat{\tau}$ such
that $(\widehat{S},\widehat{\tau})$ is topologically conjugate to $(S,\tau) $.
\end{coro}
\begin{proof}
The surface $\widehat{S}$ is uniformized by $\ker(\eta)$ and we use
$\eta=\zeta$, $A=C_{2n}$, in the above lemma.
\end{proof}
Note that if $\widehat{S}$ is the surface given in the above Corollary
$D_{2n}\leq\mathrm{Aut}(\widehat{S})$.
For every pseudo-real Riemann surface $S$, Corollary 1 implies ${\mathcal{O}%
}(S,\tau)\cap{\mathcal{M}}_{g}(\mathbb{R})\neq\varnothing$ and that
$\mathrm{Fix}(\sigma)$ is connected.
| {
"timestamp": "2017-11-13T02:09:16",
"yymm": "1710",
"arxiv_id": "1710.04627",
"language": "en",
"url": "https://arxiv.org/abs/1710.04627",
"abstract": "The moduli space ${\\mathcal{M}}_{g}$, of genus $g\\geq2$ closed Riemann surfaces, is a complex orbifold of dimension $3(g-1)$ which carries a natural real structure i.e. it admits an anti-holomorphic involution $\\sigma$. The involution $\\sigma$ maps each point corresponding to a Riemann surface $S$ to its complex conjugate $\\overline{S}$. The fixed point set of $\\sigma$ consists of the isomorphism classes of closed Riemann surfaces admitting an anticonformal automorphism. Inside $\\mathrm{Fix}(\\sigma)$ is the locus ${\\mathcal{M}}_{g}(\\mathbb{R})$, the set of real Riemann surfaces, which is known to be connected by results due to P. Buser, M. Seppälä and R. Silhol. The complement $\\mathrm{Fix}(\\sigma)-{\\mathcal{M}}_{g}(\\mathbb{R})$ consists of the so called pseudo-real Riemann surfaces, which is known to be non-connected. In this short note we provide a simple argument to observe that $\\mathrm{Fix}(\\sigma)$ is connected.",
"subjects": "Complex Variables (math.CV)",
"title": "On the connectedness of the set of Riemann surfaces with real moduli",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9854964194566753,
"lm_q2_score": 0.7185943925708562,
"lm_q1q2_score": 0.7081722009202234
} |
https://arxiv.org/abs/1407.3481 | When is the multiplicative group of a field indecomposable? | The multiplicative group of a finite field is well known to be cyclic; in this note, we determine the finite fields whose multiplicative groups are direct sum indecomposable. We obtain our classification using a direct argument and also as a corollary to Catalan's Conjecture. Turning to infinite fields, we prove that any infinite field whose characteristic is not equal to 2 must have a decomposable multiplicative group. We conjecture that this is also true for infinite fields of characteristic 2 and we narrow the class of possible counter-examples. Finally, using the classification of finite commutative primary rings with cyclic multiplicative groups, we determine all finite commutative rings with indecomposable multiplicative groups. | \section{Introduction}
Fermat primes and Mersenne primes are two central classes of prime numbers which have enjoyed great esteem in number theory. Spreading from the blackboards of professional mathematicians to the notebooks of amateurs, these primes and the various problems surrounding them have been a source of great inspiration and fascination. As we investigate the question posed in the title of this paper, both classes will make a surprising entry onto the stage. The related problem of determining which abelian groups can occur as the multiplicative group of a field was raised by L\'{a}szl\'{o} Fuchs more than 50 years ago in \cite{Fuchs}. Since then, much progress has been made, but the problem remains unsolved (see, for example, \cite{May, Dicker, Eugene1, Hua}). We refer the reader to \cite{CMN} for a survey of results in this area. Fuchs also asked whether the torsion subgroup of the multiplicative group of a field is necessarily a summand; this question was answered negatively by Cohn in \cite{Cohn}. In this paper, the question we ask is very much in the spirit of the aforementioned work: which fields have indecomposable multiplicative groups? (Recall that a group is said to be indecomposable if it cannot be written as direct sum of two non-trivial subgroups.) In \S \ref{ff}, we classify the finite fields with indecomposable multiplicative groups. Our argument is simple and direct. However, the main result may also be obtained as a corollary to Catalan's Conjecture, described in \S \ref{catalan}, which was proved in 2002 by the Swiss mathematician Preda Mih\u{a}ilescu. In \S \ref{if}, we consider infinite fields, where we show that any infinite field whose characteristic is not equal to 2 must have a decomposable multiplicative group. We are unable to resolve the characteristic 2 case, though we have narrowed the class of possible counter-examples. Finally, in \S \ref{fr}, we use the classification of finite commutative primary rings with cyclic unit groups (found in \cite{PS}) to determine all finite commutative rings with indecomposable unit groups. Throughout, we will use standard elementary facts from number theory, algebra, and group theory which may be found in \cite{Burton}, \cite{DummitFoote}, and \cite{Robinson}, respectively.
The structure of the group of units in a ring has been studied extensively, especially for finite rings and group rings. Examples where the unit group is saddled with a similarly strong simplifying condition include \cite{PS}, referred to above, and our recent work \cite{24, 12, CLY}, where we examined the conditions under which every non-trivial unit in a ring has order $p$ a prime.
\section{Finite Fields\label{ff}}
The goal of this section is to prove Theorem \ref{finitefields}, classifying the finite fields with indecomposable multiplicative groups. Our proof in this section uses elementary methods; in \S \ref{catalan}, we obtain this classification as a corollary to Catalan's Conjecture.
We begin by recording two basic facts about finite fields which can be found in any standard algebra textbook; see \cite{DummitFoote}, for instance. First, recall that every field has prime power order, and for every prime $p$ and positive integer $r$, there is a unique (up to isomorphism) finite field whose order is $p^r$. (The prime $p$ is the characteristic of the field.) Second, recall that the multiplicative group of a finite field $F$, written $F^\times$, is cyclic. (In fact, the multiplicative group of any field is locally cyclic (\cite[1.3.4]{Roman}); i.e., every finite subgroup is a cyclic group.) In our first proposition we make a simple observation which follows from the structure theorem for finite abelian groups. We refer to a positive integer as a prime power if it is equal to $p^r$ for some prime $p$ and integer $r \geq 1$.
\begin{prop}
If $F$ is a finite field of order $p^{r}$, then $F^{\times}$ is indecomposable if and only if $p^{r}-1$ is either 1 or a prime power.
\end{prop}
Note that $F^\times$ has order 1 if and only if $F = \mathbf{F}_2$, the finite field with two elements.
\begin{proof}
As mentioned above, the group $F^{\times}$ is isomorphic to $C_{p^{r}-1}$, the multiplicative cyclic group of order $p^r - 1$. From the structure theorem for finite abelian groups, a finite cyclic group is indecomposable if and only if it is trivial or has prime power order.
\end{proof}
Because there is a unique finite field corresponding to each prime power $p^{r}$, determining which finite fields have an indecomposable multiplicative group is equivalent to solving the following number theoretic problem: find all pairs $(p, r)$, where $p$ is prime and $r$ is a positive integer, such that $p^{r}-1$ is a prime power. We begin by determining the pairs $(p, r)$ for which $p^{r}-1$ is $1$ or a power of $2$. Recall that a Fermat prime is a prime of the form $2^{2^{n}}+1$.
\begin{prop} \label{2powers}
The quantity $p^{r} - 1$ is a power of $2$ if and only if $p$ is a Fermat prime and $r = 1$ or $p = 3$ and $r = 2$.
\end{prop}
\begin{proof} Suppose $p^{r} - 1$ is a power of $2$. Then $p^{r}- 1= 2^{n}$ for some positive integer $n$ and $p$ must be odd. Now,
$$(p -1)(p^{r-1} + \cdots + p + 1) = 2^{n},$$
and this implies that $p-1$ is a power of $2$, so $p$ is a Fermat prime. (If $2^{m}+1$ is prime, then it is well known that $m$ must be a power of 2.) If $r = 1$, then we are done, but if $r \geq 2$, then the second factor is also divisible by 2, so $r$ must be even. Thus, writing $r = 2v$, we have $$(p^v -1)(p^v + 1) = 2^{n}.$$ However, the only pair of positive integers that differ by two and are powers of two are 2 and 4, so it must be that $p^v = 3$. It follows that $(p, r) = (3, 2)$.
\end{proof}
The next proposition gives the pairs $(p, r)$ for which $p^{r}-1$ is a power of an odd prime. Recall that a Mersenne prime is a prime of the form $2^r - 1$.
\begin{prop} \label{ppowers}
Suppose $p^r - 1 = q^n$ for some odd prime $q$ and positive integer $n$. Then, $n = 1$, $p = 2$, and $q = 2^r - 1$ is a Mersenne prime.
\end{prop}
\begin{proof}
First suppose $n = 1$ and $q = p^r -1$ is an odd prime. Parity considerations imply that $p = 2$, so $q = 2^r - 1$ is a Mersenne prime.
We claim there are no other solutions to $p^r - 1 = q^n$. Indeed, assume to the contrary that $p^r - 1 = q^n$ with $n \geq 2$ and $q$ an odd prime. As above, we have $p = 2$ and $q^n + 1 = 2^r$. Since $q > 1$, we have $r \geq 2$, and hence $q^n \equiv -1 \, (4)$. This means $q \equiv -1\, (4)$ and $n$ is odd. We now have a factorization $$2^r = q^n + 1 = (q+1)(q^{n-1} - q^{n-2} + \cdots - q + 1).$$ Because $ n \geq 2$, the second factor is even. However, it is also congruent to the odd integer $n$ modulo 2, a contradiction. This proves that there are no solutions when $n \geq 2$, and the proof is complete.
\end{proof}
The following theorem now follows from the previous three propositions.
\begin{thm} \label{finitefields} Let $F$ be a finite field. The multiplicative group of $F$ is indecomposable if and only if $F$ is one of the following fields:
\begin{enumerate}
\item $\mathbf{F}_2$,
\item $\mathbf{F}_9$,
\item $\mathbf{F}_p$ where $p$ is a Fermat prime, or
\item $\mathbf{F}_{q+1}$ where $q$ is a Mersenne prime.
\end{enumerate}
\end{thm}
\noindent We will see in the next section that the odd-ball case $\mathbf{F}_9$ corresponds to the unique solution in Catalan's Conjecture (see Theorem \ref{m-theorem}).
It is natural to ask whether there are infinitely many finite fields with an indecomposable multiplicative group. In light of the above theorem, this question has an affirmative answer if and only if either the collection of Fermat primes or the collection of Mersenne primes is infinite. It is not known whether either collection is finite or infinite. As of June 2014, only 5 Fermat primes and 48 Mersenne primes are known. It is believed that there are only finitely many Fermat primes and infinitely many Mersenne primes.
\section{Catalan's Conjecture\label{catalan}}
In 1844, the French mathematician Eug\`{e}ne Catalan conjectured that $8$ and $9$ are the only consecutive perfect powers among the positive integers. More precisely, he conjectured the following statement which was proved in 2002 by the Swiss mathematician Preda Mih\u{a}ilescu.
\begin{thm}[Mih\u{a}ilescu]\label{m-theorem}
The Diophantine equation
\[ x^{u} - y^{v} = 1 \ \ \ \ (x \ge 1, y \ge 1, u \ge 2, v \ge 2)\]
has a unique solution which is given by $x^{u} = 3^{2}$ and $y^{v} = 2^{3}$.
\end{thm}
\noindent We refer the reader to \cite{Metsankyla} for the interesting history behind this theorem and an exposition of Mih\u{a}ilescu's proof.
We now make explicit the connection between Catalan's Conjecture and our eponymous problem. In \S \ref{ff}, we reduced the determination of the finite fields with (non-trivial) indecomposable multiplicative group to finding pairs $(p, r)$ (where $p$ is prime and $r$ is a positive integer) such that $p^{r}-1= q^{n}$ for some prime $q$ and positive integer $n$. The last equation rearranges to $p^r - q^n = 1$. The connection to Catalan's Conjecture is now clear: we seek solutions to the Diophantine equation $x^{u} - y^{v} = 1$, where $x$ and $y$ are prime numbers and the exponents are natural numbers. Propositions \ref{2powers} and \ref{ppowers} may be together viewed as a special case of Catalan's Conjecture; they give a complete list of the consecutive prime powers.
We will now prove Theorem \ref{finitefields} using Catalan's Conjecture. Let $F$ be a finite field of order $p^{r}$ whose multiplicative group is indecomposable. Then, as explained above, we obtain $p^{r} - q^{n} = 1$, where $q$ is prime (here, we allow $n \geq 0$). We will consider 3 cases which neatly organize the fields obtained in Theorem \ref{finitefields}.
First, suppose $r = 1$ and $n \geq 0$. This gives $p - q^{n} = 1$. If $p = 2$, then $n = 0$ and $F = \mathbf{F}_2$. If $p$ is odd, then $q = 2$ and $p$ is a Fermat prime. The corresponding finite fields are $\mathbf{F}_p$ where $p$ is a Fermat prime.
Next, suppose $r \geq 2$ and $0 \leq n \leq 1$. Here, $r \geq 2$ forces $n = 1$, so $p^{r} - q = 1$. If $q = 2$, then $p^{r} = 3$, which is not possible. If $q > 2$, then $p = 2$, and $2^{r} - 1$ is a Mersenne prime. The corresponding finite fields are $\mathbf{F}_{q+1}$ where $q$ is a Mersenne prime.
Finally, assume $r \ge 2$ and $n \ge 2$. In this case, Catalan's Conjecture implies that $p^{2}= 3^{2}$. The corresponding finite field is $\mathbf{F}_{9}$.
\section{Infinite Fields\label{if}}
Our goal in this section is to determine all infinite fields with an indecomposable multiplicative group. We are currently unable to find a single example of such a field; we can, however, narrow the possible examples to a special class of fields of characteristic 2 (see Theorem \ref{reduction}).
To begin, let $F$ be an infinite field whose multiplicative group is indecomposable. We will first argue that $F^\times$ must be torsion free. The following theorem relies upon a classical result of Pr\"{u}fer and Baer on the structure of abelian groups whose elements have boundedly finite orders (see \cite[\S 4]{Robinson} for details). Recall that a $p$-group is a group in which every non-trivial element has (finite) order a power of $p$.
\begin{thm}[{\cite[4.3.12]{Robinson}}]\label{corqc}
Let $G$ be an indecomposable abelian group that is not torsion-free. Then $G$ is either a cyclic or quasicyclic $p$-group for some prime $p$.
\end{thm}
A $p$-group is quasicyclic if it is isomorphic to $C_{p^\infty}$, the union of all cyclic groups of order a power of $p$:
\[ C_{p^{\infty}} = \bigcup_{n\ge 0} C_{p^n}.\]
This group, also called the Pr\"{u}fer group, is written additively as the direct limit $\underset{\longrightarrow}{\lim}\, \mathbb{Z}/(p^n)$. It is the injective hull of $\mathbb{Z}/(p)$ and is isomorphic to $\mathbb{Z}\left[1/p\right]/\mathbb{Z}$.
Since our multiplicative group $F^\times$ is both infinite and indecomposable, Theorem \ref{corqc} implies $F^\times \cong C_{p^{\infty}}$. We now prove that this is impossible.
\begin{prop}
There is no field $F$ whose multiplicative group is a quasicyclic $p$-group.
\end{prop}
\begin{proof}
Assume to the contrary that there is a field $F$ whose multiplicative group is isomorphic to $C_{p^{\infty}}$. Then every element in $F^\times$ is a torsion element. This implies that the characteristic of $F$ cannot be $0$.
Let $q > 0$ be the characteristic of $F$. Note that $F^\times$ contains a copy of $C_{p^i}$ for all positive integers $i$. In particular, $F$ contains $\zeta_{p^i}$, a primitive $p^i$th root of unity, for all $i \ge 1$. Consider the ascending tower of finite fields $\mathbf{F}_q( \zeta_{p^i})$ for $i \ge 1$ inside $F$. Let $q^{n_i}$ denote the orders of these finite fields. The multiplicative groups of these finite fields are finite cyclic subgroups of $C_{p^{\infty}}$. Consequently, there are integers $m_i$ such that
\label{keyequation}
\begin{equation} q^{n_i} - 1 = p^{m_i}\ \ \text{for}\ \ { i \ge 1}.\label{nozpi}\end{equation}
We now offer 3 different arguments to show that this is impossible.
1. In Equation (\ref{nozpi}), note that $\{m_i\}$ and $\{n_i\}$ are both increasing sequences. When $p=2$, this is impossible by Proposition \ref{2powers} and when $p$ is odd, this is impossible by Proposition \ref{ppowers}.
2. Note that Equation (\ref{nozpi}) can also be rewritten as
\[ q^{n_i} - p^{m_i} = 1\ \ \text{for}\ \ { i \ge 1}.\]
This shows that there are infinitely many solutions in positive integers to the equation $x^{u} - y^{v} = 1$. This contradicts Catalan's Conjecture.
3. Our final argument relies on the following special case of Zsigmondy's Theorem, proved by A. S. Bang in 1886 (see \cite{Ribenboim} for the statement of Zsigmondy's Theorem and an account of its interesting history).
\begin{thm}[Bang] Let $a$ and $t$ be integers greater than $2$. There
exists a prime divisor $l$ of $a^{t} - 1$ such that $l$ does not divide $a^{j} -1$ for all $
0 < j < t$.
\end{thm}
\noindent It is easy to see that Equation (\ref{nozpi}) contradicts this theorem.\end{proof}
Now, coupling the above work with the observation that any field of characteristic not equal to 2 has non-trivial torsion element $-1$, we have the following theorem.
\begin{thm}\label{tfthm}
If $F$ is an infinite field such that $F^\times$ is not torsion-free, then $F^\times$ is a decomposable group. In particular, infinite fields of characteristic not equal to $2$ have decomposable multiplicative groups.
\end{thm}
It is our suspicion that the characteristic 2 case is no different; we therefore make the following conjecture.
\begin{conjecture} Every infinite field has a decomposable multiplicative group. \end{conjecture}
\noindent Theorem \ref{reduction} summarizes everything we know about possible counter-examples to this conjecture. We will use the next proposition in that theorem. Recall that the rank of an abelian group $A$, written $\rank A$, is the size of a maximal linearly independent subset; equivalently, $\rank A$ is the dimension of $A\otimes \ensuremath{\mathbb{Q}}$ as a $\ensuremath{\mathbb{Q}}$-vector space. Further, since $\ensuremath{\mathbb{Q}}$ is flat over $\ensuremath{\mathbb{Z}}$, we have $\rank B \leq \rank A$ whenever $B$ is a subgroup of $A$.
\begin{prop}\label{free-summand}
Let $k$ be a field such that $k^\times$ is a free abelian group and let $K$ be a finite extension of $k$. Then, $K^\times$ has a summand isomorphic to $k^\times$.
\end{prop}
\begin{proof}
The field norm
\[ N = N_{K/k} \colon K^\times \longrightarrow k^\times \]
is a non-trivial homomorphism of $\mathbb{Z}$-modules (see \cite[8.1.3]{Roman}). The restriction of $N$ to $k^\times$ is the $d$th-power map, where $d = [K\colon k]$. The image of $N$ therefore contains $(k^\times)^d$, which is isomorphic to $k^\times$ since the latter group is a free $\ensuremath{\mathbb{Z}}$-module. We now have $$k^\times \cong (k^\times)^d \subseteq \im N \subseteq k^\times,$$ hence $\rank \im N = \rank k^\times$. Since $\im N$ is a submodule of a free module, and submodules of free modules over a principal ideal domain are always free, we conclude that the image of the norm map is a free $\mathbb{Z}$-module whose rank is the same as the rank of $k^\times$; hence, $\im N \cong k^\times$. Further, since free $\mathbb{Z}$-modules are projective, the surjection
\[ N \colon K^\times \longrightarrow \im N \]
splits. Thus $K^\times$ has a summand isomorphic to $k^\times$, as desired.
\end{proof}
\begin{thm}\label{reduction}
Let $F$ be an infinite field with indecomposable multiplicative group. Then, $F$ is an extension of $\mathbf{F}_2$, and there is an intermediate field $F \supseteq L \supsetneq \mathbf{F}_2$ such that $F$ is algebraic over $L$ and $L$ is a purely transcendental extension of $\mathbf{F}_2$. The fields $F$ and $L$ must satisfy the following properties.
\begin{enumerate}
\item The group $L^\times$ is free abelian of infinite rank.\label{Lx}
\item $[F\colon L] = \infty$.\label{degree}
\item The field $F$ is a completely transcendental extension of $\mathbf{F}_2$. In particular, $F$ is not algebraically closed.\label{ct}
\item The group $F^\times$ is a torsion-free indecomposable abelian group of infinite rank. In particular, $F^\times$ is reduced and $\Hom(F^\times, \ensuremath{\mathbb{Z}}) = 0.$\label{fxprops}
\item The group $F^\times$ is an essential infinite union of subgroups, each having a free abelian group isomorphic to $L^\times$ as a summand.\label{essential}
\end{enumerate}
\end{thm}
Recall that an abelian group is reduced if it has no divisible subgroups. A completely transcendental extension $A/B$ is one where every element of $A\setminus B$ is transcendental over $B$. We say that a set $T$ is an essential union of a collection of subsets if each subset in the collection is necessary in covering the set $T$.
\begin{proof}
Since $F$ is infinite and $F^\times$ is indecomposable, we immediately obtain that $F^\times$ is torsion-free and $F$ is an extension of $\mathbf{F}_2$ by Theorem \ref{tfthm}. The existence of the intermediate field $L$ is a standard result in field theory. It is a proper extension of $\mathbf{F}_2$, for otherwise $F$ would contain elements algebraic over $\mathbf{F}_2$, contradicting (\ref{ct}), proved below.
Now consider (\ref{Lx}). Let $S$ be an algebraically independent set over $\mathbf{F}_2$ such that $L = \mathbf{F}_2(S)$. The ring $\mathbf{F}_{2}[S]$ of polynomials with indeterminates in $S$ and coefficients in $\mathbf{F}_2$ is a unique factorization domain, and $\mathbf{F}_2(S)$ is its field of fractions. We therefore have \[ L^\times \cong \bigoplus_{f \in \Delta} \mathbb{Z},\] where $\Delta$ is the set of irreducible polynomials in $\mathbf{F}_{2}[S]$. This is a free abelian group of infinite rank.
We next compute $[F\colon L]$. Assume to the contrary that $[F \colon L] < \infty $. We may then apply Proposition \ref{free-summand} to obtain that the free abelian group $L^\times$ of infinite rank is a summand of $F^\times$. This is impossible, however, as $F^\times$ is indecomposable. So $[F\colon L] = \infty$.
Now consider statement (\ref{ct}). No element in $F \setminus \mathbf{F}_2$ can be algebraic over $\mathbf{F}_2$, for if $a$ in $F$ is algebraic over $\mathbf{F}_2$, then the subfield $\mathbf{F}_2(a)$, being a finite field different from $\mathbf{F}_2$, will contain non-trivial torsion elements, contradicting the fact that $F^\times$ is torsion-free. Thus $F$ is a completely transcendental extension of $\mathbf{F}_2$. Since $F$ contains no roots of unity, it is certainly not algebraically closed.
For (\ref{fxprops}), we have already observed that $F^\times$ is torsion-free, and it is assumed indecomposable. It has infinite rank since it contains the subgroup $L^\times$ which has infinite rank. To see that it is reduced, we first summon several facts from the homological algebra of abelian groups. See \cite[\S4]{Robinson} for details. The category of abelian groups is identical to the category of modules over the ring $\ensuremath{\mathbb{Z}}$. In this category, an abelian group is divisible if and only if it is an injective $\ensuremath{\mathbb{Z}}$-module. The salient property of injective modules here is that an injective module is a summand of any module in which it embeds. Now, suppose the indecomposable group $F^\times$ has a divisible subgroup. This subgroup must be a summand, and therefore $F^\times$ is itself divisible. The only indecomposable divisible abelian groups are the quasicyclic groups $C_{p^\infty}$ and the rational numbers $\ensuremath{\mathbb{Q}}$. Since $F^\times$ is torsion-free, we must have $F^\times \cong \ensuremath{\mathbb{Q}}$. However, $\ensuremath{\mathbb{Q}}$ has rank 1, so this is not possible. Finally, if $\Hom(F^\times, \ensuremath{\mathbb{Z}}) \neq 0$, then $F^\times$ admits a nontrivial homomorphism onto an infinite cyclic group. Such groups are projective as $\ensuremath{\mathbb{Z}}$-modules, and any surjective map to a projective module splits. We therefore obtain that $\ensuremath{\mathbb{Z}}$ is a summand of $F^\times$, so $F^\times \cong \ensuremath{\mathbb{Z}}$. This is impossible, again because $F^\times$ has infinite rank.
Finally, we turn our attention to (\ref{essential}). Since $[F\colon L] = \infty$, we can express $F$ as an essential infinite union of finite extensions over $L$. The group $F^\times$ is an essential infinite union of the multiplicative groups of these finite extensions. Now apply Proposition \ref{free-summand} to each of these finite extensions to obtain that $L^\times$ is a summand of each of these subgroups.\end{proof}
\section{Finite Commutative Rings\label{fr}}
The problem under investigation can be easily generalized to rings. Let $R$ be a commutative ring and let $R^\times$ denote the multiplicative group of units in $R$. For which rings $R$ is $R^\times$ indecomposable? We provide an answer to this question for finite commutative rings in Theorem \ref{finiterings}.
Let $R$ be a finite commutative ring. The ring $R$ obviously satisfies the descending chain conditions on its ideals. That is, $R$ is an Artinian ring. It therefore decomposes as a direct product of finite commutative Artinian local rings (see \cite[\S 8]{am}):
\begin{equation} R = R_1 \times R_2 \times \cdots \times R_k. \label{adecomp}\end{equation}
Since each $R_i$ is Artinian and local, it has has a unique prime ideal (such rings are called primary rings). Thus, $k$ is the number of prime ideals in $R$. Taking units, we obtain
\[ R^\times = R_1^\times \times R_2^\times \times \cdots \times R_k^\times.\]
The group $R^\times$ is indecomposable exactly when one factor is indecomposable and the remaining factors are trivial. Since a non-trivial indecomposable finite abelian group is a cyclic group of prime power order, we have the following proposition.
\begin{prop}
Let $R$ be a finite commutative ring. The ring $R$ has an indecomposable multiplicative group of units if and only if the multiplicative group of exactly one factor in the decomposition (\ref{adecomp}) has prime power order and all the remaining factors have trivial multiplicative groups.
\end{prop}
This proposition reduces our problem to finding all finite commutative Artinian local (primary) rings whose multiplicative group is either trivial or a cyclic group of prime power order. The more general problem of finding finite commutative primary rings with cyclic multiplicative groups was solved by Pearson and Schneider in \cite{PS}.
\begin{thm}[{\cite{PS}}]
Let $R$ be a finite commutative primary ring. The ring $R$ has a cyclic group of units if and only if $R$ is isomorphic to one of the following rings:
\begin{enumerate}
\item $\mathbf{F}_{q^{t}}$, where $q$ is prime and $t \ge 1$,
\item $\mathbb{Z}_{q^{s}}$, where $q$ is an odd prime and $s \ge 1$,
\item $\mathbb{Z}_{4}$,
\item $\mathbf{F}_{q}[x]/(x^{2})$, where $q$ is prime,
\item $\mathbf{F}_{2}[x]/(x^{3})$, or
\item $\mathbb{Z}_{4}[x]/(2x, x^{2}-2)$.
\end{enumerate}
\end{thm}
Computing the group of units for each of these rings is straightforward. We summarize the results below.
\begin{enumerate}
\item $(\mathbf{F}_{q^{t}})^\times \cong C_{q^{t}-1}$.
\item $(\mathbb{Z}_{q^{s}})^\times \cong C_{q^{s-1}(q-1)}$ (since $q$ is odd).
\item $(\mathbb{Z}_{4})^\times \cong C_2$.
\item $(\mathbf{F}_{q}[x]/(x^{2}) )^\times\cong C_{(q-1)q}$.
\item $(\mathbf{F}_{2}[x]/(x^{3}))^\times \cong C_4$.
\item $(\mathbb{Z}_{4}[x]/(2x, x^{2}-2))^\times \cong C_4$.
\end{enumerate}
We must now go through this list and isolate the rings whose multiplicative groups are either trivial or of prime power order. This, in conjunction with Theorem \ref{finitefields}, results in following theorem. (Note that the only finite primary ring with trivial multiplicative group of units is the ring $\mathbb{Z}_2$.) A ring is said to be indecomposable if cannot be expressed as a direct product of two non-zero rings.
\begin{thm} \label{finiterings}
The following is a complete list of the finite commutative indecomposable rings which have an indecomposable multiplicative group:
\begin{enumerate}
\item $\mathbf{F}_2$,\label{first}
\item $\mathbf{F}_9$,
\item $\mathbf{F}_p$, where $p$ is a Fermat prime,
\item $\mathbf{F}_{q+1}$, where $q$ is a Mersenne prime,
\item $\mathbb{Z}_{4}$,
\item $\mathbf{F}_{2}[x]/(x^{2})$,
\item $\mathbf{F}_{2}[x]/(x^{3})$, and
\item $\mathbb{Z}_{4}[x]/(2x, x^{2}-2)$.\label{last}
\end{enumerate}
A finite commutative ring $R$ has an indecomposable group of units if and only if $R$ is a (possibly empty) product of finitely many copies of $\ensuremath{\mathbb{Z}}_2$ and exactly one ring on the above list.
\end{thm}
\begin{proof}
It remains only to verify that each ring on the above list is indeed indecomposable; this is obvious for the first four rings, which are fields. For the remaining rings, observe that a ring $R$ admits a non-trivial decomposition $R \cong R_1 \times R_2$ if and only if $R$ contains an idempotent element $e \neq 0, 1$. It is straightforward to check that the last four rings contain no such idempotent elements.\end{proof}
\bibliographystyle{alpha}
| {
"timestamp": "2014-07-15T02:10:48",
"yymm": "1407",
"arxiv_id": "1407.3481",
"language": "en",
"url": "https://arxiv.org/abs/1407.3481",
"abstract": "The multiplicative group of a finite field is well known to be cyclic; in this note, we determine the finite fields whose multiplicative groups are direct sum indecomposable. We obtain our classification using a direct argument and also as a corollary to Catalan's Conjecture. Turning to infinite fields, we prove that any infinite field whose characteristic is not equal to 2 must have a decomposable multiplicative group. We conjecture that this is also true for infinite fields of characteristic 2 and we narrow the class of possible counter-examples. Finally, using the classification of finite commutative primary rings with cyclic multiplicative groups, we determine all finite commutative rings with indecomposable multiplicative groups.",
"subjects": "Number Theory (math.NT); Group Theory (math.GR)",
"title": "When is the multiplicative group of a field indecomposable?",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9854964181787612,
"lm_q2_score": 0.7185943925708561,
"lm_q1q2_score": 0.7081722000019213
} |
https://arxiv.org/abs/1307.6321 | An Uncertainty Principle for Discrete Signals | By use of window functions, time-frequency analysis tools like Short Time Fourier Transform overcome a shortcoming of the Fourier Transform and enable us to study the time- frequency characteristics of signals which exhibit transient os- cillatory behavior. Since the resulting representations depend on the choice of the window functions, it is important to know how they influence the analyses. One crucial question on a window function is how accurate it permits us to analyze the signals in the time and frequency domains. In the continuous domain (for functions defined on the real line), the limit on the accuracy is well-established by the Heisenberg's uncertainty principle when the time-frequency spread is measured in terms of the variance measures. However, for the finite discrete signals (where we consider the Discrete Fourier Transform), the uncertainty relation is not as well understood. Our work fills in some of the gap in the understanding and states uncertainty relation for a subclass of finite discrete signals. Interestingly, the result is a close parallel to that of the continuous domain: the time-frequency spread measure is, in some sense, natural generalization of the variance measure in the continuous domain, the lower bound for the uncertainty is close to that of the continuous domain, and the lower bound is achieved approximately by the 'discrete Gaussians'. |
\section{Introduction}
Fourier Transform, due to the fact that it is a global transform, is not
well-suited for the analysis of signals that exhibit transient behavior.
This is a rather significant drawback since such signals exist in
abundance.
One way to remedy this shortcoming is the use of window functions:
a window function enables us to localize the function
to some specific interval of interest that we want to look at.
This gives rise to time-frequency analysis and makes it
possible for us to study the frequency structure of functions at
varying points in time.
Just like Fourier analysis, time-frequency analysis is a fundamental
tool in science, especially in signal processing.
In this article, we define the Fourier Transform $\hat f$ of a complex-valued
function $f$ defined on the real line ${\mathbb{R}}$ via
\begin{equation}
\label{eq:continuousFourier}
\hat f(\xi) := \int_{\mathbb{R}} f(t) e^{-2\pi i \xi t}\, dt, \quad \xi\in{\mathbb{R}}.
\end{equation}
The Windowed Fourier Transform of $f$ with a given window function
$g:{\mathbb{R}} \to {\mathbb{C}}$ would then be defined as
\[
{\mathcal{V}}_g(\tau, \xi) := \int_{\mathbb{R}} f(t) \overline{g(t-\tau)} e^{-2\pi i \xi t}\, dt, \quad \tau, \xi\in{\mathbb{R}}.
\]
If $g$ and $\hat g$ are supported near the origin, one may interpret that ${\mathcal{V}}_g f(\tau, \xi)$ is the `$\xi$-frequency
content of $f$ at time $\tau$'.
Unfortunately, such an ideal interpretation cannot become a reality;
the well-known uncertainty principles expresse the idea that
there is a fundamental limit on how $g$ and $\hat g$ can be simultaneously localized in the two domains.
The most famous formulation of the uncertainty principle is given by the
Heisenberg-Pauli-Weyl inequality (see, e.g., \cite{gr01}):
\begin{thm}
\label{thm:heisenberg}
For $f\in L_2({\mathbb{R}})$, define the variance of $f$ by
\begin{equation}
\label{eq:varCont}
v_f := \min_{a\in{\mathbb{R}}} \frac{1}{\nnorm{f}_2^2} \int_{-\infty}^\infty (t-a)^2 |f(t)|^2 \, dt.
\end{equation}
Then,
\[
v_f v_{\hat f} \ge \frac{1}{16\pi^2}.
\]
Equality holds if and only if $f$ is a multiple of $\varphi_{a,b}$,
defined by
\[
\varphi_{a,b}(t) := e^{2\pi i b (t-a)} e^{-\pi (t-a)^2/c}
\]
for some $c > 0$.
\end{thm}
We may define the mean of $f$ by
\[
\mu_f := \argmin_{a\in{\mathbb{R}}} \frac{1}{\nnorm{f}_2^2} \int_{-\infty}^\infty (t-a)^2 |f(t)|^2 \, dt.
\]
Clearly, the smaller $v_f$ is, the more concentrated the function $f$
is around $\mu_f$.
In other words, $v_f$ is a measure of time-spreading of $f$.
Similarly, $v_{\hat f}$ is a frequency-spreading measure
of $f$.
Thus, the Heisenberg-Pauli-Weyl inequality expresses
the intrinsic limit on how well an $L_2({\mathbb{R}})$ function can be localized
on the time-frequency plane.
Moreover, the theorem also tells us what the minimizing functions are.
While the Heisenberg Uncertainty Principle gives us a clear picture of
what can be achieved for time-frequency localization for the continuous
functions defined on ${\mathbb{R}}$, our discussion so far is somewhat detached from reality;
we can only consider functions defined on finite intervals in real life.
Furthermore, in this day and age of computers, processing can be done
only when the signal can be stored in memory.
Therefore, the signals are discrete and finite.
A pertinent question is: what can be said about the uncertainty for
the time-frequency analysis when the Discrete Fourier Transform is used?
Is there any relation between the uncertainties for the continuous and the
discrete cases?
To our knowledge, surprisingly little is known for this problem,
and this is the area that we aim to contribute to with our work.
\section{Discrete Uncertainty Relations: Some Related Works}
In this section, we discuss some works in the literature which may serve
as an introduction to the problem that we are interested in.
\subsection{Uncertainty for Continuous Functions Defined on the Circle}
The Fourier series for periodic functions may be viewed as something
intermediate between the continuous Fourier Transform for functions on
the real line and the discrete Fourier Transform for finite signals.
It could be a good starting point of our discussion on the uncertainty
for discrete signals.
For a $2\pi$-periodic function $f$, the Fourier coefficients for $f$
is defined by
\[
\hat f(k) := \frac{1}{2\pi} \int_{0}^{2\pi} f(t) e^{-ikt} \,dt,
\quad k \in {\mathbb{Z}}.
\]
\emph{Remarks on notations: }
For lightness, we will sacrifice the precision and
use the same notation $\hat f$ to mean various different
Fourier Transforms whose meaning will become clear depending on what $f$ is.
Such a convention extends to $\nnorm{\cdot}$ as well.
We also point out that the definition of the continuous Fourier Transform $\hat f$
used in this subsection is defined without the $2\pi$-factor in \eqref{eq:continuousFourier}.
The question we are interested in is how concentrated, or conversely how
spread, $f$ and $\hat f$ are.
We note that even though $\hat f$ is a discrete sequence, there is no problem in
defining the variance of it; we need only to replace the integral in \eqref{eq:varCont}
with an analogous sum. The mean $\mu_{\hat f}$ can be similarly defined.
The situation is different for $f$.
The issue is that we cannot simply compute
\[
\frac{1}{\nnorm{f}_2^2} \int_{0}^{2\pi} t |f(t)|^2 \, dt
\]
for the mean of $f$.
Such a quantity fails to take the periodicity into account.
A different way to characterize
`the mean value' had been proposed (see \cite{Breitenberger83uncertainty}):
\[
\tau(f) := \frac{1}{\nnorm{f}_2^2} \int_0^{2\pi} e^{it} |f(t)|^2 \, dt.
\]
The periodicity is clearly reflected in $\tau(f)$.
With that, one defines `the variance' of $f$ as
\[
\frac{1}{\nnorm{f}_2^2} \int_0^{2\pi} |e^{it} - \tau(f)| |f(t)|^2 \, dt
= 1 - \tau(f)^2.
\]
With these time-frequency spread measures, the uncertainty relation for the
continuous functions on the circle was shown to be as follows:
\begin{equation} \label{eq:preUncertaintySemi}
\left(1-\tau(f)^2\right) v_{\hat f} \ge \frac{\tau(f)^2 }{4}.
\end{equation}
Note that unlike in the continuous setting, the quantity on the right-hand
side depends on the function $f$.
Therefore, if we were to use $(1-\tau(f)^2) v_{\hat f}$ as the measure of
uncertainty of $f$, the equality in \eqref{eq:preUncertaintySemi}
does not immediately imply that $f$ is a minimizer of the uncertainty.
A simple way to bypass this issue is to define the time spread of $f$ as
\[
v_{f} := \frac{1-\tau(f)^2}{\tau(f)^2}.
\]
A more precise description of the resulting uncertainty principle is given as follows
\cite{Narcowich96wavelets,Prestin99optimal}:
\begin{thm}
For a function $f \in AC_{2\pi}$ with $f' \in L_2([0,2\pi])$
where $f$ is not of the form $c e^{ikt}$ for any $c\in{\mathbb{C}}$,
$k\in {\mathbb{Z}}$, it holds that
\[
v_f v_{\hat f} > \frac{1}{4}.
\]
The lower bound is not attained by any function, but is best
possible. Here, $AC_{2\pi}$ is the class of $2\pi$-periodic absolutely continuous functions.
\end{thm}
One reservation towards this result is that the meaning of the
so-called \emph{angular spread} $v_f$ is not very intuitive.
In addition, the theorem does not give any guide on what functions may
have the uncertainty product close to the lower bound.
A result in \cite{prestin03} sheds some light on the second problem.
The authors used a process of periodization and dilation to show that
a sequence of functions achieve the uncertainty for functions defined
on the real line in the limit.
They proved:
\begin{thm}
For an admissible function $f$ (defined on the real line),
\[
\lim_{a\to\infty} \frac{1}{a^2} v_{f_a} = v_f,
\quad
\lim_{a\to\infty} a^2 v_{\hat f_a} = v_{\hat f},
\]
where
\[
f_a(t) := \sqrt{a} \sum_{k\in{\mathbb{Z}}} f(a(t+2\pi k)).
\]
Therefore,
\[
\lim_{a\to\infty} v_{f_a} v_{\hat f_a} = v_f v_{\hat f}.
\]
\end{thm}
\
We remind the reader that the definitions of $v_{f_a}$ and $v_f$
are quite different.
Since the minimum of $v_f v_{\hat f}$ is known to be $1/4$
and is achieved by (essentially) Gaussian functions,
the theorem provides a way to build periodic functions that
are asymptotically optimal in the given measure of time-frequency
spreads.
We will see that our result shares some similarity with this.
Another way to obtain periodic functions which nearly achieve the
uncertainty bound is by computing directly with numerical optimization
\cite{parhizkar2013sequences}.
In this approach, Parhizkar et al. fixed the angular spread at a
prescribed level and then searched for functions that minimize
the frequency spread with the given angular spread.
They formulated the problem as a quadratically constrained quadratic
program and hence enabled efficient computations of desired window
functions.
For more results for uncertainties for functions on the circle
which include different spread measures, refer to \cite{Ishii:1986uncertainty,calvez:1992uncer,venkateshUncertainty}.
\subsection{Sparsity and Entropy}
There are several works in the literature on the uncertainty relation for
finite discrete signals where the Discrete Fourier Transform is considered;
see, e.g., \cite{Donoho89uncertainty,Donoho01uncertaintyprinciples,Tao_anuncertainty,meshulam:uncert,GhJa:uncert,
krahmer-pfander-rashkov-08,Przebinda99using}.
Most results in these can be generically stated as $\phi(\mathbf{x}) + \phi(\hat\mathbf{x}) \ge c_s$ or $\phi(\mathbf{x})\phi(\hat\mathbf{x}) \ge c_p$ for some
constants $c_s$ and $c_p$ where $\phi(\mathbf{x})$ measures the spread of $\mathbf{x}$.
In \cite{Donoho89uncertainty,Donoho01uncertaintyprinciples,Tao_anuncertainty,meshulam:uncert},
$\phi(\mathbf{x})$ is chosen to be $\norm{\mathbf{x}}_0$, i.e., the sparsity or the number of non-zero entries of $\mathbf{x}$.
In \cite{Przebinda99using}, the entropy of $\mathbf{x}$, ${\mathcal{S}}(\mathbf{x})$, is used for $\phi(\mathbf{x})$.
For more on these and other topics regarding uncertainty principle,
refer to \cite{ricaud12survey}.
While these results are deep and important with much impact, we note that $\norm{\mathbf{x}}_0$ and ${\mathcal{S}}(\mathbf{x})$
(and other similar measures) do not reflect properly the underlying geometry.
For example, if $\mathbf{x}$ consists of two pulses, $\norm{\mathbf{x}}_0 = 2$ no matter where the pulses are.
However, in many contexts, we clearly regard $\mathbf{x}$ is more localized/concentrated if the pulses are next to
each other.
Another potential drawback is that
the minimizers of these uncertainty measures tend to be the picket-fence signals
(Dirac comb).
From the perspective of window signals, those are intuitively regarded
as poorly localized on the time-frequency plane.
These are the reasons why we insist on the definitions in Section \ref{sec:measureUnc}.
Before closing the section, we mention the work \cite{gomi:uncert}. In this work, they
consider two operators (which may not even be self-adjoint) in a Hilbert space and
derive related uncertainty relations. Since their result is general, one can apply
it in the setting that we are interested in and obtain some uncertainty relation.
For appropriate choice of operators, one may obtain a result that would be close to
ours. While interesting, we think this is not a simple task. We also point out that
our result links uncertainty relations in two different domains, which is not addressed
by \cite{gomi:uncert}.
\section{Connection between Discrete and Continuous Uncertainty Relations}
In this section, we present the main result of this paper.
\subsection{Discretized Time-Frequency Spreads Measures}
\label{sec:measureUnc}
Let us fix a positive integer $N$ and consider the space ${\mathbb{C}}^N$ of $N$-dimensional signals.
For our purposes, we will regard a vector $\mathbf{x}\in{\mathbb{C}}^N$ as defined on
$N$ uniformly spaced points
\[
{\mathcal{D}_N} := \Big\{ -\frac{N}{2}+1, -\frac{N}{2}+1,
\ldots, \frac{N}{2} \Big\}/\sqrt N.
\]
With this understanding, the Discrete Fourier Transform $\hat \mathbf{x}\in{\mathbb{C}}^N$ of $\mathbf{x}\in{\mathbb{C}}^N$
is defined by
\[
\hat\mathbf{x}(k) := \frac{1}{\sqrt N} \sum_{j\in{\mathcal{D}_N}} \mathbf{x}(j) e^{-2\pi jk}, \quad k \in {\mathcal{D}_N}.
\]
The inverse transform has the following form:
\[
\mathbf{x}(j) = \frac{1}{\sqrt N} \sum_{k\in{\mathcal{D}_N}} \hat\mathbf{x}(k) e^{2\pi jk}, \quad j \in {\mathcal{D}_N}.
\]
Next, we consider a measure of spread of a vector $\mathbf{x}\in{\mathbb{C}}^N$.
For this, we go back to \eqref{eq:varCont} and adapt it to our setting.
Viewing $|t-a|$ as the distance between $t$ and $a$, it is natural to
define the variance $v_\mathbf{x}$ of $\mathbf{x}\in{\mathbb{C}}^N$ by
\[
v_\mathbf{x} := \min_{a\in \mathcal{I}_N} \frac{1}{\norm{\mathbf{x}}_2^2} \sum_{j\in{\mathcal{D}_N}} d(j,a)^2 |\mathbf{x}(j)|^2
\]
where $\mathcal{I}_N$ denotes interval $(-\sqrt N, \sqrt N]/2$ and $d(j,a)$ is the distance between $j$ and $a$.
Now note that our definition of Discrete Fourier Transform assumes that the signals
in ${\mathbb{C}}^N$ are $\sqrt N$-periodic. Taking this into account, we define
the distance between two points $j$ and $a$ by
\[
d(j, a) := \min_{l\in\sqrt N{\mathbb{Z}}} |j-a-l|.
\]
Finally, we may define the mean $\mu_\mathbf{x}$ of $\mathbf{x}$ to be the minimizing value $a\in\mathcal{I}_N$
of the right-hand side expression above for $v_\mathbf{x}$.
Note that $v_{\hat\mathbf{x}}$ is identically defined.
\subsection{No Uncertainty?}
\label{sec:noUncertainty}
With our definition of uncertainty $v_\mathbf{x} v_{\hat \mathbf{x}}$,
there cannot be any uncertainty principle in the conventional sense.
Clearly, for any $\mathbf{x}\in{\mathbb{C}}^N$, we have
$v_\mathbf{x} \le N/4$ and $v_{\hat\mathbf{x}} \le N/4$.
On the other hand, the vector $\mathbf{x}$ that is supported at the origin satisfies $v_\mathbf{x} = 0$.
Hence, $v_\mathbf{x} v_{\hat \mathbf{x}} = 0$.
It appears that there is no uncertainty at all and that we can do as well as
we want!
Of course, such a claim is non-sense, and it runs counter to our intuition that we could not
have a signal localized simultaneously in both domains as accurate as we wanted.
A closer look at the case $v_\mathbf{x} v_{\hat\mathbf{x}} = 0$ reveals why we came to this conclusion.
The signal $\hat\mathbf{x}$ is \emph{globally} supported but $v_{\hat\mathbf{x}}$ fails to express
the badness in frequency localization since it is always bounded above by $N/4$.
In contrast, one would have had $v_{\hat f} = \infty$ in such cases.
One way to resolve this issue would be to re-define $v_{\hat\mathbf{x}}$ (and $v_{\mathbf{x}}$) in a way
so that $v_{\hat\mathbf{x}} = \infty$ in this kind of signals $\mathbf{x}$.
However, we will not take this route since the argument in Section \ref{sec:measureUnc}
shows that $v_\mathbf{x}$ is a sensible way to gauge the time spread of $\mathbf{x}$.
How can we formulate a sensible uncertainty principle then?
\subsection{Uncertainty for a Subclass of Discrete Signals}
As seen in \ref{sec:noUncertainty}, there are signals that we clearly want to exclude from
our consideration. Thus, it makes sense to
restrict our attention to a subclass of signals in ${\mathbb{C}}^N$ in order to exclude
the cases where $\mathbf{x}$ or $\hat\mathbf{x}$ are `globally supported'.
Based on the similarity between the discrete and the continuous Fourier Transforms,
it is natural to suspect that discrete finite samples of Gaussian functions
might be optimal windows for the Discrete Fourier Transform.
While this appears reasonable, it looks difficult to show its validity rigorously.
Moreover and perhaps obviously, taking discrete finite samples of Gaussian functions
would be a bad idea \emph{unless} they happen to be nearly zero outside the
sampling interval.
This leads us to introduce `admissible functions' for our discussion.
We say that
$f\in L_2({\mathbb{R}})$ is $(N,\epsilon)$-\emph{localized} if
\begin{equation}
|f(t)| \le \frac{\epsilon}{|t|^2}, \quad |t| \ge \frac{\sqrt N}{2},
\end{equation}
and that a signal
$\mathbf{x}\in {\mathbb{C}}^N$ is \emph{admissible} with constant $\epsilon$ if
\[
\mathbf{x}(j) = \mathbf{x}_f(j) := N^{-1/4} \sum_{l\in\sqrt N{\mathbb{Z}}} f(j + l), \quad j\in{\mathcal{D}_N}
\]
for a function $f$ with $(N,\epsilon)$-localized functions $f$, $f'$, $\hat f$, $\hat f'$.
That is, admissible vectors in ${\mathbb{C}}^N$ are obtained by uniformly
sampling $\sqrt N$-periodized localized functions in $L_2({\mathbb{R}})$.
Our main result is the following:
\begin{mthm}
\label{thm:main}
Suppose that $f\in L_2({\mathbb{R}})$ is localized in time-frequency domain with constant $\epsilon$.
Then,
\[
\sqrt{v_f v_{\hat f}} ( 1 - \sqrt\epsilon) \le \sqrt{v_\mathbf{x} v_{\hat \mathbf{x}}}
\le \sqrt{v_f v_{\hat f}} ( 1 + \sqrt\epsilon),
\]
where $\mathbf{x} := \mathbf{x}_f$.
Thus, if $\mathbf{x}$ is an admissible signal, then
\[
v_\mathbf{x} v_{\hat\mathbf{x}} \ge \frac{(1-\sqrt\epsilon)^2}{16\pi^2}.
\]
\end{mthm}
To give some idea of the proof, we ask first: Why do we associate $\mathbf{x}_f$ to $f$ instead of sampling the function
directly without periodizing it?
Upon some reflection, the periodization seems to be natural given the well-known
phenomenon of folding (aliasing) associated with sampling approach.
It is the periodization that makes the two endpoints of ${\mathcal{D}_N}$ to be neighbors
when the sampling is done.
Another crucial reason for us to introduce $\mathbf{x}_f$ in that way is the observation
that $\hat\mathbf{x}_f = \mathbf{x}_{\hat f}$, which is a standard consequence of Poisson Summation Formula.
Thanks to this identity, we need only to show that $v_\mathbf{x}$ and $v_{\hat\mathbf{x}}$ are good approximations of
$v_f$ and $v_{\hat f}$, respectively.
To show that $v_\mathbf{x}$ and $v_f$ are close to each other, we show that relevant moments of $\mathbf{x}$
and $f$ are very close. For this purpose, we apply the Poisson Summation Formula
and the Parseval's identity. This is also where we use $(N,\epsilon)$-localizedness of $f$, $f'$, $\hat f$, and
$\hat f'$. A detailed proof of Theorem \ref{thm:main} will be given in an up-coming work.
\section{Discussion and Conclusion}
One implication of Theorem \ref{thm:main} is that, if we were to consider only the admissible
signals in ${\mathbb{C}}^N$ as windows -- which is not unreasonable in many applications since one would
like to have `smooth' and `fast-decaying' windows for the time-frequency analysis -- thanks to Theorem \ref{thm:heisenberg},
we can easily construct nearly optimal windows for the Discrete Fourier Transforms
by periodizing Gaussian functions and taking finite uniform
samples as long as the Gaussian functions are supported essentially on the interval
of sampling.
This is a mild requirement due to the exponential decay of the Gaussian functions, especially when $N$ is large.
We must keep in mind that `discrete gaussians' above are,
a priori, nearly optimal only among admissible signals in ${\mathbb{C}}^N$;
however, we will demonstrate in the up-coming work that
the near optimality of the discrete gaussians may be valid for `all signals' in ${\mathbb{C}}^N$.
More theoretical evidence
related to the near optimality of the discrete Gaussians
will be given there.
We also show by numerical computation that the uncertainty bound
is indeed very close to $1/(16\pi^2)$.
To conclude, we asserted that the uncertainty products of admissible signals with constant $\epsilon$
in ${\mathbb{C}}^N$ are bounded below by constant close to $1/(16\pi^2)$.
Based on this claim, we derived that the discrete Gaussians are near optimal windows
among the admissible signals.
Even though the near optimality of the discrete Gaussians among \emph{all} signals is strongly
suspected, a definitive proof is still missing and remains as future work.
Also, as a side problem, it would be interesting to study the characteristics of the discrete
Gaussians that arise from Gaussian functions with wide support. For example, are those signals
near optimal in some other sense?
Finally, we mention the question of optimal windows for distinguishing, e.g., linear chirps.
In our follow-up, we take the approach of this work and try to establish, at least
formally, that modulated discrete Gaussians (so that they themselves are linear chirps) are
nearly optimal as well.
\section*{Acknowledgment}
This work is supported by the European project UNLocX (FET-OPEN, grant number 255931).
The author would like to thank Bruno Torr\'esani and Benjamin Ricaud for helpful discussions
on the subject. The author also thanks the reviewers for their constructive comments.
\nocite{Folland97uncertainty}
\bibliographystyle{IEEEtran}
| {
"timestamp": "2013-07-25T02:03:52",
"yymm": "1307",
"arxiv_id": "1307.6321",
"language": "en",
"url": "https://arxiv.org/abs/1307.6321",
"abstract": "By use of window functions, time-frequency analysis tools like Short Time Fourier Transform overcome a shortcoming of the Fourier Transform and enable us to study the time- frequency characteristics of signals which exhibit transient os- cillatory behavior. Since the resulting representations depend on the choice of the window functions, it is important to know how they influence the analyses. One crucial question on a window function is how accurate it permits us to analyze the signals in the time and frequency domains. In the continuous domain (for functions defined on the real line), the limit on the accuracy is well-established by the Heisenberg's uncertainty principle when the time-frequency spread is measured in terms of the variance measures. However, for the finite discrete signals (where we consider the Discrete Fourier Transform), the uncertainty relation is not as well understood. Our work fills in some of the gap in the understanding and states uncertainty relation for a subclass of finite discrete signals. Interestingly, the result is a close parallel to that of the continuous domain: the time-frequency spread measure is, in some sense, natural generalization of the variance measure in the continuous domain, the lower bound for the uncertainty is close to that of the continuous domain, and the lower bound is achieved approximately by the 'discrete Gaussians'.",
"subjects": "Information Theory (cs.IT)",
"title": "An Uncertainty Principle for Discrete Signals",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9854964181787612,
"lm_q2_score": 0.7185943925708561,
"lm_q1q2_score": 0.7081722000019213
} |
https://arxiv.org/abs/2003.13766 | Hybrid Projection Methods for Large-scale Inverse Problems with Mixed Gaussian Priors | When solving ill-posed inverse problems, a good choice of the prior is critical for the computation of a reasonable solution. A common approach is to include a Gaussian prior, which is defined by a mean vector and a symmetric and positive definite covariance matrix, and to use iterative projection methods to solve the corresponding regularized problem. However, a main challenge for many of these iterative methods is that the prior covariance matrix must be known and fixed (up to a constant) before starting the solution process. In this paper, we develop hybrid projection methods for inverse problems with mixed Gaussian priors where the prior covariance matrix is a convex combination of matrices and the mixing parameter and the regularization parameter do not need to be known in advance. Such scenarios may arise when data is used to generate a sample prior covariance matrix (e.g., in data assimilation) or when different priors are needed to capture different qualities of the solution. The proposed hybrid methods are based on a mixed Golub-Kahan process, which is an extension of the generalized Golub-Kahan bidiagonalization, and a distinctive feature of the proposed approach is that both the regularization parameter and the weighting parameter for the covariance matrix can be estimated automatically during the iterative process. Furthermore, for problems where training data are available, various data-driven covariance matrices (including those based on learned covariance kernels) can be easily incorporated. Numerical examples from tomographic reconstruction demonstrate the potential for these methods. | \section{Introduction} \label{sec:introduction}
For many imaging systems, the ability to obtain good image reconstructions from observed data requires the inclusion of a suitable prior.
Priors provide a systematic and efficient means to describe in probabilistic terms any prior knowledge about the unknowns. Oftentimes prior knowledge will come from a \textit{combination} of sources, and striking a good balance of information is critical. For example, priors may be learned from available training data, but bias in the reconstructions can be a big concern (e.g., when the training set is small or the desired image is very different from the training set). Thus, a safer approach is to include a prior that combines learned information with conventional smoothness properties. In other scenarios (e.g. in seismic tomography), the desired solution may consist of components with different smoothness properties, and the correct mixture of smoothness priors can be difficult to know a priori.
Using mixed Gaussian priors, where the prior covariance matrix can be represented as a convex combination of matrices, is a common approach to incorporate different prior covariance matrices. However, various computational challenges arise for problems where the number of unknowns is very large and the regularization and mixing parameter are not known in advance. We address these challenges by developing hybrid iterative projection methods for the efficient computation of solutions to inverse problems with mixed Gaussian priors. By exploiting a project-then-regularize framework, we enable statistical optimization tools for selecting the regularization parameter and the mixing parameter automatically, which would be very costly for the original problem.
We are interested in linear inverse problems of the form,
\begin{equation}
\label{eq:problem}
\mathbf{d} = \mathbf{A}\mathbf{s} + {\boldsymbol{\epsilon}}
\end{equation}
where $\mathbf{d} \in \mathbb{R}^{m}$ contains the observed data, $\mathbf{A} \in \mathbb{R}^{m \times n}$ models the forward process, $\mathbf{s} \in \mathbb{R}^{n}$ represents the desired parameters, and ${\boldsymbol{\epsilon}} \in \mathbb{R}^{m}$ represents noise in the data. We assume that ${\boldsymbol{\epsilon}} \sim \mathcal{N}({\bf0},\mathbf{R})$, where $\mathbf{R}$ is a symmetric positive definite matrix whose inverse and square root are inexpensive (e.g., a diagonal matrix). The goal of the inverse problem is to compute an approximation of $\mathbf{s}$, given $\mathbf{d}$ and $\mathbf{A}$.
Due to ill-posedness, small errors in the data may lead to large errors in the computed
approximation of $\mathbf{s}$, and regularization is required to stabilize the inversion process. We follow a Bayesian framework, where we assume a prior for $\mathbf{s}$. That is, we treat $\mathbf{s}$ as a Gaussian random variable with mean
vector
${\boldsymbol{\mu}} \in \mathbb{R}^{n}$ and covariance matrix $\mathbf{Q}\in \mathbb{R}^{n \times n}$. That is, $\mathbf{s} \sim \mathcal{N}({\boldsymbol{\mu}},\lambda^{-2}\mathbf{Q}) $, where
$\lambda$ is a scaling parameter (yet to be determined) for the precision matrix.
In many applications, the choice of $\mathbf{Q}$ is pre-determined (e.g., using expert knowledge) and is chosen to enforce smoothness or regularity conditions on the solution \cite{chung2017generalized,hochstenbach2010iterative,buccini2017iterated}. However, in some cases, there is not enough information to determine $\mathbf{Q}$ completely or expensive procedures are needed to determine an informative subset of covariates from a set of candidates (e.g., in geophysical imaging \cite{yadav2016statistical, yao1999calculating, zhang1995estimation}). These scenarios motivate us to consider mixed Gaussian priors, where the covariance matrix can be represented as a convex combination of matrices. Without loss of generality we consider prior covariance matrices of the form,
\begin{equation}
\label{eq:Qsum}
\mathbf{Q} = \gamma\mathbf{Q}_1 + (1-\gamma) \mathbf{Q}_2
\end{equation}
where $\mathbf{Q}_1$ is a symmetric positive definite matrix, $\mathbf{Q}_2$ is a symmetric positive semi-definite matrix, and mixing parameter $0 < \gamma \leq 1$. We consider the case where computing matrix-vector products with $\mathbf{Q}_1$ is easy, but accessing $\mathbf{Q}_1^{-1}$ or its symmetric factorization (e.g., Cholesky or eigenvalue factorization) is not feasible. Such scenarios arise, for example, when the prior covariance matrix is modeled entry-wise using covariance kernels. In such cases, the main challenge is that the resulting covariance matrices are large and dense, and factorizing or inverting them can be computationally prohibitive. However, matrix-vector multiplications can be done efficiently (e.g., via FFT embedding). A wide range of kernels, including nonseparable spatio-temporal kernels \cite{chung2017generalized}, can be included. We assume that matrix-vector products with $\mathbf{Q}_2$ can be done efficiently.
Covariance matrices of the form~\eqref{eq:Qsum} are becoming more common, especially in modern imaging applications where data (e.g., in the form of training images) are playing a larger role in the development of reconstruction algorithms \cite{arridge2019solving}.
Suppose we are given a dataset consisting of $N$ samples, $\mathbf{s}^{(i)} \in \mathbb{R}^{n}, i=1,2,\ldots,N$.
Then the training data can be used to obtain an unbiased estimator of an $n \times n$ sample covariance matrix,
\begin{equation}
\label{eq:covar-est}
\widehat\mathbf{Q} = \frac{1}{N}\sum_{i=1}^{N}(\mathbf{s}^{(i)}-\bar{\mathbf{s}})(\mathbf{s}^{(i)} - \bar{\mathbf{s}})\t,
\end{equation}
where $\bar{\mathbf{s}} = \frac{1}{N}\sum_{i=1}^N \mathbf{s}^{(i)}$ is the sample mean.
Notice that
$\widehat\mathbf{Q} = \mathbf{S} \mathbf{S}\t$, where the symmetric factor is defined as $\mathbf{S}=\frac{1}{\sqrt{N}}\left(\begin{bmatrix} \mathbf{s}^{(1)} & \dots & \mathbf{s}^{(N)} \end{bmatrix} -\bar{\mathbf{s}}\otimes\textbf{1}^\top\right)$ with $\textbf{1}\in\mathbb{R}^N$ denoting the vector whose elements are all $1$. For any vector $\mathbf{x} \in \mathbb{R}^n$, multiplication with $\widehat \mathbf{Q}$ can be done efficiently if $N << n$, e.g., using the following order of operations $\mathbf{S}(\mathbf{S}\t \mathbf{x})$. However, notice that $\widehat\mathbf{Q}$ is likely positive semi-definite rather than positive definite, so it is common to use $\widehat \mathbf{Q} + \gamma \mathbf{I}$ where $\gamma$ is a nudging term. Such approaches are known as sample based priors \cite{calvetti2005priorconditioners}. Another common approach is to use a convex combination, i.e., the prior covariance matrix is given as
\begin{equation}
\label{eq:convexcomb}
\mathbf{Q} = \gamma \mathbf{D} + (1-\gamma) \widehat\mathbf{Q}
\end{equation}
where $\mathbf{D}$ is chosen to be the identity matrix or a suitably chosen diagonal or correlation matrix, which ensures that $\mathbf{Q}$ is positive definite, and $\gamma \in \mathbb{R}$ is called the mixing parameter. The matrix in \eqref{eq:convexcomb} is called a shrinkage estimator of the covariance matrix \cite{schafer2005shrinkage}.
It is worth noting that covariance matrices of the form ~\eqref{eq:convexcomb} are also used in hybrid methods for data assimilation that combine an ensemble Kalman filter system with a variational (e.g., 3D-Var) system \cite{asch2016data}. These methods require careful tuning of the so-called blending parameter $\gamma$, and many of the existing approaches require $\gamma$ to be fixed in advance. We do not assume this.
Previous works on combining training data with regularization techniques typically follow an optimal experimental design or empirical Bayes risk minimization framework \cite{haber2003learning, chung2011designing}. More recently, there has been significant work on using training data in the context of machine learning to learn regularization functionals (e.g., \cite{li2018nett,schwab2018deep}) or to learn the ``invisible'' regions (e.g., \cite{bubba2019learning}). The area of data-driven machine learning is currently a hot topic \cite{lucas2018using,arridge2019solving}, where the main goal is to determine new ways to combine physical models with deep learning techniques. In this work, we incorporate training data in a Bayesian framework and exploit tools from numerical linear algebra not only to compute solutions efficiently but also to determine the appropriate weighting of the training data.
In this paper we develop a hybrid iterative projection method that is based on a mixed, generalized Golub-Kahan process to approximate the MAP estimate,
\begin{equation}
\label{eq:MAP}
\mathbf{s}_{\rm MAP} = \mathrm{argmin}_\mathbf{s} \frac{1}{2}\norm[\mathbf{R}^{-1}]{\mathbf{A}\mathbf{s} - \mathbf{d}}^2 + \frac{\lambda^2}{2} \norm[\mathbf{Q}^{-1}]{\mathbf{s} - {\boldsymbol{\mu}}}^2.
\end{equation}
where $\mathbf{Q}$ is of the form~\eqref{eq:Qsum}. Our approach can handle a wide range of scenarios, including data-informed regularization terms that use training or test images to define the prior. We assume that $\gamma$ is not known in advance and neither the inverse nor the factorization of $\mathbf{Q}$ is available. The proposed method has two distinctive features. First, we assume that \emph{both} $\gamma$ and $\lambda$ are unknown a priori and we estimate them during the solution process. For problems where $\gamma$ is fixed in advance, generalized hybrid methods \cite{chung2017generalized} can be directly applied. However, developing a hybrid method where $\gamma$ can be selected adaptively is not an obvious extension.
We develop an iterative hybrid approach
where the problem is projected onto generalized Krylov subspaces of small but increasing dimension and the regularization parameter and mixing parameter can be simultaneously and automatically selected. Second, we describe and investigate various scenarios where training data can be used to define $\mathbf{Q}_1$ and $\mathbf{Q}_2$, so
our approach can be considered a learning approach for the regularization term.
An outline for the paper is as follows. In Section~\ref{sec:background} we provide some background on Gaussian priors and focus on various data-driven prior covariance matrices. Then in Section~\ref{sec:agenGK}, we describe mixed, generalized hybrid projection methods for approximating the MAP estimate~\eqref{eq:MAP}, where $\mathbf{Q}$ is of the form~\eqref{eq:Qsum}. The approach consists of two-steps: (1) Project the problem onto a subspace of small but increasing dimension using an extension of the generalized Golub-Kahan bidiagonalization approach. (2) Solve the projected problem where the regularization parameter $\lambda$ and mixing parameter $\gamma$ can be selected automatically. Various regularization paremeter selection techniques will be investigated, and some theoretical results will be provided.
In Section~\ref{sec:numerics} numerical results on various image processing applications show the potential benefits and flexibility of these methods. Conclusions are provided in Section~\ref{sec:conclusions}.
\section{Mixed Gaussian priors}
\label{sec:background}
In this section, we motivate the need for mixed Gaussian priors and draw some connections to existing works on multi-parameter Tikhonov regularization and shrinkage estimation.
To begin, we focus on using Gaussian random fields to represent prior information and summarize some common choices for the (unscaled) prior covariance matrix $\mathbf{Q}$. Oftentimes, the covariance matrix is generated using a covariance function (also called a kernel function). Covariance functions are crucial in many fields and
encode assumptions about the form of the function that we are modeling. In most cases, the prior covariance matrix $\mathbf{Q}$ is large and dense with entries directly computed as $\mathbf{Q}_{ij} = \kappa(\mathbf{z}_{i}, \mathbf{z}_j)$, where $\{\mathbf{z}_{i}\}_{i=1}^{n}$ are the spatial points in the domain and $\kappa(\cdot,\cdot)$ is a covariance kernel function. Some commonly used parametric covariance functions \cite{rasmussen2003gaussian} are provided in Table~\ref{cov_table}.
\begin{table}[bthp]
\centering
\begin{tabular}{ |c|c|}
\hline
& covariance kernel function \\
\hline
squared exponential & $\text{exp}\left(-\frac{r^2}{2\ell^2}\right)$\\[5pt]
Mat$\acute{\text{e}}$rn & $\frac{1}{2^{\nu -1}\Gamma(\nu)} \left( \frac{\sqrt{2\nu }r}{\ell}\right)^{\nu}K_{\nu}\left(\frac{\sqrt{2\nu} r}{\ell}\right)$ \\[5pt]
$\gamma-$exponential & $\text{exp}\left(-\left(\frac{r}{\ell}\right)^{\gamma}\right)$ \\[5pt]
rational quadratic & $\left(1 + \frac{r^2}{2\nu\ell^2}\right)^{-\nu}$ \\[5pt]
sinc & $\frac{\sin(\nu r)}{\nu r}$ \\[5pt]
\hline
\end{tabular}
\caption{Summary of commonly-used covariance functions. The covariance functions are written either as functions of $\mathbf{z}_{i}$ and $\mathbf{z}_{j}$, or as a function of $r = |\mathbf{z}_{i} - \mathbf{z}_{j}|$ and depend on $\ell$ or $\ell$ and $\nu$. $\Gamma$ is the Gamma function and $K_{\nu}(\cdot)$ is the modified Bessel function of the second kind of order $\nu$.}
\label{cov_table}
\end{table}
For some kernel choices, the precision matrix (i.e., the inverse of the covariance matrix) is sparse or structured, so working with $\mathbf{Q}^{-1}$ or its symmetric factorization has obvious computational advantages.
However, in many applications, the precision matrix is not readily available, and the aim is to develop computational methods that can work with $\mathbf{Q}$ directly and avoid the need for the inverse or symmetric factorization. Such covariance kernels may arise
in dynamic scenarios with nonseparable, spatio-temporal priors \cite{chung2018efficient, long2011state, galkaspatiotemporal} or from spatially-variant priors \cite{dong2018tomographic, yang2016spatially}.
It is worth mentioning that in a truly Bayesian framework, the regularization parameter and the covariance kernel parameters could be included as hyperparameters and explored using MCMC methods \cite{bardsley2018computational}, but the computational costs of this approach would be very high.
One reason to use Gaussian mixtures as prior distributions is that it allows greater flexibility in the definition of the prior. In this paper, we consider a mixture of two Gaussians, but one could consider more general mixtures.
From a statistical viewpoint, a general formulation with $N$ Gaussian random vectors would correspond to a sum of covariance matrices.
That is, let $\mathbf{x}_1, ..., \mathbf{x}_N$ be $N$ mutually independent $n\times 1$ normal random vectors having means ${\boldsymbol{\mu}}_1, ... {\boldsymbol{\mu}}_N$ and covariance matrices $\mathbf{V}_1, ...\mathbf{V}_N.$ Let $\mathbf{B}_1,...\mathbf{B}_N$ be real $L \times n$ full rank matrices. Then the $L \times 1$ random vector
\begin{equation}
\mathbf{y} = \sum_{i=1}^N \mathbf{B}_i \mathbf{x}_i
\end{equation}
has a normal distribution with mean $\mathbb{E} \mathbf{y} = \sum_{i=1}^N \mathbf{B}_i {\boldsymbol{\mu}}_i$ and covariance matrix of the form $Cov(\mathbf{y}) = \sum_{i=1}^N \mathbf{B}_i \mathbf{V}_i \mathbf{B}_i\t.$ Thus, a Gaussian mixture prior corresponds to an assumption that the desired solution can be represented as a linear combination of Gaussian realizations (e.g., with different smoothness properties).
In the context of inverse problems, we point out a connection between mixed Gaussian priors and multi-parameter Tikhonov regularization. The basic idea of multi-parameter Tikhonov regularization, see e.g. \cite{Wang2012,LuPereverzev2011,BazanBorgesFrancisco2012,GazzolaNovati2013}, is to solve a problem of the form,
\begin{equation}
\min_\mathbf{s} \norm[\mathbf{R}^{-1}]{\mathbf{A} \mathbf{s} - \mathbf{d}}^2
+ \sum_{i=1}^N \lambda_i^2 \norm[2]{\mathbf{L}_i \mathbf{s}}^2,
\end{equation}
where $\lambda_i \in \mathbb{R}$ is the regularization parameter corresponding to regularization matrix $\mathbf{L}_i$ for $i=1, \ldots, N$. By including multiple penalty terms, this approach can enforce different smoothness properties (e.g, at different frequency bands) and avoid difficulties in having to select just one regularization matrix.
In a Bayesian framework, the multi-parameter Tikhonov solution can be interpreted as a MAP estimate, under the assumption of a Gaussian prior with mean ${\bf0}$ and covariance matrix
$ \left(\sum_{i=1}^N \lambda_i^2\mathbf{L}_i\t \mathbf{L}_i \right)^{-1}
$. Notice that except for in very limited scenarios, this is not the same as using mixed Gaussian priors, since here the precision matrix (not the covariance matrix) is represented as a sum of matrices.
\subsection{Data-driven prior covariance matrices}
With the increasing amount of and access to data in many applications, an important and challenging task is to determine how to efficiently and effectively incorporate prior knowledge in the form of training data both in the solution computation process and the subsequent data analyses. In this section, we describe various examples where training data can be used to define the prior covariance matrix. For all cases, we assume that training data is provided and the sample covariance matrix \eqref{eq:covar-est} has the form $\widehat\mathbf{Q} = \mathbf{S} \mathbf{S}\t$.
As described in the introduction, the most common approach is to take $\mathbf{Q}_2 = \widehat\mathbf{Q}$ and $\mathbf{Q}_1 = \mathbf{D}$ where $\mathbf{D}$ is easy to invert (e.g., diagonal or identity matrix). In this case, a very popular approach called shrinkage estimation of covariance matrices, or more general biased estimation, can be used to reduce the variance of the estimator. Typical shrinkage targets are diagonal matrices (e.g., including the identity matrix), and approaches to estimate the optimal shrinkage intensity $\gamma$ have been proposed by Ledoit and Wolf, Rao and Blackwell, and others \cite{LW04,asch2016data,schafer2005shrinkage,chen2009shrinkage}.
Another approach to incorporate training data is to force some structure or functional form on the prior covariance kernel function. For kernel functions that depend on a few parameters, the training data can be used to estimate these parameters. A similar idea was considered in \cite{haber2003learning} where training data was used to learn parameters defining the regularization functional. However, that approach requires solving an expensive constrained optimization problem, and the learned regularization functional is tailored to the forward operator and the noise level. We consider the case where the training data come from a prior defined by a covariance kernel function (e.g., for simplicity, we consider Mat\'ern kernels). We use the training data to learn the parameters defining the prior. This reduces to an optimization problem where the goal is to learn two parameters $\nu$ and $\ell$ from the training data by solving the optimization problem,
\begin{equation}
(\hat \nu, \hat \ell) = \mathrm{argmin}_{\nu>0,\ell>0}\norm[F]{\mathbf{Q}(\nu,\ell) - \widehat \mathbf{Q}}^2.
\end{equation}
Once the parameters are computed, they can be used to define $\mathbf{Q}_1 = \mathbf{Q}(\hat\nu,\hat\ell)$, which can be used directly in generalized hybrid methods, or they can be combined with the sample covariance matrix, i.e., $\mathbf{Q}$ as in~\eqref{eq:Qsum} with $\mathbf{Q}_1 = \mathbf{Q}(\hat\nu,\hat\ell)$ and $\mathbf{Q}_2 = \widehat \mathbf{Q}$, and solvers described in Section \ref{sec:agenGK} can be used.
Next, we describe some computationally efficient methods to estimate $\hat \nu$ and $\hat \ell.$
Notice that
\begin{align}
\|\mathbf{Q}(\nu,\ell) - \widehat{\mathbf{Q}}\|^2_F & = {\mathop{\mathrm{tr}}}(\mathbf{Q}(\nu,\ell) - \widehat{\mathbf{Q}})^\top(\mathbf{Q}(\nu,\ell) - \widehat{\mathbf{Q}}) \\
& = \mathbb{E}( \| (\mathbf{Q}(\nu,\ell) - \widehat{\mathbf{Q}}) {\boldsymbol{\xi}} \|^2_2)
\end{align}
where ${\boldsymbol{\xi}}$ is a random variable such that $\mathbb{E} {\boldsymbol{\xi}} = {\bf0}$ and $\mathbb{E} ({\boldsymbol{\xi}} {\boldsymbol{\xi}}\t) = \mathbf{I}$. Although stochastic optimization methods \cite{shapiro2009lectures} could be use here, we follow an approximation approach where we use a Hutchinson trace estimator. That is, we let ${\boldsymbol{\xi}}^{(i)} \in \mathbb{R}^n$ for $i=1,2,\ldots,M$ be realizations of a Rademacher distribution (i.e., ${\boldsymbol{\xi}}$ consists of $\pm1$ with equal probability), and we consider the approximate optimization problem,
\begin{equation}
\label{ref:RademacherEst}
(\check{\nu},\check{\ell}) =\mathrm{argmin}_{\nu>0,\ell>0}
\frac{1}{M}\sum_{i=1}^{M} \|(\mathbf{Q}(\nu,\ell) - \widehat\mathbf{Q}){\boldsymbol{\xi}}^{(i)} \|^2_2.
\end{equation}
We mention that for problems without training data, semivariogram hyperparameters were investigated in \cite{bardsley2018semivariogram} to estimate Mat{\' e}rn parameters from the data.
\section{Hybrid projection methods for mixed Gaussian priors}
\label{sec:agenGK}
In this section, we describe a hybrid projection method to approximate the MAP estimate~\eqref{eq:MAP}. The distinguishing factor of this approach compared to generalized Golub-Kahan (genGK) hybrid methods \cite{chung2017generalized} is that we address problems where the prior covariance matrix is of the form~\eqref{eq:Qsum}. That is, we consider priors of the form $\mathbf{s} \sim\mathcal{N}({\boldsymbol{\mu}}, \lambda^{-2}(\gamma \mathbf{Q}_1 + (1-\gamma) \mathbf{Q}_2))$, and exploit a hybrid projection framework to enable tools for selecting both the regularization parameter $\lambda$ and the mixing parameter $\gamma$ simultaneously.
Using the following change of variables,
$$\mathbf{x} = \mathbf{Q}^{-1}(\mathbf{s} -{\boldsymbol{\mu}}), \quad \mathbf{b} = \mathbf{d} - \mathbf{A} {\boldsymbol{\mu}},$$
we see that solving~\eqref{eq:MAP} is equivalent to solving
\begin{equation}
\label{eq:transformed}
\min_\mathbf{x} \frac{1}{2}\norm[\mathbf{R}^{-1}]{\mathbf{A}\mathbf{Q}\mathbf{x} - \mathbf{b}}^2 + \frac{\lambda^2}{2} \norm[\mathbf{Q}]{\mathbf{x}}^2.
\end{equation}
If $\gamma$ is known in advance, we can directly apply the genGK hybrid method and estimate $\lambda$ automatically \cite{chung2017generalized}. However, in many cases, we don't know $\gamma$ in advance, so we want to estimate $\gamma$ during the iterative process. For this, we develop a variant of the genGK bidiagonlization which we call a \emph{mixed} Golub-Kahan (mixGK) process.
Each iteration of the mixGK process requires two steps. The first step is to run one iteration of the genGK bidiagonalization process with $\mathbf{Q}_1$. The second step incorporates $\mathbf{Q}_2$ so that the regularized problem can be iteratively projected onto a smaller subspace, and $\gamma$ and $\lambda$ can \textit{both} be selected automatically. Next we describe the mixGK process in detail.
Given matrices $\mathbf{A}$, $\mathbf{R}$, $\mathbf{Q}_1$, and vector $\mathbf{b},$ with initializations $\beta_1 = \norm[\mathbf{R}^{-1}]{\mathbf{b}}$, $\mathbf{u}_1 = \mathbf{b}/\beta_1$ and $\alpha_1 \mathbf{v}_1 = \mathbf{A}\t \mathbf{R}^{-1} \mathbf{u}_1$,
the $k$th iteration of the genGK bidiagonalization procedure with $\mathbf{Q}_1$ generates vectors $\mathbf{u}_{k+1}$ and $\mathbf{v}_{k+1}$ such that
\begin{align*}
\beta_{k+1} \mathbf{u}_{k+1} & = \mathbf{A} \mathbf{Q}_1 \mathbf{v}_k -\alpha_k \mathbf{u}_k\\
\alpha_{k+1} \mathbf{v}_{k+1} & = \mathbf{A}\t \mathbf{R}^{-1} \mathbf{u}_{k+1} -\beta_{k+1} \mathbf{v}_k,
\end{align*}
where scalars $\alpha_i, \beta_i \geq 0$ are chosen such that $\norm[\mathbf{R}^{-1}]{\mathbf{u}_i} = \norm[\mathbf{Q}_1]{\mathbf{v}_i} = 1$. At the end of $k$ steps, we have
\[ \mathbf{B}_k \equiv \> \begin{bmatrix}
\alpha_1 \\ \beta_2 & \alpha_2 \\ & \beta_3 & \ddots \\ & & \ddots & \alpha_k \\ & & & \beta_{k+1}
\end{bmatrix}\,, \qquad \mathbf{U}_{k+1} \equiv [\mathbf{u}_1,\dots,\mathbf{u}_{k+1}],\quad \mbox{and} \quad \mathbf{V}_k \equiv [\mathbf{v}_1,\dots,\mathbf{v}_k],\]
where the following relations hold up to machine precision,
\begin{align}\label{e_bk}
\mathbf{U}_{k+1}\beta_1 \mathbf{e}_1 = &\> \mathbf{b} \\ \label{e_vk}
\mathbf{A} \mathbf{Q}_1 \mathbf{V}_k = & \>\mathbf{U}_{k+1} \mathbf{B}_k \\ \label{e_uk}
\mathbf{A}\t \mathbf{R}^{-1} \mathbf{U}_{k+1} = & \> \mathbf{V}_k \mathbf{B}_k\t + \alpha_{k+1}\mathbf{v}_{k+1}\mathbf{e}_{k+1}\t\,.
\end{align}
Furthermore, in exact arithmetic, matrices $\mathbf{U}_{k+1}$ and $\mathbf{V}_k$ satisfy the following orthogonality conditions
\begin{equation}
\label{eq:orthog}
\mathbf{U}_{k+1}\t \mathbf{R}^{-1} \mathbf{U}_{k+1} = \mathbf{I}_{k+1} \qquad \mbox{and} \qquad \mathbf{V}_k\t \mathbf{Q}_1 \mathbf{V}_k = \mathbf{I}_k.
\end{equation}
If we let $\widetilde{\mathbf{U}}_{k+1} = \mathbf{L}_{\mathbf{R}}\mathbf{U}_{k+1}$ where $\mathbf{R}^{-1} = \mathbf{L}\t_{\mathbf{R}}\mathbf{L}_{\mathbf{R}}$, then $\widetilde{\mathbf{U}}\t_{k+1}\widetilde{\mathbf{U}}_{k+1} = \mathbf{I}_{k+1}$.
Next, in order to incorporate $\mathbf{Q}_2$, we additionally compute $m \times k$ matrix $\mathbf{L}_{\mathbf{R}}\mathbf{A} \mathbf{Q}_2 \mathbf{V}_k$. Assuming that the columns of $\widetilde{\mathbf{U}}_{k+1}$ and $\mathbf{L}_{\mathbf{R}}\mathbf{A} \mathbf{Q}_2 \mathbf{V}_k$ are linearly independent, we can compute the skinny QR factorization, $(\mathbf{I} - \widetilde{\mathbf{U}}_{k+1} \widetilde{\mathbf{U}}_{k+1}\t)\mathbf{L}_{\mathbf{R}}\mathbf{A} \mathbf{Q}_2 \mathbf{V}_k = \mathbf{Y}_k \mathbf{R}_k$ where $\mathbf{Y}_{k}\in\mathbb{R}^{m \times k}$
contains orthonormal columns and $\mathbf{R}_k \in \mathbb{R}^{k\times k }$ is upper triangular. Notice that since column vectors in $\mathbf{Y}_k$ and $\widetilde \mathbf{U}_{k+1}$ are orthogonal, we get the skinny QR factorization,
\begin{equation}
\label{eq:skinnyQR}
\begin{bmatrix}
\widetilde{\mathbf{U}}_{k+1} & \mathbf{L}_{\mathbf{R}}\mathbf{A} \mathbf{Q}_2 \mathbf{V}_k
\end{bmatrix} = \begin{bmatrix}
\widetilde{\mathbf{U}}_{k+1} & \mathbf{Y}_k
\end{bmatrix}
\begin{bmatrix}
\mathbf{I}_{k+1} & \widetilde{\mathbf{U}}_{k+1}\t \mathbf{L}_{\mathbf{R}}\mathbf{A} \mathbf{Q}_2 \mathbf{V}_k \\
{\bf0} & \mathbf{R}_k
\end{bmatrix}.
\end{equation}
The mixGK process is summarized in Algorithm~\ref{alg:agenGK}.
\begin{algorithm}[bthp]
\begin{algorithmic}[1]
\REQUIRE Matrices $\mathbf{A}$, $\mathbf{R}$, $\mathbf{Q}_1$ and $\mathbf{Q}_2$, and vector $\mathbf{b}$.
\STATE $\beta_1 \mathbf{u}_1 = \mathbf{b},$ where $\beta_1 = \norm[\mathbf{R}^{-1}]{\mathbf{b}}$
\STATE $\alpha_1 \mathbf{v}_1 = \mathbf{A}\t \mathbf{R}^{-1}\mathbf{u}_1$
\FOR {$k=1, 2, \dots$}
\STATE $\beta_{k+1}\mathbf{u}_{k+1} = \mathbf{A}{\mathbf{Q}_1}\mathbf{v}_k - \alpha_k \mathbf{u}_k$, where $\beta_{k+1} = \norm[\mathbf{R}^{-1}]{\mathbf{A}{\mathbf{Q}_1}\mathbf{v}_k - \alpha_k \mathbf{u}_k}$
\STATE $\alpha_{k+1}\mathbf{v}_{k+1} = \mathbf{A}\t \mathbf{R}^{-1} \mathbf{u}_{k+1} - \beta_{k+1} \mathbf{v}_k$, where $\alpha_{k+1} = \norm[\mathbf{Q}_1]{\mathbf{A}\t \mathbf{R}^{-1} \mathbf{u}_{k+1} - \beta_{k+1} \mathbf{v}_k}$
\STATE $[\mathbf{Y}_k, \mathbf{R}_k] = qr((\mathbf{I} - \widetilde{\mathbf{U}}_{k+1}\widetilde{\mathbf{U}}_{k+1}\t)\mathbf{L}_{R}\mathbf{A} \mathbf{Q}_2 \mathbf{V}_k, 0);$
\ENDFOR
\end{algorithmic}
\caption{mixed Golub-Kahan (mixGK) process}
\label{alg:agenGK}
\end{algorithm}
Notice that in addition to the computational cost of the genGK bidiagonalization, which includes one matrix-vector product with $\mathbf{A}$, one with $\mathbf{A}\t$, two with $\mathbf{Q}_1$, and two solves with $\mathbf{R}$, each iteration of the mixGK process requires one matrix-vector product with $\mathbf{Q}_2$ and a QR factorization in step 6. Instead of performing a standard QR factorization on an $m$-by-$k$ matrix, an efficient rank-one update strategy can be used to alleviate the computational cost. More specifically, we will describe it using mathematical induction. Let
\begin{equation}
\label{eq:rankone}
(\mathbf{I} - \widetilde{\mathbf{U}}_{k}\widetilde{\mathbf{U}}_{k}\t)\mathbf{L}_{R}\mathbf{A} \mathbf{Q}_2 \mathbf{V}_{k-1} = \mathbf{Y}_{k-1}\mathbf{R}_{k-1}
\end{equation}
be the skinny QR factorization,
where $\mathbf{Y}_{k-1}\t \mathbf{Y}_{k-1} = \mathbf{I}_{k-1}$ and $\mathbf{R}_{k-1}$ is an upper triangular matrix.
Define $\widetilde{\mathbf{U}}_{k+1} = \begin{bmatrix}
\widetilde{\mathbf{U}}_{k} & \widetilde{\mathbf{u}}_{k+1}
\end{bmatrix}$ and $\mathbf{V}_{k} = \begin{bmatrix}
\mathbf{V}_{k-1} & \mathbf{v}_{k}
\end{bmatrix}$. Then by (\ref{eq:rankone}), we have
\begin{align*}
(\mathbf{I} - \widetilde{\mathbf{U}}_{k+1}\widetilde{\mathbf{U}}_{k+1}\t)\mathbf{L}_{R}\mathbf{A} \mathbf{Q}_2 \mathbf{V}_{k} &= \begin{bmatrix}
(\mathbf{I} - \widetilde{\mathbf{U}}_{k+1}\widetilde{\mathbf{U}}_{k+1}\t)\mathbf{L}_{R}\mathbf{A} \mathbf{Q}_2 \mathbf{V}_{k-1} & (\mathbf{I} - \widetilde{\mathbf{U}}_{k+1}\widetilde{\mathbf{U}}_{k+1}\t)\mathbf{L}_{R}\mathbf{A} \mathbf{Q}_2 \mathbf{v}_{k}
\end{bmatrix}\\
&=\begin{bmatrix}
(\mathbf{I} - \widetilde{\mathbf{U}}_{k}\widetilde{\mathbf{U}}_{k}\t - \widetilde{\mathbf{u}}_{k+1}\widetilde{\mathbf{u}}_{k+1}\t )\mathbf{L}_{R}\mathbf{A} \mathbf{Q}_2 \mathbf{V}_{k-1} & (\mathbf{I} - \widetilde{\mathbf{U}}_{k+1}\widetilde{\mathbf{U}}_{k+1}\t)\mathbf{L}_{R}\mathbf{A} \mathbf{Q}_2 \mathbf{v}_{k}
\end{bmatrix}\\
&=\begin{bmatrix}
\mathbf{Y}_{k-1}\mathbf{R}_{k-1} - \widetilde{\mathbf{u}}_{k+1}\widetilde{\mathbf{u}}_{k+1}\t \mathbf{Y}_{k-1}\mathbf{R}_{k-1} & (\mathbf{I} - \widetilde{\mathbf{U}}_{k+1}\widetilde{\mathbf{U}}_{k+1}\t)\mathbf{L}_{R}\mathbf{A} \mathbf{Q}_2 \mathbf{v}_{k}
\end{bmatrix}.
\end{align*}
Since the first matrix is a rank-one update of a QR factorization, its QR factorization can be obtained in
$\mathcal{O}(mk)$ operations \cite{daniel1976reorthogonalization}. That is, we have
$$\mathbf{Y}_{k-1}\mathbf{R}_{k-1} - \widetilde{\mathbf{u}}_{k+1} (\mathbf{R}_{k-1}\t\mathbf{Y}_{k-1}\t\widetilde{\mathbf{u}}_{k+1})\t = \widehat{\mathbf{Y}}_{k-1}\widehat{\mathbf{R}}_{k-1}$$
where $\widehat{\mathbf{Y}}_{k-1}\t\widehat{\mathbf{Y}}_{k-1} = \mathbf{I}_{k-1}$ and $\widehat{\mathbf{R}}_{k-1}$ is an upper triangular matrix.
Finally, let $\widehat{\mathbf{v}}_{k} = (\mathbf{I} - \widetilde{\mathbf{U}}_{k+1}\widetilde{\mathbf{U}}_{k+1}\t)\mathbf{L}_{R}\mathbf{A} \mathbf{Q}_2 \mathbf{v}_{k}$, then one step of the Gram-Schmidt process gives the desired QR factorization, $$\begin{bmatrix}
\widehat{\mathbf{Y}}_{k-1}\widehat{\mathbf{R}}_{k-1} & \widehat{\mathbf{v}}_{k}
\end{bmatrix} = {\mathbf{Y}}_{k}{\mathbf{R}}_{k}.$$
\subsection{Solving the projected problem}
Using the mixGK process described above, we now describe a hybrid iterative projection method to solve~\eqref{eq:transformed}. In particular, we consider the projected problem,
\begin{equation}
\label{prob:projected}
\min_{\mathbf{x} \in \mathcal{R}(\mathbf{V}_k)}\frac{1}{2} \norm[\mathbf{R}^{-1}]{\mathbf{A}\mathbf{Q}\mathbf{x} - \mathbf{b}}^2 + \frac{\lambda^2}{2} \norm[\mathbf{Q}]{\mathbf{x}}^2
\end{equation}
where $\mathcal{R}(\cdot)$ denotes the column space. Let $\mathbf{x} = \mathbf{V}_k \mathbf{y}$ where $\mathbf{y}\in\mathbb{R}^k$. Then using the relationships from the mixGK process, we obtain the equivalent problems,
\begin{align}
\min_\mathbf{y} & \frac{1}{2} \norm[\mathbf{R}^{-1}]{\gamma\mathbf{A} \mathbf{Q}_1 \mathbf{V}_k \mathbf{y} + (1-\gamma) \mathbf{A} \mathbf{Q}_2 \mathbf{V}_k \mathbf{y} - \mathbf{b}}^2 + \frac{\lambda^2}{2} \mathbf{y}\t \mathbf{V}_k\t (\gamma\mathbf{Q}_1+ (1-\gamma) \mathbf{Q}_2) \mathbf{V}_k \mathbf{y}\\
\min_\mathbf{y} & \frac{1}{2}\norm[2]{\gamma\widetilde{\mathbf{U}}_{k+1} \mathbf{B}_k \mathbf{y} + (1-\gamma) \mathbf{L}_{\mathbf{R}}\mathbf{A} \mathbf{Q}_2 \mathbf{V}_k \mathbf{y} - \mathbf{L}_{\mathbf{R}}\mathbf{b}}^2 + \frac{\lambda^2\gamma}{2} \mathbf{y}\t \mathbf{y} + \frac{\lambda^2 (1-\gamma)}{2} \mathbf{y}\t \mathbf{V}_k\t \mathbf{Q}_2 \mathbf{V}_k \mathbf{y} \\
\min_\mathbf{y} & \frac{1}{2}\norm[2]{\begin{bmatrix}
\widetilde{\mathbf{U}}_{k+1} & \mathbf{L}_{\mathbf{R}}\mathbf{A} \mathbf{Q}_2 \mathbf{V}_k \end{bmatrix} \begin{bmatrix}\gamma\mathbf{B}_k \\ (1-\gamma) \mathbf{I}_k \end{bmatrix}\mathbf{y} - \mathbf{L}_{\mathbf{R}}\mathbf{b}}^2 + \frac{\lambda^2\gamma}{2} \norm[2]{\mathbf{y}}^2 + \frac{\lambda^2 (1-\gamma)}{2} \mathbf{y}\t \mathbf{V}_k\t \mathbf{Q}_2 \mathbf{V}_k \mathbf{y}.
\label{eq:notorthog}
\end{align}
Using equation (\ref{eq:skinnyQR}) and
the fact that
\begin{equation}
\begin{bmatrix}
\widetilde{\mathbf{U}}_{k+1} & \mathbf{Y}_{k}
\end{bmatrix} \begin{bmatrix}
\beta_1 \mathbf{e}_1 \\ \bf0
\end{bmatrix} = \widetilde{\mathbf{U}}_{k+1} (\beta_1 \mathbf{e}_1) = \mathbf{L}_{\mathbf{R}}\mathbf{b}
\end{equation}
where
$\begin{bmatrix}\widetilde{\mathbf{U}}_{k+1} & \mathbf{Y}_k \end{bmatrix}$ contains orthonormal columns (so it can be taken out of the norm), the projected, regularized problem becomes
\begin{equation}
\label{eq:projectedproblem}
\min_\mathbf{y} \frac{1}{2}\norm[2]{\begin{bmatrix}
\mathbf{I}_{k+1} & \widetilde{\mathbf{U}}_{k+1}\t \mathbf{L}_{\mathbf{R}}\mathbf{A} \mathbf{Q}_2 \mathbf{V}_k \\
{\bf0} & \mathbf{R}_k
\end{bmatrix}\begin{bmatrix}\gamma\mathbf{B}_k \\ (1-\gamma) \mathbf{I}_k \end{bmatrix}\mathbf{y} - \begin{bmatrix}
\beta_1 \mathbf{e}_1 \\ \bf0
\end{bmatrix}}^2 + \frac{\lambda^2\gamma}{2} \norm[2]{\mathbf{y}}^2 + \frac{\lambda^2 (1-\gamma)}{2} \mathbf{y}\t \mathbf{V}_k\t \mathbf{Q}_2 \mathbf{V}_k \mathbf{y}.
\end{equation}
Note that the solution subspace for $\mathbf{x}$ does not depend on $\gamma$ and $\lambda$, but the solution of the projection problem depends on both $\gamma$ and $\lambda$. Let $\mathbf{y}_k(\lambda,\gamma)$ denote the solution to~\eqref{eq:projectedproblem}, then the $k$ iterate of the mixGK method is given as
\begin{equation}
\label{eq:iterates}
\mathbf{s}_k (\lambda, \gamma) = {\boldsymbol{\mu}} + (\gamma\mathbf{Q}_1 + (1-\gamma) \mathbf{Q}_2) \mathbf{V}_k \mathbf{y}_k (\lambda,\gamma).
\end{equation}
In Section~\ref{sub:param} we describe some techniques for selecting $\lambda$ and $\gamma$ at each iteration, but first we provide a theoretical result. We show that for fixed regularization parameter $\lambda$ and fixed mixing parameter $\gamma$, the proposed mixGK method converges in exact arithmetic to the desired regularized solution.
\begin{theorem}
\label{thm:convergence}
Assume $\lambda> 0$ and $0<\gamma\leq 1$. Let $\mathbf{y}_k(\lambda, \gamma)$ be the exact solution to projected problem (\ref{eq:projectedproblem}). Then the kth iterate of the mixGK approach, written as
\begin{equation}
\mathbf{s}_k = {{\boldsymbol{\mu}}} + \mathbf{Q}\mathbf{V}_k\mathbf{y}_k(\lambda, \gamma)
\end{equation}
converges to the MAP estimate given by
\begin{equation}
\mathbf{s}_{\rm MAP} = {{\boldsymbol{\mu}}} + \mathbf{Q}(\mathbf{A}\t\mathbf{R}^{-1}\mathbf{A}\mathbf{Q} + \lambda^2\mathbf{I}_n)^{-1}\mathbf{A}\t\mathbf{R}^{-1}\mathbf{b}.
\end{equation}
\end{theorem}
\begin{proof}
The proof is provided in Appendix \ref{sec:appendix}.
\end{proof}
\subsection{Regularization parameter selection methods}
\label{sub:param}
In this section, we describe two extensions of existing regularization parameter selection methods that can be used for selecting $\gamma$ and $\lambda$ at each iteration of the mixGK hybrid method. Notice that the solution at the $k$-th iteration can be written as
\begin{equation}
\label{eq:regsoln}
\mathbf{s}_k (\lambda, \gamma) = {\boldsymbol{\mu}} + (\gamma\mathbf{Q}_1 + (1-\gamma) \mathbf{Q}_2) \mathbf{V}_k \mathbf{y}_k (\lambda,\gamma),
\end{equation}
where
\begin{equation}
\label{eq:proj-solution}
\begin{array}{rcl}
\mathbf{y}_k (\lambda,\gamma) & = & \left(\mathbf{D}_k(\gamma)\t \mathbf{D}_k(\gamma) + \lambda^2 \gamma \mathbf{I}_k + \lambda^2(1-\gamma)\mathbf{V}_k\t\mathbf{Q}_2\mathbf{V}_k \right)^{-1} \mathbf{D}_k(\gamma)\t \begin{bmatrix} \beta_1\mathbf{e}_1 \\ \textbf{0} \end{bmatrix} \\
& =& \mathbf{C}_k(\gamma, \lambda) \begin{bmatrix} \beta_1\mathbf{e}_1 \\ \textbf{0} \end{bmatrix}
\end{array}
\end{equation} with
\begin{align}
\label{eq:D}
\mathbf{D}_k(\gamma) & = \begin{bmatrix}
\mathbf{I}_{k+1}& \widetilde{\mathbf{U}}_{k+1}\t\mathbf{L}_{\mathbf{R}}\mathbf{A} \mathbf{Q}_2 \mathbf{V}_k \\ \textbf{0} & \mathbf{R}_k \end{bmatrix} \begin{bmatrix}\gamma\mathbf{B}_k \\ (1-\gamma) \mathbf{I}_k \end{bmatrix} = \begin{bmatrix}
\gamma\mathbf{B}_{k} + (1-\gamma)\widetilde{\mathbf{U}}_{k+1}\t\mathbf{L}_{\mathbf{R}}\mathbf{A} \mathbf{Q}_2 \mathbf{V}_k \\
(1-\gamma) \mathbf{R}_{k}
\end{bmatrix}\\
\mathbf{C}_k(\gamma, \lambda) & = \left(\mathbf{D}_k(\gamma)\t \mathbf{D}_k(\gamma) + \lambda^2\gamma\mathbf{I}_k + \lambda^2(1-\gamma)\mathbf{V}_k\t\mathbf{Q}_2\mathbf{V}_k \right)^{-1} \mathbf{D}_k(\gamma)\t.
\end{align}
As with regularization parameter selection methods for standard hybrid methods, there is not one method that will work for all problems, so it is advised to try various approaches in practice.
In order to provide a comparison, we provide ``optimal'' parameters which are computed as
\begin{equation}\label{eq:reg_opt}
(\gamma_{\rm opt}, \lambda_{\rm opt}) = \mathrm{argmin}_{0< \gamma \le 1,\,\lambda} \norm[2]{\mathbf{s}_k(\gamma, \lambda) - \mathbf{s}_{\rm true}}^2,
\end{equation}
where $\mathbf{s}_{\rm true}$ is the true solution (that is not available in practice).
\paragraph{Unbiased predictive risk estimation (UPRE).} We can select parameters $\gamma, \lambda$ such that \begin{equation}
(\gamma_{\rm u}^{\rm proj},\lambda_{\rm u}^{\rm proj}) = \mathrm{argmin}\limits_{0 <\gamma\le 1,\,\lambda} \mathcal{U}_{\rm proj}(\gamma,\lambda) = \dfrac{1}{2k+1}\|\mathbf{r}_k^{\rm proj}(\gamma,\lambda)\|^2_2 + \dfrac{2\sigma^2}{2k+1 } {\rm tr}(\mathbf{D}_k(\gamma)\mathbf{C}_k(\gamma,\lambda)) -\sigma^2
\label{eq:projUpre}
\end{equation} where $\sigma^2$ is noise level, and \begin{equation}
\label{eq:proj_res}
\mathbf{r}_k^{\rm proj}(\gamma,\lambda) = \mathbf{D}_k(\gamma) \mathbf{y}_k(\gamma,\lambda) - \begin{bmatrix} \beta_1\mathbf{e}_1 \\ \textbf{0}\end{bmatrix}
\end{equation}
and
\begin{equation}
\label{eq:trace_res}
\begin{array}{rcl}
{\rm tr}(\mathbf{D}_k(\gamma)\mathbf{C}_k(\gamma,\lambda)) & = & {\rm tr}(\mathbf{C}_k(\gamma,\lambda)\mathbf{D}(\gamma)) \\
& = & {\rm tr}(\left((\mathbf{D}_k(\gamma))\t \mathbf{D}_k(\gamma) + \lambda^2 \gamma \mathbf{I}_k + \lambda^2(1-\gamma)\mathbf{V}_k\t\mathbf{Q}_2\mathbf{V}_k \right)^{-1} (\mathbf{D}_k(\gamma))\t \mathbf{D}_k(\gamma)). \\
\end{array}
\end{equation} When the noise level $\sigma^2$ is not provided, a noise level estimation algorithm (e.g., based on a wavelet decomposition of the observation) can be utilized \cite{donoho1995denoising}.
\paragraph{Generalized cross validation (GCV).} Without a priori knowledge of the noise level, another option is to use an extension of the GCV method \cite{golub1979generalized,hansen2010discrete}. The basic idea is to select parameters, \begin{equation}\label{eq:proj_gcv}
(\gamma_{\rm g}^{\rm proj}, \lambda_{\rm g}^{\rm proj}) = \mathrm{argmin}\limits_{0 < \gamma \le 1,\,\lambda} {\cal G}_{\rm proj}(\gamma,\lambda) = \dfrac{\|\mathbf{r}_k^{\rm proj}(\gamma,\lambda)\|_2^2}{({\rm tr}(\mathbf{I}_{2k+1} - \mathbf{D}_k(\gamma)\mathbf{C}_k(\gamma,\lambda)))^2}
\end{equation} where $\mathbf{r}_k^{\rm proj}(\gamma,\lambda)$, $\mathbf{D}_k(\gamma)$, and $\mathbf{C}_k(\gamma,\lambda)$ are same as \eqref{eq:projUpre}. \\
Notice that $\mathbf{r}_k^{\rm proj}$ and ${\rm tr}(\mathbf{D}_k(\gamma)\mathbf{C}_k(\gamma,\lambda))$ are functions of $k$ in both the GCV and UPRE functions.
In order to prove convergence of the parameters chosen by UPRE and GCV, we begin with a lemma that shows convergence of the projected residual $\mathbf{r}_k^{\rm proj}$ and trace term ${\rm tr}(\mathbf{D}_k(\gamma)\mathbf{C}_k(\gamma,\lambda))$ to their full counterparts.
\begin{lemma}
\label{lemma:res_proj}
With \eqref{eq:proj_res}, \eqref{eq:trace_res}, if $k \rightarrow n$, then \begin{equation}
\begin{array}{rcl}
\mathbf{r}_k^{\rm proj}& \rightarrow & \mathbf{r}^{\rm full}(\gamma,\lambda)\\
{\rm tr}(\mathbf{D}_k(\gamma)\mathbf{C}_k(\gamma,\lambda)) & \rightarrow & {\rm tr}(A(\gamma,\lambda))
\end{array}
\end{equation} where \begin{equation}
\begin{array}{rcl}
\mathbf{r}^{\rm full}(\gamma,\lambda) & =& \mathbf{L}_{\mathbf{R}}\mathbf{A}\mathbf{Q}\mathbf{x}(\gamma,\lambda)-\mathbf{L}_{\mathbf{R}}\mathbf{b} \\
A(\gamma,\lambda) & = & \mathbf{L}_{\mathbf{R}}\mathbf{A}\mathbf{Q}(\mathbf{Q}\t\mathbf{A}\t\mathbf{R}^{-1}\mathbf{A}\mathbf{Q} + \lambda^2\mathbf{Q})^{-1}\mathbf{Q}\t\mathbf{A}\t\mathbf{L}_{\mathbf{R}}\t.
\end{array}
\end{equation} and $\mathbf{r}^{\rm full}(\gamma,\lambda)=\mathbf{r}_n^{\rm proj}(\gamma,\lambda)$.
\end{lemma}
\begin{proof}
The proof is provided in Appendix \ref{sec:appendix2}.
\end{proof}
Next we provide convergence results for the UPRE and GCV selected parameters that are similar to results provided in \cite{renaut2017hybrid} but are extended to the mixed hybrid methods. In particular, we show in Theorem \ref{thm:Convergence_U_G} that the UPRE parameters
for the projected problem converge to the UPRE parameters for the full problem. Then, we show that with an additional weighting parameter, the same result holds for GCV parameters.
\begin{theorem}
\label{thm:Convergence_U_G}
From \eqref{eq:transformed}, the UPRE for the full problem is given \begin{equation}
(\gamma_{\rm u}^{\rm full},\lambda_{\rm u}^{\rm full}) = \mathrm{argmin}\limits_{0 < \gamma \le 1,\,\lambda}\mathcal{U}_{\rm full}(\gamma,\lambda) = \dfrac{1}{m}\|\mathbf{r}^{\rm full}(\gamma,\lambda)\|^2_2+\dfrac{2\sigma^2}{m}{\rm tr}(A(\gamma,\lambda))-\sigma^2.
\label{eq:fullUpre}
\end{equation} Then, \begin{equation}
(\gamma_{\rm u}^{\rm proj}, \lambda_{\rm u}^{\rm proj}) \rightarrow (\gamma_{\rm u}^{\rm full}, \lambda_{\rm u}^{\rm full})
\end{equation} as $k\rightarrow n$.
\end{theorem}
\begin{proof}
Since $\|\mathbf{r}^{\rm proj}_k\|_2^2 \rightarrow \|\mathbf{r}^{\rm full}\|^2_2$ and ${\rm tr}(\mathbf{D}_k(\gamma)\mathbf{C}_k(\gamma,\lambda)) \rightarrow {\rm tr}(A(\gamma,\lambda))$ as shown in Lemma \ref{lemma:res_proj}, \begin{equation*}
\mathrm{argmin}_{0 < \gamma \le 1,\,\lambda}{\cal U}_{\rm proj}(\gamma,\lambda) \rightarrow \mathrm{argmin}_{0 < \gamma \le 1,\,\lambda}{\cal U}_{\rm full}(\gamma,\lambda)
\end{equation*} as $k\rightarrow n$ for the same noise level $\sigma^2$.
\end{proof}
For the full problem, the GCV parameters are given by \begin{equation}
(\gamma_{\rm g}^{\rm full},\lambda_{\rm g}^{\rm full}) = \mathrm{argmin}\limits_{0 < \gamma \le 1,\,\lambda} {\cal G}_{\rm full}(\gamma,\lambda) = \dfrac{\|\mathbf{r}^{\rm full}(\gamma,\lambda)\|_2^2}{({\rm tr}(\mathbf{I}_m - A(\gamma,\lambda)))^2}.
\label{eq:fullGCV}
\end{equation} In contrast with UPRE, $(\gamma_{\rm g}^{\rm proj}, \lambda_{\rm g}^{\rm proj})$ does not minimize \eqref{eq:fullGCV} as $k\rightarrow n$ because the trace of $\mathbf{I}_{2k+1} -\mathbf{D}_k(\gamma)\mathbf{C}_k(\gamma,\lambda)$ does not converge to the trace of $\mathbf{I}_m-A(\gamma,\lambda)$. To compensate for this, we include an additional parameter $\omega$ in \eqref{eq:proj_gcv} as, \begin{equation}
\label{eq:proj_wgcv}
(\lambda^{\rm proj}_{\rm w},\gamma^{\rm proj}_{\rm w})=\mathrm{argmin}_{0 < \gamma \le 1,\,\lambda}{\cal W}(\gamma,\lambda)_{\rm proj}=\frac{\|\mathbf{r}_k^{\rm proj}(\gamma,\lambda)\|^2_2}{({\rm tr} (\mathbf{I}_{2k+1}-\omega\mathbf{D}_k(\gamma)\mathbf{C}_k(\gamma,\lambda)))^2}
\end{equation} where $\omega = \frac{2k+1}{m}$. Since \begin{equation}
({\rm tr} (\mathbf{I}_{2k+1}-\omega\mathbf{D}_k(\gamma)\mathbf{C}_k(\gamma,\lambda)))^2 = \frac{2k+1}{m}({\rm tr} (\mathbf{I}_{m}-\mathbf{D}_k(\gamma)\mathbf{C}_k(\gamma,\lambda)))^2,
\end{equation} ${\cal G}_{\rm full}(\gamma,\lambda)$ is minimized by $(\lambda^{\rm proj}_{\rm w},\gamma^{\rm proj}_{\rm w})$ as $k\rightarrow n$. Similar modified GCV functions were considered in \cite{chung2008weighted, renaut2017hybrid}.
\section{Numerical results} \label{sec:numerics}
In this section, we provide various numerical results from tomography to investigate our proposed hybrid method based on the mixGK process, which we denote as `mixHyBR'. First, in Section \ref{sec:numericEx1} we investigate data-driven mixed Gaussian priors where we assume that training data are available, and we compare various hybrid methods to existing shrinkage algorithms. Then, we consider a seismic crosswell tomography reconstruction problem in Section \ref{sec:numericEx2}, where we show that using a combination of covariance kernels can result in improved reconstructions. For the stopping criteria for mixHyBR, we use a combination of approaches described in \cite{chung2008weighted, chung2015hybrid, chung2017generalized}, where the iterative process is terminated if either of the following three criteria is satisfied: (i) a maximum number of iterations is reached, (ii) depending on the chosen regularization parameter selection method,
the function (\ref{eq:projUpre}) for UPRE,
(\ref{eq:proj_gcv}) for GCV, or (\ref{eq:proj_wgcv}) for WGCV attains a minimum or flattens out,
and (iii) tolerances on residuals are achieved.
\subsection{Spherical tomography example} \label{sec:numericEx1}
For our first example, we use a spherical means tomography reconstruction problem from the IRTools toolbox \cite{IRtools,hansen2018air}. Such models are often used in imaging problems from photoacoustic or optoacoustic imaging, which is a non-ionizing biomedical imaging modality.
The true image $\mathbf{s}_{\rm true}$ consists of $128 \times 128$ pixels, and the forward model matrix $\mathbf{A}$ represents a ray-tracing operation along semi-circle curves where the angle of centers range from $0^\circ$ to $90^\circ$ at steps of $(90/64)^\circ$. The number of circles at each angle is $90$. Thus the dimension of $\mathbf{A}$ is $5,760\times16,384$ and the sinogram is $90 \times 64$. The simulated observed sinogram was obtained as in (\ref{eq:problem}), where we have included $3\%$ additive Gaussian white noise, i.e., $\frac{\norm{{\boldsymbol{\epsilon}}}}{\norm{\mathbf{A}\mathbf{s}_{\rm true}}} = 0.03$. Other conditions are chosen as the default settings provided by the toolbox; see \cite{IRtools} for details. In the left panel of Figure \ref{fig:SphericalForwardModel}, we provide the true image along with some of the integration curves.
\begin{figure}[b!]
\begin{center}
\begin{tabular}{ccc}
\raisebox{-.5\height}{\includegraphics[width=.5\textwidth]{SphericalForwardModel.eps}}&
\hspace{-0.16in}\includegraphics[width=.2\textwidth]{training1.eps}
& \hspace{0.27in}\includegraphics[width=.2\textwidth]{training2.eps}\\[-16.5ex]
& \hspace{-0.16in}\includegraphics[width=.2\textwidth]{training3.eps}&
\hspace{0.27in}\includegraphics[width=.2\textwidth]{training4.eps}
\end{tabular}
\caption{Spherical tomography example. On the left, the true image is provided, along with a few of the integration curves whose centers are located at $45^\circ$. Four sample images from the training dataset are provided on the right.}\label{fig:SphericalForwardModel}
\end{center}
\end{figure}
Next, we assume that we have a dataset of training images for this problem consisting of $49$ images; four of the training images are provided in the right panel of Figure \ref{fig:SphericalForwardModel}. All of the images contain a circular mask to denote the region of interest or region of visibility. The inner regions of the images are generated using a linear combination of sine-squared functions, where the coefficients are random numbers uniformly distributed between $0.5$ and $1$, and the random numbers in sine-squared functions are uniformly distributed between $0$ and $128$. Furthermore, each image is contaminated by at most $8$ ``freckles'' generated as white disks, where $5$ of them have radius $3$ and the rest have radius $4$. The freckles are randomly placed, where the origins of the freckles are uniformly distributed. Notice that the freckles do no appear in the true image.
Given the training dataset $\{\mathbf{s}^{(1)}, \ldots, \mathbf{s}^{(49)}\}$, we first compute the (vectorized) mean image $\bar{\mathbf{s}}$ and the sample covariance matrix $\widehat{\mathbf{Q}}$. Next, assuming that the prior covariance matrix represents a Mat\'ern kernel, we solve optimization problem \eqref{ref:RademacherEst} to obtain ``learned'' Mat\'ern parameters $\check{\nu}$ and $\check{\ell}$ and consider the covariance matrix $\mathbf{Q}_{\text{learn}} = \mathbf{Q}(\check{\nu},\check{\ell}).$
\begin{figure}[b!]
\centering
\begin{tabular}{cc}
\includegraphics[width=0.45\textwidth]{opt-rel.eps} &
\includegraphics[width=0.45\textwidth]{upre-rel.eps} \\
\includegraphics[width=0.45\textwidth]{gcv-rel.eps} &
\includegraphics[width=0.45\textwidth]{wgcv-rel.eps}\\
\end{tabular}
\caption{Comparison of relative reconstruction error norms for various iterative hybrid approaches for spherical tomography reconstruction. The top left plot corresponds to using the optimal regularization parameters. Other plots correspond to different methods to choose the regularization parameters, including UPRE, GCV, and WGCV.}
\label{fig:other_com}
\end{figure}
We consider four hybrid iterative reconstruction methods, all with initial vector $\bar{\mathbf{s}}$. Given the training data, we run the genHyBR algorithm with $\mathbf{Q} = \mathbf{Q}_{\text{learn}}$ which we denote as `genHyBR-data-driven'. We also provide results for `mixHyBR' where $\mathbf{Q} = \gamma\mathbf{Q}_{\text{learn}} + (1-\gamma)\widehat{\mathbf{Q}}$ where $\gamma$ and $\lambda$ are selected during the iterative process.
For comparison, we provide results for genHyBR with $\mathbf{Q} = \gamma\mathbf{I} + (1-\gamma)\widehat{\mathbf{Q}}$ where $\gamma$ was pre-selected using the Rao-Blackwell Ledoit and Wolf estimator (rblw) \cite{LW04, chen2009shrinkage,chen2010shrinkage}. We also provide results for HyBR where $\mathbf{Q} = \mathbf{I}$, but remark that this approach only uses the training data for the initial (sample mean) vector. Note that for all considered methods, the regularization parameter $\lambda$ must be selected, and we investigate various approaches to do this.
In Figure \ref{fig:other_com}, we provide relative reconstruction error norms computed as $\norm[2]{\mathbf{s}_k - \mathbf{s}_{\rm true}}/\norm[2]{\mathbf{s}_{\rm true}},$ where $\mathbf{s}_k$ is the reconstruction at the $k$th iteration. Each plot corresponds to a different method for selecting the regularization parameters. For comparison, we provide in the top left plot results corresponding to the optimal regularization parameter, although these parameters cannot be computed in practice.
We observe that both genHyBR-data-driven and mixHyBR result in small error norms and that even with the optimal regularization parameter $\lambda$, the rblw approach performs poorly because of the poorly-estimated mixing parameter $\gamma.$
We remark that we also compared these results to a shrinkage algorithm based on the oracle approximating shrinkage (OAS) estimator \cite{chen2009shrinkage,chen2010shrinkage} for obtaining $\gamma$. However, we observed very similar results as rblw, so we do not include them here.
For the automatic parameter selection methods, we observe that mixHyBR reconstructions with GCV and WGCV and genHyBR-data-driven reconstructions with UPRE have the smallest relative reconstruction error norms per iteration, compared to the other methods. Thus, we observe that including a data-driven covariance matrix, if done properly, can be beneficial. The black dots denote the (automatically-selected) stopping iteration for mixHyBR. Although one may wish to tweak the stopping criteria, all of the examples with mixHyBR resulted in a good reconstruction with the described stopping criteria. For a better comparison of the different parameter selection methods, we provide all relative reconstruction errors for mixHyBR in Figure \ref{fig:compare_com}, where it is evident that relative errors for WGCV are very close to those for the optimal regularization parameter for this example.
\begin{figure}[t!]
\centering
\includegraphics[height=8cm, width=.8\textwidth]{compare-rel.eps}
\caption{Relative reconstruction error norms per iteration of mixHyBR, for various regularization parameter choice methods. Black dots denote the automatically computed stopping iteration.}
\label{fig:compare_com}
\end{figure}
\begin{figure}[t!]
\centering
\includegraphics[width=1\textwidth]{optmix-recon.eps}
\caption{Absolute error images (in inverted colormap), with relative reconstruction error norms provided in the titles. The top row compares reconstructions using optimal regularization parameters, and the bottom row compares mixHyBR reconstructions with different parameter choice methods.}
\label{fig:recon1}
\end{figure}
Absolute error images, computed as $|\mathbf{s}_k - \mathbf{s}_{\rm true}|$, reshaped as an image, and displayed in inverted colormap, are provided in Figure \ref{fig:recon1}. For better comparison, all error images have been put on the same scale, and dark regions corresponds to larger absolute errors. Relative reconstruction error norms are provided in the titles. In the top row, we compare reconstructions at iteration $140$ using the optimal regularization parameter. Absolute error images in the bottom row correspond to mixHyBR reconstructions with automatic regularization parameter selection and correspond to the iteration determined by the stopping criteria. We notice that even with the optimal regularization parameter, the HyBR-optimal reconstruction suffers from the lack of sufficient prior information and the rblw-optimal reconstruction contains large errors due to the poor choice of $\gamma$ and disruptions due to freckles in the training data.
The mixHyBR and genHyBR-data-driven reconstructions have overall smaller absolute errors in the image. For this example, all parameter selection methods combined with the stopping iteration performed reasonably well.
\subsection{Seismic tomography example}
\label{sec:numericEx2}
In this experiment, we consider a linear inversion problem from crosswell tomography \cite{ambikasaran2013large}. Crosswell tomography is used to image the seismic wave speed in some region of interest, given data collected from multiple source-receiver pairs. The sources send out a seismic wave, and the receivers measure the travel time taken by the seismic wave to hit the receiver. The goal of the inverse problem is to image the slowness (reciprocal wave velocity) of the medium in the domain. We consider an example from Continuous Active Source Seismic Monitoring (CASSM) \cite{Daley2011}, where the goal is to monitor the spatial development of a small scale injection of CO$_2$ into a high quality reservoir. We consider reconstruction at a single time point and investigate the impact of including mixed Gaussian priors on the reconstruction.
\begin{figure}[b!]
\begin{center}
\begin{tabular}{cc}
\includegraphics[width=0.4\textwidth]{true_img2.eps} \quad & \quad
\includegraphics[width=0.15\textwidth]{observed_cassm.eps}
\end{tabular}
\end{center}
\caption{CASSM example. In the left panel, we provide the true slowness field image, along with some of the locations of the sources and the detectors. Seven of the source-receiver pairs are highlighted in the figure.
In the right panel, we provide the observations corresponding to $20$ sources and $50$ receivers.}
\label{fig:CASSM_seismic_true}
\end{figure}
The inverse problem can be represented as (\ref{eq:problem}) where the goal is to reconstruct the slowness $\mathbf{s} \in \mathbb{R}^{n\times1}$ of the medium from the measured travel times $\mathbf{d} \in \mathbb{R}^{m\times1}$ which are assumed to be corrupted by Gaussian white noise ${\boldsymbol{\epsilon}} \in \mathbb{R}^{m\times1}$.
In our problem setup, the true slowness field was discretized into $n=188,356$ cells, where the slowness within each cell is assumed to be constant. The true image (normalized between $0$ and $1$) is of size $434 \times 434$ and was obtained from \cite{cassmrawdata}.
For the observations, there were $m_s = 20$ sources and $m_r = 50$ receivers, so a total of $m = m_r m_s$ measurements. Each row of the forward model matrix $\mathbf{A} \in \mathbb{R}^{m\times n}$ corresponds to a source-receiver pair. Since the wave travels along a straight line from source to receiver, only the cells lying on the straight line contribute to the non-zero entries. Hence, $\mathbf{A}$ is very sparse with $\mathcal{O}(\sqrt{m}n)$ non-zero entries. The true image along with a schematic of the source-detector pairs are given in the left panel of Figure \ref{fig:CASSM_seismic_true}. The observations, which contain $1\%$ noise, are provided in the right panel of Figure \ref{fig:CASSM_seismic_true}.
\begin{figure}[t!]
\begin{center}
\includegraphics[height=6cm,width=.8\textwidth]{CASSM_RAW_ErrorPlot_opt.eps}
\end{center}
\caption{Comparison of relative reconstruction error norms for genHyBR and mixHyBR with optimal parameters $\gamma$ and $\ell$.}
\label{fig:CASSM_opt}
\end{figure}
\begin{figure}[b!]
\includegraphics[width=\textwidth]{CASSM_RAW_Recons_opt.eps}
\caption{Reconstructions with zoomed subimages for CASSM example. All of the reconstructions use the optimal regularization parameter and relative reconstruction errors are provided in the titles.}
\label{fig:CASSM_seismic_recons_opt}
\end{figure}
Next we investigate the impact of different choices of $\mathbf{Q}$ on the reconstruction. First, we consider the genHyBR method with three different prior covariance matrices $\mathbf{Q}_1, \mathbf{Q}_2,$ and $\mathbf{Q}_3$ defined by a Mat\'{e}rn kernel with $\nu=0.5$ and $\ell=.25$, a rational quadratic with $\nu=2$ and $\ell=0.1$, and a sinc function with $\nu = 30\pi$, respectively. These approaches are denoted by `genHyBR1', `genHyBR2', and `genHyBR3' respectively. Then we consider two mixHyBR approaches that include mixed Gaussian priors, where $\text{mixHyBR}(Q_1,Q_2)$ uses covariance matrix $\mathbf{Q} = \gamma\mathbf{Q}_1+(1-\gamma)\mathbf{Q}_2$ and
$\text{mixHyBR}(Q_1,Q_3)$ uses covariance matrix $\mathbf{Q} = \gamma\mathbf{Q}_1+(1-\gamma)\mathbf{Q}_3$, where the mixing parameter $\gamma$ is selected during the reconstruction process. For the optimally selected regularization parameters, we provide in Figure \ref{fig:CASSM_opt} the relative reconstruction error norms per iteration.
We observe that if a good covariance matrix (in this case, $\mathbf{Q}_1$) is known in advance, stand-alone genHyBR can perform well and result in small relative reconstruction errors. Otherwise, the relative reconstruction errors may remain large, and multiple solves with different covariance matrices would be needed to determine a good prior. In this case, the mixHyBR approach can prove beneficial. The mixHyBR approaches produce reconstructions with overall smaller relative reconstruction errors than genHyBR with each covariance matrix alone.
Image reconstructions, including a zoomed subregion, are provided in Figure \ref{fig:CASSM_seismic_recons_opt}. Notice that the mixed Gaussian priors are better able to resolve some details of the true image.
Thus, incorporating mixed Gaussian priors can lead to improved reconstructions.
\begin{figure}[t!]
\begin{center}
\begin{tabular}{cc}
\includegraphics[width=0.48\textwidth]{CASSM_RAW_ErrorPlot_mix1_new.eps}&
\includegraphics[width=0.48\textwidth]{CASSM_RAW_ErrorPlot_mix2.eps}
\end{tabular}
\end{center}
\caption{Comparison of relative reconstruction errors for $\text{mixHyBR}(Q_1,Q_2)$ (left) and $\text{mixHyBR}(Q_1,Q_3)$ (right) for different parameter choice methods. The automatically detected stopping iteration is marked with a black circle.}
\label{fig:CASSM_seismic_true_forward_err}
\end{figure}
Next we investigate the performance of different regularization parameter selection methods within the mixHyBR methods. Relative reconstruction errors for the GCV, WGCV, and UPRE methods with stopping iterates are provided in Figure \ref{fig:CASSM_seismic_true_forward_err}, along with results for the optimal parameters. We used a tolerance of $10^{-6}$ for the residual errors. We observe that all of the parameter selection methods work well for this example.
\section{Conclusions}
\label{sec:conclusions}
This paper describes a hybrid iterative projection method, dubbed mixHyBR, that is based on an extensions of the generalized Golub-Kahan bidiagonalization and that can be used for solving inverse problems (i.e., computing MAP estimates) with mixed Gaussian priors. The main advantage of this approach is that the mixing or blending parameter does not need to be known a priori, but rather can be estimated during the iterative process along with the regularization parameter. Various methods for selecting these parameters were considered and evaluated. Furthermore, mixHyBR methods can easily incorporate data-driven priors where training data are used to define the prior covariance matrix itself (e.g., sample based priors) or to learn parameters for the covariance kernel function. Comparisons to widely-used shrinkage algorithms reveal that the mixed hybrid approaches are more robust under the presence of noise or artifacts in the data and enable greater flexibility when selecting suitable priors. Numerical results from both spherical and seismic tomography show the potential of these methods.
\section*{Acknowledgments}
This work was partially supported by NSF DMS 1723005 and NSF DMS 1654175. J. Chung would also like to acknowledge support from the Alexander von Humboldt Foundation.
\bibliographystyle{siamplain}
| {
"timestamp": "2020-04-01T02:02:13",
"yymm": "2003",
"arxiv_id": "2003.13766",
"language": "en",
"url": "https://arxiv.org/abs/2003.13766",
"abstract": "When solving ill-posed inverse problems, a good choice of the prior is critical for the computation of a reasonable solution. A common approach is to include a Gaussian prior, which is defined by a mean vector and a symmetric and positive definite covariance matrix, and to use iterative projection methods to solve the corresponding regularized problem. However, a main challenge for many of these iterative methods is that the prior covariance matrix must be known and fixed (up to a constant) before starting the solution process. In this paper, we develop hybrid projection methods for inverse problems with mixed Gaussian priors where the prior covariance matrix is a convex combination of matrices and the mixing parameter and the regularization parameter do not need to be known in advance. Such scenarios may arise when data is used to generate a sample prior covariance matrix (e.g., in data assimilation) or when different priors are needed to capture different qualities of the solution. The proposed hybrid methods are based on a mixed Golub-Kahan process, which is an extension of the generalized Golub-Kahan bidiagonalization, and a distinctive feature of the proposed approach is that both the regularization parameter and the weighting parameter for the covariance matrix can be estimated automatically during the iterative process. Furthermore, for problems where training data are available, various data-driven covariance matrices (including those based on learned covariance kernels) can be easily incorporated. Numerical examples from tomographic reconstruction demonstrate the potential for these methods.",
"subjects": "Numerical Analysis (math.NA)",
"title": "Hybrid Projection Methods for Large-scale Inverse Problems with Mixed Gaussian Priors",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES\n\n",
"lm_q1_score": 0.9854964173268184,
"lm_q2_score": 0.7185943925708562,
"lm_q1q2_score": 0.7081721993897201
} |
https://arxiv.org/abs/2201.02865 | Further results on angular equivalence of norms | Angular equivalence of norms is introduced by Kikianty and Sinnamon (2017) and is a stronger notion than the usual topological equivalence. Given two angularly equivalent norms, if one norm has a certain geometrical property, e.g. uniform convexity, then the other norm also possesses such a property. In this paper, we show further results in this direction, namely angular equivalent norms share the property of uniform non-squareness, and that angular equivalence preserves the exposed points of the unit ball. A discussion on the (equivalence of the) dual norms of angularly equivalent norms is also given, giving a partial answer to an open problem as stated in the paper by Kikianty and Sinnamon (2017). | \section{Introduction}
In the paper \cite{angular}, a new notion of norm equivalence, namely angular equivalence, is introduced. Two norms are angularly equivalent on a real vector space, if over all pairs of nonzero vectors, the angle of the pair with respect to one norm is comparable to the angle of the same pair with respect to the other norm. Any two norms that are angularly equivalent are also topologically equivalent. Angular equivalence preserves certain properties, e.g. uniform convexity, that the usual equivalence does not.
\medskip
One needs a concept of angle in normed space to define such an equivalence. In a real normed space $(X, \norm{\cdot})$, the mapping $g^\pm \colon X\times X\to \mathbb{R}$ given by
$$g^\pm(x,y):=\|x\|\lim_{t\rightarrow 0^\pm} \frac1t\left(\|x+ty\|-\|x\|\right)$
exists. The $g$-functional relative to $\norm{\cdot}$ is defined as the map $g\colon X\times X\to \mathbb{R}$ given by
$$g(x,y):=\frac12(g^+(x,y)+g^-(x,y)), \quad x,y\in X. $$
We note that $g$ is not symmetric in general. If $x$ and $y$ are non-zero vectors in $X$, the norm angle from $x$ to $y$ is $\theta=\theta(x,y)$, defined by $0\leq \theta\leq \pi$ and
$$\cos\theta(x,y)=\frac{g(x,y)}{\|x\|\|y\|}.$$
\medskip
\noindent With this norm angle, angular equivalence is defined as follows.
\begin{definition}[Kikianty and Sinnamon \cite{angular}]
Two norms $\norm{\cdot}_1$ and $\norm{\cdot}_2$, on a real vector space $X$ are angularly equivalent provided there exists a constant $C$ such that for all non-zero $x,y\in X$
$$\tan\left(\frac{\theta_2(x,y)}{2}\right)\leq C \tan\left(\frac{\theta_1(x,y)}{2}\right).$$
Here $\theta_1(x,y)$ and $\theta_2(x,y)$ are the norm angles from $x$ to $y$ relative to $\norm{\cdot}_1$ and $\norm{\cdot}_2$, respectively. Also $\tan(\pi/2)$ is taken to be $+\infty$.
\end{definition}
\noindent It is straightforward to see that angular equivalence is both reflexive and transitive. Despite appearances, angular equivalence is a symmetric relation (cf. \cite[p. 944]{angular}) and thus it is an equivalence relation. In what follows, we recall some results concerning angular equivalence, specifically the preservation of geometrical properties by this equivalence. For further results, we refer the readers to the paper \cite{angular}.
\begin{proposition}[Kikianty and Sinnamon \cite{angular}]\label{prop:results}
Let $\norm{\cdot}_1$ and $\norm{\cdot}_2$ be two angularly equivalent norms on the real vector space $X$. Then, the following statements are true.
\begin{enumerate}[$(\mathrm{AE}1)$]
\item Both norms $\norm{\cdot}_1$ and $\norm{\cdot}_2$ are topologically equivalent.
\item The norm $\norm{\cdot}_1$ is induced by an inner product if and only if $\norm{\cdot}_2$ is induced by an inner product.
\item For $0\neq x\in X$, then $x/\norm{x}_1$ is an extreme point of $B_{(X,\norm{\cdot}_1)}$ if and only if $x/\norm{x}_2$ is an extreme point of $B_{(X,\norm{\cdot}_2)}$.
\item The space $(X,\norm{\cdot}_1)$ is strictly convex (uniformly convex), if and only if $(X,\norm{\cdot}_2)$ is strictly convex (uniformly convex).
\item If $p,q\in [1,\infty]$ and $n\in\mathbb{N}$ with $n\geq 2$, then the $\ell^p$ and $\ell^q$ norms on $\mathbb{R}^n$ are angularly equivalent, if and only if $p\neq q.$
\end{enumerate}
\end{proposition}
In this paper, we further showcase how angular equivalent norms share other geometrical properties, similar to results (AE3) and (AE4) in Proposition \ref{prop:results}. In Section \ref{section:unsq}, we see that angular equivalence also preserves uniform non-squareness, and in Section \ref{section:exposed}, we also show that angular equivalence preserve exposed points of a unit ball. In \cite{angular}, a counter example is given to the following question: If $X$ is a real normed spaces with two angularly equivalent norms $\norm{\cdot}_1$ and $\norm{\cdot}_2$, are their dual norms $\norm{\cdot}^*_1$ and $\norm{\cdot}^*_2$ equivalent on $X^*$? In Section \ref{section:duality}, extra conditions to the underlying space $X$ are given to obtaint an affirmative answer, namely strict convexity, smoothness, and reflexivity.
\section{Preliminary}
Let $(X, \norm{\cdot})$ be a normed space. Throughout the paper, we use the standard notation of $S_X$ and $B_X$ for the unit sphere and unit ball, respectively, of the normed space $X$. Let $x_0 \in X$. The one-sided G\^ateaux derivatives
$$G^{\pm}(x_0,y)=\lim_{t\rightarrow 0^\pm} \frac1t\left(\|x_0+ty\|-\|x_0\|\right)$$
exist for all $y\in X$ \cite[Lemma 5.4.14]{Megginson}. Furthermore, Lemma 5.4.14 of Megginson \cite{Megginson} also gives the result that $G^{\pm}$ is sub-(super-)additive with respect to the second argument, as summarised in the following proposition.
\begin{proposition}\label{prop:subadditive}
Let $(X, \norm{\cdot})$ be a normed space. For any $x,y,z\in X$, we have
$$G^+(x,y+z)\leq G^+(x,z)+G^+(y,z)$$
and
$$G^-(x,y+z)\geq G^-(x,z)+G^-(y,z).$$
\end{proposition}
Let $(X, \norm{\cdot})$ be a real normed space. For any $x,y\in X$,
$$g^\pm(x,y)=\|x\|\lim_{t\rightarrow 0^\pm} \frac1t\left(\|x+ty\|-\|x\|\right)=\norm{x}G^\pm(x,y).$$
We recall the following result (see \cite[Lemma 1]{Milicic-convexity}) which readily follows from the definition of the mapping $g^\pm$.
\begin{proposition}\label{prop:useful
Let $(X,\norm{\cdot})$ be a real normed space. For any $x,y\in X$, we have the following inequality
\[
-\norm{x}\norm{y}\leq \norm{x}(\norm{x}-\norm{x-y})\leq g^-(x,y)\leq g^+(x,y)\leq \norm{x}( \norm{x+y}-\norm{x})\leq \norm{x}\norm{y}.
\]
\end{proposition}
We note that there is a connection between the $g$-functional with the notion of semi-inner product. We recall the definition of semi-inner product.
\begin{definition}\label{dfn:sip}
Let $X$ be a vector space over the field $\mathbb{K}$. The mapping $[\cdot, \cdot]: X\times X \rightarrow \mathbb{K}$ is called a semi-inner product, if for all $x,y,z\in X$ and $\alpha\in \mathbb{K},$ the following properties are satisfied:
\begin{enumerate}[(S1)]
\item $[x+y,z]=[x,z]+[y,z]$;
\item $[\alpha x, y]=\alpha [x,y]$;
\item $[x,x]\geq 0$ and $[x,x]=0$ implies $x=0$;
\item $|[x,y]|^2 \leq [x,x] [y,y]$;
\item $[x,\alpha y]= \bar{\alpha}[x,y]$.
\end{enumerate}
\end{definition}
\noindent Lumer \cite{Lumer} introduced this concept without (S5) which was later added by Giles \cite{Giles}.
\begin{remark}
Let $X$ be a vector space equipped with a semi-inner product $[\cdot,\cdot]$. Then,
$$\|x\|:=[x,x]^{\frac12}, \quad (x\in X),$$
is a norm on $X$ (see \cite[Proposition 3]{Dragomir}). We therefore say that on a normed space $(X,\norm{\cdot})$ with a semi-inner product $[\cdot, \cdot]$, that $[\cdot, \cdot]$ generates the norm $\norm{\cdot}$ if $\|x\|=[x,x]^{\frac12}$, for all $x\in X.$ We note that such a semi-inner product always exists on a normed space $X$ (\cite[Theorem 1]{Giles}). The next proposition provides a condition for uniqueness.
\end{remark}
\begin{proposition}[Dragomir \cite{Dragomir}, Proposition 4, p. 21]\label{prop:unique}
Let $(X,\norm{\cdot})$ be a normed space. Then $X$ is smooth if and only if there exists a unique semi-inner product which generates $\norm{\cdot}$.
\end{proposition}
Let $(X,\norm{\cdot})$ be a real normed space. Recall that the $g$-functional relative to $\norm{\cdot}$ is the map $g\colon X\times X\to \mathbb{R}$ given by
$$g(x,y)=\frac12(g^+(x,y)+g^-(x,y)), \quad x,y\in X.$$
We are in a position to specify the construction of a (unique) semi-inner product, using the $g$-functional relative to $\norm{\cdot}$, which generates $\norm{\cdot}$.
\begin{proposition}\label{prop:unique-g}
Let $(X,\norm{\cdot})$ be a real normed space. Define
$[\cdot,\cdot]\colon X\times X \to \mathbb{R}$ by
$$[y,x]:=g(x,y), \quad \text{for all } x,y\in X.$$
Then,
\begin{enumerate}[(i)]
\item $[\cdot, \cdot]$ satisfies properties (S2)-(S5) of Definition \ref{dfn:sip};
\item If $X$ is smooth, then $[\cdot,\cdot]$ is the unique semi-inner product on $X\times X.$
\end{enumerate}
\end{proposition}
\begin{proof}
First we note that $g(x,x)=\norm{x}^2$ for all $x\in X$. We omit the proof of {\it (i)}, as the proof for (S2), (S3), and (S5) readily follows from the definition of $g$, and (S4) follows from Proposition \ref{prop:useful}. We prove {\it (ii)}. First, note that since $X$ is assumed to be smooth, then $g^+\equiv g^-$, i.e. $$g(x,y)=\|x\|\lim_{t\rightarrow 0} \frac1t\left(\|x+ty\|-\|x\|\right), \quad \text{for all } x,y\in X.$$
By Proposition \ref{prop:subadditive}, $g^\pm$ is also sub-(super-)additive with respect to the second argument, and thus $$ g^-(x,y)+g^-(x,z)\leq g^-(x,y+z)=g(x,y+z)=g^+(x,y+z)\leq g^+(x,y)+g^+(x,z)$$ and since $g^+\equiv g^-$, we get equality, and therefore, $$[y+z,x]=g(x,y+z)=g(x,y)+g(x,z)=[y,x]+[z,x].$$
This shows (S1) of Definition \ref{dfn:sip} and together with {\it (i)}, we conclude that $[\cdot,\cdot]$ is a semi-inner product which generates $\norm{\cdot}$. Uniqueness follows from Proposition \ref{prop:unique}.
\end{proof}
\begin{example}\label{ex:ell-p}[Mili\v{c}i\'c \cite{Milicic-parallelogram}, p. 72] From Proposition \ref{prop:unique-g}, we note that the smoothness of the normed space implies the linearity of the $g$-functional (in the second argument). Let $x=(x_i), y=(y_i)\in \ell^p$ with $1 < p < \infty$. The functional
\begin{equation}
[y,x]_{\ell^p}= g_{\ell^p}(x,y) = \left\{
\begin{array}{ll}
\|x\|_{\ell^p}^{2-p}\sum_{i} |x_i|^{p-1}\mathrm{sgn}(x_i)y_i, & x\neq 0;\\
0, &x=0;
\end{array}\right.
\end{equation}
is the unique semi-inner product on $\ell^p \times \ell^p$. We note that
\begin{equation}
g_{\ell^1}(x,y) = \|x\|_{\ell^1}\sum_{i}\mathrm{sgn}(x_i)y_i
\end{equation}
is linear in the second argument, and thus is a semi-inner product on $\ell^1\times \ell^1$, although the space is not smooth.
\end{example}
\section{Uniform non-squareness} \label{section:unsq}
Let $(X,\norm{\cdot})$ be a normed space. Recall that $X$ is said to be uniformly convex if for all $\varepsilon \in (0,2)$ there exists $\delta \in (0,1)$ such that the following holds:
$$\text{if }x,y\in S_X \text{ with }\norm{x-y}\geq \varepsilon, \text{ then } \norm{\frac{x+y}2}\leq 1-\delta.$$
\noindent The notion of uniform non-squareness is introduced by James \cite{James} as a weaker form of uniform convexity. In particular, James showed that a Banach space is reflexive provided that the unit ball is uniformly non-square and thus it gave a refinement to the implication of reflexivity by uniform convexity, that is,
\begin{center}
Uniform convexity \quad $\Rightarrow$ \quad Uniform non squareness \quad $\Rightarrow$ \quad Reflexivity.
\end{center}
\begin{definition}\label{dfn:unsq}
Let $(X,\norm{\cdot})$ be a normed space. The space $X$ is said to be uniformly non-square if there exists $\delta \in (0,1)$ such that
$$\text{if }x,y\in S_X \text{ with }\norm{\frac{x-y}{2}}\geq 1-\delta, \text{ then } \norm{\frac{x+y}2}\leq 1-\delta.$$
\end{definition}
\begin{remark}
\begin{enumerate}
\item Definition \ref{dfn:unsq} is rewritten from its original definition in \cite{James}.
\item In $\mathbb{R}^2$, if $1<\lambda<\sqrt{2}$, then the norm $\norm{\cdot}_\lambda$ defined by
$$\norm{(x,y)}_\lambda:=\max\big\{(x^2+y^2)^\frac12,\lambda\max\{|x|,|y|\}\big\}, \quad (x,y)\in \mathbb{R}^2,$$
is uniformly non-square but not strictly convex (hence, not uniformly convex). This example is due to Kato and Takahashi \cite[p. 1058]{Kato-Takahashi}.
\end{enumerate}
\end{remark}
Our aim is to show that uniform non-squareness is shared by angularly equivalent norms. We start with two lemmas which provide characterisations of uniform non-squareness using norm angles. We follow the main idea of the proof of Theorem 2.6 of \cite{angular}.
\begin{lemma}\label{lemma:unsq-delta}
Let $(X,\norm{\cdot})$ be a normed space. Then $X$ is uniformly non-square if and only if there exists $\delta \in (0,1)$ such that the following holds:
$$\text{if }x,y\in S_X \text{ with } \norm{\frac{x-y}2}\geq 1-\delta, \text{ then } \tan\left(\frac{\theta(x,y)}2\right)\geq \sqrt{\delta}.$$
\end{lemma}
\begin{proof}
Assume that $X$ is uniformly non-square, i.e. there exists $\eta\in (0,1)$ such that
$$\text{if }x,y\in S_X \text{ with } \norm{\frac{x-y}2}\geq 1-\eta, \text{ then } \norm{\frac{x+y}2}\leq 1-\eta.$$
Set $\delta:=\eta.$ Let $x,y\in S_X$ with $\norm{\frac{x-y}2}\geq 1-\delta=1-\eta.$ Since $x,y\in S_X$, we have the following inequality
$$-1\leq 1-\norm{x-y}\leq g(x,y)\leq \norm{x+y} -1\leq 1, $$
from Proposition \ref{prop:useful}. Therefore, we have $1+g(x,y)\leq 2$ and $1-g(x,y)\geq 2-\norm{x+y}$. Then,
\begin{align*}
\tan\left(\frac{\theta(x,y)}2\right) &\geq \sqrt{\frac{1-g(x,y)}{1+g(x,y)}}\\
&\geq \sqrt{\frac{1-g(x,y)}2}\geq \sqrt{1-\norm{\frac{x+y}2}}\geq\sqrt{\eta}=\sqrt{\delta}.
\end{align*}
Conversely, assume there exists $\eta\in (0,1)$ such that
$$\text{if }x,y\in S_X \text{ with } \norm{\frac{x-y}2}\geq 1-\eta, \text{ then } \tan\left(\frac{\theta(x,y)}2\right)\geq \sqrt{\eta}.$$
Choose $\delta:=\min\{\frac{\eta}2, \frac{\eta}{1+\eta}\}>0$. Let $x,y\in S_X$ with $\norm{\frac{x-y}2}\geq 1-\delta\geq 1-\eta$, since $\delta\leq \frac{\eta}2< \eta.$
If $\norm{x+y}=0$, then $\norm{\frac{x+y}2}=0\leq 1-\delta$.
We consider the case $\norm{x+y}\neq 0$. Now,
\begin{align*}
&\norm{(2-\norm{x+y})x-\norm{x+y}\left(\frac{x+y}{\norm{x+y}}-x\right)}\\
&=\norm{2x-\norm{x+y}x+\norm{x+y}x-x+y}=\norm{x-y}\geq 2(1-\delta).
\end{align*}
Thus, either
$$\norm{(2-\norm{x+y})x}\geq 2\delta$$
or
$$\norm{\norm{x+y}\left(\frac{x+y}{\norm{x+y}}-x\right)}\geq 2(1-\delta)-2\delta=2-4\delta,$$
which follows from the triangle inequality.
In the first case, we have
$$2-\norm{x+y}=\norm{(2-\norm{x+y})x}\geq 2\delta$$
that is
$$\norm{\frac{x+y}2}\leq 1-\delta, $$
and we are done. In the second case, we have
$$\norm{\frac{x+y}{\norm{x+y}}-x}\geq \frac{2-4\delta}{\norm{x+y}}\geq 1-2\delta\geq 1-\eta$$
by our choice of $\delta\leq \frac\eta2.$ Therefore, by our assumption,
$$\sqrt{\eta}\leq \tan\left(\frac{\theta(x,y)}2\right)=\sqrt{\frac{1-g(\frac{x+y}{\norm{x+y}},x)}{1+g(\frac{x+y}{\norm{x+y}},x)}},$$
and by rearranging we obtain
$$g\left(\frac{x+y}{\norm{x+y}},x\right)\leq\frac{1-\eta}{1+\eta}.$$
By Proposition \ref{prop:useful} with $x+y$ and $x$, we have
$$\norm{x+y}-1\leq \frac{g(x+y,x)}{\norm{x+y}}$$
and thus
\begin{align*}
\norm{\frac{x+y}2}\leq \frac12\left(1+\frac{g(x+y,x)}{\norm{x+y}}\right)&\leq \frac12\left(1+\frac{1-\eta}{1+\eta}\right)\\
&=\frac{1}{1+\eta}=1-\frac{\eta}{1+\eta}\leq 1-\delta
\end{align*}
as we choose $\delta\leq \frac{\eta}{1+\eta}.$
This completes the proof.
\end{proof}
\begin{lemma}\label{lemma:unsq-equivalence}
Let $(X,\norm{\cdot})$ be a normed space. Then the following are equivalent.
\begin{enumerate}[(i)]
\item $X$ is uniformly nonsquare.
\item there exists $\delta \in (0,1)$ such that the following holds:
$$\text{if }x,y\in S_X \text{ with } \norm{\frac{x-y}2}\geq 1-\delta, \text{ then } \tan\left(\frac{\theta(x,y)}2\right)\geq \sqrt{\delta}.$$
\item there exists $\varepsilon\in (0,2)$ and $\delta\in (0,1)$ such that the following holds:
$$\text{if }x,y\in S_X \text{ with }\norm{x-y}\geq \varepsilon, \text{ then } \tan\left(\frac{\theta(x,y)}2\right)\geq \delta.$$
\end{enumerate}
\end{lemma}
\begin{proof}
The equivalence of {\it (i)} and {\it (ii)} follows from Lemma \ref{lemma:unsq-delta}. We show that {\it (ii)} and {\it (iii)} are equivalent. Assume that there exists $\eta \in (0,1)$ such that the following holds:
$$\text{if }x,y\in S_X \text{ with } \norm{\frac{x-y}2}\geq 1-\eta, \text{ then } \tan\left(\frac{\theta(x,y)}2\right)\geq \sqrt{\eta}.$$
Set $\varepsilon:=2(1-\eta)>0$ and $\delta:=\sqrt{n}>0$. Let $x,y\in S_X$ be such that $\norm{x-y}\geq \varepsilon.$ Thus, $\norm{x-y}\geq 2(1-\eta)$, that is $\norm{\frac{x-y}2}\geq 1-\eta.$ By assumption, $$\tan\left(\frac{\theta(x,y)}2\right)\geq \sqrt{\eta}=\delta.$$
Now we assume that there exists $\varepsilon \in (0,2)$ and $\eta \in (0,1)$ such that the following holds:
$$\text{if }x,y\in S_X \text{ with }\norm{x-y}\geq \varepsilon, \text{ then } \tan\left(\frac{\theta(x,y)}2\right)\geq \eta.$$
Set $\delta:=\min\{1-\frac{\varepsilon}2,\eta^2\}>0$. Let $x,y\in S_X$ be such that $\norm{\frac{x-y}2}\geq 1-\delta.$ Thus, by our choice of $\delta\leq 1-\frac{\varepsilon}2$, we have
$$\norm{\frac{x-y}2}\geq 1-\delta\geq \frac{\varepsilon}2, \quad \text{and so} \quad \norm{x-y}\geq\varepsilon.$$
By assumption, we have $\tan\left(\frac{\theta(x,y)}2\right)\geq \eta\geq\sqrt{\delta},$
by our choice of $\delta\leq \eta^2.$
\end{proof}
Now we prove our main result of the section.
\begin{theorem}
Let $X$ be a real normed space with two angularly equivalent norms $\norm{\cdot}_1$ and $\norm{\cdot}_2$. Then $X$ is uniformly non-square with respect to $\norm{\cdot}_1$ if and only if $X$ is uniformly non-square with respect to $\norm{\cdot}_2$.
\end{theorem}
\begin{proof}
We need to only prove one side of the implication, as the other side follows by reversing the roles of $\norm{\cdot}_1$ and $\norm{\cdot}_2$. Let $C>1$ be such that
$$\tan\left(\frac{\theta_1(x,y)}2\right)\leq C \tan\left(\frac{\theta_2(x,y)}2\right)$$
for all $x,y\in X$, where $\theta_i(x,y)$ is the norm angle from $x$ to $y$ with respect to $\norm{\cdot}_1$. Since angular equivalence implies norm equivalence, let $M,m>0$ be such that
$$m\norm{x}_1\leq \norm{x}_2\leq M\norm{x}_1, $$
for all $x\in X.$
Let $X$ be uniformly non-square with respect to $\norm{\cdot}_1$. By Lemma \ref{lemma:unsq-equivalence} part {\it (iii)} there exist $\nu,\eta>0$ such that
$$\text{if }x,y\in S_{(X, \norm{\cdot}_1)} \text{ with }\norm{x-y}_1\geq \nu, \text{ then } \tan\left(\frac{\theta_1(x,y)}{2}\right)\geq \eta.$$
Set $\varepsilon:=2\frac{M\nu}{m}>0$ and $\delta:=\frac{\eta}{C}>0$. Let $x,y\in S_{(X, \norm{\cdot}_2)}$ with
$\norm{x-y}_2\geq \varepsilon. $
Let $\hat{x}=\frac{x}{\norm{x}_1}$ and $\hat{y}=\frac{y}{\norm{y}_ 1}$. Note that $\norm{\hat{x}}_1=1=\norm{\hat{y}}_1$. Also, since $x\in S_{(X, \norm{\cdot}_2)}$, we have $\norm{\hat{x}}_2=\frac{\norm{x}_2}{\norm{x}_1}=\frac{1}{\norm{x}_1}$, and thus
$$x=\norm{x}_1\hat{x}=\frac{\hat{x}}{\norm{\hat x}_2}, \quad
\text{and similarly,} \quad
y=\frac{\hat y}{\norm{\hat y}_2}.$$
Using Dunkl-Williams inequality, we get
\begin{align*}
\varepsilon\leq \norm{x-y}_2
&\leq \norm{\frac{\hat x}{\norm{\hat x}_2}-\frac{\hat y}{\norm{\hat y}_2}}_2\\
&\leq \frac{4\norm{\hat x-\hat y}_2}{\norm{\hat x}_2+\norm{\hat y}_2} \leq \frac{4M\norm{\hat x-\hat y}_1}{m\norm{\hat x}_1+m\norm{\hat y}_1}=\frac{2M}{m}\norm{\hat x-\hat y}_1
\end{align*}
Thus,
$$\norm{\hat x-\hat y}_1\geq \frac{m\varepsilon}{2M}=\nu.$$
Therefore,
$$\eta \leq \tan\left(\frac{\theta_1(\hat x,\hat y)}{2}\right)=\tan\left(\frac{\theta_1( x, y)}{2}\right)\leq C \tan\left(\frac{\theta_2( x, y)}{2}\right), $$
that is,
$$ \tan\left(\frac{\theta_2( x, y)}{2}\right)\geq \frac{\eta}{C}=\delta, $$
and this completes the proof.
\end{proof}
\section{Exposed points}\label{section:exposed}
Our aim in this section is to prove a similar result to that of Proposition 2 part (AE3), by considering exposed points instead of extreme points. First we recall the following definitions.
\begin{definition}
Let $(X,\norm{\cdot})$ be a real normed space and $A$ be a subset of $X$. A nonzero $f\in X^*$ is a support functional for $A$ if there is an $x_0\in A$ such that $f(x_0)=\sup\{f(x):x\in A\}$, in which case $x_0$ is a support point of $A$, the set $\{x:x\in X,\ f(x)=f(x_0)\}$ is a support hyperplane for $A$ and the functional $f$ and the support hyperplane are both said to support $A$ at $x_0$.
\end{definition}
\begin{remark}
Note that as a consequence of the Hahn-Banach theorem, for any $x\in X$ there exists $f\in S_{X^*}$ such that $f(x)=\norm{x}$. Also, $f\in S_{X^*}$ supports $B_X$ at $x_0\in S_X$ if and only if $f(x_0)=1.$
\end{remark}
\begin{definition}
Let $(X,\norm{\cdot})$ be a real normed space and $C$ be a nonempty closed convex subset of $X$. A point $x\in C$ is said to be an exposed point of $C$ if there is $f\in X^*$ such that $f$ is bounded from above on $C$ and attains its supremum on $C$ at $x$ and only at $x$. In this case we call $f$ an exposing functional of $C$ and exposing $C$ at $x$.
\end{definition}
\begin{remark}
If $x_0$ is an exposed point of a nonempty closed convex subset $C$ of $X$, then it is also an extreme point. The converse is not true. For instance, the point $A$ in Figure 1 is an extreme point that is not an exposed point of the bounded region.
\begin{figure}[h!]
\centering
\begin{tikzpicture
\begin{axis}[axis equal,
xmin=-1.1,xmax=1.1,
ymin=-1.1,ymax=1.2,
ticks = none,
xlabel={$\phantom{x}$}, ylabel={$\phantom{y}$},
]
\addplot [domain=-0.6:0.6,samples=100,very thick]({x},{-0.8});
\addplot [domain=-0.6:0.6,samples=100,very thick]({x},{0.8});
\addplot [domain=-53:53,samples=100,very thick]({cos(x)},{sin(x)});
\addplot [domain=127:233,samples=100,very thick]({cos(x)},{sin(x)});
\node at (170,189) {\small{$\bullet$}};
\node at (172,203) {A};
\end{axis}
\end{tikzpicture}
\caption{An extreme point that is not an exposed point}
\end{figure}
\end{remark}
We recall the following result and refer the readers to Lemma 5.4.16 from Megginson \cite[p. 486]{Megginson} for its proof. We reformulate this for any real normed space.
\begin{proposition}\label{prop:gateaux-ineq}
Let $X$ be a real normed space, $x_0\in S_X$ and $f\in S_{X^*}$. Then $f$ supports $B_X$ at $x_0$ if and only if
\[
\lim_{t\to0^-}\frac{\norm{x_0+ty}-\norm{x_0}}{t}=G_{-}(x_0,y)\leq f(y)\leq G_{+}(x_0,y)= \lim_{t\to0^+}\frac{\norm{x_0+ty}-\norm{x_0}}{t}
\]
for all $y\in X.$
\end{proposition}
We provide a characterisation of an exposed point of the unit ball using the $g$-functional.
\begin{lemma}\label{lemma:exposed-g}
Let $(X,\norm{\cdot})$ be a real normed space.
Then $x_0\in S_X$ is an exposed point of $B_X$ if and only if $\{y\in S_X: g(x_0,y)=1\}=\{x_0\}$.
\end{lemma}
\begin{proof}
Let $x_0\in S_X$ be an exposed point of $B_X$ with exposing functional $f$. Thus, $f(x_0)=1$ and $f(x_0)>f(y)$ for all $y\in S_X$. By Proposition \ref{prop:gateaux-ineq}, we have the following inequality
$$g^-(x_0,y)=G_{-}(x_0,y)\leq f(y)\leq G_+(x_0,y)=g^+(x_0,y),$$
for all $y\in X$. Suppose that there exists $y_0\in S_X$ with $y_0\neq x_0$ such that $g(x_0,y_0)=1$, i.e. $g^-(x_0,y_0)+g^+(x_0,y_0)=2$. Since $f(y_0)<1,$ by assumption, we have $g^-(x_0,y_0)\leq f(y_0)<1$ and thus $$g^+(x_0,y_0)=2-g^-(x_0,y_0)>1,$$
contradicting Proposition \ref{prop:useful}. Conversely, assume that $\{y\in S_X: g(x_0,y)=1\}=\{x_0\}$ and suppose that $x_0\in S_X$ is not an exposed point of $B_X$. Thus, if $f\in S_{X^*}$ with $f(x_0)=\norm{x_0}=1$, there exists $y_0\in S_X$ distinct from $x_0$ such that $f(y_0)=\norm{y_0}=1.$ Note that for any $t\in [0,1]$, we have
$$f\left((1-t)x_0+ty_0\right)=tf(x_0)+(1-t)f(y_0)=1.$$
Since $f\in S_{X^*}$, we have $\norm{(1-t)x_0+ty_0}=1$ for all $t\in [0,1]$. Now,
\begin{align*}
g^{\pm}(x_0,y_0)&=\lim_{t\to0^\pm}\frac1t\left(\norm{x_0+ty_0}-1\right)\\
&=\lim_{s\to0^\pm}\frac{(1-s)}{s}\left(\norm{x_0+\frac{s}{1-s}y_0}-1\right)\\
&=\lim_{s\to0^\pm}\frac1{s}\left(\norm{(1-s)x_0+sy_0}-1+s\right)=1.
\end{align*}
Thus, $g(x_0,y_0)=1$ which contradicts the assumption. Therefore, $x_0\in S_X$ must be an exposed point of $B_X$.
\end{proof}
\begin{theorem}
Let $X$ be a real normed space with two angularly equivalent $\norm{\cdot}_1$ and $\norm{\cdot}_2$. Then, $x/\norm{x}_1$ is an exposed point of $B_{(X,\norm{\cdot}_1)}$ if and only if $x/\norm{x}_2$ is an exposed point of $B_{(X,\norm{\cdot}_2)}$.
\end{theorem}
\begin{proof}
Let $C>0$ such that
$$\frac{1-\cos \theta_1(x,y)}{1+\cos \theta_1(x,y)}\leq C\ \frac{1-\cos \theta_2(x,y)}{1+\cos \theta_2(x,y)}$$
for all $x,y\in X$.
It is sufficient to prove one side of the implication as the reverse implication follows from swapping the roles of $\norm{\cdot}_1$ and $\norm{\cdot}_2$. We argue the contrapositive. Assume that $x_0\in S_{(X,\norm{\cdot}_2)}$ is not an exposed point of $B_{(X,\norm{\cdot}_2)}$. By Lemma \ref{lemma:exposed-g}, there exists $y_0\in S_{(X,\norm{\cdot}_2)}$ distinct from $x_0$ such that $g_2(x_0,y_0)=1,$ i.e. $\cos \theta_2(x_0,y_0)=1$ since $x_0,y_0\in S_{(X,\norm{\cdot}_2)}$. Thus, by angular equivalence,
$$\cos \theta_1(x_0,y_0)=1$$
that is,
$$g_1\left(\frac{x_0}{\norm{x_0}_1},\frac{y_0}{\norm{y_0}_1}\right)=1.$$
By Lemma \ref{lemma:exposed-g} again, since $\frac{x_0}{\norm{x_0}_1}\neq \frac{y_0}{\norm{y_0}_1}$, $\frac{x_0}{\norm{x_0}_1}$ is not an exposed point of $ B_{(X,\norm{\cdot}_1)}$.
\end{proof}
\section{Dual norms}\label{section:duality}
The following theorem is due to Giles \cite[Theorem 6]{Giles}.
\begin{theorem}[Giles, 1967]\label{thm:riesz}
Let $(X, \norm{\cdot})$ be a smooth and uniformly convex Banach space and $[\cdot, \cdot]$ be a semi-inner product which generates $\norm{\cdot}$. Then for all $f\in X^*$, there exists a unique $x\in X$ such that $f(y)=[y,x]$ for all $y\in X.$
\end{theorem}
One of the tools that is used in proving Theorem \ref{thm:riesz} is that every closed convex subset in a uniformly convex space is a Chebyshev set. Recall that a non-empty subset $A$ of a metric space $(M,d)$ is a Chebyshev set if for every element $x\in M$, there exists exactly one element $y\in A$ such that
$$d(x,y)=d(x,A):=\inf_{z\in A} d(x,z).$$
However, the assumption of uniform convexity may be replaced by a weaker assumption. This result is due to MM Day (cf. \cite[Corollary 5.1.19]{Megginson}):
\begin{lemma}[Day, 1941]\label{lemma:chebyshev}
If a normed space is strictly convex and reflexive, then each of its nonempty closed convex subsets is a Chebyshev set.
\end{lemma}
\noindent Recall that uniform convexity implies strict convexity and reflexivity. We prove a version of Theorem \ref{thm:riesz} by replacing uniform convexity with strict convexity and reflexivity and reformulate it in terms of the $g$-functional. We first state some results from \cite{Dragomir} and \cite{Giles} which are reformulated in terms of the $g$-functional, with the aid of Proposition \ref{prop:unique-g}. Recall that, from Proposition \ref{prop:unique-g}, when $X$ is a smooth normed space, then the $g$-functional gives rise to a unique semi-inner product given by
$$[x,y]=g(y,x),\quad x,y\in X.$$
In a normed space $(X, \norm{\cdot})$ over the field $\mathbb{K}$, $x\in X$ is said to be $B$-orthogonal to $y\in X$ if $\norm{x+\lambda y}\geq \norm{x}$ for all $\lambda\in \mathbb{K}$. In the usual manner, we say that $x\in X$ is $B$-orthogonal to a subset $Y\subseteq X$, if $x$ is $B$-orthogonal to every $y\in Y.$ We restate the following results from \cite{Giles}, in terms of the $g$-functional, instead of a semi-inner product (via Proposition \ref{prop:unique-g}).
\begin{lemma}[Giles \cite{Giles}, Theorem 2]\label{lemma:birkhoff}
If $(X,\norm{\cdot})$ is smooth normed space over $\mathbb{K}$, then $g(x,y)=0$ if and only if $x$ is $B$-orthogonal to $y$.
\end{lemma}
\begin{lemma}[Giles \cite{Giles}, Lemma 5]\label{lemma:sc}
Let $(X,\norm{\cdot})$ be a smooth normed space over reals. Then $X$ is strictly convex if and only if for any nonzero $x,y\in X$, if $g(x,y)=\norm{x}\norm{y}$, then $y=\lambda x$ for some real number $\lambda>0.$
\end{lemma}
We now restate Theorem 6 of Giles \cite{Giles} (Theorem \ref{thm:riesz} above) with a weaker assumption of strict convexity and reflexivity in place of uniform convexity.
\begin{theorem}\label{thm:riesz-g}
Let $(X, \norm{\cdot})$ be a smooth, strictly convex, and reflexive space. Then for all $f\in X^*$, there exists a unique $x\in X$ such that $f(y)=g(x,y)$ for all $y\in X.$ Furthermore, $\norm{f}=\norm{x}$.
\end{theorem}
\begin{proof}
If $f(y)=0$ for all $y\in X$, then we choose $x=0.$ If $f(y)\neq 0$ for some $y\in X$, then the null space of $N$ of $f$ is a proper closed subspace of $X$. Thus, by Lemma \ref{lemma:chebyshev} there exists a unique nonzero vector $z_0\in N$ such that $\norm{y-z_0}=\inf_{z\in N}\norm{y-z}.$ Writing $x_0=y-z_0$, we get
$\norm{x_0}\leq \norm{x_0+z}$
for all $z\in N$, that is $x_0$ is $(B)$-orthogonal to $z$ for all $z\in N$. By Lemma \ref{lemma:birkhoff}, $g(x_0,z)=0$ for all $z\in N.$ We make the following observations:
\begin{enumerate}[(1)]
\item If $z_0\in N$, then $f(z)=0=g(x,z_0)$, for any $x=\alpha x_0$ with $\alpha \in \mathbb{R}$.
\item Observe that
$$f(x_0)=g\left(\frac{f(x_0)}{\norm{x_0}^2}x_0,x_0\right).$$
So $f(x_0)=g(x,x_0)$ for $x=\frac{f(x_0)}{\norm{x_0}^2}x_0.$
\end{enumerate}
Thus, any $y\in X$ can be written as $y=z_0+ x_0$, where $z_0\in N$, and $0\neq x_0\in X$ is such that $g(x_0, z)=0$ for all $z\in N$. Set $x=\frac{f(x_0)}{\norm{x_0}^2}x_0$. Since $z_0\in N,$ observation (1) gives us $f(z_0)=g(x,z_0)$ and (2) give us
$f(x_0)=g(x,x_0)$. Therefore,
\begin{eqnarray*}
f(y)&=&f(z_0+ x_0)\\
&=&f(z_0)+f(x_0)\\
&=&g(x,z_0)+g(x,x_0)=g(x,z_0+ x_0)=g(x,y).
\end{eqnarray*}
To prove uniqueness, let $x,x'\in X$, $x\neq x'$ such that $f(y)=g(x,y)$ and $f(y)=g(x',y)$ for all $y\in X.$ Then,
$$\norm{x}^2=|g(x,x)|=|g(x',x)|\leq \norm{x'} \norm {x}$$
so $\norm{x}\leq \norm{x'}$ and
$$\norm{x'}^2=|g(x',x')|=|g(x,x')|\leq \norm{x} \norm {x'}$$
so $\norm{x'}\leq \norm{x}$. Thus, $\norm{x'}= \norm{x}$, and
$$\norm{x}^2=g(x',x)$$
gives us
$$\norm{x}\norm{x'}=g(x',x)$$
and so by Lemma \ref{lemma:sc}, we conclude that $x=\lambda x'$. Combining this with $\norm{x'}= \norm{x}$, we conclude that $x=x'.$ Finally,
$$|f(y)|=|g(x,y)|\leq \|x\|\|y\|$$
and so
$$\norm{f}=\sup_{0\neq y\in X}\frac{|f(y)|}{\norm{y}}\leq\norm{x},$$
and $$\norm{x}^2=|g(x,x)|=|f(x)|\leq \norm{f}\norm{x}$$
so $\norm{x}\leq \norm{f}.$ This completes the proof.
\end{proof}
We now restate Theorem 7 of Giles \cite{Giles} in terms of the $g$-functional.
\begin{corollary}\label{cor:sip-dual-g}
Let $(X,\norm{\cdot})$ be a normed space. Assume that $X$ is smooth, strictly convex, and reflexive. Then, the dual space $X^*$ is smooth, strictly convex, and reflexive; and the $g$-functional on $X^*$, is given by
$$g(\phi,\psi)=g(x_{\psi},x_{\phi}), \quad \text{ for any } \phi,\psi\in X^*,$$
where $x_\phi$ and $x_\psi$ in $X$ are associated to $\phi$ and $\psi$, respectively, as given in Theorem \ref{thm:riesz-g}.
\end{corollary}
\begin{proof}
By reflexivity of $X$, it follows that $X^*$ is reflexive, and since $X$ is smooth and strictly convex, $X^*$ is smooth and strictly convex. Let $\phi,\psi\in X^*$. By Theorem \ref{thm:riesz-g}, there exist $x_\phi,x_\psi\in X$ such that
$$\phi(z)=g(x_{\phi},z) \quad\text{and}\quad \psi(z)=g(x_{\psi},z), \quad \text{for all }z\in X,$$
with $\norm{\phi}=\norm{x_\phi}$ and $\norm{\psi}=\norm{x_\psi}$. Define $[\cdot,\cdot]\colon X\times X \to\mathbb{R}$ by
$$[\phi,\psi]:=g(x_\phi,x_\psi),\quad \text{for any }\phi,\psi\in X^*. $$
It is sufficient to show that $[\cdot,\cdot]$ is a semi-inner product on $X^*$, since smoothness of $X^*$, implies that $[\cdot,\cdot]$ is the unique semi-inner product on $X^*$ which in turn implies that the $g$-functional in $X^*$ is given by
$$g(\psi,\phi)=[\phi,\psi]=g(x_\phi,x_\psi),\quad \text{for any } \phi,\psi\in X^*,$$
as desired. Let $\phi,\psi,\tau\in X^*$ and $\alpha,\beta\in \mathbb{R}$. Firstly we note the following,
$$[\phi,\psi]=g(x_\phi,x_\psi)=\phi(x_\psi).$$
Now we show that $[\cdot,\cdot]$ satisfies the properties of semi-inner product. We have
$$[\phi+\psi,\tau]=(\phi+\psi)(x_\tau)=\phi(x_\tau)+\psi(x_\tau)=g(x_{\phi},x_{\tau})+g(x_{\psi},x_{\tau})=[\phi,\tau]+[\psi,\tau].$$
Next, we note that for all $z\in X$,
$$(\alpha\phi)(z)=\alpha\phi(z)=\alpha g(x_{\phi},z)=g(\alpha x_{\phi},z),$$
that is, a one-to-one correspondence between $\alpha\phi\in X^*$ with $\alpha x_{\phi}\in X.$
Thus
$$[\alpha\phi,\beta\psi]=g(\alpha x_{\phi},\beta x_{\psi})=\alpha\beta g(x_\phi,x_{\psi})=\alpha \beta [\phi,\psi].$$
Next, we have
$$[\phi,\phi]=g(x_\phi,x_\phi)=\norm{x_\phi}^2=\norm{\phi}^2.$$
Thus, $[\phi,\phi]=\norm{\phi}^2\geq 0$ and $[\phi,\phi]=0$ implies $\norm{\phi}^2=0$, so $\norm{\phi}=0.$
Finally,
$$|[\phi,\psi]|=|g(x_\phi,x_\psi)|\leq\norm{x_\phi} \norm{x_\psi}=\norm{\phi}\norm{\psi}.$$
This completes the proof.
\end{proof}
\begin{theorem}\label{thm:AE-dual}
Let $X$ be a normed space with two norms $\norm{\cdot}_1$ and $\norm{\cdot}_2$ that are both strictly convex, smooth, and reflexive, and that both norms are angularly equivalent. Then, the dual norms
$\norm{\cdot}^*_1$ and $\norm{\cdot}^*_2$ are also angularly equivalent.
\end{theorem}
\begin{proof}
Denote by $g_1$ and $g_2$, the $g$-functional associated to the norm $\norm{\cdot}_1$ and $\norm{\cdot}_2$, respectively. By the assumption of angular equivalence, there exists $C>0$ such that
$$\frac{1-g_2(x,y)}{1+g_2(x,y)}\leq C \frac{1-g_1(x,y)}{1+g_1(x,y)},$$
for any $x,y\in X$. Take two elements $\phi$ and $\psi$ of the dual space $X^*$. By Theorem \ref{thm:riesz-g}, there exists $x_{\phi}$ and $x_{\psi}$ in $X$ such that
$$\phi(y)=g_1(x_{\phi},y)\quad \text{and}\quad \psi(y)=g_2(x_{\psi},y), \quad \text{for all } y\in X.$$
Thus, we have
\begin{equation}\label{eq:ineq-ae}
\frac{1-g_2(x_{\psi},x_{\phi})}{1+g_2(x_{\psi},x_{\phi})}\leq C \frac{1-g_1(x_{\psi},x_{\phi})}{1+g_1(x_{\psi},x_{\phi})}.
\end{equation}
By Corollary \ref{cor:sip-dual-g}, we have the $g$-functionals on $(X^*,\norm{\cdot}^*_1)$ and $(X^*,\norm{\cdot}^*_2)$, denoted by $g^*_1$ and $g^*_2$, are given by
$$g^*_i(\phi,\psi)=g_i(x_\psi,x_\phi), \quad i=1,2.$$
Consequently, \eqref{eq:ineq-ae} becomes
$$\frac{1-g^*_2(\phi,\psi)}{1+g^*_2(\phi,\psi)}\leq C \frac{1-g^*_1(\phi,\psi)}{1+g^*_1(\phi,\psi)}$$
which shows that the dual norms $\norm{\cdot}^*_1$ and $\norm{\cdot}^*_2$ are also angularly equivalent. \end{proof}
\section{Discussion}
The assumptions of Theorem \ref{thm:AE-dual} are as follows.
\begin{enumerate}[({A}1)]
\item A real vector space $X$ with two angularly equivalent norms $\norm{\cdot}_1$ and $\norm{\cdot}_2.$
\item Both $\norm{\cdot}_1$ and $\norm{\cdot}_2$ are strictly convex.
\item Both $\norm{\cdot}_1$ and $\norm{\cdot}_2$ are smooth.
\item Both $\norm{\cdot}_1$ and $\norm{\cdot}_2$ are reflexive.
\end{enumerate}
Corollary 2.2 of \cite{angular} states that angular equivalence preserves strict convexity and thus (A2) may be weakened to only requiring one of the norms to be strictly convex. This led to the following questions:
\begin{enumerate}[(Q1)]
\item Does angular equivalence preserves smoothness?
\item Does angular equivalence preserves reflexivity?
\end{enumerate}
Note also that the statement of Theorem \ref{thm:AE-dual} remains true, when the assumptions (A2)-(A4) are changed to the following.
\begin{enumerate}[({A}1*)]\setcounter{enumi}{1}
\item Both $\norm{\cdot}_1$ and $\norm{\cdot}_2$ are uniformly convex.
\item Both $\norm{\cdot}_1$ and $\norm{\cdot}_2$ are uniformly smooth.
\end{enumerate}
By Corollary 2.7 of \cite{angular}, since angular equivalence preserves uniform convexity, (A1*) may be weakened to only requiring that one of the norms to be uniformly convex. This led to the question:
\begin{enumerate}[(Q1)]\setcounter{enumi}{2}
\item Does angular equivalence preserves uniform smoothness?
\end{enumerate}
An affirmative answer to (Q1)-(Q3) will strengthen the result of Theorem \ref{thm:AE-dual}.
| {
"timestamp": "2022-02-01T02:39:30",
"yymm": "2201",
"arxiv_id": "2201.02865",
"language": "en",
"url": "https://arxiv.org/abs/2201.02865",
"abstract": "Angular equivalence of norms is introduced by Kikianty and Sinnamon (2017) and is a stronger notion than the usual topological equivalence. Given two angularly equivalent norms, if one norm has a certain geometrical property, e.g. uniform convexity, then the other norm also possesses such a property. In this paper, we show further results in this direction, namely angular equivalent norms share the property of uniform non-squareness, and that angular equivalence preserves the exposed points of the unit ball. A discussion on the (equivalence of the) dual norms of angularly equivalent norms is also given, giving a partial answer to an open problem as stated in the paper by Kikianty and Sinnamon (2017).",
"subjects": "Functional Analysis (math.FA)",
"title": "Further results on angular equivalence of norms",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9854964224384745,
"lm_q2_score": 0.718594386544335,
"lm_q1q2_score": 0.7081721971238124
} |
https://arxiv.org/abs/1906.09392 | Prefix palindromic length of the Thue-Morse word | The prefix palindromic length $PPL_u(n)$ of an infinite word $u$ is the minimal number of concatenated palindromes needed to express the prefix of length $n$ of $u$. In a 2013 paper with Puzynina and Zamboni we stated the conjecture that $PPL_u(n)$ is unbounded for every infinite word $u$ which is not ultimately periodic. Up to now, the conjecture has been proven for almost all words, including all words avoiding some power $p$. However, even in that simple case the existing upper bound for the minimal number $n$ such that $PPL_u(n)>K$ is greater than any constant to the power $K$. Precise values of $PPL_u(n)$ are not known even for simplest examples like the Fibonacci word.In this paper, we give the first example of such a precise computation and compute the function of the prefix palindromic length of the Thue-Morse word, a famous test object for all functions on infinite words. It happens that this sequence is $2$-regular, which raises the question if this fact can be generalized to all automatic sequences. | \section{Introduction}
By the usual definition, a palindrome is a finite word $p=p[1]\cdots p[n]$ on a finite alphabet such that $p[i]=p[n-i+1]$ for every $i$. We consider decompositions of a finite word $s$ to a minimal number of palindromes which we call a {\it palindromic length} of $s$: for example, the palindromic length of $abbaba$ is equal to 3 since this word is not a concatenation of two palindromes, but $abbaba=(abba)(b)(a)=(a)(bb)(aba)$. A decomposition to a minimal possible number of palindromes is called {\it optimal}.
In this paper, we are interested in the palindromic length of prefixes of an infinite word $u=u[1]\cdots u[n]\cdots$, denoted by $PPL_u(n)$.
The length of the shortest prefix of $u$ of palindromic length $k$ is denoted by $SP_u(k)$ and can be considered as a kind of an inverse function to $PPL_u(n)$. Clearly, $SP_u(k)$ can be infinite: for example, if $u=abababab\cdots$, $SP_u(k)=\infty$ for every $k \geq 3$.
The following conjecture was first formulated, in slightly different terms, in our 2013 paper with Puzynina and Zamboni \cite{fpz}.
\begin{conjecture}\label{c1}
For every non ultimately periodic word $u$, the function $PPL_u(n)$ is unbounded, or, which is the same, $SP_u(k)<\infty$ for every $k \in \mathbb N$.
\end{conjecture}
In fact, there were two versions of the conjecture considered in our paper \cite{fpz}, one with the prefix palindromic length and the other with the palindromic length of any factor of $u$. However, Saarela \cite{saarela} later proved the equivalence of these two statements.
In the same initial paper \cite{fpz}, the conjecture was proved for the case when $u$ is $p$-power-free for some $p$, as well as for the more general case when a so-called $(p,l)$-condition holds for some $p$ and $l$. Due to the above-mentioned result by Saarela, this means that the conjecture is proven for almost all words, since almost all words contain as long $p$-power-free factors as needed. However, for some cases, the conjecture remains unsolved, and, for example, its proof for all Sturmian words \cite{frid} required a special technique.
Most published papers on palindromic length concern algorithmic aspects; in particular, there are several fast effective algorithms for computing $PPL_u(n)$ \cite{fici,shur2,shur1}.
The original proof of Conjecture \ref{c1} for the $p$-power-free words is not constructive. The upper bound for a length $N$ such that $PPL(N)\geq K$ for a given $K$ is given as a solution of a transcendental equation and grows with $K$ faster than any exponential function. However, this does not look the best possible bound. So, it is reasonable to state the following conjecture.
\begin{conjecture}\label{c2} If a word $u$ is $p$-power free for some $p$, then $$\lim \sup \frac{PPL_u(n)}{\ln n}>0,$$ or, which is the same, $SP_u(k)\leq C^k$ for some $C$. The constant $C$ can be chosen independently of $u$ as a function of $p$.
\end{conjecture}
In this paper, we consider in detail the case of the Thue-Morse word \seqnum{A010060}, a classical example of a word avoiding powers greater than 2 \cite{a_sh_ubi}. We give precise formulas for its prefix palindromic length and discuss its properties. This is a simple but necessary step before considering all $p$-power-free words, or all fixed points of uniform morphisms, or any other family of words containing the Thue-Morse word.
The results of this paper, in less detail, have been announced in the proceedings of DLT 2019 \cite{dlt}, together with some other results on the prefix palindromic length.
Throughout this paper, we use the notation $w(i..j]=w[i+1]..w[j]$ for a factor of a finite or infinite word $w$ starting at position $i+1$ and ending at $j$.
The following lemma is a particular case of a statement by Saarela \cite[L.\ 6]{saarela}. We give its proof for the sake of completeness.
\begin{lemma}\label{WRITEIT}
For every word $u$ and for every $n\geq 0$, we have
\[PPL_u(n)-1\leq PPL_u(n+1) \leq PPL_{u}(n)+1.\]
\end{lemma}
\begin{proof} Consider the prefixes $v$ and $va$ of $u$ of length $n$ and $n+1$ respectively. Clearly, for any decomposition $u=p_1\cdots p_k$ to $k$ palindromes $ua=p_1\cdots p_ka$ is a decomposition of $ua$ to $k+1$ palindrome. On the other hand, for any palindromic decomposition $ua=q_1\cdots q_k$, we have either $q_k=a$, and then $u=q_1\cdots q_{k-1}$, or $q_k=ap_k a$, for a (possibly empty) palindrome $p_k$, and then $u=q_1\cdots q_{k-1}ap_k$ is a decomposition of $u$ to $k+1$ palindromes. If initial decompositions were optimal, this gives $PPL_u(n+1)\leq PPL_u(n)+1$ and $PPL_u(n)\leq PPL_u(n+1)+1$. \end{proof}
So, the first differences of the prefix palindromic length can be equal only to -1, 0, or 1, and the graph never jumps.
In this paper, it is convenient to consider the famous Thue-Morse word \seqnum{A010060}
\[t=abbabaabbaababba\cdots\]
as the fixed point starting with $a$ of the morphism
\[\tau: \begin{cases} a \to abba,\\ b \to baab. \end{cases}\]
Both images of letters under this morphism, which is the square of the usual Thue-Morse morphism $a \to ab, b \to ba$, are palindromes.
It is thus easy to see that every prefix of the Thue-Morse word of length $4^k$ is a palindrome, so that $PPL_t(4^k)=1$ for all $k\geq 0$. The first values of $PPL_t(n)$ and of $SP_t(k)$ are given below, see also the \seqnum{A307319} entry of the OEIS \cite{oeis}.
\medskip
\noindent
\begin{center}
\begin{tabular}{|c||c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|}
\hline
$n$ &1 & 2& 3&4&5&6&7&8&9&10&11&12&13&14&15&16 \\
\hline
$PPL_t(n)$ & 1& 2& 2&1& 2& 3& 3&2& 3&4& 3& 2& 3& 3& 2& 1\\
\hline
\end{tabular}
\end{center}
\medskip
As for the shortest prefix of a given palindromic length, we give its length in decimal and quaternary notation; see also the \seqnum{A320429} entry of the OEIS \cite{oeis}.
\medskip
\begin{center}
\begin{tabular}{|c||c|c|c|c|c|c|c|c|}
\hline
$k$ &1 & 2& 3&4&5&6&7&8 \\
\hline
$SP_t(k)$ & 1& 2& 6&10& 26& 90& 154&410\\
\hline
4-ary & 1&2&12&22&122&1122&2122&12122\\
\hline
\end{tabular}
\end{center}
Now we are going to prove the self-similarity properties which we observe.
\section{Recurrence relations}
\begin{theorem}\label{t:tm}
The following identities hold for all $n\geq 0$:
\begin{align}
PPL_t(4n)&=PPL_t(n), \label{e:4n}\\
PPL_t(4n+1)&=PPL_t(n)+1,\label{e:4n+1}\\
PPL_t(4n+2)&=\min(PPL_t(n),PPL_t(n+1))+2,\label{e:4n+2}\\
PPL_t(4n+3)&=PPL_t(n+1)+1.\label{e:4n+3}
\end{align}
\end{theorem}
\bigskip
To prove Theorem \ref{t:tm}, we need several observations. First of all, the shortest non-empty palindrome factors in the Thue-Morse word are $a$, $b$, $aa$, $bb$, $aba$, $bab$, $abba$, $baab$. All palindromes of length more than 3 are of even length and have $aa$ or $bb$ in the center: if $t(i..i+2k]$ is a palindrome, then $t(i+k-1,i+k+1]=aa$ or $bb$.
Let us say that an occurrence of a palindrome $t(i..j]$ is of type $(i', j')$ if $i'$ is the residue of $i$ and $j'$ is the residue of $j$ modulo 4. For example, the palindrome $t(5..7]=aa$ is of type $(1,3)$, the palindrome $t(4,8]=baab$ is of type $(0,0)$, and the palindrome $t(7..9]=bb$ is of type $(3,1)$.
\begin{proposition}\label{p:4-m}
Every occurrence of a palindromic factor of length not equal to one or three to the Thue-Morse word is of a type $(m,4-m)$ for some $m\in \{0,1,2,3\}$.
\end{proposition}
\begin{proof} Every such a palindrome in the Thue-Morse word is of even length which we denote by $2k$, and every occurrence of it is of the form $t(i..i+2k]$. Its center $t(i+k-1,i+k+1]$ is equal to $aa$ or $bb$, and these two words always appear in $t$ at positions of the form $t(2l-1,2l+1]$ for some $l \geq 1$. So, $i+k-1=2l-1$, meaning that $i=2l-k$ and $i+2k=2l+k$. So, modulo $4$, we have $i+(i+2k)=4l \equiv 0$, that is, $i \equiv -(i+2k)$. \end{proof}
\medskip
Note that the palindromes of odd length in the Thue-Morse word are, first, $a$ and $b$, which can be of type $(0,1)$, $(1,2)$, $(2,3)$ or $(3,0)$, and second, $aba$ and $bab$, which can only be of type $(2,1)$ or $(3,2)$.
\begin{proposition}\label{p:+-1}
Let $t(i..i+k]$ for $i>0$ be a palindrome of length $k>0$ and of type $(m,4-m)$ for some $m \neq 0$. Then $t(i-1..i+k+1]$ is also a palindrome, as well as $t(i+1..i+k-1]$.
\end{proposition}
\begin{proof} The type of the palindrome is not $(0,0)$, meaning that its first and last letters $t[i+1]$ and $t[i+k]$ are not the first the last letters of $\tau$-images of letters. Since these first and last letters are equal and their positions in $\tau$-images of letters are symmetric and determine their four-blocks $abba$ or $baab$, the letters $t[i]$ and $t[i+k+1]$ are also equal, and thus $t(i-1..i+k+1]$ is a palindrome. As for $t(i+1..i+k-1]$, it is a palindrome since is obtained from the palindrome $t(i..i+k]$ by erasing the first and the last letters. \end{proof}
\medskip
Let us say that a decomposition of $t(0..4n]$ to palindromes is a {\it $0$-decomposition} if all palindromes in it are of type $(0,0)$. The minimal number of palindromes in a 0-decomposition is denoted by $PPL^0_t(4n)$.
\begin{proposition}
For every $n \geq 1$, we have $PPL_t(n)=PPL^0_t(4n)\geq PPL_t(4n)$.
\end{proposition}
\begin{proof} It is sufficient to note that $\tau$ is a bijection between all palindromic decompositions of $t(0..n]$ and 0-decompositions of $t(0..4n]$. \end{proof}
\begin{proposition}\label{p:+2+4}
If \eqref{e:4n+2} holds for $n=N-1$, then
\begin{equation}\label{e:+2>+4}
PPL_t(4N-2)>PPL_t(4N).
\end{equation}
\end{proposition}
\begin{proof} The equality \eqref{e:4n+2} means that $PPL_t(4N-2)=\min(PPL_t(N-1),PPL_t(N))+2$, but since due to Lemma \ref{WRITEIT} we have $PPL_t(N)\leq PPL_t(N-1)+1$, we also have
$\min(PPL_t(N-1),PPL_t(N))+2\geq PPL_t(4N)+1$. \end{proof}
Now we can start the main proof of Theorem \ref{t:tm}.
The proof is done by induction on $n$. Clearly, $PPL_t(0)=0$, $PPL_t(1)=PPL_t(4)=1$, and $PPL_t(2)=PPL_t(3)=2$, the equalities \eqref{e:4n}--\eqref{e:4n+3} hold for $n=0$, and moreover, \eqref{e:4n} is true for $n=1$. Now suppose that they all, and, by Proposition \ref{p:+2+4}, the equality \eqref{e:+2>+4}, hold for all $n<N$, and \eqref{e:4n} holds also for $n=N$. We fix an $N>0$ and prove for it the following sequence of propositions.
\begin{proposition}
An optimal decomposition to palindromes of the prefix $t(0..4N+1]$ cannot end by a palindrome of length 3.
\end{proposition}
\begin{proof} Suppose the opposite: some optimal decomposition of $t(0..4N+1]$ ends by the palindrome $t(4N-2..4N+1]$. This palindrome is preceded by an optimal decomposition of $t(0..4N-2]$. So, $PPL_t(4N+1)=PPL_t(4N-2)+1$; but by \eqref{e:+2>+4} applied to $N-1$, which we can use by the induction hypothesis, $PPL_t(4N-2)>PPL_t(4N)$. So, $PPL_t(4N+1)>PPL_t(4N)+1$, contradicting to Lemma \ref{WRITEIT}.
\end{proof}
\begin{proposition}
There exists an optimal decomposition to palindromes of the prefix $t(0..4N+2]$ which does not end by a palindrome of length 3.
\end{proposition}
\begin{proof} The opposite would mean that all optimal decompositions of $t(0..4N+2]$ end by the palindrome $t(4N-1..4N+2]$ preceded by an optimal decomposition of $t(0..4N-1]$. So, $PPL_t(4N+2)=PPL_t(4N-1)+1$; by the induction hypothesis, $PPL_t(4N-1)=PPL_t(4N)+1$. So, $PPL_t(4N+2)=PPL_t(4N)+2$, and thus another optimal decomposition of $t(0..4N+2]$ can be obtained as an optimal decomposition of $t(0..4N]$ followed by two palindromes of length 1. A contradiction.
\end{proof}
\begin{proposition}\label{p:min123}
For every $m\in \{1,2,3\}$, the equality holds
\[PPL_t(4N+m)=\min(PPL_t(4N+m-1),PPL_t(4N+m+1))+1.\]
\end{proposition}
\begin{proof} Consider an optimal decomposition $t(0..4N+m]=p_1\cdots p_k$, where $k=PPL_t(4N+m)$. Denote the ends of palindromes as $0=e_0<e_1<\cdots < e_k=4N+m$, so that $p_i=t(e_{i-1},e_i]$ for each $i$. Since $m\neq 0$ and due to Proposition \ref{p:4-m}, there exist some palindromes of length 1 or 3 in this decomposition. Let $p_j$ be the last of them.
Suppose first that $j=k$. Then due to the two previous propositions, $p_k$ can be taken of length 1 not 3, so that $t(0..4N+m-1]=p_1\cdots p_{k-1}$ is decomposable to $k-1$ palindromes. Due to Lemma \ref{WRITEIT}, we have $PPL_t(4N+m-1)=k-1$, and thus $PPL_t(4N+m)=PPL_t(4N+m-1)+1$. Again due to Lemma \ref{WRITEIT}, we have $PPL_t(4N+m+1)\geq PPL_t(4N+m)-1=PPL_t(4N+m-1)$, and so the statement holds.
Now suppose that $j<k$, so that $e_j\equiv -e_{j+1} \equiv e_{j+2} \equiv \cdots \equiv (-1)^{k-j} e_k \bmod 4$. Here $p_j$ is the last palindrome in an optimal decomposition of $p_1\cdots p_j$ and it is of length 1 or 3. But if $e_j\equiv 1$ or $2 \bmod 4$, $p_j$ can be taken of length 1 due to the two previous propositions applied to some smaller length; and if $e_j\equiv 3 \bmod 4$, it is of length 1 since the suffix of length $3$ of $t(0..4n+3])$ is equal to $abb$ or to $baa$, so, it is not a palindrome. So, anyway, we can take $p_j$ of length one: $p_j=t(e_{j}-1,e_j]$.
Since $e_j\equiv \pm e_k$ and $e_k \equiv m \neq 0 \bmod 4$, we may apply Proposition \ref{p:+-1} and see that $p_j'=t(e_j-1..e_{j+1}+1]$ is a palindrome, as well as $p_{j+1}'=t(e_{j+1}+1..e_{j+2}-1]$ and so on up to $p_{k-1}'=t(e_{k-1}+(-1)^{k-j}..e_k-(-1)^{k-j}]$. So, $p_1 \cdots p_{j-1} p_j' \cdots p_{k-1}'$ is a decomposition of $t(0..4N+m-(-1)^{k-j}]$ to $k-1$ palindromes. So, as above, $PPL_t(4N+m)=PPL_t(4N+m-(-1)^{k-j})+1$, and since $PPL_t(4N+m+(-1)^{k-j})\geq PPL_t(4N+m)-1=PPL_t(4N+m-(-1)^{k-j})$, the proposition holds. \end{proof}
\medskip
\begin{proposition}\label{p:4n}
Every optimal palindromic decomposition of $t(0..4N+4]$ is a 0-decomposition, and thus $PPL_t(4N+4)=PPL_t(N+1)$.
\end{proposition}
\begin{proof} Suppose the opposite; then the last palindrome in the optimal decomposition which is not of type (0,0) is of type $(m,0)$ and thus is of length 1 not 3. Since the proof of Theorem \ref{t:tm} proceeds by induction on $N$, this proposition is true for all $n<N$, and thus the palindrome of type $(m,0)$ is the very last palindrome of the optimal decomposition. Since the suffix of length $3$ of $t(0..4N+4]$ is equal to $bba$ or $aab$ and thus is not a palindrome, the last palindrome of the optimal decomposition is of length $1$, meaning that $PPL_t(4N+4)=PPL_t(4N+3)+1$. Now let us use Proposition \ref{p:min123} applied to $m=3,2,1$; every time we get $PPL_t(4N+m)=PPL_t(4N+m-1)+1$. Summing up these inequalities, we get $PPL_t(4N+4)=PPL_t(4N)+4$, which is impossible since $PPL_t(4N)=PPL_t(N)$ and $PPL_t(4N+4)\leq PPL_t(N+1)\leq PPL_t(N)+1$. A contradiction. \end{proof}
\medskip
We have proven
\eqref{e:4n} for $n=N+1$. It remains to prove \eqref{e:4n+1}--\eqref{e:4n+3} for $n=N$. Indeed, we know that
\begin{equation}\label{e:-1+1}
-1 \leq PPL_t(4N+4)-PPL_t(4N)=PPL_t(N+1)-PPL_t(N)\leq 1.
\end{equation}
Now to prove \eqref{e:4n+1} suppose by contrary that $PPL_t(4N+1)\leq PPL_t(4N)=PPL_t(N)$. Due to Proposition \ref{p:min123}, this means that $PPL_t(4N+1)=PPL_t(4N+2)+1$, that is, $PPL_t(4N+2)<PPL_t(4N)$, and, again by Proposition \ref{p:min123}, $PPL_t(N+1)=PPL_t(4N+2)-2$. Thus, $PPL_t(N)-PPL_t(N+1)\geq 3$, a contradiction to \eqref{e:-1+1}.
So, \eqref{e:4n+1} is proven.
The equality \eqref{e:4n+3} is proven symmetrically. Now \eqref{e:4n+2} follows from both these equalities in combination with Proposition \ref{p:min123}, completing the proof of Theorem \ref{t:tm}.
\section{Corollaries}
The first differences $(d_t(n))_{n=0}^{\infty}$ of the prefix palindromic length are defined as $d_t(n)=PPL_t(n+1)-PPL_t(n)$; here we set $PPL_t(0)=0$. Due to Lemma \ref{WRITEIT}, $d_t(n)\in \{-1,0,+1\}$ for every $n$; so, it is a sequence on a finite alphabet which we prefer to denote $\{
\begin{tikzpicture}[baseline=1pt,scale=0.3]
\draw[thick] (0,1) -- (1,0);
\node[circle,fill=black,inner sep=0pt,minimum size=4pt] (a) at (0, 1) {};
\node[circle,fill=black,inner sep=0pt,minimum size=4pt] (a) at (1, 0) {};
\end{tikzpicture},
\begin{tikzpicture}[baseline=-3pt,scale=0.3]
\draw[thick] (0,0) -- (1,0);
\node[circle,fill=black,inner sep=0pt,minimum size=4pt] (a) at (0, 0) {};
\node[circle,fill=black,inner sep=0pt,minimum size=4pt] (a) at (1, 0) {};
\end{tikzpicture},
\begin{tikzpicture}[baseline=1pt,scale=0.3]
\draw[thick] (0,0) -- (1,1);
\node[circle,fill=black,inner sep=0pt,minimum size=4pt] (a) at (0, 0) {};
\node[circle,fill=black,inner sep=0pt,minimum size=4pt] (a) at (1, 1) {};
\end{tikzpicture}
\}$. We write these symbols joining the ends of intervals from left to right, so that the sequence $(d_t(n))$ becomes the plot of $PPL_t(n)$.
The following corollary of Theorem \ref{t:tm} is more or less straightforward.
\begin{corollary}
The sequence $(d_t(n))$ is the fixed point of the morphism
\[\delta: \begin{cases}
\begin{tikzpicture}[baseline=1pt,scale=0.3]
\draw[thick] (0,0) -- (1,1);
\node[circle,fill=black,inner sep=0pt,minimum size=4pt] (a) at (0, 0) {};
\node[circle,fill=black,inner sep=0pt,minimum size=4pt] (a) at (1, 1) {};
\end{tikzpicture}
\to
\begin{tikzpicture}[baseline=1pt,scale=0.3]
\draw[thick] (0,0) -- (1,1)-- (2,2)--(3,2)--(4,1);
\node[circle,fill=black,inner sep=0pt,minimum size=4pt] (a) at (0, 0) {};
\node[circle,fill=black,inner sep=0pt,minimum size=4pt] (a) at (1, 1) {};
\node[circle,fill=black,inner sep=0pt,minimum size=4pt] (a) at (2, 2) {};
\node[circle,fill=black,inner sep=0pt,minimum size=4pt] (a) at (3, 2) {};
\node[circle,fill=black,inner sep=0pt,minimum size=4pt] (a) at (4, 1) {};
\end{tikzpicture}
\\
\begin{tikzpicture}[baseline=1pt,scale=0.3]
\draw[thick] (0,0) -- (1,0);
\node[circle,fill=black,inner sep=0pt,minimum size=4pt] (a) at (0, 0) {};
\node[circle,fill=black,inner sep=0pt,minimum size=4pt] (a) at (1, 0) {};
\end{tikzpicture}
\to
\begin{tikzpicture}[baseline=2pt,scale=0.3]
\draw[thick] (0,0) -- (1,1)-- (2,2)--(3,1)--(4,0);
\node[circle,fill=black,inner sep=0pt,minimum size=4pt] (a) at (0, 0) {};
\node[circle,fill=black,inner sep=0pt,minimum size=4pt] (a) at (1, 1) {};
\node[circle,fill=black,inner sep=0pt,minimum size=4pt] (a) at (2, 2) {};
\node[circle,fill=black,inner sep=0pt,minimum size=4pt] (a) at (3, 1) {};
\node[circle,fill=black,inner sep=0pt,minimum size=4pt] (a) at (4, 0) {};
\end{tikzpicture}\\
\begin{tikzpicture}[baseline=1pt,scale=0.3]
\draw[thick] (0,1) -- (1,0);
\node[circle,fill=black,inner sep=0pt,minimum size=4pt] (a) at (0, 1) {};
\node[circle,fill=black,inner sep=0pt,minimum size=4pt] (a) at (1, 0) {};
\end{tikzpicture}
\to
\begin{tikzpicture}[baseline=1pt,scale=0.3]
\draw[thick] (0,1) -- (1,2)-- (2,2)--(3,1)--(4,0);
\node[circle,fill=black,inner sep=0pt,minimum size=4pt] (a) at (0, 1) {};
\node[circle,fill=black,inner sep=0pt,minimum size=4pt] (a) at (1, 2) {};
\node[circle,fill=black,inner sep=0pt,minimum size=4pt] (a) at (2, 2) {};
\node[circle,fill=black,inner sep=0pt,minimum size=4pt] (a) at (3, 1) {};
\node[circle,fill=black,inner sep=0pt,minimum size=4pt] (a) at (4, 0) {};
\end{tikzpicture}
\end{cases}\]
\end{corollary}
{\sc Proof.} Theorem \ref{t:tm} immediately means that $PPL_t(4n),\ldots,PPL_t(4n+4)$ are determined by $PPL_t(n)$ and $PPL_t(n+1)$, and moreover, $d_t(4n)$, $\ldots$, $d_t(4n+3)$ are determined by $d_t(n)$. This means exactly that the sequence $d_t(n)$ is a fixed point of a morphism of length 4. The equality \eqref{e:4n+1} means that the first symbol of any morphic image of $\delta$ is $\begin{tikzpicture}[baseline=1pt,scale=0.3]
\draw[thick] (0,0) -- (1,1);
\node[circle,fill=black,inner sep=0pt,minimum size=4pt] (a) at (0, 0) {};
\node[circle,fill=black,inner sep=0pt,minimum size=4pt] (a) at (1, 1) {};
\end{tikzpicture}$; the equality \eqref{e:4n+3} means that the last symbol of any morphic image of $\delta$ is $ \begin{tikzpicture}[baseline=1pt,scale=0.3]
\draw[thick] (0,1) -- (1,0);
\node[circle,fill=black,inner sep=0pt,minimum size=4pt] (a) at (0, 1) {};
\node[circle,fill=black,inner sep=0pt,minimum size=4pt] (a) at (1, 0) {};
\end{tikzpicture} $; the two symbols in the middle can be found from \eqref{e:4n+2} and depend on $d_t(n)$. \hfill $\Box$
\medskip
With the previous corollary, we can draw the plot of $PPL_t(n)$ as the fixed point of $\delta$.
\begin{figure}[h]
\centering
\begin{tikzpicture}[baseline=1pt,scale=0.1]
\draw [<->] (0,7) node (yaxis) [above] {$PPL_t(n)$}
|- (105,0) node (xaxis) [right] {$n$};
\foreach \x/\xtext in {4/4, 16/16, 64/64}
\draw[shift={(\x,0)}] (0pt,5pt) -- (0pt,-5pt) node[below] {\tiny $\xtext$};
\draw[thick] (0,0)--(1,1)--(2,2)--(3,2)--(4,1)--(5,2)--(6,3)--(7,3)--(8,2)--(9,3)--(10,4)--(11,3)--(12,2)--(13,3)--(14,3)--(15,2)--(16,1)--(17,2)--(18,3)--(19,3)--(20,2)--(21,3)--(22,4)--(23,4)--(24,3)--(25,4)--(26,5)--(27,4)--(28,3)--(29,4)--(30,4)--(31,3)--(32,2)--(33,3)--(34,4)--(35,4)--(36,3)--(37,4)--(38,5)--(39,5)--(40,4)--(41,5)--(42,5)--(43,4)--(44,3)--(45,4)--(46,4)--(47,3)--(48,2)--(49,3)--(50,4)--(51,4)--(52,3)--(53,4)--(54,5)--(55,4)--(56,3)--(57,4)--(58,4)--(59,3)--(60,2)--(61,3)--(62,3)--(63,2)--(64,1)--(65,2)--(66,3)--(67,3)--(68,2)--(69,3)--(70,4)--(71,4)--(72,3)--(73,4)--(74,5)--(75,4)--(76,3)--(77,4)--(78,4)--(79,3)--(80,2)--(81,3)--(82,4)--(83,4)--(84,3)--(85,4)--(86,5)--(87,5)--(88,4)--(89,5)--(90,6)--(91,5)--(92,4)--(93,5)--(94,5)--(95,4)--(96,3)--(97,4)--(98,5)--(99,5)--(100,4);
\node[circle,fill=black,inner sep=0pt,minimum size=2pt] (a) at (0,0) {};
\node[circle,fill=black,inner sep=0pt,minimum size=2pt] (a) at (1,1) {};
\node[circle,fill=black,inner sep=0pt,minimum size=2pt] (a) at (2,2) {};
\node[circle,fill=black,inner sep=0pt,minimum size=2pt] (a) at (3,2) {};
\node[circle,fill=black,inner sep=0pt,minimum size=2pt] (a) at (4,1) {};
\node[circle,fill=black,inner sep=0pt,minimum size=2pt] (a) at (5,2) {};
\node[circle,fill=black,inner sep=0pt,minimum size=2pt] (a) at (6,3) {};
\node[circle,fill=black,inner sep=0pt,minimum size=2pt] (a) at (7,3) {};
\node[circle,fill=black,inner sep=0pt,minimum size=2pt] (a) at (8,2) {};
\node[circle,fill=black,inner sep=0pt,minimum size=2pt] (a) at (9,3) {};
\node[circle,fill=black,inner sep=0pt,minimum size=2pt] (a) at (10,4) {};
\node[circle,fill=black,inner sep=0pt,minimum size=2pt] (a) at (11,3) {};
\node[circle,fill=black,inner sep=0pt,minimum size=2pt] (a) at (12,2) {};
\node[circle,fill=black,inner sep=0pt,minimum size=2pt] (a) at (13,3) {};
\node[circle,fill=black,inner sep=0pt,minimum size=2pt] (a) at (14,3) {};
\node[circle,fill=black,inner sep=0pt,minimum size=2pt] (a) at (15,2) {};
\node[circle,fill=black,inner sep=0pt,minimum size=2pt] (a) at (16,1) {};
\node[circle,fill=black,inner sep=0pt,minimum size=2pt] (a) at (17,2) {};
\node[circle,fill=black,inner sep=0pt,minimum size=2pt] (a) at (18,3) {};
\node[circle,fill=black,inner sep=0pt,minimum size=2pt] (a) at (19,3) {};
\node[circle,fill=black,inner sep=0pt,minimum size=2pt] (a) at (20,2) {};
\node[circle,fill=black,inner sep=0pt,minimum size=2pt] (a) at (21,3) {};
\node[circle,fill=black,inner sep=0pt,minimum size=2pt] (a) at (22,4) {};
\node[circle,fill=black,inner sep=0pt,minimum size=2pt] (a) at (23,4) {};
\node[circle,fill=black,inner sep=0pt,minimum size=2pt] (a) at (24,3) {};
\node[circle,fill=black,inner sep=0pt,minimum size=2pt] (a) at (25,4) {};
\node[circle,fill=black,inner sep=0pt,minimum size=2pt] (a) at (26,5) {};
\node[circle,fill=black,inner sep=0pt,minimum size=2pt] (a) at (27,4) {};
\node[circle,fill=black,inner sep=0pt,minimum size=2pt] (a) at (28,3) {};
\node[circle,fill=black,inner sep=0pt,minimum size=2pt] (a) at (29,4) {};
\node[circle,fill=black,inner sep=0pt,minimum size=2pt] (a) at (30,4) {};
\node[circle,fill=black,inner sep=0pt,minimum size=2pt] (a) at (31,3) {};
\node[circle,fill=black,inner sep=0pt,minimum size=2pt] (a) at (32,2) {};
\node[circle,fill=black,inner sep=0pt,minimum size=2pt] (a) at (33,3) {};
\node[circle,fill=black,inner sep=0pt,minimum size=2pt] (a) at (34,4) {};
\node[circle,fill=black,inner sep=0pt,minimum size=2pt] (a) at (35,4) {};
\node[circle,fill=black,inner sep=0pt,minimum size=2pt] (a) at (36,3) {};
\node[circle,fill=black,inner sep=0pt,minimum size=2pt] (a) at (37,4) {};
\node[circle,fill=black,inner sep=0pt,minimum size=2pt] (a) at (38,5) {};
\node[circle,fill=black,inner sep=0pt,minimum size=2pt] (a) at (39,5) {};
\node[circle,fill=black,inner sep=0pt,minimum size=2pt] (a) at (40,4) {};
\node[circle,fill=black,inner sep=0pt,minimum size=2pt] (a) at (41,5) {};
\node[circle,fill=black,inner sep=0pt,minimum size=2pt] (a) at (42,5) {};
\node[circle,fill=black,inner sep=0pt,minimum size=2pt] (a) at (43,4) {};
\node[circle,fill=black,inner sep=0pt,minimum size=2pt] (a) at (44,3) {};
\node[circle,fill=black,inner sep=0pt,minimum size=2pt] (a) at (45,4) {};
\node[circle,fill=black,inner sep=0pt,minimum size=2pt] (a) at (46,4) {};
\node[circle,fill=black,inner sep=0pt,minimum size=2pt] (a) at (47,3) {};
\node[circle,fill=black,inner sep=0pt,minimum size=2pt] (a) at (48,2) {};
\node[circle,fill=black,inner sep=0pt,minimum size=2pt] (a) at (49,3) {};
\node[circle,fill=black,inner sep=0pt,minimum size=2pt] (a) at (50,4) {};
\node[circle,fill=black,inner sep=0pt,minimum size=2pt] (a) at (51,4) {};
\node[circle,fill=black,inner sep=0pt,minimum size=2pt] (a) at (52,3) {};
\node[circle,fill=black,inner sep=0pt,minimum size=2pt] (a) at (53,4) {};
\node[circle,fill=black,inner sep=0pt,minimum size=2pt] (a) at (54,5) {};
\node[circle,fill=black,inner sep=0pt,minimum size=2pt] (a) at (55,4) {};
\node[circle,fill=black,inner sep=0pt,minimum size=2pt] (a) at (56,3) {};
\node[circle,fill=black,inner sep=0pt,minimum size=2pt] (a) at (57,4) {};
\node[circle,fill=black,inner sep=0pt,minimum size=2pt] (a) at (58,4) {};
\node[circle,fill=black,inner sep=0pt,minimum size=2pt] (a) at (59,3) {};
\node[circle,fill=black,inner sep=0pt,minimum size=2pt] (a) at (60,2) {};
\node[circle,fill=black,inner sep=0pt,minimum size=2pt] (a) at (61,3) {};
\node[circle,fill=black,inner sep=0pt,minimum size=2pt] (a) at (62,3) {};
\node[circle,fill=black,inner sep=0pt,minimum size=2pt] (a) at (63,2) {};
\node[circle,fill=black,inner sep=0pt,minimum size=2pt] (a) at (64,1) {};
\node[circle,fill=black,inner sep=0pt,minimum size=2pt] (a) at (65,2) {};
\node[circle,fill=black,inner sep=0pt,minimum size=2pt] (a) at (66,3) {};
\node[circle,fill=black,inner sep=0pt,minimum size=2pt] (a) at (67,3) {};
\node[circle,fill=black,inner sep=0pt,minimum size=2pt] (a) at (68,2) {};
\node[circle,fill=black,inner sep=0pt,minimum size=2pt] (a) at (69,3) {};
\node[circle,fill=black,inner sep=0pt,minimum size=2pt] (a) at (70,4) {};
\node[circle,fill=black,inner sep=0pt,minimum size=2pt] (a) at (71,4) {};
\node[circle,fill=black,inner sep=0pt,minimum size=2pt] (a) at (72,3) {};
\node[circle,fill=black,inner sep=0pt,minimum size=2pt] (a) at (73,4) {};
\node[circle,fill=black,inner sep=0pt,minimum size=2pt] (a) at (74,5) {};
\node[circle,fill=black,inner sep=0pt,minimum size=2pt] (a) at (75,4) {};
\node[circle,fill=black,inner sep=0pt,minimum size=2pt] (a) at (76,3) {};
\node[circle,fill=black,inner sep=0pt,minimum size=2pt] (a) at (77,4) {};
\node[circle,fill=black,inner sep=0pt,minimum size=2pt] (a) at (78,4) {};
\node[circle,fill=black,inner sep=0pt,minimum size=2pt] (a) at (79,3) {};
\node[circle,fill=black,inner sep=0pt,minimum size=2pt] (a) at (80,2) {};
\node[circle,fill=black,inner sep=0pt,minimum size=2pt] (a) at (81,3) {};
\node[circle,fill=black,inner sep=0pt,minimum size=2pt] (a) at (82,4) {};
\node[circle,fill=black,inner sep=0pt,minimum size=2pt] (a) at (83,4) {};
\node[circle,fill=black,inner sep=0pt,minimum size=2pt] (a) at (84,3) {};
\node[circle,fill=black,inner sep=0pt,minimum size=2pt] (a) at (85,4) {};
\node[circle,fill=black,inner sep=0pt,minimum size=2pt] (a) at (86,5) {};
\node[circle,fill=black,inner sep=0pt,minimum size=2pt] (a) at (87,5) {};
\node[circle,fill=black,inner sep=0pt,minimum size=2pt] (a) at (88,4) {};
\node[circle,fill=black,inner sep=0pt,minimum size=2pt] (a) at (89,5) {};
\node[circle,fill=black,inner sep=0pt,minimum size=2pt] (a) at (90,6) {};
\node[circle,fill=black,inner sep=0pt,minimum size=2pt] (a) at (91,5) {};
\node[circle,fill=black,inner sep=0pt,minimum size=2pt] (a) at (92,4) {};
\node[circle,fill=black,inner sep=0pt,minimum size=2pt] (a) at (93,5) {};
\node[circle,fill=black,inner sep=0pt,minimum size=2pt] (a) at (94,5) {};
\node[circle,fill=black,inner sep=0pt,minimum size=2pt] (a) at (95,4) {};
\node[circle,fill=black,inner sep=0pt,minimum size=2pt] (a) at (96,3) {};
\node[circle,fill=black,inner sep=0pt,minimum size=2pt] (a) at (97,4) {};
\node[circle,fill=black,inner sep=0pt,minimum size=2pt] (a) at (98,5) {};
\node[circle,fill=black,inner sep=0pt,minimum size=2pt] (a) at (99,5) {};
\end{tikzpicture}
\caption{$PPL_t(n)$}\label{f:plot}
\end{figure}
The next proposition can be obtained from Theorem \ref{t:tm} by elementary computations. Recall that $SP(k)=SP_t(k)$ is the length $n$ of the shortest prefix of $t$ such that its palindromic length $PPL_t(n)$ is equal to $k$.
\begin{proposition}\label{p:sp}
We have $SP_t(1)=1$, $SP_t(2)=2$, $SP_t(3)=6$ and for all $k>0$,
\[SP_t(k+3)=16 SP_t(k)-6.\]
\end{proposition}
{\sc Proof.} Let us introduce $SP_2(k)$ as the minimal number $n$ such that $PPL_t(n)=PPL_t(n+1)=k$. By definition, $SP_2(k)\geq SP(k)$. The first values of $SP(k)$ and $SP_2(k)$ are given below.
\medskip
\begin{center}
\begin{tabular}{|c||c|c|c|c|c|}
\hline
$k$ &1 & 2& 3&4&5 \\
\hline
$SP(k)$ & 1& 2& 6&10& 26\\
\hline
$SP_2(k)$ & $\infty$ &2&6&22&38\\
\hline
\end{tabular}
\end{center}
\medskip
From the definition of the morphism $\delta$ we immediately see that a new value $n=SP(k)$ can appear either in the middle of the $\delta$-image of $d(n')=d(SP_2(k-2))$, or in the middle of the $\delta$-image of $d(n'')$, where $n''=SP(k-1)-1$. The latter case is also the only possible way to get a new value $n=SP_2(k)$. So,
\begin{equation}\label{e:min}
SP(k)=\min(4SP_2(k-2)+2, 4SP(k-1)-2),
\end{equation}
\begin{equation}\label{e:sp2}
SP_2(k)=4SP(k-1)-2.
\end{equation}
As we see from the table, for $3\leq k\leq 5$, we have
$SP(k-1)\leq SP_2(k-1)<SP(k)$. The first inequality is obvious, but let us prove the second one by induction. Its base is observed for $3\leq k\leq 5$, so, consider $k\geq 6$ such that for all $k'=k-3,k-2,k-1$ we have $SP_2(k'-1)<SP(k')$.
In particular, $SP_2(k-4)<SP(k-3)$, so due to \eqref{e:min}, we have $SP(k-2)=4SP_2(k-4)+2$, and so due to \eqref{e:sp2},
\begin{equation}\label{e:sp22}
SP_2(k-1)=16SP_2(k-4)+6.
\end{equation}
On the other hand, we have $SP_2(k-2)<SP(k-1)$, so, \eqref{e:min} becomes $SP(k)=4SP_2(k-2)+2$, and together with \eqref{e:sp2} this gives
\begin{equation}\label{e:spk}
SP(k)=16 SP(k-3)-6.
\end{equation}
Combining \eqref{e:sp22}, \eqref{e:spk}, the induction base $SP_2(k-4)<SP(k-3)$ and the fact that all the values are integers, we obtain that $SP_2(k-1)<SP(k)$ for all $k\geq 3$. We also see that \eqref{e:spk} is true for all $k\geq 4$, proving this proposition. \hfill $\Box$
\medskip
The following corollary of the previous proposition can be proved by straightforward induction.
\begin{corollary}
In the 4-ary numeration system, we have $SP(3k+2)=((12)^k2)_4$ for all $k \geq 0$; $SP(3k)=(1(12)^{k-1}2)_4$ for all $k \geq 1$; $SP(3k+1)=(2(12)^{k-1}2)_4$ for all $k \geq 1$.
\end{corollary}
Another direct consequence of Proposition \ref{p:sp} is
\begin{corollary}
We have $$\displaystyle \lim \sup \frac{PPL_t(n)}{\ln n}=\frac{3}{4 \ln 2},$$ whereas $\displaystyle \lim \inf \frac{PPL_t(n)}{\ln n}=0$ since $PPL_t(4^m)=1$ for all $m$.
\end{corollary}
\section{Regularity}
The sequence $(PPL_t(n))$ is closely related to the Thue-Morse word, the most classical example of a 2-automatic sequence. In general, a sequence $w=(w[n])$ is called $k$-automatic if there exists a finite automaton such that for the input equal to the $k$-ary representation of $n$, the output is equal to $w[n]$. Equivalently, due to a theorem by Cobham \cite{cobham72}, a sequence is $k$-automatic if and only if it is an image under a coding $c: \Sigma \to \Delta$ of a fixed point of a $k$-uniform morphism $\varphi$: $w=c(w')$, where $w'=\varphi(w')$ \cite[Ch.\ 6]{a_sh}. So, the Thue-Morse word is 2-automatic since it is a fixed point of the $2$-uniform morphism $a \to ab, b \to ba$, and the sequence $(d(n))$ is 4-automatic since it is a fixed point of $\delta$. In both cases, the coding can be taken to be trivial: $c(x)=x$ for every letter $x$. It is also well-known that a sequence is $k$-automatic if and only if it is $k^m$-automatic for any integer $m$, so, the Thue-Morse word is also 4-automatic and the sequence $(d(n))$ is 2-automatic.
A more general notion of a {\it $k$-regular sequence} was introduced by Allouche and Shallit \cite{a_sh_kreg}, see also \cite[Ch.\ 16]{a_sh}. A sequence $(a(n))$ is called $k$-regular (on $\mathbb Z$) if there exists a finite number of sequences $\{(a_1(n)),\ldots,(a_s(n))\}$ such that for every integer $i \geq 0$ and $0\leq b <k^i$ there exist $c_1,\ldots,c_s \in \mathbb Z$ such that for all $n\geq 0$ we have
\[a(k^i n +b)=\sum_{1\leq j \leq s} c_j a_j(n).\]
It is also known that a sequence is $k$-automatic if and only if it is $k$-regular and takes on finitely many values \cite[Thm.\ 16.1.4]
{a_sh}. Moreover, a sequence $a=(a(n))$ is $k$-regular if and only if there exist $r$ sequences $a_1=a,\ldots,a_r$ and a matrix-valued morphism $\mu$ such that if
\[V(n)=\begin{pmatrix} a_1(n) \\ a_2(n) \\ \cdots \\ a_r(n) \end{pmatrix},\]
then
\[V(kn+b)=\mu(b)V(n)\]
for $0\leq b < k$ \cite[Thm.\ 16.1.3]{a_sh}.
Many sequences related to $k$-automatic words are $k$-regular, as it was shown by Charlier, Rampersad and Shallit \cite{crs}. However, it seems that the general approach from the mentioned paper does not directly work for the palindromic length, so, we have to prove the following corollary only for the Thue-Morse word.
\begin{corollary}
The sequence $PPL_t(n)$ is 4-regular.
\end{corollary}
{\sc Proof.} The sequence $(d(n))$, here considered on the alphabet $\{-1,0,1\}$, is 4-automatic as a fixed point of $\delta$ and thus 4-regular, as well as its image $(e(n))$ under the following coding:
$$e(n)=\begin{cases} 1, d(n)=-1,\\ 0, \mbox{~otherwise}.\end{cases} $$
So, for each $b=0,1,2,3$ it is sufficient to combine the respective matrices $\mu$ for $d$ and $e$ with the following equalities equivalent to Theorem \ref{t:tm}:
\begin{align*}
PPL_t(4n)&=PPL_t(n),\\
PPL_t(4n+1)&=PPL_t(n)+1,\\
PPL_t(4n+2)&=PPL_t(n)+2-e(n),\\
PPL_t(4n+3)&=PPL_t(n)+d(n)+1. ~~~~~~~~\hfill ~ \Box
\end{align*}
\begin{remark}{\rm
In fact, every sequence with $k$-automatic first differences is $k$-regular, which can be proven with a similar construction, perhaps with several auxiliary sequences like $(e(n))$.
}
\end{remark}
\section{Conclusion}
Up to my knowledge, the results of this paper are thus far only precise formulas for the prefix palindromic length of a non-trivial infinite word not constructed especially for that.
Even for famous and simple examples like Toeplitz words or the Fibonacci word \seqnum{A003849}, lower bounds for the prefix palindromic length are difficult \cite{f:num,dlt}. The only more or less universal lower bounds for all $p$-power-free words are those from the first paper on the subject \cite{fpz}, with $SP(k)$ growing faster than any constant to the power $k$. Later \cite{f:num}, some calculations allowed a reasonable exponential conjecture on the $SP(k)$ of the Fibonacci word, but it is not clear how to prove it. So, the following more particular open questions can be added to general Conjectures \ref{c1} and \ref{c2}.
\begin{problem}
Find a precise formula for the prefix palindromic length of the period-doubling word \seqnum{A096268}, or a lower bound for its $\lim \sup$.
\end{problem}
\begin{problem}
Find a precise formula for the prefix palindromic length of the Fibonacci word \seqnum{A003849}, or a lower bound for its $\lim \sup$.\end{problem}
\begin{problem}
Is it true that the function $PPL_u(n)$ is $k$-regular for any $k$-automatic word $u$? Fibonacci-regular for the Fibonacci word?
\end{problem}
| {
"timestamp": "2019-08-29T02:10:35",
"yymm": "1906",
"arxiv_id": "1906.09392",
"language": "en",
"url": "https://arxiv.org/abs/1906.09392",
"abstract": "The prefix palindromic length $PPL_u(n)$ of an infinite word $u$ is the minimal number of concatenated palindromes needed to express the prefix of length $n$ of $u$. In a 2013 paper with Puzynina and Zamboni we stated the conjecture that $PPL_u(n)$ is unbounded for every infinite word $u$ which is not ultimately periodic. Up to now, the conjecture has been proven for almost all words, including all words avoiding some power $p$. However, even in that simple case the existing upper bound for the minimal number $n$ such that $PPL_u(n)>K$ is greater than any constant to the power $K$. Precise values of $PPL_u(n)$ are not known even for simplest examples like the Fibonacci word.In this paper, we give the first example of such a precise computation and compute the function of the prefix palindromic length of the Thue-Morse word, a famous test object for all functions on infinite words. It happens that this sequence is $2$-regular, which raises the question if this fact can be generalized to all automatic sequences.",
"subjects": "Discrete Mathematics (cs.DM); Combinatorics (math.CO)",
"title": "Prefix palindromic length of the Thue-Morse word",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9854964220125032,
"lm_q2_score": 0.718594386544335,
"lm_q1q2_score": 0.7081721968177118
} |
https://arxiv.org/abs/1807.02333 | Reflexivity of Rings via Nilpotent Elements | An ideal $I$ of a ring $R$ is called left N-reflexive if for any $a\in$ nil$(R)$, $b\in R$, being $aRb \subseteq I$ implies $bRa \subseteq I$ where nil$(R)$ is the set of all nilpotent elements of $R$. The ring $R$ is called left N-reflexive if the zero ideal is left N-reflexive. We study the properties of left N-reflexive rings and related concepts. Since reflexive rings and reduced rings are left N-reflexive, we investigate the sufficient conditions for left N-reflexive rings to be reflexive and reduced. We first consider basic extensions of left N-reflexive rings. For an ideal-symmetric ideal $I$ of a ring $R$, $R/I$ is left N-reflexive. If an ideal $I$ of a ring $R$ is reduced as a ring without identity and $R/I$ is left N-reflexive, then $R$ is left N-reflexive. If $R$ is a quasi-Armendariz ring and the coefficients of any nilpotent polynomial in $R[x]$ are nilpotent in $R$, it is proved that $R$ is left N-reflexive if and only if $R[x]$ is left N-reflexive. We show that the concept of N-reflexivity is weaker than that of reflexivity and stronger than that of left N-right idempotent reflexivity and right idempotent reflexivity which are introduced in Section 5. | \section{Introduction}
Throughout this paper, all rings are associative with identity. A
ring is called {\it reduced} if it has no nonzero nilpotent
elements. A weaker condition than ``reduced'' is defined by Lambek
in \cite{Lam}. A ring $R$ is said to be {\it symmetric} if for any
$a$, $b, c\in R$, $abc=0$ implies $acb=0$. Equivalently, $abc=0$
implies $bac=0$. An equivalent condition on a ring to be symmetric
is that whenever a product of any number of elements is zero, any
permutation of the factors still yields product zero.
In \cite{Mi}, a right ideal $I$ of $R$ is said to be {\it
reflexive} if $aRb \subseteq I$ implies $bRa \subseteq I$ for any
$a$, $b\in R$. $R$ is called a {\it reflexive ring} if $aRb = 0$
implies $bRa = 0$ for any $a$, $b\in R$. And in \cite{ZZG}, $R$
is said to be a {\it weakly reflexive ring} if $aRb = 0$ implies
$bRa\subseteq \mbox{nil}(R)$ for any $a$, $b\in R$. In \cite{KUH},
a ring $R$ is said to be {\it nil-reflexive} if $aRb\subseteq$
nil$(R)$ implies that $bRa\subseteq$ nil$(R)$ for any $a$, $b\in
R$. Let $R$ be a ring. In \cite{JAKL}, $R$ is called a {\it
reflexivity with maximal ideal axis ring}, in short, an {\it RM
ring} if $aMb = 0$ for a maximal ideal $M$ and for any $a$, $b\in
R$, then $bMa = 0$, similarly, $R$ has {\it reflexivity with
maximal ideal axis on idempotents}, simply, {\it RMI}, if $eMf =
0$ for any idempotents $e$, $f$ and a maximal ideal of $M$, then
$fMe = 0$. In \cite{KL1}, $R$ has {\it
reflexive-idempotents-property}, simply, {\it RIP}, if $eRf = 0$
for any idempotents $e$, $f$, then $fRe = 0$, A left ideal $I$ is
called {\it idempotent reflexive} \cite{Ki} if $aRe\subseteq I$
implies $eRa\subseteq I$ for $a$, $e^2 = e\in R$. A ring $R$ is
called {\it idempotent reflexive} if $0$ is an idempotent
reflexive ideal. And Kim and Baik \cite{KB}, introduced the left
and right idempotent reflexive rings. A two sided ideal $I$ of a
ring $R$ is called {\it right idempotent reflexive} if
$aRe\subseteq I$ implies $eRa\subseteq I$ for any $a$, $e^2 = e\in
R$. A ring $R$ is called {\it right idempotent reflexive} if $0$
is the right idempotent reflexive ideal. Left idempotent reflexive
ideals and rings are defined similarly. If a ring $R$ is left and
right idempotent reflexive then it is called an idempotent
reflexive ring.
In this paper, motivated by these classes of types of reflexive
rings, we introduce left N-reflexive rings and right N-reflexive
rings. We prove that some results of reflexive rings can be
extended to the left N-reflexive rings for this general setting.
We investigate characterizations of left N-reflexive rings, and
that many families of left N-reflexive rings are presented.
In what follows, $\Bbb{Z}$ denotes the ring of integers and for a
positive integer $n$, $\Bbb{Z}_n$ is the ring of integers modulo
$n$. We write $M_n(R)$ for the ring of all $n\times n$ matrices,
$U(R)$, nil$(R)$ will denote the group of units and the set of
all nilpotent elements of $R$, $U_n(R)$ is the ring of upper
triangular matrices over $R$ for a positive integer $n\geq 2$, and
$D_n(R)$ is the ring of all matrices in $U_n(R)$ having main
diagonal entries equal.
\section{N-reflexivity of rings} In this section, we introduce a class
of rings, so-called left N-reflexive rings and right N-reflexive
rings. These classes of rings generalize reflexive rings. We
investigate which properties of reflexive rings hold for the left
N-reflexive case. We supply an example to show that there are left
N-reflexive rings that are neither right N-reflexive nor reflexive
nor reversible. It is shown that the class of left N-reflexive
rings is closed under finite direct sums. We have an example to
show that homomorphic image of a left N-reflexive ring is not left
N-reflexive. Then, we determine under what conditions a
homomorphic image of a ring is left N-reflexive.
\noindent We now give our main definition.
\begin{df}{\rm Let $R$ be a ring and $I$ an ideal of $R$. $I$ is called {\it left N-reflexive} if for any
$a\in$ nil$(R)$, $b\in R$, being $aRb \subseteq I$ implies $bRa
\subseteq I$. The ring $R$ is called {\it left N-reflexive} if the
zero ideal is left N-reflexive. Similarly, $I$ is called {\it
right N-reflexive} if for any $a\in$ nil$(R)$, $b\in R$, being
$bRa \subseteq I$ implies $aRb \subseteq I$. The ring $R$ is
called {\it right N-reflexive} if the zero ideal is right
N-reflexive. The ring $R$ is called {\it N-reflexive} if it is
left and right N-reflexive.}
\end{df}
Clearly, every reflexive ring and every semiprime ring are
N-reflexive. There are left N-reflexive rings which are not
semiprime and there are left N-reflexive rings which are neither
reduced nor reversible.
Let $F$ be a field and $R = F[x]$ be the polynomial ring over
$F$ with $x$ an indeterminate and $\alpha : R\rightarrow R$ be a
homomorphism defined by $\alpha(f(x)) = f(0)$ where $f(0)$ is the
constant term of $f(x)$. Let $D^{\alpha}_2(R)$ denote skewtrivial
extension of $R$ by $R$ and $\alpha$. So $D^{\alpha}_2(R) =
\left\{\left(\begin{array}{cc}f(x)&g(x)\\0&f(x)\end{array}\right)\mid
f(x), g(x)\in R\right\}$ is the ring with componentwise addition
of matrices and multiplication: \begin{center}
$\left(\begin{array}{cc}f(x)&g(x)\\0&f(x)\end{array}\right)\left(\begin{array}{cc}h(x)&t(x)\\0&h(x)\end{array}\right)
= \left(\begin{array}{cc}f(x)h(x)&\alpha(f(x))t(x) +
g(x)h(x)\\0&f(x)h(x)\end{array}\right)$.\end{center} There are
left N-reflexive rings which are neither reflexive nor semiprime. The N-reflexive property of rings is not left-right symmetric.
\begin{ex}{\rm
Let $D^{\alpha}_2(R)$ denote skewtrivial extension of $R$ by $R$
and $\alpha$ as mentioned above. Then by \cite[Example 3.5]{ZZG},
$D^{\alpha}_2(R)$ is not reflexive. We show that $D^{\alpha}_2(R)$
is left N-reflexive. Note that the set of all nilpotent elements
of $D^{\alpha}_2(R)$ is the set
$\left\{\left(\begin{array}{cc}0&f(x)\\0&0\end{array}\right)\mid
f(x)\in R\right\}$. Let $A =
\left(\begin{array}{cc}0&f(x)\\0&0\end{array}\right)$ be a
nilpotent in $D^{\alpha}_2(R)$ and $B =
\left(\begin{array}{cc}h(x)&g(x)\\0&h(x)\end{array}\right)$ any
element in $D^{\alpha}_2(R)$. Assume that $AD^{\alpha}_2(R)B = 0$.
We may assume $f(x)\neq 0$. Then an easy calculation,
$AD^{\alpha}_2(R)B = 0$ reveals that $h(x) = 0$, and also
$BD^{\alpha}_2(R)A = 0$. Hence $D^{\alpha}_2(R) $ is left
N-reflexive. Next we show that $D^{\alpha}_2(R)$ is not right
N-reflexive. Let $A =
\left(\begin{array}{cc}0&f(x)\\0&0\end{array}\right)$ be a
nilpotent and $B =
\left(\begin{array}{cc}xh(x)&g(x)\\0&xh(x)\end{array}\right)$ any
element in $D^{\alpha}_2(R)$ with both $f(x)$ and $h(x)$ nonzero.
By definitions $BD^{\alpha}_2(R)A = 0$. Since $xh(x)f(x)$ is
nonzero, $AD^{\alpha}_2(R)B =
\left(\begin{array}{cc}0&f(x)r(x)xh(x)\\0&0\end{array}\right)$ is
nonzero for some nonzero $r(x)\in R$. So $D^{\alpha}_2(R)$ is not
right N-reflexive.
On the other hand, nil$(D^{\alpha}_2(R))$
is an ideal of $D^{\alpha}_2(R)$ and
$(\mbox{nil}(D^{\alpha}_2(R)))^2=0$ but nil$(D^{\alpha}_2(R))\neq
0$. Therefore $D^{\alpha}_2(R)$ is not semiprime.}
\end{ex}
\begin{prop}\label{ilk} Let $R$ be a left N-reflexive ring. Then for any
idempotent $e$ of $R$, $eRe$ is also left N-reflexive.
\end{prop}
\begin{proof} Let $eae\in eRe$ be a nilpotent and $ebe\in eRe$
arbitrary element with $eaeRebe = 0$. Then we have $ebeReae = 0$ since $R$ is left N-reflexive.
\end{proof}
\begin{exs}\label{ex}{\rm (1) Let $F$ be a field and $R = M_2(F)$.
In fact, $R$ is a simple ring, therefore prime. Let $A$, $B\in R$ with $ARB = 0$. Since $R$ is prime, $A = 0$ or
$B = 0$. Hence $BRA = 0$. So
$R$ is reflexive. Therefore $R$ is N-reflexive.\\
(2) Let $F$ be a field and consider the ring $R = D_3(F)$. Then
$R$ is neither left N-reflexive nor right N-reflexive. Let $A =
\left(\begin{array}{ccc}0&0&1\\0&0&1\\0&0&0\end{array}\right)$ be
nilpotent in $D_3(F)$ and $B =
\left(\begin{array}{ccc}0&1&1\\0&0&1\\0&0&0\end{array}\right)\in
D_3(F)$. Then $ARB = 0$. For $C =
\left(\begin{array}{ccc}1&1&1\\0&1&1\\0&0&1\end{array}\right)$,
$BCA =
\left(\begin{array}{ccc}0&0&1\\0&0&0\\0&0&0\end{array}\right)\neq
0$. Hence $R$ is not left N-reflexive.\\ Next we show that $R$ is
not right N-reflexive either. Now assume that $A =
\left(\begin{array}{ccc}0&1&1\\0&0&0\\0&0&0\end{array}\right)\in
\mbox{nil}(R)$ and $B =
\left(\begin{array}{ccc}0&1&1\\0&0&1\\0&0&0\end{array}\right)\in
R$. It is clear that $BRA = 0$. For $C =
\left(\begin{array}{ccc}1&1&1\\0&1&1\\0&0&1\end{array}\right)\in
R$, we have $ACB\neq 0$. Hence $R$ is not right N-reflexive.}
\end{exs}
\begin{lem}\label{iso} N-reflexivity of rings is preserved under isomorphisms.
\end{lem}
\begin{thm} Let $R$ be a ring. Assume that $M_n(R)$ is left N-reflexive. Then $R$ is left N-reflexive.
\end{thm}
\begin{proof
Suppose that $M_n(R)$ is a left N-reflexive ring. Let $e_{ij}$
denote the matrix unit which $(i,j)$-entry is $1$ and the other
entries are $0$. Then $R\cong Re_{11}=e_{11}M_{n}(R)e_{11}$ is
N-reflexive by Proposition \ref{ilk} and Lemma \ref{iso}.
\end{proof}
\begin{prop}\label{reversible} Every reversible ring is left and right N-reflexive.
\end{prop}
\begin{proof} Clear by the definitions.
\end{proof}
The converse statement of Proposition \ref{reversible} may not be
true in general as shown below.
\begin{ex}{\rm By Examples \ref{ex}(1), $M_2(F)$ is both left and right N-reflexive. But it
is not reversible. }\end{ex}
\begin{thm} Let $R$ be a ring. Then the following are equivalent.
\begin{enumerate}
\item $R$ is left N-reflexive.
\item $IRJ=0$ implies $JRI=0$ for any ideal $I$ generated by a nilpotent element and
any nonempty subset $J$ of $R$.
\item $IJ=0$ implies $JI=0$ for any ideal $I$ generated by a nilpotent element and any
ideal $J$ of $R$.
\end{enumerate}
\end{thm}
\begin{proof} (1) $\Rightarrow$ (2) Assume that $R$ is left N-reflexive. Let $I=RaR$ with $a\in R$
nilpotent and $\emptyset\neq J\subseteq R$ such that $IRJ=0$. Then
for any $b\in J$, $aRb=0$. This implies that $bRa=0$, hence
$bR(RaR)=bRI=0$ for any $b\in J$. Thus $JRI=0$.\\
(2) $\Rightarrow$ (3) Let $I=RaR$ with $a\in R$ nilpotent and $J$
be an ideal of $R$ such that $IJ=0$. Then $J=RJ$, so $IRJ=0$. By
(2), $JRI=0$, thus $JI=0$.\\
(3) $\Rightarrow$ (1) Let $a\in R$ be nilpotent and $b\in R$ with
$aRb=0$. Then $(RaR)(RbR)=0$. By (3), $(RbR)(RaR)=0$. Hence
$bRa=0$. Therefore $R$ is left N-reflexive.
\end{proof}
For any element $a\in R$, $r_{R}(a)=\{b\in R\mid ab=0\}$ is called
the {\it right annihilator of $a$ in $R$}. The {\it left
annihilator of $a$ in $R$} is defined similarly and denoted by
$l_{R}(a)$.
\begin{prop}
Let $R$ be a ring. Then $R$ is N-reflexive if and only if for any
nilpotent element $a$ of $R$, $r_{R}(aR)=l_{R}(Ra)$.
\end{prop}
\begin{proof}
For the necessity, let $x\in r_{R}(aR)$ for any nilpotent element
$a\in R$. We have $(aR)x=0$. The ring $R$ being N-reflexive
implies
$xRa=0$. So $x\in l_{R}(Ra)$. It can be similarly showed that $l_{R}(Ra)\subseteq r_{R}(aR)$.\\
\noindent For the sufficiency, let $a\in$ nil$(R)$ and $b\in R$
with $aRb=0$. Then $b\in r_{R}(aR)$. By hypothesis, $b\in
l_{R}(Ra)$, and so $bRa=0$. Thus $R$ is N-reflexive.
\end{proof}
For a field $F$, $D_3(F)$ is neither left N-reflexive nor right
N-reflexive. Subrings of left N-reflexive rings or right
N-reflexive rings need not be left N-reflexive or right
N-reflexive, respectively. But there are some subrings of $D_3(F)$
that are left N-reflexive or right N-reflexive as shown below.
\begin{prop}\label{saturday} Let $R$ be a reduced ring (i.e., it has no nonzero nilpotent elements). Then the following hold.
\begin{enumerate}
\item Let $S =
\left\{\left(\begin{array}{ccc}a&b&c\\0&a&0\\0&0&a\end{array}\right)\mid
a, b, c\in R\right\}$ be a subring of $D_3(R)$. Then $S$ is
N-reflexive.
\item Let $S =
\left\{\left(\begin{array}{ccc}a&0&c\\0&a&b\\0&0&a\end{array}\right)\mid
a, b, c\in R\right\}$ be a subring of $D_3(R)$. Then $S$ is
N-reflexive.
\end{enumerate}
\end{prop}
\begin{proof} (1) Let $A = \left(\begin{array}{ccc}0&b&c\\0&0&0\\0&0&0\end{array}\right)\in S$ be
any nonzero nilpotent element and $B =
\left(\begin{array}{ccc}u&v&t\\0&u&0\\0&0&u\end{array}\right)\in
S$. Assume that $ASB = 0$. This implies $AB=0$, and so $bu=0$ and
$cu=0$. For any
$C=\left(\begin{array}{ccc}x&y&z\\0&x&0\\0&0&x\end{array}\right)\in
S$,
$BCA=\left(\begin{array}{ccc}0&uxb&uxc\\0&0&0\\0&0&0\end{array}\right)$.
Being $bu=cu=0$ implies $(uxb)^2=(uxc)^2=0$. Since $R$ is reduced,
$uxb=uxc=0$. Then $BCA=0$. Hence $BSA = 0$. Thus $R$ is left
N-reflexive. A similar proof
implicates that $S$ is right N-reflexive.\\
(2) By \cite[Proposition 2.2]{ZZG}.
\end{proof}
The condition $R$ being reduced in Proposition \ref{saturday} is
not superfluous as the following example shows.
\begin{ex}\label{örn}{\rm Let $F$ be a field and $R = F<a, b>$ be the free algebra with noncommuting indeterminates $a$,
$b$ over $F$. Let $I$ be the ideal of $R$ generated by $aRb$ and
$a^2$. Consider the ring $\overline {R} = R/I$. Let $\overline a$,
$\overline b\in \overline R$. Then $\overline a\overline
R\overline b = 0$. But $\overline b\overline R\overline a\neq 0$
since $ba\notin I$. Note that $\overline R$ is not reduced.
Consider the ring $S=\left\{\left(\begin{array}{ccc} \overline
x&\overline
y&\overline z\\ \overline 0&\overline x&\overline 0\\
\overline0&\overline 0&\overline x\end{array}\right)\mid \overline
x, \overline y, \overline z\in \overline R\right\}$. Let
$A=\left(\begin{array}{ccc} \overline a&\overline
1&\overline 1\\ \overline 0&\overline a&\overline 0\\
\overline0&\overline 0&\overline a\end{array}\right) \in nil(S) $ and
$B=\left(\begin{array}{ccc} \overline 0&\overline
b&\overline b\\ \overline 0&\overline 0&\overline 0\\
\overline0&\overline 0&\overline 0\end{array}\right) \in S$. Then
$ASB=0$ since $\overline a \overline R \overline b=0$. However
$BA\neq 0$ since $\overline b \overline a\neq 0$. Hence $S$ is not
left N-reflexive.}
\end{ex}
\begin{df} Let $R$ be a ring. We call that $R$ is {\it left N-reversible} if for any nilpotent
$a\in R$ and $b\in R$, $ab = 0$ implies $ba = 0$.
\end{df}
In \cite{MMZ}, a ring $R$ is called {\it nil-semicommutative} if
for every nilpotent $a$, $b\in R$, $ab = 0$ implies $aRb = 0.$
\begin{thm} If a ring $R$ is N-reversible, then $R$ is nil-semicommutative and N-reflexive.
\end{thm}
\begin{proof} Assume that $R$ is N-reversible. Let $a\in R$ be nilpotent and $b\in R$
with $aRb = 0$. Then $ab=0$. For any $r\in R$, being $abr=0$
implies $bra=0$, and so $bRa=0$. Hence $R$ is left N-reflexive. By
a similar discussion, $R$ is right N-reflexive. So $R$ is
N-reflexive. In order to see that $R$ is nil-semicommutative, let
$a, b\in R$ be nilpotent with $ab=0$. N-reversibility of $R$
implies $ba=0$, and so $bar=0$ for any $r\in R$. Again by the
N-reversibility of $R$, we have $arb=0$. Thus $aRb=0$.
\end{proof}
Note that in a subsequent paper, N-reversible rings will be
studied in detail by the present authors.
Let $R$ be a ring and $I$ an ideal of $R$. Recall by \cite{CKL},
$I$ is called {\it ideal-symmetric} if $ABC\subseteq I$ implies
$ACB\subseteq I$ for any ideals $A, B, C$ of $R$. In this vein, we
mention the following result.
\begin{prop} Let $R$ be a ring and $I$ an ideal-symmetric ideal of $R$.
Then $R/I$ is an N-reflexive ring.
\end{prop}
\begin{proof} Let $\overline a \in R/I$ be nilpotent and $\overline b\in
R/I$ with $\overline a (R/I) \overline b=0$. Then $aRb\subseteq
I$. So $R(RaR)(RbR)\subseteq I$. By hypothesis,
$R(RbR)(RaR)\subseteq I$. Therefore $bRa\subseteq I$, and so
$\overline b(R/I)\overline a=0$. It means that $R/I$ is left
N-reflexive. Similarly, it can be shown that $R/I$ is also right
N-reflexive.
\end{proof}
Let $R$ be a ring and $I$ an ideal of $R$. In the short exact
sequence $0\rightarrow I\rightarrow R\rightarrow R/I\rightarrow
0$, $I$ being N-reflexive (as a ring without identity) and $R/I$
being N-reflexive need not imply that $R$ is N-reflexive.
\begin{ex}{\rm Let $F$ be a field and consider the ring $R = D_3(F)$. Let
$I = \begin{pmatrix}0&F&F\\0&0&F\\0&0&0\end{pmatrix}$. Then $I$ is
N-reflexive since $I^3 = 0$. Also, $R/I$ is N-reflexive since
$R/I$ is isomorphic to $F$. However, by Examples \ref{ex} (2), $R$
is not N-reflexive. }\end{ex}
\begin{thm}\label{böl} Let $R$ be a ring and $I$ an ideal of $R$. If $I$ is
reduced as a ring (without identity) and $R/I$ is left
N-reflexive, then $R$ is left N-reflexive.
\end{thm}
\begin{proof} Let $a$ be nilpotent in $R$ and $b\in R$ with
$aRb=0$. Then $\overline a (R/I) \overline b=0$ and $\overline a$
is nilpotent in $R/I$. By hypothesis, $\overline b (R/I) \overline
a=0$. Hence $bRa\subseteq I$. Since $I$ is reduced and $bRa$ is
nil, $bRa=0$.
\end{proof}
The reduced condition on the ideal $I$ in Theorem \ref{böl} is
not superfluous.
\begin{ex}{\rm Let $F$ be a field and $I = \left\{\left(\begin{array}{ccc}0&a&b\\0&0&c\\0&0&0\end{array}\right)\mid a,b,c\in F\right\}$
denote the ideal of $D_3(F)$. Then by Examples \ref{ex}(2),
$D_3(F)$ is not left and right N-reflexive. The ring $D_3(F)/I$ is
isomorphic to $F$ and so it is left and right N-reflexive. Note
that $I$ is not reduced since $I^3 = 0$. }\end{ex}
The class of left (or right) N-reflexive rings are not closed under homomorphic images.
\begin{ex} \label{örnek}{\rm Consider the rings $R$ and $\overline {R} = R/I$ in the Example \ref{örn} where $I$ is the ideal of $R$ generated by $aRb$ and $a^2$. Then $R$ is
reduced, hence left (and right) N-reflexive. Let $\overline a$,
$\overline b\in \overline R$. Then $\overline a\overline
R\overline b = 0$. But $\overline b\overline R\overline a\neq 0$
since $ba\notin I$. Hence $R/I$ is not left N-reflexive. }\end{ex}
Let $e$ be an idempotent in $R$. $e$ is called {\it left
semicentral} if $re = ere$ for all $r\in R$. $S_l(R)$ is the set
of all left semicentral elements. $e$ is called {\it right
semicentral} if $er = ere$ for all $r\in R$. $S_r(R)$ is the set
of all right semicentral elements of $R$. We use $B(R)$ for the
set of central idempotents of $R$. In \cite{BKP}, a ring $R$ is
called {\it left(right) principally quasi-Baer} (or simply, {\it
left(right) p.q.-Baer}) ring if the left(right) annihilator of a
principal right ideal of $R$ is generated by an idempotent.
\begin{thm}\label{semicentral} The following hold for a ring $R$.
\begin{enumerate}
\item[(1)] If $R$ is right N-reflexive, then $S_l(R)=B(R)$.
\item[(2)] If $R$ is left N-reflexive, then $S_r(R)=B(R)$.
\end{enumerate}
\end{thm}
\begin{proof} (1) Let $e\in S_l(R)$ and $a\in R$. Then
$(1-e)Re=0$. It follows that $(1-e)Re(a-ae)=(1-e)R(ea-eae)=0$.
Since $ea-eae$ is nilpotent and $R$ is right N-reflexive,
$(ea-eae)R(1-e)=0$. Hence $(ea-eae)(1-e)=0$. This implies
$ea-eae=0$. On the other hand, $(1-e)R(a-ea)e\subseteq (1-e)Re=0$.
Thus $(1-e)R(ae-eae)=0$, and so $(1-e)(ae-eae)=0$. Then
$ae-eae=0$. So we have $ea=ae$, i.e., $e\in B(R)$. Therefore
$S_l(R)\subseteq B(R)$. The reverse inclusion is obvious. \\
(2) Similar to the proof of (1).
\end{proof}
\begin{thm}\label{Baer} Let $R$ be a right p.q-Baer ring. Then the following
conditions are equivalent.
\begin{enumerate}
\item[(1)] $R$ is a semiprime ring.
\item[(2)] $S_l(R)=B(R)$.
\item[(3)] $R$ is a reflexive ring.
\item[(4)] $R$ is a right N-reflexive ring.
\end{enumerate}
\end{thm}
\begin{proof} (1) $\Leftrightarrow$ (2) By \cite[Proposition
1.17(i)]{BKP}.\\
(1) $\Leftrightarrow$ (3) By \cite[Proposition 3.15]{KL}.\\ (3)
$\Rightarrow$ (4) Clear by definitions.\\
(4) $\Rightarrow$ (2) By Theorem \ref{semicentral}(1).
\end{proof}
\begin{thm} Let $R$ be a left p.q-Baer ring. Then the following
conditions are equivalent.
\begin{enumerate}
\item[(1)] $R$ is a semiprime ring.
\item[(2)] $S_r(R)=B(R)$.
\item[(3)] $R$ is a reflexive ring.
\item[(4)] $R$ is a left N-reflexive ring.
\end{enumerate}
\end{thm}
\begin{proof} Similar to the proof of Theorem \ref{Baer}.
\end{proof}
\begin{prop}
Let $R$ be a ring and $e^2=e\in R$. Assume that $R$ is an N-reflexive. Then $aRe=0$ implies $ea=0$ for any nilpotent element $a$ of $R$.
\end{prop}
\begin{proof} Suppose that $aRe=0$ for any $a\in$ nil$(R)$.
Since $R$ is N-reflexive, $eRa=0$, and so $ea=0$.
\end{proof}
\noindent \textbf{\emph{Question:}} If a ring $R$ is N-reflexive, then is $R$ a $2$-primal ring?\\
\noindent There is a $2$-primal ring which is not N-reflexive.
\begin{ex}{\rm
Consider the $2$ by $2$ upper triangular matrix ring $R=\left(
\begin{array}{cc}
\Bbb{Z}_{2} & \Bbb{Z}_{2} \\
0 & \Bbb{Z}_{2} \\
\end{array}
\right)$ over the field $\Bbb{Z}_{2}$ of
integers modulo $2$. For $A=\left(
\begin{array}{cc}
0 & 1 \\
0 & 0 \\
\end{array}
\right)\in$ nil$(R)$ and $B=\left(
\begin{array}{cc}
1 & 1 \\
0 & 0 \\
\end{array}
\right)\in R$,
we have $ARB=0$ but $BRA\neq 0$. But $R$ is $2$-primal by
\cite[Proposition 2.5]{BHL}. }\end{ex}
\begin{prop} Let $\{R_i\}_{i\in I}$ be a class of rings. Then $R=\prod\limits_{i\in I}
R_i$ is left N-reflexive if and only if $R_i$ is left N-reflexive
for each $i\in I$.
\end{prop}
\begin{proof}
Assume that $R=\prod\limits_{i\in I} R_i$ is left N-reflexive. By
Proposition \ref{ilk}, for each $i\in I$, $R_i$ is left
N-reflexive. Conversely, let $a=(a_i)\in R$ be nilpotent and
$b=(b_i)\in R$ with $aRb=0$. Then $a_iR_ib_i=0$ for each $i\in I$.
Since each $a_i$ is nilpoent in $R_i$ for each $i\in I$, by
hypothesis, $b_iR_ia_i=0$ for every $i\in I$. Hence $bRa=0$. This
completes the proof.
\end{proof}
\section{Extensions of N-reflexive rings}
In this section, we study some kinds of extensions of N-reflexive
rings to start with, the Dorroh extension $D(R,\Bbb{Z})=\{(r,n)\mid r\in R, n\in \Bbb{Z}\}$ of a ring $R$ is a ring with operations
$(r_1,n_1)+(r_2,n_2)=(r_1+r_2, n_1+n_2)$ and $(r_1,n_1)(r_2,n_2)=(r_1r_2+n_1r_2+n_2r_1, n_1n_2)$, where $r_i\in R$ and $n_i\in \Bbb{Z}$
for $i=1,2$.
\begin{prop} A ring $R$ is left N-reflexive if and only if the Dorroh extension $D(R,\Bbb
Z)$ of $R$ is left N-reflexive.
\end{prop}
\begin{proof} Firstly, we note that nil$(D(R,\Bbb
Z))=\{(r,0)\mid r\in \mbox{nil}(R)\}$. For the necessity, let
$(a,b)\in D(R,\Bbb Z)$ and $(r,0)\in $ nil$(D(R,\Bbb Z))$ with
$(r,0)D(R,\Bbb Z)(a,b)=0$. Then $(r,0)(s,0)(a,b)=0$ for every
$s\in R$. Hence $rs(a+b1_R)=0$ for all $s\in R$, and so
$rR(a+b1_R)=0$. Since $R$ is left N-reflexive, $(a+b1_R)Rr=0$.
Thus $(a,b)(x,y)(r,0)=((a+b1_R)(x+y1_R)r,0)=0$ for any $(x,y)\in
D(R,\Bbb Z)$. For the sufficiency, let $s\in R$ and $r\in $
nil$(R)$ with $rRs=0$. We have $(r,0)\in \mbox{nil}(D(R,
\Bbb{Z}))$. This implies $(r,0)D(R,\Bbb Z)(s,0)=0$. By hypothesis,
$(s,0)D(R,\Bbb Z)(r,0)=0$. In particular, $(s,0)(x,0)(r,0)=0$ for
all $x\in R$. Therefore $sRr=0$. So $R$ is left N-reflexive.
\end{proof}
Let $R$ be a ring and $S$ be the subset of $R$ consisting of
central regular elements. Set $S^{-1}R=\{s^{-1}r\mid s\in S, r\in
R\}$. Then $S^{-1}R$ is a ring with an identity.
\begin{prop} For a ring $R$, $R[x]$ is left N-reflexive if and
only if $(S^{-1}R)[x]$ is left N-reflexive.
\end{prop}
\begin{proof} For the necessity, let $f(x) = \displaystyle\sum_{i=0}^m s^{-1}_ia_ix^i$ be nilpotent and
$g(x)=\displaystyle\sum_{i=0}^n t^{-1}_ib_ix^i\in (S^{-1}R)[x]$
satisfy $f(x)(S^{-1}R)[x]g(x)=0$. Let $s = s_0s_1 \dots s_m$ and
$t = t_0t_1t_2 \dots t_n$. Then $f_1(x) = sf(x)$ is nilpotent and
$g_1(x)=tg(x)\in R[x]$ and $f_1(x)R[x]g_1(x) = 0$. By hypothesis,
$g_1(x)R[x]f_1(x) = 0$. Then $g(x)(S^{-1}R)[x]f(x) = 0$. The
sufficiency is clear.
\end{proof}
\begin{cor} For a ring $R$, $R[x]$ is left N-reflexive if and
only if $R[x;x^{-1}]$ is left N-reflexive.
\end{cor}
According to \cite{Hi}, a ring $R$ is said to be {\it
quasi-Armendariz} if whenever $f(x)=\sum_{i=0}^m a_ix^i$ and
$g(x)=\sum_{j=0}^n b_jx^j\in R[x]$ satisfy $f(x)R[x]g(x)=0$, then
$a_iRb_j=0$ for each $i,j$.
The left N-reflexivity or right N-reflexivity and the quasi-Armendariz property of rings do
not imply each other.
\begin{exs}{\rm (1) Let $F$ be a field and consider the ring $R=\left(
\begin{array}{cc}
F & F \\
0 & F \\
\end{array}
\right)$. Then $R$ is quasi-Armendariz by \cite[Corollary 3.15]{Hi}.
However, $R$ is not left N-reflexive. For $A=\left(
\begin{array}{cc}
0 & 1 \\
0 & 0 \\
\end{array}
\right)\in$ nil$(R)$ and $B=\left(
\begin{array}{cc}
1 & 1 \\
0 & 0 \\
\end{array}
\right)\in R$, we have $ARB=0$ but $BA\neq 0$.\\
(2) Consider the ring $R=\left\{ \left(\begin{array}{cc} a & b \\0 & a \\\end{array}\right)\mid a, b\in \Bbb{Z}_{4} \right\}$. Since $R$ is commutative, $R$ is N-reflexive. For $f(x)=\left(
\begin{array}{cc}
0 & 1 \\
0 & 0 \\
\end{array}
\right)+\left(
\begin{array}{cc}
2 & 1 \\
0 & 2 \\
\end{array}
\right)x$ and $g(x)=\left(
\begin{array}{cc}
0 & 1 \\
0 & 0 \\
\end{array}
\right)+\left(
\begin{array}{cc}
2 & 3 \\
0 & 2 \\
\end{array}
\right)x\in R[x]$, we have $f(x)Rg(x)=0$, and so by \cite[Lemma 2.1]{Hi} $f(x)R[x]g(x)=0$, but
$\left(
\begin{array}{cc}
2 & 1 \\
0 & 2 \\
\end{array}
\right)R\left(
\begin{array}{cc}
0 & 1 \\
0 & 0 \\
\end{array}
\right)\neq 0$. Thus $R$ is not quasi-Armendariz.}
\end{exs}
\begin{prop}\label{pol} Let $R$ be a quasi-Armendariz ring. Assume that coefficients of any nilpotent polynomial in $R[x]$ are nilpotent in $R$. Then $R$ is left N-reflexive if and only if $R[x]$ is left N-reflexive.
\end{prop}
\begin{proof} Suppose that $R$ is left N-reflexive and $f(x)=\sum_{i=0}^m
a_ix^i$, $g(x)=\sum_{j=0}^n b_jx^j\in R[x]$ with $f(x)R[x]g(x)=0$
and $f(x)$ nilpotent. The ring $R$ being quasi-Armendariz implies
$a_iRb_j = 0$ for all $i$ and $j$, and $f(x)$ being nilpotent
gives rise to all $a_0$, $a_1$, $a_2$, $\cdots$, $a_m$ nilpotent.
By supposition $b_jRa_i=0$ for all $i$ and $j$. Therefore
$g(x)R[x]f(x) = 0$, and so $R[x]$ is left N-reflexive. Conversely,
assume that $R[x]$ is left N-reflexive. Let $a\in R$ be nilpotent
and $b\in R$ any element with $aRb = 0$. Then $aR[x]b = 0$. Hence
$bR[x]a = 0$. Thus $bRa = 0$ and $R$ is left N-reflexive.
\end{proof}
Note that in commutative case, the coefficients of any nilpotent
polynomial are nilpotent. However, this is not the case for
noncommutative rings in general. Therefore in Proposition
\ref{pol} the assumption ``coefficients of any nilpotent
polynomial in $R[x]$ are nilpotent in $R$" is not superfluous as
the following example shows.
\begin{ex} {\rm Let $S = M_n(R)$ for a ring $R$.
Consider the polynomial $f(x) = e_{21} + (e_{11} - e_{22})x -
e_{12}x^2\in S[x]$, where the $e_{ij}$'s are the matrix units.
Then $f(x)^2 = 0$, but $e_{11} - e_{22}$ is not nilpotent.}
\end{ex}
\section{Applications}
In this section, we study some subrings of full matrix rings
whether or not they are left or right N-reflexive rings.
{\bf The
rings $H_{(s,t)}(R)$ :} Let $R$ be a ring and $s, t$ be in the
center of $R$. Let\begin{center} $H_{(s,t)}(R) = \left
\{\begin{pmatrix}a&0&0\\c&d&e\\0&0&f
\end{pmatrix}\in M_3(R)\mid a, c, d, e, f\in R, a - d = sc, d - f = te\right \}$.\end{center}
Then $H_{(s,t)}(R)$ is a subring of $M_3(R)$. Note that any
element $A$ of $H_{(s,t)}(R)$ has the form
$\begin{pmatrix}sc+te+f&0&0\\c&te+f&e\\0&0&f\end{pmatrix}$.
\begin{lem}\label{nil} Let $R$ be a ring, and let $s, t$ be in the center of $R$. Then the set of all nilpotent elements of $H_{(s, t)}(R)$ is \begin{center} nil$(H_{(s, t)}(R)) = \left
\{\begin{pmatrix}a&0&0\\c&d&e\\0&0&f\end{pmatrix}\in H_{(s,
t)}(R)\mid a, d, f\in nil(R), c, e\in R\right \}$.\end{center}
\end{lem}
\begin{proof} Let $A = \begin{pmatrix}a&0&0\\c&d&e\\0&0&f\end{pmatrix}\in$ nil$(H_{(s, t)}(R))$
be nilpotent. There exists a positive
integer $n$ such that $A^n = 0$. Then $a^n = d^n = f^n = 0$.
Conversely assume that $a^n = 0$, $d^m = 0$ and $f^k= 0$ for some
positive integers $n,m,k$. Let $p = max\{n, m, k\}$. Then $A^{2p}
= 0$.
\end{proof}
\begin{thm} The following hold for a ring $R$.
\begin{enumerate}
\item[(1)] If $R$ is a reduced ring, then $H_{(0, 0)}(R)$ is
N-reflexive but not reduced.
\item[(2)] If $R$ is reduced, then $H_{(1, 0)}(R)$ is N-reflexive but not reduced.
\item[(3)] If $R$ is reduced, then $H_{(0, 1)}(R)$ is N-reflexive but not reduced.
\item[(4)] $R$ is reduced if and only if $H_{(1, 1)}(R)$ is reduced.\end{enumerate}
\end{thm}
\begin{proof} (1) Let $A = \begin{pmatrix}a&0&0\\c&a&e\\0&0&a\end{pmatrix}\in$
nil$(H_{(0, 0)}(R))$ be nilpotent. By Lemma \ref{nil}, $a$ is
nilpotent. By assumption, $a = 0$. Let $B =
\begin{pmatrix}k&0&0\\l&k&n\\0&0&k\end{pmatrix}\in H_{(0,
0)}(R)$ with $AH_{(0, 0)}(R)B = 0$. $AB = 0$ implies $ck = 0$ and
and $ek = 0$. For any $X =
\begin{pmatrix}x&0&0\\y&x&u\\0&0&x\end{pmatrix}\in H_{(0,
0)}(R)$, $AXB =
\begin{pmatrix}0&0&0\\cxk&0&exk\\0&0&0\end{pmatrix} = 0$. Then $cxk = 0$ and $exk = 0$ for all $x\in R$.
The ring $R$ being reduced implies $kxc = 0$ and $kxe = 0$ for all
$x\in R$. Then $BXA =
\begin{pmatrix}0&0&0\\kxc&0&kxe\\0&0&0\end{pmatrix} = 0$ for all
$X\in H_{(0, 0)}(R)$. Hence $H_{(0, 0)}(R)$ is left N-reflexive. A
similar discussion reveals that $H_{(0, 0)}(R)$ is also right
N-reflexive. Note that being $R$ reduced does not imply $H_{(0,
0)}(R)$ is reduced because $A =
\begin{pmatrix}0&0&0\\1&0&1\\0&0&0\end{pmatrix}\in
H_{(0, 0)}(R)$ is a nonzero nilpotent element.\\
(2) Let $A =\begin{pmatrix} 0&0&0\\0&0&e\\0&0&0\end{pmatrix}\in $
nil$(H_{(1, 0)}(R))$ and $B =
\begin{pmatrix} f+c&0&0\\c&f&d\\0&0&f\end{pmatrix}\in H_{(1,
0)}(R)$ with $AH_{(1, 0)}(R)B=0$. For any $C=\begin{pmatrix}
m+n&0&0\\n&m&u\\0&0&m\end{pmatrix}\in H_{(1, 0)}(R)$, $ACB=0$.
Then $emf=0$ and $fme=0$. This implies $BCA=0$. Therefore $H_{(1,
0)}(R)$ is left N-reflexive. Similarly, $H_{(1, 0)}(R)$ is also
right N-reflexive.
\\(3) Let $A =
\begin{pmatrix}0&0&0\\c&0&0\\0&0&0\end{pmatrix}\in
$ nil$(H_{(0, 1)}(R))$ and $B =
\begin{pmatrix} e+f&0&0\\a&e+f&e\\0&0&f\end{pmatrix}\in H_{(0,
1)}(R)$ with $AH_{(0, 1)}(R)B=0$. For any $C=\begin{pmatrix}
m+n&0&0\\k&m+n&m\\0&0&n\end{pmatrix}\in H_{(0, 1)}(R)$, $ACB=0$.
Then $c(m+n)(e+f)=0$ and $(e+f)(m+n)c=0$. This implies $BCA=0$.
Therefore $H_{(0, 1)}(R)$ is left N-reflexive. Similarly, $H_{(0,
1)}(R)$ is also right N-reflexive.\\(4) Let $A =
\begin{pmatrix}c+e+f&0&0\\c&e+f&e\\0&0&f\end{pmatrix}\in
$ nil$(H_{(1, 1)}(R))$ be nilpotent. Then $f$ is nilpotent and so
$f = 0$. In turn, it implies $e = c = 0$. Hence $A = 0$.
Conversely, assume that $H_{(1, 1)}(R)$ is reduced. Let $a\in R$
with $a^n = 0$. Let $A =
\begin{pmatrix}a&0&0\\0&a&0\\0&0&a\end{pmatrix}\in H_{(1, 1)}(R)$.
Then $A$ is nilpotent. By assumption $a = 0$.
\end{proof}
\section{Generalizations, Examples and Applications}
In this section, we introduce left N-right idempotent reflexive rings and right N-left idempotent
reflexive rings generalize reflexive idempotent rings in Kwak and Lee \cite{KL}, and Kim
\cite{Ki}, Kim and Baik in \cite{KB}. An ideal $I$ of a ring $R$
is called {\it idempotent reflexive} if $aRe\subseteq I$ implies
$eRa\subseteq I$ for any $a\in R$ and $e^2 = e\in R$. A ring $R$
is said to be {\it idempotent reflexive} if the ideal 0 is idempotent reflexive.
In \cite{KL1}, a ring $R$ is called to have the {\it reflexive-idempotents-property}
if $R$ satisfies the property that $eRf = 0$ implies $fRe = 0$ for any idempotents $e$ and $f$ of $R$. We introduce following some classes of rings to produce counter examples related to left N-reflexive rings. These classes of rings will be studied in detail in a subsequent paper by authors.
\begin{df}{\rm Let $I$ be an ideal of a ring $R$. Then $I$ is called
{\it left N-right idempotent reflexive} if being
$aRe\subseteq I$ implies $eRa\subseteq I$ for any nilpotent $a\in R$ and $e^2 = e\in R$. A ring $R$ is called
{\it left N-right idempotent reflexive} if $0$ is a left N-right idempotent reflexive ideal.
Left N-right idempotent reflexive ideals and rings are defined similarly. If a ring $R$ is left N-right
idempotent reflexive and right N-left idempotent reflexive, then it is called an {\it N-idempotent reflexive ring}.}
\end{df}
Every left N-reflexive ring is a left N-right idempotent reflexive
ring. But there are left N-right idempotent reflexive rings that
are not left N-reflexive.
\begin{exs}{\rm (1) Let $F$ be a field and $A = F\textless X, Y\textgreater$ denote the free algebra generated by noncommuting indeterminates $X$ and $Y$ over $F$.
Let $I$ denote the ideal generated by $YX$. Let $R = A/I$ and $x =
X + I$ and $y = Y + I\in R$. It is proved in \cite[Example 5]{Ki}
that $R$ is abelian and so $R$ has reflexive-idempotents-property
but not reflexive by showing that $xRy\neq 0$ and $yRx = 0$.
Moreover, $xyRx = 0$ and $xRxy \neq 0$. This also shows that $R$
is not left N-reflexive since $xy$ is nilpotent in $R$.
(2) Let $F$ be a field and $A = F\textless X, Y\textgreater$
denote the free algebra generated by noncommuting indeterminates
$X$ and $Y$ over $F$. Let $I$ denote the ideal generated by $X^3$,
$Y^3$, $XY$, $YX^2$, $Y^2X$ in $A$. Let $R = A/I$ and $x = X + I$
and $y = Y + I\in R$. Then in $R$, $x^3 = 0$, $y^3 = 0$, $xy = 0$,
$yx^2 = 0$, $y^2x = 0$. In \cite[Example 2.3]{JAKL}, $xRy = 0$,
$yRx\neq 0$ and idempotents in $R$ are 0 and 1. Hence for any
$r\in$ nil$(R)$ and $e^2 = e\in R$, $rRe = 0$ implies $eRr = 0$.
Thus $R$ is left N-right idempotent reflexive. We show that $R$ is
not a left N-reflexive ring. Since any $r \in R$ has the form $r =
k_0 + k_1x + k_2x^2 + k_3y + k_4y^2 + k_5yx$ and $x$ is nilpotent,
as noted above, $xRy = 0$. However, $yRx\neq 0$ since $yx \neq 0$.
Thus $R$ is not left N-reflexive.
(3) Let $F$ be a field of characteristic zero and $A = F\textless
X, Y, Z\textgreater$ denote the free algebra generated by
noncommuting indeterminates $X$, $Y$ and $Z$ over $F$. Let $I$
denote the ideal generated by $XAY$ and $X^2 - X$. Let $R = A/I$
and $x = X + I$, $y = Y + I$ and $z = Z + I\in R$. Then in $R$,
$xRy = 0$ and $x^2 = x$. $xy = 0$ and $yx$ is nilpotent and $x$ is
idempotent and $xRyx = 0$. But $yxRx \neq 0$. Hence $R$ is not
right N-left idempotent reflexive. In \cite[Example 3.3]{KL}, it
is shown that $R$ is right idempotent reflexive. }
\end{exs}
| {
"timestamp": "2018-07-09T02:07:12",
"yymm": "1807",
"arxiv_id": "1807.02333",
"language": "en",
"url": "https://arxiv.org/abs/1807.02333",
"abstract": "An ideal $I$ of a ring $R$ is called left N-reflexive if for any $a\\in$ nil$(R)$, $b\\in R$, being $aRb \\subseteq I$ implies $bRa \\subseteq I$ where nil$(R)$ is the set of all nilpotent elements of $R$. The ring $R$ is called left N-reflexive if the zero ideal is left N-reflexive. We study the properties of left N-reflexive rings and related concepts. Since reflexive rings and reduced rings are left N-reflexive, we investigate the sufficient conditions for left N-reflexive rings to be reflexive and reduced. We first consider basic extensions of left N-reflexive rings. For an ideal-symmetric ideal $I$ of a ring $R$, $R/I$ is left N-reflexive. If an ideal $I$ of a ring $R$ is reduced as a ring without identity and $R/I$ is left N-reflexive, then $R$ is left N-reflexive. If $R$ is a quasi-Armendariz ring and the coefficients of any nilpotent polynomial in $R[x]$ are nilpotent in $R$, it is proved that $R$ is left N-reflexive if and only if $R[x]$ is left N-reflexive. We show that the concept of N-reflexivity is weaker than that of reflexivity and stronger than that of left N-right idempotent reflexivity and right idempotent reflexivity which are introduced in Section 5.",
"subjects": "Rings and Algebras (math.RA)",
"title": "Reflexivity of Rings via Nilpotent Elements",
"lm_name": "Qwen/Qwen-72B",
"lm_label": "1. YES\n2. YES",
"lm_q1_score": 0.9854964177527898,
"lm_q2_score": 0.7185943865443349,
"lm_q1q2_score": 0.7081721937567057
} |
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